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arXiv:2106.14765 · Physical Review Research

Exact description of quantum stochastic models as quantum resistors

Tony Jin, João S. Ferreira, Michele Filippone, Thierry Giamarchi

Published Feb 11, 2022• 41 citations
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Abstract

We study the transport properties of generic out-of-equilibrium quantum systems connected to fermionic reservoirs. We develop a perturbation scheme in the inverse system size, named $1/N$ expansion, to study a large class of out of equilibrium diffusive/ohmic systems. The bare theory is described by a Gaussian action corresponding to a set of independent two level systems at equilibrium. This allows a simple and compact derivation of the diffusive current as a first-order pertubative term. In addition, we obtain exact solutions for a large class of quantum stochastic Hamiltonians (QSHs) with time and space dependent noise, using a self-consistent Born diagrammatic method in the Keldysh representation. We show that these QSHs exhibit diffusive regimes, which are encoded in the Keldysh component of the single particle Green's function. The exact solution for these QSHs models confirms the validity of our system size expansion ansatz, and its efficiency in capturing the transport properties. We consider in particular three fermionic models: (i) a model with local dephasing, (ii) the quantum simple symmetric exclusion process model, and (iii) a model with long-range stochastic hopping. For (i) and (ii) we compute the full temperature and dephasing dependence of the conductance of the system, both for two- and four-points measurements. Our solution gives access to the regime of finite temperature of the reservoirs, which could not be obtained by previous approaches. For (iii), we unveil a ballistic-to-diffusive transition governed by the range and the nature (quantum or classical) of the hopping. As a byproduct, our approach equally describes the mean behavior of quantum systems under continuous measurement.

DOI
10.1103/physrevresearch.4.013109

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We study the transport properties of generic out-of-equilibrium quantum systems connected to fermionic reservoirs. We develop a perturbation scheme in the inverse system size, named 1/N1/N1/N expansion, to study a large class of out of equilibrium diffusive/ohmic systems. The bare theory is described by a Gaussian action corresponding to a set of independent two level systems at equilibrium. This allows a simple and compact derivation of the diffusive current as a first-order pertubative term. In addition, we obtain exact solutions for a large class of quantum stochastic Hamiltonians (QSHs) with time and space dependent noise, using a self-consistent Born diagrammatic method in the Keldysh representation. We show that these QSHs exhibit diffusive regimes, which are encoded in the Keldysh component of the single particle Green's function. The exact solution for these QSHs models confirms the validity of our system size expansion ansatz, and its efficiency in capturing the transport properties. We consider in particular three fermionic models: (i) a model with local dephasing, (ii) the quantum simple symmetric exclusion process model, and (iii) a model with long-range stochastic hopping. For (i) and (ii) we compute the full temperature and dephasing dependence of the conductance of the system, both for two- and four-points measurements. Our solution gives access to the regime of finite temperature of the reservoirs, which could not be obtained by previous approaches. For (iii), we unveil a ballistic-to-diffusive transition governed by the range and the nature (quantum or classical) of the hopping. As a byproduct, our approach equally describes the mean behavior of quantum systems under continuous measurement.
In this work we study quantum transport phenomena in two different types of systems. The first type corresponds to so-called complex quantum systems, such as chaotic ballistic cavities and disordered quantum wires, which have a complex distribution of energy levels and transmission eigenvalues. This class of system does not admit a simple description in terms of a fixed Hamiltonian, and therefore we will make a statistical approach of its transport properties through Random Matrices Theory. We obtain exact expressions for the first three moments of the heat conductance of a quantum chain that crosses over from a superconducting quantum dot to a superconducting disordered quantum wire. Our analytic solution provides exact detailed descriptions of some smooth transitions that can be observed in the system as a function of its length, which include ballistic-metallic and metallic-insulating transitions. The two Bogolyubov de Gennes symmetry classes with time-reversal symmetry are accounted for. The striking effect of total suppression of the insulating regime in systems with broken spin-rotation invariance is observed at large length scales. For a single channel system, this anomalous effect can be interpreted as a signature of the presence of the elusive Majorana fermion in a condensed matter system. The second type of system corresponds to qubits, which, unlike complex systems, can be described by very simple Hamiltonians. In this case, we study the properties of open systems using master equations. We present a study of non-equilibrium thermodynamics of qubit systems submitted to quantum control. More specifically, we performed a comparative study of two types of simple two-level non-interacting quantum transport systems coupled to two bosonic and fermionic reservoirs respectively. Each system is submitted to a Wiseman-Milburn type feedback scheme in the formulation of stochastic thermodynamics. We see the effects of finite temperature and time delay on two specific feedback applications: a heat pump and a purification protocol. We observed a clear signature of the purity of the qubits in the Full Counting Statistics observed in the current flowing through the system.
We devise a semi-classical model to describe the transport properties of low-dimensional fermionic lattices under the influence of external quantum stochastic noise. These systems behave as quantum stochastic resistors, where the bulk particle transport is diffusive and obeys the Ohm/Fick's law. Here, we extend previous exact studies beyond the one-dimensional limit to ladder geometries and explore different dephasing mechanisms that are relevant to different physical systems, from solid-state to cold atoms. We find a non-trivial dependence of the conductance of these systems on the chemical potential of the reservoirs. We then introduce a semi-classical approach that is in good agreement with the exact numerical solution and provides an intuitive and simpler interpretation of transport in quantum stochastic resistors. Moreover, we find that the conductance of quantum ladders is insensitive to the coherence of the dephasing process along the direction transverse to transport, despite the fact that the system reaches different stationary states. We conclude by discussing the case of dissipative leads affected by dephasing, deriving the conditions for which they effectively behave as Markovian injectors of particles in the system.
We devise a semiclassical model to describe the transport properties of low-dimensional fermionic lattices under the influence of external quantum stochastic noise. These systems behave as quantum stochastic resistors, where the bulk particle transport is diffusive and obeys the Ohm/Fick's law. Here, we extend previous exact studies beyond the one-dimensional limit to ladder geometries and explore different dephasing mechanisms that are relevant to different physical systems, from solid-state to cold atoms. We show how the semiclassical description is useful to explain the nontrivial dependence of the conductance of these systems on the chemical potential of the reservoirs. This description provides an intuitive and simpler interpretation of transport in quantum stochastic resistors in good quantitative agreement with the exact numerical solution. Moreover, we find that the conductance of quantum ladders is insensitive to the coherence of the dephasing process along the direction transverse to transport, despite the fact that the system reaches different stationary states.
We study the evolution of a system of free fermions in one dimension under the simultaneous effects of coherent tunneling and stochastic Markovian noise. We identify a class of noise terms where a hierarchy of decoupled equations for the correlation functions emerges. In the special case of incoherent, nearest-neighbour hopping the equation for the two-point functions is solved explicitly. The Green's function for the particle density is obtained analytically and a timescale is identified where a crossover from ballistic to diffusive behaviour takes place. The result can be interpreted as a competition between the two types of conduction channels where diffusion dominates on large timescales.
The motivation for this thesis is to find a fluctuating hydrodynamic description of quantum coherent effects in mesoscopic quantum systems with diffusive transport properties. Coherent effects are inscribed into the coherences (two-point Green's function) on which we focus as the building blocks of such a theory. Our approach is rather mathematical and not related to concrete experiments. We study the quantum symmetric simple exclusion process (QSSEP), a potentially iconic model describing transport of noisy free fermions on a 1D lattice, which has the minimal structure to capture what we are interested in: Long-ranged coherences and diffusive transport. In mean, QSSEP reduces to the symmetric simple exclusion process (SSEP), a classical toy model that was important in the development of the macroscopic fluctuation theory, a fluctuating hydrodynamic description of diffusive transport in classical systems. Studying QSSEP we hope to make progress towards a quantum coherent extension of the macroscopic fluctuation theory. The thesis summarizes many results we have obtained about QSSEP in the last three years, such as the dynamical equation and an exact stationary solution for correlation functions of coherences at hydrodynamic scales, the distribution of entanglement in QSSEP and an argument why QSSEP might be an effective noisy description for more generic mesoscopic quantum systems. Many of these results are due to a relation between the statistical properties of coherences in QSSEP and free probability theory. We devote a whole chapter to this relation and present, as a by-product, a method to characterize the spectrum of subblocks of a large class of structured random matrices.
The progress in micro-fabrication technologies makes it possible to address individual, tailored, mesoscopic structures with sizes on a nanometer to micrometer scale. The transport properties of such quasi zero- or one-dimensional systems are considerably influenced by the electron-electron interaction. In theoretical studies this often becomes apparent in the emergence of divergences in a perturbative treatment of the interaction. A successful approach may then be given by renormalization group (RG) methods which are based on the idea of treating energy scales successively from high to low. In the functional RG (fRG) an exact infinite hierarchy of flow equations for multi-particle Green or vertex functions is derived, that allows for systematic approximations on different levels. In numerous studies the fRG has been found to be a powerful tool for the investigation of low dimensional Fermi systems. Usually it is formulated within Matsubara formalism which applies to a system in thermal equilibrium. A central topic of this thesis is the implementation of the fRG within Keldysh formalism, providing the possibility to treat equilibrium and non-equilibrium situations on equal footing. The resulting Keldysh fRG is used in this thesis to explore the transport properties of quantum wires with several barriers and of the single impurity Anderson model. The thesis starts with an introduction to Keldysh formalism and to related concepts. Then the structure of the Green and vertex functions of Keldysh formalism is investigated. This includes studying the consequences of causality, characterizing thermal equilibrium, and formulating corresponding relations for the multi-particle functions. After that, the implementation of the fRG within Keldysh formalism is presented in detail. The flow equations for the vertex functions are derived via their diagrammatic expansion. Two especially useful flow parameters are determined: one of them cuts off the imaginary frequencies that appear as poles of the Fermi functions of the reservoirs. The second flow parameter is an artificially enhanced reservoir hybridization; on one hand it has the advantage to preserve the fluctuation dissipation theorem in equilibrium even in frequency dependent approximations; on the other hand, its use can be more cumbersome. One of the models studied in this thesis is a tight-binding chain with nearest-neighbour interaction. At weak and intermediate interactions it belongs to the class of Luttinger liquids (LLs) whose low energy properties are characterized by power laws. The transport properties of LLs have been studied experimentally for example on carbon nanotubes. However, the presence of transport barriers at the contacts make the interpretation of the experimental data difficult. A theoretical treatment with the fRG permits a rather large flexibility to model the barrier configuration. In this thesis the fRG is first used in its equilibrium Matsubara implementation in order to analyse the temperature dependence of the linear conductance in the presence of two, three or four barriers. It is shown that temperature induces an averaging over quantum phases of propagation, leading to combinations of the power law exponents of the individual barriers. An experimental set-up that allows to verify this behaviour is proposed. In a second step the Keldysh version of the fRG is applied to the problem of transport through a quantum wire at finite bias voltage. For a wire with two high contact barriers it is found that the non-equilibrium particle distribution in the wire is responsible for the appearance of a two-fold power law in the density of states. An additional weakly attached probe lead can be used to make the two different power laws visible in the conductance. The second model which is analysed is the one of the single Anderson impurity, often considered as a paradigm for the investigation of strong local electron correlations. The latter become noticeable in the appearance of the Kondo scale, an energy exponentially small in the interaction strength, that characterizes the low energy physics of the model in the regime of spin fluctuations. On the basis of this model an advanced frequency dependent approximation for the Keldysh fRG is developed in this thesis. Such an approximation can account for subleading effects of the interaction, namely the fact that two-particle interactions can generate decay rates for single-particle states. The derivation of a consistent approximation scheme is presented, that turns out to yield a good description of the model at moderate interaction strength.