We report the first direct experimental characterization of continuous
variable quantum Stokes parameters. We generate a continuous wave light beam
with more than 3 dB of simultaneous squeezing in three of the four Stokes
parameters. The polarization squeezed beam is produced by mixing two quadrature
squeezed beams on a polarizing beam splitter. Depending on the squeezed
quadrature of these two beams the quantum uncertainty volume on the
Poincar\'{e} sphere became a `cigar' or `pancake'-like ellipsoid.Comment: 4 pages, 5 figure

A. Aspect

A. P. Alodjants

B. A. Robson

D. F. Walls

G. G. Stokes

G. S. Agarwal

Hans-A. Bachor

J. Hald

J. M. Jauch

M. Kaku

P. Grangier

Ping Koy Lam

Roman Schnabel

Warwick P. Bowen

English

arXiv

We report the first direct experimental characterization of continuous variable quantum Stokes parameters. We generate a continuous wave light beam with more than 3 dB of simultaneous squeezing in three of the four Stokes parameters. The polarization squeezed beam is produced by mixing two quadrature squeezed beams on a polarizing beam splitter. Depending on the squeezed quadrature of these two beams the quantum uncertainty volume on the Poincaré sphere becomes a “cigarlike” or “pancakelike” ellipsoid

Bowen, Warwick P.

Schnabel, Roman

Bachor, Hans A.

Lam, Ping Koy

University of Queensland eSpace

VOLUME 88, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 4 MARCH 2002
093601-1Polarization Squeezing of Continuous Variable Stokes Parameters
Warwick P. Bowen, Roman Schnabel, Hans-A. Bachor, and Ping Koy Lam
Department of Physics, Faculty of Science, Australian National University, ACT 0200, Australia
(Received 22 October 2001; published 13 February 2002)
We report the first direct experimental characterization of continuous variable quantum Stokes parame-
ters. We generate a continuous wave light beam with more than 3 dB of simultaneous squeezing in three
of the four Stokes parameters. The polarization squeezed beam is produced by mixing two quadrature
squeezed beams on a polarizing beam splitter. Depending on the squeezed quadrature of these two beams
the quantum uncertainty volume on the Poincaré sphere becomes a “cigarlike” or “pancakelike” ellipsoid.
DOI: 10.1103/PhysRevLett.88.093601 PACS numbers: 42.50.Dv, 42.65.TgThe quantum properties of the polarization of light have
received much attention in the single photon regime where
fundamental problems of quantum mechanics, related to
Bell’s inequality and the Einstein-Podolski-Rosen (EPR)
paradox, have been examined [1]. In comparison quan-
tum polarization states in the continuous variable regime
have received little attention. Grangier et al. [2] generated
a polarization squeezed beam using an optical parametric
process to improve the sensitivity of a polarization interfer-
ometer. Other schemes using Kerr-like media and optical
solitons in fibers have also been proposed [3,4] but have
not yet been produced experimentally. This paper presents
the generation of a new polarization squeezed state. A
stably locked beam with better than 3 dB of squeezing in
three Stokes parameters simultaneously was produced uti-
lizing two bright quadrature squeezed beams. We present
the first direct characterization of the polarization quantum
noise on a continuous wave light beam allowing an experi-
mental observation of polarization commutation relations
in the continuous variable regime.
Both the production of new continuous variable quantum
polarization states and the ability to accurately characterize
them are necessary for these states to fulfill their potential
in the field of quantum information. They can be carried by
a bright laser beam providing high bandwidth capabilities
and therefore faster signal transfer rates than single photon
systems. Perhaps surprisingly they retain the single pho-
ton advantage of not requiring the universal local oscillator
necessary for other proposed continuous variable quantum
networks. Mapping of quantum states from photonic to
atomic media is a crucial element in most proposed quan-
tum information networks. Continuous variable polariza-
tion states are the only continuous variable state for which
this mapping has been experimentally demonstrated [5].
The polarization state of a light beam in classical op-
tics can be visualized as a Stokes vector on a Poincaré
sphere (Fig. 1) and is determined by the four Stokes pa-
rameters [6]: S0 represents the beam intensity whereas S1,
S2, and S3 characterize its polarization and form a Carte-
sian axes system. If the Stokes vector points in the di-
rection of S1, S2, or S3 the polarized part of the beam
is horizontally, linearly at 45±, or right-circularly polar-
ized, respectively. The quantity S  S21 1 S22 1 S23120031-90070288(9)093601(4)$20.00is the radius of the classical Poincaré sphere and the frac-
tion SS0 0 , SS0 , 1 is called the degree of polariza-
tion. For quasimonochromatic laser light which is almost
completely polarized S0 is a redundant parameter, com-
pletely determined by the other three parameters (S0  S
in classical optics). All four Stokes parameters are acces-
sible from the simple experiments shown in Fig. 2.
The Stokes parameters are observables and therefore can
be associated with Hermitian operators. Following [7,8]
we define the quantum-mechanical analog of the classical
Stokes parameters for pure states in the commonly used
notation:
Sˆ0  aˆ
y
HaˆH 1 aˆ
y
V aˆV , Sˆ2  aˆ
y
HaˆVe
iu 1 aˆ
y
V aˆHe
2iu,
Sˆ1  aˆ
y
HaˆH 2 aˆ
y
V aˆV , Sˆ3  iaˆ
y
V aˆHe
2iu 2 iaˆ
y
HaˆVe
iu, (1)
where the subscripts H and V label the horizontal and
vertical polarization modes, respectively, u is the phase
shift between these modes, and the aˆH,V and aˆ
y
H,V are
annihilation and creation operators for the electromagnetic
field in frequency space. The commutation relations of
these operators
aˆk , aˆ
y
l   dkl, with k, l [ H ,V  , (2)
directly result in Stokes operator commutation relations,
(a)
S1 S2
S3 right-circularly 
    polarized
horizontally
  polarized 45° polarized
(b) S3
S2
S1
FIG. 1. (a) Classical and (b) quantum Stokes vectors mapped
on a Poincaré sphere; the ball at the end of the quantum vector
visualizes the quantum noise in Sˆ1, Sˆ2, and Sˆ3; and the quantum
sphere thickness visualizes the quantum noise in Sˆ0.© 2002 The American Physical Society 093601-1
VOLUME 88, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 4 MARCH 2002(b)
Input beam
S1
-
PBS
(d)
Input beam
S3
λ/
2@
22
.5
o
λ/
4@
45
o
-
PBS
(a)
Input beam
PBS
S0
+
(c)
Input beam
S2
λ/
2@
22
.5
o
-
PBS
FIG. 2. Apparatus required to measure each of the Stokes pa-
rameters. PBS: polarizing beam splitter; l2 and l4: half-
and quarter-wave plates, respectively; the plus and minus signs
imply that an electrical sum or difference has been taken.
Sˆ1, Sˆ2  2iSˆ3, Sˆ2, Sˆ3  2iSˆ1, Sˆ3, Sˆ1  2iSˆ2 .
(3)
Apart from a normalization factor, these relations are iden-
tical to the commutation relations of the Pauli spin matri-
ces. In fact the three Stokes parameters in Eq. (3) and
the three Pauli spin matrices both generate the special uni-
tary group of symmetry transformations SU(2) of Lie al-
gebra [9]. Since this group obeys the same algebra as the
three-dimensional rotation group, distances in three dimen-
sions are invariant. Accordingly the operator Sˆ0 is also
invariant and commutes with the other three Stokes opera-
tors (Sˆ0, Sˆj  0, with j  1, 2, 3). It can be shown from
Eqs. (1) and (2) that the quantum Poincaré sphere radius
is different from its classical analog, Sˆ	  Sˆ20 1 2Sˆ0	12.
The noncommutability of the Stokes operators Sˆ1, Sˆ2 and
Sˆ3 precludes the simultaneous exact measurement of their
physical quantities. Their mean values Sˆj	 and vari-
ances Vj  Sˆ2j 	 2 Sˆj	2 are restricted by the uncertainty
relations [7]
V1V2 $ jSˆ3	j2, V2V3 $ jSˆ1	j2, V3V1 $ jSˆ2	j2. (4)
In general this results in nonzero variances in the individual
Stokes parameters as well as in the radius of the Poincaré
sphere [see Fig. 1(b)]. Recently it has been shown that
these variances may be obtained from the frequency spec-
trum of the electrical output currents of the setups shown
in Fig. 2 [4].
It is useful to express the Stokes operators of Eq. (1)
in terms of quadrature amplitudes of the horizontally and
vertically polarized components of the beam. The creation
and annihilation operators can be expressed as sums of real
classical amplitudes aH,V and quadrature quantum noise
operators dXˆ1H,V and dXˆ2H,V [10]
aˆH,V  aH,V 1
1
2
dXˆ1H,V 1 idXˆ
2
H,V  . (5)
If the variances of the noise operators are much smaller
than the coherent amplitudes then, to first order in the noise
operators, the Stokes operator mean values are093601-2Sˆ0	  a2H 1 a
2
V  nˆ	, Sˆ2	  2aHaV cosu,
Sˆ1	  a2H 2 a
2
V , Sˆ3	  2aHaV sinu,
(6)
where nˆ is the photon number operator. For a coherent
beam the expectation value and variance of nˆ have the
same magnitude, this magnitude equals the conventional
shot-noise level. If dXˆ6H and dXˆ6V are uncorrelated the
variances of the Stokes parameters are given by
V0  V1  a2H dXˆ
1
H 
2	 1 a2V dXˆ
1
V 
2	 ,
V2u  0  V3u  p2  a2V dXˆ
1
H 
2	
1 a2H dXˆ
1
V 
2	 ,
V3u  0  V2u  p2  a2V dXˆ
2
H 
2	
1 a2H dXˆ
2
V 
2	 .
(7)
The variances of the noise operators in the above equation
are normalized to one for a coherent beam. Therefore the
variances of the Stokes parameters of a coherent beam are
all equal to the shot noise of the beam. For this reason a
Stokes parameter is said to be squeezed if its variance falls
below the shot noise of an equal power coherent beam. Al-
though the decomposition to the H,V polarization axis of
Eqs. (7) is independent of the actual procedure of gener-
ating a polarization squeezed beam, it becomes clear that
two overlapped quadrature squeezed beams can produce a
single polarization squeezed beam. We choose the two spe-
cific angles of u  0 andp2 in Eqs. (7) corresponding to
our actual experimental setup. If the polarization squeezed
beam is generated from two amplitude squeezed beams Sˆ0
and two additional Stokes parameters can in theory be per-
fectly squeezed while the fourth is antisqueezed. In this
case the uncertainty volume formed by the noise variances
of the Stokes parameters in the Poincaré space becomes a
“cigarlike” ellipsoid. If the polarization squeezed beam is
generated from two phase squeezed beams the uncertainty
volume becomes a “pancakelike” ellipsoid (see Fig. 5).
The experimental setup used to generate the polariza-
tion squeezed beams is shown in Fig. 3. Two quadrature
squeezed beams were produced in a pair of spatially
separated optical parametric amplifiers (OPAs). The
OPAs were optical resonators constructed from hemilithic
MgO:LiNbO3 crystals and output couplers. The reflec-
tivities of the output couplers were 96% and 6% for
the fundamental (1064 nm) and the second harmonic
(532 nm) laser modes, respectively. The OPAs were
pumped with single-mode 532 nm light generated by
a 1.5 W Nd:YAG nonplanar ring laser and frequency
doubled in a second harmonic generator (SHG). The
SHG was of identical structure to the OPAs but with
92% reflectivity at 1064 nm. Each OPA was seeded with
1064 nm light after spectral filtering in a modecleaner.
This seed enabled control of the length of the OPAs. The
coherent amplitude of each OPAs output was a deam-
plified/amplified version of the seed coherent amplitude;093601-2
VOLUME 88, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 4 MARCH 2002H
SHG
DC
λ/2
50/50 BSOPA
O
PA
DC
DC
θ
   Polarization 
squeezed beam
λ/2
λ/2
λ/2
φsh
φsh PBS
50/50 BS
70/30 BSNd:YAG
Laser
MC
V
H
H
H
V
V
FIG. 3. Experimental layout; BS: beam splitter; DC: dichroic
beam splitter; l2: half-wave plate; fsh: phase shift between
532 and 1064 nm light at the OPAs; u: phase shift between
quadrature squeezed beams; PBS: polarizing beam splitter.
the level of amplification was dependent on the phase
difference between pump and seed fsh. Locking to
deamplification or amplification provided amplitude or
phase squeezed outputs, respectively.
The two quadrature squeezed beams were combined
with orthogonal polarization on a polarizing beam splitter
[4]. This produced an output beam with Stokes parameter
variances as given by Eqs. (7). The power of each input
beam was 0.4 mW aH  aV  a
p
2 and both beams
were squeezed in the same quadrature. The Stokes pa-
rameters and their variances were determined as shown in
Fig. 2. The beam was split on a polarizing beam splitter
and the two outputs were detected on a pair of high quan-
tum efficiency photodiodes with 30 MHz bandwidth; the
resulting photocurrents were added and subtracted to yield
the means and variances of Sˆ0 and Sˆ1. To measure Sˆ2	
and V2 the polarization of the beam was rotated by 45±
with a half-wave plate before the polarizing beam splitter
and the detected photocurrents were subtracted. To mea-
sure Sˆ3	 and V3 the polarization of the beam was rotated
by 45± with a half-wave plate and a quarter-wave plate was
introduced before the polarizing beam splitter such that a
horizontally polarized input beam became right circular.
Again the detected photocurrents were subtracted. The
relative phase between the quadrature squeezed input
beams u was locked to p2 rads producing in all cases
a right-circularly polarized beam with Stokes parameter
means of Sˆ1	  Sˆ2	  0 and Sˆ0	  Sˆ3	  jaj2. There
is no fundamental bias in the orientation of the quantum
Stokes vector. By varying the angle of an additional
half-wave plate in the polarization squeezed beam and
by varying u any orientation may be achieved. In fact
the experiment discussed here was also carried out with
u locked to 0 rads, in this case the Stokes vector was
rotated to point along Sˆ2 and the quantum noise was
similarly rotated. Near identical results were obtained but
on alternative Stokes parameters.
Each quadrature squeezed beam had an overall detec-
tion efficiency of 73%. The loss came primarily from four
sources: loss in escape from the OPAs 14%, detector093601-3inefficiency 7%, loss in optics 5%, and mode overlap
mismatch between the beams 4%. Depolarizing effects
are thought to be another significant source of loss for some
polarization squeezing proposals [11]. In our scheme the
nonlinear processes (OPAs) are divorced from the polariza-
tion manipulation (wave plates and polarizing beam split-
ters), and depolarizing effects are insignificant.
As discussed earlier the Stokes parameters of a beam are
squeezed if their variances fall below the shot-noise level
of an equally intense coherent beam. This shot-noise level
was used to calibrate the measurements presented here,
it was determined by operating a single OPA without the
second harmonic pump. The seed power was adjusted so
that the output power was equal to that of the polarization
squeezed beam. In this configuration the detection setup
for Sˆ2 [see Fig. 2(c)] functions exactly as a homodyne
detector measuring vacuum noise scaled by the OPA output
power the variance of which is the shot noise. During each
experimental run the sum of output powers from the two
OPAs was always within 2% of the power of the coherent
calibration beam. This led to a conservative error in our
frequency spectra of 60.05 dB.
The shot-noise and Stokes parameter variances pre-
sented here were taken with a Hewlett-Packard E4405B
spectrum analyzer over the range from 3 to 10 MHz. The
dark noise of the detection apparatus was always more
than 4 dB below the measured traces and was taken into
account. Each displayed trace is the average of three
measurement results normalized to the shot noise and
smoothed over the resolution bandwidth of the spectrum
analyzer which was set to 300 kHz. The video bandwidth
of the spectrum analyzer was set to 300 Hz.
Figure 4 shows the measurement results obtained with
both input beams amplitude squeezed. Sˆ0, Sˆ1, and Sˆ3 are
all squeezed from 4.5 MHz to the limit of our measure-
ment, 10 MHz. Sˆ2 is antisqueezed throughout the range
of the measurement. Between 7.2 and 9.6 MHz, Sˆ0, Sˆ1,
and Sˆ3 are all more than 3 dB below the shot noise. The
squeezing of Sˆ0 and Sˆ3 is degraded at low frequency due
to phase noise on the second harmonic pump coupling into
the amplitude quadrature of the OPA outputs. Since this
noise is correlated it cancels in the variance of Sˆ1. The
maximum squeezing of Sˆ0 and Sˆ2 was 3.8 and 3.5 dB, re-
spectively, and was observed at 9.3 MHz. The maximum
squeezing of Sˆ1 was 4.3 dB at 5.7 MHz. The repetitive
structure at 4, 5, 6, 7, 8 and 9 MHz was caused by electri-
cal pickup in our SHG resonator emitted from a separate
experiment operating in the laboratory.
By locking both OPAs to amplification we obtained re-
sults similar to those in Fig. 4 but with the input beams
phase squeezed. In this case Sˆ0, Sˆ1, and Sˆ3 were all anti-
squeezed and Sˆ2 was squeezed throughout the range of
the measurement. The optimum noise reduction of Sˆ2 was
2.8 dB below shot noise and was observed at 4.8 MHz.
In Fig. 5 we visualize our experimental data at 8.5 MHz
assuming Gaussian noise statistics. Given this assump-
tion, the standard deviation contour surfaces shown here093601-3
VOLUME 88, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 4 MARCH 2002(a)
4 5 6 7 8 9
Frequency (MHz)
-4
-3
-2
-1
0
N
oi
se
 p
ow
er
 (d
B)
S0 (b)
4 5 6 7 8 9
Frequency (MHz)
-4
-3
-2
-1
0
N
oi
se
 p
ow
er
 (d
B)
S1
(d)
4 5 6 7 8 9
Frequency (MHz)
-4
-3
-2
-1
0
N
oi
se
 p
ow
er
 (d
B)
S3(c)
4 5 6 7 8 9
Frequency (MHz)
0
2
4
6
8
10
N
oi
se
 p
ow
er
 (d
B)
S2
Shot-noise level
3 dB squeezed
3 dB squeezed
Shot-noise level
3 dB squeezed
Shot-noise level
FIG. 4. Measured variance spectra of quantum noise on (a) Sˆ0,
(b) Sˆ1, (c) Sˆ2, and (d) Sˆ3 for locked amplitude squeezed input
beams; normalized to shot noise.
provide an accurate representation of the states’ three-
dimensional noise distribution. The quantum polariza-
tion noise of a coherent state forms a sphere of noise
as portrayed in Fig. 5(a). The noise volume formed by
our experimental results with two phase or two amplitude
squeezed beams, respectively, was found to be a pancake-
like [Fig. 5(b)] or cigarlike ellipsoid [Fig. 5(c)].
In all experiments prior to this work involving polariza-
tion squeezing the squeezing was generated by combin-
ing a strong coherent beam with a single weak amplitude
squeezed beam [2,5]. Utilizing only one squeezed beam
it is not possible to simultaneously squeeze any two of
Sˆ1, Sˆ2, and Sˆ3 to quieter than 3 dB below the shot noise
(Vi 1 Vj $ nˆ	, with i, j [ 1, 2, 3; i fi j . This can
be seen from the more general form of Eqs. (7) which in-
cludes arbitrary phase angle u between the beams.
To conclude, we present the first direct experimental
characterization of continuous variable quantum Stokes pa-
rameters. We generate a new quantum state with better
than 3 dB squeezing of three Stokes parameters (Sˆ0, Sˆ1,
and Sˆ3) simultaneously, a quality not possible from only
one quadrature squeezed beam. We represent our results on
a quantum version of the Poincaré sphere and experimen-
tally demonstrate that in this space a polarization squeezed
beam generated from two amplitude squeezed beams has
a cigarlike quantum noise distribution; and one from two
phase squeezed beams has a pancakelike quantum noise
distribution.
The authors acknowledge the Alexander von Humboldt
Foundation for support of R. Schnabel; the Australian Re-093601-4(a) σS2
σS3 σS1
12
0
1 2
1
2
0
1 2
σS1
σS2
0
1
2
12σS3
(c)
σS3
σS112
0
1 2
σS2
1
2
0
1 2
σS1
σS2
0
1
2
1
2σS3
(b)
0
σS3
σS1
1
2
1 2
σS2
1
2
0
1 2
σS1
0
1
2
12
σS2
σS3
S2
S1
S3
FIG. 5. Measured quantum polarization noise at 8.5 MHz
mapped onto the Poincaré sphere: (a) coherent beam; (b) beam
from two phase squeezed inputs; (c) beam from two amplitude
squeezed inputs. The surface of the ellipsoids defines the
standard deviation of the noise normalized to the shot noise
sSi 
p
Vi .
search Council for financial support; M. B. Gray and B. C.
Buchler for technical assistance; and T. C. Ralph for in-
sightful discussion.
[1] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 49,
91 (1982).
[2] P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, Phys.
Rev. Lett. 59, 2153 (1987).
[3] A. P. Alodjants, S. M. Arakelian, and A. S. Chirkin, Appl.
Phys. B 66, 53 (1998).
[4] N. V. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and
C. Silberhorn, quant-ph/0108098.
[5] J. Hald and J. L. Sørensen, C. Schori and E. S. Polzik, Phys.
Rev. Lett. 83, 1319 (1999).
[6] G. G. Stokes, Trans. Cambridge Philos. Soc. 9, 399 (1852).
[7] J. M. Jauch and F. Rohrlich, The Theory of Photons and
Electrons (Springer, Berlin, 1976).
[8] B. A. Robson, The Theory of Polarization Phenomena
(Clarendon, Oxford, 1974).
[9] See, for example, M. Kaku, Quantum Field Theory (Oxford
University Press, New York, 1993).
[10] D. F. Walls and G. J. Milburn, Quantum Optics (Springer,
Berlin, 1995).
[11] G. S. Agarwal and R. R. Puri, Phys. Rev. A 40, 5179
(1989).093601-4


Polarization squeezing of continuous variable stokes parameters

Crossref

Polarization Squeezing of Continuous Variable Stokes Parameters

http://espace.library.uq.edu.au/view/UQ:152217/UQ152217.pdf

Polarization Squeezing of Continuous Variable Stokes Parameters

Abstract

Similar works

Full text

Available Versions

University of Queensland eSpace

Crossref