PHYSICAL REVIEW A VOLUME 31, NUMBER 3 MARCH 1985 Influence-functional theory of metastability in a dissipative quantum system W. Zwerger' Department of Physics, Uniuersity of Illinois at Urbana Ch— ampaign, 1110 West Green Street, Urbana, Illinois 6I801t and Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, California 93106 (Received 30 July 1984} The problem of metastability due to quantum and thermal fluctuations in a weakly damped sys- tem is treated within Feynman s influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable po- tential it has the form of a Feynman propagator with a complex Lagrangian containing second- order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as T" +' for a spectrum of bath excitations vanishing as co . It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate pro- vides a method to measure the dissipation mechanism in the quantum regime. I. INTRODUCTION The problem of a unified description of quantum and thermal fluctuations in a metastable system has — in spite of its considerable interest in many areas — so far been treated only under the assumption that there is complete thermodynamic equilibrium in the initial configuration and that apart from maintaining this, the dissipative in- teraction with the environment does not further influence the dynamics. ' Actually this assumption fails already in the classical case both for small and for high damping as Kramers has shown long ago. It is only by the recent work of Caldeira and Leggett, however, that it has been realized that the situation for tunneling is even worse. In fact, the dissipation then does not only influence the pre- factor, but much more pronounced, it strongly decreases the exponent of the decay rate. Thus any theory which aims at a closed description of metastable decay has to in- corporate explicitly the interaction with the environment. It is the intention of this work to demonstrate that Feynman's influence-functional formalism provides a method to treat the problem in general and, although we have not been able to give a complete solution, the results obtained so far contain at least some of the essential features, which are expected in a comprehensive theory. The paper is organized as follows. In Sec. II the influence-functional theory is used to determine the exact time development of the nonequilibrium density matrix. It is shown that quite generally the problem can be re- duced to the calculation of a single path integral which al- ready indicates the way in which formally the equilibrium distribution is established for large times. In particular, for a cubic potential one obtains an expression similar to Feynman's single-particle propagator, but with an effec- tive Lagrangian which contains second-order time deriva- tives, a retarded contribution representing the effects of the fluctuating force, and an irreversible term due to the systematic influence of the bath. In Sec. III it is demon- strated that both classical activation and tunneling are contained in the nonequilibrium density matrix as dif- ferent solutions of the associated Lagrange equations. The exponent of the tunneling rate is determined to lowest order in the damping and temperature. The question of a proper definition of the decay rate is discussed in Sec. IV. Finally in Sec. V a different method is used to show that its temperature dependence may be understood in terms of "phonon-assisted" tunneling processes. Under the as- sumption of small energy transfers the problem reduces essentially to a driven oscillator, and the transition ampli- tude into a final continuum state exhibits infrared diver- gencies similar to that in a dissipative two-level system. A simple factorization approximation taking into account the emission of an arbitrary number of low-lying excita- tions then gives a result which essentially agrees with the one obtained by the influence-functional method. For an excitation spectrum vanishing as co", the corrections in the exponent of the decay rate are proportional to T" +' in the limit T +0. Thus the — low-frequency behavior of the environment density of states may be determined by measuring the decay rate at small temperatures. In the Conclusion, the results obtained are discussed and pros- pects for future work are indicated. II. INFLUENCE-FUNCTIONAL THEORY FOR THE TIME-DEPENDENT DENSITY MATRIX Let us consider a quantum-mechanical particle of mass M 1n a potential Vo(q) with one degree of freedom, which may represent, for instance, the phase difference in a current biased Josephson junction or the magnetic flux in a SQUID (superconducting quantum interference device). Any linearly behaving environment may be modeled by an equivalent set of linearly coupled harmonic oscillators with frequencies m corresponding to the possible transi- tion energies, which have to form a continuum in order to describe dissipation. Since the environment then has an infinite specific heat and also, as a purely harmonic sys- tem, temperature-independent frequencies, this approxi- 31 1745 1985 The American Physical Socj.ety