Cavity-QED simulation of qubit-oscillator dynamics in the ultrastrong-coupling regime

We propose a quantum simulation of a two-level atom coupled to a single mode of the electromagnetic field in the ultrastrong-coupling regime based upon resonant Raman transitions in an atom interacting with a high finesse optical cavity mode. We show by numerical simulation the possibility of realizing the scheme with a single rubidium atom, in which two hyperfine ground states make up the effective two-level system, and for cavity QED parameters that should be achievable with, for example, microtoroidal whispering-gallery-mode resonators. Our system also enables simulation of a generalized model in which a nonlinear coupling between the atomic inversion and the cavity photon number occurs on an equal footing with the (ultrastrong) dipole coupling and can give rise to critical-type behavior even at the single-atom level. Our model takes account of dissipation, and we pay particular attention to observables that would be readily observable in the output from the system.

Figure 1

(a) Energy-level diagram for the $D_1$ transitions (not to scale). The two laser fields (blue dashed lines) are ${\sigma }_{-}$ and ${\sigma }_{+}$ polarized, respectively. The cavity mode (red solid lines) is $\pi$ polarized and participates in resonant (or near-resonant) Raman transitions from both the $F=1$ and $F=2$ ground states. (b) Detailed view of the leftmost part of the energy-level diagram. Horizontal dashed lines indicate detunings of the fields from the excited states.

Figure 2

Effective parameters in units of $2\pi$ MHz: $g_{\mathrm{eff}}$ (blue solid line), ${\omega }_0$ (blue dashed line), $\omega$ (black dash-dotted line) and $U$ (black, dotted line). Top panel: ${\Delta }_1/\left(2\pi \right)=-26$ GHz, ${\Delta }_2/\left(2\pi \right)=-20$ GHz, while ${\Omega }_1$ is varied (${\Omega }_2$ is varied simultaneously in such a way that $g_{\mathrm{eff}}^{\left(1\right)}=g_{\mathrm{eff}}^{\left(2\right)}$ for all ${\Omega }_1$). Bottom panel: ${\Omega }_1/\left(2\pi \right)=320$ MHz, while ${\Delta }_2$ is varied, with ${\Omega }_2$ chosen such that $g_{\mathrm{eff}}^{\left(1\right)}=g_{\mathrm{eff}}^{\left(2\right)}$ for all ${\Delta }_2$. For both panels $g_{\mathrm{cav}}/\left(2\pi \right)=200$ MHz and $\delta ={\delta }_{\mathrm{cav}}=0$ (so ${\Delta }_2-{\Delta }_1={\omega }_2-{\omega }_{21}^{\prime }$).

Figure 3

Time evolution of the probability of the system being in the initial state $|e0\rangle$ and the mean intracavity photon number $\langle a^{†}a\rangle$. The two top rows are for the parameter sets denoted I in Table , the two middle rows for the sets II, and the two bottom rows for the sets III. The left column corresponds to sets denoted by $a$, and the right column to $b$. The green solid lines are the solutions of Eq. (9), and the dashed black lines are those of the effective model, Eq. (22). The two solutions coincide perfectly to the precision of the figure. The dotted lines are for the effective model with the same parameters, but with $U$ set to zero. Each panel shows results for two different $\kappa /\left(2\pi \right)=0.1,0.01$ MHz—the higher curves correspond to smaller $\kappa$.

Figure 4

Population decay of the $\left\{\right|g\rangle ,|e\rangle \}$ atomic subspace in the simulation of the full system, with $\gamma /\left(2\pi \right)=5.7$ MHz and parameter sets I$a$ (green solid line), II$a$ (blue dashed line), and III$a$ (black dotted line). Exponential fits of the form $c+a\exp (-\Gamma t)$ yield $\Gamma =4.7\times {10}^{-5}$ MHz (I$a$), $1.9\times {10}^{-4}$ MHz (II$a$), and $7.8\times {10}^{-4}$ MHz (III$a$). Note that $\Gamma$ scales linearly with ${\Omega }_i^2$ ($i=1,2$).

Figure 5

Time evolution of the probability of the system being in the initial state $|e0\rangle$ and the mean intracavity photon number $\langle a^{†}a\rangle$. The two top panels are for the parameter set denoted I in Table , the two middle panels for the set II, and the two bottom panels for the set III. The green solid lines are the solutions of Eq. (9), and the dashed black lines are those of the effective model, Eq. (22). Each panel shows results for two different $\kappa /\left(2\pi \right)=0.1,0.01$ MHz—the higher curves correspond to smaller $\kappa$.

Figure 6

Evolution of the logarithmic negativity $E_N\left(\rho \right)$ for the effective model, Eq. (22), with initial state $|g0\rangle$. We consider three different coupling strengths: $g_{\mathrm{eff}}/\left(2\pi \right)=2.0$ (solid lines), $1.0$ (dashed lines), and $0.5$ (dotted lines) MHz. Top panel: $\kappa /\left(2\pi \right)=0.1$ MHz; middle panel: $\kappa /\left(2\pi \right)=0.01$ MHz; bottom panel: $\kappa /\left(2\pi \right)=0.0$ MHz. Other parameters are ${\omega }_0=0.0$, $\omega /\left(2\pi \right)=1.0$ MHz, and $U/\left(2\pi \right)=-0.18$ MHz.

Figure 7

Contour plots of the Wigner function $W\left(\alpha \right)$, with $\alpha =(x+iy)/\sqrt{2}$, showing snapshots of the cavity field's time evolution. $g_{\mathrm{eff}}/\left(2\pi \right)=2.0$ MHz, $\kappa /\left(2\pi \right)=0.1$ MHz, and other parameters as given in the text. Starting from the top left and traversing rows first, the times for the snapshots are $t=\{0,0.25,0.5,0.75,1.0\}\mu$s. The corresponding mean photon numbers are $\langle a^{†}a\rangle =\{0,6.8,11.8,5.5,0.9\}$.

Figure 8

Steady-state atomic inversion, photon number, and intensity correlation function $g^{\left(2\right)}\left(0\right)$ as a function of $U$ for $g_{\mathrm{eff}}/\left(2\pi \right)=0.5$ MHz (dotted lines), $1.0$ MHz (dashed lines), and $2.0$ MHz (solid lines). Other parameters are $\omega /\left(2\pi \right)={\omega }_0/\left(2\pi \right)=1.0$ MHz and $\kappa /\left(2\pi \right)=0.2$ MHz.

Figure 9

Contour plots of the Wigner function $W\left(\alpha \right)$, with $\alpha =(x+iy)/\sqrt{2}$, for varying values of $U$, with ${\omega }_0/\left(2\pi \right)=\omega /\left(2\pi \right)=1.0$ MHz, $g_{\mathrm{eff}}/\left(2\pi \right)=2.0$ MHz, and $\kappa /\left(2\pi \right)=0.2$ MHz.

Figure 10

Time evolution of the atomic inversion $\langle {\sigma }_z\rangle$ (solid green lines in left column), and the intracavity photon number $\langle a^{†}a\rangle$ (solid green lines in right column) for Eq. (9). The dashed lines are the corresponding results for the effective model, Eq. (22). The solutions are nearly indistinguishable. The effective parameters are in each case set to ${\omega }_0/\left(2\pi \right)=\omega /\left(2\pi \right)=g_{\mathrm{eff}}/\left(2\pi \right)=1.0$ MHz, $\kappa /\left(2\pi \right)=0.2$ MHz, $\gamma /\left(2\pi \right)=5.7$ MHz, while $U$ is varied. The corresponding cavity and laser parameters used are given in Table , with sets I–V starting from the top to bottom row.