Quantum dynamics of a driven parametric oscillator in a Kerr medium

In this paper, we first analyze a parametric oscillator with both mass and frequency time-dependent. We show that the evolution operator can be obtained from the evolution operator of another parametric oscillator with a constant mass and time-dependent frequency followed by a time transformation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow \int _0^t dt'\,1/m(t')$$\end{document}t→∫0tdt′1/m(t′). Then we proceed by investigating the quantum dynamics of a parametric oscillator with unit mass and time-dependent frequency in a Kerr medium under the influence of a time-dependent force along the motion of the oscillator. The quantum dynamics of the time-dependent oscillator is analyzed from both analytical and numerical points of view in two main regimes: (i) small Kerr parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi $$\end{document}χ, and (ii) small confinement parameter k. In the following, to investigate the characteristics and statistical properties of the generated states, we calculate the autocorrelation function, the Mandel Q parameter, and the Husimi Q-function.


Introduction
A momentous concept of coherent states with the eigenvalue relation â|α⟩ = α|α⟩ as, a very convenient foundation for studying and describing the radiation field, was first introduced by Schrödinger in 1926 which appeared from the investigation of the quantum harmonic oscillator [1][2][3][4] .But the quantum theory of coherence based on coherent states and photodetection had been developed by Glauber, Wolf, Sudarshan, Mandel, Klauder, and many others in the early 1960s that are most resembling quantum states in classical radiation fields and are therefore considered as the boundary between classical mechanics and quantum mechanics.Glauber innovative work was acknowledged by awarding him the Nobel Prize in 2005 5,6 .Indeed, coherent states have become one of the most commonly used instruments in quantum physics which performed a very significant role in various fields particularly in quantum optics and quantum information.The coherent states allowed us to describe the behavior of light in phase space, using the quasi-probabilities developed much earlier by Wigner and others 7 .The significance of coherent states is because of their generalizations that have been demonstrated to have the capacity to present non-classical radiation field characteristics [8][9][10] .The manifestation of the laser as a great potential coherent light marked the start of an extensive study of non-linear interactions between light and matter 11 .This can be attained experimentally by crossing a coherent state via a Kerr medium as a result of the advent of recognizable macroscopic superpositions of coherent states, the so-called cat states 12 .Kerr states as the output of a Kerr medium had been introduced by Kitagawa and Yamamoto, when the state at the way in of the Kerr medium is a canonical coherent state 13 .The Kerr effect generates quadrature squeezing but does not modify the input field photon statistics, i.e. it remains Poissonian, which is a characteristic of the canonical coherent state input and was used for the generation of a superposition of coherent states [14][15][16] .Here it is worth noting that diffusion of light in a Kerr medium is also characterized by the anharmonic oscillator sample and the anharmonic term is taken to be equal to np , where p is an integer (p > 1) 17,18 .This oscillator mode can be evaluated as describing the evolution of a coherent state injected into a transmission line with a nonlinear susceptibility, an optical fiber for example.A laser beam that is quantum mechanically depicted by a coherent state, while passing via non-linear media, can undergo a diversity of complex alterations containing collapses and revivals of the quantum state.In any evolution of linear or non-linear, dissipation is always ready.The dissipative effects classically conduce to decreasing in the amplitude, however, if the interactions befall at atomic scales, quantum effects are significant 19 .Nonlinear coherent states are one of the most prominent generalizations of the standard coherence states 20 .
An appropriate question has been appointed: What will occur if the temporal evolution of an initial coherent state is influenced by a time-dependent harmonic-oscillator Hamiltonian with the coupling of time-dependent external additive potentials [21][22][23][24] ?There are miscellaneous sorts of time-dependent harmonic oscillators such as parametric oscillators 11,25 , Caldirola-Kanai oscillators 26,27 , and harmonic oscillators with a strongly pulsating mass 28 .
Here we first investigate the quantum dynamics of a parametric oscillator with both mass and frequency time-dependent Eq. ( 1) and show that the corresponding time-evolution operator can be obtained from another parametric oscillator with a constant mass and time-dependent frequency followed by a time transformation t → t 0 dt ′ 1/m(t ′ ).Therefore, we mainly focus on a parametric oscillator described by the Hamiltonian Ĥ(t) = p2 /2 + ω 2 (t) q2 /2, in a Kerr medium and under the influence of a arXiv:2306.02249v1[quant-ph] 4 Jun 2023 classical external source.

Quantum harmonic oscillator with both mass and frequency time-dependent
To set the stage, let us first consider a parametric oscillator with time-dependent mass and frequency The Hamiltonian Eq. ( 1) can be written as where we have defined ω(t) = m(t)Ω(t).Let E * n (t) and ψ * n (q,t) be eigenvalues and eigenfunctions of the Hamiltonian Ĥ * (t) respectively then, one easily finds Therefore, from Eq. ( 2) we deduce that the Hamiltonian Ĥ(t) has the same eigenfunctions as Ĥ * (t) but the corresponding eigenvalues E n (t) are given by that is time-dependent mass does not affect the eigenvalues but eigenfunctions, as expected.Now let us find a connection between the time evolution operators of the Hamiltonians Ĥ(t) and Ĥ * (t).
Let Û(t) be the time-evolution operator corresponding to Ĥ(t) = Ĥ * (t)/m(t), then by using we deduce that if we define a new variable τ as τ = ρ(t) = t 0 1/m(t ′ ) dt ′ which is an increasing function of t for m(t) > 0, then the time-ordering will does not change and we can rewrite Eq. ( 6) as Therefore, as the first step, we replace the parameter t in the Hamiltonian Ĥ * (t) with ρ −1 (τ), and define the transformed Hamiltonian as H * (τ) = Ĥ * (ρ −1 (τ)).Let us denote the corresponding time-evolution operator by Ũ * (τ), then Now we can prove that the time-evolution operator Û(t) corresponding to the Hamiltonian Ĥ(t) can be obtained from Ũ * (τ) by replacing τ with ρ(t 2/13

Example
As an example let us find the time-evolution operator for the Hamiltonian we have The quantum propagator for Hamiltonians of type Eq. ( 11) has been investigated in 29 , let us denote the quantum propagator of H * (τ) in position space by K * (q, τ|q ′ , 0), then the quantum propagator corresponding to the main Hamiltonian Ĥ(t) is Also, the position and momentum operators in the Heisenberg picture are given by where q * (τ) and p * (τ) are position and momentum operators in the Heisenberg picture corresponding to the Hamiltonian H * (τ).From the Heisenberg equation for q * (τ) one finds The Eqs. (14), have the following solutions where  = 4ω 2 0 − γ 2 , tan θ = /γ, and the constant operators Ĉ1 and Ĉ2 can be obtained from the initial conditions q * (0) = q(0) and p * (0) = p(0).After straightforward calculations, we obtain Therefore, the Hamiltonian Ĥ * (t) is the main ingredient in Eq. ( 2).In the next section, we will focus on the Hamiltonians of the type Ĥ * (t) in the presence of an external time-dependent classical source in a Kerr medium.

The model
The model that we will investigate in the following is a generalization of the Hamiltonian Eq. ( 2) given by describing the quantum dynamics of a time-dependent harmonic oscillator in a Kerr medium and under the influence of a time-dependent force −e(t) along the motion of the oscillator.In Eq. (17), Ω(t) is a time-dependent frequency and Ĥkerr = χ n2 .The Kerr parameter χ is a constant proportional to the third-order nonlinear susceptibility χ 3 which is, in general, a small parameter.To be specific, in what follows we will choose where k is also a small confinement parameter 30 .To this end, the annihilation, creation, and number operators are defined respectively by where for notational simplicity we have set h = 1.The time-dependent operators given in Eq. ( 18) fulfill the Heisenberg algebra at any time In the absence of a Kerr medium (χ = 0), the Hamiltonian Eq. ( 17) reduces to Ĥf (t) given by + e(t) The Hamiltonian Ĥf (t) can be diagonalized.To this end, let us define the time-dependent displacement operator as By making use of the Baker-Campbell-Hausdorff (BCH) formula we find where Ĥ0 (t) is given in Eq. ( 20) and for convenience we defined Therefore, the Hamiltonian Ĥf (t) is obtained from Ĥ0 (t) trough a similarity transformation followed by a translation as Let |n⟩ 0 t and E 0 n (t) be the eigenstates and eigenvalues of the Hamiltonian Ĥ0 (t) respectively Ĥ0 (t)|n⟩ by using Eq. ( 25) one easily finds that the states |n⟩ t = D † t (λ t )|n⟩ 0 t are the eigenstates of the Hamiltonian Ĥf (t) with eigenvalues

4/13
Position representation of the eigenfunctions of Ĥf (t) The position representation of the eigenfunctions of the Hamiltonian Ĥf (t) can be obtained as follows where we made use of Eqs.(18).The eigenfunction ψ 0 n (q,t) of the Hamiltonian Ĥ0 (t) can be obtained from ψ 0 n (q,t) = ( Â † t ) n / √ n! ψ 0 0 (q,t), where Ât ψ 0 0 (q,t) = 0, the explicit form of the eigenfunction ψ 0 n (q,t) is where H n (z) is a Hermite polynomial of order n Therefore, in the presence of an external source (λ t ̸ = 0), the eigenfunction ψ f n (q,t) is obtained by shifting q → q − λ t 2/Ω(t) in the free eigenfunction ψ 0 n (q,t).

Linearization of the Hamiltonian
In this section, in the framework of the Heisenberg picture, we will find approximate solutions for the time-evolution of the ladder operators â(t) and â † (t) using a linearization process.For this purpose, we assume that the confinement parameter is negligible (k ≪ 1), so Ω(t) ≈ Ω 0 .The time-dependent Hamiltonian Ĥ(t) now becomes where From Heisenberg equation we have where ν = Ω 0 − χ.By inserting â(t) = e −iνt b(t) into Eq.( 33) we find In Eq. ( 33) the term 2χ â n can be ignored up to the first order approximation since χ ≪ 1, then where

Time-evolution operator
In this section, we reconsider the Hamiltonian Eq. ( 45) and try to find the corresponding time-evolution operator approximately.

Properties of the states |β ⟩ ξ
Let us define a new set of ladder operators as where the function f ( n) is defined by The operators B, B † and n, fulfil the usual Heisenberg algebra

Phase-space (quasi)probability distributions
where n is the photon number operator and g (2) (0) is the normalized second-order correlation function.
For the state |β ⟩ ξ we have therefore, Q = 0, indicating that the statistical distribution of excitations is Poissonian.In the next section, we will study the autocorrelation function to find out how the evolved state resembles the original state.

Autocorrelation function
The autocorrelation function is the overlap between the evolved and the initial state 38 , and shows the possibility of total or partial resemble of the initial state when the overlap is complete or partial, respectively.The overlap or the scalar product of the initial and the evolved state |ψ α ,t⟩ is (see Eq.

Husimi distribution function
The Husimi function, which can be measured using quantum tomographic techniques, is always positive so it is a distribution on phase space.It has been found that the Husimi distribution function is linked to classical information entropy, which can be used to measure non-classical correlations in composite systems, through the Wehrl entropy.In the phase space, the Husimi distribution function has been used to measure and study the erasing information, coherence loss, relaxation processes and adjustable phase-space information 39 .The Husimi function is defined by and [ f (0)]! = 1.One can easily show that the state |β ⟩ ξ is a coherent state for the new annihilation operator B with eigenvalue β B|β ⟩ ξ = β |β ⟩ ξ .(59) If we define the modified displacement operator DB (β ) = e β B † − β B, then |β ⟩ ξ = DB |0⟩, note that |0⟩ ξ = |0⟩, and the parameter ξ = χt is hidden in the definition of B and B † .The state |β ⟩ ξ can also be considered as a Kerr state if we consider the Hamiltonian Ĥ = hχ â † â + hχ â † â † â â, = hχ n2 , (60) with the corresponding time-evolution operator Û(t) = e −itχ n2 = e −iξ n2 .(61) If the system is initially prepared in the coherent state |β ⟩, then the evolved state is the state |β ⟩ ξ given by Û(t)|β ⟩ = e −iξ n2 |β ⟩ = |β ⟩ ξ , n 2 |n⟩.