Can optical squeezing be generated via polarization self-rotation in a thermal vapour cell?

Abstract

The traversal of an elliptically polarized optical field through a thermal vapour cell can give rise to a rotation of its polarization axis. This process, known as polarization self-rotation (PSR), has been suggested as a mechanism for producing squeezed light at atomic transition wavelengths. In this paper, we show results of the characterization of PSR in isotopically enhanced Rubidium-87 cells, performed in two independent laboratories. We observed that, contrary to earlier work, the presence of atomic noise in the thermal vapour overwhelms the observation of squeezing. We present a theory that contains atomic noise terms and show that a null result in squeezing is consistent with this theory. Comment: 10 pages, 11 figures, submitted to PRA. Please email author for a PDF file if the article does not appear properly

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arXiv:quant-ph/0510165v2 24 Oct 2005
Can optical squeezing be generated via polarization self-rotation in a thermal vapour
cell?
M. T. L. Hsu, G. H´ etet, A. Peng, C. C. Harb, H.-A. Bachor, M. T. Johnsson, J. J. Hope and P. K. Lam∗
Australian Centre for Quantum-Atom Optics, Department of Physics,
Australian National University, ACT 0200, Australia
A. Dantan, J. Cviklinski, A. Bramati and M. Pinard
Laboratoire Kastler Brossel, Universit´ e Pierre et Marie Curie, case 74, 75252 Paris cedex 05, France
(Dated: February 1, 2008)
The traversal of an elliptically polarized optical field through a thermal vapour cell can give rise
to a rotation of its polarization axis. This process, known as polarization self-rotation (PSR), has
been suggested as a mechanism for producing squeezed light at atomic transition wavelengths. In
this paper, we show results of the characterization of PSR in isotopically enhanced Rubidium-87
cells, performed in two independent laboratories. We observed that, contrary to earlier work, the
presence of atomic noise in the thermal vapour overwhelms the observation of squeezing. We present
a theory that contains atomic noise terms and show that a null result in squeezing is consistent with
this theory.
I.INTRODUCTION
Squeezing is the reduction of the noise variance of
an optical field below the quantum noise limit (QNL).
Many applications, ranging from increased sensitivity of
interferometric measurements [1] to quantum entangle-
ment based information protocols [2, 3, 4], are reliant on
squeezed light. Recently, Duan et al. [5] proposed a long-
distance quantum communication network that is based
on the interaction of atomic ensembles with squeezed and
entangled light beams. To achieve such goals, squeezed
light at atomic wavelengths is required.
Conventionally, squeezing can be generated via effi-
cient non-linear optical processes, such as χ(2)paramet-
ric down-conversion [2, 6, 7].
dows of non-linear optical crystals, however, may not
coincide with some atomic transitions.
commonly used Sodium and Rubidium atomic transition
wavelengths are difficult to access via χ(2)crystals. An-
other method of generating squeezed light is to utilize
the χ(3)atomic Kerr effect at the required atomic wave-
length. These experiments, however, require ultra-cold
atoms confined in cavities and are therefore technically
challenging [8, 9].
The transparency win-
For example,
Recently, there has been a proposal for generating
atomic wavelength squeezing via the single traversal of
an optical field through a thermal vapour cell [10]. This
proposal promises a simple, scalable and cost-effective
means of generating squeezed light for Rb and potentially
for other atomic species. Due to the ac Stark shift and
optical pumping-induced refractive index changes of the
atomic vapour, an elliptically polarized input field will
experience an intensity dependent rotation of the optical
polarization axes [11]. This effect, known as polariza-
∗Email: ping.lam@anu.edu.au
tion self-rotation (PSR), was suggested as a non-linear
mechanism for squeezing [10, 12]. Assuming negligible
atomic spontaneous emission noise, Matsko et al. [10]
developed a phenomenological model that treats PSR as
a cross-phase modulation mechanism. In the situation of
a linearly polarized input field propagating through the
vapour cell, a non-linear cross-phase interaction occurs
between the two circularly polarized field components.
This results in the squeezing of the output vacuum field
mode that is orthogonally polarized to the input field.
Analogous to cross-phase modulation squeezing in opti-
cal fibres [13, 14, 15, 16, 17, 18], it was suggested that 6
dB of PSR squeezing is possible with thermal Rb vapour
cell. Subsequently Ries et al. [19] reported an observa-
tion of 0.85 dB maximum squeezing from a Rb vapour
cell and attributed their squeezing to PSR.
The phenomenological model of PSR squeezing by
Matsko et al. [10] ignored effects such as atomic sponta-
neous emission. In contrast, Josse et al. [20] pointed out
the importance of noise terms arising from the atomic
dynamics that could possibly degrade, if not totally de-
stroy, squeezing.The model of Josse et al.
based on the interaction of a linearly polarized field with
4-level atoms. They showed that in the high saturation
regime, the atomic noise contribution could potentially
be larger than the squeezing term. Nevertheless, in the
low saturation regime and at sideband frequencies larger
than the atomic relaxation rate, squeezing on the vacuum
mode can be generated via the cross-Kerr effect induced
by the bright field. Such a regime, however, can only be
obtained with ultra-cold trapped atoms enclosed in an
optical cavity [9].
This paper is structured as follows - In Section II, we
review the theoretical works of Matsko et al. [10] and
Josse et al. [20]. We modified the analysis of Josse et
al.
[20] to the case of a single traversal optical field
through a thermal vapour cell. In Section III, we report
measurements of both the transmittivity and the PSR of
an elliptically polarized field through an isotopically en-
[20] was

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hanced87Rb vapour cell on both the D1and D2lines. We
then study the noise properties of the outgoing vacuum
field. The parameter regime investigated extends beyond
the squeezing regime reported in Ref. [19]. In contradic-
tion to the results in Ref. [19], no optical squeezing was
observed. Instead, we observed excess quadrature noise
above the QNL for a wide range of parameters. Finally,
in Section IV we relate experimental results to the theory
and show that under our experimental conditions, where
atomic spontaneous emission is significant, squeezing is
overwhelmed by atomic noise terms.
II. THEORY
A. Cross-phase modulation squeezing
For cross-phase modulation squeezing in fibers, a
bright input optical pulse in the x-polarization is deliv-
ered into a weakly birefringent optical fiber. As a result
of the χ(3)non-linearity in the fiber, the annihilation (ˆ ay)
and creation (ˆ a†
y) operators for the y-polarized vacuum
field become coupled [13, 14, 15, 16]. The equation of
motion for the y-polarized field, in the rotating frame, is
given by
∂
∂zˆ ay(z,t) = iκ
3(2|?ˆ ax?|2ˆ ay+ ?ˆ ax?2ˆ a†
y) (1)
where κ = n2¯ hω2
non-linear index coefficient of the medium, ω0is the car-
rier frequency and A is the effective transverse area of the
propagating field. The last term of Eq. (1) describes the
cross-Kerr coupling between the bright x- and vacuum y-
linearly polarized fields, and is responsible for generating
squeezing in the y-polarized field.
Matsko et al. [10] proposed that the PSR effect in
atomic vapour can be used to generate vacuum squeez-
ing. Their proposal was related to the mechanism of
cross-phase coupling between two orthogonal polariza-
tion fields. We consider the PSR effect [11], where an
elliptically polarized field undergoes a rotation in its po-
larization ellipse upon propagation through an atomic
medium. For an optically thin medium, the rotation an-
gle is given by
0/(cA) is the Kerr coefficient, n2is the
φ = Gǫ(0)l (2)
where G is the PSR parameter (dependent on the input
field intensity and frequency), ǫ(0) is the input field ellip-
ticity (assumed to be small and constant during propaga-
tion, ǫ(0) = ǫ(l)) and l is the length of the medium. One
could take the analogy of the PSR effect to the quantum
regime by considering a bright linearly x-polarized input
field. The PSR effect projects fluctuations of the bright
x-polarized field onto the y-polarized vacuum field. The
relative phase between the x- and y-polarized fields then
provides amplification or attenuation of the y-polarized
field. This effect could potentially result in the reduction
of the quantum fluctuations of the y-polarized field.
We will now introduce a methodical representation for
our optical field. For a measurement performed in an
exposure time T, a freely propagating single-mode optical
field can be described by the electric field operator given
by
ˆE(z,t) = E0
?
ˆ a(z,t)ei(kz−ωt)+ ˆ a†(z,t)e−i(kz−ωt)?
¯ hω
2ǫ0cTA, ˆ a(z,t) and ˆ a†(z,t) are the slowly
varying field envelope annihilation and creation opera-
tors, respectively. z is the field propagation axis, ω is the
field carrier frequency and A is the quantisation cross-
section area. We can simplify the expression by intro-
ducing χ = kz − ωt and phenomenologically extend the
classical PSR to the quantum regime. The resulting y-
polarized field at the output of the PSR medium is given
by
(3)
where E0=
?
ˆEy(l) = E0
?
ˆ ay(0)?eiχ− iGlcosχ?
+ˆ a†
y(0)?e−iχ+ iGlcosχ??
(4)
The noise variance for theˆEy(l) field, taking into account
a phenomenogical absorption parameter α [10], is given
by
?ˆE†
y(l)ˆEy(l)? = E0
??
1 − 2Glsinχcosχ
+G2l2cos2χ
?
e−αl+ (1 − e−αl)
?
(5)
where for appropriate values of the phase χ, squeezing of
the y-polarized field can be observed. Such a model pre-
dicts squeezing values of 6-8 dB below the QNL. However,
crucial details such as spontaneous emission and atomic
noise are completely ignored, the effects of which can re-
duce, if not completely destroy, squeezing.
B.Squeezing in a 4-level System
Since optical pumping is the main cause of PSR in
the high saturation regime [10, 23], which is the relevant
regime in our experiment, we can approximate the D1
and D2lines of87Rb using a 4-level atom model. In such
a regime, the influence of atomic coherences are negligi-
ble. We thus explore the alternative cross-Kerr squeezing
model proposed by Josse et al. [20]. In the model, 4-level
atoms interact with two orthogonal circularly polarized
fields, as shown in Fig. 1. In the experiment of Ref. [9],
squeezing was obtained in the vacuum field (orthogo-
nally polarized to the bright input field) from ultra-cold
trapped atoms, enclosed in a cavity. The 4-level squeez-
ing model approximated the level structure of ultra-cold
Cesium atoms (|6S1/2,F = 4? to |6P3/2,F = 5?), used
in the experiment. In this section, we extend this cavity
model to a single-propagation scenario for a single-mode
bright x-polarized input field. We derive the equation

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FIG. 1: Two orthogonal σ+ and σ− circularly polarized light
fields interacting with a 4-level atomic system.
of motion describing the noise fluctuations of the output
y-polarized vacuum field. The interaction Hamiltonian is
given by
ˆ Hint = ¯ hnAeff
?l
0
dz
?
∆ˆ σ44(z,t) + ∆ˆ σ33(z,t)
−g
+ˆ a−(z,t)ˆ σ32(z,t) + ˆ a†
?
ˆ a+(z,t)ˆ σ41(z,t) + ˆ a†
+(z,t)ˆ σ14(z,t)
−(z,t)ˆ σ23(z,t)
??
(6)
where ˆ a+(z,t) and ˆ a−(z,t) are the respective slowly vary-
ing field envelope operators for the σ+and σ−circularly
polarized fields, n is the atomic density and g is the atom-
field coupling constant. The atomic dipole operator at
position z in the rotating frame is defined by locally av-
eraging over a transverse slice containing many atoms
ˆ σij(z,t) =
1
nAδz
?
zk∈δz
e
i(ωi−ωj)zk
c
|i?k?j|k
(7)
The optical Bloch equations for the atomic variables
are then given by
∂
∂tˆ σ14 = −(γ + i∆)ˆ σ14+ igˆ a+(ˆ σ11− ˆ σ44) +ˆF14
∂
∂tˆ σ23 = −(γ + i∆)ˆ σ23+ igˆ a−(ˆ σ22− ˆ σ33) +ˆF23
∂
∂tˆ σ11 = γ(ˆ σ33+ ˆ σ44) − igˆ a+ˆ σ41+ igˆ a†
∂
∂tˆ σ22 = γ(ˆ σ33+ ˆ σ44) − igˆ a−ˆ σ32+ igˆ a†
∂
∂tˆ σ33 = −2γˆ σ33+ igˆ a−ˆ σ32− igˆ a†
∂
∂tˆ σ44 = −2γˆ σ44+ igˆ a+ˆ σ41− igˆ a†
+ˆ σ14+ˆF11
−ˆ σ23+ˆF22
−ˆ σ23+ˆF33
+ˆ σ14+ˆF44
(8)
where we have introduced the spontaneous decay term
γ and Langevin noise operatorsˆFij that arise from the
coupling of atoms to a vacuum reservoir. The Maxwell
wave equations describing the σ+and σ−-polarized opti-
cal fields are given respectively by
?∂
?∂
∂t+ c∂
∂t+ c∂
∂z
?
?
ˆ a+(z,t) = igNˆ σ14(z,t)(9)
∂z
ˆ a−(z,t) = igNˆ σ23(z,t)(10)
where N is the total number of atoms. To deduce the
noise properties of the field, we linearize the equations
around the semi-classical steady state, and write the op-
erators in the form ˆ a = ?ˆ a? + δˆ a. Transforming into the
Fourier domain and linearizing Eqs. (8)-(10) yields the
equation of motion for the quantum fluctuations of the
y-polarized vacuum mode ˆ ay= −i(ˆ a++ ˆ a−)/√2, given
by
∂
∂¯ zδˆ ay= −Γ(ω)δˆ ay+ κ(ω)
where ¯ z = z/l and
?
δˆ ay− δˆ a†
y
?
+ˆFy
(11)
κ(ω) = κ(0)Λ(ω)
Γ(ω) = −iωl
(12)
c+ κ(ω) + κ(0)∗Λ′(ω)
Cγ
2(γ + i∆)1 + s
Ix(γ − iω)(2γ − iω)
2Ix(γ − iω)2− iω(2γ − iω)[(γ − iω)2+ ∆2]
Λ′(ω) = iωIx(γ − iω) − (γ − i∆)(γ − i∆ − iω)(2γ − iω)
2Ix(γ − iω)2− iω(2γ − iω)[(γ − iω)2+ ∆2]
where C = g2Nl/γc is the cooperativity parameter, Ix=
|g?ˆ ax?|2is the mean field intensity and s = Ix/(γ2+∆2)
is the saturation parameter. The last term of Eq. (11)
represents the atomic Langevin noise term and is respon-
sible for a loss or degradation of squeezing. Its exact form
and noise spectrum are given and discussed in Sec. IV.
Note that for ω = 0, the imaginary part of κ(0) from
the second term on the right hand side of Eq. (11) equates
to the first term on the right hand side of Eq. (1). This
turns out to be the parameter Gl given in Eq. (2). In the
4-level atom model, the PSR for one velocity class in-
creases with the number of atoms and is maximum when
∆2= γ2+ Ix. For a Doppler-broadened vapour, Gl can
be obtained by summing Eq. (12) over all the velocity
classes. Note that κ(ω) also gives the amplitude of the
cross-Kerr squeezing term in δˆ a†
ever, the associated atomic noise contribution must be
evaluated in order to obtain the total noise spectrum for
the output y-polarized field.
κ(0) =
1
Λ(ω) =
y, as in Eq. (11). How-
III.EXPERIMENT
In this section, we present experimental results ob-
tained from the two authoring institutions. Both exper-

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iments have similar experimental arrangement. In our
experiments, a coherent beam at 795 nm (or 780 nm)
was delivered from a Ti:Sapphire laser (Coherent MBR-
110), as shown in Fig. 2.
sured to be quantum noise limited at sideband frequen-
cies ≥ 1 MHz. A small fraction of the beam was sent
through another Rubidium (Rb) vapour cell for satu-
rated absorption spectroscopy. This provided us with
a fine frequency reference for the laser and also allowed
the possibility of laser frequency stabilisation. The ma-
jority of the beam was sent through a polarizer which
transmitted the x-polarized field. In order to measure
The laser beam was mea-
FIG. 2: Schematic of experimental setup. All polarising optics
are of the Glan-Thompson type. FI: Faraday isolator, BS:
beam-splitter. Pol.: Polarizer, PBS: polarising beam-splitter,
λ/4: quarter wave-plate, λ/2: half wave-plate, PZT: piezo-
electric actuator.
the PSR and absorption of an input elliptically polarized
beam through the vapour cell, the orange-shaded con-
figuration of Fig. 2 was used. The x-linearly polarized
beam was converted into an elliptically polarized beam
using a λ/4 wave-plate. The beam (collimated to a waist
size of ∼ 425µm) then passed through an isotopically
enhanced87Rb vapour cell (75mm length), which was
temperature stabilised at 72◦C (which corresponded to
an atomic density of 1011atoms/cm3). The vapour cell
was enclosed in a two-layer µ-metal alloy cylinder, with
end caps. The stray magnetic fields within the shield-
ing region were measured to be < 2 mG in all three
spatial axes. The output beam from the cell was then
analysed using a balanced polarimeter setup, which con-
sisted of a λ/2 wave-plate, a polarising beam-splitter and
two balanced photo-detectors. The λ/2 wave-plate was
adjusted to balance the powers in the x- and y-linearly
polarized beams from the outputs of the polarising beam-
splitter, when the frequency of the laser was tuned far
off-resonance. Thus any rotation of the axis of the input
elliptically polarized beam could be measured using the
relationship [11]
φ =
V1− V2
2(V1+ V2)
(13)
where V1 and V2 are the DC signals from the photo-
detectors.
To measure the quadrature noise properties of the y-
linearly polarized vacuum beam, we then performed ho-
modyne detection, as shown in Fig. 2, using the x-linearly
polarized output of the polarising beam-splitter as a local
oscillator.
A.Classical results
The PSR and transmission of an input elliptically po-
larized beam through the Rb vapour cell were measured
by scanning the laser frequency across the energy lev-
els of interest, for a fixed input beam intensity.
the D2line, the relevant levels were |52S1/2,Fg= 2? to
|52P3/2,Fe= 1,2,3? and for the D1line, |52S1/2,Fg= 2?
to |52P1/2,Fe= 1,2?. We repeated the measurement for
varying input beam powers and obtained a contour map
of PSR and transmission as a function of laser frequency
detuning and input beam intensity, shown in Figs. 3, 4,
6 and 7.
The transmission results for the D2line are shown in
Fig. 3. The region of lowest transmission < 10 % oc-
For
FIG. 3: False colour contour plot of the normalised trans-
mission results for the D2 line, as a function of input beam
intensity and laser frequency detuning. Zero frequency cor-
responds to the |52S1/2,Fg = 2? to |52P3/2,Fe = 3? energy
levels.
curred at input beam intensities ≤ 15 mW, around laser
frequencies close to zero detuning. For input beam pow-
ers ≥ 30 mW greater transmission (≥ 30 %) was ob-
served. However, power broadening effects were also ob-

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served for higher input beam powers, with reduced trans-
mission at frequencies ≤ −1 GHz. The transmission was
non-symmetric with high transmission (> 90 %) for fre-
quencies ≥ 1 GHz, whilst reduced transmission (> 60 %)
for frequencies ≤ −1 GHz. This was due to the level
structure of the excited states of the D2 line, where
the separations between the hyperfine levels are small
(within a frequency band of ∼ 0.5 GHz). Power broad-
ening effects were also observed for input beam intensities
≥ 30 mW.
The PSR results for the D2line are shown in Fig. 4.
The regions of largest PSR were 0.3 GHz and -0.6 GHz.
FIG. 4: False colour contour plot of Gl for the D2 line, nor-
malised to the input beam ellipticity of 2◦, as a function of
input beam intensity and laser frequency detuning. Zero fre-
quency corresponds to the |52S1/2,Fg = 2? to |52P3/2,Fe = 3?
energy levels.
The input beam powers which gave the largest Gl mag-
nitudes of 8 and 13 were ∼ 8 mW and ∼ 30 mW, re-
spectively. Zero Gl around zero detuning for input beam
powers ≤ 15 mW was due to the low transmission of the
input beam for the optically thick87Rb vapour cloud.
However, at frequency detunings ≥ 0.5 and ≤ −0.5 GHz,
significant PSR was observed even though the transmis-
sion was reduced. For input beam intensities ≥ 20 mW,
the PSR was preferentially larger with positive frequency
detunings as opposed to negative frequency detunings.
In order to explain the asymmetry present in the PSR
results, we modelled the hyperfine energy levels of the D2
line and took into account Doppler broadening. The the-
oretical fits to the experimental data are shown in Fig. 5.
The reduction in PSR in the negative frequency detuning
region was due to reduced transmission, as observed in
Fig. 3. Broadening of the PSR profile was observed for
higher input beam powers.
The transmission results for the D1line are shown in
Fig. 6. The region of lowest transmission (< 50 %) oc-
cured for input beam intensities ≤ 3 mW. These regions
FIG. 5: The normalised transmission and Gl results for the
D2 line are shown in Figures (i) and (ii), respectively. The
red curves are the theoretical fits to the experimental results
(green curve).Input beam intensity= 31.5 mW, and zero
frequency corresponds to the |52S1/2,Fg = 2? to |52P3/2,Fe =
3? energy levels.
FIG. 6: False colour contour plot of the normalised trans-
mission results for the D1 line, as a function of input beam
intensity and laser frequency detuning. Zero frequency cor-
responds to the |52S1/2,Fg = 2? to |52P1/2,Fe = 1? energy
levels.
were confined around two frequency detuning bands, the
−0.2 to 0.25 GHz band and the 0.4 to 0.8 GHz band.
The two frequency bands corresponded to the absorption
lines centred at the |52S1/2,Fg= 2? to |52P1/2,Fe= 1?
and |52S1/2,Fg= 2? to |52P1/2,Fe= 2? energy levels, re-
spectively. For input beam powers ≥ 5 mW, significant
transmission was observed (> 70 %). For most input
beam powers, the transmission of the D1line was signif-

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icantly higher than that of the D2line. This was due to
the weaker atom-field coupling in the D1line compared
to the D2line.
The PSR results for the D1line are shown in Fig. 7.
The regions of largest PSR occurred at frequency detun-
FIG. 7: False colour contour plot of Gl for the D1 line, nor-
malised to the input beam ellipticity of 2◦, as a function of
input beam intensity and laser frequency detuning. Zero fre-
quency corresponds to the |52S1/2,Fg = 2? to |52P1/2,Fe = 1?
energy levels.
ings -0.15 GHz and 0.6 GHz. The input beam powers
that gave the largest Gl magnitudes of 10 and 11 were
∼ 35 mW and ∼ 22 mW, respectively. Significant PSR
was observed for input beam powers > 3 mW since the
transmission was always > 50 %. The Gl magnitude was
almost equal in both frequency bands corresponding to
the two absorption lines centred at the |52S1/2,Fg= 2? to
|52P1/2,Fe= 1? and |52S1/2,Fg= 2? to |52P1/2,Fe= 2?
energy levels, for most input beam powers. This was due
to the excited state level structure of the D1line, where
the two excited state levels have a large separation of
∼ 0.8 GHz. This is illustrated by modelling the hyper-
fine excited state level structure of the D1line. The the-
oretical fits to the experimental data are shown in Fig. 8.
The two transmission dips are of approximately the same
magnitude, resulting in the two PSR peaks to be of equal
magnitudes.
B.Quantum results
The input field was linearly polarized in the x-axis
and we measured the quadrature noise of the outgoing
y-polarized vacuum field, using the homodyne detection
setup shown in Fig. 2. The bright x-polarized output
field was used as a local oscillator. The fringe visibil-
ity of the interferometer was 99 %. The two outputs of
FIG. 8: The normalised transmission and Gl results for the
D1 line are shown in Figures (i) and (ii), respectively. The
red curves are the theoretical fits to the experimental results
(green curve). Input beam intensity= 22.3 mW, and zero
frequency corresponds to the |52S1/2,Fg = 2? to |52P1/2,Fe =
1? energy levels.
the interferometer were then detected using two balanced
Silicon photo-detectors (which consisted of Hamamatsu
S3883 photo-diodes with measured quantum efficiency
values of 94.6%) with bandwidths of ∼ 20MHz. Blocking
the weak field provided a measurement of the QNL. The
QNL was checked for linearity with beam power and the
common mode rejection was optimised to ∼ 30-40 dB
from 100 kHz to 10 MHz.
polarising beam-splitter was well aligned such that neg-
ligible amounts of the x-polarized field emerged at the
y-polarized output port. The result of the noise mea-
surement for various sideband frequencies at various laser
frequency detunings and input beam powers, are shown
in Figs. 9 (D2line) and 11 (D1line).
The largest quadrature noise observed for the D2line
was 10 dB at a detuning of -70 MHz as shown in Fig. 9 (c).
A time scanned quadrature noise measurement is shown
in Fig. 10. In the noise plots of Figs. 9 (a), (c) and (d),
we observed large levels of excess noise of typically 5 dB
above the QNL. In Fig. 9 (b) the excess noise level was
0.8 dB above the QNL. This was the lowest noise level
observed around zero detuning. The largest values of the
phase quadrature noise level corresponded to the regions
of maximum PSR as shown in Fig. 4. At large frequency
detunings from resonance, both quadrature noise levels
were reduced to the QNL.
The noise measurements of the output vacuum field,
for the D1 line, are shown in Fig. 11. The largest noise
modulation observed was 7 dB which occurred at a fre-
quency detuning of 150 MHz as shown in Fig. 11 (c). In
Figs. 11 (a)-(d), the phase quadrature noise level around
zero detuning was always above the QNL due to the pres-
ence of large excess noise (3-4 dB). The largest values of
We also checked that the

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FIG. 9: (i) Amplitude and (ii) phase quadrature noise re-
sults for the D2 line, normalised to the QNL and dark noise-
subtracted. Figures (a) and (b) correspond to an input beam
power of 21 mW at sideband frequencies of 3 MHz and 6 MHz,
respectively. Figures (c) and (d) are results for an input
beam power of 35 mW, at sideband frequencies of 3 MHz
and 6 MHz, respectively. Zero frequency corresponds to the
|52S1/2,Fg = 2? to |52P3/2,Fe = 3? energy level. ResBW:
100 kHz and VBW: 30 Hz.
FIG. 10: Scanned quadrature noise for the D2 line measured
in zero span at a sideband frequency of 3 MHz. The input
beam power was 35 mW and the laser frequency was -70 MHz
from the |52S1/2,Fg = 2? to |52P1/2,Fe = 3? energy level. All
plots are dark noise subtracted. ResBW: 100 kHz and VBW:
30 Hz.
the amplitude noise level corresponded to the regions of
maximum PSR as shown in Fig. 7. At large frequency
detunings from resonance, both quadrature noise levels
were reduced to the QNL.
FIG. 11: (i) Amplitude and (ii) phase quadrature noise re-
sults for the D1 line, normalised to the QNL and subtracted
by the dark noise. Figures (a) and (b) correspond to an input
beam power of 21 mW at sideband frequencies of 3 MHz and
6 MHz, respectively. Figures (c) and (d) are results for an in-
put beam power of 35 mW, at sideband frequencies of 3 MHz
and 6 MHz, respectively. Zero frequency corresponds to the
|52S1/2,Fg = 2? to |52P1/2,Fe = 1? energy level. ResBW:
100 kHz and VBW: 30 Hz.
The noise measurement results presented do not vary
qualitatively with varying beam focussing geometry, in-
cident power or temperature. A large amount of excess
noise was systematically observed close to resonance. We
also performed similar experiments using paraffin-coated
cells and cells containing buffer gas, none of which re-
sulted in the observation of squeezing.
PSR and transmission results measured were in a very
similar regime to that of Ref. [19], the quantum noise
results are not in agreement with either the predictions
of Ref. [10] or the observations of Ref. [19]. We use the
model presented in Sec. IIB to discuss our experimental
observations.
Although the
IV. DISCUSSION AND CONCLUSIONS
A. Langevin noise analysis
In order to contrast the effect of the atomic noise terms
with the squeezing term in Eq. (11), we consider the
Langevin term given by
ˆFy=gNl
c
?
(A + B)ˆfy+ Bˆf†
y+ i
?
IxA
?
ˆfz
−iω+
ˆf′
z
2γ − iω
(14)
??

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8
where
A =
(γ − i∆ − iω)(−iω)(2γ − iω)
D
Ix(γ − iω)
D
D = 2Ix(γ − iω)2− iω(2γ − iω)[(γ − iω)2+ ∆2]
with Ix = |g?ˆ ax?|2,ˆfy = (ˆF14+ˆF23)/√2,ˆfz = (ˆF22−
ˆF11)/√2 andˆf′
of this noise term, which depends on the sideband fre-
quency, is to be compared with the cross-Kerr squeezing
term κ(ω). As shown in Ref. [20], in the low satura-
tion regime, large excess atomic noise associated with
optical pumping on the y-polarized field dominates at
sideband frequencies lower than the spontaneous emis-
sion rate (ω ≪ γ). In the low sideband frequency regime
(assuming ∆ ≫ γ), one obtains
∂
∂¯ zδˆ ay = i
B =
z= (ˆF44−ˆF33)/√2. The contribution
δ0
1 + sδˆ a†
y+gNl
2γc(ˆfy+ˆf†
y+
∆
2√Ix
ˆfz)(15)
where δ0= Cγ/2∆ denotes the linear dephasing. Ignor-
ing depletion of the mean x-polarized field, the Langevin
noise contribution is shown to be proportional to C/gl at
least. For the experiment, this quantity is greater than
the QNL, such that large excess noise is present in all
quadratures for low sideband frequencies, even when ab-
sorption is ignored. One therefore cannot observe squeez-
ing in this regime.
In the experiment, the quantum noise of the vacuum
field was measured only for sideband frequencies greater
than the excited state decay rate (ω ≥ γ). In this high
sideband frequency regime (assuming ∆ ≫ γ), we obtain
succinct expressions for κ(ω), Γ(ω) and˜Fygiven by
κ(ω) =
−iδ0s
(1 + s)(1 + 2s),
gNl
∆c(1 + 2s)
Γ(ω) =−iδ0
1 + s
(16)
ˆFy ≃ −i
?(1 + Ix/ω∆)ˆfy+ (Ix/ω∆)ˆf†
−(
y
?
Ix/ω)(ˆfz+ˆf′
z)?.(17)
The above equations describe the atomic noise contri-
bution that may degrade the squeezing of the output y-
polarized vacuum field.
The optimization of squeezing is dependent on find-
ing a regime that has low absorption and strong non-
linearity. We now proceed by dividing the discussion into
low and high atomic transition saturation regimes.
B.Low saturation regime with ultra-cold atoms
Since cold atoms have higher atomic density, one can
operate in the low saturation regime (s ≪ 1) and still
obtain strong PSR with minimal atomic noise [9], when
off-resonance. In the Kerr limit (∆ ≫√Ix ≫ γ), the
equation of motion for the vacuum field fluctuations is
given by
∂
∂¯ zδˆ ay= iδ0δˆ ay− iδ0s(2δˆ ay− δˆ a†
y) − igNl
c∆
ˆfy.(18)
One recovers in the equation above the same terms as in
the cavity model of Ref. [20] under the same approxima-
tions. The term in δ0δˆ ay corresponds to the linear de-
phasing, the second term gives the cross-Kerr squeezing
term and the Langevin noise contribution correspond-
ing to the last term can be shown to be proportional
to Cγ2/∆2, which can be small in the off-resonant sit-
uation (∆ ≫ γ). In accordance with the prediction of
Ref. [20] and the experimental observations of Ref. [9]
vacuum squeezing can be generated when δ0s ∼ 1 and
Cγ2/∆2.
C.High saturation regime with thermal vapour cell
Contrary to the situation of cold atoms, the Doppler
broadening in a thermal vapour makes it impossible to
work in the low saturation regime while simultaneously
having low absorption or high non-linearity. It is how-
ever possible to observe strong PSR in the high saturation
regime. In this regime, the atomic noise term is signifi-
cantly different to that given in Eq. (18). For Ix≫ ∆2,
the equation of motion is given by
∂
∂¯ zδˆ ay=iδ0
2s(δˆ ay+ δˆ a†
y) − igNl
cω(ˆfy+ˆf†
y).(19)
As we have seen experimentally with the PSR measure-
ments, the non-linear term in δ0/(2s) = Gl can still
be significant when the number of atoms are increased.
However, the optical pumping processes associated with
PSR now produce a lot of excess noise even in the high
sideband frequency regime. The contribution of the last
term in Eq. (19) can be shown to be proportional to
Cγ2/ω2≫ 1.
atomic noise prevents the observation of squeezing at all
sideband frequencies.
For our experimental parameters, the
D.Further considerations
We now discuss the possible discrepancies between the
theoretical models and the experiment. Due to the com-
plexity of the problem, many effects have not been taken
into account in the various models discussed in this pa-
per.
Firstly, the presence of resonance fluorescence has not
been considered in Ref. [10]. In Ref. [20], it was shown
that PSR cannot generate squeezing in the low satura-
tion regime because of optical pumping processes. We
have shown in this paper that this is also true in the
high saturation regime where the resonance fluorescence
noise dominates over the cross-Kerr squeezing term, even

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9
at high detection frequencies. This conclusion is in agree-
ment with other observations [21, 22, 23].
Secondly, none of the models presented have included
the Doppler effect. Since we are dealing with thermal
atoms, the passage of light through the atoms will give
rise to a range of observed atomic detuning. The inte-
grated effect due to Doppler broadening will be detrimen-
tal to the observation of squeezing.
Thirdly, the multi-level hyperfine structure of the ex-
cited states of87Rb have only been considered for the
theoretical fits to the classical PSR results, but have
not been included in any of the squeezing model. The
experimental PSR data presented in this paper clearly
shows that the multi-level hyperfine structure causes ob-
servable asymmetry in the PSR spectrum. This feature
cannot be explained by any of the theoretical models pre-
sented in Sec. II. The multi-level theory can be expanded
to include Langevin noise terms. However, a simple 4-
level atom model is sufficient to demonstrate the lack of
squeezing. The multi-level structure is also certainly less
favourable to the generation of squeezing when compared
with a simplified 4-level model. Different hyperfine levels
will not contribute constructively towards a collective in-
teraction that will generate squeezing. The added noise
from these different levels will add up significantly. The
inclusion of Doppler broadening and multi-level effects
would only result in a dominance of the atomic noise
term over the squeezing term.
Finally, the propagation of the transverseintensity pro-
file of the input field has been totally ignored in all mod-
els. A full treatment of the process should include the
multi-modal analysis of the evolution of the transverse
field modes during propagation through the vapour cell.
In the high saturation regime and for high atomic densi-
ties, self-focussing is readily observed. This is due to the
atom induced Kerr lens-effect on the optical field. Thus
the centre of the field intensity distribution will undergo
greater PSR than the edges. The cross-Kerrnon-linearity
and the atomic absorption used in our calculations is a
result of an “integrated” effect of the various transverse
modes. It therefore does not model accurately the situa-
tion of the experiment. Similar to the previous argument,
it is unlikely that the multi-modal consideration of the
process will yield better squeezing.
E. Conclusion
We have presented experimental results of PSR from
two independent laboratories and have observed no
squeezing. Instead we have observed excess noise in the
output field spectrum at all sideband frequencies. We
have modelled semi-classically the multi-level hyperfine
structure of87Rb and obtained theoretical fits to the ex-
perimental PSR data. Our multi-level modelling can pre-
dict the asymmetry in the PSR, that is due to the pres-
ence of other hyperfine excited states. We considered a
quantum mechanical 4-level atomic model and showed
that the squeezing term is overwhelmed by atomic noise
terms in the situation of a thermal vapour.
fects of resonance fluorescence, the Doppler effect and
the multi-level hyperfine structure of87Rb all contribute
to overwhelm the squeezing term. Therefore, it is ex-
pected that a full quantum mechanical treatment of a
multi-level87Rb atom will yield a result where squeezing
cannot be generated. In spite of this, the 4-level atom
model shows that squeezing can be generated in the situ-
ation of cold atoms where the Doppler effect is negligible.
When the input field is off-resonance, the non-linearity is
large but the absorption low, such that the atomic noise
term does not overwhelm the squeezing term.
The ef-
Acknowledgments
We would like to thank P. Drummond, W. P. Bowen,
J. J. Longdell and A. Lvovsky for fruitful discussions
and Paul Tant for technical support. This research was
funded under the Australian Research Council Centre of
Excellence Programmeand the European Project n◦FP6-
511004 (COVAQIAL).
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