Coherent ground-state transport of neutral atoms

Quantum state transport is an important way to study the energy or information flow. By combining the unconventional Rydberg pumping mechanism and the diagonal form of van der Waals interactions, we construct a theoretical model via second-order perturbation theory to realize a long-range coherent transport inside the ground-state manifold of neutral atoms systems. With the adjustment of the Rabi frequencies and the interatomic distance, this model can be used to simulate various single-body physics phenomena such as a Heisenberg $XX$ spin chain restricted in the single-excitation manifold, coherently perfect quantum state transfer, the parameter-adjustable Su-Schrieffer-Heeger model, and chiral motion of atomic excitation in a triangle by varying the geometrical arrangement of the three atoms, which effectively avoids the influence of atomic spontaneous emission at the same time. Moreover, the influence of atomic position fluctuation on the fidelity of quantum state transmission is discussed in detail, and the corresponding numerical results show that our work provides a robust and easily implemented scheme for quantum state transport with neutral atoms.

Figure 1

Experimental setup. (a) $N$ Rydberg atoms arranged as a linear chain at equal intervals with same energy-level configuration. Atoms are driven by two global laser fields propagating along the $z$ axis from two sides. Meanwhile, local laser fields ${\mathrm{\Omega }}_{i1}$ and ${\mathrm{\Omega }}_{i2}$ propagating along the $x$ axis are focused onto individual sites. (b) Level structure for the proposed atomic system. We consider ${}^{87}\mathrm{Rb}$ with $|g\rangle =|5S_{1/2},F=1,m_F=0\rangle$, $|e\rangle =|5S_{1/2},F=2,m_F=1\rangle$, and $|r\rangle =|73S_{1/2},m_J=1/2\rangle$. (c) The equivalent system after adiabatically eliminating the intermediate states.

Figure 2

(a) Populations of the two-atom states $|eg\rangle$ (solid) and $|ge\rangle$ (dashed) governed by the full Hamiltonian of Eq. (1) and the effective Hamiltonian of Eq. (8), respectively. (b) Population evolutions dominated by the full master equation (5) and effective master equation (9) with $\gamma =2\pi \times 0.005\mathrm{MHz}$, respectively. (c) Populations of states $|eg\rangle$ and $|ge\rangle$ together with the singly (purple) and doubly (green) excited states governed by the full Hamiltonian of Eq. (1), where “excited state” refers to the Rydberg state $|r\rangle$. The other parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$. (d) Populations of states $|eg\rangle$ and $|ge\rangle$ together with the singly (purple) and doubly (green) excited-states under the protocol in Ref. [49] with parameters ${\mathrm{\Delta }}_s={\mathrm{\Delta }}_p=50\mathrm{MHz}$, ${\mathrm{\Omega }}_s={\mathrm{\Omega }}_p=7\mathrm{MHz}$, and $U=92.5\mathrm{MHz}$.

Figure 3

Transport dynamics of the multi-atomic model with $N=5$ without considering spontaneous emission. (a) Populations of ground states with same separation and coupling strength governed by the full Hamiltonian of Eq. (1) with ${\mathrm{\Omega }}_j=0.05{\mathrm{\Omega }}_p$. (b) The evolution of ground states governed by the full Hamiltonian under the perfect transmission condition ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_5=\mathrm{\Omega }$, ${\mathrm{\Omega }}_2={\mathrm{\Omega }}_4=2\mathrm{\Omega }$, ${\mathrm{\Omega }}_3=\sqrt{3/2}\mathrm{\Omega }$ with $\mathrm{\Omega }=0.025{\mathrm{\Omega }}_p$, while the other parameters are taken as ${\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\delta ={\mathrm{\Omega }}_p$, and ${\mathcal{U}}_{i,i+1}=\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$.

Figure 4

The mismatch of the unconventional Rydberg pumping condition governed by full Hamiltonian (1). The parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }U={\mathcal{U}}_{j,j+1}-\mathrm{\Delta }$.

Figure 5

(a) The effective coupling processes for topological spin model. (b) The ratio of $J_a$ to $J_b$ is shown as a function of detuning $\mathrm{\Delta }$. (c) Populations of ground states $|eggg\rangle$ (solid), $|gegg\rangle$ (dot-dashed), $|ggeg\rangle$ (dotted), and $|ggge\rangle$ (dashed) governed by full and effective Hamiltonian shown as Eqs. (1) and (17) with $\mathrm{\Delta }=2\pi \times 330\mathrm{MHz}$. (d) The population evolutions of ground states with $\mathrm{\Delta }=2\pi \times 310\mathrm{MHz}$. (e) The energy spectrum with topological phase with $N=100$, where $\mathrm{\Delta }=2\pi \times 310\mathrm{MHz}$. The other parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, ${\mathrm{\Omega }}_j=0.05{\mathrm{\Omega }}_p$, $r_{2i-1,2i}=4\mu \mathrm{m}$, and $r_{2i,2i+1}=4.1\mu \mathrm{m}$.

Figure 6

The transport dynamics and the edge states of the system with $N=8$. Panels (a) and (b) correspond to the probability amplitudes of two edge states governed by the effective Hamiltonian (17). (c) Populations of the edge state $|A\rangle$ governed by the full Hamiltonian (1) (solid line) and the populations of states $|geg\cdots g\rangle$ and $|gge\cdots g\rangle$ governed by the full (dashed and dotted) and effective (diamonds and circles) Hamiltonian initially excited at the second particle, respectively. The parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, $\mathrm{\Delta }=2\pi \times 310\mathrm{MHz}$, $r_{2i-1,2i}=4\mu \mathrm{m}$, and $r_{2i,2i+1}=4.1\mu \mathrm{m}$.

Figure 7

(a) Schematic representations of Rydberg atoms arranged as an equilateral triangle form, $J_{jk}$ describes the effective coupling between the $j\mathrm{th}$ and $k\mathrm{th}$ atoms. (b) Periodic modulated pulses for realizing the chiral motion of atomic excitation. (c) The effective coupling processes have been chosen for realizing the chiral motion.

Figure 8

(a) The quasienergy spectrum of the effective Hamiltonian $H_{\mathrm{eff}}=iln\left(e^{-i{H}_3\tau }{e}^{-i{H}_2\tau }{e}^{-i{H}_1\tau }\right)/3\tau$ under various time intervals $\tau$. Panels (b) and (c) show the enlarged view of the quasienergy spectrum, which respectively show the energy-splitting process and the position the chiral motion generated. Panels (d)–(f) show the populations of the ground states under different time intervals, respectively corresponding to the quasienergy spectrum with two degenerate eigenenergies, arbitrary split three eigenenergies, and the eigenenergy with the same separations, where $\tau =0.01\mu \mathrm{s}$, $0.1\mu \mathrm{s}$, and $0.12425\mu \mathrm{s}$. The parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$.

Figure 9

The chiral motion of atomic excitation under the equilateral-triangle structure. Panels (a) and (b) respectively show the average value of the effective coupling strength and induced phases under different time intervals. Panels (c) and (d) intuitively represent the transfer of the ground state between three positions. Panels (e) and (f) further show the populations of ground states as a function of time measured for values of ${\mathrm{\Phi }}_z=\pi /2,-\pi /2$ while the time interval is respectively taken as 0.12425 μs and $0.15025\mu \mathrm{s}$. The parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, ${\mathrm{\Omega }}_j=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$. Panels (g) and (h) show the time interval $\tilde{\tau }$ as functions of parameters $\delta$ and ${\mathrm{\Omega }}_p$, respectively.

Figure 10

The system dynamics with and without considering the wave vectors (marked with subscript $k$ after the ket symbol). Panels (a) and (b) correspond to the dynamics of an isometric chain structure with $N=2$ and $N=3$, respectively, where $r=4.1\mu \mathrm{m}$. The evolutions are governed by Eqs. (1) and (27). (c) The ratio of $J_a$ to $J_b$ is shown as a function of the detuning $\mathrm{\Delta }$. (d) The energy spectrum of the SSH model with $N=100$, where $r_{2i-2,2i}=4\mu \mathrm{m}$ and $r_{2i,2i+1}=4.1\mu \mathrm{m}$. The other parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$.

Figure 11

The feasible experimental structure for the equilateral triangle structure, where the global laser fields ${\mathrm{\Omega }}_{p1}$ and ${\mathrm{\Omega }}_{p2}$ are counterpropagating along the $z$ axis. (a) The effective wave vectors ${\mathbf{k}}_i$ of weak fields ${\mathrm{\Omega }}_i$ are orthogonal to the atomic position ${\mathbf{r}}_i$. (b) The weak fields ${\mathrm{\Omega }}_i$ propagate along a straight line perpendicular to atoms 1 and 2.

Figure 12

The dynamics of the chiral motion of atomic excitation with ${\tilde{\tau }}_{ac}=0.12425\mu \mathrm{s}$. Panel (a) corresponds to the experimental structure shown in Fig. 11. Panel (b) corresponds to the experimental structure shown in Fig. 11. The other parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$, and $r=4.1\mu \mathrm{m}$.

Figure 13

Transport dynamics considering the position fluctuations. Panels (a) and (e) correspond to populations of states $|eg\rangle$ and $|ge\rangle$ together with considering the random fluctuation obeys the uniform density and standard normal density distribution, respectively. The parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$, and ${\mathcal{U}}_{j,j+1}^{\prime }\simeq 300{\mathrm{\Omega }}_p+F\left(t\right)$. (b)–(d) Populations of states $|eg\rangle$, $|ge\rangle$, $|ee\rangle$, and $|gg\rangle$ under the adjusted parameters with $F\left(t\right)$ uniformly distributed, respectively. (f)–(h) The dynamics of the system with $F\left(t\right)$ obeying the standard normal density distribution. The optimized parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 10\mathrm{MHz}$, $\mathrm{\Omega }=0.1{\mathrm{\Omega }}_p$, $\mathrm{\Delta }=30{\mathrm{\Omega }}_p$, and ${\mathcal{U}}_{j,j+1}^{\prime }\simeq 10{\mathrm{\Omega }}_p+F\left(t\right)$.

Figure 14

The derivation of $J_{j,j+1}\left(F\right)$ to $F$ under the original and optimized parameters. The original parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$, and ${\mathcal{U}}_{j,j+1}=300{\mathrm{\Omega }}_p$. The optimized parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 10\mathrm{MHz}$, $\mathrm{\Omega }=0.1{\mathrm{\Omega }}_p$, $\mathrm{\Delta }=30{\mathrm{\Omega }}_p$, and ${\mathcal{U}}_{j,j+1}=10{\mathrm{\Omega }}_p$.

Figure 15

The pair potentials of ${}^{87}\mathrm{Rb}$ atoms around the defined pair state $|rr\rangle =|73S_{1/2},m_j=1/2;73S_{1/2},m_j=1/2\rangle$. The red color denotes the overlap of the eigenstates with $|rr\rangle$.

Figure 16

Transport dynamics of states $|eg\rangle$ (red) and $|ge\rangle$ (blue) with and without states $|{\varphi }_2\rangle$, $|{\varphi }_3\rangle$. Panel (a) corresponds to the system with $\mathrm{\Delta }U=0$. Panel (b) corresponds to the system with $\mathrm{\Delta }U=-2\pi \times 50\mathrm{MHz}$. The other parameters are taken as ${\mathrm{\Omega }}_j=0.05{\mathrm{\Omega }}_p$, $E_1=2\pi \times 300\mathrm{MHz}$, $E_2=-2\pi \times 511.25\mathrm{MHz}$, $E_3=-2\pi \times 1258.83\mathrm{MHz}$, ${\alpha }_1=\sqrt{0.72}$, ${\alpha }_2=\sqrt{0.126}$, and ${\alpha }_3=\sqrt{0.088}$.

Figure 17

Transport dynamics of diatomic model governed by the full Hamiltonian (A2) after shifting the energy of $|e_i\rangle$ and $|g_i\rangle$ and the effective Hamiltonian of Eq. (A4), respectively. The initial state is $|\mathrm{\Psi }\left(0\right)\rangle =1/\sqrt{3}\left(\right|eg\rangle +|gg\rangle +|ee\rangle )$ and the parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$.

Figure 18

The effective coupling strengths and phases between arbitrary ground states induced by periodical driving. The parameters are taken as $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, ${\mathrm{\Omega }}_i=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$.

Figure 19

Panels (a) and (d) respectively show the change of $\tilde{\tau }$ and $J_{\mathrm{eff}}$ with $\mathrm{\Omega }$. The other parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$. Panels (b) and (e) respectively show the change of $\tilde{\tau }$ and $J_{\mathrm{eff}}$ with $\delta$. The other parameters are ${\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=2\pi \times 0.05\mathrm{MHz}$, $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$. Panels (c) and (f) respectively show the change of $\tilde{\tau }$ and $J_{\mathrm{eff}}$ with ${\mathrm{\Omega }}_p$. The other parameters are $\delta =2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=2\pi \times 0.05\mathrm{MHz}$, $\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$.

Figure 20

Populations of ground states as a function of time corresponding to ${\mathrm{\Phi }}_z=\pi /2,-\pi /2$ while the time interval is respectively $0.124\mu \mathrm{s}$ and $0.15\mu \mathrm{s}$. The parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, ${\mathrm{\Omega }}_j=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=2\pi \times 300\mathrm{MHz}$.

Figure 21

The chiral motion of atomic excitation under a tight-binding model with $N=4$ arranged in a square. Panels (a) and (b) respectively show the average value of the effective coupling strength and induced phases under different time intervals. Panels (c) and (d) show the chiral current $\langle I_{1,2}\rangle$ of ground-state governed by the periodical Hamiltonian from the initial state $|eggg\rangle$. The corresponding time intervals are $\tau \in [0.001,0.096]\mu \mathrm{s}$ and $\tau \in [0.13,0.199]\mu \mathrm{s}$, respectively. The other parameters are $\delta ={\mathrm{\Omega }}_p=2\pi \times 1\mathrm{MHz}$, $\mathrm{\Omega }=0.05{\mathrm{\Omega }}_p$, and $\mathrm{\Delta }=300{\mathrm{\Omega }}_p$.