Progress towards macroscopic spin and mechanical superposition via Rydberg interaction

This paper is a proposal for the generation of a many-body entangled state in atomic and mechanical systems. Here the detailed feasibility study shows that application of a strong Rydberg dressing interaction and a fast bifurcation scheme in a Bose-Einstein condensate of Rb atoms, results in the formation of large cat states. By detailed study of the decoherence effects using the quantum jump Monte Carlo approach and taking into account the obstacles like collective decoherence and level mixing, this proposal predicts the formation of a 700-atom cat state. Subsequent transfer of the generated superposition to far separated mechanical oscillators is proposed, using dipole coupling between Rydberg atoms and charged cantilevers.

Figure 1

Proposed scheme for creation of large spin superposition. (a) Level scheme in rubidium. The pseudospin states are the hyperfine levels in the ground state. An off-resonant laser field ${\mathrm{\Omega }}_r$ dresses the excited state with the Rydberg level $|r\rangle$ via a ladder two excitation process. This creates a Kerr-type interaction between the atoms in the excited state. (b) The vector field representation of Hamiltonian [Eq. (4)] under the simultaneous application of ${\mathrm{\Omega }}_c$ and ${\mathrm{\Omega }}_r$ and optimal condition $\mathrm{\Lambda }=2$ and $\delta =0$; see Fig. 2. The created bifurcation and bistabilities in (b) explain how an initial coherent spin state CSS (c), squeezes initially (d) and eventually splits into two CSS pointing in opposite directions on the Bloch sphere (e). The final state could be approximated as the superposition of all the atoms being in the ground state and excited state. (f) Revival of the initial state could be seen after $2{\tau }_c$. The generated figure shows the case for 80 atoms.

Figure 2

Optimized bifurcation parameters. (a) The trajectories and (b) Husimi Q functions of 30 atoms are plotted for different parameters $\mathrm{\Lambda }=1.15,2,3$ in the columns (1–3). Considering the cat state wave function of $|\mathrm{cat}\left(\mathrm{\Delta }\theta \right)\rangle =\left(\right|\frac{\pi -\mathrm{\Delta }\theta }{2},{\varphi }^{\prime }\rangle +|\frac{\pi +\mathrm{\Delta }\theta }{2},{\varphi }^{\prime }\rangle )/\sqrt{2}$, the angular separation in columns 1–3 are $\mathrm{\Delta }\theta =\{0.4\pi ,0.98\pi ,0.8\pi \}$ and ${\varphi }^{\prime }=\{0,0,\pi \}$ with the fidelity of Fid={0.94, 0.42, 0.31} and Fisher information of F/N={10.5, 20.3, 18}. Odd orders of $J_z$ disturb the symmetry in the two hemispheres. This fact can be seen in column 4 with $\mathrm{\Lambda }=2$, ${\delta }^{\prime }=0.3$.

Figure 3

Effects of dressing strength and decoherence on the entanglement generation. (a) The evolution of Fisher information over cat state generation of N=300 atoms, for different de-excitation probabilities of $P_{de}\left({\tau }_c\right)=\{0,0.01,0.3,0.8\}$. Fisher information of the generated cat states is plotted as a function of (b) dressing strength and also Poisson probabilities of (c) de-excitation and (d) de-phasing. The lines from bottom to top are corresponding to the ensemble sizes of $N=\{16,36,64,100,144,300,700,1000\}$ atoms. The Monte Carlo simulations have been averaged over 160 separate runs. (e) The trend of optimal dressing strength $w$ as well as de-excitation and de-phasing rates that reduce the Fisher information by 10% are plotted as a function of cat size.

Figure 4

Maximum achievable cat size and corresponding Fisher information as a function of the principal number $n$ of the Rydberg state. The de-excitation and de-phasing probabilities and dressing strength are set according to Fig. 3. Blockade radius is set to be three times larger than the trap diameter to ensure the homogeneity of interaction [19, 42]. The inset shows the required cat creation time. Without the cryogenic environment achievable cat size would reduce from 700 to 550 atoms.

Figure 5

Level mixing and Blockade infidelity. (Left column) Eigenenergies of dressing Hamiltonian [Eq. (7)] close to the ground-state dressing energy [Eq. (2)] over close interatomic separation. First and second rows are corresponding to interatomic orientations of $\theta =\{0,90\}$ relative to quantization axis. (Right column) Represents the perseverance of blockade, mainly due to the dilute intensity of the optically accessible pair states. (e) A zoomed-in version of (c) and (d) where the black line is the map of infidelity on the eigenenergy red (light gray) lines to match the position of minor infidelity peaks with the avoided crossing points. The graph is for dressing to $|50S_{1/2}1/2\rangle$ with ${\mathrm{\Omega }}_r=3.3\mathrm{MHz}$ and $\mathrm{\Delta }=-23\mathrm{MHz}$.

Figure 6

Proposed scheme for mapping the spin cat state to the mechanical system. (a) Setup includes two electrically charged bridge cantilevers coupled with the Rydberg excited superatom via dipole-dipole interaction. (b) and (c) Level scheme. Rubidium clock states are labeled by $|e\rangle$ and $|g\rangle$. Collective Rydberg excitation with S and P orbitals are labeled by $|s\rangle$ and $|p\rangle$. (b) Accommodating the ensemble within the blockade radius, applying the first $\pi$ pulse creates single Rydberg excitation in a collective form (superatom). The existing superatom provokes a single phonon in resonant mechanical oscillator over $\pi /g$ atom-cantilever coupling time. Following that a $\pi$ pulse transfers the Rydberg atom to the ground state. Consequent repetition of this cycle transfers the initial spin cat state to the mechanical system, resulting in the superposition of the presence and absence of N phonons on one cantilever. (c) The population of $|e\rangle$ and $|g\rangle$ would transfer to a third long-lived state via different intermediate Rydberg levels that are in resonance with different cantilevers. Consequent repetition of this process results in the superposition of $N$ phonons being in the right or left cantilever.