Rotating Wave Approximation Research Articles

Detecting quantum vacuum fluctuations of the electromagnetic field

Abstract Quantum vacuum fluctuations of the electromagnetic field result in two signatures on a harmonically trapped charged particle: a shift from the natural trap frequency and generation of quantum coherences. We assess the role of the long-wavelength and rotating-wave approximations in estimating this frequency shift. We estimate the magnitude of the frequency shift using parameters from a single-electron cyclotron experiment and also demonstrate how the dependence of the frequency shift on the magnetic field of the cyclotron is tied to the rotating-wave approximation. We expect the frequency shift to be observable in future experiments. We also suggest a possible route to detecting vacuum-generated quantum coherences. These experiments should settle the debate on
the choice of approximations and gauge in capturing the effect of the quantum vacuum fluctuations.

Superradiant phase transitions in ultrastrong coupling regime

Abstract One of the long-standing problems in quantum optics and laser physics is the observation of both equilibrium and non-equilibrium (Dicke) superradiant states in two-level (qubit) systems interacting with the quantized electromagnetic (EM) field. We demonstrate that the superradiant phase transition (PT) at thermal equilibrium in such a system is close to ultrastrong light–matter coupling (USC) condition, which is currently available for superconducting devices. We examine the USC regime for EM fields interacting with two-level systems located in highly connected graph nodes of photonic networks beyond the rotating wave approximation. We find that a critical matter–field coupling parameter reduces ⟨ k ⟩ times due to network peculiarities providing suitable conditions for equilibrium PT observation; ⟨ k ⟩ is a network average node degree. We study the non-equilibrium (Dicke) superradiance for the same network system and show that superradiance appears as a non-equilibrium PT to the superradiant laser operating within a bad cavity limit. We show that population inversion in this case plays the same role as the reciprocal temperature inherent to the equilibrium superradiant PT. Our results open new perspectives for experimental observation of superradiance in cavity-free systems in the gigahertz photonic regime and beyond.

Microwave-optical spectroscopy of Rydberg excitons in the ultrastrong driving regime

Abstract We study the ultrastrong driving of Rydberg excitons in Cu$_2$O by a microwave field. The effect of the field is studied using optical absorption spectroscopy, and through the observation of sidebands on the transmitted laser light. A model based on Floquet theory is constructed to study the system beyond the rotating wave approximation. We obtain near quantitative agreement between theory and experiment across a 16-fold range of microwave field strengths spanning from the perturbative to the deep strong driving regime. Compared to Rydberg atoms, the large non-radiative widths of Rydberg excitons leads to new behaviour such as the emergence of an absorption continuum without ionization.

Nonlinearity-induced symmetry breaking in a system of two parametrically driven Kerr-Duffing oscillators

Abstract We study the classical dynamics of a system comprising a pair of Kerr-Duffing nonlinear oscillators, which are coupled through a nonlinear interaction and subjected to a parametric drive. Using the rotating wave approximation (RWA), we analyze the steady-state solutions for the amplitudes of the two oscillators. For the case of almost identical oscillators, we investigate separately the cases in which only one oscillator is parametrically driven and in which both oscillators are simultaneously driven. In the latter regime, we demonstrate that even when the parametric drives acting on the two oscillators are identical, the system can transition from a stable symmetric solution to a broken-symmetry solution as the detuning is varied.

CoOMBE: A suite of open-source programs for the integration of the optical Bloch equations and Maxwell-Bloch equations

The programs described in this article and distributed with it aim (1) at integrating the optical Bloch equations governing the time evolution of the density matrix representing the quantum state of an atomic system driven by laser or microwave fields, and (2) at integrating the 1D Maxwell-Bloch equations for one or two laser fields co-propagating in an atomic vapour. The rotating wave approximation is assumed. These programs can also be used for more general quantum dynamical systems governed by the Lindblad master equation. They are written in Fortran 90; however, their use does not require any knowledge of Fortran programming. Methods for solving the optical Bloch equations in the rate equations limit, for calculating the steady-state density matrix and for formulating the optical Bloch equations in the weak probe approximation are also described. Program summaryProgram Title: CoOMBECPC Library link to program files:https://doi.org/10.17632/5wsg9d52dk.1Developers' repository link:https://github.com/durham-qlm/CoOMBELicensing provisions: GPLv3Programming language: Fortran 90Nature of problem: The present programs can be used for the following operations: (1) Integrating the optical-Bloch equations within the rotating wave approximation for a multi-state atomic system. At the choice of the user, the calculation will return either the time-dependent density matrix at given times or the density matrix in the long time limit if the system evolves into a steady state in that limit. The calculation can be done with or without averaging over the thermal velocity distribution of the atoms. The number of atomic states which can be included in the calculation is limited only by the CPU time available and possibly by memory requirements. An arbitrarily large number of laser or microwave fields can be included in the calculation if these fields are all CW. This number is currently limited to one or two for fields that are not all CW. The calculation can be done in the weak probe approximation, or in the rate equations approximation, or without assuming either of these two approximations. Calculating refractive indexes, absorption coefficients and complex susceptibilities is also possible. (2) Integrating the 1D Maxwell-Bloch equations in the slowly varying envelope approximation for one or two fields co-propagating in a single-species atomic vapour. Although geared towards the case of atoms interacting with laser fields, this code can also be used for more general quantum systems with similar equations of motion (e.g., molecular systems, spin systems, etc.).Solution method: The Lindblad master equation is expressed as a system of homogeneous first order linear differential equations, which are transformed as required and solved to obtain the density matrix representing the state of the atomic system. A variety of methods are offered to this end. The same approach is also used in the calculation of the polarisation of the medium when integrating the Maxwell-Bloch equations. The latter are integrated over space using predictor-corrector methods. The library includes a general driving program making it possible to use these codes without additional program development. The distribution also includes examples of the use of a container for running these programs without a pre-installed Fortran compiler.

Analytical solution for Ξ-type qutrit interacting nonlinearly with a cavity field through Kerr-like medium with intrinsic noise present

This paper presents an analytical solution for the Milburn equation describing the interaction between a 3-level atom in a [Formula: see text]-configuration, a cavity field, and a nonlinear Kerr-like medium. The Hamiltonian of the model is presented in the rotating wave approximation (RWA) under specific conditions. The obtained analytical solution accounts for the impact of intrinsic decoherence and nonlinear terms on the dynamics of various physical phenomena. Specifically, we explore the behavior of atomic population, entropy functions, entanglement, negativity, and correlation, highlighting their dependence on the physical parameters of the [Formula: see text]-level atom-cavity system.

Jaynes–Cummings model breaks down when the cavity geometry significantly reduces free-space emission

Strong coupling between a single resonator mode and a single quantum emitter is key to a plethora of experiments and applications in quantum science and technology and is commonly described by means of the Jaynes–Cummings model. Here, we show that the Jaynes–Cummings model only applies when the cavity does not significantly change the emitter’s decay rate into free-space. Most notably, the predictions made by the Jaynes–Cummings model become increasingly wrong when approaching the ideal emitter-resonator systems with no free-space decay channels. We present a Hamiltonian that provides, within the validity range of the rotating wave approximation, a correct theoretical description that applies to all regimes. As minimizing the coupling to free-space modes is paramount for many cavity-based applications, a correct description of strong light-matter interaction is therefore crucial for developing and optimizing quantum protocols.

Spectral and dynamical validity of the rotating-wave approximation in the quantum and semiclassical Rabi models [Invited

Ultrastrong coupling (USC) in the quantum Rabi model, characterized by the breakdown of the rotating-wave approximation (RWA), has emerged as a topic of considerable interest and study. This critical reevaluation of the validity of the RWA concludes that the accepted definition of USC in terms of a fixed ratio of coupling to field frequency is inadequate. Connecting an improved spectral validity criterion with the derivation of the semiclassical limit suggests that the dynamical validity of the quantum RWA should be linked to that of the corresponding semiclassical model. This, however, is not supported by numerical calculations of coherent-state dynamics, which unambiguously demonstrate that spectral validity does not imply dynamical validity and reveal surprisingly complicated dependence on coupling and field amplitude.

Non-Gaussian feature of the output field from a double qubit-cavity ultrastrong coupling system

We investigate the non-Gaussian feature of radiation in a circuit quantum electrodynamics (QED) system where two qubits are strongly coupled to a single-mode cavity. In the regime of ultrastrong coupling (USC), the rotating-wave approximation is not valid, and the Rabi Hamiltonian contains counter-rotating wave terms, leading to level crossing and avoided crossings in the energy spectrum. We further analyze the intensity-amplitude correlation of the output field in these two novel scenarios. In the USC regime, the creation and annihilation operators in the correlation function are replaced, allowing for the identification of non-Gaussian features in the output field. Our findings reveal that despite the absence of squeezing effects in the output light, significant non-Gaussian characteristics are present. Additionally, we demonstrate that as the driving or coupling strength increases, the non-Gaussian features of the output field become more pronounced. This suggests that USC systems hold broad potential applications in the realms of nonlinear optics and the generation of non-Gaussian states.

On the relevance of weak measurements in dissipative quantum systems

We investigate the impact of dissipation, including energy relaxation and decoherence, on weak measurements. While weak measurements have been successful in signal amplification, dissipation can compromise their usefulness. More precisely, we show that in systems with a unique steady state, weak values always converge to an expectation value of the measured observable as dissipation time tends to infinity, in contrast to systems with multiple steady states, where the weak values can remain anomalous, i.e. outside the range of eigenvalues of the observable, even in the limit of an infinite dissipation time. In addition, we propose a method for extracting information about the dissipative dynamics of a system using weak values at short dissipation times. Specifically, we explore the amplification of the dissipation rate in a two-level system and the use of weak values to differentiate between Markovian and non-Markovian dissipative dynamics. We also find that weak measurements operating around a weak atom-cavity coupling can probe the atom dissipation through the weak value of non-Hermitian operators within the rotating-wave approximation of the weak interaction.

Application of the quantum model of the rotating wave approximation to the generation of spin waves in nanowires

The paper simulates systems in which two types of spin waves can occur, propagating in opposite directions. To achieve this, the quantum slow-wave approximation is employed, allowing for the determination of the probability distribution of occupied states in a quantum two-particle system subjected to perturbations, commonly referred to as Rabi oscillations. This model is specifically applied to a configuration of parallel nanowires exposed to an external electromagnetic wave within the microwave range. The study explores a potential application of the investigated nanostructures in the development of a device for detecting electromagnetic fields.

Linear response of molecular polaritons.

In this article, we show that the collective light-matter strong coupling regime where N molecular emitters couple to the photon mode of an optical cavity can be mapped to a quantum impurity model where the photon is the impurity that is coupled to a bath of anharmonic transitions. In the thermodynamic limit where N ≫ 1, we argue that the bath can be replaced with an effective harmonic bath, leading to a dramatic simplification of the problem into one of the coupled harmonic oscillators. We derive simple analytical expressions for linear optical spectra (transmission, reflection, and absorption) where the only molecular input required is the molecular linear susceptibility. This formalism is applied to a series of illustrative examples, showing the role of temperature, disorder, vibronic coupling, and optical saturation of the molecular ensemble, explaining that it is useful even when describing an important class of nonlinear optical experiments. For completeness, we provide Appendixes A-C that include a self-contained derivation of the relevant spectroscopic observables for arbitrary anharmonic systems (for both large and small N) within the rotating-wave approximation. While some of the presented results herein have already been reported in the literature, we provide a unified presentation of the results as well as new interpretations that connect powerful concepts in open quantum systems and linear response theory with molecular polaritonics.

Instability of multi-mode systems with quadratic Hamiltonians

We present a novel geometric approach for determining the unique structure of a Hamiltonian and establishing an instability criterion for quantum quadratic systems. Our geometric criterion provides insights into the underlying geometric perspective of instability: A quantum quadratic system is dynamically unstable if and only if its Hamiltonian is non-elliptic (i.e., hyperbolic or lineal). By applying our geometric method, we analyze the stability of two-mode and three-mode optomechanical systems. Remarkably, our approach demonstrates that these systems can be stabilized over a wider range of system parameters compared to the conventional rotating wave approximation (RWA) assumption. Furthermore, we reveal that the systems transit their phases from stable to unstable, when the system parameters cross specific critical boundaries. The results imply the presence of multistability in the optomechanical systems.

Adiabatically Manipulated Systems Interacting with Spin Baths beyond the Rotating Wave Approximation

The Stimulated Raman Adiabatic Passage (STIRAP) on a three-state system interacting with a spin bath is considered, focusing on the efficiency of the population transfer. Our analysis is based on the perturbation treatment of the interaction term evaluated beyond the Rotating Wave Approximation, thus focusing on the limit of weak system–bath coupling. The analytical expression of the correction to the efficiency and the consequent numerical analysis show that, in most of the cases, the effects of the environment are negligible, confirming the robustness of the population transfer.

Controllable spontaneous emission spectrum in an artificial giant atom: Dark lines and bound states

We study the spontaneous emission spectra of a few-level giant atom strongly coupled to a one-dimensional waveguide by employing variational and analytical approaches, which are beyond the rotating-wave approximation and the Markovian approximation. We show that the interference effects due to the multiple coupling points result in dark lines in spontaneous emission spectra of either two-level or three-level giant atoms regardless of whether there is a bound state or not. We illustrate that the generation of dark lines depends on the distance between the coupling points and their number. In the absence of a bound state, the dark lines can be used to suppress emission at a certain frequency in the spectrum. In the presence of a bound state, the total intensity of the emission spectrum can be significantly suppressed. The present results offer the possibility of control over the spontaneous emission with the dark lines due to the destructive interference in giant atoms. Published by the American Physical Society 2024

Periodically driven three-dimensional Kitaev model

We study the dynamics of a three-dimensional generalization of Kitaev’s honeycomb lattice spin model (defined on the hyperhoneycomb lattice) subjected to a harmonic driving of J z , one of the three types of spin-couplings in the Hamiltonian. Using numerical solutions supported by analytical calculations based on a rotating wave approximation, we find that the system responds nonmonotonically to variations in the frequency ω (while keeping the driving amplitude J fixed) and undergoes dynamical freezing, where at specific values of ω, it gets almost completely locked in the initial state throughout the evolution. However, this freezing occurs only when a constant bias is present in the driving, i.e., when Jz=J′+Jcosωt , with J′≠0 . Consequently, the bias acts as a switch that triggers the freezing phenomenon. Dynamical freezing has been previously observed in other integrable systems, such as the one-dimensional transverse-field Ising model.

Bright soliton propagation with loss and gain phenomena in transversely connected nonlinear pendulum pairs

In this work, we seek to investigate the dynamics of bright soliton in a chain of coupled pendulum pairs. After deriving the linear dispersion relation from the equation of the model, we find that among the obtained modes, the fast mode is the one on which we are going to be focused. Since the discrete simultaneous equation describing the dynamics of the model has not been extensively studied in the literature, we assume that the two lines of the model are proportional to each other. We use the rotating wave approximation method to derive a NLS equation governing the propagation of waves in the network. Depending on the choice of wave number, we deduce that the system supports bright and hole-soliton solutions. We use the obtained bright soliton as the initial condition for numerical computation, which demonstrates the significant role of the transverse coupling parameter in the system. That is, it affects the behavior of the forward-bright soliton generated in the system. The lattice allows gain and loss phenomena during the propagation of the waves.

Surface Topological Plexcitons: Strong Coupling in a Bi2Se3 Topological Insulator Nanoparticle-Quantum Dot Molecule

Strong coupling of quantum states with electromagnetic modes of topological matter offer an interesting platform for the exploration of new physics and applications. In this work, we report a novel hybrid mode, a surface topological plexciton, arising from strong coupling between the surface topological plasmon mode of a Bi2Se3 topological insulator nanoparticle and the exciton of a two-level quantum emitter. We study the power absorption spectrum of the system by working within the dipole and rotating-wave approximations, using a density matrix approach for the emitter, and a classical dielectric-function approach for the topological-insulator nanoparticle. We show that a Rabi-type splitting can appear in the spectrum suggesting the presence of strong coupling. Furthermore, we study the dependence of the splitting on the separation of the two nanoparticles as well as the dipole moment of the quantum emitter. These results can be useful for exploring exotic phases of matter, furthering research in topological insulator plasmonics, as well as for applications in the far-infrared and quantum computing.

Taming the Rotating Wave Approximation

The interaction between light and matter is one of the oldest research areas of quantum mechanics, and a field that just keeps on delivering new insights and applications. With the arrival of cavity and circuit quantum electrodynamics we can now achieve strong light-matter couplings which form the basis of most implementations of quantum technology. But quantum information processing also has high demands requiring total error rates of fractions of percentage in order to be scalable (fault-tolerant) to useful applications. Since errors can also arise from modelling, this has brought into center stage one of the key approximations of quantum theory, the Rotating Wave Approximation (RWA) of the quantum Rabi model, leading to the Jaynes-Cummings Hamiltonian. While the RWA is often very good and incredibly useful to understand light-matter interactions, there is also growing experimental evidence of regimes where it is a bad approximation. Here, we ask and answer a harder question: for which experimental parameters is the RWA, although perhaps qualitatively adequate, already not good enough to match the demands of scalable quantum technology? For example, when is the error at least, and when at most, 1%? To answer this, we develop rigorous non-perturbative bounds taming the RWA. We find that these bounds not only depend, as expected, on the ratio of the coupling strength and the oscillator frequency, but also on the average number of photons in the initial state. This confirms recent experiments on photon-dressed Bloch-Siegert shifts. We argue that with experiments reporting controllable cavity states with hundreds of photons and with quantum error correcting codes exploring more and more of Fock space, this state-dependency of the RWA is increasingly relevant for the field of quantum computation, and our results pave the way towards a better understanding of those experiments.

Chain-mapping methods for relativistic light-matter interactions

The interaction between localized emitters and quantum fields, both in relativistic settings and in the case of ultra-strong couplings, requires non-perturbative methods beyond the rotating-wave approximation. In this work we employ chain-mapping methods to achieve a numerically exact treatment of the interaction between a localized emitter and a scalar quantum field. We extend the application range of these methods beyond emitter observables and apply them to study field observables. We first provide an overview of chain-mapping methods and their physical interpretation, and discuss the thermal double construction for systems coupled to thermal field states. Modelling the emitter as an Unruh-DeWitt particle detector, we then calculate the energy density emitted by a detector coupling strongly to the field. As a stimulating demonstration of the approach's potential, we calculate the radiation emitted from an accelerated detector in the Unruh effect, which is closely related to the thermal double construction as we discuss. We comment on prospects and challenges of the method.