Elsevier

Physics Reports

Volume 304, Issues 5–6, 1 October 1998, Pages 229-354
Physics Reports

Driven quantum tunneling

https://doi.org/10.1016/S0370-1573(98)00022-2Get rights and content

Abstract

A contemporary review on the behavior of driven tunneling in quantum systems is presented. Diverse phenomena, such as control of tunneling, higher harmonic generation, manipulation of the population dynamics and the interplay between the driven tunneling dynamics and dissipative effects are discussed. In the presence of strong driving fields or ultrafast processes, well-established approximations such as perturbation theory or the rotating wave approximation have to be abandoned. A variety of tools suitable for tackling the quantum dynamics of explicitly time-dependent Schrödinger equations are introduced. On the other hand, a real-time path integral approach to the dynamics of a tunneling particle embedded in a thermal environment turns out to be a powerful method to treat in a rigorous and systematic way the combined effects of dissipation and driving. A selection of applications taken from the fields of chemistry and physics are discussed, that relate to the control of chemical dynamics and quantum transport processes, and which all involve driven tunneling events.

Introduction

During the last few decades we could bear witness to an immense research activity, both in experimental and theoretical physics, as well as in chemistry, aimed at understanding the detailed dynamics of quantum systems that are exposed to strong time-dependent external fields. The quantum mechanics of explicitly time-dependent Hamiltonians generates a variety of novel phenomena that are not accessible within ordinary stationary quantum mechanics. In particular, the development of laser and maser systems opened the doorway for creation of novel effects in nonlinear quantum systems which interact with strong electromagnetic fields 1, 2, 3, 4, 5, 6, 7. For example, an atom exposed indefinitely to an oscillating field eventually ionizes, whatever the values of the (angular) frequency and the intensity of the field. The rate at which the atom ionizes depends on both, the driving frequency and the intensity. Interestingly enough, in a pioneering paper by H. R. Reiss in 1970 [8], the seemingly paradoxical result was established that extremely strong field intensities lead to smaller transition probabilities than more modest intensities, i.e., one observes a declining yield with increasing intensity. This phenomenon of stabilization that is typical for above threshold ionization (ATI) is still actively discussed, both in experimental and theoretical groups 9, 10. Other activities that are in the limelight of current topical research relate to the active control of quantum processes; e.g. the selective control of reaction yields of products in chemical reactions by use of a sequence of properly designed coherent light pulses 11, 12, 13.

Our prime concern here will focus on the tunneling dynamics of time-dependently driven nonlinear quantum systems. Such systems exhibit an interplay of three characteristic components, (i) nonlinearity, (ii) nonequilibrium behavior (as a result of the time-dependent driving), and (iii) quantum tunneling, with the latter providing a paradigm for quantum coherence phenomena.

By now, the physics of driven quantum tunneling has generated widespread interest in many scientific communities 1, 2, 3, 4, 5, 6, 7 and, moreover, gave rise to a variety of novel phenomena and effects. As such, the field of driven tunneling has nucleated into a whole new discipline.

Historically, first precursors of driven barrier tunneling date back to the experimentally observed photon-assisted-tunneling (PAT) events in 1962 in the Al–Al2O3–In superconductor–insulator–superconductor hybrid structure by Dayem and Martin [14]. A clear-cut, simple theoretical explanation for the step-like structure in the averaged voltage–tunneling current characteristics was put forward soon afterwards by Tien and Gordon [15] in 1963, who introduced the physics of driving-induced sidechannels for tunneling across a uniformly, periodically modulated barrier. The phenomenon that the quantum transmission can be quenched in arrays of periodically arranged barriers, e.g. semiconductor superlattices, leading to such effects as dynamic localization or absolute negative conductance have theoretically been described over twenty years ago 16, 17, 18; but these have been verified experimentally only recently 19, 20, 21, 22.

The role of time-dependent driving on the coherent tunneling between two locally stable wells [23] has only recently been elaborated [24]. As an intriguing result one finds that an appropriately designed coherent cw-drive can bring coherent tunneling to an almost complete standstill, now known as coherent destruction of tunneling (CDT) 24, 25, 26. This driving induced phenomenon in turn yields several other new quantum effects such as low frequency radiation and/or intense, nonperturbative even harmonic generation in symmetric metastable systems that possess an inversion symmetry 27, 28, 29.

We shall approach this complexity of driven quantum tunneling with a sequence of sections. In the first half of seven sections we elucidate the physics of various novel tunneling phenomena in quantum tunneling systems that are exposed to strong time-varying fields. These systems are described by an explicitly time-dependent Hamiltonian. Thus, solving the time-dependent Schrödinger equation necessitates the development of novel analytic and computational schemes which account for the breaking of time translation invariance of the quantum dynamics in a nonperturbative manner. Beginning with Section 8we elaborate on the effect of weak, or even strong dissipation, on the coherent tunneling dynamics of driven systems. This extension of quantum dissipation 30, 31, 32, 33, 34, 35 to driven quantum systems constitutes a nontrivial task: Now, the bath modes couple resonantly to differences of quasienergies rather than to unperturbed energy differences. The influence of quantum dissipation to driven tunneling is developed theoretically in 8 Driven dissipative tunneling, 9 Floquet–Markov approach for weak dissipation, 10 Real-time path integral approach to driven tunneling, 11 The driven dissipative two-state system (general theory), and applied to various phenomena in the remaining 12 The driven dissipative two-state system (applications), 13 The driven dissipative periodic tight-binding system, 14 Dissipative tunneling in a driven double-well potential.

The authors made an attempt to comprise in this review many, although necessarily not all important developments and applications of driven tunneling. In doing so, this review became rather comprehensive.

As an inevitable consequence, the authors realize that not all readers will wish to digest the present review in its entirety. We trust, however, that a reader is able to choose from the many methods and applications covered in the numerous sections which he is interested in.

There is the consistent underlying theme of driven quantum tunneling that runs through all sections, but nevertheless, each section can be considered to some extent as self-contained. In this spirit, we hope that the readers will be able to enjoy reading from the selected fascinating developments that characterize driven tunneling, and moreover will become invigorated doing own research in this field.

Section snippets

Floquet theory

In presence of intense fields interacting with the system it is well known 37, 38, 39 that the semiclassical theory (i.e., treating the field as a classical field) provides results that are equivalent to those obtained from a fully quantized theory, whenever fluctuations in the photon number (which, for example, are of importance for spontaneous radiation processes) can safely be neglected. We shall be interested primarily in the investigation of quantum systems with their Hamiltonian being a

Two-state approximation to driven tunneling

In this section we shall investigate the dynamics of driven two-level-systems (TLS), i.e., of quantum systems whose Hilbert space can be effectively restricted to a two-dimensional space.

The most natural example is that of a particle of total angular momentum J=ℏ/2, as for example a silver atom in the ground state. The magnetic moment of the particle is μ=12ℏγσ, where γ is the gyromagnetic ratio and σ=(σx,σy,σz) are the Pauli spin matrices.1

Driven tight-binding models

In the previous section we have considered driven two-state systems. Here we shall consider the driven, infinite tight-binding (TB) system. Similar to the case of a two-state system, the infinite TB system can result from the reduction of the tunneling dynamics of a particle moving in an infinite periodic potential V0(q)=V0(q+d) within its lowest energy band. To be definite, if one fixes the potential depth such that the Wannier states |n building up the lowest band have appreciable overlap

Driven quantum wells

With the recent advent of powerful radiation sources, which are able to deliver extremely strong and coherent electric driving fields, the quantum transport in solid state devices, such as in superconductors and semiconductor heterojunctions, has been revolutionalized. The latter, by now, can even be nanostructured. Of particular interest is the role of photon-assisted tunneling (PAT) in single, double-barrier and periodic superlattice-like structures. The mechanism of PAT has originally been

Tunneling in driven bistable systems

In this section we address the physics of coherent transport in bistable systems. These systems are abundant in the chemical and physical sciences. On a quantum mechanical level of description bistable, or double-well potentials, are associated with a paradigmatic coherence effect, namely quantum tunneling. Here we shall investigate the influence of a spatially homogeneous monochromatic driving on the quantal dynamics in a symmetric, quartic double well. This archetype system is particularly

Sundry topics

The physics of driven tunneling, as demonstrated with the foregoing sections, exhibits a rich structure and carries a potential for various prominent applications. In the following, we shall report on some special applications that provide the seed for interesting new physics. Our selection is by no means complete but has been determined mainly by our knowledge and prejudices only. For example, we do not discuss here in detail the role of nonadiabatic CDT that controls the experimentally

Driven dissipative tunneling

Up to this point in our review we have investigated the effects of driving on a quantum system with few degrees of freedom, under the hypothesis that this system could be considered as isolated from its surroundings. This idealization often fails to describe thermal and dynamical properties of real physical or chemical systems when the quantum system is in mutual contact with a thermal reservoir. In this case, the system has to be considered as an open system. The coupling with a heat bath

Floquet–Markov approach for weak dissipation

In this section we shall describe the Floquet–Markov–Born approximation to the dynamics, which can be applied to strongly nonlinear systems under the assumption of weak coupling between the system and the reservoir, and whenever the external driving field E(t) is periodic, i.e., if E(t+T)=E(t),T=2π.This method combines the Markov–Born approach to quantum dissipation (leading to a master equation for the reduced density operator) with the Floquet formalism, cf. Section 2, that allows to treat

The influence-functional method

A very useful tool to describe non-equilibrium time-dependent dissipative phenomena is the real-time path integral approach. Within the simple phenomenological model described by the Hamiltonian in , , it enables to evaluate the trace operation involved in the definition of the reduced density matrix ρ(t)=TrB{W(t)} of the relevant system exactly. Here, we shall assume the factorized intial condition (232) for the total density matrix W(t), and refer to 34, 187 for the case of more general

The driven spin-boson model

In this section we shall investigate the driven dynamics of a dissipative two-level system (TLS). As discussed in Section 3, the driven TLS can model the physics of intrinsic two-level systems, such as spin 12 particles in external magnetic fields or, more generally, the motion of a quantum particle at low temperatures in an effective double-well potential in the case that only the lowest energy doublet is occupied. In this latter case, the dissipative TLS can describe, for example, hydrogen

Tunneling under ac-modulation of the bias energy

To make quantitative predictions the detailed form of the driving field has to be specified. In this section we shall study the effects of a driving field which periodically modulates the bias energy of the undriven TLS. To be definite, we consider a time-dependent TLS of the form HTLS(t)=−12ℏ{Δ0σx+[ε0+ε̂cos(Ωt)]σz}.

In the following, we shall focus our attention to the investigation of the position expectation value P(t)≔σzt which is the relevant quantity in the context of control of tunneling.

The driven dissipative periodic tight-binding system

In this section we consider the effect of dissipation on the driven infinite tight-binding (TB) model introduced in Section 4. The dissipative TB system can serve as an idealized model for the diffusion of a quantum particle on a surface of a single-crystal, or among interstitials inside a crystal [280]. It can also be invoked to investigate quantum effects in the current–voltage characteristic of a small Josephson junction [281] or of superlattices driven by strong dc- and ac-fields 18, 19, 20

Dissipative tunneling in a driven double-well potential

In Section 6the nondissipative tunneling motion of a particle moving in a driven double well, cf. , , has been discussed, while in 11 The driven dissipative two-state system (general theory), 12 The driven dissipative two-state system (applications)we investigated the driven and dissipative dynamics restricting our attention only to the lowest doublet of the potential, i.e., in the two-level-system (TLS) approximation. Here we shall investigate the dissipative motion within the full double-well

Conclusions

With this long review we have attempted to present a “tour of horizon” of the physics that is governed by driven quantum tunneling events. With the present day availability of strong radiation sources such as e.g. free-electron laser systems yielding strong and coherent electric driving fields, together with the advances in micro- and nano-technology, the challenges involving driven quantum transport have moved into the limelight of present day scientific activities. In doing so, several novel

Acknowledgements

M. Grifoni and P. Hänggi greatly acknowledge the support of the Deutsche Forschungsgemeinschaft via the Schwerpunktprogramm “Zeitabhängige Phänomene und Methoden in Quantensystemen der Physik und Chemie” (HA 1517/14-2). We also thank L. Hartmann for preparing some figures for this review. Moreover, we thank L. Hartmann, M. Holthaus, N. Makri, M. Thorwart, R. Utermann, M. Wagner and M. Winterstetter for providing original figures relating to their works. In the course of writing this review we

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