Topological frequency conversion in a driven dissipative quantum cavity

Recent work [Martin et al., Phys. Rev. X 7, 041008 (2017)] shows that a spin coupled to two externally supplied circularly polarized electromagnetic modes can effectuate a topological, quantized transfer of photons from one mode to the other. Here, we study the effect in the case when only one of the modes is externally provided, while the other is a dynamical quantum mechanical cavity mode. Focusing on the signatures and stability under experimentally accessible conditions, we show that the effect persists down to the few-photon quantum limit and that it can be used to generate highly entangled “cat states” of cavity and spin. By tuning the strength of the external drive to a “sweet spot,” the quantized pumping can arise starting from an empty (zero-photon) cavity state. We also find that inclusion of external noise and dissipation does not suppress but rather stabilizes the conversion effect, even after multiple cavity modes are taken into account.

Figure 1

(a) We study a model of a magnetic particle with angular moment $L$ (black dot, with arrow indicating magnetization) in a cavity, coupled to a circularly polarized cavity mode (blue) and a circularly polarized driving mode (red). Interpreting the particle's spin $s$ and the cavity photon number $n_1$ as an orbital and lattice degrees of freedom, respectively, the system is equivalent to a driven tight-binding model (b). In suitable parameter regimes, the tight-binding model acts as a Thouless pump, and the magnetic particle effectuates a transfer of energy to the cavity at the universal rate $\omega \mathrm{\Omega }L/\pi$.

Figure 2

(a) Region of parameter space which supports topological frequency conversion (red). Here, $B_d$ and $B_c$, respectively, denote the amplitudes of the driving and cavity modes, while $B_m$ indicates the strength of the Zeeman field. (b) Numerically obtained wave function of the system (absolute squared) as a function of photon number $n$ and time, in the topological regime (see main text for further details). The boundaries of the topological regime [corresponding to the boundaries of the highlighted region in (a)] are indicated by dashed lines, and the quantized pumping rate is indicated by the slope of the blue line.

Figure 3

Unitary evolution of the system's wave function (see main text for further details). (a) Evolution of the system's wave function for the model depicted in Fig. 2, when the spin is initially antialigned with the net magnetic field. (b) Evolution of the system's wave function for the same setup as in (a), when the spin is initially perpendicular to the net field. (c) Evolution of the system in (a) over 5000 periods. (d) Evolution of the model at the “sweet spot” in the phase diagram [see Fig. 2] $B_d=B_m=10B_0$, when the cavity is initially empty. Black dashed lines indicate phase boundaries, and blue dashed lines indicate the theoretically expected rate of $\langle \dot{n}\rangle =\frac{1}{T}$.

Figure 4

Evolution of the system in the presence of noise and dissipation. Each panel plots a randomly picked example of the system's wave function (absolute squared), as a function of photon number $n$ and time, obtained by stochastic integration of the Floquet-Lindblad master equation in Eqs. (20, 21, 22). Parameters for the respective panels are given in Sec. 4b. Black dashed lines indicate the topological phase boundaries, and blue dashed lines indicate the expected steady-state photon number consistent with a quantized emission rate (see Sec. 4b). (a) Example of the wave function when spin and cavity dissipation are present. (b) Example of the wave function when spin and cavity dissipation are present at the “sweet spot” in the phase diagram where $B_m=B_d$, and the system is initialized from an empty cavity. (c) Example of the wave function when only spin dissipation is present. (d) Example of the wave function when only cavity dissipation is present.

Figure 5

(a) Evolution of the cavity field energy for the classical model in Eq. (23), with parameters and initialization corresponding to the quantum system in Fig. 2 (see Sec. 5 for details). Blue line indicates slope consistent with the topological pumping rate ${\dot{E}}_c=L\omega \mathrm{\Omega }/\pi$. Dashed lines indicate topological phase boundaries identified for the quantum case in Sec. 3. (b) Classical evolution of the system in the presence of cavity and spin dissipation (see main text for details). Blue line indicates steady state at $E_c=300\hslash \omega$ expected from emission of energy at the topological pumping rate.

Figure 6

Energy stored in the cavity modes as a function of time, in a setting where the first harmonic is in the topological regime, while the remaining modes are initially only weakly excited. See main text for details. (a) Energies of modes $\pm 2...\pm 15$ (green-purple curves) and mode $-1$ (red curve). (b) Energy of mode 1 (green curve).