Quantum systems in a mechanical embedding, the breathing mode of a small particles, optomechanical system, etc. are far not the full list of examples in which the volume exhibits quantum behavior. Traditional consideration suggests strain in small systems as a result of a collective movement of particles, rather than the dynamics of the volume as an independent variable. The aim of this work is to show that some problem here might be essentially simplified by introducing periodic boundary conditions. At this case, the volume is considered as the independent dynamical variable driven by the internal pressure. For this purpose, the concept of quantum volume based on Schrödinger’s equation in [Formula: see text] manifold is proposed. It is used to explore several 1D model systems: An ensemble of free particles under external pressure, quantum manometer and a quantum breathing mode. In particular, the influence of the pressure of free particle on quantum oscillator is determined. It is shown also that correction to the spectrum of the breathing mode due to internal degrees of freedom is determined by the off-diagonal matrix elements of the quantum stress. The new treatment not using the “force” theorem is proposed for the quantum stress tensor. In the general case of flexible quantum 3D dynamics, quantum deformations of different type might be introduced similarly to monopole mode.
Quantum systems in a mechanical embedding, the breathing mode of a small particles, optomechanical system, etc. are far not the full list of examples in which the volume exhibits quantum behavior. Traditional consideration suggests strain in small systems as a result of a collective movement of particles, rather than the dynamics of the volume as an independent variable. The aim of this work is to show that some problem here might be essentially simplified by introducing periodic boundary conditions. At this case, the volume is considered as the independent dynamical variable driven by the internal pressure. For this purpose, the concept of quantum volume based on Schrödinger’s equation in [Formula: see text] manifold is proposed. It is used to explore several 1D model systems: An ensemble of free particles under external pressure, quantum manometer and a quantum breathing mode. In particular, the influence of the pressure of free particle on quantum oscillator is determined. It is shown also that correction to the spectrum of the breathing mode due to internal degrees of freedom is determined by the off-diagonal matrix elements of the quantum stress. The new treatment not using the “force” theorem is proposed for the quantum stress tensor. In the general case of flexible quantum 3D dynamics, quantum deformations of different type might be introduced similarly to monopole mode.
Heat and work are fundamental concepts for thermodynamical systems. When these are scaled down to the quantum level they require appropriate embeddings. Here we show that the dependence of the particle spectrum on system size giving rise to a formal definition of pressure can, indeed, be correlated with an external mechanical degree of freedom, modelled as a spatial coordinate of a quantum oscillator. Under specific conditions this correlation is reminiscent of that occurring in the classical manometer.
The conventional understanding of pressure in physics is strictly tied to physical space, representing the force exerted per unit area by particles colliding with boundaries or interacting within a spatial volume. In this paper, we propose a radical extension of thermodynamic concepts to the domain of quantum mechanics by introducing the framework of the Intrinsic Pressure of Quantum States (IPQS). We postulate that quantum states possess an intrinsic internal pressure, denoted as P_psi, which operates independently of physical space volume and is fundamentally defined by the sensitivity of the system energy E to changes in the accessible state space volume V_Hilbert. By formalizing V_Hilbert as the measure of the accessible volume in the underlying Hilbert space or projective state space, we establish a rigorous mathematical foundation that links quantum state configuration changes to thermodynamic work and stability. We explore the thermodynamic analogies of IPQS, derive the equations of state for closed and open quantum systems, and analyze how modifications to state accessibility influence system stability, phase transitions, and quantum fluctuations. Furthermore, we discuss the profound implications of IPQS for quantum information theory, entanglement dynamics, and many-body quantum systems, paving the way toward a unified thermodynamics of quantum states.
We study a microscopic model, the Mirror-Oscillator-Field (MOF) model proposed in [1], for describing optomechanical interactions. In contrast with the conventional approach where the mirror-field interaction is understood as arising from the radiation pressure of an optical field inducing the motion of the mirror’s CoM, the MOF model incorporates the dynamics of the internal degrees of freedom of the mirror that couple to the optical field directly. Considering the mirror’s internal and mechanical degrees of freedom as two separate degrees of freedom we derive the optomechanical properties of the coupled mirror and field system. The major advantage in this approach is that it provides a self-consistent treatment of the three relevant subsystems (the mirror’s motion, its internal degrees of freedom and the field) including their back-actions on each other, thereby giving a more accurate account of the coupled internal and external dynamics. The optical and the mechanical properties of a mirror arising from its dynamical interaction with the field are obtained without imposing any boundary conditions on the field additionally, as is done in the conventional way. We find that our results agree with those from the boundary condition approach in the appropriate limits and more generally the model provides a framework within which one can study optomechanical elements with different internal structures and mechanical properties, which makes it suited for studying hybrid systems. Considering the quantum dynamics of the coupled subsystems we look at the entanglement between the mirror’s motion and the field, showing that the internal degrees of the mirror, in the appropriate parameter regimes, can act as a means to coherently transfer quantum correlations between the field and the mechanics thus leading to a larger optomechanical entanglement. We then use the MOF model to study the entanglement between the motion of an atom and a field for the setup in [95] and find a larger optomechanical entanglement when the field is closer to the internal resonance. We also study the interaction between two mirrors as described by the MOF model, specifically looking at the entanglement between the motion of their centers of mass. Conclusively, we see that including the dynamics of the internal degrees of freedom of a mirror, which is the quintessential mediator of interaction between the mirror center of mass and the field, leads to qualitatively different physics, specifically in the quantum regime, thus giving a physically more complete treatment of mirror-field interactions.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.
Small (but still containing many, about 10 2 -10 4 , atoms) quantum systems (traditionally termed nano-systems) are dramatically different from their macroscopic or genuine microscopic (atomic) cousins.Microscopic molecular systems (with a few atoms) obey a regular quantum dynamics (described by time dependent Schrodinger equation), whereas in macroscopic systems with continuous energy spectra, one can expect, also regular, although typically relaxation, dynamic behavior.The topic of our paper is in-between these limits.Nano-scale systems are characterized by small (but finite) mean inter-level spacing.In such a case with recurrence periods in pico-second range, Loschmidt echo and double resonance phenomena come into the game.System behavior becomes non-trivial and manifests a sort of transitions between regular and chaotic dynamics.We show that such dynamic transitions occur when the Loschmidt echo time of life exceeds the typical recurrence cycle period.We illustrate this behavior in the frame work of a few versions of the exactly solvable quantum problem, proposed long ago by Zwanzig.It is based on the study of time evolution of the initially prepared vibrational state coupled to a reservoir with dense spectrum of its vibrational states.In the simplest version of the Zwanzig model, the reservoir has an equidistant spectrum, and the system -reservoir coupling matrix elements are independent of the reservoir states.We generalize the model to include into consideration the coupling of the initially prepared single state to system phonon excitations.The coupling results to temperature dependent broadening and decay of the echo components.Another generalization is to replace a single level by two states coupled to the Zwanzig reservoir.We anticipate that the basic ideas inspiring our work can be applied to a large variety of interesting for the applications nano-systems (e.g., dissipative free propagation of excitations along molecular chains, or as a model for exchange reactions).