In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.
In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.
We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength. As an illustration of these new results we solve the corresponding master equation and calculate, among other physical quantities, the uncertainty function whose late time behavior can be obtained fully. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that any initial state always settles down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. For more general spectra we show that the solution of the master equation can be reduced to solving for the motion of a classical parametric oscillator with parametric frequency determined by the unsolved for master equation coefficients. States in these systems experience evolution that is parametrically similar to the simpler evolution explicitly determined for in the case of Laurent-series spectra.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
The effect of the environment on a quantum system is studied on an exactly solvable model: a harmonic oscillator interacting with a one-dimensional massless scalar field. We show that in an open quantum system, dissipation can cause decorrelation on a time scale significantly shorter than the relaxation time which characterizes the approach of the system to thermodynamic equilibrium. In particular, we demonstrate that the density matrix decays rapidly toward a mixture of ``approximate eigenstates'' of the ``pointer observable,'' which commutes with the system-environment interaction Hamiltonian. This observable can be regarded as continuously, if inaccurately, monitored by the scalar field environment. Both because in a harmonic oscillator the state of the system rotates in the phase space and because the effective environment ``measurement'' is weak, the system, on the short ``collision'' time scale (1/\ensuremath{\Gamma}), maintains a coherence in this pointer observable on time scales of order [\ensuremath{\gamma}/\ensuremath{\Omega}ln(\ensuremath{\Gamma}/\ensuremath{\Omega})]1/2 and on longer time scales settles into a mixture of coherent states with a dispersion approximately consistent with the vacuum state. The master equation satisfied by the exact solution differs from the other master equations derived both for the high-temperature limit and for T=0. We discuss these differences and study the transition region between the high- and low-temperature regimes. We also consider the behavior of the system in the short-time ``transient'' regime. For T=0, we find that, in the long-time limit, the system behaves as if it were subject to ``1/f noise.'' The generality of our model is considered and its predictions are compared with previous treatments of related problems. Some of the possible applications of the results to experimentally realizable situations are outlined. The significance of the environment-induced reduction of the wave packet for cosmological models is also briefly considered.
The rather new field of quantum thermodynamics intends to derive thermodynamics from quantum mechanics, as well as to study thermodynamical behavior of quantum systems. It turns out that, indeed, thermalization can be obtained even for very small quantum systems due to the interaction with an appropriate environment. The consideration of an ensemble, as in statistical mechanics, is not necessary. However, the total system - consisting of system and environment - in quantum thermodynamics is usually considered to be closed. In particular, there exists no interaction with an external observer. Since quantum measurements typically influence the measured system, one may wonder which influence such an observation will have in the quantum thermodynamical setting. In this thesis, we study this influence of measurements in quantum thermodynamical systems. It turns out that various concepts known from statistical mechanics arise from such a "quantum thermodynamics under observation", although they do not have any meaning in the isolated quantum thermodynamical setting. The model discussed in the first part consists of a two-level system coupled to a large spin environment, the magnetization of which is measured periodically. This leads to quasi-classical trajectories for the system and - as in statistical mechanics - thermalization and temperature only have a meaning in terms of an ensemble average or a long-time average. Calculating these averages, we prove the ergodicity of the system. Interestingly, the entropy of the obtained attractor state depends on the time between the measurements and can be shown to be connected with the subjective lack of knowledge of the observer. In case of optimal measurements, the ensemble averaged state of the system under periodic measurements equals the state of the system in the quantum thermodynamical model without measurements. Introducing a detuning between system and environment, we find that the system can be prepared in almost arbitrary thermal states due to the periodic measurements. Even the creation of inversion becomes possible. As shown in numerical simulations, our results, which were analytically obtained for large environments, are even applicable to describe the behavior of the system coupled to small environments consisting of only a few spins. Besides measurements of the environment, we also study direct projective measurements of the system, as well as measurements with finite duration, obtaining that the dynamics of the system is almost independent of the concrete realization of the measurements. In the second part of the thesis, we discuss the meaning of temperature fluctuations in quantum thermodynamical systems. In statistical mechanics, there exist different approaches to the problem of temperature fluctuations, which lead to the well-known temperature fluctuation formula. The nature and operational accessibility of these fluctuations, however, remains rather unclear. We argue that according to quantum thermodynamics a system relaxes into a quasi-stationary thermal state and therefore the temperature does not fluctuate. However, since temperature is not an observable, it can only be determined indirectly. For modular systems, we show that the fluctuations of the temperature estimate obtained from a single energy measurement, indeed, are given by the temperature fluctuation formula derived in statistical mechanics, but only within a certain temperature range. The validity range of this formula increases with increasing size of the system. Thus, one can conclude that these "temperature fluctuations" can be interpreted as representing a fundamental limit for the instantaneous determination of temperature.
Entropy of a quantum oscillator in the presence of a quantum environment (or heat bath) is studied here. The spectral density, which comprises the environmental influences, here corresponds to a quasi-monochromatic thermal harmonic noise. This further implies that the environment is a structured one compared to the usually employed unstructured strictly ohmic environment. The entropy is found to be vanishing as the temperature approaches its absolute zero value, which thereby guarantees the third law of thermodynamics. At very high temperatures the classical behavior is observed. Similar to the usually employed ohmic-Drude bath model, the coefficient of the leading temperature dependent term in the entropy (at low temperatures) is proportional to the damping strength but not on other parameters of the spectral density characterising the bath.
This work studies the relationship between parametric amplification (or particle creation), adiabaticity and irreversibility in the non-quasi-static regime of a time-dependent quantum harmonic oscillator (TDHO) that evolves unitarily. We provide analytical results for the evolution of the TDHO valid for any functional value of the frequency, which enables us to monitor the behavior of the thermodynamical magnitudes in the non-quasi-static regime. In the latter, the largest modes of the energy eigenstates commonly undergo a process of spontaneous thermalization, where the concept of temperature naturally arises from the unitary evolution of the oscillator, i.e. without relation to any external source of temperature or thermal bath. As the evolution is unitary, this thermalization process can be reversible. We extend the standard definitions of quantum heat and work to account for the change in the populations of the energy levels and relate it to the diagonal entropy to obtain an effective temperature from the associated second principle of thermodynamics. The qualitative behavior of this temperature is similar to the temperature of the largest modes and is only significant in the non quasi-static region, returning to the original value T→0 afterwards. This effect is briefly analyzed in the case of a periodically driven quantum system, where the energy eigenstate mixing and the temperature grow and return to the original value periodically.