It is often argued that a small non-degenerate quantum system coupled to a bath has a fixed energy in its ground state since a fluctuation in energy would require an energy supply from the bath. We consider a simple model of a harmonic oscillator (the system) coupled to a linear string and determine the mean-squared energy fluctuations. We also analyze the two-time correlator of the energy and discuss its behavior for a finite string.
It is often argued that a small non-degenerate quantum system coupled to a bath has a fixed energy in its ground state since a fluctuation in energy would require an energy supply from the bath. We consider a simple model of a harmonic oscillator (the system) coupled to a linear string and determine the mean-squared energy fluctuations. We also analyze the two-time correlator of the energy and discuss its behavior for a finite string.
We investigate the role of vacuum (zero-point) fluctuations in generating decoherence in a number of simple models. First we discuss a harmonic oscillator coupled to a semi-infinite elastic string and discuss the irreversible nature of such a bath. We investigate the fluctuations in energy of the oscillator and discuss the trace the oscillator leaves in the bath. Most of the work deals with two-level systems coupled to a bosonic bath (a transmission line). For two-level systems with a Hamiltonian that commutes with the total Hamiltonian (system plus coupling plus bath) the ground state is a pure state. The energy of the system is a constant of motion. For the general case, the energy of the two-level system fluctuates, and the ground state is only partially coherent. A particular realization of such a two level system consists of a mesoscopic ring with a quantum dot coupled capacitively to a transmission line. In the presence of an Aharonov-Bohm flux this system exhibits a persistent current. This current is a measure of the coherence of the ground state. As a function of the coupling strength the ground state undergoes a crossover from a state characterized by a time-averaged persistent current which is much larger than its time-averaged mean squared fluctuations to a state characterized by a persistent current with an average amplitude that is much smaller than its mean squared fluctuations.
The nonzero ground-state energy of the quantum mechanical harmonic oscillator implies quantum fluctuations around the minimum of the potential with the mean-square value proportional to Planck's constant. On the other hand, thermal fluctuations occur when the oscillator is coupled to a heat bath. Using quantum statistical mechanics, this paper describes the transition from pure quantum fluctuations at zero temperature to classical thermal fluctuations in the high-temperature limit. It was early pointed out by Peierls that the classical mean-square thermal fluctuations in a harmonic chain—a linear chain of coupled harmonic oscillators—increase linearly with the distance of the masses from the fixed end of the chain, destroying long-range crystalline order. As shown here, the corresponding quantum fluctuations lead to a much slower logarithmic increase with the distance from the end. It is also shown that this behavior implies, for example, the absence of sharp Bragg peaks in x-ray scattering in an infinite chain at zero temperature, which instead shows power-law behavior typical for one-dimensional quantum liquids (called Luttinger liquids).
We study the bath dynamics in the dephasing model of a two-state quantum system (qubit) coupled to an environment of harmonic oscillators.This model was shown [Morozov et al., Phys.Rev. A, 2012 85, 022101] to admit the analytic solution for the qubit and environment dynamics.Using this solution, we derive the exact expression for the bath reduced density matrix in the presence of initial qubit-environment correlations.We obtain the non-equilibrium phonon distribution function and discuss in detail the time behavior of the bath energy.It is shown that only the inclusion of dynamic correlations between the qubit and the bath ensures the proper time behavior of the quantity which may be interpreted as the "environment energy".
The quantum dynamics of a low-dimensional system in contact with a large but finite harmonic bath is theoretically investigated by coarse-graining the bath into a reduced set of effective energy states. In this model, the couplings between the system and the bath are obtained from statistically averaging over the discrete, degenerate effective states. Our model is aimed at intermediate bath sizes in which non-Markovian processes and energy transfer between the bath and the main system are important. The method is applied to a model system of a Morse oscillator coupled to 40 harmonic modes. The results are found to be in excellent agreement with the direct quantum dynamics simulations presented in the work of Bouakline et al. [J. Phys. Chem. A 116, 11118-11127 (2012)], but at a much lower computational cost. Extension to larger baths is discussed in comparison to the time-convolutionless method. We also extend this study to the case of a microcanonical bath with finite initial internal energies. The computational efficiency and convergence properties of the effective bath states model with respect to relevant parameters are also discussed.
Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also for small values of the couplings the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree-Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.
Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim, we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also, for small values of the couplings, the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree–Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.