Spectroscopic Measurement of the Softness of Ultra-Cold Atomic Collisions

TL;DR

This paper introduces spectroscopic methods to measure the collisional softness of ultra-cold atoms, revealing its value and impact on coherence decay, with implications for understanding atomic interactions and quantum systems.

Contribution

It combines Ramsey and echo spectroscopy to directly measure collisional softness and demonstrates its application in determining scattering lengths and analyzing decay dynamics.

Findings

Collisional softness measured as 2.5 (3).

Decay dynamics transition related to softness.

Methods applicable to other quantum systems.

Abstract

The softness of elastic atomic collisions, defined as the average number of collisions each atom undergoes until its energy decorrelates significantly, can have a considerable effect on the decay dynamics of atomic coherence. In this paper we combine two spectroscopic methods to measure these dynamics and obtain the collisional softness of ultra-cold atoms in an optical trap: Ramsey spectroscopy to measure the energy decorrelation rate and echo spectroscopy to measure the collision rate. We obtain a value of 2.5 (3) for the collisional softness, in good agreement with previously reported numerical molecular dynamics simulations. This fundamental quantity was used to determine the $s$-wave scattering lengths of different atoms but has not been directly measured. We further show that the decay dynamics of the revival amplitudes in the echo experiment has a transition in its functional decay. The transition time is related to the softness of the collisions and provides yet another way to approximate it. These conclusions are supported by Monte Carlo simulations of the full echo dynamics. The methods presented here can allow measurements of a generalized softness parameter for other two-level quantum systems with discrete spectral fluctuations.

Equations

$$s={\Gamma }_{\text{coll}}/\Gamma .$$

$$C_{\text{R}}(t)={\left[1+0.95(t/\tau {)}^2\right]}^{-3/2}$$

$$C_{\text{R}}(t)\sim \exp \left[-\frac{2.86}{{\Gamma }^2{\tau }^2}\left(e^{-\Gamma t}+\Gamma t-1\right)\right],$$

$$C_{\text{E}}(2t_{\pi })\sim \exp (-2{\Gamma }_{\text{coll}}{t}_{\pi }).$$

$$s=\left(\frac{n_{\text{high}}}{n_{\text{low}}}\sqrt{\frac{T_{\text{high}}}{T_{\text{low}}}}\right)\frac{{\Gamma }_{\text{coll}}^{\text{low}}}{{\Gamma }^{\text{high}}}.$$

$$C_{\text{E}}(2t_{\pi })\sim \exp \left[-\frac{\Gamma {\left(2t_{\pi }\right)}^3}{6{\tau }^2}\right].$$

$$C_{\text{E}}(2t_{\pi })\sim \exp \left\{-2{\Gamma }_{\text{coll}}{t}_{\pi }\left[1-\sqrt{\frac{\pi }{2}}\frac{t_{\text{tr}}}{2t_{\pi }}\text{erf}\left(\frac{\sqrt{2}{t}_{\pi }}{t_{\text{tr}}}\right)\right]\right\}.$$

Full text

Elastic collisions are of great importance in atomic physics, both from a theoretical and a practical point of view. They are relevant for atomic clocks, metrology, quantum information, evaporative cooling, atom-ion hybrid systems and more Gibble and Gallagher (1991); Gibble and Chu (1993); Anderson et al. (1995); Swallows et al. (2011); Meir et al. (2016); Hohmann et al. (2016). Collisions may also have a significant effect on the coherence properties of an ensemble of atoms, providing either elongation Dicke (1953); Schmidt et al. (1973); Berman et al. (1975); Rothberg and Bloembergen (1984); Firstenberg et al. (2013, 2007); Shuker et al. (2007); Sagi et al. (2010a, b, 2011) or shortening Carr and Purcell (1954); Deutsch et al. (2010) of the atomic coherence time.

Considering a rapid collisional process compared to other dynamical timescales 111the collision time can be estimated by the range of atomic interaction divided by the relative velocity of the colliding atoms. For dilute ultra-cold atoms and excluding mean-field interactions the collision time is on the nanosecond scale, much shorter than the mean time between collisions and the oscillation time in the trap ($\gtrsim 10$ msec and $\gtrsim 1$ msec, respectively for our experiment)., there exist two extremities for a colliding atom in the center-of-mass frame of the interacting ensemble: “hard collisions”, in which the energy of the atom is completely randomized after a single collision, and “soft collisions” in which the atomic energy remains almost unchanged after each collision Sagi et al. (2010a). We therefore define the “collisional softness” parameter, $s$, as the number of times an atom has to collide in order for the correlation between its initial and final energies to drop to $1/e$ 222This definition of $s$ is related to the strength parameter of velocity changing collisions, $\alpha$, defined in Berman et al. (1975), by $s=\frac{1}{1-\alpha^{2}}$. The collisional softness of hard collisions is one, since the energy correlation drops to zero after a single collision. Collisions are considered “soft” if their softness parameter is much larger than unity. Even though the $s$-wave collisional process considered here is itself is of universal nature, the softness of the collisions can be affected by the confining potential. This can be intuitively understood by considering that only the kinetic energy changes due to a collision whereas the potential energy does not, carrying a “memory” of the energy prior to the collision.

More formally, an ensemble of colliding trapped thermal atoms has two relevant characteristic rates. First, the atomic collisions, treated as a Poisson process energy-randomization events, occur at an average collision rate $\Gamma_{\text{coll}}$. Second, the single atom temporal energy autocorrelation function, averaged over the atomic ensemble, decays exponentially with an energy decorrelation rate $\Gamma$. The collisional softness is then defined as:

[TABLE]

Collisions with $s=1$ are hard, and collisions with $s\gg 1$ are soft.

The softness of $s$-wave elastic collisions of ultra-cold bosonic atoms trapped in a harmonic potential and far from a Feshbach resonance was evaluated using molecular dynamics simulations and found to be 2.5 Monroe et al. (1993); Newbury et al. (1995); Wu and Foot (1996); Arndt et al. (1997); DeMarco et al. (1999); Dalibard (1999); Hopkins et al. (2000); Bohn and Jin (2014). This value of the softness has been used to determine the elastic collision cross sections of different atoms Monroe et al. (1993); Newbury et al. (1995); Arndt et al. (1997); Hopkins et al. (2000), but has not been measured directly. For a given number of collisions the softness is the number of collisions required for thermalization in a perturbed trap, having immediate repercussions on the physics of evaporative cooling Monroe et al. (1993); Wu and Foot (1996).

In this paper we present a direct spectroscopic measurement of the softness of ultra-cold atomic collisions. We do so using a combination of two spectroscopic methods (Ramsey and echo Hahn (1950)), in two opposite regimes of low and high collision rates. First we show that the coherence of an atomic ensemble in an echo experiment at low density asymptotically depends only on $\Gamma_{\text{coll}}$. We then show that in a high density Ramsey experiment the decay is independent of the collision rate and can be fitted to reveal the energy decorrelation rate $\Gamma$. By combining these two measurements, we are able to quantitatively extract the $s$-wave collisional softness of cold 87Rb atoms in an optical dipole trap. We obtain good agreement with previously reported theoretical results from molecular dynamics simulations, validating our method and laying the foundations for its application in measuring the softness of other collisional processes.

We further show that the coherence decay in an echo experiment is qualitatively different for short and long times Schmidt et al. (1973); Berman et al. (1975). This can be used to approximate the softness by combining a Ramsey measurement and an echo measurement in a single, intermediate density regime. These methods may allow measurements of a softness parameter for other two-level quantum systems that have discrete energy fluctuations Dicke (1953); Koch et al. (2007); Wodkiewicz et al. (1984); Ambrose and Moerner (1991); Uren et al. (1985).

Spectroscopic signatures of collisional softness

${}^{87}\text{Rb }$atoms trapped in an optical dipole trap experience a differential AC-Stark shift imposed by the different detuning of the trapping laser from their two ground state hyperfine levels. If the mean time between atomic collisions is larger than the oscillation period in the trap, the fast oscillations can be averaged. The rate of the phase accumulated by the wavefunction, determined by the detuning, then depends on the average energy Kuhr et al. (2005). Effectively this creates a stationary inhomogeneous broadening of the spectrum, decreasing the coherence time of the ensemble.

Due to this effect, the dynamics of the hyperfine coherence in a Ramsey ($\pi/2-\pi/2$) experiment with no collisions is given by

[TABLE]

for an ensemble of two-level atoms in thermal equilibrium in an optical harmonic potential Kuhr et al. (2005). The bare Ramsey time is given by $\tau\approx 2\hbar/\eta k_{B}T$. Here $T$ is the temperature of the cloud and $\eta$ is the ratio between the hyperfine splitting and the detuning of the trapping laser 333For 87Rb and a YAG 1064 nm trapping laser $\eta\approx 7\times 10^{-5}$. In an echo experiment ($\pi/2-\pi-\pi/2$), where the echo-pulse is given at time $t_{\pi}$, the echo coherence $C_{\text{E}}(2t_{\pi})\equiv C(t=2t_{\pi})$ fully revives in the absence of elastic atomic collisions due to the stationarity of the trap perturbation 444For strong trap perturbations and for chaotic traps this may not be the case Andersen et al. (2003, 2006).

Factoring in the effect of elastic atomic collisions, the Ramsey and echo coherences have a complicated behaviour Firstenberg et al. (2013). However, they both have some useful, simple limits. For high density $n_{\text{high}}$, the spectrum is collisionally-narrowed, resulting in an elongated Ramsey coherence, that can be approximated by the generalized Gumbel function Sagi et al. (2010b)

[TABLE]

dependent only on $\tau$ and the energy decorrelation rate $\Gamma$, and not on the collision rate $\Gamma_{\text{coll}}$.

In the opposite regime, of low density $n_{\text{low}}$, the asymptotic long-time echo coherence behaves as

[TABLE]

In this regime the coherence depends solely on the collision rate $\Gamma_{\text{coll}}$ and not on the energy decorrelation rate $\Gamma$. This is due to the fact that every collision, no matter how soft, causes a finite deflection in the atomic trajectory. As $t\to\infty$ this deflection will fully decohere the atom. This implies that the coherence in this long-time regime is nothing but the fraction of atoms that did not collide.

Fig. 1 illustrates the role of collisions in an echo experiment. It presents the normalized upper hyperfine state population as a function of the time between the Ramsey pulses. In (a) the measurement is performed with very low collision rate ($\Gamma_{\text{coll}}\tau\approx 0.15$) 555Low/high collision rates are defined, throughout this paper, with respect to the bare Ramsey time $\tau$. In both regimes the collision rate is much smaller than the trapping frequencies.. The coherence decays fast, and the echo revival amplitude is significant. On the other hand, in (b) the collision rate is much higher ($\Gamma_{\text{coll}}\tau\approx 10$). The Ramsey decay is slower (partially due to collisional narrowing) and the echo revival is negligible, manifesting the failure of the echo due to atomic collisions.

Both the collision rate and the energy decorrelation rate are proportional to the multiplication of the atomic density $n$ with the average velocity $v_{\text{rel}}\sim\sqrt{T}$: $\Gamma_{\text{coll}}=n\sigma v_{\text{rel}}$. Therefore, knowing the ratios $n_{\text{high}}/n_{\text{low}}$ and $T_{\text{high}}/T_{\text{low}}$ at the two different experimental conditions allows for the normalization of the collision rate, measured at low density, and the energy decorrelation rate, measured at high density, and extraction of the collisional softness

[TABLE]

We note that in the intermediate regime $\Gamma\tau\approx 1$ it was theoretically shown that the spectrum of an ensemble of colliding atoms depends weakly on the softness Sagi et al. (2010a). It was further suggested that it may be possible to distinguish between the two extreme cases of hard and soft collisions, by measuring a Dicke narrowed spectrum in a Ramsey experiment. Practically, this task turns out to be challenging. Small uncertainties in the experimental conditions, such as the collision rate and the inhomogeneous broadening of the spectrum, may cause an incorrect model to fit well to the experimental data Rothberg and Bloembergen (1984); Firstenberg et al. (2013).

Measuring the collisional softness

Our apparatus is described in detail in Sagi et al. (2010a). Briefly, the experiment consists of 87Rb atoms trapped in a 1064 nm far-detuned crossed-beam optical dipole trap. The atoms are evaporatively-cooled down to two distinct regimes: high density with $n_{\text{high}}=3.5~{}(2)\times 10^{12}$ cm*-3* and $T_{\text{high}}=0.56~{}(2)~{}\mu$K, and low density with $n_{\text{low}}=3.6~{}(3)\times 10^{11}$ cm*-3* and $T_{\text{low}}=6.8~{}(3)~{}\mu$K 666The temperature is measured using time-of-flight and the peak atomic density using $n=\omega_{x}\omega_{y}\omega_{z}N\left(\frac{m}{2\pi k_{B}T}\right)^{3/2}$, where $\omega_{i}$ are the trap frequencies, $N$ is the total number of atoms, $m$ is the atomic mass, $k_{B}$ is the Boltzmann constant and $T$ the measured temperature. Measurement of the total number of atoms, and hence the peak density, is susceptible to common systematic errors and obtaining an exact value for it is challenging Weller et al. (2012). However, as our method relies only on the knowledge of the ratio between densities [Eq. (5)], systematic errors are common-mode rejected. All errors stated in throughout the paper represent a 1$\sigma$ confidence level.

The coherence is measured between the first-order Zeeman insensitive hyperfine $\left|1\right\rangle\equiv\left|F=1,m_{F}=0\right\rangle$ and $\left|2\right\rangle\equiv\left|F=2,m_{F}=0\right\rangle$ states of the $5^{2}S_{1/2}$ manifold. The atoms are prepared by optical pumping and microwave transitions in state $\left|1\right\rangle$. We then use a microwave $\sim 6.8$ GHz control to perform Ramsey ($\pi/2-\pi/2$) or phase-scanned echo ($\pi/2-\pi-\pi/2$) manipulations on the atoms. At the end of each experiment we use a state-selective fluorescence-detection scheme to evaluate $N_{2}/(N_{1}+N_{2})$, the fraction of atoms at state $\left|2\right\rangle$. The coherence is defined as the normalized amplitude of the fringes of the Ramsey and echo data.

We measure the collisional softness using Eq. (5), by first obtaining $\Gamma_{\text{coll}}^{\text{low}}$ from an echo measurement in the low density regime by fitting the asymptotic decay described by Eq. (4) and then obtaining $\Gamma^{\text{high}}$, given by Eq. (3) from a Ramsey measurement in the opposite regime.

Focusing first on the low density regime, the resulting echo coherence and an additional Ramsey measurement at the same experimental conditions are presented in Fig. 2(a). The echo decay time is indeed much longer than that of the Ramsey experiment (by about a factor of four). The extracted low-density bare Ramsey time is $\tau_{\text{low}}=37~{}(1)$ ms, compared to 32 (1) ms obtained directly from the measured temperature. The echo measurement exhibits a long-time linear decay in a semi-logarithmic scale [Fig. 2(b)], confirming the expected exponential decay of Eq. (4). The slope, excluding short times, gives a collision rate of $\Gamma_{\text{coll}}^{\text{low}}=9.4~{}(3)~{}\text{s}^{-1}$. Next we obtain the energy decorrelation rate, $\Gamma^{\text{high}}$, from the collisional narrowing of a high-density Ramsey measurement. The atomic coherence is shown in Fig. 2(c). From the measured temperature, we expect $\tau_{\text{high}}=390\ (15)$ ms. We use this value as a fixed parameter and fit the coherence data to Eq. (3), extracting $\Gamma^{\text{high}}=10.6~{}(1)~{}\text{s}^{-1}$.

The softness is then calculated using Eq. (5) to be $s=2.5~{}(3)$ 777This value is correct for harmonic trapping potential that describes well our crossed Gaussian beam optical potential. For other trap shapes and energy distributions it may vary, e.g. for a flat box potential $s=1.5$ Wu and Foot (1996), in excellent agreement with molecular dynamics simulations Wu and Foot (1996).

We perform Monte Carlo simulations to study the effect of the softness of the collisions on the full dynamics of the echo decay. The simulation calculates the ensemble coherence of $2\times 10^{4}$ two-level atoms with the energy distribution corresponding to a 3D harmonic potential Kuhr et al. (2005) as a function of time. The collision rate $\Gamma_{\text{coll}}$ is drawn from a Poisson distribution and the collisional softness $s$ is generated by introducing controlled correlations between the energy jumps of successive collisional events using the Cholesky decomposition method of the required correlation matrix Hardoon et al. (2004). Typical energy trajectories for $s=1$ and $s=10$ are illustrated in the insets of Fig. 3. A comparison between the experimental data and simulation results is presented in Fig. 3 for the echo experiment with hard ($s=1$), soft ($s=10$) and $s=2.5$ collisions, using the measured energy decorrelation rate, rescaled using Eq. (5), $\Gamma^{\text{low}}=\left(\frac{n_{\text{high}}}{n_{\text{low}}}\sqrt{\frac{T_{\text{high}}}{T_{\text{low}}}}\right)^{-1}\Gamma^{\text{high}}=3.76~{}\text{s}^{-1}$. The simulation of the $s=1$ and $s=10$ clearly disagrees with the data, whereas the $s=2.5$ collisions agrees best with the experimental data for all times and with no fit parameters. The echo decays of the hard and of the soft collisions, with the same energy decorrelation rate, clearly do not agree with the experimental results.

Measuring the softness using a transition in the functional decay of the echo dynamics

In the low density regime of $\Gamma\tau\ll 1$, valuable information can be extracted by observing the entire dynamics of the decay of the echo coherence. Eq. (4) gives the long-time limit of the coherence $C_{\text{E}}(2t_{\pi})\sim\exp\left(-2\Gamma_{\text{coll}}t_{\pi}\right)$, depending solely on the collision rate $\Gamma_{\text{coll}}$. The short-time limit, however, depends on the energy decorrelation rate $\Gamma$ and the bare Ramsey time $\tau$ and is given by Carr and Purcell (1954)-

[TABLE]

In both limits the echo coherence decays as $C_{\text{E}}(2t_{\pi})\sim\exp\left(-\beta(2t_{\pi})^{\alpha}\right)$, with $(\alpha,\beta)=\left(1,\Gamma_{\text{coll}}\right)$ for long times or $(\alpha,\beta)=\left(3,\Gamma/6\tau^{2}\right)$ for short times. Defining $t_{\text{tr}}\equiv\tau/\sqrt{s}$, a transition time between the two regimes, an interpolating function can be written to describe the entire dynamics, which is the accurate solution in the limit of soft collisions Berman et al. (1975):

[TABLE]

We measure the echo coherence decay with a moderate collision rate ($\Gamma_{\text{coll}}t_{\text{tr}}\approx 1$). The resultant coherence is summarized in Fig. 4. The transition of $\alpha$ is evident from the fit to Eq. (7). An indication for the $\alpha=3$ decay is shown in Fig. 4(b), where we plot the echo revival amplitudes in a logarithmic scale against $(2t_{\pi})^{3}$. Fig. 4(c) shows the same data in a semi-logarithmic scale as a function of $2t_{\pi}$. Here the linear dependence at long times indicates the $\alpha=1$ decay. A similar transition was observed for warm molecular gases in the limit of soft collisions Schmidt et al. (1973).

More quantitatively, extracting $t_{\text{tr}}$ from the echo decay using Eq. (7), and $\tau$ from the Ramsey decay under the same experimental conditions, we can evaluate the softness by the definition of the transition time $s=(t_{\text{tr}}/\tau)^{2}$. Fitting the echo decay of Eq. (7) to the data of Fig. 4, using $\Gamma_{\text{coll}}=5.1~{}(3)~{}\text{s}^{-1}$ obtained from fitting the long-time decay of Fig. 4(c), we get $2t_{\text{tr}}=205~{}(17)~{}$ms 888To evaluate the error $2\Delta t_{\text{tr}}$ we use the value $\Gamma_{\text{coll}}\pm\Delta\Gamma_{\text{coll}}$ as a fitting parameter. The upper $1\sigma$ confidence bound on $t_{\text{tr}}$ is that which is obtained for $\Gamma_{\text{coll}}+\Delta\Gamma_{\text{coll}}$ and the lower bound is that which is obtained for $\Gamma_{\text{coll}}-\Delta\Gamma_{\text{coll}}$. This, in addition with a Ramsey experiment under the same experimental conditions [inset of Fig. 4(a)] that gives $\tau=76~{}(3)$ ms, yields $s=(t_{\text{tr}}/\tau)^{2}=1.8~{}(3)$. The agreement with the theoretical value of 2.5 is not as good as for the measurement described in the previous section.

We investigate the use of the interpolation function of Eq. (7) as a fitting function for extracting the softness using the Monte Carlo simulations described previously. We find that for a Gaussian energy distribution the method is quite precise. Fig 5 presents a summary of the softness obtained for a Gaussian energy distribution (full symbols) by fitting Eq. (7) to the numerically simulated decay of coherence as a function of the input softness for different values of $\Gamma\tau$. For most cases, the output softness is very close to the input softness. The relative error is $<10\%$ as long as $\Gamma_{\text{coll}}t_{\text{tr}}$ is within the range $0.1<\Gamma_{\text{coll}}t_{\text{tr}}<10$. Outside this range, the value of the coherence at the transition time $C_{\text{E}}(2t_{\text{tr}})\approx\exp\left(-2\Gamma_{\text{coll}}t_{\text{tr}}\right)$ is either too high or too low for the fitting procedure to accurately extract the transition time. For the case of an energy distribution of a 3D harmonic trap similar to the one we have in the experiment the situation is different [Fig. 5, empty symbols]. The approximate solutions for the Ramsey [Eq. (3)] and echo [Eq. (7)] fit only qualitatively, yielding significant errors similar to the ones observed in the experiment.

Summary and outlook

We have measured the softness of ultracold elastic atomic collisions using a combination of two spectroscopic methods: a measurement of the energy decorrelation rate obtained from the collisional narrowing of a Ramsey experiment with a high collision rate, and a direct measurement of the collision rate obtained from an echo experiment with a low collision rate. The obtained collisional softness is 2.5 (3) - in excellent agreement with the value obtained by molecular dynamics simulations. We have also demonstrated a transition in the functional decay of the echo coherence, from $\exp\left(-t^{3}\right)$ at short times, to $\exp\left(-t\right)$, at long times Schmidt et al. (1973). This transition occurs at a softness-dependent time $t_{\text{tr}}=\tau/\sqrt{s}$. We show that this transition in the functional decay can be used to approximate the softness at a single, intermediate density regime.

Our results validate the spectroscopic method, allowing for its use in measuring the softness of other, non-trivial, collisional processes such as extensions to higher temperature involving the inclusion of more partial waves into the scattering process, fermionic collisions and thermalization Brantut et al. (2013), inter-species hybrid collisions Ferrier-Barbut et al. (2014); Delehaye et al. (2015) and atom-ion collisions Meir et al. (2016). It can also be useful as a tool in simulating the efficiency of evaporative cooling Wu and Foot (1996), the investigation of high-density atom interferometers Fogarty et al. (2013) and slow and stored light Firstenberg et al. (2013). Our methods may allow measurements of a generalized softness parameter for other two-level quantum systems with discrete spectral jumps Koch et al. (2007); Wodkiewicz et al. (1984); Ambrose and Moerner (1991); Uren et al. (1985).

Our work can be extended to warm vapor systems where the bare Ramsey time is dominated by Doppler broadening and the presence of a buffer gas induces collisional narrowing Dicke (1953). As in our system, the effect of collisional softness on the Ramsey signal is in itself very small and challenging to detect Rothberg and Bloembergen (1984); Firstenberg et al. (2013). Combining Raman Rabi, Ramsey and echo spectroscopy at high and low collisional-rate regimes can provide an accurate measure of the collisional softness for different buffer gases. In this case, it is possible to change $\Gamma_{\text{coll}}\tau$ by simply altering the angle between the Raman beams without actively changing the density Firstenberg et al. (2013).

Acknowledgements.

The authors would like to thank Yoav Sagi, Hagai Edri and Noam Matzliah for discussions. This work was supported in part by the ICore Israeli excellence center Circle of light and the Weizmann Institute Texas A&M collaboration program.