Floquet theory for temporal correlations and spectra in time-periodic open quantum systems: Application to squeezed parametric oscillation beyond the rotating-wave approximation
C. Navarrete-Benlloch, R. Garcés, N. Mohseni, G. J. de Valcárcel
Open quantum systems can display periodic dynamics at the classical level either due to external periodic modulations or to self-pulsing phenomena typically following a Hopf bifurcation. In both cases, the quantum fluctuations around classical solutions do not reach a quantum-statistical stationary state, which prevents adopting the simple and reliable methods used for stationary quantum systems. Here we put forward a general and efficient method to compute two-time correlations and corresponding spectral densities of time-periodic open quantum systems within the usual linearized (Gaussian) approximation for their dynamics. Using Floquet theory we show how the quantum Langevin equations for the fluctuations can be efficiently integrated by partitioning the time domain into one-period duration intervals, and relating the properties of each period to the first one. Spectral densities, like squeezing spectra, are computed similarly, now in a two-dimensional temporal domain that is treated as a chessboard with one-period x one-period cells. This technique avoids cumulative numerical errors as well as efficiently saves computational time. As an illustration of the method, we analyze the quantum fluctuations of a damped parametrically-driven oscillator (degenerate parametric oscillator) below threshold and far away from rotating-wave approximation conditions, which is a relevant scenario for modern low-frequency quantum oscillators. Our method reveals that the squeezing properties of such devices are quite robust against the amplitude of the modulation or the low quality of the oscillator, although optimal squeezing can appear for parameters that are far from the ones predicted within the rotating-wave approximation.
Squeezed Phonon Lasing via Floquet-Controlled Solid-State Defects
Universidad de La Frontera（弗雷罗纳塔大学）；Universita di Trento（特伦托大学）；INFN-TIFPA, Trento Institute for Fundamental Physics and Applications（意大利国家核物理研究所-特伦托基础与应用物理研究所）；Khalifa University of Science and Technology（哈利法科技大学）；University of Electronic Science and Technology of China（电子科技大学）；Universidad Mayor（麦奥尔大学）
The position variance of a single-mode Yuen states can go below the standard quantum limit. For two-mode squeezed states, it is shown that the time-dependent evolution of the entanglement of formation can be contractive, going below that of the squeezed state with minimum Einstein-Podolsky-Rosen dispersions, and increasing thereafter. The rate of change of the Einstein-Podolsky-Rosen dispersions as a function of the two-mode phases control this process. Contractive entanglement is shown to be equivalent to a rotating phase space accompanied by time-dependent single-mode squeezing.
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-…
Properties of time-periodic Hamiltonians can be exploited to increase the dephasing time of qubits and to design protected one and two-qubit gates. Recently, Huang et al. [Phys. Rev. Applied 15, 034065 (2021)] have shown that time-dependent Floquet states offer a manifold of working points with dynamical protection larger than the few usual static sweet spots. Here, we use the framework of many-mode Floquet theory to describe approaches to robustly control Floquet qubits in the presence of multiple drive tones. Following the same approach, we introduce a longitudinal readout protocol to measur…
We show that a collection of two-level atoms in an optical cavity beyond the rotating wave approximation and in the dispersive and strong-coupling regime constitutes a nonlinear medium and is capable of generating squeezed state of light. It is found that squeezing produced in the strong-coupling regime is significantly higher compared to that produced in the dispersive limit. On the other hand, we also show that it could be possible to observe the Dicke superradiant quantum phase transition in the dispersive regime where the vector potential term is negligible. Such a system can be an essenti…
We show that a thermal reservoir can effectively act as a squeezed reservoir on atoms that are subject to energy-level modulation. For sufficiently fast and strong modulation, for which the rotating-wave-approximation is broken, the resulting squeezing persists at long times. These effects are analyzed by a master equation that is valid beyond the rotating wave approximation. As an example we consider a two-level-atom in a cavity with Lorentzian linewidth, subject to sinusoidal energy modulation. A possible realization of these effects is discussed for Rydberg atoms.
We engineer the fast rotation of a quantum particle confined in an effectively one-dimensional, harmonic trap, for a predetermined rotation angle and time, avoiding final excitation. Different schemes are proposed with different speed limits that depend on the control capabilities. We also make use of trap rotations to create squeezed states without manipulating the trap frequencies.
The extended quantum Rabi models make a significant contribution to understand the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, anisotropic Rabi model and asymmetric Rabi model, into rotating frame, and regard them as periodically driven quantum systems. The analytical solutions of the quasi-energy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the non-stroboscopic dynamics of physical observables such as atomic inversion, transvers…
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Center for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）；Dipartimento di Fisica “A. Volta”, Università di Pavia（帕维亚大学“A.沃尔塔”物理学系）
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs t…
Cavity optomechanics and electromechanics form an established field of research investigating the interactions between electromagnetic fields and the motion of quantum mechanical resonators. In many applications, linearised form of the interaction is used, which allows for the system dynamics to be fully described using a Lyapunov equation for the covariance matrix of the Wigner function. This approach, however, is problematic in situations where the Hamiltonian becomes time dependent as is the case for systems driven at multiple frequencies simultaneously. This scenario is highly relevant as …
A periodically driven quantum system, when coupled to a heat bath, relaxes to a non-equilibrium asymptotic state. In the general situation, the retrieval of this asymptotic state presents a rather non-trivial task. It was recently shown that in the limit of an infinitesimal coupling, using so-called rotating wave approximation (RWA), and under strict conditions imposed on the time-dependent system Hamiltonian, the asymptotic state can attain the Gibbs form. A Floquet-Gibbs state is characterized by a density matrix which is diagonal in the Floquet basis of the system Hamiltonian with the diago…
Generalized squeezing rotating-wave approximation (GSRWA) is proposed by employing both the displacement and the squeezing transformations. A solvable Hamiltonian is reformulated in the same form as the ordinary RWA ones. For a qubit coupled to oscillators experiment, a well-defined Schrödinger-cat-like entangled state is given by the displaced-squeezed oscillator state instead of the original displaced state. For the isotropic Rabi case, the mean photon number and the ground-state energy are expressed analytically with additional squeezing terms, exhibiting a substantial improvement of the GS…
Recent experimental advances enable the manipulation of quantum matter by exploiting the quantum nature of light. However, paradigmatic exactly solvable models, such as the Dicke, Rabi or Jaynes-Cummings models for quantum-optical systems, are scarce in the corresponding solid-state, quantum materials context. Focusing on the long-wavelength limit for the light, here, we provide such an exactly solvable model given by a tight-binding chain coupled to a single cavity mode via a quantized version of the Peierls substitution. We show that perturbative expansions in the light-matter coupling have …
Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be non-trivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet…
It is widely believed that ground-state time crystals are not realizable in realistic macroscopic systems. In particular, Watanabe and Oshikawa proved a theorem that implies the absence of the time-dependent long-range order (TDLRO) in the ground states of short-range many-body systems. However, this theorem does not forbid the presence of the ground-state TDLRO for macroscopic quantities. In this work, we investigate a simple bosonic model with a squeezed ground state and point out that the time-dependence of the ground-state TDLRO for the number operator is proportional to the square of the …
We study the dissipative dynamics of a two-level system (TLS) exposed to strong ac driving. By combing Floquet theory with Van Vleck perturbation theory in the TLS tunneling matrix element, we diagonalize the time-dependent Hamiltonian and provide corrections to the renormalized Rabi frequency of the TLS, which are valid for both a biased and unbiased TLS and go beyond the known high-frequency and rotating-wave results. In order to mimic environmental influences on the TLS, we couple the system weakly to a thermal bath and solve analytically the corresponding Floquet-Bloch-Redfield master equa…
We study the transverse-field Ising model with interactions that are modulated in time. In a rotating frame, the system is described by a time-independent Hamiltonian with many-body interactions, similar to the cluster Hamiltonians of measurement-based quantum computing. In one dimension, there is a three-body interaction, which leads to string order instead of conventional magnetic order. We show that the string order is robust to power-law interactions that decay with the cube of distance. In two and three dimensions, there are five- and seven-body interactions. We discuss adiabatic preparat…
The operation of autonomous finite-time quantum heat engines rely on the existence of a stable limit cycle in which the dynamics becomes periodic. The two main questions that naturally arise are therefore whether such a limit cycle will eventually be reached and, once it has, what is the state of the system within the limit cycle. In this paper we show that the application of Floquet's theory to Lindblad dynamics offers clear answers to both questions. By moving to a generalized rotating frame, we show that it is possible to identify a single object, the Floquet Liouvillian, which encompasses …
The striking nonlinear effects exhibited by cavity QED systems make them a powerful tool in modern condensed matter and atomic physics. A recently discovered example is the quantized pumping of energy into a cavity by a strongly-coupled, periodically-driven spin. We uncover a remarkable feature of these energy pumps: they coherently translate, or boost, a quantum state of the cavity in the Fock basis. Current optical cavity and circuit QED experiments can realize the required Hamiltonian in a rotating frame. Boosting thus enables the preparation of highly-excited non-classical cavity states in…
We study the manipulation of Majorana zero modes in a thin disk made from a $p$-wave superconductor in order to understand their use as a building block for topological quantum computers. We analyze the second-order topological corner modes that arise when an in-plane magnetic field is applied, and calculate their dynamical evolution when rotating the magnetic field, with special emphasis on non-adiabatic effects. We characterize the phase transition between high-frequency and near-adiabatic evolution using Floquet analysis. We show that oscillations persist even in the adiabatic phase because…
I introduce and analyse chiral light--matter interaction in the ultrastrong coupling limit where the rotating-wave approximation cannot be made. Within this limit, a two-level system (TLS) with a circularly polarized transition dipole interacts with a copolarized mode through rotating-wave terms. However, the counter-rotating terms allow the TLS to couple to a counter-polarized mode with the same coupling strength, i.e., one that is completely decoupled within the rotating-wave approximation. Although such a Hamiltonian is not particle number conserving, the conservation of angular momentum ge…
We present the exact analytical solution of the explicitly time-periodic quantum Langevin equation describing the dual-beam backaction-evasion measurement of a single mechanical oscillator quadrature due to Braginsky, Vorontsov and Thorne beyond the commonly used rotating-wave approximation. We show that counterrotating terms lead to extra sidebands in the optical and mechanical spectra and to a modification of the main peak. Physically, the backaction of the measurement is due to periodic coupling of the mechanical resonator to a light field quadrature that only contains cavity-filtered shot …
We study the evolution of the hybrid entangled states in a bipartite (ultra) strongly coupled qubit-oscillator system. Using the generalized rotating wave approximation the reduced density matrices of the qubit and the oscillator are obtained. The reduced density matrix of the oscillator yields the phase space quasi probability distributions such as the diagonal P-representation, the Wigner W-distribution and the Husimi Q-function. In the strong coupling regime the Q-function evolves to uniformly separated macroscopically distinct Gaussian peaks representing 'kitten' states at certain specifie…
The ground state of the photon-matter coupled system described by the Dicke model is found to be perfectly squeezed at the quantum critical point of the superradiant phase transition (SRPT). In the presence of the counter-rotating photon-atom coupling, the ground state is analytically expressed as a two-mode squeezed vacuum in the basis of photons and atomic collective excitations. The variance of a quantum fluctuation in the two-mode basis vanishes at the SRPT critical point, with its conjugate fluctuation diverging, ideally satisfying the Heisenberg uncertainty principle.
Tunable Ultrastrong Magnon-Magnon Coupling and Squeezed States in a Triaxial Van der Waals Antiferromagnet
National Key Laboratory of Spintronics and School of Physics, Nanjing University（南京大学自旋电子学国家重点实验室和物理学院）；School of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院）；Laboratory of Solid State Microstructures, Nanjing University（南京大学固体微结构实验室）；State Key Laboratory of Spintronics, Nanjing University（南京大学自旋电子学国家重点实验室）
Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；University of the Basque Country UPV/EHU（巴斯克大学）；EHU Quantum Center, University of the Basque Country UPV/EHU（巴斯克大学EHU量子中心）；Centro de Física de Materiales (CFM-MPC), CSIC-UPV/EHU（材料物理中心，西班牙国家研究委员会-巴斯克大学）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基克科学基金会）；Department of Physics, University of the Basque Country UPV/EHU（巴斯克大学物理系）
We focus on quantum systems that can be effectively described as a localized spin-$s$ particle subject to a static magnetic field coplanar to a coexisting elliptically rotating time-periodic field. Depending on the values taken on by the static and rotating components, the total magnetic field shows two regimes with different topological properties. Along the boundary that separates these two regimes, the total magnetic field vanishes periodically in time and the system dynamics becomes highly nonadiabatic. We derive a relation between two time-averaged quantities of the system which is linked…
For adiabatically and periodically manipulated dissipative quantum systems we derive, using Floquet theory, a simple Markovian master equation. Contrary to some previous works we explicitly take into account the time dependence of the Hamiltonian and, therefore, obtain a master equation with a time-dependent dissipative part. We illustrate our theory with two examples and compare our results with the previously proposed master equations. In particular, we consider the problem of Cooper pair pumping and demonstrate the inadequacy of the secular (rotating wave) approximation when calculating the…
We investigate the out-of-equilibrium properties of a system of interacting bosons in a ring lattice. We present a Floquet driving that induces clockwise (counterclockwise) circulation of the particles among the odd (even) sites of the ring which can be mapped to a fully connected model of clocks of two counter-rotating species. The clock-like motion of the particles is at the core of a period-$n$ discrete time crystal where $L=2n$ is the number of lattice sites. In presence of a "staircase-like" on-site potential, we report the emergence of a second characteristic timescale in addition to the…
The Mollow triplet is a fundamental signature of quantum optics, and has been observed in numerous quantum systems. Although it arises in the 'strong driving' regime of the quantized field, where the atoms undergo coherent oscillations, it can be typically analyzed within the rotating wave approximation. Here we report the first observation of high-order effects in the Mollow triplet structure due to strong driving. In experiments, we explore the regime beyond the rotating wave approximation using concatenated continuous driving that has less stringent requirements on the driving field power. …
Intermittency in Quantum Graviton-Phonon Conversion
Ecole Normale Supérieure - PSL（巴黎高等师范学院-PSL）；Kobe University（神户大学）；Kyushu University（九州大学）；Quantum and Spacetime Research Institute, Kyushu University（九州大学量子与时空研究所）
Ultracold two-level atoms in optical lattices offer an excellent experimental platform to explore magnon excitations [Fukuhara et al 2013 Nat. Phys. 9, 235; Fukuhara et al 2013 Nature 502, 76]. Here, we investigate how gradient magnetic field and periodically modulated tunneling strength affect the two-magnon excitations in these ultracold atomic systems. In the resonant condition where the driving frequency matches and smooths the potential bias, the system gains translational invariance in both space and time in the rotating frame, and thus we can develop a Floquet-Bloch band theory for two …
We study creation of bi- and multipartite continuous variable entanglement in structures of coupled quantum harmonic oscillators. By adjusting the interaction strengths between nearest neighbors we show how to maximize the entanglement production between the arms in a Y-shaped structure where an initial single mode squeezed state is created in the first oscillator of the input arm. We also consider the action of the same structure as an approximate quantum cloner. For a specific time in the system dynamics the last oscillators in the output arms can be considered as imperfect copies of the ini…
Engineered non-Hermitian systems featuring exceptional points can lead to a host of extraordinary phenomena in diverse fields ranging from photonics, acoustics, opto-mechanics, electronics, to atomic physics. Here we introduce and present non-Hermitian dynamics of coupled optical parametric oscillators (OPOs) arising from phase-sensitive amplification and de-amplification, and show their distinct advantages over conventional non-Hermitian systems relying on laser gain and loss. OPO-based non-Hermitian systems can benefit from the instantaneous nature of the parametric gain, noiseless phase-sen…
