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IEEE Conference on Nanotechnology (IEEE NANO)

Quantum evolution of charge and phase on a capacitor in a dissipative nanoscale circuit

Wayne H. Richardson

Published Jan 24, 2004• 0 citations
Publisher

Abstract

Recently, the kinetic equation for the reduced density matrix that describes a capacitor coupled to a resistor was derived from a fundamental hamiltonian. In this work, a solution of the Markoffian Master equation that describes the capacitor is presented. A preliminary look at a solution that takes into account the nonMarkoffian terms is also included. The total solution reveals for the first time the quantum dynamical features of the RC circuit and the results are dramatic. The solution for the quasi probability distribution shows that even at zero temperature, the expectation value of the square of the phase, varies linearly with time, for times longer than the characteristic time constant of the coupled system. This yields a finite value for the diffusion constant at zero temperature. Using the formula it is now possible to answer a question that has been asked by researchers since 1905: is there a relationship between diffusion and mobility at absolute zero temperature.

DOI
10.1109/nano.2003.1231728

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Recently, the kinetic equation for the reduced density matrix that describes a capacitor coupled to a resistor was derived from a fundamental hamiltonian. In this work, a solution of the Markoffian Master equation that describes the capacitor is presented. A preliminary look at a solution that takes into account the nonMarkoffian terms is also included. The total solution reveals for the first time the quantum dynamical features of the RC circuit and the results are dramatic. The solution for the quasi probability distribution shows that even at zero temperature, the expectation value of the square of the phase, varies linearly with time, for times longer than the characteristic time constant of the coupled system. This yields a finite value for the diffusion constant at zero temperature. Using the formula it is now possible to answer a question that has been asked by researchers since 1905: is there a relationship between diffusion and mobility at absolute zero temperature.
An equation for the reduced density matrix which describes a capacitor, that is coupled to a resistor, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the resistance (or the corresponding fluctuation–dissipation relation). Unexpectedly, through the charge density–density correlation, the longitudinal dielectric function is shown to play a critical role.
An equation for the reduced density matrix which describes a free particle, that is interacting with a linearly dissipative medium, is derived using the total Hamiltonian, and without resorting to any artificial model. A Master equation is also obtained for the dissipative medium. The theory erases the notion of the reservoir. It is shown that the dynamical interaction with the medium is not completely determined by the friction constant (or the corresponding fluctuation-dissipation relation). Unexpectedly, through the density-density correlation, the longitudinal dielectric function is shown to play a critical role.
Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time ttt is connected with the reduced density at initial time t0t_0t0​ for $t_0 \leq t
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
For common condensed phase problems described by a low-dimensional system coupled to a harmonic bath, Feynman’s path integral formulation of time-dependent quantum mechanics leads to expressions for the reduced density matrix of the system where the effects of the harmonic environment enter through an influence functional that is nonlocal in time. In a recent Letter [Chem. Phys. Lett. 221, 482 (1994)], we demonstrated that the range of the nonlocal interactions is finite even at zero temperature, such that the nonlocal kernel extends over only a few time steps if the path integral is expressed in terms of accurate quasiadiabatic propagators. This feature arises from disruption of phase coherence in macroscopic environments and leads to Markovian dynamics for an augmented reduced density tensor, permitting iterative time evolution schemes. In the present paper we analyze the structure and properties of the relevant tensor propagator. Specifically, we show that the tensor multiplication scheme rigorously conserves the trace of the reduced density matrix, and that in cases of short-range nonlocality it leads to Redfield-type equations which are correct to all orders in perturbation theory and which take into account memory effects. We also argue that a simple eigenvector analysis reveals (without actual iteration) the nature of the dynamics and of the equilibrium state, and directly yields quantum reaction or relaxation rates.
Six major theories of quantum dissipative dynamics are compared: Redfield theory, the Gaussian phase space ansatz of Yan and Mukamel, the master equations of Agarwal, Caldeira-Leggett/Oppenheim-Romero-Rochin, and Louisell/Lax, and the semigroup theory of Lindblad. The time evolving density operator from each theory is transformed into a Wigner phase space distribution, and classical-quantum correspondence is investigated via comparison with the phase space distribution of the classical Fokker-Planck (FP) equation. Although the comparison is for the specific case of Markovian dynamics of the damped harmonic oscillator with no pure dephasing, certain inferences can be drawn about general systems. The following are our major conclusions: (1) The harmonic oscillator master equation derived from Redfield theory, in the limit of a classical bath, is identical to the Agarwal master equation. (2) Following Agarwal, the Agarwal master equation can be transformed to phase space, and differs from the classical FP equation only by a zero point energy in the diffusion coefficient. This analytic solution supports Gaussian solutions with the following properties: the differential equations for the first moments in p and q and all but one of the second moments (q2 and pq but not p2) are identical to the classical equations. Moreover, the distribution evolves to the thermal state of the bare quantum system at long times. (3) The Gaussian phase space ansatz of Yan and Mukamel (YM), applied to single surface oscillator dynamics, reduces to the analytical Gaussian solutions of the Agarwal phase space master equation. It follows that the YM ansatz is also a solution to the Redfield master equation. (4) The Agarwal/Redfield master equation has a structure identical to that of the master equation of Caldeira-Leggett/Oppenheim-Romero-Rochin, but the two are equivalent only in the high temperature limit. (5) The Louisell/Lax HO master equation differs from the Agarwal/Redfield form by making a rotating wave approximation (RWA), i.e., keeping terms of the form ââ†,â†â and neglecting terms of the form â†â†,ââ. When transformed into phase space, the neglect of these terms eliminates the modulation in time of the energy dissipation, modulation which is present in the classical solution. This neglect leads to a position-dependent frictional force which violates the principle of translational invariance. (6) The Agarwal/Redfield (AR) equations of motion are shown to violate the semigroup form of Lindblad required for complete positivity. Considering the triad of properties: complete positivity, translational invariance and asymptotic approach to thermal equilibrium, AR sacrifices the first while Lindblad’s form must sacrifice either the second or the third. This implies that for certain initial states Redfield theory can violate simple positivity; however, for a wide range of initial Gaussians, the solution of the AR equations does maintain simple positivity, and thus for these states appears to be distinctly more physical than the solution of the semigroup equations.