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arXiv:1705.02325 · Physical review. B./Physical review. B

Analyzing the spectral density of a perturbed analog quantum simulator using the Keldysh formalism

Sebastian Zanker, Iris Schwenk, Jan-Michael Reiner, Juha Leppäkangas, Michael Marthaler

Published Jun 4, 2018• 3 citations
arXivHTMLPDF

Abstract

Simulation of interacting electron systems is one of the great challenges of modern quantum chemistry and solid state physics. Controllable quantum systems offer the opportunity to create artificial structures which mimic the system of interest. An interesting quantity to extract from these quantum simulations is the spectral function. We map a noisy quantum simulator onto a fermionic system and investigate the influence of decoherence on the simulation of the spectral density using a diagrammatic approach on the Keldysh contour. We show that features stronger than the single-qubit decoherence rate can be resolved, while weaker features wash out. For small systems, we compare our Keldysh approach to master-equation calculations.

DOI
10.1103/physrevb.97.214301

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Simulation of interacting electron systems is one of the great challenges of modern quantum chemistry and solid state physics. Controllable quantum systems offer the opportunity to create artificial structures which mimic the system of interest. An interesting quantity to extract from these quantum simulations is the spectral function. We map a noisy quantum simulator onto a fermionic system and investigate the influence of decoherence on the simulation of the spectral density using a diagrammatic approach on the Keldysh contour. We show that features stronger than the single-qubit decoherence rate can be resolved, while weaker features wash out. For small systems, we compare our Keldysh approach to master-equation calculations.
We investigate the possibility to assist the numerically ill-posed calculation of spectral properties of interacting quantum systems in thermal equilibrium by extending the imaginary-time simulation to a finite Schwinger-Keldysh contour. The effect of this extension is tested within the standard maximum entropy approach to analytic continuation. We find that the inclusion of real-time data improves the resolution of structures at high energy, while the imaginary-time data are needed to correctly reproduce low-frequency features such as quasiparticle peaks. As a nonequilibrium application, we consider the calculation of time-dependent spectral functions from retarded Green function data on a finite time interval, and compare the maximum entropy approach to direct Fourier transformation and a method based on Padé approximants.
We study an analog quantum simulator coupled to a reservoir with a known spectral density. The reservoir perturbs the quantum simulation by causing decoherence. The simulator is used to measure an operator average, which cannot be calculated using any classical means. Since we cannot predict the result, it is difficult to estimate the effect of the environment. Especially, it is difficult to resolve whether the perturbation is small or if the actual result of the simulation is in fact very different from the ideal system we intend to study. Here, we show that in specific systems a measurement of additional correlators can be used to verify the reliability of the quantum simulation. The procedure only requires additional measurements on the quantum simulator itself. We demonstrate the method theoretically in the case of a single spin connected to a bosonic environment.
We investigate the spectral properties of polaronic excitations within the framework of an analog quantum simulator based on inductively coupled superconducting transmon qubits and microwave resonators. This system emulates a lattice model that describes a nonlocal coupling of an itinerant spinless-fermion excitation to dispersionless (Einstein-type) phonons through the Peierls and breathing-mode interaction mechanisms. The model is characterized by a sharp, level-crossing transition at a critical value of the effective excitation-phonon coupling strength; above the transition point, the ground state of this model corresponds to a heavily dressed (small-polaron) excitation. Using the kernel-polynomial method, we evaluate the momentum-frequency resolved spectral function of this system for a broad range of parameters. In particular, we underscore the ramifications of the fact that the zero-quasimomentum Bloch state of a bare excitation represents the exact eigenstate of the Hamiltonian of this system for an arbitrary excitation-phonon coupling strength. We also show that---based on the numerically evaluated spectral function and its well-known relation with the survival probability of the initial, bare-excitation Bloch state (the Loschmidt echo)---one can make predictions about the system dynamics following an excitation-phonon interaction quench. To make contact with anticipated experimental realizations, we utilize a previously proposed method for extracting dynamical-response functions in systems with local (single-qubit) addressability using the multiqubit (many-body) version of the Ramsey interference protocol.
We consider quantum decoherence in solid-state systems by studying the transverse dynamics of a single qubit interacting with a fermionic bath and driven by external pulses. Our interest is in investigating the extent to which the lost coherence can be restored by the application of external pulses to the qubit. We show that the qubit evolution under various pulse sequences can be mapped onto Keldysh path integrals. This approach allows a simple diagrammatic treatment of different bath excitation processes contributing to qubit decoherence. We apply this theory to the evolution of the qubit coupled to the Andreev-fluctuator bath in the context of widely studied superconducting qubits. We show that charge fluctuations within the Andreev-fluctuator model lead to a 1/f1/f1/f noise spectrum with a characteristic temperature depedence. We discuss the strategy for suppression of decoherence by the application of higher-order (beyond spin echo) pulse sequences.
The fff-electron spectral function of the Falicov-Kimball model is calculated via a Keldysh-based many-body formalism originally developed by Brandt and Urbanek. We provide results for both the Bethe lattice and the hypercubic lattice at half filling. Since the numerical computations are quite sensitive to the discretization along the Kadanoff-Baym contour and to the maximum cutoff in time that is employed, we analyze the accuracy of the results using a variety of different moment sum rules and spectral formulas. We find that the fff-electron spectral function has interesting temperature dependence, becoming a narrow single-peaked function for small UUU and developing a gap, with two broader peaks for large UUU.
Nonequilibrium quantum mechanics can be solved with the Keldysh formalism, which evolves the quantum mechanical states forward in time in the presence of a time-dependent field, and then evolves them backward in time, undoing the effect of the time-dependent field. The Feynman path integral over the Keldysh contour is employed to calculate the strongly correlated Green's function. We examine the accuracy of this procedure for the simplest problem that requires a nonequilibrium formulation: the electron spectral function of the spinless Falicov-Kimball model.