The oscillator-bath model of the environment.

Transcription

1 Statistical Mechanics Phys504 Fall 2006 Lecture #21 Anthony J. Leggett Department of Physics, UIUC The oscillator-bath model of the environment. In the last lecture I introduced the idea of representing the environment of a macroscopic system (in this case a Brownian particle) by a set of simple linear harmonic oscillators, with a coupling which is linear in the oscillator 1 coordinate x i. This idea, which goes back (at least) to the well-known paper of Feynman and Vernon (Ann. Phys. 24, 118 (1963)), is extremely attractive; it not only permits the concise derivation of (classical) Brownian motion given in l.20, but enables us to do a host of other analytical calculations of the effect of the environment on various classical and quantum aspects of the behavior of the system. It is therefore of some importance to determine how general is the validity of this oscillatorbath model. To be sure, in some cases the microscopic Hamiltonian already explicitly has this form; for example, this would be true for the current in an antenna (the coupling is of the form (in the Lagrangian) A(rt), and the vector potential can be directly expanded in terms of the oscillator-like modes of the EM field). However, in the case of a Brownian particle, and many other cases of physical interest, the explicit Hamiltonian looks nothing like an oscillator-bath model, so it is not obvious that it can be converted into this form. Let us start by considering a system (S) described by a single coordinate q and conugate momentum p. Although we will be primarily interested in the case that S is macroscopic, this is not in itself essential to the argument which follows. We suppose that in the absence of coupling to the environment the system is described by the Hamiltonian Ĥ 0 (p, q) = p 2 /2M + V (q) (1) (where M need not have the physical meaning of a mass ; for example, in the case of an electrical LC-circuit, where q might be the flux through the circuit, p would be the charge on the capacitor plates and M thus the capacitance). The total Hamiltonian of the universe is then of the generic form Ĥ = Ĥ0(p, q) + Ĥenv(ξ) + Ĥint(q, ξ) (2) 1 And, in the case of the Brownian particle of l.20, also in the system coordinates, but this is not the most general case (see below). 1

2 where ξ is shorthand for all the variables (coordinates and momenta) of the environment. We do not at this stage put any specific constraints on the environment Hamiltonian Ĥenv(ξ); nor on the S-E interaction term Ĥint(q, ξ), except to specify that the coupling of S is to a very large number N of environmental degrees of freedom (see below for a more precise formulation). Note in particular that we do not need at this stage to assume that the characteristic frequency scale of the system is much less than that of the environment, although in many cases of interest this is in fact so. We proceed by converting our problem to the adiabatic (Born-Oppenheimer) basis; note that we do not make below the Born-Oppenheimer approximation. The BO basis is defined by a set of functions of the form 2 Ψ ik (q, ξ) = φ ik (q)χ k (q, ξ) (3) where the functions χ k (q, ξ) satisfy the Schrödinger equation (in which q enters as a parameter) {Ĥenv(ξ) + Ĥint(q, ξ)}χ k (q, ξ) = Ũk(q)χ k (q, ξ) (4) (which defines the eigenvalue Ũk(q)) and the φ ik (q) satisfy the Schrödinger equation ) ( 2 2 2M q + U k(q) φ 2 ik (q) = E ik φ ik (q) (5) where U k (q) V (q) + Ũk(q) (6) The BO basis diagonalizes the truncated Hamiltonian Ĥ (0) BO Ĥtr(q) + Ĥenv(ξ) + Ĥint(q, ξ) (7) where Ĥtr(q) is defined by its matrix elements ik Ĥtr l ( 2 /2M) δ kl φ ik(q) ( 2 / q 2 ) φ k (q) dq (8) To restore the complete Hamiltonian (2) we need to add the terms in which the operator / q operates on the χ k (q, ξ). Explicitly, the correction term Ĥ is given by its matrix 2 We have a slight notational problem here: while in Ĥenv(ξ) and Ĥint(q, ξ) the variable ξ denotes all variables (coordinates plus momenta) of E, in χ k (q, ξ) it should of course refer only to (whichever set of variables we have chosen as) the coordinates. 2

3 elements as 2 ik Ĥ l = 2M dq { dξ 2φ ik(q) q φ l(q) χ k(q, ξ) q χ l(q, ξ) } (9) + φ ik(q)φ l (q) χ k(q, ξ) 2 q 2 χ l(q, ξ) Up to this point everything is exact: we have simply rewritten the original problem in the adiabatic (BO) basis. We now come to a delicate but crucial point: While the total S-E coupling may not be at all small, the coupling to any one individual mode k is, by hypothesis, of order N 1/2 (so that the probability of excitation is N 1 ); hence, the dependence of the quantity U k (q) on the index k is similarly of this order 3, and thus the functions φ k (q) are effectively independent of k. To the same approximation, E ik can be written as E (S) i +E (E) k, where E (E) k is independent of i and hence of q. This simplifies the ensuing formulae considerably. We construct a linear operator ˆK(q) in the space spanned by the functions χ k (q, ξ) by giving its matrix elements in this basis: k ˆK(q) l ( χ k(q, ξ) i ) χ l (q, ξ) dξ (10) q The operator so defined is Hermitian and is parametrically dependent on q. Since it is a linear operator and the basis set χ k (q, ξ) is complete for given q, it follows after a little algebra that the perturbation terms in (9) can be written ik Ĥ l = 1 2M i p ˆK kl (q) + ˆK kl (q)p + ( ˆK 2 ) kl (q) (11) where as usual i p ( φ i (q) i ) φ (q) dq (12) q and we have written for clarity k ˆK(q) l ˆK kl (q), etc. Thus the total Hamiltonian can be written Ĥ = 1 2M (p + ˆK(q)) 2 + U(q) + Ĥ env (13) where Ĥ env is an operator (defined by its matrix elements E (E) k δ kl ) which is a function only of the environment coordinates ξ, and ˆK(q) is another operator on these coordinates which is parametrically dependent on q. The cross-term in brackets is symmetrized as in (11). 3 U k (q) may of course have a q-independent term of order 1; subtracted by redefining the zero of E ik. 3

4 Apart from a factor of the coupling constant ( e) the expression (13) is formally identical to the familiar Hamiltonian of a charged particle interacting with an electromagnetic vector potential in one dimension. Note that all the ˆK kl are (at least) of order N 1/2. Let us now turn to the ustification of the oscillator-bath representation of the environment. We will regard the problem posed by eqn. (13) as a special case of a more general problem in which some external agency (in this case the system S) interacts with the system of interest (the environment), and we shall make essential use of the assumption that any one degree of freedom of the environment is only weakly perturbed by the external agency. (Note that this is exactly the condition we needed in l.15 to ustify the responsefunction technique). Let the coordinates of the environment (E) be denoted by ξ, and its Hamiltonian be Ĥ env. The eigenfunctions ψ n (ξ) satisfy the equation Ĥ envψ n (ξ) = E n ψ n (ξ) (14) Let the interaction of E with an external agency described by some set of coordinates q be given by some real function V of ξ, q and their conugate momenta, and define V nm (p, q) ψn(ξ) ˆV (p, q, p ξ, ξ)ψ m (ξ) dξ = Vmn(p, q) (15) where p is the momentum conugate to q and p ξ that conugate to ξ. We suppose now that the following condition is satisfied for all relevant values of p and q: V mn (p, q)/(e n E m ) ɛ 1 (16) Let us at this point specialize to the case of zero temperature. If the system E then starts in the groundstate, the probability of a process in which n factors of V enter will be proportional to ɛ 2n. Hence, if we neglect terms of order ɛ 3 and higher, the only matrix elements we need be concerned with are the elements V 0. We define Hermitian operators ˆx and ˆp in the space of the eigenfunctions ψ (ξ) by their matrix elements ˆx 0 = 0 ˆx = ( /2m ω ) 1/2 (17) all other elements zero, ˆp 0 = 0 ˆp = im ω ˆx 0 (18) all other elements zero, where ω (E E 0 )/ and the masses m are arbitrary. We now add, for each index, a (possibly fictitious) infinite set n, n = 0, 1, 2,..., of oscillator 4

5 states: in this notation the groundstate of E is a product of 0, and the states called above are now denoted 1 k 0 k. The operators ˆx and ˆp are extended into the new space so constructed by assigning to them the conventional oscillator matrix elements ( n + 1 ˆx n = (n + 1) 1/2 ( /2m ω ) 1/2, etc.). We also construct an effective Hamiltonian Ĥ eff = ( ˆp ) 2m 2 m ω 2 ˆx 2 (19) Finally we add to Ĥeff an interaction Hamiltonian of the form Ĥ int = (F (p, q)ˆx + G (p, q)ˆp ) (20) where ( ) 1/2 2m ω F (p, q) Re V 0(p, q) (21a) ( ) 1/2 2 G (p, q) Im V 0(p, q) (21b) m ω Since the matrix elements of Ĥ int and the eigenvalues of Ĥ eff reproduce the true matrix elements V nm (q) and true energy level differences E n E m for all singly excited states, the combination Ĥeff + Ĥint will generate the correct dynamics of the system to the extent that double excitation is negligible. It is only to the extent that this condition holds that the response of E will be linear in the external perturbation, so we can say that for our purposes the replacement of the true Hamiltonian by that appropriate to a set of harmonic oscillators is effectively equivalent to the assumption that the response of the environment to the perturbation exerted by the system is linear (or more precisely, that the response to a classical perturbation whose magnitude is less than or equal to the maximum value of the interaction function Ĥint(p, q) obtained for values of p and q of interest in the tunneling problem, is linear). Whether or not this condition holds for any specific physical system considered is of course a matter for detailed argument in each case; however, it should again be stressed that the condition of linearity of environment response is quite different from (and much weaker than) the condition of weak damping - ust as the fact that a beam of light may be almost totally absorbed in a gas by no means implies that nonlinear effects have to be taken into account. We now apply the above general result to the perturbation corresponding to eqn. (11). First, consider the part of the perturbation which is linear in q, (p/m) ˆK(q) (or more accu- 5

6 rately the correctly symmetrized version of this). According to eqns. (10), (21b) and (15) we have 4 F (p, q) 0, G (p, q) = (2/m ω ) 1/2 χ (ξ, q) q χ 0(ξ, q) dξ K (q) (22) so this part becomes M 1 p K (q). What about the term in ˆK 2 (q)? This is a bit more tricky, since putting ˆK 2 (q) = ( p K (q)) 2 is equivalent to the assumption that in the right-hand side of the identity l ˆK 2 (q) k = n l ˆK(q) n n ˆK(q) k (23) the only intermediate state which contributes to the sum is the groundstate 0. However, since the only relevant values of k and l correspond precisely to states which are generated from the ground state by application of the momentum operator ˆp, this assumption seams extremely plausible (in fact, its violation would seem to be equivalent to the occurrence of double excitation and thus in general of violation of the linearity hypothesis). So I shall make it, and the Hamiltonian now takes the simple form [ Ĥ(p, q, {p, x }) = 1 p + 2M where H env {p, x } has the SHO form (19). K (q)p ] 2 + U(q) + H env {p, x } (24) The final step in this rather tedious argument consists in showing that the momentummomentum coupling in the first (KE) term of (24) can be converted into a coordinatecoordinate coupling plus a counterterm. We relegate this step to the appendix and simply quote the final result: Ĥ = ĤS + ĤE + Ĥint + Ĥct (25) 4 I implicitly assume that Ĥ env is time reversal invariant, so that the χ k (ξ) can be chosen real. In the more general case the argument needs to be supplemented. 6

7 where Ĥ S p 2 /2M + V (q) (26a) Ĥ E 1 (p 2 i /m i + m i ωi 2 x 2 i ) 2 (26b) i Ĥ int F (q)x (26c) Ĥ ct 1 F 2 (q)/m ω 2 2 (26d) where F (q) is defined in terms of K (q) by eqn. (A.6) of the appendix. Although the derivation of eqn. (25) has been quite tedious, we shall see that it pays off handsomely. Before applying the oscillator-bath model described by eqns. (25) and (26), however, let us note that it has been explicitly derived above only (a) for a single macroscopic (system) degree of freedom, and (b) at T = 0. It is clear that the generalization to a number of system DOF s is straightforward; Ĥ S is now of the form Ĥ S = α p 2 α/2m + V {q α } (27) Ĥ E is unchanged, and eqns. (26c-d) remain valid with, in the most general case, F now a function of all the q s. However, in many cases of practical interest we can argue on physical grounds that the dissipation and noise associated with the different q s is uncorrelated; this is, for example, rather obviously the case when the q s are different Cartesian components of the position vector and the dissipation is of simple ohmic form (so that the frictional form is ust ηṙ; in this case, as we saw in the last lecture, we have F (q) = qc ). In such a case it is natural to break up H E, H int and H ct so that each q interacts with its own bath only 5. However, one cannot exclude situations in which two different system DOF s interact with the same bath; an obvious example would be a pair of spins in a solid, each coupled to the same bath of electrons. For simplicity, in the following I shall usually discuss the case of a single system DOF. The generalization of the fundamental result (26) to finite T is less clear-cut. It is at least plausible that for any given initial energy eigenstate n of the system and m of 5 In the case where the q i are Cartesian components of a vector, it is perhaps more natural to treat the bath of consisting of 3D harmonic oscillators with coordinates r i and the set Ĥint = r i C ir i 7

8 the environment the argument goes through as in the zero-temperature case, and that the dispersion of both S and E energies around their most probable values is small that the result would essentially be to reproduce (25) with F (q) in (26) in general a function of T. To be sure, one might worry about the fact that if k B T is a given ω i, then the oscillatorbath model implies population of oscillator states n i > 1 which do not necessarily have any counterpart in the original system. It seems likely that this and similar worries could be resolved by an appropriate reformulation, perhaps in terms of groups of energy levels rather than single modes; however, to my knowledge this has not been done, and the tendency in the literature has been to simply assume that the oscillator-bath model retains its validity at nonzero T, with the quantities F (q) (and hence in the linear case J(ω)) being possibly explicitly temperature-dependent. One final remark: Is there any case when we can be sure that the oscillator-bath model of dissipation and noise will fail? In general, this is most likely to happen when a single degree of freedom of the environment is excited with a probability 1, and this in turn is most likely when the coupling is not very small, something most likely to be the case when the system variable q is macroscopic. In fact, it has been suggested 6 that a clear counterexample to the validity of the OB model is a single electron spin interacting with a relatively small number of nuclear spins; this seems plausible, though it is not entirely clear to me that the possibility of a proper BO treatment recovering the model is excluded. I know of no case of a genuinely macroscopic system variable for which there is particular reason to suspect that the model fails, either at T = 0 or (with the above qualifications) at finite T. Once we have convinced ourselves that the oscillator-bath model described by eqns. (25) and (26) is a reasonable description of our system, we can move ahead fast. The first thing we can do is to derive a generalized Langevin equation. In the general case, while the principle is straightforward, the notation is rather messy, so I ust treat two simpler cases (a) dissipation linear but frequency-dependent. In this case we can introduce a quantity J(ω) ust as in l.20; the equation of motion of the system is M q(t) + ˆη[q(t)] + V q = F n (t) (28) where ˆη[q(t)] is the linear integrodifferential operator whose Fourier transform is η(ω) = J(ω)/ω, and the Fourier transform of the noise correlations F n (t) is in the 6 N.V. Prokofiev and P.C. Stamp, Reps. Prog. Phys. 63, 669 (2000). 8

9 classical limit F n (t)f n (t ) ω = 2k B T η(ω) (29) Evidently the results of l.20 are a special case of this general result. (b) dissipation nonlinear but frequency-independent. In this case we can define a nonlinear friction coefficient η(q) by η(q) π (m ω 2 ) 1 ( F (q)/ q) 2 δ(ω ω ) (30) 2 and a rather messy argument 7 then shows that the equation of motion is M q(t) + η(q) q + V q = F n (q, t) (31) where the correlations of the noise now depend on the system coordinate: again assuming the classical limit, F n (q, t)f n (q, t ) ω = 2k B T η(q) (32) Again, the results of l.20 are a special case. APPENDIX A: PROOF OF EQUIVALENCE OF EQNS. (24) AND (25) (1) We make in (24) a trivial exchange of the oscillators coordinates and momenta: p m ω x (A.1) Then the KE term is (1/2M)(p + m ω x K (q)) 2, and the other terms in (24) are obviously unaffected. (2) We go back from this form of Hamiltonian to an equivalent Lagrangian L = 1 2 M q2 q K (q)m ω x U(q) + L env L env 1 (m ẋ 2 m ω 2 x 2 2 ) (A.2) 7 See A.O. Caldeira and AJL, Ann. Phys. 199, 374, appendix C. 9

10 (3) We add to (25) the total time derivative L = d dt ( m ω x Λ (q)) Λ (q) q 0 K (q ) dq (A.3) an operation which has no effect at all in classical mechanics, and in QM generally 8 merely adds physically irrelevant phase to the propagators, etc. The resulting Lagrangian is L = 1 2 M q2 + m ω Λ (q)ẋ U(q) + L env (A.4) (4) We go back to the Hamiltonian formulation: with the above definition of L env H = p 2 /2M U(q) + { (p + m ω Λ (q)) 2 /2m + 1 } 2 m ω 2 x 2 (A.5) (5) Finally, we reverse the transformation (A.1) (with a sign for convenience) and define q F (q) (m ω ) 2 Λ (q) (m ω ) 2 K (q ) dq 0 (A.6) In this way we recover eqn. (25) of the text. 8 Some care is needed if the space of the system variable q is non-compact (e.g. if q is the phase difference in a Josephson device.) 10