ABSTRACT: The concept of open quantum system is very broad and it is related to the ability of measuring only certain degree of freedoms of a particular system. Although this idea is relatively simple, the separation between the system of interest, the degree of freedoms that are accesible experimentally, and the reservoir, the degree of freedoms that are not accesible experimentally, is not always clear. For instance, in molecular systems, electronic spectroscopy has access only to electronic degree of freedoms so that the nuclear and vibrational degree of freedom become the reservoir, this makes its description not trivial. This scenario leads to have non-trivial and structured reservoirs and to develop powerful tools to analyze them. The hallmark of non-trivial and structured reservoirs is the non-Markovian dynamics. By translating the Feynman and Vernon influence functional approach into phase-space representation, we develop two theories of semiclassical evolution of the Wigner function of the system of interest that incorporate non-Markovian dynamics and highly non-trivial quantum effects such as non-locality of quantum mechanics: (i) We translate the Caldeira-Leggett model into phase-space representation of quantum mechanics and (ii) we consider the possibility of having non-linear baths and therefore, truly quantum reservoirs. During the last forty years the study of energy loss and coherence in quantum systems has been based on the Ullersma-Caldeira-Leggett model, a model that describes the environment of quantum systems of interest as a collection of harmonic oscillators with classical evolution. We constructed this model in the Wigner-Weyl representation of quantum mechanics and discuss the classical nature of the evolution of the bath modes and the semiclassical evolution of the central system. As an application of the semiclassical Wigner propagator, the non-Markovian time evolution under the Morse potencial is analyzed. There, it is clear how decohering processes shrink the propagator to smaller regions of phase space implying that the dynamics become more local, i.e., more classical. The current level of experimentation and control of physical systems have called into question the validity of the model in, one hand, molecular systems (e.g. photosynthetic complexes immersed in solvents, chemical systems in liquid phase or gas and manipulated with intense laser pulses) and, in the other hand, solid state systems (e.g. Josephson junctions, spins in quantum dots or "spinning ice"). Given the importance of these systems in the development of new quantum technologies and in the understanding of quantum phenomena in mesoscopic systems, it has become necessary to develop new models of the environment and efficient methodologies with quantitative prediction power. However, some of them are artificial modifications of the Ullersma-Caldeira-Leggett model without solid and clean physical support. Here we formulated a general framework that allows the study of quantum correlations in quantum systems in the presence of non-harmonic thermal baths (e.g. baths formed by strongly coupled diatomic molecules). This formulation will allow a more precise and quantitative description of processes such as the transport of excitons in photosynthetic complexes, the transfer of heat in solid state devices, among others. Results clearly show the non-classical time dynamics of the bath modes. The implementation of this particular theory remains, however, as a challenge.
ABSTRACT: The concept of open quantum system is very broad and it is related to the ability of measuring only certain degree of freedoms of a particular system. Although this idea is relatively simple, the separation between the system of interest, the degree of freedoms that are accesible experimentally, and the reservoir, the degree of freedoms that are not accesible experimentally, is not always clear. For instance, in molecular systems, electronic spectroscopy has access only to electronic degree of freedoms so that the nuclear and vibrational degree of freedom become the reservoir, this makes its description not trivial. This scenario leads to have non-trivial and structured reservoirs and to develop powerful tools to analyze them. The hallmark of non-trivial and structured reservoirs is the non-Markovian dynamics. By translating the Feynman and Vernon influence functional approach into phase-space representation, we develop two theories of semiclassical evolution of the Wigner function of the system of interest that incorporate non-Markovian dynamics and highly non-trivial quantum effects such as non-locality of quantum mechanics: (i) We translate the Caldeira-Leggett model into phase-space representation of quantum mechanics and (ii) we consider the possibility of having non-linear baths and therefore, truly quantum reservoirs. During the last forty years the study of energy loss and coherence in quantum systems has been based on the Ullersma-Caldeira-Leggett model, a model that describes the environment of quantum systems of interest as a collection of harmonic oscillators with classical evolution. We constructed this model in the Wigner-Weyl representation of quantum mechanics and discuss the classical nature of the evolution of the bath modes and the semiclassical evolution of the central system. As an application of the semiclassical Wigner propagator, the non-Markovian time evolution under the Morse potencial is analyzed. There, it is clear how decohering processes shrink the propagator to smaller regions of phase space implying that the dynamics become more local, i.e., more classical. The current level of experimentation and control of physical systems have called into question the validity of the model in, one hand, molecular systems (e.g. photosynthetic complexes immersed in solvents, chemical systems in liquid phase or gas and manipulated with intense laser pulses) and, in the other hand, solid state systems (e.g. Josephson junctions, spins in quantum dots or "spinning ice"). Given the importance of these systems in the development of new quantum technologies and in the understanding of quantum phenomena in mesoscopic systems, it has become necessary to develop new models of the environment and efficient methodologies with quantitative prediction power. However, some of them are artificial modifications of the Ullersma-Caldeira-Leggett model without solid and clean physical support. Here we formulated a general framework that allows the study of quantum correlations in quantum systems in the presence of non-harmonic thermal baths (e.g. baths formed by strongly coupled diatomic molecules). This formulation will allow a more precise and quantitative description of processes such as the transport of excitons in photosynthetic complexes, the transfer of heat in solid state devices, among others. Results clearly show the non-classical time dynamics of the bath modes. The implementation of this particular theory remains, however, as a challenge.
The Wigner phase space formulation of quantum mechanics is a complete framework for quantum dynamic calculations that elegantly highlights connections with classical dynamics. In this series of two articles, building upon previous efforts, we derive the full hierarchy of approximate semiclassical (SC) dynamic methods for adiabatic and non-adiabatic problems in Wigner phase space. In Paper I, focusing on adiabatic single surface processes, we derive the well-known double Herman-Kluk (DHK) approximation for real-time correlation functions in Wigner phase space and connect it to the linearized SC (LSC) approximation through a stationary phase approximation. We exploit this relationship to introduce a new hybrid SC method, termed Adiabatic Hybrid Wigner Dynamics (AHWD) that allows for a few important "system" degrees of freedom (dofs) to be treated at the DHK level, while treating the rest of the dofs (the "bath") at the LSC level. AHWD is shown to accurately capture quantum interference effects in models of coupled oscillators and the decoherence of vibrational probability density of a model I2 Morse oscillator coupled to an Ohmic thermal bath. We show that AHWD significantly mitigates the sign problem and employs reduced dimensional prefactors bringing calculations of complex system-bath problems within the reach of SC methods. Paper II focuses on extending this hybrid SC dynamics to nonadiabatic processes.
ABSTRACT: We use the divisibility approach to describe non-Markovian dynamics (memory effects) in open quantum systems coupled to an environment (reservoir). Specifically, sufficient and necessary criteria were incorporated in the energy domain that play a relevant role in dynamics of the system. In this case, we adopt the Feshbach partition in the space of (Hilbert-Liouville). On the other hand, to investigate both Markovianity and non-Markovianity, we study the divisibility of the Wigner function propagator. For this, within the framework of path integrals, we carry out an analysis of non-classical paths in phase space. Furthermore, we have used tools of topology to propose theorems intimately related to measures of non-Markovianity that are experementally accessible.
We discuss a phase space representation of quantum dynamics of systems with many degrees of freedom. This representation is based on a perturbative expansion in quantum fluctuations around one of the classical limits. We explicitly analyze expansions around three such limits: (i) corpuscular or Newtonian limit in the coordinate-momentum representation, (ii) wave or Gross-Pitaevskii limit for interacting bosons in the coherent state representation, and (iii) Bloch limit for the spin systems. We discuss both the semiclassical (truncated Wigner) approximation and further quantum corrections appearing in the form of either stochastic quantum jumps along the classical trajectories or the nonlinear response to such jumps. We also discuss how quantum jumps naturally emerge in the analysis of non-equal time correlation functions. This representation of quantum dynamics is closely related to the phase space methods based on the Wigner-Weyl quantization and to the Keldysh technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal product, Bopp operators, and others automatically emerge from the Feynmann's path integral representation of the evolution in the Heisenberg representation. We illustrate the applicability of this expansion with various examples mostly in the context of cold atom systems including sine-Gordon model, one- and two-dimensional Bose Hubbard model, Dicke model and others.
In the present Thesis we study the behavior of multi-time correlation functions and of thermodynamical quantities such as heat in open quantum systems undergoing an evolution generally affected by the presence of memory effects, i.e. non-Markovian. In the last decade, a large part of the scientific community in this field has dedicated its efforts to the understanding, precise definition and quantification of non-Markovianity in the quantum realm and now we have at our disposal several benchmark results and a plethora of different estimators that allow to determine the degree of non-Markovianity of a given dynamics. It comes therefore natural to investigate how other different dynamical quantities relate to such estimators also in order to understand the physical implications of memory effects on the statistics of observable quantities. In the first part of this work, a quantitative test of the violation of the so-called quantum regression theorem in presence of a non-Markovian dynamical regime is investigated. The quantum regression theorem represents a procedure that, whenever valid, allows to reconstruct two-time correlation functions of system's operators from the sole knowledge of the dynamics of mean values. It is worth stressing that two-time correlation functions are necessary in order to fully characterize the statistical properties of a quantum system, since they are able to catch aspects of the dynamics, such as fluorescence spectrum, in general not accessible looking at mean values. Despite their relevance however, obtaining two-time correlation functions often represents a formidable task, since the knowledge of the full "system+environment" dynamics is required, a generally too demanding request in the context of open quantum systems theory. The quantum regression theorem represents in this regard the easiest route to determine two-time correlation functions, this highlighting its importance. In this work we show that, in a pure-dephasing spin-boson model, the quantum regression theorem represents a stronger condition than non-Markovianity, in the sense that any presence of memory effects in the reduced dynamics inevitably results in violations to the former. These results have been published in [G.Guarnieri, A. Smirne, B. Vacchini, Phys. Rev. A 90, 022110 (2014)]. The second part of the Thesis is devoted to the characterization of heat ow at the microscopic level in open quantum systems, both finite and infinite dimensional. In particular we begin by studying the time behavior of its mean value in a non-Markovian dynamical regime, showing that, at variance with what happens in the Born-Markov semigroup limiting case, heat can backflow from the environment to the system. After providing a condition for the occurrence of such phenomenon and a measure for its amount for a given dynamics, the relationship with suitable non-Markovianity estimators is sought in two paradigmatic models, namely the spin-boson and the quantum brownian motion. The results, collected in [ G. Guarnieri, C. Uchiyama, B. Vacchini, Phys. Rev. A 93, 012118 (2016); G. Guarnieri, J. Nokkala, R. Schmidt, S. Maniscalco, B. Vacchini, Phys. Rev. A 94, 062101 (2016)], on the one hand allow for the identification of parameter-regions where the heat backflow is absent or maximum. On the other hand they show that the occurrence of heat backflow represents a stricter condition than non-Markovianity, in the sense that non-Markovianity allows for the observation of heat flowing back from the environment to the system and, vice versa, a Markovian dynamics prevents its occurrence. This Thesis concludes with the formulation of a new family of lower bounds to the mean dissipated heat in an environmental-assisted erasure-protocol scenario where Landauer's principle applies. As originally conceived for classical systems, this principle states that every irreversible erasure of information stored in a system inevitably carries along an amount of heat dissipated into the environment which is expended to perform the action. Within the framework recently put forward in [D. Reeb, M. M. Wolf, New J. Phys. 16, 103011 (2014)], which guarantees the validity of Landauer's principle in an open quantum systems scenario, we provide an asymptotically tight family of lower bounds to the dissipated heat which are also valid in the non-equilibrium setting. This construction is applied to an open system consisting of a three-level V-system, in which one transition is externally pumped by a laser field while the other is coupled through an XX-interaction to an environment consisting of a spin chain. Beside calculating all these quantities, an exact solution for the dynamics of such system is also provided. These results are collected in [G. Guarnieri, S. Campbell, J. Goold, S. Pigeon, M. Paternostro, B. Vacchini, in preparation].
We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.
Quantum dissipation has been the object of study within the physics and chemistry communities for many years. Despite this, the field is in constant evolution, largely due to the fact that novel systems where the understanding of dissipation and dephasing processes is of crucial importance have become experimentally accessible in recent years. Among the ongoing research themes, we mention the defeat of decoherence in solid state-based quantum bits (qubits) (e.g. superconducting qubits or quantum dot based qubits), or dissipation due to non-equilibrium Fermi reservoirs, as is the case for quantum transport through meso- and nanoscale structures. A close inspection of dissipation in such systems reveals that one has to deal with 'unconventional' environments, where common assumptions of, for example, linearity of the bath and/or equilibrium reservoir have to be abandoned. Even for linear baths at equilibrium it might occur that the bath presents some internal structure, due, for example, to the presence of localized bath modes. A large part of this focus issue is devoted to topics related to the rapidly developing fields of quantum computation and information with solid state nanodevices. In these implementations, single and two-qubit gates as well as quantum information transmission takes place in the presence of broadband noise that is typically non-Markovian and nonlinear. On both the experimental and theory side, understanding and defeating such noise sources has become a crucial step towards the implementation of efficient nanodevices. On a more fundamental level, electron and spin transport through quantum dot nanostructures may suffer from 'unconventional' dissipation mechanisms such as the simultaneous presence of spin relaxation and fermionic dissipation, or may represent themselves out of equilibrium baths for nearby mesoscopic systems. Finally, although not expected from the outset, the present collection of articles has revealed that different 'unconventional' questions were still open on the standard harmonic oscillator and spin baths. This includes both fundamental issues, such as the possibility of estimating the specific heat for a free particle in the presence of dissipation, and the development of methods suitable to dealing with long range correlations at zero temperature and with quantum chaotic environments. We believe that the present focus issue on Quantum Dissipation in Unconventional Environments , although certainly not exhaustive, provides an important open-access resource that presents the latest state of the art of research in this field along its different lines. Focus on Quantum Dissipation in Unconventional Environments Contents Coherent control of an effective two-level system in a non-Markovian biomolecular environment J Eckel, J H Reina and M Thorwart Charge transfer dynamics in driven molecular ratchets: quantum Monte Carlo results and rate models L Mühlbacher and J Ankerhold Decoherence in qubits due to low-frequency noise J Bergli, Y M Galperin and B L Altshuler Dephasing by electron–electron interactions in a ballistic Mach–Zehnder interferometer Clemens Neuenhahn and Florian Marquardt Quantum frustration of dissipation by a spin bath D D Bhaktavatsala Rao, Heiner Kohler and Fernando Sols A random matrix theory of decoherence T Gorin, C Pineda, H Kohler and T H Seligman Dissipative dynamics of a biased qubit coupled to a harmonic oscillator: analytical results beyond the rotating wave approximation Johannes Hausinger and Milena Grifoni Dissipative dynamics of a two-level system resonantly coupled to a harmonic mode Frederico Brito and Amir O Caldeira Spin correlations in spin blockade Rafael Sánchez, Sigmund Kohler and Gloria Platero Landau–Zener tunnelling in dissipative circuit QED David Zueco, Peter Hänggi and Sigmund Kohler Quantum oscillations in the spin-boson model: reduced visibility from non-Markovian effects and initial entanglement F K Wilhelm Dynamics of dissipative coupled spins: decoherence, relaxation and effects of a spin-boson bath P Nägele, G Campagnano and U Weiss Spin chain model for correlated quantum channels Davide Rossini, Vittorio Giovannetti and Simone Montangero Finite quantum dissipation: the challenge of obtaining specific heat Peter Hänggi, Gert-Ludwig Ingold and Peter Talkner Dynamics of large anisotropic spin in a sub-ohmic dissipative environment close to a quantum-phase transition Frithjof B Anders Effects of low-frequency noise cross-correlations in coupled superconducting qubits A D'Arrigo, A Mastellone, E Paladino and G Falci From coherent motion to localization: dynamics of the spin-boson model at zero temperature Haobin Wang and Michael Thoss Phonon distributions of a single-bath mode coupled to a quantum dot F Cavaliere, G Piovano, E Paladino and M Sassetti