No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Quantum Noise, A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics
... In this section we introduce a discrete time description of the system-emission dynamics coarse grained over the time scales in which the master equation in Eq. (1) is valid. This approach, based on the input-output formalism [84], provides a convenient way to analyze the information transferred from the system to the output field [19,27,32,85,86]. An illustration of the idea behind this formalism is presented in Fig. 2. The input-output field is discretized in time bins of length ∆t, the so-called time-bin modes, each of them corresponding to an independent bosonic mode. ...
... We employ the so-called quantum input-output formalism [84], in which the Hamiltonian in the laboratory frame for the system and environment is given by the sum of the following terms: ...
... We consider the dynamics coarse grained on a timescale much larger than B −1 . At this scale, the field operators a(t) satisfy [a(t), a † (t ′ )] = δ(t − t ′ ) [84]. A discrete-time dynamics can be derived by considering discrete time steps that are small compared to the relaxation timescales of Eq. (1), but much larger than the coarse graining timescale ∆t ≫ B −1 [19,32,85,86]. ...
Many-body quantum systems hosting emergent collective behavior bear the promise to enable quantum enhanced parameter estimation. Formally this means that the variance of the parameter to be estimated decreases faster than , where N is the number of particles forming the quantum system. In practice such scaling is challenging to achieve as the underlying many-body correlations are fragile. Moreover, devising the optimal measurements that indeed tap the quantum enhancement is often rather involved. Here we show that the inefficient detection of the photo emission from a dissipative quantum many-body system is sufficient to reach quantum enhanced parameter estimation even when some loss channels remain completely unmonitored. We illustrate our approach by considering the so-called boundary time-crystal, which is a nonequilibrium many-body system that has been realized recently experimentally in cold atomic gases. By analyzing the structure of the temporal correlations of its emission field, we are able to construct a family of near optimal parameter estimation measurements with a simple interferometric setup.
... Quantum systems invariably interact with their environments, which typically consist of a macroscopic number of degrees of freedom [1][2][3]. These interactions give rise to a diverse range of phenomena, including decoherence, dissipation-induced phase transitions [4,5], and emergence of classical physics in quantum systems [6] among many others [3]. ...
... These interactions give rise to a diverse range of phenomena, including decoherence, dissipation-induced phase transitions [4,5], and emergence of classical physics in quantum systems [6] among many others [3]. Understanding and accurately describing the dynamics of open quantum systems is of fundamental importance in various fields, ranging from quantum information science and quantum computing [7][8][9][10] to condensed-matter physics [11] and quantum optics [1,12]. Recent technological advances [13][14][15][16][17][18][19][20] have opened up new possibilities for experimental studies of open quantum systems, necessitating the development of accurate and computationally efficient theoretical models. ...
... Traditional Markovian approaches [1,[21][22][23] for treating open quantum systems rely on assumptions of weak coupling and timescale separation of the system and the bath, or substantially high temperature, which may not always hold in experimentally relevant setups leading to non-Markovian * Contact author: vasilii.1.vadimov@aalto.fi dynamics [24][25][26][27][28][29][30]. ...
We introduce a numerically exact and computationally feasible nonlinear-response theory developed for lossy superconducting quantum circuits based on a framework of quantum dissipation in a minimally extended state space. Starting from the Feynman-Vernon path-integral formalism for open quantum systems with the system degrees of freedom being the nonlinear elements of the circuit, we eliminate the temporally nonlocal influence functional of all linear elements by introducing auxiliary harmonic modes with complex-valued frequencies coupled to the nonlinear elements. In our work, we propose a concept of time-averaged observables, inspired by experiment, and provide an explicit formula for producing their quasiprobability distribution. We illustrate the consistency of our formalism with the well-established Markovian input-output theory by applying them the dispersive readout of a superconducting transmon qubit. For an important demonstration of our approach beyond weak coupling, we analyze the low-frequency linear response of a capacitively and resistively shunted Josephson junction and observe signatures of a much-debated quantum phase transition at a finite temperature. The developed framework enables a comprehensive fully quantum-mechanical treatment of nonlinear quantum circuits coupled to their environment, without the limitations of typical approaches to weak dissipation, high temperature, and weak drive. This versatile tool paves the way for accurate models of quantum devices and increased fundamental understanding of quanutm mechanics such as that of the quantum measurement.
Published by the American Physical Society 2025
... Viewing SGD from the perspective of dynamic systems, it can be considered as a specific classical open dissipative system with stochastic noise, named as classical Langevin system [32,63,69]. This observation inspired us to harness a distinct category of open dissipative quantum systems, known as quantum Langevin system [31], with the explicit purpose of constructing an optimization algorithm. ...
... The analogous equations for quantum systems have been formulated and have found applications in numerous instances of physical interest [31]. The quantum case under consideration involves a model of a heat bath comprising harmonic oscillators characterized by distinct masses m n and spring constants k n . ...
... The quantum case under consideration involves a model of a heat bath comprising harmonic oscillators characterized by distinct masses m n and spring constants k n . The Hamiltonian governing the entire system is expressed as follows [25,31]: ...
We initiate the study of utilizing quantum Langevin dynamics (QLD) to solve optimization problems, particularly those nonconvex objective functions that present substantial obstacles for traditional gradient descent algorithms. Specifically, we examine the dynamics of a system coupled with an infinite heat bath. This interaction induces both random quantum noise and a deterministic damping effect to the system, which nudge the system towards a steady state that hovers near the global minimum of objective functions. We theoretically prove the convergence of QLD in convex landscapes, demonstrating that the average energy of the system can converge to zero in the low temperature limit with an exponential convergence rate. Numerically, we first show the energy dissipation capability of QLD by retracing its origins to spontaneous emission. Furthermore, we conduct detailed discussion of the impact of each parameter. Finally, based on the observations when comparing QLD with the classical Fokker-Plank-Smoluchowski equation, we propose a time-dependent QLD by setting temperature and as time-dependent parameters, which can be theoretically proven to converge better than the time-independent case and also outperforms a series of state-of-the-art quantum and classical optimization algorithms in many nonconvex landscapes.
... The study of quantum systems in optical cavities has played a pivotal role in the advancement of quantum optics [1,2]. Optical cavities provide a controlled environment that enables exploration of nonclassical behaviors of quantized electromagnetic fields, making them ideal platforms for observing phenomena such as spontaneous emission, quantum decoherence, and system thermalization [3][4][5][6]. In the absence of dissipation, the field modes within these cavities remain isolated, and their dynamics can be accurately described using the Schrödinger equation [7][8][9][10][11][12]. ...
... These studies have shown that, at zero temperature, the dynamics of the system are dominated solely by cavity dissipation, while at finite temperatures, thermal excitations from the reservoir introduce significant modifications [17]. Furthermore, exact solutions are critical for benchmarking numerical methods and for exploring phenomena such as thermalization, decoherence, and entanglement dynamics in more complex systems [6,18,19]. Entanglement, in particular, stands out due to its fundamental role in quantum technologies such as quantum computing, cryptography, and communication [18,[20][21][22]. Entangled states arise from the interaction of quantized fields and exhibit correlations that challenge classical explanations while also being highly susceptible to environmental interactions, which can induce quantum decoherence and even lead to the loss of entanglement as the system interacts with a thermal reservoir [4,19]. ...
... To contextualize this study within the larger literature, we note that previous work has focused primarily on specific cases of initial states, often neglecting a comparative analysis between entangled and unentangled scenarios [30][31][32][33][34]. Furthermore, while numerical approaches have been widely employed, there is a scarcity of analytical solutions that provide fundamental insights into the interaction between dissipation and thermal effects [6,13,35]. This gap highlights the need for a detailed theoretical and numerical exploration of these systems, addressing both entangled and nonentangled initial conditions. ...
In this study, we present an exact solution to the Lindblad master equation describing the interaction of two quantized electromagnetic fields in a decaying cavity coupled to a thermal reservoir at a finite temperature. The solution is obtained using the superoperator technique, leveraging commutation relations to factorize the exponential of the Lindblad superoperators into a product of exponentials. To demonstrate the applicability of this approach, we analyze the dynamics of the system both analytically and numerically for two initial conditions: nonentangled and entangled coherent states, exploring their temporal evolution. Additionally, we employ entropy and quantum discord analysis to characterize quantum correlations and analyze the behavior of entanglement (or lack thereof) during the evolution. This comprehensive analysis provides valuable insights into the behavior of open quantum systems and their interaction with the environment.
... To derive the main result, we use a quasi-probability representation, where the system state is described by a real-valued function of phase variables, known as the Wigner function [50,51]. Using the eigenvalue α of the annihilation operatorâ as the phase variable, we map the time-evolution equations (3) and (5) to a Fokker-Planck equation for the complex-valued variables α and α * (see Sec. V for details). ...
... A coherent state |α⟩ is an eigenstate of the annihilation operator with a complex eigenvalue α, i.e.,â|α⟩ = α|α⟩. The set of coherent states forms a non-orthogonal overcomplete basis [51]. ...
... Quasi-probability distributions are defined through the inverse Fourier transform of a moment-generating function in analogy to classical cases. One typical example is the Wigner function [50,51], which is defined by f (α, α * , t) = 1 π 2 tr ρ(t)e −i(ξâ(t)+ξ * â † (t)) e i(ξα+ξ * α * ) d 2 ξ. ...
Thermodynamic constraints impose a trade-off between power and efficiency in heat engines, preventing the simultaneous achievement of high power and high efficiency. For classical microscopic engines, explicit inequalities have been discovered, demonstrating the inherent inevitability of this power-efficiency trade-off. However, extensions of these results to quantum engines have so far been limited to cases of slow operation. In this study, we derive a power-efficiency trade-off relation for a paradigmatic quantum engine operating within a finite time, specifically the Otto cycle of a quantum harmonic oscillator. By utilizing a phase-space approach based on quasi-probability representations, we establish a universal trade-off relation applicable to arbitrary time-dependent protocols during the adiabatic processes. Our results reveal that the power of the quantum engine vanishes as the efficiency approaches the quantum mechanical efficiency bound, which is stricter than the Carnot bound. Furthermore, we identify the conditions under which the upper bound is attained, which indicate maximum power is achieved when the generation of quantum coherence is reduced, and the difference in time durations of the isochoric processes increases. These findings are validated through numerical calculations, which confirm their applicability across various types of protocols for heat engine cycles.
... ] T contains operators representing Langevin noise sources [40], [41], and M ∈ ℂ 2N×2N is the spatial evolution matrix. It is important to point out that, according to our notation, M = −iH with respect to the Hamiltonian description of NH waveguides, with the corresponding consequences in the physical interpretation of real and complex eigenvalues. ...
... mn . Therefore, for a photonic mode m experiencing linear loss , we can define the Langevin noise vector componentF m in terms of a bosonic noise operator: [40], [42]. Similarly, ...
... for a mode subject to amplification with linear gain g, and the associated commutator is given by [40], [42]. On the other hand, gain and loss can also arise from nonlinear or parametric phenomena leading to squeezing transformations and mixing creation and destruction photonic operators. ...
Non-Hermitian (NH) photonic systems leverage gain and loss to open new directions for nanophotonic technologies. However, the quantum and thermal noise intrinsically associated with gain/loss affects the eigenvalue/eigenvector structure of NH systems, and thus the existence of exceptional points, as well as the practical noise performance of these systems. Here, we present a comparative analysis of the impact of different gain and loss mechanisms on the noise generated in gain–loss compensated NH waveguide systems. Our results highlight important differences in the eigenvalue/eigenvector structure, noise power, photon statistics and squeezing. At the same time, we identify some universal properties such as the occurrence of phase-transition points in parameter space and intriguing phenomena related to them, including coalescence of pairs of eigenvectors, gain–loss compensation, and linear scaling of the noise with the length of the waveguide. We believe that these results contribute to a better understanding of the impact of the gain/loss mechanism on the noise generated in NH systems.
... Laser quantum noise is commonly modeled [63][64][65] using SDEs in a normally ordered quantum phase-space representation. Consider a model for the quantum noise of a single mode laser as it turns on, near threshold: ...
... This approach started when Schrödinger [68] pointed out that quantum oscillators can have classical equations. This was extended to other systems [69][70][71], especially including lasers and quantum optics [2,[63][64][65]. ...
... The spectrum of an internal field variable is not the one that is usually measured. An important application of stochastic equations is therefore in calculating output, measured spectra of lasers, quantum optics, opto-mechanics and quantum circuits [65,80]. These have the feature that the measured output spectrum may also include noise from reflected fields at the input/output ports. ...
This is the fourth major release of the xSPDE toolbox, which solves stochastic partial and ordinary differential equations, with applications in biology, chemistry, engineering, medicine, physics and quantum technologies. It computes statistical averages, including time-step and sampling error estimation. xSPDE can provide higher order convergence, Fourier spectra and probability densities. The toolbox has graphical output and statistics, as well as weighted, projected, or forward-backward equations. It can generate input-output quantum spectra. The equations can have independent periodic, Dirichlet, and Neumann or Robin boundary conditions in any dimension, for any vector component, and at either end of any interval. xSPDE has functions that can numerically solve both ordinary and partial differential stochastic equations of any type, obtaining correlations, probabilities and averages. The toolbox has a core treating stochastic differential equations, with averages, probability distributions and full error estimates. There are stochastic extensions treating applications to partial differential equations, projected equations, quantum stochastic equations, master equations and quantum phase-space simulations including Gaussian boson sampling experiments.
... At first, we make a unitary transformation W = e −i ∑ 2 j=1 j b † j b j t and transfer the master equation into the interaction picture. Then, we use the relation between the density matrix and the corresponding probability distribution function in terms of the Bargmann states [59], given by ...
... is the probability distribution function with j = Q j + iP j and * j being its complex conjugate [59]. Next, for simplicity, we assume g1 = g2 = g , A t1 = A t2 = A t and make use of Eqs. ...
We propose a scheme to create and unidirectionally transport thermal squeezed states and random-phase coherent states in a system of two interacting levitated nanoparticles. In this coupled levitated system, we create a thermal squeezed state of motion in one of the nanoparticles by parametrically driving it and then transporting the state to the other nanoparticle by making use of a unidirectional transport mechanism. This mechanism is based on inducing a nonreciprocal type of coupling in the system by suitably modulating the phases of the trapping lasers and the interparticle distance between the levitated nanoparticles. A nonreciprocal coupling creates a unidirectional channel where energy flows from one nanoparticle to the other nanoparticle but not vice versa, thereby allowing for the transport of mechanical states between the nanoparticles. We also affirm this unidirectional transport mechanism by creating and efficiently transporting a random-phase coherent state in the coupled levitated system. In both instances of mechanical state transport, the final nanoparticle showed similar characteristics to the original nanoparticle, depicting a high-fidelity unidirectional transport mechanism. Further, we make use of the feedback nonlinearity and parametric driving to create simultaneous bistability in the coupled levitated system also via this unidirectional mechanism. Our results may have potential applications in tunable sensing, metrology, quantum networks, and in exploring many-body physics under a controlled environment.
... The reason why we chose Eq. (5) over the other is that, in the first place, the former is intimately connected to Mandel's photon-counting theory [24,26] and represents the probability of detecting 2 photons, one in each mode, when (n > 2)-photon events can be ruled out. Secondly, these 2-fold (or n-fold in general) integrals can be efficiently computed from the equations of motion that we will derive in subsequent sections. ...
... For instance, which is the probability of detecting both photons in (I)? To answer this question we make use of the photon-counting theory [24,26]. ...
Frequency-filtered photon correlations have been proven to be extremely useful in grasping how the detection process alters photon statistics. Harnessing the spectral correlations also permits refinement of the emission and unraveling of previously hidden strong correlations in a plethora of quantum-optical systems under continuous-wave excitation. In this work, we investigate such correlations for time-dependent excitation and develop a methodology to compute efficiently time-integrated correlations, which are at the heart of the photon-counting theory, and subsequently apply it to analyze the photon emission of pulsed systems. By combining this formalism with the sensor method -- which facilitates frequency-resolved correlations -- we demonstrate how spectral filtering enhances single-photon purity and suppresses multi-photon noise in time-bin-encoded quantum states. Specifically, filtering the central spectral peak of a dynamically driven two-level system boosts temporal coherence and improves the fidelity of time-bin entanglement preparation, even under conditions favoring multi-photon emission. These results establish spectral filtering as a critical tool for tailoring photon statistics in pulsed quantum light sources.
... While Fock states are a natural choice due to their experimental relevance, they present challenges for simulating classical lattice dynamics, primarily because the resulting populations are generally complex. Similar to phase-space methods [68], alternative lattice representations ofρ exist that ensure real-valued and non-negative populations. ...
... The FSL of the JC model, Eq.(68). The red circles represent sites where the atom is in the excited state, while the blue circles correspond to the atom in the ground state. ...
We introduce Liouville Fock state lattices (LFSLs) as a framework for visualizing open quantum systems through matrix representations of the Lindblad master equation (LME). By vectorizing the LME, the state evolves in a doubled Hilbert space, naturally forming a synthetic lattice. Unlike the unitary evolution of pure states, LFSL states exhibit nontrivial dynamics due to the non-Hermitian Liouvillian, featuring population drifts, sources, and sinks--paralleling stochastic classical lattices. We explore these "classical simulators" in both the Fock representation and alternative positive semidefinite representations, which more closely resemble classical probability distributions. We further demonstrate how infinite steady state manifolds can derive from frustration in the LFSL.
... and ρ 01 = (ρ 10 ) * . To solve the differential equations above we express the density matrix using the positive-P representation [57]: ...
... By applying these identities to Eq. (B7), using partial integration and assuming that the boundary terms vanish (P (±∞, ±∞) = 0), we obtain the following correspondences [57]: ...
Direct dipole coupling between a two-level system and a bosonic mode describes the interactions present in a wide range of physical platforms. In this work, we study a coupling that is mixed between two pairs of quadratures of a bosonic mode and a spin. In this setting, we can suppress the dispersive shift while retaining a nonzero Kerr shift, which remarkably results in a cubic relationship between shot noise dephasing and thermal photons in the oscillator. We demonstrate this configuration with a simple toy model, quantify the expected improvements to photon shot-noise dephasing of the spin, and describe an approach to fast qubit readout via the Kerr shift. Further, we show how such a regime is achievable in superconducting circuits because magnetic and electric couplings can be of comparable strength, using two examples: the Cooper pair transistor and the fluxonium molecule.
... Representative examples include, e.g., ultra-and deep-strong light-matter interactions in cavity and circuit quantum electrodynamics (QED) [1][2][3], time retardation effects caused by light propagation in waveguide QED [4][5][6] and giant atoms [7][8][9][10], nonlinear optomechanical interactions induced by radiation pressure [11][12][13][14], and nonlinear phononics [15][16][17], to name a few. A faithful and accurate characterization of the properties of physical systems in said regimes requires the improvement of existing or the invention of new analytical and numerical methods [18,19]. ...
... The common starting point of devising such methods is often a microscopic description of the environment and its interaction with the system of interest, and a variety of perturbative or non-perturbative treatments can be derived depending on the parameter regime of interest. For example, perturbation theory can lead to an effective description of the system alone in terms of quantum master equations [18][19][20][21][22] in which the environmental effects are implicitly accounted for. On the other hand, non-perturbative methods can be grounded on a variety of ideas, ranging from discretization of the the continous environment [23][24][25] or the Feymann-Vernon influence functional [26][27][28][29][30][31][32][33] for tensor-network simulations to solving unitarily equivalent models obtained from the reaction coordinate mapping [34][35][36][37][38][39], the chain mapping [40][41][42][43] and the polaron transformation [44][45][46][47][48][49][50], or constructing effective models with auxiliary degrees of freedom, e.g., * nwlambert@gmail.com ...
The theory of purified pseudomodes [arXiv:2412.04264 (2024)] was recently developed to provide a numerical tool for the analysis of the properties of a quantum system and the environment it couples to via linear system-bath interactions. Here we extend this theory to allow for the description of general nonlinear system-bath interactions. We demonstrate the validity of our method by considering the spontaneous decay of a two-level atom placed inside a single-mode lossy cavity and furthermore, its potential application to nanophotonics by calculating the resonance fluorescence spectrum of a quantum dot in the presence of a phonon environment. Our method provides a useful tool for the study of phonon-assisted emission in quantum dots and holds the the promise for broad applications in fields like quantum biology, nonlinear phononics, and nanophotonics.
... where the phase factor e − √ ξ W (a)dϕ (Stratonovich integral ) determines noise-driven deviations from classical trajectories, and W (a) is a Wiener process [51] satisfying ⟨W (a)W (a ′ )⟩ = δ(a − a ′ ). The Stratonovich integral preserves the chain rule and is appropriate for multiplicative noise in physical systems under the specific Hamiltonian structure. ...
... Near a → 0, W (a) induces exponential suppression Ψ(a, ϕ) ∼ e −S(a,ϕ)/ℏ , where S(a, ϕ) diverges as a −3 for k = 1. This parallels loop quantum cosmology's repulsive quantum geometry [51], but here suppression arises naturally from the stochastic action without ad hoc boundary conditions. ...
The problem of time in canonical quantum gravity remains one of the most significant challenges, primarily due to the ``frozen'' formalism emerging from the Wheeler--DeWitt equation. Within the ADM formalism, we introduce a novel approach in which a scalar field is treated as a stochastic clock. By imposing a divergence-free condition on the scalar momentum, we integrate out quantum gravitational fluctuations and derive an effective noise term via the Hubbard--Stratonovich transformation. This noise drives dynamic adjustments in spacetime foliations, enabling a Schr\"{o}dinger-like evolution that preserves diffeomorphism invariance and, upon noise averaging, maintains unitary evolution. Interestingly, by introducing stochastic variations in the foliations, the quantum indeterminacy of the clock recasts time as a diffusive process emerging from quantum fluctuations -- where correlations between matter and geometry replace an absolute time parameter. This is considerably a potential pathway for understanding quantum time evolution while maintaining background independence in canonical quantum gravity.
... The measurement of the oscillator positionb 1 is enabled by the linear, momentum-exchange optomechanical interactionû 1b1 [2,4,24,30]. This leads to the asymmetrical Langevin equations where only the oscillator's momentumb 2 is driven by the light fluctuationsû 1 in Eq. (2c) [31]. These fluctuations are transduced by the mechanical oscillator and contribute to the phase quadrature of the output field, causing back action noise. ...
... photon loss), or additive noise (e.g. dark noise) [31]. ...
Optomechanical systems subjected to environmental noise give rise to rich physical phenomena. We investigate entanglement between a mechanical oscillator and the reflected coherent optical field in a general, not necessarily Markovian environment. For the input optical field, we consider stationary Gaussian states and frequency dependent squeezing. We demonstrate that for a coherent laser drive, either unsqueezed or squeezed in a frequency-independent manner, optomechanical entanglement is destroyed after a threshold that depends only on the environmental noises -- independent of the coherent coupling between the oscillator and the optical field, or the squeeze factor. In this way, we have found a universal entangling-disentangling transition. We also show that for a configuration in which the oscillator and the reflected field are separable, entanglement cannot be generated by incorporating frequency-dependent squeezing in the optical field.
... To effectively describe these dynamic processes, researchers frequently employ the Lindblad master equation, which models large environments as memoryless baths under the Markovian approximation [1,2]. This framework provides sufficient generality to capture a wide range of open-system phenomena across various fields such as quantum optics [3,4], quantum biology [5][6][7][8], and quantum chemistry [9,10]. Beyond its role in scientific inquiry, Lindbladian simulation has proven transformative for quantum computing tasks, including thermal and ground-state preparation [11][12][13][14], dissipative quantum state engineering [15], autonomous quantum error correction [16][17][18], and the solution of differential equations [19]. ...
... Proposition 4. The compensation W(τ ) in Eq. (C5) can be represented as a (µ, ϵ)-LCS formula on Pauli-conjugate basis such that µ ≤ e e(2∥L∥τ ) 3 , and ϵ ≤ 2e∥L∥ pauli τ K W + 1 ...
Quantum computers can efficiently simulate Lindbladian dynamics, enabling powerful applications in open system simulation, thermal and ground-state preparation, autonomous quantum error correction, dissipative engineering, and more. Despite the abundance of well-established algorithms for closed-system dynamics, simulating open quantum systems on digital quantum computers remains challenging due to the intrinsic requirement for non-unitary operations. Existing methods face a critical trade-off: either relying on resource-intensive multi-qubit operations with experimentally challenging approaches or employing deep quantum circuits to suppress simulation errors using experimentally friendly methods. In this work, we challenge this perceived trade-off by proposing an efficient Lindbladian simulation framework that minimizes circuit depths while remaining experimentally accessible. Based on the incoherent linear combination of superoperators, our method achieves exponential reductions in circuit depth using at most two ancilla qubits and the straightforward Trotter decomposition of the process. Furthermore, our approach extends to simulate time-dependent Lindbladian dynamics, achieving logarithmic dependence on the inverse accuracy for the first time. Rigorous numerical simulations demonstrate clear advantages of our method over existing techniques. This work provides a practical and scalable solution for simulating open quantum systems on quantum devices.
... To find the squeezing and phase-matching conditions in finite-bandwidth "squeezed bath", we adapt ref. [58] derivation of output squeezed vacuum from a source OPO. We start with a source OPO in modeâ, which has output modeâ out (t) whose correlation functions are ...
... Finally, the phase of M , according to ref. [58], is M = |M |e 2iϕ , where ϕ = arg(E a ). In our simulation, we set ϕ = 0, so according to phase-matching condition θ +θ e = θ + 2ϕ = π, we have θ = π. ...
Light-matter interaction with squeezed vacuum has received much interest for the ability to enhance the native interaction strength between an atom and a photon with a reservoir assumed to have an infinite bandwidth. Here, we study a model of parametrically driven cavity quantum electrodynamics (cavity QED) for enhancing light-matter interaction while subjected to a finite-bandwidth squeezed vacuum drive. Our method is capable of unveiling the effect of relative bandwidth as well as squeezing required to observe the anticipated anti-crossing spectrum and enhanced cooperativity without the ideal squeezed bath assumption. Furthermore, we analyze the practicality of said models when including intrinsic photon loss due to resonators imperfection. With these results, we outline the requirements for experimentally implementing an effectively squeezed bath in solid-state platforms such as InAs quantum dot cavity QED such that \textit{in situ} control and enhancement of light-matter interaction could be realized.
... A process is said to be non-Markovian if there exists a pair of initial states ρ 1 (0) and ρ 2 (0) and a certain time t such that [17,[37][38][39][40][41][42] σ(t) = d dt D(ρ 1 (t), ρ 2 (t)) > 0. ...
In the framework of a novel dissipative scheme, we have investigated the quantum dynamics of an oscillating system interacting with two reservoirs with different temperatures trough different time-dependent coupling functions. The reduced density matrix, quantum optical characteristic functions, and (quasi) distribution functions like Husimi, Glauber-Sudarshan and Wigner functions on the phase space of the oscillator are obtained. The problem has been generalized to the case where the oscillator is interacting with n distinctive reservoirs, and a quantum current and an effective reservoir is introduced. Finally, the quantum dynamics of a two-level system interacting with two reservoirs has been investigated, and the exact reduced density matrix is obtained.
... In addition, a in (m in j ) is the noise operators for the cavity (j th magnon) mode. Despite the fact that these noise operators' mean values are zero, their nonzero correlation functions are given as 58 : ...
We theoretically propose a scheme to generate and control continuous variable bipartite entanglement and Gaussian quantum steering in an optical whispering gallery mode (WGM)-based cavity optomagnonical system that consists of two macroscopic YIG resonator. Owing to the well-known Faraday effect, both the magnon modes are coupled to single-mode optical WGM through nonlinear interaction. We investigate in detail the impact of several physical parameters such as effective cavity detuning, input laser power, environment temperature, optomagnonical coupling strengths and magnon decay rate on different bipartite entanglement. We also found a suitable parameter regime to obtain maximum cavity-magnon and magnon-magnon bipartite entanglement in our proposed system. It is interesting to note that the numerical simulation result shows that magnon-magnon entanglement persists up to 60K. With a proper choice of optomagnonical coupling strengths and normalized effective cavity detuning, we can effectively control the nature and strength of Gaussian quantum steering. In the WGM-based cavity optomagnonical system, our current work will offer a new way of greatly controlling a variety of nonclassical quantum correlations of macroscopic objects, which may find use in a number of contemporary quantum technology fields.
... Efforts are being made to find generalizations of the SSE in the non-Markovian regime while possessing a reasonable physical interpretation. One method for achieving this involves generalizing the Markovian Lindblad approaches in Ref. [16] to non-Markovian master equations [17] and finding physically meaningful unravelings, as demonstrated for jump processes in Ref. [18]. In contrast, we leverage the approach in Ref. [19] and start from a generalized SSE. ...
Environmental noise affecting controlled quantum systems is typically described by a dissipative Lindblad equation, which captures the system's average state through the density matrix ρ . One approach to deriving this equation involves a stochastic operator evolving under white noise in the Schrödinger equation; however, white noise fails to accurately depict real-world noise profiles, where lower frequencies often dominate. This study proposes a method to determine the of qubit fidelities in significant stochastic Schrödinger equation scenarios, with qubits evolving under more realistic noise profiles such as noise. This method enables the prediction of the mean, variance, and higher-order moments of qubit fidelities, offering insights crucial for assessing permissible noise levels in prospective quantum computing systems and guiding decisions about control systems procurement. Additionally, these methodologies are essential for optimizing qubit state control affected by classical control system noise.
Published by the American Physical Society 2025
... Monte Carlo simulations are used to study the process of photon emission. In the Monte-Carlo wavefunction formalism [36,37], a single quantum trajectory consists of coherent evolution of the state |ψ(t)⟩ under the influence of H ef f (eq. 11), which is interspersed with random quantum jumps, as mentioned in section III. ...
Two-photon states are essential for quantum technologies such as metrology, lithography, and communication. One of the main methods of two-photon generation is based on parametric down-conversion, but this suffers from low efficiency and a large footprint. This work is a detailed investigation of an alternative approach: two-photon generation from an atom in a doubly resonant cavity. The system, consisting of an atom interacting with two modes of the cavity, is modelled by the Lindblad Master Equation. An approximate analytical solution is derived, using a novel approximation method, to determine the practically achievable limits on efficiency and brightness. The model also predicts the optimal cavity parameters for achieving these limits. For experimentally feasible parameters, the maximum efficiency turns out to be approximately , which is about three orders of magnitude greater than that of parametric down-conversion-based methods. The optimal rate and efficiency for two-photon generation are achieved when the outcoupling rate of the cavity mode at the two-photon emission frequency matches the single-photon atom-field coupling strength. Moreover, the outcoupling rate of the cavity mode at the one-photon emission frequency for single photons should be minimized. The cavity field properties are also examined by studying the second-order correlation function at zero time delay and the Fano Factor. The quantum-jump framework, combined with Monte Carlo simulations, is used to characterize the mechanism of two-photon emission and the emission spectra of the cavity. Two-photon emission is demonstrated to be a rapid cascade process of quantum jumps, and the spectrum exhibits distinct peaks that correspond to transitions between the manifolds of the system.
... Finally, we connect our derivations to an input-output approach [22][23][24], which also illustrates the role of the total field in the system. We use both the weak-coupling approximation (κ rad ≪ ω 0 and ω rad ≈ ω 0 ) and the rotating-wave approximation (only frequency components around the resonance will be considered), as is commonly assumed in the input-output ...
The Lorentz-Drude model for electric dipoles is a classical framework widely used in the study of dipole dynamics and light-matter interactions. Here we focus on the behaviors of Lorentz-Drude dipoles when their radiative rate dominates their energy loss. We show that dipole radiation losses do not count toward phenomenological dipole losses if the driving field is interpreted as the total field at the dipole. In particular, if the dipole does not contain non-radiative losses, then the Lorentz-Drude damping term should be removed. This is verified by self-consistent implementations of point dipoles in finite-difference time-domain simulations, which also provide a method to directly compute the transport properties of light when dipoles are present.
... In Appendix A 2, we show that this can be accomplished by using the input-output formalism [61] and the quantum regression theorem [62,63]. In particular, we show that the correlator C (n) ...
Quantum optical systems comprising quantum emitters interacting with engineered optical modes generate non-classical states of light that can be used as resource states for quantum-enhanced interferometry. However, outside of well-controlled systems producing either single-mode states (e.g. Fock states or squeezed states) or highly symmetric multi-mode states (e.g. superradiant states), their potential for quantum advantage remains uncharacterized. In this work, we develop a framework to analyze quantum enhanced interferometry with general Markovian quantum light sources. First, we show how to compute the quantum Fisher Information (QFI) of the photons emitted by a source efficiently by just tracking its internal dynamics and without explicitly computing the state of the emitted photons. We then use this relationship to elucidate the connection between the level structure and spectrum of the source to a potential quantum advantage in interferometry. Finally, we analyze optimal measurement protocols that can be used to achieve this quantum advantage with experimentally available optical elements. In particular, we show that tunable optical elements with Kerr non-linearity can always be harnessed to implement the optimal measurement for any given source. Simultaneously, we also outline general conditions under which linear optics and photodetection is enough to implement the optimal measurement.
... Indeed, dissipation [67][68][69] is an important phenomenon which is often neglected in most theoretical models of strongly coupled light-matter systems. Dissipative processes suppress quantum correlations, and may destroy any collective phenomena that rely on the coherent coupling of matter to photons. ...
We study the properties of photons in a cryogenic cavity, made by cryo-cooled mirrors surrounded by a room temperature environment. We model such a system as a multimode cavity coupled to two thermal reservoirs at different temperatures. Using a Lindblad master equation approach, we derive the photon distribution and the statistical properties of the cavity modes, finding an overall non-thermal state described by a mode-dependent effective temperature. We also calculate the dissipation rates arising from the interaction of the cavity field with the external environment and the mirrors, relating such rates to measurable macroscopic quantities. These results provide a simple theory to calculate the dissipative properties and the effective temperature of a cavity coupled to different thermal reservoirs, offering potential pathways for engineering dissipations and photon statistics in cavity settings.
... . According to the input-output theory, the operators for the output fields are related to the cavity and to the input noise operators by the relation [71] out ( ) = in ( ) − √ 2 0 ( ) ( B 1 ) ...
An enhancing exceptional‐point‐based (EP‐based) sensing scheme with the pump gain in the reversed‐dissipation regime is proposed. It is showed that EP‐based sensing has excellent signal‐to‐noise ratio (SNR) at the balanced total gain and loss. Meanwhile, this scheme is adaptable to a wide range of sensing targets represented by the perturbations, encompassing the isolated resonance, the loss (gain), and the coupling mechanism. Compared with conventional sensing schemes, this scheme significantly boosts the sensitivity coefficient across all types of perturbations, especially in small perturbations. Particularly, the readout process can be simplified for perturbations involving loss (gain) and coupling mechanism. Moreover, the robustness against noise for EP‐based sensing is confirmed through an analysis of the output spectra. This work reveals the condition for realizing the EP‐based sensing in reversed‐dissipation regime, and paves the way for the development of multifunctional and high‐sensitivity sensors within various non‐Hermitian quantum systems. It also opens new avenues for the creation of compact, integrated, and readout‐friendly optomechanical sensors.
... The Tavis-Cummings (TC) model is a prototypical model for lasing [7,65] which finds applications in modern AMO research, in particular, to study ultra-narrow linewidth lasing in a bad cavity [66][67][68][69][70] and dynamical phase transitions between non-radiative, lasing and superradiant lasing regimes [18,71]. TC consists of a single bosonic mode, representing photons, coupled to an ensemble of spin-1/2 degrees of freedom, modeling two-level atoms confined within an optical cavity. ...
We put forward a user-friendly framework of the truncated Wigner approximation (TWA) for dissipative quantum many-body systems. Our approach is computationally affordable and it features a straightforward implementation. The leverage of the method can be ultimately traced to an intimate connection between the TWA and the semi-classical limit of the quantum Langevin equation, which we unveil by resorting to a path integral representation of the Lindbladian. Our approach allows us to explore dynamics from early to late times in a variety of models at the core of modern AMO research, including lasing, central spin models, driven arrays of Rydbergs and correlated emission in free space. Notably, our TWA outperforms the method of cumulant expansion in terms of efficiency, ease of use, and broad applicability. We therefore argue that TWA could become in the near future a primary tool for a fast and efficient first exploration of driven-dissipative many-body dynamics on consumer grade computers.
... where tr e denotes partial trace over the environment. We take the continuum limit for the bath modes and consider that the coupling constants follow an Ohmic distribution, given by [32] ...
A unitary evolution in time may be treated as a curve in the manifold of the special unitary group. The length of such a curve can be related to the energetic cost of the associated computation, meaning a geodesic curve identifies an energetically optimal path. In this work, we employ sub-Riemannian geometry on the manifold of the unitary group to obtain optimally designed Hamiltonians for generating single-qubit gates in an environment with the presence of dephasing noise as well as a two-qubit gate under a time-constant crosstalk interaction. The resulting geodesic equation involves knowing the initial conditions of the parameters that cannot be obtained analytically. We then introduce a random sampling method combined with a minimization function and a cost function to find initial conditions that lead to optimal control fields. We also compare the optimized control fields obtained from the solutions of the geodesic equation with those extracted from the well-known Krotov method. Both approaches provide high fidelity values for the desired quantum gate implementation, but the geodesic method has the advantage of minimizing the required energy to execute the same task. These findings bring new insights for the design of more efficient fields in the arsenal of optimal control theory.
... When the system is in contact with a thermal reservoir at temperature T , it follows the Lindblad dynamics [41][42][43], given by ...
Quantum heat engines have undergone extensive studies over the last two decades. Simultaneously, the studies of the applications of stochastic resetting in various fields are on the rise. We explore the effect of stochastic resetting on the dynamics of a two-level and a three-level quantum heat engine. The extracted work is shown to increase with the resetting rate. However, the effective efficiency that includes the work expended in resetting is shown to exhibit a steady decay with the increase in resetting rate. The efficient power is observed to increase beyond that obtained in the absence of resetting, and is shown to be higher for a three-level engine.
... From a fundamental perspective, despite significant experimental strides in isolating quantum systems, a finite coupling to the environment is unavoidable, imparting dynamic characteristics that encompass a diverse range of features not observed in equilibrium systems [1,2]. In practical terms, these systems offer a platform for employing controlled dissipation channels to engineer captivating quantum states as the stationary outcome of their dynamics, thus holding potential applications in quantum information tasks [3][4][5]. Diverging from closed quantum systems, where a wave function is commonly used to represent the quantum state, the focus of study in open quantum systems shifts to the density operator ρ. Effectively describing interacting open quantum many-body systems presents a significant challenge for both theoretical and numerical approaches [2]. ...
The transformer architecture, known for capturing long-range dependencies and intricate patterns, has extended beyond natural language processing. Recently, it has attracted significant attention in quantum information and condensed matter physics. In this work, we propose the transformer density operator ansatz for determining the steady states of dissipative quantum many-body systems. By vectorizing the density operator as a many-body state in a doubled Hilbert space, the transformer encodes the amplitude and phase of the state's coefficients, with its parameters serving as variational variables. Our design preserves translation invariance while leveraging attention mechanisms to capture diverse long-range correlations. We demonstrate the effectiveness of our approach by numerically calculating the steady states of dissipative Ising and Heisenberg spin chain models, showing that our method achieves excellent accuracy in predicting steady states.
... The Lindblad master equation is the most general type of completely positive trace-preserving Markovian master equation (Gorini, Kossakowski & Sudarshan 1976;Lindblad 1976;Gardiner & Zoller 2004;Stéphane, Joye & Pillet 2006;Pearle 2012;Manzano 2020). It can be written in dimensionless form as ...
Quantum algorithms have been proposed to accelerate the simulation of the chaotic dynamical systems that are ubiquitous in the physics of plasmas. Quantum computers without error correction might even use noise to their advantage to calculate the Lyapunov exponent by measuring the Loschmidt echo fidelity decay rate. For the first time, digital Hamiltonian simulations of the quantum sawtooth map, performed on the IBM-Q quantum hardware platform, show that the fidelity decay rate of a digital quantum simulation increases during the transition from dynamical localization to chaotic diffusion in the map. The observed error per CNOT gate increases by as the dynamics varies from localized to diffusive, while only changing the phases of virtual RZ gates and keeping the overall gate count constant. A gate-based Lindblad noise model that captures the effective change in relaxation and dephasing errors during gate operation qualitatively explains the effect of dynamics on fidelity as being due to the localization and entanglement of the states created. Specifically, highly delocalized states that are entangled with random phases show an increased sensitivity to dephasing and, on average, a similar sensitivity to relaxation as localized states. In contrast, delocalized unentangled states show an increased sensitivity to dephasing but a lower sensitivity to relaxation. This gate-based Lindblad model is shown to be a useful benchmarking tool by estimating the effective Lindblad coherence times during CNOT gates and finding a consistent 2\unicode{x2013}3{\times } shorter time than reported for idle qubits. Thus, the interplay of the dynamics of a simulation with the noise processes that are active can strongly influence the overall fidelity decay rate.
... However, things become subtle when considering retrodiction: inference about the past based on current knowledge, here specifically past state inference [5][6][7], and recovery of irreversible processes [8][9][10][11][12][13][14]. The common tool for retrodiction is Bayes' rule, for which several quantum extensions have been proposed [15][16][17][18][19][20][21][22][23][24][25][26][27]. Essentially, Bayes' rule is a belief update, and it requires the agent to choose a prior belief about the initial state of the system. ...
A mixed quantum state can be taken as capturing an unspecified form of ignorance; or as describing the lack of knowledge about the true pure state of the system ("proper mixture"); or as arising from entanglement with another system that has been disregarded ("improper mixture"). These different views yield identical density matrices and therefore identical predictions for future measurements. But when used as prior beliefs for inferring the past state from later observations ("retrodiction"), they lead to different updated beliefs. This is a purely quantum feature of Bayesian agency. Based on this observation, we establish a framework for retrodicting on any quantum belief and we prove a necessary and sufficient condition for the equivalence of beliefs. We also illustrate how these differences have operational consequences in quantum state recovery.
... The two components α,ᾱ above have the usual signal/idler interpretation [35], see Figure 4. By using the sub-block structure of the Green's function, we can actually simplify the response signal to the expression ...
We consider a linear array of trapped ions subjected to local parametric modulation of the trapping potential and continuous laser cooling. In our model, the phase of the parametric modulation varies linearly along the array, breaking time-reversal symmetry and inducing non-trivial topological effects. The linear response to an external force is investigated with the Green's function formalism. We predict the appearance of topological amplification regimes in which the trapped ion array behaves as a directional amplifier of vibrational excitations. The emergence of topological phases is determined by a winding number related to non-Hermitian point-gap topology. Beyond its fundamental interests as a topological driven-dissipative system, our setup can be used for quantum sensing of ultra-weak forces and electric fields. We consider a scheme in which a trapped ion at one edge of the array acts as a sensor of an ultra-weak force, and the vibrational signal gets amplified towards the last trapped ion, which acts as a detector. We consider arrays of 2-30 Mg ions, assuming that the detector ion's displacement is measured via fluorescence with a spatial resolution of 200-500 nm, and predict sensitivities as small as 1 yN Hz. Our system has the advantage that the detected force frequency can be tuned by adjusting the frequency of the periodic drive.
... Introduction.-The theory of open quantum systems [1][2][3] represents the backbone of quantum simulation engineering in the so-called NISQ (Noisy Intermediate Scale Quantum) era. Having exactly solvable models of strongly correlated (open) quantum systems, in particular of the type of digital quantum circuits, is of unprecedented value for the control, benchmarking and science demonstration purpose of NISQ devices (see e.g. ...
We propose spatially inhomogeneous matrix product ansatz for an exact many-body density operator of a boundary driven XXZ quantum circuit. The ansatz has formally infinite bond-dimension and is fundamentally different from previous constructions. The circuit is driven by a pair of reset quantum channels applied on the boundary qubits, which polarize the qubits to arbitrary pure target states. Moreover, one of the reset channels can be replaced by an arbitrary local unitary gate, thus representing a hybrid case with coherent/incoherent driving. Analyzing the ansatz we obtain a family of relatively robust separable nonequilibrium steady states (NESS), which can be viewed as a circuit extension of spin-helix states, and are particularly suited for experimental investigations.
... A better description may be found in terms of a density matrix ρ(t), which fully characterizes the state of an (open) quantum system [47]. In particular, the diagonal elements of the matrix (ρ i,i = Tr (ρ |i⟩ ⟨i|)) are known as populations, and represent the probability of occupation of a (pure) state |i⟩, while the off-diagonal elements of ρ, the coherences, quantify the superposition between states [48,49]. The quantum evolution is recast into many equivalent subcircuits based on voltage-controlled current sources (VCCSs), while the model specifies how to convert between quantum and classical variables (panel c). ...
Scalable solid-state quantum computers will require integration with analog and digital electronics. Efficiently simulating the quantum-classical electronic interface is hence of paramount importance. Here, we present Verilog-A compact models with a focus on quantum-dot-based systems, relevant to semiconductor- and Majorana-based quantum computing. Our models are capable of faithfully reproducing coherent quantum behavior within a standard electronic circuit simulator, enabling compromise-free co-simulation of hybrid quantum devices. In particular, we present results from co-simulations performed in Cadence Spectre, showcasing coherent quantum phenomena in circuits with both quantum and classical components using an industry-standard electronic design and automation tool. Our work paves the way for a new paradigm in the design of quantum systems, which leverages the many decades of development of electronic computer-aided design and automation tools in the semiconductor industry to now simulate and optimize quantum processing units, quantum-classical interfaces, and hybrid quantum-analog circuits.
... For these processes, the average of the stochastic variable is zero and the average of the two-time correlation is given by φ (t)φ (t ′ ) = 2Dδ (t − t ′ ); where D is the diffusion coefficient. Often in the literature, the stochastic process chosen to represent this phase is the Ornstein-Uhlenbeck [22,23], which includes an extra term to the Wiener process to make it mean-reversible. ...
We investigate the influence of the distribution of atom velocities in a hot rubidium sample on the correlation between field-intensity fluctuations of two independently generated four-wave mixing signals and between the transmission signals. The nonlinear process is driven by a single cw laser in a pure two-level system due to the forward geometry with circular and parallel polarization of the input fields. The intensity cross-correlations of the four-wave mixing signals and the transmission signals present an oscillatory behavior with a clear dependence on the power of the incident fields, which indicates a connection with Rabi oscillations. A two-level theoretical model using stochastic differential equations to account for the mechanism of conversion of phase noise into amplitude noise shows good agreement with our experimental results. Moreover, we show how the response of the system is affected by the different atomic velocity groups.
... The study of the mathematical structure of Markovian dynamical maps is the foundation underlying many areas of physics stemming from open quantum systems [1][2][3][4][5], to foundational issues of quantum mechanics [6][7][8][9][10], to hybrid models of classical-quantum interactions [11][12][13][14][15][16][17][18]. The essence of the investigation is, broadly, to identify which maps comply with the statistical interpretation of quantum mechanics, which requires that the density matrixρ of any system to be a positive operator. ...
Markovian master equations underlie many areas of modern physics and, despite their apparent simplicity, they encode a rich and complex dynamics which is still under active research. We identify a class of continuous variable Markovian master equations for which positivity and complete positivity become equivalent. We apply this result to characterize the positivity of the partially transposed evolution of bipartite Gaussian systems, which encodes the dynamics of entanglement. Finally, the entangling properties of models of classical gravity interacting with quantum matter are investigated in the context of the experimental proposals to detect gravitationally induced entanglement. We prove that entanglement generation can indeed take place within these models. In particular, by focusing on the Di\'osi-Penrose model for two gravitationally interacting masses, we show that entanglement-based experiments would constrain the free parameter of the model up to values six orders of magnitude stronger than the current state of the art.
... These quantities encode all the information about the quantum fluctuations of an observable. Understanding such fluctuations is not only essential at a fundamental level but also finds important applications in, for example, the development of optical and electrical devices [1][2][3], atomic and particle physics [4,5], random number generation [6], and materials science [7], to name a few. ...
In quantum mechanics, the probability distribution function (PDF) and full counting statistics (FCS) play a fundamental role in characterizing the fluctuations of quantum observables, as they encode the complete information about these fluctuations. In this letter, we measure these two quantities in a trapped-ion quantum simulator for the transverse and longitudinal magnetization within a subsystem. We utilize the toolbox of classical shadows to postprocess the measurements performed in random bases. The measurement scheme efficiently allows access to the FCS and PDF of all possible operators on desired choices of subsystems of an extended quantum system.
... We assume that the interaction between the system and the environment satisfies the Markovian approximation. [63] Hence, the correlation functions of the noise term can be characterised by ...
In this study, a scheme is proposed to generate nonreciprocal strong mechanical squeezing in a cavity‐magnon optomechanical system based on the Kerr nonlinearity of the yttrium‐iron garnet (YIG) sphere. It is found that the magnon squeezing can be directly produced by the Kerr effect and it can induce strong mechanical squeezing through the squeezing transfer achieved by the cavity‐magnon interaction and the optomechanical interaction. The results further show that the mechanical mode coupled to the cavity can achieve squeezing effect in the selected magnetic field direction, but not in the opposite one. Moreover, by properly adjusting the magnon effective detuning and the Kerr effect strength, the degree of squeezing of the mechanical mode can even beyond the 3‐dB limit, and the optimal conditions for generating stronger squeezing are obtained. This work has important applications in the nonreciprocal quantum devices, quantum information processing, and quantum precision measurement.
... Since non-Markovian noise arises due to quantum memory effects, the environment can no longer be regarded as merely a passive reservoir [43,44]. One way to model such noise is to treat it as a semi-classical stochastic process [45][46][47][48]. The system's response to this noise is characterized by its noise spectrum, which describes how different frequency components affect the system [36,49,50]. ...
Non-classical resources enable measurements to achieve a precision that exceeds the limits predicted by the central limit theorem. However, environmental noise arising from system-environment interactions severely limits the performance of such resources through decoherence. While significant progress has been made in mitigating Markovian noise, the extent to which non-Markovian noise can be mitigated remains poorly understood. We demonstrate that Heisenberg Scaling, the ultimate quantum limit on measurement precision, can be recovered in quantum metrology under non-Markovian noise by leveraging carefully designed Dynamical Decoupling Techniques. Importantly, our approach does not rely on assumptions of Markovian dynamics. By imposing appropriate conditions on the control Hamiltonian, we show that HS can be achieved irrespective of whether the noise is Markovian or non-Markovian. We also prove necessary and sufficient conditions for the existence of such control Hamiltonians. As an illustrative example, we apply our framework to the damped Jaynes-Cummings model, successfully mitigating memory effects and maintaining measurement precision in complex, non-Markovian environments. These findings highlight the power of quantum control to overcome decoherence challenges and enhance metrological performance in realistic, noisy quantum systems.
... Many important experimental observables, such as emission spectra, can be computed from multitime expectation values. Systems described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation [15,16] often exhibit the useful property that these multitime observables satisfy the same equation of motion as the density matrix -this is the essence of the Quantum Regression Theorem (QRT) [17][18][19]. The validity and limitations of the QRT have been extensively investigated in recent years, leading to an understanding of when the theorem applies and when it breaks down [20][21][22][23][24][25][26][27][28]. ...
We investigate a quantum dot (QD) system coupled to a vibrational environment with a super-Ohmic spectral density and weakly to a leaky cavity mode, a model relevant for semiconductor-based single-photon sources. The phonon coupling induces dephasing and broadens the absorption and emission line shapes, while the weakly coupled cavity mode leads to effective driving of the QD. To capture non-Markovian effects, we use non-Markovian Quantum State Diffusion and its hierarchical extension the Hierarchy of Pure States to compute multitime correlation functions underlying absorption and resonance fluorescence spectra. We present numerical results for the absorption spectra at strong phonon coupling and finite temperature, as well as for resonance fluorescence spectra at varying phonon coupling strengths and temperatures, and analyse the visibility of the resonance fluorescence spectra to provide insights into how phonon coupling and thermal effects influence the spectral features.
... Notice that the damping term in Eq. (3) is Markovian, i.e., damping depends only on the system operators at the moment. It is one of the core assumptions underlying the quantum input-output theory 65,66 . ...
... If the external environment changes, such as an increase in network latency or an attacker attempts brute-force intercepted messages, the timestamp mechanism resolves these issues. Although this may affect the probability of transition, the future state depends only on the current state, not on past states and historical information, so the execution process of the proposal conforms to the Markovian effect and does not conform to the non-Markovian effect [56]. These are left for a future in-depth analysis. ...
In a distributed wireless sensor network (DWSN), sensors continuously perceive the environment, collect data, and transmit it to remote users through the network so as to realize real-time monitoring of the environment or specific targets. However, given the openness of wireless channels and the sensitivity of collecting data, designing a robust user authentication protocol to ensure the legitimacy of user and sensors in such DWSN environments faces serious challenges. Most of the current authentication schemes fail to meet some important and often overlooked security features, such as resisting physical impersonation attack, resisting smartcard loss attack, and providing forward secrecy. In this work, we put forward a practically secure two-factor authentication scheme using a physically unclonable function to prevent a physical impersonation attack and sensor node capture attack, utilize Chebyshev chaotic mapping to provide forward secrecy, and improve the efficiency and security of session key negotiation. Furthermore, we use the fuzzy verifier technique to prevent attackers from offline guessing attacks to resist smartcard loss attacks. In addition, a BAN logic proof and heuristic security analysis show that the scheme achieves mutual authentication and key agreement as well as prevents known attacks. A comparative analysis with state-of-the-art schemes shows that the proposal not only achieves desired security features but also maintains better efficiency.
... over relatively short times, we can estimate the correlation function as well as the transition rate of equilibrium fluctuations as the time derivative of the correlation functioṅ C(t). The importance of evaluating C(t) andĊ(t) has been emphasised in the study of classical molecular dynamics [38][39][40], and also its quantum contexts [41][42][43][44]. However, stochastic numerical computations are in general requested for long-time simulation to analyse open quantum systems. ...
Estimating transition rates in open quantum systems is a significant challenge in physics and chemistry due to the exponential increase in computational resources required with the system sizes. Addressing this issue, we introduce a novel and efficient quantum simulation method to compute dynamical transition rates in Markovian open quantum systems. This method reformulates the transition rate expression, represented as the time derivatives of correlation functions, into directly measurable quantities in parametrised mixed-state quantum circuits. This approach allows us to estimate the transition rate more accurately without requiring the computation of the correlation function in a dynamical open quantum system. To validate our approach, we perform quantum simulations experimentally on IBMQ quantum processors for a decohering spin-1/2 model, compared with analytic solutions and quantum numerical simulation. Motivated by quantum chemistry-related applications, we examine our method in one-dimensional Caldeira-Leggett quantum Brownian motion model. Through theoretical and numerical analyses, we address the scalability of our scheme and the promise of significant computational advantages over its classical counterpart. Thus, our new approach holds the potential to surpass the bottlenecks of current quantum chemical research, particularly in accurately predicting molecular reaction rates and designing novel materials on the current and the near-term realisable quantum hardware.
... where κ a , γ b , and κ m , respectively, represent the decay rates of cavity mode, phonon mode, and Kittel mode. σ in is the input noise operator corresponding to the three modes with the nonzero correlation functions [58] ...
We propose a scheme to achieve nonreciprocal mechanical squeezing in a hybrid Kerr-modified cavity magnomechanical system, where the magnon mode is driven by two-tone microwave fields. The nonreciprocity originates from the magnon Kerr effect. Strong mechanical squeezing beyond the 3 dB limit can be nonreciprocally generated by adjusting the magnon frequency detuning, effective magnomechanical coupling strength, as well as the damping of the oscillator and the dissipation of the cavity. Importantly, the proposed scheme is robust against environmental thermal noise and system dissipation, ensuring its feasibility under current experimental conditions. This work may pave the way for the development of nonreciprocal quantum devices, such as isolators and circulators. Furthermore, the ability to achieve such robust mechanical squeezing has significant implications for advancing quantum precision measurements in metrology and sensing, offering new opportunities for exploring quantum-enhanced technologies.
... where s j (s † j ) is the annihilation (creation) operator for the jth spin. This describes an inverted (i.e., upside down) ensemble of harmonic oscillators [51]. A spin deexcitation creates an excitation in the inverted oscillator system, which lowers its energy. ...
Masers once represented the state of the art in low-noise microwave amplification technology but eventually became obsolete due to their need for cryogenic cooling. Masers based on solid-state spin systems perform most effectively as amplifiers, since they provide a large density of spins and can, therefore, operate at relatively high powers. While solid-state maser oscillators have been demonstrated at room temperature, continuous-wave amplification in these systems has only ever been realized at cryogenic temperatures. Here, we report on a continuous-wave solid-state maser amplifier operating at room temperature. We achieve this feat using a practical setup that includes an ensemble of nitrogen-vacancy center spins in a diamond crystal, a strong permanent magnet, and a simple laser diode. We describe important amplifier characteristics including gain, bandwidth, compression power, and noise temperature and discuss the prospects of realizing a room-temperature near-quantum-noise-limited amplifier with this system. Finally, we show that in a different mode of operation the spins can be used to reduce the microwave noise in an external circuit to cryogenic levels, all without the requirement for physical cooling.
Published by the American Physical Society 2024
In this paper, we study the synchronization of dissipative quantum harmonic oscillators in the framework of a quantum open system via the active–passive decomposition (APD) configuration. We show that two or more quantum systems may be synchronized when the quantum systems of interest are embedded in dissipative environments and influenced by a common classical system. Such a classical system is typically termed a controller, which (1) can drive quantum systems to cross different regimes (e.g., from periodic to chaotic motions) and (2) constructs the so-called active–passive decomposition configuration, such that all the quantum objects under consideration may be synchronized. The main finding of this paper is that we demonstrate that the complete synchronizations measured using the standard quantum deviation may be achieved for both stable regimes (quantum limit circles) and unstable regimes (quantum chaotic motions). As an example, we numerically show in an optomechanical setup that complete synchronization can be realized in quantum mechanical resonators.
We study the entanglement dynamics of multi-qubit systems coupled to a common dissipative environment, focusing on systems with one or two initially excited qubits. Using the Lindblad master equation, we derive the time evolution of the density matrix and analyze the entanglement between qubit pairs via the concurrence measure. The solution method involves applying the Lindblad super-operator to the initial density matrix, generating a subspace of non-duplicate states. We extend our analysis to n-qubit systems, including three-qubit and four-qubit configurations, and explore the effects of thermal noise on entanglement dynamics. Our results demonstrate how initial conditions, system size, and environmental interactions shape entanglement, providing valuable insights for quantum information processing applications.
A novel and efficient method is proposed for generating mechanical squeezing in a double‐cavity optomechanical system by introducing an optical parametric amplifier (OPA) with pump modulation (PM). The combined effects of periodic amplitude modulation of the driving laser and the OPA enable mechanical squeezing in both the resolved‐sideband and unresolved‐sideband regimes, with significantly enhanced squeezing observed in the resolved‐sideband regime. The individual contributions of the OPA and PM to mechanical squeezing is systematically investigated, with results indicating that PM plays a more dominant role. Additionally, the proposed scheme demonstrates excellent robustness against environmental thermal noise. These findings highlight the potential of this approach for advancing quantum precision measurements and exploring macroscopic quantum phenomena.
Observing non-classical properties of light is a long-standing interest to advance a wide range of quantum application from computing to metrology. Optical cavities are essential to generate and manipulate non-classical light. However, detecting changes in cavity properties induced by the quantum state remains a critical challenge in the optical domain due to the weak material nonlinearity, limiting our ability to observe quantum states generated in optical cavities. Here, we propose a framework for observing the dynamics of quantum states generated inside nonlinear optical cavities. We utilize symmetry-breaking to obtain high sensitivity to small perturbations introduced to the quantum state, resulting in an asymmetric equilibrium of a macroscopic observable. With a nonlinear response at the single photon level, our approach directly imprints the field distribution of the cavity quantum state onto the statistics of bistable cavity steady-states. We experimentally demonstrate our approach in a degenerate optical parametric oscillator, generating and reconstructing the quasi-probability distribution of different quantum states. As a validation, we reconstruct the Husimi Q function of the cavity squeezed vacuum state. In addition, we observe the evolution of the quantum vacuum state inside the cavity as it undergoes phase-sensitive amplification. By enabling generation and measurement of quantum states in a single nonlinear optical cavity, our method paves a way for studying exotic dynamics of quantum optical states in nonlinear driven-dissipative systems such as soliton generation and Kerr frequency combs.
We investigate quantum metrology in a degenerate down-conversion system composed of a pump mode and two degenerate signal modes. In the conventional parametric approximation, the pump mode is assumed to be constant, not a quantum operator. We obtain the measurement precision of the coupling strength beyond the parametric approximation. Without a dissipation, the super-Heisenberg limit can be obtained when the initial state is the direct product of classical state and quantum state. When the pump mode suffers from a single-photon dissipation, the measurement uncertainty of the coupling strength is close to 0 as the coupling strength approaches 0 with a coherent driving. The direct photon detection is proved to be the optimal measurement. This result has not been changed when the signal modes suffer from the two-photon dissipation. When the signal modes also suffer from the single-mode dissipation, the information of the coupling strength can still be obtained in the steady state. In addition, the measurement uncertainty of the coupling strength can also be close to 0 and become independent of noise temperature as a critical point approaches. Finally, we show that a driven-dissipation down-conversion system can be used as a precise quantum sensor to measure the driving strength.
Optical Schr\"{o}dinger cat states are non-Gaussian states with applications in quantum technologies, such as for building error-correcting states in quantum computing. Yet the efficient generation of high-fidelity optical Schr\"{o}dinger cat states is an outstanding problem in quantum optics. Here, we propose using squeezed superpositions of zero and two photons, , as ingredients for protocols to efficiently generate high-fidelity cat states. We present a protocol using linear optics with success probability that can generate cat states of size with fidelity . The protocol relies only on detecting single photons and is remarkably tolerant of loss, with detection loss still achieving for cats with . We also show that squeezed states are ideal candidates for nonlinear photon subtraction using a two-level system with near deterministic success probability and fidelity for cat states of size . Schemes for generating states using quantum emitters are also presented. Our protocols can be implemented with current state-of-the-art quantum optics experiments.
36, 347 van Harlingen, D. 347, 348 van Kampen
- R C Tolman
Tolman, R.C. 36, 347
van Harlingen, D. 347, 348
van Kampen, N.G. 56, 79, 348, 349,
350