Entanglement generation between spinor Bose-Einstein condensates using Rydberg excitations

We propose an experimental scheme of generating entangled states between two spinor Bose-Einstein condensates (BECs) using Rydberg excitations. Due to the strong interaction between Rydberg atoms, the Rydberg excitation creates an interaction between two closely located BECs. The method is suitable particularly for atom chip and permanent magnetic trap systems, which can create many BECs with an arbitrary two-dimensional geometry. We show two schemes of entangled state generation, based on stimulated Raman adiabatic passage (STIRAP) methods. The first method produces a symmetric state with total $S^x$ spin zero between ground and excited states of the atoms using a single STIRAP pair, while the second produces a NOON state between hyperfine ground states using two STIRAP pairs. We show that despite the additional complexity of the BECs, it is possible to identify the initial and final adiabatic states exactly. We verify our theoretical predictions using numerical simulations on small boson number systems.

Figure 1

(a) Representation of the energy levels used for the STIRAP schemes. $g_i$ and $f_i$ are the energy levels of the ground states of the atoms forming the BECs for $i=1,2$. $e_i$ are the energy levels for the excited states, and $r_i$ are the Rydberg levels. (b) Pulses for the STIRAP sequence. The pulses are in counterintuitive order: the sequence starts with the pulse coupling the unpopulated states $|e_i\rangle$ and $|r_i\rangle$, not affecting the initial state until the pulse coupling the states $|e_i\rangle$ and $|g_i\rangle$ is turned on. (c) Pulses for the two-STIRAP sequences. The second pulse sequence is in reverse order compared to the first one. The relative amplitude $tan\theta =|\frac{{\mathrm{\Omega }}_e}{{\mathrm{\Omega }}_r}|$ is also plotted for the one- and two-STIRAP sequences.

Figure 2

Energy eigenvalues $\varepsilon$ of the Hamiltonian Eq. (15) for (a, b) $N_1=N_2=2$ and (c, d) $N_1=N_2=3$. (b) and (d) are zoomed in versions of (a) and (c) near the zero-energy region. We use pulses for ${\mathrm{\Omega }}_{e,r}$ as defined in Eqs. (30) and (31). Parameters used are ${\epsilon }_0=\hslash /\tau , \mathrm{\Delta }t/\tau =0.11, {\mathrm{\Omega }}_r^{\max }\tau =500\pi , {\mathrm{\Omega }}_e^{\max }\tau =500\pi$, and $U\tau /\hslash =200\pi$.

Figure 3

Probability amplitude $|{\psi }_{\mathit{n}}{|}^2$ evolution during the STIRAP sequence for states $|k_1,k_2,r_1,r_2\rangle$ with (a) $N=2$ and (b) $N=3$. In (a), horizontal lines show populations of labeled states in the ideal adiabatic case. Parameters used are (a) $\mathrm{\Delta }t/\tau =0.36, {\mathrm{\Omega }}_r^{\max }\tau =60\pi , {\mathrm{\Omega }}_e^{\max }\tau =60\pi , U\tau /\hslash =200\pi$; (b) $\mathrm{\Delta }t/\tau =0.36, {\mathrm{\Omega }}_r^{\max }\tau ={10}^4, {\mathrm{\Omega }}_e^{\max }\tau ={10}^4, U\tau /\hslash =2{\mathrm{\Omega }}_r^{\max }$.

Figure 4

Fidelity of the final state after adiabatic evolution with respect to the ideal NOON state Eq. (28). Particle numbers correspond to (a) $N=2$ and (b) $N=3$. Parameters used are $\mathrm{\Delta }t/\tau =0.36, {\mathrm{\Omega }}_r^{\max }={\mathrm{\Omega }}_e^{\max }=\mathrm{\Omega }$, and $U\tau /\hslash =2\mathrm{\Omega }$.

Figure 5

Entanglement, as measured by the logarithmic negativity, for different pulse amplitudes and (a) $N=2$, (b) $N=3$. Parameters used are $\mathrm{\Delta }t/\tau =0.36, {\mathrm{\Omega }}_r^{\max }={\mathrm{\Omega }}_e^{\max }=\mathrm{\Omega }$, and $U\tau /\hslash =2\mathrm{\Omega }$.

Figure 6

(a) Typical time evolution for the geometric phase gate with two STIRAP sequences for $N_1=N_2=2$ from the initial state $|k_1=0,r_1=0\rangle |k_2=0,r_2=0\rangle$. Parameters used are $\mathrm{\Delta }t/\tau =0.40, {\mathrm{\Omega }}_r^{\max }\tau =1000, {\mathrm{\Omega }}_e^{\max }\tau =1000, U\tau /\hslash =2{\mathrm{\Omega }}_e^{\max }, \omega \tau =-0.05$, and $\mathrm{\Delta }T/\tau =2$. (b) Fidelity of the initial state after the two STIRAP pulses for various values of ${\mathrm{\Omega }}_r^{\max }={\mathrm{\Omega }}_e^{\max }=\mathrm{\Omega }$.

Figure 7

(a) Phases picked up during two STIRAP pulse procedure by various states as labeled by $|k_1,k_2\rangle =|k_1{,0,0\rangle }_z{|k_2,0,0\rangle }_z$ in Eq. (49). The phases picked up by $|k_2,k_1\rangle$ are the same as $|k_1,k_2\rangle$. Parameters used are the same as in Fig. 6. Phases are with references to the state $|0,0\rangle$, which does not pick up any phase under the evolution. (b) Entanglement created by evolving the two STIRAP sequence starting from the state $|\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\rangle \rangle |\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\rangle \rangle$. Solid line shows the result for the phases picked up as in subfigure (a). Dashed lines shows the ideal result Eq. (60).