Dynamics of Low-Density Ultracold Rydberg Gases

Dynamics of Low-Density Ultracold Rydberg Gases

J. O. Day    E. Brekke    T. G. Walker tgwalker@wisc.edu Department of Physics, University of Wisconsin-Madison, Madison, WI 53706
July 12, 2019
Abstract

Population dynamics in weakly-excited clouds of ultracold Rb Rydberg atoms were studied by means of trap loss, fluorescence detection, and state dependent stimulated emission. Rydberg atoms were excited to various Rydberg states via continuous two-photon excitation from a magneto-optical trap. A stimulated emission probe laser was then used to bring the Rydberg atoms down to the 6P state, allowing state-dependent detection of the Rydberg atoms. Measurements of trap loss and fluorescent emission reveal information about the evolution of the Rydberg populations. In particular, population in the initial Rydberg state quickly transfers to other Rydberg states by a non-collisional mechanism, likely superradiant emission. The trap-loss measurements are consistent with black-body ionization as the dominant loss mechanism.

pacs:
32.80.Ee,42.50.Nn,34.50.Cx

In the past 10 years it has become possible to study the interesting properties of dense clouds of ultracold Rydberg atoms, usually produced by laser excitation from magneto-optical traps (MOTs) Mourachko et al. (1998); Anderson et al. (1998). A variety of interesting collisional phenomena Afrousheh et al. (2004, 2006); Vogt et al. (2006); Bohlouli-Zanjani et al. (2007); Farooqi et al. (2003); Li et al. (2005); Singer et al. (2005); Amthor et al. (2007); Carroll et al. (2006) have been observed, including surprisingly rapid spontaneous conversion of dense ultracold Rydberg clouds into plasmas Robinson et al. (2000); Li et al. (2005); Walz-Flannigan et al. (2004). It is clear that resonant Anderson et al. (1998); Afrousheh et al. (2004, 2006); Vogt et al. (2006); Bohlouli-Zanjani et al. (2007) and near resonant energy transfer collisions Li et al. (2005); Singer et al. (2005); Amthor et al. (2007) play important roles in the population dynamics of clouds of ultracold Rydberg atoms.

In addition, intriguing proposals to use the strong, long-range interactions between ultracold Rydberg atoms to perform conditional quantum manipulations Jaksch et al. (2000) are driving a number of groups to study coherent manipulations with small numbers of atoms and small samples. In addition, the concept of Rydberg blockade Lukin et al. (2001) brings up the possibility of using spatially confined samples of atoms Sebby-Strabley et al. (2005); Yavuz et al. (2006) for mesoscopic quantum manipulations. Using such collective Rydberg excitations might allow novel quantum applications such as fast quantum gates Lukin et al. (2001); Brion et al. (2007), single atom and directed single photon sources Saffman and Walker (2002), and fast quantum state detection Saffman and Walker (2005).

The ultracold Rydberg cloud experiments, with many Rydberg atoms excited at once, and the quantum manipulation experiments, with only one Rydberg atom excited at a time, operate in very different regimes. Still, it is interesting to note that in both cases their success is reliant on the strong interatomic forces between Rydberg atoms. For quantum manipulations, one wants the Rydberg-Rydberg interactions to be so strong that multi-atom excitations are suppressed Walker and Saffman (2008). A number of experiments Tong et al. (2004); Singer et al. (2004); Liebisch et al. (2005); Heidemann et al. (2007) report suppression signatures suggesting that this regime should be achievable. Most recently, collisional dephasing of coherent Rydberg Rabi flopping was observed for only two atoms confined in a dipole trap Johnson et al. (2008), and dephasing of coherent Rydberg excitation in a BEC was also observedHeidemann et al. (2008).

The experiment reported in this paper studies a transition regime between the two types of experiments. We weakly excite ultracold Rydberg atoms in an un-blockaded regime and study their evolution using three tools: trap loss, spontaneous emission, and stimulated emission. Using these diagnostics, we find that under our weak excitation conditions the probability is very small that the Rydberg atoms experience an inelastic collision sufficiently strong to leave the trap. They predominantly return to the atomic ground state by spontaneous emission. Using the stimulated emission diagnostic, we also observe rapid population transfer out of the initial Rydberg state–the initial Rydberg state is depopulated on a time scale substantially shorter than expected for transfer from black-body radiation or single-atom spontaneous emission. The transfer rate slowly decreases with increasing principal quantum number, strongly suggesting that the transfer process is not collisional in nature. We build upon the recent observation of superradiance by Wang et al. Wang et al. (2007) and argue that superradiance is the likely mechanism for this transfer process. Finally, we show that trap loss rates are consistent with expectations from black-body ionization. We conclude with implications of these results for other experiments.

I Trap Loss Studies

In this section, we describe observations of loss of atoms from the MOT due to Rydberg excitation. Other groups Dutta et al. (2001); Singer et al. (2004) have reported observations of Rydberg trap loss under various conditions, without attempting to explain the mechanisms for trap loss, especially under the low excitation conditions reported here. We find that trap loss rates are much less than excitation rates, showing that, once excited, the Rydberg atoms primarily decay back to the ground state by spontaneous emission processes.

We excite magneto-optically trapped Rb Rydberg atoms using continuous-wave 5S 5Pnl two-photon excitation. The 780 nm (20 mW) and 480 nm (10 mW) lasers used are tuned MHz above the intermediate 5P F=3 state, thus avoiding direct excitation of the 5P state (Fig. 1). Both excitation beams have spatial dimensions on the order of the 1 mm size of the MOT. Typical two-photon excitation rates vary from 10 to 100/s depending on n level. These rates are deduced from calculated atomic matrix elements, measured laser intensities, and the observed transition line-widths, and are averaged over the atomic and laser spatial distributions. The effective excitation rate is , where is the two-photon Rabi frequency, and are the single photon Rabi frequencies for the 780 nm and 480 nm lasers, and is the observed transition line-width which is typically 8 to 10 MHz for these scans as described below. This results in 10 Rydberg atoms at a relatively low density of 10/cm. This density is an order of magnitude or more smaller than the densities at which ultracold plasmas are formed Walz-Flannigan et al. (2004); Li et al. (2005). By changing the frequency of the 480 nm laser, we excite to Rydberg states at 28D, 43D, 58D, and 30S; in most cases the D-state excitations are to the J=5/2 state. These particular states are chosen to provide a range of different spontaneous decay rates (which scale roughly as ) and a wider range of Rydberg-Rydberg van der Waals interactions , with GHz mWalker and Saffman (2008).

Figure 1: Diagram of the Rydberg excitation scheme.

A very simple and effective way to observe excitation of Rydberg atoms in a MOT is via trap loss, with a resulting decrease in the observed fluorescence from the MOT. Processes such as black-body ionization, photo-ionization by the applied lasers, and inelastic collisions between Rydberg atoms and ground state atoms either ionize the atoms or give the atoms sufficient ( K) kinetic energy to leave the trap even if they radiatively return to the ground state. Radiative processes such as photon emission or absorption, elastic or inelastic collisions with cold electrons, and near-resonant Rydberg-Rydberg collisions do not transfer enough kinetic energy to the atoms to cause trap loss. Trap loss is very sensitive; it is easy to observe a 0.1/s change in the trap loss rate from the MOT. With modest Rydberg excitation rates of 10/s, a 1% inelastic or photo-ionization channel can easily be detected.

Since the excitation lasers are sufficiently off-resonance to cause negligible perturbations to the MOT trapping and cooling processes, the loss due to production of Rydberg atoms affects the MOT only through a change in the ejection or loss rate from the trap. If the MOT lasers produce a loading rate and a loss rate , the steady-state number of trapped atoms is . Excitation of Rydberg atoms using lasers tuned to frequency adds a new loss rate which then changes the number of atoms to . The loss rate can then be found from

(1)

This is operationally simpler than taking separate MOT loading transients at each frequency and determining the loss rate from the time constant. We have checked that the two procedures give the same results.

Figure 2

Figure 2: Excitation beams are scanned over the Rydberg resonance at the 28D state. The peak excitation rate is 110/s. The figure shows the loss rate from the trap induced by the excitation beams as well as the photon count rate from the spontaneous emission cascade.

shows the results of sweeping the two-photon excitation frequency through a typical Rydberg resonance. Loss rates induced by the excitation lasers are around 0.2/s, a rate much smaller than the 110/s Rydberg excitation rate. We may infer from this that the probability of inelastic collisions occurring with sufficient energy transfer for the atoms to leave the trap is less than 1/500. For the other states studied, we find 1/50 for 43D and 58D, and 1/1000 for the 30S state.

Noting that the trap loss probabilities are significantly higher for 43D and 58D as compared to 28D, a logical hypothesis would be that inelastic collisions are responsible for the trap loss. To this end, we compared the trap loss rates from 41D and 43D, and found little difference. Since the 43D van der Waals interactions are anomalously large due to a near resonance in the 41P+45F and 43D+43D potentials Reinhard et al. (2007); Bohlouli-Zanjani et al. (2007); Walker and Saffman (2008), then very different results should be obtained when comparing 41D and 43D if collisions were responsible. Thus we again find no evidence that resonant energy transfer collisions cause significant trap loss.

The following estimate supports these conclusions. The inelastic collision rate should be a capture rate multiplied by a probability of energy transfer. We estimate the capture rate as , where , , and is the Rydberg atom flux. We then deduce a capture rate of

(2)

for an density of /cm. This rate, already small, is further reduced by the energy transfer probability. For states like 43D, where the van der Waals interactions are repulsive at long range, there are no thermally accessible curve crossings and we expect the rate to be suppressed by at least a Boltzmann factor , which even for 43D with small is a factor of .

The line-widths observed are significantly broader than the laser line-widths, and exceed what we expect from drifts of our reference cavity used to stabilize the 480 nm laser frequency. This is consistent with observations by others Teo et al. (2003); Singer et al. (2004).

Since the atoms are only rarely leaving the trap due to Rydberg-Rydberg collision or ionization, they must primarily return to the ground state by emission of one or more photons. In the next section we introduce a direct probe of the Rydberg state population that will give more information about the details of this process.

Ii Cascade Fluorescence

As a complementary diagnostic to trap loss observations, we use a photon counting module and a narrowband interference filter to detect the 420 nm decay photons from the 6P state to the 5S ground state. These photons are only observed coming from the MOT cloud under conditions of Rydberg excitation. The fluorescent branching ratio to the 6P from a high S or D Rydberg state is calculated to be about , varying only slightly with principal quantum number. (The predominant channel is emission to the 5P state due to its much larger factor in the emission rate, which easily compensates for a slightly smaller dipole matrix element. This channel is difficult to observe in the presence of strong MOT fluorescence at the same wavelength.) The probability of multiple photon cascade into the 6P state is small because the long-wavelength photons required are disfavored. P and F Rydberg states predominantly cascade into 5S, 6S, and 4D levels, all of which lie energetically below the 6P state. Even for states slightly above the 6P, such as the 5D, the predominant decay channel is to states below the 6P. Thus the cascade fluorescence is likely a reliable probe of the S or D Rydberg state populations, and is relatively insensitive to P or F Rydberg states. Figure 2 shows the observed cascade counts as the excitation lasers are scanned across a Rydberg resonance.

The observed cascade signal has a background, mainly from dark counts, of around 250/s and reaches a peak signal of about 10000/s for the 28 D excitation. We can compare these observed rates with our expected rates from the calculated excitation rates and fluorescent branching ratios. Accounting for the finite collection solid angle , % detection efficiency of the photomultiplier tube, and a calculated 6P–5S emission branching ratio we expect to observe a cascade count rate

(3)

at the peak of the 28D excitation resonance. The ratio of spontaneous to total decay rates accounts for the effects of black-body radiation causing radiative transitions to nearby Rydberg states. These states are assumed, by the argument above, not to result in detected cascade photons. For the data seen in Fig. 2 we observe 10000/s, only 55% of the expected count rate. This suggests that roughly 1/2 of the Rydberg atoms are being transferred out of the excitation state by some other process. The ratio of expected to detected cascade counts for the other excitation states (30S, 43D, and 58D) are (0.5, 0.6, 0.6). Again, by the arguments from the previous section, the explanation for this cannot be inelastic Rydberg-Rydberg collisions, which would either produce an extremely large trap loss rate in contradiction to observations, or would necessarily vary greatly with principal quantum number.

Iii Stimulated Emission Probe

For dipole blockade applications, one is particularly interested in the evolution of the blockaded Rydberg state. Furthermore, in the particular applications of dipole blockade to single atom and single photon sourcesSaffman and Walker (2002), stimulated emission is used to couple the blockaded Rydberg level to an intermediate atomic level. Thus it is natural for us to pursue the development of a stimulated emission probe of Rydberg dynamics. The intensity dependence of the signals produced by the stimulated emission probe allows further information to be obtained about the population dynamics. Such a probe is non-destructive, and has inherently high spectral resolution. While being less general and less sensitive than field ionization, it does have the potential to be applied to a number of different states, subject to dipole selection rules.

As shown in Fig. 1, we apply a tunable diode laser in the range of 1013 to 1027 nm to perform stimulated emission probing of the Rydberg states produced by the two-photon excitation. This “state probe” de-excites atoms from the Rydberg states to the 6P state, which subsequently decays via a 420 nm photon to the 5S ground state. The 6P state is either detected in the same manner as the cascade light mentioned above, or by a reduction in trap loss (explained below). As with the excitation lasers, the spatial size of the state-probe laser is comparable to the MOT cloud size. We deliver nearly 100 mW of light to the atoms, corresponding to stimulated emission rates from 510 to 510/s depending upon the Rydberg level involved. The natural decay rate of the 6P level of 9.2/s is fast enough that population cannot accumulate there. The state-probe laser is stabilized to a 300 MHz optical spectrum analyzer that is itself locked to a 780 nm Rb saturated absorption resonance. It is tuned by changing the frequency of a double-passed acousto-optic modulator.

Figure 3: Stimulated emission probe scan across the 6P manifold. The cascade count background is around 1000 counts.

When the stimulated emission probe is on resonance, atoms are returned to the ground state more quickly than they would otherwise spontaneously radiate from the long-lived Rydberg levels. This reduces the loss rate from the trap as the Rydberg atoms do not stay excited long enough for loss mechanisms such as black-body ionization and inelastic collisions to remove many of them from the trap. Keeping the excitation beams on resonance with the two-photon excitation, we can scan the frequency of the stimulated emission probe to observe the 6P hyperfine manifold. Such a scan is shown in Fig. 3. When tuned on resonance with the 6P F=3 hyperfine state, the MOT loss rate is reduced from 0.8/s to 0.2/s, with the amount of reduction depending on probe intensity.

The state-probe laser produces a reduction in the trap loss rates. An example of this for the 28D Rydberg state is shown in Fig. 4. There the trap loss is shown as a function of the stimulated emission rate from the probe beam. The figure clearly demonstrates that by using a sufficiently high state-probe intensity the atoms can be returned to the ground state before ionization or inelastic collisions can occur, thus reducing the loss rate of the MOT. The surprising feature of the data is that much higher intensities are needed than would be expected if the dominant population transfer from the Rydberg level were spontaneous decay or black-body transfer (rates of 410/s and 210/s, respectively)Gallagher (1994). The residence time of the Rydberg atoms is deduced to be nearly 10 s.

Alternatively, we can measure the state probe laser effects by detecting the number of 6P decay photons from stimulated emission from the Rydberg state. The count rate data, scaled by the number of ground state MOT atoms, is shown in Fig. 6 for the 28D state. The cascade and background signal have been subtracted for this data, thus there are no counts when the stimulated emission rate is zero. As with the repletion data of Fig. 4, the counts saturate at a stimulated emission rate of around 10/s, which implies that the other rates out of the excitation state must be on this order.

Figure 4: Loss rate dependence on stimulated emission probe intensity for the 28D state, showing short residence times for the Rydberg state produced by two-photon excitation.
Figure 5: Loss rate dependence on stimulated emission probe intensity for the 28D state, with the MOT magnetic field switched off during excitation.
Figure 6: Dependence of 6P decay counts on stimulated emission probe intensity for the 28D state.

From the trap loss or state probe data, the residence time of the Rydberg state varies from about 10 s at to about 50 s at . Thus the population is being transferred out of the initial Rydberg state faster than can be accounted for by spontaneous decay and black-body transfer (decay times of 25 s and 50 s at , respectively). The transfer times increase with , just the opposite dependence as would be expected for energy transfer collisions between Rydberg states. These would be expected to decrease with due to rapid increase in van der Waals interaction strengths. This counter trend thus implies that Rydberg-Rydberg collisions are not the process responsible for the fast transfer out of the initial Rydberg state.

Interestingly, the 30S state has the highest transfer rate to other Rydberg states. This again implies that atom-atom interactions do not dominate the process, as the strengths of van der Waals interactions tend to be larger for D states than for S states.

We have checked that the transfer rates increase with increasing excitation rate. This confirms that some transfer process besides black-body radiation is occurring.

As can be seen from Fig. 4, there is a non-zero loss rate at high probe intensities where the stimulated emission laser should fully deplete the original excitation state. There are several possible explanations for this. If there were a loss mechanism that was being enhanced by the state-probe laser, this would produce such a behavior. However, processes such as photoionization or light-induced 6P-5S collisions can be estimated to be far too weak to account for this effect. Another possibility has to do with Zeeman precession in the nd states causing population to accumulate in inaccessible magnetic sublevels. The magnetic field gradient used to confine the MOT atoms is large enough that precession between magnetic sublevels of the Rydberg atoms at the edges of the MOT occurs at a rate of several MHz, which is on the order of the stimulated emission rate. The result of this precession is that a fraction of the atoms - up to 1/3 - move to a state that is inaccessible to the state-probe laser because of the dipole selection rules determined by the polarization of the state-probe laser. For linearly polarized state-probe light, Zeeman levels cannot be excited to the 6P state. Thus population that accumulates in these levels cannot be de-excited by the state-probe laser.

This effect was verified by repeating the experiment with the MOT magnetic field being switched off for 10 ms intervals and only switching the Rydberg excitation lasers during the times the field was off. This data, shown in Fig. 5 for shows that Rydberg populations at high state probe intensities are markedly reduced as compared to when the magnetic field is on. The reduced signal-to-noise for this experiment made it possible to do this only for .

Iv Model of Rydberg Population Dynamics

In the previous sections, we have described the basic processes that are evidently at work under the conditions of our experiment, and their experimental signatures. To further analyze the results, we present here a simplified model of the Rydberg dynamics and use it to extract the values of a few simple parameters from the data. Confirming the interpretation in the previous section, we find that some process that does not cause trap loss nevertheless transfers population out of the excitation Rydberg state on a time scale short compared to spontaneous or black-body lifetimes. The lengthening of this time scale with principal quantum number leads us to believe that it is not inelastic collisions between Rydberg atoms.

The processes included in the model are illustrated in Fig. 7. We describe the system with a three state model: A ground state with atoms, the excitation Rydberg state with atoms, and an additional effective Rydberg state that accounts for other states that are populated from state .

The dynamics of the excitation state depend on laser excitation and de-excitation, spontaneous and black-body radiation, and transfer to the other Rydberg states . Population enters by excitation from the ground state at a rate where is calculated as described in Section I. Spontaneous decay to low-lying levels occurs at a rate . Black-body radiation and other potential processes that transfer atoms to other Rydberg states occur at a rate . There is also the possibility of trap loss (through ionization, for example) at a rate directly from state . Finally, de-excitation from the state-probe laser occurs at a rate . Thus

(4)

is the rate equation for the excitation state population.

The other Rydberg states are produced by collisional or radiative transfer from state at the rate and have an effective radiative lifetime . We also assume that these states can cause trap loss at a rate due to black-body ionization and other collisional processes. Thus they obey

(5)

We are assuming that transfer from back to is unlikely.

In addition to the radiative de-excitation and excitation processes with the Rydberg levels, the ground state population is affected by MOT loading () and loss () processes that we assume are not materially changed when the Rydberg excitation lasers are on. The resulting rate equation for the ground state population is

(6)

Plugging in the steady-state solutions to Eq. 4 and 5 gives

(7)

The loss-rate from the trap is increased by an amount

(8)

This is a product of two factors. The first essentially measures the excitation rate of Rydberg atoms, modified by the de-excitation due to the state-probe laser. The second factor is the probability that the excited Rydberg atoms experience trap loss as opposed to radiatively decaying back to the ground state.

The model similarly predicts the count rate of 420 nm photons produced by the state-probe laser:

(9)

Thus the state-probe-induced 420 nm count rate can also be used to determine the transfer rate , with the data and fit for 28D shown in Fig. 6.

Figure 7: Diagram of the simplified model of Rydberg population dynamics.

The results of fitting our experimental data to this model are summarized in Table 1. We list there the deduced values of the primary unknowns and , as well as the assumed values for the input parameters and . We note that both trap loss and 420 nm count rates can be used to extract , an important internal consistency check. The primary results are these:

  • The atoms transfer out of the excitation Rydberg state at a rate that is substantially faster than spontaneous decay or black-body transfer rates.

  • The mechanism for population transfer, to an excellent approximation, does not cause trap loss.

  • The population transfer rate decreases slowly with increasing principle quantum number, as opposed to the expected rapid increase if near-resonant energy transfer collisions were the relevant mechanism.

  • The probability of trap loss is very small; most Rydberg excitations result in radiative repopulation of the ground state without trap loss.

These conclusions from fitting the experimental data to the model are consistent with the simplified analyses presented in Sections IIII. In the next two sections we will discuss their implications.

State (counts) (loss)
28D 1.210 1.310 2.610 265
43D 7.410 7.210 1.110 1.110 602
58D 2.610 2.010 6.110 4.810 433
30S 3.910 5.010 2.310 4.410 83
Table 1: A summary of transfer rate data using both count rate and loss rate methods. As a basis for comparison, the black-body transfer rate A, spontaneous emission rate A, the estimated mean spontaneous emission rate from other Rydberg states, and the inferred trap loss rate are also included. All rates are in units of s.

V Superradiant Population Transfer

We have argued above that the transfer mechanism responsible for population transfer out of the excitation state in a few microseconds cannot be due to near-resonant energy transfer collisions between excited Rydberg atoms. L-changing collisions between Rydberg atoms and free electrons are another possible mechanism. Such collisions are unlikely to be energetic enough for the recoil to eject atoms from the trap and so would agree with much of the observed behavior. However, at our low excitation rates it is unlikely that electrons are present in sufficient quantities to cause the observed state transfer. With black-body ionization rates of 400/s (see Section VI) acting on a population of 10 Rydberg atoms, free electrons are produced at a rate of 410/s. Free electrons are typically fast-moving and dissipate at a rate of 10/s Killian et al. (1999), giving an average free electron population of 400, which would not have a large impact on the 10 atoms in the excitation state. We estimate that there is insufficient ionization to cause an electron trap Killian et al. (1999). In addition, the rate of l-changing collisions should increase with principle quantum number, in contrast with our observations.

Having argued against collisional phenomena being responsible for Rydberg energy transfer, we need a radiative mechanism to explain our results. Recently, Wang et al. Wang et al. (2007) observed superradiance in the measured lifetimes of Rydberg states. That superradiance could play an important role in Rydberg population dynamics can be understood by the following arguments. For principal quantum numbers , the size of the MOT is less than the wavelength for radiative emission from the excitation state to nearby dipole-allowed states. If there are atoms initially in a particular Rydberg state, the collective dipole moment is enhanced by a factor of (assuming the atom cloud is much smaller than the wavelength of the emitted light). The emission rate is enhanced by a factor of , or by a factor of on a per-atom basis. Since is on the order of 10 for our experiment, the superradiant decay rate will be on the order of

(10)

which is /s for 10 Rydberg atoms. The low emission frequency is compensated for by the large number of cooperatively radiating atoms. Since the spontaneous decay rate is proportional to , the relative importance of superradiance and spontaneous decay goes only as , a factor of only 4 for this experiment.

Wang et al. Wang et al. (2007) developed a sophisticated theory of superradiance with application to cold Rydberg gases. Here we present a simplified model of superradiance to use as a interpretive guide. We base this model on Dicke’s original work Dicke (1954) as elucidated by Gross and Haroche Gross and Haroche (1982) and Rehler and Eberly Rehler and Eberly (1971).

In considering the superradiant decay of an initial state to a lower energy state , the Dicke approach introduces an effective collective spin state of the 2N-level system as , with atoms in state and atoms in state . The radiation rate is found to be . To extend the two-level case to our multi-level case, we assume that we can model superradiance with a set of rate equations

(11)

This model reproduces the key features of superradiance: a large initial inversion radiates at (per atom) at first, then as the inversion is reduced the rate accelerates to a maximum of , occurring over a time Rehler and Eberly (1971). As an additional check on the model, we have simulated the experiment of Ref. F Gounand and Berlande (1979) (which was performed at high temperatures and much smaller ) and our model reproduces the dominant features of the data shown there.

The emission rates are the rates for spontaneous emission multiplied by a cooperativity parameter :

(12)

where the are the calculated absorption oscillator strengths. For a system of atoms whose spatial extent is on the order of the wavelength of the transition, superradiance will occur at a reduced rate. The wavelength of the transition to the nearest lower lying Rydberg state varies from 0.17 cm for 30S-29P to 2.8 cm for the 58D-59P transition. For low , this is quite close to the 1 mm spatial extent of the MOT, and lower levels will have even shorter transition wavelengths. Following Ref. Rehler and Eberly (1971), the cooperativity parameter for a uniform density system of N atoms in a volume V radiating in direction is (in the limit)

(13)
(14)

for a spherical uniform density cloud, where is the product of the wavenumber and the radius of the atomic sample. The cooperativity parameter is 1 for and decreases to for . In practice, the cooperativity parameter becomes small around for a 1 mm MOT.

To account for superradiance, we replace the level of the model of Section IV with a set of levels near in energy to that are coupled to each other and to by black-body radiation and by superradiance from Eq. 11. We find steady-state solutions of the resulting non-linear equations and from them deduce the effective Rydberg-Rydberg transfer rate from Eq. 11 with .

Table II compares the observed Rydberg transfer rates and those predicted by our simulation. These rates do not drop off as quickly as would be expected from Equation 10. This is a result of the spatial factor approaching unity for the higher n-levels, balancing out the decrease in natural emission rate. Additionally as a result of this increasing spatial factor, our simulation indicates that the atoms are transferred predominantly to f-states for the 43d and 58d levels. This transfer to f-states could explain the higher total loss rate from the trap for these levels. We emphasize that there are no adjustable parameters in our simplified superradiance model, and yet it naturally predicts the order of magnitude of the Rydberg-Rydberg transfer rates. A more sophisticated model would be expected to explain the variation seen.

State (calculated) (expt)
28D 1.710 1.310
43D 2.410 7.410
58D 1.210 2.010
30S 2.210 5.010
Table 2: A comparison of Rydberg-Rydberg transfer rates deduced by comparing measurements deduced from the model of Section IV with those predicted by the superradiance model. All units are s.

Vi Black-Body Ionization

So far we have not discussed the actual mechanism for trap loss. The deduced trap loss rates are quite modest, typically 400/s on a per Rydberg atom basis. This is close to what would be expected from black-body ionization. Black-body ionization rates were recently calculated in Ref. Beterov et al. (2007) for the various Rydberg levels. Assuming Rydberg -level distributions from our superradiance model, we find expected photoionization rates from black-body radiation to be as shown in Table 3. For the D states, there is reasonable agreement between our deduced experimental rates and the predicted values, suggesting that black-body ionization probably composes a large portion of the total loss from the trap. For the s-state, the predicted trap loss is greater than observed by about a factor of 3, for which we have no explanation.

State
28D 212 322
43D 470 720
58D 329 457
30S 77 265
Table 3: Comparison of inferred Rydberg trap loss rates with black-body ionization rates. is the rate of black-body ionization for each of these states calculated from Ref. Beterov et al. (2007). All rates are in units of s.

Vii Discussion

In this paper, we have presented results on the dynamics of low-density ultracold Rydberg clouds excited using two-photon absorption from a MOT. By looking at trap loss, radiative cascade, and state-selective stimulated emission we find that the probability of the Rydberg atoms undergoing collisional loss before decaying back to the ground state is small. Additionally, the Rydberg state produced by the two-photon excitation is depleted by some mechanism on a time scale significantly shorter than can be explained by black-body transfer or single-atom radiative decay. This time scale increases with increasing principal quantum number, in contradiction to expectations if inelastic Rydberg-Rydberg collisions were responsible for the rapid state transfer. On the other hand, estimates of collective superradiant light emission and a simplified model thereof suggest that this mechanism can explain the rapid population transfer observed in this experiment. The overall trap loss rates are consistent with expectations from black-body ionization.

Before the experiment of Wang et al.Wang et al. (2007) the effects of superradiance were not appreciated for the population dynamics of high density ultracold Rydberg atoms. Superradiance can happen on very rapid time scales, especially under strong Rydberg excitation as achieved in a number of previous experiments. In fact, superradiance can easily occur on sub-microsecond time scales. A key consequence of superradiance is that it populates Rydberg states lying energetically below the state being excited by the laser. Indeed, in the experiments of Ref. Walz-Flannigan et al. (2004) and Ref. Li et al. (2004), population clearly moves to lower lying Rydberg levels on a fast time scale, consistent with the hypothesis of superradiant transfer.

An additional consequence of fast superradiant population transfer is that it provides a mechanism for rapid population of states of neighboring orbital angular momentum from the excitation state. Pairs of atoms with interact at long range via the resonant dipole-dipole interaction, not the usual van der Waals interaction. The much stronger collision interactions between these atoms may explain the very rapid time scales for plasma formation in a number of experiments where resonant dipole-dipole interactions were not purposely produced using external fields.

Acknowledgements.
J. Strabley contributed to the early stages of this work. We appreciate helpful discussions with M. Saffman and D. Yavuz. This work was supported by the National Science Foundation and NASA.

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