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Coherent-state propagator of two coupled generalized time-dependent parametric oscillators
16June
1997
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 230 (1997) 144-152
Coherent-state propagator of two coupled generalized time-dependent parametric oscillators KM. Ng, C.F. Lo Deportment ofPhysics. The Chinese Uniuerxity offlong Kong. Shark, New Territories, Hong Kong Received
16 May 1996; revised manuscript received 8 January 1997; accepted for publication Communicated by P.R. Holland
7 March 1997
Abstract
We have derived the coherent-state propagator of a pair of coupled generalized time-dependent parametric oscillators using the Lie algebraic approach. The results are for the most general pair of coupled time-dependent oscillators, and thus will be useful for future studies in quantum optics as well as in atomic and molecular physics. 0 1997 Elsevier Science B.V. PAC.9
03.65.Fd
Recently a Lie-algebraic approach based upon the Wei-Norman formalism [ll was developed by Lo [2,3] to derive the propagator (in both coordinate representation and coherent-state representation) of the generalized time-dependent parametric oscillator. The method is simple and gives the same results as those obtained by the path-integral approach. Later Lo and Wong [4] applied this technique together with a canonical decoupling transformation to a system of two coupled driven time-dependent oscillators and obtained the propagator in coordinate representation. In this paper we shall extend the method to find the coherent-state propagator of a system of two coupled generalized time-dependent parametric oscillators, whose Hamiltonian takes the form
H(t)=
i:
j=l
_!L +rnj( t) Wj( Q2x; 2mj(r)
2
+~(r>~j+,j(r)Pj+h,(r)x,x2+A2(t)x,~2
XlPl
+kdt)%P,
+h(~)PIP2+4(f)
fP,Xl
2
+%(f)
x2Pz+P2x2
2
.
w,(t), mj(t>, fj(t>. gj(t) and Aj(t) are arbitrary functions of time. This coupled oscillator system plays a very significant role in physics because it provides the mathematical basis for many soluble models in many different branches of physics such as the Lee model in quantum field theory [5,61, the Bogoliubov transformation in superconductivity [7-91, relativistic models of elementary particles [lo,1 11, the covariant harmonic oscillator model for the parton picture [ 12,131, models in molecular physics [14], and the two-mode squeezed states of 03759601/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00212-0
KM. Ng, C.F. Lo/Phy.sics
Letter., A 230 (1997) 144-152
145
light [15- 191. Also, since the propagator is for the most general pair of coupled time-dependent parametric oscillators, results for any special case can be easily deduced from it. These results will be useful for future studies in quantum optics as well as in atomic and molecular physics. To begin with, let us consider the Hamiltonian H,(t),
ff,(
r)
=
A_ +mj(t)wj(r)Zx;
i
2m,(r)
j=l
f W) Introducing
XI PI
2
+Plx, 2
xzP2+Pzx2 *
+W)
the raising and lowering
+~,~~~~,~,+~,~~~~,~,+~,~~~~*~,+~,~~~P,~,
.
(2)
operators
mjO ojO xi + i pj (3)
with wjO = ~~(0) and m,, = mj(0),H,(t)can be rewritten
as
H,=A,(r)Y,+A;(r)Y,+A,(r)Y2fA;(r)Y,+B,(r)K,+B,(r)K~+C,(r)K++Cz(r)K~ +C,(r)K+C,(r)K_,
(4)
where
B,(r) = h
m,(r) 4(r)
l
mlOWl0
+-
hA,(r) Cd
C2(r) =
m,(r)
i’
m2(r)4(r)
B,(r) =h l
11220“20
+~~~~+i~~~-~Jm,ow,,m,,w,o,
1) = 2
mlOWl0
mlom20w10w20
fiAl(r>
-~~~~+i~~~+~~m,,w,,m,,w,,,
2~~,0~20%0~20
C,(f) =
fiAl(r) 2 mlom20@
IO w20
+~~~~-~~~~+~l/m,ow,om,o~~o,
+ m20w20 m*(r)
1 .
146
KM.
The operators
K,,
K,,
Kh, K’,
C.F. Lo/Physics
and 5 (j=
K_=
a;a2 + ala;
K;=
4
alal +a,a,
K, =
ala2,
144-152
1, 2, 3, 4) defined by i
K,=a’;ai,
A 230
t K\ = aja2,
’
4
KL=a,a’;,
_L
y, = +a;a;,
Y, = iala;,
’
form the closed de Sitter-Lie is
Y, = ta,a,,
algebra (in the bosonic realization)
Y, = fa2u2
[lo]. The corresponding
Schriidinger
Ha(t)l$(t))=iti~ls(r)).
equation
(7)
As usual, we shall define the time evolution operator U&t, 0) such that 11,5(t)) = I/&t, 0) 1$(O)), where I G(O)> is the wave function at time c = 0. To find the evolution operator U&r, O), we need to solve the following evolution equation, Ho(f)U,(t,
0) = ihiUO(r,
U,(O, 0) = I.
0),
Since the de Sitter algebra is a closed algebra, the evolution form [ll, &(tV 0) = exp[c,(t)K+]
exp[c,(t)Y,]
Xexp[c&)G]
(8) operator can be expressed
exp[c,(r)Y2]
exp[c,(r)K:]
exp[&)Y,]
exp[c&)K:] exp[c&)Y41
in the following
product
exp[c,(WGl
ex~h<~>~-l~
(9)
As in Refs. [2-41, by direct substitution and comparing the two sides of Eq. (8), we shall obtain ten coupled ordinary differential equations in the ten unknowns cj(t). Then the coefficients cj(t) will be determined by solving the set of coupled differential equations with the initial conditions c,(O) = 0. However, since U,, [201. First of all, we define the associated normal function lJ(Jn”‘
s b,,
a2 IQ(t* 0) I aI, 4
=ev(-I=,
12{1-Gus,
ev[cdf)] -exp[c,(tf]))exp(-I*,1211
-exp[&)l})
Xexp[c,(t)a~+~,(t)cr,*~+c~(t)~~+c~(t)~~~] Xexp{c,0(r)oI~2
+crWo,*o;
XexP{exP[c,(t)l[c,(t)a,*a,
+fMt)
+c,wv;l}.
+c&)l)
(‘0)
to see that for any arbitrary operator consisting of aj and ~3, there is a one to one correspondence between the associated normal function and the normal-ordered form of the operator, namely the normal-ordered form of the operator can be obtained by replacing the oj* and crj in the associated normal function by the operators a: and aj respectively, and vice versa. This property is very helpful for solving operator equations such as the evolution equation in Eq. (8). Next, we convert the evolution equation to a c-number equation by evaluating its expectation value with respect to the coherent state I a,, cr,),
It is not difficult
(a,,
(11)
KM. Ng, C.F. Lo/Physics
Letters A 230 f 1997) 144-152
I’he right-hand side of the equation is given by a
+
I a, I
2$exp( c5)+
c4c7
+al.a,-+[c, exp(c,)]
evtc6)] + I a2 I2 d%
+c++a,a2
+a::
exp(c,)% +aTa,$[c, e&,)1 dc,o
dr
)
U,j”)(&
a;,
a,> ff2, r),
(12)
and the left-hand side is (a,.
a2 I H,( f>&( r, 0) I aI 9 4
=
A,a,‘a;
+A,a;ff;
+A;
(%+&)(a,++)+A+2+-&)(%+-$)
i +B,[a;(a,+&)+i]+B2[n(n2+-&)+~]+C:+~
+C2[a+2+-$)]+C3[+?+-$)]
(13) Then, comparing the coefficients of the quadratic and bilinear terms on both sides of the equation, we can obtain ten coupled differential equations, ~~~~=~A;C,C,+~A;C,~~+(B,+B~)C,+C,+~C~C~+~C~C~+C~(C~+~C~~~).
(14)
dc, ih~=A,+4A~c~+A~c~+2B,c2+C2c,+2C,c,c2.
(15)
ih-
dc, =A;c~+A2+4A;c~+2B2~3+Cj~,+2C~~,~3, dr
dU,
(16)
ihx
= 4A; c,U, + 2A; c,U, + B,U, + C2U2 + C,( c,U, + 2c2U2),
(17)
ifi$
=2A;c,U,
(18)
+4A;c,U,+B,U,+C,U,
+C,(2c,U,
+c,U2),
ih~=2A~c,U~+4A~c,Li,+B,U,+C,U~+C,(c,U3-t2c,U~),
ifi?
= 4A;c2U,+2A;c,U3
% ‘fidr
=A;U,“+A;U,2fC&J,U,,
+B,U,
+C,U,
(19) +C4(2c2U3
+ c,U4),
(20)
(21)
KM. Ng, C.F. Lo /Physics Letters A 230 (1997) 144-152
148
(22) ifi-
dc,, dt
= 2 A; U,U, + 2 A; U,lJ, + C,(U,U3
(23)
+ &U,),
where the functions U,(t) are defined by u,(t)
=exp[c,(t)l
+cAt)cAt)
U&I = exp[c&)l
I
exp[c~(r)l,
G(r)
=cAf>
exp[c~Lr)l,
exP[%wl~
U4,(t) = cdr>
(24)
Of these ten differential equations, the first three equations are completely independent of all the others and each of them is of the nonlinear Riccati type. Thus, we can first solve these three coupled nonlinear differential equations and obtain the coefficients cj(t), j = 1, 2, 3. Knowing these coefficients, we can then apply the standard technique of coupled linear differential equations to determine the functions Uj(t), j = 1, 2, 3, 4. Note that Eqs. (171, (18) and Eqs. (19), (20), in fact, form two independent sets of two coupled linear first-order differential equations. With all the functions U,(t) being known, the coefficients Cj(t), j = 4, 5, 6, 7 can be evaluated readily. Furthermore, the remaining three equations can be solved by straightforward integration to give the coefficients c,(t), j = 8, 9, 10. Since U,(t, 0) is known, the evolution operator I/( t, 0) describing H(t) = H,(t) + H,(t) will be given by U(t, 0) = Uo(t, O&,(t, 01, with U,(t, 0) satisfying the evolution equation H,(r)U,(t,O)=ifi~U,(t,O),
U,(O, 0) = I,
(25)
where H,(t)=U,-‘(t,O)H,(t)U,(t,O)=
&f!(t)a:+di(t)aj= j=
ihj(t)
(26)
I
j=
I
and d;(t)
=
(77, +27?;c*
+rl;c$J3
-
u,u3
(r/2
d:(t)
+
77,*c,
-
+2772*c&
77;c,
f2712C3P4
-
(77,
+2rl;c2
+
172w72
=
1 - c,,,
=
772+771*c,
u,
71,
-
rl2+77;c,
l-c,,
+ (
-
-
u, + 2c,
u2u4
1
-
4v
u2u4
771+27?;c* + 7?;c, u2 UP3
-
u2u4
U
u2”4
u3
77;c,
+2r);c,
u,u,
rll+2rl;c2+7?;c, u,u3
-
2c,
+
o;u4
+27l;c,
+
u2 u,r/,
r/2+r)l*c, u3
u,“,
77,+27?,+c2
(
+27?;c2+7);c, u,u3
= -2c,
1 + c,,)
+
1
u2u4
2c,
-
u2u4
+27/;c3
u,u3
i
d:(t)
+
u2u4
U,U, d!(t)
(772
-
1 + c,o
+ (
u2u4
r/2
+
77;
c,
u,u, -
+
2.6
c3
u2u4
“’
(27)
Lerrers A 230 (1997)
K.M. Ng, C.F. Lo/Physics
In terms of the generators of the Heisenberg-Weyl hj( r) = b:‘(t)
algebra,
149
144-152
can be written as [211
hj(r)
e{ + bi( r) e:’ + b{e{,
(28)
where e{ =
e;
ai.
=
at. I’
e{
b:‘(r)
=d:(r),
=
I
(29)
and b:(r)
=d’_,
bj‘( r) = 0.
(30)
The operators ei form a closed Lie algebra, [e{, ei] =e{,
[CT/, e{] = [ei, e$] =0,
(31)
and thus by following the procedure shown above, the evolution operator U,(f, 0) can be easily found to be U,(r,O)
= ,fiexp[h:(t)ej]
exp[hi(t)e{]
exp[h{(t)e!],
h:‘(r) = &bj(
u) du,
(32)
with h{(r) = &b{(u) h;(r) = ;
du,
jdh:(u)bj(
u) du.
(33)
Hence, we have obtained an exact form of the time evolution operator U(r, 0) of the two coupled generalized time-dependent parametric oscillators. Now we shall determine the coherent-state propagator associated with the evolution operator U(t, 0). Let {I a], CQ)) be the coherent states of the initial Hamiltonian H(t = 0). Then the coherent-state propagator K(cr,, ff7.. r; p,, p2, 0) is given by the matrix element ( (Y , , a2 1U(r, 0) I pi, pz > which denotes the probability amplitude that the system undergoes a transition from the state 1 P, , & > at time f = 0 to the state 1a I, a2 > at time r > 0. From Eqs. (91, (10) and (32), we have Cc+ (Y,IU(l,O)I = (a,,
P,, Pz>
a2 I&(r,O)I
P, +h$)*
/32+$(r))
~,~exp[~~~~(r)+P,12~~l~j12+P,h:(r)+h~(r)] = exp(--(yI* [ P, + G(t)]{1- c4(f)c7(f)exp[c&)l-exp[c~(r)l}) Xexp( -4 [ P2+GWl{l -expMf)l)) xexp(c,(r)[~,+h:(r)]2+c2(r)~~2+c~(r)[P2+h:(r)l2+c~(r)a;2)
Xexp{c,,(r)[
PI
xexp[exp[c,(r)]{c,(r)a;[
X ~exp(~lh~(r)+/3jI’-il
+hi(r)][
/32+%(r)]
P2+G(r)]
+cdr)~ltG
+cAr)[
P,12+P,h:‘(t)+h:(r)]*
+3[4t>
PI
+Cdr)l}
+G(r)la;)]
(34)
Iy.M. Ng, C.F. Lo/Physics
150
Lerters A 230 (19971 144- I52
This coherent-state propagator is for the most general pair of coupled time-dependent parametric oscillators, and thus results for any special case can be easily deducedefrom it. To illustrate the validity of our approach, we consider a charged oscillator with a time-dependent mass and frequency moving in the q-plane under the influence of a time-dependent electromagnetic field, whose Hamiltonian takes the form [22] H(r)
=
b-q‘wN2 2m( f)
(35)
++n(f)O&)2(X:+X:)+q+(f).
We shall restrict ourselves to the cases where the electromagnetic potentials are given by A(r) = B(t)(x, and &I) = 0. With this choice the Hamiltonian H(t) becomes
P, -
x, P,)/2
where w = \i,o0 + h and A = qB/2m. Here we have set fi = c = 1. In terms of the generators of the closed de Sitter-Lie algebra, H(t) can be re-written as H(t) =A,(r)Y,
+A~(t)Y~+A2(r)Y;!+A;(t)Yq+B,(t)Ko+B2(t)K;,+C,(t)K++C2(t)K:
+ C,(t)K’+
C,(t)K_,
(37)
where mt0> w(O)
=A2(t)
‘A(t).
2m( f)
m(0) w(O) m(t)
c, = 0, C,=
= B2(r) = B(r),
(38) (39)
(40)
-ih,
(4’)
C, = ih,
(42)
c, = 0,
(43)
and the corresponding evolution operator U(t, 0) can be expressed in the form of Eq. (9). The coefficients ci are determined by solving the ten coupled ordinary differential equations in Eqs. (14)-(23), subject to the initial conditions c;(O) = 0. From Eqs. (14)-(16), it is not difficult to see that c, = 0 and c2 = cs = ?. Thus, we are left with only one equation, that is, dZ ix = A + 4Ac”*+ 2B?.
(44)
By a simple Ricatti transformation defined by . ;=
_‘”
4A w’
(45)
the Ricatti equation in Eq. (44) can be cast in the form of a linear second-order ordinary differential equation
K.M. Ng, C.F. Lo/Physics
which can be solved by the standard approaches. dU, ix = (4Ac”+B)U,
Lerters A 230 (1997)
Furthermore,
144-152
Eqs. (17)-(20)
151
can be simplified
to give
- iMY,,
dU, ix=(4AF+B)U,+ihU,, = (4AFf
ix
dU,
ix
B)U, - ihll,,
= (4A?+B)U,+ihU,.
Obviously, one can conclude that U, + CJ2= U, - U, = X, and U, - U, = U, + Ud = X_. To determine we need to solve the following two coupled ordinary differential equations,
dX+
i---z
dt i%
(5’)
(4AZ+B)X++iAX_, = (4AFf
B)X_-
X,,
(52)
iAX+.
Defining f+(t) = X, exp{ij,$4A(r)Z(r) independent linear second-order ordinary
+ B(T)] dr}, the two coupled equations differential equations
can be decoupled
to give two
(53) Since both equations are of the same form, we need to solve one of them only. Finally, the remaining three equations in Eqs. (21)--(23) can be solved readily to yield cs = cg = - i/,$A(r)[Uf(r> + U,~(T)]} d7 and cIO = 0. As a result, the time evolution operator of the system will be completely determined once we solve the two linear second-order ordinary differential equations in Eq. (46) and Eq. (53). In summary, we have investigated the algebraic structure of the Schradinger equation associated with a pair of coupled generalized time-dependent parametric oscillators. Using the Lie algebraic technique we have derived the time evolution operator of the system, which in turn yields the coherent-state propagator readily. We believe that these results will be useful for future studies in quantum optics as well as in atomic and molecular physics. To illustrate tire validity of the approach, we have also applied the results to the case of a charged oscihator with a time-dependent mass and frequency in a time-dependent electromagnetic field.
References [I] [2] [3] [4] [S] [6] [7] [8] [9] IJO]
J. Wei and E. Norman, J. Math. Phys. 4 (1963) 575. C.F. Lo, Phys. Rev. A 47 (1993) I IS. C.F. Lo, Europhys. Lett. 24 (1993) 319. CF. Lo and Y.J. Wong, Europhys. Lett. 32 (1995) 193. T.D. Lee, Phys. Rev. 95 (1954) 1329. S.S. Schweber. An introduction to relativistic quantum field theory (Row-Peterson. Elmsford, NY, 1961). N.N. Bogoliubov, Nuovo Cimento 7 (1958) 843. A.L. Fetter and J.D. Welecka, Quantum theory of many particle systems (McGraw-Hill, New York, 1971). M. Tinkham, Introduction to superconductivity (Krieger, Malabar, FL, 1975). Y.S. Kim and M.E. Noz, Phase space picture of quantum mechanics (World Scientific, Singapore, 1991). ,I I] H. van Dam, Y.J. Ng and L.C. Biedenham, Phys. Lett. B 158 (1985) 227. 1121 Y.S. Kim and M.E. Noz, Theory and applications of the Poincare group (Reidel, Dordrecht, 1986).
152
K.M. Ng, C.F. Lo /Physics
Letters A 230 (1997) 144-152
1131 Y.S. Kim, Phys. Rev. Len. 63 (1989) 348. 1141 F. lachello and S. Oss, Phys. Rev. Lett. 66 (1991) 2976. 1151 D. Han, Y.S. Kim and M.E. Noz, Phys. Rev. A 41 (1990) 6233. [16] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [17] C.M. Caves and B.L. Schumaker, Phys. Rev. A 31 (1985) 3068; B.L. Schumaker and C.M. Caves, Phys. Rev. A 31 (1985) 3093. 1181B.L. Schumaker, Phys. Rep. 135 (1986) 317. [19] B. Yurke. S. McCall and J.R. Klauder, Phys. Rev. A 33 (1986) 4033. [20] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [21] H. Weyl, Gruppentheorie und Quantenmechanik (Hitzel, Stuttgart, 1928); P. Cartier, Proc. Symp. Pure Math., Vol. 9, Algebraic groups and discontinuous sub-groups (American Mathematical Society. Providence, RI, 1966). [22] C.F. Lo, Phys. Rev. A 45 (1992) 5262.
1997
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 230 (1997) 144-152
Coherent-state propagator of two coupled generalized time-dependent parametric oscillators KM. Ng, C.F. Lo Deportment ofPhysics. The Chinese Uniuerxity offlong Kong. Shark, New Territories, Hong Kong Received
16 May 1996; revised manuscript received 8 January 1997; accepted for publication Communicated by P.R. Holland
7 March 1997
Abstract
We have derived the coherent-state propagator of a pair of coupled generalized time-dependent parametric oscillators using the Lie algebraic approach. The results are for the most general pair of coupled time-dependent oscillators, and thus will be useful for future studies in quantum optics as well as in atomic and molecular physics. 0 1997 Elsevier Science B.V. PAC.9
03.65.Fd
Recently a Lie-algebraic approach based upon the Wei-Norman formalism [ll was developed by Lo [2,3] to derive the propagator (in both coordinate representation and coherent-state representation) of the generalized time-dependent parametric oscillator. The method is simple and gives the same results as those obtained by the path-integral approach. Later Lo and Wong [4] applied this technique together with a canonical decoupling transformation to a system of two coupled driven time-dependent oscillators and obtained the propagator in coordinate representation. In this paper we shall extend the method to find the coherent-state propagator of a system of two coupled generalized time-dependent parametric oscillators, whose Hamiltonian takes the form
H(t)=
i:
j=l
_!L +rnj( t) Wj( Q2x; 2mj(r)
2
+~(r>~j+,j(r)Pj+h,(r)x,x2+A2(t)x,~2
XlPl
+kdt)%P,
+h(~)PIP2+4(f)
fP,Xl
2
+%(f)
x2Pz+P2x2
2
.
w,(t), mj(t>, fj(t>. gj(t) and Aj(t) are arbitrary functions of time. This coupled oscillator system plays a very significant role in physics because it provides the mathematical basis for many soluble models in many different branches of physics such as the Lee model in quantum field theory [5,61, the Bogoliubov transformation in superconductivity [7-91, relativistic models of elementary particles [lo,1 11, the covariant harmonic oscillator model for the parton picture [ 12,131, models in molecular physics [14], and the two-mode squeezed states of 03759601/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00212-0
KM. Ng, C.F. Lo/Phy.sics
Letter., A 230 (1997) 144-152
145
light [15- 191. Also, since the propagator is for the most general pair of coupled time-dependent parametric oscillators, results for any special case can be easily deduced from it. These results will be useful for future studies in quantum optics as well as in atomic and molecular physics. To begin with, let us consider the Hamiltonian H,(t),
ff,(
r)
=
A_ +mj(t)wj(r)Zx;
i
2m,(r)
j=l
f W) Introducing
XI PI
2
+Plx, 2
xzP2+Pzx2 *
+W)
the raising and lowering
+~,~~~~,~,+~,~~~~,~,+~,~~~~*~,+~,~~~P,~,
.
(2)
operators
mjO ojO xi + i pj (3)
with wjO = ~~(0) and m,, = mj(0),H,(t)can be rewritten
as
H,=A,(r)Y,+A;(r)Y,+A,(r)Y2fA;(r)Y,+B,(r)K,+B,(r)K~+C,(r)K++Cz(r)K~ +C,(r)K+C,(r)K_,
(4)
where
B,(r) = h
m,(r) 4(r)
l
mlOWl0
+-
hA,(r) Cd
C2(r) =
m,(r)
i’
m2(r)4(r)
B,(r) =h l
11220“20
+~~~~+i~~~-~Jm,ow,,m,,w,o,
1) = 2
mlOWl0
mlom20w10w20
fiAl(r>
-~~~~+i~~~+~~m,,w,,m,,w,,,
2~~,0~20%0~20
C,(f) =
fiAl(r) 2 mlom20@
IO w20
+~~~~-~~~~+~l/m,ow,om,o~~o,
+ m20w20 m*(r)
1 .
146
KM.
The operators
K,,
K,,
Kh, K’,
C.F. Lo/Physics
and 5 (j=
K_=
a;a2 + ala;
K;=
4
alal +a,a,
K, =
ala2,
144-152
1, 2, 3, 4) defined by i
K,=a’;ai,
A 230
t K\ = aja2,
’
4
KL=a,a’;,
_L
y, = +a;a;,
Y, = iala;,
’
form the closed de Sitter-Lie is
Y, = ta,a,,
algebra (in the bosonic realization)
Y, = fa2u2
[lo]. The corresponding
Schriidinger
Ha(t)l$(t))=iti~ls(r)).
equation
(7)
As usual, we shall define the time evolution operator U&t, 0) such that 11,5(t)) = I/&t, 0) 1$(O)), where I G(O)> is the wave function at time c = 0. To find the evolution operator U&r, O), we need to solve the following evolution equation, Ho(f)U,(t,
0) = ihiUO(r,
U,(O, 0) = I.
0),
Since the de Sitter algebra is a closed algebra, the evolution form [ll, &(tV 0) = exp[c,(t)K+]
exp[c,(t)Y,]
Xexp[c&)G]
(8) operator can be expressed
exp[c,(r)Y2]
exp[c,(r)K:]
exp[&)Y,]
exp[c&)K:] exp[c&)Y41
in the following
product
exp[c,(WGl
ex~h<~>~-l~
(9)
As in Refs. [2-41, by direct substitution and comparing the two sides of Eq. (8), we shall obtain ten coupled ordinary differential equations in the ten unknowns cj(t). Then the coefficients cj(t) will be determined by solving the set of coupled differential equations with the initial conditions c,(O) = 0. However, since U,,
s b,,
a2 IQ(t* 0) I aI, 4
=ev(-I=,
12{1-Gus,
ev[cdf)] -exp[c,(tf]))exp(-I*,1211
-exp[&)l})
Xexp[c,(t)a~+~,(t)cr,*~+c~(t)~~+c~(t)~~~] Xexp{c,0(r)oI~2
+crWo,*o;
XexP{exP[c,(t)l[c,(t)a,*a,
+fMt)
+c,wv;l}.
+c&)l)
(‘0)
to see that for any arbitrary operator consisting of aj and ~3, there is a one to one correspondence between the associated normal function and the normal-ordered form of the operator, namely the normal-ordered form of the operator can be obtained by replacing the oj* and crj in the associated normal function by the operators a: and aj respectively, and vice versa. This property is very helpful for solving operator equations such as the evolution equation in Eq. (8). Next, we convert the evolution equation to a c-number equation by evaluating its expectation value with respect to the coherent state I a,, cr,),
It is not difficult
(a,,
(11)
KM. Ng, C.F. Lo/Physics
Letters A 230 f 1997) 144-152
I’he right-hand side of the equation is given by a
+
I a, I
2$exp( c5)+
c4c7
+al.a,-+[c, exp(c,)]
evtc6)] + I a2 I2 d%
+c++a,a2
+a::
exp(c,)% +aTa,$[c, e&,)1 dc,o
dr
)
U,j”)(&
a;,
a,> ff2, r),
(12)
and the left-hand side is (a,.
a2 I H,( f>&( r, 0) I aI 9 4
=
A,a,‘a;
+A,a;ff;
+A;
(%+&)(a,++)+A+2+-&)(%+-$)
i +B,[a;(a,+&)+i]+B2[n(n2+-&)+~]+C:+~
+C2[a+2+-$)]+C3[+?+-$)]
(13) Then, comparing the coefficients of the quadratic and bilinear terms on both sides of the equation, we can obtain ten coupled differential equations, ~~~~=~A;C,C,+~A;C,~~+(B,+B~)C,+C,+~C~C~+~C~C~+C~(C~+~C~~~).
(14)
dc, ih~=A,+4A~c~+A~c~+2B,c2+C2c,+2C,c,c2.
(15)
ih-
dc, =A;c~+A2+4A;c~+2B2~3+Cj~,+2C~~,~3, dr
dU,
(16)
ihx
= 4A; c,U, + 2A; c,U, + B,U, + C2U2 + C,( c,U, + 2c2U2),
(17)
ifi$
=2A;c,U,
(18)
+4A;c,U,+B,U,+C,U,
+C,(2c,U,
+c,U2),
ih~=2A~c,U~+4A~c,Li,+B,U,+C,U~+C,(c,U3-t2c,U~),
ifi?
= 4A;c2U,+2A;c,U3
% ‘fidr
=A;U,“+A;U,2fC&J,U,,
+B,U,
+C,U,
(19) +C4(2c2U3
+ c,U4),
(20)
(21)
KM. Ng, C.F. Lo /Physics Letters A 230 (1997) 144-152
148
(22) ifi-
dc,, dt
= 2 A; U,U, + 2 A; U,lJ, + C,(U,U3
(23)
+ &U,),
where the functions U,(t) are defined by u,(t)
=exp[c,(t)l
+cAt)cAt)
U&I = exp[c&)l
I
exp[c~(r)l,
G(r)
=cAf>
exp[c~Lr)l,
exP[%wl~
U4,(t) = cdr>
(24)
Of these ten differential equations, the first three equations are completely independent of all the others and each of them is of the nonlinear Riccati type. Thus, we can first solve these three coupled nonlinear differential equations and obtain the coefficients cj(t), j = 1, 2, 3. Knowing these coefficients, we can then apply the standard technique of coupled linear differential equations to determine the functions Uj(t), j = 1, 2, 3, 4. Note that Eqs. (171, (18) and Eqs. (19), (20), in fact, form two independent sets of two coupled linear first-order differential equations. With all the functions U,(t) being known, the coefficients Cj(t), j = 4, 5, 6, 7 can be evaluated readily. Furthermore, the remaining three equations can be solved by straightforward integration to give the coefficients c,(t), j = 8, 9, 10. Since U,(t, 0) is known, the evolution operator I/( t, 0) describing H(t) = H,(t) + H,(t) will be given by U(t, 0) = Uo(t, O&,(t, 01, with U,(t, 0) satisfying the evolution equation H,(r)U,(t,O)=ifi~U,(t,O),
U,(O, 0) = I,
(25)
where H,(t)=U,-‘(t,O)H,(t)U,(t,O)=
&f!(t)a:+di(t)aj= j=
ihj(t)
(26)
I
j=
I
and d;(t)
=
(77, +27?;c*
+rl;c$J3
-
u,u3
(r/2
d:(t)
+
77,*c,
-
+2772*c&
77;c,
f2712C3P4
-
(77,
+2rl;c2
+
172w72
=
1 - c,,,
=
772+771*c,
u,
71,
-
rl2+77;c,
l-c,,
+ (
-
-
u, + 2c,
u2u4
1
-
4v
u2u4
771+27?;c* + 7?;c, u2 UP3
-
u2u4
U
u2”4
u3
77;c,
+2r);c,
u,u,
rll+2rl;c2+7?;c, u,u3
-
2c,
+
o;u4
+27l;c,
+
u2 u,r/,
r/2+r)l*c, u3
u,“,
77,+27?,+c2
(
+27?;c2+7);c, u,u3
= -2c,
1 + c,,)
+
1
u2u4
2c,
-
u2u4
+27/;c3
u,u3
i
d:(t)
+
u2u4
U,U, d!(t)
(772
-
1 + c,o
+ (
u2u4
r/2
+
77;
c,
u,u, -
+
2.6
c3
u2u4
“’
(27)
Lerrers A 230 (1997)
K.M. Ng, C.F. Lo/Physics
In terms of the generators of the Heisenberg-Weyl hj( r) = b:‘(t)
algebra,
149
144-152
can be written as [211
hj(r)
e{ + bi( r) e:’ + b{e{,
(28)
where e{ =
e;
ai.
=
at. I’
e{
b:‘(r)
=d:(r),
=
I
(29)
and b:(r)
=d’_,
bj‘( r) = 0.
(30)
The operators ei form a closed Lie algebra, [e{, ei] =e{,
[CT/, e{] = [ei, e$] =0,
(31)
and thus by following the procedure shown above, the evolution operator U,(f, 0) can be easily found to be U,(r,O)
= ,fiexp[h:(t)ej]
exp[hi(t)e{]
exp[h{(t)e!],
h:‘(r) = &bj(
u) du,
(32)
with h{(r) = &b{(u) h;(r) = ;
du,
jdh:(u)bj(
u) du.
(33)
Hence, we have obtained an exact form of the time evolution operator U(r, 0) of the two coupled generalized time-dependent parametric oscillators. Now we shall determine the coherent-state propagator associated with the evolution operator U(t, 0). Let {I a], CQ)) be the coherent states of the initial Hamiltonian H(t = 0). Then the coherent-state propagator K(cr,, ff7.. r; p,, p2, 0) is given by the matrix element ( (Y , , a2 1U(r, 0) I pi, pz > which denotes the probability amplitude that the system undergoes a transition from the state 1 P, , & > at time f = 0 to the state 1a I, a2 > at time r > 0. From Eqs. (91, (10) and (32), we have Cc+ (Y,IU(l,O)I = (a,,
P,, Pz>
a2 I&(r,O)I
P, +h$)*
/32+$(r))
~,~exp[~~~~(r)+P,12~~l~j12+P,h:(r)+h~(r)] = exp(--(yI* [ P, + G(t)]{1- c4(f)c7(f)exp[c&)l-exp[c~(r)l}) Xexp( -4 [ P2+GWl{l -expMf)l)) xexp(c,(r)[~,+h:(r)]2+c2(r)~~2+c~(r)[P2+h:(r)l2+c~(r)a;2)
Xexp{c,,(r)[
PI
xexp[exp[c,(r)]{c,(r)a;[
X ~exp(~lh~(r)+/3jI’-il
+hi(r)][
/32+%(r)]
P2+G(r)]
+cdr)~ltG
+cAr)[
P,12+P,h:‘(t)+h:(r)]*
+3[4t>
PI
+Cdr)l}
+G(r)la;)]
(34)
Iy.M. Ng, C.F. Lo/Physics
150
Lerters A 230 (19971 144- I52
This coherent-state propagator is for the most general pair of coupled time-dependent parametric oscillators, and thus results for any special case can be easily deducedefrom it. To illustrate the validity of our approach, we consider a charged oscillator with a time-dependent mass and frequency moving in the q-plane under the influence of a time-dependent electromagnetic field, whose Hamiltonian takes the form [22] H(r)
=
b-q‘wN2 2m( f)
(35)
++n(f)O&)2(X:+X:)+q+(f).
We shall restrict ourselves to the cases where the electromagnetic potentials are given by A(r) = B(t)(x, and &I) = 0. With this choice the Hamiltonian H(t) becomes
P, -
x, P,)/2
where w = \i,o0 + h and A = qB/2m. Here we have set fi = c = 1. In terms of the generators of the closed de Sitter-Lie algebra, H(t) can be re-written as H(t) =A,(r)Y,
+A~(t)Y~+A2(r)Y;!+A;(t)Yq+B,(t)Ko+B2(t)K;,+C,(t)K++C2(t)K:
+ C,(t)K’+
C,(t)K_,
(37)
where mt0> w(O)
=A2(t)
‘A(t).
2m( f)
m(0) w(O) m(t)
c, = 0, C,=
= B2(r) = B(r),
(38) (39)
(40)
-ih,
(4’)
C, = ih,
(42)
c, = 0,
(43)
and the corresponding evolution operator U(t, 0) can be expressed in the form of Eq. (9). The coefficients ci are determined by solving the ten coupled ordinary differential equations in Eqs. (14)-(23), subject to the initial conditions c;(O) = 0. From Eqs. (14)-(16), it is not difficult to see that c, = 0 and c2 = cs = ?. Thus, we are left with only one equation, that is, dZ ix = A + 4Ac”*+ 2B?.
(44)
By a simple Ricatti transformation defined by . ;=
_‘”
4A w’
(45)
the Ricatti equation in Eq. (44) can be cast in the form of a linear second-order ordinary differential equation
K.M. Ng, C.F. Lo/Physics
which can be solved by the standard approaches. dU, ix = (4Ac”+B)U,
Lerters A 230 (1997)
Furthermore,
144-152
Eqs. (17)-(20)
151
can be simplified
to give
- iMY,,
dU, ix=(4AF+B)U,+ihU,, = (4AFf
ix
dU,
ix
B)U, - ihll,,
= (4A?+B)U,+ihU,.
Obviously, one can conclude that U, + CJ2= U, - U, = X, and U, - U, = U, + Ud = X_. To determine we need to solve the following two coupled ordinary differential equations,
dX+
i---z
dt i%
(5’)
(4AZ+B)X++iAX_, = (4AFf
B)X_-
X,,
(52)
iAX+.
Defining f+(t) = X, exp{ij,$4A(r)Z(r) independent linear second-order ordinary
+ B(T)] dr}, the two coupled equations differential equations
can be decoupled
to give two
(53) Since both equations are of the same form, we need to solve one of them only. Finally, the remaining three equations in Eqs. (21)--(23) can be solved readily to yield cs = cg = - i/,$A(r)[Uf(r> + U,~(T)]} d7 and cIO = 0. As a result, the time evolution operator of the system will be completely determined once we solve the two linear second-order ordinary differential equations in Eq. (46) and Eq. (53). In summary, we have investigated the algebraic structure of the Schradinger equation associated with a pair of coupled generalized time-dependent parametric oscillators. Using the Lie algebraic technique we have derived the time evolution operator of the system, which in turn yields the coherent-state propagator readily. We believe that these results will be useful for future studies in quantum optics as well as in atomic and molecular physics. To illustrate tire validity of the approach, we have also applied the results to the case of a charged oscihator with a time-dependent mass and frequency in a time-dependent electromagnetic field.
References [I] [2] [3] [4] [S] [6] [7] [8] [9] IJO]
J. Wei and E. Norman, J. Math. Phys. 4 (1963) 575. C.F. Lo, Phys. Rev. A 47 (1993) I IS. C.F. Lo, Europhys. Lett. 24 (1993) 319. CF. Lo and Y.J. Wong, Europhys. Lett. 32 (1995) 193. T.D. Lee, Phys. Rev. 95 (1954) 1329. S.S. Schweber. An introduction to relativistic quantum field theory (Row-Peterson. Elmsford, NY, 1961). N.N. Bogoliubov, Nuovo Cimento 7 (1958) 843. A.L. Fetter and J.D. Welecka, Quantum theory of many particle systems (McGraw-Hill, New York, 1971). M. Tinkham, Introduction to superconductivity (Krieger, Malabar, FL, 1975). Y.S. Kim and M.E. Noz, Phase space picture of quantum mechanics (World Scientific, Singapore, 1991). ,I I] H. van Dam, Y.J. Ng and L.C. Biedenham, Phys. Lett. B 158 (1985) 227. 1121 Y.S. Kim and M.E. Noz, Theory and applications of the Poincare group (Reidel, Dordrecht, 1986).
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K.M. Ng, C.F. Lo /Physics
Letters A 230 (1997) 144-152
1131 Y.S. Kim, Phys. Rev. Len. 63 (1989) 348. 1141 F. lachello and S. Oss, Phys. Rev. Lett. 66 (1991) 2976. 1151 D. Han, Y.S. Kim and M.E. Noz, Phys. Rev. A 41 (1990) 6233. [16] H.P. Yuen, Phys. Rev. A 13 (1976) 2226. [17] C.M. Caves and B.L. Schumaker, Phys. Rev. A 31 (1985) 3068; B.L. Schumaker and C.M. Caves, Phys. Rev. A 31 (1985) 3093. 1181B.L. Schumaker, Phys. Rep. 135 (1986) 317. [19] B. Yurke. S. McCall and J.R. Klauder, Phys. Rev. A 33 (1986) 4033. [20] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973). [21] H. Weyl, Gruppentheorie und Quantenmechanik (Hitzel, Stuttgart, 1928); P. Cartier, Proc. Symp. Pure Math., Vol. 9, Algebraic groups and discontinuous sub-groups (American Mathematical Society. Providence, RI, 1966). [22] C.F. Lo, Phys. Rev. A 45 (1992) 5262.