Coherent State Representation Research Articles

Gaussian representation of extended quantum states

Excited energy eigenstates and their superpositions typically exhibit a fine oscillatory structure near caustics. Semiclassical theory accesses these, but depends on detailed geometrical knowledge of the caustics. Here we show that a finite placement of coherent states on the classical region efficiently fits such extended states, reproducing structures that are much finer than the Gaussian width of the basis states. An extended state, evolved such that it becomes fully distinguishable from the original state, can also be faithfully reproduced by this finite basis. The ideal fitting follows from the projection of the extended state on the finite Hilbert space spanned by the Gaussians, rather than any discretization of the continuous (overcomplete) coherent state representation.

Effective action for strongly correlated electron systems

The su ( 2 | 1 ) coherent-state path-integral representation of the partition function of the t – J model of strongly correlated electrons is derived at finite doping. The emergent effective action is compared to the one proposed earlier on phenomenological grounds by Shankar to describe holes in an antiferromagnet [R. Shankar, Nucl. Phys. B 330 (1990) 433]. The t – J model effective action is found to have an important “extra” factor with no analogue in Shankarʼs action. It represents the local constraint of no double electron occupancy and reflects the rearrangement of the underlying phase-space manifold due to the presence of strong electron correlation. This important ingredient is shown to be essential to describe the physics of strongly correlated electron systems.

Wigner Function of a Special Type of Squeezed Coherent State

By using the technique of integration within an ordered product of operators and the coherent state representation of the Wigner operator, the Wigner function of a special type of squeezed coherent state [Phys. Lett. A 220, 81 (1996)] is presented. The properties of the Wigner function are discussed based on the numerical calculation.

Semiclassical propagator for SU(n) coherent states

We present a detailed derivation of the semiclassical propagator in the SU(n) coherent state representation. In order to provide support for immediate physical applications, we restrict this work to the fully symmetric irreducible representations, which are suitable for the treatment of bosonic dynamics in n modes, considering systems with conservation of total particle number. The derivation described here can be easily extended to other classes of coherent states, thus representing an alternative approach to previously published methods.

Spin Coherent State Representation of the Crow-Kimura and Eigen Models of Quasispecies Theory

We present a spin coherent state representation of the Crow-Kimura and Eigen models of biological evolution. We deal with quasispecies models where the fitness is a function of Hamming distances from one or more reference sequences. In the limit of large sequence length N, we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes. The results are obtained by constructing a path integral for the propagator on the coset SU(2)/U(1) and taking the classical limit. The classical limit gives a Hamiltonian function on a circle for one reference sequence, and on the product of 2m−1 circles for m reference sequences. We apply our representation to study the Schuster-Swetina phenomena, where a wide lower peak is selected over a narrow higher peak. The quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation-fitness parameter phase space. Furthermore, we use our method to investigate more biologically relevant system, a model of escape from adaptive conflict through gene duplication, and find three different phases for the asymptotic population distribution.

Gaussian channel in a quantum information network

Starting from the quantum Fokker–Planck equation, a viable quantum Gaussian channel is constructed in a quantum information network. It is shown that the solution of this equation satisfies a Gaussian distribution channel. Transforming the solution of the Fokker–Planck equation and substituting it into the information formula, a quantum dynamical mutual information equation is obtained in the coherent state representation. On the basis of this equation, a new scheme is proposed to implement parallel quantum information processing in the quantum Gaussian channel. The proposed scheme takes the coefficients related to the encryption states of photons as the signal, and encodes the information that will be passed in the quantum Gaussian channel from the input terminal. Information is then accessed by extracting and decoding the coefficients from the output terminal. Compared with a classical Gaussian channel, this approach offers the advantage of quantum parallelism.

Stabilizing simulations of complex stochastic representations for quantum dynamical systems

Path integral representations of quantum dynamics can often be formulated as stochastic differential equations (SDEs). In a series of papers, Corney and Drummond (2004 Phys. Rev. Lett. 93 260401), Deuar and Drummond (2001 Comput. Phys. Commun. 142 442–5), Drummond and Gardnier (1980 J. Phys. A: Math. Gen. 13 2353–68), Gardiner and Zoller (2004 Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer Series in Synergetics) 3rd edn (Berlin: Springer)) and Gilchrist et al (1997 Phys. Rev. A 55 3014–32) and their collaborators have derived SDEs from coherent states representations for density matrices. Computationally, these SDEs are attractive because they seem simple to simulate. They can be quite unstable, however. In this paper, we consider some of the instabilities and propose a few remedies. Particularly, because the variances of the simulated paths typically grow exponentially, the processes become de-localized in relatively short times. Hence, the issues of boundary conditions and stable integration methods become important. We use the Bose–Einstein Hamiltonian as an example. Our results reveal that it is possible to significantly extend integration times and show the periodic structure of certain functionals.

New Formulation of Weyl Quantization Scheme in Coherent State Representation

By virtue of the technique of integration within an ordered product of operators we present a new formulation of the Weyl quantization scheme in the coherent state representation, which not only brings convenience for calculating the Weyl correspondence of normally ordered operators, but also directly leads us to find both the coherent state representation and the Weyl ordering representation of the Wigner operator.

Entanglement dynamics via coherent-state propagators

The dynamical generation of entanglement in closed bipartite systems is investigated in the semiclassical regime. We consider a model of two particles, initially prepared in a product of coherent states, evolving in time according to a generic Hamiltonian, and derive a formula for the linear entropy of the reduced density matrix using the semiclassical propagator in the coherent-state representation. The formula is explicitly written in terms of quantities that define the stability of classical trajectories of the underlying classical system. The formalism is then applied to the problem of two nonlinearly coupled harmonic oscillators and the result is shown to be in remarkable agreement with the exact quantum measure of entanglement in the short-time regime. An important byproduct of our approach is a unified semiclassical formula which contemplates both the coherent-state propagator and its complex conjugate.

Quantum diffusion in a fermionic bath

We propose a scheme for quantum brownian motion of a particle in a fermionic bath. Based on the spin coherent-state representation of the noise operators and a canonical thermal distribution of the associated c numbers, we derive a quantum analog of generalized Langevin equation for quantum-mechanical mean position of the particle subjected to an external force field. The approach allows us to map the quantum problem on a classical setting. The quantum dispersion around the mean can be estimated order by order by a set of quantum correction equations up to a desired degree of accuracy for a given nonlinear potential. We derive a quantum diffusion equation for free particle and show that quantization, in general, enhances the mean-square displacement. Increase in temperature leads to suppression of mean-square displacement. The method is based on canonical quantization procedure and may be used for understanding diffusive transport and thermally activated processes in a fermionic bath.

Single-qubit lasing in the strong-coupling regime

Motivated by recent ``circuit QED'' experiments we study the lasing transition and spectral properties of single-qubit lasers. In the strong coupling, low-temperature regime quantum fluctuations dominate over thermal noise and strongly influence the linewidth of the laser. When the qubit and the resonator are detuned, amplitude and phase fluctuations of the radiation field are coupled, and the phase diffusion model, commonly used to describe conventional lasers, fails. We predict pronounced effects near the lasing transition, with an enhanced linewidth and non-exponential decay of the correlation functions. We cover a wide range of parameters by using two complementary approaches, one based on the Liouville equation in a Fock state basis, covering arbitrarily strong coupling but limited to low photon numbers, the other based on the coherent-state representation, covering large photon numbers but restricted to weak or intermediate coupling.

Application of the two-mode squeezed coherent state representation in deriving generalized optical Collins formula

We propose a new two-mode squeezed coherent state representation | z 1, z 2〉 g which is characteristic of the correlation between the squeezing and the displacement. Based on it and using the technique of integration within an ordered product of operators we obtain a generalized two-mode Fresnel operator (GTFO), which is an image of the mapping from ( z 1, z 2) to ( s z 1 + r z 2 * , r z 1 * + s z 2 ) in | z 1, z 2〉 g representation. The matrix element of GTFO in the coordinate representation leads to a generalized two-dimensional Collins formula (Huygens–Fresnel integration transformation describing optical diffraction) in entangled form.

Spin Path Integrals, Berry Phase, and the Quantum Phase Transition in the Sub-Ohmic Spin-Boson Model

The quantum critical properties of the sub-Ohmic spin-1/2 spin-boson model and of the Bose-Fermi Kondo model have recently been discussed controversially. The role of the Berry phase in the breakdown of the quantum-to-classical mapping of quantum criticality in the spin-isotropic Bose-Fermi Kondo model has been discussed previously. In the present article, some of the subtleties underlying the functional integral representation of the spin-boson and related models with spin anisotropy are discussed. To this end, an introduction to spin coherent states and spin path integrals is presented with a focus on the spin-boson model. It is shown that, even for the Ising-anisotropic case as in the spin-boson model, the path integral in the continuum limit in the coherent state representation involves a Berry phase term. As a result, the effective action for the spin degrees of freedom does not assume the form of a Ginzburg-Landau-Wilson functional. The implications of the Berry-phase term for the quantum-critical behavior of the spin-boson model are discussed. The case of arbitrary spin is also considered.

High magnetic field theory for the local density of states in graphene with smooth arbitrary potential landscapes

We study theoretically the energy and spatially resolved local density of states (LDoS) in graphene at high perpendicular magnetic field. For this purpose, we extend from the Schr\"odinger to the Dirac case a semicoherent-state Green's-function formalism, devised to obtain in a quantitative way the lifting of the Landau-level degeneracy in the presence of smooth confinement and smooth disordered potentials. Our general technique, which rigorously describes quantum-mechanical motion in a magnetic field beyond the semiclassical guiding center picture of vanishing magnetic length (both for the ordinary two-dimensional electron gas and graphene), is connected to the deformation (Weyl) quantization theory in phase space developed in mathematical physics. For generic quadratic potentials of either scalar (i.e., electrostatic) or mass (i.e., associated with coupling to the substrate) types, we exactly solve the regime of large magnetic field (yet at finite magnetic length, formally, this amounts to considering an infinite Fermi velocity) where Landau-level mixing becomes negligible. Hence, we obtain a closed-form expression for the graphene Green's function in this regime, providing analytically the discrete energy spectra for both cases of scalar and mass parabolic confinement. Furthermore, the coherent-state representation is shown to display a hierarchy of local energy scales ordered by powers of the magnetic length and successive spatial derivatives of the local potential, which allows one to devise controlled approximation schemes at finite temperature for arbitrary and possibly disordered potential landscapes. As an application, we derive general analytical nonperturbative expressions for the LDoS, which may serve as a good starting point for interpreting experimental studies. For instance, we are able to account for many puzzling features of the LDoS recently observed by high magnetic field scanning tunneling spectroscopy experiments on graphene, such as a roughly $\sqrt{m}$ increase in the $m\text{th}$ Landau-level linewidth in the LDoS peaks at low temperatures, together with a flattening of the spatial variations in the Landau-level effective energies at increasing $m$.

Normally-Ordered Time Evolution Operator for Mass-Varying Harmonic Oscillator and Wigner Function of Squeezed Number State

For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique of integration within an ordered product of operators. The normally ordered time evolution operator is thus obtained. We then derive the Wigner function of u(t)|n⟩, where |n⟩ is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.

Pseudo-Classical Partition Function of Spin One-Half Derived by Coherent State Method

Using the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum we find the formula for the quantum Hamiltonian H = ia†iU̇ijU†jlal for SU(2) rotation U, in this way we further specify the angular velocity , where is the Pauli matrix. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation |z⟩g and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens–Fresnel integration transformation describing optical diffraction). The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz - rz* in the |z⟩g representation, while |z⟩g in phase space is graphically denoted by an ellipse.

Off-Center Coherent-State Representation and an Application to Semiclassics

By using the overcompleteness of coherent states we find an alternative form of the unit operator for which the ket and the bra appearing under the integration sign do not refer to the same phase-space point. This defines a new quantum representation in terms of Bargmann functions, whose basic features are presented. A continuous family of secondary reproducing kernels for the Bargmann functions is obtained, showing that this quantity is not necessarily unique for representations based on overcomplete sets. We illustrate the applicability of the presented results by deriving a semiclassical expression for the Feynman propagator that generalizes the well-known van Vleck formula and seems to point a way to cope with long-standing problems in semiclassical propagation of localized states.

New representation of the multimode phase shifting operator and its application

Based on the rotation transformation in phase space and the technique of integration within an ordered product of operators, the coherent state representation of the multimode phase shifting operator and one of its new applications in quantum mechanics are given. It is proved that the coherent state is a natural language for describing the phase shifting operator or multimode phase shifting operator. The multimode phase shifting operator is also a useful tool to solve the dynamic problems of the multimode coordinate–momentum coupled harmonic oscillators. The exact energy spectra and eigenstates of such multimode coupled harmonic oscillators can be easily obtained by using the multimode phase shifting operator.

Entanglement Involved in Pair Coherent State Studied via Wigner Function Formalism

We calculate Wigner function, tomogram of the pair coherent state by using its Schmidt decomposition in the coherent state representation. It turns out that the Wigner function can be seen as the quantum entanglement (QE) between two two-variable Hermite polynomials (TVHP) and the tomogram is further simplified as QE of two single-variable Hermite polynomials. The Husimi function of pair coherent state is also calculated.