The response of a quantum system to a time-dependent periodic external field is investigated in connection with the dynamical symmetry breaking and level dynamics of the adiabatic states of the system. The main results are as follows. (A) When the periodic external field preserves the dynamical symmetry of the system, its response is like that of elastic matter. (B) When the periodic external field breaks the dynamical symmetry, several cases may occur: (a) in the adiabatic limit, the system still responds elastically; (b) if the initial state is an eigenstate of the evolution operator U ( T ) for one period T of the external field, the system evolves in time cyclically and responds quasi-elastically; (c) if the initial state is not an eigenstate of the evolution operator U ( T ), the system evolves in time non-cyclically and responds non-elastically. The detailed non-elastic behaviour depends on the statistical nature of the adiabatic eigenstates of the system. If the adiabatic spectrum is chaotic, the non-elastic response is expected to be strongly dissipative. The avoided level crossings of the adiabatic eigenstates play a crucial role in both producing chaoticity of the adiabatic levels and causing dissipation of the non-elastic response. The non-elastic role played by the adiabatic progressive phase is also addressed. Computer experiments are performed for the su(2) dynamical model to illustrate the above general results.
The response of a quantum system to a time-dependent periodic external field is investigated in connection with the dynamical symmetry breaking and level dynamics of the adiabatic states of the system. The main results are as follows. (A) When the periodic external field preserves the dynamical symmetry of the system, its response is like that of elastic matter. (B) When the periodic external field breaks the dynamical symmetry, several cases may occur: (a) in the adiabatic limit, the system still responds elastically; (b) if the initial state is an eigenstate of the evolution operator U ( T ) for one period T of the external field, the system evolves in time cyclically and responds quasi-elastically; (c) if the initial state is not an eigenstate of the evolution operator U ( T ), the system evolves in time non-cyclically and responds non-elastically. The detailed non-elastic behaviour depends on the statistical nature of the adiabatic eigenstates of the system. If the adiabatic spectrum is chaotic, the non-elastic response is expected to be strongly dissipative. The avoided level crossings of the adiabatic eigenstates play a crucial role in both producing chaoticity of the adiabatic levels and causing dissipation of the non-elastic response. The non-elastic role played by the adiabatic progressive phase is also addressed. Computer experiments are performed for the su(2) dynamical model to illustrate the above general results.
We will discuss on the quantum dynamics on the following three cases. (1) First, we will study structure of adiabatic energy levels for a triangle magnetic cluster with the Dzyaloshinsky-Moriya (DM) interaction. Generally, in single molecular magnets, the magnetic energy levels are discrete and we can find various interesting magnetic behavior when we sweep the external magnetic field. For the dynamics, the adiabatic energy level structure, in particular, avoided level structures at the energy level crossings, plays an important role. The mechanism to open the gap is attributed to some interactions which do not commute with the magnetization. We will discuss dependence of the dynamical magnetization process under the field cycling on the angle between the DM interaction vector and the external magnetic fields.[1,2] We also discuss a hidden conserved quantity due to the chiral symmetry.[3] The dynamical DM effect will be also discussed. (2) We will study the quantum dynamics of one-dimensional transverse Ising model which exhibits a quantum phase transition at zero temperature. There, we find a metastable state and a kind of spinodal decomposition phenomena in the sense of the quantum dynamics. We also refer to the cases of very fast sweep where some size independent collective motions are observed.[4] (3) Finally, we will also study an itinerant electron system in which the Mott insulator-Nagaoka ferromagnetism transition. There, we study the adiabatic change from the Mott insulator to the Nagaoka ferromagnetic state under removal of an electron from the lattice. We also study the magnetization process as a function of the magnetic field.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
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.
We describe a general method to solve a time-dependent Hamiltonian system exhibiting dynamical symmetry under the Lie algebra SU(1,1). The instantaneous eigenstates can be obtained by applying a displacement operator which is constructed explicitly. The construction proves to be equivalent to the solution of the classical equations of motion. One can also obtain the exact solution of the quantum Hamiltonian using this displacement operator. We define and discuss the differences between the cyclic evolution in the space of the physical states and in the parameter space. This leads to a method to define and find the T-cyclic and periodic states and their phases for a system with dynamical symmetry SU(1,1) just by using its time evolution operator. For SU(1,1) one can show that T-cyclic states only exist for some values of T which can be determined through an inequality which is found explicitly and discussed. The example used to check these results has been the degenerate optical parametric oscillator with dynamical symmetry SU(1,1). The system has no periodic states but it does possess T-cyclic states. The characteristic interaction times are distributed in forbidden and allowed regions as in the band theory of particles in a periodic potential. We also define exact phases and adiabatic phases along the lines of the above discussion on T-cyclic and periodic states. For sufficiently low frequencies the initial eigenstates of the system acquire non-trivial Berry phases in each cycle of the evolution of the functional parameters on which H depends. Such a phase can be measured experimentally and from the discussion of its possible values under various circumstances emerges an interesting relationship with the golden mean.
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).
In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.