Quantum simulation of multiphoton and nonlinear dissipative spin-boson models

We present a framework for the realization of dissipative evolutions of spin-boson models, including multiphoton exchange dynamics, as well as nonlinear transition rates. Our approach is based on the implementation of a generalized version of a dissipative linear quantum Rabi model. The latter comprises a linearly coupled spin-boson term, spin rotations, and standard dissipators. We provide numerical simulations of illustrative cases supporting the good performance of our method. Our work allows for the simulation of a large class of fundamentally different quantum models where the effect of distinct dissipative processes can be easily investigated.

Figure 1

Sketch of the scheme for the simulation of nonlinear, multiphoton, and dissipative spin-boson models. In (a) a linearly coupled spin-boson system (a generalized quantum Rabi model with spin bias ${\delta }_0$, spin drivings with amplitude ${\mathrm{\Omega }}_{j>0}$, and frequency ${\mathrm{\Delta }}_{j>0}$) with Hamiltonian $H_{\mathrm{G}}$ [cf. Eq. (1)] undergoes dissipation characterized by the rates ${\gamma }_k$ and the jump operators $F_k$. This allows us to simulate a nonlinear $n$-boson spin-boson model undergoing dissipation described by the ${\tilde{F}}_k$'s operators. In (b) the Hamiltonian is $H_{\mathrm{n}}$ [cf. Eq. (15)], which comprises interaction terms such as ${\sigma }^{+}{a}^n+\mathrm{H}.\mathrm{c}.$ and/or ${\sigma }^{+}{\left(a^{†}\right)}^n+\mathrm{H}.\mathrm{c}.$ In addition, the transition rates may strongly depend on the Fock-state label $m$ through a function $f_n\left(m\right)$ whose behavior is shown in (c) for different cases [cf. Sec. 2 and Eq. (20)]. Note that we show $f_n\left(m\right)=\left.〈m\right|f_n\left(a^{†}a\right)\left|m\right.〉$, rescaled by $f_n\left(0\right)$, for various choices of $n$ and $\eta$. While $f_n\left(m\right)\approx f_n\left(0\right)$ for small values of $\eta$ and $m,f_n\left(m\right)$ significantly depends on $m$ for larger values of $\eta$, so that Eq. (14) is no longer well approximated by Eq. (15) and, thus, a correct description requires its nonlinear counterpart, as described in Sec. 2a.

Figure 2

The dynamics induced by the nJCM embodied by the Hamiltonian $H_{\mathrm{nJCM}}$ in Eq. (33) is depicted by lines and compared to their simulation through $H_{\mathrm{G}}$ (shown by the points) as given in Sec. 3b1. In the plots reported in the left column we have taken $n=1$, while $n=2$ has been used for the right column. In (a) and (b) [(c) and (d)] we show the evolution of $〈{\sigma }_z〉$ ($〈a^{†}a〉$), while in (e) and (f) we plot the infidelity $1-F({\rho }_{\mathrm{nJCM}}\left(t\right),\mathrm{\Gamma }{\rho }_{\mathrm{G}}\left(t\right){\mathrm{\Gamma }}^{†})$ that quantifies the quality of the simulation obtained starting from ${\rho }_{\mathrm{G}}\left(t\right)$. The initial state ${\left|\psi \left(0\right)\right.〉}_{\mathrm{nJCM}}=\left|0\right.〉\left|e\right.〉$ evolves under $H_{\mathrm{nJCM}}$ without dissipation (solid red line), and with dissipators corresponding to the transformed boson leakage (dashed blue line) and spin dephasing (dotted green line). The parameters used in the simulations are $\tilde{\omega }=n\tilde{\nu },{\tilde{g}}_1=\tilde{\nu }/2,{\tilde{g}}_2=\tilde{\nu }/10,{\mathrm{\Omega }}_0=\nu /100,\tilde{\nu }=\nu /2000,{\gamma }_{\mathrm{bl}}=\tilde{\nu }/2$, and ${\gamma }_{\mathrm{sd}}=\tilde{\nu }/40$.

Figure 3

Dynamics of a nQRM, $H_{\mathrm{nQRM}}$ [see Eq. (34)] for $n=1$ (left) and $n=2$ (right column), depicted by lines and their simulation using $H_{\mathrm{G}}$ (points) as described in Sec. 3b2. In (a) and (b) [(c) and (d)] we show the evolution of the expectation value $〈{\sigma }_z〉$ ($〈a^{†}a〉$), while in (e) and (f) we plot the infidelity $1-F({\rho }_{\mathrm{nQRM}}\left(t\right),\mathrm{\Gamma }{\rho }_{\mathrm{G}}\left(t\right){\mathrm{\Gamma }}^{†})$ between the targeted state and its simulated counterpart. For the 1QRM (left column) we choose a pure initial state ${\left|\psi \left(0\right)\right.〉}_{1\mathrm{QRM}}=\left|\alpha =1/2\right.〉\left|e\right.〉$ where $\left|\alpha \right.〉$ represents a coherent state, while for the 2QRM the initial state reads as ${\rho }_{2\mathrm{QRM}}\left(0\right)={\rho }_b^{\mathrm{th}}\left(0.25\right)\otimes \left|+\right.〉\left.〈+\right|$ where ${\rho }_b^{\mathrm{th}}\left(〈a^{†}a〉\right)$ is a thermal state with $〈a^{†}a〉$ average number of bosons. Different styles correspond to distinct coupling constants (left) or dissipation rates (right), as indicated in (e) and (f), respectively. The parameters are $\tilde{\nu }=\nu /5000$ and $\mathrm{\Omega }=\nu /100$, while ${\gamma }_{\mathrm{bl}}=2{\gamma }_{\mathrm{sd}}=\tilde{\nu }/50$ and $\tilde{\omega }=0$ (1QRM) and ${\tilde{g}}_2=\tilde{\nu }/10$ with $\tilde{\omega }=2\tilde{\nu }$ (2QRM). See Sec. 3c for further details.

Figure 4

Dissipative dynamics towards the steady state in (a) $H_{\mathrm{G}}$ and (b) $H_{1\mathrm{aJCM},\eta }$ [see Eq. (33)], where $H_{\mathrm{G}}$ is such that allows for the realization of the nonlinear 1JCM (see Sec. 3b3), where $\tilde{\omega }=\tilde{\nu }={\mathrm{\Omega }}_0{f}_1\left(0\right)/4$ (which would be equivalent to ${\tilde{g}}_1=\tilde{\nu }/2$ for $H_{1\mathrm{aJCM}}$) and $\tilde{\nu }={10}^{-3}\nu$ with $\eta =0.8$. The initial state ${\rho }_{\mathrm{G}}\left(0\right)={\rho }_b^{\mathrm{th}}\left(0.75\right)\otimes \left|g\right.〉\left.〈g\right|$ for $H_{\mathrm{G}}$ evolves under boson damping with rate ${\gamma }_{\mathrm{bl}}=\tilde{\nu }/2$ for (a), and its transformed form for (b) [Eq. (30)]. The insets show the short-time dynamics. (c) Shows the time evolution of the purity for ${\rho }_{\mathrm{G}}\left(t\right)$ (solid lines) and ${\rho }_{1\mathrm{aJCM},\eta }$ (dashed lines) while in (d) we plot the infidelity between $\mathrm{\Gamma }{\rho }_{\mathrm{G}}\left(t\right){\mathrm{\Gamma }}^{†}$ and ${\rho }_{1\mathrm{aJCM}}$ (dashed lines) and ${\rho }_{1\mathrm{aJCM},\eta }$ (solid lines) for different rates, which for the latter may drop below our numerical precision ${10}^{-5}$. See Sec. 3c for further details and a discussion.

Figure 5

Dissipative preparation of a Fock state via $H_{1\mathrm{aJCM},\eta }$ and spontaneous emission ${\left|\psi \right.〉}_{1\mathrm{aJCM},\eta }^{ss}=\left|m_s\right.〉\left|g\right.〉$, which also corresponds to a pure state in the $H_{\mathrm{G}}$ counterpart but of the form ${\left|\psi \right.〉}_{\mathrm{G}}^{ss}=\mathcal{D}(i\eta /2)\left|m_s\right.〉\left|+\right.〉$. In (a) we show the dynamics toward the steady state of ${\rho }_{1\mathrm{aJCM},\eta }$ (solid lines) and its simulation from ${\rho }_{\mathrm{G}}$ (points), starting from ${\left|\psi \left(0\right)\right.〉}_{1\mathrm{aJCM},\eta }=\left|0\right.〉\left|g\right.〉$ and with $\tilde{\omega }=\tilde{\nu }={\mathrm{\Omega }}_0{f}_1\left(0\right)$ (which would be equivalent to ${\tilde{g}}_1=2\tilde{\nu }$ for $H_{1\mathrm{aJCM}}$) and ${\gamma }_{\mathrm{se}}=4\tilde{\nu }$. For $\eta \approx 0.64,f_1(m_s=8)$ vanishes, so that the state $\left|m_s=8\right.〉\left|g\right.〉$ becomes steady [see (b)]. (c), (d) Show the evolution of the purity (green), fidelity with respect to its steady and pure state ${\rho }^{ss}$ (red) for ${\rho }_{1\mathrm{aJCM},\eta }$ (lines) and ${\rho }_{\mathrm{G}}$ (points), and the infidelity between ${\rho }_{1\mathrm{aJCM},\eta }\left(t\right)$ and $\mathrm{\Gamma }{\rho }_G\left(t\right){\mathrm{\Gamma }}^{†}$.

Figure 6

Dissipative dynamics when ${\tilde{D}}_{\mathrm{sd}}\left[\rho \right]$ (top) and ${\tilde{D}}_{\mathrm{se}}\left[\rho \right]$ (bottom) can not be approximated by the simple equations (23) and (25), respectively. In (a) we show the time evolution of ${〈a^{†}a〉}_{1\mathrm{JCM},\eta }$ (lines) and its reconstruction using ${\rho }_{\mathrm{G}}$ (points), when ${\tilde{D}}_{\mathrm{sd}}\left[\rho \right]$ is taken as in Eq. (22) (solid lines) or approximated as in Eq. (23) (dashed lines), for different parameters, namely, $\eta =0.5$ with ${\gamma }_{\mathrm{sd}}={\gamma }_{\mathrm{bl}}/2=5\tilde{\nu }$ (blue) and $\eta =0.4$ and 0.8 with ${\gamma }_{\mathrm{sd}}=2{\gamma }_{\mathrm{bl}}=\tilde{\nu }$ (red and green). The rest of the parameters are $\tilde{\omega }=\tilde{\nu }=f_1\left(0\right){\mathrm{\Omega }}_0/2$ with $\tilde{\nu }={10}^{-2}\nu$, and initial state ${\left|\psi \left(0\right)\right.〉}_{\mathrm{G}}=\left|0\right.〉\left|+\right.〉$. With the same format, in (b) we show the infidelity $1-F$ between the ${\rho }_{\mathrm{G}}$ and ${\rho }_{1\mathrm{JCM},\eta }$ considering ${\tilde{D}}_{\mathrm{sd}}\left[\rho \right]$ as in Eq. (22) or (23). The same is plotted in the bottom panels (c) and (d), but with ${\tilde{D}}_{\mathrm{se}}\left[\rho \right]$ and ${\gamma }_{\mathrm{se}}$ instead of ${\tilde{D}}_{\mathrm{sd}}\left[\rho \right]$ and ${\gamma }_{\mathrm{sd}}$ and ${\left|\psi \left(0\right)\right.〉}_{\mathrm{G}}=\left|0\right.〉\left|e\right.〉$.