Dynamic formation of Rydberg aggregates at off-resonant excitation

The dynamics of a cloud of ultracold two-level atoms is studied at off-resonant laser driving to a Rydberg state. We find that resonant excitation channels lead to strongly peaked spatial correlations associated with the buildup of asymmetric excitation structures. These aggregates can extend over the entire ensemble volume, but are in general not localized relative to the system boundaries. The characteristic distances between neighboring excitations depend on the laser detuning and on the interaction potential. These properties lead to characteristic features in the spatial excitation density, the Mandel $Q$ parameter, and the total number of excitations. As an application an implementation of the three-atom cswap or Fredkin gate with Rydberg atoms is discussed. The gate not only exploits the Rydberg blockade, but also utilizes the special features of an asymmetric geometric arrangement of the three atoms. We show that continuous-wave off-resonant laser driving is sufficient to create the required spatial arrangement of atoms out of a homogeneous cloud.

Figure 1

(a) Pair correlation function $g^{\left(2\right)}\left(r\right)$ as a function of interparticle distance $r$ and detuning $\Delta$. Dashed lines are theoretical prediction for resonance positions as explained in the main text. (b) Sections through (a) at detunings $\Delta /\Omega =3,5,7$. Parameters are $L=30\mu$m, $N=30$, $\Omega =10$ MHz, and $C_6=6.6$ THz $\mu$m${}^6$.

Figure 2

(a) Illustration of the resonant excitation channels. Bare state energies [diagonal part of the Hamiltonian (1)] as a function of the laser detuning. We used a sample of 16 atoms randomly placed in a one-dimensional trap. The energy of $m$-fold excited states decreases with slope $m$. The solid red (upper gray) and black lines mark states that are resonantly coupled to the ground state for a certain detuning $\Delta$. Note that in a dressed state picture (eigenstates of the Hamiltonian) all the crossings would become avoided ones. (b) Dependence of the two-photon excitation linewidth on the detuning. The solid line is obtained from Eq. (3), red crosses are obtained from the numerical simulations. The inset shows the detuning-dependent transformation of energetic resonances into spatial resonances via the interaction potential $V\left(r\right)=C_6/r^6$.

Figure 3

(a) Total number of excited atoms and (b) Mandel $Q$ parameter as a function of the detuning. $Q^{*}$ is the corresponding parameter evaluated without the ground state fraction. Solid lines show $N=30$; dashed lines show $N=45$. $C_6$ and $L$ are as in Fig. 1.

Figure 4

(a) Rydberg excitation density in excitations per micrometer vs. position in the 1D trap and detuning. Dashed line: Resonance line corresponding to $2\Delta =C_6/r^6$ as explained in the text. (b) Excitation density for $\Delta /\Omega =7$ split into excitation-number subspaces.

Figure 5

Illustration of the aggregate formation: Positions of the excitations are determined from the exact quantum state by a Monte Carlo procedure mimicking a measurement. The distances of the excitations to the leftmost excitation are collected in a histogram. Note that the scale of the ordinate in (c) and (d) differs from (a) and (b). The height of the two-photon resonance is 12 in (c) and 19 in (d). Parameters are as in Fig. 1, $\Delta /\Omega =6$. In (d) the steady state has been reached. Excitation times are given in microseconds.

Figure 6

(a) Illustration of the gate protocol and level scheme of a single atom. The truth table is given for two exemplary initial states $|CAB\rangle =|010\rangle$ and $|110\rangle$. Depending on the control qubit $C$, qubits $A$ and $B$ interchange their states, or not. (b) Hamiltonian of the detuned pulse (blue in gate protocol). The thickness of the bars stands for the value $E$ of the diagonal element of the corresponding state (with $m$ Rydberg excitations). Double arrows indicate laser couplings. In the blue subspace, the state $|0e0\rangle$ evolves into $|00e\rangle$ under the $\sqrt{2}\pi$ pulse.

Figure 7

(a) Fidelity $F_{\text{gate}}$ of the cswap gate against detuning $\Delta$ and imperfections $\delta r$ of the spatial arrangement of the three atoms. (b) $F_{\text{gate}}$ and $F_{\text{bell}}$ for the optimal $r_{AB}$ and averaged over a Lorentzian distribution of distances $r_{AB}$. The fidelity is relatively insensitive to variations in $r_{CA}$.