Quantum chaotic fluctuation-dissipation theorem: effective Brownian motion in closed quantum systems

We analytically describe the decay to equilibrium of generic observables of a non-integrable system after a perturbation in the form of a random matrix. We further obtain an analytic form for the time-averaged fluctuations of an observable in terms of the rate of decay to equilibrium. Our result shows the emergence of a Fluctuation-Dissipation theorem corresponding to a classical Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions can be tested in quantum simulation experiments, thus helping to bridge the gap between theoretical and experimental research in quantum thermalization. We test our analytic results by exact numerical experiments in a spin-chain. We argue that our Fluctuation-Dissipation relation can be used to measure the density of states involved in the non-equilibrium dynamics of an isolated quantum system

Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension

Quantum devices, such as quantum simulators, quantum annealers, and quantum computers, may be exploited to solve problems beyond what is tractable with classical computers. This may be achieved as the Hilbert space available to perform such `calculations' is far larger than that which may be classically simulated. In practice, however, quantum devices have imperfections, which may limit the accessibility to the whole Hilbert space. We thus determine that the dimension of the space of quantum states that are available to a quantum device is a meaningful measure of its functionality, though unfortunately this quantity cannot be directly experimentally determined. Here we outline an experimentally realisable approach to obtaining the required Hilbert space dimension of such a device to compute its time evolution, by exploiting the thermalization dynamics of a probe qubit. This is achieved by obtaining a fluctuation-dissipation theorem for high-temperature chaotic quantum systems, which facilitates the extraction of information on the Hilbert space dimension via measurements of the decay rate, and time-fluctuations

Quantum chaos and the emergence of statistical physics

The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results

Off-diagonal observable elements from random matrix theory: distributions, fluctuations, and eigenstate thermalization

We derive the eigenstate thermalization hypothesis (ETH) from a random matrix Hamiltonian by extending the model introduced by Deutsch (1991 Phys. Rev. A 43 2046). We approximate the coupling between a subsystem and a many-body environment by means of a random Gaussian matrix. We show that a common assumption in the analysis of quantum chaotic systems, namely the treatment of eigenstates as independent random vectors, leads to inconsistent results. However, a consistent approach to the ETH can be developed by introducing an interaction between random wave-functions that arises as a result of the orthonormality condition. This approach leads to a consistent form for off-diagonal matrix elements of observables. From there we obtain the scaling of time-averaged fluctuations of generic observables with system size for which we calculate an analytic form in terms of the inverse participation ratio. The analytic results are compared to exact diagonalizations of a quantum spin chain for different physical observables in multiple parameter regimes

Quantum dynamics in a single excitation subspace: deviations from eigenstate thermalization via long time correlations

In this work we study a scenario where unitary quantum dynamics in a many-body interacting system is restricted to a single excitation subspace. We ask how dynamics within to such a subspace may in general differ from predictions of the eigenstate thermalization hypothesis (ETH). We show that for certain initial states and observables, if thermalization occurs, it will not fulfil other key predictions of the ETH; instead following differing generic behaviours. We show this by analysing long-time fluctuations, two-point correlation functions, and the out-of-time-ordered correlator; analytically detailing deviation from ETH predictions. We derive instead an ETH-like relation, with non-random off-diagonals for matrix elements of observables, with correlations which alter long-time behaviour and constrain dynamics. Further, we analytically compute the time-dependence of the decay to equilibrium, showing it is proportional to the survival probability of the initial state. We finally note the conditions studied are common in many physical scenarios, such as under the rotating-wave approximation. We show numerically our predictions are robust to perturbations which break this approximation

A 100 kHz time-resolved multiple-probe femtosecond to second infrared absorption spectrometer

We present a dual-amplifier laser system for time-resolved multiple-probe infrared (IR) spectroscopy based on the ytterbium potassium gadolinium tungstate (Yb:KGW) laser medium. Comparisons are made between the ytterbium-based technology and titanium sapphire laser systems for time-resolved IR spectroscopy measurements. The 100 kHz probing system provides new capability in time-resolved multiple-probe experiments, as more information is obtained from samples in a single experiment through multiple-probing. This method uses the high repetition-rate probe pulses to repeatedly measure spectra at 10 μs intervals following excitation allowing extended timescales to be measured routinely along with ultrafast data. Results are presented showing the measurement of molecular dynamics over >10 orders of magnitude in timescale, out to 20 ms, with an experimental time response o

Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories

14 pags., 10 figs., 6 apps.We investigate theoretically the emergence of classical statistical physics in a finite quantum system that is either totally isolated or otherwise subjected to a quantum measurement process. We show via a random matrix theory approach to nonintegrable quantum systems that the set of outcomes of the measurement of a macroscopic observable evolve in time like stochastic variables, whose variance satisfies the celebrated Einstein relation for Brownian diffusion. Our results show how to extend the framework of eigenstate thermalization to the prediction of properties of quantum measurements on an otherwise closed quantum system. We show numerically the validity of the random matrix approach in quantum chain models.Funding from Project No. PGC2018-094792-BI00 (MCIU/AEI/FEDER, UE), EPSRC Grant No. EP/M508172/1, and COST Action No. CA17113

Quantum dynamics in a single excitation subspace: deviations from eigenstate thermalization via long time correlations

25 pags., 9 figs.In this work we study a scenario where unitary quantum dynamics in a many-body interacting system is restricted to a single excitation subspace. We ask how dynamics within to such a subspace may in general differ from predictions of the eigenstate thermalization hypothesis (ETH). We show that for certain initial states and observables, if thermalization occurs, it will not fulfil other key predictions of the ETH; instead following differing generic behaviours. We show this by analysing long-time fluctuations, two-point correlation functions, and the out-of-time-ordered correlator; analytically detailing deviation from ETH predictions. We derive instead an ETH-like relation, with non-random off-diagonals for matrix elements of observables, with correlations which alter long-time behaviour and constrain dynamics. Further, we analytically compute the time-dependence of the decay to equilibrium, showing it is proportional to the survival probability of the initial state. We finally note the conditions studied are common in many physical scenarios, such as under the rotating-wave approximation. We show numerically our predictions are robust to perturbations which break this approximation.We acknowledge funding by EPSRC Grant No. EP/M508172/1, Project PGC2018-094792-B-I00 (MCIU/AEI/FEDER, UE), the Gordon and Betty Moore Foundation Projects GBMF8820 and TIPICQA (COST Action CA17113).Peer reviewe