Nonlinear quantum Rabi model in trapped ions

We study the nonlinear dynamics of trapped-ion models far away from the Lamb-Dicke regime. This nonlinearity induces a blockade on the propagation of quantum information along the Hilbert space of the Jaynes-Cummings and quantum Rabi models. We propose to use this blockade as a resource for the dissipative generation of high-number Fock states. Also, we compare the linear and nonlinear cases of the quantum Rabi model in the ultrastrong and deep strong-coupling regimes. Moreover, we propose a scheme to simulate the nonlinear quantum Rabi model in all coupling regimes. This can be done via off-resonant nonlinear red- and blue-sideband interactions in a single trapped ion, yielding applications as a dynamical quantum filter.

Figure 1

(a) Logarithm of the absolute value of the operator $f_1\left(\widehat{n}\right)$ evaluated for different Fock states $|n\rangle$ and LD parameters $\eta$. Dark (blue) regions represent cases where $f_1\left(\widehat{n}\right)|n\rangle \approx 0$. (b) Nonlinear function $f_1\left(n\right)$ for a fixed value of the LD parameter $\eta =0.5$ (oscillating blue curve). Zero value (horizontal orange line).

Figure 2

Average value of ${\sigma }_z$ operator versus time for a coherent initial state $|\alpha =\sqrt{30}\rangle$ after (a) linear JC and (b) nonlinear JC evolution, both with the same coupling strength $g$ and $\eta =0.5$ for the nonlinear case. As shown in (a), there exists an approximate collapse and subsequent revival in the JCM dynamics, while for the nonlinear JCM this is not the case.

Figure 3

(a) The nonlinear function $f_1$ evaluated at different Fock states $n$ for the case of $\eta =0.4518$ (decreasing blue curve). Zero value (horizontal orange line). For this value of the LD parameter, $f_1|17\rangle =0$. (b) Phonon statistics of the initial thermal state with $\langle n\rangle =1$. (c) Time evolution of the average value of the number operator $\widehat{n}$ starting from the state in (b) and following the evolution for the preparation of Fock state $|17\rangle$, that is, during a nonlinear anti-JCM with spontaneous decay of the two-level system. (d) Phonon statistics at the end of the protocol, $t=100\times 2\pi /g$, with all the population concentrated in Fock state $|17\rangle$.

Figure 4

(a) Fidelity with respect to the initial state $P\left(t\right)=|\langle {\psi }_0|\psi \left(t\right)\rangle {|}^2$ versus time. As an initial state we choose $|0,\mathrm{g}\rangle$ and the evolution occurs under the NQRM with LD parameter $\eta =0.67898$, where $f_1|7\rangle =0,g/{\omega }^{\mathrm{R}}=4$, and ${\omega }_0^{\mathrm{R}}=0$. (b) Phonon statistics at different times for the NQRM evolved from the initial state $|0,\mathrm{g}\rangle$. The propagation of population through Fock states stops at $|n=7\rangle$, with Fock states of $n>7$ never getting populated.

Figure 5

(a) Overlap of the instantaneous state with the initial state $P\left(t\right)=|\langle {\psi }_0|\psi \left(t\right)\rangle {|}^2$ versus time for a coherent initial state $|\alpha =1,\mathrm{g}\rangle$ evolving under the linear QRM. Collapses and revivals are observed, as expected in the DSC regime of the linear QRM. (b) Phonon statistics at different times, where we see the round trip of a phonon-number wave packet.

Figure 6

(a) Overlap with the initial state $P\left(t\right)=|\langle {\psi }_0|\psi \left(t\right)\rangle {|}^2$ versus time for initial state $|\alpha =1,\mathrm{g}\rangle$ evolving under the NQRM with LD parameter $\eta =0.57838$, where $f_1|10\rangle =0,g/{\omega }^{\mathrm{R}}=3.7$, and ${\omega }_0^{\mathrm{R}}=0$. (b) Phonon statistics at different times for the NQRM evolved from the initial state $|\alpha =1,\mathrm{g}\rangle$. The Fock state $|10\rangle$ is never surpassed because $f_1|10\rangle =0$.