Download Experimental Characterization of Nonclassical

Experimental Characterization of
Nonclassical Polarization States
of Intense Light
Den Naturwissenschaftlichen Fakultäten
der Friedrich-Alexander-Universität Erlangen-Nürnberg
zur
Erlangung des Doktorgrades
vorgelegt von
Joel Heersink
aus Edmonton, Kanada
ii
Als Dissertation genehmigt von den naturwissenschaftlichen Fakultäten der Universität Erlangen-Nürnberg
Tag der mündlichen Prüfung:
21. Juli 2006
Vorsitzender der Promotionskommission:
Erstberichterstatter:
Zweitberichterstatter:
Prof. Dr. D.-P. Häder
Prof. Dr. G. Leuchs
Prof. Dr. P. Kumar
Zusammenfassung
Experimentelle Charakterisierung
nichtklassischer
Polarisationszustände
hellen Lichts
Die in dieser Arbeit präsentierten Ergebnisse beschreiben die experimentelle Herstellung und Charakterisierung neuartiger Quellen nichtklassischen Lichtes. Diese
Quellen reduzieren die Fluktuationen der Quadratur- und Polarizationsvariablen
heller ultrakurzer Laserpulse mittels der optischen Kerr-Nichtlinearität in Glasfasern.
Die Anwendung dieses Effektes in einigen unterschiedlichen Konfigurationen eines
asymmetrischen Faser-Sagnac-Interferometers ermöglichte die Erzeugung von Quantenzuständen mit verringertem Amplitudenrauschen. Eine variable, faserintegrierte Konfiguration dieses Gerätes wurde benutzt, um auf experimenteller Weise das
optimale Teilungsverhältnis, 93:7, des Faser-Sagnac-Interferometers zu bestimmen,
was frühere Arbeiten bestätigte. Mit diesem Teilungsverhältnis wurde in weiteren
Experimenten polarisations-gequetschtes Licht hergestellt, indem zwei amplitudengequetschte, orthogonal-polarisierte Lichtpulse überlagert wurden. Zusätzlich wurde
Polarisationsverschränkung mit diesem resourceneffizienten Aufbau gezeigt.
Im Rahmen dieser Arbeit wurde eine neuartige und vereinfachte silikatfaserbasierte
Quelle gequetschten Lichts in Form eines einfachen Durchgangs durch die Faser realisiert. Polarisationsquetschung wird erzeugt, indem zwei orthogonal-polarisierte
Laserpulse zusammen durch eine Faser propagieren, und die daraus entstehenden
quadratur-gequetschten Pulse nach der Glasfaser überlagert werden. Der Aufbau,
welcher ohne den Einsatz eines Interferometers Polarisationsquetschung erzeugt, ist bemerkenswert einfacher, sowie effizienter als vorherige Schemata, wie in der maximalen
gemessenen Polarisationsquetschung −5.1 ± 0.3 dB erkennbar ist. Der multimodige
Charakter der Polarisationsvariablen, welche als Vakuumsignal mit einem zusammen-
iv
propagierenden Lokaloszillator interpretiert werden können, ermöglichte die experimentelle Bestimmung der Wignerfunktion der glasfasergequetschten Zustände.
Dank der Eleganz dieser Quelle gequetschten Lichts wurde eine hervorragende
Übereinstimmung zwischen einer grundlegenden Quanten-Propagationssimulation
und den experimentell gemessenen Ergebnissen beobachtet. Hier wurde festgestellt,
dass thermisches Rauschen in der Glasfaser (das sogenannte Guided Acoustic Wave
Brillouin Scattering - GAWBS) der Effekt ist, der die Quetschung bei niedriger Pulsenergie limitierte, wobei Raman-Streueffekte den beschränkenden Faktor bei hoher Energie darstellen. In diesen Rechnungen musste nur ein einziger Parameter angepasst
werden, um die Effekte von GAWBS in die Propagation einzubringen. Außerdem
wurde das effiziente Schema zur Polarisationsquetschung in einem Quanteninformationsprotokoll für die Destillation von Quetschung, die von nicht-Gaußschem Rauschen
gestört wurde, ausgenutzt. Dieses Rauschen kann, zum Beispiel, in der Erzeugung oder
Transmission quetschter Zustände vorkommen. Die Quetschung des Systems konnte
auf probabilistische Weise mittels eines Postselektionsvorgangs, welcher auf der Messung eines kleinen, vom Strahlteiler abgezweigten, Teil des Signals basierte, wiederhergestellt werden.
Summary
Experimental Characterization of
Nonclassical Polarization States
of Intense Light
The results presented in this thesis describe the experimental production and characterization of new sources of nonclassical light. These devices reduce the fluctuations in
the quadrature and polarization variables of intense trains of ultrashort laser pulses by
exploiting the optical Kerr nonlinearity in silica fibers. Employing this effect in several
different asymmetric fiber Sagnac interferometer configurations, states with reduced
amplitude noise were generated. A variable, all-in-fiber configuration was used to experimentally determine the optimum splitting ratio of the Sagnac loop, 93:7, confirming
previous work. With this splitting ratio polarization squeezed light was generated by
overlapping two orthogonally polarized, amplitude squeezed pulses. Additionally, polarization entanglement was demonstrated using this Sagnac loop setup in a resourceefficient scheme.
A novel and simplifying improvement on fiber based squeezing sources, the single
pass method, was developed in this thesis. In this setup two orthogonally polarized
pulses copropagate through an optical fiber and polarization squeezing is generated by
overlapping the quadrature squeezed pulses after the fiber. Without the need for an interferometer to generate squeezing, it is noticeably simpler and more efficient, producing a maximum measured polarization squeezing of −5.1 ± 0.3 dB. Taking advantage of
the multimode nature of the polarization variables, which can be interpreted as a vacuum signal with a copropagating local oscillator, it was possible to measure the Wigner
function of fiber squeezed states.
Due to the elegance of this squeezing source, first principles quantum propagation
simulations of the experiments agreed very well with the measured results. Thus it was
vi
observed that Guided Acoustic Wave Brillouin Scattering (GAWBS) limits fiber squeezing at low pulse energies, whereas Raman scattering is the restricting factor at high
energies. In these calculations only one fitting parameter was necessary, accounting for
the GAWBS. Further, the efficient polarization squeezing scheme was leveraged in a
quantum information protocol for the distillation of squeezing. The nonclassicality of
a squeezed beam afflicted by non-Gaussian noise, acquired for example during generation or transmission, was probabilistically recovered via a post selection process based
on a tap measurement.
Contents
1 Introduction
1
2 Characterizing the state of light
2.1 Classical description of light . . . . . . . . . . . . . .
2.1.1 Polarization of light . . . . . . . . . . . . . .
2.2 Quantum mechanical description of light . . . . . .
2.2.1 Basic quantum states . . . . . . . . . . . . . .
2.2.2 Quasi-probability distributions . . . . . . . .
2.2.3 Quantum polarization . . . . . . . . . . . . .
2.3 Quantum noise detection . . . . . . . . . . . . . . . .
2.3.1 Direct detection . . . . . . . . . . . . . . . . .
2.3.2 Homodyne detection . . . . . . . . . . . . . .
2.3.3 Polarization measurements . . . . . . . . . .
2.3.4 Quasi-probability distribution reconstruction
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4 Propagation of light in optical fibers
4.1 Semi-classical effects and propagation . . . . . . . . . . . . . . . . . . . . .
4.1.1 Linear effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Nonlinear birefringence . . . . . . . . . . . . . . . . . . . . . . . . .
49
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3 Exploiting the quantum properties of light
3.1 Quantum noise reduction . . . . . . .
3.1.1 Quadrature squeezing . . . . .
3.1.2 Polarization squeezing . . . . .
3.2 Polarization entanglement . . . . . . .
3.3 Distillation of squeezing . . . . . . . .
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viii
Contents
4.2
4.1.3 Nonlinear Schrödinger equation and solitons
4.1.4 Squeezing in a semi-classical picture . . . . .
4.1.5 Scattering effects . . . . . . . . . . . . . . . .
Quantum propagation model . . . . . . . . . . . . .
4.2.1 Interaction Hamiltonians . . . . . . . . . . .
4.2.2 Quantum nonlinear Schrödinger Equation .
4.2.3 Methodology . . . . . . . . . . . . . . . . . .
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5 Experimental setup
5.1 Femtosecond laser . . . . . . . . . .
5.2 Optical fibers . . . . . . . . . . . . . .
5.3 Asymmetric Sagnac loop . . . . . . .
5.4 Single pass squeezing method . . . .
5.5 Polarization entanglement . . . . . .
5.6 Tomography . . . . . . . . . . . . . .
5.7 Distillation of non-Gaussian noise .
5.8 Detection . . . . . . . . . . . . . . . .
5.8.1 Squeezing and entanglement
5.8.2 Tomography and Distillation
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6 Results and Discussion
6.1 Amplitude squeezing . . . . . . . .
6.1.1 Free space Sagnac loop . . .
6.1.2 All-in-fiber Sagnac loop . .
6.2 Polarization squeezing . . . . . . .
6.2.1 Sagnac loop setup . . . . .
6.2.2 Single pass setup . . . . . .
6.2.3 Simulations . . . . . . . . .
6.3 Polarization Entanglement . . . . .
6.4 Quantum state tomography . . . .
6.5 Distillation of non-Gaussian noise
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7 Conclusion and Outlook
119
A Characterization of fiber output pulses
123
Bibliography
123
Curriculum vitae
153
Chapter 1
Introduction
Quantum communication and information science are an undeniably important area
in modern physics, highlighted by its rapid expansion in recent years. This field has
its roots in the investigation of nonclassical optical states beginning with experiments
in photon anti-bunching [1], squeezing (or noise reduction) [2] and entanglement (or
quantum correlations) [3]. Since these beginnings, research in this field has grown to
include many novel and disparate topics such as quantum optical teleportation, quantum computing with single atoms and quantum cryptography with coherent states [4].
While these might appear to be more closely related to ”traditional” physics, nonclassical states, atomic, optical or otherwise, remain at the heart of this branch of physics.
Experiments investigating these and many other topics today can be broadly divided
into two major groups: those using discrete variables, i.e. single photon polarization or
single nuclear spin, and those employing continuous variables (CV) [5], i.e. the quadrature variables of intense light or the spin of atomic ensembles.
Nonclassical, continuous variable, optical states are the focus of this thesis. The
experiments presented here investigate and characterize several methods for the generation of such states. These states, examples of which are squeezed or entangled states,
exhibit photon statistics and correlations in, for example, the quadrature or polarization variables which have no classical counterpart [5]. The so-called squeezed states are
characterized by a reduction in their noise fluctuations below a classical bound [6]. Such
quantum limits, the basis of quantum science, take the form of Heisenberg uncertainty
relations, similar to those of position and momentum. These are related to the fluctuation levels of the coherent state, which is often taken as the boundary between quantum
an classical physics. Entangled states, typically derived from squeezed states and thus
often called two-mode squeezed states, exhibit correlations between, for example, two
2
1. Introduction
spatially separated beams which are stronger than classically allowed. This bound is
also quantified by the uncertainty relationships governing the chosen observables.
The generation of squeezed or noise reduced optical states requires a nonlinear interaction to produce intra-beam correlations. Such a state was measured for the first time in
1985 using atomic samples as the interaction medium [2]. Optical fibers, among others,
were soon investigated as a robust and flexible nonlinear medium for the production
of these nonclassical optical states [7, 8]. Early experiments used continuous wave laser
light [8], but they were plagued by excess phase noise arising from acoustic vibrations in
the silica fibers [9]. The introduction of ultrashort pulse laser systems provided a means
of significantly decreasing this obscuring noise, as demonstrated in 1991 [10]. This has
formed a fruitful basis for many ensuing fiber experiments producing squeezing and
entanglement, for example [11, 12, 13, 14, 15].
Silica fibers are the work horses of the schemes generating quadrature and polarization squeezing and entanglement in this thesis. The well known asymmetric Sagnac
fiber interferometer [16, 17, 13, 18] used to produce amplitude squeezing was optimized
and characterized to serve as the foundation of experiments generating polarization
squeezing and entanglement. These nonclassical polarization states have been generated not only in fiber systems but also by Optical Parametric Oscillators (OPO) [19, 20]
and cold atomic samples [21, 22]. Additionally, it has been found that bright polarization squeezing is equivalent to vacuum squeezing in the mode orthogonal to the bright
excitation [23]. This fact shows the great promise for the integration of nonclassical
polarization states into the existing quantum information protocols and networks.
In real world implementations of quantum communication, signal degradation during generation and transmission is inevitable. A number of protocols for the distillation
of nonclassical beams afflicted by Gaussian noise have been developed [24, 25, 26, 27]
and first step towards realization have been taken [28]. However these rely on experimentally difficult non-Gaussian operations. In contrast, the distillation of non-Gaussian
noise can be accomplished using the readily available techniques of linear optics and
homodyne detection. Non-Gaussian noise is naturally occurring, found in mixture
or phase noise sources, a prime example of which is turbulent atmospheric transmission [29, 30]. Such a distillation protocol has been successfully demonstrated here using
the exemplary case of a squeezed beam subject to on/off noise.
An offshoot of the schemes for the generation of polarization squeezing has been
the novel single pass method, which bears a certain resemblance to earlier experiments [31, 32]. Here this setup has been developed as a highly efficient source of polarization squeezed states. This method is similar to previous experiments using symmetric Saganc interferometers [10, 11] insofar as both require the production of two quadrature squeezed states which are mutually interfered. In this manner one of the limiting
factors of the asymmetric loop is circumvented, enabling a more efficient squeezing
3
production. Compared to these experiments, the present scheme exploiting the polarization variables has the further advantage of generating a beam containing both the
squeezing as well as the perfectly matched local oscillator. The elegant simplicity of
this scheme allowed a thorough characterization of fiber squeezed optical pulses. These
experimental results were found to agree very well with first principles quantum propagation simulations [33]. This represents a long deserved triumph for quantum soliton
propagation theory, which, since it inception in 1987 [34], has seen a number of less
successful attempts to unite experiment and fundamental theory [17, 13].
4
1. Introduction
Chapter 2
Characterizing the state of light
The present chapter is devoted to describing the basic notions of the classical and quantum mechanical descriptions of electromagnetic radiation and the relevant detection
schemes. It should provide a basis for the following theoretical and experimental discussions of the generation and exploitation of nonclassical optical states. The first section, Sec. 2.1, provides an outline of the classical theory of electromagnetic waves, including a discussion of the phenomenon of polarization. The Sec. 2.2 discusses the
different quantum mechanical representations of optical modes and applies these to the
fundamental quantum optical states. The polarization operators are then introduced
and their salient points are highlighted. The idea of a quantum joint quasi-probability
distribution describing a state’s density matrix is outlined for the single mode operators
as well as the polarization operators. The final section (Sec. 2.3) considers a number of
different detection schemes used in quantum optics, again for both the single mode and
polarization operators. The implementation of these schemes to reconstruct the state’s
quasi-probability distribution or density matrix closes this chapter.
2.1 Classical description of light
Maxwell’s equations, which describe the propagation of oscillating, classical electromagnetic fields in free space as well as in media, are well known (see for example [35, 36]). They are:
δB
,
δt
∇ · D = ρf,
∇×E = −
∇ × H = Jf +
∇ · B = 0,
δD
,
δt
(2.1)
6
2. Characterizing the state of light
where E and H are the electric and magnetic field vectors, and D and B are the electric
and magnetic flux densities. J f and ρ f are the free current density vector and the free
charge density respectively, representing the source of the electromagnetic field. The
flux densities are related to the field vectors by:
D = ǫ0 E + P,
B = µo H + M,
(2.2)
where P and M are the induced electric and magnetic polarizations, ǫ0 the electric permittivity and µ0 the magnetic permeability of free space. The solution to these equations
for a plane wave in vacuum can be written in terms of a scalar A0 (t, r) and a vector potential A(t, r) [35, 36]. Using the Coulomb gauge (∇ · A(t, r) = 0) has the effect, among
others, of associating A0 with the charge density. Thus assuming a charge free vacuum
(i.e. where P = M = J f = ρ f = 0) eliminates A0 from the discussion and it is found that
the vector field obeys the wave equation:
∇2 A(t, r) −
1 ∂2 A(t, r)
= 0,
c2 ∂t2
(2.3)
where c is the speed of propagation in a vacuum. The solution to this equation can be
written as:
ik·r
∗
−ik·r
,
(2.4)
p
A
(
t,
r
)
e
+
A
(
t,
r
)
e
A(t, r) = ∑ ι,k
ι
ι
ι
where ω is the oscillation frequency associated with a given wave vector k where
2
k2 = ωc2 . The magnitudes of this wave vector are given by k j = 2πn
L where j = 1, 2, 3 rep3
resenting the three spatial dimensions, n is an integer and L the mode volume. Further,
pι,k is the unit polarization vector of ι-polarization for the k mode where k · pι,k = 0.
Finally Aι is the complex field envelope of the ι-polarized mode described by the mode
spectrum:
Z
1
Aι (t, r) = √
dk Ãι (ωk , r)ei (ωk t+φk ) ,
(2.5)
2π
where the integration over k arises from allowing the mode volume to go to infinity.
The Ãι is the strength of the spectral component ωk and φk is the phase shift of the k
mode. This definition allows a straightforward description of multimode pulses. For
this case the observed fields E and B depend only on A:
E(t, r) = −
1 ∂A(t, r)
,
c ∂t
B(t, r) = ∇ × A(t, r).
(2.6)
For the electric field this leads to:
Z
i
dk ωk Ãι (ωk , r)ei (k·r−ωk t−φk ) − Ã∗ι (ωk , r)e−i (k·r−ωk t−φk ) ;
E(t, r) = ∑ pι,k √
c 2π
ι
(2.7)
2.1. Classical description of light
7
a similar equation can be found for B. At this point it should be noted that the approximation of a charge free vacuum is appropriate for the description of most experimental
situations in an input-output formalism, such as interference on a beam splitter. These
simple arguments must be significantly altered if the medium affects the transmitted
light, i.e. if P 6= 0 leading to, for example, absorption or a nonlinear response [37, 38].
This is vital when considering the propagation of ultrashort laser pulses in optical fibers
where the material nonlinearity plays a significant role in classical and quantum propagation [39, 40, 34]. This point is revisited in Sec. 4.2 where a quantum model of light
propagation in optical fibers is discussed.
The complex amplitude wave of Eq. 2.7 can be considered as combination of cosine
and sine waves, or the real (Re) and imaginary (Im) amplitudes of the complex electric
field. For a single frequency mode it is found that:
ω
√
(2.8)
E(t, r) =
∑ pι,k Pι (t, r) sin(ωt) + Qι (t, r) cos(ωt) ,
c 2π ι
where the electric field quadratures for a single polarization mode are:
P(t, r) = A(ω, r) + A∗ (ω, r) ∝ Re(E(t, r)),
Q(t, r) = i ( A(ω, r) − A∗ (ω, r)) ∝ Im(E(t, r)).
2.1.1
(2.9)
Polarization of light
The polarization of a wave refers to the behavior of the appropriate field vector with
time [41]. In the case of electromagnetic waves, the electric field is used as the reference
vector, given in Eq. 2.7 by pι,k . This vector can be described in terms of the orthogonal
spatial of the unit vectors x, y. Considering a coherent plane wave, the electric field vector will generally trace out an ellipse, shown in Fig. 2.1 in terms of x, y. The polarization
ellipse is characterized by the set of three parameters a, b, θ, the semi-major, semi-minor
axes and the azimuth angle respectively [41, 42]. The ellipticity of such a given polarization can be defined as e = ± ba = ± tan(ǫ), where ǫ is the ellipticity angle. The ±
correspond to right- and left-handed polarizations.
Without loss of generality the direction of propagation can be assumed to be in z.
Thus pι,k lies in the x − y plane. The polarization vector of a completely polarized light
beam assuming a single mode k, decomposed in the laboratory frame, is given by:
∑ pι = px eiφx x + py eiφy y.
(2.10)
ι
Here x and y are the unit vectors of the laboratory frame and φx , φy are the absolute
phase shifts of the x and y modes with amplitudes p x and py , derived from A. The
8
2. Characterizing the state of light
y
minor
axis
major
axis
q
x
b
a
Figure 2.1: A general elliptical polarization state is characterized by the lengths a and b, the
semi-major and semi-minor axes lengths respectively and the angle of rotation of the ellipse
relative to + x, θ. The ellipticity angle is given by ǫ = tan−1 ( ba )
polarization state of light can then be described by another set of three independent
parameters: p x , py , (φx − φy ).
A further alternative description of the polarization state uses the Stokes parameters.
These are a set of four parameters {S0 , S1 , S2 , S3 } defined by G. G. Stokes in 1852 [43].
They describe the preference of light for a given polarization, i.e. x (horizontal) or righthand circular (σ+ ) polarization [41, 44, 35]:
2
2
2
2
S0 = hE2x (t)i + hEy2 (t)i = hE+
45◦ (t)i + h E−45◦ (t)i = h Eσ+ (t)i + h Eσ− (t)i , (2.11)
S1 = hE2x (t)i − hEy2 (t)i
= S0 cos(2ǫ) cos (2θ ) ,
2
2
S2 = 2hEx (t)Ey (t) cos(φy − φx )i = hE+
45◦ (t)i − h E−45◦ (t)i
= S0 cos(2ǫ) sin(2θ ) ,
S3 = 2hEx (t)Ey (t) sin(φy − φx )i = hEσ2+ (t)i − hEσ2− (t)i
= S0 sin(2ǫ) ,
where ǫ and θ are the polarization ellipticity and azimuth; Eι (t) represents the strength
of the electric field in the ι-polarized mode. The parameters are given in terms of the
RT
time averaged electric fields given by hEi = T1 0 dt E such that the integral is independent of T. S0 is the total intensity, S1 is the difference between the amount of light
polarized in the x and y directions, S2 is the difference between the ±45◦ polarization
2.1. Classical description of light
9
basis and S3 is the difference between the right- and left-hand circular basis, represented
by σ± . Also shown is the link between the Stokes parameters and the parameters describing the polarization ellipse. For fully polarized light S0 is given by:
q
(2.12)
S0 = S12 + S22 + S32 .
Leading to a natural definition for the degree of polarization P of a light beam [41]:
q
S12 + S22 + S32
P=
,
(2.13)
S0
where P = 1 corresponds to fully polarized light and P = 0 to completely unpolarized
light.
S3
(s+)
S
2e
2q
S1
(x)
S2
(+45°)
Figure 2.2: The classical Poincaré sphere showing the polarized light beam defined by the vector
S which has the corresponding polarization ellipse given by polarization ellipticity ǫ and azimuth
θ.
The Stokes parameters are handily visualized on the Poincaré sphere introduced by
a French scientist of the same name in 1892 [45]. This sphere is based on the orthonormal
basis given by S1 , S2 , S3 and is shown in Fig. 2.2 [41, 46, 44, 35]. Here the link between
the polarization vector ellipse and the Stokes vector is shown. Fully polarized light
is represented as a point on the Poincaré sphere. Thus a given polarization state is
described by the total intensity S0 and the projections of the polarization vector onto
the three axes S = (S1 , S2 , S3 ).
What is needed to measure the Stokes parameters is a gadget able to separate appropriate pairs of orthogonal polarizations, the difference of which gives the desired Stokes
10
2. Characterizing the state of light
S1
S2
+/-
+/-
x
x
i(x-y)/√2
y
y
-i(x+y)/√2
PBS
PBS
l/2
Q=22.5°
S3
+/x
x
(ix+y)/√2
y
iy
-(ix+y)/√2
l/4
F=0°
l/2
Q=22.5°
PBS
Figure 2.3: Detection setups for the Stokes parameters. The unknown polarization state is fully
characterized by the difference currents of the two detectors in the three measurement apparatus.
S0 is the sum in all measurements.
parameter. Polarization beam splitters (PBS) are designed exactly for such tasks, splitting for example the x and y polarizations. Combinations of wave plates are known to
transform the polarization of passing light beams. In particular, it has been shown that
two quarter wave-plates and a half wave-plate are sufficient to perform all polarization transformations [47]. This is a fact commonly exploited in telecommunications in
the form of the ubiquitous fiber polarization controllers [48]. Splitting this transformed
light on a PBS can then give the projection of the transformed eigenmode onto the PBS
basis, for example x, y. From this the Stokes parameter which is the eigenmode of the
wave plate combination is measured.
It is postulated that the combination of a single half- and quarter-wave plate is capable of transforming to the PBS basis, representing an optimized measurement setup. Let
the quarter-wave plate be at angle Φ, and the half-wave plate at angle Θ relative to the
following PBS, from which the sum and difference signals can be measured (Fig. 2.4(b)).
The Jones matrices [41, 42] of this system give the following results for the Stokes parameters:
1
cos 4Θ + cos 4(Φ − Θ) ,
2
1
sin 4Θ + sin 4(Φ − Θ) ,
=
2
= − sin 2Φ ,
S1 =
S2
S3
(2.14)
2.1. Classical description of light
11
verifying that all Stokes parameters can be measured in this setup. Thus three efficient
measurements corresponding to the three independent Stokes parameters (in the laboratory reference frame) are defined. These setups, seen in Fig. 2.3, are used as the
standard Stokes measurements in this thesis [42, 49].
(a)
(b)
0.5
S3
l/4, F
l/2, Q
1.0
+/-
0.0
-0.5
-1.0
1.0
0.5
S1
0.0
0.0
-0.5
-0.5
-1.0
0.5
1.0
S2
-1.0
Figure 2.4: a) Visualization of the eigenvalues for a half- and quarter-wave plate pair in Poincaré
space. The gray curves are the trajectory (or projections thereof) for the rotation of a quarterwave plate (Φ = 0◦ → 45◦ ). The black trajectories describe the action of a half-wave plate
(Θ = 0◦ → 45◦ ) placed before a quarter-wave plate, the latter at (from above): Φ = 40◦ , 15◦ ,
0◦ , −15◦ and −40◦ . b) Schematic of the general polarization measurement device.
While the three setups of Fig. 2.3 represent easy measurements of three orthogonal
Stokes parameters, what form does a general polarization measurement take? Using
the general Stokes measurement setup of Fig. 2.4(b), all polarization variables can be
measured. This is shown in the five rings of Fig. 2.4(a). These correspond to rotations of
Θ = 0◦ → 45◦ for Φ = ±40◦ , ±15◦ , 0◦ . It is seen that the rotation of a quarter-wave plate
traces out a figure-of-eight on the surface of the Poincaré sphere, assuming constant S0 .
Physically this gives rise to a growing minor axis with π/2 relative phase shift to the
major axis, with the extreme cases being circular and linear polarized light. In contrast,
a half wave-plate changes the polarization along circles orthogonal to the S3 axis, i.e.
rotates the major axis of the polarization ellipse without altering the ellipticity.
12
2. Characterizing the state of light
2.2 Quantum mechanical description of light
A complete description of optical fields requires the consideration of the quantized nature of the electromagnetic field [50, 51]. Only in such a framework can many optical
phenomenon be explained. Thus, via the correspondence principle, an electric field
operator can be defined in analogy to Eq. 2.7 [50]:
s
h̄ω
i (k ·r− ωt)
†
−i (k ·r− ωt)
p
(
t,
r
)
â
(
t
)
e
−
â
(
t
)
e
.
(2.15)
Ê(t, r) = i
k
∑ ι,k
k
2ǫ0 V ∑
ι k
Here the photon creation/annihilation operators have been associated with the complex
amplitudes of vector potential modes:
s
s
h̄
h̄
Ãι (ω, r) ↔
âι (t, k), Ã∗ι (ω, r) ↔
↠(t, k).
(2.16)
2ωǫ0 V
2ωǫ0 V ι
These operators also have the same time dependence as the complex amplitudes A:
âι (t, k) = âι (0, k)e−iωt ,
â†ι (t, k) = â†ι (0, k)eiωt ,
and additionally obey the commutation relations:
h
i
3
âι (t, k), â†ι′ (t′ , k′ ) = δkk
′ διι ′ δtt ′ ,
âι (t, k), âι′ (t′ , k′ ) = 0,
h
i
â†ι (t, k), â†ι′ (t′ , k′ ) = 0.
(2.17)
(2.18)
(2.19)
(2.20)
From the creation and annihilation operators one can construct the photon number operator:
n̂ι (t, k) = â†ι (t, k)âι (t, k),
(2.21)
which gives the number of photons in the mode k as its eigenvalue.
It can often be useful in calculations involving bright quantum states to describe
these operators by an time averaged classical component and a quantum noise operator:
âι (t, r, k) = | αι (r, k)| + δ âι (t, r, k),
(2.22)
considering an ι-polarized beam where |αι | is the complex field amplitude. The noise
operator is the analog of the photon creation/annihilation operator and contains only
the fluctuating terms of âι where hδ âι (t, r, k)i = 0
Again assuming propagation only in z, r → z and k → kz ≡ k and pι,k is decomposed
in the laboratory x − y basis. A slowly varying photon-density operator Ψ̂ι (t, z) can
2.2. Quantum mechanical description of light
13
then be defined, similar to the classical field envelope of Eq. 2.5. Similar to this previous
equation, the mode volume has been allowed to go to infinity to give a continuous mode
distribution. Thus photon-density operator can be written as [50, 52]:
1
Ψ̂ι (t, z) = √
2π
Z
dk âι (t, k)ei (k−k0 )z+iωt ,
(2.23)
where k0 is the field propagation vector of the ι-polarized mode. The wave vectors are
2
related to the mode frequency by k2 = ωc2 . The photon number operator n̂(t) is given in
terms of the photon density operator by:
n̂(t) =
Z
dz Ψ̂† (t, z)Ψ̂ (t, z),
(2.24)
assuming a unit incidence cross-section. The equal-time commutation relations of these
operators are:
h
i
†
′
Ψ̂ι (t, z), Ψ̂ι′ (t, z ) = δ(z − z′ )διι′ .
Quadrature amplitude operators can be defined similar to Eq. 2.9:
P̂ι (θ, t, k) = â†ι (t, k)eiθ + âι (t, k)e−iθ
= Re(α),
π
Q̂ι (θ, t, k) = i â†ι (t, k)eiθ − âι (t, k)e−iθ = P̂ι θ + , t, k
2
= Im(α),
(2.25)
where α is a complex phase space amplitude. This is given in terms of a phase θ and an
amplitude |α|, as depicted in Fig. 2.5 for a single polarization mode:
α = |α| eiθ .
(2.26)
A general quadrature amplitude in terms of α is given by:
P̂ (θ ) = Re(α) cos(θ ) + Im(α) sin (θ ).
(2.27)
The quadrature operators do not commute:
[ P̂(θ, t, k), Q̂ (θ, t′ , k′ )] = 2iδtt′ δkk′ ,
(2.28)
which gives rise to a Heisenberg uncertainty relation:
∆2 P̂ (θ, t, k) ∆2 Q̂(θ, t, k) ≥ 1,
(2.29)
where ∆2 P̂ = Var ( P̂ ) = h P̂2 i − h P̂ i2 is the variance of P̂. This relation thus describes a
fundamental uncertainty in the preparation of a state. It represents the limiting accuracy
14
2. Characterizing the state of light
of separate measurements of P̂ and Q̂ on identically prepared quantum states, see for
example Ref. [53]. P̂, Q̂ define a global phase space. It is often convenient to define a
relative reference frame:
P̂(θ, t, k) → X̂ (θ, t, k),
Q̂(θ, t, k) → Ŷ (θ, t, k).
(2.30)
In keeping with standard usage X̂ and Ŷ are defined in the optical state’s reference
frame in which X̂ (0, t, k) is the amplitude (radial) quadrature; Ŷ (0, t, k) is the phase
(azimuthal) quadrature.
Q
(X)
W’
(Y)
W’
Y(0)
ˆ
X(0)
ˆ
|a|
q
P
Figure 2.5: The phase space representation of the quadrature amplitudes of light in terms of P̂
and Q̂ or the real and imaginary parts of the electric field. The projections of a coherent state’s
probability distribution onto the amplitude (X̂) and phase (Ŷ) quadratures (i.e. θ ′ = 0) are also
depicted.
Using these constructs, optical states can be illustrated in a semi-classical phasor
diagram as shown in Fig. 2.5. Shown in the figure is a coherent state of amplitude | α|
and phase θ. The circle centered at the expectation value of the state represents the fullwidth at half maximum of the measured probability distribution for all quadratures
X̂ (θ ′ ) (see Sec. 2.2.2). This picture is well suited to the visualization of the interference
of different optical modes, as in for example Fig. 4.1(b).
2.2.1
Basic quantum states
A given state is completely described by it density matrix ρ̂. This is written as a sum of
states:
(2.31)
ρ̂ = ∑ wι,ιι |ιih j|,
ι,j
2.2. Quantum mechanical description of light
15
where |ιi and | ji are pure states in a given orthonormal basis and wι,j are the expansion coefficients of the matrix. Those with nonzero off-diagonal elements are quantum
entities exhibiting nonclassical traits. The basis typically used in the expansion of the
density matrix is the Fock or number state.
Fock states
Possibly the most basic quantum state is the Fock or number state. This is an eigenstate
of the number or photon operator (Eq. 2.21) [50, 54, 51]. Considering a given polarization, n̂ acting on a Fock state has the effect:
n̂ |ni = n |ni ,
(2.32)
where n is the photon number of the state. These states form an orthogonal and complete set in which all optical states can be expanded. In phase space, the average values
of X̂ and Ŷ are always zero. Fock states |0i where n 6= 0 have been investigated in
the laboratory although their production is involved and only low photon numbers are
achievable, for example [55, 56]. A special and indeed ubiquitous Fock state is the vacuum state |0i where the photon occupation is zero:
n̂ |0i = 0 |0i .
(2.33)
Nonetheless, the variances of the amplitude quadratures of this state are nonzero:
∆2 X̂ = ∆2 Ŷ = 1,
(2.34)
a fact which arises from the noncommutation of the photon operators. Thus there is an
intrinsic zero-point or vacuum energy in contrast to classical predictions. The |0i state
is also a minimum uncertainty state in the quadrature amplitudes, as the uncertainty
relation of Eq. 2.29 is saturated with symmetrically distributed uncertainties.
Coherent states
The coherent states developed by R. Glauber in 1963 [57, 58] are a set of states that are
also minimum uncertainty states in the quadrature operators. These are given by a
coherent superposition of Fock states [50, 54, 51]:
1 2
|αi = e− 2 |α |
∞
αn
√
∑ n! |ni ,
n =0
(2.35)
where αn is the complex amplitude of |ni in phase space. These states are eigenstates of
the photon annihilation operator and can be created by the action of the displacement
operator D̂ (α) on the vacuum state:
D̂ (α) |0i = eαâ
† − α∗ â
| 0i = | α i .
(2.36)
16
2. Characterizing the state of light
The photon number distribution of coherent states is Poissonian, which for large photon
numbers can be approximated by a Gaussian distribution, where the average photon
number is |α|2 . They do not form an orthogonal or complete basis. Most importantly,
coherent states can be considered the most ”classical” quantum state. The output of shot
noise limited lasers, although phase randomized, is often approximated by the coherent
state.
2.2.2
Quasi-probability distributions
In analogy to classical probability distributions there exist probability distributions predicting the behavior of quantum mechanical systems [59, 50, 54]. A close correspondence between classical joint probability distributions describing the quadrature amplitudes and quantum probability distributions for the quadrature operators is impossible,
not least because of the operators’ noncommutation. Nevertheless it is often advantageous to use such semiclassical quasi-probability distributions to describe a state’s density matrix ρ̂ in phase space.
P-function
Considering the set of coherent states, a first distribution could be a projection onto this
set of states [60, 61]:
Z
ρ̂ =
d2 α P(α) | αihα| ,
(2.37)
where P(α) is the normalized probability of the P-representation over all phase space
(d2 α ≡ dRe(α) dIm(α)) described by the complex phase space parameter α. This distribution corresponds to normally ordered operators1 and diagonalizes the density operator in terms of coherent states, making it a popular distribution for calculations in
quantum optics [54, 62]. A pure coherent state in this representation is a delta function,
seen in Fig. 2.6(a), which allows experimental determination of P by the deconvolution
of the Wigner function (see the next section). This poses the question of how to describe
a squeezed state, which should then be more singular than a delta function due to its
having a quadrature with a variance reduced below that of a coherent state.
There are also a number of further mathematical problems with this representation [58, 54], which has prompted the extension of this function, resulting in the development of the positive-P representation (P+ ) [63, 64]. Here the parameter α and its
complex conjugate α∗ are exchanged for α, β which are independent complex parame1 Normally
ordered operators, i.e. ↠â, are fundamentally related to detection processes in quantum
mechanics [50].
2.2. Quantum mechanical description of light
17
ters only complex conjugate in the mean2 . The P+ distribution is [62]:
ρ̂ =
Z
2
d α
Z
| αi h β∗ |
d β P (α, β) ∗
,
h β | |αi
2
+
(2.38)
from which it is clear that P+ (α, β) projects onto diagonal as well as nondiagonal entries
of the density matrix. It is possible to choose α, β such that P+ (α, β) is always positive,
normalizable and well-behaved i.e. it is a real probability function. Additionally, the
evolution equations of P+ are true Fokker-Planck equations and so this distribution can
be used in solving for the system’s quantum evolution [62]. However, a given ρ̂ can be
in general described by more than one P+ by virtue of the increased degrees of freedom.
All distributions will however produce the same observable properties and this can be
viewed as an artifact of the nonorthogonality of coherent states. A physical interpretation of P+ is not immediate, though it has been shown that the canonical form [62] can
be thought of as the statistics of four detectors [65], not dissimilar to the interpretation
of the Q-function.
Wigner function
A mathematically better behaved quasi-probability distribution is the Wigner function
introduced in 1932 by E. Wigner [66] given here in the notation of Ref. [59]:
1
W (α ) =
π
Z
2
d ς Tr ρ̂ D̂ (ς) e
ας∗ − α∗ ς
1
=
π
Z
2
d 2 ς e − 2| α − ς | P ( ς ),
(2.39)
where ς is a complex parameter in phase space and D̂ is the displacement operator
(Eq. 2.36). The Wigner distribution can be described as a convolution of P(α) with a
Gaussian of unit width, i.e. the vacuum state [59], seen in Fig. 2.6(b). This convolution has the effect of smoothing the sharp characteristics of P(α), making W (α) a better
behaved function.
This function is defined to be real-valued and normalized but can be negative and
as such it is still a quasi-probability distribution. However, the Wigner distribution
is closely related to experimental measurements, insofar that the marginal integrals of
W (α) for a particular quadrature give the true probability density of that quadrature [67,
54]. Expressing W in terms of the θ = 0 quadratures, represented by the parameters x
and y, it is found:
′
W (X (θ )) =
+∞
Z
dx
−∞
2 This
+∞
Z
−∞
dy δ(X (θ ) − x cos(θ ) − y sin(θ ))W ( x, y),
function is similar to the R-function defined by R. Glauber in Ref. [60].
(2.40)
18
2. Characterizing the state of light
where W ′ (X (θ )) is the marginal probability distribution associated with the quadrature
given by θ. Alternatively this can be written in terms of rotated quadratures:
′
W (X (θ )) =
+∞
Z
dY (θ ) W (X (θ ), Y (θ )).
(2.41)
−∞
Such distributions are visualized in Fig. 2.5. Here the circle represents the phase space
contour of a coherent state, given by the full-width at half-maximum of W ′ . It is possible
to uniquely reconstruct the Wigner function of a given state from these experimentally
measured distributions (Sec. 2.3.4).
Q-function
A final semi-classical quasi-probability distribution which is frequently used is the Qfunction:
Z
2
1
1
Q (α ) =
d2 ς e−|α−ς| P(ς).
(2.42)
hα |ρ̂| αi =
2π
2π
This function is also normalized and real-valued. Similar to W, the Q-function can also
be thought of as a convolution of P(α) with a Gaussian but of width two, or equivalently
of W (α) with a unit Gaussian [59], visualized in Fig. 2.6(c). This distribution is then the
best behaved of the three functions presented here. It also represents the result of the
optimal strategy for the simultaneous measurement of two conjugate quadratures in
homodyne detection [68, 69].
(a)
(b)
(c)
0.3
0.2
P 0.1
0
0.3
0.2
W 0.1
0
0.3
0.2
Q 0.1
0
0
X
0 Y
0 Y
0
X
0 Y
0
X
Figure 2.6: The representation of a coherent state in phase space for the a) P-function, b) Wigner
function and c) Q-function.
The differences of the above discussed quasi-probability functions is illustrated in
Fig. 2.6 where a coherent state is shown for the P-function (a), Wigner function (b) and
the Q-function (c). These three semi-classical distributions are three standardly used
cases of a spectrum of distributions, as suggested by the relation of W (α) and Q(α) to
P(α). The larger family of functions is the so-called s-parameterized quasi-probability
2.2. Quantum mechanical description of light
19
distribution described by [59]:
W (α, s) =
1
π
Z
It follows that:
W (α, 1) ≡ P(α),
h
i
2
∗
∗
d2 ς Tr ρ̂D (ς)es| ς| eας −α ς .
W (α, 0) ≡ W (α),
W (α, −1) ≡ Q(α).
(2.43)
(2.44)
How do these functions compare in their description of nonclassical states, the focus
of this thesis? The P-function, which uses the coherent states as its basis, has severe
difficulties describing states with noise properties reduced below those of the coherent
states. Its extension to the P+ -function largely overcomes these problems but at the cost
of introducing further degrees of freedom. Further, these quasi-probability distributions
are either less or not feasible for experimental measurement. The Wigner distribution,
the convolution of P, can be easily determined from experiment. It has the problem of
exhibiting negative probabilities for certain states, e.g. Fock states [54, 55]. Squeezed
states however have strictly positive Wigner functions and the Wigner function is then
well-suited to such measurements. The Q-function is always positive regardless of input, but as the convolution of W, this comes at the cost of a loss of information about
the state.
2.2.3
Quantum polarization
The characterization of quantum polarization states relies on the measurement of the
quantum Stokes operators [49]. These Hermitian operators, as in the classical case, are
well suited to the description of two modes. As such, they are closely linked to the
definition of the (relative) quantum phase operator, for example Ref. [70, 71, 72]. They
are defined in analogy to their classical counterparts of Eq. 2.12 [73, 42]:
Ŝ0 = â†x âx + â†y ây ,
Ŝ2 = â†x ây + â†y âx ,
Ŝ1 = â†x âx − â†y ây ,
(2.45)
Ŝ3 = i (â†y âx − â†x ây ),
where âx and ây are two orthogonally polarized modes corresponding to, for example,
the laboratory reference frame. The dependence upon t, k and r is implicit. From this it
is seen that, within a factor of h̄2 , the operators Ŝ1 , Ŝ2 , Ŝ3 coincide with the angular momentum operators and thus obey the SU(2) Lie algebra [70, 72, 74]. The Stokes operators
commutators, for the same time, mode and position, are given by:
(2.46)
Ŝ0 , Ŝι = 0 and
Ŝι , Ŝ j = 2iǫι,j,k Ŝk ,
where ι, j, k = 1, 2, 3. These follow from the the Stokes operators’ definitions and the
noncommutation of the photon operators. The great advantage of this SU(2) algebra is
20
2. Characterizing the state of light
that manipulations in this algebra can be intuitively understood as unitary rotations or
phase shifts. Changes in linear polarization are carried out by movement along lines of
constant latitude, i.e. rotations about the Ŝ3 axis. In contrast changes in the ellipticity
occur as translations along lines of constant longitude, i.e. toward or away from the
poles. These transformations are most easily described in SU(2) space, in which particle number is conserved, and for polarization this corresponds to the Poincaré sphere.
Interestingly, it has been shown that the action of beam splitters and phase shifters (and
therefore also interferometers) as well as wave plates can be represented by rotations in
SU(2) group theory [75, 76, 77, 78, 47]. This indicates the equivalence of polarization dependent and independent components in optical applications, as described for example
in [23].
S3
(s+)
S1
(x)
S2
(+45°)
Figure 2.7: Representation of the variances of a polarization squeezed (lower right) and a coherent state (upper left) on the Poincaré sphere.
The commutation relations of Eq. 2.46 lead to Heisenberg inequalities and therefore
to the presence of intrinsic quantum fluctuations in analog to those in the quadrature
variables. However this fundamental noise depends on the mean polarization state:
2
∆2 Ŝι ∆2 Ŝ j ≥ ǫι,j,k hŜk i .
(2.47)
Accounting for this it is apparent that in a quantum picture the polarization state of
light can not be represented as a point on the Poincaré sphere. The intrinsic uncertainty
in the Stokes operators implies that, as with the quadrature operators, the measurement
of an ensemble of identically prepared states will lead to a measurement distribution in
Stokes space. This is visualized in Fig. 2.7, analogous to the phase space representation
2.2. Quantum mechanical description of light
21
of a coherent state in Fig. 2.5. Here the variances, i.e. full-width at half-maximum of the
marginal distributions, of a coherent and a polarization squeezed state are shown.
The very different form of the polarization operator uncertainty relation suggests
that the SU(2) minimum uncertainty or coherent state deviates from the the coherent
state as defined by R. Glauber (Eq. 2.35) [79]. The SU(2) state of minimum uncertainty,
i.e. the SU(2) coherent state, fulfills the Heisenberg uncertainties of Eq. 2.47 by an equal
sign for all operator combinations. Thus this basic quantum polarization state equivalently to satisfies ∆2 Ŝ1 + ∆2 Ŝ2 + ∆2 Ŝ3 = 2hŜ0 i. It is described in the number basis
by [80, 72]:
12
n 1
n
(2.48)
ξ n x |nx , n − nx i ,
|n, θ, ǫi =
2 n2 ∑
n
x
(1 + | ξ | ) n x = 0
where n is the total photon number, nx and n − nx are the photon numbers in the
modes x and y respectively. The angles of the Poincaré sphere θ, ǫ are related by
ξ = tan(θ/2)eiǫ . The mean values of such a state are:
hŜ1 i = n cos(θ ),
hŜ2 i = −n sin(θ ) sin(ǫ),
hŜ3 i = n sin(θ ) cos (ǫ).
(2.49)
Such states are perhaps more familiar in the context of spin where they are often referred
to as atomic coherent states. In optics they are not readily accessible in experiment. This
is highlighted by their relation to standard coherent states. It has been shown that two
mode quadrature coherent states α x , αy in the x and y polarization modes respectively,
can be written as a superposition of SU(2) minimum uncertainty states [72]:
where:
β=
q
−| β|2
α x , αy = e 2
| α x |2 + | α y |2
αy
|αy |
∞
βn
√
∑ n! |n, θ, ǫi ,
n =0
and
eiǫ tan(θ/2) =
(2.50)
αx
.
αy
(2.51)
It is then to be expected that the polarization variables of light exhibit noticeably different behavior compared with the quadrature variables. This and its advantages will
become more obvious during the discussions of detection (Sec. 2.3) and nonclassical
polarization states (Sec. 3).
A physical interpretation of the uncertainty relation of Eq. 2.47 is readily found for
bright fields where the fluctuations are small compared to the mean values [81]. Classically the polarization of a beam is given by the azimuthal angle θ and the ellipticity
angle ǫ of the polarization ellipse, Eq. 2.12. For intense beams, the Stokes operators can
be described by a classical mean value and a fluctuating noise operator, i.e. Ŝι = Sι + δŜι
where , hŜι i ≪ δŜι . Consider a beam with hŜ1 i 6= 0 and hŜ2 i = hŜ3 i = 0, which implies hθ i = hǫi = 0 where however the fluctuations are nonzero δθ, δǫ 6= 0. Using the
22
2. Characterizing the state of light
quantum counterpart of Eq. 2.12 and taking only the first order terms of the expanded
trigonometric functions, it is found that the fluctuations of the Stokes parameters can be
expressed as:
δŜ1 ≃ δŜ0 , δŜ2 ≃ 2Ŝ0 δθ and δŜ3 ≃ 2Ŝ0 δǫ.
(2.52)
A physical interpretation of these fluctuations, in reference to Figs. 2.1, 2.2, is the
following: The δŜ2 fluctuations are the jitter in the linear polarization coming from inphase (φx − φy = 0) photon fluctuations of the orthogonally polarized vacuum mode.
The fluctuations are described by δθ. The δŜ3 fluctuations represent the noise in the polarization ellipticity. These arise from out-of-phase (φx − φy = π2 ) photon fluctuations of
the vacuum mode given by δǫ. The fluctuations of Ŝ1 are not related to the polarization
properties but to the fluctuations of the polarization vector length or intensity of the
bright mode. From this analysis a more intuitive uncertainty relation for the polarization fluctuations can be derived:
∆2 θ ∆2 ǫ &
1
1
=
.
2
16hn̂i2
16hŜ0 i
(2.53)
The fluctuations of a coherent state are depicted in the spirit of Ref. [82] in Fig. 2.8.
Here the propagation direction is in z and the mean polarization is in x or +Ŝ1 . Such
a state has coherent photon number fluctuations in both the bright mode as well as in
the orthogonally polarized vacuum mode which gives rise to equal uncertainties in all
Stokes parameters. An equivalent visualization of a coherent state (of different mean
polarization) on the Poincaré sphere is shown in Fig. 2.7 for comparison.
dS1
z
p/2
x
de
dq
y
Figure 2.8: Jitter of the mean field’s polarization (in x) due to the fluctuations of the orthogonally
polarized vacuum mode in y for (a) a coherent state and (b) a polarization squeezed state.
Given the noncommutation of the Stokes operators, for completely polarized light it
is found that:
Ŝ0 (Ŝ0 + 2) = Ŝ12 + Ŝ22 + Ŝ32 .
(2.54)
2.3. Quantum noise detection
23
This is in contrast to the classical Stokes parameters which have only Ŝ02 on the left of
this equation (Eq. 2.12). The classical degree of polarization (Eq. 2.13) gives P = 1 for
coherent states, which clearly does not reflect the intrinsic noise in the Stokes operators [83]. P must be redefined and a number of measures of varying complexity have
been proposed [84, 85, 86]. The simplest and perhaps most fitting definition for a quantum degree of polarization for bright beams is [85, 72]:
q
hŜ1 i2 + hŜ2 i2 + hŜ3 i2
′
q
,
(2.55)
P =
hŜ0 (Ŝ0 + 2)i
Only for hŜ0 i → ∞ can a beam considered to be completely polarized and for this case
this quantum and the classical (Eq. 2.13) degree of polarizations coincide.
2.3 Quantum noise detection
In quantum optics the primary interest lies in manipulating and then determining the
quantum noise statistics of optical beams. The most complete state description is given
by the different quasi-probability distributions discussed in Sec. 2.2.2. Experimentally,
one cannot measure the noise properties at all frequencies and instead measurements
are made at a given frequency Ω with a bandwidth δΩ. These measurements are not at
the optical carrier frequency ω, which oscillates much too fast to be measured directly,
but instead at the optical sidebands separated from ω by ±Ω [51, 87]. Mathematically,
for monochromatic light, ignoring δΩ and assuming |α| ≫ |δα|, the measured mode
can be described by a classical carrier with quantum sidebands [88, 89]:
 (t) ∝ αeiωt + δ â− ei (ω −Ω)t + δ â+ ei (ω +Ω)t ,
(2.56)
where ± represent the sidebands which have been up-shifted and down-shifted relative to ω. The noise properties of the frequency mode at Ω are given by the the time
dependent evolution of the energy, or photon number, in the sideband. Expressing this
in terms of the measured quadrature operator X̂ ′ , this is given by [88]:
n̂(t) = †  ∝ α2 + αδ X̂ ′ ,
(2.57)
where this quadrature is described in terms of the up- and down-shifted quadrature
operators:
δ X̂ ′ = δ X̂+ + δ X̂− cos (Ωt) + α δŶ+ − δŶ− sin (Ωt) .
(2.58)
From this equation it is seen that what is measured is the beat frequency of the optical
sidebands with the optical carrier [87]. It also indicates what the physical meaning of
the experimental measurement of an optical quadrature is.
24
2. Characterizing the state of light
The most fundamental measurement setup is the direct detection of a beam’s intensity using a single photodetector. The latter can take many forms [51], but the most
practical for use with bright beams are based on pin-photodiodes using the photoelectric effect. These semiconductors absorb the incident light (up to a given maximum
wavelength) and produce an electrical current proportional to the incident optical field.
Classical and quantum treatments of this system give similar results [50, 51], namely:
P (t)
,
(2.59)
h̄ω
where P is the incident optical power (the classical counterpart of the photon density
operator) of a beam at frequency ω incident on a photodiode of quantum efficiency
η; e is the electron charge and h̄ is Planck’s constant divided by 2π. It is important
to note that the electrical signal can be treated classically, but nonetheless reflects the
quantum statistics of the measured optical quadrature. In real experiments η < 1 and
the information contained in the electrical signal is an imperfect (but most often still
reliable) reflection of the measured optical signal. It is important that the response of
the detector, the electrical signal, is linearly proportional to the optical signal for the light
intensities measured. An example of such a characterization is seen in Fig. 5.9, where
the detector AC signal at 17.5 MHz has been plotted against incident optical power. The
effect of photodiode efficiency is addressed in Sec. 2.3.1.
The noise spectrum, typically the parameter of interest, can be measured in a number
of ways (Sec. 5.8), the most simple of which is to use an electronic spectrum analyzer.
This device measures the spectral power density, i.e. the power in a given spectral
component of the signal. The time dependent photocurrent is defined by the Fourier
transform of its spectrum:
i (t) = eη
1
i T (t) = √
2π
+∞
Z
dω i T (ω )eiωt ,
(2.60)
−∞
where here it is understood that a measurement has been made over time T such that
i T (t) 6= 0 for |t| ≤ 21 T and otherwise i T (t) = 0. This parameter can be thought of as being equivalent to the resolution bandwidth (RBW) of a spectrum analyzer. To determine
the true average signal power hi2 (t)i an infinitely long measurement time is required to
include the contributions of all spectral components [90, 88]:
2
hi (t)i =
√
+∞
Z
hi2T (ω )i
,
T
T →∞
dω lim
2π
−∞
(2.61)
from which the spectral power density can be defined:
S(ω ) =
√
hi2T (ω )i
.
T
T →∞
2π lim
(2.62)
2.3. Quantum noise detection
25
The strength of this spectrum is generally proportional to the noise power in a given
optical sideband. In particular, Eq. 2.62 shows that the spectral power density S(ω )
is equivalent to a time domain measurement of the photocurrent’s noise variance. All
real observations of i (t) are made with a given bandwidth B or correspondingly time T,
where the total power is S(ω ) · B.
The photocurrent signal can be described by two orthogonal quadratures, i.e. a sine
and a cosine wave. Thus two measurements must be carried out to fully characterize
the electronic signal. Spectrum analyzers however most often scan over all electronic
quadratures (or phases) and display the maximum, minimum or average power measured S(ω ) [91] for the given frequency band Ω ± 12 B of the measured optical quadrature X̂ (θ ). This presents an efficient method for relative measurements of the spectral
power density [92], i.e. the observation of squeezing. In practice spectrum analyzers
measure the power of all electronic quadratures or phases and then record the maximum, minimum or average power. This method is however often insufficient for sensitive measurements of modulated signals where the electronic phase should be resolved.
In the following discussion it will be assumed that the optical beams are characterized
at a given sideband frequency which will not be stated explicitly.
2.3.1
Direct detection
Above the simple case of direct detection by a single detector has been outlined. There
are a number of more advanced schemes which can provide more information about
a quantum optical state. A flexible measurement setup is shown in Fig. 2.9. Here a
(lossless) beam splitter plays a pivotal role in extending the measurement capabilities.
This beam splitter is characterized by intensity transmission T and reflection R where
T + R = 1.The output modes of such a beam splitter can be written as:
ĉ =
√
T â +
√
Rb̂,
dˆ =
√
R â −
√
T b̂.
(2.63)
The limiting case of single detector, called ”direct detection”, with unity quantum efficiency is given by, for example, T = 1 resulting in the input of the mode ĉ = â to
detector 1. Replace detector 1 with a real detector with nonunity quantum efficiency
η1 < 1. It is well known that this is equivalent to assuming detector 1 to be perfect
(η1 = 1) and setting T = η where b̂ = δv̂ [54]. The latter represents the vacuum fluctuations of the empty port which are mixed with â, corrupting the measured signal [54].
This method can also be used to model the imperfect interference or mode matching of
two beams [93].
26
2. Characterizing the state of light
Det 2
+/d̂
ˆa (= a +dˆa)
a
ĉ
Det 1
ˆb (= a +dˆb)
b
Figure 2.9: A generalized measurement apparatus based on a beam splitter with intensity transmission T, reflection R.
Balanced direct detection
A setup frequently used to measure the intensity fluctuations of bright optical beams
is ”balanced direct detection” where the second input port is left empty. Referring to
Fig. 2.9, this corresponds to T = R = 12 , â is the signal under investigation and b̂ = δv̂ is
a vacuum state. It is found that the detected photon number in the linearized approximation3 and expressed in terms of the quadrature operators and the notation of Ref. [88]
is:
s
"
#
1
2
n̂1 = η1 α2a + α a δ X̂a + α a δ X̂v +
(2.64)
(1 − η1 )α a δ X̂v1 ,
2
η1
s
#
"
2
1
n̂2 = η2 α2a + α a δ X̂a − α a δ X̂v +
(1 − η2 )α a δ X̂v2 .
2
η2
Here δ X̂v1/2 are the vacuum contributions arising from the respective non-unity detector efficiencies η1/2 . Assuming η = η1 = η2 and that all vacuum contributions are
uncorrelated, the fluctuations of sum and difference photon numbers, δn̂+ and δn̂− , as
detected by the photodetectors are given succinctly by:
q
(2.65)
δn̂+ = δn̂1 + δn̂2 = ηα a δ X̂a + η (1 − η )α a δ X̂v+ ,
δn̂− = δn̂1 − δn̂2 = ηα a δ X̂v− .
The vacuum fluctuations have been collected together where X̂v+ depends on
X̂v1 , X̂v2 , X̂v while X̂v− depends only on X̂v1 , X̂v2 .It is apparent that δn+ is proportional
to the optical signal fluctuations, degraded by an amount determined by η. In contrast,
3 Terms
in δ ↠δ â are neglected.
2.3. Quantum noise detection
27
δn− contains only vacuum fluctuations. Thus the balanced detector has a great experimental advantage over the single detector variant insofar as it allows simultaneous
measurement of the signal and the corresponding quantum or shot noise level of a coherent state [94, 95]. This is also a handy way to determine if a given beam is shot noise
limited [96]. More rigorous investigations of this measurement scheme have shown
that the result obtained above holds well even for slightly unequal quantum efficiencies, small deviations in the detector balancing or slightly asymmetric beam splitting
ratios [94, 93].
In experiments the variance of these signals is usually measured. The variance of
a coherent or vacuum state is typically set to ∆2 (δ X̂coh ) ≡ 1 in phase space, making
relative noise measurements a natural choice:
∆2 n̂+
Rvar = 2 − = η∆2 (δ X̂a ) + 1 − η.
(2.66)
∆ n̂
This parameter gives the amount of excess or reduced fluctuations in a given beam.
It is the ubiquitous relative noise value found in all ”squeezing” or noise reduction
experiments (Sec. 3.1). This equation also serves as a basis for testing the authenticity
of the noise reduction observed from a given source. In particular, it can be excluded
that a given measured noise characteristic is of electronic origin, i.e. detector saturation.
Attenuating a squeezed signal beam, the measured relative noise should exhibit a linear
behavior when plotted against the linear attenuation [97, 98]. An experimental plot of
this relation is found in Fig. 6.10.
2.3.2
Homodyne detection
An entirely new class of detection schemes is found by setting b̂ to be a bright beam
of the same frequency coherent with â. This is the ”homodyne detector” originally developed in 1946 for use in microwave detection [99]. It exploits the interference of a
bright local oscillator beam b̂ with the weak signal â to allow measurement of all optical
quadratures of â. It was later suggested and investigated by Yuen and co-workers for
the detection of the quantum noise of optical beams [100, 101, 102, 103]. These early versions of the homodyne detector used only one detector, giving rise to the name ”singleport homodyne detector”. In reference to Fig. 2.9, this would use, for example, detector
1 when T ⋍ 1 and a very strong auxiliary beam, R · | αb |2 ≫ T · |α a |2 [102]. In this
limit the auxiliary beam can be treated classically, though it must be quantum noise
limited [103]. Mathematically this can be described by applying the displacement operator (Eq. 2.36) with αb to the signal such that αb ≫ α a . The signal beam is scanned by
altering the quadrature to which the displacement operator is applied, i.e. that relative
phase between signal and displacement, such that the signal quadrature of interest is
identical to the displaced quadrature.
28
2. Characterizing the state of light
The homodyne detector standardly used today is modified from the original by
use of a balanced beam splitter, earning the scheme the name ”balanced homodyne
detection”. This setup has the primary advantage that the local oscillator noise
is always canceled by taking the difference of two balanced detectors, assuming
the
|αb |2 ≫ | α a |2 [104, 105, 106]. Developing
description in the same manner as for the
balanced detector, but with b̂ =
fluctuations are:
αb + δb̂ eiθ , the sum and difference photon number
δn̂+ = ηαb δ X̂b ,
δn̂− = ηαb δ X̂a (θ ) +
q
(2.67)
η (1 − η )αb δ X̂v ,
where X̂a (θ ) is the quadrature given by the angle θ. In this limit of a bright local oscillator it is seen that all quadratures of the signal beam â can be measured in δn̂− regardless
of the precise state of the local oscillator. Additionally, if the local oscillator is known to
be coherent, then the δn̂+ channel gives the quantum noise limit, similar to the balanced
detector.
2.3.3
Polarization measurements
The measurement setups for the quantum polarization noise are identical to those used
for the classical determination of the polarization state of light [49, 107]. One need only
replace the classical modes by the corresponding operators in Fig. 2.3. To gain insight
into the nature of polarization measurements, it is instructive to draw parallels between
polarization measurements and the different quadrature detection schemes.
Preparing the mean polarization of a beam to coincide with the eigenmodes of a
given polarization measurement device, the entire intensity of the beam falls on one
detector. Consider a circularly polarized beam where only hŜ3 i 6= 0. This beam can be
described in terms of the right- and left-hand circular basis in the laboratory frame:
1
âσ+ = − √ (âx − i ây ) ↔ +Ŝ3 with hâσ+ i = −α,
2
1
√ (âx + i ây ) ↔ −Ŝ3 with hâσ− i = 0.
âσ− =
2
(2.68)
Quantum mechanically this mode contains a bright âσ+ = ασ+ + δ âσ+ mode and a vacuum âσ− = +δ âσ− mode. When this state is incident on a Ŝ3 measurement device the
âσ± modes are separated and detected independently. The sum and difference photon
number fluctuations are:
δn̂+ = δn̂− = ασ+ δ X̂σ+ .
(2.69)
2.3. Quantum noise detection
29
This is equivalent to direct detection via a single detector, where the sum and difference
channels give only information about the beam intensity fluctuations.
Another limiting case is when the beam’s intensity is divided equally between the
two detectors. This occurs when the measurement is of a polarization mode orthogonal
to the beam’s mean polarization, for example measurements of Ŝ1 and Ŝ2 on a mode of
mean polarization along Ŝ3 . Indeed all measurements in polarization bases orthogonal
to the mean Ŝ3 polarization will be balanced:
Ŝ⊥ = Ŝ1 sin θ + Ŝ2 cos θ,
(2.70)
where θ is an angle in phase space. The sum and difference photon number fluctuations
for such measurements are, using the Jones matrices and expressing the result in the
linear approximation:
δn̂+ = ασ+ δ âσ+ + ασ+ δ â†σ+ = ασ+ δ X̂σ+ ,
δn̂− = ασ+ δ âσ− e−iθ + ασ+ δ â†σ− eiθ
= ασ+ δ X̂σ− (θ )
= ασ+ δ X̂σ− (4Θ).
(2.71)
This can be interpreted by analogy to balanced homodyne detection: the bright beam σ̂+
is the local oscillator for the orthogonally polarized dark mode σ̂− . The phase between
σ̂+ and σ̂− varies with the rotation of the half-wave plate angle to give the phase space
angle θ = 4Θ. This is a unique feature of polarization measurements and has used to
great advantage in many experiments [23, 108, 109, 110, 32, 111, 20, 112, 113, 114, 115].
Measurements between the limiting cases outlined above are equivalent to a homodyne
detector wherein the beam splitting ratio and the relative phase is altered while conserving the total intensity [23]. Thus a general polarization measurement has the effect
of mixing quadrature noises of orthogonal modes.
2.3.4
Quasi-probability distribution reconstruction
Quantum state tomography is the process of reconstructing a function containing all
knowable information about a quantum state [54]. The result typically takes the form of
either a quasi-probability function [67, 78] or the density matrix ρ̂ [116, 117, 118]. Both
discrete (for example [117, 55, 56, 119, 120]) and continuous variable optical states [109,
121, 122, 123, 114] have been characterized in this fashion. Refs. [54, 124, 118] offer
comprehensive overviews of both experimental and theoretical work to date.
The process of tomography begins with the recording of a set of data corresponding
to measurements of different projections in the parameter space of interest. Typically
this is accomplished by homodyne detection of the quadrature variables, rigorously
30
2. Characterizing the state of light
investigated in Refs. [125, 126]. The reconstruction of a quantum state is accomplished
by applying one of a number of transformations or algorithms to the recorded data set,
outlined below. These techniques have been implemented in many quantum systems
besides optical states, for example: single atoms [127], molecular vibrational states [128]
and spatial beam profiles [129].
From Eq. 2.39, the Wigner function is uniquely related to the density matrix of the
state ρ̂. Thus a linear inversion via the inverse 2D Radon Transformation can produce
the Wigner function from a set of quadrature projection measurements [67, 54], implemented in the first quantum state tomography [109]. Wigner functions derived in this
manner unfortunately often exhibit some numerical artifacts arising from a necessary
low pass filtering in the algorithm [54, 118]. Still, the inverse Radon transformation
generates good results when using large numbers of quadrature projections and many
data samples. It also enjoys the particular advantage of being a standard and therefore extremely optimized algorithm, which is often used in, for example, medical imaging [54, 130]. The density matrix can also be derived in this manner, either from the
Wigner function [109] or directly from the data [116, 54]. The latter is more efficient and
less prone to statistical noise. The resultant density matrix can however be unphysical,
i.e. contain negative diagonal entries [118].
Two procedures, the maximum likelihood (MaxLik) and maximum entropy (MaxEnt), are more efficient at determining ρ̂. MaxLik finds the density matrix most likely
to generate the observed statistics in an iterative search process [131, 132, 117, 133]. It is
more physical than the inverse Radon transformation as it truncates the Hilbert space,
i.e. the Fock basis, rather than low pass filtering the integration kernel (see below). It
can also incorporate detector efficiency, as well as enforce unity trace and and positivity
of the diagonal elements of ρ̂ [119]. A further advantage is that high quality reconstructions are possible with much less data and computation time than the inverse Radon
scheme for systems containing few photons. This is most often the scheme of choice
for reconstructing a state’s density matrix [118]. The MaxEnt algorithm searches for
the ρ̂ which reproduces the observed data and maximizes the von Neumann entropy:
S(ρ̂ ) = −Tr(ρ̂ ln(ρ̂ )) [134]. This procedure also ensures unity trace and a physical density matrix. It can generate reasonable results with extremely little data, but the calculations become very involved for larger data samples [134, 118].
The fiber squeezed states of this thesis, which are inflicted with a large phase noise
(Sec. 4.1.5), typically contain hundreds of photons in their RF sidebands [9] compared
to the tens of photons in other experiments [109, 123, 55]. Thus the Wigner function of
these states, calculated via the inverse Radon transformation, is computationally more
convenient to determine than the density matrix. The tomography process begins by
measuring many different optical quadratures X (θ ) on an ensemble of identically prepared states. Histograms are derived from the measured photocurrents and these cor-
2.3. Quantum noise detection
31
respond to the marginal distributions of the Wigner function W ′ ( x (θ )) (Eq. 2.39). Sufficient data must be recorded to allow small bin sizes with large occupation numbers to
minimize statistical errors in the transformation [118]. The detected Wigner function is
reconstructed from the marginals by the inverse Radon transformation [54]:
1
W ( x, y) =
2π 2
Zπ
0
dθ
+∞
Z
−∞
dX (θ ) W ′ (X (θ )) · K ( P(θ ) − x ) ,
(2.72)
with the integration kernel:
1
K(x) =
2
+∞
Z
−∞
dς |ς| eiςx .
(2.73)
The kernel is unfortunately singular at x (θ ) = 0, and thus a mathematical trick must
be applied for the numerical calculation of the Wigner function. The infinite integration
limits in the kernel are replaced by ±kc , effectively low pass filtering the function. The
value of kc , is chosen to minimize numerical artifacts in W (Q, P) which take the form
of spurious ripples in the reconstruction. This however also has the effect of smoothing
the distribution. After evaluation of the integral over ±kc , the kernel can be expressed
as a power series near the origin and the singularity thus be circumvented. This method
is called the filtered back-projection [130].
The effect of detector loss on this procedure is to average out high frequency features
or convolute the quasi-probability distribution with a vacuum state. This is equivalent
to measuring a different s-parameterized quasi-probability distribution [135]. The original state could be obtained by performing a deconvolution, thereby compensating for
the measurement inefficiencies. Due to measurement noise and systematic errors, this
would lead however to significant statistical error and is thus impractical [125].
32
2. Characterizing the state of light
Chapter 3
Exploiting the quantum properties
of light
Quantum optics relies on the controlled manipulation of the quantum properties of optical beams. Such operations, based on the nonlinear interactions of light [136, 51], give
rise to a host of quantum states and phenomena without classical counterparts. These
include nonclassical ”squeezed” and ”entangled” states [89, 51] as well as quantum
memory [111] and quantum teleportation, among many others [5, 137, 4, 138]. In the
field of continuous variable quantum optics the most fundamental building block is the
squeezed state [5]. As such, this is the subject the first section where both the quadrature and polarization variables are considered (Sec. 3.1). Squeezed states can be used to
generate entanglement; the second part of this chapter discusses the implementation of
this in the polarization of light (Sec. 3.2). The final section (Sec. 3.3) introduces a quantum information protocol for the distillation of nonclassicality from a squeezed beam
which has been degraded by non-Gaussian noise acquired during, for example, generation or transmission. These discussions provide the background for the experimental
demonstrations found in Chapter 6.
3.1 Quantum noise reduction
Squeezing is a noise reduction of one of a set of conjugate variables (i.e. position or momentum) below a limit given by a Heisenberg uncertainty relation [139]. The Heisenberg relation is still fulfilled as the other conjugate variables exhibit a corresponding
increase in noise. As an idea ”squeezing” has existed for many decades [6] but was
first introduced to quantum optics in 1975-76 by Yuen [140]. Up to then observations
of the quantum nature of the electromagnetic field had focused on the generation of
34
3. Exploiting the quantum properties of light
sub-Poissonian statistics and antibunching of photons, for example [1]. Squeezing presented a new optical manifestation of quantum mechanics [141] and fired imaginations [136, 142], birthing the field of quantum optics. The discussion of squeezing here
focuses on the optical quadrature operators (Sec. 3.1.1) and the quantum polarization
variables (Sec. 3.1.2).
3.1.1
Quadrature squeezing
The first experimental realization exploiting the optical quadrature variables was accomplished in 1985 using nondegenerate backward four-wave mixing in Na atoms [2].
Quadrature squeezed states have since been demonstrated in many systems the most
important of which are: i) optical parametric oscillators (OPO) which have a χ(2) nonlinearity [96, 143], ii) atomic samples [2, 144, 145] or cold atoms [146] and iii) optical
fibers [147, 10] both of which have χ(3) nonlinearities.
The quadrature operators form a conjugate pair, and as such do not commute
(Eq. 2.29). The coherent states exhibit equal noise in all quadratures and thus saturates this uncertainty relation. There exist however states for which the noise in a given
quadrature lies below the coherent noise level, which can be seen as a natural boundary between classical and quantum optical statistics for the quadrature variables. These
noise reduced states obey:
∆2 X̂ (θ ) < 1 < ∆2 Ŷ (θ ),
(3.1)
where X̂ (θ ) exhibits a noise level reduced below that of a coherent state. X̂ (θ ) is thus
often referred to as being ”squeezed” below the classical or the shot noise limit. The
Heisenberg uncertainty relation in Eq. 2.29 is not violated by such states as the noise in
the orthogonal quadrature Ŷ (θ ) necessarily and correspondingly increases and thus it is
called the antisqueezed quadrature. For θ = 0, amplitude or photon number squeezing
is observed and the optical phase is antisqueezed. A squeezed state can still remain a
minimum uncertainty state if the product of the squeezed and antisqueezed variances
is one, as is the case for a pure Kerr effect or as often in OPOs [148]. The squeezed
states produced in optical fibers are however not minimum uncertainty states due to
the many other effects arising during pulse propagation, typically resulting in a large
excess phase noise (Sec. 4.1.5).
Mathematically squeezing is generated by the unitary squeezing operator [82, 6]:
1 ∗ â2 − 1 ξ â †2
2
Ŝ(ξ ) = e 2 ξ
where
ξ = reiθ .
(3.2)
Here r describes the degree of squeezing and θ the orientation of this squeezed ellipse
relative to the amplitude quadrature. Squeezing is produced by the action of Ŝ(ξ ) on
the vacuum state which can then be displaced in phase space by D̂. A single mode
3.1. Quantum noise reduction
35
W’(X)
Q
Y(
ˆ q)
W’(Y)
X(
ˆ q)
P
Figure 3.1: Representation of a prototypical squeezed state (bold lines) in phase space where
r ≈ 0.2 and θ = π8 , relative to the amplitude direction. The coherent state limit is also shown
(thin lines).
squeezed state is then described by:
|α, ξ i = D̂ (α)Ŝ (ξ ) | 0i .
(3.3)
The angle θ of Eq. 3.2 determines the quadrature which is squeezed by the factor r. The
squeezed and antisqueezed variances are:
∆2 X̂ (θ ) = e−2r ,
∆Ŷ (θ ) = e2r ,
(3.4)
where the variance of a coherent beam is set to unity. The mean photon number in a
squeezed state is [89]:
1 2
hn̂ i = |α|2 + sinh2 (r ) = |α|2 +
∆ X̂ + ∆2 Ŷ − 2 ,
(3.5)
4
from which it is clear that squeezing adds photons or energy to a field. As a result,
a squeezed vacuum has a nonzero average photon number, a distinctly nonclassical
phenomenon.
3.1.2
Polarization squeezing
The quantum polarization variables of light can also exhibit noise reduction, as first
suggested by A. S. Chirkin et al. in 1993 [149]. Their proposal for polarization squeezing
is similar to earlier remarks by D. F. Walls and P. Zoller concerning atomic spin squeezing [139] due to the similar mathematics, the SU(2) algebra. The 1990s saw many further
36
3. Exploiting the quantum properties of light
proposals involving quadratic [150, 151, 152, 153, 154, 155] and cubic [149, 156, 157, 158]
nonlinearities for the production of polarization squeezing.
The first experiment to exploit the quantum properties of polarization was performed by P. Grangier et al. in 1987 in a squeezed-light-enhanced polarization interferometer [108]. The first explicit demonstration was achieved by Sørensen et al. in the
context of quantum memory [110]. Such a promising application sparked intensified
interest, resulting in a series of theoretical investigations [85, 159, 49, 86, 160], reviewed
in [74, 161]. In the ensuing years polarization squeezing has been demonstrated in a
variety of systems: OPOs [19, 20, 162], optical fibers [163, 113] and cold atomic samples [21].
As discussed in Sec. 2.2.3, the polarization operator uncertainty relations are state
dependent, unlike those for the quadrature variables. That is, the uncertainty in two
parameters depend on the mean value of the third parameter. Since all polarization
transformations are unitary, one can always define a Stokes parameter basis in which
only one polarization operator has a nonzero mean value. Consider for example a
polarization state given by the polarization vector Ŝ, where hŜι i = hŜ j i = 0 and
hŜ i = hŜk i = hŜ0 i 6= 0. The only nontrivial Heisenberg inequality then reads:
∆2 Ŝι ∆2 Ŝ j ≥ ǫι,j,k |hŜk i|2 = |hŜ0 i|2 .
(3.6)
This mirrors the quadrature uncertainty relation, and polarization squeezing is then
similarly defined for ι 6= j 6= k:
∆2 Ŝι < |hŜk i| < ∆2 Ŝ j .
(3.7)
The Ŝι and Ŝ j polarization operators are conjugate because of the third operator Ŝk has
a nonzero mean value. Their definition in Stokes space is however not unique and
there exist an infinite set of bases Ŝ⊥ (θ ), Ŝ⊥ (θ + π2 ) that are perpendicular to the state’s
classical excitation Ŝ, for which hŜ⊥ (θ )i = hŜ⊥ (θ + π2 )i = 0. All these operator pairs
exist in the Ŝι − Ŝ j ”dark plane” where θ is an angle in this plane defined relative to Ŝι .
A general dark-plane operator is described by:
Ŝ⊥ (θ ) = cos(θ )Ŝι + sin(θ )Ŝ j .
(3.8)
Polarization squeezing is then generally given by:
∆2 Ŝ⊥ (θ ) < |hŜ0 i| < ∆2 Ŝ⊥ (θ + π/2),
(3.9)
where Ŝ⊥ (θ ) is the squeezed parameter and Ŝ⊥ (θ + π2 ) the antisqueezed parameter.
As shown in Sec. 2.3.3, the difference signal of a polarization detector measuring
dark Stokes parameters observes the noise of the mode orthogonally polarized to the
3.1. Quantum noise reduction
37
classical excitation, above given by ±Ŝk . As an example consider a bright x-polarized
beam (+Ŝ1 ) with âx = α + δ âx and an orthogonally polarized (−Ŝ1 ) dark mode in a
vacuum state ây = δ ây . The dark plane parameters correspond to the quadratures of
the dark y-polarization mode:
δŜ2 = α x (δ â†y + δ ây ) ≡ α x δ X̂y ,
(3.10)
δŜ3 = iα x (δ â†y − δ ây ) ≡ α x δŶy ,
where the Stokes operator definitions of Eq. 2.46 have been used in a linearized form.
Considering the physical interpretation of polarization squeezing in this example, it
then becomes clear that polarization squeezing is equivalent to vacuum squeezing in the orthogonal polarization mode:
∆2 Ŝ⊥ (θ ) ≤ |hα x i|2
⇔
∆2 X̂y (θ ) ≤ 1.
(3.11)
This is pictorially represented in Fig. 3.2, in the spirit of early quadrature squeezing
pictures [82], where a coherent and Ŝ2 squeezed state, both of which are Ŝ1 polarized, are
shown. In the case of the coherent state, Fig. 3.2(a), the orthogonally polarized vacuum
mode (in y) has an equal, coherent fluctuations for all phases. In contrast, the orthogonal
vacuum mode of the polarization squeezed state, Fig. 3.2(b), exhibits a phase dependent
noise. The out-of-phase (relative the bright carrier in x) quadrature, the Ŝ2 operator,
shows a reduced noise. The in-phase quadrature, Ŝ3 , exhibits increased or antisqueezed
noise. For clarity the noise in the photon number of the mean field has been omitted
from both figures.
(a)
(b)
dS1
dS1
z
p/2
z
p/2
x
x
de
dq
y
dq
y
de
Figure 3.2: Visualization of the polarization noise of (a) a coherent state and (b) a Ŝ2 polarization
squeezed state.
From this analysis it follows that polarization squeezing can be achieved by mixing
a bright beam with an independently produced squeezed vacuum beam of orthogonal
polarization [108, 110, 20]. On the other hand, it has also been shown that polarization
38
3. Exploiting the quantum properties of light
squeezing can also be produced directly, either in optical fibers [164, 32, 113] or in cold
atomic media [21, 165]. Fiber based squeezers are however less suited to the generation
of vacuum squeezing. It is then instructive to investigate the polarization noise properties of different combinations of coherent and amplitude squeezed states, which are
readily generated using optical fibers. The consideration of five different cases will serve
to illuminate the experimental results presented in Sec. 6. Whilst cases of particular polarization are considered, a straightforward generalization to all other polarizations is
readily made as polarization transformations are simply unitary rotations in the SU(2)
space.
Case 1
A bright x-polarized (+Ŝ1 ), coherent state is given by, for example: âx = α + δ âx with
∆2 X̂x (θ ) = ∆2 X̂ = 1 and α ∈ R. The Stokes variances of this state are found to be:
∆2 Ŝ0 = ∆2 Ŝ1 = ∆2 Ŝ2 = ∆2 Ŝ3 = |hαi|2 ∆2 X̂ = |hαi|2 ,
(3.12)
where the quadrature variances of a coherent state are defined as being unity. The variances of this state are pictorially represented in the Stokes space in Fig. 3.3(a), generating
a circular polarization uncertainty shape. While this state is of minimum uncertainty in
the quadrature variables, in the polarization variables minimum uncertainty is only
seen in the Stokes operators orthogonal to the mean field, here the Ŝ2 − Ŝ3 plane. The
other commuting operator pairs exhibit excess noise relative to the polarization minimum uncertainty state. This highlights the difference between quadrature and SU(2)
minimum uncertainty states, as discussed in Sec. 2.2.3.
Case 2
The second case under examination is that of a single, x-polarized (+Ŝ1 ), amplitude
squeezed beam: âx = α + δ âx where ∆2 X̂x < 1 < ∆2 Ŷx and α ∈ R. The orthogonally
polarized y mode is in a vacuum state: ây = δ ây where ∆2 X̂y (θ ) = 1. The only nonzero
Heisenberg inequality is:
∆2 Ŝ2 ∆2 Ŝ3 ≥ |hŜ1 i|2 = |hαi|4 .
(3.13)
Thus polarization squeezing can only be seen in the Stokes plane given by Ŝ2 − Ŝ3 ; all
other Stokes operator pairs commute. The Stokes variances are:
∆2 Ŝ0 = ∆2 Ŝ1 = |hαi|2 ∆2 X̂x < |hαi|2 ,
∆2 Ŝ2 = ∆2 Ŝ3 = |hαi|2 ∆2 X̂y = |hαi|2 .
(3.14)
3.1. Quantum noise reduction
39
These variances are represented in Fig. 3.3(b), where they take the form of a pancake.
The inequality of Eq. 3.13 is thus an equality and additionally no polarization squeezing is observed. Reduced noise is seen in Ŝ0 and Ŝ1 which are commuting operators,
meaning that only amplitude squeezing is present. This follows from the definition of
polarization squeezing as squeezing of the orthogonal dark polarization mode; here the
orthogonal mode is coherent and thus no polarization squeezing is seen. In the limit
of infinite squeezing (∆2 X̂x → 0), this state tends toward a polarization minimum uncertainty state. This demonstrates the intuitive conclusion that merely adding an extra
vacuum mode to a quadrature squeezed beam does not lead to multi-mode effects such
as polarization squeezing.
(a)
(b)
S3
S3
S2
S1
S1
S2
S1
S3
S3
S2
S2
S1
(c)
S3
S3
S1
S2
S2
S1
Figure 3.3: Shapes of the Stokes parameter variances of a) a coherent state, b) an amplitude
squeezed beam, and c) a polarization squeezed state.
Case 3
In this example a x-polarized (+Ŝ1 ), amplitude squeezed beam, âx =
√α
2
+ δ âx where
∆2 X̂x < 1 < ∆2 Ŷx , is overlapped with a y-polarized (−Ŝ1 ), equally intense, coherent
beam, ây = √α + δ ây where ∆2 X̂y (θ ) = 1. For α ∈ R, i.e. zero relative phase shift, the
2
state is Ŝ2 polarized and following nonzero uncertainty relation exists:
∆2 Ŝ1 ∆2 Ŝ3 ≥ |hŜ2 i|2 = |hαi|4 .
(3.15)
As hŜ2 i 6= 0, the conjugate variables are Ŝ1 and Ŝ3 . The corresponding variances of the
Stokes operators are:
∆X̂x + 1
< |hαi|2 ,
2
∆Ŷx + 1
= |hαi|2
> |hαi|2 .
2
∆2 Ŝ0 = ∆2 Ŝ1 = ∆2 Ŝ2 = |hαi|2
∆2 Ŝ3
(3.16)
40
3. Exploiting the quantum properties of light
Similar to Case 2 two parameters, Ŝ0 and Ŝ2 , exhibit amplitude squeezing. The difference lies in the fact that the variance of the noncommuting operator Ŝ1 lies below the
coherent state limit. Thus polarization squeezing is observed in Ŝ1 , and Ŝ3 must be
correspondingly antisqueezed. Equivalently, one can now speak of an orthogonally polarized squeezed vacuum mode. This is a result of the interference of beams of different
polarization. The state’s variances are a cigar shape as depicted in Fig. 3.3(c). It must
be noted that, despite exhibiting squeezing in three of the four Stokes parameters, this
squeezing is less than that of the initial amplitude squeezed beam. This is because of the
mixing with the coherent state, the effect of which is clear from Eq. 3.16. For example,
∆2 Xx = 12 , i.e. squeezing of −3 dB, gives normalized variances of only 14 in Ŝ0 , Ŝ1 and
Ŝ2 , i.e. -1.25 dB squeezing.
Case 4
The fourth case is similar to the previous one, but here both beams are
√α + δ â x/y where
amplitude squeezed with equal magnitude:
âx/y
=
∆2 X̂
∆2 X̂
∆2 Ŷx/y
2
∆2 Ŷ
<1<
=
and α ∈ R. The resulting beam has a mean
x/y =
polarization along Ŝ2 , and thus the inequality of Eq. 3.15 holds. The Stokes variances
are then:
∆2 Ŝ0 = ∆2 Ŝ1 = ∆2 Ŝ2 = |hαi|2 ∆X̂ < |hαi|2 ,
∆2 Ŝ3 = |hαi|2 ∆Ŷ > |hαi|2 .
(3.17)
This can also be depicted by the cigar shape of Fig. 3.3(c), where the cigar is now more
elongated than in Case 3. Polarization squeezing is again observed in Ŝ1 with Ŝ3 as its
antisqueezed conjugate. The important difference here is that Ŝ0 , Ŝ1 and Ŝ2 are squeezed
to the same extent as the individual amplitude squeezed beams.
Case 5
The final case considers again two equally bright, independent beams polarized in x and
y respectively (±Ŝ1 ). They are quadrature (instead of amplitude or number) squeezed
for X̂x/y (θsq ) where θsq is an angle relative to the amplitude quadrature. These beams
are described by: âx/y = √α + δ âx/y where α x = iαy ∈ R, i.e. with π2 relative phase shift.
2
In correspondence with experiment ∆2 X̂ (0) = 1 and a relative phase of π2 is introduced
between the squeezed modes âx and ây , depicted in Fig. 3.4. This results in a circularly
polarized mode with âσ+ as the mean field and âσ− as the orthogonal dark or vacuum
mode:
1
1
âσ+ = − √ (âx − i ây ) âσ− = √ (âx + i ây ).
(3.18)
2
2
3.1. Quantum noise reduction
41
Q
qsq
X
ˆ x,q
sq
ây
âx
P
qsq
X
ˆ y,q
sq
Figure 3.4: Diagram of two orthogonally polarized (x, y), quadrature squeezed states with a π2
relative phase shift; the squeezed quadrature (defined with respect to the mean field of each mode)
is the same for the two modes.
In the Poincaré sphere the mean vector is along Ŝ3 and the relevant Stokes uncertainty
relation is:
∆2 Ŝ1 ∆2 Ŝ2 ≥ |hŜ3 i|2 = |hαi|4 .
(3.19)
Because the input beams are quadrature squeezed the squeezed Stokes operator is not
Ŝ1 or Ŝ2 . The squeezed and antisqueezed parameters, generally given by Ŝ(θ ), are located in the dark plane defined by Ŝ1 − Ŝ2 :
Ŝ(θ ) = cos(θ ) Ŝ1 + sin(θ ) Ŝ2 .
(3.20)
The assumption of quadrature squeezed states, here with the initial condition of
∆2 X̂ (0) = 1, implies that polarization squeezing is observed for θsq 6= 0, since
∆2 Ŝ1 = ∆2 Ŝ2 = |hαi|2 . The shape of variances is similar to that of Cases 2 and 3,
Fig. 3.3(c), but the shape has been rotated in the Ŝ1 − Ŝ2 plane by the squeezing angle
θsq .
From previous analysis (Eqs. 2.71, 3.11), the Ŝ(θ ) defined here is related to the
quadrature fluctuations of the dark mode âσ− :
α
δŜθ = −αδ X̂σ− (θ ) = − √ δ P̂x (θ ) − δ Q̂y (θ ) ,
2
(3.21)
where the absolute quadrature variables P and Q have been used to allow for the
beams’ relative phase shift. Assuming the two quadrature noises are uncorrelated and
squeezed for the same angle θsq relative to the amplitude quadrature, one can write:
X̂x/y (θ ) = X̂ (θ ). The resulting polarization or vacuum squeezing is then characterized
42
3. Exploiting the quantum properties of light
by:
∆2 Ŝ(θsq ) = |hαi|2 ∆2 X̂ (θsq ) < |hαi|2 ,
∆2 Ŝ(θsq + π/2) = |hαi|2 ∆2 Ŷ (θsq ) > |hαi|2 .
(3.22)
3.2 Polarization entanglement
Continuous variable entanglement, also called two-mode squeezing, is a quantum phenomenon which has become a key resource in quantum information processing [137].
Historically the concept of entanglement arose from the consideration of nonlocal correlations between systems which have in the past interacted, but since have been spatially separated. Confronted with this nonintuitive physical situation, A. Einstein,
B. Podolsky and N. Rosen were forced to question the reality of quantum mechanics,
summarized in their ”EPR Paradox” of 1935 [166]. Early experiments generating entangled states focused on violations of Bell’s inequalities [167] as a signature of these
states [168, 169]. In contrast to the original proposal, these experiments exploited discrete variables of photon polarization. The first entanglement criterion for the optical
quadrature operators was proposed in 1988, the so-called EPR criterion [170, 171]. This
criterion checks if one can infer the value of a noncommuting observable of one subsystem from a measurement on the other subsystem of the pair to a precision better than
given by the respective Heisenberg uncertainty relation [170]. This was the basis of the
first experiment in the continuous variable regime [3]. A later, second entanglement criterion is rooted in the fact that separable, i.e. not entangled, systems can be written as
the convex sum of product states of two subsystems A and B. Describing the nth component of the ι-system state by the density matrix ρι,n , the condition for separability can
be written as [172]:
(3.23)
ρ̂ = ∑ pn ρ̂ A,n ⊗ ρ̂ B,n .
n
First developed for discrete variables [173, 174, 175] and then extended to the continuous quadrature variables [24, 176], it represents a more comprehensive definition of
entanglement. Whilst further entanglement criteria have been developed [177, 178, 179,
180, 181] with overviews given in [182, 137], the EPR and nonseparability criteria remain the experimentally most feasible witnesses in symmetric Gaussian systems [183].
These have been demonstrated in the many different experimental systems also used to
generate squeezed states, for example [3, 14, 22].
Of interest here is the extension of these ideas to the polarization variables [49, 184,
185, 161] and demonstrated by a number of groups [20, 186, 112, 22]. Polarization entanglement of intense light fields can be characterized by the EPR and nonseparability
criteria, both extended from the characterization of quadrature entanglement.
3.2. Polarization entanglement
43
Output A
Input I
50:50
Output B
Input II
Figure 3.5: Scheme for the generation of entanglement.
The EPR criterion is broken if a measurement on one subsystem gives information
about a second subsystem more accurately than classically possible given the governing
uncertainty relations. The measure of the precision of such inference is the conditional
variance ∆2cond (Ŝι,A |Ŝι,B ) for the Stokes operator Ŝι measured in subsystems A and B:
∆2cond
Ŝι,A |Ŝι,B =
D
δŜι,A
2 E
−
|hδŜι,A δŜι,B i|
2
2
h(δŜι,B ) i
,
(3.24)
where it is assumed possible to write the ιth Stokes operator of subsystem A as:
Ŝι,A = hŜι,A i + δŜι,A .
(3.25)
A polarization state is EPR entangled if [49]:
2
∆2cond (Ŝι,A |Ŝι,B )∆2cond (Ŝ j,A |Ŝ j,B ) < ǫι,j,k |hŜk,B i| ,
(3.26)
for ι, j, k = 1, 2, 3.
The nonseparability criterion for the polarization operators takes the form of [49]:
∆2 Ŝι,A − Ŝι,B + ∆2 Ŝ j,A + Ŝ j,B < 2ǫι,j,k | Ŝk,A | + | Ŝk,B | .
(3.27)
The sum (difference) variances of the stokes operators on the left hand side quantifies the quantum correlations between subsystems A and B in the respective conjugate
variables. The right hand side provides the reference quantum limit according to the
uncertainty relation Eq. 2.47, see also [14, 20].
States which are entangled by the criteria given in Eqs. 3.26, 3.27 can be generated
by the interference of a polarization squeezed light field with an auxiliary mode on a
polarization independent 50:50 beam splitter [187], as shown in Fig. 3.5. The auxiliary
mode can take many forms including a vacuum, a bright coherent beam or a polarization squeezed beam. There is a certain analogy to the discussion of polarization squeezing in Sec. 3.1.2, where the polarization properties of different beam combinations were
considered.
44
3. Exploiting the quantum properties of light
The polarization entanglement presented in this thesis was generated by the particular case in which Input I was a polarization squeezed beam and Input II was a vacuum
state. The advantage of this configuration is experimental ease and resource-efficient
production, grounded in the significant effort required to implement a second bright
(squeezed) beam. The bright beam could be composed of two orthogonally polarized
amplitude squeezed beams of equal intensity and zero relative phase, resulting in a
Ŝ2 polarized beam, as described in Case 4 of Sec. 3.1.2. The conjugate operators of the
polarization squeezed mode are Ŝ1 < Ŝ3 and thus outputs A and B exhibit quantum correlations in these operators. It is seen that the variance of the sum of the Ŝ3 parameters
is:
(3.28)
∆2 Ŝ3,A + Ŝ3,B = |hαi|2 ,
which corresponds to the shot noise of a coherent beam. The Ŝ2 variance term is, using
the notation of Case 4 where ∆2 X̂ < 1 is a squeezed quadrature variance:
∆2 Ŝ1,A − Ŝ1,B = |hαi|2 ∆2 X̂ < |hαi|2 ,
(3.29)
which lies below the shot noise level. The nonseparability criterion in this case is given
by:
(3.30)
∆2 Ŝ3,A + Ŝ3,B + ∆2 Ŝ1,A − Ŝ1,B = |hαi|2 ∆2 X̂ + 1 < 2|hαi|2 .
Thus the sum of the two variances lies below the shot noise limit and the beam is polarization entangled.
By replacing the vacuum of Input II by an intense coherent field or a polarization
squeezed light field, ∆2 Ŝ3,A + Ŝ3,B would also show nonclassical correlations. This
can be interpreted in analogy to the polarization squeezing Cases 2-4 discussed in
Sec. 3.1.2. Using a bright coherent state for Input II allows observation of quantum
correlations in both conjugate polarization variables, here Ŝ1 , Ŝ3 . These correlations are
however less than the degree of squeezing in the polarization squeezed Input I. If both
inputs are polarization squeezed with similar photon statistics observation of subshot
noise correlations in both conjugate variables equal to the degree of input squeezing is
possible.
3.3 Distillation of squeezing
Nonclassical states such as continuous variable entangled and squeezed states serve as
enabling resources for many quantum information protocols [137] as well as for highly
sensitive measurements beyond the shot noise limit [188]. The efficiency of these applications relies crucially on the state’s nonclassicality (i.e. the degree of single- or twomode squeezing). Therefore, uncontrolled and unavoidable interaction of the system
3.3. Distillation of squeezing
45
with the environment and the resultant loss of squeezing in generation or transmission
should be combated. This can be done by using a distillation protocol which probabilistically selects out squeezed states from a mixture, hereby increasing the output state’s
squeezing.
Various protocols exploiting either a cross Kerr nonlinearity [24, 27] or single photon
projection [25, 26] have been proposed to probabilistically distill two-mode squeezed
Gaussian states. Although a first step toward the implementation of a Gaussian state
distillation protocol using single photon projection has been taken [28], full realization
remains practically challenging since the protocols rely on non-Gaussian transformations. In fact it has been proven that the distillation of two-mode squeezed Gaussian
states by means of local Gaussian operations is impossible [189, 190, 191]. Likewise, it
has been shown that it is impossible to increase the squeezing of a single-mode Gaussian
squeezed state using only Gaussian operations and homodyne detection [192].
There has however been no work devoted to the distillation of CV Gaussian states
corrupted by non-Gaussian noise. This occurs naturally in channels with fluctuating
properties, i.e. gain or phase. The time dependent noise on a system in say the X
quadrature can be written as:
X̂ (t) = X̂ (0) · η (t)eiφ(t) + N̂.
(3.31)
The time dependent gain η and the phase φ factors result in a non-Gaussian output, even
when the noise in these parameters is Gaussian distributed. The factor N is an additive
noise which, when Gaussian, will give rise to a Gaussian output. Transmission through
a channel subject to noisy gain or phase shifts will produce mixture noise [193, 194, 30],
an example of which is the fading channel [29, 195, 196]. Therefore, as an extension to
the work on Gaussian noise, the distillation of single-mode Gaussian squeezed states
with superimposed non-Gaussian noise using linear optics and homodyne detectors has
been investigated and demonstrated.
The set of non-Gaussian noise sources is large. Thus, in the spirit of early investigations of mixture noise [193], a simple yet exemplary case of discrete noise is
treated. The methods discussed here are later experimentally demonstrated and generalized to the case of continuous noise in Sec. 6.5. In particular, the continuous noise
described by the so-called fading channel, a model of turbulent atmospheric channels [29, 30, 195, 196, 197]. The theoretical considerations here begin by considering
a squeezed vacuum state (experimentally implementable as a polarization squeezed
beam) perturbed by phase space kicks or jitter, attributable to imperfect generation or
transmission through a noisy channel. Assuming these perturbations cause a linear
phase space displacement, a convex mixture of two Gaussian squeezed states is created:
W ( x, y) = (1 − γ)W0 ( x, y) + γW1 ( x, y),
(3.32)
46
3. Exploiting the quantum properties of light
where γ is the probability of displacement of the disturbed state, described by the
Wigner function W1 , where W0 is the Wigner function of the initial state. It should
be noted that this state is a classical mixture in contrast to quantum superposition states
such as Schrödinger cat states [198, 199]. The individual constituents of the above mixture (ι = 0, 1) are described by the Wigner functions:
(y− ȳ )2
( x − x̄ )2
exp − 2∆2 Xι − 2∆2 Yι
sq
sq
q
Wι ( x, y) =
.
(3.33)
2π ∆2 Xsq ∆2 Ysq
Here x and y are the amplitude and phase quadratures. ∆2 Xsq and ∆2 Ysq are the corresponding variances of the input state. x̄0 = ȳ0 = 0 and x̄1 , ȳ1 are the mean values of the
initial and displaced squeezed states respectively. It is assumed that the two individual Gaussian states are equally squeezed in x: ∆2 Xsq < 1, where ∆2 Xsq ∆2 Ysq ≥ 1. The
first two moments of the amplitude quadrature of the mixture W ( x, y) are h x i = γ x̄1
and h x2 i = ∆2 Xsq + γ x̄12 , and thus the variance of the amplitude quadrature of the corrupted state of Eq. 3.32 is:
∆2 X = ∆2 Xsq + γ(1 − γ) x̄12 .
(3.34)
The second term here originates from the noise and degrades the squeezing. The aim of
the protocol is to recover the squeezing by distilling the initial squeezed state from this
mixture.
Preparation
Squeezer
Phase
modulator
Distillation
Signal
Verification
Conditioning
Tap
Figure 3.6: Schematic of the distillation of a non-Gaussian mixture of squeezed states. The
protocol consists of preparation, distillation and verification and steps.
The distillation protocol is shown in Fig. 3.6, and consists of three steps: i) preparation, ii) distillation and iii) verification. This demonstration uses a non-Gaussian mixed
state generated from an appropriately modulated polarization squeezed state, the latter of which is mathematically equivalent to a squeezed vacuum state (Sec. 3.1.2). This
prepared state is incident upon a beam splitter with intensity reflection R and transmission T, producing correlated output states, referred to as the tap and signal beams.
3.3. Distillation of squeezing
47
A given quadrature of the tap beam is measured. In the distillation, the signal beam is
conditioned on the tap measurement. Here the signal is selected only if the outcome lies
above a given threshold value, similar to the procedure used in Ref. [200]. Due to the
correlations between the signal and tap beams, the scheme accomplishes a probabilistic
distillation of the noisy input state, which can be ascertained in the verification step. A
similar strategy was proposed to purify decohered Schrödinger cat states [201].
Both the original and displaced squeezed states are squeezed in the same absolute
phase space quadrature. Thus the amplitude and phase quadrature components of the
input and output Wigner functions can be written separately:
W ( xs , xt , ys , yt ) = W ( xs , xt )W (ys , yt ).
(3.35)
The Wigner function of the beam splitter output is found by replacing x and y in
Eq. 3.33 by expressions in terms of the beam splitter outputs, i.e. the
√ tap and
√ signal
variables.
√ The√tap beam splitter generates a signal beam with xs = Tx + Rxv and
ys = Ty + Ryv and a corresponding tap beam where xv and yv are uncorrelated
vacuum contributions. The Wigner function of the output is then given by:
√
√
√
√
( Tys + Ryt − ȳι )2
( Txs + Rxt − x̄ι )2
exp − 2(∆2 X +∆2 X ) − 2(∆2 Y +∆2Y )
s
s
t
q t
Wι ( xs , xt , ys , yt ) =
.
(3.36)
2
2
2π ∆ Xsq ∆ Ysq
The tap measurement on the Y quadrature is described by the projector |yt ihyt |. The
measurement of yt on the beam splitter output followed by a post selection of the signal
beam with threshold yth is described by:
1
W ( xs , ys ) =
Π
+∞
Z
dyt
yth
+∞
Z
dxt W ( xs , xt , ys , yt ).
(3.37)
−∞
In this process all amplitude information in the tap beam is lost. The phase quadrature
distribution is obtained via the measurement over yt which additionally contains the
post selection by setting the lower integration bound to yth . The conditioned signal
Wigner function can also be written as:
i
1h
W ( xs , ys ) =
(3.38)
(1 − γ) G0 W0 ( xs , ys ) + γG1 W1 ( xs , ys ) ,
Π
where Π is the success probability:
Π=
+
R∞
dyt W ( xs , xt , ys , yt )
yth
+
R∞
−∞
dyt W ( xs , xt , ys , yt )
= (1 − γ)G0 + γG1 .
(3.39)
48
3. Exploiting the quantum properties of light
It then follows that Gι the filter function of the ιth stat, is:
"
√
1
yth − Rȳι
p
Gι = Erfc
2
2∆2 Ŷt
#
=
+
R∞
dyt Wι ( xs , xt , ys , yt )
yth
+
R∞
,
(3.40)
dyt Wι ( xs , xt , ys , yt )
−∞
which incorporates the effect of the tap measurement and conditioning in dependence
on the threshold value, measured quadrature and the variance of this quadrature. It
determines how much of the data of original and displaced states is retained after the
post selection process. Since the goal of the distillation is to recover the initial squeezing, only the marginal quadrature distribution associated with the squeezed quadrature
x must be considered. Measuring the phase quadrature in the tapped signal yt , the resulting probability distribution of the squeezed quadrature in the signal is found by
integrating over xs and reads:
i
1h
′
′
W ( xs ) =
(1 − γ) g0 W0 ( xs ) + γg1 W1 ( xs ) ,
Π
′
(3.41)
where Π = (1 − γ) g0 + γg1 and the individual marginals W0′ ( xs ) and W1′ ( xs )
are Gaussian
functions with variance ∆2 Xs = T∆2 Xsq + R∆2 Xv and centered at
√
x̄0 = 0, T x̄1 , respectively. Analog to Eq. 3.40, the filter function for the marginal distributions is:
"
#
√
1
yth − Rȳι
p
gι = Erfc
.
(3.42)
2
2∆2 Yt
2
Here yth is the post selection threshold and ∆2 Yt = R∆2 Y√
Afsq + T∆ Yv .
ter distillation the first two moments of the signal are h xs i = T x̄1 /(1 + r ) and
h x2s i = ∆2 Xs + T x̄12 /(1 + r ) where r = (1 − γ) g0 /γg1 . Thus the distilled squeezing is
given by:
∆2 Xsdistill = ∆2 Xs + T x̄12
r
(1 + r )2
= T∆2 Xsq + R∆2 Xv + T x̄12
(3.43)
r
.
(1 + r )2
The signal variance can be decreased or even the squeezing recovered (taking into account the loss of the tap beam splitter) by minimizing the third term. The probability γ
and the displacement x̄1 are parameters of the noisy process and thus can not be altered
in the distillation optimization. However through the choice of the threshold value yth ,
the ratio between the filter functions r can be controlled to yield efficient distillation,
corresponding to r → ∞ or r → 0.
Chapter 4
Propagation of light in optical
fibers
Optical fibers are a natural choice for both classical and quantum communication. The
interaction of light with silica fibers, linear as well as nonlinear, is well understood from
many years of research and use in telecommunications [39]. Thus production processes
have been optimized and extremely low transmission losses are standard. Glass fibers
are not only useful in communication but also in nonlinear and quantum optics, the
focus of this thesis. Nonlinear effects in fibers can be used to alter photon statistics,
necessary for the generation of bright squeezed and entangled states. Such nonclassical
states form the foundation of much of continuous variable quantum communication
and information [202, 5, 137].
Although the nonlinearity in silica fibers is weak compared with other media, i.e.
atomic samples [145, 203, 204], with a value for the second order refractive index of
n2 ≈ 2 × 10−20 m2 /W [205] they are nonetheless good candidates for nonlinear optics.
Using ultrashort laser pulses (∽100 fs) and confining the light to a very small mode field
area (.100 µm2 ) over a long interaction length (10s of meters), relatively large effective
nonlinearities are attainable.
The discussion of propagation phenomena in optical fibers is divided into two main
sections. The first describes light in fibers in general terms based on Maxwell’s equations. A phenomenological approach is taken in the remainder of this section where
linear effects and the nonlinear birefringence are presented. These serve as the basis for
a discussion of soliton propagation and as well as an intuitive picture of the squeezing
process in optical fibers. In practice the situation is however more complicated due to
photon-phonon interactions, topic of the final subsection.
50
4. Propagation of light in optical fibers
The second section discusses a first principles quantum mechanical model of light
propagation in optical fibers. This is based on the work of P. D. Drummond and J. F. Corney which was used in a collaborative simulation of one of the experiments presented in
this thesis. It is a complete model of quantum soliton propagation in silica fibers including linear effects, the third-order nonlinearity and time dependent photon scattering
created with the explicit goal of simulating and predicting squeezing experiments.
4.1 Semi-classical effects and propagation
The ability of optical fibers to transmit light is based on the principle of total internal
reflection. Light is guided in a fiber core which is surrounded by a cladding layer with
a lower refractive index than that of the core (typically 1-3%) which gives rise to total
internal reflection. Decreasing the core diameter toward the order of the wavelength
of the transmitted light requires modification of this simple description of fiber propagation. One then speaks of guided light modes rather than totally internally reflected
rays. The form of these modes can be determined by applying Maxwell’s equations
with the correct boundary conditions [36]. It is then found that for a given set of fiber
parameters (core and cladding refractive indices, core size, etc.) there exists a limiting
wavelength above which the given fiber guides only a single mode of the electromagnetic field. This is the so-called TEM00 mode; in the far field it can be well approximated
as a Gaussian [206]. In this thesis fibers supporting only one mode, referred to as single
mode fibers, were used exclusively and any effects stemming from the possible multimode nature of optical fibers are not considered.
The interaction of electromagnetic wave with a medium is completely described by
Maxwell’s equations (Eqs. 2.1, 2.2) when the magnetic M and electric P polarizations
are known. As almost all materials of interest in optics are nonmagnetic, it is assumed
that M = 0. Thus P completely describes the response of a medium to an applied
electromagnetic field. If it is known, with appropriate boundary conditions, all optical
phenomena in the given medium can be predicted. This is seldom possible, and approximations of this function must often be made. Thus, for sufficiently weak E, P can
4.1. Semi-classical effects and propagation
51
be expanded as a power series in terms of E [37]:
P(r, t) = ǫ0
+∞
Z
dr1 dt1 χ(1) (r − r1 , t − t1 ) · E(r1 , t1 )
−∞
+∞
Z
+ǫ0
+ǫ0
−∞
Z∞
−∞
dr1 dt1 dr2 dt2 χ(2) (r − r1 , r − r2 , t − t1 , t − t2 ) : E(r1 , t1 )E(r2 , t2 )
dr1 dt1 dr2 dt2 dr3 dt3 χ(3) (r − r1 , r − r2 , r − r3 , t − t1 , t − t2 , t − t3 )
..
. E ( r1 , t 1 ) E ( r2 , t 2 ) E ( r 3 , t 3 ) . . . ,
(4.1)
where χ(n) is the nth material susceptibility which is a three-dimensional tensor of rank
n+1 when including the polarization of light. Its precise form is determined by the material’s (crystalline) structure and thus the number of independent terms is often small.
A complete characterization of this function requires a full quantum mechanical treatment of the material’s reaction to an incident electromagnetic wave. However for small
perturbations far from resonance, a valid assumption for many materials including silica fibers in the near infrared [39], this response can be treated as being the same as that
of a classical anharmonic oscillator. Physically this corresponds to electrons bound to a
nucleus or infrared-active molecular vibrations [37].
Constraining the discussion to the propagation of light in silica glass fibers, a number of further simplifications can be made. Firstly, there are no free charges in the fibers,
i.e. J f = ρ f = 0 in Eq. 2.1. Secondly, for optical effects in glass fibers only the first and
third orders of P(r, t) require consideration. This is because fused silica glass is amorphous1 causing all even order terms of P(r, t) to go to zero [38, 207]. Higher, odd order
terms typically have a negligibly small effect and so are not considered [39]. The first
order term of Eq. 4.1, χ (1) , is responsible for linear optics, e.g. refraction, absorption
and dispersion, the subject of Sec. 4.1.1. The third order term produces nonresonant effects such as self-phase modulation, four-wave mixing and cross-phase modulation, all
elastics processes that do not exchange energy with the medium (Sec. 4.1.2). Combining
these phenomena, soliton formation and fiber propagation (Sec. 4.1.3) as well as a semiclassical picture of fiber squeezing (Sec. 4.1.4) are introduced. Resonant phenomena such
as stimulated Raman scattering and Brillouin scattering also result from the third order
electric polarization. An overview of these inelastic interactions with phonons in the
medium is found in Sec. 4.1.5.
1 The
SiO2 molecules exhibits inversion symmetry, i.e. the molecules are randomly ordered in the solid
and thus no bulk symmetry axes are observed.
52
4. Propagation of light in optical fibers
4.1.1
Linear effects
Linear optical effects arise from a material’s electric permittivity, given in terms of the
first order electric susceptibility as:
(1)
(1)
exx (ω ),
ǫ(ω ) = 1 + χ
(4.2)
(1)
where χexx (ω ) is the Fourier transform of χ xx (t) for modes x, degenerate in space, polarization and frequency. In a homogeneous, inversion symmetric medium there is no
dependence upon position or direction and ǫ is only a function of frequency. The linear
absorption (or attenuation) coefficient α(ω ) and the refractive index n(ω ) are derived
from the imaginary (Im) and real (Re) parts of Eq. 4.2:
ω
(1)
Im χexx (ω )
n(ω )c
1
(1)
n(ω ) = 1 + Re χexx (ω ) .
2
α (ω ) =
(4.3)
Typical values for the attenuation for a wavelength of 1500 nm lie between 3 and
0.2 dB/km; 0.18 dB/km being the physical minimum due to intrinsic Rayleigh scattering [36]. Values for α of the fibers used in this thesis are found in Table 5.1.
The refractive index of bulk fused silica at 1500 nm is 1.445 [208, 209] and the value
for an optical fiber is generally similar to this. However, doping and variation of the
geometrical structure allows some engineering of this aspect of an optical fiber, particularly the higher moments of n. Breaking the typically cylindrical symmetry of n can
give rise to two orthogonal axes exhibiting slightly different refractive indices. This is
usually achieved by applying anisotropic strain to the fiber core by embedding stress
elements in the cladding2 , as first demonstrated in 1978 [210]. Light propagating along
one of the polarization axes of such a birefringent fiber will tend to maintain its polarization state despite environmental changes and employment conditions. The birefringence is quantified by the beat length Lb , the distance after which the optical waves in
the slow and fast polarization axes exhibit a relative phase shift of 2π. Values of as little
as 1 mm are possible at 1500 nm and generally the greater the birefringence, the purer
the polarization state after propagation for a given length [39].
Of particular interest for the propagation of short optical pulses in glass fibers is
the frequency dependence of n(ω ) which results in chromatic dispersion. This causes
a change in the relative speed between co-propagating spectral components which can
temporally distort an optical pulse. Mathematically this effect is described by the mode2 The
stress members have a different thermal expansion coefficient, and after drawing and cooling the
fiber they then apply a constant strain to the fiber core.
4.1. Semi-classical effects and propagation
53
propagation constant β(ω ) which can be expanded as a Taylor series about the frequency ω0 :
β(ω ) = n(ω )
ω
c
1
1
= β 0 + ( ω − ω 0 ) β 1 + ( ω − ω 0 )2 β 2 + ( ω − ω 0 )3 β 3 + . . . ,
2
6
(4.4)
where β n is defined as:
βn =
dι β
dω ι
with ι = 1, 2, 3 . . . .
ω = ω0
For pulses & 100 fs the cubic and higher order terms are generally negligible and are
not considered further. The first order term β 1 can be interpreted as the group velocity
v g the speed at which the pulse envelope travels [39]:
dn
1
1
n+ω
= .
(4.5)
β1 =
c
dω
vg
The second order term gives rise to a frequency-dependent propagation speed, or chromatic dispersion, and thus can lead to pulse broadening or distortion:
1
dn
d2 n
ω d2 n
λ3 d 2 n
β2 =
2
+ω 2 ⋍
⋍
,
(4.6)
c
dω
dω
c dω2
2πc2 dλ2
where λ is the wavelength corresponding to ω. For applications in optical fibers
the most important feature of β(ω ) in fused silica glass is a sign change of β 2 at
λD =1.27 µm. This is referred to as the zero-dispersion wavelength. It splits the spectrum into the ”normal” (λ < λD , β 2 > 0) and ”anomalous” (λ > λD , β 2 < 0) dispersion
regimes. This feature is important as it determines whether solitons, stably propagating light packets, can exist at a given wavelength (Sec. 4.1.3). The value of β 2 can be
influenced by many factors including material doping and fiber geometry [39].
4.1.2
Nonlinear birefringence
The richness of photon-photon phenomena observed in silica fibers is in a large part
due to their nonlinearity. Generally P and χ (n) are tensors and thus also the electric
permittivity. This allows the coupling of different optical waves to generate new frequencies [37, 38]. Considering the χ(3) term, this process is called four-wave mixing, as
four distinct frequencies can be coupled. For efficient four-wave mixing the light beams
involved must be phase matched, which is often difficult for different frequencies in
glass fibers. As such, it is generally a very inefficient process and will be neglected. The
54
4. Propagation of light in optical fibers
discussion here will focus on effects originating from the nonlinear refraction (i.e. the
degenerate case) which requires no phase matching. With several simplifying assumptions Eq. 4.2 can be extended to qualitatively describe nonlinear refraction in fibers.
These are namely that the nonlinearity is a small perturbation, that the slowly varying
envelope approximation is valid, and that the material response is instantaneous. Thus,
using the notation of Eq. 4.2, a first order, nonlinear electric permit
(1)
ǫ(ω ) = 1 + χexx (ω ) + ǫχ(3) , where
ǫχ ( 3 ) =
3 (3)
χ xxxx |E(r, t)|2 .
4
(4.7)
(3)
ǫχ(3) is dependent upon the third order electric susceptibility χ xxxx and can be treated
as a constant, i.e. frequency and time independent.
However, the nonlinear term generally has electronic, nuclear, and electrostrictive
components [211, 212, 213]. It is assumed that these phenomena are nonresonant, and it
is found that each has a distinct response time. The electronic response time to an optical
field has been found to be ≤1 fs [214]. The nuclear and molecular vibrational responses,
also broadly referred to as Raman scattering, are slower due to the larger masses involved, around 70 fs [215, 205]. Electrostriction, the increase in density of a material
due to an applied electric/optical field, is of the order of 1 ns, approximately the transit time of sound waves across the optical mode diameter [212, 216]. In this work only
the electronic and nuclear contributions require consideration because the experiments
used 130 fs laser pulses. The electronic component is then effectively instantaneous.
The Raman or vibrational contribution, however, has a time dependence and thus can
not be modeled simply [217, 215]. This point will be revisited in Secs. 4.1.5, 4.2.
Thus the solution of Eq. 4.7 is incomplete for systems using pulses shorter than
⋍10 ps in that the time dependence of the nonlinear refraction is neglected. It is however valid as a first approximation since the electronic contribution to the nonlinearity
is typically 85%, with Raman scattering making up the remainder [205]. This model
can then be used to gain physical insight into pulse propagation in optical fibers and in
particular into the mechanism generating squeezing. Based upon Eq. 4.7, an expression
for the nonlinear refractive index to second order can be derived [39]:
(3)
3 Re χ xxxx
n = n0 + n2 I with n2 =
,
(4.8)
4
n20 ǫ0 c
where the refractive index has been expressed using the optical intensity,
(3)
I = 12 n0 ǫ0 c| E|2 . The imaginary component of χ xxxx , as for the linear term, is associated
with the material interaction as elaborated upon in Sec. 4.1.5.
The nonlinear refractive index causes an intensity dependent phase shift of the optical beam. As such, this effect is generally termed the optical Kerr effect by virtue of
4.1. Semi-classical effects and propagation
55
its similarity to the ”traditional” Kerr effect. This effect, discovered in 1875, refers to a
change in the refractive index of a material due to the application of an external electric
field [36, 44]. In the optical domain a distinction is made between phase shifts arising
from the influence of auxiliary beams of different wavelengths, polarizations or propagation directions - Cross Phase Modulation (XPM) - and those arising from the intensity
of the pulse itself - Self Phase Modulation (SPM). The former was observed in a single mode fiber in 1973 using a pump/probe setup [40] and the latter was shown soon
thereafter [218]. SPM was found to be the cause of the spectral broadening of ultrashort laser pulses [218], the formation of optical solitons in the anomalous dispersion
regime [219] and can be used to create all-optical switches (for example [220, 221]) and
generate squeezed light [16, 10]. Although squeezing using XPM has been shown [164],
SPM or the Kerr effect is the typical mechanism by which squeezing is generated in
fibers (Sec. 4.1.4).
The phase shift of a single mode, i.e. continuous wave beam, after propagation along
a distance z due to SPM is described by [207]:
ωn2 φ(z) = φ0 (z) + φ NL (z) = n0 +
I z,
(4.9)
c
where φ0 (z) is the linear phase shift and φ NL (z) refers to the phase shift arising from
the nonlinearity. Here losses have been neglected, a good assumption for short fibers at
wavelengths around 1500 nm. The Eq. 4.9 is often expressed using the effective nonlinearity γ:
ωn2
.
(4.10)
φ NL (z) = γPz where γ =
cAeff
Here P0 = I Aeff has been used, where P0 is the pulse peak power and Aeff is the effective
mode area defined as the beam area contained by the 1e (where e is the natural number)
amplitude contour. Thus the SPM phase shift can be increased by either i) increasing the
laser power, or by ii) decreasing the effective mode field area. The phase shifts of CW
beams are easily calculated as above. For pulsed laser systems the nonlinear phase shift
can be approximated by calculating the nonlinear shift due to the pulse peak power.
For a pulse centered at the wavelength λ0 the peak power is determined by assuming
soliton-like pulses with a sech amplitude envelope [207] (Sec. 4.1.3):
P0 =
0.882
Pavg ,
τ0 f rep
(4.11)
where τ0 is the pulse’s full-width at half-maximum and f rep is the pulse repetition rate.
4.1.3
Nonlinear Schrödinger equation and solitons
Solitons, or stably propagating wave packets, are a ubiquitous phenomenon in modern
physics - from Bose-Einstein condensates to lasers. Their appearance in the propagation
56
4. Propagation of light in optical fibers
of light in optical fibers has been the topic of numerous theoretical and experimental
works. These began with the theoretical prediction of bright optical solitons in the early
1970’s [219, 222] and the ensuing experimental observation by L. F. Mollenauer et al. in
1980 [223]. Bright fiber solitons occur only at wavelengths for which optical fibers exhibit anomalous dispersion, i.e. β 2 < 0. Here the spectral broadening due to dispersion
can be counter-acted by the fiber’s nonlinear refraction. This is seen in the well-known
nonlinear Schrödinger equation (NLSE), derived from the Maxwell equations, which
describes the propagation of pulses in the slowly varying envelope approximation [39]:
∂
iβ 2 ∂2
′
Zι (t′ , z)2 Zι (t′ , z) + α Zι (t′ , z),
Zι (t′ , z) =
Z
(
t
,
z
)
+
iγ
ι
∂z
2 ∂t′2
2
(4.12)
where the equation has been written in terms of the ι-polarized electric field envelope
Zι (t′ , z) in the propagative frame: t′ = t − z/β 1 , where t and z are time and position
in the laboratory frame. β 1 and β 2 are the group velocity and dispersion respectively, α
is the absorption parameter and γ is the effective nonlinearity. Each term can thus be
associated with a given effect, from left: i) the spatial envelope evolution, ii) the dispersion, iii) the nonlinearity and iv) the attenuation of the pulse. Eq. 4.12 does not account
for the Raman effect, though this can be included with suitable modifications [39, 224].
It is a classical equation and thus unsuited for the calculation of quantum effects such as
quantum noise reduction. Nonetheless, Eq. 4.12 is sufficient for simulating the propagation of pulses of τ0 & 5 ps in most fibers as well as the qualitative prediction of many
fiber propagation phenomena.
Using the inverse scattering method, as demonstrated by V. E. Zakharov and
A. B. Shabat [222], Eq. 4.12 can be solved. Families of eigenvalues or stationary solutions of Eq. 4.12 are then found [225]. In a glass fiber, if the absorption term is small,
the Nth order solution is of the form [219]:
Zι (t′ , z) = N
p
P0 · sech(t′ /t0 )e
iβ2
z
2t20
,
(4.13)
where P0 is the peak
power and t0 is the 1e temporal pulse width, related to
√
τ0 by τ0 = 2ln(1 + 2)t0 ≈ 1.76t0 . These solutions are referred to as solitons since they
exhibit stable propagation when injected into an ideal lossless fiber, neglecting Raman
effects. The N 6= 1 solutions are high-order solitons and these display a periodic evolution over z, whereas the fundamental soliton N = 1 propagates without changing
its shape. These solutions have been found to be robust with respect to disturbances.
This makes solitons, particularly for N=1, of great interest in telecommunications and
fundamental research.
4.1. Semi-classical effects and propagation
57
More elegant and practical forms of Eqs. 4.12, 4.13 can be written by introducing a
set of dimensionless variables:
1
Uι (t′ , z) = √ Zι (t′ , z),
P0
ζ = z/z0 =
| β2 |
z,
t20
τ = t′ /t0 .
(4.14)
Here Uι (t′ , z), ζ and τ are the normalized amplitude, length and time of the propagative
|β |
frame respectively. The dispersion length z0 = t22 has also been introduced to provide
0
a measure of the length at which the fiber dispersion becomes significant. The NLSE is
then rewritten in a form more suitable for simulations:
i
∂
1 ∂2
Uι (τ, ζ ) = sgn( β 2 )
Uι (t′ , z) − N 2 |Uι (τ, ζ )| 2 Uι (τ, ζ ),
∂ζ
2 ∂τ 2
(4.15)
where sgn( β 2 ) gives the sign of the dispersion parameter and propagation losses have
been assumed negligible, a valid assumption for the short fibers used in this thesis.
Further, setting Uι (0, 0) = 1, the solution for the fundamental soliton (N = 1) is:
Uι (τ, ζ ) = Nsech(τ ) exp (iζ/2) .
(4.16)
The N parameter of Eqs. 4.15, 4.16 is found to be a measure of the relative strengths of
the fiber dispersion and nonlinearity:
t20
z0
N =
.
=
γP0
γP0 | β 2 |
2
(4.17)
This is called the soliton order, from its appearance in Eq. 4.16, as it is found that solitons
exist only for integer values of N.
It follows that the soliton amplitude and width scale inversely, a vital trait in squeezing experiments where a high peak power but low average power is desired [226]. What
however happens in pulse-fiber combinations with noninteger values of N? For N > 0.5
the pulse will alter its height, width and central wavelength, in the process shedding energy in dispersive waves, until a soliton has been asymptotically attained. Fundamental
solitons will form for 0.5 < N < 1.5 [39], which corresponds to most of the energies investigated in this thesis. It is possible that a pulse will exhibit N = 1, but could still
not display the sech pulse envelope. In this case it too will evolve over time into a soliton. Pulses with too little energy, N < 0.5, will not form solitons and their propagation
is governed by the fiber dispersion. These phenomena have been simulated using the
full quantum model of Sec. 4.2, where Fig. 4.3(a) depicts the propagation of a weak,
dispersive pulse and Fig. 4.3(b) shows the evolution toward a soliton.
58
4.1.4
4. Propagation of light in optical fibers
Squeezing in a semi-classical picture
The first proposals for the generation of nonclassical light using the χ(3) nonlinearity
were made in 1979. These schemes exploited a nonlinear Kerr interferometer [227] or
degenerate four-wave mixing [102]. Early theoretical and experimental efforts focused
on four-wave mixing in atomic samples [228, 229, 230, 231, 232, 2]. It was however
quickly realized that optical fibers also present a suitable medium [7, 233, 234]. The first
experimental demonstration used a continuous wave laser to produce squeezing [147].
Shortly after this experiment, it was found that fiber squeezing could be explained more
simply by the Kerr effect than by degenerate four-wave mixing [16, 235, 8].
The Kerr effect, described in Sec. 4.1.2, alters the refractive index, which, when
neglecting other effects, results in an intensity-dependent phase shift as visualized in
Fig. 4.1(a). The input to the Kerr medium, from a pulsed laser, is a coherent state consisting of a superposition of number states. Under the Kerr effect these states are rotated
relative to one another in phase space (Eq. 4.10). The initially symmetric phase space
distribution characteristic of coherent states is thereby distorted into an ellipse. This
corresponds to quadrature squeezing, the squeezed quadrature rotated by θsq relative
to the amplitude quadrature, or radial direction. Because of energy conservation, the
phase space distribution is altered such that the statistics in the amplitude remain constant. Thus the squeezing cannot be detected directly in amplitude or intensity measurements. A detection scheme sensitive to the angle of the squeezed ellipse θsq is required.
This took the form of a phase shifting cavity in the first fiber squeezing experiment [147].
As an alternative observation method, an interferometer based setup was suggested [16]. The first experiment to leverage this in a balanced configuration was carried out by M. Rosenbluh and R. M. Shelby in a landmark experiment [10]. Another
improvement in this experiment was the use of ultrashort laser pulses to limit the large
noise due to thermal effects which severely limited early experiments. This is primarily
the so-called Guided Acoustic Wave Brillouin Scattering (GAWBS) [9, 236] (Sec. 4.1.5).
All fiber squeezers since have used ultrashort pulses, though the observation techniques
have varied greatly.
Schemes used to observe Kerr squeezing in single mode fibers include: i) phaseshifting cavities [147], ii) spectral filtering [12, 237, 238, 239, 240], iii) balanced interferometers [10, 11, 241, 242, 243], iv) asymmetric interferometers [13, 18, 31, 163, 244]
and v) a two-pulse, single pass method generating squeezed vacuum [32] or polarization squeezing [113]. The new technology of photonic crystal fibers (PCF), novel
fibers manufactured with specially designed light-guiding air-silica structures along
their length [245], have also been used in squeezing experiments using asymmetric interferometers [15] as well as spectral filtering [246, 247].
4.1. Semi-classical effects and propagation
(a)
59
(b)
Q
qsq
X
ˆq
(c)
Q
Q
sq
qsq
X
ˆ x,q
sq
olu
Ev
X
ˆq
sq
n
tio
ây
P
P
âx
P
qsq
X
ˆ y,q
sq
Figure 4.1: a) Representation in phase space of the evolution of a coherent beam and the Kerr
nonlinearity (bottom right), which generates a quadrature (or Kerr) squeezed state (upper left).
The arrow indicates the direction of state evolution with propagation. b) Production of amplitude
squeezing using the interference of a strong and weak beam in an asymmetric interferometer.
c) Polarization squeezing is generated by overlapping two orthogonally (x- and y-) polarized
quadrature squeezed states.
In this thesis two methods have been utilized in the measurement of squeezing,
namely asymmetric interferometers and a two-pulse, single pass method. In the
first, also referred to as the asymmetric Sagnac loop or nonlinear optical loop mirror,
two pulses of strongly asymmetric intensity are counter-propagated through a fiber
(Sec. 5.3). The stronger of the two pulses experiences the Kerr nonlinearity as shown
in Fig. 4.1(a) while the weak pulse remains approximately coherent. Interference of the
two pulses after the fiber, for certain relative phase differences, results in a light beam
squeezed in the amplitude quadrature, Fig. 4.1(b). The relative phase depends on many
experimental parameters including γ, fiber length, pulse characteristics, etc. A more
accurate treatment of this setup must however include the nonlinearity experienced by
the weak beam as well as the quality of the interference, both of which decrease the
squeezing. The former unavoidably mixes an amount of the antisqueezed quadratures
into the measurement [244]. The latter is determined by the frequency and time evolution of the two pulses, leading to degraded interference and a resultant introduction of
vacuum noise [93].
The second scheme, the single pass method is in some aspects similar to a balanced
interferometer setup. There two equally intense beams are counter-propagated and then
interfered to produce vacuum squeezing. In the single pass method two identically
bright, orthogonally polarized and equally quadrature squeezed beams are overlapped
spatially and temporally with a constant relative phase shift (Sec. 5.4). A phase space
60
4. Propagation of light in optical fibers
representation of these beams is shown in Fig. 4.1(c) where the relative phase is π/2.
The resultant beam is polarization squeezed as outlined in Sec. 3.1.2. There it is shown
that this is equivalent to the simple overlap of a squeezed vacuum with a bright, orthogonally polarized local oscillator. One can also consider the two beams acting as
local oscillators for each other in the measurements, displacing each other so as to allow
observation of the squeezing. The advantage of the single pass method compared with
the balanced interferometer is that the perfectly matched local oscillator is already included in the output beam. The squeezing can then be characterized in direct detection
via Stokes measurements (Sec. 2.1.1). Under certain assumptions, this scheme allows a
complete characterization of the bright nonclassical states produced in glass fibers.
4.1.5
Scattering effects
Thus far only photon-photon interactions have been considered. Interaction with the
optical fiber can not be ignored, as it often plays an important role in the propagation
of ultrashort pulses. The fiber-light interactions primarily take the form of scattering
effects in which the phonons or vibrational modes of the fiber are coupled to the photons traveling through the fiber. These vibrational modes can be divided into two categories: acoustic and optical phonons. The former are slow macroscopic vibrations of
the medium (approximately 1 MHz-1 GHz) which depend on the physical structure of
the medium; interaction with these phonons is referred to as Brillouin scattering. The
optical phonons are much higher frequency thermal and spontaneous atomic and molecular displacements (approximately 1-40 THz) giving rise to Raman scattering. Both
phenomena have exchange energy with the medium thereby generating optical modes
at up- and down-shifted frequencies. Due to the fundamentally similar nature of these
two processes in fibers, they can be described in a similar manner [226].
Brillouin scattering is caused by acoustic waves which modulate the density and
thus the refractive index of the fiber. This effect is usually observed as a backward
propagating mode which has experienced a small shift from the carrier frequency. This
form is however negligible for ultrashort pulses [39] and of more importance for the
discussion here is the special form called Guided Acoustic Wave Brillouin Scattering or
GAWBS. This effect, first observed in 1985 [9], can occur in the forward direction only
in waveguides. The scattering on low frequency waves alters the propagation direction of the scattered light by momentum conservation. But if the mode into which the
light is scattered is supported by the waveguide, the light can be guided further along
the fiber in the forward direction [9]. The spectrum of the scattered light consists of a
number of peaks spread across the MHz-GHz frequency range [248], each of which corresponds to an acoustic eigenmode of the waveguide. GAWBS causes a small frequency
shift compared to the spectral width of ultrashort laser pulses, but it nonetheless has a
4.1. Semi-classical effects and propagation
61
detrimental effect upon squeezing. This arises from the fact that the scattering incidents
introduce a phase noise to the optical pulses which scales linearly with total power and
fiber length [9]. Indeed, GAWBS presented a great hurdle for the first squeezing experiments with CW lasers [9, 249, 234].
The limitations imposed by GAWBS on squeezing experiments was circumvented
by the use of ultrashort soliton pulses [10]. For such pulses it is possible to have a
very high peak power while simultaneously a very low total power. Since the thermal
noise is proportional to the total power whereas the nonlinearity scales with the peak
pulse power, the relative strengths of these effects can be altered so that squeezing can
be observed [226]. This advance however does not completely eliminate this excess
noise source and thus GAWBS has been the subject of much further experimental research [250, 251, 252, 253, 254, 255, 256, 257, 258, 259]. GAWBS has also been observed
to have a depolarizing effect, thereby introducing polarization specific noise [236, 258].
One promising development in this field is the advent of photonic crystal fibers. These
fibers have a light guiding silica-air structure which can be tailored to decrease or eliminate GAWBS in a given frequency window [258, 259], though they come with a number
of practical disadvantages.
Raman scattering can generate scattered light in both the forward and backward directions. The scattering excites an atomic or molecular vibrational mode and the light
is down- or up-shifted in frequency to form the Stokes or anti-Stokes waves respectively [39]. The measured spectrum of this frequency shift α R (ω ) for a fused silica fiber
is seen in Fig. 4.2 taken from Ref. [260]. The large width of this response is due to the
amorphous nature of silica. The lack of a crystalline structure causes the vibrational
modes to spread over a wide frequency range. The best known Raman effect is stimulated Raman scattering where an incident laser beam can amplify a given lower frequency if the frequency difference between the modes lies within the frequency shift
band relative to the pump beam. This fact can be used to produce laser light at otherwise unreachable wavelengths [39]. Another well known Raman phenomenon is the
soliton self-frequency shift [261]. Similar to GAWBS, Raman scattering also introduces
excess phase noise to quantum pulses [262]. However, unlike GAWBS, Raman has a
spontaneous as well as a thermal term [263], and thus does not go to zero with decreasing temperature [262, 264].
Raman effects can also contribute to the nonlinear birefringence of optical fibers.
However due to the finite bandwidth of the Raman spectrum, this contribution is much
slower than the electronic response. For pulses longer than ⋍10 ps Raman scattering
appears instantaneous and contributes to the nonlinearity in the NLSE in the same way
as the electronic nonlinearity. For shorter pulses the time dependence of the vibrational
62
4. Propagation of light in optical fibers
response becomes important and the nonlinear refractive index is better described by:
n2 (t) = n2E + n2R (t),
(4.18)
where n2E , n2R are the electronic and Raman contributions, respectively, to the second
order nonlinear refractive index. The time dependence of Raman scattering is complex
as indicated by the form of Fig. 4.2 and thus accurate Raman models use a quantum
mechanical model which is fitted to empirical data [262, 33] although for longer pulses
approximations to this function can be made [264, 265, 266].
The Raman gain spectrum of an optical fiber α R (ω ) depends on the core material
and doping. Almost all commercial single-mode fibers, including those used here, are
doped with small amounts of GeO2 used to increase the core’s refractive index. The
results for pure silica in Fig. 4.2 are however still valid for low doping levels, where the
relative spectral structure of the Raman effects changes only marginally although the
absolute magnitude increases [205, 267].
To find the temporal response of Raman scattering the starting assumption is that the
Fourier transform of the third-order electric susceptibility can be described as a complex
number [37, 268, 269]:
(3)
χ̃ xxxx (ω ) = χ̃′ (ω ) + i χ̃′′ (ω ).
(4.19)
Raman gain
pulse spectrum
0.5
αR = 2|h"|
0.4
0.3
0.2
0.1
0
0
50
100
ω (2π× THz)
150
Figure 4.2: Parallel component of the Raman gain for a pure silica fiber, taken from Ref. [260].
The asterisks are the points sampled by the numerical algorithm. The dashed line is the spectrum
of a τ0 =130 fs sech pulse.
4.2. Quantum propagation model
63
(3)
Due to causality the real part of χ̃ xxxx (ω ) can be determined via the Kramers-Kronig
relation. Associating χ̃′′ (ω ) with the Raman gain spectrum α R (ω ), the time domain
response of the Raman effect can be determined [215, 269]. A simplification can be
made by observing that the real part is symmetric and the imaginary is asymmetric to
give:
n2R (t
′
−t ) ∝
Z∞
0
dΩ χ̃′′ (Ω) sin(Ω(t − t′ )),
(4.20)
where causality requires t ≥ 0 and Ω = (ω − ω0 )t0 is the dimensionless frequency shift
(3)
as defined by Gordon [224]. n2R (t) is related to χ̃ xxxx (ω ) [37, 224, 215] by:
α R (Ω ) =
3h̄ω02
(3)
Im
(
Ω
)
.
χ̃
xxxx
4c2 n20 ǫ0 β21 Aeff
(4.21)
The total nonlinear refractive index is defined as:
n2 =
n2E
+
+∞
Z
−∞
dt′ n2R (t − t′ ).
(4.22)
In this model only Raman scattering has been considered.
4.2 Quantum propagation model
The above discussion of light in fused silica fibers has been primarily classical and
a number of approximations were made, particularly concerning Raman scattering.
To accurately predict the propagation and resultant evolution of the quantum noise
properties of ultrashort pulses in optical fibers, the quantized nature of light must be
taken into account. Much has been published on this topic beginning with the seminal work of S. J. Carter and P. D. Drummond in 1987 [34, 52]. These and other early
works [270, 271, 272] were extended in investigations of the effect on squeezing of
photon-dielectric interactions [273] in the form of GAWBS [226, 250, 255] or Raman scattering [262, 274, 264, 265, 266, 275].
These and further works focused on predicting the output characteristics of specific
experimental setups using optical pulses: spectral filtering [266, 275, 276, 277, 278], the
symmetric Sagnac loop [250, 279], the asymmetric Sagnac loop [17, 13, 31, 280, 277] and
the single pass method [115, 281]. A quantum soliton propagation model is presented
here which has been used to simulate the single pass polarization experiments in this
thesis. This theory is based on first principle considerations of the interaction of a photon flux with a dielectric medium as developed by P. D. Drummond and J. F. Corney,
summarized recently [33].
64
4. Propagation of light in optical fibers
This model, which began with the seminal work by S. J. Carter and P. D. Drummond
in 1987 [34], has resulted in a quantum propagation theory that includes the material
dispersion and χ(3) nonlinearity as well as the nonresonant coupling to phonons. The
phonons provide a non-Markovian, or frequency dependent, reservoir that generates
additional, delayed nonlinearity, as well as spontaneous and thermal noise [262, 264,
265]. The simulations performed with this model were carried out by P. D. Drummond
and J. F. Corney in a most fruitful collaboration, the results of which are presented in
Sec. 6.2.
4.2.1
Interaction Hamiltonians
The quantum propagation model employed in the simulation of the experiments of this
thesis has its roots in the quantization of macroscopic electromagnetic theory specifically for application in dielectrics [282]. Accounting for phonon effects, the system
Hamiltonian can be described as the sum of fiber and Raman Hamiltonians respectively: Ĥ = ĤF + ĤR . Using this quantization for a single-mode fiber with dispersion
and nonlinearity ĤF can be found [273]. This Hamiltonian can be written in terms of
photon flux operators (Sec. 2.2). Expanding the frequency dependent terms about the
central frequency ω0 , in the slowly varying envelope approximation the fiber Hamiltonian for a single polarization is (in the laboratory frame) [33]:
ĤF
+∞
Z
iβ 1 ∇Ψ̂† (t, z)Ψ̂ (t, z) − Ψ̂† (t, z)∇Ψ̂ (t, z)
2
−∞
β2
χ′ †2
†
2
+ ∇Ψ̂ (t, z)∇Ψ̂ (t, z) − Ψ̂ (t, z)Ψ̂ (t, z) ,
2
2
h̄
=
2
dz
(4.23)
where the fiber is assumed to be one dimensional and uniform in in z and χ′ is the
instantaneous, frequency independent electronic contribution to the third order electric susceptibility (Eq. 4.1). Considering only the electronic contribution, the nonlinearity parameter in propagative, normalized units is related to the γ of Eq. 4.10 by:
0
γ [52]. This Hamiltonian generates a Heisenberg equation for the photon-flux
χ′ = h̄ω
β2
1
operators which is identical in form to the classical NLSE in the laboratory frame.
To the Hamiltonian of Eq. 4.23 must be added the coupling to phonon reservoirs, i.e.
Raman scattering [262]. Here linear gain and absorption are ignored as these effects are
negligible for the short fibers in the experiments presented here. The phonon-photon
results not only in additional, delayed nonlinearity but also in extra noise. In the macroscopic quantization used here, there exist phonon operators b̂(ω, z) and b̂† (ω, z) and the
4.2. Quantum propagation model
65
initial phonon state is thermal:
nth =
1
.
exp(h̄ω/kT )
The Hamiltonian for the Raman phonon reservoir is [262]:
ĤR = h̄
+∞
Z
dz
−∞
Z∞
dω
0
Ψ̂† (z)Ψ̂ (z) R(ω, z)
×[b̂ (ω, z) + b̂(ω, z)] + ω b̂† (ω, z)b̂ (ω, z) .
†
(4.24)
This model treats the atomic vibrations in the fused silica as a continuum of localized
oscillators coupled to the photons by R(ω, z). This coupling is determined empirically
through measurements of the Raman gain spectrum [262], shown in Fig. 4.2. The atomic
displacement is proportional to b̂ + b̂† , where the equal-time commutator is:
[b̂(t, ω, z), b̂† (t, ω ′ , z′ )] = δ(ω − ω ′ )δ(z − z′ ).
(4.25)
Thus the Raman scattering is treated as localized in space and frequency, which however requires proper cut-off to retain the slowly varying envelope assumption.
4.2.2
Quantum nonlinear Schrödinger Equation
A quantum NLSE in the laboratory frame can now be derived from the Raman-modified
fiber Hamiltonian Ĥ. For convenience, the scaled variables in the propagative frame can
be used, as introduced in Sec. 4.1.3, with the additional definitions:
φ̂ι = Ψ̂ι
p
β 1 t0 /n
with
n=
| β2 | Aeff λ2
,
(4π 2 cn2 h̄t0 )
(4.26)
where 2n is the photon number of a fundamental soliton. Thus the quantum NLSE for
the photon flux field as derived from the Heisenberg equation when integrating over
the Raman reservoirs is:
∂
i ∂2
φ̂ι (τ, ζ ) + i Γ̂ιR (τ, ζ )φ̂ι (τ, ζ )
φ̂ι (τ, ζ ) =
2
∂ζ
2 ∂τ
+i
Z∞
−∞
dτ ′ h(τ − τ ′ )φ̂ι† (τ ′ , ζ )φ̂ι (τ ′ , ζ )φ̂ι (τ, ζ ).
(4.27)
The ι-polarized scattering operator is described in terms of the dimensionless frequency
Ω = ωt0 by:
Γ̂ιR (τ, ζ )
=−
Z∞
0
dΩ R(Ω, ζ )[b̂† (t, Ω, ζ ) + b̂(t, Ω, ζ )].
(4.28)
66
4. Propagation of light in optical fibers
The correlations of its reservoir fields are:
D
Γ̂†ι (ω, ζ )Γ̂ι′ (ω ′ , ζ ′ )
E
α R (|Ω|)
[nth (|ω |) + Θ(−Ω)]
n
×δ(ζ − ζ ′ )δ(Ω − Ω′ )διι′ ,
=
(4.29)
The Stokes (Ω < 0) and anti-Stokes (Ω > 0) contributions to the Raman noise are
included by means of the step function Θ.
R The causal nonlinear material response function is normalized such that
dτ h(τ ) = 1 and is given by a combination of the electronic and vibrational components, as outlined in Sec. 4.1.5:
h (τ ) = h E (τ ) + h R (τ ) =
where:
χ(t) = χ′ +
+∞
Z
dt
−∞
Z∞
nz0
χ(τt0 ),
β21
dω R2 (ω ) sin (ωt),
(4.30)
(4.31)
0
where χ′ , the electronic component, is instantaneous. Using the arguments of Sec. 4.1.5,
the imaginary part of the Fourier transform of the response function h′′ (Ω) can be associated with the Raman gain spectrum. To find the precise relation α R (Ω), seen in
Fig. 4.2, is fitted by the sum of 11 Lorentzians using a least squares fit in the dimensionless parameters Ω′j = ωt0 . Taking the Fourier transform of this function [33]:
n
h R (τ ) = Θ(τ ) ∑ Fj ∆ j e−∆ j τ sin(Ω j τ ).
(4.32)
j =0
∆ j are the widths and Ω j the center frequencies, in normalized units, and Θ(τ ) is the
step function ensuring causality. Comparing these equations with the dimensionless
Raman gain α R (Ω) it is found that:
R2 ( ω ) =
χ R
α (ωt0 ),
2π
(4.33)
using the integrated nonlinearity χ of Eq. 4.31 [215, 33]. It therefore follows that:
α R (Ω) = 2|h”(Ω)|.
4.2.3
(4.34)
Methodology
In the experiments simulated in this thesis there are over 108 photons in more than
103 modes for a given pulse. This corresponds to an enormously large Hilbert space.
4.2. Quantum propagation model
67
Such complexity means that the operator equation cannot be solved directly in the timedomain. Thus, the quantum dynamics are sampled with a phase space representation
method. Such sampling has been performed exactly using the positive-P (P+ ) representation [62, 34, 52]. However, for large photon numbers n and short propagation distances, it is known that the P+ method gives squeezing predictions in agreement with a
truncated Wigner phase space method [274], allowing faster calculations. In effect, the
Wigner representation maps a field operator to a stochastic field:
φ̂ι (τ, ζ ) → φι (τ, ζ ).
(4.35)
Stochastic averages involving this field correspond to symmetrically ordered correlations of the quantum system. Because of the symmetric-ordering correspondence, quantum effects enter via vacuum noise and a direct correspondence to measured variables
can be made [50]. The Kerr effect amplifies or diminishes this noise in a phase-sensitive
manner, making the Wigner approach ideally suited for squeezing calculations.
After the mapping, we obtain a Raman-modified stochastic nonlinear Schrödinger
equation for the photon flux of the same form as Eqs. 4.27, 4.15 [33, 283]. The correlations
of the stochastic Raman noise fields Γι and the initial vacuum noise are:
1
α R (|ω |)
′ ′
nth (|ω |) +
Γι (ω, ζ )Γι′ (ω , ζ ) =
n
2
′
′
×δ(ζ − ζ )δ(ω − ω )διι′ ,
1
δ(τ − τ ′ )διι′ .
(4.36)
∆φι (τ, 0) ∆φι∗′ (τ ′ , 0) =
2n
Exemplary simulations of the pulse propagation reveal the importance of including
the laser pulses’ multimode nature, which affect weak and intense pulses in different
ways. As Fig. 4.3(a) shows, the evolution of the amplitude profile of a weak pulse is
dominated by dispersion. In contrast, an intense pulse reshapes into a soliton, whose
subsequent evolution reveals the effect of the Raman self-frequency shift, Fig. 4.3(b).
The range of input pulse energies in the experiment spanned both of these cases.
In the single pass polarization squeezing which was simulated using the presented model, two orthogonally polarized laser pulses were transmitted simultaneously
through a fiber. Due to the fiber birefringence, the two polarization components do not
temporally overlap for most of the propagation length3 , so the cross-polarization component of the Raman gain can be neglected. Both pulses can then be simulated independently. Classical, low-frequency phase noise should be largely common to both modes
as their temporal separation is less than 100 ps, and as such it can be mostly ignored.
3 Considering the 3M FSPM-7811 fiber of Table 5.1, τ =130 fs pulses at 1.5 µm only overlap significantly
0
over ≈ 15 cm.
68
4. Propagation of light in optical fibers
Figure 4.3: Simulations of the propagation of 4.8 pJ (left) and E = 53.5 pJ (right) pulses,
with initial an initial width of t0 = 74 fs in the 3M FSPM-7811 fiber (see Table 5.1). |φ| is
the dimensionless pulse amplitude envelope.
For the measurement of the propagated beams generalized Stokes operators can be
defined in the propagative variables defined in Sec. 4.1.3 (compare with Sec. 2.2.3):
Ŝ0 (ζ ) = N̂xx (ζ ) + N̂yy (ζ ),
Ŝ2 (ζ ) = N̂xy (ζ ) + N̂yx (ζ ),
where:
N̂ιι′ (ζ ) =
Z
Ŝ1 (ζ ) = N̂xx (ζ ) − N̂yy (ζ ),
Ŝ3 (ζ ) = i N̂yx (ζ ) − i N̂xy (ζ ),
dτ φ̂ι† (τ, ζ )φ̂ι′ (τ, ζ ).
(4.37)
(4.38)
To simulate the experiment, two orthogonally polarized pulses were propagated
through the fiber, generating an S3 polarized beam at the output. The pulses were mixed
after propagation, as in the experiment (Sec. 5.4):
âxθ = âx cos(θ/2) + ây sin(θ/2),
âyθ = âx sin(θ/2) − ây cos(θ/2).
(4.39)
By rotation of the phase space angle θ, the photon statistics of the original x and y
modes and their squeezing or antisqueezing can be detected in the number difference
(Secs. 2.3.3, 3.1.2):
Ŝθ = cos(θ )Ŝ1 + sin(θ )Ŝ2 = â†xθ âxθ − â†yθ âyθ .
(4.40)
The simulations were performed by means of the XMDS code-generating package [284]. The phase-space equations were discretized on a regular grid where ∆τ ≈ 0.2
4.2. Quantum propagation model
69
and ∆ζ ≈ 0.03 are the steps in the normalized propagative variables. They were solved
by a split-step Fourier method with an iterative semi-implicit algorithm for the nonlinear and stochastic terms. The non-Markovian phonon reservoirs were included by
explicitly integrating a set of phonon variables at each point in the propagation grid.
Physical quantities were calculated by stochastic averages over 10,000 runs and noise
was introduced via appropriate random number generation.
70
4. Propagation of light in optical fibers
Chapter 5
Experimental setup
This chapter is devoted to the description of the different setups and apparatus used
in the presented experiments. The first two sections (Secs. 5.1, 5.2) introduce the fundamental building blocks of the experiments, namely the femtosecond laser systems
and different optical fibers. This is followed by a discussion of the the methods employed in the generation of squeezing. These are namely the asymmetric Sagnac loop,
in a free beam as well as all-in-fiber configuration, to generate amplitude squeezing
(Sec. 5.3) and a single pass method to produce polarization squeezing (Sec. 5.4). The
two ensuing sections include setups for the polarization tomography of light and the
distillation of squeezing from a non-Gaussian mixed state. The final section outlines the
detector properties and detection processes exploited in characterizing and exploiting
the squeezed amplitude and polarization squeezed pulsed light beams (Sec. 5.8).
5.1 Femtosecond laser
The laser system primarily used in the experiments presented here is a home-made
solid state laser producing soliton-like, band-width limited pulses with temporal widths
of τ0 =130-150 fs at a central wavelength λ0 between 1495-1500 nm [285]. A typical
spectrum and autocorrelation are shown in Fig. 5.1 and a schematic of the laser is seen
in Fig. 5.2. A water cooled Cr4+ :YAG crystal serves as the active medium and it is
pumped by a commercial thin disk laser system1 . This package is comprised of a pair
of semiconductor laser diodes emitting up to 40 W of continuous wave power at 908 nm
which are used to pump a Yb-glass thin disk. The end stage produces more than 20 W
1 ELS
VersaDisk-1030-20.
72
5. Experimental setup
of continuous laser light at 1030 nm with an excellent spatial mode (M2 ≈ 1.0), of which
⋍11 W are used as a pump for the Cr4+ :YAG laser.
(a)
(b)
Figure 5.1: The output a) auto-correlation and b) spectrum of the Cr4+ :YAG laser.
The pump laser is focused into the Cr4+ :YAG crystal, the beam passing through two
focusing mirrors (HR 1500 nm, AR 1064 nm) surrounding the crystal. These mirrors are
part of a folded-Z geometry resonator near the center of which the crystal is located. An
antiresonant Fabry-Pérot saturable absorber (A-FSPA) serves as the end mirror of one
arm, the beam reflected on to its surface by a further focusing mirror. The other arm
contains a Brewster angle quartz prism pair (to balance the dispersion of the resonator)
followed by a 1.7% transmission out-coupling mirror. The combination of the saturable
absorber and the prism pair allows self-starting, passive mode-locking. Following the
laser is an optical isolator to shield the laser from spurious back reflections as well as
a further quarz prism pair to restore the output beam. The ultrashort pulses emitted
exhibit a bandwidth limited secant-hyperbolic spatial amplitude envelope and can thus
be assumed to be solitons (Sec. 4.1.3). The laser repetition rate is 163 MHz and the
average emitted power lies between 60 and 90 mW, that is pulse energies of 370 and
550 pJ. This system was the product of two diploma thesis at the institute from 199597 [286, 287, 288] and was internationally one of the first of its kind.
In the intervening years comparable commercial systems based on optical parametric oscillators have been developed, notably by the companies Spectra-Physics and
Coherent. A Millenia, Tsunami, OPAL laser chain for generating ultra-short pulses at
telecommunication wavelengths from Spectra Physics was also used in one of the experiments, namely the all-in-fiber squeezing experiments presented later. The Millenia is a
frequency doubled Nd:YAG with an output of 10 W to pump the ultrashort, Gaussian
pulse emitting Ti:Sa Tsunami. The output wavelength was set to 810 nm to feed the
5.2. Optical fibers
73
HR, 75 mm
HR, 75 mm
4+
Cr :YAG
ELS
11 W (cw) @ 1030 nm
laser diode pumped
HR, 100 mm
Quarz
prisms
Faraday
isolator
A-FPSA
OC, 1.7%
Quarz prisms
Figure 5.2: Schematic of the femtosecond Cr4+ :YAG laser.
OPAL. The latter exploits an optical parametric oscillator to produce bandwidth limited
pulses with a τ0 =130-350 fs for central wavelengths λ0 between 1490 and 1560 nm.
The pulse repetition rate is 82 MHz and the maximum average power is 250 mW corresponding to a pulse energy of 3.0 nJ.
5.2 Optical fibers
Three glass fibers were used in this thesis, in order of importance: i) the FS-PM-7811
from 3M, ii) the HB15002 from Fibercore and iii) the Panda3 type fiber SM15-PS-U25A
from Corning-Fujikura. The most relevant features of these fibers are listed in Table 5.1;
the different parameters have been introduced in Sec. 4.1. All of these fibers are polarization maintaining due to the birefringence of their cores. This is generated by applying
mechanical stress to the core via stress elements embedded in the fiber cladding 4 . This
is a necessary feature allowing the success of the experiments described in Secs. 5.3, 5.4.
The FS-PM-7811 was primarily used in this work because of its high effective nonlinearity (Sec. 4.1.4). This arises from the fiber’s small mode field diameter (d)5 , the ef2 High
Birefringence.
maintaining AND Absorption reduced fibers.
4 The FS-PM-7811 is a ”tiger’s eye” fiber due to the appearance of it’s elliptical cladding. The HB1500
and SM15-P fibers have their stress elements, two trapezoids (bow-tie) or circles respectively, arranged
either side of the core.
5 That is, compared to traditional and standard fibers. The introduction of photonic crystal fibers
allows much smaller diameters but not without a number of technical drawbacks.
3 Polarization
74
5. Experimental setup
Parameter
Mode field diameter
Nonlinear refractive
index (×10−20 )
Effective nonlinearity
(×10−3 )
Soliton energy
Dispersion
Attenuation @ 1550 nm
Beat length
Polarization crosstalk
per 100m
Symbol
FS-PM-7811
HB1500 SM15-P
Units
d
n2
5.5
2.9
8.1
2.9
10.5
2.9
µm
m 2 /W
γ
5.1
2.4
1.4
1/(m·W)
ESol
β2
α
Lb
∆P
56
-10.5
1.9
1.67
< −23
150
-13.7
1.4
3.2
< −33
250
-20.3
0.5
4.0
< −30
pJ
fs2 /mm
dB/km
mm
dB
Table 5.1: Average values for the material parameters for the 3M FS-PM-7811, Fibercore
HB1500 and Corning-Fujikura SM15-PS-U25A fibers. All values (when not otherwise stated)
are for λ0 = 1500 nm and τ0 = 130 fs.
fective nonlinearity since γ scales with d−2 (Sec. 4.1.3). The characteristics of the pulses
after propagating through this fiber were investigated as a function of pulse energy, the
results of which are shown in Appendix A.
Despite their lower γ, both the HB1500 and SM15-P fibers proved useful as they can
be used in all-in-fiber couplers. Thus it was possible to extend previous work on the
asymmetric Sagnac loop [13, 31] by investigating extremely robust all-in-fiber configurations. The FS-PM-7811 can unfortunately not be incorporated into such devices by
virtue of its extreme internal stress.
5.3 Asymmetric Sagnac loop
Squeezing generated by the asymmetric Sagnac loop has typically utilized a free beam
configuration requiring the careful alignment of two beams and use of a fixed beam
splitter [13, 18]. A further similar experiment allowed variation of the splitting ratio but
at the cost of an active phase lock [31]. In this thesis the traditional free beam approach
was implemented as well as a new method leveraging fixed and variable all-in-fiber
couplers was also investigated. Such devices have to date only been used in symmetric
Sagnac loops, for example [241].
The free beam Sagnac loop was used to generate amplitude squeezed light, as seen
in Fig. 5.3(a). A linearly polarized pulse chain is incident upon the asymmetric beam
splitter, of splitting ratio 93:7 which is polarization independent by virtue of the almost
5.3. Asymmetric Sagnac loop
75
(a)
(b)
Fiber
loop
Fiber
loop
93
:7
x:
1x
l/2
l/2
l/2
y
x
l/2
Cr4+:YAG laser
Figure 5.3: Schematic setups of a) the free beam asymmetric Sagnac loop and b) the all-in-fiber
asymmetric Sagnac loop. With the correct input power/splitting ratio, the emerging beam is
amplitude squeezed.
normal incidence: ≤ 5◦ . The two resulting beams are individually coupled into the
same polarization axis at opposite ends of the optical fiber. Upon emerging from the
fiber a stable interference is guaranteed by the fact that the pulses have simultaneously
passed through the same fiber. Thus the interference is determined by the relative nonlinear phase shift experienced by the pulses in the fiber, a function of the relative pulse
intensities (Sec. 4.1.2). In this configuration only the FS-PM-7811 fiber was used due to
its large γ. The amplitude noise of the resulting beam was characterized in balanced
detection, Sec. 2.3.
All-in-fiber asymmetric Sagnac setups were also used to produce amplitude
squeezed light, Fig. 5.3(b). These are very similar to the free beam Sagnac loop, but
here the asymmetric beam splitter is integrated into the fiber loop. Two unique schemes
were used, the first with a variable fiber coupler (coupling ratio 100:0 to 50:50)6, using
HB1500 optical fiber. Insertion loss was quoted to be 0.1 dB with a polarization isolation
better than -20 dB. The fiber loop length was variable as two of the coupler pigtails had
FC/PC connectors which introduced a linear loss of 14.4%. Two loop lengths, 4.8 and
9.5 m, were investigated. This device was used to investigate the maximum squeezing as a function of the splitting ratio. A second all-in-fiber setup was investigated
as an optimized version of the varibale coupler with minimal losses and a fixed cou6 Canadian
Instrumentation Ltd., Model 905 P.
76
5. Experimental setup
pler for improved interference. The device 7 had a measured splitting ratio of 93:7 at
1530 nm [289]. SM15-P fiber was used and a 30 m loop was made by connecting two
of the coupler pigtails. These pigtails were outiftted with FC/PC connectors which exhibited the standard 14.4% linear loss. The coupler loss was stated to be 0.03 dB and
the polarization isolation better than -24 dB. Detection was accomplished in the same
manner as for the free beam setup for the generation of amplitude squeezing.
(a)
Fiber
loop
Birefringence
compensator
l/2
piezo
l/4
93:7
l/4
l/2
y
PBS
variable delay
l/2
x
(b)
l/2
l/2
Fiber
Cr4+:YAG Laser
Figure 5.4: Schematic setups of a) the free beam asymmetric Sagnac loop and b) the single
pass method, which in conjugation with the Birefringence Compensator are used to produce
polarization squeezed states.
The free beam asymmetric Sagnac loop setup was also implemented to generate polarization squeezing by temporally and spatially overlapping two amplitude squeezed
beams (Sec. 2.2.3) as shown in Fig. 5.4(a). To compensate for the fiber birefringence a
Michelson-like polarization interferometer was placed before the fiber. By precompensating for the fiber birefringence, the number of optical elements after the fiber could
be reduced. This served to minimize losses to the squeezed beams emerging from the
fiber. The ”Birefringence Compensator”8 splits an incoming linearly polarized beam
into two components of variable intensity and introduces a relative pulse delay. This interferometer acts much like a Faraday mirror. Here the combination of passage through
7 Canadian
8 The
Instrumentation Ltd., Model 954 P.
birefringence compensator is also lovingly referred to as the ”BiFi”.
5.4. Single pass squeezing method
77
a quarter-wave plate, a reflection and passage through the same wave plate rotates the
polarization by 45◦ . Rotation of the quarter-wave plates alters the power of the x or
y polarizations transmitted through the compensator by the polarization beam splitter
(PBS).
The stability of the interferometer is ensured by an active feedback loop controlling
a piezo-mounted end mirror. The control signal is taken by tapping off less than 0.1% of
the light after the fiber. This is fed into a phase sensitive feedback measurement. To generate a beam of mean S3 or circular polarization, for example, this measurement should
be of S2 . Here, pairs of detectors with a bandwidth 0-10 kHz were used. Their electronic
signals were given to a PI controller which then controlled the measured Stokes parameter to zero, that is the intensity on each detector to the same value. This was accomplished by sending the controller correction signal to the piezo in the compensator via a
high voltage amplifier9 . Characterization of the resulting polarization states was carried
out using a Stokes detection system to determine the noise variance (Secs. 2.1.1, 5.8).
5.4 Single pass squeezing method
A significant improvement in the fiber based production of polarization came with the
implementation of the single pass method [113]. This novel scheme has a number of advantages compared with previous experiments producing bright squeezing. One being
that with this setup it is possible to produce squeezing at any given power, in contrast
to the asymmetric Sagnac loop based scheme. There is thus a certain similarity to experiments using a Mach Zehnder interferometer as a flexible asymmetric Sagnac loop [31].
The interference of a strong squeezed and a weak ”coherent” beam in asymmetric loops
however gives rise to a loss in squeezing due to the dissimilarity of the pulses as well
as losses due to the asymmetric beam splitter.
In the present setup this destructive effect is avoided by mutually interfering two
strong Kerr-squeezed pulses. These co-propagate on orthogonal polarization axes and
for equal power they have been found to be virtually identical within measurement uncertainties in, e.g. spectrum and squeezing (Appendix A). This presents the potential to
measure greater squeezing and provides a greater robustness against input power fluctuations. Formally this interference of equally squeezed pulses is reminiscent of earlier
experiments producing vacuum squeezing, for example [10, 11, 32]. The advantage here
is that no local oscillator is needed in the measurement of polarization squeezing.
In the present configuration, pictured in Fig. 5.4(b), laser pulses are coupled into
only one end of the glass fiber (3M FS-PM-7811). This produces quadrature squeezing rather than amplitude squeezing which is not directly detectable (compare with
9 Piezo-Mechanik
SVR 1000/3.
78
5. Experimental setup
Fig. 4.1(a)). However, overlapping two such squeezed pulses after the fiber by use of
the birefringence compensator allows access to this quadrature squeezing by measurement of the Stokes parameters (Sec. 2.2.3). The magnitude of the squeezing produced
in this configuration is increased over that using an asymmetric Sagnac loop. This is
primarily because of the circumvention of the imperfect interference of the strong and
weak beams in the asymmetric Sagnac loop. Measurement of the polarization squeezed
states was carried out in the standard fashion.
5.5 Polarization entanglement
As discussed in Sec. 3.2, splitting a nonclassical beam on a beam splitter generates an
entangled pair of output beams. Exploiting this resource-efficient method, a polarization squeezed beam was incident on a polarization independent 50:50 beam splitter as
depicted in Fig. 5.5. The input polarization squeezing was generated using the Sagnac
loop configuration, outlined previously in this chapter. This scheme for entanglement
production is efficient in that only one polarization squeezed beam must be fed into the
entangler. The experimental effort is however more than halved as not only is a single
squeezer required, but no active phase lock between two squeezed beams is necessary.
Coherent
vacuum
0
:5
50
Polarization
squeezer
A
l/4 l/2
B
+/-
l/4
l/2
+/+/-
Figure 5.5: Setup used for the resource efficient generation of polarization entanglement.
5.6. Tomography
79
5.6 Tomography
17.5 MHz
oscillator
l/4, F
l/2, Q
To more fully characterize the fiber squeezed states generated in this thesis, a tomography of such a state was made in the quadrature variables. This was accomplished by
exploiting the equivalence of homodyne measurements (typically used in tomography
experiments) and measurements in the dark Stokes plane (Sec. 2.3.3). As polarization
squeezing is squeezing in the dark polarization mode (Sec. 3.1.2), the properties of the
individual (assumed identical) Kerr-squeezed states could be determined by making
dark plane measurements of the polarization squeezed state. The general optical setup
used is shown in Fig. 5.6; it is the simple combination of a half- and quarter-wave plate
before a polarization beam splitter. As discussed in Sec. 2.1.1 this setup is capable of
measuring all Stokes vectors by systematic rotation of the wave-plates at angles Θ and
Φ for the half- and quarter-wave plates respectively.
Mixer,
low pass
Amplifier
16-bit AD
Figure 5.6: Schematic of the setup used to carry out the tomography of a Kerr-squeezed state.
Generally both wave plates would be rotated to ensure the measurements lies in the
Stokes parameter plane orthogonal to the bright excitation. In practice this was simplified by using a beam of mean S3 or circularly polarization which makes the quarterwave plate redundant. Thus by rotation of the half-wave plate, all quadratures X (θ )
of the Kerr-squeezed states comprising the polarization squeezed state could be made.
Due to the large number of quadratures or Stokes parameters measured, typically 128,
the measurement was completely automated. The wave plates were mounted in rotation mounts driven by computer controlled step motors10 . The detection setup was the
standard Stokes measurement, but with more advanced electronic systems which are
outlined later in this chapter.
10 Newport,
DMT 40 SM24 with PC-SM32 Control Board; repeatability ±0.01◦ .
80
5. Experimental setup
5.7 Distillation of non-Gaussian noise
The experiment for the distillation of corrupted squeezed states consists of three parts as
seen in Fig. 5.7: the preparation, distillation and verification. As the squeezing of sidebands at 17.5 ± 0.5 MHz is observed, the preparation of the mixed state is accomplished
by appropriate modulation of a squeezed beam. Here this modulation occurs before
the squeezer; this is a technical detail. Viewing the preparation stage as a black box,
only the emerging, noisy state is important. The single pass polarization squeezer is exploited using 13.3 m of 3M FS-PM-7811 glass fiber. In this experiment the equivalence
of polarization and vacuum squeezing is implicit (Sec. 3.1.2).
Signal
l/
2
,F
1
7
Tap
:9
3
Polarization
squeezer
l/
2,
df
Non-Gaussian
modulator
Verification
F
Distillation
2
df
l/2
l/2
Preparation
Mixer
Amplifier
500 kHz
RF switch
16-bit A/D
17.5 MHz
oscillator
Digital post
selection
Figure 5.7: Schematic of the preparation, distillation and verification of the distillation of squeezing from a non-Gaussian mixture of polarization squeezed states. The non-Gaussian noise was
generated by turning on and off a modulation to an EOM before the fiber to shift the state in
phase space (Fig. 5.8).
The non-Gaussian noise source is implemented by executing a fixed phase space
displacement of the squeezed state with a probability 0.5. The displacement is generated
by a phase modulation in one of the linear polarization modes at the fiber input using
an electro-optic modulator (EOM)11 fed a 17.5 MHz sine wave12 . Assume the EOM
makes a small modulate in +Ŝ1 and that two pulses of equal intensity, polarized in ±Ŝ1
respectively with a π2 relative phase shift emerge from the fiber. The mean polarization is
11 Lysop,
12 Rhode
LiNbO3 custom for 1550 nm.
& Schwarz, 100 kHz-1000 MHz SMX signal generator.
5.7. Distillation of non-Gaussian noise
81
then along Ŝ3 but the modulation will cause a small displacement in the Ŝ2 polarization
in this beam (Fig. 5.8). This process adds noise power to the given sideband. This can
be equivalently thought of as a shifting of photons from the carrier to the sideband.
The phase space displacement is governed by the strength of the 17.5 MHz sine wave.
The noisy phase kicks are simulated by periodically switching this modulation on and
off, the displacement is toggled from maximum to zero at a frequency of 500 kHz13 ,
generating a noisy non-Gaussian mixture of squeezed states. In principle choosing a
periodic or a random signal is equivalent, since in both cases the the modulation pattern
will be known only to the verifier.
S1
X
Y
df
S2
Figure 5.8: The phase space representation of the non-Gaussian state experimentally produced
using polarization squeezed input states. The polarization squeezed state is shifted periodically
by a fixed displacement along the S2 axis in the dark polarization plane. X and Y indicate the
squeezed and antisqueezed quadratures of the states.
This non-Gaussian state is fed into the distiller. It consists of two operations: i) the
tap measurement of a certain quadrature on R=7% of the beam to be distilled; ii) the
signal post selection conditioned on the tap measurement. The latter could be implemented electro-optically to probabilistically generate a freely propagating distilled signal state. To simplify this proof of principle experiment the conditioning is instead
based on data post selection using a verification measurement. The tap and the signal
are recorded simultaneously, yielding data pairs, and the signal is selected dependent
on the tap value. These measurements are implemented as Stokes measurements at a
given angle in the ’dark’ plane.
As in the tomography experiment, a motorized half-wave plate was placed either in
the tap or signal setup to allow systematic observation of all quadratures given by β in
the dark S1 − S2 plane. In the tap setup this allowed investigation of the effect of the
observed tap quadrature on the distillation. Placing the wave plate in the verification
setup, the Wigner functions of both the mixed and the distilled states could be measured
for a constant tap measurement. Thus a tomography of the non-Gaussian state was
13 Agilent,
33250A 80 MHz waveform generator.
82
5. Experimental setup
possible. Here 128 equally spaced projections in phase space were made by rotating
the half-wave plate. In each, 3.5 · 106 data points were recorded and from this data the
Wigner function was reconstructed.
5.8 Detection
To avoid technical noise at low frequencies, the squeezing of RF sidebands of the optical
carrier frequency were characterized. In the process of photodetection, these sidebands
are mixed with the carrier to produce a Radio Frequency (RF) photocurrent at the beatfrequency of Ω =17.5 MHz. From the fluctuation of this RF current the quantum noise
in the optical sidebands of the carrier light beam can be deduced.
The optical signals were converted into a current signal using balanced pairs of specially designed detectors [290] based on InGaAs pin-photodiodes14 . The detector output took two forms: a DC (.100 Hz) and a RF or AC (5-40 MHz) component. The DC
component was high pass filtered and then given to an operational amplifier which converted the current signal into a voltage signal and simultaneously amplified this. The
output signal, now of the order of several Volts, was directly proportional to the incident
optical power. The AC component was first passed through a low pass Chebyschev filter to block the large RF signal caused by the laser repetition frequency. This ensured
that the AC amplification stage was not saturated by the large signal at the repetition
frequency. The filtered signal was fed to a transimpedance amplifier which converted
the input current to an amplified voltage signal. The resulting RF signal was directly
proportional to the noise spectrum of the optical sidebands Ω. All measurements were
made at at 17.5 MHz where the detector dark noise was minimal (.-86 dBm) and the detector balancing optimal. Typical optical noise powers measured were 8-15 dBm above
this intrinsic detection noise.
In all experiments the detectors were used in pairs from which, for example, the sum
and difference photocurrents were taken. It was thus necessary to balance the output
signal strength such that the outputs of two detectors were identical for the same optical
signal. Fig. 5.9 shows a typical AC versus incident optical power characterization of
a photodetector when detecting a coherent state. The squeezing measurements were
carried out in this regime, and from the plot it can be seen that the detector response is
linear. Slight nonlinearity is however possible in the regime of antisqueezing where the
optical power is similar but the RF noise much greater.
14 JDS
Uniphase, AMS Technologies, ETX-500.
5.8. Detection
83
Noise power (dBm)
-64
-68
-72
-76
-80
-84
1
10
Optical power (mW)
Figure 5.9: Plot of the noise power measured at 17.5 MHz by a detector against the incident
optical power for the measurement of a coherent beam, corrected for -87.7 dBm of electronic
noise.
5.8.1
Squeezing and entanglement
Both amplitude and polarization squeezing use the same detection principle: the optical
beam under investigation is, possibly after transmission through a series of wave plates,
incident on a (polarization) beam splitter. These methods have been outlined previously
in Sec. 2.3, and can be seen in Fig. 2.9 (quadrature detection) and Fig. 2.3 (polarization detection). The beam splitter outputs are detected by two balanced photodetectors
of the type described above. The sum and difference RF photocurrents give different
measures of the quantum noise of the input states, as described in Sec. 2.3. These RF
photocurrents are fed into a spectrum analyzer15 to measure the spectral power density
at 17.5 MHz with a resolution bandwidth of 300 kHz. This parameter acts as a bandpass filter of the signal, allowing measurement of signals in the range 17.5±0.3 MHz.
The spectrum analyzer is phase insensitive and scans over all electronic quadratures.
The maximum signal of this scan is recorded and displayed with a video bandwidth of
30 Hz. This low pass filters the displayed data and thus has an averaging effect on the
signal. A given data series contained 401 points, usually measured over 8 s. In this manner the relative noise levels of the optical sidebands of different beams were compared.
Both sum and difference RF signals, as well as the detector DC values were recorded
simultaneously. The DC values were recorded on a digitizing oscilloscope16 .
The detection of polarization entanglement proceeds analogously to that of polarization squeezing, except that two beams are measured. Each Stokes measurement of
15 Spectrum
16 Tektronix
analyzer Hewlett-Packard 8595E and 85951E, used interchangeably.
TDS 420A.
84
5. Experimental setup
Fig. 5.5 is carried out in the standard fashion, measuring the same parameter at both
detection setups at the same time. The sum and difference of these photocurrents is
then taken. To optimize the data, the cable lengths between the measurement setups
must be balanced. Via a linear attenuator the gain between the setups could also be
fine tuned. The resulting signal is recorded on a spectrum analyzer. In this manner the
(anti-)correlations of the two optical beams can be determined. The DC values are also
recorded at the same time as the RF (or AC) signals.
5.8.2
Tomography and Distillation
In contrast to the squeezing experiments, for the tomography experiment it was crucial to record the entire photocurrent statistics rather than simply measure the signal
variance. The electronics are shown schematically in Fig. 5.6. The full signal data was
obtained by electronically down-mixing both detector signals using a single sinusoidal
17.5 MHz signal from a master oscillator17 . Following each mixer was a low pass filter
with its 3 dB roll-off point at 1.9 MHz. This avoids aliasing effects in the following digital sampling process, and can be considered equivalent to the resolution bandwidth of
a spectrum analyzer.
Prior to sampling, the filtered signals were amplified by a AC-coupled
(10 Hz-10 MHz), low noise, variable gain, voltage amplifier18 . The final signal, now
typically with an amplitude just under 1 V, was measured on a computer using a 16 bit
analog to digital (AD) card19 . It was operated at 107 samples per second with one data
run consisting of a maximum of 5 × 105 samples. The AD card was was also AC-coupled
(≥10 Hz), and the final bandwidth of the data saved to the computer was 10 Hz-1.9 MHz
(3 dB). In a typical experiment eight separate data runs of were 5 × 105 samples taken
to provide sufficient data for the statistical analysis. Using further AD card, the detector DC values were also digitally recorded for each measurement using one sample per
measurement20 . Finally, a digital high pass filtering at 2 kHz was performed to subdue
further spurious electronic noise.
The detection scheme used in the distillation experiments is very similar to that of
the tomography, with two exceptions. Firstly, four instead of two detectors were used
(see Fig. 5.7), two for the distillation signal and two for the tap measurement. Secondly, the relative phase between the photocurrents as well as between these signals
and the electronic local oscillator were crucial. The former ensures that the same electronic quadrature was measured for all four detectors. The latter guarantees that the this
17 Rhode
& Schwarz, 100 kHz-1000 MHz SMX signal generator.
DHPVA-100 MHz.
19 Gage, CompuScope 1610-1M 16-bit, 10 MS/s, 2x2 channels.
20 National Instruments, NI-PCI-6014, 16 Channel IO Board.
18 Femto,
5.8. Detection
85
electronic quadrature contains the modulated signal. For example, measurement of the
electronic ”sine”-quadrature allowed measurement of the non-Gaussian mixed state,
whilst the orthogonal ”cosine”-quadrature averages over this signal and no modulation
was visible. For experimental ease, not any principle reason, the 17.5 MHz modulation
signal used in the state preparation was used as the electronic oscillator in the measurement setup. Thus, constant measurement of the modulated electronic quadrature was
ensured.
At a sample rate of 107 data points per second, the signal is significantly oversampled. This allows the 500 kHz EOM modulation signal to be well resolved. Further, the
detection system time bins could then be synchronized with the this modulation after
each measurement in the computer. Thus each 1 µs time bin could be defined to be
entirely during a ’displacement on’ or a ’displacement off’ period. The quantum state
of interest is then a string of 10 consecutive data samples recorded by the AD converter.
Again, in the digitization eight runs were typically taken to ensure sufficient statistics
for the analysis of the post selected data.
86
5. Experimental setup
Chapter 6
Results and Discussion
The fiber based squeezing sources and the distillation protocol previously described
are experimentally investigated in this chapter. The discussion focuses first on the generation and optimization of amplitude squeezing using the asymmetric Sagnac loop
in both free space and all-in-fiber configurations (Sec. 6.1). Such an optimized amplitude squeezer is exploited in the first of two schemes for the production of polarization
squeezed light, Sec. 6.2.1. These results serve to highlight the fundamental differences
between amplitude and quadrature squeezed light beams. The second setup, an elegantly simple extension of fiber based squeezers, is characterized and represents a
significant improvement on the previous scheme (Sec. 6.2.2). This setup was simulated
using the first principles quantum propagation model summarized in Sec. 4.2 and excellent agreement is found with experiment, Sec. 6.2.3. A resource efficient source of polarization entanglement is demonstrated in Sec. 6.3, and the Wigner function of the fiber
squeezed states is found in Sec. 6.4. The final experiment is the demonstration of the
distillation of squeezing from a polarization squeezed beam afflicted by non-Gaussian
noise (Sec. 6.5).
6.1 Amplitude squeezing
A number of experiments presented here are based on the asymmetric fiber Sagnac
loop. This device has been the subject of much research over the years, for example [16, 17, 13, 18, 291]. The results of this section investigate free space [163, 292] and allin-fiber configurations [244] of this robust amplitude squeezing source. The free space
setup was characterized with the goal of using it as a building block for generating po-
88
6. Results and Discussion
larization squeezing (Sec. 6.2). The all-in-fiber scheme was exploited to experimentally
determine the optimum splitting ratio in the asymmetric fiber loop.
6.1.1
Free space Sagnac loop
The setup used here is described in Sec. 5.3; the fiber loop was composed of 13.4 m
of 3M FS-PM-7811 polarization maintaining fiber (soliton energy 56 pJ). The Cr4+ :YAG
laser served as the source of ultrashort pulses. The behavior of this setup, archetypical
of asymmetric Sagnac fiber loops, is shown in Fig. 6.1. Here the AC noise at the loop
output is plotted as a function of the input pulse energy. The two traces in the plot
represent the sum (signal) and difference (shot noise) of the two photodetectors in the
balanced direct detection system (Secs. 2.3.1, 5.8). The AC noise was measured on a
spectrum analyzer at the sideband frequency of 17.5 MHz; further settings were as in
Sec. 5.8.
(a)
(b)
x-polarization
Signal
-65
-70
-75
Shot noise
-80
-85
y-polarization
-60
Noise power (dBm)
Noise power (dBm)
-60
0
25
50
75
Input pulse energy (pJ)
100
Signal
-65
-70
-75
Shot noise
-80
-85
0
25
50
75
100
Input pulse energy (pJ)
Figure 6.1: Output noise presented as a function of input energy for the two orthogonal axes
of a polarization maintaining, asymmetric Sagnac loop using 13.4 m of FS-PM-7811 fiber from
3M. The signal (sum channel) and shot noise (difference channel) traces are displayed.
It is seen in Fig. 6.1 that for certain ranges of the input energy that the signal falls
below the corresponding shot noise level, i.e. 36-48 pJ. This indicates that the emerging pulse train is amplitude squeezed. Other energies exhibit excess noise stemming
from the beam’s antisqueezing and the GAWBS phase noise. The systematic rise and
fall of the traces is a result of the interference of the strong and weak pulses after
their counterpropagation through the fiber. This classical effect is highlighted by the
representative experimental trace of output against input pulse energy in Fig. 6.3(a).
Measured plots for both the x- and y-polarizations are shown in Fig. 6.1(a) and (b) respectively. It is noted that the output noise function is virtually identical for the two
6.1. Amplitude squeezing
89
polarizations. In particular the squeezing in the first minimum and second minima are
−3.7/ − 3.8 ± 0.3 dB and −4.2/ − 4.1 ± 0.3 dB for the x- and y-polarizations respectively.
This similarity is important for the generation of polarization squeezing (Sec. 6.2) and
entanglement (Sec. 6.3).
(a)
Q
a
b
c
P
p
d
Noise
power
Shot
noise
fNL
fNL
Output
noise
noise > shot noise
(c)
Q
(b)
p
P
noise = shot noise
Q
Q
(d)
p
Input pulse energy
fNL
p
noise < shot noise
P
fNL
P
noise = shot noise
Figure 6.2: Schematic explanation of the noise power at the Sagnac loop output in dependence
on the loop input power. Here φ NL is the relative phase between the weak and strong pulses; the
constant phase shift of π of the strong pulse on the beam splitter is also shown. a) φ NL < π
exhibits output noise above the shot noise, b) φ NL = π leads to shot noise at the fiber loop output,
c) φ NL ≈ 1.5π shows maximal squeezing, d) φ NL = 2π leads to shot noise at the output. Note:
the quadrature variables shown here are unnormalized.
The dependence of the relative AC noise power on the input energy of the asymmetric Sagnac interferometer, depicted in Fig. 6.2, can be qualitatively explained by building
upon the discussions in Secs. 4.1.2, 4.1.4. Increasing the pulse energy for a fixed splitting
ratio from zero, a small nonlinear phase shift between the strong and the weak pulse
φ NL < π is found for low energies. This leads to a partially constructive interference
between the pulses, bearing the π phase shift of the bright beam on the beam splitter in
mind. Here the noise level after the Sagnac loop lies above the shot noise, depicted in
Fig. 6.2(a). This is because the projection of the ellipse onto the amplitude quadrature is
greater than that of the corresponding coherent state or the shot noise limit.
Further increasing the input energy to the fiber loop increases φ NL due to the Kerr
nonlinearity. When this phase shift becomes π complete constructive interference occurs as seen in Fig. 6.2(b). Thus the uncertainty ellipse is simply shifted to a larger
90
6. Results and Discussion
amplitude and the output pulse amplitude noise equals the shot noise. In the case of
φ NL = 2π destructive interference is observed, but again the noise is equal to the shot
noise (Fig. 6.2(d)). For a specific input energy, however, reduced or squeezed amplitude
noise can be obtained. If the relative nonlinear phase shift is φ ≈ 1.5π, the ellipse is
shifted such that the minor axis of the ellipse lies parallel to the amplitude of the pulse
(Fig. 6.2(c). This specific input pulse energy coincides with the middle of the plateau of
the nonlinear input-output energy transfer characteristic of the fiber loop. This behavior is repeated for φ NL ≈ 2πn where n is an integer, as seen in the experimental data of
Fig. 6.1. The double dip which is seen for the higher energy squeezing minima is a classical effect arising from the interference of two independently squeezed and rotating
ellipses.
6.1.2
All-in-fiber Sagnac loop
Quantitative descriptions of the output noise characteristics of the asymmetric Sagnac
loop have been developed, all of which were based on numerical simulations of the
lowest order terms of the quantum nonlinear Schrödinger equation [17, 13, 277, 280, 31].
These attempts to predict the output noise of the Sagnac loop have exhibited good qualitative agreement agreement with experiment, including the effect the asymmetric splitting ratio [13, 31]. There has however not been a comprehensive experimental investigation of the splitting ratio to fully test these predictions. In this section the asymmetry
of an all-in-fiber Sagnac interferometer with a variable ratio coupler was thoroughly
investigated [244]. The setup is shown schematically in Sec. 5.3.
The nonlinear input-output energy transfer characteristic of an all-in-fiber interferometer consisting of 9.5 m HB1500 fiber and a splitting ratio of 93:7 is shown in
Fig. 6.3(a). The Spectra Physics OPAL was the laser source used. For input energies
between 140 and 160 pJ the plot exhibits a distinct plateau, a range containing the 150 pJ
first order soliton energy. The corresponding relative noise power to this energy transfer characteristic is depicted in Fig. 6.3(b), where multiple measurement runs are shown.
These results are uncorrected for linear losses or electronic dark noise. For input powers
in the plateau region (140-160 pJ), the AC noise at the output decreases below the shot
noise, i.e. amplitude squeezing is observed. At the first plateau the measured squeezing
is −1.8 ± 0.3 dB. The energy of this plateau is significantly greater than for the results in
Fig. 6.1, a fact arising from the larger core diameter of the fiber used here. At the second
plateau a squeezing value of −2.0 ± 0.3 dB is observed. Plotting the relative noise value
that corresponds to the middle of the first plateau in the transfer characteristic against
splitting ratio for a loop of 4.8 m of fiber gives Fig. 6.4. The maximum squeezing was
found for a splitting ratio of 93:7, as predicted in earlier numerical simulations [13].
6.1. Amplitude squeezing
91
(b)
(a)
Figure 6.3: a) Nonlinear input-output power transfer characteristic of an all-in-fiber Sagnac interferometer with a variable coupler. b) The corresponding raw noise power at the interferometer
output normalized to the shot noise; squeezing of −1.8 ± 0.3 dB is observed when correcting for
detector noise. This data was taken using 9.5 m of HB1500 fiber for the fiber loop and a splitting
ratio of 93:7.
Relative noise power (dB)
2
0
-2
-4
-6
-8
-10
-12
0
2
4
6
8
10
12
14
16
18
Splitting coefficient,
Figure 6.4: Relative noise power at the Sagnac loop output in the plateau middle as a function
of the splitting ratio (η : 1 − η). Shown are measurements (empty circles), simulation values
(squares) and simulation values corrected for 30% linear loss (triangles) for 4.8 m of HB1500
fiber.
Varying the splitting ratio changes not only the observed squeezing and the energies
in which this squeezing is observed. A schematic explanation of the splitting ratio dependence of the noise power at the Sagnac interferometer output is shown in Fig. 6.5.
For a beam splitting ratio with a large asymmetry the Kerr squeezed state is reorien-
92
6. Results and Discussion
tated to yield amplitude squeezing (Fig. 6.5(a)). This is possible because the weak beam
is nearly coherent and thus the visualization is identical to that of Fig. 6.2(b). As the
beam splitter becomes more symmetric, the weak beam also becomes squeezed. This
leads to an inappropriate displacement of the Kerr-squeezed beam (Fig. 6.5(b)). In such
cases the amplitude noise of the resultant field is does not reflect the full squeezing of
the strong beam and can even lie above the quantum noise limit.
(a)
Shot noise
(b)
Q
Q
Shot noise
P
P
Figure 6.5: Schematic of the splitting ratio dependence of the Sagnac interferometer output:
a) A high asymmetry reorients the strong beam to yield amplitude squeezing, b) more symmetric
splitting ratios lead to an inappropriate displacement of the Kerr-squeezed beam.
Based on the single mode pictorial squeezing model (Fig. 6.5), rough estimates were
calculated to verify the experimental results. To simplify the derivation only the middle
of the plateau of the input-output energy plot of the Sagnac loop was considered. This
corresponds to a relative phase shift between the two beams of ≈1.5π. The acquired
nonlinear phase shifts are then:
φ NL1 = (1 − η )
3π
,
2(2η − 1)
φ NL2 = η
3π
2(2η − 1)
(6.1)
for the strong (1) and weak (2) beams respectively. η denotes the reflectivity of the beam
splitter of ratio (1 − η ) : η. As a result of the nonlinear phase shifts both beams become
quadrature squeezed and can be described by the variances [207]:
∆2ι X (θ ) = sin2 θ (cot θ + 2φ NLι )2 + 1
(6.2)
where ι = 1, 2 and θ is the quadrature angle. θ = 0 corresponds to the variance of
the amplitude of the individual beams and for this case it is seen that ∆2i X (0) = 1
which is the quantum noise limit as expected from a Kerr squeezed beam. The Kerr
6.1. Amplitude squeezing
93
squeezed ellipses are projected onto the radial direction, or the amplitude quadrature,
of the resultant field with the angles θ = ± arctan(η/(1 − η )) respectively, and the
relative noise power is determined by adding the two noise contributions incoherently,
weighted by the beam splitting ratio.
Using this model calculations of the maximum squeezing for splitting ratios ranging
from 100:0 to 82:18 were performed (Fig. 6.4). The shape of this plot agrees qualitatively
with the experimental data. The model predicts optimum squeezing for a beam splitting ratio of 92.5:7.5, which is in good agreement with the measured optimum of 93:7.
It is predicted that the noise power exceeds the shot noise limit for ratios more symmetric than 88:12. Experimentally this intersection was measured at the ratio of 89:11
for the 4.8 m of HB1500, very similar to the theoretical value. Quantitative agreement
between the squeezing values of earlier calculations and the simplistic model presented
here with the experimentally results is however lacking. Earlier fully quantum calculations predicted, for example, −11.0 dB [17]; the simplistic model presented here predicts
−7.7 dB of squeezing. Even after correcting for the 30% linear losses of the setup, both of
the theoretical predictions differ significantly from the maximum measured squeezing
of −2.4 ± 0.3 dB.
The deviation of the simple theory from experiment could lie in the many assumptions made in simplifying it, namely: it a) neglects the multimode nature of the optical
pulses, calculating in a single mode picture, b) assumes the uncertainty region to be of
elliptical form, c) uses a linearized nonlinear phase shift, d) assumes a constant relative
phase of 1.5π as the angle for optimal squeezing, e) neglects various nonlinear effects
(apart from self-phase-modulation), f) ignores important fiber effects including Guided
Acoustic Wave Brillouin Scattering (GAWBS), dispersion and Raman scattering, and
g) disregards imperfect interference of the strong and weak beams. Because of these
assumptions it must be stressed that the simplistic model yields only a rough estimate
of the noise power of the output field. For a more complete description of the experiment one must resort to the more rigorous numerical quantum calculations presented
in [17, 13]. These calculations used the quantum nonlinear Schrödinger equation, implementing only the lowest order terms.
A possible experimental source of squeezing degradation is a poor interference in
the variable coupler, since other amplitude squeezing experiments using the Sagnac
interferometer in a free space configuration have seen noticeably more squeezing,
i.e. [13, 18, 31, 163]. This is a fundamental limitation of the asymmetric Sagnac loop arising from the large power difference between the strong and weak beams. The complex
temporal and spectral evolution of ultrashort laser pulses in optical fibers will cause
significant differences in the output forms of the strong and weak pulses. This is illustrated in Appendix A where the auto-correlations and spectra of pulses of different
energy which were propagated through 8 m of 3M FS-PM-7811 are shown.
94
6. Results and Discussion
Noise power (dBm)
-60
-65
Signal
-70
-75
Shot noise
-80
0
100
200
300
400
500
Input pulse energy (pJ)
Figure 6.6: Squeezing presented as a function of input power for the a fused coupler (93:7)
in an all-in-fiber Sagnac loop using 30 m of SM15-P Panda fiber. The signal (sum) and shot
noise (difference) photocurrents are displayed, where -86.3 and -86.1 dBm dark noise have been
subtracted.
In an attempt to improve the interference quality in the all-in-fiber setups a fixed
coupler Sagnac loop was tested. Here a coupler with a fixed splitting ratio of 93:7 was
used with a loop of 30 m of SM15-P Panda fiber from Corning-Fujikura. The results,
found in Fig. 6.6, also exhibit weak squeezing [289, 293]. It should be noted that the
first squeezing minima occurs at pulse energies even higher than those for the variable
coupler setup due to the yet larger fiber core diameter. Despite its shortcomings, this
setup nevertheless represents an extremely robust and simple fiber based squeezer.
It is thus unlikely that the interference quality is solely responsible, as the interference effects of the all-in-fiber Sagnac loop in the output noise power are as apparent as
in free space experiments using this device. A final possibility is that Guided Acoustic
Wave Brillouin Scattering (GAWBS) decreased the squeezing. This could be possible in
via the compound effect of the Kerr and GAWBS effects, in which the acoustic vibrations of the fiber were coupled into the amplitude quadrature. The exact cause of the
deviation between theory and experiment cannot be named based only on these results.
Nevertheless, the qualitative agreement between both the earlier rigorous calculations,
the simplistic model and the experimental results is good, all giving an optimum splitting ratio for the asymmetric Sagnac loop of approximately 93:7. This result serves as
the basis of the following experiment producing squeezing in the polarization variables
of light.
6.2. Polarization squeezing
95
6.2 Polarization squeezing
Polarization squeezing is an intrinsically multimode phenomenon, requiring a minimum of two (nontrivial) independent modes for its generation. This as a differentiating
feature compared to amplitude squeezing is the subject of the first set of experimental
results [163]. These measurements were based on the asymmetric Sagnac fiber loop.
In stark contrast, the second set of results produced polarization squeezing using two
copropagating pulses in a single pass setup [113]. This allowed quantitatively better
results. The elegant simplicity and quality of this experiment was such that it exhibited excellent agreement with first principle quantum simulations of the setup [115], the
focus of the closing section.
6.2.1
Sagnac loop setup
The asymmetric Sagnac fiber loop, as described in Sec. 6.1.1, was exploited in the generation of states with squeezed polarization noise [163, 292]. In these experiments 13.4 m
of the 3M FS-PM-7811 fiber was used in a loop with a splitting ratio of 93:7, corresponding to results in Fig. 6.1. Polarization squeezing was produced by overlapping two such
orthogonally polarized, amplitude squeezed pulses. Thus it is important that the two
polarization axes, x and y in Fig. 6.1, can simultaneously generate optical pulse trains
with equal amplitude squeezing. The Sagnac loop setup of Sec. 5.3 could then be used
to experimentally investigate the polarization noise traits of different beam combinations. These are namely the Cases 2 and 4 of Sec. 3.1.2, that is of a single amplitude and
of two overlapped amplitude squeezed beams.
The results shown in Fig. 6.7 are for Case 2: an x-polarized, amplitude squeezed
pulse was combined with an orthogonally polarized vacuum state. As expected, both
Ŝ0 and Ŝ1 are squeezed relative to the corresponding coherent light limit: by −3.7 ± 0.3
and −3.6 ± 0.3 dB respectively. That is, these parameters exhibit the amplitude squeezing of the beam. However no polarization squeezing, as defined in Eq. 3.9, is seen.
The remaining parameters, Ŝ2 and Ŝ3 both lie +0.1 ± 0.3 dB above the coherent level,
an effect primarily due to electronic noise. Similar results were seen for ±45◦ and circularly polarized beams, whereby the squeezed parameters were Ŝ0 & Ŝ2 and Ŝ0 & Ŝ3
respectively.
The case outlined in Case 4 - the overlap of two equivalent, amplitude squeezed
pulses - was also experimentally investigated. The results thereof are displayed in
Fig. 6.8. In these data series the relative phase of the pulses was locked to zero relative phase giving rise to a linearly polarized beam at 45◦ , or a +S2 polarized beam.
Here three of the four Stokes parameters are squeezed: Ŝ0 and Ŝ1 by −3.4 ± 0.3 dB and
Ŝ2 by −2.8 ± 0.3 dB. Ŝ1 is polarization squeezed as its noise value has been brought un-
96
6. Results and Discussion
(a)
(b)
^
S
^
S
0
1
(d)
(c)
^
S
2
^
S
3
Figure 6.7: The variances of the Stokes parameters: a) Ŝ0 , b) Ŝ1 , c) Ŝ2 and d) Ŝ3 of a bright,
+S1 polarized, amplitude squeezed pulse measured over 8 s; the subtracted electronic noise was
at -86.2 dBm.
der the corresponding Heisenberg uncertainty bound, ∆2 Ŝ1 < h|Ŝ2 |i, the shot noise of
the beam. The variance of the conjugate antisqueezed parameter is h|Ŝ2 |i < ∆2 Ŝ3 has
a value of +23.5 ± 0.3 dB. It is so large as it contains the additional phase noise due to
GAWBS inherent to squeezing in fibers. It should be remembered that the squeezing in
Ŝ0 and Ŝ2 is only amplitude squeezing.
The increase in the noise of the traces in Fig. 6.8 compared with Fig. 6.7 has two
primary roots. First, the phase noise between the x- and y-polarizations or ±Ŝ1 modes,
is seen in the (here) phase sensitive parameters Ŝ2 and Ŝ3 . Whilst the average relative phase between the output polarization modes is zero, there is significant noise on
the beam from thermal and acoustic sources which the phase control was unable to
fully cancel. This phase jitter was in general amplified by any slightly misaligned wave
6.2. Polarization squeezing
97
(a)
(b)
^
S
^
S
0
1
(d)
(c)
^
S
2
^
S
3
Figure 6.8: The variances of the Stokes parameters: a) Ŝ0 , b) Ŝ1 (polarization squeezed), c) Ŝ2
and d) Ŝ3 for two phase locked, orthogonally polarized, equally bright, equivalently amplitude
squeezed pulses of mean +S2 polarization measured over 8 s; the subtracted electronic noise was
at -86.2 dBm.
plates. The second error source, fluctuations in the laser power can produce noise in all
parameters. This stems from the fact that squeezing is strongly dependent on pulse energy. Further complicating the matter, the four beams (two strong, two weak) coupled
into the fiber generally could not be coupled in with identical efficiency giving rise to
small squeezing differences between the x- and y-polarizations.
6.2.2
Single pass setup
The single pass squeezing method is a novel tool in the production of polarization
squeezed states. This elegantly simple scheme, depicted in Sec. 5.4, allows the mea-
98
6. Results and Discussion
surement of greater squeezing as well as the direct characterization of the bright Kerrsqueezed beams [113]. Both of these traits are visible in Fig. 6.9. Here the measured AC
noise as a function of the rotation of a half-wave plate (by the angle Θ) in a dark plane
Stokes measurement is seen. An oscillation between very large noise and squeezing is
observed, as expected from the rotation of a fiber squeezed state. Plotted on the x-axis is
the projection angle θ, i.e. the angle by which the state has been rotated in phase space,
inferred from the wave plate angle (θ = 4Θ). The pulse energy was 83.7 pJ (soliton
energy 56 pJ), the fiber was 13.3 m of 3M FS-PM-7811 and the Cr4+ :YAG laser was used.
q
Noise
qsq
Figure 6.9: Noise against phase-space rotation angle for the rotation of the measurement halfwave plate for a pulse energy of 83.7 pJ using 13.3 m 3M FS-PM-7811 fiber. Inset: Schematic of
the projection principle for angle θ. Results are corrected for −86.1 ± 0.1 dBm electronic noise.
For θ = 0, an Ŝ1 measurement, a noise value equal to the shot noise is found. This
corresponds to the amplitude quadrature X (0) of the Kerr-squeezed states emerging
from the fiber. Rotation of the state by θsq makes the state’s squeezing observable by
projecting out only the minor axis of the uncertainty ellipse. Further rotation brings a
rapid increase in noise as the excess phase noise, composed of the antisqueezing and
the classical thermal noise arising from GAWBS, becomes visible. The maximum noise
is observed for θ = θsq + π2 given by Y (θsq ). As discussed in Case 5 of Sec. 3.1.2, this
measurement is equivalent to the characterization of the individual component Kerrsqueezed states. Here it has been assumed that the two orthogonally polarized Kerrsqueezed states are identical, i.e. exhibit similar but uncorrelated photon statistics. This
parallels experiments using local oscillators in homodyne detection, however here no
stabilization is needed after production of the polarization squeezed state. This may be
important for experiments with long acquisition times, i.e. state tomography.
It was crucial to ensure that the measured squeezing was not due to detector saturation or other spurious effects. This was accomplished by measuring a squeezed beam
6.2. Polarization squeezing
99
1.0
Linear relative noise
0.9
0.8
0.7
0.6
0.5
0.4
0.0
0.2
0.4
0.6
0.8
1.0
Optical transmision
Figure 6.10: Linear noise reduction against optical transmission for the polarization squeezing
generated by pulses of an energy of 81 pJ in 3.9 m of 3M FS-PM-7811.
after subjecting it to different optical attenuations. Measured signals due only to the
optical noise should have a linear plot of optical transmission against linear relative
noise (Sec. 2.3). A representative plot for 3.9 m of fiber is shown in Fig. 6.10. Here an
input beam with −3.9 ± 0.3 dB of squeezing was subject to the test. The result is a line
which tends to a unity relative noise value for infinite attenuation. Based on this and
other similar results, it was concluded that the measured squeezing in this thesis was
genuine.
The squeezed and antisqueezed quadratures as well as the squeezing angle θsq of
such polarization states were investigated as a function of pulse energy for different
lengths of 3M FS-PM-7811 fiber (Figs. 6.11-6.13). The experimental values are shown
as diamonds. The figures are organized into pairs of lengths: Fig. 6.11 shows 3.9 and
13.3 m, Fig. 6.12 shows 20 and 30 m and Fig. 6.13 shows 50 and 166 m. For each length
the squeezing angle (±0.3◦ ), squeezing (±0.3 dB) and antisqueezing (±0.3 dB) form
a column. The simulation results for each fiber are also plotted; these are the subject
of the next section. The x-axis shows the total pulse energy, i.e. the sum of the two
orthogonally polarized pulses comprising the polarization squeezed pulse. Due to the
technical limitations of the photodetectors it was not possible to measure above 125 pJ
or 20 mW.
Consider first the 13.3 m fiber as a representative example which showed the greatest squeezing. The squeezing angle begins at 7.0 pJ with a relatively large 12.1◦ . It then
steadily decreases with increasing energy, leveling off for energies above 80 pJ to near
1.5◦ . This behavior is generally as expected. Considering only the Kerr nonlinearity,
increasing the pulse peak energy also increases the relative nonlinear phase shift be-
100
6. Results and Discussion
(b)
35
35
30
30
Squeezing angle (degrees)
Squeezing angle (degrees)
(a)
25
20
15
10
5
25
20
15
10
5
0
0
0
20
40
60
80
100
120
0
20
Pulse energy (pJ)
80
100
120
100
120
100
120
(d)
0
0
-1
-1
Squeezing (dB)
Squeezing (dB)
60
Pulse energy (pJ)
(c)
-2
-3
-4
-2
-3
-4
-5
-5
0
20
40
60
80
100
120
0
20
Pulse energy (pJ)
40
60
80
Pulse energy (pJ)
(e)
(f)
45
45
40
40
35
Antisqueezing (dB)
Antisqueezing (dB)
40
30
25
20
15
10
35
30
25
20
15
10
5
5
0
0
0
20
40
60
80
Pulse energy (pJ)
100
120
0
20
40
60
80
Pulse energy (pJ)
Figure 6.11: Experiments (corrected for dark noise) and simulations of the polarization squeezing, antisqueezing and squeezing angle for 3.9 (a, c, e) and 13.3 m (b, d, f) fiber lengths.
6.2. Polarization squeezing
101
tween the high and low intensity regions of the uncertainty ellipse (Fig. 4.1(a)). Thus
the angle of the squeezed quadrature θsq relative to the amplitude quadrature should
decrease exponentially towards zero as the energy tends to infinity. The curve however
exhibits an unanticipated structure, indicating that the GAWBS and Raman scattering
effects cannot be neglected. GAWBS in particular will alter the squeezing angle as it
adds significant phase noise to the squeezed beam.
The squeezing of the quadrature X (θsq ), beginning near 0 dB, is seen to increase
with pulse energy. It is found to level off for energies above 80 pJ around a value of
-5 dB. Further, above energies of 40 pJ the squeezing curve exhibits nontrivial structure.
Similarly, the antisqueezed quadrature Y (θsq ) also exhibits structure above 40 pJ not
attributable to the Kerr effect alone. However at low energies the curve is nearly an
exponential, as predicted by simple theories of the Kerr nonlinearity. These deviations
from simple considerations indicate the importance of the scattering effects, GAWBS
and Raman, on qualitative predictions of the noise output from optical fibers. GAWBS
is likely to be the more significant factor at low energies as it scales linearly with pulse
energy. Raman effects, arising from the third order nonlinearity, will gain importance at
high energies.
Due to the copropagation of two pulses it is likely that certain classical phase noise
sources in the fiber, i.e. unpolarized GAWBS, should have been common to both modes
and were canceled in the difference signal. That the antisqueezed quadrature nevertheless exhibits such a large noise value indicates the presence of a significant source of
depolarizing or polarization specific noise in the fiber. This could arise from depolarizing GAWBS, polarizing optics mounted under strain (i.e. half-wave plates) or from
high frequency noise introduced in the birefringence compensator.
The maximum observed squeezing was −5.1 ± 0.3 dB at an energy of 83.7 pJ. The
losses of the setup were found to be 24%: 4% from the fiber end, 7.8% from optical
elements and 14% from the photodiodes. Thus a maximum polarization squeezing of
−10.4 ± 1.4 dB is infered. From the simple considerations of the Kerr nonlinearity one
would however expect an exponential asymptotically approaching infinite squeezing,
or accounting for losses a value of −6.20 dB. Nevertheless it should be noted that the
measured squeezing is very close to the values predicted by fundamental theory, which
lie around −11 dB at room temperature [274, 265].
The behavior of the different fiber lengths is qualitatively similar for all measured
parameters: squeezing angle, squeezing and antisqueezing. Considering the effect of
increasing fiber length, a qualitative agreement with the predictions of the simple theory is observed. It is seen that the squeezing angle decreases for increasing fiber length,
i.e. increasing nonlinearity for a constant pulse energy. This effect is seen clearly in
Fig. 6.14(a) where the squeezing angles for a pulse energy of 46.5 pJ are summarized.
102
6. Results and Discussion
(b)
35
35
30
30
Squeezing angle (degrees)
Squeezing angle (degrees)
(a)
25
20
15
10
5
25
20
15
10
5
0
0
0
20
40
60
80
100
0
120
20
Pulse energy (pJ)
80
100
120
100
120
100
120
(d)
0
0
-1
-1
Squeezing (dB)
Squeezing (dB)
60
Pulse energy (pJ)
(c)
-2
-3
-4
-2
-3
-4
-5
-5
0
20
40
60
80
100
0
120
20
Pulse energy (pJ)
40
60
80
Pulse energy (pJ)
(e)
(f)
45
45
40
40
35
Antisqueezing (dB)
Antisqueezing (dB)
40
30
25
20
15
10
35
30
25
20
15
10
5
5
0
0
0
20
40
60
80
Pulse energy (pJ)
100
120
0
20
40
60
80
Pulse energy (pJ)
Figure 6.12: Experiments (corrected for dark noise) and simulations of the polarization squeezing, antisqueezing and squeezing angle for 20 (a, c, e) and 30 m (b, d, f) fiber lengths.
6.2. Polarization squeezing
103
(b)
35
35
30
30
Squeezing angle (degrees)
Squeezing angle (degrees)
(a)
25
20
15
10
5
25
20
15
10
5
0
0
0
20
40
60
80
100
0
120
20
Pulse energy (pJ)
80
100
120
100
120
100
120
(d)
0
0
-1
-1
Squeezing (dB)
Squeezing (dB)
60
Pulse energy (pJ)
(c)
-2
-3
-4
-2
-3
-4
-5
-5
0
20
40
60
80
100
0
120
20
Pulse energy (pJ)
40
60
80
Pulse energy (pJ)
(e)
(f)
45
45
40
40
35
Antisqueezing (dB)
Antisqueezing (dB)
40
30
25
20
15
10
35
30
25
20
15
10
5
5
0
0
0
20
40
60
80
Pulse energy (pJ)
100
120
0
20
40
60
80
Pulse energy (pJ)
Figure 6.13: Experiments (corrected for dark noise) and simulations of the polarization squeezing, antisqueezing and squeezing angle for 50 (a, c, e) and 166 m (b, d, f) fiber lengths.
104
6. Results and Discussion
(a)
(b)
4
Squeezing angle (degrees)
Squeezing angle (degrees)
12
10
8
6
4
2
0
0
20
40
60
80
100
120
140
160
3
2
1
0
180
0
20
40
Fiber length (m)
60
80
100
120
140
160
180
140
160
180
140
160
180
Fiber length (m)
(d)
(c)
0
0
-1
Squeezing (dB)
Squeezing (dB)
-1
-2
-2
-3
-4
-5
-3
0
20
40
60
80
100
120
140
160
180
0
20
40
Fiber length (m)
100
120
(f)
40
40
35
35
Antisqueezing (dB)
Antisqueezing (dB)
80
Fiber length (m)
(e)
30
25
20
15
10
5
0
60
30
25
20
15
10
5
0
20
40
60
80
100
120
Fiber length (m)
140
160
180
0
0
20
40
60
80
100
120
Fiber length (m)
Figure 6.14: Polarization squeezing, antisqueezing (both corrected for dark noise) and squeezing
angle against fiber length for 46.5 pJ pulses (a, c, e) and the maximum squeezing (b, d, f).
6.2. Polarization squeezing
105
Here an exponential decrease with fiber length is found, as expected from basic considerations.
The antisqueezing of all fibers is also similar, including the fine structure above 40 pJ.
It is seen to increase with fiber length for a constant energy, as expected (Fig. 6.14(e)).
An exponential-like curve is seen, although deviations for long fibers are apparent, suggesting the rise of Raman scattering. For the particularly long fibers and high energies
the excess noise becomes extremely large and effective phase locking of the orthogonally polarized pulses was infeasible. This is apparent in the data shown in Fig. 6.13 for
the 50 and 166 m fibers.
The measured squeezing at low energies for all fiber lengths is similar (Fig. 6.14(c)),
although the most deviations between fibers is seen in this parameter. Plotting the greatest measured squeezing against length in Fig. 6.14(d), the optimum fiber length for the
measured energy range is found to be 15±5 m. The corresponding angles and antisqueezing values are also shown (Fig. 6.14(b,f)). Considering only the fiber nonlinearity,
this fact is surprising, once again showing that GAWBS and Raman effects are vital in
making accurate predictions.
Noise power (dBm)
-71
-72
Shot noise
-73
-74
-75
-76
0
20
40
60
80
100
Time (minutes)
Figure 6.15: Plot showing a stable squeezing of −4.0 ± 0.3 dB over 100 minutes corrected for
85.8 ± 0.1 dBm dark noise. A 30 m 3M FS-PM-7811 fiber with a pulse energy of 80 pJ was
used.
In the measurements of the 20 m fiber the maximum noise reduction was not observed in the dark plane. These results, shown in Figs. 6.12(c), 6.14(d) as gray diamonds,
are are not polarization squeezing. The root of this deviation is uncertain but could stem
from a noisy laser mode. Further, it should be noted that the 13.3 m length was taken
from a different production run than the other fibers. As such it mode field diameter
was smaller: 5.42 µm compared with 5.69 µm of the others. While this alters the effective
106
6. Results and Discussion
nonlinearity, it does not appear to have a significant effect on the qualitative behavior
of the data.
The single pass polarization squeezer exhibits a good temporal stability, highlighted
by the results in Fig. 6.15. Here the sum (shot noise) and difference (polarization squeezing) channels have been plotted. An average of -4.0 dB of squeezing was measured over
100 minutes, a result which has been corrected for 85.8 ± 0.1 dBm of dark noise. The
squeezer was comprised of 30 m of 3M FS-PM-7811 optical fiber into which two orthogonally polarized pulses of 80 pJ each were coupled. The most sensitive factor in this
setup is the locking of the birefringence compensator. Further important parameters
are the coupling of light into the fiber and the laser power stability. All of these could
be managed using fast and precise control systems which are commercially available.
6.2.3
Simulations
Simple models based on only the Kerr effect can be used for simple predictions of the
trends in fiber squeezing experiments, as in Sec. 6.1.2. However the inclusion of GAWBS
and Raman effects is necessary for a quantitative treatment of ultrashort pulse propagation, both of which have been long considered to be important in fiber propagation [147, 8, 262, 264, 265]. Such a model was presented in Sec. 4.2, and was applied to
the single pass polarization squeezer presented in the previous section. Based on similar
models, simulations of the propagation of ultrashort pulses in asymmetric fiber Sagnac
loops have made good qualitative predictions of experiments [17, 13, 31, 277, 280]. However quantitative agreement has been elusive.
The elegance of the single pass squeezing scheme has changed this. This setup not
only allows more efficient squeezing production but also simplifies the corresponding
simulations, also noted by [281]. This lies in the fact that there is no interference of
strong and weak pulses after propagation through the fiber. There is thus a certain
similarity to symmetric Sagnac interferometers which have been more successfully predicted, although these have not been quantum propagation simulations and have neglected Raman effects [11, 250].
The quantum propagation simulations of the single pass experiments of Sec. 6.2.2
were carried out by J. F. Corney and P. .D. Drummond in a very fruitful collaboration.
The parameters used were t0 = 74 fs, z0 = 0.52 m, n = 2 × 108 , and λ0 = 1.51 µm.
The fiber parameters were derived from the data for the 3M FS-PM-7811 fiber used
in the experiments given in Table 5.1. Fibers from two production runs were used in
the experiments with slightly different mode field diameter: the 13.3 m had a 5.42 µm
diameter while the other fibers had a diameter of 5.69 µm. This was accounted for in
the simulations. The remaining parameters were measured only for the first fiber roll;
those of the second were assumed to be identical. For technical reasons no experimental
6.2. Polarization squeezing
107
data was available about the Guided Acoustic Wave Brillouin Scattering (GAWBS) of
this fiber. Thus a fitting procedure was implemented to integrate this into the model,
as described later in this section. Simulation results for the different fibers are seen
in Figs. 6.11-6.13(a-f) where the dashed (continuous) lines show the simulated results
without (with) fitted excess phase noise, i.e. GAWBS. The dotted lines in the squeezing
(c, d) indicate the simulation sampling error. The ±0.3 dB squeezing measurement error
is also shown. The theoretical results for both squeezing (c, d) and antisqueezing (e, f)
have been corrected for the measured linear losses of 24%.
The omission of the classical noise produces results that clearly deviate from experiment (dashed lines). This is particularly visible at low pulse energies, where GAWBS is
a significant source of phase noise since it scales linearly with energy. At higher powers
(or for long fibers) the experimental and theoretical values for both the squeezing angle
(a, b) and the antisqueezing (e, f) coincide, while those for the squeezing still exhibit a
discrepancy. This is explained the fact that the fraction of the total phase noise stemming
from GAWBS decreases with increasing power as Raman effects begin to dominate. Further, the ellipse produced by the Kerr effect is both less eccentric and at a greater angle
to the amplitude quadrature at low energies and thus will be more altered by the addition of excess phase noise. The deviation of theory from experiment in the squeezing for
both high and low energies indicates the great sensitivity of this parameter to classical
phase noise.
(a)
(b)
X
X
Y
Y
Figure 6.16: Representation of the principle used to add the excess phase noise to the simulated
results. A single fit parameter was used per fiber and the fit was made to the squeezing angle,
which reacted more sensitively to this noise at low powers (b) compared to high powers (b).
As it was not measured, the classical phase noise arising from, e.g. GAWBS, was
fitted to the experimental data. In this process only one fit parameter was used per fiber
and it was assumed that this noise scaled linearly with pulse power [249, 226]. These
scaling laws have been previously checked against numerical simulations [226], which
allowed the fitting procedure to be applied after propagation. This fitting procedure is
justified in that the frequency shift due to GAWBS over these fiber lengths is negligible
compared to the mode binning and thus the extra phase noise does not significantly
108
6. Results and Discussion
alter the pulse propagation. A schematic of the fitting process is shown in Fig. 6.16 for
low powers (a) and high powers (b). In this figure it is assumed that the mean amplitude is very much larger than the fluctuations and thus the curvature of the constant
amplitude lines are neglected. It was seen that the squeezing angle for low energies
reacts very sensitively to this noise source, Figs. 6.11-6.13(a, b).
The phase noise was estimated by fitting the simulated squeezing angles to the
experimentally measured squeezing angles following the method shown in Fig. 6.16.
Mathematically this was accomplished by minimizing the variance between the fitted
angles θfit and measured angles θsq in a nonlinear least squares fit for all energies. The
fitting parameter was ∆2 Ŝ p which describes the magnitude of the phase noise. In this
process it was assumed that the phase noise scales linearly with energy [226]. The phase
noise was included in the simulation results by calculating a new relative noise variance
for the dark plane Stokes parameters Ŝ⊥ (θfit ) using:
π
∆2 Ŝ⊥ (θ ) = ∆2 Ŝ p sin2 (θ ) + ∆2 Ŝ(θsim ) cos2 (θ − θsim ) + ∆2 Ŝ(θsim + ) sin2 (θ − θsim ),
2
(6.3)
where θsim is the simulated squeezing angle without phase noise for a given energy
and ∆2 Ŝ p , ∆2 Ŝ(θsim ), ∆2 Ŝ(θsim + π2 ) are the variances of the phase quadrature, squeezing
and antisqueezing variances for this energy. This resulted in convincing fits of theory
to all three measured variables for all fibers and energies, see Figs. 6.11-6.13. This is
the first time such quantitative agreement has been achieved between first principles
quantum propagation theory and the measured data of squeezed states produced by
optical fibers.
8
Relative phase variance
7
6
5
4
3
2
1
0
0
10
20
30
40
50
60
Fiber length (m)
Figure 6.17: Plot of the fitting parameter used to account for excess phase noise in the quantum
fiber propagation simulations. This phase factor is shown as a function of length.
The phase-noise variance determined from the fitting process was seen to scale with
the fiber length, Fig. 6.17. This indicates that it is largely a fiber-induced effect. At
6.3. Polarization Entanglement
109
the measurement frequency of 17.5 MHz the most likely cause is depolarizing guided
acoustic wave Brillouin scattering (GAWBS), as noise not specific to a given polarization
will be largely common to both polarization modes and thus canceled. However, there
also seems to be some birefringent noise sources outside the fiber, possibly arising from
the polarization elements used or in the birefringence compensator and lock loop.
Despite including excess phase noise in the simulations, there is still some residual
discrepancy in the squeezing for many fiber lengths for some pulse energies, with the
notable exception of the 13.3 m fiber. This is most likely due to a variation in the fiber
parameters, stemming from the use of fibers from two different production runs. The
data in Table 5.1 are the measured values for the production run from which the 13.3 m
fiber was taken. It is quite possible that the fiber properties vary slightly between runs,
highlighted by the difference in the mode field diameter. Another reason could be that
some effects which have been neglected, such as cross-polarization Raman effects or
higher-order dispersion, do play an important role in the pulse evolution. A thorough
investigation of the fiber properties, perhaps to also include higher order effects, could
help to clarify this matter.
6.3 Polarization Entanglement
The Sagnac loop source of polarization squeezing source described in Sec. 6.2.1 was
employed in a scheme for the resource efficient production of polarization entanglement [186]. The setup for this experiment is discussed in Sec. 5.5 and consists of one
polarization squeezed beam incident on a 50:50 beam splitter. The squeezer had a
mean polarization along S2 which allowed observation of −3.4 ± 0.3 dB of polarization
squeezing in the Ŝ1 parameter, while its conjugate, the Ŝ3 parameter was antisqueezed
by +23.5 ± 0.3 dB (compare with Fig. 6.8). The noise traces characterizing the polarization squeezing and entanglement were corrected for electronic noise of -86.9 dBm.
As the polarization squeezed state in this experiment had a nonzero S2 mean value,
Eq. 3.27 can be used to check for nonseparability. The nonclassical correlations in the
conjugate Stokes operators were observed by measuring the respective Stokes parameters at the two output ports of the beam splitter and taking the variance of the sum and
the difference signals. In Fig. 6.18(a) the variances of the Stokes parameters of the individual modes at the output ports A and B of the beam splitter and the corresponding
shot noise level are plotted. Each individual mode is already squeezed in Ŝ1 , but only
by -1.3 dB due to the contribution of the vacuum fluctuations from the empty beam
splitter input. The variance of Ŝ1,A − Ŝ1,B and its corresponding quantum noise level is
shown in Fig. 6.18(b); the difference signal lies -2.9 dB below the shot noise. Thus, nonclassical correlations are observed in the Ŝ1 parameter, in particular it is found that the
110
6. Results and Discussion
normalized variance is ∆2 (Ŝ1,A − Ŝ1,B ) = 0.52. The polarization squeezing source characterized by the noise traces of Fig. 6.8 was used for the generation of the polarization
entanglement shown in Figs. 6.18, 6.19. Although the setup is identical, the noise traces
can not be directly compared as additional electronic RF-splitters/combiners were used
in the entanglement experiment. These devices attenuate the detected photocurrents
resulting in a difference in the measured absolute noise values.
(a)
(b)
-75.5
Noise power (dBm)
Noise power (dBm)
-73
-76.0
Shot noise
-76.5
Beam A
-77.0
Shot noise
-74
-75
^)
D (^
S1A-S
1B
2
Beam B
-77.5
-76
Time (s)
Time (s)
Figure 6.18: a) Polarization squeezing and b) correlations in the Ŝ1 parameter of the beam
splitter outputs A and B. The difference noise ∆2 (Ŝ1,A − Ŝ1,B ) is -2.9 dB below shot noise.
The noise traces of the Ŝ3 parameter are rather different. Each individual signal at
the two output ports has a high degree of noise (Fig. 6.19(a)) as the initial beam was antisqueezed in the Ŝ3 parameter. Nevertheless, the variance of the sum signal Ŝ3,A + Ŝ3,B
(Fig. 6.19(b)) coincides with the shot noise level. Thus, it is determined that the squeezing normalized variance ∆2 (Ŝ3,A + Ŝ3,B ) = 1. The application of the nonseparability
criterion of equation Eq. 3.27:
∆2 (Ŝ1,A − Ŝ1,B ) + ∆2 (Ŝ3,A + Ŝ3,B )
= 0.52 + 1 < 2,
hŜ2,A i2 + hŜ2,B i2
(6.4)
proves that a highly correlated, nonseparable quantum state in the Stokes variables has
been generated.
It has been shown that the source described above produces two intense light fields
which are entangled in their polarization variables. Thus the entanglement was detected and manipulated without the need of a stable phase reference as is the case for
quadrature entanglement. All relevant parameters were checked in direct detection. In
contrast to sources using, e.g. optical parametric oscillators, only one phase has to be
locked to achieve good stability. Only one nonlinear device (a polarization maintain-
6.3. Polarization Entanglement
(a)
111
(b)
-50
Beam A
Beam B
Noise power (dBm)
Noise power (dBm)
-55
-60
-65
-70
-75
2
-55
-60
-65
-70
Shot noise
^)
D (^
S3A-S
3B
^ ) = Shot noise
^ +S
D (S
3A
3B
2
-75
Time (s)
Time (s)
Figure 6.19: a) Polarization antisqueezing and b) correlations in the Ŝ3 parameter of the beam
splitter outputs A and B. The sum noise ∆2 (Ŝ3,A + Ŝ3,B ) is at the shot noise level.
ing fiber) was needed to produce the entanglement, making the source compact and
efficient.
The degree of entanglement using this fiber based system can be further improved in
two ways. To generate subshot noise quantum correlations in both conjugate variables
one needs to combine the polarization squeezed beam with a bright coherent beam instead of a vacuum state. Both variances in equation Eq. 6.4 would then fall below unity
as is desirable for many quantum communication protocols. The complexity of the experiment would increase only moderately, as one additional phase would be locked.
However the total nonclassicality will not increase; it is only differently distributed. To
increase the degree of polarization entanglement, the interference of two polarization
squeezed beams at the 50:50 beam splitter is a possibility. This would result in a polarization entanglement equal to the degree of the squeezing invested, in our case more
than −3 dB. The price one has to pay is the need for a second birefringent compensator
and fiber Sagnac interferometer. An entangler using the single pass setup is also possible, here the conjugate operators would be different, but the general behavior would be
identical.
Even though all the possible polarization generation schemes mentioned above satisfy the condition of equation Eq. 6.4 they exhibit different properties. A simple criterion
like equation Eq. 3.27 gives information on the overall amount of entanglement, but can
not take into account the exact state properties. These properties determine the applicability of the source for a particular protocol.
112
6. Results and Discussion
6.4 Quantum state tomography
The Wigner function of bright, ultrashort, Kerr-squeezed pulses produced by propagation through an optical fiber was measured for the first time [294]. This was made
possible by the equivalence of homodyne detection and polarization measurements in
the ”dark plane”. Rotating the half-wave plate in the measurement setup by an angle Θ
(Sec. 5.6) allowed observation of the squeezed state’s X (θ ) quadrature given by θ = 4Θ.
Two orthogonally polarized pulses each of 80 pJ energy were launched into 13.3 m of
3M FS-PM-7811 fiber to produce an S3 polarized beam. The squeezed pulses were measured in the dark S1 − S2 plane in 128 equally spaced (δθ = 1.41◦ ) projections in phase
space each consisting of 3.5 · 106 data points.
(a) Raw marginal distribution data
q
X(q)
(b) Reconstructed Wigner function
X
Y
Figure 6.20: a) Scatter plot of the raw marginal distribution data as a function of the phase
space angle θ and b) the reconstructed Wigner density plot of the quantum state of an ensemble
of ultra-short pulses of 80 pJ energy after propagation in 13.3 m of FS-PM-7811 glass fiber
(vertically rescaled by a factor of two for clarity).
The resulting histograms, shown in Fig. 6.20(a) as scatter plots in X (θ ) against θ,
are the marginal distributions of the measured quadratures W ′ (X (θ )). The raw marginal distributions were measured through a phase space angle of 180◦ . The inverse
Radon transformation [130] was applied to these distributions to derive the Wigner
function of the Kerr-squeezed state, a density plot of which is found in Fig. 6.20(b).
The depicted state has a squeezed variance ∆2 X = 03.2 ± 0.2 dB and an antisqueezing
of ∆2 Y = 26.3 ± 0.3 dB. The most striking feature of the quasi-probability distribution
6.5. Distillation of non-Gaussian noise
113
is the strong excess noise arising from the antisqueezing and excess phase noise in the
fiber.
6.5 Distillation of non-Gaussian noise
Corruption of nonclassical states, such as those characterized in this chapter, is inevitable during generation and transmission in real systems. Many such destructive
processes produce Gaussian noise while a number of others are non-Gaussian, for example free space transmission. The distillation of nonclassical resources afflicted by nonGaussian noise using readily available linear optics and homodyne detection is demonstrated here [114]. This was realized in an experimental setup composed of three steps
(Sec. 5.7): preparation, distillation and verification. The preparation consisted of a single
pass polarization squeezer as investigated in Sec. 6.2.2. In this experiment the source exhibited squeezing of ∆2 Xsq = −3.1 ± 0.3 dB and antisqueezing of ∆2 Ysq = +27 ± 0.3 dB
relative to the quantum noise level. These signals were characterized using a computer
based detection system sampling the sidebands at 17.5 ± 0.5 MHz relative to the optical
carrier frequency.
The first step, the preparation of the mixed state of Eq. 3.32, i.e. the noisy generation and transmission, was accomplished by combining a polarization squeezer with
a controllable noise source in the form of a discrete phase shift which gave rise to a
phase space displacement in the dark polarization plane. The squeezed state, now corrupted by non-Gaussian noise such that ∆2 X = +1.4 ± 0.3 dB, was fed into the distiller. The remaining two steps of the protocol were: i) a tap measurement on 7% of
the beam, and ii) a signal post selection conditioned on the tap measurement. Measuring the (anti-)squeezed quadrature in the (tap) signal on an ensemble of identically
prepared noisy states the distributions in Fig. 6.21(a,b) were constructed. The modulation was chosen such that the variance of the noisy signal was just greater than that
of the shot noise. Performing post selection of the simultaneously measured singal
data by conditioning it on the tap measurement, a recovery of the squeezing is observed. That is, the distilled signal distribution is narrower than that of the shot noise
(Fig. 6.21(b)). The measured variance of the tap is ∆2 Yt = +17.5.0 ± 0.3 dB relative to
the shot noise. Conditioning on the tap, the noisy signal variance ∆2 Xs = +1.1 ± 0.3 dB
fell to −2.6 ± 0.3 dB.
Using the data shown in Fig. 6.21, the distilled signal variance was investigated as a
function of the post selection threshold. In Fig. 6.22 it is seen that an increasing threshold decreases the signal variance, ultimately approaching the input squeezing. This
agrees well with the exponential increase in squeezing predicted by Eq. 3.44, given by
the dashed line. As the threshold increases, the success probability or amount of dis-
114
6. Results and Discussion
0.6
0.4
Distilled
signal
0.5
Probability
Probability
Shot noise
0.3
0.2
Threshold
0.4
Shot noise
0.3
0.2
Noisy
signal
0.1
0.1
Tap
0.0
0.0
-20
-10
0
10
Amplitude (shot noise units)
20
-4
-3
-2
-1
0
1
2
3
4
Amplitude (shot noise units)
Figure 6.21: Experimentally measured marginal distributions, centered at zero for convenience,
outlining the distillation of a squeezed state from a non-Gaussian mixture of squeezed states;
a) tap measurement yt , b) signal measurement xs . The inset shows phase space representation
of the input, tap and signal mixed states; the projected quadratures used in the measurements is
also seen. Note: the excess noise in the experiment was much greater than depicted here.
tilled data decreases to zero causing an increase in the statistical error on the variance.
Thus a compromise between the post selected variance and probability of success must
be made. The success probability was easily determined by the verifier (here the experimenter) as he controls the noise source and thus can determine the success probability
by comparing the input and output signals.
The effectiveness of a given threshold depends on i) the projection of the displacement (which is along S2 ) onto the measured quadrature and ii) the variance of the measured quadrature. Fig. 6.23(a,b), each with a different threshold, shows this effect. The
measured tap quadrature X ( β) which has been rotated by an angle β, effectively changing the displacement size. The best distillation is observed for small angles where the
displacement (x̄0 − x̄1 ) to threshold (yt ) difference is largest. It is seen that the quality
6.5. Distillation of non-Gaussian noise
115
S1
X
Y
S2
0
old
Thresh
Figure 6.22: Experimentally and theoretically distilled squeezing (left) and success probability
(right) as a function of post selection threshold for two displacements. The threshold is given
relative the center of the marginal distributions, as depicted in the inset.
of the distillation decreases with increasing β as the projection of the displacement onto
the measured quadrature increases. It is however also seen that for large thresholds
the distillation quality is approximately independent of β or the measured quadrature
X ( β ).
The Wigner function of both the mixed and the distilled states was also measured.
Fig. 6.24(a) shows the density plots of the Wigner function associated with the noisy
state; its non-Gaussian nature is evident. In Fig. 6.24(b) the Wigner function of the post
selected signal is shown and a distribution very similar to that of a single squeezed
state (Fig. 6.20) is observed. Thus the purity of the state relative to the non-Gaussian
state is increased and a corresponding recovery of the squeezing is seen. The shift to
the right of the distilled Wigner function reflects the post selection process as well as the
normalization.
The discrete noise source experimentally implemented here can be generalized to
continuously distributed noise sources. As a first example consider a top hat distribution of displaced squeezed vacuum states, a direct extension of the experiment. In these
calculations the two state discrete case of Sec. 3.3 and the experiment was extended by
simply allowing the mean displacements of the states ȳι to be continuously distributed.
The probability of any ȳι was equal and the maximum and minimum mean displacements correspond to the discrete displacements of the experiment, ȳ0 = 0, ȳ1 = 39
shot noise units. Using these and the other experimental parameters it is found that a
noisy signal of ∆2 X = 0.71 is successfully distilled to ∆2 Xsdistill = 0.64 with a probability
Π = 0.15.
116
6. Results and Discussion
(b)
(a)
S1
X
Y
S2
b
Figure 6.23: Distilled variance (left axis) and success probability (right axis) as a function of the
quadrature angle relative to the squeezed quadrature (angle β) in the tap measurement for a post
selection threshold of a) 1.3 and of b) 5.3 shot noise units. The inset shows a given thresholds for
two angles β = 0◦ , 45◦ ; a different threshold would follow a circle of a different radius.
Further, the distillation of i) vacuum and ii) displaced squeezed states subject to a
fading channel were investigated. This noise source is described by a log-Gaussian,
or log-normal, distributed attenuation factor which is typical of optical transmission
through a turbulent atmosphere [29, 195, 196]. Here spatial variations in the atmospheric temperature and pressure gives rise to fluctuations in the refractive index
along the transmission path of the form n(t, r) = n0 + δn(t, r ). The noise factor δn can
cause fluctuations in the intensity and phase of the transmitted optical beam. Considering the spatial coherence of the transmitted beam, the optical field can be expanded
as [195]:
E(r ) = E0 (r)eiφ0 (r) · eΦ(r) ,
(6.5)
where E0 , φ0 are the field amplitude and phase without turbulence and Φ is the perturbation factor:
′ E (r )
+ i φ(r) − φ(r) = ∆E(r ) + i∆Φ(r ).
(6.6)
Φ(r) = ln
E0 (r)
6.5. Distillation of non-Gaussian noise
117
(a) Noisy signal state
X
Y
(b) Distilled signal state
X
Y
Figure 6.24: Density plots of the Wigner function distributions for a) the non-Gaussian mixed
state measured at the verification setup and b) the Wigner function of the corresponding distilled
state. Note: To aid visualization the plots have been vertically rescaled by a factor of two.
∆E represents the amplitude fluctuations whilst ∆Φ describes the phase variations. Assuming these factors are independent Gaussian random variables, as is the case in the
turbulent atmosphere [29, 195], it is then obvious that the amplitude (or gain) and phase
noise sources are multiplicative. They thus give rise to intensity and phase probability
distributions which are non-Gaussian. Of primary interest here are the fluctuations in
the amplitude. In atmospheric transmission this will take the form of a fluctuating attenuation factor which will disturb any squeezed optical beams transmitted through it.
The marginal probability distribution of the attenuation in such a case is described by
the log-Gaussian function:
−(ln( a)− µ a )2
exp
2∆2 a
√
,
(6.7)
f ( a) =
a 2π∆2 a
where the variance ∆2 a and mean µa describe the Gaussian distribution of the exponent, i.e. α or L. In the theoretical calculations of the fading channel the following set of
values was considered: ∆2 a = 0.16 (corresponding to strong turbulence) and µ = −1.
Additionally, the distribution was truncated above zero attenuation (and correspondingly normalized) to avoid amplification during transmission (Fig. 6.25).
Vacuum states with the same properties as in the above experiment was transmitted through such a channel to produce a noisy state with ∆2 X = 0.75. This arises
from the log-Gaussian attenuation which produced a continuous mixture of states between the squeezed vacuum and vacuum state. This state was successfully distilled to
118
6. Results and Discussion
3.0
2.5
f(a)
2.0
1.5
1.0
0.5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Attenuation factor a
Figure 6.25: Plot of the marginal probability distribution of a log-Gaussian function with a
log-variance of 0.16 and a log-mean of -1.
∆2 Xsdistill = 0.64 with a success probability of Π = 0.002. The distillation of displaced
squeezed states was similarly simulated. As in the case of a vacuum squeezed state,
the transmitted states were a continuous mixture of states between squeezed and vacuum states, but the displacement ȳι of the states varied correspondingly. Here a noisy
variance of 0.81 was reduced to 0.74 with a probability of 0.078. In further calculations
successful distillation was observed independent of the attenuation factor distribution.
In these, as in the experiments, it was found that higher thresholds lead to improved
distillation, highlighting the versatility of the distillation protocol in realistic applications.
Chapter 7
Conclusion and Outlook
The focus of this thesis was the generation and characterization of optical pulses with
nonclassical statistics measured in the quadrature and polarization variables. Their production was accomplished by exploiting the nonlinear refractive index or Kerr effect experienced by ultrashort laser pulses in optical fibers. The resultant devices represent deterministic, high frequency sources of noise reduced (squeezed) or quantum correlated
(entangled) intense pulse trains. Such resources are a prerequisite of many quantum
communication and information protocols, i.e. teleportation and quantum cryptography. While the generation and transmission inevitably suffers loss, a method for the
distillation of nonclassicality from a squeezed beam afflicted by non-Gaussian noise
was demonstrated.
The first experimental results characterized and optimized asymmetric fiber Sagnac
loops as a source of amplitude squeezed light. A particular focus was on variable and
fixed all-in-fiber setups which are compact and highly reliable. Such systems are effectively plug-and-play sources of squeezed light, which when coupled with a fiber based
femtosecond laser would represent the simplest, most robust portable source of nonclassical light. This source can be further extended to generate quadrature entangled
light [293], making it a promising resource for field employments of quantum communication and information protocols.
The nonlinear effects in silica fibers were also harnessed in the production of light
with reduced polarization noise. Two such schemes were examined, with a single pass
method proving to be more efficient than Sagnac loop based devices. An excellent
−5.1 ± 0.3 dB of polarization squeezing was produced in this novel setup, the greatest measured to date. From this value it is inferred that −10.4 ± 1.2 dB were generated
in the fiber. To further improve the measured noise reduction, losses after the fiber must
be minimized by for example employing more efficient photodiodes in a minimal detec-
120
7. Conclusion and Outlook
tion setup using highest quality optics. It is speculated that net losses of as little as 5%
should be possible, thereby allowing the measurement of squeezing in excess of -8 dB.
This would open the possibility many application requiring very strong nonclassicality due to lossy operations. Exploiting the equivalence of polarization squeezing and
vacuum squeezing, highly squeezed dark modes could be generated as a versatile tool
for further operations. Further, the developement of photonic crystal fibers with fewer
low frequency acosutic vibrations is also expected to improve this result by minimizing
destructive GAWBS noise [257, 259]. Such an advance would bring the squeezed states
closer to minimum uncertainty states, an important feature in quantum information.
The characterization of the single pass setup fit very well with the theoretically predicted behavior based on a first principles quantum fiber propagation model [33]. These
confirmed that Guided Acoustic Wave Scattering (GAWBS) and Raman scattering are
the factor limiting squeezing at low and high pulse energies respectively. It would be
interesting to carry out the experiment as even higher powers to further check the theoretical predictions. The simulations could be improved by further and more accurate
measurements of the fiber properties including GAWBS, the nonlinear refractive index
and higher order dispersion coefficient. In a minimal loss setup, the agreement between theory and experiment would then be expected to be exceptional and without
the need for any fitting parameter. Such a measurement has motivated experiments for
over 20 years and would be a remarkable achievement of the theory of many body dynamics. The calculations could conversely be used to support experiment finding the
optimal parameters for squeezing including fiber length and pulse duration.
The presented polarization entanglement source exploiting the Sagnac loop is well
suited for future quantum communication experiments as it produces entangled states
deterministically and at a high repetition rate limited only by the laser repetition rate.
The fiber integration of the source would increase its stability and simplify its introduction into existing communication networks. Free space applications might however be
preferable as the polarization mode dispersion has to be compensated in fiber applications. Using the single pass scheme to generate entanglement would increase the degree
of entanglement, providing possibly the best entanglement employing optical fibers.
Transmitting any nonclassical state, for example through free space, inevitably subjects it to noise sources. The distillation method presented here can be implemented
to recover the nonclassicality of squeezed beams suffering from non-Gaussian noise.
It was shown theoretically that many such noise sources, e.g. discrete and continuous
phase space displacements and attenuations, can be counteracted. Future experiments
could investigate naturally occurring noise sources such as turbulent atmospheric transmission or fluctuating pump laser power. They could also generate freely propagating
resources after the conditioning for use in further quantum operations. Another extension of this work would be to perform a conditional optical operation, i.e. phase shift or
121
displacement, on the signal beam. Thus not only the distillation demonstrated here but
also a purification of the large classical noise arising in fiber pulse propagation could
be implemented [88]. This would generate an even purer nonclassical resource than
produced here. Whilst the results of this thesis have focused on single-mode squeezed
states, these techniques can assuredly be extended to two-mode squeezed or entangled
beams.
122
7. Conclusion and Outlook
Appendix A
Characterization of fiber output
pulses
The plots contained in this appendix show the intensity auto-correlations and spectra
of pulses of different input powers (typical fiber coupling of 80%) after propagating
through 8 m of 3M FS-PM-7811 fiber. The laser pulses originated from the mode-locked
femtosecond Cr4+ :YAG laser described in Sec. 5.1. Its output were sech shaped pulses
with τ=130-150 fs and λ0 =1495-1500 nm; the output characterization corresponding to
the results here is seen in Fig. 5.1. Light was coupled onto only one of the fiber’s birefringent axes; propagation in the orthogonal axis was seen to exhibit identical behavior
within the measurement accuracies1 . The figures clearly show the energy dependence
of the pulse evolution during propagation. The fiber’s soliton energy is 56 pJ. The maxima of the auto-correlations have been normalized to make relative changes in pulses’
temporal properties easily visible. The spectra have only been rescaled and thus show
the increase in pulse energy. For technical reasons, low power pulses (< 46 pJ) could
not be experimentally characterized.
1 APE
Pulsescope
124
A. Characterization of fiber output pulses
(a)
(b)
46.0 pJ
46.0 pJ
(d)
(c)
49.1 pJ
(e)
49.1 pJ
(f)
55.2 pJ
55.2 pJ
Figure A.1: Auto-correlations (a, c, e) and spectra (b, d, f) of pulses of input energies 46.0, 49.1
and 55.2 pJ, respectively, after propagating through 8 m of 3M FS-PM-7811 fiber.
125
(a)
(b)
61.3 pJ
61.3 pJ
(d)
(c)
67.5 pJ
(e)
67.5 pJ
(f)
73.6 pJ
73.6 pJ
Figure A.2: Auto-correlations (a, c, e) and spectra (b, d, f) of pulses of input energies 61.3, 67.5
and 73.6 pJ, respectively, after propagating through 8 m of 3M FS-PM-7811 fiber.
126
A. Characterization of fiber output pulses
(a)
(b)
79.8 pJ
79.8 pJ
(d)
(c)
85.9 pJ
(e)
85.9 pJ
(f)
92.0 pJ
92.0 pJ
Figure A.3: Autocorrelations (a, c, e) and spectra (b, d, f) for pulses of input energies 79.8, 85.9
and 92.0 pJ, respectively, after propagating through 8 m of 3M FS-PM-7811 fiber.
127
(a)
(b)
98.2 pJ
98.2 pJ
(d)
(c)
104.3 pJ
(e)
104.3 pJ
(f)
107.4 pJ
107.4 pJ
Figure A.4: Auto-correlations (a, c, e) and spectra (b, d, f) for pulses of input energies 98.2,
104.3 and 107.4 pJ, respectively, after propagating through 8 m of 3M FS-PM-7811 fiber
128
A. Characterization of fiber output pulses
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Curriculum vitae
Personal information
Name:
Born:
Marital status:
Nationality:
Parents:
Joel Roland Heersink
21. April 1979 in Edmonton, Canada
Married
Canadian, Dutch
Elizabeth Joanne Heersink (born Drost)
Roland Edward Heersink
Further education
Since 09/2001:
Doctorate student with Prof. Dr. G. Leuchs at the Max-Planck
Research Group, the Institute for Information, Optics and
Photonics at the Friedrich-Alexander-Universität (FAU)
Erlangen-Nürnberg, Germany
09/1997–06/2001: Master of Science at Imperial College, London, UK
10/1999–06/2000: ERASMUS year at the FAU Erlangen-Nürnberg
10/1999–06/2000: Masters project ”Investigation of the effects of magnetic fields on a
multi-channel photomultiplier and the construction and testing of a
scintillating fibre hodoscope” at the Physikalisches Institut IV,
Prof. Dr. W. Eyrich at the FAU Erlangen-Nürnberg
School education
1997:
1994–1997:
1992–1994:
International Baccalaureate in Amsterdam, Netherlands
International School of Amsterdam, Netherlands
International School of Hamburg, Germany
1989–1992:
1986–1989:
1985–1986:
Grace Brethern Schools, Simi Valley, California, USA
Timothy Christian School, Elmhurst, Illinois, USA
Arcadia Christian School, Arcadia, California, USA
Work experience
07–09/2000:
07–09/1999:
07–09/1998:
Work as a programmer at Siemens, Erlangen
Work as a programmer at Honeywell Hi-Spec Solutions, Phoenix,
Arizona, USA
Work as a programmer at Honeywell Hi-Spec Solutions, Phoenix,
Arizona, USA
Acknowledgements
The work presented in this thesis would not have been possible without the support of
my colleagues and family. In particular I would like to thank my supervising professor,
Gerd Leuchs, who provided the possiblity and foundation for this thesis at the Max
Planck Reserach Group at the University of Erlangen-Nünrberg. Further, this project
was carried out under the supervision of Natalia Korolkova and Ulrik Andersen who
gave both inspiration and guidance.
I appreciate the discussion and help as well as the light-hearted moments provided
by the many coworkers at the Insitute for Optics. In particular the colleagues, past
and present, from the Quantum Information Processing group deserve my thanks for
their advice and assistance in making the experiments here a reality. There are the ”old
ones” who innitiated me in the secret arts of the optics lab: Oliver Glöckl, Stefan Lorenz
and Christine Silberhorn. I am very grateful for those with whom I also worked with
in in the lab: Tobias Gaber, Aška Dolińska, Metin Sabuncu, Christoph Marquardt and
Ruifang Dong. The fruits of their labor are apparent in this thesis. The further aid of
Klaus Sponsel, Kristian Cveček and Freidrich König in taming the Cr4+ :YAG laser were
of great value. The linguistic consultation of Jessica Schneider and Christoph Marquardt
was much appreciated.
Collaborations with Vincent Josse, Radim Filip and Maria Chekhova in the lab, and
Joel Corney and Peter Drummond in theory, opened many opportunities for my work
which would otherwise have been unachievable. The excellent and timely labor of the
mechanical and electronic workshops both at the Max-Planck Forschunggruppe and at
the department of Physics also greatly extended the possibilities of my work. Manfred
Eberler deserves thanks for his support in the area of infromation technology.
To my wife Claudia, as well as my family, I am very much indebted for cheering me
up when things were not running ”sub-optimally” and for supporting me every day
anew.