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Feb 1, 1987 - generally believed to be obtainable only from a long and at times tedious —recently developed —instanton-type analysis of path integrals (be it ...
PHYSICAL REVIEW A

FEBRUARY 1, 1987

VOLUME 35, NUMBER 3

Noninteracting-blip

approximation

for a two-level system coupled to a heat bath H. Dekker

Physics and Electronics

I.aboratory FEI.-TNO,

P. O. Box 96864, Den Haag, The Netherlands (Received 29 August 1986)

A very simple yet novel derivation is presented in the "noninteracting-blip approximation.

"

I present a very simple yet novel derivation of the dynamics of the dissipative two-state system in the celebrated "noninteracting-blip which is approximation, generally believed to be obtainable only from a long and at times tedious recently developed instanton-type analysis of path integrals (be it for the free energy in theror for the density matrix modynamic imaginary time in real time' ). Notable exceptions are Refs. 10 and 11. The two-state system is interesting as the fundamental model for the study of thermal relaxation and quantum tunneling properties of various chemical and physical systems (e.g. , molecular transitions in liquids and macroscopic quantum coherence in Josephson solid-state devices' ). In many cases it can be considered as a subspace of more complicated systems (e.g. , the description double-well potential ). I begin with the well-known spin-boson Hamiltonian in the displaced bath oscillators basis

"'



+tr

2

dard algebra of the boson creation and annihilation operators, it is then not difficult to calculate the commutator

[fl(t), Q(t')] =i



'

~ = — A'i-tp(tr+e

of the dynamics of the dissipative two-state system

e

)+

2

g (pk +cakxk

)

g (ck/Reek

)si

cak(t

n[

—t')]

.

(4)

k

'

The next step is to average (3) with respect to the bath and sense' '' (in a weak-coupling-limit ) to decouple the environmental exponentials from the spin; this is my second The thermal average of exp i [A(t) approximation. —II(t')]I is most easily evaluated by means of theI usual cumulant (or cluster) expansion. ' Since the environment is a linear Gaussian system, only the second cumulant contributes. Inserting then (4) and

'

0( t)II(t')+ Q(t')II(t)

= g (ci, /fRi)k

, APcttk—)cos[cpk(t—t')]

)coth(

(5)

k

into the equation of motion convolution-type result

for

o, (t), I

find

the

k

~I

= g (ck /~k

(6)

(2)

)pk,

k

'

the o. + where Ao represents the tunneling frequency, are the usual auxiliary Pauli matrices, the ck are coupling coefficients, and the boson masses mk =1 for convenience. Obviously, the Hamiltonian (1) is diagonal when b.p — 0 (or in the strong-coupling limit). In the Heisenberg picture the equations of motion for cr+(t) are easily obtained. Integrating and substituting them into the equation of motion for the polarization o, (t) which for the particle's the double-well represents potential coordinate I readily obtain the exact formula

f (r) = bp cos[Q, (r)/i') exp[ —Qp(r)/iraqi], Q,

—f (r):

f

— Q~(~) = where





(t)

t

g2

f

t iritt) in(—

[

+

(tr)

)(ti)'

in(t

)in(t— )]dtt'

(3)

I now invoke the simple Baker-Hausdorff theorem' for operators which commute with their commutator, which is appropriate here if in Q(t). I insert the free-bath Using the. standynamics; this is my first approximation

'A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. (to be published). 2S. Coleman, in The Whys of Subnuclear Physics, edited by A. Zichichi (Plenum, New York, 1979), p. 805.

sin(car)

J (ca)des/cp

[1 —cos(car) ]cath(

0

'

—, A'13at )

J (at)dat/cp

I have introduced the spectral density

J(cp) =

—g (ck/cpk)5(cp

—cak)

.

(10)

k

once transforming (6) at yields cr, (s) ', where (s) is the transform of (r). The formulas identical to the (10) are exactly (6) — "noninteracting-blip" results [see, especially, Ref. 1, formulas (4.31)— (4.33), (4.22a), and (4.22b), and (3.2)], which is what I set out to show.

Laplace

=[s + f (s)]

A discussion acknowledged.

f

f

with Professor Dr.

R. Silbey

is gratefully

S. Chakravarty, Phys. Rev. Lett. 49, 681 (1982). J. Bray and M. A. Moore, Phys. Rev. Lett. 49, 1545 (1982). 58. Carmeli and D. Chandler, J. Chem. Phys. 82, 3400 (1985). A. T. Dorsey, M. P. A. Fisher, and M. S. Wartak, Phys. Rev.

4A.

1436

1987

The American Physical Society

1437

BRIEF REPORTS

35

A 33, 1117 (1986). 7S. Chakravarty and A. J. Leggett, Phys. Rev. Lett. 52, 5 (1984). H. Grabert and U. Weiss, Phys. Rev. Lett. 54, 1605 (1985). M. P. A. Fisher and A. T. Dorsey, Phys. Rev. Lett. 54, 1609

(1985).

R. Silbey and R. A. Harris, J. Chem. Phys. 80, 2615 (1984). C. Aslangul, N. Pottier, and D. Saint- James, Phys. Lett. 110A, 249 (1985). A. J. Leggett and A. Cxarg, Phys. Rev. Lett. 54, 857 (1985). ' H. Dekker, Phys. Lett. 114A, 295 (1986) ~

' W. Zwerger, Z. Phys. B 53, 53 (1983). ~sE. Merzbacher, Mechanics Quantum

(Wiley, New York, 1970), p. 167. D. Waxman, J. Phys. C 18, L421 (1985). H. Dekker, FEL-TNO Report No. HD/86-07-07, Den Haag, 1986 (unpublished). L. E. Reichl, 3 Modern Course in Statistical Physics (University of Texas, Austin, 1980). N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981) ~