1. Introduction
With the continuous development of experimental technologies, such as laser cooling and evaporative cooling, ultracold atomic physics has become the frontier of current physics [1, 2]. Compared with traditional condensed-matter physical systems, the ultracold atomic system has distinct characteristics and advantages due to its highly experimental controllability and observable macroscopic quantum properties. It not only provides an ideal platform for the research of atomic and molecular physics, condensed matter physics, quantum information, quantum optics and other fields, but it also has potential application prospects in precision measurement technology, high-precision atomic clock, atomic lasers, and so on [3–5].
Generally speaking, the interactions between atoms can be divided into short- and long-range interactions. The short-range interaction is usually referred to as a hard-core interaction, which can be modulated by using the Feshbach resonance with a tunable time-dependent magnetic field [6, 7]; while the long-range one can be realized through dipole-dipole and Rydberg interactions, which can be regulated by magnetic fields [8] and the Rydberg dressing technology [9–11], respectively. Compared with the hard-core contact interaction, where the interaction potential will tend to infinity when the atoms are infinitely close, the long-range Rydberg interaction potential no longer tends to infinity, making the Rydberg-dressed ultracold atomic system an ideal platform for exploring exotic quantum phases displaying simultaneously different types of order [12–18]. Previous studies have shown that when the long-range isotropic Rydberg interaction is strong enough, the roton gap is gradually closed, leading to spontaneous breaking of translational symmetry, which provides another way for the experimental formation of supersolids [19–21].
In real experiments, most of the ultracold atomic gases are trapped by an external potential. It is well known that the dimension and geometric configuration of the external potential play an important role in determining the ground state of such a system [22–25]. Due to its unique topology, the toroidal trap has attracted great attention, and is used to simulate various physical phenomena in traditional condensed matter physics [26–32]. Previous work has shown that the interplay between the spin–orbit coupling (SOC) and the toroidal trap results in a rich variety of ground states, such as a modified stripe, an alternately arranged stripe, and countercircling states [33]. In the presence of non-local Rydberg interactions, the system will exhibit various novel supersolid states under the combined effects of SOC and non-local Rydberg interactions [34, 35].
To the best of our current knowledge, there is little work on the effects of Rydberg blockade radius on the ground-state phases of Rydberg-dressed Bose gas confined in toroidal trap, which is what we attempt to do. In this work, we consider a Rydberg-dressed Bose gas confined in a quasi-two-dimensional (Q2D) toroidal trap, with emphasis on the effects of the Rydberg interaction, external potential, and especially the Rydberg blockade radius, on the ground-state structure of such a system. Our results show that the Rydberg blockade radius, which can be regarded as another controllable parameter, can be used to obtain a variety of ground-state phases.
2. Model and method
We begin with a two-component Bose–Einstein condensate with both the contact interactions and Rydberg interactions confined in a Q2D toroidal trap on the x-y plane. Under the mean-field approximation at zero-temperature, the ground state and dynamics of such a system can be described by the coupled Gross–Pitaevskii equations, which can be written as:
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{\hslash }\displaystyle \frac{\partial {{\rm{\Psi }}}_{\uparrow }}{\partial t}=\left(-\displaystyle \frac{{{\hslash }}^{2}{{\rm{\nabla }}}^{2}}{2m}+V({\boldsymbol{r}})+{{\rm{g}}}_{\uparrow \uparrow }| {{\rm{\Psi }}}_{\uparrow }{| }^{2}\right.\\ \,\left.+\,{{\rm{g}}}_{\uparrow \downarrow }| {{\rm{\Psi }}}_{\downarrow }{| }^{2}+{A}_{\uparrow \uparrow }| {{\rm{\Psi }}}_{\uparrow }{| }^{2}+{A}_{\uparrow \downarrow }| {{\rm{\Psi }}}_{\downarrow }{| }^{2}\Space{0ex}{3.5ex}{0ex}\right){{\rm{\Psi }}}_{\uparrow },\\ {\rm{i}}{\hslash }\displaystyle \frac{\partial {{\rm{\Psi }}}_{\downarrow }}{\partial t}=\left(-\displaystyle \frac{{{\hslash }}^{2}{{\rm{\nabla }}}^{2}}{2m}+V({\boldsymbol{r}})+{{\rm{g}}}_{\downarrow \downarrow }| {{\rm{\Psi }}}_{\downarrow }{| }^{2}\right.\\ \,\left.+\,{{\rm{g}}}_{\downarrow \uparrow }| {{\rm{\Psi }}}_{\uparrow }{| }^{2}+{A}_{\downarrow \downarrow }| {{\rm{\Psi }}}_{\downarrow }{| }^{2}+{A}_{\downarrow \uparrow }| {{\rm{\Psi }}}_{\uparrow }{| }^{2}\Space{0ex}{3.5ex}{0ex}\right){{\rm{\Psi }}}_{\downarrow },\end{array}\end{eqnarray}$
where ψi(j) is the order parameter of the i(j)-th component (i(j) = ↑, ↓), and satisfies the normalization condition ∫d2r∣ψi(j)∣2 = Ni(j) with Ni(j) being the atom number of each component. ${{\rm{\nabla }}}^{2}=\tfrac{{\partial }^{2}}{\partial {x}^{2}}+\tfrac{{\partial }^{2}}{\partial {y}^{2}}$ is the two-dimensional Laplace operator, m is the atomic mass, and r = (x, y). We assume that the system is tightly confined by a strong harmonic trap along the z-direction with frequency ωz and characteristic length ${a}_{z}=\sqrt{{\hslash }/m{\omega }_{z}}$. In this case, the effective contact interaction parameters are given by ${{\rm{g}}}_{{ij}}=\sqrt{8\pi }{{\hslash }}^{2}{a}_{{ij}}/({{ma}}_{z})$. The aij is the corresponding s-wave scattering lengths for intra- and inter-component interactions, which can be adjusted by Feshbach resonance technology [7]. The non-local Rydberg interaction potential can be written as ${A}_{{ij}}({\boldsymbol{r}})=\tfrac{{U}^{{ij}}}{{R}_{c}^{6}+| {\boldsymbol{r}}{| }^{6}}$ with Uij and Rc being the Rydberg interaction strength and the Rydberg blockade radius, respectively [36].The trapping potential in the x-y plane considered in this work can be written as [33]
$\begin{eqnarray}V({\boldsymbol{r}})=\displaystyle \frac{1}{2}m{\omega }_{\perp }^{2}\left[{V}_{0}{\left(\displaystyle \frac{{r}^{2}}{{a}_{0}}-{a}_{0}{r}_{0}\right)}^{2}\right],\end{eqnarray}$
where $r=\sqrt{{x}^{2}+{y}^{2}}$, ω⊥ is the radial trap frequency of the harmonic potential, ${a}_{0}=\sqrt{{\hslash }/m{\omega }_{\perp }}$, V0 and r0 are the width and center height of the toroidal trap, respectively.By using the Crank-Nicolson finite difference and the backward Euler pseudospectral schemes within an imaginary time propagation approach, we can obtain the mean-field ground states of such a system [37–40]. In our numerical simulations, we work in dimensionless units by scaling with the characteristic oscillator length $\sqrt{{\hslash }/m{\omega }_{z}}$ and trap energy ℏω⊥. We start from proper initial wave-functions to obtain the real ground-state. The lowest-energy states in the different parameter space are obtained until the fluctuation in the norm of the wave function becomes smaller than 10−6. And the spatial steps h = 0.1 and time steps t = 0.001. For the initial data, we prepare the two-dimensional random functions as initial wave functions, and our results show that such random initial data always converges to the ground state of the system considered here.
3. Results and discussions
In what follows, we will perform a detailed study of the effects of Rydberg interactions, external potential, and the Rydberg blockade radius in particular, on the ground-state structure of the system. To this end, we fix the parameters of the external toroidal trap as V0 = 10 r0 = 3 a0 = 5, and assume contact interactions g↑↑ = g↓↓ = g↑↓ = g↓↑ = g = 0. Additionally, for convenience, we rescale the Rydberg interaction strength as ${U}^{{ij}}/{R}_{c}^{4}$ and denote it as Uij (the tilde is omitted for simplicity), and focus on the equal case with U↑↑ = U↓↓ = U↑↓ = U↓↑ = U.
We first consider the effects of the Rydberg interaction on the ground state of the system. It is known that when the Rydberg interaction potential is strong enough, the roton gap of the system is closed, leading to spontaneous translational symmetry breaking of the system. Figure 1 shows the typical ground-state density and phase distributions of the system for fixed Rydberg blockade radius, and for varied Rydberg interactions. Figures 1(a)–(d) show the density distributions of each component (the first and second columns) and the total density distribution of the system (the third column), while figures 1(e)–(g) exhibit the phase distributions corresponding to component ψ↑ in figures 1(b)–(d), respectively. It is easy to see that for the fixed Rydberg blockade radius, the system exhibits continuous-to-discrete rotational symmetry breaking with the increases of Rydberg interactions, as shown in figure 1 for Rc = 3. The density distribution of such a system gradually evolves from a continuous distributed ring structure to a unit cell structure distributed discretely along the azimuth angle, and the size of the unit cell decreases with the Rydberg interaction, which is in a sense reminiscent of our previous results in [34].
Figure 1. Typical ground-state density and phase distributions of a two-component Rydberg-dressed Bose gas confined in a toroidal trap for varied Rydberg interactions. The strength of the Rydberg interactions are U = 400, 600, 800, 1000 for (a)–(d), respectively, while the Rydberg blockade radius is fixed to Rc = 3. (e)–(g) show the corresponding phase distributions of component ψ↑ in (b)–(d), respectively. |
From a physics viewpoint, with the increase of the Rydberg interaction, the gap excited by the roton excitation will be gradually closed, leading to the spontaneous breaking of translational symmetry of the system [9]. In addition, the Q2D toroidal trap can be approximately regarded as a quasi-one-dimensional one under the limit case, the system will find it energetically favorable to form crystals with relatively large numbers of particles per unit cell along the azimuth angle direction. Here we want to note that for fixed Rydberg blockade radius, the discrete rotational symmetry of the system remains unchanged, leading to the fixed number of the unit cell. Figures 1(e)–(g) show the phase distributions of component ψ↑ corresponding to figures 1(b)–(d), respectively. It is easy to see that the phase distribution of the system is uniform along the ring and thus there is no phase modulation. This also explains why there is no topological excitations, such as vortex and Skyrmion, in figure 1. Here we want to note that the phase distribution of the system is also uniform in the case of varied Rydberg blockade radius as shown in the following discussion.
We next discuss the effects of the Rydberg blockade radius on the ground-state structure of the system. Figure 2 exhibits the typical ground-state density and phase distributions of the system for the fixed strength of Rydberg interactions, and for the varied Rydberg blockade radius, where the first and third columns show the density distribution and the second and fourth columns show the corresponding phase distribution. For weak Rydberg interaction and small Rydberg blockade radius, it is observed that the density distributions of such two components are uniform along the ring, as shown in figure 2(a) for U = 200 and Rc = 1. In this case, the system presents continuously rotational symmetry. It is interesting to see that the continuously rotational symmetry breaking occurs if we further increase the Rydberg blockade radius, and the density distribution of each component forms unit cell structures distributed discretely along the azimuth angle, as shown in figure 2(b) for U = 200 and Rc = 7. This indicates that when the strength of the Rydberg interaction is not enough to close the roton gap, the spontaneous rotational symmetry breaking of the system can be realized by modulating the Rydberg blockade radius.
Figure 2. Typical ground-state density and phase distributions of a two-component Rydberg-dressed Bose gas confined in a toroidal trap for fixed Rydberg interaction strength and for varied Rydberg blockade radius. The Rydberg blockade radii are Rc = 1, 7 in (a)–(b), respectively. The strength of Rydberg interaction is fixed to U = 200. |
With the increase of the strength of Rydberg interaction, the effects of the Rydberg blockade radius on the ground-state structure of the system become more and more interesting. Figure 3 shows the typical ground-state density and phase distributions of the system for a larger Rydberg interaction strength U = 800, and for varied Rydberg blockade radius. The results clearly indicate that with the increase of the Rydberg blockade radius Rc, the size of the unit cell gradually increases, and the number of unit cells along the ring decreases until it reaches four. Consequently, the associated order of rotational symmetry breaking gradually decreases from an initial 12-order to the 4-order, that is C12 → C6 → C4. We can thus conclude that the Rydberg blockade radius Rc, which can be regarded as another controllable parameter, can be used to control the order of spontaneous rotational symmetry breaking of the system.
Figure 3. Typical ground-state density and phase distributions of a two-component Rydberg-dressed Bose gas confined in a toroidal trap for fixed Rydberg interaction strength and for varied Rydberg blockade radius. The Rydberg blockade radii are Rc = 3, 6, 9, in (a)–(c), respectively. The strength of the Rydberg interaction is fixed to U = 800. (d)–(f) show the corresponding phase distributions of component ψ↑ in (a)–(c), respectively. |
Another interesting question is how the critical value of the Rydberg blockade radius (defined as the occurrence of spontaneous rotational symmetry breaking for fixed Rydberg interaction), changes with the Rydberg interaction. Figure 4 shows the typical density distributions of the system in the presence of spontaneous rotational symmetry breaking for fixed Rydberg interactions. It is easy to see that such critical values of the Rydberg blockade radius decrease with the strength of the Rydberg interactions, as shown in figures 4(a)–(c) for U = 200, 400, 600, respectively, where the corresponding critical values of the Rydberg blockade radius decrease from 7 to 3. In addition, with the increase of Rydberg interactions, the number of unit cells increases while its size decreases, which is consistent with the above results.
Figure 4. Typical density distributions of the system in the presence of spontaneous rotational symmetry breaking for fixed Rydberg interactions. The strength of the Rydberg interactions are U = 200, 400, 600, in (a)–(c), respectively. The Rydberg blockade radii are Rc = 7, 4, 3, respectively. |
Finally, we want to note that while the results presented above are representative of the possible range of strength of Rydberg interactions and the Rydberg blockade radii, they do not in any way exhaust all the phases. The richness of the possible ground-state phase lies in the large number of controllable parameters, including the aspect ratios of the external potential, the strength of contact interactions, and so on. However, for the two-dimensional parameter space expanded by the strength of Rydberg interactions and the Rydberg blockade radius, the above ground-state structures are qualitatively unchanged. On the other hand, since the non-local Rydberg interaction potential does not introduce the phase changes, the internal structure of the unit cell is relatively simple. A natural extension of our work is to include a variety of SOC, and to consider the effects of the Rydberg blockade radius on the internal structures and the formation mechanism of topological excitations.
4. Conclusions
In summary, we have investigated the ground-state structure of Rydberg-dressed Bose gas confined in a toroidal trap. The presence of the Rydberg interactions break the continuous rotational symmetry of the system, leading to the formation of unit cells distributed discretely along the azimuth angle. In the case of fixed Rydberg blockade radii, the size of the unit cell decreases with the strength of the Rydberg interaction, while its number is fixed. For the weak Rydberg interaction, the spontaneous rotational symmetry breaking of the system can be realized by modulating the Rydberg blockade radius, and its critical value decreases with the strength of the Rydberg interaction. It is interesting to find that the Rydberg blockade radius can be used to realize discrete rotational symmetry breaking with different order for the strongly interacting system. Our results open up alternate ways for the quantum control of Rydberg-dressed Bose gas.