Coherent and Squeezed States: Introductory Review of Basic Notions, Properties, and Generalizations

2020

Phys. Scr.

Self Cite

We consider the relations between nonstationary quantum oscillators and their stationary counterpart in view of their applicability to study particles in electromagnetic traps. We develop a consistent model of quantum oscillators with timedependent frequencies that are subjected to the action of a time-dependent driving force, and have a time-dependent zero point energy. Our approach uses the method of point transformations to construct the physical solutions of the parametric oscillator as mere deformations of the well known solutions of the stationary oscillator. In this form, the determination of the quantum integrals of motion is automatically achieved as a natural consequence of the transformation, without necessity of any ansätz. It yields the mechanism to construct an orthonormal basis for the nonstationary oscillators, so arbitrary superpositions of orthogonal states are available to obtain the corresponding coherent states. We also show that the dynamical algebra of the parametric oscillator is immediately obtained as a deformation of the algebra generated by the conventional boson ladder operators. A number of explicit examples is provided to show the applicability of our approach.

Section: Coherent Statesmentioning

confidence: 99%

“…Using the conventional disentangling formulae, see e.g. [32,70], using a 2 (t) and a † 2 (t), one obtains the operator…”

Section: Coherent Statesmentioning

confidence: 99%

Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

Zelaya

2020

Phys. Scr.

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“…has the classical period τ osc = 2π ω 0 . That is, in the phase-space, the point x(t) = λ(t) describes a circumference that passes through x(t 0 ) over and over as the time reaches any integer multiple of the period τ osc [50].…”

Section: Oscillator Wave-packetsmentioning

confidence: 99%

Coherent states for exactly solvable time-dependent oscillators generated by Darboux transformations

Cruz¹,

Razo²,

et al. 2020

Phys. Scr.

The Darboux method is commonly used in the coordinate variable to produce new exactly solvable (stationary) potentials in quantum mechanics. In this work we follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to include the time-variable as a parameter of the transformation. The new potentials are nonstationary and define Hamiltonians which are not integrals of motion for the system under study. We take the stationary oscillator of constant frequency to produce nonstationary oscillators, and also provide an invariant that serves to define uniquely the state of the system. In this sense our approach completes the program of the BSS method since the eigenfunctions of the invariant are an orthonormal basis for the space of solutions of the related Schrödinger equation. The orthonormality holds when the involved functions are evaluated at the same time. The dynamical algebra of the nonstationary oscillators is generated by properly chosen ladder operators and coincides with the Heisenberg algebra. We also construct the related coherent states and show that they form an overcomplete set that minimizes the quadratures defined by the ladder operators. These states are not invariant under time-evolution since their time-dependence relies on the basis of states and not on the complex eigenvalue that labels them. Some concrete examples are provided.

“…Quite remarkably, scalable quantum computing is possible by using single photon sources [3]. In the same context, as the squeezed states of light have less noise in one of their quadratures than the quantum noise limit dictates [4], they are useful to improve the precision of interferometric measurements dealing with very low intensity light signals [5,6]. The prototypical example is the squeezed-vacuum state, which consists entirely of even-photon number states [7].…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, the even and odd coherent states [9] are constructed as opposite phase superpositions of the fully coherent states introduced by Glauber [10]. This issue is of great interest because the Glauber states, formed by superpositions of photon-number states, tolerate a description in terms of the Maxwell theory [5], so they may describe the states of macroscopic systems. The creation of cat states, a theoretical description introduced by Schrödinger to show the way in which quantum mechanics contradicts our everyday experience for systems as great as a 'cat' [11], therefore, is available in the laboratory using even and odd coherent states [12,13], which are, therefore, called optical cat states [14] (although the term optical kitten is also found [15]).…”

Section: Introductionmentioning

confidence: 99%

Theory of Photon Subtraction for Two-Mode Entangled Light Beams

Zelaya

2021

Quantum Reports

Self Cite

Photon subtraction is useful to produce nonclassical states of light addressed to applications in photonic quantum technologies. After a very accelerated development, this technique makes possible obtaining either single photons or optical cats on demand. However, it lacks theoretical formulation enabling precise predictions for the produced fields. Based on the representation generated by the two-mode SU(2) coherent states, we introduce a model of entangled light beams leading to the subtraction of photons in one of the modes, conditioned to the detection of any photon in the other mode. We show that photon subtraction does not produce nonclassical fields from classical fields. It is also derived a compact expression for the output field from which the calculation of conditional probabilities is straightforward for any input state. Examples include the analysis of squeezed-vacuum and odd-squeezed states. We also show that injecting optical cats into a beam splitter gives rise to entangled states in the Bell representation.