A two-dimensional generalized oscillator with time-dependent parameters is
considered to study the two-mode squeezing phenomena. Specific choices of the
parameters are used to determine the dispersion matrix and analytic
expressions, in terms of standard hermite polynomials, of the wavefunctions and
photon distributions. (to be publish in the Third Workshop on Squeezed States
and Uncertainty Relations, Baltimore, USA, (August 1993))Comment: LaTeX, 5 pages, preprint of Instituto de Ciencias Nucleares, UNAM
  Departamento de F\ii sica and Matem\'aticas Aplicadas, No. 02 (1994

Castaños, O.

López-Peña, R.

Man'ko, V. I.

English

arXiv

A two-dimensional generalized oscillator with time-dependent parameters is considered to study the two-mode squeezing phenomena. Specific choices of the parameters are used to determine the dispersion matrix and analytic expressions, in terms of standard hermite polynomials, of the wavefunctions and photon distributions

Castanos, Octavio

Lopez-Pena, Ramon

Manko, Vladimir I.

NASA Technical Reports Server

N95- 13907
SQUEEZING IN A 2-D GENERALIZED OSCILLATOR
Octavio Castafios and Ram6n L6pez-Pefia
Instituto de Ciencias Nucleates, UNAM
Circuito Exterior, C.U., Apdo. Postal 70-5_3
0_510 Mdxico, D. F., Mdxico
Vladimir I. Man'ko
Physical Institute of the Russian Republic
Leninsky Prospect 53. Moscow, Russia
Abstract
A two-dimensional generalized oscillator with time-dependent parameters is considered
to study the two-mode squeezing phenomena. Specific choices of the parameters are used
to determine the dispersion matrix and analytic expressions, in terms of standard hermite
polynomials, of the wavefunctions and photon distributions.
1 Introduction
In the middle of the sixties and beginning of the seventies a set of quantum states of the elec-
tromagnetic field were observed which have less uncertainty in one quadrature than a coherent
state [1-3]. These one-mode squeezed states have generated big expectations in optical commu-
nication systems [4]. In some quantized fields, the interaction hamiltonians occur only between
pairs of modes and then to understand the main features of the system, one restricts to study
one and two normal modes . In the last decade two-mode squeezing phenomena have attracted
attention to study properties of noise and correlations [5-8]. Recently the accidental degeneracy
of a two-dimensional (2-D) harmonic oscillator with frequency 0a0 plus an interaction proportional
to the z-th projection of the angular momentum was studied [9]. This system was called the
generalized 2-D harmonic oscillator because presents a bigger accidental degeneracy depending on
the strength A of the angular nlonmntum interaction. This model was generalized [10] to include
time-dependent parameters, m = roof(t) and ,_ = wo_o(t). If we take f(0) = A0(0) = 1; the
hamiltonian, for t = 0, represents a charged particle moving ill a constant magnetic field .
The aim of this work is to study two-mode squeezing phenomena with this model because it
demonstrates the change of dispersions due to variation of the mass and coupling constant during
the evolution. In the framework of quantum optics the hamiltonian is built by: the operator
(1/f + f)_ • if, that causes a time-dependent exchange of kinetic and potential energies within
each mode; the interaction 1/2(1/ f - f )( _ • a_ + _ . g), which describes a degenerate two-photon
interaction; and the potential iAo(t)(a_al - a[a2), that is a mode mixing operator.
The solution of the corresponding time dependent Schroedinger equation is obtained through
the theory of integrals of motion [11]. By means of Noether's theorem, using a special variation
87
https://ntrs.nasa.gov/search.jsp?R=19950007494 2020-06-16T10:28:19+00:00Z
[10], we construct the linear time dependent integrals of the motion. The resulting quantum
invaxiants are given in terms of the positions and momenta operators [10,11] by
(i)
with the initial conditions/3(0) =/_and Q(0) = _', so that the 2 x 2 matrices previously introduced
satisfy ,kl(0) = ,k4(0) =/2 and ,k3(0) = ,k2(0) = 0. The operators ,4(t) = 1/v/'2[(_(t)/l + il/hP(t)]
and its hermitean conjugate, can be constructed with the matrices
)_p-_- _3"1- _)_1, ,_q-- _/_4"}-_/_2, (2)
with l = _ defining the oscillator length. These integrals of motion also are given in terms
of the creation and annihilation photon operators
A(t) = MIE + M2# t, At(t) = M3_ + M4ff t • (3)
With the initial conditions A(0) = # and At(0) = a-'t, the matrices defined in (3) comply with
MI(0) = M4(0) =/2 and M3(0) = M2(0) = 0. The )_k's, Mk's, )_p and Aq are entries of symplectic
matrices in four dimensions because the invaxiants (1) and (3) satisfy the commutation relations
of Heisenberg-Weyl algebras.
In the present work, we study the behavior of the model for ,k0(t) an arbitrary function of time
and considering two kinds of varying masses, i.e., two choices for the function f(t), namely:
/(t) = exp(Tt) ; (4)
1/(t) = cosh2riot
{flo(t - T) sinh rioT + cosh floT} 2
For these two cases the Ak matrices take the general form
, t<_0
, 0<t<T
, T<_t
cos8 sin0_.,kk=#kR=#k --sin0 cos/9]' k=1,2,3,4; (6)
where the definition 0 = fg tJo,ko(r)dT was used. The analytic expressions for the #k's functions
axe given in Ref. [10]. In the next sections we determine the coherent and Fock-like states, the
photon distributions and the dispersion matrices in terms of these #k's.
2 Squeezed Coherent and Fock States
The coherent-like states are obtained by solving the differential equation A(t)ff0(_,t) = 0 with
A(t) given in Eq. (3). This solution yields the vacuum state of the physical system, and its phase
is chosen to guarantee that satisfies the time dependent Schroedinger equation. The expression
for the ground state wavefunction is
q)o(_,t) - v/_-_#pl exp{ 2-hi#'0""q'}pp (7)
88
To get the last expression the relation (2) was used and the functions pp = _ (-_#, + 1/23) and
#q - _2 (_#2 + 1/24) were defined. To obtain the general expression for the eigenstates in the
coordinate representation one needs to apply the unitary operator/)(a) = exp{_ • )_t _ _.. Pt},
which is an invariant, to the vacuum wavefunction (7), i.e.,
ff_(q, t) = exp --T-lal= + 21_#p a. 5 + _-p-_#pq Ra _¢0(q, ) (8)
These are expressed in terms of multi-dimensional Hermite polynomials [12] through the relation
) .., (v)exp(-- u_*._*+v_*R_ = _ al a*n2H{UI2}n,,,,2=o nl! _ "''= 1_'7 (9)
Substituting the last expression into (8) and using the form of the coherent-like states in the
Fock-like representation, we get the Fock-like eigenstates in the coordinate representation:
(q"ln,n2) "0(,, t)H !,_n_ I2} i _.
These multi-dimensional Hermite polynomials are rewritten as a product of two standard one-
dimensional Hermite polynomials [12] as follows:
[ _I,,,_ ( ) = (_ /-tp _ (rq+nx,/2 ( 1 )_ I_p ) i . Hn, [COSe ql +sine q2]
n.,,.= - rtq 2/2/ vSgl#,l
(' )x 8. 2 vF2h[/2,[[-sine ql +cose q2] , (11)
where we use the explicit expression of matrix R. These Fock (10) and coherent (8) -like states
represent squeezed and correlated eigenstates of the system as it will be shown further.
3 Propagator
The propagator in the coherent state representation is given by the matrix elements of the evolution
operator U(t), which will be obtained by means of the theory of time dependent integrals of
motion [11]. If f(t) is an integral of motion then satisfies f(t)_f(t) = U(t)f(0). Taking its matrix
elements with respect to the coherent states, we get a linear system of differential equations, which
can be solved. Thus the propagator takes the form
G( .,.Lt)=exp(-I lU2-1 lU2)ex i' la.M-,M. ) (12)
For the cases (4) and (5), the following relations are satisfied
= --_ l#q- -Tl.tp _ gl, Mll]t42- vf_g 1 ll2q + -_/2p I2 -- g212 , (13)
89
ih .\M_ x = 1R M3M11 1 l#*q + T#p)I2 -- g412 (14)
gx ' - x/_gl
Substituting these relations into the Eq.(12) we get the propagator, which through Eq. (9) can
be expressed in terms of multi-dimensional Hermite polynomials. If we compare with the power
series expansion of the propagator we get the probability amplitude for having n: and n2 photons
in the coherent-like state ]'7, t), i.e.,
(nln2l'7, t) -
By means of the Eq.(12) this amplitude can be rewritten in terms of standard Hermite poly-
nomials [12]. The squared absolute value of this amplitude yields the photon distribution function
of the system, w.,.2(q,t) = I(nln21"7,t)lL This will let us calculate, at least formally, the mean,
(Nk), and the mean squared fluctuation of the number of photons, (ANk) 2, in direction k, which
axe present in the coherent state 1"7,t). The expectation values of Nk and N_ are evaluated directly
using the expressions of the creation and annihilation photon operators in terms of the integrals
of the motion (3), and the commutation properties for these invariants. For the vacuum state one
has 1 2
(Nk) = (.: - m) 2+ m0_0m+ , (16)
mowo
(Ul -- U4) 2 "_- (Tn0W0U3 -I- --U2) 2
//_0030
{(.l q_ /z4)2 _l_(m0w0. 3 1 .2)2} -i-
mOO.) 0
i{ l ,},(_ m) • (:7)
m0o)0
With these expressions, we evaluate the ratio of the mean squared fluctuation (ANk) 2 and the
mean number of photons (Nk), which determines the nature of the distribution function of the
system:
(Ark) 2 m0w0
For the cases (4) and (5) the ratio is greater than one when t > 0, which implies that we have
a super-Poissonian photon distribution function. For t = 0, there is a discontinuity in the ratio,
which is obtained by comparing the following limiting procedures: making t ---+0 and then ff --+ 0,
and conversely.
4 Dispersion Matrices
The dispersion matrix can be written in terms of 2 x 2 matrices characterizing the dispersions
in the positions and momenta operators and the correlation between them. Besides for the cases
under study, due to (6), they take the form
a_(t) _hmowo 1 2 #])i, ' (19)= ((mo_0)2.2 +
9O
o_,.(t)= -_ .,._ + _o_o._l,,,I. , (20)
1 h
o_q(t) = 2 mo02o(#_ + (m°02°)2#32)I2 (21)
The corresponding correlation matrices for the creation and annihilation operators are obtained
immediately from the last expressions; they are given by
2 I{ 2 "_0°°= 3 "' - .2 + (=o02o)2._ 1 =0020"3"4)} (22)(=0020)2 2iL=-----_ow.o"'"2 +
Oo,o= _., + .2 + (=o02o)_._+ (=0020)2
2 1{ 2 "5 2i[ 1--.1.2 =0020.3#4)}{Tatot = 4 "1 -- .2 Jr" (=0020)2.32 (=0020)2 "Jr- F/10020 4-
(23)
(24)
4O
35
30
25
20
15
lO
5
0
0 10 20 30 40
time
50
o.sl.0.40.3
0.2
0"1 I
0
&
g
0
0
-0.0:
-0.0:
-0.03
-0.04
-0.05
S 10 15 20 25 30
time
10 15 20 25 30
time
16
12
10
0
0 5 10 15 20
time
0.5.--
0.4"0.3'
0.2 _
O.1
0
°i-
-0.01
-0.02'
-0.03'
-0.04
-0.05
-0.06
-0.07
S i0 1S 20
tlme
S I0 lS 20
time
(a) (b}
Fig. 1. Dispersion and correlation matrices behavior in positions and momenta
space for the studied cases in this paper: (a) corresponds to Eq.(4), and (b), to Eq.(5).
91
The behavior of the dispersion matrices is illustrated in Fig. 1. For the case (4), we choose the
parameters "_ = 0.1 and m0 = w0 = 1. It is seen that there is squeezing for the coordinates and
stretching for the momenta. Also one notes that a m is a negative function and therefore there
are one-mode correlations between the coordinates and the momenta. If we reverse the sign of 7,
the roles between the dispersion for coordinates and momenta are interchanged, and a m becomes
positive. In the case (5), we use the parameters gt0 = 0.15, T = 10, and m0 = w0 = 1. In spite
of the mass is different that in the previous example, the general trends are similar. For example,
the a_ is an increasing function of time starting from its minimum value at t < 0, and there is
squeezing for the aqq. The main difference appears in the correlation apq: in this case, it can be
positive for large times, while in the previous one is negative or zero for any time.
5 Acknowledgments
Work supported in part by projects UNAM-DGAPA IN-103091 and CONACYT 1570-E9208.
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92


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