Squeezing in the quantum Rabi model with parametric nonlinearity

Abstract

The squeezing effect arising in the interacting qubit-oscillator system is studied with the presence of a parametric oscillator in the Rabi model. To solve the system, we employ the generalized rotating wave approximation which works well in the wide range of coupling strength as well as detuning. The analytically derived approximate energy spectrum portrays good agreement with the numerically determined spectrum of the Hamiltonian. For the initial state of the bipartite system, the dynamical evolution of the reduced density matrix corresponding to the oscillator is obtained by partial tracing over the qubit degree of freedom. The oscillator’s reduced density matrix yields the nonnegative phase space quasi-probability distribution known as Husimi Q-function which is utilized to compute the quadrature variance. It is noticed that the squeezing produced in the oscillator sector gets reduced with increasing coupling strength. We demonstrate the role of the parametric term to obtain adequate squeezing in the strong coupling regime. We also study the revival–collapse phenomenon and the generation of squeezed coherent state.

Appendices

Appendix A

The integrals employed to arrive at (28) are

$$\begin{aligned}&\int _{0}^{\infty } \mathrm {d}|\beta | \; |\beta | \exp \left( -|\beta |^{2}-\frac{\nu }{2\mu }(\beta ^{2}+\beta ^{*2}) \mp \frac{\eta }{\mu } (\beta + \beta ^{*}) \right) \mathrm {H}_{n}\left( i \frac{\beta _{\pm }^{*}}{\sqrt{2 \mu \nu }}\right) \mathrm {H}_{m}\left( -i \frac{\beta _{\pm }}{\sqrt{2 \mu \nu }}\right) \nonumber \\&\quad = \frac{\mu }{\tau _{_{\theta }}} (-1)^{m} \left( \pm i \frac{2 \eta (\mu -\nu )}{\sqrt{2\mu \nu }}\right) ^{n+m} \sum _{k=0}^{n} \sum _{\ell =0}^{m} \left( \pm \frac{\nu \sqrt{\mu \tau _{_{\theta }}}}{2\eta (\mu -\nu )}\right) ^{k+\ell } k! \ell ! \; \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}m\\ \ell \end{array}}\right) \nonumber \\&\qquad \times \exp (i\theta (k-\ell )) \sum _{p=\lfloor \frac{k+1}{2} \rfloor }^{n} \sum _{q=\lfloor \frac{\ell }{2} \rfloor }^{m} \left( \frac{2}{ \nu \tau _{_\theta }}\right) ^{p+q} \frac{(2p+2q+1-k-\ell )! }{p!q!} \left( {\begin{array}{c}p\\ k-p\end{array}}\right) \left( {\begin{array}{c}q\\ \ell -q\end{array}}\right) \nonumber \\&\qquad \times \exp \left( -2i\theta (p-q)\right) \mathrm {H}_{k+\ell -2p-2q-2} \left( \pm \frac{\eta \cos \theta }{\sqrt{\mu \tau _{_\theta }}} \right) . \end{aligned}$$

(A.1)

Appendix B

To obtain the weight function in (34), we make use of the following integrals

$$\begin{aligned}&\int \mathrm {d}^{2}\beta \; \beta ^{k} \; \exp \left( -|\beta |^{2}-\frac{\nu }{2\mu }(\beta ^{2}+\beta ^{*2}) \mp \frac{\eta }{\mu } (\beta + \beta ^{*}) \right) \mathrm {H}_{n}\left( i \frac{\beta _{\pm }^{*}}{\sqrt{2 \mu \nu }}\right) \mathrm {H}_{m}\left( -i \frac{\beta _{\pm }}{\sqrt{2 \mu \nu }}\right) \nonumber \\&\quad = \frac{\pi }{2}\, \mu \, i^{n} \, (-i)^{m} \, \sqrt{ n! m!} \left( \frac{2 \mu }{\nu }\right) ^{\frac{n+m}{2}} \exp \left( \frac{\eta ^{2}}{\mu } (\mu - \nu )\right) \left( G^{(k,+)}_{n,m} \pm G^{(k,-)}_{n,m} \right) . \end{aligned}$$

(B.1)

We also note that when $k=0$ in the above integrals, the finite summation series $G^{(k,\pm )}_{n,m}$ takes the values $G^{(0,+)}_{n,m} = 2 \; \delta _{n,m}$ and $G^{(0,-)}_{n,m} = 0$.

$$\begin{aligned} I_{n,m}^{(k,\ell )}\equiv & {} \int \mathrm {d}^{2}\beta \; \beta ^{k} \beta ^{*\ell } \exp \left( -|\beta |^{2}-\frac{\nu }{2\mu }(\beta ^{2}+\beta ^{*2}) \right) \mathrm {H}_{n}\left( i \frac{\beta ^{*}}{\sqrt{2 \mu \nu }}\right) \mathrm {H}_{m}\left( -i \frac{\beta }{\sqrt{2 \mu \nu }}\right) \nonumber \\= & {} \pi \; \mu \; i^{n}(-i)^{m} n! m! \left( \frac{\mu \nu }{2}\right) ^{\frac{k+\ell }{2}} \sum _{p=0}^{n} \sum _{q=0}^{m} \frac{1}{p!q!} \left( {\begin{array}{c}p\\ n-p\end{array}}\right) \left( {\begin{array}{c}q\\ m-q\end{array}}\right) \sum _{s=0}^{\min (\ell +2p-n,k+2q-m)} \left( \frac{2 \mu }{\nu } \right) ^{s} s! \quad \nonumber \\&\times \left( {\begin{array}{c}2p+\ell -n\\ s\end{array}}\right) \left( {\begin{array}{c}k+2q-m\\ s\end{array}}\right) \mathrm {H}_{k+2q-m-s}(0) \mathrm {H}_{\ell +2p-n-s}(0). \end{aligned}$$

(B.2)