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Quantum Matter

Coherent States Formalism Applied to the Quantum Well Model

Dušan Popov, Vjekoslav D. Sajfert, Jovan P. Šetrajčić, Nicolina Pop

Published Feb 15, 2014• 3 citations
Publisher

Abstract

The different nanostructures (quantum dots, wires and wells) have specific properties which differ from those of the corresponding bulk structures. This fact leads to many applications in several branches of physics. In the paper we examine the behavior of a quantum system (a free particle and a quantum free particle gas) trapped inside the infinite quantum well, by using the coherent states formalism. Our approach is based on the fact that the dynamical group associated with the infinite quantum well is SU(1, 1), with the Bargmann index k = 1/2. These coherent states always have the sub-Poissonian behavior. There is a similarity between the energy main quantum number n and the absolute value of the complex variable z which labels the coherent states. This allows us to make a useful approximation in order to calculate the partition function and some thermodynamic characteristics of a canonical non interacting particle gas embedded in the infinite square quantum well.

DOI
10.1166/qm.2014.1137

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The different nanostructures (quantum dots, wires and wells) have specific properties which differ from those of the corresponding bulk structures. This fact leads to many applications in several branches of physics. In the paper we examine the behavior of a quantum system (a free particle and a quantum free particle gas) trapped inside the infinite quantum well, by using the coherent states formalism. Our approach is based on the fact that the dynamical group associated with the infinite quantum well is SU(1, 1), with the Bargmann index k = 1/2. These coherent states always have the sub-Poissonian behavior. There is a similarity between the energy main quantum number n and the absolute value of the complex variable z which labels the coherent states. This allows us to make a useful approximation in order to calculate the partition function and some thermodynamic characteristics of a canonical non interacting particle gas embedded in the infinite square quantum well.
Since, unlike “traditional” physics, computational physics uses the discretization formalism, in this paper we have focused our attention on the discrete approach in order to solve the Schrödinger as well as the Bloch equations for a free particle and the “quantum gas” of free particles in an infinite quantum well with the finite length, respectively. By applying the so called continuous quantum mechanics limit we recover the corresponding results in continuous-variable quantum mechanics. For the infinite quantum well model we formulated the associated coherent states, which lead to the expression for the coherent qubits, in counterpart to the thermal qubits, obtained as a result of using the thermofield dynamics model. Thus, a “bridge” is created between the infinite quantum well model, coherent states and thermofield dynamics, which ultimately converge towards Quantum Computing.
We develop generalized coherent states based on the Gazeau–Klauder formalism for a particle with position-dependent mass trapped in an infinite square well. We study the quantum statistical properties of these states by means of the Mandel parameter and the second-order correlation function. Our analysis reveals that the constructed coherent states exhibit sub-Poissonian statistics. Moreover, theoretical investigations of wave packet revivals and fractional revivals for the pertaining system have been performed by means of the autocorrelation function and temporal evolution of probability density.
The 3D infinite square well: quantum dots, wells, and wires In the preceding discussion of the Schrödinger Equation the particle of interest was assumed to be “moving in the x -direction.” Of course, it is not possible for a particle to be moving in one spatial direction only. If that were true, according to the HUP it could be anywhere in the y - and z -directions and therefore be undetectable with finite volume detectors. Now, we consider the more realistic case of motion in all three spatial directions. For this purpose, we start with the 3D infinite square well. This model provides a useful conceptual framework for understanding a burgeoning “nano-industry,” namely the production and use of “quantum dots.” The 3D infinite square well is a “rigid rectangular solid box.” The potential energy of a particle trapped inside is given by U(x,y,z) = 0 , if 0 < x < L1, 0 < y < L2, 0 < z < L3, and ∞, otherwise. The particle’s momentum now has three components and its kinetic energy is K = (px2 + py2 + pz2) /2m.
The system under study consists of an interacting ultracold Bose gas confined by a finite n -well potential in one dimension ( n = 2, 3 and 4). By numerically solving the time-dependent Schrödinger equation for the effective Hamiltonian that describes the gas confined in each potential, we determine the mean population of particles in each well as a function of time. From this analysis, we obtain a continuous transition from a coherent state to a self-trapped state as a function of the parameter Λ ≈ Ng , which takes into account the number of particles N and the interaction strength g in each system. Three different behaviours as a function of Λ are found: a coherent oscillation regime, a partial-trapped state and a self-trapped state. The partial trapping regime appears to be a novel phase for finite potentials in one dimension. A systematic quantitative study allows us to conclude that the relaxation time observed in the coherent oscillation regime for a two-well potential scales as N 1/2 . Finally, a comparison with an experimental realization of a Bose–Einstein condensate in a two-well potential (Albiez et al 2005 Phys. Rev. Lett. 95 010402) is presented, exhibiting good agreement.
Abstract Since radiation field states are very important in the interaction with the various atomic structures, in this article we introduce a new type of quantum states, which we call ‘Mth nonlinear coherent states’. We define these states based on acting of the number operator M times n ˆ M on a nonlinear coherent state. By analyzing the photon distribution function and its overlap with the nonlinear coherent states, it becomes clear that these states are very similar to the corresponding nonlinear coherent states. We also investigate the nonclassical features of such states for different values of M, and it is determined that the occurrence of sub-Poissonian photon statistics, normal squeezing, and amplitude squared squeezing are considerable nonclassical properties of the Mth nonlinear coherent states. Also, examining the Husimi function shows a local behavior for the introduced states. Furthermore, by calculating the Wigner function, we find out that in some points of the phase space, the value of this function is negative which could be considered as a nonclassical feature for these states.