Spectroscopic measurement of the softness of ultracold atomic collisions

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 ultracold 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 is 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 measurement of a generalized softness parameter for other two-level quantum systems with discrete spectral fluctuations.

Figure 1

Comparison between echo revival in the regimes of (a) low collision rate (${\mathrm{\Gamma }}_{\text{coll}}\tau \approx 0.15$) and (b) high collision rate (${\mathrm{\Gamma }}_{\text{coll}}\tau \approx 10$). The Ramsey signal without the application of the echo (thin blue line) is compared to the echo signal after the application of the pulse (thick red line). Black dashed lines represent the times of the application of the $\pi$ pulses. The envelope of the obtained Ramsey fringes is an indication of the atomic coherence. (a) At low collision rates the amplitude of the echo revival is high. The value of the coherence at the peak of the revival $C(t=2t_{\pi })$ is defined as the echo coherence $C_{\text{E}}\left(2t_{\pi }\right)$. (b) At high collision rates the echo pulse essentially has no effect on the coherence. The vertical axis represents the fraction of atoms in the upper hyperfine state after the last Raman pulse.

Figure 2

Ramsey and echo experiments. (a) Atomic coherence as a function of time in Ramsey (blue circles) and echo (red squares) experiments at low density. The decay of the Ramsey signal yields a coherence time of ${\tau }_{\text{low}}=37\left(1\right)$ ms. (b) Linear fit to the tail of the echo decay, in logarithmic scale, gives ${\mathrm{\Gamma }}_{\text{coll}}^{\text{low}}=9.4\left(3\right){\text{s}}^{-1}$. Crosses are short-time data points excluded from the fit. Solid lines represent the fitted functions (see the text). Time in the echo experiment corresponds to $2t_{\pi }$. (c) Ramsey measurement of the energy decorrelation rate ${\mathrm{\Gamma }}^{\text{high}}$ at high density. The measured coherence is fit to a Gumbel function [Eq. (3)] with the energy decorrelation rate as a fitting parameter. This yields ${\mathrm{\Gamma }}^{\text{high}}=10.6\left(1\right){\text{s}}^{-1}$.

Figure 3

Echo experimental results (circles) compared to Monte Carlo simulations for atoms with an energy distribution corresponding to a 3D harmonic potential with hard ($s=1$) (dashed purple line), soft ($s=10$) (dash-dotted green line), and moderate ($s=2.5$) (solid red line) collisions. The simulation uses the calculated bare Ramsey time of ${\tau }_{\text{low}}=32$ ms and a constant energy decorrelation rate of the measured ${\mathrm{\Gamma }}^{\text{high}}=10.6{\text{s}}^{-1}$, rescaled using Eq. (5). Insets illustrate the energy of a sample atom as a function of time for hard ($s=1$) (purple, top right) and soft ($s=10$) (green, bottom left) collisions, with the same energy decorrelation rate $\mathrm{\Gamma }$.

Figure 4

Echo measurement, with ${\mathrm{\Gamma }}_{\text{coll}}{t}_{\text{tr}}\approx 1$, showing a transition of $\alpha$. (a) Data are fit to the function given in Eq. (7), yielding $2t_{\text{tr}}\approx 205$ ms [dashed vertical lines in (a)–(c)], using ${\mathrm{\Gamma }}_{\text{coll}}=5.1\left(3\right){\text{s}}^{-1}$ obtained from (c) as a set parameter. The inset shows a Ramsey experiment performed under the same experimental conditions, fitted (solid line) to Eq. (2), yielding a Ramsey time $\tau =76\left(3\right)$ ms. (b) Same data with the time axis rescaled to ${\left(2t_{\pi }\right)}^3$, showing the transition clearly on a semilogarithmic scale. The linear behavior at short times (the solid line represents a linear fit) indicates the $e^{-{\left(2t_{\pi }\right)}^3}$ dependence. (c) Same data on a semilogarithmic scale as a function of $2t_{\pi }$. The linear dependence at long times (the solid line represents a linear fit) indicates the $e^{-2t_{\pi }}$ decay.

Figure 5

Numerical investigation of the effectiveness of using Eq. (7) as a fitting function for extracting the softness for a Gaussian energy distribution (closed symbols) and the energy distribution corresponding to a 3D harmonic potential (open symbols). The simulation calculates, for a predetermined input softness, a Ramsey decay curve as well as a full echo coherence decay curve. To approximate the obtained softness, we simultaneously fit the Ramsey decay to Eq. (3) and the echo decay to Eq. (7) with three fitting parameters: $\mathrm{\Gamma }, \tau$, and $s$. The black dotted line is where the obtained softness is equal to the real one.