Semiclassical Coherent State Research Articles

A simulator of optical coherent-state evolution in quantum key distribution systems

Quantum key distribution (QKD) is believed to represent a viable solution to achieve theoretically unconditionally secure key generation. However, the available optical systems for experimental QKD, based on photon transmission, are flawed by non-idealities that ultimately limit the achievable performance. Classical simulation of the optical hardware employed in these systems may take on a determining role in engineering future QKD networks. In this article, attempts for developing a QKD simulator based on low-computational-cost models of the employed hardware are presented. In particular, the simulation infrastructure targets polarization-based QKD setups with faint laser sources, whose behaviour can be described by semiclassical coherent states and Mean Photon Number (MPN) per beam. The effects of passive optical components on the photonic qubit evolution are described by Jones matrices, whose coefficients, for some commercial devices, are stored in an ad-hoc library. Realistic eavesdropping attacks and non-idealities, such as optical losses, fibre attenuation, polarization misalignment and limited efficiency of single-photon detectors, are also taken into account. The infrastructure allows the user to describe the desired QKD configuration and it provides in output the MPN at the receiver and two fiducial performance parameters: Quantum Bit Error Rate (QBER) and secure key rate. The comparison of the simulation results with experimental data in the state-of-the-art literature highlights that this work is a step forward towards the definition of compact models for the hardware-dependent simulation of quantum-assisted communication networks.

Performance of coherent‐state quantum target detection in the context of asymmetric hypothesis testing

Due to the difficulties of implementing joint measurements, quantum illumination schemes that are based on signal-idler entanglement are difficult to implement in practice. For this reason, one may consider quantum-inspired designs of quantum lidar/radar where the input sources are semi-classical (coherent states) while retaining the quantum aspects of the detection. The performance of these designs could be studied in the context of asymmetric hypothesis testing by resorting to the quantum Stein’s lemma. However, here the authors discuss that, for typical finite-size regimes, the second- and third-order expansions associated with this approach are not sufficient to prove quantum advantage.

Spinning-off stringy electro-magnetic memories

We extend and generalise the string corrections to the EM memory to the Type I superstring including spin effects. Very much as in the simpler bosonic string context, the relevant corrections are non-perturbative in α′, slowly decaying (as 1/R) at large distances and modulated in retarded time u=t−R. For spin N states in the first Regge trajectory they entail a sequence of N derivatives wrt u on the ‘parent’ N=0 amplitude. We also briefly discuss how to include loop effects, that broaden and shift the string resonances, and how to modify our analysis for macroscopic semi-classical quasi-BPS coherent states, whose collisions may lead to detectable string memory signals in viable Type I models.

Lack of Null-Controllability for the Fractional Heat Equation and Related Equations

We consider the equation $(\partial_t + \rho(\sqrt{-\Delta}))f(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb R$ or $\mathbb T$. We prove it is not null-controllable if $\rho$ is analytic on a conic neighborhood of $\mathbb R_+$ and $\rho(\xi) = o(|\xi|)$. The proof relies essentially on geometric optics, i.e.\ estimates for the evolution of semiclassical coherent states. The method also applies to other equations. The most interesting example might be the Kolmogorov-type equation $(\partial_t -\partial_v^2 + v^2\partial_x)f(t,x,v) = \mathbf 1_\omega u(t,x,v)$ for $(x,v)\in \Omega_x\times \Omega_v$ with $\Omega_x = \mathbb R$ or $\mathbb T$ and $\Omega_v = \mathbb R$ or $(-1,1)$. We prove it is not null-controllable in any time if $\omega$ is a vertical band $\omega_x\times \Omega_v$. The idea is to remark that, for some families of solutions, the Kolmogorov equation behaves like the rotated fractional heat equation $(\partial_t + \sqrt i(-\Delta)^{1/4})g(t,x) = \mathbf 1_\omega u(t,x)$, $x\in \mathbb T$.

Multi-level description of the vibronic dynamics of open quantum systems.

A new approximate coherent state path integral approach, which enables accurate and efficient dynamical treatment of model Hamiltonians that incorporate excited electronic states of multiple chromophores that are coupled to discrete high frequency harmonic vibrational modes, is presented. The approach is based on the mapping Hamiltonian formalism for the electronic states together with semiclassical coherent state expressions for the forward and backward propagators describing the quantum bath modes. The density matrix dynamics is propagated in the full coherent state basis for the electronic mapping and discrete vibrational mode oscillators using ensembles of weighted trajectories. An effective scheme for projecting the ensemble onto selected vibronic basis states is presented enabling the evolution of the reduced system density matrix to be monitored as well as exploring the importance of selected vibronic relaxation pathways in the multichromophore system dynamics. The approach is demonstrated for simple model Hamiltonians, and we show how this coherent state density matrix propagation approach for high frequency discrete harmonic vibrational modes can be combined with partial linearized density matrix propagation to treat an additional continuum bath of low frequency environmental modes that could, in principle, include anharmonicity.

The semiclassical coherent state propagator in the Weyl representation

It is shown that the semiclassical coherent state propagator takes its simplest form when the quantum mechanical Hamiltonian is replaced by its Weyl symbol in defining the classical action, in that there is then no need for a Solari-Kochetov correction. It is also shown that such a correction exists if a symbol other than the Weyl symbol is chosen and that its form is different depending on the symbol chosen. The various forms of the propagator based on different symbols are shown to be equivalent provided the correspondingly correct Solari-Kochetov correction is included. All these results are shown for both particle and spin coherent state propagators. The global anomaly in the fluctuation determinant is further elucidated by a study of the connection between the discrete fluctuation determinant and the discrete Jacobi equation.

Corrections to the continuous time semiclassical coherent state path integral

By returning to the underlying discrete time formalism, we relate spurious results in coherent state path integral calculations to the high frequency structure of their propagators. We show how to modify the standard expressions for thermodynamic quantities to yield correct results. These expressions are relevant to a broad range of physical problems, from the thermodynamics of Bose lattice gases to the dynamics of spin systems.

Coherent States of Systems with Quadratic Hamiltonians

Different families of generalized coherent states (CS) for one-dimensional systems with general time-dependent quadratic Hamiltonian are constructed. In principle, all known CS of systems with quadratic Hamiltonian are members of these families. Some of the constructed generalized CS are close enough to the well-known due to Schrödinger and Glauber CS of a harmonic oscillator; we call them simply CS. However, even among these CS, there exist different families of complete sets of CS. These families differ by values of standard deviations at the initial time instant. According to the values of these initial standard deviations, one can identify some of the families with semiclassical CS. We discuss properties of the constructed CS, in particular, completeness relations, minimization of uncertainty relations and so on. As a unknown application of the general construction, we consider different CS of an oscillator with a time dependent frequency.

Communication: Overcoming the root search problem in complex quantum trajectory calculations

Three new developments are presented regarding the semiclassical coherent state propagator. First, we present a conceptually different derivation of Huber and Heller's method for identifying complex root trajectories and their equations of motion [D. Huber and E. J. Heller, J. Chem. Phys. 87, 5302 (1987)]. Our method proceeds directly from the time-dependent Schrödinger equation and therefore allows various generalizations of the formalism. Second, we obtain an analytic expression for the semiclassical coherent state propagator. We show that the prefactor can be expressed in a form that requires solving significantly fewer equations of motion than in alternative expressions. Third, the semiclassical coherent state propagator is used to formulate a final value representation of the time-dependent wavefunction that avoids the root search, eliminates problems with caustics and automatically includes interference. We present numerical results for the 1D Morse oscillator showing that the method may become an attractive alternative to existing semiclassical approaches.

Deformations of polyhedra and polygons by the unitary group

We introduce the set of framed (convex) polyhedra with N faces as the symplectic quotient \documentclass[12pt]{minimal}\begin{document}${\mathbb {C}}^{2N}//\mathrm{SU}(2)$\end{document}C2N// SU (2). A framed polyhedron is then parametrized by N spinors living in \documentclass[12pt]{minimal}\begin{document}${\mathbb {C}}^{2}$\end{document}C2 satisfying suitable closure constraints and defines a usual convex polyhedron plus extra U(1) phases attached to each face. We show that there is a natural action of the unitary group U(N) on this phase space, which changes the shape of faces and allows to map any (framed) polyhedron onto any other with the same total (boundary) area. This identifies the space of framed polyhedra to the Grassmannian space U(N)/ (SU(2)×U(N−2)). We show how to write averages of geometrical observables (polynomials in the faces' area and the angles between them) over the ensemble of polyhedra (distributed uniformly with respect to the Haar measure on U(N)) as polynomial integrals over the unitary group and we provide a few methods to compute these integrals systematically. We also use the Itzykson-Zuber formula from matrix models as the generating function for these averages and correlations. In the quantum case, a canonical quantization of the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners (or, in other words, SU(2)-invariant states in tensor products of irreducible representations). The total boundary area as well as the individual face areas are quantized as half-integers (spins), and the Hilbert spaces for fixed total area form irreducible representations of U(N). We define semi-classical coherent intertwiner states peaked on classical framed polyhedra and transforming consistently under U(N) transformations. And we show how the U(N) character formula for unitary transformations is to be considered as an extension of the Itzykson-Zuber to the quantum level and generates the traces of all polynomial observables over the Hilbert space of intertwiners. We finally apply the same formalism to two dimensions and show that classical (convex) polygons can be described in a similar fashion trading the unitary group for the orthogonal group. We conclude with a discussion of the possible (deformation) dynamics that one can define on the space of polygons or polyhedra. This work is a priori useful in the context of discrete geometry but it should hopefully also be relevant to (loop) quantum gravity in 2+1 and 3+1 dimensions when the quantum geometry is defined in terms of gluing of (quantized) polygons and polyhedra.

Nonlinear dynamics of semiclassical coherent states in periodic potentials

We consider nonlinear Schrödinger equations with either local or nonlocal nonlinearities. In addition, we include periodic potentials as used, for example, in matter wave experiments in optical lattices. By considering the corresponding semiclassical scaling regime, we construct asymptotic solutions, which are concentrated both in space and in frequency around the effective semiclassical phase-space flow induced by Bloch’s spectral problem. The dynamics of these generalized coherent states is governed by a nonlinear Schrödinger model with effective mass. In the case of nonlocal nonlinearities, we establish a novel averaging-type result in the critical case.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Complex invariants in two-dimension for coupled oscillator systems

In order to extract some insight into the features of a dynamical system, we present here the possibility of its complex dynamical invariant. There are many systems which admit complex invariants. To achieve this we use Lie algebraic method to study two dimensional complex systems (coupled oscillator system) on the extended complex phase plane characterized by \(x=x_{1}+ip_{3}, y=x_{2} + ip_{4} , p_x=p_{1} + ix_{3}, p_y=p_{2}+ix_{4}\). Such invariants play an important role in the analysis of complex trajectories with regard to the calculation of semi-classical coherent state propagator in the path integral method.

An initial value representation for the coherent state propagator with complex trajectories

We present an initial value representation for the semiclassical coherent state propagator based on complex trajectories. We map the complex phase space into a real one with twice as many dimensions and introduce initial valued trajectories in this double phase space. We use a procedure to eliminate non-contributing trajectories that allows for the automation of the entire calculation, rendering it simple. The resulting semiclassical formulas do not show divergences due to caustics and provide accurate results in short computational times.

Evaluation of the semiclassical coherent state propagator in the presence of phase space caustics

A uniform approximation for the coherent state propagator, valid in the vicinity of phase space caustics, was recently obtained using the Maslov method combined with a dual representation for coherent states. In this paper we review the derivation of this formula and apply it to two model systems: the one-dimensional quartic oscillator and a two-dimensional chaotic system.

Controlling phase space caustics in the semiclassical coherent state propagator

The semiclassical formula for the quantum propagator in the coherent state representation 〈 z ″ | e - i H ˆ T / ℏ | z ′ 〉 is not free from the problem of caustics. These are singular points along the complex classical trajectories specified by z′, z″ and T where the usual quadratic approximation fails, leading to divergences in the semiclassical formula. In this paper, we derive third order approximations for this propagator that remain finite in the vicinity of caustics. We use Maslov’s method and the dual representation proposed in Phys. Rev. Lett. 95, 050405 (2005) to derive uniform, regular and transitional semiclassical approximations for coherent state propagator in systems with two degrees of freedom.

Quantum mechanics on a circle: Husimi phase-space distributions and semiclassical coherent state propagators

We discuss some basic tools for an analysis of one-dimensional quantum systems defined on a cyclic coordinate space. The basic features of the generalized coherent states, the complexifier coherent states, are reviewed. These states are then used to define the corresponding (quasi)densities in phase space. The properties of these generalized Husimi distributions are discussed, in particular their zeros. Furthermore, the use of the complexifier coherent states for a semiclassical analysis is demonstrated by deriving a semiclassical coherent state propagator in phase space.

Semiclassical states in homogeneous loop quantum cosmology

Semiclassical states in homogeneous loop quantum cosmology (LQC) are constructed in two different ways. In the first approach, we firstly construct an exponentiated annihilation operator. Then a kind of semiclassical (coherent) state is obtained by solving the eigenequation of that operator. Moreover, we use these coherent states to analyse the semiclassical limit of the quantum dynamics. It turns out that the Hamiltonian constraint operator employed currently in homogeneous LQC has a correct classical limit with respect to the coherent states. In the second approach, the other kind of semiclassical state is derived from the mathematical construction of coherent states for compact Lie groups due to Hall.

The semiclassical coherent state propagator for systems with spin

We derive the semiclassical limit of the coherent state propagator for systems with two degrees of freedom of which one degree of freedom is canonical and the other a spin. Systems in this category include those involving spin-orbit interactions and the Jaynes-Cummings model in which a single electromagnetic mode interacts with many independent two-level atoms. We construct a path integral representation for the propagator of such systems and derive its semiclassical limit. As special cases we consider separable systems, the limit of very large spins and the case of spin 1/2.

SL(2) spin chain and spinning strings on [formula omitted

We derive the coherent state representation of the integrable spin chain Hamiltonian with symmetry group SL ( 2 , R ) . By passing to the continuum limit, we find a spin chain sigma model describing a string moving on the hyperboloid SL ( 2 , R ) / U ( 1 ) . The same sigma model is found by considering strings rotating with large angular momentum in AdS 5 × S 5 . The spinning strings are identified with semiclassical coherent states built out of SL ( 2 , R ) spin chain states.

Semiclassical representations of electronic structure and dynamics.

We use a new formulation of the semiclassical coherent state propagator to derive and evaluate several different approximate representations of electron dynamics. For each representation we examine: (1) its ability to treat quantum effects and electron correlation, (2) its expected scaling with system size, and (3) the types of systems for which it can be used. We also apply two of the methods to a pair of model problems, namely the minimal basis electron dynamics in H2 and the magnetization dynamics in a cluster model of the Kagome lattice, in order to verify the feasibility of these approaches for realistic systems. Based on all these criteria, we find that the representation that takes the electron spins as the classical variables is particularly promising for the quantitative and qualitative description of large systems.