Calculation of Matrix Elements for One-Dimensional Quantum-Mechanical Problems and the Application to Anharmonic Oscillators

Harris, David O.; Engerholm, Gail G.; Gwinn, William D.

doi:10.1063/1.1696963

Cited by 636 publications

(278 citation statements)

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“…It turns out that this is conveniently performed in the discrete eigenbasis of the position operator q. This representation is the so-termed discrete variable representation (DVR) [58]. The reason for this basis transformation is that only then can the influence phase, Eq.…”

Section: The Reduced Density Matrix In the Discrete Variable Repmentioning

confidence: 99%

“…Next we perform a basis transformation to the so-called discrete variable representation (DVR) [58]. The new basis is chosen as the eigenbasis of that operator which couples the bare system to the harmonic bath.…”

Section: B Real-time Paths In the Dvr Basismentioning

confidence: 99%

“…The starting point is the generalized master equation (58). The inhomogeneities I µ (t, t 0 ) on the r.h.s.…”

Section: A Markovian Approximationmentioning

confidence: 99%

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Strong Coupling Theory for Tunneling and Vibrational Relaxation in Driven Bistable Systems

2001

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A study of the dynamics of a tunneling particle in a driven bistable potential which is moderatelyto-strongly coupled to a bath is presented. Upon restricting the system dynamics to the Hilbert space spanned by the M lowest energy eigenstates of the bare static potential, a set of coupled non-Markovian master equations for the diagonal elements of the reduced density matrix, within the discrete variale representation, is derived. The resulting dynamics is in good agreement with predictions of ab-initio real-time path integral simulations. Numerous results, analytical as well as numerical, for the quantum relaxation rate and for the asymptotic populations are presented. Our method is particularly convenient to investigate the case of shallow, time-dependent potential barriers and moderate-to-strong damping, where both a semi-classical and a Redfield-type approach are inappropriate.PACS: 82.20.Pm

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Section: The Reduced Density Matrix In the Discrete Variable Repmentioning

confidence: 99%

Section: B Real-time Paths In the Dvr Basismentioning

confidence: 99%

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Strong Coupling Theory for Tunneling and Vibrational Relaxation in Driven Bistable Systems

2001

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“…Starting from the system-plusreservoir model of a driven ratchet system bilinearly coupled to a harmonic oscillator bath, we proceed in three steps: (i) We consider the regime of temperature and driving parameters which allow the truncation of the full dynamics to the Hilbert space spanned by the M lowest bands of the periodic ratchet potential. (ii) A rotation to the so-called discrete variable representation (DVR) [12], being the eigenbasis of the discrete position operator coupling to the bath, is performed. In this basis, the isolated ratchet problem is described by a tightbinding model with a periodicity of M sites, and with further nearest-neighbor coupling.…”

mentioning

confidence: 99%

“…This requires one, however, to express the full system-plusbath Hamiltonian H t in the basis of the eigenstates of the position operator, the so-termed DVR [12,13]. The contributions H ext t and H B to H t are already diagonal in this basis, but not the isolated ratchet Hamiltonian.…”

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confidence: 99%

Quantum Ratchets with Few Bands below the Barrier

et al. 2002

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We investigate directed motion in nonadiabatically rocked ratchet systems sustaining few bands below the barrier. Upon restricting the dynamics to the lowest M bands, the total system-plus-bath Hamiltonian is mapped onto a discrete tight-binding model containing all the information both on the intrawell and interwell tunneling motion. A closed form for the current in the incoherent tunneling regime is obtained. In effective single-band ratchets, no current rectification occurs. We apply our theory to describe rectification effects in vortex quantum ratchets devices. Current reversals upon variation of the ac-field amplitude or frequency are predicted. DOI: 10.1103/PhysRevLett.89.146801 PACS numbers: 73.23.-b, 05.30.-d, 05.40.-a, 85.25.-j A ratchet, i.e., a periodic structure with broken spatial symmetry, yields the possibility to obtain a directed current in the presence of noise and unbiased nonequilibrium forces [1]. As such, ratchets belong to the class of so-termed Brownian motors [2], being devices operating far from equilibrium, and which combine noise and asymmetry to generate a particle's transport. A huge amount of experimental and theoretical work exists on ratchets ruled by classical thermal fluctuations [1,2], also due to their possible application in biological systems [3,4]. In contrast, little is known about the ratchet effect in the quantum realm. This is partly due to the challenge in the experimental realization of suitable ratchet potentials. Only recently has rectification of quantum fluctuations in triangularly shaped semiconductor heterostructures [5] and in quasi-one-dimensional Josephson junction arrays [6] been observed. Theoretically, this originates from the complexity in the investigation of the quantum ratchet dynamics. By now, the quantum ratchet effect has been tackled only for adiabatically rocked ratchets in a mostly numerical work [7], for peculiar single-band models [8], in the presence of an external force of on-off type [9], and for a weakly corrugated potential [10]. Typical quantum features, as, e.g., a current inversion upon temperature decrease, were predicted [7] and demonstrated [5]. So far, the band structure of the potential was not considered.In this Letter for the first time a microscopic theory for the interplay among tunneling, vibrational relaxation, and nonadiabatic driving in ratchet potentials sustaining few bands below the barrier (cf. Fig. 1) is presented. For these potentials the semiclassical requirement [7] of having many bands below the barrier is not met. Our treatment, mostly analytical, is based on the real-time pathintegral formalism for open quantum systems [11]. Noticeably, in the temperature and driving regime in which the dynamics is effectively restricted to the lowest band of the periodic potential, no current rectification occurs. In fact, a reduction to the lowest band of the ratchet potential retains only information about the periodicity of the original Hamiltonian, but not about its reflection properties. To take into account the vibra...

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A theoretical procedure for determining rovibrational eigenstates of van der Waals complexes

Xie

Gao

1998

Int. J. Quant. Chem.

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ABSTRACT:A theoretical procedure for determining rovibrational eigenstates of van der Waals complexes is presented. In this procedure, the self-consistent field method is used to optimize the basis sets for bending and stretching motions in the van der Waals complex; then, the configuration-interaction method is employed to produce converged results. Numerical radial basis functions are used in solving the vibrational eigenvalue problems. Comparison with other approaches for the Ar-HCl complex is made. The results show that it is possible to obtain a comparable level of accuracy by using fewer configurations.

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Calculation of Matrix Elements for One-Dimensional Quantum-Mechanical Problems and the Application to Anharmonic Oscillators

Cited by 636 publications

References 4 publications

Strong Coupling Theory for Tunneling and Vibrational Relaxation in Driven Bistable Systems

Strong Coupling Theory for Tunneling and Vibrational Relaxation in Driven Bistable Systems

Quantum Ratchets with Few Bands below the Barrier

A theoretical procedure for determining rovibrational eigenstates of van der Waals complexes

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