Boosting the Quantum State of a Cavity with Floquet Driving

The striking nonlinear effects exhibited by cavity QED systems make them a powerful tool in modern condensed matter and atomic physics. A recently discovered example is the quantized pumping of energy into a cavity by a strongly coupled, periodically driven spin. We uncover a remarkable feature of these energy pumps: they coherently translate, or boost, a quantum state of the cavity in the Fock basis. Current optical cavity and circuit QED experiments can realize the required Hamiltonian in a rotating frame. Boosting thus enables the preparation of highly excited nonclassical cavity states in near-term experiments.

Figure 1

(a) Model. A spin coupled to a quantum cavity with frequency $\omega$ and subject to an external periodic drive of frequency $\mathrm{\Omega }$, such that $\mathrm{\Omega }/\omega \notin \mathbb{Q}$. The frequencies $\hslash \omega$ and $\hslash \mathrm{\Omega }$ are smaller than all other energy scales in the problem. (b) Cavity state boosting in a Fock state. A plot of the Fock state occupation $P(n)=⟨n|{\rho }_{\mathrm{cav}}(t)|n⟩$, where ${\rho }_{\mathrm{cav}}(t)$ is the reduced density matrix of the cavity, shows rephasings, marked by blue arrows. These represent the cavity state becoming near-Fock with a larger occupation number than the initial state. Parameters in model (4) are $\mathrm{\Omega }/\omega =(1+\sqrt{5})/2$, $\mu B_m/\hslash \omega =\mu B_d/\hslash \omega =6$, $\mu B_0/\hslash \omega =1.5$, and ${\theta }_{01}=3\pi /2$, initial state $|{\psi }_0⟩=|+{⟩}_{\widehat{\mathbf{x}}}|n_0⟩$ being a product of $|+{⟩}_{\widehat{\mathbf{x}}}$ (the $+S$ eigenstate of $S_x$), and $|n_0⟩$ (a Fock state) with $n_0=10$, and spin $S=1/2$ (that is, a two-level system).

Figure 2

(a)–(d) The photon number distribution $P(n)=⟨n|{\rho }_{\mathrm{cav}}(t)|n⟩$ in Fig. 1 at multiples of the period of the classical drive $T=2\pi /\mathrm{\Omega }$. The distribution broadens from the initial Fock state (a), but narrows again at special times to produce a near-Fock state again (d). (e)–(h) The Husimi $Q$ function $Q(\alpha )=(1/\pi )⟨\alpha |{\rho }_{\mathrm{cav}}(t)|\alpha ⟩$. Initially (e) the cavity is in a Fock state, with a circularly symmetric $Q$ function. At most times (f),(g), the $Q$ function is displaced from the center of the quadrature plane, and is not circular. At special times (h) the $Q$ function is again centered and approximately circularly symmetric about the origin, but now with a larger radius. The initial radius ($n=10$, red) and predicted final radius ($n=22$, blue) are marked by dashed circles for reference. Parameters are as in Fig. 1.

Figure 3

Semiclassical rephasings. The prediction for the Fock occupation number $n(t)$ (10) for an ensemble of initial phases ${\overrightarrow{\theta }}_0$ and a (a) quasiperiodic and (b) periodic drive. Both show rephasings at their almost periods and periods respectively. (c) Inspecting the variance of $n(t)$ between $N_{\theta }=32$ different values of ${\theta }_{02}$ shows that the rephasings improve in quality with increasing $T_N$ for quasiperiodic drives, but decay linearly for periodic drives.

Figure 4

Alignment of spin and field. (a)–(c) Cavity $Q$ functions for different initial states, $|+{⟩}_{\widehat{\mathbf{x}}}|{\psi }_0⟩$, with (a) $|{\psi }_0⟩=|n=10⟩$ a Fock state, (b) $|{\psi }_0⟩=|\alpha =\sqrt{10}⟩$ a coherent state, and (c) $|{\psi }_0⟩\propto |\alpha =\sqrt{10}⟩+|\alpha =-\sqrt{10}⟩$ a Schrödinger cat state. (d) The expectation value $M=⟨\overrightarrow{B}·\overrightarrow{S}⟩/\sqrt{⟨{\overrightarrow{B}}^2⟩}$ quantifies how closely aligned the spin is to an effective cavity field in a basis of coherent states. We see that $M$ remains close to its extremal value of $-S$. Parameters are as in Fig. 1.