Introduction

A momentous concept of coherent states with the eigenvalue relation \(\hat{a}|\alpha \rangle = \alpha |\alpha \rangle \) 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 oscillator1,2,3,4. 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 20055,6. 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 others7. The significance of coherent states is because of their generalizations that have been demonstrated to have the capacity to present non-classical radiation field characteristics8,9,10. 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 matter11. 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 states12. 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 state13. 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 states14,15,16. 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 \(\hat{n}^p\), 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 significant19. Nonlinear coherent states are one of the most prominent generalizations of the standard coherence states20.

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 potentials21,22,23,24? There are miscellaneous sorts of time-dependent harmonic oscillators such as parametric oscillators11,25, Caldirola-Kanai oscillators26,27, and harmonic oscillators with a strongly pulsating mass28.

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 \(t\rightarrow \int _0^t dt'\,1/m(t')\). Therefore, we mainly focus on a parametric oscillator described by the Hamiltonian \(\hat{H}(t)=\hat{p}^2/2+\omega ^2(t)\hat{q}^2/2\), in a Kerr medium and under the influence of a 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

$$\begin{aligned} \hat{H}(t)=\frac{\hat{p}^2}{2m(t)} + \frac{1}{2}\,m(t)\Omega ^2(t)\hat{q}^2. \end{aligned}$$
(1)

The Hamiltonian Eq. (1) can be written as

$$\begin{aligned} \hat{H}(t)=\frac{1}{m(t)}\bigg (\underbrace{\frac{\hat{p}^2}{2} + \frac{1}{2}\,\omega ^2(t)\hat{q}^2}_{\hat{H}^{*}(t)}\bigg ), \end{aligned}$$
(2)

where we have defined \(\omega (t)=m(t)\Omega (t)\). Let \(E_n^{*}(t)\) and \(\psi _n^{*}(q,t)\) be eigenvalues and eigenfunctions of the Hamiltonian \(\hat{H}^{*}(t)\) respectively

$$\begin{aligned} \hat{H}^{*}(t)\,\psi _n^{*}(q,t)=E_n^{*}(t)\,\psi _n^{*}(q,t), \end{aligned}$$
(3)

then, one easily finds

$$\begin{aligned}{} & {} E_n^{*}(t)=\hbar \omega (t)\,(n+1/2),\nonumber \\{} & {} \psi _n^{*}(q,t)=\frac{1}{\sqrt{2^n n!}}\,(\omega (t)/\pi )^{1/4}\,e^{-\frac{1}{2\hbar }\omega (t)q^2}\,H_n (\sqrt{\omega (t)/\hbar }\,q). \end{aligned}$$
(4)

Therefore, from Eq. (2) we deduce that the Hamiltonian \(\hat{H}(t)\) has the same eigenfunctions as \(\hat{H}^{*}(t)\) but the corresponding eigenvalues \(E_n (t)\) are given by

$$\begin{aligned} E_n (t)=\frac{1}{m(t)}\,E_N^{*}(t)=\hbar \Omega (t)\,(n+1/2), \end{aligned}$$
(5)

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 \(\hat{H}(t)\) and \(\hat{H}^{*}(t)\).

Let \(\hat{U} (t)\) be the time-evolution operator corresponding to \(\hat{H}(t)=\hat{H}^* (t)/m(t)\), then by using

$$\begin{aligned} \hat{U}(t)=\text{ T }\left[ e^{-\frac{i}{\hbar }\int _0^t dt'\,\frac{1}{m(t')}\hat{H}^* (t')}\right] , \end{aligned}$$
(6)

we deduce that if we define a new variable \(\tau \) as \(\tau =\varrho (t)=\int _0^t 1/m(t')\,dt'\) 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

$$\begin{aligned} \hat{U}(t)=\text{ T }\left[ e^{-\frac{i}{\hbar }\int _0^{\tau (t)} d\tau \,\hat{H}^* (t(\tau ))}\right] . \end{aligned}$$
(7)

Therefore, as the first step, we replace the parameter t in the Hamiltonian \(\hat{H}^* (t)\) with \(\varrho ^{-1}(\tau )\), and define the transformed Hamiltonian as \(\tilde{\hat{H}}^* (\tau )=\hat{H}^* (\varrho ^{-1}(\tau ))\). Let us denote the corresponding time-evolution operator by \(\tilde{\hat{U}}^* (\tau )\), then

$$\begin{aligned} i\hbar \,\frac{d}{d\tau }\tilde{\hat{U}}^*(\tau )=\tilde{\hat{H}}^* (\tau )\,\tilde{\hat{U}}^*(\tau ). \end{aligned}$$
(8)

Now we can prove that the time-evolution operator \(\hat{U}(t)\) corresponding to the Hamiltonian \(\hat{H}(t)\) can be obtained from \(\tilde{\hat{U}}^* (\tau )\) by replacing \(\tau \) with \(\varrho (t)=\int _0^t 1/m(t')\,dt'\), that is \(\hat{U}(t)=\tilde{\hat{U}}^* (\varrho (t))\). We have

$$\begin{aligned} i\hbar \,\frac{d}{dt}\hat{U}(t)= \, & {} i\hbar \,\frac{d}{dt}\tilde{\hat{U}}^* (\varrho (t))=i\hbar \,\frac{d\varrho }{dt}\frac{d}{d\varrho }\tilde{\hat{U}}^* (\varrho )=\frac{1}{m(t)}i\hbar \,\frac{d}{d\varrho }\tilde{\hat{U}}^* (\varrho ),\nonumber \\= \, & {} \frac{1}{m(t)}\,\tilde{\hat{H}}^{*}(\varrho )\,\tilde{\hat{U}}^* (\varrho )=\frac{1}{m(t)}\,\hat{H}^{*}(t)\,\tilde{\hat{U}}^* (\varrho )=\hat{H}(t)\,\hat{U}(t). \end{aligned}$$
(9)

Example

As an example let us find the time-evolution operator for the Hamiltonian

$$\begin{aligned} \hat{H}(t)= \, & {} \frac{\hat{p}^2}{2m_0\exp (\gamma t)}+\frac{1}{2}m_0\exp (\gamma t)\,\omega _0^2 \,\hat{q}^2,\nonumber \\= \, & {} \frac{1}{\exp (\gamma t)}\left( \frac{\hat{p}^2}{2m_0}+\frac{1}{2}m_0\omega _0^2 \exp (2\gamma t)\,\hat{q}^2\right) , \end{aligned}$$
(10)

we have

$$\begin{aligned} \tau= \, & {} \int _0^t dt'\,\frac{1}{\exp (\gamma t')}=\frac{1}{\gamma }(1-\exp (-\gamma t)),\nonumber \\{} & {} \tilde{\hat{H}}^* (\tau )=\frac{\hat{p}^2}{2m_0}+\frac{1}{2}m_0\omega _0^2 \frac{1}{(1-\gamma \tau )^2}\,\hat{q}^2. \end{aligned}$$
(11)

The quantum propagator for Hamiltonians of type Eq. (11) has been investigated in29, let us denote the quantum propagator of \(\tilde{\hat{H}}^* (\tau )\) in position space by \(\tilde{K}^*(q,\tau |q',0)\), then the quantum propagator corresponding to the main Hamiltonian \(\hat{H}(t)\) is

$$\begin{aligned} K(q,t|q',0)= \tilde{K}^*(q,\frac{1}{\gamma }(1-e^{-\gamma t})|q',0). \end{aligned}$$
(12)

Also, the position and momentum operators in the Heisenberg picture are given by

$$\begin{aligned} \hat{q}(t)= \, & {} \hat{U}^\dag (t)\,\hat{q}(0)\,\hat{U}(t)={\tilde{\hat{U}}^*}^\dag (\tau )\,\hat{q}(0)\, {\tilde{\hat{U}}^*(\tau )}=\hat{q}^*(\tau )\big |_{\tau =\varrho (t)},\nonumber \\ \hat{p}(t)= \, & {} \hat{U}^\dag (t)\,\hat{p}(0)\,\hat{U}(t)={\tilde{\hat{U}}^*}^\dag (\tau )\,\hat{p}(0)\, {\tilde{\hat{U}}^*(\tau )}=\hat{p}^*(\tau )\big |_{\tau =\varrho (t)}, \end{aligned}$$
(13)

where \(\hat{q}^*(\tau )\) and \(\hat{p}^*(\tau )\) are position and momentum operators in the Heisenberg picture corresponding to the Hamiltonian \(\tilde{\hat{H}}^*(\tau )\). From the Heisenberg equation for \(\hat{q}^*(\tau )\) one finds

$$\begin{aligned}{} & {} \frac{d^2}{d\tau ^2}\,\hat{q}^*(\tau )+\frac{\omega _0^2}{(1-\gamma \tau )^2}\,\hat{q}^*(\tau )=0,\nonumber \\{} & {} \hat{p}^*(\tau )=m_0\,\frac{d}{d\tau }\,\hat{q}^*(\tau ). \end{aligned}$$
(14)

The Eqs. (14), have the following solutions

$$\begin{aligned} \hat{q}^*(\tau )= \, & {} \hat{C}_1\,(\tau -1/\gamma )^{\frac{\gamma +i\digamma }{2\gamma }} +\hat{C}_2\,(\tau -1/\gamma )^{\frac{\gamma -i\digamma }{2\gamma }},\nonumber \\ \hat{p}^*(\tau )= \, & {} \hat{C}_1\,(\frac{\gamma +i\digamma }{2\gamma })(\tau -1/\gamma )^{-\frac{\gamma -i\digamma }{2\gamma }}+\hat{C}_2\,\frac{\gamma -i\digamma }{2\gamma }(\tau -1/\gamma )^{-\frac{\gamma +i\digamma }{2\gamma }}, \end{aligned}$$
(15)

where \(\digamma =\sqrt{4\omega _0^2-\gamma ^2}\), \(\tan \theta =\digamma /\gamma \), and the constant operators \(\hat{C}_1\) and \(\hat{C}_2\) can be obtained from the initial conditions \(\hat{q}^*(0)=\hat{q}(0)\) and \(\hat{p}^*(0)=\hat{p}(0)\). After straightforward calculations, we obtain

$$\begin{aligned}{} & {} \hat{q}(t)=\hat{q}^*(\tau )\big |_{\tau =\varrho (t)}=\frac{2}{m_0 \digamma }\,e^{-\frac{\gamma t}{2}}\big (\sin (\digamma t/2)\,\hat{p}(0)+m_0\omega _0\,\sin (\digamma t/2+\theta )\,\hat{q}(0)\big ),\nonumber \\{} & {} \hat{p}(t)=\hat{p}^*(\tau )\big |_{\tau =\varrho (t)}=-\frac{2\omega _0}{\digamma }\,e^{-\frac{\gamma t}{2}}\big (\sin (\digamma t/2-\theta )\,\hat{p}(0)+m_0\omega _0\,\sin (\digamma t/2)\,\hat{q}(0)\big ). \end{aligned}$$
(16)

Therefore, the Hamiltonian \(\hat{H}^*(t)\) is the main ingredient in Eq. (2). In the next section, we will focus on the Hamiltonians of the type \(\hat{H}^*(t)\) 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

$$\begin{aligned} \hat{H}(t)=\frac{1}{2}[\hat{p}^2 + {{\Omega }^2(t)\hat{q}^2}]+e(t)\hat{q}+\hat{H}_{kerr}, \end{aligned}$$
(17)

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), \(\Omega (t)\) is a time-dependent frequency and \(\hat{H}_{kerr} ={\chi \hat{n}^2}\). The Kerr parameter \(\chi \) is a constant proportional to the third-order nonlinear susceptibility \(\chi ^{3}\) which is, in general, a small parameter. To be specific, in what follows we will choose \(\Omega (t)=\Omega _0[1+2k\cos (2\Omega _0 t)]\) where k is also a small confinement parameter30. To this end, the annihilation, creation, and number operators are defined respectively by

$$\begin{aligned} \hat{A_t}= \, & {} \frac{1}{\sqrt{2\Omega (t)}}(\Omega (t)\hat{q}+i\hat{p}),\nonumber \\ \hat{A}^\dagger _t= \, & {} \frac{1}{\sqrt{2\Omega (t)}}(\Omega (t)\hat{q}-i\hat{p}),\nonumber \\ \hat{n}_t= \, & {} \hat{A}^\dagger _t \hat{A_t}, \end{aligned}$$
(18)

where for notational simplicity we have set \(\hbar =1\). The time-dependent operators given in Eq. (18) fulfill the Heisenberg algebra at any time

$$\begin{aligned}{} & {} [\hat{A_t},\hat{A}^\dagger _t] = 1,\nonumber \\{} & {} [\hat{n}_t,\hat{A}^\dagger _t] = \hat{A}^\dagger _t,\nonumber \\{} & {} [\hat{n}_t,\hat{A_t}] = \hat{A_t}. \end{aligned}$$
(19)

In the absence of a Kerr medium (\(\chi =0\)), the Hamiltonian Eq. (17) reduces to \(\hat{H}_f(t)\) given by

$$\begin{aligned} \hat{H}_f(t)= \, & {} \frac{1}{2}[\hat{p}^2 + {{\Omega }^2(t)\hat{q}^2}]+e(t)\hat{q},\nonumber \\= \, & {} \underbrace{\Omega (t)(\hat{A}^\dagger _t \hat{A_t}+ \frac{1}{2})}_{\hat{H}_0 (t)}+\frac{e(t)}{\sqrt{2\Omega (t)}}(\hat{A}^\dag _t+\hat{A}_t). \end{aligned}$$
(20)

The Hamiltonian \(\hat{H}_f(t)\) can be diagonalized. To this end, let us define the time-dependent displacement operator as

$$\begin{aligned} \hat{D}_t (\alpha )=e^{\alpha \hat{A}^\dag _t-\bar{\alpha }\hat{A}_t}. \end{aligned}$$
(21)

By making use of the Baker-Campbell-Hausdorff (BCH) formula we find

$$\begin{aligned}{} & {} \hat{D}^\dagger _t(\alpha )(\hat{A}^\dagger _t)\hat{D}_t(\alpha )=\hat{A}^\dagger _t+\bar{\alpha },\nonumber \\{} & {} \hat{D}^\dagger _t(\alpha )(\hat{A}_t)\hat{D}_t(\alpha )=\hat{A}_t+\alpha , \end{aligned}$$
(22)

also

$$\begin{aligned} \hat{D}^\dagger _t(\alpha )\hat{H}_0(t)\hat{D}_t(\alpha )= \, & {} \Omega _t\Big [(\hat{A}^\dagger _t\hat{A_t}+\frac{1}{2})+\alpha \hat{A}^\dagger _t+\bar{\alpha }\hat{A_t}+|\alpha |^2\Big ],\nonumber \\= \, & {} \hat{H}_f(t)+\frac{e^2 (t)}{2\Omega ^2 (t)}, \end{aligned}$$
(23)

where \(\hat{H}_0(t)\) is given in Eq. (20) and for convenience we defined

$$\begin{aligned} \alpha =\bar{\alpha }=\lambda _t=\frac{e(t)}{\Omega (t)\sqrt{2\Omega (t)}}. \end{aligned}$$
(24)

Therefore, the Hamiltonian \(\hat{H}_f(t)\) is obtained from \(\hat{H}_0 (t)\) trough a similarity transformation followed by a translation as

$$\begin{aligned} \hat{H}_f (t)= \, & {} \hat{D}^\dagger _t(\lambda _t)\hat{H}_0 (t)\hat{D_t}(\lambda _t)-\Omega (t)\,\lambda _t^2. \end{aligned}$$
(25)

Let \(|n\rangle ^0_t\) and \(E^0_n (t)\) be the eigenstates and eigenvalues of the Hamiltonian \(\hat{H}_0 (t)\) respectively

$$\begin{aligned} \hat{H}_0 (t)|n\rangle ^0_t= \, & {} E^0_n (t) |n\rangle ^0_t,\,\,n=0,1,2,\cdots ,\nonumber \\ E^0_n (t)= \, & {} (n+1/2)\Omega (t), \end{aligned}$$
(26)

by using Eq. (25) one easily finds that the states \(|n\rangle _t=D^{\dag }_t(\lambda _t)|n\rangle ^0_t\) are the eigenstates of the Hamiltonian \(\hat{H}_f (t)\) with eigenvalues \(E_n (t)\)

$$\begin{aligned} \hat{H}_f (t)|n\rangle _t= \, & {} E_n (t)|n\rangle _t,\nonumber \\ E_n (t)= \, & {} E^0_n (t)-\Omega (t)\,\lambda _t^2,\nonumber \\= \, & {} (n+1/2-\lambda _t^2)\,\Omega (t). \end{aligned}$$
(27)

Position representation of the eigenfunctions of \(\hat{H}_f (t)\)

The position representation of the eigenfunctions of the Hamiltonian \(\hat{H}_f (t)\) can be obtained as follows

$$\begin{aligned} \psi _n^f (q,t)=\langle q|n\rangle _t= \, & {} \langle q|\hat{D}^\dag _t (\lambda _t)|n\rangle ^0_t,\nonumber \\= \, & {} \langle q|e^{\lambda _t(\hat{A}_t-\hat{A}^\dag _t)}|n\rangle ^0_t,\nonumber \\= \, & {} \langle q|e^{i\lambda _t\sqrt{\frac{2}{\Omega (t)}}\hat{p}}|n\rangle ^0_t,\nonumber \\= \, & {} \langle q-\lambda _t\sqrt{2/\Omega (t)}||n\rangle ^0_t,\nonumber \\= \, & {} \psi _n^0 (q-\lambda _t\sqrt{2/\Omega (t)},t), \end{aligned}$$
(28)

where we made use of Eq. (18). The eigenfunction \(\psi _n^0 (q,t)\) of the Hamiltonian \(\hat{H}_0 (t)\) can be obtained from \(\psi _n^0 (q,t)=(\hat{A}^\dag _t)^n/\sqrt{n!}\,\psi _0^0 (q,t)\), where \(\hat{A}_t\,\psi _0^0 (q,t)=0\), the explicit form of the eigenfunction \(\psi _n^0 (q,t)\) is

$$\begin{aligned} \psi _n^0 (q,t)=\langle q|n\rangle _t^0=\frac{1}{\sqrt{2^n n!}}\left( \frac{\Omega (t)}{\pi }\right) ^{1/4}\,e^{-\frac{1}{2}\Omega (t)q^2}\,H_n (\sqrt{\Omega (t)}\,q), \end{aligned}$$
(29)

where \(H_n (z)\) is a Hermite polynomial of order n

$$\begin{aligned} H_n (z)=(-1)^n e^{z^2}\frac{d^n}{dz^n}e^{-z^2}. \end{aligned}$$
(30)

Therefore, in the presence of an external source \((\lambda _t\ne 0)\), the eigenfunction \(\psi _n^f (q,t)\) is obtained by shifting \(q\rightarrow q-\lambda _t\sqrt{2/\Omega (t)}\) in the free eigenfunction \(\psi _n^0 (q,t)\).

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 \(\hat{a} (t)\) and \(\hat{a}^\dag (t)\) using a linearization process. For this purpose, we assume that the confinement parameter is negligible (\(k\ll 1\)), so \(\Omega (t)\approx \Omega _0\). The time-dependent Hamiltonian \(\hat{H}(t)\) now becomes

$$\begin{aligned} \hat{H}_{k=0} (t)=\Omega _0\,(\hat{a}^\dag \hat{a}+1/2)+\frac{e(t)}{\sqrt{2\Omega _0}}\, (\hat{a}^\dag +\hat{a})+\chi \,(\hat{a}^\dag \hat{a})^2, \end{aligned}$$
(31)

where

$$\begin{aligned}{} & {} \hat{a}=\hat{A}_0=\frac{1}{\sqrt{2\Omega _0}}\,(\Omega _0\,\hat{q}+i\hat{p}),\nonumber \\{} & {} \hat{a}^\dag =\hat{A}^\dag _0=\frac{1}{\sqrt{2\Omega _0}}\,(\Omega _0\,\hat{q}-i\hat{p}),\nonumber \\{} & {} \hat{n}=\hat{A}^\dag _0\hat{A}_0=\hat{a}^\dag \hat{a}. \end{aligned}$$
(32)

From Heisenberg equation we have

$$\begin{aligned} i\dot{\hat{a}}= \, & {} [\hat{a},\hat{H}_{k=0}(t)],\nonumber \\= \, & {} \nu \,\hat{a}+2\chi \hat{a}\hat{n}+\frac{e(t)}{\sqrt{2\Omega _0}}, \end{aligned}$$
(33)

where \(\nu =\Omega _0-\chi \). By inserting \(\hat{a}(t)=e^{-i\nu t}\,\hat{b}(t)\) into Eq. (33) we find

$$\begin{aligned}{} & {} \dot{\hat{b}}(t)=-2i\chi \,\hat{b}(t)\,\hat{n}_t-i\frac{e(t)}{\sqrt{2\Omega _0}}e^{-i\nu t}. \end{aligned}$$
(34)

In Eq. (33) the term \(2\chi \hat{a}\hat{n}\) can be ignored up to the first order approximation since \(\chi \ll 1\), then

$$\begin{aligned} \hat{a}(t)\approx & {} e^{-i\nu t}\hat{a}(0)-\zeta (t),\nonumber \\ \hat{a}^\dag (t)\approx & {} e^{i\nu t}\hat{a}^\dag (0)-\bar{\zeta }(t), \end{aligned}$$
(35)

where

$$\begin{aligned} \zeta (t)=i\frac{1}{\sqrt{2\Omega _0}}\int _{0}^{t}dt^\prime e^{-i\nu (t-t^\prime )}e(t^\prime ). \end{aligned}$$
(36)

To proceed, let the initial state of the system be a number state \(|\psi (0)\rangle =|n\rangle \) then we can linearize Eq. (34) by replacing \(\hat{n}_t\) with its average value \(\bar{n}(t)\)

$$\begin{aligned} \bar{n}(t)= \, & {} \langle n|\hat{a}^\dag (t)\hat{a}(t)|n\rangle ,\nonumber \\= \, & {} \langle n|\hat{b}^\dag (t)\hat{b}(t)|n\rangle ,\nonumber \\= \, & {} \langle n|\hat{a}^\dag (0)\hat{a}(0)-e^{i\nu t}\varphi _t\hat{a}^\dag (0)-e^{-i\nu t}\varphi ^\star (t)\hat{a}(0)+|\varphi _t|^2|n\rangle ,\nonumber \\= \, & {} n+|\varphi _t|^2, \end{aligned}$$
(37)

we find

$$\begin{aligned} \dot{\hat{b}}(t)=-2i\chi \hat{b}(t)[n+|\varphi _t|^2]-i\frac{e(t)}{\sqrt{2\Omega _0}}\,e^{-i\nu t}. \end{aligned}$$
(38)

The solution of Eq.(38) is

$$\begin{aligned} \hat{b}(t)=e^{-i\gamma (t)}\hat{b}(0)+\delta (t), \end{aligned}$$
(39)

where

$$\begin{aligned}{} & {} \gamma (t)=\int _{0}^{t}A(t^\prime )dt^\prime , \end{aligned}$$
(40)
$$\begin{aligned}{} & {} A(t)=2\chi [n+|\varphi _t|^2], \end{aligned}$$
(41)
$$\begin{aligned}{} & {} \delta (t)=e^{-i\gamma (t)}\int _{0}^{t}dt^\prime e^{i\gamma (t^\prime )}\frac{e(t^\prime )}{i\sqrt{2\Omega _0}}. \end{aligned}$$
(42)

Now using Eq. (39) we find a better solution for the ladder operators

$$\begin{aligned} \hat{a}(t)= \, & {} e^{-i(\nu t+\gamma (t))}\hat{a}(0)+e^{-i\nu t}\,\delta (t), \end{aligned}$$
(43)
$$\begin{aligned} \hat{a}^\dag (t)= \, & {} e^{i(\nu t+\gamma (t))}\hat{a}^\dag (0)+e^{i\nu t}\,\bar{\delta }(t). \end{aligned}$$
(44)

Time-evolution operator

In this section, we reconsider the Hamiltonian equation (45) and try to find the corresponding time-evolution operator approximately. For this purpose, we make use of the properties of Heisenberg algebra \(\{1,\hat{a}, \hat{a}^\dag \}\). Let us rewrite the Hamiltonian equation (45) in the following form

$$\begin{aligned} \hat{H}_{k=0} (t)= \, & {} \Omega _0\,(\hat{a}^\dag \hat{a}+1/2)+\chi \,(\hat{a}^\dag \hat{a})^2+\frac{e(t)}{\sqrt{2\Omega _0}}\,(\hat{a}^\dag +\hat{a}),\nonumber \\= \, & {} \hat{H}_0+\frac{e(t)}{\sqrt{2\Omega _0}}\,(\hat{a}^\dag +\hat{a}), \end{aligned}$$
(45)

where \(\hat{H}_0=\Omega _0 (\hat{n}+1/2)+\chi \hat{n}^2\). The evolution operator \(\hat{U}(t)\) corresponding to the Hamiltonian \(\hat{H}_{k=0}\) can be written as

$$\begin{aligned} \hat{U}(t)=\hat{U}_0(t)\hat{U}_I(t), \end{aligned}$$
(46)

where \(\hat{U}_0(t)\) is the evolution operator corresponding to the time-independent Hamiltonian \(\hat{H}_0\) given by

$$\begin{aligned} \hat{U}_0(t)= \, & {} e^{-i\hat{H}_0t},\nonumber \\= \, & {} e^{-i\Omega _0t(\hat{n}+\frac{1}{2})-it\chi \hat{n}^2}. \end{aligned}$$
(47)

The time-dependent part \(\hat{U}_I(t)\) fulfills

$$\begin{aligned} i\frac{d\hat{U}_I(t)}{dt}=\hat{V}_I(t)\,\hat{U}_I(t), \end{aligned}$$
(48)

where

$$\begin{aligned} \hat{V}_I(t)= \, & {} \hat{U}_0^\dag (t)\hat{V}\hat{U}_0(t),\nonumber \\= \, & {} \frac{e(t)}{\sqrt{2\Omega _0}}(e^{-i\Omega _0t-i\chi t(2\hat{n}+1)}\hat{a}+\hat{a}^\dag e^{i\Omega _0t+i\chi t(2\hat{n}+1)}),\nonumber \\= \, & {} g(t)\hat{a}+\hat{a}^\dag \bar{g}(t), \end{aligned}$$
(49)

and

$$\begin{aligned} g(t)=\frac{e(t)}{\sqrt{2\Omega _0}}\,e^{-it(\Omega _0+\chi )}e^{|\alpha |^2(e^{-2i\chi t}-1)}. \end{aligned}$$
(50)

In deriving Eq. (49), to simplify the calculations, we assumed that the system is initially prepared in a coherent state \(|\alpha \rangle \), and replaced the nonlinear term \(e^{[\pm i\Omega (\hat{n})t]}\) by its average value \(\langle \alpha |e^{[\pm i\Omega (\hat{n})t]}|\alpha \rangle \), see31,32,33,34,35.

To find \(\hat{U}_I(t)\), we assume

$$\begin{aligned} \hat{U}_I(t)=\prod _{n=1}^{3}\exp (X_n(t)\hat{\gamma }_n), \end{aligned}$$
(51)

where we have defined \(\hat{\gamma }_1=1, \hat{\gamma }_2=\hat{a}, \hat{\gamma }_3=\hat{a}^\dag \) and the initial conditions are \(X_1(0)=X_2(0)=X_3(0)=0\). By inserting Eq. (51) into Eq. (48), we find36,37

$$\begin{aligned} \dot{X}_1(t)= \, & {} \dot{X}_3(t){X}_2(t), \nonumber \\ \dot{X}_2(t)= \, & {} -i\bar{g}(t), \nonumber \\ \dot{X}_3(t)= \, & {} -i g(t). \end{aligned}$$
(52)

The Eq. (52) can be solved either analytically or numerically and from now on we assume that the solutions are known functions. Therefore, if we denote the temporal evolution of the initial state \(|\alpha \rangle \) by \(|\alpha ,t\rangle \), we have

$$\begin{aligned} |\psi _\alpha ,t\rangle= \, & {} \hat{U}_0(t)\hat{U}_I(t)|\alpha \rangle ,\nonumber \\= \, & {} G_\alpha \sum _{n=0}^{\infty }e^{-i\Omega _0tn-itn^2\chi }\frac{(X_2(t)+\alpha )^n}{\sqrt{n!}}|n\rangle . \end{aligned}$$
(53)

where \(G_\alpha =e^{X_3(t)\alpha +X_1(t)-i\frac{\Omega _0t}{2}-\frac{|\alpha |^2}{2}}\), and from the normalization condition of the wave function \(\langle \psi _\alpha ,t |\psi _\alpha ,t\rangle =1\), we find \(|G_\alpha |^2=\exp (-|(X_2(t)+\alpha )|^2)\). For convenience, let us define the dimensionless parameters \(\xi =\chi t\) and \(X_2(t)+\alpha =\eta _t\), then Eq. (53) can be rewritten as

$$\begin{aligned} |\psi _\alpha ,t\rangle= \, & {} e^{\frac{-|\eta _t|^2}{2}}e^{-i\xi \hat{n}^2}\sum _{n=0}^{\infty }\frac{(e^{-i\Omega _0t}\eta _t)^n}{\sqrt{n!}}|n\rangle ,\nonumber \\= \, & {} e^{-i\xi \hat{n}^2}|e^{-i\Omega _0t}\eta _t\rangle ,\nonumber \\= \, & {} |e^{-i\Omega _0 t}\eta _t\rangle _\xi , \end{aligned}$$
(54)

where we have defined \(|\zeta \rangle _\xi =e^{-i\xi \hat{n}^2}|\zeta \rangle \) for an arbitrary coherent state \(|\zeta \rangle \). Therefore, the evolved state \(|\psi _\alpha ,t\rangle \) is of the kind \(|\beta \rangle _{\xi }\) where \(\beta =e^{-i\Omega _0 t}\eta _t\). In the next section we will study the properties of these states.

Properties of the states \(|\beta \rangle _{\xi }\)

Let us define a new set of ladder operators as

$$\begin{aligned}{} & {} \hat{B}=\hat{a}\,f(\hat{n})=f(\hat{n}+1)\,\hat{a},\nonumber \\{} & {} \hat{B}^\dag =f^\dag (\hat{n})\,\hat{a}^\dag =\hat{a}^\dag \,f^\dag (\hat{n}+1), \end{aligned}$$
(55)

where the function \(f(\hat{n})\) is defined by

$$\begin{aligned} f(\hat{n})=e^{i \xi (2 \hat{n}-1)}. \end{aligned}$$
(56)

The operators \(\hat{B}\), \(\hat{B}^\dag \) and \(\hat{n}\), fulfil the usual Heisenberg algebra

$$\begin{aligned}{} & {} [\hat{B},\hat{B}^\dag ]=1,\nonumber \\{} & {} [\hat{n},\hat{B}]=-\hat{B},\nonumber \\{} & {} [\hat{n},\hat{B}^\dag ]=\hat{B}^\dag . \end{aligned}$$
(57)

The state \(|\beta \rangle _{\xi }\) can be expanded in number states basis as

$$\begin{aligned} |\beta \rangle _{\xi }= \, & {} e^{-i\xi \hat{n}^2}\left( e^{-\frac{|\beta |^2}{2}}\sum _{n=0}^\infty \frac{\beta ^n}{\sqrt{n!}}\,|n\rangle \right) ,\nonumber \\= \, & {} e^{-\frac{|\beta |^2}{2}}\sum _{n=0}^\infty \frac{\beta ^n}{\sqrt{n!}}e^{-i\xi n^2}\,|n\rangle ,\nonumber \\= \, & {} e^{-\frac{|\beta |^2}{2}}\sum _{n=0}^\infty \frac{\beta ^n}{\sqrt{n!}}\frac{1}{[f(n)]!}\,|n\rangle , \end{aligned}$$
(58)

where \([f(n)]!=\displaystyle \Pi _{k=1}^n f(n)\), and \([f(0)]!=1\). One can easily show that the state \(|\beta \rangle _{\xi }\) is a coherent state for the new annihilation operator \(\hat{B}\) with eigenvalue \(\beta \)

$$\begin{aligned} \hat{B} |\beta \rangle _{\xi }=\beta \,|\beta \rangle _{\xi }. \end{aligned}$$
(59)

If we define the modified displacement operator \(\hat{D}_B (\beta )=e^{\beta \,\hat{B}^\dag -\bar{\beta }\,\hat{B}}\), then \(|\beta \rangle _{\xi }=\hat{D}_B |0\rangle \), note that \(|0\rangle _{\xi }=|0\rangle \), and the parameter \(\xi =\chi t\) is hidden in the definition of \(\hat{B}\) and \(\hat{B}^\dag \).

The state \(|\beta \rangle _{\xi }\) can also be considered as a Kerr state if we consider the Hamiltonian

$$\begin{aligned} \hat{H}= \, & {} \hbar \chi \,\hat{a}^\dag \hat{a}+\hbar \chi \,\hat{a}^\dag \hat{a}^\dag \hat{a}\hat{a},\nonumber \\= \, & {} \hbar \chi \hat{n}^2, \end{aligned}$$
(60)

with the corresponding time-evolution operator

$$\begin{aligned} \hat{U} (t)=e^{-it\chi \,\hat{n}^2}=e^{-i\xi \hat{n}^2}. \end{aligned}$$
(61)

If the system is initially prepared in the coherent state \(|\beta \rangle \), then the evolved state is the state \(|\beta \rangle _{\xi }\) given by

$$\begin{aligned} \hat{U}(t) |\beta \rangle= \, & {} e^{-i\xi \hat{n}^2}|\beta \rangle =|\beta \rangle _{\xi },\nonumber \\= \, & {} e^{-\frac{|\beta |^2}{2}}\sum _{n=0}^\infty \frac{\beta ^n}{\sqrt{n!}} e^{-i\xi n^2} |n\rangle . \end{aligned}$$
(62)

The probability of having n excitation in the evolved state \(|\beta \rangle _{\xi }\) is a Poissonian distribution

$$\begin{aligned} P_{\xi }(n)=|\langle n|\beta \rangle _{\xi }|^2=e^{-|\beta |^2}\frac{|\beta |^{2n}}{n!}. \end{aligned}$$
(63)

To study the squeezing effects, let us find the normalized variances of the position \(\hat{x}=(\hat{a}+\hat{a}^\dag )/2\) and momentum \(\hat{p}= (\hat{a}-\hat{a}^\dag )/2i \) defined by \((\triangle q)_\xi / (\triangle q)_{\xi =0}\) and \((\triangle p)_\xi / (\triangle p)_{\xi =0}\), respectively. We have

$$\begin{aligned}{} & {} \frac{(\Delta q)_\xi }{(\Delta q)_{\xi =0}} \nonumber \\{} & {} = \sqrt{2|\beta |^2+1+2|\beta |^2\,e^{-2|\beta |^2\sin ^2(2\xi )}\cos (2\phi -4\xi -|\beta |^2\sin (4\xi ))-4|\beta |^2\,e^{-4|\beta |^2\sin ^2(\xi )}\cos ^2 (\phi -\xi -|\beta |^2\sin (2\xi ))},\nonumber \\{} & {} \frac{(\Delta p)_\xi }{(\Delta p)_{\xi =0}} \nonumber \\{} & {} = \sqrt{2|\beta |^2+1-2|\beta |^2\,e^{-2|\beta |^2\sin ^2(2\xi )}\cos (-2\phi +4\xi +|\beta |^2\sin (4\xi ))-4|\beta |^2\,e^{-4|\beta |^2\sin ^2(\xi )}\sin ^2 (\phi -|\beta |^2\sin (2\xi ))}. \end{aligned}$$
(64)

where \(\beta =|\beta |e^{i\phi }\) and \((\Delta q)_{\xi =0}=1/\sqrt{2\Omega _0}\) and \((\Delta p)_{\xi =0}=\sqrt{\Omega _0/2}\). Note that \(|\beta \rangle _{\xi =0}=|\beta \rangle \) ia a coherent state for \(\hat{a}\) leading to a minimal uncertainty. In Fig. 1, the variances are depicted for \(\beta =0.5\).

Figure 1
figure 1

The normalized variances of position (left) and momentum (right) depicted for \(\beta =0.5\) in dimensionless parameter \(\xi =\chi t\). Squeezing occurs only for position in time intervals where \(\frac{(\Delta q)_\xi }{(\Delta q)_{\xi =0}}<1\).

Full size image

In the next section, we will focus on statistical properties of the state \(|\beta \rangle _{\xi }\) using (quasi)probability distribution functions.

Phase-space (quasi)probability distributions

Mandel Q parameter

The Mandel Q parameter measures the deviation of the occupation number distribution from Poissonian statistics. The quantum state \(|\psi \rangle \) has a sub-Poissonian, Poissonian or super-Poissonian statistics if \(Q< 0\), \(Q=0\) or \(Q>0\), respectively. The Mandel Q parameter is defined by35

$$\begin{aligned} Q_M= \, & {} \frac{\langle (\Delta \hat{n})^2\rangle -\langle \hat{n}\rangle }{\langle \hat{n}\rangle },\nonumber \\= \, & {} \frac{\langle \hat{n}^2\rangle -\langle \hat{n}\rangle ^2}{\langle \hat{n}\rangle }-1,\nonumber \\= \, & {} \langle \hat{n}\rangle (g^{(2)}(0)-1), \end{aligned}$$
(65)

where \(\hat{n}\) is the photon number operator and \(g^{(2)}(0)\) is the normalized second-order correlation function.

For the state \(|\beta \rangle _{\xi }\) we have

$$\begin{aligned} \langle \hat{n}^2\rangle _{\xi }= \, & {} \langle |\beta |e^{i\xi \hat{n}^2}\,\hat{n}^2\,e^{-i\xi \hat{n}^2}|\beta \rangle =\langle \beta |\hat{n}^2|\beta \rangle ,\nonumber \\= \, & {} |\beta |^4+|\beta |^2,\nonumber \\ \langle \hat{n}\rangle _{\xi }= \, & {} \langle \beta |\hat{n}|\beta \rangle =|\beta |^2, \end{aligned}$$
(66)

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 state38, 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 \(|\psi _\alpha ,t\rangle \) is (see Eq. 54)

$$\begin{aligned} F(t)= \, & {} \langle \psi _\alpha ,0|\psi _\alpha ,t\rangle ,\nonumber \\= \, & {} \langle \alpha |\psi _\alpha ,t\rangle ,\nonumber \\= \, & {} e^{-\frac{(|\alpha |^2+|\eta _t|^2)}{2}}\sum _{n=0}^{\infty } e^{-i\chi t n^2}\frac{(e^{-i\Omega _0t}\bar{\alpha }\eta _t)^n}{n!}. \end{aligned}$$
(67)
Figure 2
figure 2

The function \(|F(t)|^2\) in the presence of external force \(e(y)=\cos (y),\,\,\,(y=\Omega _0 t)\) for the parameters: \(\Omega _0=1\), \(\alpha = 3\), and Kerr parameters \(\chi =0\), \(\chi =0.25\) and \(\chi =1\).

Full size image

In Fig. 2, the function \(|F(t)|^2\) has been depicted in the presence of the external source \(e(\tau )=\cos (\tau )\) for different values of the Kerr parameter \(\chi \). In the absence of the Kerr nonlinearity (\(\chi =0\)), we have a driven oscillator, and in this case \(|F(t)|^2\) is decreasing in time without a considerable revival. In the presence of Kerr nonlinearity (\(\chi =0.25\), \(\chi =1\)), there are periodic fractional and complete revivals with a period decreasing with increasing the Kerr parameter \(\chi \).

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 information39. The Husimi function is defined by40

$$\begin{aligned} Q(\gamma ,t)=\frac{1}{\pi }\langle \gamma |\hat{\rho }(t)|\gamma \rangle . \end{aligned}$$
(68)

Having the Husimi Q-function, we can obtain the expectation value of an arbitrary observable \(A(\hat{a},\hat{a}^\dag )\) as

$$\begin{aligned} \langle A(\hat{a},\hat{a}^\dag )\rangle _t=\int d^2 \alpha \,Q(\alpha ,t)\,A(\alpha ,\alpha ^*), \end{aligned}$$
(69)

where \(A(\hat{a},\hat{a}^\dag )\) is anti-normally ordered

$$\begin{aligned} A(\hat{a},\hat{a}^\dag )=\sum _{n,m} c_{nm}\,\hat{a}^n\hat{a}^{\dag m}. \end{aligned}$$
(70)

For the pure state \(\hat{\rho }(t)=|\psi _\alpha ,t\rangle \langle \psi _\alpha ,t|\), (see Eq. 54), we have

$$\begin{aligned} Q(\gamma ,t)= \, & {} \frac{1}{\pi }\langle \gamma |\psi _\alpha ,t\rangle \langle \psi _\alpha ,t|\gamma \rangle ,\nonumber \\= \, & {} \frac{1}{\pi }\,e^{-(|\gamma |^2+|\eta _t|^2)}\left| \sum _{n=0}^\infty \frac{(\bar{\gamma }e^{-i\Omega _0 t}\eta _t)^n\,e^{-i\chi t n^2}}{n!}\right| ^2. \end{aligned}$$
(71)
Figure 3
figure 3

The Husimi distribution function depicted in \(x=\text{ Re }[\gamma ]\) and \(y=\text{ Im }[\gamma ]\) at the scaled times (\(\tau =\Omega _0 t=0, \pi /4, \pi , 2\pi , 4\pi , 8\pi \)) for the values: \(\Omega _0=1\), \(\chi =0.25\), \(\alpha =3\) and \(e(\tau )=\cos (\tau )\).

Full size image

In Fig. 3, the Husimi distribution function is depicted in terms of the variables \(x=\text{ Re }[\gamma ]\) and \(y=\text{ Im }[\gamma ]\) at six different times (\(\tau =\Omega _0 t=0, \pi /4, \pi , 2\pi , 4\pi , 8\pi \)). By using the definition \(X_2(t)+\alpha =\eta _t\), at \(t=0\), we have \(\eta _0=\alpha \), and the Husimi distribution function represents a coherent state (\(Q(\gamma ,0)=e^{-|\gamma -\alpha |^2}\)) with a Gaussian distribution. At the time \(\tau =\pi \), Husimi distribution has four small picks, at \(\tau =2\pi \), there are two picks, at \(\tau =8\pi /2\), the single peak is revived but displaced in phase space, and finally, at the revival time \(T_{rev}=8\pi \), the distribution is exactly revived.

Conclusions

We found that the time-evolution operator of a parametric oscillator with both mass and frequency time-dependent can be obtained from the time-evolution operator of another parametric oscillator with a constant mass but time-dependent frequency followed by a time-transformation \(t\rightarrow \int _0^t dt'\,1/m(t')\). We considered a driven parametric oscillator in the absence of the Kerr parameter (\(\chi =0\)), and by making use of a new set of ladder operators (\(\hat{A}_t^\dag , \hat{A}_t\)), we found the eigenvalues and eigenfunctions of the corresponding Hamiltonian \(\hat{H}_f (t)\). The eigenfunctions were obtained from free eigenfunctions \(\psi _n^0 (q,t)\) by shifting \(q\rightarrow q-\lambda _t\sqrt{2/\Omega (t)}\), that is \(\psi _n^f (q,t)=\psi _n^0 (q-\lambda _t\sqrt{2/\Omega (t)},t)\). By setting the confinement parameter \(k=0\), we investigated the Hamiltonian \(\hat{H}_{k=0} (t)\) perturbatively, considering the Kerr parameter \(\chi \) as the perturbation parameter. Also, by linearizing the Hamiltonian, we obtained approximate solutions for the evolved ladder operators given in Eq. (35). We also studied the (quasi) probability distribution functions on phase space for the Hamiltonian \(\hat{H}_{k=0} (t)\). The Kerr states \(|\beta \rangle _{\xi }\) and their relation to deformed coherent states were introduced. The photon distribution in the Kerr state \(|\beta \rangle _\xi \) was Poissonian and the Mandel Q parameter for this state was zero since the Mandel Q parameter is not sensitive to the phase of a state. The normalized variances of the position \(\hat{x}=(\hat{a}+\hat{a}^\dag )/2\) and momentum \(\hat{p}= (\hat{a}-\hat{a}^\dag )/2i \) were obtained and squeezing occurred only for position in the periodic short time-intervals. In the following, we found the Husimi distribution function. The Husimi distribution evolved from a single-pick state (coherent state) to a four-picks state at the scaled time \(\tau =\pi \), and to a two-picks state at \(\tau =2\pi \), and finally, after the revival time \(T_{rev}=8\pi \), the distribution was revived.