Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0910.2939

Approaches in evaluating two-time correlation function

Sintayehu Tesfa

Published Oct 15, 2009• 0 citations
arXivHTMLPDF

Abstract

Derivation of the procedures that can be applied in evaluating two-time correlation function in terms of coherent-state propagator and corresponding Q-function is presented. On the basis that the involved functions are generally exponential in nature, obtaining the two-time second-order correlation function is essentially claimed to be reduced to carrying out relatively simple integrations. Fundamentally, the time dependence of the operators is transferred to the density operator. Moreover, manipulation in reordering the operators is performed by applying the usual trace operation. With all details, it is basically observed that the two-time correlation can be readily determined once the pertinent coherent-state propagator or Q-function is known. Since working with c-number equation is far more handy than the associated operator equation, it is expected that the results derived in this contribution can aid in easing the otherwise involving mathematical rigor.

DOI
10.48550/arxiv.0910.2939

Related papers

Two-time second-order correlation function

Sintayehu Tesfa

Derivation of two-time second-order correlation function by following approaches such as stochastic differential equation, coherent-state propagator, and quasi-statistical distribution function is presented. In the process, the time dependence of the operators is transferred to the density operator by making use of trace operation in which the coherent state propagator and $Q$-function that represent the quantum system under consideration are expressed in terms of different time parameters. Even though the number of resulting integrations are found to be large, the accompanying implementation turns out to be straightforward in view that the associated $c$-number functions are Gaussian by nature. In relation to the asserted possibility of rewriting the result of one of the approaches in terms of the other, the presented derivation is expected to lay a strong foundation for viable technique of calculating correlations of various moments at different times that can be deployed in revealing quantum correlations.

arXiv:2406.10546•Jun 15, 2024•0 citations

Quantum Dynamics

Uri Peskin

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Cambridge University Press eBooks•May 11, 2023•0 citations

Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. III. A Generalized Wick Theorem and Multitime Mapping

G. S. Agarwal, E. Wolf

The new $c$-number calculus for functions of noncommuting operators, developed in Paper I and employed in Paper II to formulate a general phase-space description of boson systems, deals with situations involving equal-time operators only. In the present paper extensions are presented for the treatment of problems involving boson operators at two or more instants of time. The mapping of time-ordered products onto $c$-number functions is studied in detail. The results make it possible to evaluate time-ordered products of boson operators by phase-space techniques. The usual Wick theorem for boson systems is obtained as a special case of a much more general theorem on time ordering. Our method of derivation appears to provide the first direct proof of Wick's theorem as well as a clear insight into its true meaning. A closed expression is also obtained for the time-evolution operator in terms of the solution of the $c$-number differential equation for the phase-space equivalent of this operator. The new calculus is also applied to the problem of evaluating normally ordered time-ordered, and also the antinormally ordered time-ordered, correlation functions.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1970•173 citations

A method for calculating two-time correlation functions in the quantum trajectory approach

A. Karpati, Петер Адам, József Janszky

We developed a technique for calculating two-time correlation functions based on a quantum trajectory approach. Our method requires the simulation of the evolution of a given set of density matrices only once. The evaluation can be carried out from the density matrix data obtained for any quantities, time arguments and initial conditions. The computational efficiency of the method is estimated and an example of application is presented.

Journal of Optics B Quantum and Semiclassical Optics•Mar 1, 2004•0 citations

Mathematical aspects of quantum charge transport and current fluctuations

Sven Bachmann

The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.

•Jan 1, 2009•0 citations

Feynman path integral representation for many‐fermion interacting systems

Fernando Calamante, Roberto C. Bochicchio, Horacio Grinberg

Abstract A many‐fermion interacting system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a basis set, closely related to the coherent states, are used to obtain the many‐body analytic propagator. A second‐quantized Hamiltonian involving a restricted set of two‐body interactions and the whole set of Coulomb interactions are separately and shown to lead to an exact and an approximate propagator, respectively. In the latter case, use of a grand canonical ensemble allows the grand partition function and the density operator matrix to be readily obtained. No further approximations are required in the calculation of the trace of the evolution operator involved in the evaluation of statistical expectation values. © 1994 John Wiley & Sons, Inc.

International Journal of Quantum Chemistry•Mar 5, 1994•4 citations
Derivation of the procedures that can be applied in evaluating two-time correlation function in terms of coherent-state propagator and corresponding Q-function is presented. On the basis that the involved functions are generally exponential in nature, obtaining the two-time second-order correlation function is essentially claimed to be reduced to carrying out relatively simple integrations. Fundamentally, the time dependence of the operators is transferred to the density operator. Moreover, manipulation in reordering the operators is performed by applying the usual trace operation. With all details, it is basically observed that the two-time correlation can be readily determined once the pertinent coherent-state propagator or Q-function is known. Since working with c-number equation is far more handy than the associated operator equation, it is expected that the results derived in this contribution can aid in easing the otherwise involving mathematical rigor.
Derivation of two-time second-order correlation function by following approaches such as stochastic differential equation, coherent-state propagator, and quasi-statistical distribution function is presented. In the process, the time dependence of the operators is transferred to the density operator by making use of trace operation in which the coherent state propagator and QQQ-function that represent the quantum system under consideration are expressed in terms of different time parameters. Even though the number of resulting integrations are found to be large, the accompanying implementation turns out to be straightforward in view that the associated ccc-number functions are Gaussian by nature. In relation to the asserted possibility of rewriting the result of one of the approaches in terms of the other, the presented derivation is expected to lay a strong foundation for viable technique of calculating correlations of various moments at different times that can be deployed in revealing quantum correlations.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
The new ccc-number calculus for functions of noncommuting operators, developed in Paper I and employed in Paper II to formulate a general phase-space description of boson systems, deals with situations involving equal-time operators only. In the present paper extensions are presented for the treatment of problems involving boson operators at two or more instants of time. The mapping of time-ordered products onto ccc-number functions is studied in detail. The results make it possible to evaluate time-ordered products of boson operators by phase-space techniques. The usual Wick theorem for boson systems is obtained as a special case of a much more general theorem on time ordering. Our method of derivation appears to provide the first direct proof of Wick's theorem as well as a clear insight into its true meaning. A closed expression is also obtained for the time-evolution operator in terms of the solution of the ccc-number differential equation for the phase-space equivalent of this operator. The new calculus is also applied to the problem of evaluating normally ordered time-ordered, and also the antinormally ordered time-ordered, correlation functions.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.
Abstract A many‐fermion interacting system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a basis set, closely related to the coherent states, are used to obtain the many‐body analytic propagator. A second‐quantized Hamiltonian involving a restricted set of two‐body interactions and the whole set of Coulomb interactions are separately and shown to lead to an exact and an approximate propagator, respectively. In the latter case, use of a grand canonical ensemble allows the grand partition function and the density operator matrix to be readily obtained. No further approximations are required in the calculation of the trace of the evolution operator involved in the evaluation of statistical expectation values. © 1994 John Wiley & Sons, Inc.