Robust quantum switch with Rydberg excitations

We develop an approach to realize a quantum switch for Rydberg excitation in atoms with Y- typed level configuration. We find that the steady population on two different Rydberg states can be reversibly exchanged in a controllable way by properly tuning the Rydberg-Rydberg interaction. Moreover, our numerical simulations verify that the switching scheme is robust against spontaneous decay, environmental disturbance, as well as the duration of operation on the interaction, and also a high switching efficiency is quite attainable, which makes it have potential applications in quantum information processing and other Rydberg-based quantum technologies.

where the atomic operators σ α β = αβ k k k ( ) , α β ∈ g m s r , { , , , }. The properties of the interaction between atoms are dependent on the Rydberg states we chose. For nS Rydberg states, in the absence of electrostatic field the interaction is dominant by the second-order dipole-dipole interaction (i.e. vdWs interaction) 31  For a single atom, there exist two different pathways for excitation, I: → → g m r and II: → → g m s . We consider the condition ω Ω > so state r is the strongly-coupled state and the excitation pathway I is preferred, while s is the weakly-coupled state and the pathway II is less taken. Considering the long lifetime of states s and r , their spontaneous decays γ s and γ r are far less than the decay Γ for state m . For simplicity, we first assume Ω = Ω p , Δ = 0 and γ γ γ = = s r . As shown in Fig. 1(c), if ω Ω  , it is easy to envision that there is a steady state that almost 1/2 population transfers from g to r through pathway I with no population on m or s , which is labeled as "OFF" state. However, we will show that considerable population would counter-intuitively , the atom will be excited to the strongly-coupled state r through pathway I, defined as the "OFF" status of the switch; if  ≠ 0 rr 0, the atom prefers the excitation to the weakly-coupled state s through pathway II, defined as the "ON" status. For both cases the interaction strength = 0 sr 0,  and  ss 0, is arbitrary. . This process is found to be fully irrespective of the exact interaction strength  ss 0, and can serve as a controllable switch between the two status "OFF" and "ON" corresponding to different Rydberg excitations.

Single-atom case
We begin with the status "OFF" which can be analyzed in single-atom frame due to the absence of Rydberg interaction. The analytical expression for steady state can be obtained by solving the master equation 2) with ρ k and k  the single-atom density matrix and Hamiltonian, respectively. Here the Lindblad superoperator satisfying < P P s r due to ω < 1. In the limit ω  1 and due to γ Γ  , Eq. (4) reduces to → P 1/2 r and → P 0 s , coinciding with our previous predictions about the status "OFF". In Fig. 2, we plot P r and P s as functions of ω, which shows that P r decays and P s grows up as ω increases and they become equal at ω = 1. For further increased ω, both of them decrease but at different rates. The monotonous decrease of P r is easy to understand, while the variation of P s is ascribed to the electromagnetically induced transparency (EIT) effect in pathway II. A unique feature of the effect is that the excitation probability decreases as enhancing the coupling laser strength 32 .

Two-atom Case
Turning to the picture of two interacting atoms, if the initial state is | 〉 gg , the total Hamiltonian  can be expanded by the ten symmetric two-atom bases only, mm  ss  rr  gm  gs  gr  ms  mr  sr  {  ,  ,  ,  ,  ,  , , , , } where αβ αβ βα = ± ± ( ) / 2, with the asymmetric states αβ − safely ignored 33 . The coupling strategie and strength among them are presented in Fig. 1   is nonzero, giving rise to an energy shift on state rr , the transition from + mr to rr will be affected. If the condition for strong blockade, , is satisfied 24 , the transition to the doubly Rydberg excited state will be fully suppressed. Instead, the population moves towards + sr when the interstate interaction = 0  1 2 , in which ρ is replaced by a two-atom density matrix. The steady populations + P gr and + P gs for states

The Switch Efficiency
To investigate the performance of the quantum switch, we first define the switching efficiency as In Fig. 4(a-c) we show the dependence of η on the interaction strength  rr 0, under the different relative couplings ω (ω is already normalized by Ω). For comparison, the steady populations + P gs (blue dashed) and + P gr (black dotted) are presented in the same frame. η reaches a saturation value and no longer changes with rr , satisfying the two-atom strong blockade condition 24 . This brings us a big advantage at selections of state r in practice, especially for the atoms with multiple Rydberg energy levels. Besides, the saturation value of η is observed to be enhanced with the increase of ω, which is attributed to the slight changes of  and ω, we also explore the influence of other parameters on the switching scheme, including the interaction ss 0,  of the weakly-coupled Rydberg state, the spontaneous decay Γ of the middle state, and the Rabi frequency Ω p . In Fig. 5(a) we find that η keeps constant with ss 0,  . This is due to the isolation of state ss from the transfer pathway as presented in Fig. 1(b), so that the energy shift of ss induced by interaction ss 0,  has no effect on η. However, an off-resonant transition → g m characterized by detuning Δ will reduce the steady population of Rydberg state, resulting in a decrease of η. We find that the switching scheme is robust to the middle-state decay Γ. As displayed in Fig. 5(b), η keeps almost unvaried in a broad regime of Γ . < < 1 0 1 0 and start to slowly decrease only when Γ > 10. In contrast, a larger decay γ means a quick decay from Rydberg states, which directly causes a drop of η.   . The steady-state population + P gr off and + P gs on as a function of the driving laser Ω p are shown in (d). All frequencies are scaled by Ω.
absence of applied electrostatic fields, the interstate interactions are negligible compared with the intrastate interactions, which has been confirmed theoretically 34,35 as well as experimentally 36 .
Finally, we consider a more general case Ω ≠ . is unable to maintain in this case, the definition of η is no longer rigorous. We then show + P gr off and + P gs on versus Ω p individually in Fig. 5(d). As the increase of Ω p , + P gr off (black dashed) exhibits a clear reduction after reaching its maximum value 0.5 at Ω = .
1 0 p . Similar trends are observed in + P gs on (red solid) but the maximum 0.6 appears at about Ω ≈ . 0 6 p . The reductions are because that for a large Ω p the transition of → + gm mm is enhanced which results in a decrease of excitations to + gs and + gr , see Fig. 1(b). Hence, we conclude that Ω = . 1 0 p is an optimized value for our switching scheme, because the population of status "OFF" and "ON" are asymmetry for other values.

Experimental Implementation
After carefully researching the steady state of the switching system, we now turn to study the switching dynamics by numerically simulation with a series of experimental parameters. We assume two 87 Rb atoms are respectively confined in two independent optical dipole traps whose separation R can be adjusted from μ 15 m to μ . 4 0 m by changing the incidence angle of the optical beams in a duration τ of the orders of several μs 24,37  6 due to the large difference in principle quantum numbers of the two Rydberg states 39 , so that the interstate interaction  sr 0, is largely suppressed and can be safely neglected. The Rabi frequencies, π Ω = /2 10 MHz and ω π = /2 2 MHz, are typical of current experiments. To simulate the variation of the interaction under control, we introduce a time-dependent pulse sequence of  t ( ) rr ss 0, ( ) for a complete switching cycle consisting of three status: OFF, ON, and OFF,  Fig. 6(a), the larger the τ is, the slower and smoother the switching operation between  rr off 0, and rr on 0,  is. The total duration of the switching cycle is μs 100 which is less than the lifetime of the Rydberg states. As discussed before, the excitation is  Fig. 6(b). Initially prepared in the two-atom ground state gg , the system tends to be a steady state with → . is tuned down again, but at this time a longer period is required for the retrieval of the population for status "OFF" due to the different transition pathways. The significant oscillation of population appears only at the period < < + t t 0 0 with a larger duration τ. This is because in status "OFF" except + gr , gg and rr are also stably occupied, the resonant excitation between gg and + gr will give rise to a Rabi-like oscillation if the duration of the switch is long enough. As an opposite example, in status "ON" only + sr is dominantly occupied except for + gs , so the resonant excitation between them is not isolated but suffers from a strong decoherence. Hence, when > − t t 0 , even if τ is large there is no oscillation but a smooth redistribution of the population via decay process, which requests a longer time determined by the lifetime of Rydberg states.
Based on our numerical simulations with practical parameters, a realistic switching efficiency is estimated as η = ≈ . with the total operating time μ = t 100 s cyc . We stress this optimal value is obtained by theoretical calculations with practical parameters however it excludes the influences from all the technical errors in a real experiment, e.g. the instrument precision, the stability of system and so on.

Applications and Conclusions
The quantum switch we present here based on the controllable strong interaction between two Rydberg atoms, but different from the single-photon transistor with Rydberg blockade 18 , it enables an efficient and compact transition between two symmetric singly Rydberg excited states | 〉 + gr and | 〉 + gs . With appropriate applications and developments, this will broaden exciting perspectives on quantum information processing with Rydberg atoms. For example, owing to its long lifetime and entanglement 40 , the singly excited states can become an excellent carrier of quantum information. Then the reversible and swift switch of these states is a requisite operation for implementation of information transfer and quantum computation. Especially, the considerable separation ( μ ∼ m 10 ) between two Rydberg atoms in our design allows local operations on one of them individually, served with our switching on two-atom states, various quantum logic gates are hopefully realizable [41][42][43][44] . Besides, the Rydberg atomic pair-state interferometer has been experimentally realized 45 recently. A high-precision quantum switch between different Rydberg excitations can enrich its measurement objects and develop the application of Rydberg atoms in quantum metrology. Finally, Rydberg dressing has been proposed to realize a number of interesting phases in ultra-cold gases, such as rotons and solitons 46,47 . An extension of our switch in a many-atom case will allow a more complex structure of Rydberg dressing, which makes it possible to simulate various and exotic spin-dependent phases by Rydberg atoms 48 .
To conclude, our work presents a robust and experimentally feasible scheme of quantum switch, implemented in a system of two interacting Rydberg atoms. Each atom has a Y-typed level structure with two highly-excited Rydberg states. We show that which Rydberg state to be excited can be simply and effectively controlled by opening or closing the intrastate interaction of the strongly-coupled Rydberg state. After systematically investigating the steady state and the dynamics of the system in a numerical way, we verify the robustness of the scheme by presenting its insensitivity to the self-interaction of the weakly-coupled Rydberg state, the decay of intermediate state, and the duration time for switching. Our method is suitable for two Rydberg nS states in which the interstate exchange interaction between them can be totally suppressed by considering two nS states with large different principle quantum numbers. More possibilities for the implementation with other energy levels may work, e.g. by applying an external electrostatic field 49 . We show a numerical simulation of switch operation in 87 Rb atoms under realistic experimental conditions and find the switch efficiency approaching as high as 0.92. A many-atom case maybe treated as a good extension to the current scheme in the future, requiring more attentions to complex energy levels and transitions. We also plan to develop the applications of such particular switch in the fields of quantum information processing and other quantum devices.