Driven Quantum Systems Research Articles

Simulating the Landau-Zener transitions and Landau-Zener-Stückelberg interferometers with compacted optical waveguides: An invariant method

Experimental demonstrations of coherent dynamics for the driven quantum system are usually limited by their shorter coherent times (due to the inevitable environment noises). Given the Maxwell equation for the electromagnetic waves (EMWs) prorogating in the optical waveguides takes the similar form of the Schr\"odinger equation for the driven quantum system, the quantum dynamics in time domain can be simulated by the light propagating along the waveguide in spatial domain. To simulate the fast time evolutions of quantum states during the Landau-Zener (LZ) transitions and Landau-Zenner-St\"uckelberg (LZS) interferences for the periodically driven two-level systems, we develop an invariant method to design the compacted curved wavguides and demonstrate the LZ transitions and LZS interferometery in the spatial domain. Due to significantly long coherent lengths of the EMWs propagating along the waveguides, the proposed optical-quantum analogies should be feasible with the current integrated optical devices.

Exact solution to a kind of multi-level Rosen-Zener models

Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.

Statistical periodicity in driven quantum systems: General formalism and application to noisy Floquet topological chains

Much recent experimental effort has focused on the realization of exotic quantum states and dynamics predicted to occur in periodically driven systems. But how robust are the sought-after features, such as Floquet topological surface states, against unavoidable imperfections in the periodic driving? In this work, we address this question in a broader context and study the dynamics of quantum systems subject to noise with periodically recurring statistics. We show that the stroboscopic time evolution of such systems is described by a noise-averaged Floquet superoperator. The eigenvectors and -values of this superoperator generalize the familiar concepts of Floquet states and quasienergies and allow us to describe decoherence due to noise efficiently. Applying the general formalism to the example of a noisy Floquet topological chain, we re-derive and corroborate our recent findings on the noise-induced decay of topologically protected end states. These results follow directly from an expansion of the end state in eigenvectors of the Floquet superoperator.

Avoided Crossing and sub-Fourier-sensitivity in Driven Quantum Systems.

The response of a linear system to an external perturbation is governed by the Fourier limit, with the inverse of the interaction time constituting a lower limit for the system bandwidth. This does not hold for nonlinear systems, which can thus exhibit sub-Fourier-behavior. The present Letter identifies a mechanism for sub-Fourier-sensitivity in driven quantum systems, which relies on avoided crossing between Floquet states. Features up to three orders of magnitude finer than the Fourier limit are presented.

Low-frequency spectroscopy for quantum multilevel systems

A periodically driven quantum system with avoided-level crossing experiences both non-adiabatic transitions and wave-function phase changes. These result in coherent interference fringes in the system's occupation probabilities. For qubits, with repelling energy levels, such interference, named after Landau-Zener-Stuckelberg-Majorana, displays arc-shaped resonance lines. We demonstrate that in the case of a multi-level system with an avoided-level crossing of the two lower levels, the shape of the resonances can change from convex arcs to concave heart-shaped and harp-shaped resonance lines. In this way, the shape of such resonance fringes is defined by the whole spectrum, providing insight on the slow-frequency system spectroscopy. As a particular example, we consider this for valley-orbit silicon quantum dots.

Bloch-like energy oscillations

We identify a new type of periodic evolution that appears in driven quantum systems. Provided that the instantaneous (adiabatic) energies are equidistant we show how such systems can be mapped to (time-dependent) tilted single-band lattice models. Having established this mapping, the dynamics can be understood in terms of Bloch oscillations in the instantaneous energy basis. In our lattice model the site-localized states are the adiabatic ones, and the Bloch oscillations manifest as a periodic repopulation among these states, or equivalently a periodic change in the system's instantaneous energy. Our predictions are confirmed by considering two different models: a driven harmonic oscillator and a Landau-Zener grid model. Both models indeed show convincing, or even perfect, oscillations. To strengthen the link between our energy Bloch oscillations and the original spatial Bloch oscillations we add a random disorder that breaks the translational invariance of the spectrum. This verifies that the oscillating evolution breaks down and instead turns into a ballistic spreading.

A Necessary Condition for Quantum Adiabaticity Applied to the Adiabatic Grover Search

Numerous sufficient conditions for adiabaticity of the evolution of a driven quantum system have been known for quite a long time. In contrast, necessary adiabatic conditions are scarce. A practicable necessary condition well-suited for many-body systems has been proven recently in [Phys. Rev. Lett. 119, 200401 (2017)]. Here we tailor this condition for estimating run times of quantum adiabatic algorithms. As an illustration, the condition is applied to the adiabatic algorithm for searching in an unstructured database (adiabatic Grover search algorithm). We find that thus obtained lower bound on the run time of this algorithm reproduces $\sqrt N$ scaling ($N$ being the number of database entries) of the explicitly known optimal run time. This observation highlights the merits of the new adiabatic condition and its potential relevance to adiabatic quantum computing.

Floquet analysis of a fractal-spectrum-generating periodically driven quantum system

We employ Floquet analysis to study the spectral properties of a double-kicked top (DKT) system. This is a classically nonintegrable dynamical system, which also shows chaos. However, even for the underlying classically chaotic dynamics, the quantum quasienergy spectrum of this system does not follow the random matrix conjecture which was proposed for the quantum spectrum of any classically chaotic systems. Instead the quasienergy spectrum of the DKT system shows a butterfly-like self-similar fractal spectrum. Here we investigate the relation between the quasienergy spectrum and the energy spectrum of the corresponding time-independent Floquet Hamiltonian. This Hamiltonian is determined by factorizing the Floquet time-evolution operator into three terms: an initial kick and a final kick, and in between a time-independent evolution dictated by a time-independent Hermitian operator which is called the Floquet Hamiltonian. Like any other generic systems, the Floquet Hamiltonian of the DKT system is also not possible to determine exactly. We apply a recently proposed perturbation theory to obtain the approximate Floquet Hamiltonian at the high-frequency driving limit. We then study the parameter regime where the quasienergy spectrum of the Floquet time-evolution operator matches the energy spectrum of the approximate Floquet Hamiltonian. We have also done a comparative analysis of how the two butterfly spectra disappear with the variation of a system parameter. Finally, we also explore the self-similar property of the energy spectrum of the approximate Floquet Hamiltonian and find its connection with the Farey sequence. Unlike all previous studies, here we have extensively investigated the self-similar property of the whole DKT butterfly.

Floquet–Gibbs state in open quantum systems

We study long-time asymptotic states of periodically driven quantum systems coupled to a thermal bath. In order to describe a class of such a system, we introduce the Floquet-Gibbs state, i.e. the state whose density matrix is diagonalized in the basis of the Floquet state of the system Hamiltonian, and its diagonal element obeys the Boltzmann distribution over its Floquet quasienergy. We obtain sufficient conditions for the realization of the Floquet-Gibbs state in a system with infinitesimal system-bath coupling [T. Shirai, et al., Phys. Rev. E 91, 030101 (2015)]. These conditions severely restrict a class of suitable physical models attaining the Floquet-Gibbs state. We also show that some of the conditions can be lifted by imposing conditions on timescales of the thermal bath with the aid of the truncated Floquet Hamiltonian in the Floquet-Magnus expansion [T. Shirai, et al., New Journal of Physics 18, 053008 (2016)]. In this paper we give a overview of this theory and reconsider it by looking at the dynamics from a rotating frame.

Floquet perturbation theory: formalism and application to low-frequency limit

We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger–Rabi and the Landau-Zener–Stückelberg–Majorana models. We reproduce some known expressions for transition probabilities in a simple and systematic way and clarify and extend their regime of applicability. We then apply the theory to a periodically-driven system of fermions on the lattice and obtain the spectral properties and the low-frequency dynamics of the system.

Quantum work statistics, Loschmidt echo and information scrambling

A universal relation is established between the quantum work probability distribution of an isolated driven quantum system and the Loschmidt echo dynamics of a two-mode squeezed state. When the initial density matrix is canonical, the Loschmidt echo of the purified double thermofield state provides a direct measure of information scrambling and can be related to the analytic continuation of the partition function. Information scrambling is then described by the quantum work statistics associated with the time-reversal operation on a single copy, associated with the sudden negation of the system Hamiltonian.

Schrödinger–Koopman quasienergy states of quantum systems driven by classical flow

We study the properties of the quasienergy states of a quantum system driven by a classical dynamical system. The quasienergies are defined in the same manner as in light-matter interaction but where the Floquet approach is generalized by the use of the Koopman approach of dynamical systems. We show how the properties of the classical flow (fixed and cyclic points, ergodicity, chaos) influence the driven quantum system. This approach of Schrödinger–Koopman quasienergies can be applied to quantum control, quantum information in presence of noises, and dynamics of mixed classical-quantum systems. We treat the example of a kicked spin ensemble where the kick modulation is governed by discrete classical flows as the Arnold’s cat map and the Chirikov standard map.

Asymptotic theory of quasiperiodically driven quantum systems.

The theoretical treatment of quasiperiodically driven quantum systems is complicated by the inapplicability of the Floquet theorem, which requires strict periodicity. In this work we consider a quantum system driven by a biharmonic driving and examine its asymptotic long-time limit, the limit in which features distinguishing systems with periodic and quasiperiodic driving occur. Also, in the classical case this limit is known to exhibit universal scaling, independent of the system details, with the system's reponse under quasiperiodic driving being described in terms of nearby periodically driven system results. We introduce a theoretical framework appropriate for the treatment of the quasiperiodically driven quantum system in the long-time limit and derive an expression, based on Floquet states for a periodically driven system approximating the different steps of the time evolution, for the asymptotic scaling of relevant quantities for the system at hand. These expressions are tested numerically, finding excellent agreement for the finite-time average velocity in a prototypical quantum ratchet consisting of a space-symmetric potential and a time-asymmetric oscillating force.

Relations between the single-pass and double-pass transition probabilities in quantum systems with two and three states

In the experimental determination of the population transfer efficiency between discrete states of a coherently driven quantum system it is often inconvenient to measure the population of the target state. Instead, after the interaction that transfers the population from the initial state to the target state, a second interaction is applied which brings the system back to the initial state, the population of which is easy to measure and normalize. If the transition probability is $p$ in the forward process, then classical intuition suggests that the probability to return to the initial state after the backward process should be $p^2$. However, this classical expectation is generally misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) and SU(3) symmetries of the propagators describing the evolution of quantum systems with two and three states, resulting in explicit analytic formulas that link the two-step probabilities to the single-step ones. Explicit examples are given with the popular techniques of rapid adiabatic passage and stimulated Raman adiabatic passage. The present results suggest that quantum-mechanical probabilities degrade faster in repeated processes than classical probabilities. Therefore, the actual single-pass efficiencies in various experiments, calculated from double-pass probabilities, might have been greater than the reported values.

Coherence-enhanced phase-dependent dissipation in long SNS Josephson junctions: Revealing Andreev bound state dynamics

One of the best known causes of dissipation in ac driven quantum systems stems from photon absorption. Dissipation can also be caused by the retarded response to the time-dependent excitation, and in general gives insight into the system's relaxation times and mechanisms. We address the dissipation in a mesoscopic normal wire with superconducting contacts, that sustains a supercurrent at zero frequency and that may be expected to remain dissipationless at frequency lower than the superconducting gap. We probe the high frequency linear response of a Normal/Superconductor ring to a time-dependent flux by coupling it to a highly sensitive multimode microwave resonator. Far from being the simple derivative of the current-phase relation, the ring's ac susceptibility also displays a dissipative component whose phase dependence is a signature of the dynamical processes occurring within the Andreev spectrum. We show how dissipation is driven by the competition between the two aforementioned mechanisms. Depending on the relative strength of those contributions, dissipation can be maximal at $\pi$, when the minigap closes, or can be maximal near $ \pi/2$, when the dc supercurrent is maximal. We also find that the dissipative response increases at low temperature and can even exceed the normal state conductance. The results are confronted with predictions of the Kubo linear response and time-dependent Usadel equations. This experiment shows the power of the ac susceptibility measurement of individual hybrid mesoscopic systems in probing in a controlled way the quantum dynamics of ABS. By spanning different physical regimes, our experiments provide a unique access to inelastic scattering and spectroscopy of an isolated quantum coherent system. This technique should be a tool of choice to investigate topological superconductivity and detect the topological protection of edge states.

Quantum ring with the Rashba spin-orbit interaction in the regime of strong light-matter coupling

We developed the theory of electronic properties of semiconductor quantum rings with the Rashba spin-orbit interaction irradiated by an off-resonant high-frequency electromagnetic field (dressing field). Within the Floquet theory of periodically driven quantum systems, it is demonstrated that the dressing field drastically modifies all electronic characteristics of the rings, including spin-orbit coupling, effective electron mass and optical response. Particularly, the present effect paves the way to controlling the spin polarization of electrons with light in prospective ring-shaped spintronic devices.

Controlling symmetry and localization with an artificial gauge field in a disordered quantum system

Anderson localization, the absence of diffusion in disordered media, draws its origins from the destructive interference between multiple scattering paths. The localization properties of disordered systems are expected to be dramatically sensitive to their symmetries. So far, this question has been little explored experimentally. Here we investigate the realization of an artificial gauge field in a synthetic (temporal) dimension of a disordered, periodically driven quantum system. Tuning the strength of this gauge field allows us to control the parity–time symmetry properties of the system, which we probe through the experimental observation of three symmetry-sensitive signatures of localization. The first two are the coherent backscattering, marker of weak localization, and the recently predicted coherent forward scattering, genuine interferential signature of Anderson localization. The third is the direct measurement of the β(g) scaling function in two different symmetry classes, allowing to demonstrate its universality and the one-parameter scaling hypothesis.

Quantum work fluctuations versus macrorealism in terms of non-extensive entropies

Fluctuations of the work performed on a driven quantum system can be characterized by the so-called fluctuation theorems. The Jarzynski relation and the Crooks theorem are famous examples of exact equalities characterizing non-equilibrium dynamics. Such statistical theorems are typically formulated in a similar manner in both classical and quantum physics. Leggett–Garg inequalities are inspired by the two assumptions referred to as the macroscopic realism and the non-invasive measurability. Together, these assumptions are known as the macrorealism in the broad sense. Quantum mechanics is provably incompatible with restrictions of the Leggett–Garg type. It turned out that Leggett–Garg inequalities can be used to distinguish quantum and classical work fluctuations. We develop this issue with the use of entropic functions of the Tsallis type. Varying the entropic parameter, we are often able to reach more robust detection of violations of the corresponding Leggett–Garg inequalities. In reality, all measurement devices suffer from losses. Within the entropic formulation, detection inefficiencies can naturally be incorporated into the consideration. This question also shows advantages that are provided due to the use of generalized entropies.

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

We introduce ${\mathbb{Z}}_{2}$-valued bulk invariants for symmetry-protected topological phases in $2+1$-dimensional driven quantum systems. These invariants adapt the ${W}_{3}$ invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a nonzero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and particle-hole symmetric graphene model with periodic driving, where they successfully predict the appearance of boundary states that exist despite the trivial topological character of the Floquet bands. Especially for particle-hole symmetry, the combination of the ${W}_{3}$ and the ${\mathbb{Z}}_{2}$ invariants allows us to distinguish between weak and strong topological phases.

Time-translation-symmetry breaking in a driven oscillator: From the quantum coherent to the incoherent regime

We study the breaking of the discrete time-translation symmetry in small periodically driven quantum systems. These systems are intermediate between large closed systems and small dissipative systems, which both display such symmetry breaking but have qualitatively different dynamics. As a nontrivial example, strongly different from the familiar case of parametric resonance, we consider period tripling in a quantum nonlinear oscillator. We develop theoretical methods of the analysis of period tripling, including the theory of multiple-state resonant tunneling in phase space with the account taken of the involved geometric phase. For moderately strong driving, the period tripling persists for a time, which is exponentially long compared with all dynamical times. This time is further extended by an even weak decoherence.