Time-dependent pseudofermionic systems and coherent states

Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that various classes of coherent states such as the canonical coherent states, pure squeezed states which are obtained by the sole action of the squeeze operator on the vacuum state, fermionic coherent states can be defined with all the desired properties on a right quaternionic Hilbert space. Furthermore, we shall also demonstrate that squeezed states, which are obtained by the action of the squeeze operator on canonical coherent states, in other words, they are obtained by the action of the squeeze operator followed by the action of the displacement operator on the vacuum state, can be defined on the same Hilbert space, but the noncommutativity of quaternions prevents us in getting the desired results.

We apply the techniques of canonical transforms to equations of the type \[\left[ {A( t )\mathbb{P}^2 + B( t )\{ \mathbb{P}\mathbb{Q} + \mathbb{Q}\mathbb{P} \} + C( t )\mathbb{Q}^2 + D( t )\mathbb{Q} + E( t )\mathbb{P} + F( t )\mathbb{1}} \right]\psi ( q,t ) = - i\partial _t \psi ( q,t ),\]where $\mathbb{Q}$ and $\mathbb{P}$ are the quantum position and momentum operators. The time-dependent parameters of the ${\text{W}} \wedge {\text{SL}}( {{\text{2,}}R} )$ evolution operator are found through linear differential equations. In terms of these we give explicitly the Green’s function, all separating coordinates and similarity solutions of the equation. We analyze the behavior of Gaussian and coherent-state initial conditions in closed form and present a new interpretation of all the Lewis-Riesenfeld constants of motion.

We present a possible construction of coherent states on the unit circle as configuration space. Our approach is based on Borel quantizations on S1 including the Aharonov–Bohm-type quantum description. Coherent states are constructed by Perelomov’s method as group-related coherent states generated by Weyl operators on the quantum phase space . Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction, we use the analogy with our quantization and coherent states over a finite periodic chain where the quantum phase space was . The coherent states constructed in this work are shown to satisfy the resolution of unity. To compare them with canonical coherent states, some of their further properties are also studied demonstrating similarities as well as substantial differences.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

We discuss the coherent state of an attractive superfluid Fermi gas. In order to explain the time evolution of the Fermi atomic gas in the optical lattice, we present mathematical relations for the autocorrelation function and first-order correlation function. The analytical results show a dependence of the oscillation time on the controlling parameters, which may be achieved experimentally.

Grassmann phase space methods for fermions. I. Mode theory

In this paper we investigate the time evolution of a general driven time-dependent oscillator using the evolution operator method developed by Cheng et al. We obtain an exact form of the time evolution operator which, in turn, enables us to find the exact wave functions and coherent states at any time t. Our analyses indicate that the time-dependent coherent state is equivalent to the well-known squeezed state, while the time-dependent number state is equivalent to the displaced and squeezed number state. Besides, we also calculate the time-dependent transition probabilities among the coherent states and number states of a simple harmonic oscillator associated with the initial Hamiltonian H(0).

The relationship between quantum degeneracies and the emergence of classical periodic orbits in the Fock-Darwin (FD) system is systematically explored by using the ladder operators in the Cartesian coordinates. The quantum-classical connection is analytically developed with the time-dependent coherent state that is theoretically verified to correspond to the Gaussian wave packet state with unitary transformation. The time-dependent coherent state is further used to derive the stationary coherent state that can be expressed as a superposition of degenerate eigenstates. More importantly, the stationary coherent state can alternatively be expressed as an integral of the Gaussian wave packet over a periodic orbit. With the integral representation, the quantum vortex structures of the stationary coherent states can be precisely characterized up to extremely high order. The vortex arrays are generally observed when the phase structures are near the crossing points and cusps of the classical trajectories. Finally, the quantum Fourier transform is employed to verify that the circularly symmetric eigenstate physically corresponds to an ensemble of classical hypotrochoid orbits. This work can provide a pedagogical insight into the formation of Laguerre-Gaussian eigenmodes and hypotrochoid geometric modes in degenerate spherical cavities.

The purpose of this paper is to generalize fermionic coherent states for two-level systems described by pseudo-Hermitian Hamiltonian (Cherbal et al 2007 J. Phys. A: Math. Theor. 40 1835) to n-level systems. Central to this task is the expression of the coherent states in terms of generalized Grassmann variables. These kinds of Grassmann coherent states satisfy the bi-overcompleteness condition instead of the overcompleteness one, as it is reasonably expected because of the biorthonormality of the system. Choosing an appropriate Grassmann weight function, the resolution of identity is examined. Moreover Grassmannian coherent and squeezed states of the deformed group SUq(2) for the three-level pseudo-Hermitian system are presented.

Streamwise rolls and accompanying streamwise streaks are ubiquitous in wall-bounded shear flows, both in natural settings, such as the atmospheric boundary layer, as well as in controlled settings, such as laboratory experiments and numerical simulations. The streamwise roll and streak structure has been associated with both transition from the laminar to the turbulent state and with maintenance of the turbulent state. This close association of the streamwise roll and streak structure with the transition to and maintenance of turbulence in wall-bounded shear flow has engendered intense theoretical interest in the dynamics of this structure. In this work, stochastic structural stability theory (SSST) is applied to the problem of understanding the dynamics of the streamwise roll and streak structure. The method of analysis used in SSST comprises a stochastic turbulence model (STM) for the dynamics of perturbations from the streamwise-averaged flow coupled to the associated streamwise-averaged flow dynamics. The result is an autonomous, deterministic, nonlinear dynamical system for evolving a second-order statistical mean approximation of the turbulent state. SSST analysis reveals a robust interaction between streamwise roll and streak structures and turbulent perturbations in which the perturbations are systematically organized through their interaction with the streak to produce Reynolds stresses that coherently force the associated streamwise roll structure. If a critical value of perturbation turbulence intensity is exceeded, this feedback results in modal instability of the combined streamwise roll/streak and associated turbulence complex in the SSST system. In this instability, the perturbations producing the destabilizing Reynolds stresses are predicted by the STM to take the form of oblique structures, which is consistent with observations. In the SSST system this instability exists together with the transient growth process. These processes cooperate in determining the structure of growing streamwise roll and streak. For this reason, comparison of SSST predictions with experiments requires accounting for both the amplitude and structure of initial perturbations as well as the influence of the SSST instability. Over a range of supercritical turbulence intensities in Couette flow, this instability equilibrates to form finite amplitude time-independent streamwise roll and streak structures. At sufficiently high levels of forcing of the perturbation field, equilibration of the streamwise roll and streak structure does not occur and the flow transitions to a time-dependent state. This time-dependent state is self-sustaining in the sense that it persists when the forcing is removed. Moreover, this self-sustaining state rapidly evolves toward a minimal representation of wall-bounded shear flow turbulence in which the dynamics is limited to interaction of the streamwise-averaged flow with a perturbation structure at one streamwise wavenumber. In this minimal realization of the self-sustaining process, the time-dependent streamwise roll and streak structure is maintained by perturbation Reynolds stresses, just as is the case of the time-independent streamwise roll and streak equilibria. However, the perturbation field is maintained not by exogenously forced turbulence, but rather by an endogenous and essentially non-modal parametric growth process that is inherent to time-dependent dynamical systems.

The Wigner quasi-probability distribution function is an important tool for estimating the quantumness of quantum states. The Wigner distribution function (WDF) is related monotonically to the coherent field state density operator and their probability distributions. In this paper, we analyze the time-dependent resonator coherent field states produced by a flux qubit’s interaction with a resonator-field through a two-photon coupling. The WDF dynamics of the coherent field states is investigated under the effects of the qubit-resonator detuning and the intrinsic decoherence. The WDF non-classicality associated with the mixedness resonator entropy dynamics can be controlled by varying the qubit-resonator interaction, detuning as well as the decoherence. The intrinsic decoherence clearly affects the WDF non-classicality, identified by the negative values of the WDF. Finally, by fixing one or both of the complex phase space components, we are able to explore the dynamics of the Wigner-distribution functions.

Comment on “Delayed luminescence of biological systems in terms of coherent states” [Phys. Lett. A 293 (2002) 93

Double-Fano resonance in a two-level quantum system coupled to zigzag Phosphorene nanoribbon

In this paper, we propose a new theoretical scheme for generating a macroscopic Schrödinger cat state of a mechanical oscillator in a hybrid optomechanical system where a beam of two-level atoms passes through the cavity. In the model under consideration, the cavity field couples to the macroscopic mirror through the optomechanical interaction while it couples to the atom through a generalized Jaynes–Cummings interaction that involves the cavity-mode structure. The motion of the mirror modifies the cavity-mode function and therefore modulates the atom-field interaction, leading to the three-mode atom-field-mirror coupling or, equivalently, polariton-mirror coupling in a dressed picture. This interaction induces a controllable anharmonicity in the energy spectrum of the mechanical oscillator, which provides the possibility of generating a superposition of two time-dependent coherent states of the mechanical oscillator just by performing a conditional measurement on the internal states of the atoms exiting the optomechanical cavity. We also investigate the tripartite atom-field-mirror entanglement, which is controllable by adjusting the parameters of the system. In addition, we explore the effects of the mechanical dissipation and thermal noise on the tripartite quantum correlation in the system as well as the generated mechanical superposition state.

When an electromagnetic field in a coherent or quasiclassical (e.g., squeezed) state is used to simultaneously drive an ensemble of two-level atoms, the quantum nature of the field will, in general, cause the final state of the atoms to differ from the one predicted for a totally classical field. This is a potential source of error in quantum logic gates in which the gate is the rotation of the atoms by a laser. In this paper, we use second-order perturbation theory to find how this error scales with the number of atoms, $N$, being driven simultaneously, for an arbitrary rotation angle. The result depends on the initial atomic state: for some highly entangled states, and a field in a coherent state, the error may scale as ${N}^{2}$, yet we find that the average over a random distribution of initial states only scales as $N$. We discuss possible ways to mitigate the error, including the use of squeezed states, as well as adjusting the interaction time between the field and atoms to be different from what would be expected from the classical-field treatment.

Coherent states for isospectral Hamiltonians