PHYSICAL REVIEW E
VOLUME 55, NUMBER 1
JANUARY 1997
Exact time evolution and master equations for the damped harmonic oscillator Robert Karrlein and Hermann Grabert Fakulta¨t fu¨r Physik der Albert-Ludwigs-Universita¨t, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany ~Received 26 August 1996! Using the exact path integral solution for the damped harmonic oscillator it is shown that in general there does not exist an exact dissipative Liouville operator describing the dynamics of the oscillator for arbitrary initial bath preparations. Exact nonstationary Liouville operators can be found only for particular preparations. Three physically meaningful examples are examined. An exact master equation is derived for thermal initial conditions. Second, the Liouville operator governing the time evolution of equilibrium correlations is obtained. Third, factorizing initial conditions are studied. Additionally, one can show that there are approximate Liouville operators independent of the initial preparation describing the long-time dynamics under appropriate conditions. The general form of these approximate master equations is derived and the coefficients are determined for special cases of the bath spectral density including the Ohmic, Drude, and weak coupling cases. The connection with earlier work is discussed. @S1063-651X~97!04001-4# PACS number~s!: 05.40.1j, 05.30.2d
I. INTRODUCTION
Recently the problem of the reduced dynamics of a quantum system in contact with a reservoir has gained renewed interest @1–3#. While the foundations of quantum dissipative processes were layed already in the 1960s @4#, this early work was mainly concerned with weakly damped systems and has relied on the Born and Markov approximations. In this conventional approach the dynamics of the dissipative quantum system is described in terms of quantum master or Langevin equations. Important progress beyond the limitations of the weak coupling approach was made in the 1980s, in particular path integral techniques were shown to be powerful means to describe quantum dissipative systems @5#. This has led to unexpected results, such as, for instance, the nonexponential decay of correlation functions in the low temperature range @6,7#, not directly available within the conventional master equation approach. The insight gained also provided a basis for profound critique @1,3,8,9# of concepts developed in the context of the quantum master equation approach such as complete positivity or the quantum regression theorem. However, the results of the path integral approach were rarely used to derive improved master equations valid in the range of strong damping and/or low temperatures. In this paper we shall address this problem for the ubiquitous quantum dissipative system, the damped harmonic oscillator. Based on available exact results @10# we derive generalized quantum master equations describing the exact relaxation of mean values and the time evolution of equilibrium correlation functions. Whenever appropriate, the results will be confronted and compared with earlier findings. More specifically, this paper is organized as follows. In the following section the microscopic model of a harmonic oscillator coupled to a harmonic bath is introduced. The exact time evolution of the reduced density matrix of the oscillator @10# will briefly be summarized. In Sec. III we examine whether the exact time evolution of the density matrix is compatible with a generalized quantum master equation with a time-dependent Liouville operator. It is shown that in gen1063-651X/97/55~1!/153~12!/$10.00
55
eral there is no Liouville operator independent of the initial preparation. In Sec. IV we study specific types of initial preparations. Exact time-dependent Liouville operators describing the time evolution of thermal initial conditions and of correlation functions are derived. In the classical limit these Liouville operators are shown to reduce to the Adelman Fokker-Planck operator @11#. We also examine the case of factorizing initial conditions. In Sec. V we investigate particular types of damping leading to a time evolution of the density matrix governed approximately by a time-independent Liouville operator. Earlier results by Talkner @9# and by Haake and Reibold @12# are recovered. We discuss in detail the limit of a weakly damped oscillator and derive a generalization of the Agarwal equation @13#. Additional time coarse graining is shown to lead to the quantum-optical master equation by Weidlich and Haake @14#. Finally in Sec. VI we present our conclusions. II. MODEL HAMILTONIAN AND EXACT TIME EVOLUTION
The standard microscopic model @15–17# for the damped harmonic oscillator starts out from a Hamiltonian H composed of the oscillator part H O5
M p2 1 v 20 q 2 , 2M 2
~1!
the bath Hamiltonian
(S N
H R5
n51
p 2n 2m n
1
D
mn 2 2 v x , 2 n n
~2!
and an interaction part N
H OR 52q
(
n51
N
c n x n 1q 2
(
n51
c 2n 2m n v 2n
.
~3!
This model describes an oscillator with the associated classical equation of motion 153
© 1997 The American Physical Society
154
ROBERT KARRLEIN AND HERMANN GRABERT
M q¨ ~ t ! 1
E
t
0
ds g ~ t2s ! q˙ ~ s ! 1 v 20 q ~ t ! 50,
~4!
55
¯,r ¯! 5i S ~ x f ,r f ,t,x i ,r i ,x
D
\¯ r2 ^ p 2& 2 ¯ x 1M @ x f r f f 1 ~ t ! 2 1 2^q & 2\
1x i r i f 2 ~ t ! 2x i r f f 3 ~ t ! 2x f r i f 4 ~ t !#
where
i 1 M @ x 2i R 11 ~ t ! 12x f x i R 12 ~ t ! 2
N
c 2n 1 g~ t !5 cos~ v n t ! M n51 m n v 2n
(
~5!
1x 2f R 22 ~ t !# 1M $¯ r @ x iC 1 1 ~t!
is the damping kernel. The dynamics of the density matrix of the entire system ~oscillator and bath! is given by W ~ t ! 5exp~ 2iHt/\ ! W ~ 0 ! exp~ iHt/\ ! ,
r ~ t ! 5TrR W ~ t ! .
E
¯dq ¯8 J ~ q f ,q 8f ,t,q i ,q 8i ,q ¯,q ¯8 ! dq i dq 8i dq
¯,q ¯8 ! , 3l ~ q i ,q i8 ,q
1x f C 2 2 ~ t !# %
N~ t !5
~8!
(j O j W b O 8j .
f 1~ t ! 5 f 2~ t ! 5
f 3~ t ! 5
E
˙ ~t! G , G~ t !
~9!
1 , G~ t !
~16!
R 11 ~ t ! 5
H
~17!
M ^ p 2& f 3~ t ! 1 $ 2 ^ q 2 & S˙ ~ t ! \ M 2 ^ q 2&
J
1 f 3 ~ t !@ ^ q 2 & 2 2S 2 ~ t !# % , ¯dq ¯8 l ~ q,q 8 ,q ¯,q ¯8 ! ^¯ dq q u W b u¯ q 8& , ~10!
R 12 ~ t ! 5
where ¯,q ¯8 ! 5 l ~ q,q 8 ,q
(j
H
~11!
¯,q ¯8 ) are discussed in Several relevant examples for l(q,q 8 ,q @10# and special cases will be considered below. ¯,q ¯8 ) is the propagating function describJ(q f ,q 8f ,t,q i ,q 8i ,q ing the time evolution of the reduced density matrix. In terms of the transformed coordinates r5(q1q 8 )/2, x5q2q 8 , it reads
F
G
i 1 ¯,r ¯! 5 ¯,r ¯! , J ~ x f ,r f ,t,x i ,r i ,x exp S ~ x f ,r f ,t,x i ,r i ,x N~ t ! \ ~12!
R 22 ~ t ! 5
~18!
M ¨ f 3~ t ! S ~ t ! 2 f 1 ~ t ! S˙ ~ t ! 2 2 $ S˙ ~ t ! S ~ t ! \ ^q &
J
1 f 1 ~ t !@ ^ q 2 & 2 2S 2 ~ t !# % , q &^¯ q 8 u O 8j u q 8 & . ^ q u O j u¯
~14!
~15!
˙ 2 ¨ ~ t !1 G~ t ! , f 4 ~ t ! 52G G~ t !
We then have
^ q u W ~ 0 !u q 8& 5
2p\ ~ 2 p ^ q 2 & ! 1/2. M u f 3~ t !u
The functions f i (t), R 66 (t), and C 6 i (t) are given in terms of the symmetrized part S(t) and the antisymmetrized part A(t)52(\/2M )G(t) of the equilibrium coordinate autocorrelation function C(t)5 ^ q(t)q & 5S(t)1iA(t) in the following way:
¯,q ¯8 ) is the so-called preparation function where l(q i ,q i8 ,q defining the initial nonequilibrium state W(0) as a modification of the equilibrium state W b 5exp(2bH)/Trexp(2bH) in the subspace of the oscillator. This modification can be produced by operators O j , O 8j that act upon the particle only leaving the reservoir coordinates unchanged: W~ 0 !5
~13!
and
~7!
The path integral technique allows for a complete solution to this problem for a large class of initial states. Since the derivation is expounded in @10#, we merely state the result
r ~ q f ,q 8f ,t ! 5
1 ¯ 1x f C 2 1 ~ t !# 2ix @ x i C 2 ~ t !
~6!
from where the reduced density matrix of the oscillator is obtained by tracing out the bath modes
where
S
~19!
H
J
M ^ p 2& 1 @ S˙ ~ t ! 2 f 1 ~ t ! S ~ t !# 2 1 ^ q 2 & f 21 ~ t ! , 2 2 \ M ^ q 2& ~20! C1 1 ~ t ! 5 f 3~ t !
C2 1 ~ t !5
S~ t ! 2 f ~ t !, ^ q 2& 1
~21!
S˙ ~ t ! S~ t ! 1 f ~ t !, 2 2 f 1~ t ! q ^ & ^ q 2& 4
C1 2 ~ t !5
F
G
M ^ p 2& 1 f 3 ~ t ! S˙ ~ t ! , \ M2
~22!
~23!
55
EXACT TIME EVOLUTION AND MASTER EQUATIONS . . .
M ¨ @ S ~ t ! 2 f 1 ~ t ! S˙ ~ t !# . \
C2 2 ~ t !5
~24!
ˆ ~ z ! 5 @ z 2 1 v 2 1z gˆ ~ z !# 21 , G 0 `
05Z rr ~ t ! 1X xx ~ t ! f 1 ~ t ! 2 2iY rx ~ t ! f 1 ~ t ! , 052iY rx ~ t ! 12X xx ~ t ! f 1 ~ t ! ,
In the following we shall give results mainly in terms of S(t) and G(t), which both remain finite in the classical limit while A(t) vanishes. The Laplace transforms of G(t) and S(t) read in terms of the Laplace transform gˆ (z) of the damping kernel @10,18#
1 Sˆ ~ z ! 5 Mb
z
( 2 2 @ Gˆ ~ z ! 2Gˆ ~ u n nu !# , n52` n n 2z
~25!
05X xx ~ t ! f 4 ~ t ! 2 . Inserting the solution X xx (t)5Y rx (t)5Z rr (t)50 into the 12 remaining equations, one finds that the conditions obtained by comparing the coefficients of 1 and r f x i imply L c (t)5Y rr (t). Taking this into account, we are left with the following set of 11 equations: ˙f 1 ~ t ! 52iX xr ~ t ! f 1 ~ t ! 2 1 @ Y rr ~ t ! 1Y xx ~ t !# f 1 ~ t ! 12iZ xr ~ t ! , ~29!
where n n 52 p n/\ b . Finally,
^ q 2 & 5S ~ 0 ! 5
1 Mb
`
(
@ v 20 1 n 2n 1 u n n u gˆ ~ u n n u !# 21 ,
n52`
^ p 2 & 52M 2 S¨ ~ 0 ! 5
~26!
v 20 1 u n n u gˆ ~ u n n u ! 2 2 ˆ ~ u n nu ! n52` v 0 1 n n 1 u n n u g
M b
(
~31!
˙f 4 ~ t ! 5 @ Y xx ~ t ! 12iX xr ~ t ! f 1 ~ t !# f 4 ~ t ! ,
~32!
22iX xr ~ t ! R 22 ~ t ! 1iY xr ~ t !# f 3 ~ t ! ,
S
D
] ] ] r ~ x,r,t ! 5L x,r, , ,t r ~ x,r,t ! ]t ]x ]r
S DS
\ ] ] L5L c ~ t ! 1 , M ]x ]r
X xx ~ t !
X xr ~ t !
X rx ~ t !
X rr ~ t !
Y xx ~ t !
Y xr ~ t !
Y rx ~ t !
Y rr ~ t !
Z xx ~ t ! M ~ x,r ! \ Z rx ~ t !
D
SD
Z xr ~ t ! Z rr ~ t !
D
SD ] ]x ] ]r
] ]r
x r
22iY xr ~ t ! f 1 ~ t ! 12X rr ~ t ! f 21 ~ t ! ,
~35!
2 C˙ 1 1 ~ t ! 522iX xr ~ t ! f 3 ~ t ! C 1 ~ t ! ,
~36!
2 ˙2 C 1 ~ t ! 5 @ Y xx ~ t ! 12iX xr ~ t ! f 1 ~ t !# C 1 ~ t ! ,
~37!
2 ˙1 C 2 ~ t ! 522iX xr ~ t ! f 3 ~ t ! C 2 ~ t ! ,
~38!
2 ˙2 C 2 ~ t ! 5 @ Y xx ~ t ! 12iX xr ~ t ! f 1 ~ t !# C 2 ~ t ! .
~39!
Since only seven functions of the ansatz ~28! remain to be determined, the set ~29!–~39! will be seen to have no solution in general. To demonstrate this explicitly, let us first disregard Eqs. ~36!–~39!, which stem from comparing coefficients involving the coordinates ¯ x and ¯ r of the preparation function. The remaining set of equations ~29!–~35! has a unique solution and, the resulting Liouville operator can be written in the form
] ]x
DS D
~34!
R˙ 22 ~ t ! 52 @ Y xx ~ t ! 12iX xr ~ t ! f 1 ~ t !# R 22 ~ t ! 12Z xx ~ t !
~27!
with a generally time-dependent Liouville operator L. Since S is quadratic in x and r, it is sufficient to make the ansatz
S
˙f 3 ~ t ! 5 @ Y rr ~ t ! 12iX xr ~ t ! f 1 ~ t !# f 3 ~ t ! ,
R˙ 12 ~ t ! 5 @ Y xx ~ t ! 12iX xr ~ t ! f 1 ~ t !# R 12 ~ t ! 1 @ 22X rr ~ t ! f 1 ~ t !
We now want to investigate the conditions under which the time evolution of the reduced density matrix can be described by a master equation
2
~30!
R˙ 11 ~ t ! 524iX xr ~ t ! f 3 ~ t ! R 12 ~ t ! 12X rr ~ t ! f 23 ~ t ! , ~33!
III. MASTER EQUATION
S
˙f 2 ~ t ! 52iX xr ~ t ! f 4 ~ t ! f 3 ~ t ! ,
`
are the equilibrium variances.
1 ~ x,r !
155
~28!
with complex parameter L c (t) and complex matrices X(t), Y(t), and Z(t). Without loss of generality we may put X rx (t)5X xr (t) and Z rx (t)5Z xr (t). Hence, there are eleven time-dependent functions in the ansatz ~28!. Using Eqs. ~8!, ~12!, ~13!, and ~28! to write the left-hand and right-hand sides of Eq. ~27! in explicit form, we find that a master equation exists for arbitrary preparation functions provided that 15 ordinary differential equations are fulfilled. Three of these, obtained by comparing the coefficients of r 2f ,r f r i ,r 2i read
iM ] iM ] i\ ] 2 2 D q~ t ! x L~ t ! 5 g q ~ t ! rx2 g p ~ t ! x 2 M ]x]r \ ]x \ ]r 2
M2 D ~ t !x 2. \2 p
~40!
To see this one first notes that Eqs. ~30!, ~31!, and ~33! give X rr ~ t ! 5Y rr ~ t ! 50,
X xr ~ t ! 5i/2.
~41!
The remaining four functions are then readily determined. For later convenience they are expressed in terms of the four functions introduced in Eq. ~40!, which are given by
156
ROBERT KARRLEIN AND HERMANN GRABERT
S ~ t ! 5d 1 e 2l 1 t 1d 2 e 2l 2 t
¨ 2 ~ t ! 2G ^ ~t! ˙ ~ t !G G g q ~ t ! 522iZ xr ~ t ! 5 2 , ¨ ~t! ˙ ~ t ! 2G ~ t ! G G
g p ~ t ! 52Y xx ~ t ! 5
¨ ~t! ^ ~ t ! 2G ˙ ~ t !G G~ t !G , ¨ ~t! ˙ 2 ~ t ! 2G ~ t ! G G
~42!
i\ ^p & D q ~ t ! 5 Y xr ~ t ! 5 g q ~ t ! ^ q 2 & 2 2 M M
D p~ t ! 5
S ~ t !@ g p ~ t ! X ~ t ! 1X˙ ~ t !# 1 g p ~ t ! Y ~ t ! 1Y˙ ~ t ! , ^ q 2& G ~ t !
\ ^ p 2 & S˙ ~ t !@ g p ~ t ! X ~ t ! 1X˙ ~ t !# Z xx ~ t ! 5 g p ~ t ! 2 1 M M ^ q 2& G ~ t ! 1
˙ ~ t !@ g p ~ t ! Y ~ t ! 1Y˙ ~ t !# G , G~ t !
where we have introduced ~43!
Inserting X xr (t)5i/2 and Y rr (t)50 into Eqs. ~36!-~39!, it is readily seen that Eq. ~36! and ~38! are already satisfied. However, Eqs. ~37! and ~39! are only fulfilled provided that S¨ ~ t ! 1 g p ~ t ! S˙ ~ t ! 1 g q ~ t ! S ~ t ! 50,
~44!
^ S ~ t ! 1 g p ~ t ! S¨ ~ t ! 1 g q ~ t ! S˙ ~ t ! 50.
~45!
Differentiating Eq. ~44! and subtracting Eq. ~45! we obtain
g˙ p ~ t ! S˙ ~ t ! 1 g˙ q ~ t ! S ~ t ! 50.
~46!
Now for g˙ p (t)Þ0 this gives ˙ ~t! S˙ ~ t ! g˙ q ~ t ! G 52 5 , S~ t ! g˙ p ~ t ! G ~ t !
~47!
where the last equation follows by means of Eq. ~42!. This implies S ~ t ! 5cG ~ t !
~ t>0 ! ,
~48!
where c is a real constant. Clearly, this condition, which is equivalent to Onsager’s regression hypothesis @3#, is never met exactly whatever the form of the damping kernel. To see this explicitly we note that the Taylor series of S(t) and G(t) start according to S ~ t !5 ^ q 2& 2
^ p 2& 2M 2
t 2 1O ~ t 4 ! ,
~ t>0 ! ,
~51!
where c 1 and c 2 are complex constants. It is now easily seen that Eqs. ~50! and ~51! never hold exactly except for g p 50, which means in the absence of damping. Hence there is no exact master equation for the damped harmonic oscillator with a Liouville operator L independent of the preparation function. In the remainder of this work we first consider specific initial preparations for which the time evolution is described exactly by a time-dependent Liouville operator. We then give examples for approximate Liouville operators valid for particular types of damping.
In the previous section we have shown that there is no exact Liouville operator that is independent of the preparation function. However, for certain preparations the set of equations ~29!–~39!, which determine the time-dependent parameters in L can be reduced allowing for an exact solution. In the following three types of initial states will be considered. A. Thermal initial condition
Let us first consider a system that is initially in a state
^ q u W ~ 0 ! u q 8 & 5r ~ q,q 8 ! ^ q u W b u q 8 & ,
~52!
where r(q,q 8 ) is an arbitrary function of q and q 8 . This initial condition allows in Eq. ~9! only for operators O j and O 8j that are diagonal in position space. It can be used to describe initial states resulting from position measurements but excludes measurements of velocities or variables that couple to the position and the momentum. Following Hakim and Ambegaokar @19# we call Eq. ~52! a thermal initial condition. Inspection of Eq. ~10! shows that the corresponding preparation function is given by ¯,r ¯! 5r ~ x i ,r i ! d ~¯ l ~ x i ,r i ,x x 2x i ! d ~¯ r 2r i ! .
~53!
This form of the preparation function has the consequence that only the difference of Eqs. ~37! and ~32! and likewise the difference of Eqs. ~39! and ~34! must be fulfilled. Since Eqs. ~36! and ~38! are again satisfied as a consequence of Eq. ~41!, there is indeed an exact solution
~49!
g q~ t ! 5
¨ ~ t ! S˙ ~ t ! ˙ ~ t ! S¨ ~ t ! 2G G , ˙ ~ t ! S ~ t ! 2G ~ t ! S˙ ~ t ! G
On the other hand, in the case g˙ p (t)50, the condition ~46! is only fulfilled if g p (t) and g q (t) are both independent of time. Further S(t) must be of the form
g p~ t ! 5
¨ ~ t !S~ t ! G ~ t ! S¨ ~ t ! 2G , ˙ ~ t ! S ~ t ! 2G ~ t ! S˙ ~ t ! G
G ~ t ! 5t1O ~ t 3 ! .
~50!
IV. LIOUVILLE OPERATORS FOR SPECIAL INITIAL PREPARATIONS
˙ ~ t ! S ~ t ! 2G ~ t ! S˙ ~ t ! , X ~ t ! 5G ˙ ~ t ! S˙ ~ t ! . Y ~ t ! 5G ~ t ! S¨ ~ t ! 2G
~ t>0 ! ,
where d 1 and d 2 are complex constants and l 1/25 g p /26i Ag q 2 g 2p /4. Moreover, from Eq. ~42! we see that g p (t) is constant provided that G(t) is of the form G ~ t ! 5c 1 e 2l 1 t 1c 2 e 2l 2 t
2
1
55
~54!
55
EXACT TIME EVOLUTION AND MASTER EQUATIONS . . .
D q~ t ! 5 g q~ t ! ^ q 2& 2
^p &
^ p 2& M
2
157
with a noise force j (t) satisfying
,
^ j ~ t ! & 50
~62!
^ j ~ t ! j ~ 0 ! & 5M k B T g ~ t ! .
~63!
2
D p~ t ! 5
M2
g p~ t ! .
and
When these coefficients are inserted into Eq. ~40!, we obtain a time-dependent Liouville operator valid for a large class of initial states. Using the relations 1 r→ $ q,• % , 2
x→ @ q,• # , i ] → $ p,• % , ] x 2\
The time-dependent coefficients of the Adelman equation are given by
g q~ t ! 5 ~55!
g p~ t ! 5
F
iM 1 i g ~ t ! q, $ q, r ~ t ! % 1 ^ q 2 & @ p, r ~ t !# \ q 2 \
F
i 1 i 2 g p ~ t ! q, $ p, r ~ t ! % 2 ^ p 2 & @ q, r ~ t !# \ 2 \
F
G
G
i 1 i 2 p, $ p, r ~ t ! % 2 ^ p 2 & @ q, r ~ t !# . M\ 2 \ Here @ A,B # 5AB2BA
G ~56!
denotes the commutator and
S
i dxexp 2 x p r ~ x,q,t ! . \
D
~57!
\ ] , i ]p
$ q,• % →2q,
~58!
Using the rules @ q,• # →2
@ p,• # →
\ ] , i ]q
we find from Eq. ~56!
H
F
$ p,• % →2p,
] ] ˙ ~ p,q,t ! 5 M g q ~ t ! q1 ^ q 2 & w ]p ]q 1
F
GF
] ] 1 ] g ~ t !2 p1 ^ p 2& ]p p ]q M ]p
~59!
E
t
0
ds g ~ t2s ! q˙ ~ s ! 5
~60!
1 j~ t !, M
~65!
B. Liouville operator for the time evolution of equilibrium correlations
In this section we investigate the time evolution of equilibrium correlation functions
^ A ~ t ! B & 5Tr~ Ae 2iHt/\ BW b e iHt/\ ! ,
~66!
where A and B are variables of the oscillator and Tr denotes the trace over the Hilbert space of the entire system ~oscillator and bath!. Inserting three partitions of unity and using Tr5
E
dqTrR ^ q u • u q & ,
~67!
where TrR is the trace over the Hilbert space of the bath, Eq. ~67! can be written as a fourfold integral:
E
dq 1 dq 2 dq 3 dq 4 A ~ q 2 ,q 1 ! B ~ q 3 ,q 4 ! ~68!
where A(q,q 8 )5 ^ q u A u q 8 & and
This is of the form of a generalized Fokker-Planck equation. For the classical harmonic oscillator an equation of similar form was found by Adelman @11# based on the generalized classical Langevin equation q¨ ~ t ! 1 v 20 q ~ t ! 1
M C˙ ~ t ! k B T cl
3 P ~ q 1 ,q 2 ,t,q 3 ,q 4 ! ,
GJ
3w ~ p,q,t ! .
~64!
the coefficients ~54! reduce to ~64!. This shows that we have derived an exact quantum mechanical generalization of the Adelman equation.
^A~ t !B&5
G
^ cl~ t ! 2C˙ cl~ t ! C¨ cl~ t ! C cl~ t ! C . C˙ 2cl~ t ! 2C cl~ t ! C¨ cl~ t !
G ~ t ! 52
To see the connection of this master equation with earlier results, we rewrite it in terms of the Wigner transform of the reduced density matrix defined by
E
cl
Since in the classical limit S(t) reduces to C cl(t), and
$ A,B % 5AB1BA the anti-commutator.
w ~ p,q,t ! 5
cl
cl
i ] → @ p,• # ]r \
the resulting exact master equation can be written in the form
r˙ ~ t ! 52
^ cl~ t ! ˙ cl~ t ! C C¨ 2cl~ t ! 2C , 2 C˙ ~ t ! 2C ~ t ! C¨ ~ t !
~61!
P ~ q 1 ,q 2 ,t,q 3 ,q 4 ! 5TrR ~ ^ q 1 u e 2iHt/\ u q 3 &^ q 4 u W b e iHt/\ u q 2 & ! . ~69! This function contains complete information about equilibrium correlation functions. P(q 1 ,q 2 ,t,q 3 ,q 4 ) satisfies an exact master equation. To see this we first note that an equilibrium correlation function may be calculated in the following way @10#. One propagates the initial reduced ‘‘density matrix’’ r B (0)5B r b and takes the expectation value of A after time t. Hence,
158
ROBERT KARRLEIN AND HERMANN GRABERT
^ A ~ t ! B & 5tr@ A r B ~ t !# 5
E
dq f dq 8f A ~ q 8f ,q f ! r B ~ q f ,q 8f ,t ! , ~70!
where tr denotes the trace over the Hilbert space of the oscillator. Since the initial reduced ‘‘density matrix’’ B r b corresponds to the ‘‘density matrix’’ BW b of the entire system, the preparation function reads ¯,q ¯8 ! 5B ~ q i ,q ¯ ! d ~¯ l B ~ q i ,q 8i ,q q 8 2q 8i ! .
~71!
Now, using
r B ~ q f ,q 8f ,t ! 5
E
¯dq ¯8 J ~ q f ,q 8f ,t,q i ,q 8i ,q ¯,q ¯8 ! dq i dq 8i dq
¯,q ¯8 ! 3l B ~ q i ,q 8i ,q
P ~ x f ,r f ,t,x i ,r i ! 5
1
A2 p N ~ t ! 2
exp
S
~72!
H F
we arrive at
^A~ t !B&5
E
¯dq ¯8 A ~ q f ,q 8f ! B ~ q i ,q ¯! dq f dq 8f dq i dq i8 dq
¯,q ¯8 ! d ~¯ 3J ~ q f ,q 8f ,t,q i ,q 8i ,q q 8 2q 8i ! , which yields by comparison with Eq. ~68! P ~ q 1 ,q 2 ,t,q 3 ,q 4 ! 5
E
dyJ ~ q 1 ,q 2 ,t,q 3 ,y,q 4 ,y ! . ~74!
Inserting the explicit form of the propagating function and after the Gaussian integration over y, we are left with an expression containing S(t) and G(t) only in the combination S(t)2i\G(t)/2M 5C(t). One finds
F GJ
D
¨ ~t! ˙ 2~ t ! ^ q 2& iM N N i\ C ~ t ! i\ ^ q 2 & 2 2 1 rr x x 1 ~ r f 1r i ! 22 f i 3 \ C ~ t ! 2N ~ t ! C ~ t ! M N~ t ! M N~ t ! i f
S
P˙ ~ x f ,r f ,t,x i ,r i ! 5L x f ,r f ,
D
] ] , ,t P ~ x f ,r f ,t,x i ,r i ! , ]x f ]r f ~76!
with a Liouville operator L of the form Eq. ~40! with the coefficients
^ ~ t ! 2C˙ ~ t ! C¨ ~ t ! C~ t !C , ¨ ~t! ˙ 2 ~ t ! 2C ~ t ! C C
D q~ t ! 5 g q~ t ! ^ q & 2
^ p 2& M2
F
P˙ ~ x f ,r f ,t,x i ,r i ! 5 2
M2
S
\ ] 2 g p ~ t ! x1 iM ] r
DS
D
] ^ p 2& 1 2 x ]x \
3 P ~ x f ,r f ,t,x i ,r i ! .
~77!
,
g p~ t ! .
Hence P(x f ,r f ,t,x i ,r i ) satisfies the exact evolution equation
DG ~78!
To illuminate the virtue of this equation we note that it can be used to calculate correlation functions in a quasiclassical manner. Introducing the double Wigner transform ˜ P ~ p f ,q f ,t,p i ,q i ! 5
1 ~ 2p\ !2
F
˜ A ~ p f ,q f ! 5
E
dx f dx i P ~ x f ,q f ,t,x i ,q i !
G
i ~ 2x f p f 1x i p i ! . \
E
S
dx f exp 2
D
i x p A ~ x f ,q f ! , \ f f
~79! and
~80!
equilibrium correlations may be written as a double phasespace integral
^A~ t !B&5 ^p &
S
~75!
,
and the Wigner-Moyal transforms of A(x f ,r f ) B(x i ,r i ) according to
2
D p~ t ! 5
G
] iM g q ~ t ! x r1 ^ q 2 & \ ]r
3exp
¨ 2 ~ t ! 2C ^~t! ˙ ~ t !C C g q~ t ! 5 2 , ¨ ~t! ˙ ~ t ! 2C ~ t ! C C
2
~73!
˙ ~t! iM N N˙ ~ t ! ^ q 2 & iM ^ p 2 & N˙ ~ t ! 2 ^ q 2 & 2 2 1 ~ x f r f 2x i r i ! 1 ~ x i r f 2x f r i ! 2 ~ x 2f 1x 2i ! 2\ N ~ t ! N~ t !C~ t ! \ M 4C ~ t ! 2 N ~ t !
where r f 5(q 1 1q 2 )/2, x f 5q 1 2q 2 , r i 5(q 3 1q 4 )/2, and x i 5q 4 2q 3 . Further, we have introduced N(t)5 ^ q 2 & 2 2C(t) 2 . In view of the d function in Eq. ~71! there are again fewer conditions that must be satisfied by the time-dependent coefficients of the Liouville operator. In fact, it is easily seen that
g p~ t ! 5
55
E
d p f dq f d p i dq i ˜ A ~ p f ,q f ! ˜ B ~ p i ,q i !
3˜ P ~ p f ,q f ,t,p i ,q i ! .
~81!
This means that we can view ˜ P (p f ,q f ,t,p i ,q i ) as a quantum mechanical generalization of the classical joint probability.
55
EXACT TIME EVOLUTION AND MASTER EQUATIONS . . .
The Wigner form of the master equation ~78!, which is again a generalized Fokker-Planck equation, was first derived by Schramm, Jung, and Grabert @20# on the basis of phenomenological reasoning. Note that the time-dependent coefficients are complex. However, in the classical limit they become real, since the imaginary part of C(t) vanishes, and the generalized Fokker-Planck equation reduces again to the Adelman equation. C. Factorizing initial preparation
In earlier work it has been frequently assumed @12,17,21,22# that the initial density matrix W 0 of the entire system factorizes according to W 05 r 0W R ,
~82!
where r 0 is the density matrix of the oscillator, while W R 5Z R exp(2bHR) is the canonical density matrix of the unperturbed heat bath. Within our approach this situation cannot be described by a special form of the preparation ¯,q ¯8 ), however, factorizing initial states function l(q i ,q 8i ,q are easily gained by disregarding contributions coming from the imaginary time path integral ~see @10# for details!. The time evolution of the density matrix is then given by
r ~ x f ,r f ,t ! 5
E
dr i dx i J FV~ x f ,r f ,t,x i ,r i ! r ~ x i ,r i ,0! ,
The function K 8 (t) is the real part of the real-time influence kernel. Its Laplace transform is related to the Laplace transform of the damping kernel gˆ (z) by @10# M Kˆ 8 ~ z ! 5 \b
F
G
i M u f 3~ t !u exp S FV~ x f ,r r ,t,x i ,r i ! 2p\ \ ~84!
and S FV~ x f ,r f ,t,x i ,r i ! 5M @ x f r f f 1 ~ t ! 1x i r i f 2 ~ t ! 2x i r f f 3 ~ t ! i 11 2x f r i f 4 ~ t !# 1 M @ x 2i R FV ~t! 2 22 12 12x f x i R FV ~ t ! 1x 2f R FV ~ t !# .
~85!
The index FV refers to Feynman and Vernon @21#. Note that the definitions of R 66 (t) are now modified: 11 R FV ~ t ! 5 f 23 ~ t ! K q ~ t ! , 12 ˙ q ~ t ! 2 f 1 ~ t ! K q ~ t !# , R FV ~ t ! 5 f 3 ~ t !@ 21 K
~86!
22 ˙ q ~ t ! 1 f 21 ~ t ! K q ~ t ! , R FV ~ t ! 5K p ~ t ! 2 f 1 ~ t ! K
where the f i (t) are given by Eqs. ~15!–~17! and 1 M
E E
1 K p~ t ! 5 M
E E
K q~ t ! 5
t
t
ds
0
0
t
~87! t
ds
0
duK 8 ~ s2u ! G ~ s ! G ~ u ! ,
0
˙ ~ s !G ˙ ~ u !. duK 8 ~ s2u ! G
z
( 2 2 @ z gˆ ~ z ! 2 u n nu gˆ ~ u n nu !# . n52` z 2 n n
g q~ t ! 5
g p~ t ! 5
D q~ t ! 5
~88!
¨ 2 ~ t ! 2G ^ ~t! ˙ ~ t !G G ¨ ~t! ˙ 2 ~ t ! 2G ~ t ! G G
¨ ~t! ^ ~ t ! 2G ˙ ~ t !G G~ t !G , ¨ ~t! ˙ 2 ~ t ! 2G ~ t ! G G
F
G
g p~ t ! \ 1¨ K ~ t ! 2K p ~ t ! 1 g q ~ t ! K q ~ t ! 1 K˙ q ~ t ! , M 2 q 2 ~89!
D p~ t ! 5
J FV~ x f ,r r ,t,x i ,r i ! 5
`
The integrals in Eq. ~87! cannot be expressed in terms of S(t) and G(t) as is the case for R 66 (t). The quadratic ansatz ~28! now leads to Eqs. ~29!–~35! with the functions 66 R 66 (t) replaced by R FV (t). There are no equations replacing ~36!–~39! since oscillator and bath are uncorrelated in the initial state. This set allows for a solution by the timedependent parameters
~83!
where
159
F
G
\ 1 g q~ t ! K˙ p ~ t ! 1 K˙ q ~ t ! 1 g p ~ t ! K p ~ t ! . M 2 2
The resulting master equation is equivalent to the result by Haake and Reibold @12,23# who derived it directly from microscopic dynamics. Their formulas for K q (t) and K p (t) contain frequency integrals that may be evaluated to obtain Eq. ~87!. Later, this equation was rederived by Hu, Paz, and Zhang @22# from the path integral representation. The equivalence can most easily be seen using the simplified derivation given by Paz @24#. We mention that for factorizing initial conditions the classical limit does not yield the Adelman equation @25#. The generalized Fokker-Planck operator differs by terms that persist over times of the order of the relaxation time. This means that switching on the interaction with the bath at t50 pathologically affects also the long-time behavior of the system. Usually, the oscillator and the bath are integral parts of the same system and the factorization assumption is not appropriate. V. LIOUVILLE OPERATORS FOR PARTICULAR TYPES OF DAMPING
So far we have searched for exact master equations. Let us now turn to the question whether for particular types of damping the dynamics may be described in terms of approximate Liouville operators valid for arbitrary preparation functions. Thus, we have to find circumstances under which S(t) and G(t) take the forms of Eqs. ~50! and ~51!. Using Eq. ~42! we see that in this case the Liouville operator ~40! is time independent and the coefficients D p (t)5D p and D q (t)5D q read
160
ROBERT KARRLEIN AND HERMANN GRABERT
D q5 g q^ q 2& 2 D p5 g p
^ p 2& M
^ p 2& M2
2
,
.
~90!
Note that this result is independent of the coefficients c 1 ,c 2 ,d 1 ,d 2 in Eqs. ~50! and ~51!. The Wigner transform of the density matrix then obeys Eq. ~60! with time-independent coefficients g q and g p . This result is in accordance with the findings of Talkner @9# on the most general form of a Liouville operator in Wigner form compatible with the correct equilibrium expectation values. To see explicitly when conditions ~50! and ~51! hold, we first investigate the consequences of Eq. ~51!. By virtue of Ehrenfest’s theorem G(t) is purely classical @6#, since it is related to the response function
x~ t !5u~ t !
1 G~ t !, M
~91!
describing the mean nonequilibrium displacement in response to an applied force. Hence Eq. ~51! implies that the classical equation of motion is solved by a sum of two exponentials. This means essentially Ohmic damping.
This means that we have to find conditions under which G(t) may be disregarded. Now G(t) decays at least as exp(2nt), where n [ n 1 52 p k B T/\. Therefore, for temperatures T@\ g /4p k B , we have n @Re(l 1 ), Re(l 2 ), and G(t) decays faster than the first two terms in Eq. ~95!. Hence for t@ n 21 the last term in Eq. ~95! may be disregarded and S(t) is of the form of Eq. ~51!. However, in the strictly Ohmic case we do not have a well-defined Liouville operator since the sum ~26! for ^ p 2 & is logarithmically divergent leading to associated divergences of the coefficients D q and D p in Eq. ~90!. To avoid this divergence we have to take the high frequency behavior of the damping coefficient into account, which implies in realistic cases lim gˆ ~ z ! 50.
B. Drude regularization
A more realistic behavior of the damping coefficient is modeled by
gˆ ~ z ! 5
~99!
where l 1/25 a 6i h ,
l 35 d
~100!
are the solutions of z 3 2 v D z 2 1 ~ v 20 1 g v D ! z2 v 20 v D 50
~101!
and c 1 52
~93!
where
c 25 ˜ S ~ v ! 5Sˆ ~ 2i v ! 1Sˆ ~ i v ! , ˜ ~ v ! 5G ˆ ~ 2i v ! 2G ˆ ~ iv !. G
~98!
often referred to as Drude damping. Using Eq. ~25! we then find
~92!
where l 1/25 g /26i Av 20 2 g 2 /4 and c 1/257(l 1 2l 2 ) 21 . Thus g p 5 g and g q 5 v 20 . We still have to examine whether S(t) fulfills Eq. ~50!. To this purpose it is useful to note that Eq. ~25! gives for the Fourier transforms
S D
gvD , z1 v D
G ~ t ! 5c 1 e 2l 1 t 1c 2 e 2l 1 t 1c 3 e 2l 3 t ,
For strictly Ohmic damping gˆ (z)5 g and condition ~51! holds exactly. We have
vp ˜ i\ ˜ S~ v !5 coth G~ v !, 2M n
~97!
z→`
A. Ohmic damping
G ~ t ! 5c 1 e 2l 1 t 1c 2 e 2l 2 t ,
55
~94!
c 35
i a 2i h 1 d , 2 h a 1i h 2 d
i a 1i h 1 d , 2 h a 2i h 2 d
~102!
2a . ~ a 2i h 2 d !~ a 1i h 2 d !
The latter relations follow from the fact that S(t) is symmetric and G(t) antisymmetric. Of course, Eq. ~93! is just the familiar fluctuation-dissipation theorem. Now, performing the inverse Fourier transform we find @6#
This result holds also in the overdamped case where h becomes imaginary. To calculate S(t) we have to evaluate the inverse Fourier transform of Eq. ~93! by contour integration. Using Eq. ~99! one finds @26#
S ~ t ! 5d 1 e 2l 1 t 1d 2 e 2l 2 t 2G ~ t ! ,
S ~ t ! 5d 1 e 2l 1 t 1d 2 e 2l 1 t 1d 3 e 2l 3 t 2G ~ t ! ,
~103!
S D
~104!
~95!
where d 1/25c 1/2 `
S D
where
\ p l 1/2 cot , 2M n
n n exp~ 2 n n t ! 2g G~ t !5 . M b n51 ~ v 20 1 n 2n ! 2 2 g 2 n 2n
(
d i 5c i ~96!
\ pli cot , 2M n
i51,2,3,
`
G~ t !5
v 2D n n e 2 n n t 2g . M b n51 ~ l 21 2 n 2n !~ l 22 2 n 2n !~ l 23 2 n 2n !
(
~105!
55
EXACT TIME EVOLUTION AND MASTER EQUATIONS . . .
From these results we see that S(t) and G(t) are of the forms ~50! and ~51!, respectively, if the exponentials exp(2l3t) and exp(2nnt) decay much faster than exp(2l1/2t). This is the case for Re~ l 1 ! ,
Re~ l 2 ! ! d , n .
r˙ ~ t ! 52
2a1d5vD ,
a 2 1 h 2 5 v 20 v D / d ,
~107!
a 2 1 h 2 12 a d 5 v 20 1 g v D , we find that Re(l 1/2)! d implies
a!d.
F F
~109!
2
i 1 i p, $ p, r ~ t ! % 2 ^ p 2 & @ q, r ~ t !# , M\ 2 \
g v 2D a5 , 2 v 2D 1 v 20
h5A
v 20 12 a v 20 / v D 2 a 2 ,
S D
Hence, for the classical limit it is not sufficient to have n @ v 0 , that is k B T@\ v 0 , rather we also need g ln(vD /n)!n. Thus the strictly Ohmic limit v D →` can only be taken after the high temperature limit. With this sequence of limits we obtain the classical coefficients
g q 5 v 20 ,
g p5 g ,
D q 50,
D p5
~111!
These relations are valid for arbitrary ratios of g and v 0 . Also v D / v 0 is not necessarily large. As a consequence of this analysis we find that only under the conditions ~112!
In this section we show that S(t) and G(t) always take the forms of Eqs. ~50! and ~51! in the limit of weak damping. Let us assume that the damping kernel g (t) has a high frequency cutoff v c and that its Laplace transform is an analytic function in the vicinity of 2i v 0 . We introduce a typical damping strength by
g c5
E
`
0
ds g ~ s ! cos~ v 0 s ! .
~113!
S(t) may be approximated by the first two terms in Eq. ~103!. Hence, for sufficiently large Drude cutoff and sufficiently high temperatures, k B T@\ g , the oscillator dynamics can be described by an approximate Liouville operator with the coefficients
This combines with Eq. ~90! to yield the approximate master equation
~118!
Apart from this g (t) is not assumed to have additional properties. In the limit considered one can determine the poles of ˆ (z) from Eq. ~25! perturbatively. To first order in g the G c poles are
D
~119!
g c 1i g s 5 gˆ ~ 2i v 0 ! .
~120!
l 1/25
g q5 a 21 h 2 ~114!
~117!
Then the weak damping condition is
g c! v 0 , v c , n .
g p 52 a .
~116!
The associated Liouville operator is equivalent to the classical Fokker-Planck operator of the Kramers equation @28#. We stress again that the results in this section remain valid for strong damping provided Eq. ~112! is satisfied.
and 21 t@ v 21 . D ,n
g k BT . M
C. Weak damping with arbitrary frequency dependence
d 5 v D 22 a .
g!vD ,n
~115!
M \g vD ln . p n
~110!
Hence, it is natural to use g / v D as a small parameter to determine the roots of Eq. ~101!. Up to first order in g / v D one obtains from Eqs. ~100! and ~101!
G
first derived by Haake and Reibold @12#. The equilibrium variances can be calculated analytically @6#. In the strictly Ohmic limit v D →`, ^ q 2 & is a regular expression but ^ p 2 & diverges logarithmically. If we disregard terms of the order v 0 / v D , g / v D , n / v D , the divergent part of ^ p 2 & is given by
In view of Eq. ~108! this gives @27#
g / v D 52 a / d 28 ~ a / d ! 2 1O„~ a / d ! 3 ….
G G
2i 1 i a q, $ p, r ~ t ! % 2 ^ p 2 & @ q, r ~ t !# \ 2 \
~108!
Further, the relations ~107! yield
g a 11 ~ v 0 / d ! 2 52 . vD d ~ 112 a / d ! 2
F
iM 2 1 i ~ a 1 h 2 ! q, $ q, r ~ t ! % 1 ^ q 2 & @ p, r ~ t !# \ 2 \
2
~106!
Using the Vieta relations
161
S
gc gs 6i v 0 1 , 2 2
where
Performing the inverse Laplace transform of Eq. ~25! we find
162
ROBERT KARRLEIN AND HERMANN GRABERT
G~ t !5
i ~ e 2l 1 t 2e 2l 2 t ! 2v0
g v 2D g c5 2 , v D 1 v 20
~121!
with residues in zeroth order. Higher-order corrections would depend on the specific form of gˆ (z) but need not be determined because the diffusion constants ~90! are independent of the residues. Thus, in the weak damping limit G(t) is of the form of Eq. ~51!. From Eq. ~93! we see that ˜ S ( v ) has the ˜ same poles as G ( v ) and, in addition, poles at i n n (n integer!. As shown above, we can disregard the terms coming from the poles at i n n for times greater than the thermal relaxation time n 21 . Thus, S(t) is effectively of the form of Eq. ~50!. Inserting l 1/2 into Eq. ~90!, we find to leading order in the damping strength
55
g s5
S D FS D S
g cv 0M v 0p coth , K c5 2 n K s5
H
v0M ngs v0 vD 1 g c Re c 11i 2 c 11 p 2v0 n n
1 Mb
`
( n52`
g s v 0 2 u n n u gˆ ~ u n n u ! , v 20 1 n 2n ~122!
1 D p5 Mb
`
(
n52`
S D
g c v 20 \v0 v 0p coth . 2 2 5gc 2M n v 01 n n
\ ˆ K 8 ~ 2i v 0 ! . M2
~123!
S D
gv0M vD ln p n
F
~126!
G
S D
M v0 v 0p coth †q, @ q, r ~ t !# ‡, \ n
~127!
which was derived with the help of projection operator techniques from the same microscopic model using the Born approximation in conjunction with a short-memory approximation. As a main difference, we see that in Agarwal’s equation the K s term is absent. This term is only negligible if
v 0! v D! n Thus, D q and D p are essentially given by the sine and cosine ˆ 8 (2i v ) the master moments of K 8 (t). With K c 1iK s 5K 0 equation takes the form
.
M v 20 2 i p2 ik 1 q , r ~ t ! 2 @ q, $ p, r ~ t ! % # \ 2M 2 \
2k
D p 1i v 0 D q 5
D GJ
is logarithmically divergent. The master equation ~124! with the coefficients ~125! can be compared with the well-known Agarwal equation @13#
r˙ ~ t ! 52
This result can be expressed in terms of the Laplace transform ~88! of the real part of the influence kernel
~125!
In the strictly Ohmic limit v D →` we have g c 5 g , g s 50 but K s 52
D q5
g v 0v D , v 2D 1 v 20
~128!
Hence, the master equation ~124! is a generalization of the Agarwal equation. D. Connection to Lindblad theory
F
M ~ v 20 1 v 0 g s ! 2 i p2 1 q ,r~ t ! r˙ ~ t ! 52 \ 2M 2
G
2
igc Ks †p, @ q, r ~ t !# ‡ @ q, $ p, r ~ t ! % # 2 2\ M \v0
2
Kc †q, @ q, r ~ t !# ‡. \
~124!
This general weak coupling master equation is given in terms of four dissipation coefficients. g s leads to a frequency shift and may be absorbed by renormalizing v 0 . g c is the classical damping coefficient. The coefficients K c and K s depend on the temperature. While K c equals M g c /\ times the average energy of a quantum oscillator of frequency v 0 , K s depends on the specific form of gˆ (z) and can be calculated analytically only in certain cases. One of these is the Drude model. Then the moments g c/s and K c/s are readily evaluated to read
The approximate time-independent Liouville operators studied above describe the dynamics after the decay of fast transients. Markovian Liouville operators such as these are often discussed in the context of Lindblad theory @29#. This theory establishes the most general form of generators L of dissipative quantum dynamics r˙ (t)5Lr (t) preserving the positivity of density operators. The Lindblad master equation reads i ˜ 1 r˙ ~ t ! 52 @ H , r ~ t !# 1 \ 2\
(i @ L i r ~ t ! ,L 1i # 1 @ L i , r ~ t ! L 1i # , ~129!
˜ is a Hermitian operawhere L l are arbitrary operators and H tor. Using results by Sandulescu and Scutaru @30# it is easily seen that all the above-derived time-independent Liouville operators are not of Lindblad form. This is not too astonishing since the master equations derived hold only for times t.t 0 where t 0 is larger than an inverse cutoff frequency and n 21 . The short time dynamics for t&t 0 reduces the density
55
EXACT TIME EVOLUTION AND MASTER EQUATIONS . . .
matrix to a subspace where the fast components have decayed. The Markovian master equation holds within this subspace only, while Lindblad theory requires validity for any reduced density matrix. This is of course not necessary as has been emphasized again recently @1,31#. However, we will show that in the weak coupling limit further coarse graining will result in a Lindblad operator. To this aim we first write the weak-coupling master equation ~124! in the form
r˙ ~ t ! 5Lr ~ t ! 5L0 r ~ t ! 1 g L1 r ~ t ! .
~130!
t@ v 21 0 .
~131!
Using the operators Pn 5
(k
u k &^ k1n u ,
~132!
˜ #r~ 0 !, r ~ t ! 5e L0 t @ 11 g tL 1
(n i v 0 nPn .
~133!
Further
g L1 52i
gs † g c 2i g s †2 g c 1i g s 2 @ a a,• # 1 @ a ,• # 2 @ a ,• # 2 4 4
1 g ↓ ~@ a•,a † # 1 @ a,•a † # ! 1 g ↑ ~@ a † •,a # 1 @ a † ,•a # ! 1
K c 1iK s ~@ a † •,a † # 1 @ a † ,•a † # ! 2M v 0
˜ 5 L 1
~134!
F S D G
gc gc v 0p Kc 6 5 coth 61 . 2M v 0 4 4 n
~135!
The time evolution is formally given by
r ~ t ! 5e Lt r ~ 0 ! .
~136!
Now we rewrite this by a well-known operator identity
E
t
0
dse L0 ~ t2s ! L1 e Ls r ~ 0 ! .
~137!
For weak damping and times t! g 21 the operator e Ls in the integrand may be replaced by e L0 s . Inserting then Eq. ~133! into Eq. ~137! we find
F
r ~ t ! 5e L0 t 11 g
( n,m
E
t
0
~141!
The density matrix ~140! coincides with the solution of the master equation ˜ !r~ t ! r˙ ~ t ! 5 ~ L0 1 g L 1
~142!
t! g 21 .
~143!
for
21 Thus, within the time window v 21 the two opera0 !t! g ˜ tors L0 1 g L1 and L0 1 g L1 give the same dynamics. The ˜ may be evaluated further. It is seen that only operator L 1 ‘‘non-rotating’’ terms containing equal numbers of creation and annihilation operators survive the coarse graining in time. The resulting master equation
S
r˙ ~ t ! 52i v 0 1
D
gs @ a 1 a, r ~ t !# 1 g ↑ ~@ a 1 r ~ t ! ,a # 2 ~144!
where we have introduced
r ~ t ! 5e L0 t r ~ 0 ! 1 g
(n Pn L1 Pn .
1 @ a 1 , r ~ t ! a # ! 1 g ↓ ~@ a r ~ t ! ,a 1 # 1 @ a, r ~ t ! a 1 # !
K c 2iK s 1 ~@ a•,a # 1 @ a,•a # ! , 2M v 0
g ↓↑ 5
~140!
where we have introduced the effective dissipative Liouville operator
where u k & are the eigenstates of a † a, L0 may be written as L0 5
~139!
Then, by performing the time integral, the off-diagonal terms (nÞm) are seen to be smaller than the diagonal terms by a factor ( v 0 t) 21 . This means that in the time window 21 v 21 0 !t! g
In terms of the usual creation and annihilation operators a † , a we have L0 52i v 0 @ a † a,• # .
163
G
dse i v 0 ~ n2m ! s Pn L1 Pm r ~ 0 ! .
Further coarse graining is achieved by demanding
~138!
was first derived by Weidlich and Haake @14# from a microscopic model for the damped motion of a single mode of the electromagnetic field in a cavity. The generator defined by this master equation is of Lindblad form. However, the resulting mean value equations violate Ehrenfest’s theorem, in particular
] ^ q ~ t ! & Þ ^ p ~ t ! & /M . ]t
~145!
This is due to the fact that on the coarse-grained time scale the variables p(t)/M v 0 and q(t) exchange idenDt@ v 21 0 tity frequently and only a time-averaged version of the mean value equations must be obeyed. VI. CONCLUSIONS
Based on results of the path integral technique we have examined quantum master equations for the damped harmonic oscillator. An exact generalized master equation describing the relaxation of initial thermal conditions was derived. This equation was shown to be a quantum mechanical generalization of Adelman’s Fokker-Planck equation. We also have given an exact Liouville operator describing the time evolution of equilibrium correlation functions, which
164
ROBERT KARRLEIN AND HERMANN GRABERT
55
likewise reduces to the Adelman Fokker-Planck operator in the classical limit. The fact that two different quantum generalizations of the Adelman operator must be used for the relaxation of expectation values and the regression of fluctuations is intimately connected with the failure of the Onsager regression hypothesis in the quantum regime. Indeed, the two Liouville operators ~56! and ~76! are only identical if S(t) is proportional to G(t). Apart from these exact results we have studied in detail the range of parameters leading to quantum master equations with time-independent generator. In the case of strong damping a time-independent Liouville operator is obtained approximately for essentially frequency-independent damping. However, strictly Ohmic damping is ill behaved in the quantum case, and the appropriate generalization of the classical Fokker-Planck process is given by a low frequency Ohmic model with high frequency cutoff such as the Drude model. This is not too amazing since already in the classical limit the Adelman operator becomes time independent only for frequency-independent damping. On the other hand, in the case of weak damping the detailed frequency dependence of the damping coefficient is unimportant. We have derived a generalized master equation valid for arbitrary weak damping as long as the sine and
cosine moments of the damping kernel exist. The new weakcoupling master equation is more general than the wellknown Agarwal equation. We have explained why the Liouville operator is not of Lindblad form. However, time coarse graining leads to a generator of Lindblad form. The resulting coarse grained master equation was found to be the Weidlich-Haake equation also known as the quantum optical master equation. Due to the time coarse graining only a timeaveraged version of the mean value equation is obeyed leading to an apparent contradiction with the Ehrenfest theorem. In summary, we have derived several generalized master equations for the damped quantum oscillator for various cases of interest. In view of the results earlier findings were put in proper perspective.
@1# P. Pechukas, Phys. Rev. Lett. 73, 1060 ~1994!. @2# H.-P. Breuer and F. Petruccione, Phys. Rev. Lett. 74, 3788 ~1995!. @3# O. W. Ford and R. F. O’Connell, Phys. Rev. Lett. 77, 798 ~1996!. @4# For reviews see, e.g., F. Haake in Quantum Statistics in Optics and Solid-State Physics, edited by G. Ho¨hler, Springer Tracts in Modern Physics Vol. 66 ~Springer, Berlin, 1973!; H. Haken, Rev. Mod. Phys. 47, 67 ~1975!. @5# For a survey see, e.g., U. Weiss, Quantum Dissipative Systems ~World Scientific, Singapore, 1993!. @6# H. Grabert, U. Weiss, and P. Talkner, Z. Phys. B 55, 87 ~1984!. @7# R. Jung, G.-L. Ingold, and H. Grabert, Phys. Rev. A 32, 2510 ~1985!. @8# H. Grabert, Z. Phys. B 49, 161 ~1982!. @9# P. Talkner, Ann. Phys. ~N.Y.! 167, 390 ~1986!. @10# H. Grabert, P. Schramm, and G. L. Ingold, Phys. Rep. 168, 115 ~1988!. @11# S. A. Adelman, J. Chem. Phys. 64, 124 ~1976!. @12# F. Haake and R. Reibold, Phys. Rev. A 32, 2462 ~1985!. @13# G. S. Agarwal, Phys. Rev. 178, 2025 ~1969!; Phys. Rev. A 4, 739 ~1971!. @14# W. Weidlich and F. Haake, Z. Phys. 185, 30 ~1965!. @15# P. Ullersma, Physica 23, 27 ~1966!; 23, 56 ~1966!; 23, 74 ~1966!; 23, 90 ~1966!. @16# R. Zwanzig, J. Stat. Phys. 9, 215 ~1973!.
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ACKNOWLEDGMENTS
The authors would like to thank J. Ankerhold, J. Hainz and F. J. Weiper for valuable discussions and S. Gnutzmann and F. Haake for communicating their results prior to publication. Financial support was provided by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 276.
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A. O. Caldeira and A. J. Leggett, Physica A 121, 587 ~1983!. In Ref. @10#, G(t) is denoted by G 1 (t). V. Hakim and V. Ambegaokar, Phys. Rev. 32, 423 ~1985!. P. Schramm, R. Jung, and H. Grabert, Phys. Lett. A 107, 385 ~1985!. R. P. Feynman and F. L. Vernon, Ann. Phys. ~N.Y.! 24, 118 ~1963!. B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843 ~1992!. There are some typos in equation ~9.49! of Ref. @12#: A˙ A must ^ , A 2 by AA ^ , and the sign in front of X¨ must be replaced by A˙ A be changed. J. P. Paz, in The Physical Origins of Time Asymmetry, edited by J. J. Halliwell, J. Paˆerez-Mercader and W. H. Zurek ~Cambridge University Press, Cambridge, 1994!. P. Ha¨nggi, Z. Phys. B 31, 407 ~1978!; P. Ha¨nggi, in, Noise in Non-linear dynamics Vol. I, edited by F. Moss and P. V. E. McClintock ~Cambrige University Press, Cambridge, 1989!, pp. 307–328. The related result for a harmonic barrier is given in J. Ankerhold and H. Grabert, Chem. Phys. 204, 27 ~1996!. We assume that v D / v 0 is of order 1 or larger. H. A. Kramers, Physica 7, 284 ~1940!. G. Lindblad, Commun. Math. Phys. 48, 119 ~1976!. A. Sandulescu and H. Scutaru, Ann. Phys. ~N.Y.! 173, 277 ~1987!. S. Gnutzmann and F. Haake ~unpublished!.
