Abstract
Manybody correlations govern a variety of important quantum phenomena such as the emergence of superconductivity and magnetism. Understanding quantum manybody systems is thus one of the central goals of modern sciences. Here we demonstrate an experimental approach towards this goal by utilizing an ultracold Rydberg gas generated with a broadband picosecond laser pulse. We follow the ultrafast evolution of its electronic coherence by timedomain Ramsey interferometry with attosecond precision. The observed electronic coherence shows an ultrafast oscillation with a period of 1 femtosecond, whose phase shift on the attosecond timescale is consistent with manybody correlations among Rydberg atoms beyond meanfield approximations. This coherent and ultrafast manybody dynamics is actively controlled by tuning the orbital size and population of the Rydberg state, as well as the mean atomic distance. Our approach will offer a versatile platform to observe and manipulate nonequilibrium dynamics of quantum manybody systems on the ultrafast timescale.
Introduction
Atomic, molecular and optical physics with advanced laser technologies has recently emerged as a new platform to study and possibly simulate quantum manybody systems^{1,2,3}. One of its latest developments is the study of longrange interactions among ultracold particles, in which dipolar quantum gases^{4,5}, ion crystals^{6,7,8}, polar molecules^{9,10,11} and Rydberg atoms^{12,13,14,15} allow for revealing the effects of manybody correlations. Rydberg atoms distinguish themselves by their large dipole moments and tunability of the strength and nature of dipolar interactions^{12,13}.
The active electron in a Rydberg atom moves in a macroscopic orbital whose size could range from submicrometre to several tens of micrometres^{16}. The resulting large dipole moments yield a strong interaction U(r) of either van der Waals or resonant dipolar character between a pair of Rydberg atoms separated by r. This interaction, which can be tuned in various ways^{12}, features the emergence of atomic manybody correlations in an ultracold gas. As a prominent example, these interactions shift the energy levels of atoms encompassing a given Rydberg atom, so that additional Rydberg excitations are suppressed for U(r) larger than the excitation line width. This effect, referred to as Rydberg blockade^{13,13,17}, results in spatial or temporal correlations among the atoms, as recently demonstrated in various settings^{15,18,19,20,21,22,23,24,25,26}. Its applications include, for example, the realization of universal atomic and photonic logic gates for quantum information processing^{12,27,28,29,30,31}. The Rydberg blockade has also turned out to be an outstanding new resource to investigate manybody problems^{32}. The blockade condition determines the smallest distance between neighbouring Rydberg atoms, which is typically of the order of several micrometres. However, Rydberg excitations can also be induced at smaller interatomic distances by, for example, detuning the excitationlaser frequency from an atomic resonance^{33,34,35,36,37,38}. In this regime, the character of the correlations changes, resulting in the facilitated formation of aggregates consisting of large number of Rydberg atoms^{34,35,36,37,38}.
A complementary approach to the correlations induced by the interparticle interactions consists in studying the temporal evolution of electronic coherences of the Rydberg atoms. Using a broadband picosecond laser pulse, Rydberg excitations can be induced over a wide range of interatomic distances from <1 μm to the isolated atom limit. The number of Rydberg excitations per unit volume can thus be larger than the ones in the blockade regime by two orders of magnitude. Although picosecond and femtosecond laser pulses have been previously used to observe electronic wave packets in isolated Rydberg atoms^{39,40} and the dephasing due to twobody interactions^{41}, here we exploit them to explore how coherent dynamics evolves in a manybody regime. The strong interactions in our ultracold Rydberg gas induce an electronic dephasing on the picosecond timescale, which is directly observed in a timedomain Ramsey interferogram oscillating with a period of ∼1 fs and phaseshifted on the attosecond timescale^{42} by the Rydberg interactions. We measure this minute phase shift and the dephasing directly, and compare them with theoretical simulations based on nearestneighbour interactions, a meanfield model and manybody correlations to reveal effects indicating atomic correlations beyond a meanfield description. The twobody interaction energy at around 1 μm distance, which can be easily accessed in our approach, exceeds the average kinetic energy by many orders of magnitude. This regime compares favourably with previous experiments in thermal vapour cell experiments with nanosecond laser pulses^{38,43}, where the nearestneighbour distance is also <1 μm. In thermal cells, the Rydberg interaction energy is comparable to the average kinetic energy, so that the measurement of coherent evolution of the interaction dynamics is strongly affected by the thermal atomic motion, as will be discussed later quantitatively.
Results
Experimental setup
Figure 1a shows the schematics of our experimental setup. A cold ensemble of ^{87}Rb atoms is prepared in an optical dipole trap with temperature and highest peak atomdensity estimated to be ∼70 μK and ∼1.3 × 10^{12} cm^{−3}, respectively (see Methods sections ‘Estimation of the temperature’ and ‘Estimation of the atom density’ for these temperature and density estimations). On the timescales of interest, atomic motion can be ignored in the frozen gas regime^{44,45}. The atoms are optically pumped to the hyperfine state F=2, m_{F}=+2 of the ground state 5S_{1/2} and excited to Rydberg states via a twophoton transition using broadband picosecond laser pulses with their centre wavelengths tuned to ∼779 and ∼481 nm (Fig. 1b), hereafter referred to as the infrared and blue pulses, respectively. The dipoletrap laser is turned off 2 μs before the irradiation of the infrared and blue pulses to avoid 2+1 multiphoton ionization induced by a combination of the infrared and blue pulses with the trapping laser light. The infrared and blue pulses, and the optical pumping beam are circularly polarized in the same direction with respect to the magnetic field, suppressing excitations to the S Rydberg states, so that the state vD_{5/2}, m_{J}=+5/2 is mostly populated, where v is a principal quantum number. The population of the Rydberg state is measured by field ionization^{16}. Details of the field ionization are described in Methods section ‘Rydberg excitation and detection’. The maximum population of the vD states in the present experiment is not >5%, to suppress photoionization by the picosecond laser pulses (see Supplementary Note 1 for the effects of ions). More details on the atom preparation, the estimation of the atom density and the temperature, as well as the Rydberg excitation are described in Methods and Supplementary Note 2.
The bandwidth of our Rydberg excitation is determined from a fieldionization spectrum exemplified in Fig. 1c (see Supplementary Note 3), in which the excitation is tuned to the 42D_{5/2} state, which is the main target state of the current experiment. The fieldionization voltage is ramped up slowly enough on the 5 μs timescale to resolve neighbouring Rydberg levels in such fieldionization spectra. The bandwidth of the excitation is ∼150 GHz (full width at half maximum, FWHM) and is much larger than those of the continuouswave, nanosecond, and subnanosecond pulsed lasers employed in previous ultracold Rydberg experiments^{12,13,41,46,47}. The fieldionization spectrum indicates that the bandwidth of the Rydberg excitation is larger than the energy separation of neighbouring Rydberg states and is accordingly wide enough to remove the Rydberg blockade. As schematically shown in the upper panel of Fig. 1d, our picosecond laser pulses can excite a pair of Rb atoms simultaneously to the Rydberg states v=42 even at interatomic distances shorter than 1 μm.
Observation of manybody electron dynamics
The interaction among the Rydberg atoms is observed by timedomain Ramsey interferometry with a pair of the Rydberg excitations, hereafter referred to as the ‘pump’ and ‘probe’ excitations, whose delay was stabilized on the attosecond timescale with our homemade optical interferometer^{48} (see Methods section ‘Timedomain Ramsey interferometry’). By scanning the delay time τ between the pump and probe excitations on the attosecond timescale, we measure the Ramsey oscillation of the population integrated over all Rydberg states, which remains after the probe excitation, by the field ionization. The fieldionization voltage in these Ramsey measurements is ramped up rapidly enough on the 100 ns timescale to avoid subsequent decay processes of the Rydberg states. For the excitation tuned to the 42D_{5/2} state as shown in Fig. 1c, a single Rydberg state v〉 with v=42 is predominantly populated. The Rydberg population thus measured oscillates as a function of τ with a frequency close to the transition frequency between the 5S and Rydberg states^{42}. This is in contrast to the standard Ramsey interferometry, in which the signal oscillates with a frequency close to the detuning frequency of the excitation laser from the atomic transition^{22,23}. That is, in the absence of interactions, the population in the Rydberg state P_{v}(τ) is given by
and oscillates with the frequency E_{v}/h, where E_{v} is the energy of the Rydberg state v〉 measured from the ground state 5S (see refs 42, 49 and Supplementary Note 2) and ℏ is the Planck constant h divided by 2π. This oscillation is identical to the temporal oscillation of the Rydberg state v〉, except that the realtime t is replaced by the pumpprobe delay τ. Therefore, the Ramsey oscillation for v=42 corresponds essentially to the temporal oscillation of the Rydberg wave function v=42〉 and to the recurrence motion of an electronic wave packet, which is composed of the 5S and Rydberg state v=42 superposed coherently by the excitation pulses. Here we investigate how the Rydberg interactions affect this coherent electron dynamics.
In a simplified meanfield approach, the Rydberg interactions change only the energy of the Rydberg state v〉 and accordingly the oscillation period of P_{v}(τ). This results in a phaseshift of the Ramsey oscillation accumulated in the delay time τ. In addition, as the atoms are randomly distributed in the ensemble, their energy levels E_{v} are shifted randomly by the interactions, making the periods of their Ramsey oscillations different from each other. Therefore, the measured signal is the superposition of many oscillations with different periods, so that its contrast is expected to decay as a function of τ due to Rydberg interactions.
Figure 2a–c show examples of the Ramsey oscillations for the 42D_{5/2} state with a population of 3.3±0.1%. As mentioned above, these oscillations correspond to the ultrafast recurrence motion of the electronic wave packet with a period of ∼1 fs. For each oscillation, we measured the fieldionization signals of two atomic ensembles with different peak densities estimated to be ∼1.3 × 10^{12} and ∼4 × 10^{10} cm^{−3} alternately to suppress systematic uncertainties, scanning τ in steps of ∼30 as (see Methods section ‘Estimation of the atom density’ for these density estimations). We obtained the contrasts and phases of the measured oscillations by sinusoidal fitting, as shown in Fig. 2a–c (see Methods section ‘Timedomain Ramsey interferometry’). Figure 2d shows that the contrast is approximately constant for τ up to ∼500 ps for the lowerdensity ensemble (bluecircle data points), indicating that the effects of the interactions are negligibly small. This result is consistent with the interaction strength estimated from the present atom density and the twoatom potential curve presented in Supplementary Fig. 1. Hereafter, we take these contrasts and phases measured in the lowerdensity ensemble as references to be compared with those measured in the higherdensity ensemble. The oscillatory structures on the ∼10 ps timescale at τ∼130–170 ps seen in Fig. 2d could be partly due to the recurrence motion of a wavepacket composed of the Rydberg state v=42 and the traces of its neighbouring states seen in Fig. 1c (see Supplementary Note 4 for details).
Figure 2d shows that the contrast decays as a function of τ for the higherdensity ensemble (redcircle data points). The phase shift of the higherdensity ensemble from the lowerdensity one also changes as a function of τ as seen in Fig. 2e. The offset of this phase shift at τ=0 is essentially due to the difference between ACStark shifts of the atomic levels in the higher and lowerdensity ensembles. Slight differences between the sizes, shapes and positions of the higher and lowerdensity ensembles could lead to their different ACStark shifts (see Supplementary Note 2 for more details on the origin of this zerodelay offset of the phase shift).
Figure 3 shows the results of measurements of the contrast ratio between the two different densities given above, hereafter referred to as ‘Ramsey contrast’, and the phase shift for the 42D_{5/2} state. These results are plotted as functions of τ for the two different Rydberg populations p_{e} of 1.2±0.1% and 3.3±0.1%, showing that the Ramsey contrast decays, and the phase shift is accumulated as a function of τ.
It is evident from Fig. 3 that the contrast decay and the phase shift are enhanced when the Rydberg population is increased from ∼1.2% to ∼3.3%. The strength of the interactions is tuned by varying the Rydberg quantum number v and the atom density. Figure 4a shows Ramsey contrasts as functions of τ for three different Rydberg levels v=38, 42 and 50. It is seen from this figure that the contrast decay is accelerated by increasing the principal quantum number v of the Rydberg level. The dependence of the Ramsey contrast decay on the atom density is shown in Fig. 4b, in which the contrast decay is accelerated by increasing the atom density (see Supplementary Note 5 for the estimation of the atom densities plotted in the abscissa of this figure). From these combined measurements as functions of the Rydberg population, principal quantum number and atom density, we conclude that the observed contrast decay and the phase shift are induced by Rydberg interactions.
The origin of the observed behaviour of the contrast decay and the phase shift is further investigated in Fig. 3. Here we compare the experimental data with the Ramsey contrast decays and the phase shifts calculated for nearestneighbour interactions without considering interactions among three or more Rydberg atoms (solid lines). The zerodelay offset of the calculated phase shift is arbitrary and adjusted so that the average of the first ten data points is equal to the calculated phase shift averaged over the delay window for those ten data points. Results for nearestneighbour interactions are obtained by simple numerical calculations following those performed in previous Ramsey studies^{23,41}, in which the timedomain Ramsey oscillations modulated by nearestneighbour interactions are averaged over the distribution of nearestneighbour distances (see Supplementary Note 6 for details). Here, the Ramsey oscillation is obtained by solving a Schrödinger equation with a Hamiltonian for two interacting atoms. We assume a van der Waals interaction of the form U(r)=−C_{6}/r^{6} with the coefficient C_{6} being an adjusting parameter. It is seen in the insets of Fig. 3a,b that the calculated Ramsey contrast decay becomes slightly faster as the C_{6} value is increased, but does not grow beyond the decay thresholds 0.988 and 0.967 determined by the Rydberg population p_{e}∼1.2% and 3.3%, respectively (for details on these thresholds, see Supplementary Note 6). Similarly, the calculated phase shift is almost constant as the C_{6} value is increased, as shown in Fig. 3c,d. These features are not altered by changing the character of the interaction such as van der Waals and dipole–dipole (see Supplementary Note 6 for details). The contrast decays and phase shifts observed experimentally are thus clearly larger than expected for nearestneighbour interactions. It is therefore concluded that our experimental observation of the Ramsey contrast decay and phase shift on the attosecond timescale demonstrates interactions among more than two Rydberg atoms, and the effect of those interactions on the electron dynamics can be actively controlled by tuning the population and principal quantum number of the Rydberg level, as well as the atom density.
Test of a meanfield model
To model these observations, we first apply a meanfield model (see Supplementary Note 7 for its details). As in the calculations with nearestneighbour interactions above, we assume a van der Waals interaction of the form U(r)=−C_{6}/r^{6} with the coefficient C_{6} being the only fitting parameter, which is optimized to reproduce the measured Ramsey contrast in Fig. 3b by a leastsquares fitting. The outline of the leastsquares fitting is given in Supplementary Note 8. The results of the meanfield simulations are plotted in Fig. 3 (dashed lines). Figure 3a,b show that reasonable agreements can be found for the Ramsey contrast between the measurements and the meanfield simulations with C_{6}=1.9 GHz μm^{6}. This value for C_{6} yields the phase shifts simulated by the meanfield model in Fig. 3c,d, where the zerodelay offset is adjusted in the same way as in the calculations with the nearestneighbour interactions above. Figure 3c shows that the meanfield simulation yields a phaseshift larger than the measured one by a factor of ∼4, failing to reproduce our observations. This discrepancy between measured and simulated phase shifts is not improved by assuming a dipole–dipole interaction, a hybrid form of a dipole–dipole and a van der Waals interaction or by introducing anisotropic interactions (see Supplementary Figs 2 and 3). The closer agreement between the measured and simulated phase shifts for p_{e}∼3.3% in Fig. 3d than for p_{e}∼1.2% in Fig. 3c is understood as follows. The Gaussian atomdensity distribution of our experimental setup leads to the saturation of the phase shift at longer pumpprobe delays. This is because the rapid decrease of the atom density in the Gaussian tails results in the suppression of the contribution to the phase shift from atoms distant from the centre, and therefore the phase shift does not grow afterwards. This saturation is reached both by the measured and simulated phase shifts for p_{e}∼3.3% within our measurement time 500 ps (unlike the case with p_{e}∼1.2% and therefore weaker interactions), giving their closer agreement for p_{e}∼3.3% than for p_{e}∼1.2%. This saturation effect is explained more quantitatively in Supplementary Note 9.
Beyond meanfield analysis
Next, we apply an exactly solvable theory model^{50,51,52,53,54} to the observations. Details of this model are presented in ref. 54. Briefly, we represent each atom as a twolevel system, which is a pseudospin system, consisting of a ground state g〉 () and an excited Rydberg state e〉 () with energies E_{g} and E_{e}, respectively. The experiment consists of four stages: (i) the pump excitation, (ii) an evolution with an Natom Hamiltonian H, (iii) the probe excitation and (iv) the population measurement by field ionization. The Natom Hamiltonian in stage (ii) includes the atomic energies and the interactions U(r_{jk}) between atoms in the Rydberg states as follows:
with the atomicresonance frequency and the Pauli matrix, which is used to represent the internal states of the pseudospin at position j (see Methods section ‘Outline of the exactly solvable model simulation’ for more details)^{50,51,52,53,54}.
An exact solution for the time evolution with a Hamiltonian of the form of equation (2) has recently been presented in refs 50, 51, 52, 53, 54 (see also similar numerical approaches in refs 55, 56). This allows for deriving an expression for the exact time evolution of the Ramsey signal P(τ) for any strength of interactions^{50,51,52,53,54}. For any given atom j interacting with N–1 neighbouring Rydberg atoms, one obtains
where p_{g} and p_{e} are the ground and Rydbergstate populations, respectively, produced by the initial pump excitation, describes a frequency shift induced by the interaction between atoms j and k, and ϕ is the phase offset arising from the ACStark shifts during the picosecond pulse excitations (see Supplementary Note 2). It is seen from this equation that this model considers different clusters of interactions superposed coherently instead of averaged as performed in our preceding meanfield analysis. This coherent superposition leads to correlations among different atoms, as exemplified for a twoatom correlation in Fig. 5 (ref. 54).
Further analytical progress is possible by using an approximation, hereafter referred to as a ‘continuum approximation’, in which a continuum function n(r) is considered for the density distribution of Rydberg atoms (see Methods section ‘Continuum approximation’ for details). Briefly, we assume the density distribution to be homogeneous in a small volume around a particular position r (ref. 54). This approximation leads to the following expression for the Ramsey signal averaged over the whole atomic ensemble
The contrast decay g(τ) and phase shift α(τ) are then obtained from , which, in the case of an isotropic van der Waals interaction and for , is given by
Here, ω_{B}=2π × 75 GHz is the half width half maximum of the pump excitation, n_{p} is the peak atom density, the coefficient results from the Gaussian atomic density distribution in the experiment and is the decay constant^{54}. Equation (5) predicts that the Ramsey contrast decays approximately as a stretched exponential with a squareroot dependence on the pumpprobe delay τ. This squareroot dependence is characteristic of the van der Waals interaction^{54}. The decay constant reads , where is the average density for a Gaussian distribution.
Figure 6 shows comparisons between (i) the experimental data for the contrast decay and phase shift as functions of τ for the 42D_{5/2} state and (ii) their numerical results based on the analytical continuum approximation equation (4) (solid lines). The curves for the exact numerical results agree well with the analytical ones for , so that they are indistinguishable on the present scale of the figure. The figure demonstrates that both the exact and analytical results obtained for C_{6}=34 GHz μm^{6} agree well with the measured Ramsey contrasts and phase shifts for both of the Rydberg populations p_{e}∼1.2% and 3.3% used in the measurement. Figure 6 also shows numerical results with the continuum approximation for the Ramsey contrasts and phase shifts obtained for a finite average number of interacting Rydberg atoms N=20 and 40 (semitransparent lines; see Methods section ‘Continuum approximation’). The agreement between the numerical and experimental results improves monotonically with increasing N. Figure 7 shows the Ramsey contrast and phase shift measured (the darkgrey solid lines accompanied by lightgrey shaded areas) and simulated with the continuum approximation (the blue solid lines) at the pumpprobe delay τ∼500 ps. It is seen from Fig. 7b that the blue solid line crosses the upper boundary of the lightgrey shaded area, which is a confidence interval of the measured value, at an average number of interacting atoms around 40, showing that more than ∼40 atoms correlated are necessary to reproduce the measured phase shift.
In the theory model above, the C_{6} coefficient of the van der Waals interaction serves as the only fitting parameter for the Ramsey contrast in Fig. 6b. The C_{6} value thus obtained is used for calculating the contrast decay in Fig. 6a and the phase shifts in Fig. 6c,d. Then, the zerodelay offset of the phase shift serves as the only adjustable parameter for the phase shift in each of Fig. 6c,d. For distances shorter than the average interatomic separation, the bandwidth of our Rydberg excitation with the picosecond infrared and blue pulses covers multiple adiabatic interaction potentials that can hybridize with the one correlating asymptotically to the 42D+42D limit (see Supplementary Fig. 1). Although in principle this could affect the contrast decay, the present model based on a single effective potential captures the observed dynamics both for the Ramsey contrast and the phase shift (see Supplementary Note 10 for the effective treatment of the interaction potentials).
We note that the predicted decay constant for the Ramsey contrast for p_{e}∼3.3% is approximately given by . The pumpprobe delay to reach e^{−1}(∼0.37) of the initial contrast is thus given by , which agrees well with our measurement time of 500 ps giving the contrast reduction to 45%.
We have further employed g(τ) in equation (5) to reproduce the measured Ramsey contrasts shown in Fig. 4a for three Rydberg levels v=38, 42 and 50, finding good agreement. The phase shifts for these levels have also been measured and analysed in Supplementary Note 5. It should be noted that several Rydberg states are excited for v=50 (see Supplementary Fig. 4). However, the simulation with a single Rydberg state gives good agreement with the experimental results as shown by the solid lines in Fig. 4a.
We have also performed additional calculations of the measured Ramsey contrasts and phase shifts using dipole–dipole interactions, a hybrid form of a dipole–dipole and a van der Waals interaction, and an anisotropic van der Waals interaction. Although all results are in qualitative agreement with the experimental data, we find that the results for pure van der Waals interactions presented above reproduce the experimental data well with only a single fitting parameter. The results of these additional calculations are given in Supplementary Note 10.
Origin of the failure of a meanfield model
The overestimation of the phase shift by the meanfield approximation in Fig. 3c is intuitively understood as follows. In our actual measurement, the Ramsey contrast of each atom decays due to many different frequencies corresponding to different clusters of interactions superposed coherently, as is expected from equation (3) of the exact theory model. In the meanfield model, however, the Ramsey oscillation of each atom has its own mean frequency, interacting with ‘the other atoms’ as a whole, so that its contrast does not decay, but the decay arises only from an ensemble average of Ramsey oscillations of many atoms phase shifted from each other. Accordingly, those phase shifts need to be overestimated, as seen in Fig. 3c, to reproduce the measured contrast decay.
Figure 8 shows a comparison between the phase shifts calculated by the mean field and exact models as functions of the Rydberg population p_{e} at three different pumpprobe delays τ=20, 50 and 70 ps. It is seen from this figure that the meanfield phase shift is larger than the exact one for p_{e}<0.5. It is also seen from this figure that this difference becomes larger as the effective number of interacting atoms becomes larger at longer pumpprobe delays. The meanfield model thus overestimates the phase shift for the present p_{e}∼0.01 and 0.03 (<0.5), and τ∼50–500 ps, as is intuitively understood as described in the preceding paragraph.
Discussion
The experiments presented here bear similarities with recent investigations on coherent spinexchange dynamics with rotational states of polar molecules trapped in an optical lattice^{9,10}. In these experiments, the effect of singleparticle decoherence, which proceeds faster than the interaction timescale (∼10 ms), has successfully been circumvented by applying spin echo techniques. In our experiment, on the other hand, the interaction timescale ∼1 ns is about three orders of magnitude shorter than the timescale of singleparticle decoherence, which is induced mainly by Doppler broadening and is ∼1 μs at the temperature ∼100 μK estimated for our Rydberg gas (see Methods section ‘Estimation of the temperature’). The combination of ultrafast and ultracold approaches thus provides an effective pathway for isolating the observation and control of coherent dynamics of a system from its singleparticle decoherence processes.
Experiments similar to the ones demonstrated here could in principle be implemented in thermal cells^{38,43} if one can compensate for the relevant Doppler shift, which causes a phase shift comparable to the interactioninduced phase shift, and can also compensate the effects of atomic motions. In the thermal cell at a room temperature, the atoms are not frozen, but move with kinetic energies at that temperature, so that the motion would lead to additional phase shifts on the order of under the conditions (λ∼297 nm, τ∼500 ps, ms ^{−1}), where λ is the wavelength that corresponds to the energy difference between the ground and Rydberg states. In our current ultracold measurements, on the other hand, the atoms move only 40 pm on average during the same duration, and this distance is shorter than λ by four orders of magnitude. This distance is also shorter than the average nearestneighbour distance in our Rydberg gas by four orders of magnitude (see Methods section ‘Estimation of the atom density’). Our Rydberg gas is therefore safely regarded as a frozen gas, so that the effects of the atomic motions and collisions on the measurements of coherent dynamics are negligible.
In our present study, a combination of the contrast and phase measurements serves as a useful tool to observe the effects of manybody correlations. The correlations would be further verified by additional measurements of variance in Ramsey signals, as is performed in refs 8, 57, to observe similar manybody correlation effects. Combining a microscope^{18,19,21} with our experimental setup offers another future possibility to observe manybody correlations more directly in a spatially resolved manner.
We anticipate promising future applications of applying ultrashort coherent laser pulses to ultracold Rydberg gases. Using alternative excitation schemes, one may investigate beyond meanfield effects in Ramsey experiments with more complex Hamiltonians such as Heisenbergtype Hamiltonians of interest for polar molecules^{9,10} and atomic clocks^{57} in optical lattices. Another application could be the investigation of a scenario in which Rydberg electronic wave functions are spatially overlapped between neighbouring Rydberg atoms^{58}. This could lead to new exotic phases in which the Rydberg electrons are shared among many nuclei, and exchange interactions play key roles in their dynamical properties on the ultrafast timescales. Such a metallike manybody Rydberg state would naturally lead to Penning ionization quite rapidly. However, Jaksch and colleagues^{59} have theoretically estimated the lifetime of such a metallike Rydberg state of two ^{85}Rb atoms (v=50) to be ∼100 ns. This is longer than our measurement timescale by more than two orders of magnitude, rendering the observation of this state possible using our timedomain approach.
Methods
Atom preparation
A magnetooptical trap (MOT) of ^{87}Rb atoms was loaded from background vapour for 1.4 s. During the subsequent MOT compression for 30 ms, an optical dipole trap was turned on. The dipole trap was composed of a single 1,064 nm beam with its power and beam waist being ∼4 W and ∼30 μm (1/e^{2} radius), respectively. Polarization gradient cooling was performed for 100 ms. The trapped atoms were then transferred into the F=1 ground state by switching off the MOT repump laser. After that, we turned off the MOT trapping beams and the magnetic field. While keeping the intensity of the dipole trap laser, plain evaporative cooling was carried out for 50 ms. During the evaporation process, a 76 μT homogeneous magnetic field was turned on, pointing along the direction of the dipole trap laser. For the next 200 μs, the atoms were optically pumped to the state by using the MOT repump beam and a beam, which is resonant to the transition from to and counterpropagates with the dipole trap laser. The dipoletrap laser was turned off 2 μs before the irradiation of the picosecond infrared and blue pulses to avoid 2+1 multiphoton ionization induced by a combination of the picosecond pulses and the trapping laser beam, whereas the homogeneous magnetic field remained on. The picosecond infrared and blue pulses at ∼779 and ∼481 nm, which propagated collinearly with the dipole trap beam, had crosssections with FWHMs of ∼130(100)μm and 30(30)μm along the x (y) direction (Fig. 1a).
Estimation of the atom density
At first, the atom density in the Ramsey measurements was estimated solely from the total number of atoms and the size of the atomic ensemble obtained by insitu absorption imaging with a CCD (chargecoupled device) camera without expanding the atomic ensemble. However, in contrast to the axial size of the atomic ensemble (∼2 mm FWHM), the spatial resolution of the insitu absorption imaging with the CCD camera was not high enough mostly because of the aberrations of the imaging lens to resolve the radial size, resulting in an underestimation of the atom density.
In a later independent experiment, hereafter referred to as a ‘reference experiment’, we estimated the radial size from the temperature of the atomic ensemble and the trap frequency of the radial direction, setting the trapping conditions almost the same as those employed in the Ramsey measurements. Those trapping conditions set almost the same were (1) the same loading sequence as described in the preceding subsection; (2) the dipoletrap laser power (with an error described in the next sentence); (3) the dipoletrap laser focusing; and (4) the total number of atoms (with an error described in the next sentence). We performed two reference experiments under two different trapping conditions corresponding to the higher and lower densities in the Ramsey measurements, setting the dipoletrap laser power (the trapping condition (2)) with a difference of ∼2% and ∼5% from the power averaged over the Ramsey measurements for the higher and lower densities, respectively, and setting the total number of atoms (the trapping condition (4)) within 1 s.d. ∼14% and ∼33% of the eight and seven corresponding values measured in the Ramsey measurements for the higher and lower densities, respectively. In these reference measurements, the temperature was measured by an expansion of the atomic ensemble and parametric heating was employed to infer the radial trap frequency. The temperatures and the trap frequencies were thus obtained to be ∼67 μK and 2.2 kHz and ∼39 μK and 1.1 kHz under those two trapping conditions, respectively. These temperatures and trap frequencies gave typical radial sizes (FWHM) of the atomic ensembles in the reference experiments to be ∼14 and 20 μm, which we regarded to be the radial sizes of the atomic ensembles in the Ramsey measurements for the higher and lower densities, respectively.
The total number of atoms was also slightly underestimated by the insitu absorption imaging in the Ramsey measurements because of the spatial resolution, so that it was calibrated in a later independent experiment in which the loading sequence and the trap laser focusing (the trapping conditions (1) and (3)) were almost the same as those employed in the Ramsey measurements. In this calibration experiment, we measured the total number of atoms by the absorption imaging with the expansion of the atomic ensemble as a function of the one measured without the expansion to obtain a linear calibration curve with its slope being 1.04±0.03. This linear calibration curve gave the total numbers of atoms to be ∼6 × 10^{5} and ∼4 × 10^{4} for the higher and lower densities in the Ramsey measurements, respectively, for v=42.
These radial sizes and the total numbers of atoms were combined with the axial sizes measured in situ in the Ramsey measurements, to give the peak atom densities of ∼1.3 × 10^{12} and ∼4 × 10^{10} cm^{−3} for the higher and lower densities, respectively, for v=42. Similarly, the peak atom density and the total number of atoms for v=38 were estimated to be ∼1.2 × 10^{12} cm^{−3} and ∼5 × 10^{5} for the higher density and ∼4 × 10^{10} cm^{−3} and ∼4 × 10^{4} for the lower density, respectively, and for v=50 they were estimated to be ∼1.2 × 10^{12} cm^{−3} and ∼4 × 10^{5} for the higher density and ∼3 × 10^{10} cm^{−3} and ∼3 × 10^{4} for the lower density, respectively.
Estimation of the temperature
The temperature of the atomic ensemble was not measured in situ in a series of the Ramsey measurements; however, it was measured in a later independent experiment, hereafter referred to as a ‘temperature experiment’, in which we set the trapping conditions (1)(4) in the same way as described in the preceding subsection. In this temperature experiment, the temperature was measured by an expansion of the atomic ensemble. Two temperature experiments were performed independently under the trapping conditions corresponding to the higher density in the Ramsey measurements, giving ∼67 and ∼72 μK, respectively, so that we estimated the temperature of the higher density ensemble to be ∼70 μK. Similarly, a temperature experiment was performed independently under the trapping conditions corresponding to the lower density in the Ramsey measurements, giving ∼39 μK, so that we estimated the temperature of the lower density ensemble to be ∼39 μK.
Rydberg excitation and detection
The output of a Ti:sapphire laser system (Spectra Physics; Spitfire Ace, wavelength ∼779 nm, pulse width ∼1 ps, repetition rate 1 kHz) was used as the infrared pulse and was also used to pump an optical parametric amplifier (Spectra Physics; TOPAS) to generate the blue pulse tuned to ∼481 nm. The repetition rate was reduced using pulse pickers to synchronize it with the atom preparation sequence whose duration was 1.6 s. To reduce the number of Rydberg states to be excited, the spectra of infrared and blue pulses were cut using homemade pulse shapers in a 4fsetup (f=500 mm), respectively. The infrared and blue pulses were combined collinearly with a dichroic mirror and their relative timing was coarsely adjusted to be zero by crosscorrelation measurements based on sum frequency generation and was further optimized to maximize the Rydberg ion signals. They were introduced into a Michelsontype interferometer to produce a pair of identical double pulses, each of which was composed of the infrared and blue pulses. Those two double pulses induced the pump and probe excitations, respectively (see Fig. 1a). The relative phase between these two double pulses were tuned with attosecond precision^{48}. Those two double pulses were combined collinearly with the dipoletrap laser beam with another dichroic mirror. Those two double pulses and the trapping laser beam were focused with a planoconvex lens (f=250 mm) to the atomic ensemble.
The infrared and blue pulses, and the optical pumping beam were circularly polarized in the same direction with respect to the magnetic field, so that the state was mostly populated, and excitations to S Rydberg states were suppressed due to transition selection rules even though the effective twophoton excitation spectrum covered the S states. This excitation scheme also suppressed the Raman transition between the F=2 and F=1 hyperfine states in the ground state that could happen within the single infrared pulse to induce undesirable beating in the Ramsey contrast as a function of τ, of which an example is shown in Fig. 2d, with a period of 146 ps, which is the reciprocal of the hyperfine splitting 6.83 GHz.
The maximum population of the vD Rydberg states was not >5%, to suppress photoionization. The typical pulse energies of the infrared and blue pulses were ∼10 and 400 nJ, respectively, for the 1.2% population and were ∼30 and 600 nJ, respectively, for the 3.3% population. The population was estimated from the loss of the number of the groundstate atoms in the higherdensity ensemble induced by the irradiation of the picosecond infrared and blue pulses. We measured the number of the atoms by absorption imaging with and without the picosecond pulses and compared those numbers to evaluate the loss. See the previous subsection ‘Estimation of the atom density’ for the details of the evaluation of the atom number by absorption imaging. The loss induced by a single pair of the infrared and blue pulses was too small (a few percent) to be evaluated securely, so that we shined 30 or 50 pairs of the infrared and blue pulses, each of which was accompanied by field ionization^{16}, at a 1 kHz repetition rate to induce the loss that was clearly visible and was used to infer the loss induced by the single pair of the infrared and blue pulses. The loss for different amounts of pulse pairs followed the relation , where N_{r} is the number of atoms remaining after q pulses and N_{t} is the total atom number. p_{e} is the Rydberg state population for the atoms in the ensemble.
After the probe excitation, the populated Rydberg states were ionized by means of field ionization. The Rb^{+} ions thus produced were detected with a micro channel plate placed 5.5 cm away from the atomic ensemble. The electric field for the ionization was triggered 50 ns after the probe excitation, reaching the ionization threshold within the next 100 ns. The output of the micro channel plate was amplified with a preamplifier and sent to a gated integrator, whose output was fed into a computer. It is noteworthy that the fieldionization spectrum shown in Fig. 1c was measured using an oscilloscope with the electric field ramped up slowly on the microsecond timescale to check how many Rydberg states were populated by our broadband excitation with the picosecond pulses.
Timedomain Ramsey interferometry
The interaction among Rydberg atoms was observed by timedomain Ramsey interferometry with a pair of twophoton excitations: pump and probe. Their delay τ was coarsely tuned on the picosecond timescale with a motorized mechanical stage placed in one arm of the interferometer mentioned above and was scanned finely on the attosecond timescale with a piezoelectric transducer to measure Ramsey interferograms. A HeNe laser beam was introduced to the interferometer to check the linearity of the scan by monitoring its optical interference. The period of this optical interference was also used for the calibration of the pumpprobe delay τ. We measured the fieldionization signals of the two atomic ensembles with different densities alternately to suppress systematic uncertainties, scanning τ in steps of ∼30 as over a range of ∼3 fs at each coarse delay tuned by the mechanical stage on the picosecond timescale. The obtained Ramsey interferogram was fitted with a sinusoidal function. As the expected energy shift induced by the interaction was at most on the order of 10 GHz, much smaller than the eigenfrequency of the Rydberg state itself (∼1 × 10^{15} Hz), we fitted the interferograms with the same eigenfrequencies for the higher and lower densities to evaluate the phase shift of the higherdensity ensemble from the lowerdensity one. We have defined the contrast of the interferogram to be the ratio of the amplitude of the fitted sinusoidal function to its mean value. In Fig. 2a–c, the signal intensities are normalized by the mean value of the sinusoidal function fitted to each interferogram. We have defined Ramsey contrast to be the ratio of the contrast of the higherdensity ensemble to that of the lowerdensity one, as shown in Figs 3, 4 and 6. We assume that the interactions are negligible in the lowerdensity ensemble, which is thus taken to be a reference for measuring the contrast decay and the phase shift that may be induced by the interactions in the higherdensity ensemble.
Outline of the exactly solvable model simulation
We represent each atom as a twolevel system, which is a pseudospin system, consisting of a ground state g〉 () and an excited Rydberg state e〉 () with energies E_{g} and E_{e}, respectively. This assumption should be reasonable, as the contribution of the neighbouring Rydberg levels is small for the 42D_{5/2} state, as seen from the fieldionization spectrum shown in Fig. 1c. The Natom wave function is initially assumed to be a product of independent singleatom wave functions . The experiment consists of four stages: (i) the pump excitation, (ii) an evolution with an Natom Hamiltonian H, (iii) the probe excitation and (iv) the population measurement by field ionization. The experimental observable is the number of Rydberg excited atoms detected as a timedependent signal , which is the expectation value of the sum of projection operators , where measures the population in the Rydberg state for atom j normalized to the total atom number N.
The Natom Hamiltonian in stage (ii) includes the atomic energies and the interactions between atoms in the Rydberg states. This is justified on the picosecond timescale of the current measurement, as Rydberg interactions provide coupling strengths on the order of GHz for micrometre separations, much larger than those between Rydberg and groundstate atoms and between two groundstate atoms on the order of kilohertz and less than kilohertz, respectively, for micrometre separations. This leads to a diagonal Hamiltonian of the form in equation (2).
Continuum approximation
In this approximation we assume the density distribution to be homogeneous in a small volume around a particular position r (ref. 54). The Ramsey signal P(r, τ) is calculated for such a homogeneous region as
where and r_{0} is referred to as a cutoff radius, so that we consider interactions only among the atoms within a sphere whose radius is r_{0}. A blockade radius r_{B} is determined by the bandwidth of the pump excitation and is ∼0.88 μm in the present experiment. N_{0}(r) is the number of atoms in the local volume and depends on the local density n(r) due to . This signal is averaged over the whole ensemble whose atomdensity distribution is assumed to be Gaussian with the peak atomdensity n_{p} to obtain equation (4), in which the Ramsey contrast and phase shift can be derived from
to be the absolute value and phase of the function g(τ), respectively. In the case that the potential is given by a van der Waals interaction and in the limit , the continuum approximation leads to the expression in equation (5) for the function g(τ). We expect this approximation to describe the system well for a sufficiently low population of Rydberg excited atoms p_{e}<<p_{g} (ref. 54), which is safely satisfied in our experiments.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Takei, N. et al. Direct observation of ultrafast manybody electron dynamics in an ultracold Rydberg gas. Nat. Commun. 7, 13449 doi: 10.1038/ncomms13449 (2016).
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Acknowledgements
We thank T. Pfau, R. Löw, D. Jaksch, M. Kiffner, A. Browaeys, T. Lahaye, V. Vuletić, I. Lesanovsky, R.J. Levis, R.J.D. Miller, I. Ohmine, S. Saito, T. Yanai, S. Takeda, A. Tanaka and D. Vodola for fruitful discussions. We also acknowldge K.R.A. Hazzard and R. Mukherjee for showing us their unpublished numerical simulations of the contrast decays and phaseshifts based on the exact expression equation (3). This work was partly supported by CRESTJST, JSPS GrantinAid for Specially Promoted Research Grant Number 16H06289 and Photon Frontier Network Program by MEXT. C.G. was supported from the Austrian Science Fund (FWF) via project P24968N27. G.P. was supported by ERCSt Grant ColdSIM (Number 307688), EOARD, RySQ, UdS via IdEX and ANR via BLUESHIELD. M.W. was supported by the EU FET Network ‘Rydberg Quantum Simulators RySQ’ and Heidelberg Center for Quantum Dynamics. K.O. thanks Alexander von Humboldt foundation, University of Heidelberg and University of Strasbourg for supporting this international collaboration.
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N.T., C.S., M.W. and K.O. designed the experiment. N.T., C.S., H.G., K.K., H.C. and K.O. constructed the apparatus. N.T. and C.S. made the measurements and analysed the data. C.S., C.G. and G.P. developed the theory and performed theoretical analyses. N.T., C.S., C.G., G.P., M.W. and K.O. discussed the results and wrote the paper. K.O. conceived and presided over the project.
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Takei, N., Sommer, C., Genes, C. et al. Direct observation of ultrafast manybody electron dynamics in an ultracold Rydberg gas. Nat Commun 7, 13449 (2016). https://doi.org/10.1038/ncomms13449
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