Abstract
When two quantum systems interact strongly with each other, their simultaneous excitation by the same driving pulse may be forbidden. The phenomenon is known as blockade of excitation. Recently, extensive studies have been devoted to the socalled Rydberg blockade between neutral atoms, which appears when the atoms are in highly excited electronic states, owing to the interaction induced by the accompanying large dipole moments. Rydberg blockade has been proposed as a basic tool in quantuminformation processing with neutral atoms^{1,2,3,4,5}, and can be used to deterministically generate entanglement of several atoms. Here, we demonstrate Rydberg blockade between two atoms individually trapped in optical tweezers at a distance of 4 μm. Moreover, we show experimentally that collective twoatom behaviour, associated with the excitation of an entangled state between the ground and Rydberg levels, enhances the allowed singleatom excitation. These observations should be a crucial step towards the deterministic manipulation of entanglement of two or more atoms, with possible implications for quantuminformation science, as well as for quantum metrology, the study of strongly correlated systems in manybody physics, and fundamental studies in quantum physics.
Main
A large experimental effort is nowadays devoted to the production of entanglement, that is quantum correlations, between individual quantum objects such as atoms, ions, superconducting circuits, spins or photons. There are several ways to engineer entanglement in a quantum system. Here, we focus on a method that relies on a blockade mechanism where the strong interaction between different parts of a system prevents their simultaneous excitation by the same driving pulse. Single excitation is still possible but is delocalized over the whole system, and results in the production of an entangled state. This approach to entanglement is deterministic and can be used to realize quantum gates^{1} or to entangle mesoscopic ensembles, provided that the blockade is effective over the whole sample^{2}. Blockade effects have been observed in systems where interactions are strong such as systems of electrons using the Coulomb force^{6} or the Pauli effective interaction^{7}, as well as with photons and atoms coupled to an optical cavity^{8}. Recently, atoms held in the ground state of the wells of an optical lattice have been shown to exhibit interaction blockade, due to swave collisions^{9}. An alternative approach uses the comparatively strong interaction between two atoms excited to Rydberg states. This strong interaction gives rise to the socalled Rydberg blockade, which has been observed in clouds of cold atoms^{10,11,12,13,14,15} as well as in a Bose condensate^{16}. A collective behaviour associated with the blockade has been reported in an ultracold atomic cloud^{17}. Recently, an experiment demonstrated the blockade between two atoms 10 μm apart, by showing that when one atom is excited to a Rydberg state, the excitation of the second one is greatly suppressed^{18}. However, the enhancement of the excitation rate of one atom when two atoms are present, explained by the excitation of an entangled state in the blockade regime, has not been observed until now.
Here, we study two individual atoms, held at a few micrometres distance by two optical tweezers. The ground state g〉 and a Rydberg state r〉 of an atom are separated by an energy E (see Fig. 1a) and can be coupled by a laser. For noninteracting atoms, a and b, the twoatom spectrum exhibits two transitions at the same frequency E/ℏ, connecting states g,g〉 to r,g〉 or g,r〉, and then to r,r〉. This enables the simultaneous excitation of the two atoms to state r,r〉. However, if the two atoms interact strongly when in state r,r〉, this energy level is shifted by an amount ΔE and the laser excitation cannot bring the two atoms to state r,r〉. A fundamental consequence of this blockade is that the atoms are excited in the entangled state , where r_{a} and r_{b} are the positions of the two atoms and k is related to the wave vectors of the exciting lasers. More precisely, the laser excitation is described by the operator (ℏΩ/2)(e^{ik·r}_{a}r,g〉〈g,g+e^{ik·r}_{b}g,r〉〈g,g+complex conjugate) (ref. 19). Here, Ω is the Rabi frequency characterizing the coupling between the laser and one atom. It is then convenient to use as a basis the two entangled states , so that Ψ_{−}〉 is not coupled to the ground state, whereas Ψ_{+}〉 is coupled with an effective Rabi frequency . In the blockade regime, where the state r,r〉 is out of resonance, the two atoms are therefore described by an effective twolevel system involving collective states g,g〉 and Ψ_{+}〉 coupled with a strength of , as shown in Fig. 1b. Hence, the atoms are excited into an entangled state containing only one excited atom, with a probability oscillating times faster than the probability to excite one atom when it is alone.
In our experiment, we excite two individual rubidium 87 atoms to the Rydberg state r〉=58d_{3/2},F=3,M_{F}=3〉, with the internuclear axis aligned along the quantization axis. The state 58d_{3/2} was chosen because of the existence of a quasidegeneracy between the twoatom states (58d_{3/2},58d_{3/2}) and (60p_{1/2},56f_{5/2}), also called a Förster resonance^{20}. As detailed in Supplementary Information, the dipole–dipole interaction lifts the degeneracy and leads to two potential curves ±(C_{3}/R^{3}), as represented in Fig. 1a. For our particular geometry, we calculated C_{3}≈h×3,200 MHz μm^{3}. Accordingly, the interaction energy between two atoms at distance R=4 μm is ΔE≈h×50 MHz.
The two single rubidium 87 atoms are confined in two independent optical dipole traps, as shown in Fig. 2a. The traps are formed in the focal plane of the same large numerical aperture lens^{21}. Each trap has a waist of 0.9 μm and a depth of 0.5 mK. The distance between the two traps can be varied between 3 and 20 μm with a precision of 0.5 μm. The axis between the two traps is aligned with the magnetic field defining the quantization axis. The traps are loaded from the cold atomic cloud of an optical molasses. We collect the fluorescence emitted by the atoms, induced by the cooling lasers, on two separated singlephoton counters. A high fluorescence level indicates the presence of an atom in the respective trap and triggers the experimental sequence, described hereafter.
We prepare the two atoms in the hyperfine state g〉=5s_{1/2},F=2,M_{F}=2〉 by a 600 μs optical pumping phase (efficiency ∼90%). We then excite the atoms to the Rydberg state r〉=58d_{3/2},F=3,M_{F}=3〉 by a twophoton transition, as represented in Fig. 2b. The intermediate state 5p_{1/2},F=2,M_{F}=2〉 is connected to the ground state by a laser detuned by 400 MHz to the blue side of the 795 nm transition to avoid populating the intermediate state. The second laser connects the 5p_{1/2} to the 58d_{3/2} state and has a wavelength of 474 nm. Both laser beams illuminate the two atoms. During the excitation (<500 ns), the dipole trap is turned off to avoid an extra lightshift on the atoms. We finally detect the excitation to the Rydberg state through the loss of the atom after turning the dipole trap back on. In a Rydberg state, the atom is not trapped by the optical potential any longer. Owing to its residual velocity (∼10 cm s^{−1}) it leaves the trapping region in less than 10 μs. This time is much shorter than the lifetime of Rydberg state 58d_{3/2} induced by blackbody radiation (160 μs) and than the radiative decay time (200 μs; ref. 22).
In a first experiment, we placed the two traps at a distance of 18±0.5 μm and repeated the excitation sequence 100 times, starting each time with newly trapped atoms. We measured for each atom whether it was lost or recaptured at the end of each sequence and calculated the probability to excite it in the Rydberg state, which is equal to the probability to lose it. When only one of the two traps is filled, the excitation probability exhibits Rabi oscillations between the ground state and the Rydberg state, as shown in Fig. 3a. A fit to the data yields a twophoton Rabi frequency Ω≈2π×7 MHz, which is in agreement with the measured waists and powers of the lasers. The decay of the fringe amplitude is explained by frequency fluctuations (∼1 MHz) as well as shottoshot intensity fluctuations of the lasers (∼5%), which results in a jitter in the twophoton resonance frequency. We attribute the maximum excitation probability of ∼80% to this decay and to the imperfect optical pumping of the atoms in the Zeeman state 5s_{1/2},F=2,M_{F}=2〉.
We then repeated the sequence with two atoms trapped at the same time and measured the probability to excite the two atoms with the same laser pulse. The results are represented in Fig. 3a by the triangles. We compared this probability with the probability to excite simultaneously two noninteracting atoms, which should be equal to the product of the probabilities to excite each atom independently, measured previously. The blue circles in Fig. 3a represent this product, calculated from the data for each independent atom. The agreement between the two curves shows that the two atoms, when separated by 18 μm, behave independently and therefore have a negligible interaction. This result agrees with the theory because the blockade becomes effective at a distance between the atoms for which the interaction shift ΔE is equal to the linewidth of the excitation pulse, of the order of the Rabi frequency Ω. For our particular choice of the Rydberg state, this yields R≈8 μm.
In a second step, we repeated the previous experiment but with a distance between the traps of 3.6±0.5 μm, which is in a regime where blockade is expected. Once again we measured the probability to excite one atom when the other one is absent and got the oneatom Rabi oscillations. When two atoms were trapped, we measured the probability to excite the two atoms simultaneously, as shown in Fig. 3b by the triangles. The simultaneous excitation of the two atoms is greatly suppressed with respect to the case where the atoms are far apart. This suppression is the signature of the blockade regime. The fact that the probability of simultaneous excitation of the two atoms is not completely cancelled may be explained by the existence of extra potential curves coming from imperfect control of the atomic state and leading to smaller interaction energies^{20}. This imperfect control can be due to stray electric fields, imperfect polarizations of the lasers and random positions of the atoms in their trap (see last paragraph) meaning that the internuclear axis is not always perfectly aligned with the quantization axis.
We now come to the direct observation of collective oneatom excitation in the blockade regime, that is, with two atoms separated by R=3.6 μm. Figure 4 shows the probability to excite only one of the two atoms as a function of the duration of the excitation pulse, together with the probability to excite only one atom when the other dipole trap is empty. The two probabilities oscillate with different frequencies, the ratio of which is 1.38±0.03 (the error corresponds to one standard deviation). This value is compatible with the ratio that we expect in the blockade regime. As explained at the beginning of this letter, the oscillation of the probability to excite only one atom at a frequency is the signature that the twoatom system oscillates between the state g,g〉 and the entangled state , where k=k_{R}+k_{B} is the sum of the wave vectors of the two lasers involved in the twophoton transition.
Finally, we analyse the influence of the atoms’ motion on this entangled state. We measured a temperature of the atoms in their trap of 70 μK (ref. 23). This leads to amplitudes of the motion of ±800 nm in the longitudinal (y) direction (trap frequency 16 kHz) and ±200 nm in the radial (x and z) direction of the traps (frequency 77 kHz). As the fastest oscillation period is 13 μs and the excitation time is of the order of a hundred nanoseconds, the motion of the atoms is frozen during the excitation. The temperature results only in a dispersion of the positions of the atoms from shot to shot. Therefore, the relative phase φ=k·(r_{a}−r_{b}) between the two components of the superposition is constant during the excitation, but varies randomly from shot to shot over more than 2π. This creates an effective decoherence mechanism for the state Ψ_{+}〉, which would prevent the direct observation of the entanglement. However, this fluctuating phase can be erased in the following way: one first couples one hyperfine ground state 0〉 to a Rydberg state r〉, producing the state as described in this letter. Then a second pulse, applied before the atoms move, couples r〉 to a second hyperfine ground state 1〉. If the wave vectors of the two excitations are the same, the phase during the second step cancels the phase of the first excitation. The resulting entangled state is therefore , which involves longlived atomic qubits^{24,25}.
In conclusion, the results presented here indicate that we control the physical mechanism needed to deterministically entangle two atoms on fast timescales, compatible with submicrosecond operation of a quantum gate^{1}. Combined with our abilities to manipulate the state of a single atom^{24}, to keep and to transport its quantum state^{25}, our system is well adapted to applications of the Rydberg blockade in quantuminformation processing.
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Acknowledgements
We thank M. Saffman, T. Walker, R. Côté and T. F. Gallagher for enlightening discussions. We thank I. Liu for theoretical support and T. Puppe for technical assistance with the lasers. We thank C. Evellin for calculations and experimental assistance, as well as L. Servant. We acknowledge support from the EU through the IP SCALA, and from IARPA and IFRAF. A.G. is supported by a DGA fellowship. Y.M. and T.W. are supported by IFRAF.
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Gaëtan, A., Miroshnychenko, Y., Wilk, T. et al. Observation of collective excitation of two individual atoms in the Rydberg blockade regime. Nature Phys 5, 115–118 (2009). https://doi.org/10.1038/nphys1183
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