Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
Canonical coherent states of a quantum harmonic oscillator have been generalized by requiring the conditions of normalizability, continuity in the label and resolution of the identity operator with a positive weight function. Superpositions of these states are considered in the present scenario as a generalization of the optical Schrödinger cat states. The Fock space is assumed to be canonical or finite-dimensional. The photon number distribution of these generalized Schrödinger cat states departs from the Poisson statistics in various ways for high photon numbers. For small nonvanishing values of the label, the photon number distribution is sub-Poissonian (nonclassical) or super-Poissonian, according to the interference properties. In fact, the sub- or super-Poissonian statistics is determined by the interplay between the relative phase and a critical value of the phase. The photon number distribution is uniquely sub-Poissonian for large values of the label.
A generalization of the canonical coherent states of a quantum harmonic oscillator has been performed by requiring the conditions of normalizability, continuity in the label, and resolution of the identity operator with a positive weight function. Relying on this approach, in the present scenario, coherent states are generalized over the canonical or finite dimensional Fock space of the harmonic oscillator. A class of generalized coherent states is determined such that the corresponding distributions of the number of excitations depart from the Poisson statistics according to combinations of stretched exponential decays, power laws, and logarithmic forms. The analysis of the Mandel parameter shows that the generalized coherent states exhibit (non-classical) sub-Poissonian or super-Poissonian statistics of the number of excitations, based on the realization of determined constraints. Mittag-Leffler and Wright generalized coherent states are analyzed as particular cases.
Quantum superposition is a fundamental principle of quantum mechanics, which provides a crucial basis to observe phenomena beyond the predictions of classical physics. For example, a quantum entangled state can exhibit stronger correlation than classically possible one. In quantum state engineering, many new quantum states can be obtained from the superposition of many known states. In recent decades, the superposition of coherent states (CSs) with the same amplitude but two different phases has been a subject of great interest. This superposition state was often called Schrodinger cat state (here, we also name it 2-headed cat state (2HCS)), which becomes an important tool to study a lot of fundamental issues. Surprisingly, some studies have extended the quantum superposition to involving more than two component coherent states. In order to produce the superposition of three photons, people have considered the superposition of coherent states with three different phases (here, we also name it 3-headed cat state (3HCS)). Furthermore, in microwave cavity quantum electrodynamics of bang-bang quantum Zeno dynamics control, people have proposed the superposition of coherent states with four different phases (here, we also name it 4-headed cat state (4HCS)). In this paper, we make a detailed investigation on the quantum statistical properties of a phase-type 3HCS. These properties include photon number distribution, average photon number, sub-Poissionian distribution, squeezing effect, and Wigner function, etc. We derive their analytical expressions and make numerical simulations for these properties. The results are compared with the counterparts of the CS, the 2HCS and the 4HCS. The conclusions are obtained as follows. 1) The CS, the 2HCS, the 3HCS and the 4HCS have k, 2k, 3k and 4k photon number components, respectively (k is an integer); 2) small difference in average photon number among these quantum states in small-amplitude range can be observed, while their average photon numbers become almost equal in large-amplitude range; 3) the CS exhibits Poisson distribution, and the 2HCS, the 3HCS and the 4HCS exhibit super-Poisson distributions in most amplitude ranges, however, sub-Poisson distribution can be seen for the 3HCS and the 4HCS in some specific amplitude ranges; 4) except for the 2HCS that may have the squeezing property, no squeezing properties can be found in the CS, the 3HCS and the 4HCS; 5) negative values can exist in the Wigner functions for the 2HCS, the 3HCS and the 4HCS, while it is not found in the CS. Similar to the 2HCS and 4HCS, the Wigner function of the 3HCS has negative component, which implies its nonclassicality. Different from the 2HCS, the 3HCS exhibits sub-Poisson photon number distribution in a certain amplitude range, it is weaker than that of the 4HCS. At the same time, no squeezing is found in the 3 or 4HCS, which is another difference from the 2HCS.
We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These “k-hypercats” can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, …, k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time.
In the present scenario, coherent states of a quantum harmonic oscillator are generalized with positive Fox H auxiliary functions. The novel generalized coherent states provide canonical coherent states and Mittag-Leffler or Wright generalized coherent states, as particular cases, and resolve the identity operator, over the Fock space, with a weight function that is the product of a Fox H function and a Wright generalized hypergeometric function. The novel generalized coherent states, or the corresponding truncated generalized coherent states, are characterized by anomalous statistics for large values of the number of excitations: the corresponding decay laws exhibit, for determined values of the involved parameters, various behaviors that depart from exponential and inverse-power-law decays, or their product. The analysis of the Mandel Q factor shows that, for small values of the label, the statistics of the number of excitations becomes super-Poissonian, or sub-Poissonian, by simply choosing sufficiently large values of one of the involved parameters. The time evolution of a generalized coherent state interacting with a thermal reservoir and the purity are analyzed.
We define Schrödinger cat states as superpositions of q-deformed Barut–Girardello su (1, 1) coherent states with an adjustable angle φ in a q-deformed Fock space. We study the statistical properties of the q-deformed Barut–Girardello su (1, 1) coherent states and Schrödinger cat states. The statistical properties of photons are always sub-Poissonian for q-deformed Barut–Girardello su (1, 1) coherent states. For Schrödinger cat states in the cases φ = 0, π /2, π , the statistical properties of photons are always sub-Poissonian if φ = π /2, and the other cases are hard to determine because they depend on the parameters q and k. Moreover, we find some interesting properties of Schrödinger cat states in the limit | z | → 0, where z is the parameter of those states. We also derive that the statistical properties of photons are sub-Poissonian in the undeformed case where π /2 ≤ φ ≤ 3 π /2.
The concept of a photon added two-mode Schrödinger cat state in which both modes are independent is introduced, and their nonclassical properties and entanglement are studied. The introduced states emerge as eigenstates of f 1 f 2 a 1 a 2 , where f 1 , f 2 are nonlinear functions of the number operator, and a 1 , a 2 are annihilation operators. We study the evolution of these states under canonical transformation using the parity operator for the case of standard coherent states of the harmonic oscillator. The nonclassical properties of these states are evaluated especially by considering sub-Poissonian photon statistics and photon number distribution. Interestingly, the addition of photons leads to shifting the region in which photon number distribution shows oscillatory behavior. In addition, the entanglement of introduced states is quantitatively analyzed using concurrence. We observe that the state approaches maximum entanglement more rapidly after the addition of photons.