Abstract
A two-level quantum system coherently driven by a resonant electromagnetic field oscillates sinusoidally between the two levels at frequency Ω (refs 1, 2). In dilute gases, the inhomogeneous distributions of both the coupling strength to the field and the interactions between individual atoms reduce the visibility of these so-called Rabi oscillations and may even suppress them completely. However, in the limit where only a single excitation is present, a collective, many-body Rabi oscillation at a frequency arises that involves all N≫1 atoms, even in inhomogeneous systems3,4. When one of the two levels is a strongly interacting Rydberg level, many-body Rabi oscillations emerge as a consequence of a phenomenon known as Rydberg excitation blockade5. Here we report initial observations of coherent many-body Rabi oscillations between the ground level and a Rydberg level using several hundred cold rubidium atoms, with a 0.67(10) preparation efficiency of the singly-excited many-body state. The strongly pronounced oscillations indicate a nearly complete excitation blockade of the entire mesoscopic ensemble by a single excited atom.
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A two-level quantum system coherently driven by a quasi-resonant electromagnetic field is one of the centrepieces of modern quantum physics. Notably, Rabi oscillations in isolated single atoms or dilute gases form the basis for metrological applications such as atomic clocks and precision measurements of physical constants6. A wide array of two-level systems have been realized, with atoms, molecules, nuclei, and Josephson junctions being some of the prominent settings. More than half a century ago Dicke recognized that an atomic ensemble coupled to an electromagnetic field cannot always be treated as a collection of independent atoms3. His ground-breaking work gave rise to a rich field of collective atom–field interaction physics7.
A key prediction of Dicke’s theory is that under certain conditions atom–field coupling is enhanced by a factor when compared to one atom. Collectively enhanced atom–field coupling has since been observed in a variety of settings involving either the emission or absorption of radiation. A coherent multi-atom Rabi oscillation at a frequency is a particularly dramatic manifestation of quantum mechanics at work on mesoscopic scales, where an entire ensemble exhibits the dynamical behaviour of a single two-level system. In 2001, Lukin et al. proposed to realize many-body Rabi oscillations in ensembles of atoms driven by a laser tuned to a Rydberg level, and outlined designs for scalable quantum gates for quantum computation and simulation and generation of entangled collective states for metrology beyond the standard quantum limit5.
When an atom is promoted into a Rydberg level with principal quantum number n, the valence electron is in an orbit that is ∼n2 larger than that of the ground-level atom. The atomic dipole moment is correspondingly larger, so that the interaction of two atoms is increased by ∼n4 in the dipole–dipole regime and by ∼n11 in the van der Waals regime8. For n≃100 the interactions are sufficiently strong that for two atoms separated by a distance ∼10 μm the associated energy shift may prevent the second atom from being excited. This excitation blockade mechanism gives rise to an oscillation between the collective ground state and the state in which one of the N atoms is in the Rydberg level |r〉, with frequency (refs 4, 5, 7, 9, 10). The average number 〈N〉r of atoms in level |r〉 is given by:
This result holds for an inhomogeneous distribution of atom–light coupling Ωi with the modification and , see Methods. For two atoms, Rydberg blockade11 and the accompanying enhancement of the Rabi oscillation frequency12 have been observed. Over the past decade significant progress has been made in studying many-atom Rydberg blockade13,14,15,16,17,18,19,20,21, however neither blockade by a single atom nor the many-body Rabi oscillations have been achieved.
Here we report observations of many-body Rabi oscillations for a mesoscopic (a≃15 μm) ensemble of rubidium atoms in the regime of Rydberg excitation blockade by just one atom. To achieve this, the interaction strength ΔEB≡Δi j(a) between a pair of atoms at a distance equal to the ensemble size a must be greater than the spectral width δ ω of the exciting laser field. The duration of coherent atom–light interaction is limited to ≲2 μs by the finite coherence time of the ground-Rydberg transition caused by atomic motion22. Therefore, we choose a high-lying Rydberg level |r〉 = |102s1/2〉 for which ΔEB≳5 MHz is sufficiently large. The experimental apparatus has been described elsewhere22; the atoms were excited with laser fields of two-photon linewidth δ ω≈5 MHz (comparable to ΔEB) while relying on the dephasing of multiply-excited spin waves23,24,25 to generate high-quality single photons. To realize the excitation blockade regime, we lower the laser linewidths to <100 kHz and employ a longer (1 μs instead of 0.2 μs) excitation pulse. We reduce the impact of decreasing atomic density due to ballistic expansion of the cloud, and the concomitant smearing of the oscillations, by using a shorter, 50 μs instead of 200 μs, sequence of trials for each lattice loading (see Methods).
We prepare a gas of 87Rb atoms of temperature T≃10 μK and of peak density ρ0≃1012 cm−3 in a one-dimensional optical lattice. The lattice is shut off, and the atoms are driven in resonance between the ground |g〉 = |5s1/2〉 and a Rydberg |r〉 level with the two-photon Rabi frequency Ω(r) = Ω1(r)Ω2(r)/(2Δ1) for a duration τ = 1 μs, with the corresponding single-atom excitation pulse area θ≡Ω(0)τ, as shown in Fig. 1a,b. The transverse size (Gaussian waists wx≈wy≃6 μm) of the Rydberg excitation region is determined by the overlap of the nearly counter-propagating two-photon excitation laser fields Ω1 at 795 nm and Ω2 at 474 nm. The longitudinal extent of the ensemble is determined by the sample size of the waist wz≈11 μm along z.
We measure the population of state |r〉 by quantum state transfer onto a retrieved light field using a 1 μs long read-out field Ω3 at 474 nm, in resonance with the transition26,27. The retrieved field is coupled into a single-mode fibre followed by a beam splitter and a pair of single-photon detectors D1 and D2. Figure 2a shows the sum of the photoelectric detection event probabilities at the two detectors P≡p1+p2 as a function of the single-atom Rabi angle θ, varied by changing Ω1(0) between 0 and 5.5 MHz for a fixed Ω2(0) = 3.3 MHz. The data are fit with the sinusoidal oscillation of equation (1) modified by two Gaussians, as described in the Methods. The choice of the fit function is motivated by a physical picture in which the visibility of the oscillation is smeared by fluctuations of the atom number and the intensities of the laser fields Ω1 and Ω2. The overall decay of the retrieved signal is due to an inhomogeneous distribution of light shifts for atoms in state |R〉, NeΩ(0)2/ΔEB which couple the state |R〉 to other collective singly-excited states |R′〉, and due to population of doubly-excited states |R R〉 which are retrieved with substantially suppressed efficiency due to spin-wave dephasing22,23,24. The effective number of atoms Ne is defined as . For our experimental geometry Ωi2 = Ω2(0)exp(−2x2/wx2−2y2/wy2), and the atom density ρ = ρ0 exp(−2z2/wz2). Therefore Ne = (π/2)3/2wxwywzρ0. The efficiency ηp with which state |R〉 is prepared is obtained by normalizing the probability of a photoelectric detection event per trial by the retrieval efficiency of state |R〉 into a single photon ηr = 0.23(3) and the transmission and detection efficiency ηtd = 0.26 (see Methods). At the first oscillation maximum () we obtain ηp = 0.67(10). The uncertainty is largely due to the value of ηr measured with the |81s1/2〉 Rydberg level. ηr maybe be somewhat lower for the |102s1/2〉 level owing to longer retrieved fields and correspondingly larger motional dephasing22. Our Monte-Carlo simulations of the excitation process that include atomic interactions and motional spin-wave dephasing predict ηp≃0.75 for . We expect measured values of ηp to be closer to unity when Rydberg excitation blockade is stronger, which can be achieved by reducing the size of the ensemble or increasing the lifetime of the ground-Rydberg optical coherence.
To explore the collective character of the observed Rabi oscillations, we measure P as a function of θ while varying the peak density of the sample ρ0, as shown in Fig. 2b–d. Figure 2e shows the normalized frequency of the Rabi oscillation extracted from the data in Fig. 2a–d as a function of the number of atoms in the ensemble Na. The latter is calculated using the peak density ρ0 measured by the hyperfine state-selective fluorescence imaging of the atomic sample with magneto-optical trap cooling beams used without a repumping field to exclude the contribution of |5s1/2,F = 1〉 atoms. The absence of further peaks in Fig. 1c supports a near-unity value for the fraction of atoms f in the m = 0 Zeeman sub-level. Ideally, we expect the effective atom number Ne extracted from the Rabi oscillation period to equal the atom number Na determined by fluorescence imaging of the sample. The parameter C in the fit in Fig. 2e would equal unity, whereas we extract C = 0.74. Apart from the factor , probable causes for C<1 are alignment imperfections, uncertainties in the determined waists of the two-photon excitation laser beams, and uncertainties in the fluorescence measurements of ρ0.
We further confirm that the dynamics seen in Fig. 2 correspond to the oscillation of equation (1) by measurements of the second-order intensity correlation function at zero delay g(2)(0) as a function of θ, shown in Fig. 3. Measured values of g(2)(0) well below unity, together with substantial visibility of the oscillations, indicate that only one Rydberg excitation is present in the entire ensemble of several hundred atoms. The substantial observed values of g(2)(0)≈0.3 for in Fig. 3b suggest that the population of doubly-excited states contributes noticeably to the extracted values of α. Combining all the data points for in Fig. 3b we obtain g(2)(0) = 0.006(6), which to our knowledge is the lowest value for this quantity for any previously reported light source. It is consistent with a lower bound of gb g(2)(0) = 0.012(2) due to background counts, of which about half are due to detector dark counts. Our Monte-Carlo simulations suggest that both excitation blockade5 and spin-wave dephasing23 mechanisms contribute to the suppression of two-photon events. In contrast, in our previous study using shorter and wider linewidth excitation, numerical simulations employing spin-wave dephasing, without excitation blockade, accurately described the observed spatial spin-wave correlations28.
The importance of achieving excitation blockade, ΔEB≫δ ω, to observe many-body Rabi oscillations is checked by reducing ΔEB in measurements with n = 90 and n = 81, as shown in Fig. 4. Figure 5a shows data with increased δ ω using a shorter τ = 0.2 μs excitation. The oscillation is less pronounced both for smaller ΔEB, Fig. 4, and larger δ ω, Fig. 5a. Figure 5b shows a similarly suppressed oscillation in measurements with the |100d3/2〉 level with a τ = 1 μs excitation. This may be attributed to a blockade breakdown due to a strong angular dependence of the atomic interaction strengths for |n d〉-levels13. It should also be noted that for a Gaussian distribution of atom–field couplings single-atom Rabi oscillations are almost completely washed out29, which makes the observation of many-atom oscillations under these conditions even more remarkable.
We have demonstrated coherent many-body Rabi oscillations in an ensemble of several hundred cold rubidium atoms. The oscillations provide compelling evidence for the achievement of a collective Rydberg excitation blockade by a single excited atom. Our results pave the way towards quantum computation and simulation using ensembles of atoms5,30.
Methods
Atom preparation and laser excitation.
A magneto-optical trap of 87Rb is loaded from background vapour for 70 ms. During the following 25 ms, the detuning of cooling light is increased, the repump intensity is decreased, and the optical lattice is turned on. The lattice is composed of a single 782 nm retro-reflected linearly polarized Gaussian beam. Untrapped atoms are allowed to fall away from the experimental region during the next 15 ms period, and a B0 = 4.3 G bias magnetic field is turned on. The trapped atoms are optically pumped to the |5s1/2,F = 2,mF = 0〉 state, the optical lattice is switched off by an acousto-optical modulator (AOM), and a 3 μs long sequence of two-photon Rabi driving and retrieval is repeated for 50 μs, with a 1 μs optical pumping period included every five cycles. The overall repetition rate of the experiment is ≈8 Hz. For the data in Fig. 2 the peak density ρ0 was controlled by varying the time period between lattice loading and the two-photon excitation sequence between 15 and 90 ms. Both Ω1 (at 795 nm) and Ω2 (at 474 nm) fields are linearly polarized along the same axis. The 795 nm field is produced by an extended cavity diode laser. Light at 474 nm is generated by frequency-doubling the output of a tapered amplifier driven by a 950 nm extended cavity diode laser. Both lasers are frequency-locked to a thermally stabilized ultra-low expansion glass cavity. The transition is located by scanning the laser frequency across a resonance and measuring the photoelectric detection probability for the retrieved field. The 795 and 474 nm excitation fields are tuned to the two-photon resonance between the ground-level component |5s1/2,F = 2,mF = 0〉 and a Zeeman component of the Rydberg level |n s1/2,mj = −1/2〉. The single-photon Rabi frequency on the blue transition is , where is electric field amplitude. The radial matrix element is reduced using the Wigner–Eckart theorem. The angular part is calculated following ref. 31, while the reduced matrix element is approximated by 〈r〉 = 0.14×(50/n)3/2a0 (ref. 13).
Because Ω2 and Ω3 fields are propagating in the same spatial mode, the retrieved field is phase matched into the mode of the Ω1 field and coupled into a single mode 50/50 fibre beamsplitter followed by a pair of single-photon detectors D1 and D2. A gating AOM at the fibre beamsplitter input port is employed to avoid damage to the single photon detectors by the Ω1 field.
Photoelectric detection and data analysis.
For every experimental trial, photoelectric events on detectors D1 and D2 are recorded within a time interval determined by the length of the retrieved pulse. Photoelectric detection probabilities for both detectors are calculated as p1,2 = N1,2/N0, where N1,2 are the numbers of recorded events and N0 is the number of received triggers.
For the storage and retrieval protocol p1,2∼ηsηrηtdn, where n is the number of photons in the incident Ω1 field. We can therefore extract the retrieval efficiency of state |R〉 into a single photon ηr from ηsηr through the retrieved signal measurements, and the storage efficiency ηs through the measurements of the transmitted fraction of Ω1. Using |r〉 = |81s1/2〉 we obtain ηs = 0.0098(14) and ηsηr = 0.00222(7), resulting in ηr = 0.23(3). A higher value of the Rabi frequency Ω2(0) = 5.7 MHz is employed in these measurements. Transmission through the glass vacuum chamber is 0.92, the gating AOM diffraction efficiency is 0.7, the fibre coupling efficiency is 0.73, and the quantum efficiency of the single-photon counters is 0.55, for a combined light transmission and detection efficiency ηtd = 0.26.
The photoelectric detection probability for double coincidences is calculated as N12/N0, where N12 is the total number of simultaneous clicks on both detectors for a given experimental trial. The second-order intensity correlation function at zero delay is given by g2(0) = p12/(p1p2).
Decoherence model.
We employ the following Hamiltonian to describe our system: . The atomic operators for the atom μ are defined as , where a,b∈[g,r] with |g〉μ being the atomic ground state and |r〉μ being the addressed Rydberg level. The two-photon excitation is modelled using the effective Rabi frequency Ω = Ω1Ω2/(2Δ). The interaction between Rydberg levels is described with a single-channel model. For , the excitation blockade is operational. Adiabatic elimination of double and higher-order excitations from the equations of motion results in an effective Hamiltonian for the singly-excited part of the spectrum: . Here , Ci j = −ΩiΩj/(4Δi j), where |j〉is the many-body state with the jth atom in the Rydberg level. The first two terms of the effective Hamiltonian are due to the light shifts induced by the (off-resonant) doubly-excited states onto the single excitations.
When the interaction-induced inhomogeneous light shifts are omitted, the Hamiltonian results in an ideal Rabi oscillation between the ground state |G〉 and the single spin wave . If at time t = 0 the system is in state |G〉, the state at future times is given by |ψ(t)〉 = cos(Ω t/2)|G〉−i sin(Ω t/2)|R〉. When the light shift terms are included, the state |R〉 is coupled to a broad distribution of singly-excited states and therefore leaks into this quasi-continuum, leading to P∼|〈R|ψ(t)〉|2 decaying with a rate NeΩ02/ΔEB. The doubly-excited states are expected to be populated at a rate NeΩ02. Trial-to-trial fluctuations ΔΩ and ΔNe in Ne and Ω0, respectively, lead to a decay of the oscillation visibility. The probability of photoelectric detection per trial P as a function of θ in Figs 2–5 is therefore fit by a function , where dimensionless fit parameters αNe and β(ΔNe/2Ne)2+(ΔΩ/Ω0)2 describe the roles of the light shifts and population of doubly-excited states, and atom number and intensity fluctuations, respectively, while an amplitude A represents the overall measured retrieval and detection efficiency.
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Acknowledgements
We thank A. Radnaev, B. Kennedy and A. Zangwill for discussions. This work was supported by the Atomic Physics Program and the Quantum Memories MURI of the Air Force Office of Scientific Research and the National Science Foundation.
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Y.O.D. and A.K. designed the experiment, Y.O.D. and L.L. carried out the experiment and analysed the data, F.B. carried out the theoretical analysis. All authors contributed to writing the manuscript.
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Dudin, Y., Li, L., Bariani, F. et al. Observation of coherent many-body Rabi oscillations. Nature Phys 8, 790–794 (2012). https://doi.org/10.1038/nphys2413
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DOI: https://doi.org/10.1038/nphys2413
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