We consider the model of a nondegenerate parametric oscillator, in the adiabatic limit of strongly damped pump mode, and find an exact analytic solution for the joint two-mode Wigner function in the steady-state regime. The one-mode Wigner functions for the individual signal and idler modes are derived as well. This gives an explicit and complete phase-space representation of this two-mode nonequilibrium quantum system. Simple examples are given illustrating the phase-space images of optical bistability and the phase diffusion effect

Kheruntsyan, K. V.

Petrosyan, K. G.

University of Queensland eSpace

PHYSICAL REVIEW A, VOLUME 62, 015801Exact steady-state Wigner function for a nondegenerate parametric oscillator
K. V. Kheruntsyan1 and K. G. Petrosyan2
1Department of Physics, University of Queensland, St. Lucia, Queensland 4067, Australia
2Department of Theoretical Physics, Yerevan Physics Institute, Alikhanyan Bros 2, Yerevan 375036, Armenia
~Received 13 September 1999; revised manuscript received 4 February 2000; published 7 June 2000!
We consider the model of a nondegenerate parametric oscillator, in the adiabatic limit of strongly damped
pump mode, and find an exact analytic solution for the joint two-mode Wigner function in the steady-state
regime. The one-mode Wigner functions for the individual signal and idler modes are derived as well. This
gives an explicit and complete phase-space representation of this two-mode nonequilibrium quantum system.
Simple examples are given illustrating the phase-space images of optical bistability and the phase diffusion
effect.
PACS number~s!: 42.50.2p, 42.65.YjThe concept of quantum-mechanical phase space and as-
sociated quasiprobability distributions has proven to be ex-
tremely useful and appealing in many fundamental applica-
tions of quantum mechanics. The oldest and most famous
phase-space quasiprobability is the Wigner function @1#. Re-
cently, there have been major experimental successes in re-
constructing the Wigner function for a number of quantum-
mechanical systems. The original proposal—referred to as
quantum state tomography—was suggested by Vogel and
Risken @2# for measuring the quantum state of light in quan-
tum optics. The technique was then successfully realized in a
series of experiments @3#. This in turn has greatly increased
the interest in employing and further developing the Wigner
function and phase-space methods ~see, e.g., @4#!. The prob-
lems under investigation have now been extended to molecu-
lar vibrational states @5#, motional states of atoms in traps
and atomic beams @6#, and Bose-Einstein condensates @7#.
Despite these achievements, there is a known
disadvantage—from the methodological and theoretical
points of view—of the Wigner function method, which con-
tinues to serve as a challenging problem. The disadvantage is
in the difficulty of solving realistic problems directly in
terms of the Wigner function, which is due to the complexity
of the corresponding evolution equation. Even for the sim-
plest models involving nonlinear interactions ~such as those
due to quadratic or cubic nonlinearities in quantum optics!
and dissipation, the Wigner function evolution equation con-
tains third- or higher-order derivative terms. While numeri-
cal techniques can routinely be employed here ~see, e.g., @8#!,
no general analytic methods are known for solving this type
of partial differential equation to find the Wigner function.
This is in contrast to employing generalized P represen-
tations @9,10# of the density matrix. The P-representation
method often produces a Fokker-Planck evolution equation
~having no higher than the second-order derivatives! which
can be treated using well-known methods @10,11#. Of espe-
cial importance here are the models that possess exact ana-
lytic solutions in either the positive or complex P represen-
tation @9,12,13#. The disadvantage of the P representations,
however, is that the positive P method requires a doubled
number of phase-space variables, while the complex P rep-
resentation is defined on specific contours in complex planes
of the phase space. This is not the case of the Wigner func-
tion; hence ~together with the historical reasons! its wider
usage and popularity.1050-2947/2000/62~1!/015801~4!/$15.00 62 0158In this paper we employ an indirect method to find an
exact analytic solution for the Wigner function for one of the
basic models in nonlinear and quantum optics—the nonde-
generate parametric oscillator ~NDPO! @13–16#. In the limit
of adiabatically eliminated pump mode, and provided that
the damping rates and cavity detunings for the signal and
idler modes are equal, the reduced model has an exact
steady-state solution @13,16# to the corresponding Fokker-
Planck equation in the complex P representation.
We use this solution to derive an exact and rather simple
analytic result for the joint two-mode Wigner function de-
scribing the steady state of this nonequilibrium dissipative
quantum system. Single-mode Wigner functions for the indi-
vidual signal and idler modes are derived as well. To our
best knowledge, this is the first example of an exact analytic
Wigner function solution for a two-mode nonlinear dissipa-
tive problem. Earlier solutions of this type were only found
for single-mode models of a degenerate parametric oscillator
@17,18# and a driven anharmonic oscillator @19#. The phase-
space representation of the individual signal and idler modes
is itself a result for the model of a NDPO that has not been
discussed before to our knowledge.
The model of NDPO that we consider describes three
quantized modes of the radiation field—the pump, signal,
and idler. The modes interact via a quadratic nonlinearity in
a triply resonant cavity. The pump mode is driven externally
by a coherent driving field, and all three modes are assumed
to be coupled to zero-temperature reservoirs, resulting in the
usual single-photon losses. This system can be modeled by
the following Hamiltonian, in the rotating-wave approxima-
tion @13,15#:
H5(
i50
2
\v iai
†ai1i\k~a1
†a2
†a02a1a2a0
†!
1i\~Ee2ivLta0
†2E*eivLta0!1(
i50
2
~aiG i
†1ai
†G i!. ~1!
Here ai
† and ai are boson creation and annihilation operators
for the cavity modes at frequencies v i , where i50, 1, and 2
refer to the pump, signal, and idler, respectively. The para-
metric coupling k is due to the second-order susceptibility
(x (2)) of the nonlinear medium, and we assume a phase-©2000 The American Physical Society01-1
BRIEF REPORTS PHYSICAL REVIEW A 62 015801matching condition so that v0.v11v2. In addition, E is the
amplitude of the coherent driving field with the carrier fre-
quency vL.v0, and G i
†
, G i are the reservoir operators
which give rise to the cavity losses with decay rates g i .
In the limit of strongly damped pump mode, g0@g1,2 , the
pump mode can be adiabatically eliminated @13#. The re-
duced model can then be described by the following interac-
tion picture master equation for the density operator r of the
signal and idler modes:
]r
]t
5i (
i51,2
D i@ai
†ai ,r#1
1
i\ @He f f ,r#
1 (
i51,2
g i~2airai
†2ai
†air2rai
†ai!
1g~2a1a2ra1
†a2
†2a1
†a2
†a1a2r2ra1
†a2
†a1a2!. ~2!
Here the effective interaction Hamiltonian is given by
He f f5i\~ma1
†a2
†2m*a1a2!1\xa1
†a2
†a1a2 . ~3!
In addition, D05v02vL is the cavity detuning for the pump
mode and D1,25v1,22(vL/26e) are the cavity detunings
for the signal and the idler modes, with vL/26e being their
characteristic rotating-frame frequencies. We have also de-
fined
m5
kE
g¯ 0
, x52
k2D0
ug¯ 0u2
, g5
k2g0
ug¯ 0u2
, ~4!
where g¯ 05g01iD0.
The above form of the master equation and the effective
Hamiltonian ~3! simply implies that, after adiabatic elimina-
tion of the pump mode, we are dealing with a two-mode
model that describes two cavity modes ~signal and idler! that
are driven parametrically with a driving strength m , and are
subject to the usual one-photon ~linear! losses at rates g i and
to a nondegenerate two-photon ~nonlinear! loss mechanism
with a rate g. In addition, the two modes are coupled by a
Kerr type interaction due to an effective cubic ‘‘nonlinear-
ity’’ x , causing cross-phase modulation. This x term would
be absent in the case of exact resonance, D050. We note
that the cross-phase modulation and nondegenerate two-
photon absorption terms could explicitly be included into the
original Hamiltonian ~1!, representing the possibility of com-
bining different nonlinear optical processes in a single cav-
ity. The corresponding nonlinear constants would then be
simply added to the above ‘‘effective’’ couplings x and g,
thus providing additional degrees of freedom for the control
over the system parameters.
Using the complex P representation @9# of the density
operator, the above master equation is transformed into the
Fokker-Planck equation, which can then be solved exactly
for a steady state @13#, using the method of potential equa-
tions. The solution is available for the case of symmetric
decay rates g15g2[g and cavity detunings D15D2[D .
The resulting steady-state P function has the following form
@13,16#:01580P~b1 ,b2 ,b1
†
,b2
†!5N~b1b22«!n21~b1†b2†2«*!n*21
3exp@2~b1b1
†1b2b2
†!# . ~5!
Here b1 , b2 , b1
†
, and b2
† are independent complex vari-
ables, corresponding to the operators a1 , a2 , a1
†
, and a2
†
,
respectively, N is the normalization constant, and we have
introduced the following dimensionless parameters:
«[
m
g1ix 5
E
k
, n[
2g¯ *
g2ix 5
2g¯ *g¯ 0*
k2
, ~6!
where g¯ 5g1iD .
The two-mode Wigner function that we are interested in
is defined as follows:
W~a1 ,a2!5S 1
p2
D 2E E d2d1d2d2
3Tr~red1a1
†
2d1*a11d2a2
†
2d2*a2!
3ed1*a12d1a1*1d2*a22d2a2*. ~7!
This can be expressed in terms of the complex P repre-
sentation as
W~a1 ,a2!5S 2p D
2
e22(ua1u
21ua2u
2)
3E
Cb1
E
Cb2
E
Cb1
†
E
Cb2
†
db1db2db1
†db2
†
3P~b1 ,b1
†
,b2 ,b2
†!
3e22(b1b1
†
1b2b2
†)12(a1*b11a1b1
†)12(a2*b21a2b2
†)
,
~8!
where the integrals are contour integrals in the individual
complex planes for the independent variables b1 , b2 , b1
†
,
and b2
†
, according to the way the complex P representation
is defined @9#.
Substituting the steady-state complex P function ~5! into
Eq. ~8!, and assuming that the integrations over b j and b j
†
variables are along the same contours in the respective com-
plex planes ~i.e., the contour Cb1 is the same as Cb1†, and the
contour Cb2 is the same as Cb2†), one can express the Wigner
function W(a1 ,a2) as follows:
W~a1 ,a2!5NS 2p D
2
e22(ua1u
21ua2u
2)uIu2, ~9!
where we have defined1-2
BRIEF REPORTS PHYSICAL REVIEW A 62 015801I[I~a1* ,a2* ;« ,n*!
5E
Cb1
E
Cb2
db1db2~b1b22«!n*21e2(a1*b11a2*b2),
~10!
so that I*5@I(a1* ,a2* ;« ,n*)#*5I(a1 ,a2 ;«*,n).
To evaluate the integral I we transform to new variables
t52«a2*/b1 and t52a2*(b1b22«)/b1:
I52~«!n*E
Ct
dttn*21etE
Ct
dtt2n*21et14«a1*a2*/t. ~11!
The integrals over t and t can now be identified, respec-
tively, with contour integral representations of the gamma
function G(n*) and the Bessel function Jn*(A216«a1*a2*)
@20#, where both the integration contours Ct and Ct start at
infinity on the negative t (t) axis, encircle the origin coun-
terclockwise, and return to 2‘ . As a result, we obtain the
result that I}Jn*(A216«a1*a2*)/(216«a1*a2*)n*/2. The
steady-state Wigner function W(a1 ,a2) is then given by
W~a1 ,a2!5Ne22(ua1u
21ua2u
2)UJn~A216«*a1a2!
~216«*a1a2!n/2
U2, ~12!
where N is the normalization constant to be found from
**d2a1d2a2W(a1 ,a2)51. Using the expansion of the
Bessel function
Jn~At !5
tn/2
2nG~n!
(
k50
‘
~2t/4!kG~n11 !
k!G~k1n11 ! , ~13!
one can rewrite the above Wigner function as
W~a1 ,a2!
5N8e22(ua1u
21ua2u
2)U(
k50
‘
~4«*a1a2!k G~n11 !
k!G~k1n11 ! U
2
.
~14!
The constant N85Nu2nG(n)u22 can now be easily found
from the normalization condition by carrying out the integra-
tions in polar coordinates @a j5r jexp(iwj)# and using
*0
2pdw exp@i(k2m)w#52pdkm :
N85S 2p D
2F (
k50
‘
~4u«u2!kuG~n11 !u2
uG~k1n11 !u2 G
21
. ~15!
Equations ~12! and ~14! are the key results of the present
paper. We point out that the Wigner function W(a1 ,a2) is
positive everywhere. Therefore it can act as a local hidden
variable theory for quadrature phase measurements and thus
cannot violate a Bell inequality, in contrast to the results that
can be obtained with a nondissipative model of NDPO ~see,
e.g., @21–23#!.
We next use the two-mode Wigner function W(a1 ,a2) to
calculate the Wigner function for each of the individual ~sig-01580nal or idler! modes. This is obtained by integrating
W(a1 ,a2) with respect to one of the phase-space variables,
a2 or a1. Due to the symmetry of W(a1 ,a2) with respect to
a1 and a2, the one-mode Wigner functions for the signal and
the idler modes are equal to each other and are given, from
Eq. ~14!, by
W1~a!5E d2a2W~a ,a2!
5
pN8
2 e
22uau2 (
k50
‘
~8u«u2uau2!kuG~n11 !u2
k!uG~k1n11 !u2
. ~16!
This expression can be rewritten in a compact form, in terms
of the hypergeometric function 0F2 @20#:
W1~a!5
pN8
2 e
22r2
0F2~n11,n*11,8u«u2r2!, ~17!
where r5uau.
Examples of W1(a) are plotted in Fig. 1. The Wigner
function W1(a) depends only on the radial coordinate r
5uau, and is uniformly distributed with respect to the phase
w . The phase distribution function, defined as an integral
over the radial coordinate F1(w)5*0‘r dr W1(a5reiw), can
be found directly using the normalization condition, giving
the result that F1(w)51/(2p). This radial symmetry of
W1(a) reflects the known phenomenon of phase diffusion
occurring in NDPOs @15#, according to which the phases of
the signal and idler modes do not have well-defined values.
In Fig. 1~a!, the coexistence of the central peak together
with the outer ring represents the phenomenon of bistability
known to occur in the NDPO for DD0.gg0 and experimen-
tally observed in @24#. In terms of the semiclassical steady-
state solutions @14–16# for the intensities of the signal and
FIG. 1. One-mode Wigner function for Re n5230, Im n
5230, and two values of the driving field parameter: u«u516.5 ~a!,
and u«u545 ~b!.1-3
BRIEF REPORTS PHYSICAL REVIEW A 62 015801idler modes, n1
(0)5n2
(0)[n (0), the bistability is referred to the
overlap of the stability regions for the below-threshold solu-
tion n (0)50 and the above-threshold solution with n (0)5 0.
Using the parameters n and « , the above-threshold solution
can be written as
n (0)52Re~n/2!6Au«u22@Im~n/2!#2. ~18!
The bistability in the NDPO occurs if Re(n/2),0, and the
corresponding region—as a function of the scaled driving
field intensity u«u2—is determined by u« (2)u2,u«u2,u« thu2.
Here u« thu25unu2/4 gives the threshold value of u«u2, while
u« (2)u25@Im(n/2)#2 is the turning point where the system
returns from the above-threshold regime of oscillation to the
below-threshold regime. In Fig. 1~a! the value of u«u is cho-
sen to be within the bistability region, so that the central
peak corresponds to the below-threshold solution, while the
outer ring represents the above-threshold solution. For small
values of u«u the Wigner function only has the central peak,
while large values of u«u lead to disappearance of the central
peak, and we observe only the above-threshold outer ring, as
in Fig. 1~b!.
Thus, the Wigner function solutions found here provide
us with an explicit phase-space representation of the steady
state of this two-mode nonequilibrium quantum system, for01580arbitrary values of relevant parameters. These are the key
objects for quantum tomographic applications @25#, and the
results might be especially useful for analyzing Bell type
inequalities where the knowledge of phase-space probabili-
ties to a great degree of accuracy can often be of crucial
importance @23,26,27#. We note that the results presented
here are valid in the threshold region and in the extreme
quantum regime of operation characteristic of high nonlin-
earities and low damping rates. We also point out a remark-
able spin-off ~from the methodological point of view! of the
solution found here. This follows from the comparison of the
two-mode Wigner function with the earlier known solutions
to single-mode models of the degenerate parametric oscilla-
tor and anharmonic oscillator: Quite surprisingly, the struc-
ture of the Wigner functions in all these three quite distinct
exactly soluble models appears similar, once the solutions
are expressed @18,19# in terms of Bessel functions. This sug-
gests an intriguing possibility of finding a direct method of
solving at least a class of Wigner function evolution equa-
tions with third-order derivative terms, having the form ge-
neric to these models. Hopefully this would provide previ-
ously unknown solutions to other problems.
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Exact steady-state Wigner function for a nondegenerate parametric oscillator

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Exact steady-state Wigner function for a nondegenerate parametric oscillator

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