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arXiv:cond-mat/0405475 · Physical Review B

Fermi gas response to time-dependent perturbations

N. d’Ambrumenil, B. A. Muzykantskiǐ

Published Jan 20, 2005• 38 citations
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Abstract

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.

DOI
10.1103/physrevb.71.045326

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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.
We formulate the problem of the Fermi edge singularity in non-equilibrium states of a Fermi gas as a matrix Riemann–Hilbert problem with an integrable kernel. This formulation is the most suitable for studying the singular behavior at each edge of non-equilibrium Fermi states by means of the method of steepest descent, and also reveals the integrable structure of the problem. We supplement this result by extending the familiar approach to the problem of the Fermi edge singularity via the bosonic representation of the electronic operators to non-equilibrium settings. It provides a compact way to extract the leading asymptotes.
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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.
Bosonization technique for one-dimensional fermions out of equilibrium is developed in the framework of the Keldysh action formalism. We first demonstrate how this approach is implemented for free fermions and for the problem of nonequilibrium Fermi edge singularity. We then employ the technique to study an interacting quantum wire attached to two electrodes with arbitrary energy distributions. The nonequilibrium electron Green's functions, which can be measured via tunneling spectroscopy technique and carry the information about energy distribution, zero-bias anomaly, and dephasing, are expressed in terms of functional determinants of single-particle ``counting'' operators. The corresponding time-dependent scattering phase is found to be intrinsically related to ``fractionalization'' of electron-hole excitations in the tunneling process and at boundaries with leads. Results are generalized to the case of spinful particles as well to Green's functions at different spatial points (relevant to the problem of dephasing in Luttinger liquid interferometers). For double-step distributions, the dephasing rates are oscillatory functions of the interaction strength.
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.