Aug 1, 1991 - Jing-Bo Xu and Tie-Zheng Qian. Depar tment ofPhysics, Zhejiang University, Hangzhou 310027, China. Xiao-Chun Gao. Chinese Center ...
PHYSICAL REVIEW A
Time-evolution
VOLUME 44, NUMBER 3
1
AUGUST 1991
operator and geometric phase for a system with a su(1, 1)(+,h(4) Hamiltonian Depar tment
Jing-Bo Xu and Tie-Zheng Qian of Physics, Zhejiang University, Hangzhou 310027, China
Xiao-Chun Gao of Advanced Science and Technology (World Laboratory), P. O. Box 8730, Beijing, Institute of Theoretical Physics, Academia Sinica, P. O. Box 2735, Bejiing, China; and Department of Physics, Zhejiang University, Hangzhou 310027, China
Chinese Center
China;
(Received 8 January 1991)
In this paper, the exact solution and time-evolution operator for the system with a su(1, 1)(+,h(4) Hamiltonian are found by making use of Lewis-Riesenfeld quantum theory. The time-evolution operator is then used to derive the Aharonov-Anandan phase. Finally, the noncyclic evolution is discussed with the help of the Aharonov-Anandan phase and the Berry phase is calculated by taking the adiabatic limit.
I.
INTRODUCTION
The geometric phase in quantum adiabatic evolution was first discovered by Berry [1]. It was recognized immediately by Simon [2] that this phase is nothing but the topological invariant of the Chem class and can be interpreted as a holonomy of the Hermitian fiber bundle over the parameter space. In a fundamental generalization of Berry's idea, Aharonov and Anandan [3] have removed the adiabatic condition and studied the geometric phase for any cyclic evolution. This Aharonov-Anandan (AA) phase is related to a holonomy associated with the parallel transport around the circuit in projective Hilbert space and has been verified in optical and NMR interferometry experiments [4, 5]. Recently, in a series of papers, the geometric phase for the coherent states [6] and generalized coherent states [7] has been discussed. On the other hand, there has been a good deal of discussion on the time evolution [9-11] and geometric phase problem [8] of quantum systems whose Hamiltonians are some combinations of the generators of a Lie algebra. In particular, Dattoli and co-workers [9, 10] have made use of the Wei-Normal [11] procedure to study the time evolution of the systems with time-dependent Hamiltonians that are linear combinations of the generators of su(2), su(1, 1), and su(1, 1) C+, h(4) [C+, denotes a semidirect sum] Lie algebra. In their works, Hamiltonians associated with su(1, 1) C+, h(4) Lie algebra were used to discuss both the non-Poissonian efects in a laser-plasma scattering and the pulse propagation in a free-electron laser [10]. In this paper, the system with a su(1, 1) C+, h (4) Hamiltonian is studied by making use of the Lewis-Riesenfeld quantum theory [12]. The invariant for this system is found and the exact solution obtained. Then the timeevolution operator is derived from the exact solution. It is worth indicating that in the Wei-Norman approach the time-evolution operator should be assumed to be of the form of a product of simple exponential operators. Fernandez [13] pointed out that this assumption caused some difficulties. We would like to emphasize that in our approach the assumption is not necessary. Finally, the
time-evolution operator is used to derive the AA phase for the system. The noncyclic evolution is discussed with the help of the AA phase and the Berry phase is calculated by taking the adiabatic limit.
II. THE EXACT SOLUTION AND TIME-EVOLUTION OPERATOR
FOR THE SYSTEM According to Lewis and Riesenfeld [12], a timeHamiltonian H(t) admits a time-dependent invariant operator I(t) satisfying the relation
dependent Hermitian
dI(t)
BI(t)
dt
Bt
1
)
[
(
)]
where I(t) does not involve time differentiation and has a complete set of eigenstates I ~n, t ) ] (n =1,2, . . . ). It can be verified that the corresponding eigenvalues are time independent and the general solution of the Schrodinger equation is of the form
~g(t)) =pc„e " where the phase
~n,
t),
a„(t) is
tt„(ti=1o ttt'
i, Bt
FI(t't tt, , t
(2)
Idt-
(3)
Now we consider the su(1, 1) C+, h (4) Hamiltonian which is a combination of the generators of the su(1, 1) C+, h (4) Lie algebra
H(t)= A(t)K~+F(t)K++F*(t)K +B(t)at
+B*(t)a+G(t),
(4)
where A (t) and G(t) are real functions and F(t) and B (t) are complex functions. Furthermore, the generators
of the SU(1, 1) group are realized by
Ko=(ata+aa )/4, K+ =(at)~/2, K =a /2, and 1485
a, a
are harmonic-oscillator
1991
annihilation
(5)
and
The American Physical Society
'
JING-BO XU, TIE-ZHENG QIAN, AND XIAO-CHUN GAO
1486
creation operators, respectively. The commutation tions obeyed by (5) are immediately inferred as
[K+,K From
the
]
= —2K(),
rela-
d
[K(),K+ ] =+K+ .
of
the Casimir operator unit operator) relevant to the realization (5), we know that the Bargmann index can take on values k = —,' and —, We use the notation Im, k ) to express the eigenvector of K: expression
C=k(k —1)I= —,', I —(Iis
2y
*) B—
+i &2(B
3 —F —F* ~' —4IFI' +l F —F*) dt 2( A — +2
'.
KOIm,
k)=(m+k)
m,
k)
(m
=0, 1,2, . . . ) .
(7a)
From this it follows that
K+
m, k ) = [(m
+1)(m +2k)]'~
+ l, k ), —l, k ) .
(7b)
Im
Im, k ) = [m (m +2k —1)]''2Im
I(t)=2coshrKO+e
'~sinhrK~
+ e '~sinh&K —(z *cosh' +ze '( sinhr —(z coshr+z*e '~sinhr)a" +( IzI cosh'+ 'z* —,
e
U(g) =exp(gK+
D (z) = exp(za t — z *a
(9)
A
+l. d
2( A
F
F —F 3 —F —F'
X—A
Ua U
T — a+e
2
~sinh
—a 2
=cosh — a +e I ~sinh
(18)
—a, 2
2
I=2DUKpU D
(19)
Thus the eigenvalue equation of I(t) can be written as
I(t) z(t), g(t);m, k ) =2(m +k)Iz(t), g(t);m, k ), (20)
F*)—
~' —4IFI'
—,
which lead to the result
where x and y satisfy the auxiliary equations
2x
+ ' e '~sinh~K
—,
(10a)
(10c)
—F —F*
'(, and using the formula, we obtain
'~sinh~K+
e
UaU =cosh
—
d
&
—tanh(r/2)e
—,
2
3 —F —F* 2 xF* —2ix x'— xF — e'&sinh~= — a — X A — F —F* x (F F*)—2ixx— 2 —F —F — &2i (B — B ") z= v'21— y+i . 2y +i(F F*)+ — — F* 3 F X
=
(12b)
—,
We want to point out that the invariant in Eq. (8), when restricted to only su(1, 1), is nothing but the invariant obtained in Ref. [14]. For simplicity, cosh~, e'~sinh~, and z may be further expressed in terms of x and y:
—
),
—,
—,
cosh~=
—/*K ),
)Ko]exp(
+ cosh'TEp (13) — UK+ U = T((coshr+1)K++ '(cosh' —l)e '~K+ + e —'~sinh~E p, (14) DK, D'=K, —'za' —-'z*a+-,' IzI', (15) — 'z* DIG + D =K + z'af+ (16) DK D =K — za+ 'z DaD~=a —z, Da D =a —z*, (17)
solutions of the fol-
+x— xF —xF* —2ix
—I/I
(12a)
UKp U
—2Im(Fe'~), P= 3 —2Re(Fe'~)cothr, z =— i ( (zA +z*F+B ), z *=i( 'z* A +zF*+B*) .
l
)
=exp(gK+ )exp[in(1
r'=
—,
—(*K
where g= —(r/2)e'(, g= Baker-Campbell-HausdorA
)a
3 —F —F*
(1 lb) The eigenvalue equation of I(t) should be solved first. Introducing two unitary operators
'~sinhr+ —,'z e'('sinhr),
where r, P, and z are the c-number lowing equations:
~
F*— y F — (F F*— )(B B*)—
2
(7c)
With Eqs. (4) —(6), it can be shown that the system has Hermitian invariant which satisfies Eq. (1):
A
— —F —F
where Iz(t), g(t);m, k ) =D [z(t)]U[g(t)]Im, k ), which is referred to as the generalized coherent state. In order to find the exact solution for the system, we first calculate (z(t), g(t); m, kIi(B/r)t)Iz (t), g(t)m, k ) which is equal to
z(tl, t(t);m, k
(
m, k U
2x (1 la)
i
8
i
z(t), t(t);m, 8
Um, k
k)—
+
m, k
UiDDUm,
k
(21)
..
TIME-EVOLUTION OPERATOR AND GEOMETRIC PHASE. Using Eq. (7), the first term of Eq. (21) may be rewritten as
i
m, k U
a — at
-a
i
U
(t)
O, k
U m, k
—1, k
m
m, k U
Bt
U(g) m, k)
at
U
i
U O, k
ai
27 sinh—
+2m
2
(25) =2(m +k)P sinh— ' 2 With the help of Eqs. (7) and (8) it is not difficult to show
—1, k
U m
i
1487
(m, klUaUtlm, k ) =( mklUatUt m, k) =0 m, k U
(Uk+U
i
(
)
Um
jest
[m (m
—(,
kl (m, kl U'(~ D'D')Ulm, k
+2k —1)]'~
= ( m, k
From Eq. (14) and
U(g) o, k ) =(1 —gl')"exp(gK+ )l0, k
&
—(zz * —zz * + iza t
*a U m, k iz —
)
l
(27)
.
)
2
+ (i ~ i
(z, g;m,
Psinh~) — UKO U
.a z, k;m, k) at
k i
(23)
=2(m +k)lp slnh
U(g) o, k
at
U
=ik(jg* —g *)/(1 —lgl') =2k/
sinh
—
27 . 2
(m, k
(24)
l
U KOUlm. , k
( m, k UtK+ U
Employing Eqs. (22) —(24), we find
(z(t), g(t);m, klHlz(t), g(t);m,
—+ 2
Equations (7), (13), and (14) lead
l
l
) =(m
2
(zz
zz
)
.
«
+k)cosh',
(29a)
(m + k)e —'~sinhr m, k ) = —
.
k ) = (m, k
I
U'[&Ko+FK++F*K-+( 2~z*+F*z)a-+( ~z+ + '( lzl2A +z*~F+z F*)+B*(a+z)+&(a +z*)+G] Ulmk 2
&
= (m + k)( g cosh' Fe '&sinh~ —F*e '~sinhr—) . +( ' lzl 2+ ,'z* F+ ,'z F*+zB—'+z*8—+6)
(30)
—,
Thus, on using Eqs. (28) and (30), the Lewis phase is found to be
+k)
f
2p sinh
——3 cosh~+Fe'~sinhr
+Z'e
'&sinh
dt'
r
+
—(zz* —zz *)—( —lzl2A + —'Fz* + 2 '
p
(29b)
Jt then follows that
—,
k(t) =(m
)
Using Eqs. (21), (25), and (27), we obtain
—UK (UK + Ut)=2/ sinh 2~
o, k U"(g)
Ut l
=— '(zz — zz
&
ii follows that i
(26)
2
+zB *+z*B+ 6 dt',
2
f(t)), = g
C kexp[ia~ k(t)]lz(t), g(t);m,
k)
.
(32)
m, k
Equations (9), (31), and (32) represent the exact solution for the system with the Hamiltonian (4). The above exact solution can be used to derive the evolution operator V(t2, t, ) for the system. By applying V (tz, t, ) to Eq. (32), we get V(t2, ti)lzi, gi, m, k&=exp(ia k)lzp, g~, m, k &,
'F*z-
(33)
2
or
(31)
where r, P, and z are determined by Eq. (9). Accordingly, the general solution of the Schrodinger equation is of the form
V(t, , t, )D(z, )U(g )lm,
k&
=exp(ia
„)D(z~)U($2)lm,
The explicit form for the time-evolution found to be
k)
. (34)
operator is then
JING-BO XU, TIE-ZHENG QIAN, AND XIAO-CHUN GAO
1488
V(tp, t) ) =D(z2)U($2)
g exp(ia „)~m, k ) (m, k
~
U (g, )Dt(z,
)
m, k
l . ~ —' — (zz* zz *)— ' 2 +
=exp i
1
( —,
~z~
1
XD(zz)U($2)exp
iKO
where z(t) and g(t) = — (r/2)e (9) and z(t,
2gsinh
——A coshr+Fe'~sinhr+F*e
=exp[ig k(t, t, )] V(t, t& )~z„g, ;m, k
)=e'
~
g
) (n =1,2, . . . ),
):
Any state can be expanded in terms
X2
2(m+k)
i
2 —F —F*
—F —F'
—i. v'2-y 2
d dt
2
=
Schrodinger
equation
we get
f '&zegzmzk
=exp i
J
(z, g;m, k
i
)dt,
lHlz, g;m, k
(39)
—z, t;
,
mtk~zd„g, ;m,
)
k)
where (zt, ) =z(t )2=z„g(t, ) =g(t~) =g, . Here the AA phase factor exp(i13 k) is nothing but the eigenvalue of ),
P(t„t,
B Q
=exp i
J
z, @m, k i
=exp i
f
(m
+k)2$sinh
d dt
. &2 —F —F* +l
z, t;m, k)dz
—+ —(z*z —zz *) 2
Employing Eq. (10), the AA phase P pressed as
F —F 3 —F —F' F —F* d dt 3 —F —F*
X2
„)
exp(if3
X
B
2
k
dt .
(41) can also be ex-
X
dt
3 —F —F~
B*- B*-
i 2d y 2 dt
y
A
t& )
the
and
(38)
V= HV,
P(t„t, ) ~z„g„m, k )
For a given time interval [t &, t2] the evolution operator V(t2, t, ) has a set of eigenstates [~z„g,;m, k)] from which the complete set [ V(t, t, ) ~z„g, ; m, k —m, kt ) ]
f
)
k(tz,
operator.
p k=
Eq.
Using
i (d ldt
(38a)
(40) (36)
where ~g(t, ))We' ~P(tz) ), namely, ~g(t)) does not obey the cyclic evolution condition. Thus we may say that the study of cyclic evolution is the study of all evolutions [15]. However, no realistic example has been seen. As an example, we study the geometric phase for the su(1, 1) C+, h(4) system by making use of the above evolution
may be constructed. of the set
),
(38b)
t),
mt,
(37)
(z„g, ;m, k ~P'P ~z„g, ;m, k ) =0 .
[15]).
t,
(35)
P(t, t, )~z„g, ;m, k)
We now turn to the discussion of the AA phase for the su(1, 1) C+, h (4) system. The geometric phase acquired by a system which undergoes cyclic evolution is referred to as the AA phase [3]. Recently, Anandan [15] pointed out that a system which undergoes noncyclic evolution can also be studied with the help of the AA phase for cyclic evolution. We know that any state ~g(t) ) can always be expanded in terms of a complete set of states which undergo cyclic evolution in the time interval [t„t2] (Ref.
m,
(z&),
a unitary operator P (t2, t, ) such that
Introducing
FOR THE su(1, 1) e, h(4) SYSTEM
~m,
U ((&)D
m, k
III. GEOMETRIC PHASE
q(t))=pc
dt
C „V(t, t, )~z„g, ;m, k) .
~P(t))=g
by Eq.
g(t2) =g
,
'~sinhr
1
'~ are determined
)=z„
g(t, ) =g, , z(t, ) =z,
f
+ ,'Fz—* + ,'F*—z +zB*+z*B+6 dt
3 —F —F*
3 —F —F*
(42)
It should be pointed out that the auxiliary Eq. (11), is usually dillicult to solve, so that expression (42) for the AA phase is a formal one. However, in some special cases, the solution of the auxiliary equation may be found. As an example, we discuss the case of the displaced harmonic oscillator with H = IV(t)ata+B(t)at+B*(t)a+6'(t),
(43)
TIME-EVOLUTION OPERATOR AND GEOMETRIC PHASE. which is a su(1, 1) (+, h(4) Hamiltonian be found to be
i— W'(t)dt
f
z (t2) =exp
with A
z (t
)
&
f
2
i—
1
=2W, exp i
1
F =0.
..
1489
The solutions of the corresponding
f W(t')dt'
auxiliary equation can
B (t)dt
(44)
1
and the invariant becomes
I(t)=[a —z*(t)][a —z(t)]
.
Using this solution the evolution operator V(t2, t, ) can be diagonalized
to obtain its eigenstates
z„m, k) =D[z(t, )]Im, k),
(46)
with
z(t, )=
i exp
i
f —W(t)dt f exp f W(t')dt' i f—W(t)dt — exp i
tl
B(t)dt (47)
r
2
1
1
All the dynamical and AA phases can then be obtained. The expression for the AA phase is
P
k
=—
y,
)—
—~ (F F)(B—B* F F*—B+B*+ F F*— —
=— 2
(z*z — zz *)dt 1
8
t z t
+ —,'B
tz*t+ 'B*tz t dt.
F*)——
F g2 —4IFI2 —F —F* 1— 2 — A
2(A'
4IFI')
1/4
d
«A
F —F* F
F*——
+k (
Xd
dt
—,
(48) where z satisfies the cyclic condition z(t, )=z(t2). The problem of noncyclic evolution of the system can therefore be solved. Now, we proceed to discuss the adiabatic limit of the evolution. Using an approach similar to that used in [16], the adiabatic solution of Eq. (8) can be obtained to be (
+2l. d
(49a)
X
F —F* FQ
g
—F —F
F'— F'—
F F') — d —F 1—2i( A2 — ~ 41FI' dt 3 F
—
(49b)
In this limit, the eigenstate of I(t) becomes approximateof H(t). This means that if the system is initially in the eigenstate of I(0) it will be in the eigenstate of H(t) approximately all the time. This is what it should be in the ordinary adiabaric situation. In the adiabatic limit, with the help of (42) and (49), the Berry phase for a circuit C in parameter space can be found to be ly the eigenstate
— 2F")+B'(3 —2F) 2 F F' + B(— g— a' — 4IFI2)1/2
g
B —B*
2
4IFI'
— *( A — 2F) + . y B ( A 2F*)+B — (g 4IFI )
IV. DISCUSSION (1) It is worth pointing out that the m-independent part in the AA phase has no effect on the corresponding classical phase angle. Obviously, this conclusion should be valid in the adiabatic limit. This means that the independent part of Berry's phase should have no effect on Hannay's angle. We wish to indicate that the independent part comes from the vacuum state rejecting the topological property of the vacuum. We think that this topological property is an important issue and is
II-
B —B*
(50)
3 —F —F*
worth further exploring. (2) The fact that the study study of all evolution is very We believe that the approach be used to study other systems
of cyclic evolutions is the important and interesting. adopted in this paper may in this direction.
ACKNOWLEDGMENT
This project was supported by the Zhejiang Provincial Natural Science Foundation of China.
1490
JING-BO XU, TIE-ZHENG QIAN, AND XIAO-CHUN GAO
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J. Phys.
