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arXiv:cond-mat/0105243 · Physical review. B, Condensed matter

Fermi gas response to the nonadiabatic switching of an external potential

Yury Adamov, B. A. Muzykantskiǐ

Published Dec 4, 2001• 10 citations
arXivHTMLPDF

Abstract

We compute analytically the distribution function $P(E)$ for the energy E acquired by a Fermi gas after being subjected to an arbitrary time-dependent external potential (switching event). We relate the distribution function to a solution of a matrix Riemann-Hilbert problem and present explicit formulas for the low-order cumulants of $P(E).$ These general results are used to find the distribution of dissipated energy in a biased quantum point contact.

DOI
10.1103/physrevb.64.245318

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We compute analytically the distribution function P(E)P(E)P(E) for the energy E acquired by a Fermi gas after being subjected to an arbitrary time-dependent external potential (switching event). We relate the distribution function to a solution of a matrix Riemann-Hilbert problem and present explicit formulas for the low-order cumulants of P(E).P(E).P(E). These general results are used to find the distribution of dissipated energy in a biased quantum point contact.
We consider the Fermi gas in a nonequilibrium state obtained by applying an arbitrary time-dependent potential to the Fermi gas in the ground state. We present a general method that gives the quantum statistics of any single-particle quantity, such as the charge, total energy, or momentum, in this nonequilibrium state. We show that the quantum statistics may be found from the solution of a matrix Riemann-Hilbert problem. We use the method to study how the finite measuring time modifies the distribution of charge transferred through a biased quantum point contact.
We describe the Riemann-Hilbert (RH) approach to computing the long-time response of a Fermi gas to a time-dependent perturbation. The approach maps the problem onto a noncommuting RH problem. The method is nonperturbative, quite general, and can be used to compute the Fermi gas response in driven (out of equilibrium) as well as equilibrium systems. It has the appealing feature of working directly with scattering amplitudes defined at the Fermi surface rather than with the bare Hamiltonian. We illustrate the power of the method by rederiving standard results for the core-hole and open-line Green's functions for the equilibrium Fermi edge singularity (FES) problem. We then show that the case of the nonseparable potential can be solved nonperturbatively with no more effort than for the separable case. We compute the corresponding results for a biased (nonequilibrium) model tunneling device, similar to those used in single-photon detectors, in which a photon absorption process can significantly change the conductance of the barrier. For times much larger than the inverse bias across the device, the response of the Fermi gases in the two electrodes shows that the equilibrium Fermi edge singularity is smoothed, shifted in frequency, and becomes polarity dependent. These results have a simple interpretation in terms of known results for the equilibrium case but with (in general complex-valued) combinations of elements of the scattering matrix replacing the equilibrium phase shifts. We also consider the shot noise spectrum of a tunnel junction subject to a time-dependent bias and demonstrate that the calculation is essentially the same as that for the FES problem. For the case of a periodically driven device we show that the noise spectrum for coherent states of alternating current can be easily obtained using this approach.
The effect of a localized time-dependent potential on a Fermi gas is discussed. A simple bosonlike model for the particle-hole excitations is developed. Its use appears justified for the dissipative part of the response of the Fermi gas to localized time-dependent potentials. General expressions are obtained for the probability that the Fermi gas remains in the ground state and for the energy spectrum of excited states. Several illustrative examples are discussed. We consider the effect of an exponentially switched-on localized potential. It is shown that the state reached when the potential attains its full strength VVV is orthogonal to the ground state of the Fermi gas with the potential VVV, no matter how slow the switching. It consists of a spectrum of excited states of the full Hamiltonian with a width proportional to the switching rate. Other aspects of this failure of adiabaticity such as nonreversibility and path dependence are investigated. Some possible applications are pointed out. The potential source may have a finite mass. The effect of its recoil is calculated for the special case of a suddenly switched-on potential. The probability of finding the Fermi gas in its final ground state after the perturbation is switched on is found to be ${P}_{G}=\mathrm{exp}[\frac{{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}}^{2}\mathrm{ln}\ensuremath{\gamma}}{(1\ensuremath{-}{\ensuremath{\gamma}}^{2})}]$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}=V{\ensuremath{\rho}}_{\ensuremath{\epsilon}F}$ and is a measure of the coupling of the localized perturbation to the electron gas. $\ensuremath{\gamma}$ is the mass ratio mM\frac{m}{M}Mm​. The potential source is initially at rest. The above expression is critically discussed and corrections to it are calculated. They are found to be small. The spectrum of excitations produced is also calculated. In conclusion, connections are pointed out with other particle field problems exhibiting a similar infrared singularity.
When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.
Abstract We study a system of two non-interacting quantum wires with fermions of opposite chirality with a point contact junction at the origin across which tunneling can take place when an arbitrary time-dependent bias between the wires is applied. We obtain the exact dynamical non-equilibrium Green function by solving Dyson’s equation analytically. Both the space–time dependent two and four-point functions are written down in a closed form in terms of simple functions of position and time. This allows us to obtain, among other things, the I–V characteristics for an arbitrary time-dependent bias. Our method is a superior alternative to competing approaches to non-equilibrium as we are able to account for transient phenomena as well as the steady state. We study the approach to steady state by computing the time evolution of the equal-time one-particle Green function. Our method can be easily applied to the problem of a double barrier contact whose internal properties can be adjusted to induce resonant tunneling leading to a conductance maximum. We then consider the case of a finite bandwidth in the point contact and calculate the non-equilibrium transport properties which exhibit non-Markovian behaviour. When a subsequently constant bias is suddenly switched on, the current shows a transient build up before approaching its steady state value in contrast to the infinite bandwidth case. This transient property is consistent with numerical simulations of lattice systems using time-dependent density matrix renormalization group suggesting thereby that this transient build up is merely due to the presence of a short distance cutoff in the problem description and not on the other details.