Adiabatic flux insertion and growing of Laughlin states of cavity Rydberg polaritons

Recently, the creation of a strong magnetic field in a photonic cavity system has been demonstrated [N. Schine et al., Nature (London) 534, 671 (2016).]. Using this setup, we propose a scheme to adiabatically transfer flux quanta simultaneously to all cavity photons. The flux transfer is achieved using external light fields with orbital angular momentum and a near-resonant dense atomic medium as a mediator. Furthermore, by coupling the cavity fields to a Rydberg state, strong photon-photon interactions can be realized and fractional quantum Hall states can be prepared. To this end, a growing protocol is discussed consisting of a sequence of flux insertion and subsequent single-photon insertion. Specifically, we discuss the growing of the $\nu =1/2$ bosonic Laughlin state. First we adiabatically insert two photonic flux quanta, creating a two-quasihole excitation, and second we refill the hole with a single photon using the strong photon-photon interactions.

Figure 1

(a) Nonplanar resonator consisting of four mirrors creating an artificial magnetic field for cavity photons. Two clouds of atoms are used for adiabatic flux insertion and strong interactions mediated by Rydberg atoms. (b) Structure of the photonic Landau levels labeled by orbital angular quantum number $l$ and radial quantum number $n$. Here, ${\mathrm{\Delta }}_{n,\alpha }$ is the frequency of the mode $(n,\alpha )$. The green box indicates the lowest Landau level (LLL). Initially, photons are pumped into the cavity mode with $n=l=0$ (red arrow). Then, adiabatic flux insertion transfers orbital angular momentum (OAM) from a classical light beam to cavity photons, increasing the total angular momentum (blue arrow). (c) The density plot of the first three cavity modes in the LLL with angular momentum, from left to right, $l=0$, $l=3$, and $l=6$.

Figure 2

(a) The atomic level structure consists of two metastable levels $|g\rangle$, $|s\rangle$ and two excited levels $|e\rangle$ and $|r\rangle$. (b) System initially prepared in state with $l=0$. In the first stage of the protocol, the cavity modes $l=0$ and $l=1$ drive the transitions $|g\rangle \leftrightarrow |e\rangle$, $|g\rangle \leftrightarrow |r\rangle$ with coupling strength $g_0$ and $g_1$, while the laser fields ${\mathrm{\Omega }}_1\left(t\right)$, ${\mathrm{\Omega }}_0\left(t\right)$ couple the transitions $|e\rangle \leftrightarrow |s\rangle$, $|s\rangle \leftrightarrow |r\rangle$. We assume that the laser field ${\mathrm{\Omega }}_1$ has an OAM $-1\hslash$. As a consequence of this, the initial photon in the $l=0$ mode is transferred to the $l=1$ mode using the STIRAP technique. In the second step, the photon in the $l=1$ mode is transferred to the $l=3$ mode which belongs to the LLL manifold. This transition is performed by using the same atomic level structure, but with new Rabi frequency ${\mathrm{\Omega }}_2$ which carriers OAM $2\hslash$. (c) Density plots of the classical driving fields. We assume that the ${\mathrm{\Omega }}_0$ Rabi frequency has a Gaussian shape with $l=0$. The other two classical laser fields carry OAM and thus have vanishing intensity at the center.

Figure 3

(a) The expectation values of $a_{0,l}$ ($l=0,1,3$) vs time. The exact result for $a_{0,0}$ (blue line) and $a_{0,1}$ (dotted red line) according the coupled system (9) including the residual couplings for $a={10}^{-2}$. The dashed black line shows the time evolution of $a_{0,3}$ according to the coupled system (14). We choose time-dependent Rabi frequencies (18) and (19) with $\mathrm{\Omega }=\left(2\pi \right)12.4$ MHz and cavity couplings $g=\left(2\pi \right)0.45$ MHz. The other parameters are set to $\gamma =0$, $\delta =\left(2\pi \right)0.13$ MHz and $T=1\mu \mathrm{s}$. (b) The expectation value of $a_{0,1}$ at time $t_1$ against $\mathrm{\Omega }$ for different cutoff length.

Figure 4

Fidelity of the flux insertion scheme vs the peak Rabi frequency $\mathrm{\Omega }T$ and cavity coupling $gT$ according to Eq. (24). We assume Gaussian distribution of the atomic density. The parameters are $\gamma T=100$, $a=0.005$, and $\xi =0.25w_0$.

Figure 5

The growing scheme which is used for the preparation of the Laughlin-type states consists of two steps. (i) Coherent pump of a single photon in the ground state of the cavity by using the nonlinear photon-photon interaction. (ii) Increase of angular momentum per particle by 6 (flux insertion). Repeating the two steps lead to growing of the photonic Laughlin-type state.

Figure 6

Numerical simulation of growing scheme for the creation of Laughlin states with $N=2$ and $N=3$ photons. The system is prepared initially in a state with no photon. The red arrow indicates the time at which the coherent pump is applied. The angular momentum per photon is increased by repeating the flux insertion twice. The probabilities are $p_m=|\langle \psi |a_m^{†}{|0\rangle |}^2$, and, respectively, $p_{\mathrm{LN},N}$ and $p_{{\mathrm{qh}}_m,N}$ are the probabilities for the Laughlin state and $m\mathrm{th}$-quasihole state. The adiabatic flux insertion method for numerical simulation is explained in Appendix pp3. The parameters are set to ${\mathrm{\Delta }}_0/V_0=10$, ${\mathrm{\Omega }}_p/V_0=1/20$, and $g_a/V_0=g_b/V_0=1/5$.