Quantum tunnelling in a dissipative system

Caldeira, A. O.; Leggett, Anthony J.

doi:10.1016/0003-4916(83)90202-6

Cited by 3,535 publications

(2,806 citation statements)

References 47 publications

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“…They explicitly adopt the statistical interpretation of quantum mechanics. General statements are illustrated via the Caldeira-Leggett model [173,174,175,176]: a two-level system coupled to a bath of harmonic oscillators. This model undergoes a second-order phase transition with relatively weak decay of off-diagonal terms in the thermodynamic limit, provided that the coupling of the two-level system to the bath is sufficiently strong.…”

Section: Spontaneous Symmetry Breakingmentioning

confidence: 99%

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Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Spontaneous Symmetry Breakingmentioning

confidence: 99%

“…and choose forK(ω) the simplest expression having the required properties, namely the quasi-Ohmic form [174,175,197,121,122]…”

Section: Equilibrium State Of the Bathmentioning

confidence: 99%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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“…We wish to investigate dissipative dynamics in a more quantum mechanically rigorous manner. With this aim, we employ a model based on the Caldeira-Leggett (or Brownian) Hamiltonian, 1,2) H ¼p…”

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confidence: 99%

An Approach to Quantum Transport Based on Reduced Hierarchy Equations of Motion: Application to a Resonant Tunneling Diode

Sakurai

Tanimura

2013

J. Phys. Soc. Jpn.

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The quantum dissipative dynamics of a tunneling process through double barrier structures is investigated on the basis of a rigorous treatment for the first time. We employ a Caldeira-Leggett Hamiltonian with an effective potential calculated self-consistently, accounting for the electron distribution. With this Hamiltonian, we use the reduced hierarchy equations of motion in the Wigner space representation to study the effects of non-Markovian thermal fluctuations and dissipation at finite temperature in a rigorous manner. Hysteresis, double plateau-like behavior, and self-excited current oscillation are observed in a negative differential resistance (NDR) region of the current-voltage curve. We find that while most of the current oscillations decay in time in the NDR region, there is a steady oscillation characterized by a tornado-like rotation in the Wigner space in the upper plateau of the NDR region.KEYWORDS: quantum transport, Caldeira-Leggett, dissipative dynamics, resonant tunneling, non-MarkovianThe Caldeira-Leggett (or Brownian) Hamiltonian has been applied to the investigation of quantum dissipative dynamics in several fundamental contexts, including quantum tunneling, 1-3) chemical reactions, 4) SQUID rings, 5) nonlinear optical response, 6) and quantum ratchets. 7) Because a complete model of quantum dissipative dynamics must treat phenomena that can only be described in real time, a great deal of effort has been dedicated to the problem of numerically integrating equations of motion derived from the Hamiltonian that describe real-time behavior. [8][9][10] Although such equations are analogous to the classical kinetic equations, which have proved to be useful in the study of classical transport phenomena, they are difficult to derive in a quantum mechanical framework without approximations and/or assumptions.In this paper, we demonstrate that the reduced hierarchy equations of motion (HEOM) in the Wigner space representation provide a powerful method to study quantum dissipative dynamics in systems subject to non-Markovian and non-perturbative thermal fluctuations and dissipation at finite temperature. [11][12][13][14][15] As an example, we employ a model describing the thermal effects in resonant tunneling diodes (RTDs). 16,17) Although the validity of the Caldeira-Leggett Hamiltonian for describing electron transport phenomena has not yet been investigated thoroughly, because it has a firm quantum mechanical foundation, and because we feel that it is necessary to investigate the validity of the previous theoretical results in a fully quantum mechanical treatment, we believe that the present study will be helpful in constructing a general understanding of electron transport phenomena.Due to quantum effects, an RTD system exhibits novel negative differential resistance (NDR) in its current-voltage (I-V ) relation. 18) Moreover, current oscillations, 19) plateaulike behavior and hysteresis of the I-V curve have been observed in the NDR region. 20) Theoretically, Frensley found NDR in the I-V curve in...

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“…The first phonons to be excited are the long-wavelength phonons. In this regime the transition rates can be calculated by path integral techniques assuming that these phonons are coupled linearly to the hydrogen position [ 7,8]. Assuming Debye phonons and an effective tunnelling element, analytic results can be obtained.…”

Section: Introductionmentioning

confidence: 99%

Diffusion of hydrogen in transition metals

Schober

Stoneham

1991

Journal of the Less Common Metals

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The mobility of interstitial atoms is closely correlated with their vibrational behaviour. Not only can the classical (hopping) activation energy be evaluated with good accuracy from the vibrational spectrum, but also low temperature tunnelling rates are strongly related to vibrations. Because of their small mass interstitial hydrogen atoms constitute an ideal probe to study this correlation over a large temperature range (0 K < T < 300 K). Hydrogen atoms vibrate mainly with localized (optical) modes but also with strongly distorted lattice vibrations (short-wavelength acoustic modes). These vibrations lead to a "dressing" of the tunnelling element. More interesting, they determine the temperature dependence of the diffusion constant (at low temperatures together with electronic terms). Using a realistic description of the vibrations as input we calculate the diffusion constants quantitatively. We present results for hydrogen in niobium and in lutetium.

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Quantum tunnelling in a dissipative system

Cited by 3,535 publications

References 47 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

An Approach to Quantum Transport Based on Reduced Hierarchy Equations of Motion: Application to a Resonant Tunneling Diode

Diffusion of hydrogen in transition metals

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