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.

Introduction

Switch is a device that is capable of switching some kind of signals (e.g. current, voltage, energy, heat et.al.) between different pathways. Classical switch plays a vital role in electronics and signal processing. Extending such a concept into the quantum regime where the role of pathways is played by quantum states leads to the production of various quantum switches, such as the switchable acoustic meta-materials[1], the current switch in quantum dots[2,3], the superconducting switch[4], the fiber-optical switch[5] and so on. In particular, for achieving an all-optical quantum switch, one promising way is coupling the atoms to a microscopic high-finesse cavity[6-9] which can strongly enhance the light-atom interactions[10]. Such a quantum optical switch has many promising applications, ranging from quantum information processing to quantum metrology[11-14].

Recently, Rydberg atoms have been manifested as an ideal candidate to study single-photon all-optical switches[15-17] and transistors[18-20], mainly due to the presence of interatomic interactions[21,22]. The ultra-strong interaction between two Rydberg states gives rise to blockade effect, bringing on a strong enhancement for the light-atom interactions[23-25]. Moreover, the blockade effect can provide an efficient mechanism for controlling the quantum states of the atomic system itself. A simplest scheme can be carried out, for example in a two-atom system, it prohibits the excitation of the second atom when the first one has already been excited to the Rydberg state. That is, it allows to control one atom’s excitation or not via the status of the other[26-28], achieving a switchable excitation between two atomic states.

In the present work we propose a new scheme of quantum switch based on two Rydberg atoms of same Y-typed four-level configurations[29]. The special level configuration has two different Rydberg states: one is weakly coupled to the intermediated state and the other is strongly-coupled. This enables two different excitation pathways labeled as “OFF” and “ON” by us [see Fig. 1(c)], which can be efficiently switched via the control of intrastate interaction of the strongly-coupled Rydberg state. The interstate interaction between different Rydberg states, as main disturbance for the status switch, is greatly suppressed by employing the feature of nS Rydberg states that the strength of van der Waals (vdWs) interaction is not affected by Zeeman effect[30]. The robustness of the switching scheme is confirmed by its low sensitivity to the other parameters of the system, such as the intra-state interaction of the weakly-coupled state, the decay rate of the intermediate state, and the duration time of the switching process. We present a detailed discussion of a realistic experimental implementation of the switch with 87Rb atoms and predict that the final switching efficiency will reach as high as 0.92.

Figure 1

(a) Level structure of a single atom we adopted with the detailed descriptions about level couplings in the main text. (b) Level structure of two interacting atoms with the interstate transitions and the corresponding Rabi frequencies marked out. (c) Two possible pathways for exciting atoms into Rydberg states and . For and the interaction , the atom will be excited to the strongly-coupled state through pathway I, defined as the “OFF” status of the switch; if the atom prefers the excitation to the weakly-coupled state through pathway II, defined as the “ON” status. For both cases the interaction strength and is arbitrary.

Model Description

Our model consists of two identical Rydberg atoms in frozen-gas limit. As presented in Fig. 1(a), each atom has a Y-typed level structure that the ground state is coupled to the middle state via a laser field with Rabi frequency and detuning Δ, and is further resonantly coupled to two different Rydberg states and with Rabi frequencies and , respectively. The Hamiltonian for a single atom k reads ( everywhere)where the atomic operators , .

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]. It can be further classified as (i) the intrastate interaction with the strength , which presents if both atoms settle in a same Rydberg state or , (ii) the interstate interaction for atoms in different Rydberg states with . Here R represents the separation between atoms, are the interaction coefficients, and are the two-atom states. Then the total Hamiltonian is obtained from the sum of and the interaction terms,

For a single atom, there exist two different pathways for excitation, I: and II: . We consider the condition so state is the strongly-coupled state and the excitation pathway I is preferred, while is the weakly-coupled state and the pathway II is less taken. Considering the long lifetime of states and , their spontaneous decays and are far less than the decay for state . For simplicity, we first assume , and . As shown in Fig. 1(c), if , it is easy to envision that there is a steady state that almost 1/2 population transfers from to through pathway I with no population on or , which is labeled as “OFF” state. However, we will show that considerable population would counter-intuitively transfer into the weakly-coupled state through pathway II once the interaction . This process is found to be fully irrespective of the exact interaction strength 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 () with and the single-atom density matrix and Hamiltonian, respectively. Here the Lindblad superoperator is given bywhich describes the effect of spontaneous decays from states , , and . In the following calculations, we use () as the frequency (time) unit, leading to normalized parameters as , , , , , , , and . Then the steady populations of and are satisfying due to . In the limit and due to , Eq. (4) reduces to and , coinciding with our previous predictions about the status “OFF”. In Fig. 2, we plot P and P as functions of , which shows that P decays and P grows up as increases and they become equal at . For further increased , both of them decrease but at different rates. The monotonous decrease of P is easy to understand, while the variation of P 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].

Figure 2

For the single-atom case, the steady probabilities of Rydberg state (blue solid) and (red dashed) are plotted as functions of (i.e. ) with the decays and . Note that all frequency quantities are scaled by .

Two-atom Case

Turning to the picture of two interacting atoms, if the initial state is , the total Hamiltonian can be expanded by the ten symmetric two-atom bases only, where , with the asymmetric states safely ignored[33]. The coupling strategie and strength among them are presented in Fig. 1(b). In the absence of all Rydberg interaction, the population transfer is mainly following the approach (red arrows) due to the stronger coupling strengths (). Finally, the long-lived states are stably populated. Less population is found to accumulate in middle states for their short lifetimes, and in for their large decay rate and small coupling strength , respectively. The steady populations are and which is same as the status “OFF” analyzed in the single-atom case. Note that . Once the intrastate interaction is nonzero, giving rise to an energy shift on state , the transition from to 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 when the interstate interaction , which leads to the second transition pathway (blue arrows). Finally the steady populations are and , corresponding to the status “ON” for our quantum switch. is also dominantly occupied besides and .

The above qualitative analysis have been verified by numerically solving the master equation of two atoms, , in which is replaced by a two-atom density matrix. The steady populations and for states and are illustrated as functions of in Fig. 3 for three different cases: (i) (the black dashed curve for and the blue solid curve for ), (ii) and (the black dashed curve with circles for and the blue solid curve with circles for ), and (iii) and (the black dashed curve with triangles for and the blue solid curve with triangles for ). In case (i) we find and at and they two become equal as increases to 1, same as in Fig. 2 obtained in the single-atom frame. When the intrastate interaction is present, see the cases (ii) and (iii), there is a counterintuitive reversal of and at , indicating a large fraction of population transferred from to . In the shadow region of , and attain peak and off-peak values, respectively. Especially, by comparing cases (ii) and (iii) we find their variations are quite insensitive to the interaction strength in this region, which is ideally suited for operating this quantum switch.

Figure 3

Steady probabilities (blue) and (black) as functions of (i.e. ) with different intrastate interactions and . The interstate interaction is kept at zero. The shaded area of is suitable for switching. Relevant parameters are described in the main text.

The Switch Efficiency

To investigate the performance of the quantum switch, we first define the switching efficiency aswhere and are the steady population of in status “OFF” and of in status “ON”, respectively. The status is switched by turning up or down the interaction . For a ideal switch the population and the efficiency .

In Fig. 4(a-c) we show the dependence of on the interaction strength under the different relative couplings ( is already normalized by ). For comparison, the steady populations (blue dashed) and (black dotted) are presented in the same frame. reaches a saturation value and no longer changes with once , satisfying the two-atom strong blockade condition[24]. This brings us a big advantage at selections of state in practice, especially for the atoms with multiple Rydberg energy levels. Besides, the saturation value of is observed to be enhanced with the increase of , which is attributed to the slight changes of (green dot) and (blue dashed curve). For instance, in the case of the saturated . In the opposite case of , and rapidly falls to zero with the decrease of .

Figure 4

Steady populations (black dotted) and (blue dashed), and the switch efficiency (red solid) versus the interaction strength in the cases of (a) , (b) and (c) . is normalized by the Rabi frequency . is the value of at , marked by the green point, and is same as .

Except for and , we also explore the influence of other parameters on the switching scheme, including the interaction of the weakly-coupled Rydberg state, the spontaneous decay of the middle state, and the Rabi frequency . In Fig. 5(a) we find that keeps constant with . This is due to the isolation of state from the transfer pathway as presented in Fig. 1(b), so that the energy shift of induced by interaction has no effect on . However, an off-resonant transition characterized by detuning Δ will reduce the steady population of Rydberg state, resulting in a decrease of . We find that the switching scheme is robust to the middle-state decay . As displayed in Fig. 5(b), keeps almost unvaried in a broad regime of and start to slowly decrease only when . In contrast, a larger decay means a quick decay from Rydberg states, which directly causes a drop of .

Figure 5

The switching efficiency as a function of (a) the intrastate interaction for and the other parameters , , ; (b) the decay from intermediate state for the Rydberg-state decay and the other parameters , , ; (c) the interstate exchange interaction for the other parameters , , , . Additionally, and . The steady-state population and as a function of the driving laser are shown in (d). All frequencies are scaled by .

In addition to the parameters above, it should be stressed that the interstate interaction strongly destroys the switching efficiency. As shown in Fig. 5(c), rapidly decreases as long as . That is because a nonzero will shift level and hinder the transition from to , see Fig. 1(b). Worse, the competition between transitions of and is most serious when , resulting in a near zero . So the suppression of is a crucial condition for our approach of quantum switching. It can be guaranteed if we choose two nS states with large difference in principle quantum numbers n as Rydberg states. In the 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 (i.e. ). Since the common limit is unable to maintain in this case, the definition of is no longer rigorous. We then show and versus individually in Fig. 5(d). As the increase of , (black dashed) exhibits a clear reduction after reaching its maximum value 0.5 at . Similar trends are observed in (red solid) but the maximum 0.6 appears at about . The reductions are because that for a large the transition of is enhanced which results in a decrease of excitations to and , see Fig. 1(b). Hence, we conclude that 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 87Rb atoms are respectively confined in two independent optical dipole traps whose separation R can be adjusted from to by changing the incidence angle of the optical beams in a duration of the orders of several [24,37]. For the atomic states , , and Rydberg nS states , , the vdWs intrastate interaction coefficients are and , and the spontaneous decay rates are , , and (the effective lifetime is approximately and for and , respectively, at )[38]. In our switching operation, the initial separation is , leading to the interaction strength and . When R is reduced to , the interactions are enhanced to and . For Rydberg states and , we have due to the large difference in principle quantum numbers of the two Rydberg states[39], so that the interstate interaction is largely suppressed and can be safely neglected. The Rabi frequencies,  MHz and  MHz, are typical of current experiments.

To simulate the variation of the interaction under control, we introduce a time-dependent pulse sequence of for a complete switching cycle consisting of three status: OFF, ON, and OFF,where and take the previously estimated values of interaction strength for status “ON” and “OFF”, respectively, is the switching duration characterizing the changing speed of the interaction, and and are the critical switching moments of and , respectively. As shown in Fig. 6(a), the larger the is, the slower and smoother the switching operation between and is. The total duration of the switching cycle is which is less than the lifetime of the Rydberg states. As discussed before, the excitation is independent on the interaction , so for simplicity we assume it has a similar tendency of change with here.

Figure 6

(a) Time dependence of interaction in the switching process for different optical switching durations τ = 1.0 μs (black solid), 5.0 μs (blue dash-dotted) and 8.0 μs (red dashed); (b) The dynamical evolution of the population and under the sequence of in a normal and reverse order. The parameters adopted in the simulation are described in the main text.

The dynamical evolution of the population and , in response to the variation of interaction , are obtained by numerically solving the master equation and displayed in Fig. 6(b). Initially prepared in the two-atom ground state , the system tends to be a steady state with , when the external laser fields are carried out, so a slight tendency of increase(decrease) for is realized during the time of 0 < t < 20 μs in the case of τ = 1.0 μs (black solid line). Two conversions are clearly present at and . When the status “OFF” with the situation is maintained. A fast exchange of the population occurs around due to the switch , leading to the status “ON” with the reversed situation during the following period of time . The second conversion takes place around when the interaction 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 with a larger duration . This is because in status “OFF” except , and are also stably occupied, the resonant excitation between and 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 is dominantly occupied except for , so the resonant excitation between them is not isolated but suffers from a strong decoherence. Hence, when , even if 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 with the total operating time . 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 and . 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 () 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[41-44]. 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 87Rb 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.