Schr\"{o}dinger Intelligent States and Linear and Quadratic Amplitude Squeezing

2012

We describe a multi-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions and the corresponding photon statistics are discussed. Some applications to quantum optics, cavity quantum electrodynamics, and superfocusing in channeling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.

Journal of Physics A: Mathematical and Theoretical, 2007

Based on the nonlinear coherent states method, a general and simple algebraic formalism for the construction of 'f-deformed intelligent states' has been introduced. The structure has the potentiality to apply to systems with a known discrete spectrum as well as the generalized coherent states with known nonlinearity function f (n). As some physical appearance of the proposed formalism, a few new classes of intelligent states associated with 'center of-mass motion of a trapped ion', 'harmonious states' and 'hydrogen-like spectrum' have been realized. Finally, the nonclassicality of the obtained states has been investigated. To achieve this purpose the quantum statistical properties using the Mandel parameter and the squeezing of the quadratures of the radiation field corresponding to the introduced states have been established numerically.

2005

States which minimize the Schr\"odinger--Robertson uncertainty relation are constructed as eigenstates of an operator which is a element of the $h(1) \oplus \su(2)$ algebra. The relations with supercoherent and supersqueezed states of the supersymmetric harmonic oscillator are given. Moreover, we are able to compute gneneral Hamiltonians which behave like the harmonic oscillator Hamiltonian or are related to the Jaynes--Cummings Hamiltonian.

Physical Review A

We examine the generalized squeezed states defined as eigenstates of a linear combination of the lowering and raising operators a 2 and (a † ) 2 , respectively. This approach is entirely equivalent to the minimumuncertainty method applied to the amplitude-squared operators. We solve the eigenvalue equation in Glauber's coherent-state representation and find two independent solutions. Their Fock-state expansions, one containing only even and the other only odd number states, reveal a strong nonclassical character. We show that the calculation of the mean photon number is sufficient to obtain the expectation values of interest. Consequently, photon statistics is investigated in both cases by using the generating function of the photon-number distribution. We find the conditions under which the second-order squeezed states display photon antibunching and quadrature squeezing. Also discussed is the preservation of their amplitude-squared squeezing by linear amplification at gains exceeding 2. Analytically, our results are simple formulas in terms of Kummer and Gauss hypergeometric functions that allow straightforward numerical calculations.

Journal of Physics A: Mathematical and Theoretical

Current definitions of both squeezing operator and squeezed vacuum state are critically examined on the grounds of consistency with the underlying su(1,1) algebraic structure. Accordingly, the generalized coherent states for su(1,1) in its Schwinger two-photon realization are proposed as squeezed states. The physical implication of this assumption is that two additional degrees of freedom become available for the control of quantum optical systems. The resulting physical predictions are evaluated in terms of quadrature squeezing and photon statistics, while the application to a Mach-Zehnder interferometer is discussed to show the emergence of nonclassical regions, characterized by negative values of Mandel's parameter, which cannot be anticipated by the current formulation, and then outline future possible use in quantum technologies.

International Journal of Modern Physics A

This review is intended for readers who want to have a quick understanding on the theoretical underpinnings of coherent states and squeezed states which are conventionally generated from the prototype harmonic oscillator but not always restricting to it. Noting that the treatments of building up such states have a long history, we collected the important ingredients and reproduced them from a fresh perspective but refrained from delving into detailed derivation of each topic. By no means we claim a comprehensive presentation of the subject but have only tried to recapture some of the essential results and pointed out their interconnectivity.

The European Physical Journal Plus, 2021

A new class of states of light is introduced that is complementary to the wellknown squeezed states. The construction is based on the general solution of the three-term recurrence relation that arises from the saturation of the Schrödinger inequality for the quadratures of a single-mode quantized electromagnetic field. The new squeezed states are found to be linear superpositions of the photon-number states whose coefficients are determined by the associated Hermite polynomials. These results do not seem to have been noticed before in the literature. As an example, the new class of squeezed states includes superpositions characterized by odd-photon number states only, so they represent the counterpart of the prototypical squeezed-vacuum state which consists entirely of even-photon number states.

Journal of Physics A: Mathematical and Theoretical, 2012

A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n (k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezedcoherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed-coherent states tend to those of the standard oscillator.

Physical Review A, 1996

The coherent states of a Hamiltonian linear in SU͑1,1͒ operators are constructed by defining them, in analogy with the harmonic-oscillator coherent states, as the minimum-uncertainty states with equal variance in two observables. The proposed approach is thus based on a physical characteristic of the harmonic-oscillator coherent states which is in contrast with the existing ones which rely on the generalization of the mathematical methods used for constructing the harmonic-oscillator coherent states. The set of states obtained by following the proposed method contains not only the known SU͑1,1͒ coherent states but also a different class of states.

Physica A-statistical Mechanics and Its Applications, 2000

Superpositions of squeezed states were introduced by Sanders [Phys. Rev. A 39 (1998) 4284], Schleich et al. [Phys. Rev. A 38 (1988) 1177], Xin et al. [Phys. Rev. A 50 (1994) 2865], to investigate the occurrence of nonclassical properties of the quantized radiation field. In this report we present a generalized superposition state which interpolates between two arbitrary squeezed states. Nonclassical properties of this intermediate state as function of the interpolating parameters are studied, the previous results in the literature becoming a particularization of ours. An experimental proposal to generate this state is also presented.