Rydberg antiblockade regimes: Dynamics and applications

When the RRI is weak enough, i.e., $V\leq0.1{\Omega}$ (here, Ω denotes the Rabi frequency), the RRI almost has no influence on the excitation process, and RAB would appear naturally [3]. We consider two Rydberg atoms interacting with each other through vdW-type RRI (as shown in fig. 1(a)) and DD-type RRI (as shown in fig. 1(b)), respectively. In the interaction picture, the Hamiltonians of these two models can be written as

$$\begin{eqnarray} \hat{H}_{v}&= \left(\displaystyle\sum_{j=1}^{2}\Omega/2|1\rangle_{j}\langle R| + {\rm H.c.}\right) + V|RR\rangle\langle RR|,\\ \hat{H}_{d}&= \displaystyle\sum_{j=1}^{2}\Omega/2|1\rangle_{j}\langle R|+V|RR\rangle\langle rr| + {\rm H.c.}, \end{eqnarray} \tag{ 1 }$$

respectively. Here and throughout this paper, the subscript j denotes the j-th atom, and we take the abbreviation $|mn\rangle$ to denote $|m\rangle_{1}\otimes|n\rangle_2$. Through solving the Schrödinger equation $i\dot{\psi}=\hat{H}_{v(d)}\psi$ and using the definition of population $P_{RR}=|\langle \psi|RR\rangle|^2$, we plot $1-P_{RR}$ in fig. 2 to verify the RAB. The results show that the population of the two-excitation Rydberg state could also be higher than 0.99 with $V/\Omega$ equal to 0.1 for both vdW- and DD-type RRIs.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Reference [50] proposed another type of weak-DD-RRI-based RAB via Stark-tuned Förster resonance, and shows how to construct the quantum logic gate based on the proposed regime in ref. [50]. In quick succession, they experimentally demonstrate the resonant RRI [51] and generalize the idea in ref. [50] to a many-body case [52].

The RAB based on weak vdW-type RRI can be used to construct a quantum gate with three steps [3]. Step i): Exciting two Rydberg atoms from $|11\rangle$ to $|RR\rangle$. Step ii): Turning off the lasers and waiting for a time interval t, and then the phase $\theta = V\times t$ is accumulated on $|RR\rangle$. Step iii): Turning on the lasers again to perform the inverse operation of step i). Through this process, $|11\rangle$ gets a phase θ. The weak DD-type RRI-based RAB can also be used to construct a similar gate, but the evolution time in the subspace $\{|RR\rangle, |rr\rangle\}$ should be restricted to make the final state of step ii) be in $|RR\rangle$.

On the other hand, in order to use the RAB regime proposed in refs. [50–52] to construct quantum logic gates, the Rydberg states should be chosen to minimize the RRI initially to avoid the blockade effect. Then the excited Rydberg states are tuned to resonance and the desired phase would be accumulated through this resonant process. Finally the Rydberg states are de-excited to transfer the phase into ground states [50,52].

Under the parameter range V∼Ω, the two-excitation Rydberg states cannot be ignored in the process of evolution. On that basis, how to suppress the blockade error is discussed in refs. [58,59] in detail. In this section, we would show the population dynamics based on the intermediate RRI regime via numerically solving the Schrödinger equation. Then we would review the possible applications in quantum logic gates with the rational generalized Rabi cycles [58] or interferentially detuned Rabi cycles [59]. In fig. 3, we plot the population of $|RR\rangle$ vs. evolution time. In contrast to the case based on weak RRI, the complete occupation of $|RR\rangle$ cannot be achieved. After investigating this regime more comprehensively, refs. [58,59] proposed how to construct the robust Rydberg quantum logic gates by using the detuned Rabi oscillation.

Zoom In Zoom Out Reset image size

Now we briefly introduce the basic thoughts of refs. [58,59] to construct the quantum logic gate with the intermediate RRI. As shown in fig. 4(a), the vdW-type RRI-based scheme requires three steps [58]. Step i): Exciting atom 1 with the Hamiltonian $\hat{H}_{1} = \Omega/2|1\rangle_1\langle R| + {\rm H.c.}$ to achieve the transition $|1\rangle\rightarrow|R\rangle$. Step ii): Exciting atom 2 with the Hamiltonian $\hat{H}_{2} = \Omega/2|1\rangle_2\langle R| + {\rm H.c.} + V|R\rangle_1\langle R|\otimes|R\rangle_2\langle R|$ to achieve the detuned Rabi cycle $|1\rangle\rightarrow|\phi\rangle\rightarrow e^{i\theta}|1\rangle$ on atom 2, in which the expressions of $|\phi\rangle$ and the value of θ depend on whether RRI exists or not. Step iii): Performing the inverse operation of Step i). After these three steps, the operations $\{|00\rangle\rightarrow|00\rangle,~|10\rangle\rightarrow|10\rangle,~|01\rangle\rightarrow e^{i\theta_{1}}|01\rangle,~|11\rangle\rightarrow e^{i\theta_{2}}|11\rangle\}$ are achieved and one can modulate the Ω and evolution time t to get the desired gate. For DD interaction, the schematic diagram is shown in fig. 4(b), and the process is almost the same as that of the vdW interaction. However, since the dynamics of DD-type RRI is more complex than that of vdW-type RRI, the expressions of $\theta_{1}$ and $\theta_{2}$ are different from that of the vdW-type RRI-based scheme. And the quantum logic gate can be achieved by modulating the parameters and setting an appropriate evolution time. Then the scheme is generalized to one step case through using the interferentially detuned Rabi cycles to minimize the imperfections of the dephasing error [59].

Zoom In Zoom Out Reset image size

Before describing the RAB with simultaneous driving, we first show the Rydberg blockade regime induced by RRI. As shown in fig. 5(b), considering two Rydberg atoms driven by classical field resonantly with Rabi frequency Ω. In the interaction picture, the Hamiltonian can be written as (under two-atom basis)

$$\begin{equation} \hat{H}_{rb}=\sqrt{2}\Omega/2(|11\rangle\langle T|+ |T\rangle\langle RR| + {\rm H.c.})+V|RR\rangle\langle RR|, \end{equation} \tag{ 2 }$$

where $|T\rangle=(|1R\rangle+|R1\rangle)/\sqrt{2}$ denotes the single-excitation state and V denotes the RRI strength. From eq. (2), one can see that the process $|11\rangle\leftrightarrow|T\rangle$ is resonant while $|T\rangle\leftrightarrow|RR\rangle$ is detuned by V. If the condition V > Ω is fulfilled, the two-excitation state $|RR\rangle$ is blocked.

If we replace the resonant interactions by dispersive interactions with detuning Δ, Hamiltonian (2) can be rewritten as

$$\begin{equation} \hat{H}_{rab}+=\frac{\Omega}{\sqrt{2}}[(|11\rangle\langle T|+ |T\rangle\langle RR|)e^{i\Delta t} + {\rm H.c.}]+V|RR\rangle\langle RR|. \end{equation} \tag{ 3 }$$

Then, if $\Delta=V/2$ is satisfied, the effective Hamiltonian would be

$$\begin{equation} \hat{H}_{\rm eff}=\frac{\Omega^2}{2\Delta}|11\rangle\langle RR|+{\rm H.c.} + \hat{S}, \end{equation} \tag{ 4 }$$

where $\hat{S}$ denotes Stark shifts. In fig. 6, we plot the populations of $|RR\rangle$ with full and effective Hamiltonians, respectively. These two curves fit each other very well and both of them approach to unit at the predicted time, which means the RAB can be achieved under strong vdW-type RRI.

Zoom In Zoom Out Reset image size

This RAB dynamics with strong vdW interaction has been well studied in refs. [25,26] via the second-order perturbation theory and used to study the motional effects [27], dissipation-based quantum state engineering [28–39], periodically driving [40] as well as quantum computation [41–49]. For the dissipation-based quantum state engineering, ground state $|0\rangle$ should be considered, and the coupling between two ground state should be designed skillfully [28,60,61] to create a unique steady state of the system. The dynamical process for the dissipation-based entangled state preparation with the RAB regime is shown in fig. 7(a). For a quantum logic gate, the other computational basis $|0\rangle$ is also required, and the laser coupling should be designed carefully [41,42], as shown in figs. 7(b) and (c).

Zoom In Zoom Out Reset image size

In addition, this regime can also be generalized to resonant driving [45,46], and the many-body case [43,62]. Next, this regime is used to construct robust quantum logic gates via holonomic operations [48]. Very recently, the RAB with vdW has also been studied in many-body spin-phonon systems [56,57], open systems [63] as well as in quantum simulators [64]. Especially, in ref. [56], the authors show how to use the spin-phonon interactions and RAB to generate the tunable many-body interaction. On that basis, the phase diagram in the classical limit was also studied for the isotropic square lattice. In ref. [57], the RAB is employed to facilitate the dynamics and get the desired transitions in the many-body configuration.

The dynamics of simultaneous driving with strong vdW discussed above is based on two- or higher-order perturbation theory. If we can achieve the RAB with the resonant zero-order interaction, the required time of the excitation process would be reduced and the robustness would be enhanced with the decoherence environment.

The basic dynamical process of sequential-driving-based RAB with vdW-type RRI is discussed in detail in ref. [49], and the dynamical diagram is shown in fig. 8(a). Firstly, driving atom 1 resonantly with Rabi frequency Ω. The interaction Hamiltonian is $\hat{H}_{1}=\Omega/2|1\rangle\langle R| + {\rm H.c.}$ After a π pulse, the transition $|1\rangle\rightarrow|R\rangle$ is achieved. Secondly, exciting atom 2 with Hamiltonian $\hat{H}_2 = \Omega/2|1\rangle\langle R|e^{i\Delta t} + {\rm H.c.}$ and the RAB condition for sequential driving is $\Delta =V\gg\Omega$. If atom 1 is excited, the Hamiltonian with the two-atom basis can be written as $\hat{H}=|R1\rangle\langle RR|e^{i\Delta t}+{\rm H.c.} +V|RR\rangle\langle RR|$. Then, after a π pulse, the two-atom excitation state $|RR\rangle$ would be achieved. Otherwise, the state of atom 2 would remain invariant because of the large detuning coupling. This regime was also generalized to the multiple-qubit case [49].

Zoom In Zoom Out Reset image size

This regime can be used to construct the fast quantum logic gate with three steps, as shown in fig. 8(b). The first step is to resonantly excite $|1\rangle$ to $|R\rangle$ of the control atom via π pulse. The second step is to dispersively apply a $2\pi$ pulse on the target atom to couple $|1\rangle$ with $|R\rangle$. The detuning satisfies the RAB condition $\Delta=V$. The last step is the inverse operation of the first step to realize $|R\rangle\rightarrow|1\rangle$ of the control atom. After these three steps, only the state $|11\rangle$ gets a π phase. Besides, the Rydberg excitation superatom to minimize the blockade error can also be achieved [49] based on this sequential driving RAB. The multi-qubit Rydberg numerical results of antiblockade under sequential driving are shown in fig. 9.

Zoom In Zoom Out Reset image size

As shown in fig. 10, the resonant DD-type RRI can be roughly divided into two types, one involving three Rydberg states in each atom, as shown in fig. 10(a), while the other containing two Rydberg states, as shown in fig. 10(b). The RAB with these two kinds of RRIs is discussed detailedly in ref. [65]. In the following, we would briefly review these processes.

Reference [65] considers how to achieve the RAB with energy level shown in fig. 10(a). Two lasers are required to illuminate the Rydberg atoms to drive the transition $|1\rangle\leftrightarrow|d\rangle$ with an identical Rabi frequency Ω but opposite detuning Δ. The Hamiltonian in the interaction picture with rotating-wave approximation is $\hat{H}=\hat{H}_{\Omega}+\hat{H}_{V}$ [65], in which

$$\begin{eqnarray} \hat H_{\Omega}&= \frac{\Omega}{2}(e^{i\Delta t}+e^{-i\Delta t})(|1\rangle_{1}\langle R|+|1\rangle_{2}\langle R|)+{\rm H.c.},\nonumber\\ \hat H_{V}&= \sqrt{2}V|RR\rangle\langle r_{pf}|+{\rm H.c.}, \end{eqnarray} \tag{ 5 }$$

where $|r_{pf}\rangle\equiv(|pf\rangle+|fp\rangle)/\sqrt{2}$. In the two-atom basis, the Hamiltonian (5) can be rewritten as

$$\begin{eqnarray} {\hat H}_{\Omega}&= \frac{\Omega}{\sqrt2}\left(e^{i\Delta t}+e^{-i\Delta t}\right)\big[| 11\rangle\langle \Psi|+\frac1{\sqrt2}|\Psi\rangle(\langle +|+\langle -|)\big]\nonumber\\ & \quad+\frac{\Omega}{2}\left(e^{i\Delta t}+e^{-i\Delta t}\right)(|01\rangle\langle 0R|+|10\rangle\langle R0|)+{\rm H.c.}\nonumber\\ \hat{H}_{V}&= \sqrt{2}V(|+\rangle\langle +|-|-\rangle\langle -|), \end{eqnarray} \tag{ 6 }$$

where $|\Psi\rangle\equiv(|1R\rangle+|R1\rangle)/\sqrt2$ and $|\pm\rangle\equiv(|RR\rangle\pm|r_{pf}\rangle)/\sqrt2$. From eq. (6), we can understand the Rydberg blockade with DD interaction and can know how to get the RAB condition. When $\Delta=0$, although $\hat{H}_{\Omega}$ describes resonant interactions, the total Hamiltonian indicates that the two-excitation Rydberg states would be blocked due to the energy shift induced by RRI when V > Ω is satisfied. To achieve the RAB, we rotate eq. (6) with respect to $\hat{H}_V$. If the conditions $\Delta \gg \Omega$ and $V=\sqrt2\Delta-\Omega^2/(3\sqrt2\Delta)$ are satisfied, the effective form of eq. (6) in the rotating frame with respect to $e^{-i\hat{H}_Vt}$ can be achieved as [65]

$$\begin{eqnarray} &&\hat H_{\rm e}=\frac{\sqrt{2}\Omega^2}{4\Delta}|11\rangle(\langle +| - \langle -|)+{\rm H.c.} \end{eqnarray} \tag{ 7 }$$

From eq. (7), one can see that the state $|11\rangle$ is resonantly coupled to the two-excitation Rydberg state with the effective Rabi frequency $\Omega_{\rm eff}\equiv\Omega^2/(\sqrt{2}\Delta)$, which indicates that the RAB is achieved.

We now review the basic dynamics of the RAB with energy levels shown in fig. 10(b). Consider the Rydberg atom 1 (2) with two ground states $|0\rangle$ and $|1\rangle$ as well as two Rydberg states $|R\rangle$ and $|r\rangle~(|r'\rangle)$. Two lasers drive off-resonantly the atoms to achieve $|1\rangle\leftrightarrow|R\rangle$ with an identical Rabi frequency Ω but opposite detuning Δ. After the rotating-wave approximation, the Hamiltonian can be written as [65]

$$\begin{eqnarray} \hat H_{\Omega}&= \frac{\Omega}{2}\left(e^{i\Delta t}+e^{-i\Delta t}\right)(|1\rangle_{1}\langle R|+|1\rangle_{2}\langle R|) +{\rm H.c.},\nonumber\\ \hat H_{V}&= V|RR\rangle\langle rr'|+{\rm H.c.} \end{eqnarray} \tag{ 8 }$$

Through using the dressed states $|\widetilde{\pm}\rangle\equiv(|RR\rangle\pm|rr'\rangle)/\sqrt{2}$, one can rewrite eq. (8) as

$$\begin{eqnarray} \hat H_{\Omega}&= \frac{\Omega}{\sqrt2}\left(e^{i\Delta t}+e^{-i\Delta t}\right)[|11\rangle\langle \Phi|+\frac{1}{\sqrt{2}}|\Phi\rangle(\langle \widetilde{+}|+\langle \widetilde{-}|)]\nonumber\\ & \quad+\frac{\Omega}{2}\left(e^{i\Delta t}+e^{-i\Delta t}\right)(|01\rangle\langle 0d|+|10\rangle\langle d0|)+{\rm H.c.},\nonumber\\ \hat{H}_{V}&= V(|\widetilde{+}\rangle\langle \widetilde{+}|-|\widetilde{-}\rangle\langle \widetilde{-}|) \end{eqnarray} \tag{ 9 }$$

with $|\Phi\rangle\equiv(|1R\rangle+|R1\rangle)/\sqrt{2}$. If the relation Δ > Ω and RAB condition $V_d=2\Delta-\Omega^2/(3\Delta)$ are satisfied, one can get the effective Hamiltonian as [65]

$$\begin{equation} \hat{H}_e=\frac{\Omega^2}{2\Delta}|11\rangle\langle \widetilde{+}| -\langle \widetilde{-} |+{\rm H.c.}, \end{equation} \tag{ 10 }$$

which means the Rabi oscillation between $|11\rangle$ and the two-excitation Rydberg state emerges. If one chooses the evolution time $t=\pi\Delta/\Omega^2$, the RAB would be implemented.

Zoom In Zoom Out Reset image size

In fig. 11, we plot the populations of $|11\rangle$, $|+\rangle$ and $|-\rangle$ vs. time by using the full Hamiltonian without considering the $|0\rangle$ state. One can see that the RAB can be realized at the predicted time. As discussed in ref. [65], this regime can be used to realize geometric quantum computation, dissipative-dynamics-based entanglement preparation, and rough estimation of physical parameters. One can also apply the shortcut-to-adiabaticity technology [66–68] into the RAB to realize more robust quantum information processing tasks [33].

Zoom In Zoom Out Reset image size

In ref. [23], the authors considered to break the blockade regime through using the dark state. They consider three identical Rydberg atoms (atom 1, atom 2, atom 3) that interact with each other through DD interactions. Atom 3 is equidistant between atoms 1 and 2 with interaction strengths $V = V_{13} = V_{23}$ and $V' = V_{12}$. By diagonalizing the Rydberg DD interaction Hamiltonian, one can get the dark state as [23]

$$\begin{eqnarray} |m_{4}\rangle&= [V'(|dfp\rangle+|dpf\rangle+|fdp\rangle +|pdf\rangle),\nonumber\\ & \quad- 2V(|fpd\rangle+|pfd\rangle)]/\sqrt{{V'}^2+2V^2}, \end{eqnarray} \tag{ 11 }$$

where $|p\rangle$, $|d\rangle$, and $|f\rangle$ denote three Rydberg states. For weak fields, the triple-excitation probability monotonically decreases with increasing Förster defect and is maximal when the Förster defect is zero, which means that the RAB is achieved [23].

Differently from all of the above RAB regimes and the conventional Rydberg blockade regime [3,4], refs. [69–71] proposed schemes to realize robust unconventional pumping with vdW- and DD-type RRI, respectively. These schemes may also be feasible to accomplish the same quantum information tasks that can be accomplished by the RAB regimes.