Quantum Langevin Equation Research Articles

Optical bistability and absorption characteristic of an optomechanical system embedded with double quantum dot and nonlinear medium

This paper theoretically studies a hybrid optomechanical system embedded with two coupled quantum dots and a third-order Kerr nonlinear medium inside the cavity. The optical bistability and absorption spectrum are analyzed for the proposed system. From the Hamiltonian which describes the proposed system, a set of quantum Langevin equations are derived. Using these equations of motion, the steady-state mean field analysis is done which gives the phenomena of optical bistability. The performance of the optical switch is also analyzed in terms of gain and switching ratio. Further, the absorption spectrum of the system is derived and analyzed from the fluctuation dynamics. The optical bistability has the potential to design tunable all-optical switches. The absorption spectra display peculiar characteristics of negative absorption (transparency dip). The transparency dips are found to be strongly dependent on the frequency of the mechanical resonator. The results of our investigation reveal that the proposed system can be used as an optical switch and has numerous other applications in quantum communication systems.

Continuous variable quantum discord in a cavity electrooptical down-conversion quantum system

In this paper, we explore continuous variable quantum discord for the stationary microwave–optical output fields in an electrooptical system. We drive the covariance matrix of the system using the quantum Langevin equations for the intera-cavity microwave and optical modes. Employing the input–output theory, we obtained the output microwave and optical resonator variables in the Fourier frequency domain. The properties of the Gaussian quantum discord with respect to electrooptical cooperativity, the normalized cavity wave guides for microwave and optical resonators, and the thermal mean photon number for microwave and optical modes have been analyzed. Henceforth, we obtained that the Gaussian quantum discord can be enhanced by increasing the electrooptical cooperativity and microwave and optical wave guides. The thermal mean photon numbers of the microwave and optical modes have reduction effects on the Gaussian quantum discord.

Quantum Fluctuations in Vacuum Energy: Cosmic Inflation as a Dynamical Phase Transition

A variety of models of inflation in the early Universe have been proposed and applied to describe successfully the origin of all possible structures in the Universe. On the other hand, inflation theory is still phenomenological and needs systematic physical foundations, including the relation to dark matter and dark energy. The essence of cosmic inflation would be a dynamical phase transition and the spontaneous symmetry breaking process, which are common in ordinary physics in the laboratory. At the beginning of the phase transition, the system is often in an adiabatic ground state and produces a squeezed state. This is widely interpreted as the generation of classical structures; however, it is not. The common notion of decoherence is not sufficient to describe the inflationary phase transition: a particular trajectory must be singled out in the dynamics. When an interaction turns on, dissipation or the energy flow/cascade is possible, and the c-number random field appears. The separation of these classical statistical fluctuations from the deterministic time evolution is indicated by the secular divergence or the infrared divergence of the system. We describe this phase transition based on the closed-time-path method and derive a quantum Langevin equation with classical noise, which sources the development of a coherent state. Introducing the effective action method to describe the evolution of the coherent state, we describe the order parameter that characterizes the phase transition and the associated spontaneous symmetry breaking. Since this phase transition process is common in physics, we discuss further applications of this formalism in other physical systems.

Quantum coherence dynamics of displaced squeezed thermal state in a non-Markovian environment

The dynamical behavior of quantum coherence of a displaced squeezed thermal state in contact with an external bath is discussed in the present work. We use a Fano-Anderson type of Hamiltonian to model the environment and solve the quantum Langevin equation. From the solution of the quantum Langevin equation we obtain the Green's functions which are used to calculate the expectation value of the quadrature operators which are in turn used to construct the covariance matrix. We use a relative entropy based measure to calculate the quantum coherence of the mode. The single mode squeezed thermal state is studied in the Ohmic, sub-Ohmic and the super-Ohmic limits for different values of the mean photon number. In all these limits, we find that when the coupling between the system and the environment is weak, the coherence decays monotonically and exhibit a Markovian nature. When the system and the environment are strongly coupled, we observe that the evolution is initially Markovian and after some time it becomes non-Markovian. The non-Markovian effect is due to the environmental back action on the system. Finally, we also present the steady state dynamics of the coherence in the long time limit in both low and high temperature regime. We find that the qualitative behavior remains the same in both the low and high temperature limits. But quantitative values differ because the coherence in the system is lower due to thermal decoherence.

Uniformly accelerated Brownian oscillator in (2+1)D: Temperature-dependent dissipation and frequency shift

We consider an Unruh-DeWitt detector modeled as a harmonic oscillator that is coupled to a massless quantum scalar field in the (2+1)-dimensional Minkowski spacetime. We treat the detector as an open quantum system and employ a quantum Langevin equation to describe its time evolution, with the field, which is characterized by a frequency-independent spectral density, acting as a stochastic force. We investigate a point-like detector moving with constant acceleration through the Minkowski vacuum and an inertial one immersed in a thermal reservoir at the Unruh temperature, exploring the implications of the well-known non-equivalence between the two cases on their dynamics. We find that both the accelerated detector's dissipation rate and the shift of its frequency caused by the coupling to the field bath depend on the acceleration temperature. Interestingly enough this is not only in contrast to the case of inertial motion in a heat bath but also to any analogous quantum Brownian motion model in open systems, where dissipation and frequency shifts are not known to exhibit temperature dependencies. Nonetheless, we show that the fluctuating-dissipation theorem still holds for the detector-field system and in the weak-coupling limit an accelerated detector is driven at late times to a thermal equilibrium state at the Unruh temperature.

Nonequilibrium Electrical, Thermal and Spin Transport in Open Quantum Systems of Topological Superconductors, Semiconductors and Metals

We study nonequilibrium transport in various open quantum systems whose systems and leads/baths are made of topological superconductors (TSs), semiconductors, and metals. Using quantum Langevin equations and Green’s function method, we derive exact expressions for steady-state electrical, thermal, and spin current at the junctions between a system and leads. We validate these current expressions by comparing them with the results from direct time-evolution simulations. We then show how an electrical current injected in TS wires divides into two parts carried by single electronic excitations and Cooper pairs. We further show ballistic thermal transport in an open TS wire in the topological phase under temperature or voltage bias. The thermal current values grow significantly near the topological phase transition, where thermal conductance displays a sharp quantized peak as predicted earlier. We relate the quantized thermal conductance to the zero-frequency thermoelectric transmission coefficient of the open TS wire. We also observe a large thermoelectric current near the topological transition of the TS wires. We introduce a differential spin conductance which displays a quantized zero-bias peak at zero temperature for a spinful TS wire in the topological phase. The role of superconducting baths in transport is demonstrated by thoroughly examining the features of zero-temperature differential electrical conductance and thermal conductance in open systems with TS baths. Our new thermoelectric and spin transport findings in various two-terminal geometries are beneficial to the present challenges in probing the emergence of Majorana quasi-particles in experiments.

Quantum Langevin dynamics of a charged particle in a magnetic field: Response function, position–velocity and velocity autocorrelation functions

We use the quantum Langevin equation as a starting point to study the response function, the position–velocity correlation function and the velocity autocorrelation function of a charged quantum Brownian particle in the presence of a magnetic field and linearly coupled to a heat bath via position coordinate. We study two bath models – the Ohmic bath model and the Drude bath model and make a detailed comparison in various time–temperature regimes. For both bath models, there is a competition between the cyclotron frequency and the viscous damping rate giving rise to a transition from an oscillatory to a monotonic behaviour as the damping rate is increased. In the zero point fluctuation dominated low-temperature regime, non-trivial noise correlations lead to some interesting features in this transition. We study the role of the memory time-scale which comes into play in the Drude model and study the effect of this additional time-scale. We discuss the experimental implications of our analysis in the context of experiments in cold ions.

Frequency measurement and amplification of lidar echo signal based on optomechanical effects

In this work, we have proposed a new scheme to measure the frequency of lidar echo signals by optomechanically induced transparency or amplification. Our calculation is based on the quantum Langevin equations and the mean-field approximation. The phenomenon that the transmission probabilities of echo signal light of lidar are sensitive to its frequency gives us a method to obtain the probe frequency. The expressions of the external force that keeps the system on the blue or red sidebands are offered, which is the necessary condition for our measurements. For the blue sideband, the process leads to the amplification of the echo signal light if the frequency of the pump field satisfies certain relations. For the red sideband, the echo signal light may be fully absorbed and at this time whose frequency can be obtained by a formula that has been proved in this work. In contrast to the conventional method, our method may have higher accuracy, and cavity optomechanics can amplify the signal under the condition of the blue sideband. Our scheme is based on the quantum theory, which may be easier to associate with quantum Fisher information that can give the highest precision.

Influence functional for two mirrors interacting via radiation pressure

We study the effective dynamics of two mirrors, forming an optical cavity, and interacting with the cavity field via radiation pressure. We pursue a perturbative influence functional approach to trace out the degrees-of-freedom of the field, and obtain the second order effective action for the system composed by the mirrors. We find that the interaction between the mirrors is mediated by pairs of field modes, which combine in such a way to give rise to two different interaction channels. We find that the quantum and thermal fluctuations of the cavity field result in coloured, Gaussian stochastic noises acting on the mirrors. To each of these noises is associated a dissipative effect, and the corresponding power spectra and susceptibilities are related via generalized fluctuation-dissipation relations. We finally demonstrate that the dynamics of the mirrors admits a stochastic interpretation, and give the relative quantum Langevin equations.

NonMarkovianity in cosmology: Memories kept in a quantum field

In this work we ask how an Unruh–DeWitt (UD) detector with harmonic oscillator internal degrees of freedom Q measuring an evolving quantum matter field Φ(x,t) in an expanding universe with scale factor a(t) responds. We investigate the detector’s response which contains non-Markovian information about the quantum field squeezed by the dynamical spacetime. The challenge is in the memory effects accumulated over the evolutionary history. We first consider a detector W, the ‘Witness’, which co-existed and evolved with the quantum field from the beginning. We derive a nonMarkovian quantum Langevin equation for the detector’s Q by integrating over the squeezed quantum field. The solution of this integro-differential equation would answer our question, in principle, but very challenging, in practice. Striking a compromise, we then ask, to what extent can a detector D introduced at late times, called the ‘Detective’, decipher past memories. This situation corresponds to many cosmological experiments today probing specific stages in the past, such as COBE targeting activities at the surface of last scattering. Somewhat surprisingly we show that it is possible to retrieve to some degree certain global physical quantities, such as the resultant squeezing, particles created, quantum coherence and correlations. The reason is because the quantum field has all the fine-grained information from the beginning in how it was driven by the cosmic dynamics a(t). How long the details of past history can persist in the quantum field depends on the memory time. The fact that a squeezed field cannot come to complete equilibrium under continuous driving, as in an evolving spacetime, actually helps to retain the memory. We discuss interesting features and potentials of this ‘archeological’ perspective toward cosmological issues.

The Effect of Superposition on the Quantum Features of the Cavity Radiation of a Three-Level Laser

We study the statistical and squeezing properties of the cavity light produced by a degenerate three-level laser with the use of the solution of the pertinent quantum Langevin equation. Moreover, applying the density operator to the cavity radiation superposition, we investigated the quantum properties of the superposed cavity light beams generated by a pair of degenerate three-level lasers. Superposing the cavity radiation increases the mean and the variance of the photon number without affecting the quadrature squeezing. It is observed that the degree of squeezing of the separate cavity radiation, as well as the superposed cavity radiation, increases with the rate at which the atoms are injected into the cavity. We have also shown that the mean photon number of the superposed cavity radiation is the sum of the mean photon numbers of the individual cavity radiation. However, the variance of the photon number of the superposed cavity radiation turns out to be four times that of the component cavity radiation.

Non-equilibrium dynamics of the open quantum O(n)-model with non-Markovian noise: exact results

The collective and purely relaxational dynamics of quantum many-body systems after a quench at temperature T = 0, from a disordered state to various phases is studied through the exact solution of the quantum Langevin equation of the spherical and the O(n)-model in the limit n → ∞. The stationary state of the quantum dynamics is shown to be a non-equilibrium state. The quantum spherical and the quantum O(n)-model for n → ∞ are in the same dynamical universality class. The long-time behaviour of single-time and two-time correlation and response functions is analysed and the universal exponents which characterise quantum coarsening and quantum ageing are derived. The importance of the non-Markovian long-time memory of the quantum noise is elucidated by comparing it with an effective Markovian noise having the same scaling behaviour and with the case of non-equilibrium classical dynamics.

Two-Mode Squeezed and Entangled Light Production in Parametric Oscillations

We investigate the statistical and quadrature squeezings, as well as the entanglement properties, of a two-mode light generated by non-degenerate parametric oscillations coupled to a two-mode squeezed vacuum reservoir, by employing the solutions of the quantum Langevin equations. It is found that the two-mode light shows the two-mode squeezing and entanglement for all values of the time. Moreover, it is observed that the squeezed vacuum reservoir and the growing amplitude of the pump mode enhance the degrees of two-mode squeezing and entanglement. We have also shown that the amounts of squeezing and entanglement are significant in a region, where the mean photon number is higher, and the photon number correlation is lower.

Interaction of Two-Level Atoms with a Single-Mode Quantized Radiation Field

We have studied the statistical and squeezing properties of the cavity light generated by a two-level laser. This optical system contains N two-level atoms available in a cavity coupled to a single-mode vacuum reservoir. They are pumped to the top level from the bottom level by means of the electron bombardment. Applying the steady-state solutions of the equations of evolution of the expectation values of the atomic operators and the quantum Langevin equation, we obtained the global and local photon statistics of the single-mode light beam. We have found that, for the two-level laser operating well above the threshold, the uncertainties in the plus and minus quadratures are equal and satisfy the minimum uncertainty relation. In view of this, we have identified the light generated by the laser operating well above threshold to be coherent. On the other hand, the light generated by the laser operating at threshold is found to be chaotic. From the obtained results, we have also observed that a large part of the local mean photon number, the local photon number variance, and the local quadrature variance are confined in a relatively narrow frequency interval.

Environment-dependent vibrational heat transport in molecular junctions: Rectification, quantum effects, vibrational mismatch.

Vibrational heat transport in molecular junctions is a central issue in different contemporary research areas such as chemistry, materials science, mechanical engineering, thermoelectrics, and power generation. Our model system consists of a chain of molecules which are sandwiched between two solids that are maintained at different temperatures. We employ a quantum self-consistent reservoir model, which is built on a generalized quantum Langevin equation, to investigate quantum effects and far from equilibrium conditions on thermal conduction at nanoscale. The present self-consistent reservoir model can easily mimic the phonon-phonon scattering mechanisms. Different thermal environments are modeled as (i) Ohmic, (ii) sub-Ohmic, and (iii) super-Ohmic environments, and their effects are demonstrated for the thermal rectification properties of the system with spring graded or mass graded features. The behavior of heat current across molecular junctions as a function of chain length, temperature gradient, and phonon scattering rates are studied. Further, our analysis reveals the effects of vibrational mismatch between the solids phonon spectra on heat transfer characteristics in molecular junctions for different thermal environments.

Quantum corrections to the entropy in a driven quantum Brownian motion model

The quantum Brownian motion model is a typical model in the study of nonequilibrium quantum thermodynamics. Entropy is one of the most fundamental physical concepts in thermodynamics. In this work, by solving the quantum Langevin equation, we study the von Neumann entropy of a particle undergoing quantum Brownian motion. We obtain the analytical expression of the time evolution of the Wigner function in terms of the initial Wigner function. The result is applied to the thermodynamic equilibrium initial state, which reproduces its classical counterpart in the high temperature limit. Based on these results, for those initial states having well-defined classical counterparts, we obtain the explicit expression of the quantum corrections to the entropy in the weak coupling limit. Moreover, we find that for the thermodynamic equilibrium initial state, all terms odd in ℏ are exactly zero. Our results bring important insights to the understanding of entropy in open quantum systems.

Quantum optomechanical system in a Mach-Zehnder interferometer

We consider an oscillating micromirror replacing one of the two fixed mirrors of a Mach-Zehnder interferometer. In this ideal optical set-up the quantum oscillator is subjected to the radiation pressure interaction of travelling light waves, no cavity is involved. This configuration shows that squeezed light can be generated by pure scattering on a quantum system, without involving a cavity. The squeezing can be detected at the output ports of the interferometer either by direct detection or by measuring the spectrum of the difference current. We use the Hudson-Parthasarathy equation to model the global evolution. It can describe the scattering of photons and the resulting radiation pressure interaction on the quantum oscillator. It allows to consider also the interaction with a thermal bath. In this way we have a unitary dynamics giving the evolution of oscillator and fields. The Bose fields of quantum stochastic calculus and the related generalized Weyl operators allow to describe the whole optical circuit. By working in the Heisenberg picture, the quantum Langevin equations for position and momentum and the output fields arise, which are used to describe the monitoring in continuous time of the light at the output ports. In the case of strong laser and weak radiation pressure interaction highly non-classical light is produced, and this can be revealed either by direct detection (a negative Mandel Q-parameter is found), either by the intensity spectrum of the difference current of two photodetector; in the second case a nearly complete cancellation of the shot noise can be reached. In this last case it appears that the Mach-Zehnder configuration together with the detection of the difference current corresponds to an homodyne detection scheme, so that we can say that the apparatus is measuring the spectrum of squeezing.

Master-equation treatment of nonlinear optomechanical systems with optical loss

Open-system dynamics play a key role in the experimental and theoretical study of cavity optomechanical systems. In many cases, the quantum Langevin equations have enabled excellent models for optical decoherence, yet a master-equation approach to the fully nonlinear optomechanical Hamiltonian has thus far proven more elusive. To address this outstanding question and broaden the mathematical tool set available, we derive a solution to the Lindblad master equation that models optical decoherence for a system evolving with the nonlinear optomechanical Hamiltonian. The method combines a Lie-algebra solution to the unitary dynamics with a vectorization of the Lindblad equation, and we demonstrate its applicability by considering the preparation of optical cat states via the optomechanical nonlinearity in the presence of optical loss. Our results provide a direct way of analytically assessing the impact of optical decoherence on the optomechanical intracavity state.

A Green’s function approach to the linear response of a driven dissipative optomechanical system

In this paper, we first try to shed light on the ambiguities that exist in the literature in the generalization of the standard linear response theory (LRT) which has been basically formulated for closed systems to the theory of open quantum systems in the Heisenberg picture. Then, we investigate the linear response of a driven-dissipative optomechanical system (OMS) to a weak time-dependent perturbation using the so-called generalized LRT. It is shown how the Green’s function equations of motion of a standard OMS as an open quantum system can be obtained from the quantum Langevin equations (QLEs) in the Heisenberg picture. The obtained results explain a wealth of phenomena, including the anti-resonance, normal mode splitting and the optomechanically induced transparency (OMIT). Furthermore, the reason why the Stokes or anti-Stokes sidebands are amplified or attenuated in the red or blue detuning regimes is clearly explained which is in exact coincidence, especially in the weak-coupling regime, with the Raman-scattering picture.

A quantum Langevin equation approach for two-dimensional electronic spectra of coupled vibrational and electronic dynamics.

We present an efficient method to simulate two-dimensional (2D) electronic spectra of condensed-phase systems with an emphasis on treating quantum nuclear wave packet dynamics explicitly. To this end, we combine a quantum Langevin equation (QLE) approach for dissipation and a perturbative scheme to calculate three-pulse photon-echo polarizations based on wave packet dynamics under the influence of external fields. The proposed dynamical approach provides a consistent description of nuclear quantum dynamics, pulse-overlap effects, and vibrational relaxation, enabling simulations of 2D electronic spectra with explicit and non-perturbative treatment of coupled electronic-nuclear dynamics. We apply the method to simulate 2D electronic spectra of a displaced-oscillator model in the condensed phase and discuss the spectral and temporal evolutions of 2D signals. Our results show that the proposed QLE approach is capable of describing vibrational relaxation, decoherence, and vibrational coherence transfer, as well as their manifestations in spectroscopic signals. Furthermore, vibrational quantum beats specific for excited-state vs ground-state nuclear wave packet dynamics can also be identified. We anticipate that this method will provide a useful tool to conduct theoretical studies of 2D spectroscopy for strong vibronically coupled systems and to elucidate intricate vibronic couplings in complex molecular systems.