Two-time Correlation Functions Research Articles

Quantum time: Experimental multitime correlations

In this paper we provide an experimental illustration of Page and Wootters' quantum time mechanism that is able to describe two-time quantum correlation functions. This allows us to test a Leggett-Garg inequality, showing a violation from the "internal" observer point of view. The "external" observer sees a time-independent global state. Indeed, the scheme is implemented using a narrow-band single photon where the clock degree of freedom is encoded in the photon's position. Hence, the internal observer that measures the position can track the flow of time, while the external observer sees a delocalized photon that has no time evolution in the experiment time-scale.

Probing nonclassicality under dissipation

We investigate the nonclassicality of an open quantum system using Leggett–Garg inequalities (LGI) which test the correlations of a single system measured at different times. Violation of LGI implies nonclassical behavior of the open system. We investigate the violation of the Leggett–Garg inequality for a two level system (qubit) spontaneously decaying under a general non-Markovian dissipative environment. A robust nonclassical behavior is shown through the violation of LGI in the non-Markovian regime of the environment. Our results are exact as we have evaluated the two-time correlation functions exactly without using quantum regression theorem. We probe quantum signatures of the open system through the exact dynamics of Leggett–Garg inequalities for a wide range of system-environment parameters beyond Born–Markov regime.

Two-time correlation functions and the Lee-Yang zeros for an interacting Bose gas.

Two-time correlation functions of a system of Bose particles are studied. We find a relation of zeros of the correlation functions with the Lee-Yang zeros of the partition function of the system. The obtained relation gives the possibility to observe the Lee-Yang zeros experimentally. A particular case of Bose particles on two levels is examined, and zeros of two-time correlation functions and Lee-Yang zeros of the partition function of the system are analyzed.

Open quantum systems with delayed coherent feedback

We present an elementary derivation and generalisation of a recently reported method of simulating feedback in open quantum systems. We use our generalised method to simulate systems with multiple delays, as well as cascaded systems with delayed backscatter. In addition, we derive a generalisation of the quantum regression formula that applies to systems with delayed feedback, and show how to use the formula to compute two-time correlation functions of the system as well as output field properties. Finally, we show that delayed coherent feedback can be simulated as a quantum teleportation protocol that requires only Markovian resources, pre-shared entanglement, and time travel. The requirement for time travel can be avoided by using a probabilistic protocol.

Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes

Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time t. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times and in the persistence regime of large separations. We also investigate other quantities, such as the backward and forward recurrence times, as well as the occupation time of the process. The latter has a broad distribution which is determined exactly. Physical implications of these results for the Poland Scheraga and related models are given. These results also give exact answers to questions posed in the past in the context of stochastically evolving surfaces.

Influence of Initial Correlations on Evolution of a Subsystem in a Heat Bath and Polaron Mobility

A regular approach to accounting for initial correlations, which allows to go beyond the unrealistic random phase (initial product state) approximation in deriving the evolution equations, is suggested. An exact homogeneous (time-convolution and time-convolutionless) equations for a relevant part of the two-time equilibrium correlation function for the dynamic variables of a subsystem interacting with a boson field (heat bath) are obtained. No conventional approximation like RPA or Bogoliubov’s principle of weakening of initial correlations is used. The obtained equations take into account the initial correlations in the kernel governing their evolution. The solution to these equations is found in the second order of the kernel expansion in the electron–phonon interaction, which demonstrates that generally the initial correlations influence the correlation function’s evolution in time. It is explicitly shown that this influence vanishes on a large timescale (actually at \(t\rightarrow \infty \)) and the evolution process enters an irreversible kinetic regime. The developed approach is applied to the Frohlich polaron and the low-temperature polaron mobility (which was under a long-time debate) is found with a correction due to initial correlations.

Four-time 7Li stimulated-echo spectroscopy for the study of dynamic heterogeneities: Application to lithium borate glass

To study the nature of the nonexponential ionic hopping in solids a pulse sequence was developed that yields four-time stimulated-echo functions of previously inaccessible spin-3/2-nuclei such as 7Li. It exploits combined Zeeman and octupolar order as longitudinal carrier state. Higher-order correlation functions were successfully generated for natural-abundance and isotopically-enriched lithium diborate glasses. Four-time 7Li measurements are presented and compared with two-time correlation functions. The results are discussed with reference to approaches devised to quantify the degree of nonexponentiality in glass forming systems and evidence for the occurrence of dynamic heterogeneities and dynamic exchange were found. Additional experiments using the 6Li species illustrate the challenge posed by subensemble selection when the dipolar interactions are not very much smaller than the quadrupolar ones.

Non-Markovian Two-Time Correlation Dynamics and Nonergodicity

The two-time correlation functions of the coordinate and velocity of a non-Markovian harmonic particle are derived analytically. They are decomposed into the components of differences between the initial variances and the equilibrium of the particle; in particular, the dependence of a random force on the initial preparation of the system is included. Using those expressions, we simultaneously investigate nonstationary, nonergodic, and nonequilibrium features of a forced system. It is demonstrated that the result of combining the oscillating relaxation and the initial preparation-dependent noise leads to breakdown of both ergodicity and equilibration of a forced system. The finite-size effect of a coupled-oscillator-chain heat bath is also discussed.

Violation of the quantum regression theorem and the Leggett–Garg inequality in an exactly solvable model

Violation of the quantum regression theorem and the Leggett–Garg inequality is studied by means of the exactly solvable multi-mode Jaynes–Cummings model. An exact expression of a two-time correlation function is compared with that derived by the quantum regression theorem. It is found that the quantum regression theorem is not valid even if the reduced time evolution of the qubit is Markovian. Furthermore, it is shown that if the quantum regression theorem is applied in this model, the Leggett–Garg inequality is satisfied while it is violated by the exact correlation function.

Communication: Heterogeneous water dynamics on a clathrate hydrate lattice detected by multidimensional oxygen nuclear magnetic resonance

Previous deuteron nuclear magnetic resonance studies revealed conflicting evidence regarding the possible motional heterogeneity of the water dynamics on the hydrate lattice of an ice-like crystal. Using oxygen-17 nuclei as a sensitive quadrupolar probe, the reorientational two-time correlation function displays a clear nonexponentiality. To check whether this dispersive behavior is a consequence of dynamic heterogeneity or rather of an intrinsic nonexponentiality, a multidimensional, four-time magnetic resonance experiment was devised that is generally applicable to strongly quadrupolarly perturbed half-integer nuclei such as oxygen-17. Measurements of an appropriate four-time function demonstrate that it is possible to select a subensemble of slow water molecules. Its mean time scale is compared to theoretical predictions and evidence for significant motional heterogeneity is found.

Partition-free theory of time-dependent current correlations in nanojunctions in response to an arbitrary time-dependent bias

Working within the Nonequilibrium Green's Function (NEGF) formalism, a formula for the two-time current correlation function is derived for the case of transport through a nanojunction in response to an arbitrary time-dependent bias. The one-particle Hamiltonian and the Wide Band Limit Approximation (WBLA) are assumed, enabling us to extract all necessary Green's functions and self energies for the system, extending the analytic work presented previously [Ridley et al. Phys. Rev. B (2015)]. We show that our new expression for the two-time correlation function generalises the B\"uttiker theory of shot and thermal noise on the current through a nanojunction to the time-dependent bias case including the transient regime following the switch-on. Transient terms in the correlation function arise from an initial state that does not assume (as is usually done) that the system is initially uncoupled, i.e. our approach is partition-free. We show that when the bias loses its time-dependence, the long time-limit of the current correlation function depends on the time difference only, as in this case an ideal steady state is reached. This enables derivation of known results for the single frequency power spectrum and for the zero frequency limit of this power spectrum. In addition, we present a technique which for the first time facilitates fast calculations of the transient quantum noise, valid for arbitrary temperature, time and voltage scales. We apply this to the quantum dot and molecular wire systems for both DC and AC biases, and find a novel signature of the traversal time for electrons crossing the wire in the time-dependent cross-lead current correlations.

Nonequilibrium transient dynamics of photon statistics

We investigate the transient dynamics of photon statistics through two-time correlation functions for optical fields. We find that the transient correlations at different time t yield a smooth transition from antibunching to bunching photon statistics in the weak system-environment coupling regime. In the strong-coupling regime, the two-time correlations exhibit bunching-antibunching oscillations that persists both in the transient process and in the steady-state limit. The photon bunching-antibunching oscillations is a manifestation of strong non-Markovian dynamics, where the system remains in nonequilibrium from its environment. We also find that the antibunching to bunching transition in the weak-coupling regime and the bunching-antibunching oscillation in the strong-coupling regime are strongly influenced by the initial environment temperature.

Linear response theory for open systems: Quantum master equation approach

A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.

On the use of two-time correlation functions for X-ray photon correlation spectroscopy data analysis.

Multi-time correlation functions are especially well suited to study non-equilibrium processes. In particular, two-time correlation functions are widely used in X-ray photon correlation experiments on systems out of equilibrium. One-time correlations are often extracted from two-time correlation functions at different sample ages. However, this way of analysing two-time correlation functions is not unique. Here, two methods to analyse two-time correlation functions are scrutinized, and three illustrative examples are used to discuss the implications for the evaluation of the correlation times and functional shape of the correlations.

Green's functions technique for calculating the emission spectrum in a quantum dot-cavity system

We introduce the Green's functions technique as an alternative theory to the quantum regression theorem formalism for calculating the two-time correlation functions in open quantum systems at the steady state. In order to investigate the potential of this theoretical approach, we consider a dissipative system composed of a single quantum dot inside a semiconductor cavity and the emission spectrum is computed due to the quantum dot as well as the cavity. We propose an algorithm based on the Green's functions technique for computing the emission spectrum that can easily be adapted to more complex open quantum systems. We found that the numerical results based on the Green's functions technique are in perfect agreement with the quantum regression theorem formalism. Moreover, it allows overcoming the inherent theoretical difficulties associated with the direct application of the quantum regression theorem in open quantum systems.Received: 6 September 2016, Accepted: 5 November 2016; Edited by: J. P. Paz; DOI: http://dx.doi.org/10.4279/PIP.080008Cite as: E A Gómez, J D Hernández-Rivero, H Vinck-Posada, Papers in Physics 8, 080008 (2016)This paper, by E A Gómez, J D Hernández-Rivero, H Vinck-Posada, is licensed under the Creative Commons Attribution License 3.0.

Leggett-Garg correlation functions from a noninvasive velocity measurement continuous in time

In the Leggett-Garg approach to testing macrorealism, the two-time correlation functions, which are normally obtained by sequential measurements of a dichotomic variable Q, need to be measured in a non-invasive way in order to exclude certain types of alternative classical explanation. Here, it is shown, for a class of macrorealistic theories, that the correlation functions are readily expressed in terms of a time integral of the velocity corresponding to Q and that this expression can be determined from a single final-time measurement of an auxiliary system in continual weak interaction with the primary system. The protocol has the form of a "waiting detector" which clicks only when Q changes sign. It shares features with both ideal negative measurements and weak measurements and we argue that it is essentially non-invasive, under certain reasonable assumptions. We show that the non-invasiveness persists to a quantum model of the process.

Nonclassicality criteria: Quasiprobability distributions and correlation functions

We use the exact calculation of the quantum mechanical, temporal characteristic function $\chi(\eta)$ and the degree of second-order coherence $g^{(2)}(\tau)$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state, to study the different criteria for nonclassicality. In particular, we contrast criteria that involve only one-time functions of the dynamical system, for instance, the quasiprobability distribution $P(\beta)$ of the Glauber-Sudarshan coherent or P-representation of the density of state and the Mandel $Q_{M}(\tau)$ parameter, versus the criteria associated with the two-time correlation function $g^{(2)}(\tau)$.

Aging and related scaling during the collapse of a polymer

We present results showing aging during the collapse of a polymer chain quenched from an expanded state to the globular phase via Monte Carlo dynamics of a model homopolymer. The constructed two-time correlation function, an analogue of the density- density autocorrelation, shows a power-law scaling with respect to the size of monomer clusters formed during the collapse. The numerical estimates obtained at different quench temperatures are in agreement with a theoretically predicted bound for the decay exponent governing the scaling behavior.

Equilibration and Aging of Liquids of Non-Spherically Interacting Particles.

The nonequilibrium self-consistent generalized Langevin equation theory of irreversible processes in liquids is extended to describe the positional and orientational thermal fluctuations of the instantaneous local concentration profile n(r,Ω,t) of a suddenly quenched colloidal liquid of particles interacting through nonspherically symmetric pairwise interactions, whose mean value n(r,Ω,t) is constrained to remain uniform and isotropic, n (r,Ω, t) = n (t). Such self-consistent theory is cast in terms of the time-evolution equation of the covariance [Formula: see text] of the fluctuations [Formula: see text] of the spherical harmonics projections nlm(k;t) of the Fourier transform of n(r,Ω,t). The resulting theory describes the nonequilibrium evolution after a sudden temperature quench of both, the static structure factor projections Slm(k,t) and the two-time correlation function [Formula: see text], where τ is the correlation delay time and t is the evolution or waiting time after the quench. As a concrete and illustrative application we use the resulting self-consistent equations to describe the irreversible processes of equilibration or aging of the orientational degrees of freedom of a system of strongly interacting classical dipoles with quenched positional disorder.

Floquet approach to bichromatically driven cavity-optomechanical systems

We develop a Floquet approach to solve time-periodic quantum Langevin equations in steady state. We show that two-time correlation functions of system operators can be expanded in a Fourier series and that a generalized Wiener-Khinchin theorem relates the Fourier transform of their zeroth Fourier component to the measured spectrum. We apply our framework to bichromatically driven cavity optomechanical systems, a setting in which mechanical oscillators have recently been prepared in quantum-squeezed states. Our method provides an intuitive way to calculate the power spectral densities for time-periodic quantum Langevin equations in arbitrary rotating frames.