Download Stimulated Raman Transitions Between Hyperfine Ground States of

Stimulated Raman Transitions
Between Hyperfine Ground States of
Magnetically Trapped Rubidium-87
Atoms
Basiselementen
| Logo
Tony Hubert
huisstijlrichtlijnen Universiteit van Amsterdam
Thesis presented
LOGOfor partial fulfillment of the degree of
Master of Science (MSc.) in physics
Supervisors: Dr. R.J.C. Spreeuw and Julian Naber
Institute of Physics
University of Amsterdam
The Netherlands
April 2015
BEELDMERK
WO
Abstract
In this thesis we present an approach for coherent manipulation of the hyperfine ground states of magnetically trapped 87 Rb atoms through two-photon
stimulated Raman transitions. We describe the theory behind stimulated
Raman transitions in which light shifts of the hyperfine ground states manifest themselves naturally, and calculate the specific laser intensity ratio for
which the differential light shift vanishes. We also show how for the specific |F = 1, mF = −1i −→ |F = 2, mF = 1i transition, the two possible Raman paths through the D2 line interfere destructively, causing a significant
supression of the Raman Rabi frequency. A phase-locked laser system is described in detail and experimental results are given. Measurements of the
differential light shifts are in perfect agreement with theory and we elucidate
the importance of minimizing them in order to increase coherence. We also
discuss several Raman line broadening effects, including Doppler broadening
and power broadening. Preliminary attempts at observing Raman Rabi oscillations stress the importance of laser intensity stability and confinement
of atoms within the laser beam’s interaction region.
Contents
1 Introduction
5
2 Theory
2.1 Rubidium-87 Level Structure . . . . . . . . . . . . . . . .
2.1.1 (Zero-Field) Fine and Hyperfine Structure . . . .
2.1.2 Zeeman-Splitting . . . . . . . . . . . . . . . . . .
2.1.3 Hyperfine Ground States As Quantum Bits . . . .
2.2 Stimulated Raman Transitions . . . . . . . . . . . . . . .
2.2.1 General Theory of Stimulated Raman Transitions
2.2.2 Stimulated Raman Transitions in 87 Rb . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
8
8
8
9
10
11
12
17
3 Experimental Setup
3.1 Raman Laser System . . . . . . . . . . . . . . . . . . . .
3.1.1 Raman Lasers . . . . . . . . . . . . . . . . . . . .
3.1.2 Phase Lock . . . . . . . . . . . . . . . . . . . . .
3.1.3 Polarization Spectroscopy and Master Laser Lock
3.1.4 Pulse Mechanism . . . . . . . . . . . . . . . . . .
3.2 Sample preparation and Laser Injection . . . . . . . . . .
3.2.1 Sample preparation . . . . . . . . . . . . . . . . .
3.2.2 Laser Injection . . . . . . . . . . . . . . . . . . .
3.3 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
28
28
29
30
33
36
38
38
39
40
4 Experimental Results
4.1 Doppler- and Power Broadening
4.1.1 Doppler Broadening . .
4.1.2 Power Broadening . . . .
4.2 Light Shifts . . . . . . . . . . .
4.2.1 Differential Light Shifts .
4.3 Rabi Dynamics . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
43
44
44
47
49
49
51
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Conclusion
53
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3
Acknowledgements
I would like to thank everyone involved in aiding me in this project. Particularly Robert Spreeuw for giving me the opportunity and Julian Naber for
guiding me through it.
Thanks.
4
Chapter 1
Introduction
Coherent control of two-state quantum systems has been subject to many
recent studies, both theoretical and experimental. Applications range from
atomic clocks and magnetometers to the more popularized advances in quantum information processing. The subject discussed in this thesis concerns
mostly the latter, in which a two-state quantum system is commonly referred to as a quantum bit or q-bit. The value of the q-bit is determined by
the state the two-level system: either |0i, |1i or a superposition of both.
Among the proposed physical systems and physical phenomena that could
possibly serve as a platform for quantum bits are superconducting circuits
[14], nuclear magnetic resonance [12], trapped ions [11] and trapped neutral
atoms. The main advantages of using neutral atoms are their relatively weak
interaction with the environment and the variety of available techniques and
tools that can be used to manipulate both their external and internal degrees
of freedom.
Particularly suitable neutral atoms are alkali metals such as rubidium
(Rb) and caesium (Cs). These atoms, having two long-lived hyperfine ground
states at an energy splitting of a few GHz and their first excited state being
hundreds of THz away, can effectively be treated as two-level systems under
the right circumstances.
Recent studies have proposed a quantum information platform based on
two-dimensional on-chip lattices of magnetic microtraps for ultracold 87 Rb
atoms [20][22]. The idea behind a lattice of microtraps is that every individual trapped atomic cloud could act as a single q-bit. Interactions between
neighbouring microtraps are then facilitated by strong dipole-dipole interactions of highly excited Rydberg states, potentially allowing for Rydbergblockade based entanglement and ultimately quantum gates [16][19].
5
0
20
40
60
80
100
120
140
ricated from
Johnson noise
able single sit
fields act on t
with RF, micr
dress a single
atomic regist
which have so
and ion syste
quantum sim
30
20
10
y (µm)
150
100
50
0
0
100
200
300
400
500
600
700
x (µm)
Figure 1.1: Absorption imaging of a magnetic microtrap lattice loaded with
FIG.
7. Absorption imaging of the loaded magnetic lattice.
ultracold 87 Rb atoms. The atoms can be loaded simultaneously into hexagoThe
rubidium atoms are loaded simultaneously into hexagonal
nal (left side) and square (right side) lattices. The lattice period is 10µm, al(left
and square (right
side) lattices.
A magnified
view
of
lowingside)
for inter-microtrap
dipole-dipole
Rydberg interactions
of the
trapped
the
center
region
atomic
clouds.
[20] with the crossover between both geometries
is shown in the upper part.
Coherent manipulation of the hyperfine ground states of 87 Rb atoms has
been successfully demonstrated using a combination of radiofrequency- and
microwaves [21]. However, their relatively long wavelengths limit the spaV. CONCLUSION AND OUTLOOK
tial resolution of this approach. It would be impossible to adress a single
microtrap.
A different
might
be to
coherently
the 87 Rb
We
have approach
described
the
design
andmanipulate
construction
ofground
an
states through stimulated Raman transitions. Using a combination of two
advanced
atom chip based on a lattice of permanent magnear infrared (∼ 780 nm) lasers, we can drive the same ground state transinetic
microtraps
neutral
atoms.
Considerations
which
tion, but
with a spatialfor
resolution
high
enough to
potentially excite individual
microtraps.
we
have taken into account include: the electrical curCoherent
stimulated
Raman
transitions
hyperfine magnetic
ground states
rent load and
thermal
load
of thebetween
preliminary
of alkali atoms have been proposed before [10], and have recently been suctrap
based on current conducting wires, the possibility for
cessfully demonstrated in optically trapped single cesium atoms [2] and rubidrapid
chip
the
mechanical
the chip,
ium atoms
[1].exchange,
Experimental
reports
on similar stability
transitions of
in atomic
clouds
compactness
and space
limitations,
optical
accessibility
remain scarse, although
there have
been made attempts
on room
temperature
rubidium
[21].
and
thevapour
requirement
to image with a resolution in the
In
this
work
we
describe
our,
arguably
successful,
attempts at
the experimicrometer range, and the
needs
of ultra-high
vacuum.
mental realization of coherent stimulated Raman transitions between hyperFollowing
the ofconstruction
and initial
testing, the new
87
fine ground states
magnetically trapped
Rb atoms.
atom chip has been placed in vacuum and finally tested
by loading rubidium atoms into
the magnetic microtraps
6
after a sequence of compression and cooling stages.
Having atoms in a 10 µm spacing lattice on an atom
chip will allow us to access the Rydberg dipole-dipole
interaction regime required for quantum information experiments. At the same time we find that we are reaching
VI.
ACKNOW
The author
help in magn
Technology C
their support
the Miller In
versity of Ca
the research p
Research on M
lands Organi
UvA authors
Curie ITN CO
1 M.
Endres, M
C. Gross, L.
Kuhr, Science
2 D. Greif, T. U
Science 340,
3 I. Bloch, J. D
(2012).
4 J. Simon, W.
Greiner, Natu
5 M. Greiner,
Bloch, Natu
6 S. Seidelin, J.
J. Britton, J.
Hume, W. M
and D. J. Win
We shall begin by briefly introducing the general theory of stimulated
Raman transitions as well as describing the more specific case of 87 Rb. Next,
we explain the setup that was designed to drive Raman experiment. We
conclude with our experimental results, conclusion and outlook on future
improvement and challenges.
7
Chapter 2
Theory
2.1
Rubidium-87 Level Structure
Rubidium-87 (87 Rb) is one of two naturally occuring isotopes of Rubidium,
the other being 85 Rb. Its specific characteristics and energy level structure
make it a popular atom for quantum and atom optics experiments. We
begin by briefly outlining the 87 Rb fine- and hyperfine energy structure in
both zero-field and an externally applied magnetic field.
2.1.1
(Zero-Field) Fine and Hyperfine Structure
87
Rb is an Alkali atom with one electron in the outermost 5S shell. The
transition to the 5P excited state is split into two D-line components; the D1
line (52 S1/2 → 52 P1/2 ) and the D2 line (52 S1/2 → 52 P3/2 ). Of these two components the latter has been of much more relevance to experimental atom
physics due to the fact that its cycling transition can be exploited for the
cooling and trapping of 87 Rb. There is also a strong supply of relatively affordable and available laser diodes that can operate at the D2 line wavelength
(780nm).
The 52 S1/2 , 52 P1/2 and 52 P3/2 states are split into multiple hyperfine F
states, where F = I + J is the total angular momentum of the atom, I being
the nuclear angular momentum and J = L + S the sum of the electron’s
angular momentum L and spin S. F = |F| takes on values |I + J| ≥ F ≥
|I − J| in steps on one integer (or one quantum of angular momentum h̄).
The entire D1 and D2 fine and hyperfine level structure is shown in figure
2.1.
8
F=3
δ2,3 = 2π × 267 MHz
52 P3/2
(n2S+1 LJ )
F=2
D2 780.24 nm
5P
(nL)
F=1
F=0
δ1,2 = 2π × 157 MHz
δ0,1 = 2π × 72 MHz
F=2
2
2π × 812 MHz
5 P1/2
F=1
D1 794.98 nm
F=2
5S
2
5 S1/2
∆HFS = 2π × 6.834 673 617 GHz
F=1
Figure 2.1: Zero-field fine and hyperfine energy level structure of
2.1.2
87
Rb.
Zeeman-Splitting
Each of the hyperfine (F ) energy levels contains 2F + 1 magnetic sublevels
(mF ). These sublevels are degenerate in the absence of any externally applied magnetic fields. However, when such fields are applied, the degeneracy
is broken. This effect is known as Zeeman-splitting and the Hamiltonian describing the atomic interaction with a magnetic field B along the z-direction
is given by
µB
(gS S + gL L + gI I) · B
HZ =
h̄
(2.1)
µB
=
(gS Sz + gL Lz + gI Iz ) Bz ,
h̄
where µB = eh̄/2me is the Bohr magneton and gS , gL and gI are respectively
the electron spin, nuclear spin and electron orbital g-factors. They have all
been measured experimentally with great accuracy.
In the low-field limit where the hyperfine-splitting always dominates the
Zeeman interaction (∆E Z ∆E hfs ), the total atom angular momentum
F = J + L + S is always conserved and the Zeeman interaction Hamiltonian
can be written in the coupled representation[17]:
HZ =
1
µB gF Fz Bz ,
h̄
9
(2.2)
in which the hyperfine Landé factor gF is approximately
gF ' 1 +
F (F + 1) + J(J + 1) − I(I + 1)
.
2F (F + 1)
(2.3)
In this low-field, coupled representation, we may assume that the Zeeman
interaction perturbs the hyperfine eigenstates. To first order, the perturbation then gives rise to an energy shift on top of the hyperfine-structure and
the Zeeman interaction effectively lifts the degeneracy in mF :
Z
∆E|F,m
= µB gF mF Bz ,
Fi
(2.4)
in which mF can take all integer values limited to the interval F ≥ mF ≥ −F .
We now calculate the Zeeman splitting of the hyperfine (F ) levels of the
2
5 S1/2 ground state of 87 Rb. For the 52 S1/2 state we have J = 1/2 and
I = 3/2, so that gF becomes

1

−
for F = 1
2
(2.5)
gF '

 1 for F = 2.
2
The resulting Zeeman shifts are shown in figure 2.2.
mF = −1
mF = 2
52 S1/2 , F = 1
10
52 S1/2 , F = 2
20
∆E Z (MHz)
∆E Z (MHz)
mF = 1
5
mF = 0
0
−5
−10
mF = 1
0
5
10
15
Magnetic Field (G)
20
10
mF = 0
0
−10
mF = −1
−20
mF = −2
0
5
10
15
Magnetic Field (G)
20
Figure 2.2: Zeeman splitting of the 52 S1/2 ground states in the low field
regime.
2.1.3
Hyperfine Ground States As Quantum Bits
In the low-field regime the Zeeman sub-levels |F = 1, mF = −1i and |F = 2, mF = 1i
can be considered to be good candidates for the role of quantum-bits (|0i,
|1i) in quantum information experiments for the following reasons:
1. Both of these states are low-field seekers, making them trappable in
the local minima of magnetic potential fields.
10
2. Beyond first order approximation, the slopes of the Zeeman shifts of
|F = 1, mF = −1i and |F = 2, mF = 1i depend on the magnetic field, and
to obtain a more exact value for the energy difference between the
states one should include a second order Zeeman shift. At B = Bm =
3.228 917 G these two slopes are exactly equal. So when a cloud of
atoms is trapped in a magnetic potential with the trap bottom at
Bm , the energy shift between the two states remains nearly constant
along the atomic cloud and the effects of field fluctuations is minimized.
Bm is often termed as the magic field. The hyperfine splitting of the
|F = 1, mF = −1i and |F = 2, mF = 1i ground states including a second
m
order Zeeman shift at BM is ∆B
HFS = 6.834 678 114 GHz.
3. It is possible to coherently drive transitions between these two states,
so that we can manipulate quantum bits to be in |0i, |1i or an entangled
superposition of the two: e.g. √12 (|0i + |1i).
2.2
Stimulated Raman Transitions
In order to realize coherent transitions between the two hyperfine 87 Rb ground
states |0i = |F = 1, mF = −1i and |1i = |F = 2, mF = 1i, for which |∆mF| = 2,
one needs to drive a two-photon transition. One way to do this is by using
a combination of microwave (MW) and radio-frequency (RF) fields in a ladder transitions, as shown in figure 2.3. The transition between |0i and |1i
is driven through an intermediate state (|F = 2, mF = 0i) to which the MW
field is coupled with a fixed detuning to prevent unwanted resonant scattering. The RF field simultaneously couples this intermediate state to the final
state |1i. Although a RF+MW transition is relatively easy to establish, the
long wavelengths involved result in a very poor spatial resolution. Also, the
maximum practically atainable two-photon Rabi frequency (or atomic coupling to the field) is much lower than could potentially be had for a minimally
divergent, high intensity laser beam.
11
ωRF , ΩRF
F=2
ωMW , ΩMW
∆HFS = 2π × 6.8 GHz
F=1
-2
-1
0
mF
1
2
Figure 2.3: Two-photon excitation scheme for the |F = 1, mF = −1i →
|F = 2, mF = 1i transition in the 52 S1/2 87 Rb ground state using a combination of microwave (MW) and radiofrequency (RF) fields.
A much higher spatial resolution can be obtained through stimulated
Raman transitions by employing lasers in a Λ-configuration (figure 2.4). In
such a configuration, two atom states are coherently coupled to each other
through an energetically higher lying intermediate state. In the following
sections the dynamics of stimulated Raman transitions will be explained,
starting with a general theory of the matter and followed up by a more
detailed description for the specific case of 87 Rb.
2.2.1
General Theory of Stimulated Raman Transitions
We consider a simple three level Raman system as illustrated in figure 2.4.
The aim of such a stimulated Raman system is to coherently drive transitions between two states (|0i and |1i) through an energetically higher lying
intermediate state (|2i) by employing two coupled lasers.
One laser, which we call the pump laser, couples the state |0i to the
intermediate state |2i with frequency ωP . Another laser, called the Stokes
laser, simultaneously couples state |1i to the same intermediate state with
ωS .
To avoid driving population into the intermediate state we detune the two
lasers from resonance by ∆ (Note: in figure 2.4, ∆ is taken to be negative).
The detuning from the two-photon resonance frequency is denoted by δ,
which we take to be much smaller than ∆.
The system’s dynamics can be obtained by solving the time-dependent
(TD) Schrödinger equation for the system’s Hamiltonian. For simplicity, we
take the energy of the ground state |0i to be zero. In the rotating wave
12
Γ
2
Δ
€
€
Pump
Laser
Stokes
Laser
€
ω S , ΩS
ω P , ΩP
δ
1
€
ΔE
€0
€
€
Figure 2.4: Basic scheme of a stimulated
€ Raman system. States |0i and
|1i are coupled to each other through an intermediate level |2i. This two€ transition is being driven by two separate lasers, each coupling a
photon
separate state to the intermediate level. Both lasers are detuned from |2i by
∆ to minimize unwanted one-photon transitions. The two-photon detuning
is denoted δ.
approximation (RWA) [18], the Hamiltonian for this system in the ordered
basis {|0i , |1i , |2i} can be written as


0
0
ΩP
h̄
(2.6)
Ĥ =  0 −2δ ΩS  ,
2
ΩP ΩS −2∆
In which the off-diagonal terms represent the coupling between states due
to the pump and Stokes lasers with respective Rabi-frequencies ΩP and ΩS .
The Rabi frequency of a laser coupled to a transition is determined by the
laser-field amplitude at the atom E(r0 ) and the expectation value of the
dipole-transition operator D̂:
1
ΩLg,e = − he| D̂ · EL (r0 ) |gi ,
h̄
(2.7)
where |gi and |ei are the ground and excited state respectively and the
symbol L is used to differentiate between lasers (EP for the pump and ES
for the Stokes laser)[18].
The time dependent state Ψ(t) of the three-level atom can be expressed
as a superposition of its unperturbed eigenstates (|0i, |1i and |2i) weighed
13
by their respective time dependent probability amplitudes Ci (t):
Ψ(t) = C0 (t) |0i + C1 (t) |1i + C2 (t) |2i .
(2.8)
Solving the TD Schrödinger equation
ĤΨ(t) = ih̄
d
Ψ(t)
dt
(2.9)
gives us a set of three coupled equations for the probability amplitudes Ci (t):
1
iĊ0 (t) = ΩP C2 (t),
2
1
iĊ1 (t) = ΩS C2 (t) − δC1 (t),
2
1
iĊ2 (t) = (ΩP C0 (t) + ΩS C1 (t)) − ∆C2 (t).
2
(2.10)
Assuming that the detuning ∆ is much greater than the Rabi-frequencies,
we can adiabatically eliminate any population in state |2i. That is, the
population in |2i will undergo much faster oscillations than |0i and |1i so
that we can assume Ċ2 (t) to average out to zero over a large number of
cycles. By doing this, we can reduce our system to an effective two-level
scheme:
ΩP
(ΩP C0 (t) + ΩS C1 (t)) ,
iĊ0 (t) =
4∆
(2.11)
ΩS
iĊ1 (t) =
(ΩP C0 (t) + ΩS C1 (t)) − δC1 (t),
4∆
for which the effective Hamiltonian is
2
h̄
ΩP /∆
ΩP ΩS /∆
Ĥeff =
.
(2.12)
4 ΩP ΩS /∆ Ω2S /∆ − 4δ
Here, the diagonal terms give rise to an energy shift Λ of |0i and |1i respectively, shifting the energy levels up or down depending on the sign of the
laser detuning:
Ω2
Ω2
Λ0 = h̄ P ,
Λ1 = h̄ S .
(2.13)
4∆
4∆
Because these energy shifts are caused by the coupling of light, they are
known as light shifts (also referred to as AC Stark shifts). The effective
differential light shift δΛ of the transition frequency between the two states
is then given by
δΛ = Λ1 − Λ0 .
(2.14)
14
and the effective two-photon Raman detuning becomes
δeff = δ − δΛ
(2.15)
Note: Remember that we have assumed ∆ ΩP , ΩS . For a smaller detuning the approximation fails and this expression for the light-shifts does not
apply anymore.
Again, the off-diagonal terms represent the coupling between the two
states, for which we now define the two-photon Rabi frequency as
ΩR =
ΩP ΩS
.
2∆
(2.16)
Solving the Schrödinger equation for Heff under the assumption that the
initial population is in state |0i we arrive at the time-dependent population
densities ρi (t):
ρ0 (t) = |C0 (t)|2 = 1 +
Ω2R
[cos Ω0 t − 1] ,
2Ω20
Ω2
ρ1 (t) = |C1 (t)| = R2 [1 − cos Ω0 t] ,
2Ω0
(2.17)
2
where we defined the generalized Rabi frequency as
q
2
Ω0 = Ω2R + δeff
.
(2.18)
Equation 2.17 describes an oscillating probability of finding atoms in
states |0i or |1i, known as Rabi oscillations. As a complete population inversion only occurs for δeff = 0, it is of foremost importance that the lasers
in a Raman laser system are capable of operating at exact two-photon Raman resonance with minimal fluctuations in frequency. This can be readily
achieved by employing a phase-locked laser setup, which will be thoroughly
described later in this thesis. Also, since δΛ depends on the ratio of laser intensities, precise control and stability of laser power is essential. An unstable
laser intensity ratio will result in fluctuations of the two-photon resonance
frequency, effectively causing a decoherence of the Raman transitions.
15
1
δeff = 0
δeff = 1ΩR
δeff = 2ΩR
ρ1 (t)
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10 12
Time (seconds)
14
16
18
Figure 2.5: Rabi oscillations of the population density ρ1 (t) for ΩR =
1 rad s−1 and different δeff . Complete population inversion only occurs at
δeff = 0. For larger detuning the maximum transition probability decreases,
while the generalized Rabi frequency Ω0 increases.
Cross Coupling
One thing we have not explicitly considered is cross coupling of the lasers (see
figure 2.6). Let us assume that the lasers are on Raman resonance (δ = 0).
Although the Stokes laser is meant to couple state |1i to |2i with detuning
∆, it also unintentionally couples |0i to |2i with detuning ∆ − ∆E . The same
argument can be made for the pump beam, which inadvertently couples |1i
to |2i with detuning ∆ + ∆E .
In practice, as long as ∆E is larger than ∆, the cross coupling does not
lead to single- or two-photon transitions, and therefore should not affect the
two-photon Rabi frequency. However, it does add considerably to the light
shifts. Taking this cross coupling into account, the more complete expression
for the light shift of a ground state becomes:
Λg = ΛPg + ΛSg ,
so that:
"
#
(ΩS0,2 )2
h̄ (ΩP0,2 )2
,
+
Λ0 =
4
∆
∆ − ∆E
"
#
(ΩP1,2 )2
h̄ (ΩS1,2 )2
.
Λ1 =
+
4
∆
∆ + ∆E
16
(2.19)
(2.20)
(2.21)
Γ
Γ
2
Δ
€
Δ − ΔE
2
Δ
€
Δ + ΔE
€
ω P , ΩP0,2
€
€
Pump
Laser
Stokes
Laser
ω S, Ω
€
ω S , ΩS0,2
1
€
ΔE
ω P , ΩP0,2
S
0,2
1
€
€ 0
ΔE
0
€
€
€
Pump
Laser
Stokes€
Laser
€
(a)
€
€
(b)
€
€ pump and Stokes lasers in a stimulated
Figure 2.6: Cross coupling of the
Raman system. Although not on purpose, each individual laser effectively
couples both ground states to the intermediate excited state with a different
detuning. While the Stokes laser is intended to couple the ground state |1i
to |2i with detuning ∆, it also effectively couples |0i to |2i with detuning
∆ + ∆E (2.6a). The opposite can be said of the pump laser (2.6b).
2.2.2
Stimulated Raman Transitions in
87
Rb
The previous description of a stimulated Raman system is valid for a threelevel atom. However, as we have shown before, 87 Rb is not a three-level atom.
Both the ground state and D2 excited state are split into multiple hyperfine
levels F, which in turn have different sublevels mF . To obtain the complete
expressions for the light shifts Λg and the two-photon Rabi-frequency ΩR we
must take into account that every sublevel couples to the light fields with a
different Rabi-frequency and detuning.
A complete picture of all relevant sublevels and their respective detuning
relative to both lasers is given in figure 2.7. We will only concern ourselves
with the D2 excitation line as we are far enough away from the D1 line to
completely neglect its contributions.
17
δ2,3
52 P3/2
δ1,2
δ0,1
∆0
D2 line
780 nm
∆1
Pump
Laser
∆2
∆3
F=3
F=2
F=1
F=0
Stokes
Laser
F=2
2
5 S1/2
∆HFS 2π × 6.834 GHz
F=1
-3
-2
-1
0
1
2
3
mF
Figure 2.7: Complete level structure of the D2 line in 87 Rb, including the
detunings of all intermediate hyperfine (F) states with respect to the two
Raman (pump and Stokes) lasers. Hyperfine energy splittings of the excited
states are denoted δ0,1 , δ1,2 and δ2,3 . The |F = 1, mF = −1i and |F = 2, mF = 1i
ground states (colored red) are separated by the hyperfine splitting ∆HFS =
2π × 6.834GHz.
The Rabi frequency Ω (given by equation 2.17) with which an atomic level
couples to the light field is characterized by the matrix elements of the dipole
operator D̂: hFg , mg | D̂ |Fe , me i. In order to calculate these matrix elements,
we use the Wigner-Eckart theorem to factor out the mF dependency and
write the elements as a product of a reduced matrix element and a ClebschGordan coefficient [17]:
hFg , mg | D̂ |Fe , me i = DFg ,Fe hFe , me , 1, q | Fg mg i .
(2.22)
The reduced matrix element DFg ,Fe is independent of magnetic sublevels and
can be expressed in units of D2,3 , which is the matrix element of the closed
transition of the 87 Rb D2 line (F = 2 ←→ F = 3):
DFg ,Fe = D2,3 dFg ,Fe .
In which
s
D2,3 =
h̄Γ
30 λ3D2
= 3.58 × 10−29 C m,
8π 2
(2.23)
(2.24)
where we have used that Γ = 2π × 6.1MHz and λD2 = 780.2nm. The relative
18
coupling strength dFg ,Fe is given by
dFg ,Fe
q
Fg Fe 1
Fg +I+Je +1
= (2Fg + 1)(2Je + 1)(−1)
,
Je Jg I 6j
(2.25)
where {}6j is the 6j-Racah symbol.
The second factor in equation 2.22 is the Clebsch-Gordan coefficient and
describes the coupling between different sublevels through the absorbtion or
emission of a photon with spherical polarization q . For which q is labeled
0, 1 or -1 for π, σ + or σ − respectively. In the cartesian basis {x̂, ŷ, ẑ} this is
defined as:

√

q = −1
(x̂ − iŷ)/ 2 ,
(2.26)
q = ẑ ,
q=0

√

−(x̂ + iŷ)/ 2 , q = 1.
For tables of the 87 Rb D2 hyperfine dipole matrix elements I refer to [17].
Using the formalism above we can now write the Rabi frequency for the
coupling between specific magnetic sublevels as
r
I
(2.27)
ΩFg ,mg ,Fe ,me = Γ
dF ,F hFe , me , 1, q | Fg mg i,
2Isat g e
in which I is the laser intensity and the saturation intensity Isat is defined as
Isat
h̄2 Γ2
=
= 1.669 33 (35) mW/cm2 .
D2,3
(2.28)
Multi-level Light Shifts
As both Raman lasers couple to multiple sublevels with a different detuning,
they both produce multiple light shifts. To obtain an expression for the
light shift of a ground state, we have to sum over all intermediate sublevels.
Using equation 2.27 for the sub-level specific Rabi frequency we obtain the
following expression for the light shift of a ground state caused by a laser L
with intensity IL :
ΛLFg ,mg
2
2
Γ2 IL X dFg ,Fe hFe , me , 1, q | Fg mg i
,
=
8 Isat F ,m ,q
∆LFg ,Fe
e
(2.29)
e
Let us consider a Raman laser field with σ + + σ − polarization parallel to the
quantization (B-field) axis (as is the case in our experiment), the contribution to the light shift due to π-transitions are zero. In this case, the only
19
sublevels that do contribute to the light shifts of the |F = 1, mF = −1i and
|F = 2, mF = 1i ground states are given in figure 2.8.
F=3
F=2
F=1
F=0
52 P3/2
σ−
σ+
σ−
σ+
F=2
52 S1/2
F=1
-3
-2
-1
0
1
2
3
mF
Figure 2.8: All contributions from the D2 line for σ + +σ − polarized light that
add to the light shifts of the |1, −1i and |2, 1i ground states (coloured red).
The blue and green arrows indicate the coupling of intermediate sublevels to
|1, −1i and |2, 1i respectively.
Using equation 2.29 we can then calculate the exact light shifts of both
ground states, where we also take into account the contribution of the previously described cross coupling:
Λ1,−1 = ΛP1,−1 + ΛS1,−1
"
Γ2 IP
7
5
4
=
+
+
192 Isat ∆2 ∆1 ∆0
#
IS
7
5
4
+
+
,
+
Isat ∆2 − ∆HFS ∆1 − ∆HFS ∆0 − ∆HFS
(2.30)
Λ2,1 = ΛP2,1 + ΛS2,1
"
Γ2 IS
52
25
3
=
+
+
960 Isat ∆3 ∆2 ∆1
#
IP
52
25
3
+
+
+
.
Isat ∆3 + ∆HFS ∆2 + ∆HFS ∆1 + ∆HFS
(2.31)
and
20
Here, IP and IS are the pump and Stokes laser intensities respectively,
∆HFS = 2π × 6.834 GHz is the ground state hyperfine splitting and ∆0,1,2,3
denotes the laser detuning with respect to hyperfine levels F = 0, 1, 2, 3 respectively as shown in figure 2.7.
Figure 2.9 shows the differential light shift, δΛ = Λ2,1 − Λ1,−1 , of the
|F = 1, mF = −1i and |F = 2, mF = 1i hyperfine ground states as a function of
the intensity ratio IS /IP of the probe (IP ) and Stokes (IS ) lasers. It shows
that for a fixed detuning ∆, there is a ratio R0 for which the differential light
shifts are exactly zero, independent of the total laser intensity.
IS
IS
IS
IS
δΛ (kHz)
50
= 0.5
= 1.0
= 1.5
= 2.0
mW/mm2
mW/mm2
mW/mm2
mW/mm2
0
IS
IP
= R0
−50
∆3 = −2π × 2713 MHz
0
0.2
0.4
0.6
0.8
1
IP /IS
Figure 2.9: Differential light shift δΛ of the |F = 1, mF = −1i and
|F = 2, mF = 1i hyperfine 87 Rb ground states versus laser intensities. IS and
IP are the Stokes and pump laser intensities respectively. This calculation is
specific for a detuning ∆3 = −2π × 2713 MHz. Notice that there is a specific
laser intensity ratio, R0 , for which δΛ = 0. R0 is independent of total laser
intensity and depends on the detuning ∆.
Because a differential light shift δΛ leads to an effective Raman detuning
of the lasers (equation 2.15), it is necessary to correct for this in an experiment. One straightforward way of doing this would be to fine-tune the laser
frequencies to adjust for the induced detuning, which can be done relatively
easy and with great accuracy in a phase locked laser setup. However, as the
light shifts depend on the laser intensities, which are spatially distributed in
a Gaussian manner, this approach fails to fulfill the resonance condition for
21
0.55
2.6
0.5
R0 (IS /IP )
R0 (IS /IP )
2.8
2.4
2.2
2
0.45
0.4
0.35
∆3 < 0
1.8
∆3 > 0
0.3
−3.0 −2.5 −2.0 −1.5 −1.0 −0.5
∆3 /2π (GHz)
0.5
(a)
1.0
1.5 2.0 2.5
∆3 /2π (GHz)
3.0
(b)
Figure 2.10: Zero differential light-shift intensity ratio R0 as a function of∆3
for both (a) red and (b) blue detuning.
all atoms struck by the laser beams. Not only does this decrease the maximum population transfer, it also gives rise to unwanted light shift induced
effects such as spectral line broadening and damping of the Rabi oscillations.
Some of these effects will be discussed in the Results chapter.
Another solution is to fix both laser powers to the intensity ratio R0 for
which δΛ = 0. This way, all atoms struck by the two laser beams experience no differential light shift, provided that their Gaussian intensity profiles
overlap perfectly (this can be achieved by coupling both lasers in and out of
the same fiber). Calculating R0 amounts to equating the differential light
shift to zero and solving for the laser intensities:
δΛ = Λ2,1 − Λ1,−1 = 0
−→
IS
= R0 .
IP
(2.32)
An analytical expression for R0 can be written in terms of the laser detunings
∆ with respect to the 52 P3/2 excited hyperfine states (see figure 2.7):
52
35
25
25
3
20
− ∆3 +∆HFS + ∆2 − ∆2 +∆HFS + ∆1 − ∆1 +∆HFS + ∆0
R0 = (2.33)
52
25
35
3
25
20
+ ∆2 − ∆2 −∆HFS + ∆1 − ∆1 −∆HFS − ∆0 −∆HFS
∆3
Figure 2.10 shows R0 as a function of ∆3 .
Damping of Two-Photon Rabi-Frequency
The presence of multiple hyperfine intermediate states in 87 Rb also results
in multiple contributions two the two-photon Rabi frequency. There are
two possible paths that contribute to the |1, −1i → |2, 1i Raman transition:
22
either through the |1, 0i or |2, 0i hyperfine 52 P3/2 sublevel. Both paths are
shown in figure 2.11. Including both contributions, the complete expression
for the two-photon Raman Rabi frequency then becomes
ΩR =
Ω|1,−1i,|1,0i Ω|2,1i,|1,0i Ω|1,−1i,|2,0i Ω|2,1i,|2,0i
+
.
2∆1
2∆2
(2.34)
F=3
F=2
F=1
F=0
2
5 P3/2
σ+
σ−
F=2
2
5 S1/2
F=1
-3
-2
-1
0
1
2
3
mF
Figure 2.11: There are two possible stimulated Raman paths for the
|1, −2i −→ |2, 1i transition. Either through the |1, 0i or |2, 0i hyperfine
52 P3/2 sublevel.
Unfortunately, the transition strengths of these paths are exactly equal in
magnitude, but opposite in sign, causing both Raman paths to destructively
interfere:
h2, 1| D̂ |1, 0i h1, 0| D̂ |1, −1i = − h2, 1| D̂ |2, 0i h2, 0| D̂ |1, −1i ,
(2.35)
so that
Ω|1,−1i,|1,0i Ω|2,1i,|1,0i + Ω|1,−1i,|2,0i Ω|2,1i,|2,0i = 0.
(2.36)
Using this, we can then rewrite the two-photon Raman Rabi frequency as
1
1
1
−
,
(2.37)
ΩR = Ω|1,−1i,|1,0i Ω|2,1i,|1,0i
2
∆1 ∆2
or more explicitly as a product of a ”one-path” Raman Rabi frequency and
a damping factor D(∆1 ):
ΩR =
Ω|1,−1i,|1,0i Ω|2,1i,|1,0i
D(∆1 ),
2∆1
23
(2.38)
where
B
δ1,2
D(∆1 ) = B
,
δ1,2 − ∆1
(2.39)
B
and δ1,2
is the hyperfine splitting of between the F = 1 and F = 2 52 P3/2
hyperfine states including the zeeman shifts for an applied magnetic field
flux B (see figure 2.7). For a detuning of ∆3 = −2π × 2713 MHz, and
B=0
= 2π × 156.9 MHz we have D ≈ 0.05, causing a severe suppression of
δ1,2
the Raman Rabi frequency.
Spontaneous Raman Scattering
Despite the large detuning from the intermediate 52 S3/2 hyperfine states, the
Raman lasers may still off-resonantly scatter from them.
Every time an atom spontaneously scatters from an intermediate state
into one of the ground state sublevels, it is lost in the coherent Rabi oscillation
of the atom cloud. This atom may then again couple with the laser fields
and be driven into excitation, but it’s individual Rabi oscillation will now
be out of phase with the rest of the atoms, effectively dampening the entire
cloud’s average Rabi oscillations. We call this a decoherence effect.
To evaluate the decoherence effect of this spontaneous Raman scattering
we begin by writing the scattering rate RFsci ,mi ,Ff ,mf for scattering events from
an initial ground state |Fi , mi i to a final ground state |Ff , mf i induced by a
laser field coupling to multiple intermediate excited sublevels |Fe , me i. In the
low saturation limit, where ∆ Ω, we can write RFsci ,mi ,Ff ,mf as the absolute
square of the sum of all possible scattering paths:
2
X Ω
Γ
Fi ,mi ,Fe ,me
Asc
(2.40)
RFsci ,mi ,Ff ,mf ' Ff ,mf ,Fe ,me ,
8 F ,m
∆Fg ,Fe
e
e
for which we have defined the scattering amplitude to be
Asc
Ff ,mf ,Fe ,me = dFf ,Fe hFf , mf , 1, q | Fe me i .
(2.41)
Calculating the time dependent population densities of the ground state
sublevels then comes down to solving the master equation in Lindblad form:
d
i
ρ = − [Heff , ρ] + L(ρ),
dt
h̄
(2.42)
where ρ = |Ψi hΨ| is the system’s density matrix, Heff is the multi-sublevel
equivalent of equation 2.12 including all ground state F = 1 and F = 2 Zeeman shifts normalized with respect to |1, −1i and |2, 1i respectively (the normalization ensures that we remain on Raman resonance for |1, −1i −→ |2, 1i
24
independent of the Zeeman shift).L(ρ) is called the Lindblad superoperator,
which is obtained as follows,
1 X
†
†
†
L(ρ) =
2cj ρcj − cj cj ρ − ρcj cj ,
(2.43)
2 j=
Fi mi
Ff mf
and
1/2
c Fi mi = (Rsc
|Ff , mf i hFi , mi | .
Fi mi )
Ff mf
(2.44)
Ff mf
A numerical solution of equation 2.42 at Raman resonance (δeff = 0) and
zero differential light shift (δΛ = 0 and IIPS = R0 ) is plotted for ρ|1,−1i and
ρ|2,1i in figure 2.12. It predicts a strong scattering induced decoherence effect,
that dampens the two-photon Rabi amplitude by more than half after two
cycles.
1
B = 3.23G
I = 1.0 mW/mm2
Ω0 = 544 Hz
∆3 = −2π × 2713 MHz
ρ|F,mF i
0.8
0.6
ρ|1,−1i
ρ|2,1i
ρ|2,2i
0.4
0.2
0
0
1,000
2,000
3,000 4,000
Time (µs)
5,000
6,000
Figure 2.12: Resonant Raman Rabi oscillations of the |1, −1i, |2, 1i and |2, 2i
magnetically trappable ground states. We assumed Raman resonance and
zero differential light shift of the |1, −1i and |2, 1i ground states (δeff = 0).
The overall detuning with respect to F = 3 is ∆3 = −2π × 2713 MHz and
we included first order Zeeman shifts of the 52 S1/2 hyperfine mF sublevels
at a magnetic field of B = 3.23 Gauss. For a combined laser intensity of
IS + IP = 1.0 mW, we find a Rabi frequency of 544 Hz. The oscillations are
damped by spontaneous Raman scattering, effectively causing a decoherence
of the system.
25
Generally, the effect of scattering can be greatly reduced by simply choosing a large detuning ∆, as the scattering rate Rsc is suppressed much faster
(Rsc ∝ 1/∆2 ) than the two-photon Rabi frequency (ΩR ∝ 1/∆). However,
this does not apply to our specific case, in which we also have to take into account the previously described damping of ΩR (equation 2.37). What makes
this damping particularly problematic is that for increasingly large detuning
it causes ΩR to also scale as ∝ 1/∆2 . Effectively, this means that Rsc and
ΩR are similarly suppressed, making it less advantageous to use very large
∆.
It should be noted that in these calculations, we haven’t accounted for
other decoherence effects such as trap-losses, field fluctuations and collisional
effects.
Intensity Ratio Fluctuations
Fluctuations in the power of the individual Raman lasers may cause fluctuating differential light shifts, therefore increasing the generalized Rabi frequency (equation 2.18). To elucidate the effect of unstable individual laser
powers we have plotted the Rabi dynamics (solutions to equation 2.42) for
fluctuating intensity ratios near R0 in figure 4.3.
26
0.8
ρ|2,1i
ρ|2,2i
0.6
Population Density
Population Density
0.8
0.4
0.2
0
ρ|2,1i
ρ|2,2i
0.6
0.4
0.2
0
0
200
400
600
800
Pulse Length (µs)
1,000
0
(a) IS /IP = R0
400
600
800
Pulse Length (µs)
1,000
(b) IS /IP = 1.01 · R0
0.8
ρ|2,1i
ρ|2,2i
0.6
Population Density
Population Density
0.8
200
0.4
0.2
0
ρ|2,1i
ρ|2,2i
0.6
0.4
0.2
0
0
200
400
600
800
Pulse Length (µs)
1,000
0
(c) IS /IP = 1.02 · R0
200
400
600
800
Pulse Length (µs)
1,000
(d) IS /IP = 1.03 · R0
Figure 2.13: Simulations of F = 2 Rabi oscillations including spontaneous
Raman scattering for different Raman laser intensity ratios. The calculations
show that for minimal deviations from the optimal intensity ratio R0 , the
light shift induced Raman detuning significantly affects the generalized Rabi
frequency and maximum population transfer. The calculation is based on
an arbitrary Stokes laser intensity of IS = 3.25 mW/mm2 and detuning
∆3 = 2π × 2.7 GHz.
Clearly, the effect of power fluctuations is quite significant. A 2% increase
of power in the Stokes laser almost doubles the generalized Rabi frequency
and more than halves the maximum population transfer.
27
Chapter 3
Experimental Setup
We now turn to describe the experimental setup employed in our pursuit to
realize coherent stimulated Raman transitions. We shall begin by discussing
the Raman laser system in detail, followed by a brief explaination of the
cooling, trapping and optical pumping mechanisms used to prepare the 87 Rb
atoms for the experiment. We conclude this technical section by describing
our method of imaging and (52 S1/2 , F = 2) population detection.
3.1
Raman Laser System
The Raman laser system can be divided into four parts (figure 3.1), all serving
their particular purpose:
1. Raman lasers. The two lasers responsible for a two-photon stimulated Raman transition are arranged in a master-slave configuration.
That is, the slave laser (SL) is set up to follow the master laser (ML)
with a frequency difference exactly equal to the ground state hyperfine
splitting ∆HFS = 6.834 GHz (see figure 2.7).
2. Phase lock. In order to stabilize the frequency difference between
the master and slave laser, we employ a phase locking mechanism.
This mechanism locks the slave laser’s frequency and phase relative to
the master laser through an electronical feedback loop. The frequency
difference can then be fine-tuned with high precision on a computer.
3. Polarization spectroscopy and master laser lock. The absolute
master laser frequency is locked on a 87 Rb transition with a fixed detuning ∆ using Doppler-free polarization spectroscopy.
28
4. Pulse mechanism. Before both lasers are coupled into an optical
fibre leading to the experiment, they are coupled into an acousto-optic
modulator (AOM), allowing us to accurately pulse the laser light with
variable pulse lengths.
50:50
PBS
Slave Laser
(SL)
λ
λ
Phase Lock
Mechanism
2
2
PBS λ
€
Pulse
Mechanism
& Experiment
2
€
PBS
€
Master Laser
(ML)
λ
λ
2
PBS
2
Polarization
Spectroscopy
& ML Lock
€
Figure 3.1: Overview of the Raman laser system. Both lasers are split up into
multiple beams leading to the various parts of the system. A combination
of polarizing beam splitters€(PBS) and have-wave plater (λ/2) allow us to
carefully tune the amount of light into each section.
3.1.1
Raman Lasers
At the heart of our stimulated Raman laser system lie two commerically available DL 100 narrow linewidth tunable external cavity diode lasers (ECDL’s)
from manufacturer TOPTICA fitted with Axcel Photonics laser diodes. These
lasers can be set up to operate at a center wavelength of 780 nm with a typical course tuning range of about 3 nm and mode-hop free tuning range of
20 GHz. The typical maximum output power for our specific laser diodes is
about 150 mW and the typical linewidth lies between 100 KHz and 1 MHz.
The DL 100 provides two means for fast tuning of the lasing frequency:
1. by directly controlling the laser diode driving current.
2. by controlling the voltage over the piezo actuated grating angle.
It is important to note that they differ dramatically in bandwidth. Slower
electronic feedback up to a few tens of kHz is often fed to the grating piezo,
29
while the current may be modulated at frequencies in the order of a few tens
of MHz. In practice, this means that for a simple frequency lock the piezo
grating often suffices. For much faster locking mechanisms (such as phase
locks) it is often necessary to modulate the current also.
3.1.2
Phase Lock
In order to stabilize the frequency difference of the master and slave lasers
and maximize their phase coherence (that is, to minimize their relative phase
noise), we employ a phase locking mechanism. The electronic system is designed to lock the beat note signal produced by both lasers to a reference RF
oscillator. The error signal produced by comparing the beat note with the
reference signal can be fed back into proportional-integral-derivative (PID)
controllers which then actively adjust the slave laser frequency, thereby stabilizing and narrowing the beat note and thus fixing the lasers’ relative frequency and phase. A simplified schematic of this phase locked loop (PLL) is
depicted in figure 3.2.
Amplifier
Beat Signal
~ 6.8 GHz
ML + SL Beam
MW
6.834 GHz
+ 25 MHz
Mixer
Fast
PD
Beat Signal
~ 25 MHz
Splitter
PC
Lens
PBS
Rotated 45°
Spectrum
Analyzer
Amplifier
Error
Signal
mFalc
Phase Detector
+ PID
PID
Frequency
Detector
RF
25 MHz
Piezo
Amplifier
Slave Laser
(SL)
SL FET
SL Piezo
Figure 3.2: A simplified schematic of the phase locking electronics. See the
text for a detailed explanation.
30
The beat note signal of the ML and SL is obtained through a fast photodetector (FPD). We use a New Focus amplified free-space photoreceiver that
converts optical signals up to 12 GHz. Before the two overlapped beams
are collected on the FPD, they pass through a 45◦ rotated polarizing beam
splitter (PBS) to ensure equal linear polarizations. Since the FPD measures the intensity (∝ |E|2 ) of the combined laser field, its signal equals the
squared sum of both individual laser amplitudes. It therefore comprises of
two components: an extremely fast component equal to the sum of both
laser frequencies and a slower component equal to the difference of the laser
frequencies. The former is in the order of hundreds of THz and is left completely undetectable by the FPD. The latter is termed the beat note and
should be approximately 6.8 GHz (if we tune our lasers properly).
The detected beat note signal is then amplified and mixed down to a
secondary beat signal at a frequency more suited to our feedback electronics.
More specifically, the primary beat signal is multiplicatively mixed with a
microwave (MW) oscillator set exactly equal to the Zeeman shifted 87 Rb
hyperfine splitting of the |F = 1, mF = −1i and |F = 2, mF = 1i ground states
m
∆B
HFS = 6.834 678 114 GHz plus an additional 25 MHz. This way, if the
secondary beat signal is to be locked exactly at 25 MHz, we know that the
primary beat signal must be at exactly 6.834 678 114 GHz. Fine-tuning the
exact primary beat frequency can be done by simply tuning the MW oscillator
through the connected laboratory control computer (PC). The MW oscillator
is referenced to an external 10 MHz rubidium atomic clock.
We split part of the secondary beat signal to a spectrum analyzer, so that
we can monitor it in real time.
The remaining signal is amplified and fed into a phase- and frequency
detector, which both compare the secondary beat note to a referenced 25
MHz RF oscillator and produce an error signal accordingly. For frequency
detection we use the frequency discrimination function of an Analog Devices
AD9901 digital phase/frequency discriminator, which outputs a ”slow” error
signal in the order of a few kHz. This frequency error signal is then sent to
a PID controller which controls the slave laser’s piezo actuated grating. For
phase detection we employ a commercially available Toptica mFALC module,
which is a combined analog phase detector and PID controller. The mFalc
produces a ”fast” phase error signal in the order of a few MHz and directly
controls the slave laser diode current through the SL’s build in field effect
transistor (FET).
31
Frequency Locked
Phase Locked
Power (dBm)
−20
−40
−60
−80
20
22
24
26
Beat Frequency (MHz)
28
30
Figure 3.3: Beat note signal obtained from the spectrum analyzer, under the
effect of both the frequency lock and phase lock.
The procedure to obtaining a successful phase lock is as follows:
1. We first manually tune the SL to be approximately 6.8 GHz away from
the ML, and make sure the SL is free from any occuring mode hops.
2. Next, we activate the SL frequency feedback loop by engaging the grating piezo controlling PID. This locks the primary beat note to be centered exactly at 6.834 GHz (and thus the secondary beat to exactly
25 MHz). This frequency lock serves only as a support to the eventual
phase lock. Its role is to stabilize the beat note for the phase detector
to use and retrieve the beat in case the lock is lost for some reason.
3. Finally, the phase lock loop is activated by employing the mFALC phase
detector/PID. The fast current feedback of the mFALC significantly
reduces the ML and SL’s relative phase noise and therefore greatly
reduces the beat note’s linewidth.
Figure 3.3 shows the secondary beat note signal, obtained from the spectrum analyzer, under the effect of both the frequency lock and phase lock. We
observe a great reduction of the beat note’s linewidth: from approximately
3 MHz for the frequency lock alone, to less than 100 Hz (the spectrum analyzer’s resolution limit) with the phase lock engaged. We also observe a
significant increase of beat note signal of up to 40 dBm relative to the background.
32
3.1.3
Polarization Spectroscopy and Master Laser Lock
We use a polarization spectroscopy based electronical feedback mechanism to
lock the master laser’s frequency to an atomic Rubidium D2 transition. Polarization spectroscopy is used to create a frequency error signal that can be fed
back to the ML’s piezo actuated grating through a PID controlled feedback
loop. The advantage of this technique over alternative locking mechanisms is
that it does not rely on the modulation of the laser frequency, thus preventing
the addition of any unwanted noise on the laser light.
Polarization Spectroscopy
Polarization spectroscopy is a high resolution spectroscopic method that can
be used to probe polarization dependent optical properties of a measured
atomic medium. We use a combination of two counter-propagating laser
beams to pump and probe a atomic vapour cell containing both 87 Rb and
85
Rb.
The pump beam has a relatively high intensity and is circularly polarized
(either σ + or σ − ). It serves to optically pump the atomic ensemble into
a collective stretched state. That is, the circularly polarized light will only
induce ∆mF = 1 or ∆mF = −1 transitions (for σ + or σ − respectively) and
therefore pump the atomic population into the highest, respectively lowest,
mF hyperfine magnetic sublevels. The result is a non-uniform population of
different magnetic sublevels.
The linearly polarized, low intensity probe beam will observe this imposed
anisotropy of the atomic medium as a birefringence. As a linearly polarized
beam can be decomposed into a linear combination of σ + and σ − , the probe
beam will then experience a decrease in absorbtion for one orthogonal polarization component and an increase for the other. This effect is strongest
for closed transitions (such as F = 2 ←→ F = 3 in 87 Rb), as their excited
states never decay out of the pumping cycle (F=3 can only decay back into
F=2). Retrieving both individual polarization components from the probe
beam can be done by simply filtering them through a polarizing beam splitter
(PBS).
The counter-propagating alingment of the beams allows Doppler-free absorbtion measurements, as used in standard saturated absorbtion spectroscopy.
For a more elaborate description of polarization spectroscopy and saturated
absorbtion spectroscopy I refer to [3] and [15] respectively.
33
Master Laser
(ML)
ML Piezo /
Error
Signal
Piezo
Amplifier
Grating
λ
4
Differential
Amplifier
Pump Beam
2
Probe
Beam
€
λ
PBS
ML
Beam
PID
Controller
PD1
€
Vapour Cell
85Rb + 87Rb
PBS
PD2
Figure 3.4: A simplified schematic of the polarization spectroscopy based
master laser (ML) lock.
Polarization Spectroscopy Based Laser Lock
A simplified schematic of the polarization spectroscopy based master laser
(ML) locking mechanism is given in figure 3.4. The linearly (π) polarized
ML beam is split up into a pump- and probe beam using a polarizing beam
splitter (PBS), where the pump/probe intensity ratio may be adjusted using
a half-wave plate (λ/2). The pump beam’s polarization is then made circular
(either σ + or σ − ) using a quarter-wave plate (λ/4). Both beams are then
directed through the 87 Rb+85 Rb atomic vapour cell in a counter-propagating
alignment.
Using another PBS, the π polarized probe beam is then split into its
two orthogonal polarization components (σ + and σ − ), which are individually
collected by two photodiodes (PD1 and PD2). Both photodiode signals are
substracted from each other and the difference is amplified using a differential
amplifier, resulting in a dispersive error signal (see figure 3.5) around an
atomic transition resonance frequency. This error signal is fed back to the
ML’s piezo actuated grating through a PID loop, thus effectively locking the
laser frequency on an atomic transition.
34
F0 =2 c.o. 4
F0 =4
F0 =3 c.o. 4
F0 =3 c.o. 4
Signal (a.u.)
F0 =4
F0 =2 c.o. 4
Signal (a.u.)
F = 3 → F0
PD 1
PD 2
Laser Frequency (a.u.)
Laser Frequency (a.u.)
Figure 3.5: Saturated polarization spectroscopy of atomic 85 Rb transitions.
The left figure shows the individual photodiode signals (PD1, PD2) of the
two orthogonal polarizition components (σ + , σ − ) of the pump beam. The
figure on the right shows the dispersive error signal obtained by substracting
both photodiode signals. Closed transitions and their repective cross-overs
(c.o.) with other transitions are strongest. Each atomic transition creates
its own lockable error signal.
If we now want to lock the master laser on a 87 Rb transition with a
fixed detuning ∆, we can simply lock on a nearby 85 Rb transition instead.
For example, the 85 Rb F = 2 −→ F = 3 closed transition has its resonance
frequency -2526 MHz away from the 87 Rb F == 1 −→ F = 2 transition. So
locking the ML on this particular 85 Rb transition will be effectively equal to
locking to the87 Rb transition with a fixed detuning of ∆ = 2π × −2526 MHz.
A disadvantage of this locking method is that we are quite limited in our
accessible range of detunings. It should also be noted that with this method,
we are effectively locking on the side of a transitional line, instead of on its
center. Depending on the sign of error signal’s slope, this results in a slightly
lower or higher locked detuning than expected.
Also, as the magnetic sublevels may undergo Zeeman splitting, this locking mechanism is very sensitive to external magnetic fields. For a typical
background magnetic field of 0.5 Gauss (the approximate average magnetic
field at the earth’s surface), the zeeman shifts of both the 87 Rb and 85 Rb
magnetic sublevels will only be in the order of 2π × 1 MHz. In our magnetic
trap, the magnetic field at the trap bottom is approximately 3.2 Gauss, and
the maximum zeeman shifts will be closer to 2π × 10 MHz. For the sake
of simplicity, when we refer to a detunig ∆ in this thesis, it will be with
respect to the mF = 0 sublevels, which do not undergo a Zeeman shift in the
low-field limit.
35
3.1.4
Pulse Mechanism
Before the laser beams are coupled into an optical fibre leading to the experiment, they pass through an acousto-optic modulator, allowing us to accurately pulse the laser light in rectangular pulses of variable duration.
An acousto-optic modulator (AOM) is a device which can be used for
controlling the power, frequency or spatial direction of a laser beam. It is
based on the acousto-optic effect, i.e. the modification of the refractive index
by an oscillating mechanical pressure. The key element of an AOM is a
transparent crystal attached to a piezoelectric transducer used to excite a
acoustic wave through the crystal. The laser light then experiences Brillouin
scattering at the generated periodic refractive index and interference occurs
similar to Bragg diffraction. The frequency of the 1st order diffracted light is
Doppler shifted by an amount equal to the frequency of the acoustic wave.
By only using the 1st order diffracted laser beam and driving the transducer
with square shaped RF pulses, we can effectively produce precisely tuned
laser pulses.
Transducer
Transmitted
Beam
( 0th Order )
Acoustic
Wave
θ
€
Absorber
1st Order
Diffracted
Beam
Figure 3.6: The basic working principle of an acousto-optic modulator (AOM). A piezoelectric transducer sends acoustic waves through a
transparant crystal. An absorber prevents standing waves from forming inside the crystal. The incoming laser beam then scatters off the acoustically
generated periodic refractive index, resulting in a diffracted beam at a diffraction angle θ. By switching on and off the transducer, we can effectively pulse
the 1st order diffracted laser beam.
We use a commercially available Isomet 1205C-2 AOM driven by an 80
MHz RF oscillator at 1.2 Watts. It is designed to operate at wavelengths
between 633-830 nm with a maximum 1st order deflection efficiency of ≥ 85%
at a maximum beam diameter of 2.0 mm. Narrowing down the beam diameter results in a slightly lower deflection efficiency, but can also significantly
shorten the pulse’s rise time τ . To obtain steep edged rectangular shaped
36
pulses we opt for a minimal rise time of approximately 25 ns, at a (I = 1/e2 )
beam diameter of 100-150 µm and a maximum deflection efficiency of about
60%.
Figure 3.7 shows a schematic of the pulse mechanism. The overlapped
master laser (ML) and slave laser (SL) beams are focused down through the
AOM using a telescope, after which only the 1st order diffracted beam is
coupled into an optical fibre leading to the experiment. Driving the AOM
is an 80 MHz RF oscillator which is coupled to a TTL (transistor-transistor
logic) driven electronical switch. The switch allows us to turn on and off
the RF signal at an arbitrary pulse length, thereby effectively pulsing the
fibre-coupled laser beam.
RF 80 MHz
TTL Signal
Switch
λ
Beam
Dump
Amplifier
0th
2
1st
PC
PBS For
Polarization
Filtering
AOM
€
λ
2
Lens
ML + SL Beam
Optical
Fibre
€
To
Experiment
Figure 3.7: A simplified schematic of the AOM based laser pulsing mechanism.
To account for the latency between the TTL pulse and the actual laser
pulse we measure the laser pulse on a photodiode (PD) at the point of interest
(e.g. at the end of the optical fibre), and display both signals (TTL + PD)
on an oscilloscope. Figure 3.8 shows a measurement for a 200 ns pulse after
the optical fibre leading to the experiment. The pulse delay includes the
latency of all involved electronics and optics.
37
TTL Pulse
PD Signal
Signal (a.u.)
2.5
τ ≈ 25 ns
2
1.5
Delay
1
≈ 0.7 µs
0.5
0
0
0.2
0.4
0.6
0.8
Time (µs)
1
1.2
1.4
Figure 3.8: Oscilloscope measurement of the TTL signal and the corresponding laser pulse measured on a photodiode (PD). We observe a laser pulse delay
of approximately 0.7 µs relative to the TTL pulse. The laser pulse rise time τ
is approximated to be 25 ns. The slower rising drift of the pulse’s maximum
signal intensity is due to the thermalization of the AOM crystal, which takes
approximately 0.4µ s to settle.
A often discussed problematic feature of the AOM is the crystal’s thermalization [6], which, from the moment the transducer is turned on, takes
approximately 0.4 µs before it stabilizes. This thermalization effect leads to
beam pointing of the diffracted beam, i.e. it leads to a temporarily drifting
diffraction angle θ. This beam pointing effect causes the fibre coupling efficiency to slowly drift until the thermalization settles. For longer pulses, in
the order of tens of µs, this effect can be largely neglected. In general, the
effect may be minimized by tuning the fibre coupling to the specific pulse
length used.
3.2
3.2.1
Sample preparation and Laser Injection
Sample preparation
We initially load 7 × 107 87 Rb atoms into a vacuum chamber using an ultrapure evaporative 87 Rb dispenser. The atomic cloud is laser cooled using
a laser slightly detuned from the F = 2 ←→ F = 3 cycling transition and
trapped in a magneto optical trap (MOT) [9]. The initial cloud is then
cooled to ' 50 µK using optical molasses [22]. A repump laser is used to
38
pump any atoms that have decayed into the F = 1 state back into the MOT
cycle.
After the optical molasses, we switch off the repump lasers with the MOT
lasers still on until almost all the atoms have decayed back into the F = 1
ground state. The atoms are then loaded into a Ioffe-Pritchard type magnetic
trap, where it is further cooled to ' 10 µK using RF evaporation [4][5]. In the
magnetic trap, only atoms in the |F = 1, mF = −1i state remain, as the other
magnetic sublevels are untrappable (see figure 2.2). Finally, the magnetic
trap potential’s bottom is set at the magic field value of Bm = 3.2 G. At this
point the trap contains ' 106 atoms.
3.2.2
Laser Injection
Once the atoms are prepared in the |F = 1, mF = −1i ground state, we turn
on the Raman lasers and inject the laser light into the vacuum chamber and
through the atomic cloud.
Straight from the optical fibre, the light is arbitrarily linearly polarized.
To ensure a correct polarization with respect to the quantization axis (which
lies parallel to the bias magnetic field at the magnetic trap bottom), the laser
beam goes through a half-wave plate (λ/2) and a polarizing beam splitter
(PBS). The PBS is positioned so that the reflected beam’s linear polarization
is perpendicular to the trap bottom’s magnetic field direction, i.e. the Raman
light’s polarization is σ + +σ − with respect to the quantization axis. The halfwave plate is used to maximize intensity of the reflected beam.
The Raman laser beam is directed through the centre of the atomic cloud,
so that the position of the Gaussian laser beam’s peak intensity coincides with
cloud’s highest atom number density.
39
Mirror
From Raman
Lasers
Optical
Fibre
!
B
Atomic
Cloud
Magnetic Trap
Potential
€
λ
2
PBS
Figure 3.9: Before the Raman lasers are sent through the trapped atomic
cloud, the beams’ polarizations are adjusted to be linearly polarized perpendicular to the cloud’s quantization axis. The quantization axis is defined
by the bias magnetic field€direction B̃ at the bottom of the magnetic trap
potential. The polarizing beam splitter (PBS) only reflects light that has
a linear polarization perpendicular to B̃. A half-wave plate λ/2 is used to
maximize the intensity of the reflected beam.
3.3
Imaging
Information about the atomic cloud, and the |F = 2, mF = 1i ground state
population in particular, is obtained through absorbtion imaging. We measure the attenuation of laser light passing through the atomic cloud by imaging the cloud’s shadow on a CCD camera.
For the imaging of atoms in the F = 2 ground state we probe the cloud’s
absorbance using laser beam resonant to the F = 2 −→ F = 3 transition
(3.11) and linearly polarized with respect to the quantization axis. A 90:10
(transmition : reflection) beam splitter first reflects part of the beam through
the atoms after which it is reflected back into the cloud by a mirror mounted
inside the vacuum chamber. Part of the back-reflected beam then travels
through the beam splitter and is focused onto the CCD camera. See [13] for
a more detailed explanation of absorption imaging. To prevent imaging of
atoms in the non-trappable |F = 2, mF = (0, −1, −2)i ground states we wait
until all of these atoms have been lost from the trapped cloud before taking
an image (typically a few ms).
40
Mirror
Atomic
Cloud
90:10
Beam
Splitter
Probe Laser
Beam
CCD Camera
Figure 3.10: Simplified drawing of the imaging setup. The cloud’s absorbance
is probed by a laser beam resonant to the F = 2 −→ F = 3 transition.
The law of Bouguer-Lambert-Beer relates the attenuation of light travelling a distance l through a column of material (the atomic cloud) to the
scattering cross-section σ and number density n inside the column of the
attenuating species (the atoms) in the material. It states that the intensity
of light decreases exponentially with distance inside the material:
If = I0 e−σnl ,
(3.1)
where I0 and If are the intensity of light before and after attenuation respectively.
The optical depth (O.D.) is defined as the product of the scattering crosssection and column density:
O.D. ≡ σn
(3.2)
We may then express the optical depth in terms of the transmittance:
If
.
(3.3)
O.D. = − ln
I0
In order to retrieve the optical depth from the probe beam, we take three
images: a light image of the un-attenuated probe beam, taken in the absence
of atoms; a shadow image of the attenuated probe beam as it has passed
through the atomic cloud; and finally a dark image, in the absence of both
the atoms and probe beam. The dark image is substracted from the other
two images to correct for any unwanted background light and the optical
depth per pixel can then be calculated as follows:
Shadow − Dark
O.D. = − ln
.
(3.4)
Light − Dark
41
An example of an imaged atomic cloud with population in the F = 2 ground
state is given in figure 3.11.
(a) Light
(b) Dark
(c) Shadow
(d) Optical Depth (O.D.)
Figure 3.11: Absorption imaging of a magnetically trapped atomic cloud
with population in the F = 2 ground state. The optical depth (d) is obtained
through equation 3.3.
42
Chapter 4
Experimental Results
In the following chapter we present our experimental results of our attempt
to produce stimulated Raman transitions. In order to prove that we are
indeed dealing with coherent Raman transitions we fit the imaging results of
several different measurements to their respective theoretical expectations.
All measurements were done under the following conditions and assumptions:
1. Both Raman lasers were locked at a detuning ∆3 = −π × 2.72(5) Ghz
(see figure 2.7) and phase-locked at the exact Raman resonance frequency difference of 6.834 678 114 GHz. Scanning the lasers around the
Raman resonance frequency was done by fine-tuning the phase locked
beat note through the reference microwave oscillator (see section 3.1.2).
2. All laser intensities given in this section are the peak intensities defined
as
2P
,
(4.1)
I=
πw2
where w is the 1/e2 Raman beam waist at the atomic cloud, and P
is the laser power. In our case w was approximately 140 µm and P
could be varied anywhere between 0 and approximately 75 µW. The
exact power of the lasers at the atomic cloud turned out difficult to
determine precisely; we estimated an error of ±20%.
3. The intensity ratio of the Raman lasers was changed by manually adjusting the master laser’s power through a combination of polarizing
beam splitter and half-wave plate positioned before the optical fibre
and AOM. The total laser intensity was changed in a similar fashion.
Unfortunately, the stability of the laser intensities were not optimal.
43
To correct for power fluctuations of the individual lasers, we have estimated an individual laser power fluctuation of ±2%.
4. We assume a constant differential Zeeman shift of the |F = 1, mF = −1i
and |F = 2, mF = 1i ground states throughout the entire atomic cloud.
This can be justified by the fact that the derivative of the Zeeman shifts
of these ground states is equal at the magic field Bm .
5. Images of the F = 2 atoms were taken after all the non-trappable atom
had left the atomic cloud, so that the only atoms that were imaged
were the ones in the |F = 2, mF = (1, 2)i ground states.
Small Beam, Big Trap, Moving
Atoms
Figure 4.1 shows an imaging result of F = 2 atoms after a 5 ms Raman
laser pulse near Raman resonance, strongly suggesting a successful implementation of stimulated Raman transitions.
5 ms
Figure 4.1: F = 2 −→ F = 3 absorbtion image of atomic cloud taken after
a 5 ms Raman resonant laser pulse. The red circle represents roughly where
the Raman laser beam strikes the atomic cloud. Notice how the atomic cloud
is much larger compared to the laser beam. Due to the oscillatory nature
of the magnetic trap, the Raman transferred atoms have likely redistributed
105 msthemselves during the Raman laser pulse.
4.1
Doppler- and Power Broadening
We start by examining two common line-broadening effects: Doppler- and
power broadening.
4.1.1
Doppler Broadening
One of the peculiarities of a stimulated Raman transition is that for copropagating laser beams it is practically free of Doppler broadening, but ex44
periences maximal Doppler broadening for counter-propagating beams. This
is exactly contradictive to single-photon transitions and two-photon ladder
transitions.
An atom moving at velocity v towards a laser beam will observe a Doppler
shifted laser frequency ω 0 :
ω 0 = ω + v · k,
(4.2)
where ω is the laser frequency observed when at rest and k is the laser’s wave
vector with |k| = ω/c. The effective Doppler shift of the Raman frequency
of the combined pump (P) and Stokes (S) lasers ωR = ωP − ωS can thus be
written as
δDoppler = ωR0 − ωR
= (ωP0 − ωS0 ) − (ωP − ωS )
(4.3)
= v · (kP − kS ) ,
Clearly, the Raman Doppler shift is minimized for co-propagating beams and
is largest for counter-propagating beams.
Let us now explicitly calculate the Doppler broadened profile of the resonant Raman transition for counter-propagating laser beams. For perfectly
counter-propagating beams, the Doppler shifted pump and Stokes laser frequencies observed by an atom moving with velocity v towards the pump
beam can be written as
v
0
ωP = ωP 1 +
c
(4.4)
v
0
,
ωS = ωS 1 −
c
so that the Doppler shifted Raman frequency of the combined lasers becomes
ωR0 = ωP0 − ωS0
v
= ωP − ωS + (ωP + ωS ) .
c
(4.5)
If Pv (v)dv is the fraction of atoms with velocities v to v + dv, then the
corresponding distribution of Raman laser frequencies observed by the atom
is
dv
PωR0 (ωR0 ) dωR0 = Pv vωR0
dωR0
dωR0
0
(4.6)
c
ωR − ωR
0
=
Pv c
dωR ,
ωP + ωS
ωP + ωS
ω 0 −ω
where vωR0 = c ωRP +ωRS is the atom velocity towards the pump beam corresponding to a shifted Raman laser frequency ωR0 . Assuming a thermal
45
velocity distribution given by the Maxwell distribution:
r
m
mv 2
Pv (v)dv =
exp −
dv,
2πkB T
2kB T
(4.7)
with kB the Boltzmann constant, m the atomic mass and T the temperature.
The Raman laser frequency distribution can then be written as a Gaussian
profile:
s
"
0
2 #
2
2
1
mc
ω
−
ω
mc
R
R
PωR0 (ωR0 ) dωR0 =
exp −
dωR0 . (4.8)
ωP + ωS 2πkB T
2kB T ωP + ωS
For a temperature (of the trapped atomic cloud) in the order of 10 µK equation 4.8 predicts a Doppler broadened Gaussian line profile with a FWHM
in the order of 100 kHz.
0.04
Counter-Propagating
Lasers
experiment
Fit to Theory
Co-Propagating
Lasers
experiment
Lorentzian Fit
0.3
Optical Depth
Optical Depth
0.03
0.02
FWHM = 322 kHz
0.2
FWHM
= 9 kHz
0.1
0.01
0
-200
100
0
100
Raman Detuning (kHz)
200
-20
(a)
-10
0
10
Raman Detuning (kHz)
20
(b)
Figure 4.2: Stimulated Raman transition line measurements for counterpropagating lasers (a) and co-propagating lasers (b).
For counterpropagating lasers we clearly see a Doppler broaded profile that fits equation
4.8. For co-propagating beams the transition line is best fitted to a Dopplerfree Lorentzian profile. Notice that the maximum population transfer is much
lower in the presence of Doppler broadening.
Figure 4.2 shows the results of measurements done for both counterpropagating and co-propagating Raman lasers. The counter propagating
measurement was done by having the back-reflected Raman beams overlap
with the incident beams. The maximum optical depth (O.D.) is plotted
against the Raman scan frequency of the Raman lasers through Raman resonance. Although the optical depth is not equal to the atom number density,
it is proportional to it. We clearly see a Doppler broadened Gaussian profile
46
for the counter-propagating lasers and a practically Lorentzian Doppler free
profile for co-propagating lasers.
The Doppler broadened data is fitted to equation 4.8, while the Doppler
free data is fitted with an arbitrary Lorentzian. The data shows a good fit
to the theory. From the Doppler broadened fit, we extract an atomic cloud
temperature of 29.7 ± 1.7 µK, which is within realistic expectation.
We also observe a significant decrease in maximum population transfer
in the presence of Doppler broadening.
4.1.2
Power Broadening
Another common line-broadening effect that we have to take into account is
that of power broadening. From 2.17, we get that the maximum undamped
amplitude A of a Rabi cycle is
Ω2R
,
A= 2
2
ΩR + δeff
(4.9)
where δeff = δ + δΛ and ΩR is the two photon Raman Rabi frequency given
by equation 2.37. The amplitude drops to half maximum at δeff = ΩR , so
we expect the FWHM of a Raman transition to scale linearly with the twophoton Rabi frequency:
p
(4.10)
FWHMPower = 2 ΩR ∝ IP · IS
Figure 4.3 shows two Raman transition lines for two different laser intensities: IS = 0.92 ± 0.18 mW/mm2 and IS = 2.76 ± 0.55 mW/mm2 , and
IP = 0.42(1) · IS in both cases. Using equations 2.37 and 4.10, we calculate
for these intensities a FWHM of approximately 1.3 ± 0.3 kHz and 4.0 ± 0.8
kHz respectively. A Lorentzian fit to the experimental data yields a FWHM
of 1.7 ± 0.1 kHz and 6.8 ± 0.3 kHz respectively.
47
0.3
Experiment
Lorentzian
Optical Depth
Optical Depth
0.4
IS = 0.92 ± 0.18 mW/mm2
IP = 0.42(1) · IS
0.2
FWHM =
1698 ± 146 Hz
0.1
0.3
IS = 2.76 ± 0.55 mW/mm2
IP = 0.42(1) · IS
0.2
Experiment
Lorentzian
FWHM =
6794 ± 292 Hz
0.1
0
0
-4
-2
0
Raman Scan Frequency (kHz)
2
-15
-10
-5
0
5
10
Raman Scan Frequency(kHz)
15
Figure 4.3: Power broadening of a Doppler-free stimulated Raman transition. The Raman transition line is shown for two intensities I = IP + IS .
The transition linewidth increases with higher Raman laser intensities as
predicted.
For low intensities (but still high enough so that the power broadening
is greater than the co-propagating Doppler broadening), the linewidth produced by equation 4.10 comes close to experiment.
However, for higher intensities it consistently falls short compared to experiment. This is likely because the linewidth is also affected by light shifts,
which also depend on laser intensity. The correct expression of the linewidth
is therefore possibly better described by a combination of both effects.
The effect of light shifts on the power broadened Raman transition linewidth
becomes particularly apparant at greater differential light shifts. Figure 4.4
shows two Raman transition lines for intensity ratios considerably higher respectively lower than the optimum ratio R0 (see equation 2.33). For greater
differential light shifts we observe a strongly asymmetric line broadening
effect.
0.3
IS = 2.66 ± 0.53 mW/mm2
IS
= 1.1 · R0
IP
IS = 2.53 ± 0.51 mW/mm2
IS
= 0.7 · R0
IP
0.4
Optical Depth
Optical Depth
0.4
0.2
0.1
0
0.3
0.2
0.1
0
-15
-10
-5
0
Raman Scan Frequency (kHz)
5
-5
0
5
10
Raman Scan Frequency (kHz)
15
Figure 4.4: Inhomogeneous and asymmetric light shift induced Raman transition line broadening. The detuning from intermediate state was ∆3 =
−2π × 2.7 GHz. The data is fitted with an arbitrary asymmetric Lorentzian
function, where we have allowed the FWHM to vary sigmoidally.
48
A quantitative description of this asymmetric broadening effect is beyond this thesis, but the origin of the effect may very well be understood.
Remember that for intensity ratios unequal to R0 , the differential light shift
is proportional to the laser intensity (see figure 2.9). This means that for
a Raman laser with Gaussian intensity distribution, the induced differential
light shift should also be inhomogenous along the illuminated atomic cloud.
In fact, a very similar effect has been predicted and observed for microwave induced hyperfine transitions in optically trapped alkali atoms [8],
where the optical trap potential causes inhomogeneous light shifts in the
trapped atoms. This effect, convoluted with the spatial number density distribution of the trapped atomic cloud, could very well result in the asymmetric lineshape we also observe here.
4.2
4.2.1
Light Shifts
Differential Light Shifts
We have measured the differential light shift δΛ of the |F = 1, mF = −1i and
|F = 2, mF = 1i hyperfine ground states for different Raman laser intensities.
The Raman beams were set up in co-propagating manner with low laser intensities. The Stokes laser intensity was kept fixed at IS = 0.04(1) mW/mm2
while the pump laser’s intensity was changed through a combination of halfwave plate and polarizing beam splitter. All measurements were done with
equal pulse length of 10 ms. The detuning from the intermediate states was
kept constant at ∆3 = −2π × 2.7 GHz, for which equation 2.33 predicts
δΛ = 0 at a laser intensity ratio 1/R0 = IP /IS = 0.415.
The experimental results are given in figure 4.5. As expected, the differential light shift is minimal around R0 and becomes greater for ratios away
from R0 .
49
Optical Depth
0.15
IS = 0.04(1) mW/mm2
∆3 = −2π × 2.7 GHz
10 ms pulses
IP /IS = 0.14
IP /IS = 0.29
IP /IS = 0.42
IP /IS = 0.67
IP /IS = 0.81
0.1
0.05
-2
-1
0
1
Raman Scan Frequency (kHz)
2
3
Figure 4.5: Stimulated Raman transition lines for different laser intensity
ratios. The differential light shift δΛ of the Raman transition between |1, −1i
and |2, 1i ground states is minimized for laser intensity ratios near 1/R0 =
0.41. An increasing δΛ also causes a decrease in population transfer and an
increase of the transition linewidth.
Again, we observe an increasing linewidth for Raman transitions with
greater differential light shifts. For intensity ratios away from R0 , the differential light shift δΛ varies over the spatial Gaussian intensity profile of
the laser, therefore causing an effective inhomogeneous broadening of the
transition linewidth.
Notice also how, as expected, the maximum optical depth decreases for
greater differential light shifts. Reason for this is that for laser intensity
ratios away from R0 , it becomes impossible to tune the entire laser beam
into Raman resonance, as the produced light shifts are different along the
Gaussian spatial intensity distribution of the beam.
In order to investigate the differential light shifts more quantitatively, we
have plotted the detuned Raman resonance frequencies versus intensity pump
and Stokes laser intensity ratio IP /IS and fitted the data to δΛ = Λ2,1 − Λ1,−1
using equations 2.30 and 2.31 with free fit parameters ∆3 and IS . Not only
does the data show a perfectly linear dependency on laser intensity, the best
fit parameter values are remarkably close to expectation; we retrieve from
50
the data a best fitted Stokes laser intensity of IS = 0.047(1) mW/mm2 and
intermediate state detuning ∆3 = −2π × 2768 ± 36 MHz.
4
Experiment
Fit to Theory
3
δΛ (kHz)
2
1
0
−1
−2
0
0.2
0.4
0.6
0.8
1
IP /IS
Figure 4.6: Experimentally measured differential light shift δΛ fitted to theory
(equations 2.30 and 2.31). The experiment shows a near perfect fit to theory.
4.3
Rabi Dynamics
We conclude this chapter with the results of preliminary attempts to observe
Rabi oscillations of stimulated Raman transitions. The results of a typical
experiment is shown in figure 4.7.
The plot shows the maximum optical depths acquired from absorption
imaging of the atomic cloud after the application of resonant Raman laser
pulses with incrementally increasing pulse lengths. Assuming all other experimental parameters are the same for every measurement, the obtained data
should be consistent with the collective F = 1 ←→ F = 2 Rabi oscillations
of the atomic cloud. Note that since |F = 2, mF = 1i and |F = 2, mF = 2i are
both magnetically trapped states, the imaged atoms are a combination of
both state populations.
The specific measurement shown in figure 4.7 was done at an intermediate state detuning ∆ = −2π × 2.7 GHz and laser intensities IS = 0.04(1)
mW/mm2 and IP = 1/R0 · IS = 0.41 · IS . The two-photon Rabi frequency
calculated from equation 2.37 for these parameters is ΩR = 2π × 37 ± 7 Hz.
51
Optical Depth
0.4
0.3
0.2
20
40
60
80
Pulse Length (ms)
100
Figure 4.7: Experimental measurement of |2, 1i + |2, 2i population evolution
under the effect of Raman laser pulses. The maximum optical depth is plotted
against Raman pulse length. The laser intensities used were IS = 0.04(1)
mW/mm2 and IP = 0.41 · IS and the intermediate state detuning was ∆ =
−2π×2.7 GHz. Although we can arguably observe some degree of population
oscillations, the frequency and amplitudes are inconsistent.
Although we can arguably observe some degree of oscillations of the F = 2
population, the oscillation frequency and amplitudes are highly irregular and
inconsistent. Also, the population inversion cycle of a full Rabi oscillation
seems to be highly inefficient: on average, more than half of the atoms stay
in the F = 2 state.
This ineffiency is very likely the result of using a Gaussian laser beam
confined to a much larger atomic cloud. This results in a wide range of Rabi
oscillations driven at different frequencies, therefore impeding the observation
of synchronous Rabi dynamics [7].
Also, due to the oscillatory nature of the magnetic trap, atoms may travel
in and out of the interaction region of the Raman beam, effectively causing
a decoherence of the Rabi oscillations altogether. We have already seen in
figure 4.1 how the Raman transferred atoms redistribute themselves along
the magnetic trap.
Combined with the effects of intensity fluctuations portrayed in figure ,
it is not too far-fetched that all of these effects may be contribute to the
irregularities of the observed population oscillations.
52
Chapter 5
Conclusion
5.1
Summary
In this thesis we have discussed and demonstrated stimulated Raman transitions between hyperfine ground states of Rubidium-87 atoms.
We started by laying out the physics of Raman transitions in general
and derived a more detailed theoretical description for the specific case
of 87 Rb. We elucidated the importance of careful tuning of the Raman
laser intensities in order to minimize the differential light shift of the hyperfine 52 S1/2 ground states relevant to the transition. For the specific
|F = 1, mF = −1i −→ |F = 2, mF = 1i Raman transition we have shown that
there are two possible Raman transition paths that destructively interfere,
therefore significantly damping the two-photon Rabi frequency. Solving the
optical Bloch equations including spontaneous Raman scattering showed a
relatively strong decoherence of Raman Rabi oscillations.
Next, we introduced the experimental setup designed to drive and observe
stimulated Raman transitions in magnetically trapped 87 Rb. Essential in this
setup is the phase locked laser system, which enables the two Raman lasers
to operate with an ultra-stable frequency difference and minimal phase noise.
We have demonstrated that with this phase lock mechanism engaged we are
able to lock the combined Raman lasers’s beat note to the 52 S1/2 ground state
hyperfine splitting with Hz precision and reduce the beat note’s linewidth to
be less than 100 Hz (the spectrum analyzer’s resolution limit). Additionally,
we have demonstrated how we are able to quantify |F = 1, mF = −1i −→
|F = 2, mF = 1i population transfer using a combination of |F = 1, mF = −1i
state preparation and absorbtion imaging.
Lastly, we presented our experimental results. We have reported absorption imaging results of successful stimulated Raman transitions. We calcu-
53
lated and experimentally demonstrated the Doppler broadening of a Raman
transition line and showed that the effect is maximal for counter-propagating
laser beams. For co-propagating lasers, power broadening and light-shift induced broadening are the most prominent line broadening effects. Furthermore, we showed the dependency of the differential light shift on the Raman
lasers’s intensity ratio, and revealed how the experimental data is in perfect
agreement with theory. An important observation was that the maximum
population transfer decreased for greater differential light shifts, confirming
the importance of careful tuning of Raman laser intensities. We concluded
this work with preliminary results of Raman Rabi measurements, which arguably showed an oscillatory transferred population. However, the oscillation
frequency and amplitude were highly irregular and inconsistent, not unlikely
the result of fluctuating differential light shifts and the asynchronous effect
of the Gaussian laser intensity profile.
54
5.2
Outlook
The first step forward from here is to improve the current setup in such a
way that we are able to produce and reproduce coherent Rabi oscillations
for at least one full cycle. We must improve the stability of the individual
Raman laser intensities, perhaps by implementing a commercially available
laser noise eater. Also, we will have to find a way to minimize the effects of
having a Gaussian laser intensity profile, either by changing it to a flat-top
profile or by further confining the atomic cloud.
As for the damping of the two-photon Rabi frequency due to destructively
interfering Raman paths, an approach worth investigating is that of using the
D1 transition line instead of D2 . Although the same destructive interference
occurs for Raman paths through the D1 line, the hyperfine splitting of the
52 P1/2 F = 1 and F = 2 states is about 5 times larger than for the 52 P3/2
states, significantly reducing the damping of the two-photon Rabi frequency.
Besides accomplishing coherent stimulated Raman transitions of magnetically trapped 87 Rb in general, an entirely different challenge will present itself
when we want to manipulate individual microtraps on a chip (which we have
briefly shown in the introduction, see figure 1.1). For this specific purpose,
we are working on a technique that employs a spatial light modulator (SLM)
to transform one Gaussian laser beam spot into multiple smaller spots that
coincide exactly with individual microtrapped atom clouds. Specifically challenging will be to eliminate Doppler broadening of the Raman transitions,
as the laser light backreflected off the reflective chip surface will effectively
produce a counter-propagating Raman laser allignment.
55
Bibliography
[1] Lucas Beguin. Measurement of the van der Waals interaction between
two Rydberg atoms. PhD thesis, Institut d’Optique Graduate School,
2013.
[2] Igor Dotsenko. Raman spectroscopy of single atoms. PhD thesis, Mathematisch-Naturwissenschaftlichen Fakultät der Rheinischen
Friedrich-Wilhelms-Universität Bonn, October 2002.
[3] C.P. Pearman et al. Polarization spectroscopy of a closed atomic transition: Applications to laser frequency locking. J. Phys B: At. Mol. Opt.
Phys., 35(5141), 2002.
[4] J. Reichel et al. Atomic micromanipulation with magnetic surface traps.
Phys. Rev. Lett., 83(3398), 1999.
[5] R. Folman et al. Microscopic atom optics: from wires to an atom chip.
Adv. At., Opt. Phys., 48:263–356, 2002.
[6] T. Pfau et al.
A two-frequency acousto-optic modulator driver
to improve the beam pointing stability during intensity ramps.
arXiv:physics/0701183, February 2008.
[7] Hyosub Kim Han-gyeol Lee and Jaewook Ahn. Ultrafast rabi oscillation of a gaussian atom ensemble. arXiv:1409.5549v1 [physics.atom-ph],
2014.
[8] Sin Hyuk Yim Huidong Kim and D. Cho. Inhomogenous broadening of
a ground hyperfine transition in an optical trap. J. of Korean Phys.,
51(4):1279–1283, October 2007.
[9] A. S. Sanz J. Pérez-Rı́os.
arXiv:1310.6054, 2013.
How does a magnetic trap work?
56
[10] André Xuereb James Bateman and Time Freegarde. Stimulated raman
transitions via multiple atomic levels. Phys. Rev. A, 81(043808), April
2010.
[11] B. E. King. Quantum State Engineering and Information Processing
with Trapped Ions. PhD thesis, University of Colorado, 1999.
[12] G. Breyta C. S. Yannoni M.H. Sherwood I.L. Chuang L. M. K. Vandersypen, M. Steffen. Experimental realization of shor’s quantum factoring
algorithm using nuclear magnetic resonance. Nature, 414:883, 2001.
[13] G.B. Mulder. Towards the imaging of atoms on a 2d magnetic chip
lattice. Master’s thesis, University of Amsterdam, 2013.
[14] T. P. Orlando. Superconducting Circuits and Quantum Computation.
2004.
[15] Daryl W. Preston.
Doppler-free saturated absorbtion:
specroscopy. Amer. J. of Phys., 64:1432–1436, 1996.
Laser
[16] M. Saffman and T. G. Walker. Quantum information with rydberg
atoms. arXiv:0909.4777, 2010.
[17] Daniel A. Steck. Rubidium 87 line data. http:/steck.us/teaching, 2 May
2008.
[18] Daniel A. Steck. Quantum and Atom Optics. http:/steck.us/teaching,
2014.
[19] Mark Saffman Thad G. Walker. Entanglement of two atoms using rydberg blockade. arXiv:1202.5328, 2012.
[20] H. Schlatter L. Torralbo-Campo G.B. Mulder J. Naber M.L. Soudijn A.
Tauschinsky C. Abarbanel B. Hadad E. Golan R. Folman V.Y.F. Leung, D.R.M. Pijn and R.J.C. Spreeuw. Magnetic-film atom chip setup
with 10 µm period lattices of magnetic microtraps. arXiv:1311.4512v1
[physics.atom-ph], 19 November 2013.
[21] Amir Waxman. Coherent manipulations of the rubidium atom ground
state. Master’s thesis, Ben-Gurion University of the Negev, December
3, 2007.
[22] N.J. van Druten R.J.C. Spreeuw Y.F.V. Leung, A. Tauschinsky. Microtrap arrays on magnetic film atom chips for quantum information
science. Quantum Information Processing, 10(6):955–974, 2011.
57