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Journal of Physics C Solid State Physics

Macroscopic quantum coherence and tunnelling in a double well potential

David Waxman

Published May 30, 1985• 17 citations
Publisher

Abstract

The quantum tunnelling and coherence of a particle moving in a biased or unbiased double well potential and subject to ohmic dissipation is considered. The complete time dependence of the reduced density matrix is determined for the case where the dimensionless friction constant alpha <<1.
DOI
10.1088/0022-3719/18/15/004

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We study quantum tunneling in a biased double-well potential in the presence of Ohmic dissipation with a friction coefficient $\ensuremath{\alpha}>1$. The tunneling rate out of the metastable well is calculated in real time as a function of the temperature TTT and bias energy $\ensuremath{\epsilon}$. The T=0T=0T=0 result agrees with a droplet-model calculation in imaginary time. At low temperatures we find an enhancement to the T=0T=0T=0 rate varying as ${(\frac{T}{\ensuremath{\epsilon}})}^{2}$.
This chapter describes the analysis of the effect of dissipation on quantum properties of double-well systems. The simple qualitative observation suggests that interaction with a dissipative bath can significantly alter physical properties of a quantum mechanical particle in a double-well potential. The chapter provides the description of real-time dynamics of the double-well systems in the presence of linear Ohmic dissipation. It shows that dissipation yields total loss of coherence and, hence, real-time dynamics of the system remains incoherent at all times. Drastic simplifications occur if one assumes that the average time the system remains in a state described by the diagonal density matrix strongly exceeds the time it spends in the off-diagonal state. The collective charge variable is canonically conjugate to that of the phase, thus also describing quantum states of a macroscopically large number of electron degrees of freedom.
The real-time dynamics of a quantum-mechanical particle in a dissipative double-well potential at finite temperatures is solved ab initio by means of a multisite spin-hopping analysis. Both the macroscopic-quantum-coherence and macroscopic-quantum-tunneling regimes are covered. The theory gives a unified picture of tunneling transport and vibrational relaxation. The analysis is based on the Ohmic-model Hamiltonian [A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983); R. Zwanzig, J. Stat. Phys. 9, 215 (1973)] or the equivalent ``particle-on-a-string'' model [H. Dekker, Phys. Rev. A 31, 1067 (1985)]. The reduced quantum Liouville equation for the particle is derived and written in the energy representation. The usual truncation to the ground-state doublet of the vibrational ladder is not made. The interdoublet vibrational dynamics considerably simplifies upon coarse-graining and in the harmonic-well approximation. The intradoublet pseudospin dynamics is treated in the ``noninteracting-blip approximation'' [A. J. Leggett et al., Rev. Mod. Phys. 59, 1 (1987)]. The coarse-grained vibrational processes give rise to a Markovian one-step spin hopping between adjacent doublets (or sites). Inter alia ``inhomogeneous broadening'' and ``motional narrowing'' [P. E. Parris and R. Silbey, J. Chem. Phys. 83, 5619 (1985)] are discussed as special cases.
A quantum mechanical particle which moves in a symmetric double-well potential, and whose interaction with the environment is described in the classical regime by a phenomenological friction coefficient $\ensuremath{\eta}$, is considered. It is shown that, provided $\frac{\ensuremath{\eta}{{q}_{0}}^{2}}{\ensuremath{\hbar}}$ exceeds a critical value of order unity ($\ifmmode\pm\else\textpm\fi{}{q}_{0}$ are the locations of the potential minima), the mean rate of tunneling between the degenerate minima decreases with temperature, leading at T=0T=0T=0 to spontaneous symmetry breaking.
The macroscopic quantum tunneling in a double-well potential with dissipation is formulated without referring to instantons. This formulation, in the domain of small dissipation, goes beyond the instanton approximation which corresponds to the two-state truncated system (or, the so-called spin-boson Hamiltonian). It is first confirmed that our formulation, when applied to the two-state truncated system, reproduces all the known results of the instanton approximation. It is then shown that the ground state is mixed with excited states by dissipative interactions and this effect tends to enhance the ground-state quantum tunneling. In the Ohmic dissipation case this enhancement factor, even for a shallow double-well potential, can be at most comparable to the suppression factor which has been known from the work by Caldeira and Leggett. In the super-Ohmic case, however, the enhancement factor generally exceeds the suppression factor and the two-state approximation fails. A salient feature in the choice of counter terms in macroscopic quantum tunneling, which is essential to the above conclusion, is emphasized.
Abstract The quantum dissipative dynamics of a particle which tunnels through the barrier of a double well potential and is coupled to an Ohmic heat bath is studied. Exact formal expressions for correlation functions are presented and examined. Attention is focused on the region in which coherence effects are important, and on the algebraic long‐time tails at zero temperature. Coherence effects in the dissipative quantum transport of a particle in a tight‐binding lattice are also investigated.