We consider a model for nondegenerate cavity fields interacting through an intervening Boson field. The quantum correlations introduced in this manner are manifest through their higher-order correlation functions where a type of squeezed state is identified

Chizhov, A. V.

Haus, J. W.

Yeong, K. C.

English

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HIGHER-ORDER SQUEEZING
IN A BOSON COUPLED TWO-MODE SYSTEM
A. V. Chizhov
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research, 1_1980 Dubna, Russia
J. W. Haus and K. C. Yeong
Physics Dept., Rensselaer Polytechnic Institute, Troy, NY 12180-3590
Abstract
We consider a model for nondegenerate cavity fields interacting through an intervening
Boson field. The quantum correlations introduced in this manner are manifest through their
higher-order correlation functions where a type of squeezed state is identified.
1 Introduction
Squeezed state generation of electromagnetic fields provides a means of reducing uncertainty in one
electric field quadrature at the expense of a larger uncertainty in its conjugate partner [1, 2]. It is
one realization of nonclassical states (ideally, minimum uncertainty states) that has received wide
attention. Ordinarily, in single or multi-mode squeezing, the fluctuations of linear combinations
of the field operators are considered [1]; however, Hillery [3] introduced quadratic combinations
of the field operators as a type of higher-order squeezing [4]. The higher-order combinations are
examined to help elucidate the nature of the phase space occupied by the squeezed states.
We consider a two-mode model originally developed to study stimulated Raman scattering
[5, 6]. In a cavity environment the model has features of amplifiers [7, 8] in which quantum states
are rendered macroscopic and therefore, classically measurable, while at the same time the fields
retain some quantum mechanical correlations. The introduction of both Stokes and anti-Stokes
fields indirectly coupled through a Boson field, whose origin stems either from phonons or weak
atomic excitation of the medium, is an interesting two-mode quantum system. It differs from
several previous two-mode systems, eg. [1, 8, 9], because the two modes are coupled through the
intermediate field that acts like a rcservoir.
The emphasis of this paper is placed on higher-order squeezing found in the fields because
squeezing of the linear combinations of the operators is not present in this model. A more complete
discussion of the results can be found in [10]. The type of higher-order squeezing found is in the
variance of the variables defined by Hillery, so-called sum or difference squeezing variables; they
are used to infer that quantum correlations exist between the electromagnetic fields and the Boson
fields.
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2 Model
We investigate the model Hamiltonian for a stimulated Raman scattering process with undepleted
laser field eL, which can be treated classically. The fields in the interaction are the Stokes field,
subscript S, and anti-Stokes field, subscript A, that are coupled through a Boson field with
multiple modes [5, 6]:
T(= hwsatsas_b hwAatAaA + E hwBta_,aB,-- E(hg, eLatsatBl + h_* eLatAaBl+ h.c. ). (1)
l l
This model has a bath of Bosons, eg., phonons that have excitation energies spread over a range
of frequencies. In this model the Bosons are responsible for coupling the electromagnetic fields
and for introducing damping, as well.
In order to calculate various moments we determine the characteristic function of the operators
in normal-ordered form. The normal characteristic function after reducing the intermediate reser-
voir in the dynamical equations is expressed as an average over an initial distribution of complex
amplitudes {_s, _A}, which is the coherent-state representation for the initial field operators,
(2)
\ -
where we assume that the detuning parameter A = WL -- (ws + WA)/2 is equal to zero and define
_s(t) = _,s(t)_s + _s(t)G, _A(t)= uA(t)_A+ _A(t)_;.
The angular brackets denotes the average over the initial states of the Stokes and the anti-Stokes
fields. The coefficients in the above expressions are obtained from solution of the Heisenberg
equations of motion and the subsequent reduction of the Boson modes in the normal characteristic
function using disentangling theorems. Letting F -- ('rs - _A)IEL[2/2, where the laser field is
EL = eL exp (iwLt) and the parameters
"Is= 2_rlg(_'B)12p(_',), _'A = 2_rl'_(_8)12P(_B),
introduced from the Markoff approximation with the Boson excitation frequency ws = WL -- ws,
the results are
1 1
_ (-_er' - _); _(t) - ('Ys- "Y,J");
"fs - "YA 7s - _/A
= --VA(t)- _ (e l't- 1)ei(2t"+'_s-¢");
7s - _A
_ 1 ('y_(e 2rt- 1) q- 2"yS'YA(1 -- err)) -4-(_s "rA)2
"[S'TA (err _ 1)2 + "TAffY (1 -- e2rt);
_/A fiV
"Ts - 2A
__S_A ( 1 (ert _ l)(3,A _,Tsert) + fiv(e2rt _ l))
"TS--'TA 'TS %4
us(t)
_(t)
Bs(t)
BA(t)
DSA(t)
(e 2rt - 1);
e i(2¢L+cvs-_vA). (3)
The phases are defined by EL = IELlexp(i¢L),g = Iglexp(iCs) and _ = I_[exp(iCa).
20
The usual definition of the two-mode operators is a linear combination of the creation and
annihilation operators. However, we find that the model discussed here does not yield the usual
squeezed state correlations between the Stokes and anti-Stokes fields. The coupling through the
reservoir is also expected to degrade the coherence developed between the Stokes and the anti-
Stokes fields during evolution. It is, therefore, surprizing that the fields do display quantum
coherences in the higher-order correlations between the fields. To show this we adopt of the
definitions of sum squeezing and difference squeezing used by Hillery [3].
2.1 Sum Squeezing
For sum squeezing we define the operators
V1 =-_1 IA tsAtA -t- AsAA), V2 = -_t"sir4t AtA -- AsAA). (4)
The product of their standard deviations, AVi, satisfies the Heisenberg inequality
I (NA + Ns + I) (5)
_ v, A v2 >_-_
The operators are in a quantum state, said to be sum squeezed in the V1 direction when the
variance of V1 satisfies the inequality
1 (NA + Ns + 1) (6)(zxv, )2 < -4
To determine whether the dynamics produces a higher-order squeezed state, we define the shifted
variance 1
5V? = (AV1) 2 - _ (NA + Ns + 1); (7)
which is negative in the region of the quantum state•
The moments of these operators are calculated by using the characteristic function and the
result for the sum squeezing shifted variance of V1 is
=1
,_y? 4<[(¢S(t)_A(t))2+ 4DsA(t)¢_(t)_A(t)+ 2(D_A)_+ 2D;A¢_(t)¢,(t)+ ¢.c.]
+ 2 [l¢_(t)_A(t)l_+ Bs(t)l¢,(t)?+ B.4t)[¢_(t)?+ ID_A(*)I_+ B_(t)BA(t)]) (8)
1
4 (_s(t)(A(t) + DSA(t) + c.c.) 2 •
2.2 Difference Squeezing
For the definition of difference squeezing, define
1 t 2(AsA_ AtsAA) (9)W1 = _(AsAA + AtsAA), W2 = - •
The state is difference squeezed in the W1 operator when the variance of the operator satisfies the
inequality ((Ns} > (NA))
1 (Ns- NAI (10)(_w_) _ < _
21
The momentsare calculatedfrom the characteristicfunction, asdiscussedalreadyin the previous
subsection.We also definea shifted varianceof W1 in analogy with Eq. (7)
1 (NA- Ns)
= -
(11)
which is negative when the state is squeezed along the W1 direction. For the difference squeezing
variable Wx we have the following expression
1
- 4 ([(_s(t)_*A(t))2 + 2D*sa(t)_s(t)_a(t) + C.C.] + 2 [[_s(t)_a(t)] 2 + Bs(t)]_a(t)[ 2 (12)
1
+ BA(tli{s(tll 2 + IDsA(t)I = + Bs(tlBA(t) + I&(t)l = + Ba(t)]) -- -_ (&(tl{'A(t) + C.c.) 2.
3 Results
There are several parameters occuring in the model and appearing in Section 2. The dynamical
parameters, i.e. those appearing in the evolution equations have been previously defined. We note
that the detuning is assumed to be small in our model and this parameter is set to zero. The
initial states of the fields represent another set of important parameters. The choice of an initial
state for the Stokes and anti-Stokes fields is dictated by experimental conditions. We restrict our
discussion to combinations of two experimentally useful initial states: the coherent state and the
chaotic state. Using one of the choices, we examine the quantum correlations developed betwwen
the electromagnetic fields; of course, other situations, such as, a Fock state or a squeezed vacuum
state could also be identified. The Boson field is considered to be in a chaotic state with an average
number of excitations fiB; when the Stokes and/or anti-Stokes fields are in a chaotic state, then
their phases are randomized and their statistical properties are also represented by their average
photon number fis and fin, resp. When the Stokes and anti-Stokes fields are in coherent states,
in addition to the average photon number, the phase of the fields, q_s and CA, is also needed.
The plot of Figure 1 is a display of the shifted variance of the operator VI versus the interaction
time t for the three different values of the phase q_ = 2q_g -- _bs - _bA. The Stokes and anti-Stokes
fields are both initially in a coherent state, ns = nA = 2, and the reservoir is in the vacuum state
fly = 0. The time has been scaled to the product, 7[ELI 2, where EL is the laser field amplitude
and in the results presented here we set 7 = 7s = 7A, i.e. the damping constants are equal. The
region of the curves with negative ordinate values corresponds to the case when light is Vl-sum
squeezed. The phase value of _b= 7r/2 continues to decrease as the interaction time increases which
means that for large times squeezing occurs near the point q_= re/2. As the average number of
excitations is increased in the Boson reservoir, the region for squeezing deteriorates.
When both the Stokes and the anti-Stokes fields are initially in a chaotic state, the sum
squeezing variable V1 still shows squeezing and the phase q_= rr/2 is very robust to the values of
the initial state (Figure 2). We note that the initial value of the shifted variance has been changed
by the initial chaotic state of the variables.
No squeezing was found for the variable W1, either with coherent or chaotic initial states.
22
2 ____
0
[p=0
'5% 0'.5 1'.o 1. 2.0
t
Figure 1: Plot of the sum squeezing shifted variance versus the interaction time for initially
coherent Stokes and anti-Stokes fields. The phase ¢ = 2¢L + _bs - CA has the values 0, 7r/2 and
"_°
4 Summary
In this paper we have examined a special model for the interaction between two modes in a
cavity mediated by a Boson reservoir field [5, 6]. We find sum squeezing, a form of higher-order
squeezing, over a range of interaction times and initial states. There are two salient features of our
results; first, the intermediate field has a continuous spectrum of a reservoir, but still the two fields
develop quantum mechanical correlations; and second, the quantum nature of the correlations is
not manifest through the usual first order or even simple higher-order correlations among the
operators, but through special combinations of the field operators.
There are other models where the fields are mediated by either electronic or acoustic fields,
eg. a polariton or Brillouin scattering model [5, 6, 11]; these processes are analogous to the
present model where the directly coupled fields are not detected in an experiment. In such cases
experiments designed to measure higher-order correlations can reveal the underlying quantum
correlations induced through the fields.
Acknowledgments
This work was supported by funds from the National Science Foundation Grant number PHY92-
13449, from an NSF International Grant number INT 94-17628 and in part by Grant number
RFD 300 from the International Science Foundation and Russian Government.
References
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[2] K. Zaheer and S. Zubairy, Adv. in Atomic, Molecular and Optical Physics 28, 143 (1989).
23
-5 0 1 2 3 4 5
t
Figure 2: The sum squeezing shifted variance versus time for initially chaotic Stokes and anti-
Stokes fields.
[3] M. Hillery, Opt. Commun. 62, 135 (1987); Phys. Rev. A36, 3796 (1987); Phys. Rev. A45,
4944 (1992).
[4] C. K. Hong and L. Mandel, Phys. Rev. Lett. 54, 323 (1985); Phys. Rev. A32,974 (1985).
[5] D. F. Walls, J. Phys. A6,496 (1973).
[6] J. Pe_ina, Optica Acta 28, 325 (1981); ibid. 28, 1529 (1981).
[7] R. J. Glauber, in Proceedings of the VI International School on Coherent Optics, Ustron,
Poland 1986, ( Polish Academy of Sciences, Warsaw, 1987). Also in Group Theoretical Meth-
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[8] W. Whger, n. King, R. J. Glauber and J. W. Haus, Phys. Rev. A34, 4859 (1986).
[9] J. W. naus and H. J. Lee, Phys. Rev. A43, 4954 (1991); H. J. Lee and J. W. Haus, in
Nonlinear Optics and Materials, eds. C. D. Cantrell and C. M. Bowden, Proc. of the SPIE
1497, 228 (1991).
[10] A. V. Chizhov, J. W. Haus and K. C. Yeong, Phys. Rev. A52, 1698 (1995).
[11] A. V. Chizhov, B. B. Govorkov and A. S. Shumovsky, Mod. Phys. Lett. BT, 1233 (1993).
24


Higher-Order Squeezing in a Boson Coupled Two-Mode System

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