We present the explicit dynamics process of Rydberg cesium atoms initially experiencing a repulsive van der Waals (vdW) interaction by measuring the line width and intensity of the Rydberg ionization spectra. The signals of Rydberg atoms and free ions are recorded simultaneously within an initial 3.5 μs delay time between the excitation laser and the ramp electric field. For high-density gases, a rapid decrease of Rydberg atoms and an increase of free ions are observed, which is not found to be the case for low-density gases. The experimental results indicate that superradiance is the main cause of the redistribution of Rydberg atoms from the repulsive potential to the attractive potential for high density, which provides the initial ionization. The corresponding theoretical calculation is also given.
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As a prospective candidate to be used to accomplish quantum information and quantum gate [1,2], Rydberg atoms have attracted extensive interest in recent years. Rydberg gases excited from ultracold neutral atoms (~100 μK) appear to be frozen on the experimental interest time scale. Because the thermal motion of atoms is much smaller, the interaction between cold Rydberg atoms becomes distinct. For long-range interaction between Rydberg atoms, resonant energy transfer and blockade of multiple-Rydberg-atom excitation have been observed in many experiments [3–7]. The long-range van der Waals (vdW) interaction of Rydberg atoms is in proportion to n 11 (n is the principal quantum number of Rydberg states) and R−6 (R is the separation between Rydberg atoms) . In dense Rydberg gases, dipole–dipole interaction and vdW interaction have been proposed as the main mechanisms that induce collision and ionization [9–13]. For a pair of Rydberg atoms in the attractive potential, face-to-face motion and collision are considered as the primary causes of ionization. For a pair of atoms initially experiencing the repulsive force, collision is mainly attributed to the redistribution of Rydberg pairs induced by blackbody radiation, which leads to a transfer from the repulsive potential to the attractive potential on a time scale of tens of microseconds [12,13]. Furthermore, superradiance has been regarded as a possible reason for the state transfer within a short time scale (~1 μs) for dense Rydberg atoms [14,15]. However, the detailed dynamics of Rydberg pairs on the vdW repulsive potential for a short time scale have not been represented experimentally.
In this paper we investigate the evolution of Rydberg atoms of 133Cs initially prepared in the pair state and exhibiting purely repulsive interaction within a 3.5 μs delay time. Specially, the spectroscopic signatures in the line width and intensity of Rydberg atoms and free ion signals are presented for different delay times. It demonstrates that on a short time scale superradiance is the main cause that results in atom redistribution from the repulsive potential to the attractive potential in dense Rydberg gases. Considering the effect of the state transfer induced by superradiance, we calculate the collision time through binary interaction velocity rate equations, which is in accord with our experimental observations.
2. Experimental approach
The details of the experimental setup have been described in . Briefly, 5.2 × 107 Cs atoms with a typical temperature of 140 μK are prepared in a standard magneto-optical trap (MOT). The spatial distribution of the cold cloud is Gaussian with a diameter of ~700 μm and a corresponding peak density of about 3.5 × 1010 cm−3. The cold cloud is located between two parallel metal grids separated 15 mm apart. Excitation of Rydberg states is accomplished by a two-photon excitation scheme. The first photon of the excitation, 6S1/2 (F = 4)→6P3/2(F’ = 5), is provided by an extended-cavity diode laser (DL100, Toptica) with a wavelength of 852 nm. The second photon of the excitation, 6P3/2 (F’ = 5)→40D, is provided by a commercial laser system (TA-SHG110, Toptica) consisting of an external cavity diode laser (ECDL) subsequently amplified to a maximum power of 1 W and then frequency doubled to 509–517 nm with a line width of < 2 MHz. The first step in the excitation of the 852 nm laser illuminates the whole trapped atom cloud homogeneously, while the second step in the excitation of the 510 nm laser is focused on the atomic cloud with a waist radius of ~145 μm and a maximum power of 47 mW. The average excited volume is ~17% that of ultracold atom clouds. Here we assume the excited Rydberg sample is in a cigar shape. We use an acousto-optic modulator (AOM) to switch on the 510 nm laser and acquire a pulse of 1 μs duration. The wavelength of the 510 nm laser is calibrated by the wavelength meter (WSU-30, HF-Angstrom). While Rydberg atoms are produced, the gases can evolve freely for a variable delay time Δt. Then the Rydberg atoms are field-ionized by applying a ramp electric field with a rising time of 3 μs to the parallel nonmagnetic grids. The resulting ions are driven to a microchannel plate detector (MCP). Considering the Stark shifts, the magnitude of the ionization field is larger than the threshold field, E = 1/ (16n*4), where n* is the effective principal quantum number . The process mentioned above has a 12 Hz repetition rate. A calibrated digital CCD (IMC-82FT, IMT Tech.) is used to measure the number and distribution of the ultracold atoms by monitoring the fluorescence of 6P3/2 atoms in real time. We obtain the number of Rydberg atoms by measuring the decrease in fluorescence of 6P3/2 atoms. The delay time between the 510 nm laser pulses and the ramp electric field are controlled by a digital-delay pulse generator (DG535, SRS). When a ramp electric field is switched on, free ions in the gases reach the MCP earlier than ionized Rydberg atoms, because Rydberg atoms are ionized when the electric field reaches the corresponding ionization threshold. The signals are sampled simultaneously by two boxcar integrators, SRS250 and SRS, respectively. The data are recorded by a high-speed DAQ card (PCI-1714, Advantech) and averaged over five measurements.
3. Experimental observations and discusses
3.1 Calculation of 40D + 40D potential and observation of excitation suppression
We calculate the interactions between Rydberg atom pairs of 40D + 40D. For large inter-nuclear distances R, the potential energy of nD + nD asymptote V (R) is given as an expansion of R:
In order to investigate the dependence of the excitation production of Rydberg atoms on the power of the 510 nm laser, we insert different neutral filters in the optical path to change the power I of the second excited laser. The 510 nm laser is scanned at a rate of 2 MHz/s near to the resonance from 6P3/2(F’ = 5) to 40D3/2, 40D5/2, respectively. We apply the ramp voltage with amplitude of 210 V/cm on the grids to ionize excited Rydberg atoms. The intensities of Rydberg ionization spectra in the resonance of 6P3/2→40D3/2,40D5/2 for different powers of the 510 nm laser are shown in Fig. 2 at the delay time Δt = 100 ns. When the power of the 510 nm laser is more than 29.6 mW (~44.8 W/cm−2), the intensity ni of the 40D5/2 signal does not present a further increase. The curves show the saturation characteristics due to excitation suppression. The data are fitted by a saturation function of the form ni = pI/(1 + I/Isat) , giving the probability constant p5/2 = 0.25 for the 40D5/2 state and p3/2 = 0.08 for the 40D3/2 state. The fitted curves are shown in Fig. 2 with a solid line. The spectra of free ions and Rydberg atoms in 40D5/2 after a delay time of 2 μs are shown in the inset of Fig. 2.
The signal of free ions exhibits a blueshift relative to that of Rydberg atoms. In addition, we take note that the line width of the free ion signal is less than that of Rydberg atoms, and the intensity of the red side of the free ion signal is lower than that of the blue side. It indicates that Rydberg atoms in the blue-detuned side of the atomic resonance are easy to ionize. Similar results are obtained in an Rb MOT by Amthor et al. [11,13], in which the explicit blueshift was observed after a delay time of tens of microseconds. The results were explained as the excitation of atom pairs with a vdW repulsive potential and a sequent redistribution from the repulsive potential to the attractive potential induced by blackbody radiation. However in our experiments, the free ions are observed at ~100 ns delay time, which is not possible to explain by blackbody radiation.
3.2 The observation of evolution of Rydberg atoms in repulsive potential
We investigate the dynamics of Rydberg atoms in the 40D5/2 state by measuring the spectra of free ions and Rydberg atoms at different delay times. In order to avoid the saturation effect of atomic excitation, we apply the 510 nm laser with a power of 29.6 mW (~44.8 W/cm−2) to excite 6P3/2 atoms. The obtained Rydberg gases have an initial density of 3.5 × 1010 cm−3 and an atomic average separation of 1.7 μm. Taking into account the motion of ultracold Rydberg atoms, the atoms would move away from the interaction region after a 3.5 μs delay time. We observe just the evolution of Rydberg atoms within 3.5 μs.
The spectra of Rydberg atoms in the 40D5/2 state are present in Fig. 3(a) for different delay times of 0.5 μs (dotted blue line), 1 μs (dashed red line), and 1.5 μs (solid black line). There exists an obvious suppression of the line width of the ionization spectra when the delay time increases from 0.5 μs to 1.5 μs. To present explicitly the evolution of Rydberg atoms, the dependence of the line width of Rydberg atoms on the delay time is shown in Fig. 3(b), and the intensities of free ions (red circles) and Rydberg atoms (black squares) of nearby 6P3/2→40D5/2 resonance with different delays are shown in Fig. 3(c). In Fig. 3(b) the line width of Rydberg signals presents explicit broadening and is larger than the natural line width of 40D5/2. It indicates that Cs atoms are prepared for the pair states with vdW potential. For the Gaussian distribution of 6P3/2 atoms along the excitation laser axis, the scanned 510 nm laser excites 6P3/2 atoms with a different separation to the different positions in the potential of 40D + 40D pair states. In Fig. 3(b), the line width of the Rydberg signal decreases at delay time Δt<1.5 μs and increases at Δt>1.5 μs. The dissociation of the Rydberg atom pairs results in a decrease of the line width, and the increase of the line width is attributed to the redistribution of the Rydberg atoms after the dissociation of the atom pairs. Moreover, the intensity of the Rydberg signal in the 40D5/2 resonance decreases when the delay time increases to ~1.5 μs with the increase of the free ions, as shown in Fig. 3(c). The rapid decrease of the Rydberg atoms and the increase of the free ions are observed after 2.0 μs, which indicates that an avalanche of ionization occurs and plasmas are produced . Experimental observation shows the dynamic evolution where the 40D5/2 + 40D5/2 atom pairs in the repulsive potential are excited and disassociated from the repulsive force.
When comparing the higher density of Rydberg gases, we also measure the ionization spectra of Rydberg atoms with a 510 nm laser power of 5 mW (~7.6 W/cm−2). The corresponding density of the initial Rydberg gases is 6.2 × 109 cm−3 and the average atomic separation is 3.0 μm. The other experimental conditions are the same as those of the above measurements. In Fig. 4(a), the spectrum line width of Rydberg atoms decreases obviously as the delay time increases, but an increase in line width is not observed. Furthermore, we also take note that the intensity of 40D5/2 Rydberg atoms increases in the initial stage but is almost invariable after 1.2 μs in Fig. 4(b). The intensity of free ion signals remains almost invariable within the 3.5 μs time scale. Experimental observations indicate that the pairs of Rydberg atoms dissociate after a delay time of 1.2 μs, but the collision between Rydberg atoms and ions does not occur significantly. It is evident that the redistribution of Rydberg states does not happen, and plasma is not observed. For a different density of samples experiencing the repulsive force, different dynamic characteristics are presented in our experiments.
3.3 Calculation of collision time
Collision and ionization are apt to occur just between two atoms prepared in states exhibiting attractive interaction . For atoms in the repulsive potential, we have to consider the interaction between Rydberg atoms and the surrounding atoms in our experiments. Taking into account the Gaussian distribution of an ultracold atom cloud, we assume a higher density and a smaller atomic separation in the center of the MOT. When the atoms in the center are excited preferentially to the vdW interaction potential by a blue-detuned laser, the others in the edge of the cloud are not simultaneously excited because of the narrow line width of the 510 nm laser (~2 MHz). Rydberg atoms with variable inter-nuclear separation can be excited to the pair state by a different laser frequency and can experience different repulsive forces. The atoms in the repulsive potential will dissociate and approach their neighbor atoms . For two Rydberg atoms with an initial inter-nuclear separation R0 and initial velocity of zero, the equation of motion for the conservation of energy can be written as 
On the repulsive force, the dependence of the inter-nuclear separation R on time are calculated and shown in Fig. 5, which presents obviously different dissociation velocities for different initial separations of R0 (solid black line, R0 = 1.7 μm; dashed blue line, R0 = 3.0 μm). According to the numerical solution to the above equation, the Rydberg atom pair moves opposite from the initial separation of about 1.7 μm (~3.2 × 104a0, a0 is the Bohr radius) to the nuclear separation of 5 μm after a time of 0.3 μs. The nearest distance between a dissociated atom and its neighbor is smaller than 0.1 μm. Collision ionization is likely to happen. It could explain our observation that the Rydberg atoms in 40D5/2 decrease corresponding to the free ions that increase slightly during the delay time of 0.7 μs in Fig. 3(c). In the real ensemble, the collision probability is not big enough, so the free ions produced by collision ionization are fewer. For a significant increase of the free ions shown in Fig. 3(c) at a delay of around 2.0 μs, other dynamics need to be considered.
The rapid collision would be explained as a transfer of Rydberg pairs from the repulsive potential to the attractive potential induced by blackbody radiation , but the blackbody radiation rate is generally lower and cannot be used to explain our experimental observation. For Rydberg atoms with high density, the superradiance (∝N 2, N is the number of Rydberg atoms)  gives rise to a quick transfer from the nD states to nearby n’P (n”F) states, because the transition dipole momentum between neighboring states is high. For ~105 Rydberg atoms with a principal number n = 40 in our experiments, the superradiance decay is about 107 s−1 . When Rydberg atoms transfer to other states within the 100 ns time scale of the dissociation process, the inter-nuclear separation is about 2.5 μm. The atom pair, which consists of an atom in an initial state nD and another atom in a redistributed state n’ P (n” F), experiences an attractive force (∝R−3) with dipole–dipole interaction and collide to ionize quickly. The collision time can be calculated as . For the atomic mass of cesium Mcs = 133 a.u. and the inter-nuclear distance of atom pairs R0~2.5 μm, 40D–41P pairs collide within 1.4 μs, which is in agreement with experimental results in Figs. 3(b) and 3(c). This process affords the initial ions the chance to form the trap, capturing the subsequent electrons. When enough electrons are trapped in the ion trap, the fast collision between the electrons and the Rydberg atoms results in avalanche ionization and formation of plasmas. Here, many body effects, which result in suppression of excitation and broaden of line width , might give rise to an error in theoretical calculation.
Different dynamics are shown in Fig. 4 for dilute Rydberg gases. The line width of the Rydberg signal decreases in a time scale of 3.5 μs and the intensity of Rydberg atoms does not obviously decrease, because superradiance depends on atomic density. The dependence of nuclear separation of the Rydberg atoms in 40D with an initial separation of 3.0 μm on the delay time is shown in Fig. 5, which presents a lower dissociation velocity of the Rydberg pair. Collision with the surrounding atoms induced by dissociation can take place. Free ions are produced by the collision process. Nevertheless, a significant increase of free ions is not observed in Fig. 4(b). It means that redistribution does not occur in the dissociation process of the Rydberg pairs. The dissociation process and the diffusion of atoms result in an increase of nuclear separation. Rapid collision cannot happen, and avalanche ionization is not presented.
In conclusion, we show the dependence of ionization spectra on the delay time when Rydberg atoms are prepared in the repulsive vdW interaction. The variation of line width and intensity of the Rydberg spectra present the explicit evolutionary process of Rydberg gases. Despite the fact that the Rydberg pairs are in the repulsive potential, free ions increase rapidly after 2.0 μs in dense Rydberg gases. A similarly rapid increase is not observed in dilute Rydberg gases. For a Rydberg sample with high density, the experimental observations are explained as the state transfer from the repulsive potential to the attractive potential during the dissociation process, which is induced by superradiance. It results in the rapid collision of Rydberg atoms and produces the initial free ions. Collision time is numerically calculated using a velocity equation. The model gives a reasonable explanation and is in accord with our experimental observations.
We thank Prof. Liantuan Xiao and Lu Li for helpful discussions and suggestions. This project was supported by the 973 Program grants2006CB921603; the National Natural Science Foundation of China (NNSFC) grants 60778008, 10934004, 60978018, and 60978001; the NNSFC Project for Excellent Research Team grant 60721004 and the Key Project of Education Ministry.
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