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arXiv:0901.4518 · Physical Review B

Location- and observation time-dependent quantum tunneling

V. Fleurov, R. Schilling, B. Bayani

Published Nov 3, 2008• 0 citations

Abstract

We investigate quantum tunneling in a translation invariant chain of particles. The particles interact harmonically with their nearest neighbors except for one bond, which is anharmonic. It is described by a symmetric double-well potential. In the first step, we show how the anharmonic coordinate can be separated from the normal modes. This yields a Lagrangian which has been used to study quantum dissipation. Elimination of the normal modes leads to a nonlocal action of Caldeira-Leggett type. If the anharmonic bond defect is in the bulk, one arrives at Ohmic damping, i.e., there is a transition of a delocalized bond state to a localized one if the elastic constant exceeds a critical value ${C}_{\text{crit}}$. The latter depends on the masses of the bond defect. Super-Ohmic damping occurs if the bond defect is in the site $M$ at a finite distance from one of the chain ends. If the observation time $T$ is smaller than a characteristic time ${\ensuremath{\tau}}_{M}\ensuremath{\sim}M$, depending on the location $M$ of the defect, the behavior is similar to the bulk situation. However, for $T⪢{\ensuremath{\tau}}_{M}$ tunneling is never suppressed.

DOI: 10.1103/physrevb.78.184301

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We investigate quantum tunneling in a translation invariant chain of particles. The particles interact harmonically with their nearest neighbors except for one bond, which is anharmonic. It is described by a symmetric double-well potential. In the first step, we show how the anharmonic coordinate can be separated from the normal modes. This yields a Lagrangian which has been used to study quantum dissipation. Elimination of the normal modes leads to a nonlocal action of Caldeira-Leggett type. If the anharmonic bond defect is in the bulk, one arrives at Ohmic damping, i.e., there is a transition of a delocalized bond state to a localized one if the elastic constant exceeds a critical value

{C}_{\text{crit}}

. The latter depends on the masses of the bond defect. Super-Ohmic damping occurs if the bond defect is in the site

M

at a finite distance from one of the chain ends. If the observation time

T

is smaller than a characteristic time

{\ensuremath{\tau}}_{M}\ensuremath{\sim}M

, depending on the location

M

of the defect, the behavior is similar to the bulk situation. However, for

T⪢{\ensuremath{\tau}}_{M}

tunneling is never suppressed.

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.

As a prelude to a discussion of tunneling in the presence of dissipation, we discuss the damped harmonic oscillator. We show that for a harmonic-oscillator potential, the Caldeira-Leggett Hamiltonian can be reduced to a sum of independent harmonic-oscillator Hamiltonians by a normal-mode transformation. We then discuss tunneling through a parabolic barrier and explain the apparently contradictory results in the literature. The decay of a metastable state is then discussed. We argue that much of the published work on tunneling in dissipative systems has assumed unphysical initial conditions. We calculate the tunneling out of a metastable state, on the assumption that the environment is initially at zero temperature, and find that quantum tunneling is enhanced by dissipation.

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.

Although coupling to a super-Ohmic bosonic reservoir leads only to partial dephasing on short time scales, exponential decay of coherence appears in the Markovian limit (for long times) if anharmonicity of the reservoir is taken into account. This effect not only qualitatively changes the decoherence scenario but also leads to localization processes in which superpositions of spatially separated states dephase with a rate that depends on the distance between the localized states. As an example of the latter process, we study the decay of coherence of an electron state delocalized over two semiconductor quantum dots due to anharmonicity of phonon modes.

We suggest taking the fluctuation-dissipation theorem of Callen and Welton [Phys. Rev. 83, 34 (1951)] as a basis to study quantum dissipative phenomena (such as macroscopic quantum tunneling) in a manner analogous to the Nambu-Goldstone theorem for spontaneous symmetry breakdown [Phys. Rev. 122, 345 (1961); Nuovo Cimento 19, 154 (1961)]. It is shown that the essential physical contents of the Caldeira-Leggett model [Phys. Rev. Lett. 46, 211 (1981); Ann. Phys. (N.Y.) 149, 374 (1983)] such as the suppression of quantum coherence by Ohmic dissipation are derived from general principles only, namely, the fluctuation-dissipation theorem and unitarity and causality (i.e., dispersion relations), without referring to an explicit form of the Lagrangian. An interesting connection between quantum tunneling with Ohmic dissipation and Anderson's orthogonality theorem is also noted.