Periodically-driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics. Hence, it is of interest to understand what constraints such dynamics must satisfy. We derive a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the recurrence probability. We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters. To illustrate the relevance of these results for modern quantum systems we demonstrate our findings experimentally on a trapped-ion quantum computer, and on various IBM quantum computers. Specifically, we provide two experimental examples where these constraints surpass fundamental bounds associated with known one-cycle constraints. This scheme can potentially be used to detect the effect of the environment in quantum circuits that cannot be classically simulated. Finally, we show that, in practice, testing an n -cycle constraint requires executing only O(n) cycles, which makes the evaluation of constraints associated with hundreds of cycles realistic.
Periodically-driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins leads to complicated dynamics. Hence, it is of interest to understand what constraints such dynamics must satisfy. We derive a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the recurrence probability. We use our constraints for detecting undesired coupling to unaccounted environments and drifts in the driving parameters. To illustrate the relevance of these results for modern quantum systems we demonstrate our findings experimentally on a trapped-ion quantum computer, and on various IBM quantum computers. Specifically, we provide two experimental examples where these constraints surpass fundamental bounds associated with known one-cycle constraints. This scheme can potentially be used to detect the effect of the environment in quantum circuits that cannot be classically simulated. Finally, we show that, in practice, testing an n -cycle constraint requires executing only O(n) cycles, which makes the evaluation of constraints associated with hundreds of cycles realistic.
Periodically driven systems are ubiquitous in science and technology. In quantum dynamics, even a small number of periodically-driven spins lead to complicated dynamics. By applying the global passivity principle from quantum thermodynamics to periodically driven quantum systems, we obtain a set of constraints for each number of cycles. For pure initial states, the observable being constrained is the survival probability. We use our constraints to detect undesired coupling to unaccounted environments in quantum circuits as well as for detecting deviations from perfect periodic driving. To illustrate the applicability of these results to modern quantum systems we demonstrate our findings experimentally on on a trapped ion quantum computer, and on various IBM quantum computers. Specifically, we provide two experimental examples where these constraints surpass fundamental bounds on leakage detection based on the standard one-cycle detection scheme (including the second law of thermodynamics). The derived constraints treat the driven circuit as a black box and make no assumptions about it. Thus, by running the same circuit several times sequentially, this scheme can be used as a diagnostic tool for quantum circuits that cannot be classically simulated. Finally, we show that , in practice, testing an n-cycle constraint requires only O(n) cycles.
The Poincaré recurrence theorem shows that conservative systems in a bounded region of phase space eventually return arbitrarily close to their initial state after a finite amount of time. An analogous behavior occurs in certain quantum systems where quantum states can recur after sufficiently long unitary evolution, a phenomenon known as quantum recurrence. Periodically driven (i.e. Floquet) quantum systems in particular exhibit complex dynamics even in small dimensions, motivating the study of how interactions and Hamiltonian structure affect recurrence behavior. While most studies treat recurrence in an approximate, distance-based sense, here we address the problem of {exact}, state-independent recurrences in a broad class of finite-dimensional Floquet systems, spanning both integrable and non-integrable models. Leveraging techniques from algebraic field theory, we construct an arithmetic framework that identifies all possible recurrence times by analyzing the cyclotomic structure of the Floquet unitary's spectrum. This computationally tractable approach yields both positive results, enumerating all candidate recurrence times, and definitive negative results, rigorously ruling out candidate recurrence times for a given set of Hamiltonian parameters. We further prove that rational Hamiltonian parameters do not, in general, guarantee exact recurrences, revealing a subtle interplay between system parameters and long-time dynamics. Our findings sharpen the theoretical understanding of quantum recurrences, clarify their relationship to quantum chaos, and highlight parameter regimes of special interest for quantum metrology and control.
Recurrence time quantifies the duration required for a physical system to return to its initial state, playing a pivotal role in understanding the predictability of complex systems. In quantum systems with subspace measurements, recurrence times are governed by Anandan-Aharonov phases, yielding fractionally quantized recurrence times. However, the fractional quantization phenomenon in interacting quantum systems remains unexplored. Here, we address this gap by establishing universal lower and upper bounds for recurrence times in interacting many-body spin systems. Notably, we investigate scenarios where these bounds are approached, shedding light on the speed of quantum processes under monitoring. In specific cases, our findings reveal that the complex many-body system can be effectively mapped onto a dynamical system with a single quasi-particle, leading to integer-quantized recurrence times. Our work demonstrates a valuable link between recurrence times and the number of dark states in the system, thus providing a deeper understanding of the intricate interplay between Hilbert-space fragmentation, ergodicity breaking, measurements, and interaction effects. Finally, our findings have been implemented on an IBM quantum computer, revealing resonances and fractional quantization in agreement with theoretical predictions. This demonstrates the resilience of nonequilibrium topological fractional quantization to noise and highlights its potential use for benchmarking quantum devices and probing dark states.
Periodically driven quantum systems exhibit a wide range of complex dynamics, including chaos. Yet, at special parameter values, their evolution can recur exactly. Several quantum protocols have recently been proposed that exploit exact Floquet recurrences to demonstrate genuine quantum advantage. Such recurrences can be studied through the arithmetic structure of the Floquet unitary. In this work, we establish a necessary and sufficient arithmetic criterion for exact recurrence and determine the minimal recurrence period when one exists. Because recurrence is a global property of the unitary, it holds for every initial state. Applying our framework requires only knowledge of the number field associated with the Hamiltonian parameters. We demonstrate the method using the kicked Ising chain, establishing exact recurrences in both Clifford and non-Clifford regimes. We further show that, in the recurrent non-Clifford regime, magic--a resource for quantum computation--is generated at intermediate times and undergoes exact periodic revivals enforced by the recurrence of the full Floquet unitary. By connecting number theory, quantum many-body dynamics, and quantum information, our work shows that exact recurrence extends beyond Clifford circuits and opens new avenues for magic generation and quantum hardware benchmarking.
Driven quantum systems are systems in which a set of parameters obeys a time-dependent protocol, both periodic and non-periodic. The study of these systems bridges theoretical advances with experimental realizations in a variety of platforms—these range from quantum dot arrays to molecular junctions to trapped ions. The theoretical study is challenging due to the inherent non-equilibrium nature of the problem. These systems offer an interesting playground for quantum thermodynamics. They can be studied as quantum machines, such as refrigerators, heat pumps or engines. This poses the challenge of redefining the laws of thermodynamics in the quantum realm. This effort implies taking into account the presence of coherence, entanglement and strong correlations with the environment. In this conceptual framework, we attempt to shed new light on specific setups with applications in quantum metrology. In the first part of the Thesis, we present novel results on the thermodynamics and transport properties of quantum dots. After thoroughly introducing the main methods, namely non-equilibrium Green's functions and scattering matrix techniques, we apply these tools to adiabatic charge pumping. In this context, quantum dots, subjected to external periodic driving, act as nano-mechanical engines, transferring charge from a source to a drain. We investigate the conditions under which the pumped charge becomes quantized and how thermodynamic quantities behave in this regime. Next, we will focus on the charge shuttle mechanism in movable quantum dots as an example of a quantum clock. This physical model describes transport through a molecular state bound to two leads by van der Waals forces, with an electric field pushing the molecule from the source to the drain. The periodic oscillations arising from the system's dissipative limit cycle solution provide the basis for time measurement. We examine quantum noise primarily associated with the tunnelling process. In the second part, we propose a quantum clock that utilizes the free energy resources generated by coupling to a quantum battery, consisting of an integrable spin chain driven out of equilibrium by a quench in a chosen parameter. The operating conditions of the clock in the manifold of parameters are examined, leading to the requirement of the crossing of the critical point in the battery. The lifetime of the battery's resources is found to be extensive in its size, even when choosing a global coupling observable in the battery.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.