Generalized entropy production fluctuation theorems for quantum systems

Based on trajectory-dependent path probability formalism in state space, we derive generalized entropy production fluctuation relations for a quantum system in the presence of measurement and feedback. We have obtained these results for three different cases: (i) the system is evolving in isolation from its surroundings; (ii) the system being weakly coupled to a heat bath; and (iii) system in contact with reservoir using quantum Crooks fluctuation theorem. In Case (iii), we build on the treatment carried out by H T Quan and H Dong [arXiv/cond-mat:0812.4955], where a quantum trajectory has been defined as a sequence of alternating work and heat steps. The obtained entropy production fluctuation theorems (FTs) retain the same form as in the classical case. The inequality of second law of thermodynamics gets modified in the presence of information. These FTs are robust against intermediate measurements of any observable performed with respect to von Neumann projective measurements as well as weak or positive operator-valued measurements.     

Entropy Production in Stochastic Systems with Fast and Slow Time-Scales

The behavior of stochastic systems with interacting fast and slow degrees of freedom is investigated both for discrete and continuous processes. Effective equations that govern the process on the slow timescale are derived by asymptotic methods, both for the propagator and the entropy production of the systems. It is found that in general the result of the limiting procedure for entropy does not coincide with the one defined for the effective slow process and features an additional contribution. The specific conditions under which such a correction does or does not arise are stated and the general explicit form of this remnant entropy production is offered. Finally, the fluctuation theorems that are satisfied by this additional term are given.     

Hawking radiation and black hole entropy in a gravity’s rainbow

In the context of gravity’s rainbow, Planck scale correction on Hawking radiation and black hole entropy in Parikh and Wilczk’s tunneling framework is studied. We calculate the tunneling probability of massless particles in the modified Schwarzschild black holes from gravity’s rainbow. In the tunneling process, when a particle gets across the horizon, the metric fluctuation must be taken into account, not only due to energy conservation but also to spacetime Planck scale effect. Our results show that the emission rate is related to changes of the black hole’s quantum corrected entropies before and after the emission. In the same time, for the modified black holes, a series of correction terms including a logarithmic term to Bekenstein–Hawking entropy are obtained. Correspondingly, the spectrum of Planck scale corrected emission is obtained and it deviates from the thermal spectrum. In addition, a specific form of modified dispersion relation is proposed and applied.     

Adaptive wave-front correction by means of all-optical feedback interferometry

A novel adaptive wave-front correction system based on an all-optical feedback interferometer is described. In this system the two-dimensional output fringe intensity from a Mach-Zehnder interferometer with large radial shear is optically fed back to an optically addressed phase-only liquid-crystal spatial light modulator. Consequently, without a separate aberration-free reference wave, the modulator phase approximates the conjugate of the interferometer phase that is directly related to the phase of the input aberrated wave front, so this system is applicable in adaptive optics. We successfully achieved real-time correction of aberrated wave fronts: A diffraction pattern that was seriously distorted because of aberrations was transformed into a diffraction-limited spot immediately after the feedback loop was closed.     

Exchange fluctuation theorems for a chain of interacting particles in presence of two heat baths

The exchange fluctuation theorem for heat exchanged between two systems at different temperatures, when kept in direct contact, has been investigated by Jarzynski and Wójcik [Phys. Rev. Lett. 92, 230602 (2004)]. We extend this result to the case where two Langevin reservoirs at different temperatures are connected via a conductor made of interacting particles, and are subjected to an external drive or work source. The Langevin reservoirs are characterized by Gaussian white noise fluctuations and concomitant friction coefficients. We first derive the Crooks theorem for the ratio between forward and reverse paths, and discuss the first law in this model. Then we derive the modified detailed fluctuation theorems (MDFT) for the heat exchanged at each end. These theorems differ from the usual form of the detailed fluctuation theorems (DFT) in literature, due the presence of an extra multiplicative factor. This factor quantifies the deviation of our MFDT from the DFT. Finally, we numerically study our model, with only two interacting particles for simplicity.     

Quantum feedback control: how to use verification theorems and viscosity solutions to find optimal protocols

While feedback control has many applications in quantum systems, finding optimal control protocols for this task is generally challenging. So-called ‘verification theorems’ and ‘viscosity solutions’ provide two useful tools for this purpose: together they give a simple method to check whether any given protocol is optimal, and provide a numerical method for finding optimal protocols. While treatments of verification theorems usually use sophisticated mathematical language, this is not necessary. In this article we give a simple introduction to feedback control in quantum systems, and then describe verification theorems and viscosity solutions in simple language. We also illustrate their use with a concrete example of current interest.     

外反馈He—Ne激光器的非稳特性研究

报道了平面镜和自泵浦相位共轭镜反馈Ｈｅ－Ｎｅ激光器的非稳特性，观测了反馈镜位置对激光光强起伏及波动频率的影响，即在半腔长的整数倍附近，激光光强具有较大的起伏和秒量级的准周期振荡，并对其产生机理进行了分析。     

Improvement of two-way continuous variable quantum cryptography by using additional noise

The performance of quantum key distribution such as one-way continuous variable protocols, can be increased by adding some noise on the reference side of error correction in the error-correction phase. For this reason, we here study this possibility in the case of two-way continuous variable system. Finally, the numerical results show that the using of additional noise gives two-way schemes better security performance in terms of secret key rates and resistance to channel excess noise.     

On the Asymptotic Convergence of the Transient and Steady-State Fluctuation Theorems

Nonequilibrium molecular dynamics simulations are used to demonstrate the asymptotic convergence of the transient and steady-state forms of the fluctuation theorem. In the case of planar Poiseuille flow, we find that the transient form, valid for all times, converges to the steady-state predictions on microscopic time scales. Further, we find that the time of convergence for the two theorems coincides with the time required for satisfaction of the asymptotic steady-state fluctuation theorem.     

Position and Velocity Time Delay for Suppression Vibrations of a Hybrid Rayleigh-Van der Pol-Duffing Oscillator

In this paper, we used time delay feedback to minimize the vibrations of a hybrid Rayleigh–van der Pol–Duffing oscillator. This system is a one-degree-offreedom containing the cubic and fifth nonlinear terms and an external force. We applied the multiple scales method to get the solution from first approximation. Graphically and numerically, we studied the system before and after adding time delay feedback at the primary resonance case (Ω ≌ ω). We used MATLAB program to simulate the efficacy of different parameters and the time delay on the main system.     

Antiunitary symmetry operators in quantum mechanics

A criterion to decide that some symmetries of a quantum system must be realized as antiunitary operators is given. It is based on some mathematical theorems about the second cohomology group of the symmetry group when expressed in terms of those of a normal subgroup and the corresponding factor group. It is also shown that this criterion implies that the only possibility for the unitary subgroup in the Galilean case is that generated by the space reflection and the connected component containing the identity; otherwise only massless systems would arise.     

Stochastic equations in black hole backgrounds and non-equilibrium fluctuation theorems

We apply the non-equilibrium fluctuation theorems developed in the statistical physics to the thermodynamics of black hole horizons. In particular, we consider a scalar field in a black hole background. The system of the scalar field behaves stochastically due to the absorption of energy into the black hole and emission of the Hawking radiation from the black hole horizon. We derive the stochastic equations, i.e. Langevin and Fokker-Planck equations for a scalar field in a black hole background in the ?→0 limit with the Hawking temperature ?κ/2π fixed. We consider two cases, one confined in a box with a black hole at the center and the other in contact with a heat bath with temperature different from the Hawking temperature. In the first case, the system eventually becomes equilibrium with the Hawking temperature while in the second case there is an energy flow between the black hole and the heat bath. Applying the fluctuation theorems to these cases, we derive the generalized second law of black hole thermodynamics. In the present paper, we treat the black hole as a constant background geometry.Since the paper is also aimed to connect two different areas of physics, non-equilibrium physics and black holes physics, we include pedagogical reviews on the stochastic approaches to the non-equilibrium fluctuation theorems and some basics of black holes physics.     

Investigation of Noise-Induced Escape Rate: A Quantum Mechanical Approach

A quantum system coupled to a heat-bath in non-equilibrium environment is considered to study the problem of noise-induced escape rate from a metastable state in the moderate to strong friction limit (Kramers’ regime). It is known that starting from an initial coherent state representation of bath oscillators, one can derive a c-number generalized quantum Langevin equation where the quantum correction terms appear as a coupled infinite set of hierarchy of equations. For practical purpose, one should truncate these equations after a certain order. In our present development, we calculate the quantum correction terms in a closed analytical form based on a systematic perturbation technique and then derive the lowest order quantum correction factor exactly in the case of an Ohmic dissipative bath. Finally, to demonstrate its applicability, the effective equation of motions has been used to study the barrier crossing dynamics which incorporates the quantum correction factors.     

A modified Einstein–Nernst relation between mobility and diffusion of charges to evaluate the non-equilibrium, transient processes of ions in electrolytes

Transient correction terms to the nonlinear differential equations, describing the dynamics of migration and diffusion of the ion charges in electrolytes, have been recently defined and numerically evaluated. The system of equations has been modified in accordance with the obtained non-equilibrium corrections and the system variables have been evaluated in the case when the corrections are space-averaged. The purpose of the present work is to obtain a general solution when both the time and space dependencies of the correction terms are preserved i.e. without space-averaging of the corrections. The obtained, under these conditions, set of dynamical equations has been analytically transformed to a simpler form, which is easier to be tackled numerically. The corrected results, in contrast to the results of uncorrected equations, show much faster convergence to equilibrium of the physical system and manifest the presence of characteristic pre-electrode maxima of the transient ion currents.     

Feedback method of the noise suppression in wave-mixing amplifiers based on non-linear materials with photorefractive response in a reflection grating configuration

In the photorefractive wave-mixing system, fluctuation in the signal beam intensity of the photorefractive output with a reflection grating has been analyzed by employing pump feedback method. In this method, fluctuations of the photorefractive wave-mixing process not only induce the intensity fluctuation of the mixing waves but also induce phase fluctuation of the mixing waves. Thus, the phase of the pump and signal beams at the output surface fluctuates in time around a mean value. Using such a positive feedback method of a pump beams, the relative fluctuation in the photorefractive output signal beam intensity with respect to its mean intensity can be minimized significantly without reducing its mean intensity. The factors that control the fluctuation in the signal beam intensity, such as the phase fluctuation of the output pump beam, absorption strength of the material and the feedback reflectivity of the cavity mirrors, on the relative fluctuation of the output signal intensity in the photorefractive wave-mixing systems have been studied in detail. It has been found that the fluctuation of the output signal intensity relative to its mean intensity in the photorefractive wave-mixing system can be suppressed to larger extent by taking lower value of feedback reflectivity of the cavity mirrors which could exist at a higher value of absorption strength of the photorefractive materials.     

交流斯塔克效应对简并喇曼过程场振幅N次方压缩的影响

用改进型有效哈密顿量方法研究简并喇曼过程中辐射场的振幅N次方压缩。发现如果场初始处于相干态,则它以后不会获得任何次方的压缩。如果场初始处于压缩真空态,则它以后可获N次方压缩。讨论了场涨落对次幂N、初场平均光子数n和压缩角ξ的依赖关系。     

Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integral for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit. PACS numbers: 05.70.Ln, 05.40.-a, 05.10.Gg.     

Center manifolds,normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions

We study the instabilities known to aeronautical engineers as flutter and divergence. Mathematically, these states correspond to bifurcations to limit cycles and multiple equilibrium points in a differential equation. Making use of the center manifold and normal form theorems, we concentrate on the situation in which flutter and divergence become coupled, and show that there are essentially two ways in which this is likely to occur. In the first case the system can be reduced to an essential model which takes the form of a single degree of freedom nonlinear oscillator. This system, which may be analyzed by conventional phase-plane techniques, captures all the qualitative features of the full system. We discuss the reduction and show how the nonlinear terms may be simplified and put into normal form. Invariant manifold theory and the normal form theorem play a major role in this work and this paper serves as an introduction to their application in mechanics.Repeating the approach in the second case, we show that the essential model is now three dimensional and that far more complex behavior is possible, including nonperiodic and ‘chaotic’ motions. Throughout, we take a two degree of freedom system as an example, but the general methods are applicable to multi- and even infinite degree of freedom problems.     

Detecting characteristics of information masked by a laser-triggered microwave system via Hilbert-Huang transform

A two-wave-mixing microwave system under a delayed feedback control is proposed for chaotic communications in this study. Under the consideration of simple chaotic masking, Hilbert-Huang transform is proved to be an efficient way to detect characteristics of information signals via the spectrum of intrinsic mode functions. Based upon detrended fluctuation as well as multiscale entropy analyses on masking efficiency in the present system, we may suggest that Hilbert-Huang transform would be an alternative method to analyze complex dressed signals from nonlinear optoelectronic systems.