量子耗散系统中的隧穿效应

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"这篇文章是关于量子隧穿在耗散系统中的研究，由A.0.CALDEIRA和A.J.LEGGETT合作完成，分别来自巴西坎皮纳斯州立大学和英国苏塞克斯大学。文章探讨了耗散对量子隧穿的影响，特别是在宏观变量如SQUID中的磁通陷阱中的应用，并指出这一问题对于验证量子力学在宏观层面的适用性至关重要。" 量子隧穿是量子力学中的一种现象，允许粒子穿越经典力学中看似不可逾越的能量势垒。在理想无耗散系统中，粒子的能量低于势垒顶部，但仍然有非零概率隧穿至势垒的另一侧。然而，当考虑实际系统时，耗散效应（例如热噪声或其他环境干扰）会介入，使得问题变得复杂。作者A.0.CALDEIRA和A.J.LEGGETT在这篇1983年的《物理年鉴》论文中，尝试解答“耗散如何影响量子隧穿”的问题。他们特别关注了隧穿宏观变量，比如超导量子干涉仪（SQUID）中的磁通量子，这是一个能够展示宏观量子现象的设备。在SQUID中，磁通可以被束缚在一个近乎稳定的势阱中，其运动受到环境的摩擦力影响。在经典力学中，摩擦通常会减慢或阻止物体的运动，但在量子世界，摩擦的存在可能会对隧穿过程产生显著影响。论文中，作者引入了近似于经典运动方程的量子版本，该方程描述了粒子在势能V(q)附近受摩擦系数γ影响的运动。这个方程揭示了在考虑耗散情况下，量子隧穿的动力学行为。通过数学分析和可能的数值模拟，作者们旨在揭示耗散对隧穿概率和动力学路径的具体影响。这种分析不仅对于理解和预测SQUID等宏观量子系统的实验结果至关重要，还对于理解微小尺度上的其他隧穿过程，如分子马达或量子点中的电子隧穿，同样具有理论价值。这篇论文深入探讨了量子力学与经典物理在有耗散环境中的交汇，为理解宏观量子效应提供了理论基础，并对测试量子力学在宏观层次的适用性提出了新的视角。耗散对量子隧穿的影响是量子物理学和统计力学交叉领域的一个关键问题，其研究成果对量子计算、量子信息处理和超导技术等领域都有深远的影响。

DISSIPATION AND QUANTUM TUNNELLING

387

where B, is the WKB integral

B, E 2

% (2MV(q))“’ dq

(2.4)

and Co is a dimensionless constant of order unity which depends on the shape of the

potential V(q) and can be calculated by any of a number of standard methods.

Corrections to Eq. (2.3) are at most of relative order hw,/V,, and in almost all

situations of practical interest in the present context it is the exponential factor which

is likely to dominate the tunnelling behaviour, particularly when we are interested in

the dependence of tunnelling rate on some external control parameter (such as the

externally imposed flux in a SQUID). A case of particular interest is that of a

“quadratic-plus-cubic” potential, which is likely to describe many of the systems of

interest to us in the regime where tunnelling is appreciable (see Introduction)

(2.5)

where q. = #lw$j3 is the coordinate of the “exit point.” For such a potential it turns

out [ 71 that

B =36v,

5 wo'

Co = (60)‘:‘.

(2.6)

We now wish to enquire how the presence of dissipation affects the probability of

tunnelling. It is necessary to formulate the precise question we wish to answer rather

carefully. If we were treating the motion of the system classically using Eq. (2.1) we

would of course produce an equation of motion of the form

(2.7)

where for generality we have added a time-dependent “external” potential -qF,,,(t) to

the Lagrangian (2.1). Equation (2.7) is, of course, by Ehrenfest’s theorem also true as

an operator equation in quantum mechanics; however, in general it is not very infor-

mative, since when we take its expectation value (indicated by pointed brackets) we

cannot identify (aV/aq) with (i?V/aq)4=C9j.

Nevertheless, because of our assumptions

that the potential V(q) is reasonably smooth and that the inequality V, 3 ho, is

satisfied, we may reasonably infer that this identification is indeed approximately

valid; therefore a quantum-mechanical system, whether trapped in the metastable well

or with sufficient energy to surmount the barrier, will have an expectation value

(q(t)) of its coordinate which approximately satisfies the classical equation of motion

(2.7). (Naturally, effects which vanish in the limit A + 0, such as tunnelling itself, are

neglected in this approximation.) The (possibly hypothetical) system which behaves

388

CALDEIRA AND LEGGETT

in this way will form, as it were, the reference system with which our real-life

(damped) system will be compared.

The real-life system, then, is assumed to obey in the region in question not

Eq. (2.7) but a dumped quasiclassical equation of motion of the form

M4’ + ~4 + (dV/dq) = F,,,(t).

(2.8)

(Note that F,,,(t) is a “true” external force, i.e., such that the work done per unit time

is simply @‘,,,(t) and the power dissipated is correspondingly ~4’. In the more

general case q = q(q) this condition needs to be explicitly specified; see Appendix C.)

More precisely, the expectation value of the coordinate operator is assumed to satisfy

(2.8) (with aV/aq interpreted as (aV/8q)4z(,,, as above). That the system does indeed

obey Eq. (2.8) and the value of the (q-independent) friction coefficient q, is

something which in any particular case must be inferred from experiment; for most

cases of practical interest there are standard ways of doing this. In writing (2.8) we

have made the simplest possible assumption, namely, that the frictional force is

simply proportional to velocity, and we shall carry this assumption through the bulk

of this paper; however, the generalization to nonlinear dissipation (given by a

damping term of the form v(q)Q) is very straightforward and is given at appropriate

points in the text. The generalization to the case where the dissipation involves higher

time derivatives is also mentioned at appropriate points.

Thus, the object of this paper is to compare the tunnelling rate of a system

described (under appropriate conditions) by the damped quasi-classical equation of

motion (2.8) with that of the reference system described by the undamped Eq. (2.7).

It cannot be too strongly stressed that in making the comparison the potential V(q) is

assumed to be rhe same in the two cases (cf. below). This is our definition of the

meaning of the question “What is the effect of dissipation on quantum tunnelling?”

We suspect that the recent controversy about the sign of the effect (see Introduction)

may have been at least partly due to the lack of such a generally agreed definition. It

is appropriate, therefore, to digress for a moment at this stage to see why there is a

problem.

In the next section we are going to introduce dissipation into the behaviour of our

system by coupling it to a sufficiently complex environment. Now, if we start with an

isolated system and introduce terms in the Lagrangian (or Hamiltonian) which

couple it to its environment, then in general we will produce a mechanism of

dissipation and also other effects. For example, in the simple case in which both

system and environment are simple harmonic oscillators, most forms of coupling

(and in particular a coordinate-coordinate coupling) will shift the natural oscillation

frequency of the system; in quantum-mechanical terms this is just the familiar level-

repulsion effect. More generally, we shall find (cf. Section 4) that such coupling tends

among other things to renormalize the original potential V(q); in particular, if the

coupling is linear in the system coordinate (the case of primary interest in this paper)

V(q) acquires an extra (negative) term fMAco2q2, where the negative quantity Au*

plays the role of a correction to the squared small-oscillation frequency IX:. Such

DISSIPATION AND QUANTUM TUNNELLING

389

renormalization effects can be very large, and if ldw* / > 0: can even render the

original metastable potential minimum unstable; in some microscopic tunnelling

systems they can play a very important role (cf. [29]). We refer to these effects as

“frequency-renormalization”

effects (in a more general context “potential renor-

malisation” might be a better term).

Now, whether the frequency-renormalization phenomenon does indeed occur as a

real physical effect depends crucially on the nature of the physical system considered.

This question has of course really nothing to do with tunnelling as such, and to

emphasize this we shall illustrate the point with two systems which both perform

small oscillations around a stable equilibrium position. Consider first an example

which is not strictly macroscopic but is well suited to our purpose, the interaction of

a collective degree of freedom in a nucleus with the single-particle modes. The

simplest description of such an interaction is by a term of the general form

CjFj(S) h

x,, w ere xj is some coordinate associated with the single-particle mode j.

Clearly this type of coupling will not only provide a dissipation mechanism for the

“system” but will also shift its natural frequency; in particular the effect of those

parts of the “environment” (most of it) which have natural frequencies large

compared to wO may be taken into account by supposing that they adjust

adiabatically to the system and thereby lower the effective restoring force, i.e., the

quadratic term in the potential energy. Note that in general there is no simple

relationship between this “frequency renormalisation” and the damping constant,

although of course there are the usual Kramers-Kronig relations between the real and

imaginary parts of the complete frequency-dependent system response. In this case

the frequency-renormalization effect has a real physical significance.

Now consider a simple LC-circuit; to avoid irrelevant problems we assume that it

is constructed entirely of superconducting materials and is at zero temperature. It will

execute small undamped harmonic oscillations of the current (or flux) with frequency

wg = (LC) “*. Suppose now that we connect in parallel with the capacitor a piece of

wire made of normal metal of finite conductivity. In elementary electrical engineering

terms, we would naturally describe the effect of this element by assigning to it a

phenomenological resistance R; we might, depending on the details of the geometry,

also have to assign to it its own inductance and/or capacitance, but these are

irrelevant to the point at issue and we shall assume they are zero. (More generally,

we might have to describe the wire by a complex frequency-dependent impedance

Z(w); our assumption is that Z(w) is real and equal to a constant R for a range of

frequencies o, large compared to o,,.

The ensuing statements are true to lowest

nontrivial order in w~/w~,

cf. Appendix A.) Then the standard elementary circuit

calculation shows that the equation of motion of the circuit (where for later

convenience we take as our basic variable the magnetic flux @ passing through it) is

modified to

Csii $ d/R + Q/L = 0.

(2.9)

Hence the effect of incorporating the resistive element in the circuit is to give the

resonance frequency an imaginary part and also, in the case of under-critical

390

CALDEIRA AND LEGGETT

damping, to shift its real part (downwards). This shift, however, should be carefully

distinguished from the one occurring in the above case of the nucleus; unlike the

latter, it is entirely determined by the phenomenological dissipative coefficient (here

R) and does not require any modification of the “potential” term in the Lagrangian

(here @*/2L). Th us, on elementary physical grounds we do not in this case expect the

coupling of the circuit to the dissipative element to result in a “frequency-

renormalization effect” as we have defined the term. It follows, of course, that in

choosing an effective Lagrangian to describe the coupling of the “system” (circuit) to

the “environment” (resistor) we must take care to ensure this result. Actually, it turns

out that a sufficiently careful analysis of the standard Lagrangian technique as

applied to electromagnetic interactions will automatically guarantee this, and we

carry out such an analysis in Appendix A.

Another case in which we do not, on physical grounds, expect a frequency-

renormalization effect to occur is that of adiabatic coupling. That is, if we start from

the dynamics of a system interacting strongly with its environment but described by

the zeroth-order adiabatic approximation (which allows no dissipation), and then add

as the dissipation-producing interaction the terms omitted in this approximation, then

we would expect that these terms do not lead to a frequency-renormalization effect,

since any such effect should already have been taken into account in the zeroth-order

approximation. We shall demonstrate explicitly in Appendix C that this expectation is

correct (to lowest nontrivial order in the departure from adiabaticity).

We see, therefore, that the question “How is quantum tunnelling affected by

dissipation?” is not necessarily equivalent to the question “How is quantum

tunnelling affected by the interaction with the environment which produces the

dissipation?” In considering the latter question it may, depending on the physical

nature of the system considered, be necessary to take into account frequency-

renormalization effects. Since it is the first question we wish to consider in this paper,

it is convenient to be able to treat all cases in a unified way, irrespective of whether

or not they show physical frequency-renormalization effects (which in many

macroscopic cases of practical interest are likely to be experimentally unobservable

even if they occur). The obvious way to do this is to treat the V(q) which appears in

Eq. (2.1) not as the original “bare” potential V,(q) seen by the isolated system, but as

the renormalized potential, that is the quantity V,(q) - $I4 ldto2/ q2 (or the

appropriate nonlinear generalization, see Appendix C). In cases where there is no

physical frequency-renormalization effect,

then obviously V(q) = V,(q). Conse-

quently, our restrictions concerning smoothness, barrier height, etc. should always be

understood as referring to the renormalized potential.

To sum up, we wish to compare the tunnelling characteristics of a system whose

quasiclassical dynamics is given by the damped equation of motion (2.8) with those

of a reference system described by (2.7), for the same porential function V(q), (and,

of course, the same mass) irrespective of whether or not the potential seen by the real

(dissipative) system contains a contribution from frequency-renormalization effects.

There is, however, one respect in which our formulation of the problem is still

ambiguous. What, precisely, do we mean by the statement that Eq. (2.8) holds “under

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