1. Introduction

Rydberg atoms with the principal quantum number $n\gg 1$ are characterized by a significant separation between the electron and the ion core, leading to large polarizability [1]. This strong polarizability gives rise to a potent intra-atomic dipole-dipole interaction (DDI), which scales as $1/R^{3}$ at short distances and exhibits a van der Waals (vdW) type interaction, scaling as $1/R^{6}$ at longer ranges. At the crossover distance ($R_{c}$), the ratio of Rydberg interaction (at $100s$ state) to ground-state interaction becomes approximately $10^{12}$ [2]. The strong dipole-dipole interactions cause shifting of the Rydberg energy levels, resulting in the applied laser excitations becoming off-resonant. This forms the dipole blockade (DB) phenomenon [3,4]. The volume called the blockade sphere with blockade radius (BR), denoted by $R_b$, is defined as the region where only a single atom can be excited to a Rydberg state. In contrast, simultaneous excitation with more than one Rydberg atom is suppressed [5]. The blockaded atoms $N_b$ in the sphere are indistinguishable and comprise an effective superatom that interacts with the excitation light via collectively enhanced Rabi frequency $\sqrt {N_b}\Omega$ [6]. The Rydberg-based atomic system opens new opportunities to extend the two-qubit blockade gates [7] to multi-qubit gate operations [8,9]. Besides, the unique property of Rydberg atoms to control their interaction strength makes them excellent candidates for quantum information processing applications [2].

The two-atom Rydberg blockade was observed in 2009 using trapped Rb atoms while simultaneously showing collective Rabi oscillations as proof of a two-atom entangled state [10,11]. By combining Rydberg interactions with electromagnetically induced transparency, photon quantum gates can be realized [12–14]. A giant blockade radius creates a robust interaction between individual Rydberg atoms. It allows for more qubits to be placed at relatively more considerable distances, enabling more complex quantum computations to be performed. Therefore, finding the best transitions among Rydberg levels to achieve a giant blockade radius is essential for developing scalable and efficient quantum communication and quantum computation. To our best knowledge, the most significant BR of $23~\mu$m has been achieved so far in a hybrid ion-atom system [15]. Two Rydberg atoms transfer energy by DDI, with the efficiency depending on their distance, dipole orientation, and Rydberg state energy spacing. Over long distances, the vdW interactions become the dominant factor, while at shorter distances (usually $0.5-5~\mu$m), the magnitude of the DDI becomes comparable to the energy gaps between the states, and the interaction turns into the resonant type in a Förster resonance process [16]. The energy defect can be reduced by applying an external field, resulting in a Stark-tuned Förster resonance that dramatically enhances atomic interactions [17–21]. Additionally, an effective magnetic field was employed in a well-controlled geometry to increase the interaction [10].

In this study, we theoretically investigate the best transitions among Rydberg levels to achieve a giant blockade radius by considering different principal quantum numbers of the Rydberg states, participating in the Förster resonance. A schematic of a Förster resonance of the type $\gamma _{a}+\gamma _{b}\to \gamma _{\alpha }+\gamma _{\beta }$ is shown in Fig. 1 with initial states $|\gamma _{a}\rangle$, $|\gamma _{b}\rangle$ and final states $|\gamma _{\alpha }\rangle$, $|\gamma _{\beta }\rangle$. Here $\gamma _{a}\equiv (n_{a}, l_{a}, j_{a})$ represents the quantum numbers of the Rydberg level $a$, where $n$ is the principle quantum number, $l$ is the orbital angular momentum, and $j$ is the total electronic angular momentum [22]. We present the possible giant blockade radius formation at Förster resonances from the calculations of the magnitude of the dipole-dipole interaction in $^{87}$Rb–$^{87}$Rb atomic pairs. We introduce the methodology for calculating the blockade radii and the vdW interaction $C_6$ coefficients. The results are obtained in many cases when the difference between the initial principal quantum numbers is relatively small or large. The Rydberg blockade radius can be significantly improved and extended beyond 50 $\mu$m. Thus, achieving a giant Rydberg blockade radius through the Förster resonance is highly promising for scalable and efficient quantum communications and computations.

 

Fig. 1. Schematics of a Förster resonance transition with initial states $|\gamma _{a}\rangle$, $|\gamma _{b}\rangle$ and final states $|\gamma _{\alpha }\rangle$, $|\gamma _{\beta }\rangle$. The principal quantum numbers are $n_a$, $n_b$, $n_\alpha$, and $n_\beta$, where $n_b=n_a+\Delta _n$ in the calculation.

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2. Calculations of dipole blockade radii and DDI strengths

The dipole blockade phenomenon occurs when the energy shift caused by the DDI becomes comparable to or larger than the excitation linewidth, which is determined by various factors, including the excitation laser Rabi frequency $\Omega$. In this condition, the DDI becomes equal to the collectively enhanced Rabi frequency

In the limit where there is only one blockaded atom in the sphere, e.g., $N_b=1$, as can be seen directly from Eq. (1), this expression greatly simplifies to

We provide an overview of different Förster resonances for a $^{87}$Rb–$^{87}$Rb pair from Refs. [16,22,23] in Table 1, where the participating atoms are excited to states with the same [16], or different [22] principal quantum numbers. When the principal quantum numbers of the two atoms are the same, the strongest transition is $58d_{3/2}+58d_{3/2}{\rightarrow }60p_{1/2}+56f_{5/2}$ for any states with $n<70$, according to the calculations in Ref. [16]. By using Eq. (4), we can derive a dipole blockade radius of about $R_{b}=10.0~\mu$m for the experimentally accessible excitation laser Rabi frequency of 6 MHz (excluding the factor of $2\pi$). The further calculation of the blockade radius $R_b$ is performed in this paper using a 6 MHz Rabi frequency and comparing $R_b$ for different transition channels. When the initial principal quantum numbers are different, the DDI can be remarkably strong. For example, the $81s_{1/2}+84s_{1/2}{\rightarrow }81p_{1/2}+83p_{1/2}$ transition ($n_b=n_a+\Delta _n$ and $\Delta _n= 3$ in this case) [22] gives a dipole blockade radius of $R_b=22.4~\mu$m.

Table 1. The van der Waals coefficients $C_6$ and blockade radii for different $^{87}$Rb–$^{87}$Rb Förster resonances found in Refs. [16,22,23]. The blockade radii $R_b$ are calculated by applying the excitation Rabi frequency $\Omega = 6$ MHz. The crossover distances, $R_{c}$, are calculated by Eq. (11) with $D_{\varphi,k}$ set to 1.

View Table

We proceed to calculate the magnitude of the $C_6$ coefficients for specific Förster resonant transitions in atomic $^{87}$Rb–$^{87}$Rb pairs in order to search for the giant blockade radius condition. The Förster resonance case is depicted in Fig. 1: $\gamma _{a}+\gamma _{b}\to \gamma _{\alpha }+\gamma _{\beta }$ with two Rydberg atoms in the initial states $|\gamma _{a}\rangle$, $|\gamma _{b}\rangle$ and final states $|\gamma _{\alpha }\rangle$, $|\gamma _{\beta }\rangle$. The Förster defect $\delta _{k}$ represents the energy difference between the final and initial states participating in the Förster transition for channel $k$

The values of $\delta _0$ and $\delta _2$ for the $ns$, $np$, and $nd$ Rydberg levels are taken from the experimental study [24,25], while those for the $nf$ levels are obtained from Ref. [26].

The vdW interaction coefficient $C_{6,k}$ is defined as

Reference [23] provides a guide to calculating the coefficients and an online calculator.

The vdW interaction coefficient $C_{6,k}$ is then connected to the dipole-dipole interaction coefficient $C_{3,k}$ via the expression

The energy shift between the Förster levels follows a vdW form when the atomic distances $R$ are greater than $R_{c}$. Conversely, for $R$ values smaller than $R_{c}$, the energy shift is dominated by the dipole-dipole interaction (DDI). Note that we first assume the angular part $D_{\varphi,k}$ to be equal to 1 for all considered transitions, while in general, $D_{\varphi,k}\in [0,1]$. The crossover lengths are also shown in Table 1. Details about the calculation and analysis of $D_{\varphi,k}$ can be found in Refs. [16,22].

3. Results

Multiple cases of Förster transitions with equal initial principal quantum numbers are studied in great detail by Walker and Saffman [16]. In this paper, we study cases when the initial principal quantum numbers of the two atoms participating in the Förster transition differ, e.g., $n_{a}\neq n_{b}=n_{a}+\Delta _n$. The following two cases are discussed: (1) $n_a$ and $n_b$ differ by a small amount to at most $\pm 4$, i.e., $\Delta _n=\pm 1, \pm 2, \pm 3,\pm 4$, while those of the final states can be the same or can also be different; (2) $n_a$ and $n_b$ differ by a significant amount, e.g., $\Delta _n = 10 \sim 30$.

3.1 Small difference in initial quantum numbers

We first focus on Förster transitions involving atoms with a small difference in their initial principal quantum numbers. The strength of the long-range interaction between two atoms is quantified by the van der Waals coefficient $C_6$. A larger $C_6$ value indicates a stronger interaction and a higher potential for dipole blockade to be achieved. According to Eq. (8), a large $C_{6,k}$ coefficient is associated with a minimum in the absolute value of the Förster defect $\delta _{k}$ for channel $k$. Therefore, in order to find the transition giving the largest possible blockade radius, we plot $C_{6,k}$ and energy defect $\delta _{k}$ as a function of the principle quantum number $n_a$. Here $n_b$ is varied by $n_a\pm 1, \pm 2, \pm 3,\pm 4$, while those of the final states can be the same or different. Thirteen types of possible transitions are considered in the form of $n_{a}d_{3/2,5/2}+n_{b}d_{3/2,5/2}{\rightarrow }n_{\alpha }l_{\alpha }j_{\alpha }+n_{\beta }l_{\beta }j_{\beta }$, where $l_{\alpha, \beta }$ is $p$ or $f$ state. Thirteen transition channels from $n_{a}p_{1/2,3/2}+n_{b}p_{1/2,3/2}$ are also considered. Since the fine structure splitting of the $f$ state is small, both the $f_{5/2}$ and $f_{7/2}$ states give the same result. The excitation of $s_{1/2}$ coupled to $p_{1/2}$ or $p_{3/2}$ has been discussed in Ref. [22] for the cases when $\Delta _n < 4$. However, as $\Delta _n$ increases, the minimum Förster defect occurs at $n_a>100$, which is outside of our consideration. Consequently, we depict three transitions with higher interaction strengths in Fig. 2, and these transitions are indicated by their respective colors in the inset table. The blue ones correspond to a specific transition listed in Table 1. The set principal quantum numbers $n_{\alpha }$ and $n_{\beta }$ are varied according to the selected transitions. The results for different $\Delta _n$ up to $\pm 4$ are shown within the band area. The van der Waals coefficients $C_6$ are calculated using the Alkali Rydberg Calculator (ARC) PYTHON package [23]. Figure 2 shows diverging behavior of the absolute values of $\delta _{k}$ curves, and it does not demonstrate a minimum. The $C_6$ increases monotonically with $n_a$ and reaches its maximum around $5\times 10^4$ GHz $\mu$m$^6$ when $n_a$ approaches 100. For the transitions of $d_{3/2,5/2}+d_{3/2,5/2}{\rightarrow }p_{1/2,3/2}+f_{5/2,7/2}$ (blue ones), the strongest interaction occurs when $\Delta _n=0$ and $n_a=58$ for $n_a<70$, as indicated in Table 1 and in Ref. [16]. For $n_a>70$, the vdW coefficient $C_6$ starts to diverge. Due to this divergence behavior (i.e., no singularity point for $\Delta _n>0$ and $n_a<100$), we do not present the calculation for this transition.

 

Fig. 2. a) The van der Waals coefficient $C_6$ and the Förster defect versus principal quantum number $n_{a}$ when atoms are initially in the $p$ or $d$ states. The set principal quantum numbers $(n_{a}, n_{b}, n_{\alpha }, n_{\beta })$ and relevant transitions are shown in the inset table. The principal quantum numbers $n_a$ and $n_b$ of the initial states differ by a small amount, e.g., $\Delta _n=\pm 1, \pm 2, \pm 3, \pm 4$ (the shown number in the figures).

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3.2 Significant difference in initial quantum numbers

An interesting case of the Förster transition is when the initial principal quantum numbers of the two atoms differ significantly. A detailed study of the Förster resonance for the transition of $60p+80p{\rightarrow }59d+78d$, where $\Delta _n=20$, has been studied in Ref. [27]. Following this example, we extend the study of the Förster resonance for all possible transitions in the $p-$ and $d-$states manifolds. First, we consider the transitions of the form $n_{a}p_{1/2, 3/2}+n_{b}p_{1/2, 3/2}{\rightarrow } n_{\alpha }l_{\alpha }j_{\alpha }+n_{\beta }l_{\beta }j_{\beta }$, while keeping $\Delta _n=20$. The obtained results, shown in Fig. 3(b), indicate that all curves are crossing the $\delta _{k}=0$ line. The nearly zero Förster defect appears for special values of $n_a$, in the vicinity of $n_a=60$. For the transition $n_{a}p_{1/2}+(n_{a}+20)p_{1/2}{\rightarrow }(n_{a}-1)d_{3/2}+(n_{a}+18)d_{3/2}$ (orange points and dashed line), the minimum of the Förster defect occurs at $n_{a}=65$, which corresponds to $\delta _{k}=3.6$ MHz. Figure 3(a) shows the singularity points in the vdW coefficient $C_6$ calculation, which yields $C_6=-2.1 \times 10^5$ GHz $\mu$m$^6$. Using Eq. (4), we can achieve a blockade radius of $R_b=18~\mu$m for excitation Rabi frequency of $\Omega =6$ MHz. The distinct variations in the Van der Waals coefficient and the Förster defect, as illustrated in Fig. 3 for different Förster transitions, predominantly govern from the attributes of the radial wave functions and the energies of Rydberg levels (additionally influenced by the orbital angular momentum of the Rydberg states). It is important to note that the angular factor, $D_{\varphi }$, exhibits significant magnitude in these transitions, which do not include the Förster zeros [16]. These angular factors range between 0.44 and 0.84, encompassing the influence of Zeeman degeneracy. As a result of this resonance condition and the strong interaction, these transitions are highly promising candidates for conducting dipole blockade experiments.

 

Fig. 3. The van der Waals coefficient $C_6$ and the Förster defect versus principal quantum number $n_{a}$ with $\Delta _n=20$. The bottom tables show the the corresponding transitions, e.g., $n_a p_{1/2,3/2}+n_b p_{1/2,3/2}{\rightarrow }n_{\alpha }d_{3/2,5/2}+n_\beta d_{3/2,5/2}$ in (a)&(b). The principal quantum numbers $(n_{a}, n_{b}, n_{\alpha }, n_{\beta })=(n_{a},n_{a}+20, n_{a}-1, n_{b}-2)$ in (a), (b), (e), and (f). In (c)&(d), they are $(n_{a}, n_{b}, n_{\alpha }, n_{\beta })=(n_{a},n_{a}+20, n_{a}+1, n_{b}+2)$.

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Next, we keep the same relation between the initial and final principle quantum numbers but explore the $d$-states manifold, e.g., we study transitions of the form $n_{a}d_{3/2}+(n_{a}+20)d_{3/2}{\rightarrow }n_{\alpha }p_{1/2,3/2}+n_{\beta }p_{1/2,3/2}$, shown in Figs. 3(c) and 3(d). The $C_6$ values are much smaller than those cases in Figs. 3(a) and 3(e). Note that in these cases, we obtain the largest value for $C_6$ by setting $n_\alpha =n_a+1$ and $n_\beta =n_b+2$, which can also be derived from the calculation of radial matrix elements, $d_1$ and $d_2$. In this quantum number set, the $C_6$ values (or energy defect) for $d_{3/2}+d_{3/2}{\rightarrow }f_{5/2}+p_{1/2,3/2}$ are extremely small (or large) compared with the present ones in Fig. 3(c) (or Fig. 3(d)), and therefore, the results are not shown in the figures. However, these transition channels have at least one Förster-zero state with zero dipole-dipole interactions, which limits their use in blockade experiments.

Moreover, the cases of $n_{a}d_{3/2, 5/2}+(n_{a}+20)d_{3/2, 5/2}{\rightarrow }n_\alpha f_{5/2}+n_\beta f_{5/2}$ are shown in Figs. 3(e) and 3(f). The minimum of the Förster defect or the singularity points of $C_6$ occurs at $n_{a}=75$. The minimum $\delta _{k}=9.5$ MHz, and $C_6=-2.7\times 10^5$ GHz $\mu$m$^6$, corresponding to $R_b=19~\mu$m. Due to the significant angular factor, $D_{\varphi }$ (between 0.21 and 0.99), the transitions would be excellent candidates for dipole blockade experiments.

3.3 Varied difference in initial quantum numbers

Next, we vary the difference between the initial principle quantum numbers of the two atoms $\Delta _n$ up to 30. The strongest transitions in the $p-$ and $d-$states manifolds are shown in Fig. 4. For a given value of $\Delta _n$, it is possible to identify a singularity point, representing the Förster resonance, which significantly enhances the blockade radius. The absolute values of $C_6$ versus $n_a$ for $\Delta _n = 10, 15, 20, 25, 30$ are shown in Figs. 4(a) – 4(c). When $\Delta _n$ increases, the singularity point shifts to a higher value of $n_a$, which can be found for the same color points/lines from left to right. Take the orange samples in Fig. 4(a) as an example. The singularity points are observed at $n_a$ of 33, 49, 65, 81, and 97 for corresponding $\Delta _n$ values of 10, 15, 20, 25, and 30, respectively. The absolute values of $C_6$ increases from 15 to $3.0\times 10^7$ GHz $\mu$m$^6$, leading to an increase in the blockade radius $R_b$ from 3.7 to 41 $\mu$m for $\Omega =6$ MHz. Likewise, the cases depicted in Figs. 4(b) and 4(c) exhibit similar behavior.

 

Fig. 4. Similar to Fig. 3 but $\Delta _n$ is varied. The results for $\Delta _n = 10, 15, 20, 25$, and $30$ are shown in solid squares, solid triangles, solid circles, open squares, and open triangles, respectively. Take orange samples in (a) as an example. The singularity points occur at $n_a$ of 33, 49, 65, 81, and 97 when $\Delta _n = 10, 15, 20, 25$, and $30$, respectively. The absolute values of $C_6$ increases from 15 to $3\times 10^7$ GHz $\mu$m$^6$. In the calculation, the principal quantum numbers $(n_{a}, n_{b}, n_{\alpha }, n_{\beta })=(n_{a},n_{a}+\Delta _n, n_{a}-1, n_{b}-2)$ in the left and right columns. In the middle column, they are $(n_{a}, n_{b}, n_{\alpha }, n_{\beta })=(n_{a},n_{a}+\Delta _n, n_{a}+1, n_{b}+2)$. (d)-(f) The extracted absolute values of $C_6$ and the corresponding blockade radii from the singularity points for each $\Delta _n$ from 10 to 30. (g)-(i) The crossover distances, $R_{c}$ calculated by Eq. (11) with $D_{\varphi,k}$ set to 1 for all channels and the given set of quantum numbers are depicted in (d)-(f).

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We further extract the absolute values of $C_6$ and the corresponding blockade radii from the singularity points for each $\Delta _n$ from 10 to 30, and show them in Figs. 4(d) – 4(f). Once again, let us consider the transitions shown in orange as an example. Notably, at $\Delta _n = 15$, the magnitude of $|C_6|$ is one order of magnitude greater compared to the values at $\Delta _n =$ 14 or 16. Similar phenomena are observed in the results for $\Delta _n = 20$ and $25$. Therefore, by optimizing $n_a$ and $n_b$ ($n_a+\Delta _n$) and considering $n_a<100$, we can find the largest blockade radius for a given transition. The calculation for other transitions are shown in Figs. 4(d)–4(f). The largest blockade radius in all possible channels is 53 $\mu$m for the transition of $98d_{3/2}+124d_{3/2}{\rightarrow }97f_{5/2}+122f_{5/2}$ (red one for $\Delta _n=26$) with the Förster defect of 0.16 MHz and $C_6$ of $-1.4\times 10^8$ GHz $\mu$m$^6$.

4. Analysis

According to our systematic calculation, we have observed that when there is a small difference in the initial principal quantum numbers (e.g., $\pm 1, \pm 2, \pm 3, \pm 4$), the curves of $C_6$ values or energy defect $\delta _{k}$ versus $n_{a}$ exhibit a diverging behavior. This characteristic makes them unsuitable for achieving a dipole blockade. However, when this difference is sufficiently large, e.g., $\Delta _n>10$, giant dipole blockade radii can be achieved by using Förster resonances in the initial $p-$ or $d-$ manifolds. For any given value of $\Delta _n$, both the $C_6$ coefficient and the crossover distance $R_c$ exhibit a notable transformation at a specific principal quantum number $n_a$, marking a singularity point where the Förster defect $\delta _{k}$ approaches zero, as depicted by Eqs. (9) to (11). Alongside $\delta _{k}$, the magnitudes of $C_6$ and $R_c$ at this singularity point are also influenced by the radial matrix elements. From the optimized setting of $n_a$ and $n_b$, we can determine the maximum blockade radius for a given transition.

The significant blockade radii obtained from the relatively small Förster defects prompt a discussion about the dominance of van der Waals (vdW) interactions compared to resonant dipole interactions. The transition between these interaction types occurs at a characteristic length scale, $R_c$, as defined in Eq. (11). By examining the large observed blockade radii, we can determine the influence of non-resonant vdW interactions versus resonant dipole interactions. The calculation of crossover distances $R_c$ for specific transitions and quantum numbers (as depicted in Figs. 4(d)–4(f)) are presented in Figs. 4(g)–4(i) with $D_{\varphi,k}$ set to 1 for all channels. We discovered that in numerous transition channels, it is possible to achieve a remarkably large blockade radius surpassing $23~\mu$m, which currently stands as the largest measured blockade radius in experiments. There is a significant increase in interaction strength and distance as a result of the dipole blockade effect. When the crossover distance $R_{c}$ is larger than the corresponding blockade radius $R_{b}$ (for a given excitation Rabi frequency, $\Omega$), we conclude that the interaction is in the range of the resonant dipole interaction of $1/R^3$ type. This indicates that the interaction strength is governed by the resonant coupling between the atoms, resulting in a dipole blockade effect.

In the crossover distance $R_{c}$ calculations, we assumed a fixed value of the angular factor $D_{\varphi,k}$ as 1 in Figs. 4(g)–4(i). It is worth noting that in Ref. [16], the values of $D_{\varphi,k}$ were provided for 23 different transition channels, taking into account the initial $s$, $p$, and $d$ states along with the effects of Zeeman degeneracy. Considering the specific arrangement of angular momenta, certain transition channels can have a smaller or zero value of the angular factor, leading to an automatic vanishing of the dipole-dipole interaction strength and rendering them unsuitable for observing the dipole blockade effect. There are several Förster zero channels ($D_{\varphi,k}=0$) for the transitions in the central column of Fig. 4 and some channels have small $D_{\varphi,k}$ values in the left column. Fortunately, the green and orange channels in the left column ($p_{1/2,3/2}+p_{1/2,3/2}{\rightarrow }d_{3/2,5/2}+d_{3/2,5/2}$) and the channels in the right column ($d_{3/2,5/2}+d_{3/2,5/2}{\rightarrow }f_{5/2}+f_{5/2}$) have sufficiently large $D_{\varphi,k}$ values, ranging from 0.21 to 0.99. According to Eq. (11), the effective crossover distances would only be reduced by 23${\%}$ to 0.2${\%}$, depending on the experimental geometry and laser polarization. Applying an appropriate magnetic field to break the Zeeman degeneracy can prevent Förster zeros and consequently significantly influence the Rydberg-Rydberg interaction. The transitions we considered have significantly large values of $D_{\varphi,k}$, suggesting their potential for applications that demand a substantial blockade radius. This characteristic makes them particularly promising for areas such as quantum information processing and quantum simulation.

Furthermore, as explored in this study, the theoretical framework for adjusting Rydberg-Rydberg interaction strengths and blockade radii holds promise for practical experimental implementation. To excite initial $s$ or $d$ Rydberg states, we can apply a two-photon excitation method via an intermediate $p$ state, as demonstrated in previous works [12,13,15] for values of $n_{a,b}$ up to 90. Typically, two-photon excitation uses infrared and blue radiation, allowing implementation with readily available lasers. For the $P$ Rydberg state, experimental excitation can be achieved using either a three-photon scheme [28,29] or a single-photon scheme using an ultraviolet (UV) laser [30]. Given that the energy defect between the Rydberg pair states falls within the range of a few MHz, we can attain Förster resonance transitions by applying a weak external DC electric field to manipulate the Rydberg energy levels, relying on the Stark effect. The precise control of Rydberg interactions is important, as it provides an ideal platform for realizing coherent interactions and offers a valuable tool for simulating complex many-body quantum systems.

5. Conclusions

By conducting an in-depth investigation into various Förster resonance transitions within the $p$ and $d$ manifolds of a $^{87}$Rb atom pair, we have achieved a profound understanding of the dipole-dipole interaction strength and the corresponding dipole blockade radius for each transition. Our analysis of the dependence of the Förster defect on the small $\Delta _n$ case (where the initial principle quantum numbers of two Rydberg atoms are not significantly different) revealed that the Förster transitions are unsuitable for achieving dipole blockade. However, a remarkable breakthrough emerged when $\Delta _n\geq 10$. We observed that the Förster defect $\delta _{k}$ approaches zero in numerous cases, resulting in a singularity of $C_6$. In some transition channels, we found that a giant blockade radius exceeding $23~\mu$m (the current largest measured blockade radius in experiments) can be achieved, with the most extensive blockade radius above 50 $\mu$m present. The significance of our work lies in the substantial enhancement of the dipole blockade effect in terms of both interaction strength and distance. These outcomes will ignite interest in exploiting strong interactions in atom pairs to establish long-range coupling, opening up exciting possibilities for revolutionary advancements in quantum information processing and quantum communication technologies.