Download Universit`a degli studi di Roma “La Sapienza”

Università degli studi di Roma
“La Sapienza”
Facoltà di Scienze Matematiche, Fisiche e Naturali
Corso di Laurea in Fisica
Tesi di Laurea Magistrale
Design of the homodyne detector to
measure the squeezing level of the light
for Advanced Virgo
Laureando
Sarah Recchia
Relatore
Prof. Fulvio Ricci
Secondo Relatore
Prof. Ettore Majorana
Anno Accademico 2012/2013
Contents
Introduction
1
1 Detection of gravitational waves
1.1 Gravitational Waves: a brief introduction . . . . . . . . . . . . .
1.2 Direct Detection of Gravitational Waves: Interferometric Detectors
1.2.1 Interaction of Gravitational Waves with test masses and
Interferometric Detectors . . . . . . . . . . . . . . . . . .
1.2.2 Response of a GW interferometric detector . . . . . . . .
1.3 Quantum noise in gravitational waves interferometric detectors .
1.3.1 Photon shot noise . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Quantum radiation pressure noise . . . . . . . . . . . . .
1.3.3 Standard Quantum Limit . . . . . . . . . . . . . . . . . .
1.4 Quantum enhancement of Gravitational Waves Interferometric
Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Origin of the Standard Quantum Limit . . . . . . . . . .
1.4.2 Circumventing the SQL: injection of squeezed vacuum
states at the output port of the interferometer . . . . . .
3
4
7
8
11
15
16
18
18
19
20
22
2 Squeezing of the light
28
2.1 Field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.1.1 Electric field quadratures and modulation . . . . . . . . . 30
2.1.2 Correlation functions of the electromagnetic field . . . . . 31
2.1.3 Correlation functions and coherence properties of the electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Quantum states of light . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.1 Fock states . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2.3 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3 Generation of squeezed light . . . . . . . . . . . . . . . . . . . . . 45
2.3.1 Input-output formalism for optical cavities . . . . . . . . 48
2.3.2 Squeezing by a degenerate parametric amplifier . . . . . . 50
2.3.3 Ponderomotive squeezing . . . . . . . . . . . . . . . . . . 54
3 Homodyne detection
3.1 Theory of detection . . . . . . . . . . . . . . . . .
3.1.1 Direct detection . . . . . . . . . . . . . .
3.1.2 Balanced homodyne detection . . . . . . .
3.2 Design of the electronics of a homodyne detector
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69
CONTENTS
3.3
3.2.1 Operational amplifiers . . . . . . . . . . . . . . . . . . .
3.2.2 The photodiodes . . . . . . . . . . . . . . . . . . . . . .
3.2.3 The bias circuit of the photodiodes . . . . . . . . . . . .
3.2.4 DC, AUDIO and RADIO readout blocks. . . . . . . . .
The noise study . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Electronic noise . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Noise calculations for the homodyne detector prototype
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4 Realization and test of the homodyne detector prototype.
104
4.1 Measurement of the intensity noise power spectrum of the laser
Mefisto-InnoLight . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Conclusion
114
Bibliography
116
Ringraziamenti
117
Introduction
The direct detection of the gravitational waves (GW) is one of the most ambitious scientific targets nowadays. Its importance is both theoretical and astrophysical. In fact, the observation of the GW would confirm the predictions of
General Relativity and would provide a valuable contribution to the investigation on the Physics in a strong gravitational regime opening a new era of the
study of the exotic state of ethe matter in the inner core of the compact stars.
It would allow to study the universe from a new perspective, complementary to
that based on electromagnetic observations.
The history of the GW detectors traces back to the sixties and, starting from
the first GW antennas, arrives to the modern Michelson interferometric detectors, which have reached such an impressive sensitivity to the displacement of
test masses (∼ 1018 m) that quantum noise is expected to be the limiting factor
to the sensitivity of the next generation detectors. Thus, a farther improvement
of the detector sensitivity will require to circumvent the standard quantum
limit (SQL) of the detector. Few ideas have been proposed for the quantum
enhancement of the GW interferometers, and one of the most promising is to
inject squeezed vacuum, a non classical state of light, into the output port of
the interferometer. The quantum enhancement via squeezed vacuum has been
experimentally demonstrated on meter-scale prototype of GW interferometers.
If the squeezing angle of the injected vacuum state is constant all over the detection band (frequency independent squeezing) the SQL is beaten only
at a given detection frequency. If the squeezing angle is appropriately tailored
for each detection frequency (frequency dependent squeezing) the SQL is
beaten all over the detection band.
Squeezed states have been produced both in the few MHz band and in the
audio band of the gravitational waves detectors (10Hz-10kHz), by means of
optical parametric processes. However, technical limitations, for example photothermally driven fluctuations, reduce the squeezing level. In addition, the
squeezing produced in this way is frequency independent. In order to implement the frequency dependent squeezing few methods have been studied, such
as the use of detuned filter cavity, which allows to rotate the squeezing angle.
On the other hand, an other production method, the ponderomotive squeezing, which exploits the radiation pressure to produce squeezing as a result of
the coupling between the radiation inside an interferometer and the mechanical
motion of a suspended mirror, is being investigated. This method seems to be
able to provide frequency dependent squeezing.
These successes justify the effort spent for the design and realization of squeezing apparatus for the advanced GW detectors.
1
CONTENTS
2
Independently on their application, squeezers need a squeezing detector, which
allows to characterize the squeezing factor and angle of the produced squeezed
state. A detector for squeezed light has to be sensitive to arbitrary quadrature
of the electric field of the squeezed state. Thus, a phase dependent detection
scheme has to be implemented. Balanced homodyne detection provides such a
phase dependent scheme by mixing, through a 50/50 beam splitter, the squeezed
state with a strong coherent local oscillator (a laser), which is used as a phase
reference and it is much more intense than the squeezed field. The sum and
the difference of the photocurrents of the two beams exiting the beam splitter
are computed electronically and their spectra are measured. It can be shown
that the spectrum of the difference photocurrent contains the information about
the noise of the squeezed state at a given quadrature, whereas the spectrum of
the sum photocurrent contains only the information about the noise of the local oscillator. If the local oscillator is quantum noise limited, the spectrum of
the sum photocurrent equals the spectrum of the difference photocurrent when
the squeezed vacuum is substituted with the unsqueezed vacuum. In this case
the spectrum of the sum photocurrent is used to compare the spectrum of the
squeezed vacuum with that of the unsqueezed vacuum.
In the first chapter of this thesis, we briefly introduce the gravitational waves
and we illustrate the main characteristics of the GW interferometric detectors.
Then, we discuss the use of squeezed states of light for the quantum enhancement of the interferometer.
In the second chapter we make a review of the formal treatise of squeezing. In
particular, we summarize the fundamental concepts of quantum optics and optical coherence and apply them to the mathematical description of the squeezed
states of light. We conclude the chapter by illustrating two methods for the
production of squeezed light, i.e the production by means of optical parametric
processes and the ponderomotive squeezing.
In the third chapter, first we introduce the theory of homodyne detection, focusing on the characteristics that an homodyne detector for squeezed light should
fulfill. Then, we present the design of the electronics of a homodyne detector
prototype for the squeezer of the GW interferometer Advanced Virgo, explaining the leading criteria, such as the frequency response, followed in the design
and reporting the noise study performed for the circuit.
In the fourth chapter we illustrate the methods, which will be used to test the
prototype. Finally we show the experimental characterization of the intensity
noise of the laser, which will be used to test the homodyne detector prototype.
Chapter 1
Detection of gravitational
waves
The direct detection of gravitational waves (GW) is one of the most ambitious
experimental challenges today. It will allow to confirm the predictions of General Relativity (GR) compared to other theories of gravitation and it will open
a new era for astrophysics, providing a new and deeper insight into the universe,
complementary to the present scenario based on electromagnetic observations.
We could mention the GW stochastic background of cosmological origin, whose
detection would provide an unique view of the early Universe, and the GW
emitted by compact objects (such as neutron stars and black holes), whose detection would provide enlightening information about exotic states of matter
and strong gravitational field regimes.
This great experimental effort, started in the sixties with the first GW antennas, led today to a worldwide net of GW interferometric detectors, which have
reached such an impressive sensitivity in measuring small displacements of test
masses, that nowadays experimenters have to face the quantum limits of the
detectors. Beating those limits will be one of the challenges of the third generation detectors and some ideas have been proposed. One of the most promising
is Caves’s [4]. In 1981 he first proposed the possibility to circumvent the standard quantum limit (SQL) in Earth-based GW interferometers by injecting at
the output port of the interferometer squeezed vacuum, a non classical state
of light. Then, several experiments have been carried on to produce squeezed
light (see [23] and [5]), first in the radio-frequency band and then in the audiofrequency band interesting for advanced GW detectors. The success of these
pioneering experiments justifies the decision that the next generation GW interferometers include squeezing apparatus in their design.
The conceptual scheme of a squeezer for GW detectors is shown in Fig. 1.1. We
recognize four blocks on the diagram: the laser, i.e the light source from which
the squeezed vacuum is obtained, the squeezing block, i.e the light squeezer, the
injection system, whose central element is the Faraday Rotator, which allows
the unperturbed injection of the squeezed vacuum into the output port of the
interferometer and finally the detection block, which allows to check the obtained squeezing factor and angle before injecting the squeezed vacuum into the
3
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
4
Figure 1.1: Conceptual scheme of a squeezer for an Earth-based GW detector
inteferometer.
In order to characterize the squeezing of light the implementation of a phase
dependent detection scheme is required. A detection scheme often used for this
purpose is the homodyne detection, as described by [23], [22].
The purpose of this thesis is the design of an homodyne detection system for the
characterization of squeezed states of lights for the GW interferometer Advanced
Virgo.
1.1
Gravitational Waves: a brief introduction
In 1915 Einstein published the General Theory of Relativity (GR), its theory of
gravitation, summarized in the famous equations
8πG
1
Rµν − gµν R = 4 Tµν
2
c
(1.1)
which link the geometry of space-time (contained in the Ricci tensor Rµν and
the Ricci scalar R) to the energy-matter distribution (contained in the stressenergy tensor Tµν ).
The Ricci tensor and the Ricci scalar contain second order derivatives of the
metric tensor gµν , through which the space-time proper distance
ds2 = gµν dxµ dxν
(1.2)
is defined. The first derivation of gravitational waves from Eq. 1.1 is always due
to Einstein ([9], [10]): he showed that it is possible to linearize the equation of
GR if small perturbations of the flat space-time metric tensor ηµν are considered:
gµν = ηµν + hµν

−1 0
0 1
ηµν = 
0 0
0 0
|hµν | 1
(1.3)

0 0
0 0

1 0
0 1
(1.4)
(1.5)
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
5
He found that the solutions of the linearized equations in vacuum, i.e the components of hµν , are wave-like and exhibit some similarities with their electromagnetic counterpart.
The details of the calculation are reported in many textbooks on GR (see for
example [17]) so we only report some basic results: the derivation starts from
Eq. 1.1 in vacuum, i.e Tµν = 0. If we consider small perturbation of flat spacetime, as in Eq. 1.3, and use the gauge freedoms of GR, a simple expression for
hµν can be found:
hTµνT = 0
(1.6)
2
where = −∂ 0 + O2 while T T stands for transverse-traceless and indicates
the particular gauge choice. The solution thus found is such that
• hT0µT = 0
• hT T
i
i
=0
• ∂ j hTijT = 0
µ = 0, 1, 2, 3
i = 1, 2, 3
i, j = 1, 2, 3
(traceless condition)
(transverse condition)
Eq. 1.6 is the same wave equation of electromagnetic waves and has plane-wave
solutions
hTijT (x) = eij (k)eikx
(1.7)
where
• eij (k) is the polarization tensor, which has two independent components,
called the + and × polarizations.
• k = ( ωc , k) and ωc = |k|
so GW propagates at the speed of light
• The transverse condition ∂ j hij = 0 gives kj hij = 0
so GW are transverse waves
Those properties are in common with electromagnetic waves.
If we choose the z-axis as the propagation direction we can obtain


h+ h× 0
h z i
hTijT (t, z) = h× −h+ 0 exp ω t −
c
0
0
0 ij
(1.8)
We can recognize the retarded potential expression typical of electromagnetic
waves.
Despite those similarities, the difference between the two types of radiation
is deep. The most peculiar characteristic of GW is the metric nature, i.e the
fact that those waves are perturbations of the metric tensor, i.e are a property
of space-time. In addition we note that while the electromagnetic radiation is
vector-type field, gravitational radiation is tensor-type field. Given that the
electric charge has two signs, the lowest order mode of oscillation for electromagnetic radiation is dipolar while, given that the mass has only one sign, the
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
6
lowest order mode of oscillation of gravitational radiation is quadrupolar. Einstein derived the quadrupole formula for the wave field at distance r from the
source and for the luminosity LG of the GW produced by a source of density ρ
r
2G .. T T TT
i, j = 1, 2, 3
(1.9)
hij (t, r) = 4 Qij t −
rc
c
LG =
r ... r
G X ... h
Qij t −
Qij t −
i
5
5c ij
c
c
(1.10)
where, if qij is the quadrupole moment of the source, Qij and QTijT are, respectively, the reduced and transverse-traceless part of the quadrupole moment. The
symbol hi indicates a time average on a time much larger than the period of the
GW
Z
1
2
qij =
ρ(t, x) xi xj − δij |x| dV
(1.11)
3
V
1
m
(1.12)
Qij = qij − δij qm
3
These formulae explain why the direct detection of GW is so difficult. The
coefficient in Eq. (1.10)
G
≈ 10−54 W −1
(1.13)
5c5
is extremely small. Only systems characterized by huge masses and fast varying
quadrupole moments can produce detectable effects, i.e astrophysical systems.
The gravitational luminosity produced by those systems should be incredibly
huge for us to have a chance to detect the GW on Earth. This is evident when
we consider the gravitational flux arriving on Earth from a source at distance r
(we consider isotropic emission)
F =
LG
4πr2
(1.14)
r ∼ 10kpc for sources in our galaxy and of order of mpc for sources in the Virgo
Cluster galaxies. Thus, despite the enormous gravitational power produced by
some astrophysical events, the huge distances involved make the GW flux on
Earth extremely weak and the detection of GW so challenging.
Few examples of astrophysical objects, which are expected to produce GW, are
• binary systems in the inspiral and merger phase
the most interesting are those composed by compact object, such as neutron stars and black holes
• rotating single compact object
in particular non axisymmetric spinning objects (for example pulsars)
• cosmological and astrophysical stochastic background
the first one is the GW counterpart of the cosmological microwave background, the second one is the result of the superposition of many GW
signals coming from astrophysical sources
• supernovae explosion
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
(a)
7
(b)
Figure 1.2: Left: Orbital period variation rate of PSR 1916+13. The points
indicates the experimental data of Hulse and Taylor. The plane line indicates the
GR prediction. Right: French-Italian VIRGO Gravitational Waves detector
The first indirect evidence of the existence of GW came from the binary system
PSR 1916+13 (in which one of the companions is a pulsar) discovered in 1974 by
Hulse and Taylor. The system is composed by two neutron stars of nearly equal
masses (∼ M ) and one of these objects emits radio pulses. The two radioastronomers measured the variation rate of the orbital period of the system and
found the impressive agreement, shown in Fig. 1.2(a), with that predicted by
General Relativity if taking into account the energy loss due to the emission of
GW. Hulse and Taylor were awarded the Nobel Price in 1993.
1.2
Direct Detection of Gravitational Waves: Interferometric Detectors
After the Einstein’s paper on GW, this topic was just studied from a theoretical
point of view. It was even not obvious if they could be linked to a measurable effect. This doubt comes out from the wide gauge invariance of GR, and someone
was convinced that it was possible to cancel via a suitable gauge transformation
the gravitational-wave induced modifications of the metric tensor.
Then, in 1956-1959 Bondi demonstrated that GW carry energy, so they have
measurable effects and can be detected. Meanwhile a theoretical activity begins
aimed to better understand the processes of GW production. Pirani starts to
face the problem of their direct detection. In 1960 J. Weber found out that an
harmonic oscillator made by two test masses can couple to GW and starts to
oscillate. Weber suggested the possibility of converting those oscillations in electrical signals through piezoelectric crystals: he invented the GW antenna. Many
GW antennas, such as AURIGA, EXPLORER and NAUTILUS, were built all
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
8
around the world and some of them are still in operation. These detectors are
most sensitive only in a narrow bandwidth (∼ 50Hz) around their resonance
frequency, which is often approximately at 900 Hz, while, as for electromagnetic radiation, the GW spectrum spans over many frequency decades. Many of
the interesting signals are in audio-band frequencies or below and a broadband
detector has better detection perspectives. The Earth-based kilometer-scale
laser-interferometric detectors are wide-band detectors, whose pioneering idea
is due to Pirani, who suggested to use the light for probing the space-time between two freely gravitating test masses.
In the following we describe how GW interacts with matter and how GW interferometers work, emphasizing on the noise sources, which limit their sensitivity.
1.2.1
Interaction of Gravitational Waves with test masses
and Interferometric Detectors
In a theory as GR, invariant under general coordinate transformation, the gauge
choice can drastically change the description of the effect of the GW on test
masses. For example, the choice of the TT-gauge leads to the conclusion that
the coordinates of the test masses do not change due to the GW, but the proper
distance between them changes. On the other hand, if we choose the detector
frame (the frame that is naturally used in experiments), it can be shown that
a tidal force due to the GW is applied to the test masses. An enlightening
discussion about this topic can be found in [17].
On the other hand, whatever the reference frame choice could be, a crucial
aspect to point out is the fact that in GR it is not possible to reveal in a frameindependent way the presence of gravitational effects looking at the motion of a
single particle. In fact it is always possible to locally cancel the effect of gravity
by properly choosing the reference frame or, if a gravitational field is absent,
it is possible to simulate its presence by choosing a non-inertial frame. The
equation, which describes the motion of a single test mass freely moving in a
gravitational field, is called geodesic equation
Duµ
=0
dλ
(1.15)
duµ
Duµ
=
+ Γµνρ uν uρ
dλ
dλ
(1.16)
where
• D indicates the covariant derivative, a generalization of the concept of
derivative in differential manifolds. Γµνρ are the Christoffel symbols of the
metric. They contain first order derivatives of the metric tensor gµν and
define the covariant derivative of General Relativity.
• λ parametrizes the geodesics and can be chosen to be the proper time for
massive particles
• uµ =
dxµ
dτ
is the four-velocity of the test mass
GR tells us that Γµνρ can be locally set to zero by choosing a locally inertial
frame (see [17]). In such a frame we would obtain, locally, the geodesic equation
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
9
µ
valid for a free particle in flat space-time ( du
dλ = 0), i.e in absence of gravity.
On the other hand, we obtain the same equation of motion as Eq. 1.15 also
in absence of a gravitational field, by simply looking at the motion of the free
particle in flat space-time from an accelerated reference frame. It is not possible
to distinguish between a gravitational field and an accelerated reference frame
by looking at the motion of a single particle.
However, looking at the curvature of space-time, we can distinguish between a
gravitational effect or an accelerated frame effect. This is possible if we consider the differential motion of at least two test masses. This can be done by
considering two nearby geodesics, one with coordinates xµ (λ) the other with
xµ (λ) + ξ µ (λ), and by deriving an equation for ξ µ (λ). We obtain the so-called
equation of the geodesic deviation:
D2 ξ µ
µ
= −Rνρσ
ξ ρ uν uσ
dλ2
(1.17)
µ
where Rνρσ
is the Riemann tensor. It contains second order derivatives of the
metric tensor. The Ricci tensor and Ricci scalar, which appear in the Einstein Eq. 1.1, are derived from it. The Riemann tensor contains information
about the curvature of space-time and can be used to characterize in a frameindependent way the characteristics of space-time.
Eq. 1.17 tells us that two nearby test masses in a gravitational filed experience
a tidal force determined by the Riemann tensor.
We apply the geodesic deviation equation to the case of an Earth-based GW
detector, which we treated by now as a couple of free test masses. Given that
the gravitational field of Earth is weak, we can approximate the metric of the
space-time as flat. In addition, we assume, as is implicitly done in Earth-based
experiments, that we measure distances with rigid rulers. This is called in [17]
the proper detector frame. If in this frame we consider the arrival of a GW,
treated as a small perturbation of flat space-time metric, it can be shown that
Eq. 1.17 becomes
..i
1 .. T T
ξ = hij ξ j
(1.18)
2
This equation tells us that in the proper detector frame two test particles of
equal mass m experience the ”Newtonian force”, because of the GW,
Fi =
m .. T T j
h ξ
2 ij
(1.19)
Let us note that ξ was defined as the coordinate difference of two nearby
geodesics, so Eq. 1.18 tells us as the relative coordinates of the test masses
change, in the proper detector frame, because of the GW effect.
We can now apply this result to the simple case of a GW propagating along the
z-axis and impinging on a system of test masses disposed on the xy-plane. For
+ polarization we have (see Eq. 1.8)
1 0
hTabT = h+ sin(ωt)
(1.20)
0 −1
we express the positions of the test masses as
x(t) = (x0 + δx(t), y0 + δy(t))
(1.21)
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
10
where (x0 , y0 ) are the unperturbed positions and (δx(t), δy(t)) are the GW
induced displacements. Solving Eq. 1.18 we obtain for + polarization
h+
x0 sin(ωt)
2
h+
δy(t) = − y0 sin(ωt)
2
δx(t) =
(1.22)
(1.23)
and for × polarization
h×
x0 sin(ωt)
2
h×
δy(t) =
y0 sin(ωt)
2
δx(t) =
(1.24)
(1.25)
So we have that the displacement of the test masses from their rest positions
Figure 1.3: Effect of a gravitational wave propagating along the z-axis on a ring
of free test masses disposed on the xy-plane.
change periodically in time, in a different way for + and × polarization, because
of the GW, as shown for a ring of test masses in Fig. 1.3. Let us now substitute
the ring of test masses with a Michelson-type interferometer with suspended
end mirrors, as that shown in Fig. 1.4.
It can be shown that, if we consider GW with angular frequency higher than the
mechanical angular frequency of the suspended mirror ωGW ωm , the mirrors
can be considered as free, so we can apply the results obtained for the free test
masses. In the case of a GW exciting a differential motion of the two end mirrors
of the Michelson interferometer, the two light beams traveling in the two arms
will accumulate a different phase. This will reflect in an amplitude modulation of
the light detected by the photodetector at the output port of the interferometer.
In Fig. 1.5, we show the case in which the × polarization contributes just
to the common mode of the mirrors, which does not generate any interferometric signal.
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
11
Figure 1.4: A basic prototype of Michelson GW interferometer.
Figure 1.5: Effect of a gravitational wave propagating along the z-axis on a
Michelson interferometer with arms along the x and y axis.
1.2.2
Response of a GW interferometric detector
In this section we calculate the response of a Michelson interferometer (Fig. 1.4)
to a gravitational wave. Let us consider the + polarization
h+ (t) = h0 cos(ωGW t)
(1.26)
Following [17] we perform the computation in the TT-gauge, in which, as we
have already mentioned, the positions of the mirrors are fixed while the proper
length of the arms changes due to the GW.
The electric field of the laser is
E(t) = E0 e−i(ωt−k·x)
(1.27)
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
12
where Lx and Ly are the lengths of the x-arm and the y-arm. We chose the
beam-splitter to be an ideal 50% one and we consider it as fixed at the origin
of the xy-plane, were the two arms lie. The two arms are oriented along the y
and x axis.
In absence of the GW, the round-trip time of the light in the two arms is
tx,y = 2
Lx,y
c
(1.28)
After a round trip of the light in the two arms, the two fields (taking into
account the extra phase due to the reflection off the beam-splitter and the
mirrors) recombining at the beam splitter are
1
2Lx
x
E (t) = − E0 exp −iω t −
(1.29)
2
c
1
2Ly
E y (t) = E0 exp −iω t −
(1.30)
2
c
In presence of the GW the space-time interval is given by
ds2 = −c2 dt2 + [1 + h+ (t)]dx2 + [1 − h+ (t)]dy 2 + dz 2
(1.31)
and, considering a photon (ds2 = 0) propagating along the x arm, we have at
first order in h0 (the + sign apply for the propagation from the beam-splitter
to the end mirror, the − for the return trip)
1
dx = ±cdt 1 − h+ (t)
(1.32)
2
The photon entering the x arm at time t0 , will reach the end mirror at time t1
and again the beam-splitter at time t2 . Given that in the TT-gauge the end
mirror is fixed at position Lx , integrating Eq. 1.32 we obtain
Z
c t1 0
Lx = c(t1 − t0 ) −
dt h+ (t0 )
(1.33)
2 t0
Z
c t2 0
Lx = c(t2 − t1 ) −
dt h+ (t0 )
(1.34)
2 t1
(1.35)
The calculation leads to (see [17] for the details)
2Lx
Lx
+
h+ (t0 + Lx /c)sinc(ωGW Lx /c)
c
c
Ly
2Ly
t2 − t0 =
−
h+ (t0 + Ly /c)sinc(ωGW Ly /c)
c
c
t2 − t0 =
respectively for the x and y arm.
Thus, at time t the two fields recombining at the beam-splitter are
1
2Lx
E x (t) = − E0 exp −iω t −
+ i∆φx (t)
2
c
1
2Ly
E y (t) = E0 exp −iω t −
+ i∆φy (t)
2
c
(1.36)
(1.37)
(1.38)
(1.39)
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
13
where
ωLx
sinc(ωGW Lx /c) cos[ωGW (t − Lx /c)]
c
ωLy
sinc(ωGW Ly /c) cos[ωGW (t − Ly /c)]
∆φy (t) = −h0
c
∆φx (t) = h0
(1.40)
(1.41)
We note that in general Lx and Ly are made as close as possible in order to
cancel common mode noises of the two arms. Thus, in ∆φx and in ∆φy , which
L −L
are of order h0 , we can just replace Lx and Ly with L = x 2 y . We obtain
∆φy = −∆φx
Then, we can rewrite the two fields as
1
2L
x
E (t) = − E0 exp −iω t −
+ iφ0 + i∆φx (t)
2
c
2L
1
y
− iφ0 + i∆φy (t)
E (t) = E0 exp −iω t −
2
c
(1.42)
(1.43)
(1.44)
where φ0 = k(Lx − Ly ) and ∆φM = ∆φx − ∆φy .
The total electric field at the output of the interferometer is
Eout (t) = E x (t) + E y (t)
= −i E0 e−iω(
t− 2L
c
(1.45)
) sin[φ + ∆φ (t)]
0
x
and the corresponding output power Pout ∼ |ET OT |2 is
Pout = P0 sin2 [φ0 + ∆φx (t)]
P0
=
{1 − cos[2φ0 + ∆φM (t)]}
2
(1.46)
The phase φ0 is a parameter which can be adjusted experimentally. In particular
it is convenient to choose φ0 = π/4, in fact, with this choice the expression for
the output power becomes
Pout =
P0
{1 + sin[∆φM (t)]}
2
(1.47)
We can see from Eq. 1.47 that the effect of the GW result in a modulation of
the output power.
This is evident if we note that ∆φM (t) 1. Indeed, the sinc and cosine
functions have maximum value 1, h0 is extremely small (of order ∼ 10−20 ), the
frequency of the laser is of order ∼THz and L is of order of ∼km. Thus we can
approximate sin[∆φM (t)] ≈ ∆φM (t) and the output power becomes
P0
ωL
Pout =
1 + 2 h0
sinc(ωGW L/c) cos[ωGW (t − L/c)]]
(1.48)
2
c
The factor sinc(ωGW L/c) is ∼ 1 for ωGW L/c 1 and ∼ 0 for ωGW L/c 1: the
consequence is that the interferometer is “blind”to gravitational waves whose
wavelength is much smaller than the arm length.
For ωGW L/c 1 the GW modulate the output power at frequency ωGW and
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
14
the modulation depth depends on the GW amplitude h0 and on the arm length
L.
The gravitational wave signal which, as we have seen, modulates the output
power of the interferometer, can be readout via demodulation techniques, such
as, for example, the heterodyne readout method. See [7] for a complete review
of the readout techniques for the GW interferometric detectors.
The equation 1.48 suggests that, given h0 , the larger L the larger the modulation depth and the better we can detect the GW. On the other hand, if L
is too long, such that ωGW L/c 1, the sinc function suppresses the response
to the GW, as we have already discussed. An approximated formula for the
optimal arm length L can be found in [17]
100Hz
(1.49)
L ' 750Km
νGW
where νGW is the GW frequency.
Such values are too high for a Earth-based detector. A trick, which allows to
Figure 1.6: Fabry-Perot Michelson for Gravitational Waves detection.
overcome this problem, is to use Fabry-Perot (FP) cavities in the two arms, as
shown in Fig. 1.6.
The FP stores the light in the cavity increasing the arm length by a factor
Nef f , which represents the effective number of round trips of the light in the
FP cavity. Nef f depends on the cavity finesse F
2F
π√
π R1 R2
F =
1 − R1 R2
Nef f =
where R1 and R2 are the mirrors reflectivities.
(1.50)
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
1.3
15
Quantum noise in gravitational waves interferometric detectors
In the previous section we calculated the response of a GW Michelson interferometric detector to a GW signal without taking into account the noise sources,
which limit the sensitivity of the detector. The typical amplitudes of GW arriving on Earth are extremely small (order of magnitude h0 ∼ 10−20 − 10−22 and
below) and, in order to have a meaningful detection probability GW interferometers must be able to measure length differences smaller than 10−18 m. Thus,
noise sources, which limit the sensitivity, must be identified and filtered out as
much as possible.
Earth-base GW interferometers detect GW with frequencies in the audio-band,
for example for Virgo the band is 10Hz − 10kHz. In Fig. 1.7 we show the sensitivity curve of Advanced Virgo and its main noise contributions, as an example
of sensitivity curve of a GW interferometer. Among several noise sources cited
here, there are some dominant in a specific frequency band:
Figure 1.7: Sensitivity curve of Advanced Virgo. Contributions from all prominent noise sources are reported. The black dashed line represents the Virgo
sensitivity. If a gravitational wave at a certain frequency has amplitude (strain)
h0 bigger than the value of the sensitivity curve integrated on 1Hz band around
the GW frequency, the GW can be detected, otherwise cannot be distinguished
from the noise. The curve for AdV is more or less ten times lower than that of
Virgo, thus AdV sensitivity is ten times better.
• seismic noise
it dominates at frequencies up to order of ∼ 10 − 40Hz and is due to
the seismic-induced motion of the suspended mirrors. It can be reduced
by suspending the mirrors with a multiple-stages pendulum. Above the
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
16
suspension eigenfrequency f0 , the series of pendula attenuates the seismicinduced motion of the mirrors by a factor 1/f 2N , where f is the detection
frequency and N is the number of stages
• thermal noise
it dominates at intermediate frequencies in the detector band and is due to
the thermally induced motion of the mirrors. Thermal noise also includes
fluctuations of the refractive index of the transmissive optical elements of
the interferometer, thermally induced vibration of the reflecting surfaces
of the mirrors and thermal noise of the suspensions. Because of this noise
source materials with low mechanical losses are used for the mirrors and
their suspensions.
Another approach to reduce thermal noise, which is being investigated for
the Japanese interferometer KAGRA, could be to cool the mirrors and
the suspensions at low temperature.
• quantum noise
it is due the to the quantum nature of the light used to probe the spacetime properties inside the detector arms and it is the noise source which
ultimately limit the sensitivity of the detector. Quantum noise includes
the photon shot noise, which dominates at the high frequencies of the
detector band and the quantum radiation pressure noise which dominates at the low frequencies of the detector band
in this section we concentrate our attention on quantum noise, studying its
characteristics and deriving an expression for its contribution to the noise of the
GW interferometer.
1.3.1
Photon shot noise
We have shown already (Eq. 1.48) that the output of the GW interferometer is
ωL
P0
sinc(ωGW L/c) cos[ωGW (t − L/c)]]
(1.51)
1 + 2 h0
Pout =
2
c
P0
ωL
=
h(t)sinc(ωGW L/c)]
1+2
2
c
and in the limit sinc ∼ 1, i.e for ωGW L/c 1, we have
P0
ωL
h(t)
Pout =
1+2
2
c
(1.52)
The output power measurement can be regarded as a photon counting operation.
The number N of detected photons fluctuates following the Poisson statistics
2
with average N and variance σN
= N:
N
p(N ) =
N
e−N
N!
(1.53)
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
17
The energy of a laser photon at frequency ω is Eph = }ω and, if n is the
measured number of photon per second, we have that
n=
P out
P out
=
Eph
}ω
2
N = σN
= nT =
σN
σP
= out
N
P out
P out
T
}ω
(1.54)
(1.55)
(1.56)
The fluctuations of the output power are a noise source of the measurement of
the GW amplitude. The variance associated with the measurement of the GW
amplitude h is (also using Eq. 1.54, 1.55 and 1.56)
dh
| σPout
dPout
dh σPout
=|
P out
|
dPout P out
1 dPout
σN
=
/
|
|
N
P out dh
σh = |
(1.57)
If we set the interferometer output power at half of the input power, P out =
we have
P0
2}ω
√
r
σN
nT
2}ω
=
=
nT
P0 T
N
n=
P0
2 ,
(1.58)
(1.59)
where T is the measurement time.
The standard deviation of h is
σh =
c
L
r
}
2P0 ωT
(1.60)
Note that this kind of noise, called photon shot noise, does not depend on the
frequency of the GW and depends on the measurement time T . It corresponds
to a power spectral density that is constant in the detection bandwidth:
SHOT
Shh
(Ω) =
c2 }
L 2 P0 ω
(1.61)
Eq. 1.61 tells us that shot noise can be reduced by increasing the input power
P0 . This is done by increasing the laser power and by introducing a recycling
mirror, between the laser and the beam splitter, which redirect toward the arms
the light reflected back to the laser by the interferometer. In practice, with the
introduction of the power recycling mirror a resonant cavity is created by the
recycling mirror and the interferometer. As a consequence the power on the
P0
.
beam splitter is increased by a recycling factor K = Plaser
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
1.3.2
18
Quantum radiation pressure noise
We have seen that the fluctuation of the number of photons arriving at the
detector during the measurement time reflects in the fluctuation of the output
power of the interferometer, i.e in noise on the GW amplitude h. On the other
hand a fluctuation of the number of photons reflects in a fluctuation of the
radiation pressure, which transfers a random momentum to the mirror.
We can estimate the effect of this kind of noise by calculating the corresponding
noise power spectral density on h. The force exerted by a electromagnetic wave
of power P0 /2 to an ideal mirror is
F =
P0
2c
(1.62)
The fluctuations in the power P0 cause fluctuations of this radiation pressure
force. Using Eq. 1.54, 1.55 and 1.56 (but substituting in them P0 /2 to Pout ) we
obtain for the standard deviation of the force
r
}ωσN
}ωP0
σP
=
=
(1.63)
σF =
c
cT
2T c2
It follows that the power spectral density of the noise SF F of the force is
SF F =
}ωP0
c2
(1.64)
The suspended mirror is treated as an harmonic oscillator far from the resonance, so that the displacement induced on the mirror at angular frequency Ω
is
F
x(Ω) =
(1.65)
mΩ2
The fluctuations in the two arms are anticorrelated, i.e an extra photon in one
arm corresponds to a missing photon in the other. Thus the induced variation
of the average arm length L is 2x(Ω). The corresponding power spectral density
for h is
4SF F
1
4}ωP0
RP
Shh
(Ω) = 2 4 2 = 2 2 4
(1.66)
m Ω L
L m Ω
c2
We note that, while photon shot noise is independent on the detection frequency
Ω and decrease for increasing input power P0 , radiation pressure noise depends
on the detection frequency, in particular it decreases with the increasing of
frequency, and increases with the input power P0 .
1.3.3
Standard Quantum Limit
The GW interferometer can be regarded as a monitor of the dynamic status
of the mirror. Assuming this point of view, the shot noise and the radiation
pressure noise are interpreted as displacement and momentum fluctuations of
the mirror. This implies that both noise sources are linked to the quantum
uncertainty given by the Heisenberg principle.
The expressions for the linear noise spectral density of the photon shot noise
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
19
and of the radiation pressure noise are
hSHOT (Ω) =
hRP (Ω) =
q
q
RP
Shh
c
L
r
}
P0 ω
r
4}ωP0
1
=
LmΩ2
c2
SHOT =
Shh
We can express them in terms of the detection frequency ν =
wavelength λ = 2πc
ω
1
hSHOT (ν) =
L
hRP (ν) =
r
}λc
2πP0
r
1
mν 2 L
}P0
2π 3 cλ
(1.67)
(1.68)
Ω
2π
and the laser
(1.69)
(1.70)
The total optical noise of the interferometer is the quadrature sum of the two
contributions:
q
hT OT AL = h2SHOT (ν) + h2RP (ν)
(1.71)
By properly choosing the input power P0 we can minimize the total optical noise
for a given detection frequency, which we call ν. The resulting minimal noise
at this frequency is called the standard quantum limit (SQL) and is given,
together with the expression for the optimal input power, by
r
}
hSQL =
(1.72)
2
π mL2 ν 2
Popt = πcmλν 2
(1.73)
In Fig. 1.8 the total quantum noise for a GW Michelson interferometer is
shown, which is deduced from our calculation. We can see that an increase in
the input power reduces the contribution of the shot noise while increasing the
contribution of the radiation pressure noise. In addition, a change in the input
power also changes the SQL, i.e the minimum value of the total quantum noise,
and shifts the detection frequency corresponding to the SQL: an increase in the
power reduces the SQL and increases the corresponding frequency.
1.4
Quantum enhancement of Gravitational Waves
Interferometric Detectors
In the previous section we computed the quantum noise contributions, which
limit the sensitivity on a GW Michelson interferometer and we found that the
displacement spectral density of the shot noise is frequency independent, while
radiation pressure noise decreases for increasing detection frequency. In real
interferometers the frequency dependence of quantum noise is more complicated,
due to the introduction of Fabry-Perot cavities in the arms. In particular,
shot noise is no more frequency-independent and, above a given frequency, it
increases with frequency, as it can be seen in the solid purple line of Fig. 1.7.
In the same picture we can see that quantum noise is one of the major total
noise contributions for a GW advanced detector and reducing it would give a
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
20
Figure 1.8: Quantum noise limited sensitivity curve (total quantum noise) of a
GW Michelson interferometer in dependence of the input power. By increasing
the power, we have a reduction of the shot noise contribution and, correspondingly, an increase of the radiation pressure contribution. When changing the
input power also changes the SQL, i.e the minimal value of the total quantum
noise, and the frequency at which we have the SQL.
valuable contribution to the improvement of the detector sensitivity.
In the previous section we also deduced a SQL for the detector, which set a
”lower limit” to detector the sensitivity. However, this ”lower limit” is not
a fundamental limit and it can be circumvented. One way of doing it, first
proposed by Caves [4] and then experimentally demonstrated by [24], is to
inject squeezed vacuum, a non classical state of light, at the output port of the
interferometer.
In this section we explain in a qualitative way as this technique should work.
However, in order to better understand the squeezing potentiality we have to
better examine under which assumptions we have derived the SQL.
1.4.1
Origin of the Standard Quantum Limit
In the previous section we derived the shot noise formula assuming that the shot
noise is due to fluctuations of the number of photons arriving at the detector
during the measurement time T . We will see in the next chapter that, in a
quantized radiation field language, where the electric field acts as operator on
the radiation quantum states, “fluctuations of the number of photons arriving at
the detector during the measurement time”becomes “fluctuations of the phase
quadrature of the electric field inside the interferometer”. The shot noise can
be interpreted, in a more general way, as a result of photons arriving at the
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
21
detector at non equally spaced time intervals, i.e the radiation state associated
with the laser has not a well defined phase.
On the other hand, when we derived the radiation pressure noise formula, we
said that radiation pressure noise is due to fluctuations of the number of photons impinging on the mirrors. In a quantized radiation field language this
corresponds to “fluctuations of the amplitude quadrature of the electric field
inside the interferometer”. Indeed, fluctuations of the amplitude of the radiation impinging on the mirrors determine fluctuations of the magnitude of the
radiation pressure force acting on the mirrors, which thus randomly moves the
mirrors.
The fact that, when we try to decrease the shot noise by increasing the input
power the radiation pressure noise increases, is a manifestation of the Heisenberg uncertainty principle applied to the phase and amplitude quadrature of
the electric field, represented by non-commuting operators
[X1 , X2 ] = 2i
(1.74)
where X1 and X2 are the amplitude and phase quadrature respectively. From
the Heisenberg uncertainty principle it follows that the variances of the two
quadratures are linked by
[X1 , X2 ] 2
=1
(1.75)
VX1 VX2 − VX2 1 X2 ≥ 2
Moreover the crucial aspect of our calculation is that we assumed for both shot
and radiation pressure noise a Poissonian statistic with the same mean value.
In a quantized radiation field language this means that we are dealing with radiation states, which exhibit Poissonian photon statistic and have uncorrelated
quantum noise equally distributed between phase and amplitude quadrature.
Among the quantum radiation states, both unsqueezed vacuum states and coherent states exhibit those properties. In particular, both states have minimum
uncertainty, i.e for them the equal sign apply in Eq. 1.75, VX1 X2 = 0 (uncorrelated quadrature noise) and VX1 = VX2 = 1. We represent this status in the
X1 − X2 plane by
p an errorpcircle whose projection onto the X1 and X2 axis
are respectively VX1 and VX2 (see Fig. 1.9)
The same result can be obtained by a complete quantum description of a
Fabry-Perot Michelson GW interferometer, as it is reported in [20]. Here we
summarize just the basic results. The starting point of the treatment is the
quantum model of a Fabry-Perot Michelson GW interferometer, schematized
in Fig. 1.10. The laser is represented as a quantum coherent state (indicated
as c1 in the picture), which thus exhibits uncorrelated and equal phase and
amplitude quadrature fluctuations. The end suspended mirrors are quantized
harmonic oscillators and we indicate as c2 the vacuum noise entering via the
output port of the interferometer.
By taking into account the optomechanical interaction of the radiation field with
the movable end mirrors and the coupling of the gravitational wave with the
interferometer, it can be show that, if the vacuum field entering the output port
of the interferometer is an unsqueezed vacuum state, characterized by uncorrelated and equal phase and amplitude quadrature fluctuations, an expression
for the total quantum noise can be deduced. When this formula is optimized
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
22
Figure 1.9: Left: Error circle for unsqueezed vacuum state. Right: Error
ellipse for a squeezed vacuum state with squeezing parameter r and squeezing
angle φ.
Figure 1.10: Quantum model of a Fabry-Perot Michelson GW interferometer
used in [20]
with respect to the input power at a given detection frequency, the minimum
value is reduced to the SQL obtained in Eq. 1.72. In all these treatments the
mechanical and optical losses are not taken into account. The quantum limited
GW detector sensitivity, obtained with this computation, is reported in Fig.
1.11.
1.4.2
Circumventing the SQL: injection of squeezed vacuum states at the output port of the interferometer
In [20], the computation is continued by assuming that (always referring to Fig.
1.10) the field c2 that enters the output port is now a squeezed vacuum (see
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
23
Figure 1.11: Quantum limited GW detector sensitivity obtained in [20] for an
unsqueezed vacuum state entering the output port of the interferometer
Figure 1.12: Left: Vacuum fluctations entering the output port of the interferometer. Right: A Faraday Rotator is used to inject squeezed vacuum states
into the output port, thus substituting the ordinaray vacuum fluctuations.
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
24
Fig. 1.12).
Referring to Eq. 1.75, a squeezed vacuum state is, as for a vacuum or a coherent
state, a minimum uncertainty state (the equal sign applies in the uncertainty
relation) with VX1 6= VX2 and, in general, with VX1 X2 6= 0. A squeezed vacuum
state can be characterized by two parameters (see Fig. 1.9), the squeezing factor
r and the squeezing angle φ. In fact, the quadrature variances depends on these
parameters as
VX1 = cosh2 (r) + sinh2 (r) − 2 cosh(r) sinh(r) cos(φ)
2
2
VX2 = cosh (r) + sinh (r) + 2 cosh(r) sinh(r) cos(φ)
(1.76)
(1.77)
As shown in [20], the SQL can be circumvented if we inject squeezed states with
correlated quadrature noise VX1 X2 6= 0. Let us examine the various possibilities,
keeping in mind the results obtained in the case of unsqueezed vacuum, i.e that,
for a given specific detection frequency ν it exists the optimum input power
Popt (which depends on the particular detection frequency chosen) for which the
minimum noise hSQL occurs. In the case
• φ=0
i.e the squeezing angle is zero, once we optimize the total quantum noise
at the detection frequency ν with respect to the input power we find that
the SQL is not lowered with respect the unsqueezed case. The optimal
input power is lower than that of the unsqueezed case, as shown in Fig.
1.13
φ=0
Popt
= e−2r Popt
(1.78)
hφ=0
min
(1.79)
= hSQL
• φ = φopt
i.e. optimizing the total quantum noise at the detection frequency ν with
respect to the squeezing angle φ, we obtain an expression for the total
quantum noise at ν which depends on the input power. Optimizing this
expression with respect to the input power we find that the optimal input
power is the same as for the unsqueezed case but the minimum quantum
noise at ν is lower than that for the unsqueezed case. In this case the SQL
is beaten at detection frequency ν, as show in Fig. 1.15 and 1.14
φ
Poptopt = Popt
(1.80)
φopt
hmin
(1.81)
= e−r hSQL
An extremely important aspect to be pointed out is that all results presented
here are valid at a particular detection frequency. In other words, when we
optimize the squeezing angle and the input power, we obtain a quantum noise
below the SQL only at the particular detection frequency. At other frequencies
we can have both an increase or a decrease of the total quantum noise with
respect to the value in the unsqueezed case.
If we want to circumvent the SQL all over the detector band it is necessary to
implement a technique which allows to properly tailor the squeezing angle
of the squeezed vacuum exiting from the squeezer before injecting it into the
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
25
Figure 1.13: The minimum possible gravitational wave amplitude h detectable
as a function of power using φ=0, for three different values of the squeezing
parameter r: (curve a) r =0, (curve b) r =1, (curve c) r =2.
Figure 1.14: A comparison between using (curve a) φ = 0 and (curve b) φopt ,
in the calculation for the minimum possible value of h detectable using r =1.
The corresponding curve for no squeezing (r =0) is also shown (see curve c).
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
26
Figure 1.15: The minimum possible value of h detectable as a function of power
using φopt , for three different values of the squeezing parameter r: (curve a) r
=0, (curve b) r =1, (curve c) r =2.
output port of the intereferometer. In this way, we are able to choose the optimal squeezing angle for each detection frequency, thus beating the SQL all over
the detector band, as it is show in Fig. 1.16.
We distinguish two squeezing approaches, classified on the base of the dependence of φopt on the detection frequency:
- φopt (ν) is the frequency dependent squeezing
- φopt = const over the detection bandwidth is the frequency independent
squeezing
A way to obtain frequency dependent squeezing is to rotate the squeezing angle
of the squeezed vacuum with a filter cavity before injecting it at the output
port, as shown in Fig. 1.17 and explained in [11].
CHAPTER 1. DETECTION OF GRAVITATIONAL WAVES
27
Figure 1.16: Frequency dependent squeezing: if frequency dependent squeezed
states are injected at the output port, the sensitivity can be improved over
the complete detection bandwidth (green trace). In a quantum noise limited
interferometer the standard quantum level (red trace) can be beaten in that
case.
Figure 1.17: Frequency dependent squeezing: a filter cavity is used to tailor the
squeezing angle before injecting the squeezed vacuum in the interferometer.
Chapter 2
Squeezing of the light
In this chapter we deal with the basic aspects of squeezed light. After a brief
report of the main characteristics of both the classical and quantized electric
field, we describe the characteristics of the squeezed states of light, and how they
can be produced. In particular, after a review of the squeezing state production
by optical parametric process, which is the most used squeezing technique, we
focus our attention on the ponderomotive squeezing, the squeezing production
technique which will be used in future Advanced detectors.
2.1
Field quantization
Following [8], let us consider the classical free electromagnetic field, i.e the
solutions of the Maxwell equations in absence of sources. It can be shown that,
~ = 0), the vector potential of the field
~ ·A
with an appropriate gauge choice ( O
satisfies in vacuum the electromagnetic wave equation:
~ t) =
~ 2 A(r,
O
~ t)
1 ∂ 2 A(r,
2
c
∂t2
(2.1)
The solution of this equation can be expanded, if we integrate in a finite volume
of space, in a Fourier series of orthonormal field modes as
Â(r, t) = Â(+) (r, t) + Â(−) (r, t)
i
X } 1/2 h
=
ak ~uk (r)e−iωk t + a†k ~u∗k (r)e+iωk t
2ωk 0
(2.2)
k
where ak is the Fourier coefficient of the expansion of the mode at frequency
ωk , while ~uk (r) is the mode vector which describes the mode at frequency ωk .
It satisfies the wave equation
ωk 2
2
~
(2.3)
O + 2 ~uk (r) = 0
c
and the transverse condition
~ · ~uk (r) = 0
O
28
(2.4)
CHAPTER 2. SQUEEZING OF THE LIGHT
In addition, the mode vectors form a complete and orthonormal set, i.e
Z
~u∗k (r)~uk0 (r)dr = δkk0
29
(2.5)
V
and their expression depends on the particular boundary condition chosen when
solving Eq. 2.1.
The corresponding expressions for the electric and magnetic field can be deduced
by applying
~ =O
~
~ ×A
B
(2.6)
~
~ = − ∂A
E
∂t
The normalization coefficients are chosen in such a way that the Fourier coefficients ak e a†k are adimensional. In the classical theory the vector potential
and the electric and magnetic fields are vectors, while the Fourier components
of the expansion are complex numbers.
It is interesting to look at the expression which the Hamiltonian of the electromagnetic field takes when expressed in terms of the Fourier coefficient of Eq.
2.2
Z
1
H=
0 E 2 + µ0 H 2 d~r
(2.7)
2
X
1
=
}ωk a†k ak +
2
k
it is exactly the same expression we would have obtained for a set of quantum independent harmonic oscillators if the Fourier coefficients ak and a†k were
the creation and annihilation operators for an harmonic oscillator with angular
frequency ωk . This similarity is the starting point for the quantization of the
electromagnetic field: the Fourier coefficients of the expansion in Eq. 2.2, ak
and a†k , becomes the mutually adjoint annihilation and creation operators for the
field mode of angular frequency ωk . This is done by assigning the commutation
relations appropriate for bosons to ak and a†k
h
i
h
i
[ak , ak0 ] = a†k , a†k0 = 0
ak , a†k0 = δkk0
(2.8)
which are the same commutation rules satisfied by the creation and annihilation
operators of a quantum harmonic oscillator.
With the application of the quantization procedure the vector potential, the
electric field and the magnetic filed become operators and the electromagnetic
field is described as a set of independent quantum harmonic oscillators. The
state of the radiation field will be described by a ket of an appropriate Hilbert
space, which can be constructed as the tensor product of the Hilbert spaces of
all modes.
The expression for the quantized electric field becomes
Ê(r, t) = Ê (+) (r, t) + Ê (−) (r, t)
i
X }ωk 1/2 h
=i
ak ~uk (r)e−iωk t − a†k ~u∗k (r)e+iωk t
20
k
(2.9)
CHAPTER 2. SQUEEZING OF THE LIGHT
30
the term Ê (+) (r, t), which contains the phase dependence e−iωt , contains the
destruction operators and thus enters in absorption processes, while the term
Ê (−) (r, t), which contains the phase dependence e+iωt , contains the creation
operators and thus enters in emission processes.
2.1.1
Electric field quadratures and modulation
The expression for the electric field of Eq. 2.9 can be rewritten in a new form,
useful when we are dealing with squeezed states. This representation uses the
Hermitian quadrature operators:
X1 = a + a†
(2.10)
†
X2 = −i(a − a )
(2.11)
Let us consider a single field mode and collect all normalization factors in a
constant K. We rewrite the electric field as
E(r, t) = K[X1 sin(ωt − k · r) − X2 cos(ωt − k · r)]
(2.12)
The two quadrature operators satisfy the commutation relation
[X1 , X2 ] = 2i
(2.13)
and for them the Heisenberg uncertainty principle applies
V ar(X1 )V ar(X2 ) ≥ 1
(2.14)
which plays a crucial role in the theory of squeezing.
The X1 quadrature is called the “amplitude quadrature”, while X2 is called
the “phase quadrature”. This language is derived from the classical theory of
the signal modulation, often used in quantum optics to describe some aspects
of squeezing.
Thus, let us refer to a classical picture of electromagnetism; we consider the
electric field with only the X1 component
E(t) = K sin ωt
(2.15)
When the field is amplitude modulated with modulation depth M we obtain
(see [7])
M
(1 − cos ωm t) E(t)
(2.16)
E(t)AM = 1 −
2
M
M
M
=K
1−
sin(ω + ωm )t +
sin(ω − ωm )t ,
sin ωt +
2
4
4
Two sidebands, i.e two field components at angular frequencies (ω±ωm ), appear.
They oscillate in phase with the the main component at angular frequency ω.
Thus X1 is called amplitude quadrature.
On the other hand, if we modulate the phase of the electric field for small
modulation depth M , we obtain
E(t)F M = K sin(ωt + M cos ωm t)
M
M
≈ K sin ωt +
cos(ω + ωm )t +
cos(ω − ωm )t
2
2
(2.17)
CHAPTER 2. SQUEEZING OF THE LIGHT
31
Figure 2.1: Amplitude and phase modulation for small modulation depths in
the sideband picture. The phase of the sidebands with respect to the carrier
distinguishes amplitude modulation from phase modulation.
in this case we also have two sidebands at angular frequencies (ω ± ωm ), which
oscillate 90◦ out of phase with respect to the field component at angular frequency ω. Thus, the phase modulation yields to the X2 component, called for
this reason phase quadrature.
According to this description, if the modulation depth is small the sidebands
prevail in quadrature with the carrier, while as M increases they start to appear
also in-phase with the carrier, as for amplitude modulation. In general they are
present both in-phase and in quadrature with the carrier.
The behavior of modulation for small modulation depths is shown in Fig. 2.1.
2.1.2
Correlation functions of the electromagnetic field
Intensity measurements and interference measurements are two measurement
classes which play a crucial role both in classical and quantum optics. From
a quantum mechanical point of view those measurements can be drawn back
respectively to photon counting and photon correlation measurements.
An important concept which emerges in interference experiments is the concept of coherence, that is also linked to the possibility of obtaining interference
fringes in a Young-type experiment.
We will see that the study of the photon statistics of a radiation field and its
coherence properties is extremely important because those characteristics can
be different for a classical and for a quantum field.
Following [15], we see in Eq. 2.9 that the term E (+) (r, t) contains the annihilation operators, while E (−) (r, t) contains the creation operators. Thus,
calling the space-time point x = (r, t), we have that E (−) (x)E (+) (x) ∝ a† a,
i.e is proportional to the “number of quanta”operator which also appear in Eq.
2.7, and, being |ii the radiation field state, the intensity of the field is given by
I(x) = hi|Ê (−) (x)Ê (+) (x)|ii
(2.18)
CHAPTER 2. SQUEEZING OF THE LIGHT
32
In the general case in which the radiation field state is not a pure state |ii but
it is a statistical mixture of states with density operator ρ, the intensity is
X
I(r, t) =
Pi hi|Ê (−) (r, t)Ê (+) (r, t)|ii
(2.19)
i
= T r{ρÊ (−) (r, t)Ê (+) (r, t)}
where T r indicates the trace operation over all basis state kets. The expression
for the intensity reported in Eq. 2.19 can be drawn back to a more general
expression for the correlation of the electric field in two particular space-time
points.
We define the first order correlation function as
G(1) (x, x0 ) = T r{ρÊ (−) (x)Ê (+) (x0 )}
(2.20)
As we will see, G is sufficient for describing the Young-type interference experiments, which involve correlations between amplitudes of fields.
However, in order to describe, for example, Hanbury-Brown and Twiss-type experiments, which involve correlation between intensities of fields, higher order
correlation functions are needed.
The n-order correlation function is defined as
G(n) (x1 ...xn , xn+1 ...x2n ) = T r{ρE (−) (x1 )...E (−) (xn )E (+) (xn+1 )...E (+) (x2n )}
(2.21)
It can be shown that the n-order correlation function satisfies the relations
G(n) (x1 ...xn , xn ...x1 ) ≥ 0
(n)
G
(n)
(x1 ...xn , xn ...x1 )G
2.1.3
(2.22)
(n)
(xn+1 ...x2n , x2n ...xn+1 ) ≥ |G
(x1 ...xn , xn+1 ...x2n )|2
(2.23)
Correlation functions and coherence properties of
the electric field
We have already cited the fact that the correlation functions and the concept
of coherence play a crucial role in interference experiments (see [15]).
In particular, in the first Young-type interference experiments the concept of
coherence had been linked to the possibility of seeing interference fringes, but
it can be defined in a more general way by referring to the properties of the
correlation functions.
Let us consider a Young-type interferometer. In such apparatus two electric
fields, generated in the space-time points x1 and x2 , interfere in the space-time
point x
(+)
(+)
(2.24)
E (+) (x) = [E1 (x1 ) + E2 (x2 )]
the intensity of the field is thus
I = T r{ρE (−) (x)E (+) (x)}
(1)
=G
(1)
(x1 , x1 ) + G
(2.25)
(1)
(x2 , x2 ) + 2Re{G
(x1 , x2 )}
the first two terms are the intensities of the two interfering fields, while the third
term is the first order correlation function of the two fields and it represents the
CHAPTER 2. SQUEEZING OF THE LIGHT
33
interference term. The correlation function is a complex valued function and
we can rewrite the third term of Eq. 2.25 as
G(1) (x1 , x2 ) = |G(1) (x1 , x2 )|eiΨ(x1 ,x2 )
(2.26)
thus
I = G(1) (x1 , x1 ) + G(1) (x2 , x2 ) + 2|G(1) (x1 , x2 )| cos Ψ(x1 , x2 )
(2.27)
The interference fringes derive from the oscillation of the cosine term, and their
visibility, which we are going to define soon, is characterized by |G(1) (x1 , x2 )|.
If G(1) (x1 , x2 ) vanishes we do not observe interference fringes and we conclude
that the two fields are incoherent. On the other hand, the higher is G(1) (x1 , x2 )
the better we can see the fringes and the more coherent the fields are. However
Eq. 2.23 tells us that |G(1) (x1 , x2 )| is limited by
|G(1) (x1 , x2 )| ≤ [G(1) (x1 , x1 )G(1) (x2 , x2 )]1/2
(2.28)
so, the best fringe visibility, and thus the coherence condition, is obtained when
G(1) (x1 , x2 ) = [G(1) (x1 , x1 )G(1) (x2 , x2 )]1/2
(2.29)
We can introduce the normalized first order correlation function
g (1) (x1 , x2 ) =
G(1) (x1 , x2 )
[G(1) (x1 , x1 )G(1) (x2 , x2 )]1/2
(2.30)
and express the coherence condition as
|g (1) (x1 , x2 )| = 1
(2.31)
The fringe visibility is defined as
υ=
Imax − Imin
Imax + Imin
(2.32)
and in the Young-type experiment becomes
υ = |g (1) |
2(I1 I2 )1/2
I1 + I2
(2.33)
For interacting fields of equal intensity, it reduces to
υ = |g (1) |
(2.34)
Thus, as we already stated, the coherence condition |g (1) | = 1 corresponds to
the best fringe visibility.
A more general definition of coherence, which is implicitly contained in Eq.
2.29, is that the first order correlation function factorizes in the product of two
functions of the two space-time points x1 and x2 , which we indicate as (−) (x1 )
and (+) (x2 ),
G(1) (x1 , x2 ) = (−) (x1 )(+) (x2 )
(2.35)
CHAPTER 2. SQUEEZING OF THE LIGHT
34
This definition can also be generalized to n-order correlation functions as
G(n) (x1 ...xn , xn+1 ...x2n ) = (−) (x1 )...(−) (xn )(+) (xn+1 )...(+) (x2n )
(2.36)
The second order correlation functions, as we have already pointed out, play
a crucial role in quantum optics because they can behave quite differently for
classical states of light and quantum states of light. They enter in ”intensity correlation” experiments, which basically measure the joint probability of
detecting a photon in a space-time point and an other photo in a different spacetime point.
We restrict for simplicity our treating to the case of photon detection in the
same spatial point but at different times, i.e we concentrate on temporal coherence, but a similar discussion can be done for spatial coherence only. In addition
we treat only stationary processes, for which only the time difference between
the detection events matters.
The probability per unit time of detecting a photon at time t and an other
photon at time t + τ is given by the second order correlation function
G(2) (τ ) = hE (−) (t)E (−) (t + τ )E (+) (t + τ )E (+) (t)i
(2.37)
= hI(t + τ )I(t)iN
where N stands for normal ordering. 1 We introduce, as for the first order
correlation function, the normalized second order correlation function
g (2) (τ ) =
G(2) (τ )
|G(1) (0)|2
(2.38)
The full coherence is obtained when G(2) (τ ) factorizes, i.e when
G(2) (τ ) = (−) (t)(−) (t + τ )(+) (t + τ )(+) (t)
(1)
= |G
(0)|
(2.39)
2
thus, the coherence property reduces to
g (2) (τ ) = 1
(2.40)
For a classical field it can be shown that the second order correlation function
satisfies the two following relations
g (2) (0) ≥ 1
(2.41)
g (2) (τ ) ≤ g (2) (0)
(2.42)
Those relations are not necessarily satisfied for quantum mechanical fields. Indeed radiation fields that violate those properties are an experimental evidence
of the existence of non classical states of light.
From a quantum mechanical point of view, if we consider one single field mode, it
is easily shown that g (2) (0) can be written, in terms of creation and annihilation
operator, as
ha† a† aai
(2.43)
g (2) (0) =
ha† ai2
1 In normal ordered products we have the positive energy terms, which contain annihilation
operators, to the right and the the negative energy terms, which contain creation operators,
to the left.
CHAPTER 2. SQUEEZING OF THE LIGHT
35
and for each quantum state, which we introduce in the next section, we also
compute this expression in order to better characterize their non classical behavior.
We conclude this section with few considerations about the statistical properties
of the radiation field with respect to the photon arrival, that can be deduced by
studying g (2) (τ ).
For light generated by the superposition of independent sources (see [3]), it can
be shown that the following relation holds between the second and first order
correlation functions
g (2) (τ ) = 1 + |g (1) (τ )|2
(2.44)
In addition, as we have already pointed out, we have |g (1) (τ )| ≤ 1. Moreover,
it can be shown that in all physical situations, it vanishes at times much larger
than a characteristic time, τ tc . Thus, we have
g (2) (τ ) −→ 1
τ tc
(2.45)
For a full coherent field we have g (2) (τ ) = 1 at all τ . Thus, for a coherent filed
the probability to count a photon at t + τ if a photon had been counted at t is
independent of τ . The arrival of photon is full uncorrelated and it follows the
Poisson statistics.
On the other hand, for a field which share the classical characteristic g (2) (τ ) ≤
g (2) (0) the photons tend to arrive in pairs . This behavior is called bunching: a
photon has a higher probability (with respect to the Poissonian statistics) to be
detected at times shorter than the characteristic time tc after the arrival time
of the previous photon.
The opposite case is called antibunching and it happens when g (2) (τ ) ≥ g (2) (0).
In this case the photons tend to ”set a distance” between them, i.e a photon
has a higher probability to arrive at the detector at times larger (with respect
to the Poissonian statistics) than the characteristic time tc after the arrival
time of the previous photon. Fields, which satisfy the non classical inequality
g (2) (0) < 1, exhibit antibuching. As a matter of fact, given that g (2) (τ ) −→ 1
on large enough time scale, the condition g (2) (τ ) ≥ g (2) (0), from a given point,
will be satisfied. Obviously, as we have already pointed out, this is not possible
for classical fields. Thus, the observation of an antibunched photon statistics
would be an evidence for the existence of non classical states of light.
2.2
Quantum states of light
In the previous section we treated the quantization of the electromagnetic field
and we reported some basic results of both the quantum and classical theory of
the electromagnetic field. In this section we introduce some quantum states of
the electromagnetic field and we study their quantum fluctuations and coherence
properties, always keeping in mind the characteristics of the classical field as
comparison reference (see [8]).
2.2.1
Fock states
Fock states |nk i are the eigenstates of the Hamiltonian of Eq. 2.7, i.e they are
eigenstates of the number operator Nk = a†k ak . They have definite energy and
CHAPTER 2. SQUEEZING OF THE LIGHT
36
contain a defined number of photons:
Nk |nk i = nk |nk i
(2.46)
where nk = 0, 1, 2... and the corresponding Hamiltonian eigenvalues are }ωk nk +
The vacuum state, i.e the fundamental state of the Hamiltonian is defined by
ak |0i = 0
(2.47)
and its energy is given by
h0|H|0i =
1X
}ωk
2
(2.48)
k
There is no upper bound to the angular frequency ωk , thus the energy of the
vacuum state is infinite. This is one of the conceptual problems of the quantization of fields.
As we have already pointed out, ak and a†k are creation and annihilation operators and it can be shown from the quantum treating of the harmonic oscillator
that the Fock states satisfy
√
(2.49)
ak |nk i = nk |nk − 1i
√
†
ak |nk i = nk + 1|nk + 1i
(2.50)
(a† )nk
|0i
|nk i = √k
nk !
(2.51)
In addition, Fock states form an orthonormal base for the Hilbert space of the
radiation states, namely
∞
X
hnk |mk i = δmn
(2.52)
|nk ihnk | = 1
(2.53)
nk =0
However the Fock states basis is often not used for describing the optical fields.
This is due to the fact that, from an experimental point of view, it is difficult to
generate states with large and non-fluctuating number of photons. Only states
with few photons have been generated, while many interesting optical fields
contains a large and fluctuating number of photons. For such states, as we will
see, a representation in terms of coherent states is more convenient.
Despite those considerations, for few quantum states it is possible to find a
diagonal representation in the Fock basis, which gives useful information about
the photon statistics of the state itself. In such a representation the density
operator ρ for a single mode radiation field is expanded as
X
ρ=
Pn |nihn|
(2.54)
where Pn is the probability for the field to contain n photons. Pn simplifies the
calculation g (2) (0). As a matter of fact, referring to Eq. (2.43), we obtain
ha† a† aai
ha† ai2
V ar(n) − n
=1+
n2
g (2) (0) =
(2.55)
1
2
.
CHAPTER 2. SQUEEZING OF THE LIGHT
37
where V ar(n) = h(a† a)2 i − ha† ai2 is the variance of the number of photons, and
with the representation adopted, is also the variance of Pn .
For a field with Poisson photon statistics we obtain
e−n n
n
n!
V ar(n) = n
Pn =
(2.56)
g (2) (0) = 1
The result for g (2) (0) coincides with that presented when we have discussed the
correlation function for a Poisson distribution in the previous section.
We conclude this section by showing that the Fock states are not classical states
of light.
Let us start with the characteristics of the electric field for these states: referring to the expression for the electric field operator of Eq. 2.12, we see that the
study of the electric field characteristics reduces to the study of the operator
X̂θ = cos(θ)X̂1 + sin(θ)X̂2
(2.57)
where θ = ωt − k · r.
Given the quadrature operators 2.10 and 2.11, the average and variance of 2.57
become
hn|X̂θ |ni = 0
(2.58)
V ar(X̂θ ) = hn|X̂θ2 |ni − hn|X̂θ |ni2
(2.59)
= 2n + 1
We have already found the first non classical behavior of the Fock states: the
average electric field, which should be an oscillating function of space and time
for a classical state, is always zero for a Fock state, independently from the
number of photons.
For a Fock state, the variance of the electric field increases as the number of
photons increases and it is independent on the phase θ.
The minimum variance occurs for the vacuum state n = 0
V ar(X̂θ )||0i = 1
(2.60)
V ar(X1 )V ar(X2 )||0i = 1
(2.61)
while
and, given that the quadrature operators satisfy the Heisenberg uncertainty
principle of Eq. 2.14, we have that the vacuum state is a minimum uncertainty state.
All these properties of the Fock states with respect to the electric field quadrature operators can be represented in the X1 − X2 plane by an error circle, as
shown in Fig. 2.2. In the diagram the average electric filed is represented by a
point in the X1 − X2 plane surrounded by the error circle, which represents the
square root of the variance for all quadratures.
CHAPTER 2. SQUEEZING OF THE LIGHT
38
Figure 2.2: Error circle of a Fock state with n photons. For n = 0 we obtain the vacuum state, which is a minimum uncertainty state with V ar(X1 ) =
V ar(X2 ) = 1.
Finally, we compute g (2) (0) using Eq. 2.55. We have
ρ = |nihn|
(2.62)
P (n) = 1
V ar(n) = 0
thus
g (2) (0) = 1 −
1
n
(2.63)
i.e g (2) (0) < 1, a characteristic of non classical states of light.
2.2.2
Coherent states
In the previous section we pointed out that a basis of states exists, that of the
coherent states, which is useful for the description of many radiation fields of
physical interest. We will show that those states are also the most similar to
the sinusoidal classical field.
From a formal point of view, coherent states can be generated from the vacuum
by the displacement operator
|αi = D(α)|0i
(2.64)
where α is a complex number and
†
D(α) = eαa
=e
−α∗ a
(2.65)
−|α|2 /2 αa† −α∗ a
e
e
CHAPTER 2. SQUEEZING OF THE LIGHT
39
The coherent states satisfy the following properties
D† (α) = D−1 (α) = D(−α)
†
D (α)aD(α) = a + α
†
†
†
D (α)a D(α) = a + α
(2.66)
(2.67)
∗
(2.68)
D(α + β) = D(α)D(β)e
−iIm{αβ ∗ }
a|αi = α|αi
(2.69)
(2.70)
In addition, it can be shown that they are normalized and they form an overcomplete set. In general they are not orthogonal:
Z
|hα|αi|2 = 1
(2.71)
|αihα|d2 α = π
(2.72)
1
hβ|αi = e− 2 (|α|
2
+|β|2 )+αβ ∗
(2.73)
The property of the coherent states to be eigenstates of the annihilation operator
is worth farther attention. This property implies that all correlation functions
at all the orders 2.21 factorize for the coherent states.
In addition, as we have already outlined, the coherent states are the most similar
to a classical field. We justify this assertion by calculating the average and
variance of the electric field operator (as we did for the Fock states)
X̂θ = cos(θ)X̂1 + sin(θ)X̂2
(2.74)
We obtain
hα|X̂θ |αi = 2Re(α) cos(θ) + 2Im(α) sin(θ)
V ar(X̂θ )||αi = 1
(2.75)
(2.76)
Thus, the average behaves like a classical sinusoidal electric field. In addition,
looking at the variance we conclude that, as for the vacuum state, the coherent
states have minimum uncertainty, independent on α and θ. This means that
the coherent states have the best defined amplitude and phase allowed by the
uncertainty principle. In this respect they are the most similar to a classical
sinusoidal field, which has a defined amplitude and phase. All this properties
can be represented, as for the Fock states, in an error circle diagram, as shown
in Fig. 2.3
Finally, it can be shown that the coherent states follow the Poisson photon
statistics. In fact, a coherent state can be expanded in the Fock states basis as
|αi =
∞
X
2
1
αn
|ni √ e− 2 |α|
n!
n=0
(2.77)
It follows
|αihα| =
∞
X
P (n)|nihn|
n=0
2n
P (n) =
2
|α|
e−|α|
n!
(2.78)
CHAPTER 2. SQUEEZING OF THE LIGHT
40
Figure 2.3: Left: error circle of the vacuum state. Both a coherent state
with |α| = 0 and a Fock state with n = 0 are vacuum states. This error
circle has radius 1, which corresponds to the minimum uncertainty situation of
V ar(X1 ) = V ar(X2 ) = 1. Right:error circle of a coherent state. It is the same
error circle of the vacuum state but displaced by α.
and P (n) is indeed a Poisson distribution.
We conclude this section by describing two possible representations of quantum states trough the coherent states, useful in the following sections.
P representation
The P representation is a diagonal representation in the basis of the coherent
states (see [8])
Z
ρ = P (α)|αihα|d2 α
(2.79)
In general P (α) cannot be interpreted as a probability distribution for the parameter α, because the operator |αihα| projects on a non orthogonal set of
states. In addition, P (α) can assumes, for some of the states, negative values
or is singular.
It can be shown that those states for which P (α) is positive do not show quantum
properties such as untibunching and squeezing: for them a classical description
exists. In fact, such states can be described by a classical electric field with a
complex amplitude , which is a stochastic variable with probability distribution P (). Such fields, as for example incoherent light, can be considered as
semi-classical.
The P representation is useful in the calculation of normal ordered products
of creation and annihilation operators, and thus in the calculation of the correlation functions. In particular, it is used in the calculation of the so called
CHAPTER 2. SQUEEZING OF THE LIGHT
covariance matrices
ha2 i − hai2
C(a, a† ) = 1
†
†
†
2 haa + a ai − ha ihai
41
1
†
2 haa
+ a† ai − ha† ihai
ha†2 i − ha† i2
(2.80)
(2.81)
CN (a, a† ) =
2
2
†
ha i − hai
ha ai − ha ihai
ha† ai − ha† ihai ha†2 i − ha† i2
hX̂12 i
C(X̂1 , X̂2 ) =
†
1
2 hX̂1 X̂2
1
2 hX̂1 X̂2
2
− hX̂1 i
+ X̂2 X̂1 i − hX̂2 ihX̂1 i
(2.82)
+ X̂2 X̂1 i − hX̂2 ihX̂1 i
hX̂22 i − hX̂2 i2
where N indicates the normal ordering.
Those covariance matrices are linked by
C(a, a† ) = CN (a, a† ) +
1
2
0
1
1
0
(2.83)
(2.84)
†
T
C(X̂1 , X̂2 ) = ΩC(a, a )Ω
where
Ω=
1 1
−i i
(2.85)
Let us show how those calculations are performed in the P representation
• correlation functions:
ha†n am i =
Z
P (α)α∗n αm d2 α
(2.86)
we shifted the calculation of the correlation functions to the calculation of
the moments of P (α). For g (2) (0) we obtain
R
P (α)[|α|2 − h|α|2 i]2 d2 α
(2)
R
(2.87)
g (0) = 1 +
[ P (α)|α|2 d2 α]2
from which we see that states with negative P (α) show the non classical
characteristic g (2) (0) < 1
• covariance matrices:
the calculation is straightforward for CN (a, a† ), which only contains normal ordered products of creation and annihilation operators. We have just
to compute the elements of the covariance matrix Cp (α, α∗ ), i.e products
of the same type as that of Eq. 2.86 of (α, α∗ ) on P (α)
CN (a, a† ) = Cp (α, α∗ )
(2.88)
Once we have CN (a, a† ) it is easy to derive the other covariance matrices
by using Eq. 2.83.
It is also possible to calculate P (α) by starting from a characteristic function.
In fact, it can be shown that the density operator ρ is determined by its characteristic function
χ(η) = T r{ρ exp(ηa† − η ∗ a)}
(2.89)
CHAPTER 2. SQUEEZING OF THE LIGHT
42
we can also define a normal ordered characteristic function
χN (η) = T r{ρ exp(ηa† ) exp(−η ∗ a)}
(2.90)
If ρ has a P representation, χN (η) is given by
Z
†
∗
χN (η) = hα|eηa e−η a |αiP (α)d2 α
Z
∗
∗
= eηα −η α P (α)d2 α
(2.91)
thus χN (η) is the bidimensional Fourier transform of P (α) and, operating the
antitransformation, we obtain
Z
∗
∗
1
P (α) = 2
eηα −η α χN (η)d2 η
(2.92)
π
the problem of the existence of P (α) reduces to the problem of the existence of
the Fourier transform of χN (η).
Wigner representation
The Wigner function can be defined as the Fourier transform of the characteristic
function χ(η) given in Eq. (2.89)
Z
∗
∗
1
eηα −η α χ(η)d2 η
(2.93)
W (α) = 2
π
To the contrary of P (α), the Wigner function W (α) always exists, but it can
be negative, thus similarly to P (α) it cannot in general be considered as a
probability distribution for α. If also P (α) exists, it can be shown that
Z
2
2
W (α) =
P (β)e−2|β−α| d2 β
(2.94)
π
and
Z
W (α)d2 α = 1
(2.95)
We have already pointed out that quantum states for which a classical description exists, can only have a positive P-function, namely (looking at Eq: 2.94)
only a positive Wigner function. States with a negative Wigner function do not
admit a classical interpretation.
A relation similar to Eq. 2.88 exists
C(a, a† ) = Cw (α, α∗ )
(2.96)
The Wigner function is useful when expressed in terms of the field quadratures.
Indeed, recalling that a = 21 (X1 + iX2 ) and being |xi i the eigenstate of the
quadrature operator Xi , we can rewrite the Wigner function in terms of xi as
W (x1 , x2 ) =
1
W (α)|α=(x1 +ix2 )/2
4
(2.97)
CHAPTER 2. SQUEEZING OF THE LIGHT
43
From this expression we obtain the probability distribution associated to each
quadrature. In fact, it can be shown that
P (xk ) = hxk |ρ|xk i
Z +∞
=
dxk W (x1 , x2 )
(2.98)
−∞
where k = k − (−1)k .
Let us focus on the quantum states whose Wigner function is
Q
W (x1 , x2 ) = N e− 2
(2.99)
where the N is a normalization coefficient and Q is the quadratic form
Q = (x − hxi)T A−1 (x − hxi)
(2.100)
The contour of the Wigner function is defined by Q = 1 and corresponds to
the error circle of the state. Two examples of the Wigner functions are
• the coherent states
W (x1 , x2 ) =
02
1 − 1 (x02
e 2 1 +x2 )
2π
(2.101)
where x0i = xi − hxi i. The contour Q=1 is given by
02
x02
1 + x2 = 1
(2.102)
i.e the equation of the error circle in Fig. 2.3, which is a unity radius circle
centered on (hx1 i, hx2 i)
• the Fock states |ni
W (x1 , x2 ) =
2
2
(−1)n Ln (4r2 )e−2r
π
(2.103)
where r2 = x21 + x22 and Ln are the Laguerre’s polynomials. This Wigner
function is negative.
2.2.3
Squeezed states
In the previous section we pointed out that the coherent states are minimum
uncertainty states, i.e V ar(X1 )V ar(X2 ) = 1, and the uncertainty is the same
for both quadratures X1 e X2
V ar(X1 ) = V ar(X2 ) = 1
(2.104)
However, a more general set of minimum uncertainty states exists, the class
of squeezed states. Those states exhibit, on a particular quadrature, a smaller
uncertainty than a coherent state, but, because of the uncertainty principle, the
uncertainty on theporthogonal quadrature will be greater than that of a coherent
state. Indicating V ar(X1,2 ) = ∆X1,2 , those states can be represented on the
hyperbola of equation ∆X1 ∆X2 = 1 and the physically realizable states are
CHAPTER 2. SQUEEZING OF THE LIGHT
44
Figure 2.4: Hyperbola of the minimum uncertainty states, i.e ∆X1 ∆X2 = 1.
The unsqueezed vacuum state corresponds to ∆X1 = ∆X2 = 1 and all physically realizable states lie on the hyperbola or to its right (shaded area).
those which lie on the hyperbola and to its right, as shown in Fig. 2.4.
From a formal point of view, the squeezed states can be generated from the
vacuum state with the squeezing operator
1 ∗ 2
S() = e 2 a − 21 a†2
(2.105)
where = re2iφ and
S † ()aS() = a cosh r − a† e−2iφ sinh r
†
†
†
S ()a S() = a cosh r − ae
−2iφ
(2.106)
sinh r
Let us also introduce the rotated quadratures
Y1 + iY2 = (X1 + iX2 )e−iφ
(2.107)
S † ()(Y1 + iY2 )S() = Y1 e−r + Y2 er
With the squeezing operator we generate
• squeezed vacuum
|0, i = S()|0i
(2.108)
|α, i = D(α)S()|0i
(2.109)
• coherent squeezed states
CHAPTER 2. SQUEEZING OF THE LIGHT
45
The quadratures satisfy
hX1 + iX2 i = hY1 + iY2 ieiφ = 2α
−r
∆Y1 = e
∆Y2 = e
(2.110)
r
While for the electric field we have that the average value is the same as for a
coherent state, the variance becomes (θ = (ωt − k · r))
V ar(E(r, t)) = K{V ar(X1 ) sin2 θ + V ar(X2 ) cos2 θ − V ar(X1 , X2 ) sin 2θ}
(2.111)
For a coherent state,
V ar(X1 ) = V ar(X2 ) and V ar(X1 , X2 ) = 0,
we obtain a constant value of V ar[E(r, t)]. For a squeezed state we obtain a
variance which oscillates at frequency 2ω. The behavior of the electric field for
a coherent state and a squeezed state is shown in Fig. 2.5 .
It can be shown that the Wigner function for a squeezed state is given by
W (x1 , x2 ) =
1 − 1 (y102 e2r +y202 e−2r )
e 2
2π
(2.112)
where yi02 = (yi − hyi i). The contour Q=1 is given by
y102 e2r + y202 e−2r = 1
(2.113)
and corresponds to the error ellipses shown in Fig. 2.6
The Wigner function of a squeezed vacuum, together with that of an unsqueezed
vacuum, are reported in Fig. 2.7. As regards the photon statistics, it can be
shown that, referring to the P-representation of states, P (n) is given by (Hn
are the Hermite’s polynomials)
n
2
∗ 2 iφ
2 −iφ
1
−1 1
tanh r e−|α| − 2 tanh r((α ) e +α e ) |Hn (z)|2
P (n) = (n! cosh r)
2
(2.114)
αα∗ eiφ tanh r
z= √
2eiφ tanh r
hni = |α|2 + sinh2 r
∗ 2iφ
V ar(n) = |α cosh r − α e
(2.115)
(2.116)
2
2
2
sinh r| + 2 cosh r sinh r
(2.117)
This distribution can be both larger or thiner than the Poisson distribution,
which is that of a coherent state. Finally, we note that the squeezed vacuum
(α = 0) has a non vanishing average photon number and it contains only an
even number of photons, given that Hn (0) = 0 for odd n. Examples of P (n) for
coherent squeezed states are shown in Fig. 2.8(a) and 2.8(b).
2.3
Generation of squeezed light
In the previous section we described the properties of the squeezed states of light.
In this section we discuss how squeezed light can be produced and we report two
CHAPTER 2. SQUEEZING OF THE LIGHT
46
Figure 2.5: Electric field versus time for a: coherent state b: squeezed state
with reduced amplitude fluctuations c: squeezed state with reduced phase fluctuations.
CHAPTER 2. SQUEEZING OF THE LIGHT
47
Figure 2.6: Top left: Squeezing ellipse of a squeezed vacuum state, top right:
squeezed coherent state, bottom left: phase squeezed vacuum state, bottom
right: amplitude squeezed vacuum state.
Figure 2.7: Left: Wigner function of an unsqueezed vacuum state, right:
Wigner function of an unsqueezed vacuum state. For both states the error circle(ellipse) is shown (red line) which corresponds to the contour of the Wigner
function.
CHAPTER 2. SQUEEZING OF THE LIGHT
48
Figure 2.8: Photon number distribution for a squeezed state |α, ri a: α = 3,
r = 0, ±0.5 b: α = 3, r = 1.
example of squeezing devices. The first method is based on the degenerate
parametric amplifier, which exploits an optical cavity containing a nonlinear
medium pumped by a strong coherent field (a laser). This kind of squeezing
apparatus is one of the most popular and was used to generate squeezed light at
both radio and audio frequency band (see [23] and [5]). Then we describe the
ponderomotive squeezing, a technique which exploits the optomechanical
interaction between the radiation inside an optical cavity and movable mirror.
However, before describing this two squeezing methods, we have to report the
input-output formalism for an optical cavity, which is the starting point to the
theory of both methods.
2.3.1
Input-output formalism for optical cavities
In this section, following [14], we deal with the quantum Langevin equations
derived for an optical cavity linearly interacting with a multimode external
field, by assuming that only one cavity mode can be excited by the external
CHAPTER 2. SQUEEZING OF THE LIGHT
49
field.
The total Hamiltonian is
H = Hcav + HB + Hint
and
Z
(2.118)
+∞
dω ωb† (ω)b(ω)
HB = }
(2.119)
−∞
Z
+∞
dω k(ω) b† (ω)a − a† b(ω)
Hint = i}
(2.120)
−∞
where b(ω) are the annihilation operators for the multimode radiation field,
which satisfy the boson commutation relation
b(ω), b† (ω 0 ) = δ(ω − ω 0 )
(2.121)
a is the annihilation operator for the cavity mode excited by the external field.
We derive the quantum Langevin equations by starting from the Heisenberg
equation of motion for an operator O
.
i
O = − [O, H]
}
to b and to a:
(2.122)
.
b(ω) = −iωb(ω) + k(ω)a
Z
i
.
a = − [a, Hcav ] + dωk(ω) b† (ω) [a, a] − a, a† b(ω)
}
(2.123)
(2.124)
Solving Eq. 2.123 for t ≥ t0 we obtain
−iω(t−t0 )
b(ω) = e
Z
t
b0 (ω) + k(ω)
0
e−iω(t−t ) a(t0 )dt0
(2.125)
t0
Then, substituting in Eq. 2.124 we obtain
i
.
(2.126)
a = − [a, Hsys ]+
}
Z
n
o
dωk(ω) eiω(t−t0 ) b†0 (ω) [a, a] − a, a† e−iω(t−t0 ) b0 (ω) +
Z
Z t
n
o
0
0
dω[k(ω)]2
dt0 eiω(t−t ) a† (t0 ) [a, a] − a, a† e−iω(t−t ) a(t0 )
t0
The equation assumes a familiar form when
• we define an in field by
1
bin (t) = √
2π
which satisfies
h
Z
+∞
dωe−iω(t−t0 ) b0 (ω)
(2.127)
−∞
i
bin (t), b†in (t0 ) = δ(t − t0 )
(2.128)
CHAPTER 2. SQUEEZING OF THE LIGHT
50
• we assume k(ω) to be constant
k(ω) =
p
γ/2π
(2.129)
• we recall that a is the annihilation operator for a cavity mode, while Hcav
is the Hamiltonian for the cavity mode, i.e
[a, a† ] = 1
†
Hcav = }ω0 a a
We obtain
(2.130)
(2.131)
γ
√
(2.132)
a − γbin (t)
2
i.e, an equation similar to that of a damped harmonic oscillator, where
.
a = −iω0 a −
•
γ
2a
is a damping term and it comes from the interaction of the cavity mode
with the multimode external field
• the term bin (t) depends on b0 (ω), the value of b(ω) at t = t0 . It is interpreted as an input state. It is a noise field in the case of a incoherent
state, or a classical driving field in the case of a coherent state.
Solving Eq. 2.123 under the condition at t1 > t, we obtain
Z t1
0
−iω(t−t1 )
b(ω) = e
b1 (ω) − k(ω)
e−iω(t−t ) a(t0 )dt0
(2.133)
t
In this case we define an out field
1
bout (t) = √
2π
Z
+∞
dωe−iω(t−t1 ) b1 (ω)
(2.134)
−∞
and Eq. 2.132 becomes
.
a = −iω0 a +
γ
√
a − γbout (t)
2
(2.135)
From this equation and the previous Eq. 2.132, we obtain the input-output
relation for the cavity,
bout (t) = bin (t) +
2.3.2
√
γa(t)
(2.136)
Squeezing by a degenerate parametric amplifier
In a degenerate parametric amplifier (see [13] and [8]), a classical coherent field
at frequency ωp pumps a cavity which contains a non linear medium with χ(2)
non linearity. The cavity is tuned in such a way that it can resonate at angular
frequency ω0 = ωp /2 and the cavity mirrors are chosen to have high reflectivity
at ω0 and nearly zero at ωp .
The Hamiltonian for such a system is
Hsys = Hcav + Hint
(2.137)
CHAPTER 2. SQUEEZING OF THE LIGHT
51
Figure 2.9: Schematic picture of the cavity of an optical parametric amplifier.
where
Hcav = }ω0 a† a
(2.138)
and
1
i}χ(2) ηe−iωp t a†2 − η ∗ e+iωp t a2
(2.139)
2
where η is a measure of the effective pump field intensity and χ(2) is the nonlinear
susceptibility of the nonlinear medium.
Referring to Fig. 2.9 and to the results of the previous section for the quantum
R
Langevin equation of an optical cavity, we indicate by bL
in and bin the input
fields entering respectively on the left and right side of the cavity and by γL
and γR the damping terms of the left and right side of the cavity. For sake of
simplicity we assume that γL = γR = γ and we introduce = ηχ(2) . In addition,
it can be shown that, after a shift of the frequency axis from ω0 = ωp /2 to zero,
the interaction Hamiltonian becomes
1 (2.140)
Hint = i} a†2 − ∗ a2
2
Hint =
The Heisenberg equation of motion becomes
i
ȧ = − [a, Hint ]
}
From Eq. 2.141 we derive the quantum Langevin equation for a
√
√ R
.
a = a† − γa − γbL
γbin (t)
in (t) −
(2.141)
(2.142)
that we rewrite in a matrix form
√
√ R
ȧ = [A − γ] a − γbL
γbin (t)
in (t) −
0 0
||eiθ
A= ∗
=
0
||e−iθ
0
a
a=
a†
Applying the Fourier transform to Eq. 2.142 we have
√
√ R
− iΩa(Ω) = [A − γ] a(Ω) − γbL
γbin (Ω)
in (Ω) −
where
a(Ω) =
a(Ω)
a† (Ω)
(2.143)
(2.144)
(2.145)
CHAPTER 2. SQUEEZING OF THE LIGHT
52
and
a(Ω) =
(2.146)
√
√ R
(γ − iΩ)[ γbL
γbin (Ω)]
in (Ω) +
−
(γ − iΩ)2 − ||2
√ R†
√
γbin (−Ω)]
[ γbL†
in (−Ω) +
−
2
(γ − iΩ) − ||2
Let us refer again to Fig. 2.9. Here aout is the field to be squeezed. We assume
L
both ain and bL
in to be vacuum states. In particular, bin can be considered
as the noise associated with the coherent classical pump field.2 We show that
under these assumptions the output field comes out to be squeezed. In fact, the
input-output formalism developed in Eq. 2.136 allows to write
aout (Ω) = ain (Ω) +
−
−
[( 12 γ)2
−
√
γa(Ω) =
( 12 γ
(2.147)
2
2
− iΩ) + || ]ain (Ω) +
(γ − iΩ)2 − ||2
γa†in (−Ω)
L†
γ(γ − iΩ)bL
in (Ω) + γbin (−Ω)
(γ − iω)2 − ||2
The squeezing in the output field can be enlightened by calculating the variances of the quadratures of the output field. We recall that the pump field
amplitude is characterized by = ||eiθ , and that the definitions of the electric
field quadratures and rotated quadratures (in this case the rotation angle is θ),
are reported in Eq. 2.10-2.11 and Eq. 2.107. Indicating with N the normal
ordering, it can be shown that
||(γ/2)
δ(Ω + Ω0 )
(γ − ||)2 + Ω2
||(γ/2)
=−
δ(Ω + Ω0 )
(γ + ||)2 + Ω2
hY1,out (Ω), Y1,out (Ω0 )iN =
hY2,out (Ω), Y2,out (Ω0 )iN
(2.148)
From the previous formulae we see that the maximum squeezing is obtained on
Y2 quadrature when
|| = γ
(2.149)
and that the parametric amplifier correlates the modes at frequency ±Ω. Indeed,
the two correlation functions, which appear in 2.148, vanish unless Ω0 = −Ω.
Then we integrate over Ω0 obtaining, in correspondence of the maximum squeezing, the normally ordered spectrum
SYN2 (Ω) = −
γ2
8γ 2 + 2Ω2
(2.150)
For the resonant mode 21 ωp of the cavity, i.e for Ω = 0, we have
SYN2 (0) = −
1
8
(2.151)
2 We could also have considered the pump field as a coherent quantum field, obtaining the
same results.
CHAPTER 2. SQUEEZING OF THE LIGHT
53
Squeezing with a degenerate parametric amplifier: the Wigner function approach
In this section we briefly describe a different method for characterizing the field
exiting an optical parametric amplifier.
The starting point is again the interaction Hamiltonian for the parametric amplifier reported in Eq.2.140. We assume for simplicity to be real and we rewrite
1 Hint = i} a†2 − ∗ a2
(2.152)
2
The Heisenberg equation of motion for a and a† are given by
i
ȧ = − [a, Hint ] = a†
}
i
†
ċ = − [a† , Hint ] = a
}
It can be shown that the solutions of the two equation are
a(t) = a(0) cosh t + a† (0) sinh t
†
(2.153)
(2.154)
(2.155)
†
a (t) = a (0) cosh t + a(0) sinh t
(2.156)
The cavity field, described via the annihilation operator a, is squeezed. To show
that, let us write down the Heisenberg equation of motion for the quadrature
operators
X1 (t) = a(t) + a† (t)
†
(2.157)
X2 (t) = −i(a − a )
(2.158)
Ẋ1 = X1
(2.159)
From Eq. 2.155 we obtain
Ẋ2 = −X2
We note that those equations are different for X1 and X2 , i.e, the parametric
amplifier can ”distinguish” the two quadratures and ”treat” them in a different
way. In particular it amplifies a quadrature and attenuate the other. In the case
of a complex we change the phase of the pump field and we just attenuate and
amplify two rotated quadratures instead of X1 and X2 :
X1 (t) = et X1 (0)
−t
X2 (t) = e
X2 (0)
(2.160)
(2.161)
For the quantum noise we obtain the same equations, i.e
V ar(X1 , t) = e2t V ar(X1 , 0)
V ar(X2 , t) = e
−2t
(2.162)
V ar(X2 , 0)
When we suppose that the input field is a vacuum state, i.e V ar(X1 , 0) =
V ar(X2 , 0) = 1, the Eq. (2.162) becomes
V ar(X1 , t) = e2t
V ar(X2 , t) = e
−2t
(2.163)
CHAPTER 2. SQUEEZING OF THE LIGHT
54
obtaining a squeezed vacuum. The amount of squeezing depends on the non linear coupling, on the intensity of the pump field (both parameters are contained
in ) and on the interaction time t.
From the point of view of photon statistics, the squeezed field exhibits bunching
ha† (t)a† (t)a(t)a(t)i
ha† (t)a(t)i2
cosh 2t
=1+
sinh2 t
≥1
g (2) (0) =
(2.164)
A complete description of this squeezed state can be derived by computing its
Wigner function. We describe the initial vacuum state as a coherent state with
coherent amplitude α0 = 0, and we express he Wigner function in terms of
coherent amplitudes
2 − 1 αT Cα−1 α
e 2
π
sinh 2t cosh 2t
Cα =
cosh 2t sinh 2t
W (α, t) =
(2.165)
where αT = (α, α∗ ). This Wigner function has a Gaussian shape with vanishing
average and covariance matrix Cα .
In terms of the real variables x1 = α+α∗ and x2 = −i(α−α∗ ), which corresponds
to the quadrature operators, the Wigner function of the squeezed state becomes
1 − 1 xT Cx−1 x
e 2
2π
2t
e
0
Cx =
0
e−2t
W (x1 , x2 ) =
2.3.3
(2.166)
Ponderomotive squeezing
As we have already pointed out in the introduction to this thesis, the injection
of squeezed light into the output port of a GW Michelson interferometer is the
technique, which will be used in advanced detectors in order to beat the SQL
of the interferometer. This technique has already been experimentally tested in
few experiments [24]. Here the squeezed light was obtained with the non linear
method described in the previous section.
In all these experiments, the effect of squeezing was measured in the few MHz
frequency band, where the effects of classical noise sources, such as the laser
frequency and intensity noise, can be reduced.
Squeezed vacuum were also produced in the GW detectors band, 10Hz-10kHz,
by using optical parametric processes (see [23]). However technical limitations,
for example photothermally driven fluctuations, strongly reduce the squeezing
level.
An alternative method [6] to produce squeezed vacuum is the ponderomotive
squeezing. This technique exploits the radiation pressure to produce squeezing
as a result of the coupling between the radiation inside an interferometer and
CHAPTER 2. SQUEEZING OF THE LIGHT
55
the mechanical motion of a suspended mirror. In such a device, in order to
enhance the radiation pressure effect, the radiation power circulating in the interferometer should be high (input power of at least order of few W ), while the
mirror mass should be small (10−6 − 10−3 kg). In addition, detuned Fabry-Perot
arm cavities are used in order to generate an optical spring i.e, as it will be clear
in the following, an optomechanical rigidity, which shift the resonant frequency
of the suspended mirror. For input power and mirror mass of the indicated
order of magnitude, the optical spring can shift the resonant frequency of the
suspended mirror from the pendulum frequency (Ωp of order of Hz) to Θ of
order of kHz.
It can be shown that in the frequency band Ωp < Ω < Θ the generated ponderomotive squeezing is frequency independent. At frequencies Ω ∼ Θ the
ponderomotive squeezing is frequency dependent.
In the remaining part of this section, following [6], we illustrate a simplified
model of a Fabry-Perot cavity and we describe how ponderomotive squeezing is
generated. In addition we briefly report a realistic setup for a ponderomotive
squeezer in which an interferometer with suspended mirror is used.
Ponderomotive squeezing from an optical cavity
Let us consider an ideal Fabry-Perot cavity close to the optical resonance condition with a high reflective input mirror (IM) and a perfectly reflective suspended
end mirror (EM). In the following we list several quantities, which characterize
the cavity: Γ the linewidth, F the finesse, Π the circulating power, Φ the phase
shift gained by the carrier as it exit the cavity, ω0 the carrier angular frequency
of the incident laser
cTI
4L
2π
F =
TI
1
4I0
Π(I0 , δΓ ) =
TI (1 + δΓ2 )
Γ=
Φ(δΓ ) = −2 arctan(δΓ )
(2.167)
(2.168)
(2.169)
(2.170)
where L is the cavity length, TI is the IM power transmissivity, I0 is the input
power and δΓ is the detuning parameter, defined in terms of the difference
between the laser carrier frequency and the resonant frequency nearest to the
laser carrier frequency δ = ωres − ω0
δΓ =
δ
Γ
(2.171)
The radiation pressure force acting on the EM, expressed in terms of the circulating power is
2Π
(2.172)
c
For a particular choice of the parameters I0 and δΓ the suspended mirror is in
mechanical equilibrium due to the action of gravity and the optical spring force.
FRP =
CHAPTER 2. SQUEEZING OF THE LIGHT
56
In fact, when we shift the mirror adiabatically 3 by dx the circulating power
changes, giving rise to an additional restoring force other than that of gravity.
If Ωp is the pendulum frequency and M the mass of the EM, it can be shown
that the force variation is
dF = −M Ω2p dx +
2 ∂Π(I0 , δΓ ) dδΓ
dx
c
∂δΓ
dx
(2.173)
The coefficient, which appears in the previous formula
Kopt = −
2 ∂Π(I0 , δΓ ) dδΓ
c
∂δΓ
dx
(2.174)
is defined as optical rigidity.
From Eq. 2.167 and Eq. 2.171 it can be shown that
dδΓ
4ω0
=−
dx
cTI
(2.175)
Taking into account all the forces applied to the end mirror, its frequencydomain equation of motion is
− M Ω2 x̃ = −(M Ω2p + Kopt )x̃ +
2 ∂Π(I0 , δΓ ) ˜
I0
c
∂I0
(2.176)
The frequency dependent part of the carrier phase shift Φ̃ can be written in
terms of x̃ by using Eq. 2.170 and Eq. 2.175
dΦ(δΓ ) dδΓ
Φ̃ =
x̃
(2.177)
dδΓ
dx
Note that Eq. 2.176 and 2.177 tell us that any suspended cavity, both detuned
(Kopt is present) and not detuned (Kopt is not present) will convert fluctuations
of the input radiation into mirror motion and hence in output phase fluctuations
of the radiation field. Ad we will see, this allow the production of squeezed light
when the input field has quantum limited fluctuations.
Let us use these equations in the input-output relation for the cavity. (I 0 , δ Γ , Φ)
and (I˜0 , δ̃Γ , Φ̃) are the DC and AC components of the input power, detuning
parameter and carrier phase shift respectively.
The input field is written, using the modulation language presented in Sec.
2.1.1), as a classical real amplitude A plus quantum fluctuations of the amplitude and phase
ain (t) = (A + X1,in ) cos(ω0 t) + X2,in sin(ω0 t)
(2.178)
Indicating with SX1,in , SX2,in and SX1,in X2,in the spectral density of X1,in and
X2,in and their cross spectral density respectively, we can choose a coherent
input field and normalize it in such a way that
}ω02 A = 2I 0
SX1,in = SX2,in = 1
SX1,in X2,in = 0
(2.179)
3 the result is also valid for mirror motion band-limited at frequencies well below the cavity
linewidth, i.e in the quasistatic regime
CHAPTER 2. SQUEEZING OF THE LIGHT
57
The effect of the mirror motion is to phase shift the output filed aout (t) by Φ̃,
because the cavity length is changed adiabatically
aout (t) = (A + X1,in ) cos(ω0 t − Φ) + X2,in sin(ω0 t − Φ)
(2.180)
Using Eq. 2.177 and decomposing Φ into its DC and AC part, under the assumption that the AC part is small compared to the DC one, we rewrite the
output field
aout (t) = (A + X1,out ) cos(ω0 t − Φ) + X2,out sin(ω0 t − Φ)
(2.181)
X1,out = X1,in
(2.182)
"
X2,out = X2,in + AΦ̃ = X2,in +
4
1
TI 1 + δ 2
Γ
#
2Aω0 x̃
c
(2.183)
Then, we have, using Eq. 2.169 in Eq. 2.174
Kopt = −
4ω0 Π δ Γ
ΓLc 1 + δ 2
(2.184)
Γ
and we define the characteristic frequency
Θ2 =
Kopt
4ω0 Π δ Γ
=−
M
M ΓLc 1 + δ 2
Γ
"
#2
4ω0 I0 δ Γ
4
=−
2
M c2
TI (1 + δ )
(2.185)
Γ
The fluctuating part of the input power is
I˜0 = }ω0 AX1,in
which induces the fluctuating force on the mirror given by
#
"
2 ∂Π(I0 , δΓ ) ˜
2}ω0 A
4
I0 =
X1,in
2
˜
c
c
∂ I0
TI (1 + δ Γ )
(2.186)
(2.187)
Inserting all these expression in the EM equation of motion, Eq. 2.176, we
obtain
#
"
4
2}ω0 A
2
2
2
M [Θ + Ωp − Ω ]x̃ =
X1,in
(2.188)
2
c
TI (1 + δ Γ )
q
In the hypothesis that Ω2p + Θ2 is smaller than the cavity linewidth, Eq. 2.188
q
shows that the mechanical frequency is shifted from Ωp to Ω2p + Θ2 . If Θ is
real (δ Γ < 0) we have a resonance, if Θ is purely imaginary (δ Γ > 0) the system
is unstable.
Finally, using Eq. 2.188, we can rewrite the input-output relations 2.183 as
X1,out = X1,in
(2.189)
2
X2,out = X2,in +
Ω2
Θ
1
X1,in
2
2
− Θ − Ωp δ Γ
(2.190)
CHAPTER 2. SQUEEZING OF THE LIGHT
The input-output relations are
X1,out
1
0
X1,in
=
X2,out
−2K(Ω) 1
X2,in
where
K(Ω) =
1
1
1 − (Ω2 − Ω2p )/Θ2 δ Γ
58
(2.191)
(2.192)
K couples the output amplitude and phase quadratures, leading to squeezing.
Focusing our attention just on the fluctuations, we have for the ζ quadrature
X1,out cos ζ + X2,out sin ζ = X1,in [cos ζ − 2K(Ω) sin ζ] + X2,in sin ζ
(2.193)
Considering the spectral densities of X1,in and X2,in given in Eq. 2.179, the
spectral density of the ζ quadrature of the output field is
Sζ (Ω) = 1 + 2K 2 − 2K[sin 2ζ + K cos 2ζ] ≡ ξζ2 (Ω)
(2.194)
For a non squeezed vacuum we have Sζ (Ω) = 1.
By minimizing ξζ (Ω) we obtain that the squeezed quadrature corresponds to
1
1
arctan
2
K(Ω)
1
p
ξmin (Ω) =
|K(Ω)| + 1 + K 2 (Ω)
ζmin (Ω) =
(2.195)
(2.196)
Assuming that Ωp Ω, Ωp Θ and that |Θ| is much smaller that the cavity
linewidth we distinguish three conditions
• Ω |Θ|
K is nearly frequency independent and
1
arctan δ Γ
2
|δ |
qΓ
ξmin (Ω |Θ|) =
2
1 + δΓ + 1
ζmin (Ω |Θ|) =
(2.197)
(2.198)
• Ω |Θ|
K −→ 0 and the output field is an unsqueezed vacuum
• Ω ∼ |Θ|
the system goes trough a resonance, with a strong squeezing and highly
frequency-dependent squeezing angle in the case of a real Θ and goes
trough a smooth transition when Θ is purely imaginary
A proposed experimental set-up for a ponderomotive squeezer is illustrated in
Fig. 2.10 and in Tab 2.11 the corresponding values of the parameters are reported.
CHAPTER 2. SQUEEZING OF THE LIGHT
59
Figure 2.10: Schematic of a an interferometer for ponderomotive squeezing.
Light from a highly amplitude- and phase-stabilized laser source is incident on
the beamsplitter. High-finesse Fabry-Perot cavities in the arms of the Michelson
interferometer are used to build up the carrier field incident on the end mirrors
of the cavity. All interferometer components in the shaded triangle are mounted
on a seismically isolated platform in vacuum. The input optical path comprises
a pre-stabilized 10 Watt laser, equipped with both an intensity stabilization
servo and a frequency stabilization servo. FI is a Faraday Isolator.
Figure 2.11: Example of values of the parameters for the ponderomotive squeezing interferometer.
Chapter 3
Homodyne detection
In the introduction to this thesis we saw that a phase dependent detection
scheme is required in order to characterize squeezed light and that a detection
scheme often used for that purpose is the balanced homodyne detection. A
balanced homodyne detector, whose schematic picture is reported in Fig. 3.1,
consists of a 50/50 beam splitter which superimposes two optical fields. One
of the two fields is the squeezed field while the other one is a so-called local
oscillator, i.e a strong coherent field, which we use as a phase reference.
In this chapter, after a brief review of the theory of photodetection with a single
photodiode, we introduce the theory homodyne detection. Then we describe the
electronics needed to implement an homodyne detector. Finally, we present the
design of a homodyne detector prototype, which was developed throughout the
thesis.
3.1
Theory of detection
In this section, following [23] and [5], we introduce the theory of direct detection,
i.e detection with a single photodiode, and of homodyne detection.
From a mathematical point of view, it is useful to describe the behavior of some
optical fields, such as the coherent states of light (which can be basically seen
as classical non-fluctuating fields, which carry vacuum noise), using a linearized
version of quantum optics. In this linearized picture we consider the photon
creation and annihilation operators as formed by the sum of a complex term
α, which corresponds to the classical complex amplitudes, which appear in Eq.
2.2 as ak and which, in case of a coherent field, would represent the coherent
amplitude, and a fluctuating part containing both the quantum and classical
fluctuations of the optical field:
a = α + δa
†
∗
(3.1)
†
a = α + δa
60
(3.2)
CHAPTER 3. HOMODYNE DETECTION
61
Figure 3.1: Left: Direct detection of the optical field â. Right: homodyne
detection scheme: two optical fields â and b̂ are superimposed to a 50/50 beam
splitter. The resulting fields ĉ and dˆ are detected with two photodiodes and the
sum and difference of the corresponding photocurrents are measured.
and
hai = α
(3.3)
hδai = 0
†
ha i = α
(3.4)
∗
(3.5)
†
hδa i = 0
(3.6)
Correspondingly, the fluctuating part of the amplitude and phase quadratures
of the electromagnetic field can be written as
δX1 = δa + δa†
(3.7)
†
δX2 = i(δa − δa)
(3.8)
δXθ = δX1 cos θ + δX2 sin θ
=e
−iθ
iθ
(3.9)
†
δa + e δa
It is worth noting that, as we have seen in Chapter 2, squeezing affects the quantum noise properties of optical fields, thus, if we want to characterize squeezing
we have to be able to measure the fluctuating part of the electric field quadratures given in Eq. 3.7 , 3.8 and 3.9.
3.1.1
Direct detection
The simplest way to detect light is to use an absorption-based device, such as
a photodiode (see Fig. 3.1), which absorbs photons and produces a photocurrent proportional to the number of photons absorbed. The photocurrent thus
obtained can be electronically processed in order to extract the desired informations about the light source.
From a formal point of view, such a device produces a photocurrent, which can
be represented by an operator proportional to the photon number operator a† a:
using Eq. 3.1 and 3.2, taking α to be real and neglecting second order terms,
CHAPTER 3. HOMODYNE DETECTION
62
we have that
î(t) ∝ n̂ = a† a
2
(3.10)
†
= α + α(δa + δa)
= α2 + αδX1,a (t)
As expected, reminding the meaning of the amplitude and phase quadratures,
which we have discussed in Chapter 2, we obtain that the photocurrent is proportional to the classical intensity, given by α2 , plus a fluctuating term, which
depends only on the amplitude quadrature fluctuations and scales with the field
amplitude α. Thus the measurement of the fluctuations of the photocurrent
are enhanced by increasing the field intensity. We can Fourier transform the
photocurrent obtaining
î(ω) ∝ n̂(ω) = α2 δ(0) + αδX1,a (ω)
(3.11)
The DC term of the photocurrent, which also corresponds to the average photocurrent, can be detected by a power meter: following [5], if ne (t) is the
number of electrons produced at time t by a photodiode of quantum efficiency
ηqe when a light source of power Popt (t) impinges on it, e is the electron charge
and ∆t is the measurement interval, we have that the average photocurrent
detected during ∆t is
ηqe P opt e
ne (t)e
=
(3.12)
i(t) =
∆t
}ω
In fact, if ω is the angular frequency of the light source, }ω is the energy of
a single photon of the light source and P opt ∆t/}ω is the number of photons,
which arrive at the detector during the measurement time ∆t. The number of
electrons produced from those photons (given the quantum efficiency ηqe of the
detector) is given by ηqe P opt ∆t/}ω.
The AC term, which contains also the quantum fluctuations of the signal field
and thus is of crucial importance in squeezing measurements, can be measured
with a spectrum analyzer.
In the following we report an example of spectral measurements of the AC term
of the photocurrent 3.11 based on the measurement of the quantum noise level
of coherent light sources, which only carry vacuum noise and play a crucial role
in homodyne detection.
Spectral measurements: Shot noise of a coherent source
Let us start with the photocurrent of Eq. 3.10 and calculate its variance
V ar[î(t)] ∝ α2 V ar[X1,a (t)]
(3.13)
Fluctuations in the number of photons impinging on the detector reflects in
fluctuation of the number of electrons produced, i.e in fluctuations of the photocurrent. In particular, given that a coherent state has a Poissonian photon
statistics (as we have shown in Chapter 2 and as it can be deduced also from
Eq. 3.10 by assuming that the property of coherent states V ar[X1,a (t)] = 1
is satisfied), we have that the variance of the number of electrons equals the
average number of electron
V ar[ne (t)] = ne (t)
(3.14)
CHAPTER 3. HOMODYNE DETECTION
63
and for the photocurrent, also using Eq. 3.12 and 3.14, we have
ne (t)e
V ar[ne (t)]e2
V ar[î(t)] = V ar
=
∆t
∆t2
=
(3.15)
i(t)e
∆t
This expression can be rewritten, using the Shannon’s theorem, which links the
measurement time ∆t to the measurement bandwidth B by 1/∆t = 2B, as
V ar[î(t)] = 2ei(t)B
(3.16)
The power spectral density of the photocurrent, which we indicate with Sii (f )
can be deduced from the expression of the variance of the photocurrent 3.16 by
using the relation
Z
+∞
df Sii (f )
V ar[î(t)] =
(3.17)
0
Reminding that the quantum noise of a coherent state is equally distributed at
all frequencies, i.e it is white, we have that the power spectral density is constant
in frequency and we can simplify Eq. 3.17 as
V ar[î(t)] = Sii B
(3.18)
Finally, from Eq. 3.16 and 3.12, we obtain
Sii = 2ie = 2ηqe P opt
3.1.2
e2
}ω
(3.19)
Balanced homodyne detection
As shown in Eq. 3.11, photodetection with a single photodiode only allows to
measure fluctuations of the amplitude quadrature, while the characterization
of squeezing requires the detection of fluctuations at arbitrary quadrature, for
example of the phase quadrature. In order to achieve phase-dependent detection
we need a phase reference, as it happens in balanced homodyne detection,
whose ideal scheme is pictured in Fig. 3.1. In a balanced homodyne detector the
phase reference is provided by a laser beam, the so-called local oscillator. The
local oscillator is superimposed at a 50/50 beam splitter (if the beam splitter
is not 50/50 the homodyne detection is said unbalanced) with the field whose
quadratures have to be characterized, for example a squeezed vacuum.
Referring to Fig. 3.1 and indicating with a and b the annihilation operators of,
respectively, the local oscillator and the field, we have that the fields exiting the
beam-splitter are
1
c = √ (a + b)
2
1
d = √ (−a + b)
2
(3.20)
(3.21)
CHAPTER 3. HOMODYNE DETECTION
64
using the linearized annihilation operator that we introduced in Eq. 3.1 we can
rewrite
a = α + δa
(3.22)
b = β + δb
(3.23)
αβ
(3.24)
where α and β are complex number and, in the last expression, the local oscillator was set to be much more intense than the field. The relative phase
between α and β, i.e the relative phase between the local oscillator and the
field, is of crucial importance in homodyne detection. Thus, in order to make
the relative phase to appear clearly in the following calculation, we choose β to
be real and then we rewrite α in term of its magnitude and phase. Hence this
phase is simply the relative phase between the field and the local oscillator. For
simplicity we indicate the magnitude of α with α:
a = αeiθ + δaeiθ
(3.25)
b = β + δb
(3.26)
Following Eq. 3.10, the photocurrents produced by the two photodetectors are
proportional to the photon numbers operators c† c and d† d. Using Eq. 3.25 and
3.26 and neglecting second order terms, we obtain for the photocurrents (see
[5])
1 2
α + β 2 + 2αβ cos θ + α (δX1,a + δXθ,b ) + β (δX−θ,a + δX1,b )
2
(3.27)
1
îd ∝ d† d =
α2 + β 2 − 2αβ cos θ + α (δX1,a − δXθ,b ) − β (δX−θ,a − δX1,b )
2
(3.28)
îc ∝ c† c =
The homodyne detector performs the sum and the difference of the two photocurrents, i.e
î+ ∝ c† c + d† d = α2 + β 2 + αδX1,a + βδX1,b
†
†
î− ∝ c c − d d = 2αβ cos θ + αδXθ,b + βδX−θ,a
(3.29)
(3.30)
Let us summarize the AC components of these two photocurrents
• sum photocurrent î+
it contains the amplitude quadrature fluctuations of both local oscillator
and signal, but those of the local oscillator are scaled by β, i.e by the amplitude of the signal, while those of the signal are scaled with the classical
amplitude α of the local oscillator
• difference photocurrent î−
it contains the −θ quadrature fluctuations of the local oscillator and the
θ quadrature fluctuations of the signal, but, as in the case of the sum
photocurrent, those of the local oscillator are scaled by the amplitude β
of the signal while those of the signal are scaled by the amplitude α of the
local oscillator
CHAPTER 3. HOMODYNE DETECTION
65
• if we take the local oscillator to be much more intense of the signal, we
have that in both sum and difference photocurrents the contributions from
the fluctuations of the local oscillator are strongly suppressed
with respect to those of the signal, which are enhanced by the local
oscillator amplitude α.
All these considerations explain why balanced homodyne detection is so useful
in squeezing measurements: first, it is, in fact, a phase dependent detection
scheme. This is easily seen if we consider that the difference photocurrent
contains information of the θ quadrature of the signal, where θ is the phase
between the local oscillator and the signal. This phase can be changed, thus
allowing to span all signal quadratures. Second, the spurious contributions
deriving from the noise of the local oscillator are strongly suppressed, as we
have seen.
In order to better understand this crucial point, let us suppose that our signal is
a squeezed vacuum. In this case the average photon number is < n >= sinh2 r
(see Eq. 2.116), thus β = sinh r, while α depends on the intensity of the
laser chosen as local oscillator. δXθ,b represents the quantum fluctuations of
the squeezed vacuum in the θ quadrature (given that the squeezed vacuum, by
definition, carries only quantum noise) while δXθ,a represents the fluctuations
of the local oscillator in the θ quadrature.
It can be shown that lasers can be well approximated as coherent states
at sideband frequencies tens of MHz above the laser central frequency (see [3]).
At those sideband frequencies they only carry the quantum noise of a coherent
state, i.e the shot noise described in Sec. 3.1.1. In this case δXθ,a and δXθ,b
are of the same order of magnitude. On the other hand, at lower frequencies
lasers carry technical noises, which can be well above the shot noise. In this
case δXθ,a can be even few order of magnitudes above δXθ,b .
However, if the amplitude of the local oscillator α is large enough with respect
to the signal amplitude β, we can reach the condition
αδXθ,b βδX−θ,a
(3.31)
and the sum and difference photocurrents of Eq. 3.29 and 3.30 become
î+ ∝ α2 + αδX1,a
(3.32)
î− ∝ 2αβ cos θ + αδXθ,b
(3.33)
We have obtained that, if the approximation of Eq. 3.31 is valid, the sum
photocurrent only contains the amplitude fluctuations of the local
oscillator while the difference photocurrent contains the fluctuation in
the θ quadrature of the signal field.
Thus, in order to characterize the squeezing of the signal field it would be
sufficient to take the power spectrum of the fluctuations of the difference photocurrent, following the same procedure illustrated in Sec. 3.1.1. However, if
the laser used as local oscillator can be well approximated by a coherent state,
also the power spectrum of the fluctuations of the sum photocurrent is useful
for the characterization of squeezing.
CHAPTER 3. HOMODYNE DETECTION
66
The variance of the two photocurrents of Eq. 3.32 and 3.33:
V ar[î+ ] ∝ α2 V ar[X1,a ]
(3.34)
V ar[î− ] ∝ α2 V ar[Xθ,b ]
(3.35)
then
• if the local oscillator is a coherent state:
we have that V ar[X1,a ] = 1, thus the noise of the local oscillator can
be used as a sort of unsqueezed vacuum noise reference. The ratio of
the variance of the difference photocurrent to the variance of the sum
photocurrent simply becomes (r is the squeezing factor)
V ar[î− ]
= V ar[Xθ,b ]
V ar[î+ ]
(3.36)
and, if the squeezing spectrum is white in a certain bandwidth, the same
holds for the power spectral density of the two photocurrents (see Sec.
3.1.1)
Sii,+
= V ar[Xθ,b ]
e−2r ≤ V ar[Xθ,b ] ≤ e2r
(3.37)
Sii,−
If the field b is an usqueezed vacuum this ratio is be unity, if the field is a
squeezed vacuum it ranges from e−2r to e2r .
• if the local oscillator is not a coherent state:
in this case the power spectrum of the sum photocurrent is useless in the
characterization of squeezing because V ar[X1,a ] > 1 and the noise of the
local oscillator cannot be used as a vacuum noise reference.
However, we can obtain our vacuum noise reference by performing, before
the squeezing measurement, the so-called self-homodyne detection. In
the self-homodyne detection the squeezed vacuum is blocked and only
unsqueezed vacuum enters the beam splitter. The unsqueezed vacuum is
superimposed to the local oscillator, and, given that the vacuum is nothing
but phase-independent quantum noise, the phase appearing in Eq. 3.25
is meaningless. Indicating the vacuum with v, the sum and difference
photocurrents of Eq. 3.32 and 3.33 become
î+ ∝ α2 + αδX1,a
(3.38)
î− ∝ αδX1,v
(3.39)
and the variances become
V ar[î+ ] ∝ α2 V ar[X1,a ]
2
V ar[î− ] ∝ α V ar[X1,v ]
(3.40)
(3.41)
For the usqueezed vacuum V ar[X1,v ] = 1 and thus, taking the power spectrum
of the difference photocurrent of Eq. 3.39 is the same as taking the power
spectrum of the sum photocurrent when the local oscillator is a coherent state.
We have just to register the power spectrum Sv of the difference photocurrent
3.39 (which contain the vacuum noise power spectrum) and then we have to stop
CHAPTER 3. HOMODYNE DETECTION
67
blocking the squeezed vacuum. We thus take the power spectrum Sθ,sq of the
difference photocurrent of the squeezed vacuum for different quadratures: when
measuring the squeezed quadrature we find Sθ,sq to be below Sv , the opposite
happens when measuring the anti-squeezed quadrature.
Squeezing visibility
Up to now we have investigated the ideal balanced homodyne detection, i.e we
have not considered the possibility that both the optics and electronics of the
homodyne detector could be not perfectly balanced or that the interference of
the two fields at the beam splitter could be not perfect.
A non perfect interference can be due to a non perfect matching, in terms of
spatial mode, wave front curvature, polarization and frequency, of the two fields
which are superimposed at the beam splitter. The quality of the interference
can be characterized by the so-called fringe visibility, defined as
Imax − Imin
0 ≤ Vis ≤ 1
(3.42)
Vis =
Imax + Imin
where Imax(min) is the maximum(minimum) intensity detected by one of the
two photodiode of the homodyne detector.
From the visibility we can define an homodyne efficiency ηhom = Vis2 .
Optical unbalances are due to a non perfectly 50/50 beam splitter, which thus
would have power reflectivity R and transmittivity T slightly different from 0.5.
In this case the fields exiting the beam splitter become, instead of those of Eq.
3.20 and 3.21,
√
√
c = Ra + T b
(3.43)
√
√
d = T a − Rb
(3.44)
Electronics unbalances are due to a different quantum efficiency of the two
phtodiodes and to a different gain of the readout electronics of the two photodiodes. It can be investigated by multiplying by a gain factor G (which is unity
in case of perfect balance) one of the two photocurrents îc and îd of Eq. 3.27
and 3.28.
In all these cases, the capability of the homodyne detector to measure squeezing
is degraded.
If we take into account all these elements, the expressions, given in Eq. 3.34
and 3.35, for the variance of the sum and difference photocurrent take a slightly
different expression: after some cumbersome calculation and assuming the approximation of Eq. 3.31, we obtain
V ar[î+ ] ∝ α2 V ar[X1,a ](T + RG)2 + [1 − ηhom (1 − V ar[Xθ,b ])]RT (1 − G)2
(3.45)
2
2
V ar(î− ) ∝ α [1 − ηhom (1 − V ar[Xθ,b ])]RT (1 + G) + V ar[X1,a ](T − GR)2
(3.46)
Then, if we assume that the local oscillator is a coherent state, i.e V ar[X1,a ] = 1,
we obtain
V ar[î+ ] ∝ α2 (T + RG)2 + [1 − ηhom (1 − V ar[Xθ,b ])]RT (1 − G)2
(3.47)
2
2
2
V ar[î− ] ∝ α [1 − ηhom (1 − V ar[Xθ,b +])]RT (1 + G) + (T − GR)
(3.48)
CHAPTER 3. HOMODYNE DETECTION
68
Assuming a coherent local oscillator, if we compute the ratio V ar[î+ ]/V ar[î− ]
from Eq. 3.34 and 3.35 we obtain the “real ”squeezing factor SQreal = e2r , i.e
the squeezing factor that we would measure with an ideal homodyne detector.
If we calculate the same ratio from Eq. 3.47 and 3.48 we obtain the squeezing factor SQmeas , which we would measure in case of a non ideal homodyne
detector.
Figure 3.2: Electronic gain dependence of the squeezing detection efficiency.
Taking the reflectivity to be R=0.5, depending on the initial squeezing value
(10dB, 13dB and 20dB) and varying the electronic gain, the measured squeezing
diminishes. The situation is even worst if we allow a non perfect fringe visibility.
In Fig. 3.2 and 3.3 the dependence of the squeezing detection efficiency from the
electronic gain G, the reflectivity of the beam splitter and the fringe visibility is
shown by plotting Vreal − Vmeas while changing those parameters. In all cases
the squeezing, which we expect to measure, is lower that the real squeezing.
Figure 3.3: Beam splitter unbalance dependence of the squeezing detection efficiency. Taking the eletronic gain to be G=1, depending on the initial squeezing
value (10dB, 13dB and 20dB) and varying the beam splitter reflectivity, the
measured squeezing diminishes. The situation is even worst if we allow a non
perfect fringe visibility.
CHAPTER 3. HOMODYNE DETECTION
3.2
69
Design of the electronics of a homodyne detector
In this section we describe the design of the electronics for a homodyne detector prototype. Referring to the discussion of the previous section, we start by
itemizing the requirements that the readout circuit of a homodyne detector for
squeezed light should fulfill. Thus, we illustrate the design features, which were
adopted in order for these requirements to be fulfilled, focusing our attention
on the electronic balance, frequency response and noise performance of
the circuit.
The optics for the homodyne detector prototype was not designed in this thesis,
thus we will not discuss it.
A conceptual scheme, containing the main features of the the circuit design,
is shown in Fig. 3.4.
CHAPTER 3. HOMODYNE DETECTION
70
Figure 3.4: Conceptual scheme of the circuit design: two photodiodes are kept
in reverse bias by a 5V supply and the power of the light impinging on them
is detected through the DC blocks. The difference photocurrent is obtained
through a self-subtracion scheme and then readout at two different frequency
bands: Virgo band (10Hz-10kHz), i.e the AUDIO block; 1MHz-100MHz, i.e
theRADIO block. The photocurrent from each photodiode is also detected
in the Virgo band and the sum photocurrent is performed at this frequency
band. The blocks indicated with DC, AUDIO and RADIO are transimpedance
amplifiers for the photocurrents of each photodiode and/or for the difference
photocurrent, anb work at the appropriate frequency band.
We can summarize those features as follows
• two photodiodes
• sum and difference of photocurrents and electronic balance:
as we have seen in the previous section, the information about squeezing
is contained in the difference of the two photocurrents, as shown in Eq.
3.35.
We have also seen that every electronic gain unbalance degrades the measured squeezing level and thus we should avoid as much as possible to
introduce such unbalances, in particular when performing the difference
of the photocurrents. Because of this, a self subtraction scheme was
used, as shown in Fig. 3.4. In such a scheme, the two photocurrents,
instead of being amplified and then subtracted, are automatically subtracted and then the obtained photocurrent is amplified. The first procedure would amplify any unavoidable electronic unbalance between the
CHAPTER 3. HOMODYNE DETECTION
71
readout electronics of the two photodiodes and when finally the subtraction is performed, the amplified unbalances could dramatically degrade
the measured squeezing or even wash it out. This is avoided by adopting
the second procedure.
In order to reduce electronic unbalances, the bias of the two photodiodes
is performed by using only a 5V supply reference from which also the −5V
supply is derived (see Fig. 3.4), instead of taking two different supplies at
±5V . In this way, fluctuations in the power supply appear common-mode
for both photodiodes.
The sum of the two photocurrents is performed in audio-band (10Hz10kHz).
• low noise design:
this is a crucial aspect of the circuit design. Indeed, as we have shown in
the previous section, in order to characterize squeezing, we have to measure the power spectrum of the fluctuations of the difference (and sum)
photocurrent.
In order to better explain this point, we report the expression of the variances of the sum and difference photocurrents, as calculated in the previous section:
V ar[î+ ] ∝ α2 V ar[X1,lo ]
2
V ar[î− ] ∝ α V ar[Xθ,sig ]
(3.49)
(3.50)
If the local oscillator is a coherent state, the power spectrum of the sum
photocurrent is nothing but the power spectrum of the local oscillator
shot noise (see Sec. 3.1.1). The power spectrum of the difference photocurrent, which contains the quantum fluctuations of the squeezed field,
is below or above the shot noise level of the local oscillator depending
on which quadrature we are measuring, and equals the shot noise level
for an unsqueezed vacuum (remind that V ar[X1,lo ] = 1 for a coherent
local oscillator, that V ar[Xθ,sig ] = 1 for unsqueezed vacuum and that
e−2r ≤ V ar[Xθ,sig ] ≤ e2r for a squeezed field).
The homodyne detector circuit has to be designed in order to be able
to detect the shot noise of the local oscillator and noises below the shot
noise. A indication of how much below comes from the fact that one of the
highest squeezing level ever experimentally obtained is 11dB, i.e a factor
e−2r ∼ 3 − 4 (see [23]), thus a factor 10 below can be considered a good
starting point.
All this can be translated in the requirement that the total circuit noise
at the output of the sum and difference photocurrents readout blocks is
at least ten times below the output signals deriving from the shot noise of
the sum and difference photocurrents.
• frequency response: as shown in Fig. 3.4, the circuit prototype is
designed to have
- DC readout for each photodiodes
needed to check if both photodiodes receive the same amount of light.
This condition can be achieved using the optics of the homodyne
detector.
CHAPTER 3. HOMODYNE DETECTION
72
- AUDIO-BAND AC readout
performed for the photocurrent of each photodiode and for the difference photocurrent at sideband frequencies in Virgo band 10Hz10kHz around the local oscillator carrier frequency. The sum of the
two photodiodes photocurrents is also performed.
- RADIO-BAND AC readout
performed for the difference photocurrent in the band 1MHz-100MHz
around the local oscillator carrier frequency.
Despite the fact that this homodyne detector prototype is designed
for AdV and the interesting band for sequeezing is the Virgo band,
generation of squeezed light is easier at radio frequencies. Because
of this, usually squeezing experiments start by producing squeezing
at radio frequencies and then achieve the production at audio frequencies. Thus, the detection in the 1MHz-100MHz comes useful in
start-up squeezing experiments.
The sum of the two photocurrents was not included because in this
radio band, as we will explain in more detail, it resulted impossible
to balance the circuit transfer functions and noise performances of
the readout blocks of the sum and difference photocurrents. This
makes the sum of the two photocurrents useless for the characterization of squeezing and was thus suppressed at radio band together
with the readout blocks of the photocurrents of each photodiode. In
fact, only the readout of the difference photocurrent, obtained with
the self-subtraction scheme, is needed.
• simulation tools:
the circuit were designed with the help of Spice simulations, performed
with both CADENCE (see [1]) and TINA (see [2]). The noise analysis
was performed analytically with the help of MATLAB.
A
B
C
100n
100n
+15V
3
2
-15V
+
C22
C26
U25
6
OP27
5
C23
22u
C27
22u
R4
10k
VR2
R3
10k
R2
10k
VR1
R1
10k
100n
+15V
3
2
100n
C30
C28
U23
C24
U26
6
OP27
-15V
100n
-15V
2 -
3 +
+15V
7
5
4
1
8
D
BIAS
5
+
-
-
4
1
8
C25
22u
C31
22u
OP27
6
C16
100n
C2
33u
10u
0.5n
10u
C80
C81
R12
50k
U7
U6
R11
50k
C1
33u
C135
ETX500T
ETX500T
0.5n
C134
22u
C29
C17
22u
0
R47
R9
1k
4
R10
1k
4
50k
0
R48
C83
0.5n
5k
R44
0
10u
R58
R51
C82
5k
R43
10k
R5
10k
R7
0
R46
AUDIO_DIODO_2
5k
R45
AUDIO_DIODO_1
100n
C68
22u
C75
+15V
3
2
-15V
3
C69
22u
3
C74
100n
U16
6
AD8675
R25
150p
C12
10k
4
1
8
7
5
0
R28
1k
R26
R33
C93
6.8u
3
2
100n
C86
1k
4p
C96
18u
C11
-5V
+5V
C87
6.8u
R34
U18
6
C92
100n
OPA847
0
22u
C77
+15V
3
2
-15V
100n
C70
4
1
8
7
5
+
-
4
1
8
4
1
5
+
7
8
C71
22u
2
U17
6
C76
100n
AD8675
C20
100n
1k
R27
2
+15V
100
R29
40k
R32
C21
22u
VR1
1.8n
C13
VR2
22u
C79
+15V
3
2
-15V
100n
C72
3
2
100n
C32
7
5
+
-
7
5
R6
40k
C34
100n
2
3
U20
6
C33
22u
C73
22u
+15V
-
+
U27
R8
-15V
+5V
10k
22u
C35
OP27
6
C95
6.8u
3
2
-5V
2k
R30
C18
100n
100n
C88
C78
100n
AD8675
R31
C19
22u
U24
6
OP27
-15V
10k
C89
6.8u
50
1
U19
6
J3
1
C94
100n
J1
BNC
1
50
R50
BNC
R49
OPA847
50
R52
1
2
4
1
8
7
5
4
1
8
+
-
7
5
7
5
4
1
8
4
1
5
7
8
+
-
+
-
Figure 3.5: Electric schematics of the circuit prototype: page 1
2
J4
BNC
1
2
BNC
1
J2
2
A
B
C
D
CHAPTER 3. HOMODYNE DETECTION
73
Figure 3.6: Electric schematics of the circuit prototype: page 2
A
B
C
5
AUDIO_DIODO_2
AUDIO_DIODO_1
100n
C46
22u
C43
+15V
3
2
-15V
100n
C52
22u
C37
+15V
3
2
-15V
10k
10k
6
C53
22u
C36
100n
C42
100n
AD8675
6
U10
R16
150p
C6
C47
22u
AD8675
U8
R13
150p
C4
4
1
8
+
-
D
4
1
8
7
5
+
-
7
5
1k
R17
1k
R14
4
18u
C5
18u
C3
3
2
22u
C41
+15V
3
2
100n
C50
-15V
4
1
8
22u
C39
+15V
-15V
100n
C48
SOMMA AUDIO
4
U11
1k
C49
22u
6
6
C40
100n
AD8675
C51
22u
C38
100n
AD8675
R18
U9
1k
R15
3
3
1k
R40
1k
R39
100n
C102
22u
C105
+15V
3
2
-15V
7
5
7
5
4
1
8
+
7
5
1k
2
C103
22u
U28
6
C104
100n
AD8675
R37
2
4
1
8
+
-
+
-
5
50
R54
J6
BNC
1
1
1
A
B
C
D
CHAPTER 3. HOMODYNE DETECTION
74
2
A
B
C
D
-15V
C123
22u
C119
22u
5
U35
IN
IN
4
1
3
7
2
+15V
2
C126
100n
C118
100n
U34
100u
100u
1
-19V
C113
C111
C122
100n
+15V
X3
X2
X1
OUT
OUT
22u
C121
22u
AD586
GND
TP
TP
TP
+VIN
U36
TRIM
VOUT
NOISE
5
6
8
LM7905C/TO220
3
3
LM7805C/TO220
C112
100n
C110
100n
OUT
OUT
3
-5V
5k
R56
4
+5V
C124
100n
C120
100n
U31 LM7915C/TO220
IN
IN
3
U33 LM7815C/TO220
C127
4.7u
C125
2
1
GND
2
1
GND
+19V
GND
2
1
GND
4
C117
C115
100u
100u
3
3
2
100n
C136
-15V
C116
100n
C114
100n
+15V
C139
22u
C137
22u
+15V
6
C141
C140
OP27
5k
U32
-15V
R59
33u
33u
ALIMENTAZIONE
3
C138
100n
2
2
3
2
100n
C128
C131
U30
-15V
5k
C129
22u
+15V
6
C133
C132
OP27
33u
33u
R57
22u
7
5
4
1
8
+
-
Figure 3.7: Electric schematics of the circuit prototype: page 3
7
5
4
1
8
+
-
5
C130
100n
1
1
BIAS
A
B
C
D
CHAPTER 3. HOMODYNE DETECTION
75
CHAPTER 3. HOMODYNE DETECTION
76
Figure 3.8: a: electric symbol of a op-amp. b: equivalent circuit of a op-amp.
A op-amp takes the difference of the two voltages at its ports, called the noninverting (+) port and the inverting (−) port, and amplifies it by the open-loop
gain Av . Zi,o are the input and output impedance of theop-amp. For an ideal
op-amp Av would be infinite and frequency-independent, Zi would be infinite
and Zo = 0.
After this overview of the leading design criteria, we describe in detail the circuit
prototype, whose electric schematics is reported in Fig. 3.5 3.6 and 3.7.
We start with a brief introduction to the operational amplifiers (op-amp) and
to some filters and amplifiers, which can be built with them and were massively
employed in the circuit design.
3.2.1
Operational amplifiers
A complete description of both the ideal and real behavior of a op-amps can
be found in many electronics textbooks (see [16] and [19]), together with the
simplest examples of the their use as amplifiers or filters.
The electronic symbol and the equivalent circuit of a op-amp are shown in Fig.
3.8. Following [19], a op-amp takes the difference of the two voltages at its ports,
called the non-inverting (+) port and the inverting (−) port, and amplifies it
by the open-loop gain Av . For an ideal op-amp we would have the open-loop
gain Av and the input and output impedance Zi,o
Av = −∞
(3.51)
Zi = +∞
(3.52)
Zo = 0
(3.53)
For real op-amps Av is frequency-dependent and limited. In particular Av diminishes for increasing frequencies and its maximum value, often reached at
DC, usually ranges in ∼ 104 − 106 . The typical frequency behavior of the openloop gain of a op-amp is shown in Fig. 3.9. Regarding the input impedance,
it usually ranges from order of M Ω to order of 1012 Ω, the output impedance is
often around 10Ω.
CHAPTER 3. HOMODYNE DETECTION
77
Figure 3.9: Typical example of frequency dependence of the open-loop gain of
a op-amp.
In almost every application, op-amps are used in closed-loop, i.e the inverting port is connected to the output of the amplifier through a feedback
impedance. To understand how it works we start with the simplest example:
the inverting amplifier, shown with its equivalent circuit in Fig. 3.10.
Figure 3.10: a: Inverting amplifier. A feedback impedance connects the inverting input to the output of the amplifier. b: equivalent circuit of the inverting
amplifier. R1 and R2 can be derived from R0 using the Miller’s theorem.
Referring to this equivalent circuit, by applying the Miller’s theorem and
CHAPTER 3. HOMODYNE DETECTION
78
always indicating the open-loop gain as Av , we can substitute R0 with
R0
1 − Av
R0
R2 =
1 − A1v
R1 =
(3.54)
(3.55)
For large values of Av , R1 becomes small and tends to zero in the limit of infinite
open-loop gain while R2 tends to R0 . If we choose R0 to be much larger than Ro
and much smaller than Ri we have that Ri can be neglected with respect to R1
at the input, while Ro can be neglected with respect to R2 at the output. Thus
we have, calling R1 as RM (to remind that it derives from the Miller’s theorem)
v o = Av v i
RM
vs
vi =
RM + R0
(3.56)
(3.57)
and, for the closed-loop amplification
Av,cl ≡
vo
RM
= Av
vs
RM + R0
(3.58)
Thus, from the signal vs point of view, the op-amp with feedback impedance R0
can be seen as a device with input impedance R + RM , which amplifies by Av
the difference of the voltages at its two input ports, as shown in Fig. 3.11.
Figure 3.11: From the signal vs point of view, the op-amp with feedback
impedance R0 can be seen as a device with input impedance RM (it comes
out from the Miller’s theorem), which amplifies by Av the difference of the
voltages at its two input ports.
In the ideal case in which Av −→ −∞ we have that
RM −→ 0
(3.59)
Av RM −→ R0
(3.60)
0
Av,cl −→ −
vi −→ 0
R
R
(3.61)
(3.62)
CHAPTER 3. HOMODYNE DETECTION
79
i.e, the signal vs , which would “see”an input impedance R + RM , in this case
“sees”an input impedance R. Extremely important is the fact that vi , i.e the
voltage of the inverting input goes to zero, i.e at the same voltage of the non
inverting input, even though the two inputs are connected by a high impedance
Ri . This is due to the feedback resistance R0 and to the Miller effect, which
forces the inverting input to go to the same voltage of the non inverting input.
It is possible to recalculate the amplification of the inverting amplifier by assuming this property a priori, i.e we say that
• the voltage at the inverting (-) port is the same as that at the non inverting
(+) port
• given that the impedance between the two ports is huge, practically no
current enters the op-amp and the current circulating in R is the same
circulating in R0
In the particular case of the inverting amplifier of Fig. 3.10 we obtain
0 − v0
vs − 0
=
R
R0
R0
Av,cl = −
R
(3.63)
(3.64)
Obviously this approximation is valid in the extent to which Av can be considR0
ered practically infinite, i.e in the extent to which RM = 1−A
can neglected
v
with respect to R or equivalently, in the extent to which the two ports can be
considered to be at the same potential. In fact, as shown in Fig. 3.9, Av diminishes for increasing frequencies, i.e the op-amp becomes “less and less ideal”for
increasing frequencies and this can bring to unexpected behaviors. When the
op-amp cannot be considered ideal, calculations should be performed using the
Miller’ theorem and taking into account the real Av , which usually is plotted it
the op-amp data sheet.
Useful amplifiers
In this section we show four examples of filters/amplifiers, which are employed
in the circuit design: inverting low-pass filter, inverting high-pass filter,
transimpedance amplifier and inverting sum amplifier. We calculate
their gain by using the ideal model of op-amp.
Inverting low-pass and high-pass filter
The electric schemes of an inverting low-pass filter and an inverting high-pass
filter are shown in Fig. 3.12.
CHAPTER 3. HOMODYNE DETECTION
80
Figure 3.12: Left: inverting low-pass filter. Middle: inverting high-pass filter.
Right: general inverting filter.
They are exactly analogous to the inverting amplifier shown in Fig. 3.10 and
their closed-loop gain can be simply taken from Eq. 3.64
Av,cl = −
Z0
Z
(3.65)
we obtain
• low-pass filter
Av,cl = −
R0
1
R 1 + iωR0 C
(3.66)
iωR0 C
1 + iωRC
(3.67)
• high-pass filter
Av,cl = −
the typical shape of |Av,cl | for both filters is shown in Fig. 3.13.
CHAPTER 3. HOMODYNE DETECTION
81
Figure 3.13: Typical |Av,cl | for a low-pass and a high-pass filter.
Transimpedance amplifier
Figure 3.14: Transimpedance amplifier: a current Is is converted in an output
voltage by the transimpedance Z.
The scheme of a transimpedance amplifier is shown in Fig. 3.14: the input
current Is can only pass through Z, thus
vo = −ZIs
(3.68)
and the current is converted in a voltage signal at the output of the op-amp.
Inverting sum amplifier.
CHAPTER 3. HOMODYNE DETECTION
82
Figure 3.15: Invertin sum amplifier: the input voltages v1 , ..., vn are summed at
the output with weights −R0 /R1 , ..., −R0 /Rn respectively.
The scheme of an inverting sum amplifier is shown in Fig. 3.15:
v1
v2
vn
i=
+
+ ... +
R1
R2
Rn
vo = −R0 i
(3.69)
(3.70)
thus the input voltages v1 , ..., vn are summed at the output with weights −R0 /R1 , ..., −R0 /Rn
respectively.
If R1 = R2 = ... = Rn = R we simply obtain
vo = −
R0
(v1 + ... + vn )
R
(3.71)
i.e the sum of the input voltages amplified by R0 /R.
3.2.2
The photodiodes
We bought two different models, the EPITAXX ETX500T and the HAMAMATSU G7096. Their most useful characteristics in terms of the circuit
design are reported in Tab. 3.1.
DARK CURRENT
TIME RISE/FALL
AREA
MAX POWER
SHUNT RESISTANCE
JUNCTION CAPACITANCE
BULK RESISTANCE
EPITAXX ETX550T
12nA
2.5ns
500µm2
10 − 11mW
250M Ω
15pF
25Ω
HAMAMATSU G7096
5µA
60ps
200µm2
2mW
Table 3.1: Characteristics of the photodiode models EPITAXX ETX500T and
HAMAMATSU G7096 necessary for the circuit design.
As shown in the table, the EPITAXX photodiodes have a much lower dark
current (12nA vs 5µA) with respect to the HAMAMATSU and can receive an
CHAPTER 3. HOMODYNE DETECTION
83
higher maximum optical power (10-11mW vs 2mW). Those characteristics make
the EPITAXX photodiodes much more convenient than the HAMAMATSU in
the design of a homodyne detector because, as we have partially outlined in the
previous section, it should be a low noise circuit able to measure fluctuations
below the shot noise level of the local oscillator. For this reason a low dark
current is desirable. In addition, as we have seen, the local oscillator should be
as intense as possible in order to better enhance the detection of the squeezed
field (and, of course, the detection of the local oscillator shot noise itself). Thus,
photodiodes able to receive much intense light beams come useful.
Moreover, the EPITAXX photodiodes have a larger sensitive area than the
HAMAMATSU (500µm2 vs 200µm2 ), a characteristic which is obviously advantageous for a practical implementation of the detection on a optical table.
On the other hand, the larger is the area, the slower is the response of the photodiode and the narrower is the available detection bandwidth. In particular
the EPITAXX photodiodes have a much higher rise/fall time than the HAMAMATSU (2.5ns vs 60ps) and the detection band that they can cover is at most of
∼ 150Mz, while the HAMAMATSU can cover several hundreds of MHz. However, a ∼ 100MHz is more than enough for our purposes.
Given all these considerations, we decided to design the homodyne circuit using
the EPITAXX photodiodes.
Equivalent circuit of the EPITAXX ETX500T photodiode
In general, photodiodes can be schematized with an equivalent circuit as that
shown in Fig. 3.16,
Figure 3.16: Equivalent circuit of a photodiode.
i.e as a current generator (which “generates”the photocurrent) paralleled
with a diode with the appropriate characteristics for the photodiode model (such
as the dark current) and with a resistor and a capacitor, which schematize the
shunt resistance and the junction capacitance of the photodiode. Finally the
resistor, which schematize the bulk resistance of the photodiode is set in series
to all these other components.
This equivalent circuit was also used to schematize the photodiodes in the circuit
design and the values of its parameters are reported for the EPITAXX ETX500T
in Tab. 3.1.
3.2.3
The bias circuit of the photodiodes
The bias circuit of the photodiodes is realized starting from a high precision 5V
DC supply. This supply is obtained by filtering, through a double stage low pass
filter (cut-off frequency at 0.5Hz), the 5V supply exiting from the high-precision
CHAPTER 3. HOMODYNE DETECTION
84
Figure 3.17: Bias circuit of the two phodiodes. -VBIAS is obtained from the
VBIAS reference (5V). Note the two points, VR1 and VR2, which are respectively at voltage VBIAS and -VBIAS : they are important in the DC-readout
circuit.
5V reference device AD586 (see Fig. 3.7). The -5V supply is obtained from the
5V reference through the circuit shown in Fig. 3.17: using the properties of
ideal op-amps, we have that the current i is given by
VA − VC
VC − VB
=
2R
2R
VB
VBIAS
=−
=
2R
2R
i=
(3.72)
Thus we easily obtain that
VB = −VBIAS
VBIAS
V R1 =
2
VBIAS
V R2 = −
2
(3.73)
(3.74)
(3.75)
In this way, the photodiodes are kept in reverse bias by the two supplies at
±VBIAS (±5V ), as shown in Fig. 3.17. The advantage of that arrangement is
that the −5V supply comes from the 5V supply, thus every fluctuation of the
5V supply reflects in a equal fluctuation of the −5V supply. This greatly help
from an electronic balance point of view, a characteristic of the circuit which,
as we have seen, is extremely important when detecting squeezed light.
3.2.4
DC, AUDIO and RADIO readout blocks.
The locations of the DC, AUDIO and RADIO readout blocks are shown in Fig.
3.4 and their circuit schemes can be found in Fig. 3.5 and Fig. 3.6. All three
CHAPTER 3. HOMODYNE DETECTION
85
kind of blocks, no matter if they are deputed to the readout of a single photodiode photocurrent or of the difference photocurrent of the two photodiodes,
work as transimpedance amplifiers, analogously to that shown in Fig. 3.14,
which have feedback impedances of the same type as that of the low-pass filter
shown in Fig. 3.12. In addition, they also contain a second stage which, in the
case of the DC readout, eliminate from the output voltage the offset deriving
from the bias voltages of the photodiodes, in such a way that the output voltage
is only proportional to the photocurrent. In case of AUDIO and RADIO readout, the second stage is a high-pass filter, as that shown in Fig. 3.12, which,
together with the low-pass filter of the first transimpedance stage, shapes the
transfer function of the readout block.
Given that, as shown in Fig. 3.4, different blocks readout the same photocurrent
at different frequency bands (for example DC and AUDIO for the photocurrents
of each single photodiode and AUDIO and RADIO for the difference photocurrent) it is first of all extremely important to design the input impedances of
the various blocks in such a way that each block do not pick up signals at frequencies which have to be readout by an other block. For example, a signal at
AUDIO band (10Hz-10kHz) should not enter, as much as possible, a RADIO or
DC block.
The adopted solution is to design the transimpedance amplifiers as the general
inverting amplifier shown in Fig. 3.12: if we substitute the input voltage with
an input current we easily obtain that the output voltage only depends on the
feedback impedance, as calculated in Eq. 3.68.
The input impedance of the block, analogously to the inverting amplifier, is
given by Z + ZM , where ZM is due to the Miller’s theorem and is the same
which appear in Eq. 3.58. In the limit in which the op-amp can be considered
ideal (this is generally true for many op-amps at DC and AUDIO band, it may
not be true at RADIO band) and/or if Z ZM , the input impedance of the
amplifier is essentially given by Z.
If the same source current is connected to more than one of such transimpedance
amplifiers, the input impedances of the various blocks divide a source current
between the various amplifiers depending on their relative magnitude at the
frequency which is being considered.
CHAPTER 3. HOMODYNE DETECTION
86
Figure 3.18: Input impedances, in case of ideal op-amps, of the DC, AUDIO
and RADIO blocks.
The impedances, which were chosen in the circuit design, are shown in Fig.
3.18, while in Fig. 3.19 a plot of their magnitude vs frequency is reported.
CHAPTER 3. HOMODYNE DETECTION
87
Figure 3.19: Plot of the magnitude of the input impedances of the DC, AUDIO
and RADIO blocks, both considering ideal op-amps or taking into account the
Miller effect. Also the impedance which a photodiode sees toward the other
photodiode is plotted.
In this figure the impedance, which a photodiode sees toward the other
photodiode, is also shown. It is basically due to the shunt resistance of the photodiode and to its junction capacitance (see Fig. 3.16). In addition, the input
impedances of the DC, AUDIO and RADIO blocks, including the additional
impedance given by the Miller effect, are shown.
Remind, referring to Fig. 3.18 and Fig. 3.11, that
ZM =
Z0
1 − Av
(3.76)
where Z 0 is the feedback impedance of the transimpedance amplifier and Av is
the open-loop gain of the OP-AMP.
The choice of the feedback impedances and the characteristics of the op-amps
used for the various blocks will be discussed when talking about the design
of each block, here only the basic informations are reported which allow to
calculate the Miller part of the input impedances:
for all the three op-amps used, OP27 (DC), AD8675 (AUDIO) and OPA847
(RADIO) the open-loop gain is approximately of the type shown in Fig. 3.9
and it can be described by a function of the form
Av =
A0
1 + i ννco
(3.77)
CHAPTER 3. HOMODYNE DETECTION
88
while the feedback impedance is that of the low-pass filter shown in Fig. 3.12.
Indicating by RF and CF the feedback resistance and capacitance respectively,
we have
• DC
RF = 1kΩ
• AUDIO
RF = 10kΩ
• RADIO
RF = 1kΩ
A0 = 1.8 × 106
CF = 33µF
A0 = 106
CF = 150pF
CF = 4pF
A0 = 8 × 104
νco = 40Hz
νco = 10Hz
νco = 5 × 104 Hz
Let us note that, as shown in Fig. 3.19, in the three interesting frequency bands,
DC, 10Hz-10kHz for AUDIO and 1MHz-100MHz for RADIO, the impedance of
the corresponding detection block is well below the impedances of the other
detection blocks and of the photodiode, thus, a signal at a certain frequency, in
one of the three detection bands, will basically enter only the detection block
appropriate for its detection frequency. In addition, for a signal coming from
a photodiode the other photodiode can be considered practically an
open circuit if compared with at least one of the three readout blocks and
thus the two photodiodes can be considered independent in the frequency band
we are considering. This is also true because, even if, as shown in Fig. 3.1,
not all detection blocks detect the same photocurrent (we have only DC and
AUDIO for the photocurrent of each photodiode and only AUDIO and RADIO
for the difference photocurrent), the missing block is substituted with its input
impedance set to ground, as can be seen in Fig. 3.5. This is done in order to
balance, as much as possible, the detection circuitry of each photodiode and
that of the difference photocurrent.
Finally Fig. 3.19 shows that at up to ∼ 100MHz, given the characteristics of
the employed op-amps and of the transimpedance blocks, the effect of the Miller
impedances, due to the non ideal behavior of the op-amps, is negligible. Thus
for our purposes the op-amps can be considered ideal.
DC readout
The scheme of the DC readout block of the top photodiode of Fig. 3.17 is shown
in Fig. 3.20: the DC part of the photocurrent coming from the photodiode only
circulate in the 50kΩ resistor (which roughly is the input impedance of the DC
block) and in the transimpedance indicated with Z. The point A is at voltage
VBIAS , while the point indicated with V R1 is connected with the point V R1
shown in Fig. 3.17 and is at voltage VBIAS /2. The same description applies to
the bottom photodiode of Fig. 3.17 but in this case the sign of all voltages is
reversed.
We have
VB = VA + IP D Z
(3.78)
VA = VBIAS
(3.79)
i=
VBIAS /2 − Vo
VB − VBIAS /2
=
R
R
(3.80)
CHAPTER 3. HOMODYNE DETECTION
89
Figure 3.20: DC readot block.
thus
Vo = −IP D Z
(3.81)
The op-amp with feedback impedance Z acts as a transimpedance amplifier,
while the second stage eliminates from the output voltage the offset, due to the
bias voltage, which can be found in the expression for VB .
The op-amp model used for the DC blocks is OP27, a low noise op-amp designed for DC and radio applications. Considering that the maximum current
allowed by the photodiodes is ∼ 8mA (the maximum power is ∼ 10mW and
the responsivity is 0.8), that the transimpedance of the DC block is 1kΩ and
the expression for the output voltage of the transimpedance stage Eq. 3.78,
OP27 is a good choice for the design of the DC block. In fact it allows for
a ∼ 13.5V output voltage swing and for power dissipation higher than 50mW
and thus, it can cover the maximum output voltage expected at the output of
the transimpedance stage (∼13V) and the power dissipation expected from the
transimpedance stage (6 − 7mW).
In Fig. 3.20 the elements composing Z are shown: a 1kΩ resistor in parallel
to a 33µF capacitor, so that the first stage acts as a low-pass transimpedance
amplifier with cut-off frequency at ∼ 5Hz and transimpedance gain 1kΩ.
AUDIO readout (10Hz-10kHz)
The AUDIO blocks for the readout of both the photocurrents of each photodiode and of the difference photocurrent, are composed of a first low-pass
transimpedance stage and a second high-pass stage, as shown in Fig. 3.21.
CHAPTER 3. HOMODYNE DETECTION
90
Figure 3.21: AUDIO readot blocks: first and second stage.
All AUDIO stages uses √
the op-amp model AD8675 because its low √
input
voltage noise (vn = 2.8nV / Hz) and input current noise (in = 2.8nV / Hz)
and its high open-loop gain (106 ), allow accurate high-gain amplification of lowlevel signals. In addition, its gain-bandwidth product of 10 MHz and its 2.5
V/µs slew rate provide good dynamic accuracy in audio-band applications.
The first stage has transimpedance gain 10kΩ and low-pass cut-off frequency
∼ 11kHz. The second stage has unity gain and high-pass cut-off frequency
∼ 9Hz.
The signals exiting the second stage of AUDIO-readout blocks of the two photodiode photocurrents, are then summed with a unity gain circuit as that shown
in Fig. 3.22.
CHAPTER 3. HOMODYNE DETECTION
91
Figure 3.22: AUDIO readot blocks. Top: sum stage of the single-photodiodes
readout blocks. Bottom: third stage of the difference photocurrent readout
block.
Then, a third stage is added to the readout block of the difference photocurrent. This stage, shown in Fig. 3.22, is only a unity gain stage. It is needed
to make the output noise of the difference photocurrent readout block as much
as possible equal to the output noise of the sum photocurrent readout block.
In fact, we want that, if the local oscillator is a coherent field and if only unsqueezed vacuum enters the beam splitter, the spectra of the sum and difference
photocurrents are equal. If the sum and difference blocks have a different noise,
the presence of a squeezed field could be simulated.
RADIO readout (1MHz-100 MHz)
The design of the RADIO block requires more care than that of the other blocks.
This is due to several reasons:
• the photodiodes have a junction capacitance of ∼ 15pF and, in the considered band, it must be taken into account when designing the transimpedance amplifier, because the junction capacitance can severely affect
the frequency response of the amplifier
• op-amps, which can cover the required frequencies band (1MHz-100MHz)
are not stable for unity voltage gain. This means that the voltage amplifiers should have a gain higher than a given value (usually & 10 but
it depends on the op-amp), and this is a constraint on the ratio of the
feedback resistance to the input resistance. However, given that the opamp has its own input capacitance and that the parasitic capacitances
are unavoidable in the other circuit elements, such as resistors and wires,
a trade-off should be found in the magnitude of those resistances. Their
CHAPTER 3. HOMODYNE DETECTION
92
Figure 3.23: RADIO readot blocks: first and second stage.
values have to be not too high, otherwise, together with the parasitic capacitances, they would act as a low pass filter. Usually resistors have a
parasitic capacitance of order 0.5 − 1pF , i.e if, for example, we take a 2kΩ
feedback resistor for our amplifier, its parasitic capacitance create an unwanted low-pass filter with cut-off frequency of order ∼ 150M Hz. On the
other hand, the resistance values should not be too low because op-amps
can only drive currents up to a maximum. The lower-limit is obtained by
imposing that supply voltage of the op-amp divided by the resistance is
lower than the maximum current, which the op-amp can drive.
• unlike for the AUDIO readout, it results practically impossible to design
sum and difference readout RADIO blocks, which have the same transfer
characteristics and the same noise performance. In fact, the addition of an
extra stage can change the transfer function of the block. For that reason
we choose to install just the RADIO block for the difference photocurrent,
which is sufficient for the characterization of the squeezing level.
The difference photocurrent readout RADIO block is composed by two stage,
the first one is a low-pass transimpedance amplifier and the second one a highpass filter, as shown in Fig. 3.23. The op-amp model used for those stages is
OPA847 because it resulted the best op-amp, which we were able to find, with
respect the covered frequency band and the noise performance, at radio band.
OPA847
- is stable for voltage gain G > 12
- has a large gain-bandwidth product GBP = 3.9GHz
- has a total input capacitance of COP A = 3.7pF
The datasheet of OPA847 also includes useful advices for the use of the opamp in few applications. In particular, it provides a way to design a low-pass
transimpedance amplifier for photodiodes, which takes into account the junction
capacitance of the photodiode and the total input capacitance of OPA847 and
guarantees a flat response up to a cut-off frequency, which can be determined
once the feedback impedance has been chosen. If RF and CF are the feedback
resistance and capacitance of the amplifier, CJ is the junction capacitance of
the photodiode, COP A is the total input capacitance of the op-amp and GBP
CHAPTER 3. HOMODYNE DETECTION
93
Figure 3.24: RADIO: transimpedance amplifier transfer function.
is gain-bandwidth product, a flat response up to a cut-off frequency
s
GBP
νco =
2πRF (CJ + COP A )
(3.82)
is achieved when the relation
1
=
2πRF CF
s
GBP
4πRF (CJ + COP A )
(3.83)
is satisfied. Plugging the values of the various parameters inside those formulas, we obtain, for a transimpedance gain of ∼ 1kΩ, a cut-off frequency of
∼ 180M Hz. However, in practice this value is expected to be lower when we
take into account the contribution of the other photodiode.
The transimpedance transfer function was simulated using TINA, including
both photodiodes: the result is shown in Fig. 3.24. The second stage was
designed as a high-pass filter with gain 20 and cut-off frequency ∼ 1MHz and it
is shown in Fig. 3.23.
3.3
The noise study
In this section we report on the noise study and the performance of the homodyne detector prototype designed in this thesis. First all we briefly introduce
the basic concepts of the theory of electronic noise applied in the study of the
noise characteristics of the circuit. A thorough treating of electronic noise can
be found in several books, see [18] and [21].
CHAPTER 3. HOMODYNE DETECTION
3.3.1
94
Electronic noise
The noise can be defined as each unwanted disturbance, which overlaps with
a useful signal and degrades its information content. This definition is quite
general and includes zero-average random fluctuations, that result from
the physics of electronic devices and materials of an electronic system, as well
as disturbances due to external sources (for example cross-talks with a second
nearby electronic system) or to signal processing noises (for example the quantization noise).
In our case only the first type of noise sources matter, i.e random fluctuations intrinsic to the electronic system, because we want to determine the noise
performance of the homodyne detector prototype. Noise sources of this kind,
which are most interesting for our purposes and which are going to be described
is some detail, are thermal noise, shot noise and 1/f noise. The main
characteristics of those kind of noise is precisely the randomness, which makes
impossible to predict the magnitude of each fluctuation and which makes a statistical description of the noise itself necessary. In fact, this kind of noise sources
are in effect stochastic processes (see [12]): if x is a generic random variable,
which describes a fluctuating parameter of the electronic system, probability
distributions are associated to the ensemble of each possible configuration of
x(t), i.e to the ensemble of each possible time evolution of the random variable.
Treating such a general problem is usually difficult, but in many cases of practical interest in electronics, we can focus on stationary and ergodic stochastic
processes, i.e processes for which the statistical properties do not change with
time and can be deduced by observing just one possible time evolution of x.
Let us recall few basic operative definitions related to the stochastic processes,
useful in treating the noise, i.e the average and the autocorrelation. In the
case of the noise, the average vanishes. On the other hand, the autocorrelation
function of a stationary and ergodic stochastic processes is defined as
Z
1 +T
x(t + τ )x(t)dt
(3.84)
Rxx (τ ) = lim
T −→∞ T 0
Rxx (0) is nothing but the variance in case of zero-average fluctuations.
An equivalent description can be obtained in the frequency domain by the
power spectrum, which is defined as the Fourier transform of the autocorrelation function
Z
+∞
Rxx (τ )e−iωτ dτ
Sxx (ω) =
(3.85)
0
and is linked to the variance by
Rxx (0) =
1
2π
Z
+∞
Sxx (ω)dω
(3.86)
0
The characterization of a stochastic process through its power spectrum is extremely important in electronics because it allows to treat in a simple way the
effect of filters and amplifiers on noise. As a matter of fact, it can be shown
that if a stochastic signal with power spectrum Sxx (ω) is filtered by a linear and
stationary filter with transfer function H(iω), the power spectrum of the signal
exiting the filter is given by
Syy (ω) = Sxx (ω)|H(iω)|2
(3.87)
CHAPTER 3. HOMODYNE DETECTION
95
Thermal noise
Thermal noise is present in every physical system at temperature different from
the absolute zero and is due to thermal agitation. It is characterized by a
Gaussian statistics and, in the limit in which the quantization of energy levels
can be neglected, i.e for the condition, satisfied in almost all cases of interest in
electronics, that the temperature is not too low and the frequency not to high
(}ω kT ), its power spectrum can be considered white, i.e flat.
Thermally induced random motion of charge carriers in a conductor originates
the electronic thermal noise, also called Johnson noise: if we measure the
voltage vn (t) at the terminals of an open-circuit resistor, we find out that it
fluctuates with time around zero. Similarly, short-circuiting the terminals of
the same resistor and measuring the circulating current in (t) = vn (t)/R, we
find out that it fluctuates with time around zero.
It can be shown that, in the limit in which the power spectrum of thermal
noise can be considered white, the power spectra of the fluctuating voltage and
current only depends on the resistance and on the temperature of the resistor
through the Johnson noise formula
Svv (ω) = 4kT R
4kT
Sii (ω) =
R
(3.88)
(3.89)
Those formulae can be generalized to the thermal noise of a general twoterminal passive and linear electronic element. In fact, the fluctuationdissipation theorem tells us that, if Z(iω) is the impedance measured at
the terminals of the two-terminal device, the thermal noise voltage and current
power spectra become
Svv (ω) = 4kT <[Z(iω)]
4kT
Sii (ω) =
<[Z(iω)]
(3.90)
(3.91)
These expressions tell us that a two-terminal devices, such as ideal capacitors
and inductors, which have a purely imaginary impedance, would not be affected
by thermal noise because they would not suffer dissipation. However, every real
device incurs in dissipative processes and thus exhibit thermal noise.
In the case of capacitors, the entity of the real part of the impedance (which
is responsible for dissipation) can be inferred from the so called dissipation
factor tan δ, which can be found in the capacitor datasheet and gives
<[Z(iω)] = tan δ|=[Z(iω)]|
where =[Z(iω)] =
1
iωC
(3.92)
is the well known impedance of an ideal capacitor.
Shot noise
We have already dealt with shot noise in Sec. 3.1.1 when we talked about
photodetection. We saw that, if a coherent light beam hits the photodetector,
the resulting photocurrent fluctuates around a steady state value (which corresponds to the intensity of the light beam) following a Poisson statistics. We
CHAPTER 3. HOMODYNE DETECTION
96
then calculated the expected power spectrum of the current fluctuations and
obtained
(3.93)
Sii = 2ie
This result was obtained because light has a discrete nature, being composed
by photons, and because for a coherent light beam, photons impinge on the
detector in a completely random way, i.e every hitting event is independent
from an other and the photon arrival follows a Poisson statistics.
Shot noise in electronics has quite the same origin. In fact, if we consider
an electric current, i.e, analogously to light, a flux of discrete particles,
crossing a potential barrier in a completely random way, i.e each crossing
event is independent of an other, we obtain for the current fluctuations
power spectrum the same result of Eq. 3.19, i.e, if qe is the charge of the charge
carriers
(3.94)
Sii = 2iqe
An example of electronic shot noise is that due to a photodiode dark current.
The dark current originates from the flux of minority charge carriers in the pn junction of the photodiode, which cross the potential barrier formed at the
junction. This is a noise source that have to be taken into account in low-noise
design of photodetectors.
1/f noise
1/f noise has been observed in many different fields, from physics to electronics
and biology, but up to now no unique interpretation was found for it.
It shows a power spectrum of type
S(f ) ∝
1
fα
(3.95)
where α is a parameter near to unity.
In electronics it seems to be linked to effects of generation-recombination and/or
capture and release of charge carriers from traps and impurities in the material,
which constitutes the electronic element as well as fluctuations in the number
of charge carriers.
Wherever it comes from, 1/f noise is practically ubiquitous and represents the
limiting factor to the sensitivity of measurements at low frequencies.
Noise representation and calculation in electronics
Let us consider a noisy two-terminal electronic element. The voltage v(t) at its
terminal, in linearity regime, can be divided between a part v 0 (t), which is the
voltage that we would have measured in absence of noise and a noise part vn (t)
v(t) = v 0 (t) + vn (t)
(3.96)
Thus, using the Thevenin theorem, the noisy two-terminal component can be
represented by a noise voltage generator in series with the noiseless
component, or, equivalently, using the Northon theorem, with a noise current generator paralleled with the noiseless component, as shown in
Fig. 3.25. Both voltage and current generators are described by their power
spectrum.
CHAPTER 3. HOMODYNE DETECTION
97
Figure 3.25: Left: noisy two-terminal electronic element. Right: noise twoport device.
In case of a noisy two-port component, as that shown in Fig. 3.25, it is necessary
to specify two noise generators. A way of doing it can be to put a noise voltage
(current) generator in series (in parallel) to each port and then specify both
power spectra (real magnitudes) and the cross-spectrum (complex magnitude)
of the two noise generators. Thus, four informations are needed in order to
characterize the noise of a two-port component.
However, it can be shown that an equivalent, and easiest to deal with, representation can be obtained by placing a voltage and a current generator both at the
input port, as shown in Fig. 3.25. This representation makes the noise treating
independent on the transfer function of the two-port element and greatly simplifies noise calculations in circuits.
A crucial matter when designing low-noise electronics is how to determine the
noise contribution of a noise source placed somewhere in the circuit to the noise
at the output or at any other point of the circuit. Let us assume that in the
circuit n different noise sources are present, each one characterized by its spectrum Sk (ω). Let us now suppose that we want to compute the voltage noise
power spectrum at the output of the circuit 1 and that Hk (iω) is the transfer
function between the k-th noise source and the output. If the noise sources
are independent, i.e. no correlation exists between them, it can be shown
that the noise power spectrum at the output is given by
out
Svv
(ω) =
n
X
Sk (ω)|Hk (iω)|2
(3.97)
k1
Then, introducing the transfer function between the input signal and the output
Hsig , we can input-refer the output noise. In other words, we assume that the
output noise is due to a fictitious noise source set at the signal input and its
power spectrum is
out
Svv
in
Svv
(ω) =
(3.98)
|Hsig (iω)|2
In fact, the input-referred noise and the signal reflect on the output through the
same transfer function and thus can be easily compared.
1 A similar approach is followed to compute the current and voltage noises in any point of
the circuit
CHAPTER 3. HOMODYNE DETECTION
98
In the linear regime, there is a practical “algorithm”to deal with those noise
calculations: to obtain the transfer function Hk (iω) of a particular noise source,
we have to short-circuit every other voltage noise (or signal) source
and open-circuit every other current noise (or signal) source. Then, we
have just to treat the noise generator in question as a normal voltage or current
source and calculate its effect on the output. Once the transfer function of each
noise source has been determined and if the noise sources are independent, the
output noise can be calculated with Eq. 3.97 and, once known the transfer
function of the signal, it can be input-referred as shown in Eq. 3.98.
The procedure can be better understood with a practical example, which applies to a generic feedback amplifier and which is introductory to the noise
calculations performed for the homodyne detector prototype.
Noise in a feedback amplifier
Figure 3.26: Noise sources in a feedback amplifier
Referring to Fig. 3.26, we calculate the contribution of the noise sources in ,
vn and iF to the noise power spectrum of the output voltage vo . The first two
noise sources are those of the amplifier, which is nothing but a two-port device
as that described in the previous section, while iF is the thermal noise of the
feedback resistor. We apply the procedure described in the previous section and
we determine the transfer function of each noise source to the output:
• vn )

 v − v n = vt = − v o
A
v = v
=⇒
vo =
o
A
vn
1+A
thus
Hvn =
A
1+A
(3.99)
• in )

 v = vt = − vo
A
v − v = i R
o
n F
=⇒
vo =
ARF
in
1+A
CHAPTER 3. HOMODYNE DETECTION
99
thus
Hin =
ARF
1+A
(3.100)
• iF )

 v = vt = − vo
A
v − v = i R
o
F F
=⇒
vo =
ARF
iF
1+A
thus
HiF =
ARF
1+A
(3.101)
• vsig )
the transfer function of an input voltage signal would simply be Hsig = −A
Indicating with i2n , vn2 and i2F the power spectra of the three noise sources, the
power spectrum of the total output noise becomes
2
2
out
2 A 2
2 ARF Svv = vn + (in + iF ) (3.102)
1+A
1 + A
out
by |Hsig |2
which can be input-referred by dividing Svv
in
Svv
3.3.2
=
vn2
2
1 2
+ (i2n + i2F ) RF 1 + A
1 + A
(3.103)
Noise calculations for the homodyne detector prototype
In sec. 3.2 we have seen that we require the homodyne detector to be able
to detect noise levels at least ten times below the shot noise level of the localoscillator. As shown in Eq. 3.19, the shot noise level depends on the local
oscillator power and the maximum shot noise level reachable depends on the
maximum power that the photodiode can bare. For EPITAXX ETX500T it is
10mW.
When designing the homodyne detector, we assumed that the signals are the
shot noise levels of the two photodiodes photocurrents (which we assume to be
equal), each characterized by its power spectrum (see Eq. 3.19) and each affecting independently the readout blocks of the photocurrent of each photodiode
and of the difference photocurrent.
The photodiode intrinsic noise is due to the thermal noise of the shunt resistance and of the bulk resistance (refer to Fig. 3.27) and to the shot noise of
the dark current.
The detection blocks carry also their own noises and these noises, together
with those of the photodiodes, limit the sensitivity of the measurement of the
shot noise level of the photocurrents and thus of squeezing.
Obviously this discussion does not apply to the DC readout of the photocurrents
because in this case we are not dealing with noise measurements but with the
CHAPTER 3. HOMODYNE DETECTION
100
Figure 3.27: Noise sources of the photodiode.
detection of photocurrents of order of mA and with output voltages of order of
volts, thus the electronic noise is not a limiting factor.
In order to estimate the total amount of noise at the output of the difference
and sum photocurrents readout blocks, it is necessary to apply the procedure
described in the previous section to the AUDIO and RADIO readout blocks.
Given that these readout blocks have a first low-pass transimpedance stage and
a second high-pass stage, all noise calculations can be reduced to a prototype
calculation as that performed on the transimpendance and voltage amplifiers
shown in Eq. 3.28.
Applying the noise calculation described in the previous section, first by considering the operational amplifier non ideal and then sending the open-loop gain
to ∞ (as shown in sec. 3.2.1) we obtain:
• transimpedance amplifier
referring to Fig. 3.28, in the calculations of the noise contributions, the
signal current source is open-circuited and the calculation for vn , in and iF
are exactly the same of Eq. 3.99, 3.100 and 3.101. Thus their contributions
to the output noise power spectrum are
A 2 2
A→∞
out
vn
Svout
(3.104)
= vn2
−→
Svn = n
1+A
AZF 2 2
A→∞
−→
Siout
(3.105)
Siout
=
= |ZF |2 i2n
n
n
1 + A in
ARF 2 2
A→∞
out
iF
Svn = −→
Siout
= |ZF |2 i2F
(3.106)
F
1+A
• voltage amplifier
referring to Fig. 3.28, in the calculations of the noise contributions, the
signal voltage source is short-circuited and
vI )

 vI − vt = − vo − v t
ZI
ZF

vo = −Avt
=⇒
vo = −vI
ZF A
(1 + A)ZI + ZF
CHAPTER 3. HOMODYNE DETECTION
101
Figure 3.28: Prototype for noise calculations for Top: a transimpedance amplifier Bottom: a voltage amplifier.
thus
2
2
ZF A
−
Svout
=
I
(1 + A)ZI + ZF vI
ZF 2 2
−
Svout
=
I
Z I vI
A→∞
−→
(3.107)
in )

 in = vt + vt − vo
ZI
ZF

vo = −Avt
vo = −in
=⇒
ZF ZI A
(1 + A)ZI + ZF
thus
2
2
ZF ZI A
−
Siout
=
n
(1 + A)ZI + ZF in
A→∞
−→
2
Siout
= |−ZF | i2n
n
(3.108)
vn )

 vn + vt = vo − (vn + vt )
ZI
ZF

vo = −Avt
=⇒
vo = vn
A(ZI + ZF )
(1 + A)ZI + ZF
CHAPTER 3. HOMODYNE DETECTION
102
thus
Svout
n
A(ZI + ZF ) 2 2
v
=
(1 + A)ZI + ZF n
A→∞
−→
Svout
n
2
ZF 2
v
= 1 +
ZI n
(3.109)
iF )
(
vo − vt = iF ZF
=⇒
vo = −Avt
vo = iF
AZF
1+A
thus
AZF 2 2
Siout
=
F
1 + A iF
A→∞
−→
2
Siout
= |ZF | i2F
F
(3.110)
In both cases the total output is simply given as in Eq. 3.102.
Those calculation prototypes were applied to the homodyne detector design by
assuming that resistors and capacitors only bring thermal noise. 1/f noise was
not included because, from the informations reported in the op-amps datasheets,
it appears to be negligible for the design. With the help of MATLAB, the total
output noise were simulated for the AUDIO and RADIO difference photocurrent readout blocks and for the AUDIO sum photocurrent readout block. The
noise sources of the photodiodes as well as those of the other blocks have been
taken into account. For example, the 50kΩ input impedance of the DC blocks
contributes with its thermal noise to the output noise of the nearby AUDIO
block. Such a kind of contributions can be easily taken into account by appropriately defining the generic impedance ZI , which appears in Fig. 3.28.
The circuit noise at the output of the AUDIO sum and difference photocurrent
readout blocks has been plotted in Fig. 3.29, together with the shot noise level
at the same outputs (which represents the signal), simulated for laser powers
1mW and 10mW.
The same has been done in Fig. 3.30 for the RADIO difference photocurrents
readout block. In both figures we can see that, if we look at the shot noise
level obtained for a laser power of 10mW, the circuit noise is at least ten times
below this shot noise level, as we wanted. In addition, the noise (and signal)
for the AUDIO sum and difference appear to more or less equal, as required
for the detection of squeezing. This result was achieved by adding the third
unity-gain extra stage to the difference photocurrent readout block, as shown
in Fig. 3.22. In fact, this stage compensate the extra noise coming from the
summing amplifier and from the fact that, while the difference is performed
with the self-subtraction scheme and then amplified, the sum is performed after
that the two photocurrents have been amplified, and thus at the output of the
sum block there is more noise than at the output of the difference block.
CHAPTER 3. HOMODYNE DETECTION
103
Figure 3.29: Noise simulation for AUDIO sum and difference photocurrents’
readout blocks.
Figure 3.30: Noise simulation for RADIO difference photocurrent readout block.
Chapter 4
Realization and test of the
homodyne detector
prototype.
In the previous chapter we illustrated the design of the homodyne detector
prototype developed in this thesis.
The design have been developed with the support of LABE, the electronic lab of
INFN, which is now looking after the practical realization of the detector. The
circuit PCB has already been designed and is being produced by an external
company. However, the delivery time was too long because of bureaucratic
problems in the administrative office. Thus the circuit board is not ready yet.
Once the circuit realization is fulfilled we proceed to the testing.
The circuit test will be divided in two steps. The first step is the debug of the
electronic circuit:
• it will be checked for possible malfunctioning, due for example to wrong
or missing connections or to mistakes in the power-supply branch
• the photodiodes and their DC readout blocks will be tested by illuminating
each photodiode and looking if the corresponding voltage is provided at
the output of the DC blocks
Once this preliminary test has been completed and the mistakes have been
rectified we will also measure
• the transfer functions of the AUDIO and RADIO blocks
• the total amount of noise at the AUDIO sum and difference outputs and at
the RADIO difference output, when no light impinges on the photodiodes
and compare them with those expected from the circuit design.
The second step of the test procedure is aimed to check the prototype when
actually worked as a homodyne detector for quantum noise. In order to do that
we use a self-homodyne detection scheme, as that described in Sec. 3.1.2,
which allows a measurement of the shot noise of the laser used as local oscillator. To better understand this point, we briefly recall the main characteristics
104
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.105
of self-homodyne detection, whose ideal scheme is reported in Fig. 3.1:
a laser beam, i.e the local oscillator, is sent to one of the ports of the beam splitter while vacuum noise enters the other port. The resulting sum and difference
photocurrents, calculated in Sec. 3.1.2, are
î+ ∝ α2 + αδX1,a
(4.1)
î− ∝ αδX1,v
(4.2)
and their variance
V ar[î+ ] ∝ α2 V ar[X1,a ]
2
V ar[î− ] ∝ α V ar[X1,v ]
(4.3)
(4.4)
where v refers to the vacuum and a to the local oscillator.
V ar[X1,v ] = 1 for the unsqueezed vacuum and the power spectrum of the difference photocurrent equals the shot noise power spectrum of the laser, as shown in
Sec. 3.1.1. In the frequency bands in which the laser can be well approximated
as a coherent state V ar[X1,a ] = 1 and the power spectrum of the sum photocurrent also equals the shot noise power spectrum of the laser and the power
spectrum of the difference photocurrent. In the frequency bands in which the
laser cannot be considered a coherent state, it brings classical technical noise
which can be much larger than the quantum noise given by the shot noise. In
those cases the sum photocurrent exhibits a noise power spectrum above the
shot noise level of the laser, i.e above the power spectrum of the difference photocurrent.
Thus, by arranging a self-homodyne detection scheme for our prototype we can
check, taking the power spectrum of the difference photocurrent, if it is able to
see the shot noise of the laser at different laser powers. If it is the case, we can
also compare the circuit noise at the AUDIO sum and difference outputs and
at the RADIO difference output with the shot noise level of the laser at the
maximum power allowed for the photodiodes (10mW) and verify how much the
circuit noise is below this shot noise level (we would like a 10 factor below).
Finally, taking the power spectrum of the (AUDIO) sum photocurrent, we can
also obtain a measurement of the laser noise in the frequency band 10Hz-10kHz.
A measurement of the intensity noise in the band 100kHz-120MHz of the laser
(Mefisto-Innolight Mod S200), which will be used in the self-homodyne test,
has been performed during this thesis using a single photodiode. The procedure
used to fulfill the measurement and its results are reported in the following section and constitute an example of how the photocurrent noise measurements,
so crucial in homodyne detection for squeezed light, can be performed.
4.1
Measurement of the intensity noise power
spectrum of the laser Mefisto-InnoLight
The model of laser Mefisto (Nd:YAG, λ = 1064nm), which we are dealing with,
has a power of 200mW and a noise-eater system, which allows for the suppression of the resonant noise of the laser that appears at about 500kHz.
The noise-eater consists of an electro-optic feedback controller, embedded in the
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.106
laser device. Practical applications are the suppression of the large modulations
due to the relaxation oscillations of the laser crystal and of the technical noises
due to mechanical imperfections of the laser (see [3]).
In this section
• we report the results of the measurements of the intensity noise of the
laser Mefisto, performed both with the noise-eater turned off and on, in
the band 100kHz-150MHz
• we show the noise-suppression effect of the noise-eater in the band 100kHz1.5MHz
• we show that the laser appears to be shot noise limited, i.e quantum noise
limited, in the band 3MHz-70MHz
The measurements have been performed by using
• the spectrum analyzer GW INSTEK GSP-830
• photodetector InGaAs THORLABS PDA10CF-EC
• photodetector InGaAs NEW FOCUS 1811
Expected shot noise level at the output of a photodetector
The two photodetectors have both a readout circuit, whose transimpedance
value (TRA) is reported in the data sheet and a responsivity (RES). Thus,
if Popt is the laser power, the photodetector provides the photocurrent and the
output voltage given by
i = RES × Popt
(4.5)
v = i × T RA
(4.6)
The calculation of the shot noise level of a photocurrent has been performed in
Sec. 3.1.1 and we just report the result
Sii = ie = eRES × P opt
(4.7)
Svv = ie × T RA2 = eRES × P opt × T RA2
(4.8)
For the expression of Svv we used Eq. 3.87.
The spectrum analyzer
The used spectrum analyzer (GW INSTEK GSP-830) works in the band 9kHz3GHz. It measures the voltage at its input port in three different ways
2 V Z
• dBm= 10 log 1mW
where V is the input voltage and Z the input impedance of the analyzer
(50 Ω). The quantity V 2 Z is referred to 1mW.
V
• dBmV=20 log 1mV
in this case, the input voltage is referred to 1mV
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.107
V
• dBuV=20 log 1µV
in this case, the input voltage is referred to 1µV
Le us suppose that we want to measure a power spectrum, for example the noise
power spectrum of our laser. We can do that by registering the output voltage
of our photodetector, for example in dBmV, and then
√ calculating the square
root of the noise power spectrum, for example in nV/ Hz, by
- expressing the noise voltage exiting the photodetector and measured by
the spectrum analyzer in nV
Vnv = 10
VdBmV
20
× 106
(4.9)
- and then dividing by the square root of the resolution bandwidth RBW
of the spectrum analyzer during the measurement (the information about
the resolution bandwidth is provided by the spectrum analyzer itself)
p
Svv |(nV /√Hz) = √
Vnv
RBW
(4.10)
The resolution bandwidth tells us how much the spectral measurements are
frequency-spaced, i.e, given the measurement bandwidth, how many points the
measurement contains. The used spectrum analyzer allows only four possible
values for the resolution bandwidth, i.e 3kHz, 30kHz, 300kHz e 4MHz, thus,
even though the nominal working bandwidth of the spectrum analyzer is 9kHz3GHz, spectral measurements in the band 9kHz-100kHz contain few points and,
furthermore, the noise of the spectrum analyzer in this band came out to be
large enough to not allow the measurement of the laser noise.
Measurements with the Thorlabs photodetector
The characteristics of the photodetector necessary for the measurement are reported in Tab. 4.1.
MAGNITUDE
RES
TRA
band
max power
VALUE
0.6
5 × 103 (if terminated on 50Ω)
DC-150
1.6
UNITS
A/W
V/A
MHZ
mW
Table 4.1: Characteristics of detector Thorlabs
Given that the laser has a power of 200mW, filters have been used in order to
reduce it below the maximum power allowed by the detector (1.6mW). However,
the available filter allowed to reach at most 0.5mW.
At this laser power, using
Eq. 4.7 and 4.8, we obtain an expected shot noise
√
spectrum of ∼ 35nV / Hz.
Measurements of
• the spectrum of the intrinsic noise of the spectrum analyzer
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.108
• the spectrum of the intrinsic noise of the detector
• the spectrum of the intensity noise of the laser with noise-eater off
• the spectrum of the intensity noise of the laser with noise-eater on
have been performed as described in the previous section in the band 100kHz120MHz: the voltage at the input of the analyzer has been measured in dBmV
and registered in .txt files through the interface software between the computer
and the analyzer. From these data files the noise power spectra have been calculated as illustrated in Eq. 4.9 and 4.10.
The parameters important for the measurement are reported in Tab. 4.2.
MAGNITUDE
laser power
shot noise power spectrum expected at the
output of the photodetector
measurement band
VALUE
0.5
35
UNITS
mW√
nV / Hz
0.1-150
MHz
Table 4.2: Parameters of the measurements with detector Thorlabs
In the following figures the registered spectra have been reported. In particular we can see that
• the noise spectrum of the analyzer, in the considered band, is below
√ the
Hz at
noise spectra of the detector and the
laser.
Its
value
is
∼
40nV
/
√
500kHz, and arrives at ∼ 5 − 6nV / Hz for frequencies & 5M Hz
• in Figure 4.1 the effect of the noise-eater on the laser noise is evident in
the band 100kHz-2MHz while it has no effects at higher frequencies
• in Figure 4.1, 4.2 and 4.3 we can see that the laser appear to be shot noise
limited in the
√ band ∼ 3M Hz−70M Hz, i.e in this band the noise spectrum
is ∼ 35nV / Hz. This is more evident if we subtract in quadrature the
noise spectrum of the detector to that of the laser, as shown in Figure 4.4.
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.109
Figure 4.1: Spectra- detector Thorlabs 100kHz-5MHz. The blue curve shows
the resonant noise of the laser. The green curve shows the effect of the noise
suppretion of the resonant noise due to the noise-eater. The noise eater has no
effect at frequencies higher than 2MHz. The detector and the analyzer noise
(red curve and black curve) are below the laser noise.
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.110
Figure 4.2: Spectra- detector Thorlabs 5MHz-15MHz. The blue curve shows the
laser noise, which √
is at about the shot noise level expected at the laser power
0.5mW, i.e 35nV / Hz.
Figure 4.3: Spectra√ detector Thorlabs 16MHz-120MHz. The laser is shot noise
limited (∼ 35nV / Hz), however its noise appears to encrease because of the
detector noise, which increases and begins to dominate at frequency ∼ 120MHz.
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.111
Figure 4.4: Spectra- detector Thorlabs 5MHz-80MHz. The quadrature difference between the laser noise and the detector noise has been calculated in order
to make more evident√
that the laser is shot noise limited. In fact the quadrature
difference is ∼ 35nV / Hz.
Measurements with the New Focus photodetector
We followed the same procedure used for detector Thorlabs.
The characteristics of the photodetector necessary for the measurement are reported in Tab. 4.3.
MAGNITUDE
RES
TRA
band (AC OUT)
band (DC OUT)
max power
VALUE
0.73
40 × 103
0.025 -125
DC-50
120
UNITS
A/W
V/A
MHz
kHz
µW
Table 4.3: Characteristics of detector New Focus
With the available filters we were able to reach at most the laser power
∼ 60µW (the maximum power for this photodiode is 120µW ), thus the expected
√
shot noise power spectrum at the output of the photodetector is ∼ 95nV / Hz.
The parameters important for the measurement are reported in Tab. 4.4.
MAGNITUDE
laser power
shot noise power spectrum expected at output
of the photodetector
measurement band
VALUE
60
95
UNITS
µW √
nV / Hz
0.025-125
MHz
Table 4.4: Parameters of the measurements with detector New Focus
In the following figures the registered spectra have been reported. In particular we can see that
• in Figure 4.5 the effect of the noise-eater on the laser noise is evident in
the band 300kHz-1.5MHz while it has no effects at higher frequencies
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.112
• in Figure (4.5) and (4.6) we can see that the laser appear to be shot noise
limited in√the band 1.5MHz-40MHz , i.e in this band the noise spectrum is
∼ 95nV / Hz. At higher frequencies the noise of the detector dominates.
In conclusion the results of the measurements carried on with both the photodiodes show that the laser is to be shot noise limited in the band 3MHz-70MHz.
Figure 4.5: Spectra- detector New Focus 300kHz-5MHz. The blue curve shows
the resonant noise of the laser. The green curve shows the effect of the noise
suppretion of the resonant noise due to the noise-eater. The noise eater has no
effect at frequencies higher than 1.5MHz. The detector and the analyzer noise
(red curve and black curve) are below the laser noise.
CHAPTER 4. REALIZATION AND TEST OF THE HOMODYNE DETECTOR PROTOTYPE.113
Figure 4.6: Spectra- detector New Focus 5MHz-60MHz. The blue curve shows
the laser noise, which
√ is at about the shot noise level expected at the laser power
60µW, i.e 95nV / Hz. The detctor noise dominates at frequencies higher than
∼ 40MHz.
Conclusion
The quantum enhancement of meter-scale prototypes of GW interferometers by
injecting squeezed vacuum at the output port of the interferometer has been
experimentally demonstrated .
Frequency independent squeezing has been obtained both in the few MHz band
and in the audio band of the gravitational waves detectors (10Hz-100kHz), by
means of optical parametric processes. In addition, ponderomotive squeezing,
which is likely to provide frequency dependent squeezing, is being studied.
A squeezer in being designed for the GW interferometer Advanced Virgo. The
projects involves a first step in which the squeezing by optical parametric processes is realized. Then, the implementation of ponderomotive squeezing is
attempted. Finally, once Advanced Virgo has been completed and has started
working, the injection system for the squeezed vacuum into the output of the
interferometer will be realized.
In this thesis we designed the electronics of a homodyne detector prototype
for the the squeezer of Advanced Virgo 1 .
In the first chapter of the thesis we introduced the gravitational waves and their
direct detection with interferometric detectors. Then we illustrated the quantum enhancement of GW interferometers by injecting squeezed vacuum at their
output port.
In the second chapter, after a brief introduction to the basic concepts of quantum
optics, we describe the squeezed states of light and how they can be produced,
both by optical parametric process and by ponderomotive squeezing.
In the third chapter we discussed the theory of the homodyne detection and the
design of the electronics of the homodyne detector prototype.
In the fourth chapter we illustrated the methodology to be followed for testing
of the circuit board, once it has been realized. We concluded by reporting the
experimental characterization of the intensity noise of the laser, which will be
used to test the homodyne detector prototype.
1 Unfortunately, due to delays caused by bureaucratic problem of the administrative office,
the circuit board is still being produced by a factory.
114
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[5] Simon Chelkowski. Squeezed Light and Laser Interferometric Gravitational
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[11] H. J. Kimble et al. Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their
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[12] S. Frasca. Analisi dei segnali.
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Ringrazio...
Ettore, per la sua gentilezza, e per aver creduto in me quando io non credevo
in me stessa
Fulvio, il Grande Capo (per citare Perci), per il suo occhio attento, e per avermi
insegnato la tenacia
Giovanni Vittorio Pallottino, per la sua disponibilità, e perché senza il suo
prezioso aiuto questa tesi non sarebbe mai stata ultimata
Piero, per essere il maestro supremo e indiscusso della fantascienza e per aver
condiviso con me la sua sapienza
Sergio, per le nostre disquisizioni filosofiche che non mancheranno mai di affascinarmi, e per avermi liberato di Linux
Paola, Cristiano, Mary, Andrea, Alberto, Roberto, Ilaria, Valentina e Luca,
per la simpatia e le risate che hanno reso la vita al G23 un bello spasso
Perci, che con il suo brio e la sua inventiva non smetterà mai di sorprendermi
A tutti i ragazzi del Labe, Valerio il Boss, Manlio, Luigi, Fabrizio, Lorena, i
due Francesco, Riccardo, Daniele, Giacomo e Felice, per il loro insostituibile
aiuto nella tesi, per avermi accolto e coccolato, e per i divertentissimi pranzi
insieme
Manlio, per il controllo attento sul circuito, per i cremini, le coppe del nonno, i
magnum i dolci siciliani e il pesce, per i caffè e per le lezioni di salsa. Insomma,
per avermi viziato come una nipotina
Luigi, per essere l’ingegnere elettronico più bravo del mondo e per avermi insegnato cosı̀ tante cose di elettronica. Per non parlare poi delle lezioni di salsa
Daniele, per la pazienza e il tempo dedicato al PCB, per la simpatia, e per
avermi fatto comprendere l’importanza di non avere due panze
Massimo Testa, per essere la prova vivente che l’onniscenza esiste
Alessandro Ercoli, per essere stato un grande prof. di matematica e fisica oltre
che una persona straordinaria
117
BIBLIOGRAPHY
118
Mamma e Papà, per volermi tanto bene, e per aver sempre posto la conoscenza
sopra ogni altra cosa
Erika, stupenda sorellona, per l’affetto e il sostegno
Giulia, amica di una vita, per aver condiviso le gioie e le sconfitte di tutti
questi anni, e per avermi regalato un esempio di perseveranza
Elena, alias Madrina, per essere una amica attenta ed equilibrata e una persona di grande valore
Giovanna Chiara, amica dalla mente brillante e straordinaria cultura
Marzia, per avermi insegnato i vantaggi della misantropia e per l’umorismo
impareggiabile
Petra, per avermi aperto le porte a una nuova cultura, e naturalmente per
il ramen e il soju
Francesca, amica fidata e unica persona che io conosca capace di parlare il
greco antico (non semplicemente tradurre, parlare!)
Carlo Luciano, cofondatore del fan club ”Amici di Sailor Greco”
Nenzy, per aver accolto senza pregiudizi una vegliarda tra le sue giovani ginnaste, e averle regalato non poche soddisfazioni
Suor Marta, per aver detto ”io sapere quando Suor Elsa ha detto me che essere
tu, tu sempre fare male a te” ed essersi poi presa cura di me ogni volta che una
clavetta o un cerchio mi hanno fatto un occhio nero o il tea mi ha scottato il
braccio
Ivan, che amo tantissimissimissimo, e che, con il suo amore e la sua prorompente
simpatia, ha reso la mia vita cosı̀ dolce e ha trovato il modo di strapparmi una
risata nei momenti più tetri.