Quantum dynamics of a fully blockaded Rydberg atom ensemble

Classical simulation of quantum systems plays an important role in the study of many-body phenomena and in the benchmarking and verification of quantum technologies. Exact simulation is often limited to small systems because the dimension of the Hilbert space increases exponentially with the size of the system. For systems that possess a high degree of symmetry, however, classical simulation can reach much larger sizes. Here we consider an ensemble of strongly interacting atoms with permutation symmetry, enabling the computation of certain collective observables for hundreds of atoms at arbitrarily long evolution times. The system is realized by an ensemble of three-level atoms, where one of the levels corresponds to a highly excited Rydberg state. In the limit of all-to-all Rydberg blockade, the Hamiltonian is invariant under permutation of the atoms. Using techniques from representation theory, we construct a block-diagonal form of the Hamiltonian, where the size of the largest block increases only linearly with the system size. We apply this formalism to derive efficient pulse sequences to prepare arbitrary permutation-invariant quantum states. Moreover, we study the quantum dynamics following a quench, uncovering a parameter regime in which the system thermalizes slowly and exhibits pronounced revivals. Our results create opportunities for the experimental and theoretical study of large interacting and nonintegrable quantum systems.

Figure 1

System setup. (a) We consider atoms with two hyperfine ground states $|0\rangle$ and $|1\rangle$ and a highly excited Rydberg state $|r\rangle$. The hyperfine states are coupled by a driving field with Rabi frequency ${\mathrm{\Omega }}_1$ and detuning ${\mathrm{\Delta }}_1$. The state $|1\rangle$ is further coupled to $|r\rangle$ with Rabi frequency ${\mathrm{\Omega }}_2$ and detuning ${\mathrm{\Delta }}_2$. (b) The strong van der Waals interaction between two Rydberg states defines a blockade radius $R_b$, within which the excitation of more than two atoms to the Rydberg state is effectively forbidden. We assume that $R_b$ is greater than the largest distance between any pair of atoms. Equivalently, a sphere of radius $R_b/2$ around an atom in the Rydberg state (purple) is assumed to intersect the sphere of the same size centered at any other atom (orange).

Figure 2

Irreducible representations of $\mathfrak{su}\left(2\right)$ and $\mathfrak{su}\left(3\right)$. (a) Weight diagram of the spin $s=3/2$ representation of $\mathfrak{su}\left(2\right)$. Each dot corresponds to a basis state $|m\rangle$. The raising operator $T^{+}$ induces transitions between the states as indicated by the orange arrows. The numbers next to the arrows give the associated matrix element. (b) The three levels of a single atom form the $(p,q)=(1,0)$ representation of $\mathfrak{su}\left(3\right)$. The colored arrows show the directions in which the three distinct raising operators act. (c) In general, the weight diagram of the $(p,q)$ representation of $\mathfrak{su}\left(3\right)$ takes the form of a hexagon with side lengths $p$ and $q$. Here $(p,q)=(2,1)$. Each solid dot and each surrounding circle correspond to a basis state. The basis states are simultaneous eigenstates of $T^z$, $U^z$, and $V^z$. The respective eigenvalues $m_t$, $m_u$, and $m_v$ can be read off at the orthogonal projection onto the corresponding coordinate axis. The origin of the axis lies at the center of the diagram. (d) Matrix elements of the raising operators $T^{+}$, $U^{+}$, and $V^{+}$ for $(p,q)=(0,2)$ and the same color coding as in (b).

Figure 3

(a) Irreducible representations arising for $n=2$ atoms. The axis shows the number of Rydberg atoms. States with more than one Rydberg excitation are grayed out as they violate the blockade constraint. The table shows the partition $({\lambda }_1,{\lambda }_2,{\lambda }_3)$, the labels $(p,q)$, and the multiplicity ${\mu }_{\lambda }$ associated with each weight diagram. (b) Same as (a) but for $n=3$ atoms.

Figure 4

Preparation of the two-atom GHZ state starting from the initial state $|00\rangle$. The preparation corresponds to the time reversal of the five pulses in (a)–(e). The numbers next to the arrows indicate the product $k\mathrm{\Omega }t$ for each pulse, where $\mathrm{\Omega }$ is the Rabi frequency for a single atom, $t$ the duration of the pulse, and $k$ the magnitude of the matrix element of the collective transition. The amplitude of each basis state (up to a global phase) is indicated by the numbers next to the nodes, which are shaded according to the corresponding occupation.

Figure 5

Expected number of Rydberg excitations in a fully blockaded ensemble of $n=100$ atoms. The dynamics are shown for three different initial states (a) $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes n}$, (b) $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes n/2}{|1\rangle }^{\otimes n/2}$, and (c) $|\psi \left(0\right)\rangle ={|1\rangle }^{\otimes n}$. The system is simultaneously driven by two external fields with Rabi frequencies ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=\mathrm{\Omega }$ and detunings ${\mathrm{\Delta }}_1=\mathrm{\Omega }$ and ${\mathrm{\Delta }}_2=0$ (see Fig. 1). The plots in the top row show the rapid oscillations of the Rydberg population for times between $\mathrm{\Omega }t=0$ and $\mathrm{\Omega }t=20$.

Figure 6

(a) Revival time $t_{\text{rev}}$ and (b) revival strength ${\langle n_r\rangle }_{\mathrm{rev}}$ as a function of the number of atoms $n$. Both quantities are determined by locating the largest value of $\langle n_r\rangle$ at the first revival. The Hamiltonian parameters are the same as in Fig. 5. The solid curves in (a) are fits to the function $a\sqrt{n}+b$.

Figure 7

(a) Illustration of the spin-model approximation to the $(n,0)$ representation. By adding an unphysical state (light blue circle connected to dashed line), the Hilbert space can be viewed as the tensor product of a spin $s=n/2$ and a two-level system $\sigma$. The colored lines indicate the transitions that are being driven. The purple numbers show the matrix elements for the $|1\rangle \leftrightarrow |r\rangle$ transition. (b) Revivals of an ensemble of $n=300$ atoms initialized in the state ${|1\rangle }^{\otimes n}$. The Hamiltonian parameters are the same as in Fig. 5. The light blue curve shows the exact values of the expectation value of the number of Rydberg excitations. The blue envelope was obtained from the approximate spin model.

Figure 8

(a) To construct a spin model for the initial state ${|0\rangle }^{\otimes n}$, we add an additional, unphysical state to the left of the weight diagram (gray circle connected to dashed line). (b) Exact evolution and dynamics under the spin model (D1) of $n=300$ atoms with ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=\mathrm{\Omega }$, ${\mathrm{\Delta }}_1=\mathrm{\Omega }$, and ${\mathrm{\Delta }}_2=0$.

Figure 9

Expected number of Rydberg excitations at late times for the same parameters as in Fig. 5 ($n=100$, ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=\mathrm{\Omega }$, ${\mathrm{\Delta }}_1=\mathrm{\Omega }$, and ${\mathrm{\Delta }}_2=0$). The dynamics are shown for three different initial states (a) $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes n}$, (b) $|\psi \left(0\right)\rangle ={|0\rangle }^{\otimes n/2}{|1\rangle }^{\otimes n/2}$, and (c) $|\psi \left(0\right)\rangle ={|1\rangle }^{\otimes n}$. In (c) we also include the envelope computed according to the spin model (18).