Time-dependent oscillator with Kronig-Penney excitation

TL;DR

This paper provides exact solutions to the time-dependent Schrödinger equation for a quantum oscillator driven by periodic delta-kicks, revealing its squeezed state behavior and quantifying energy growth using Chebyshev polynomials.

Contribution

It introduces an exact analytical approach to solve the quantum oscillator with periodic delta-kick excitation and characterizes its squeezing and energy dynamics.

Findings

Oscillator enters a squeezed state under periodic delta-kicks

Squeezing coefficients are explicitly calculated

Energy increase rate is derived using Chebyshev polynomials

Abstract

Exact solutions of the time-dependent Schrodinger equation for a quantum oscillator subject to periodical frequency delta-kicks are obtained. We show that the oscillator occurs in the squeezed state and calculate the corresponding squeezing coefficients and the energy increase rate in terms of Chebyshev polynomials.

Equations

$$\dot{ϵ}(t)+{\omega }^2(t)ϵ(t)=0.$$

$${\omega }^2(t)={\omega }_0^2-2\kappa \sum _{k=0}^{n-1}\delta (t-kt),$$

$$-\frac{\ddot{ϵ}(t)}{2}+\kappa \sum _{k=0}^{N-1}\delta (t-k\tau )ϵ(t)=\frac{1}{2}{\omega }_0^2ϵ(t).$$

$${ϵ}_k(t)=A_k{e}^{i{\omega }_{0}t}+B_k{e}^{-i{\omega }_{0}t},k=0,...,N.$$

$${ϵ}_{k-1}(k\tau )={ϵ}_k(k\tau ),-\frac{1}{2}\left[{\dot{ϵ}}_k(k\tau )-{\dot{ϵ}}_{k-1}(k\tau )\right]+\kappa {ϵ}_{k-1}(k\tau )=0.$$

$$\pmatrix{A_{k}e^{i\omega_{0}t_{k}}\cr B_{k}e^{-i\omega_{0}t_{k}}}=\pmatrix{1-\frac{i\kappa}{\omega_{0}}&-\frac{i\kappa}{\omega_{0}}\cr\frac{i\kappa}{\omega_{0}}&1+\frac{i\kappa}{\omega_{0}}}\pmatrix{A_{k-1}e^{i\omega_{0}t_{k}}\cr B_{k-1}e^{-i\omega_{0}t_{k}}}.$$

$$\text{pmatrix}{A}_n\text{cr}{B}_n=S^{(n)}\text{pmatrix}{A}_0\text{cr}{B}_0=T^{-(n-1)}(MT{)}^n\text{pmatrix}{A}_0\text{cr}{B}_0,$$

$$M=\pmatrix{1-\frac{i\kappa}{\omega_{0}}&-\frac{i\kappa}{\omega_{0}}\cr\frac{i\kappa}{\omega_{0}}&1+\frac{i\kappa}{\omega_{0}}},\quad T=\pmatrix{e^{i\omega_{0}\tau}&0\cr 0&e^{-i\omega_{0}\tau}}.$$

$$S^{(1)}\equiv S=MT=\pmatrix{\Big{(}1-\frac{i\kappa}{\omega_{0}}\Big{)}e^{i\omega_{0}\tau}&-\frac{i\kappa}{\omega_{0}}e^{-i\omega_{0}\tau}\cr\frac{i\kappa}{\omega_{0}}e^{i\omega_{0}\tau}&\Big{(}1+\frac{i\kappa}{\omega_{0}}\Big{)}e^{-i\omega_{0}\tau}}$$

$$S^n=U_{n-1}(\cos \varphi )S-U_{n-2}(\cos \varphi )E,\cos \varphi =\frac{1}{2}\text{mbox}TrS=\frac{\chi }{2}.$$

$$U_n(\cos \varphi )=\frac{\sin \left[\left(n+1\right)\varphi \right]}{\sin \varphi },U_n(\chi )=\frac{{\left[\chi +{\left({\chi }^2-4\right)}^{\frac{1}{2}}\right]}^{n+1}-{\left[\chi -{\left({\chi }^2-4\right)}^{\frac{1}{2}}\right]}^{n+1}}{2^{n+1}{\left({\chi }^2-4\right)}^{\frac{1}{2}}}.$$

$$\frac{\chi }{2}=\cos \varphi =\cos {\omega }_0\tau +\frac{\kappa }{{\omega }_0}\sin {\omega }_0\tau .$$

$$S_{11}^{(n)}=\Big{(}1-\frac{i\kappa}{\omega_{0}}\Big{)}U_{n-1}\big{(}\frac{\chi}{2}\big{)}e^{-i\omega_{0}(n-2)\tau}-U_{n-2}\big{(}\frac{\chi}{2}\big{)}e^{-i\omega_{0}(n-1)\tau},$$

$$S_{12}^{(n)}=-\frac{i\kappa}{\omega_{0}}U_{n-1}\big{(}\frac{\chi}{2}\big{)}e^{-i\omega_{0}n\tau},\quad S_{21}^{(n)}=\frac{i\kappa}{\omega_{0}}U_{n-1}\big{(}\frac{\chi}{2}\big{)}e^{i\omega_{0}n\tau},$$

$$S_{22}^{(n)}=\Big{(}1+\frac{i\kappa}{\omega_{0}}\Big{)}U_{n-1}\big{(}\frac{\chi}{2}\big{)}e^{i\omega_{0}(n-2)\tau}-U_{n-2}\big{(}\frac{\chi}{2}\big{)}e^{i\omega_{0}(n-1)\tau}.$$

$$\displaystyle\sigma_{xx}(t)=\frac{\hbar}{2m\omega_{0}}\Big{(}U_{n-1}^{2}+U_{n-2}^{2}+\frac{2\kappa}{\omega_{0}}U_{n-1}^{2}\sin[2\omega_{0}(t-(n-1)\tau)]-\chi U_{n-1}U_{n-2}\Big{.}$$

$$\displaystyle+\Big{.}\frac{4\kappa^{2}}{\omega_{0}^{2}}U_{n-1}^{2}\sin^{2}[\omega_{0}(t-(n-1)\tau)]-\frac{2\kappa}{\omega_{0}}U_{n-1}U_{n-2}\sin[2\omega_{0}(t-(n-\frac{1}{2})\tau)]\Big{)}.$$

$$|{\sigma }_{xp}(t)=\frac{1}{2}\hslash {\left(1-|ϵ\dot{ϵ}{|}^2/{\omega }^2\right)}^{1/2}.$$

$$\displaystyle\sigma_{xp}=\frac{\hbar}{2}\Big{\{}\Big{[}\big{(}1+2\frac{\kappa^{2}}{\omega_{0}^{2}}\big{)}U_{n-1}^{2}+U_{n-2}^{2}-\chi U_{n-1}U_{n-2}\Big{]}^{2}-\Big{[}\frac{2\kappa}{\omega_{0}}U_{n-1}^{2}\sin[2\omega_{0}(t-(n-1)\tau)]\Big{.}\Big{.}$$

$$\displaystyle\Big{.}\Big{.}-\frac{2\kappa^{2}}{\omega_{0}^{2}}\cos[2\omega_{0}(t-(n-1)\tau)]-\frac{2\kappa}{\omega_{0}}U_{n-1}U_{n-2}\sin[2\omega_{0}(t-(n-\frac{1}{2})\tau)]\Big{]}^{2}-1\Big{\}}^{\frac{1}{2}}.$$

$$W=(m\omega_{0}/\hbar)\sigma_{xx}(t)+\sigma_{pp}(t)/\hbar m\omega_{0}=U_{n-1}^{2}\big{(}\frac{\chi}{2}\big{)}+U_{n-2}^{2}\big{(}\frac{\chi}{2}\big{)}+\frac{4\kappa^{2}}{\omega_{0}^{2}}U_{n-1}^{2}\big{(}\frac{\chi}{2}\big{)}-\chi U_{n-1}\big{(}\frac{\chi}{2}\big{)}U_{n-2}\big{(}\frac{\chi}{2}\big{)}.$$

$$r_{max}=(1-1/W^2{)}^{1/2},k_{max}^2=\frac{1+(1-1/W^2{)}^{1/2}}{1-(1-1/W^2{)}^{1/2}}.$$

$$U_{n}(x)=\frac{\sqrt{n}}{2\sqrt{n-1}}\big{(}x^{2}-1\big{)}^{-\frac{1}{2}}\Big{[}\sqrt{x+1}+\sqrt{x-1}\Big{]}\Big{[}x+\sqrt{x^{2}-1}\Big{]}^{n-\frac{1}{2}},$$

$$W=\frac{4\kappa^{2}\big{[}x+\big{(}x^{2}-1\big{)}^{\frac{1}{2}}\big{]}^{2n}}{\omega_{0}^{2}\big{(}x^{2}-1\big{)}},\quad x=\chi/2.$$

$$W=\frac{4\kappa^{2}\big{[}\frac{\kappa}{\omega_{0}}+\big{(}\frac{\kappa^{2}}{\omega_{0}^{2}}-1\big{)}^{\frac{1}{2}}\big{]}^{2n}}{\omega_{0}^{2}\big{(}\frac{\kappa^{2}}{\omega_{0}^{2}}-1\big{)}}.$$

$$W=4\big{(}2\kappa/\omega_{0}\big{)}^{2n},\quad n>>1.$$

$$W=\big{(}1+\frac{4\kappa^{2}n^{2}}{\omega_{0}^{2}}\big{)}.$$

$$W_{max}<\Big{(}1+\frac{4n^{2}\kappa^{2}}{\omega_{0}^{2}}\Big{)}.$$

Full text

The problem of quantum oscillator with a time-dependent frequency was solved in refs. [1, 2, 3]. It was shown that the wave function and, consequently, all physical characteristics of the oscillator can be expressed in terms of the solution of the classical equation of motion

[TABLE]

(for detailed review see ref.[4]). The only remaining problem is to find explicit expression for the $\epsilon$-function. A lot of papers were devoted to that in the last two decades. One can find an extensive list of references, e.g., in ref. [4]. Here we consider the interesting case of periodically kicked oscillator, where the frequency depends on time as follows

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where $\omega_{0}$ is constant frequency, $\delta$ - is the Dirac delta-function. So we have the equation for $\epsilon(t)$-function

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Obviously, upon the substitutions $t\rightarrow x,\,\epsilon\rightarrow\psi,\,\omega_{0}^{2}\rightarrow E$, this equation (3) coincides with the equation for the wave function of a quantum particle of unity mass in a Kronig-Penney potential (a sequence of delta-potentials) [5]-[7]. The case $\kappa>0$ corresponds to sequence of $\delta$-barrier, while the case $\kappa<0$ relates to sequence of $\delta$-wells. For every time interval $(k-1)\tau<t<k\tau$ the solution for (1,2) is given by

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Due to continuity conditions

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(the latter is obtained by integrating eq. (1) over the infinitely small interval $k\tau-0<t<k\tau+0$, the coefficients $A_{k}$ and $B_{k}$ must satisfy some relations which can be represented in matrix form

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After a sequence of $\delta$-kicks the coefficients $A_{n},\,B_{n}$ are connected with the initial ones $A_{0},\,B_{0}$ through the equation

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with the matrices $T,\,M$ given by

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The matrix

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is unimodular: $\mbox{det}S=1$. It can be shown (see, e.g., ref. [8]), that all the powers of two-dimensional unimodular matrices can be expressed as linear combinations of the matrices can be expressed as linear combinations of the matrices themselves and the identity matrix, the coefficients being Chebyshev polynomials of the second kind whose arguments are expressed in terms of the traces of the initial matrices ($E$ is the identity matrix)

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We use the following definitions of the Chebyshev polynomials [9]

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In the case under study

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Thus the matrix elements of the matrix $S^{(n)}$ can be written as follows

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[TABLE]

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If $\epsilon(-0)=1,\quad\dot{\epsilon}(-0)=i\omega_{0}$ at the initial time, then $A_{0}=1,\,B_{0}=0,$ so $A_{n}=S_{11}^{(n)},B_{n}=S_{21}^{(n)}.$ If at the initial time, $t<0$, a quantum oscillator was in a coherent state, a parametric excitation will transform it into a squeezed correlated state with the coordinate variance $\sigma_{xx}(t)=\frac{\hbar|\epsilon|^{2}}{2m\omega_{0}}$ [4]. So after a sequence of $\delta$-kicks one has

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The correlation between coordinate and momentum in a correlated squeezed state is not equal to zero [4]

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After a sequence of $\delta$-kicks it has the form

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The dimensionless energy of the quantum fluctuations $W=2E/\hbar\omega_{0}$ (normalized by the ground state energy $\hbar\omega_{0}/2$) after the $n$ kicks equals

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As was shown [10], the maximum coefficients of the squeezing and of the correlation are expressed through the dimensionless energy of the fluctuations as follows,

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In the case when the parameter $\chi/2$ does not belong to the interval $[-1,1]$, the asymptotic formula for the Chebyshev polynomials [9] with $n>>1$,

[TABLE]

gives rise to the asymptotic expression for the dimensionless energy of the fluctuations (we suppose $\chi>0$),

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If we choose the simplest case, when the interval between the kicks satisfies the condition $\omega_{0}\tau=\frac{\pi}{2}+2\pi k,\quad k=0,1,\,...\,$, the arguments of the Chebyshev polynomials are equal to $\kappa/\omega_{0}$. When the strength of the $delta$-kicks $\kappa$ is larger than the constant frequency $\omega_{0}$, one gets the asymptotic formula

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In the case when $\kappa/\omega_{0}>>1$, the energy of the fluctuations increases exponentially with the number of kicks

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If the $\delta$-kicks take place at moments given by $\omega_{0}\tau=2\pi m,\,m=0,1,\,...\,$, then the argument of the Chebyshev polynomials is equal to unity independently of the strength of the $\delta$-kicks. Then the energy increases much more slowly,

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In the case when the strength of the $\delta$-kicks is small, $\kappa/\omega_{0}<<1$, and the interval between the kicks is given by $\omega_{0}\tau=\frac{\pi}{2}+2\pi k,\quad k=0,1,\,...\,,$ the following inequality for the maximum total energy can be obtained,

[TABLE]