Transverse Demagnetization Dynamics of a Unitary Fermi Gas

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Science  16 May 2014:
Vol. 344, Issue 6185, pp. 722-724
DOI: 10.1126/science.1247425

Untwisting the Spin Spiral

Ultracold Fermi gases in the so-called unitary regime—where the diverging interactions between atoms make their thermodynamics universal—are an excellent test bed for an array of strongly interacting matter systems. The transport characteristics in this regime are particularly intriguing, and a discrepancy between two- and three-dimensional transport coefficients has been observed. Bardon et al. (p. 722) studied the demagnetization dynamics of a three-dimensional Fermi gas. The gas was initially polarized along a single direction and was noninteracting. An applied magnetic field gradient then caused a spin spiral to form; as the gas relaxed from this state, the authors extracted the diffusion coefficient and observed the buildup of interactions between the atoms.


Understanding the quantum dynamics of strongly interacting fermions is a problem relevant to diverse forms of matter, including high-temperature superconductors, neutron stars, and quark-gluon plasma. An appealing benchmark is offered by cold atomic gases in the unitary limit of strong interactions. Here, we study the dynamics of a transversely magnetized unitary Fermi gas in an inhomogeneous magnetic field. We observe the demagnetization of the gas, caused by diffusive spin transport. At low temperatures, the diffusion constant saturates to the conjectured quantum-mechanical lower bound Embedded Image, where m is the particle mass. The development of pair correlations, indicating the transformation of the initially noninteracting gas toward a unitary spin mixture, is observed by measuring Tan’s contact parameter.

Short-range interactions reach their quantum-mechanical limit when the scattering length that characterizes interparticle collisions diverges. A well-controlled model system that realizes this unitary regime is provided by ultracold fermionic alkali atoms tuned to a Fano-Feshbach resonance (1). These scale-invariant gases are characterized by universal parameters relevant to diverse systems such as the crust of neutron stars at 25 orders of magnitude higher density (2, 3). Experiments with ultracold atoms have already greatly contributed to the understanding of equilibrium properties of unitary gases (46). Progress has also been made in the study of unitary dynamics (711), including observations of suppressed momentum transport (7) and spin transport (810) due to strong scattering.

Spin diffusion is the transport phenomenon that relaxes magnetic inhomogeneities in a many-body system. At low temperature, where Pauli blocking suppresses collision rates, one must distinguish between diffusion driven by gradients in either the magnitude or the direction of magnetization, and quantified by longitudinal spin diffusivity Embedded Image or transverse spin diffusivity Embedded Image, respectively (12, 13). A measurement of Embedded Image in a three-dimensional (3D) unitary Fermi gas yielded a minimum trap-averaged value of Embedded Image (9). This is consistent with a dimensional argument, in which diffusivity is a typical velocity (Embedded Image for a cold Fermi gas, where Embedded Image is the Fermi momentum) times the mean free path between collisions. In the absence of localization, the mean-free path in a gas cannot be smaller than the interparticle spacing Embedded Image, which translates into a quantum lower bound of roughly Embedded Image (9, 14, 15). However, Embedded Image as low as Embedded Image was recently observed in a strongly interacting 2D Fermi gas (10). This thousand-fold range in transport coefficients remains unexplained by theory.

We measure the transverse demagnetization dynamics of a 3D Fermi gas that is initially fully spin-polarized. All of our measurements are carried out with samples of ultracold Embedded ImageK atoms in a harmonic trap. Each atom is prepared in an equal superposition of two resonantly interacting internal states, labeled Embedded Image and Embedded Image (16), which corresponds to a gas with full transverse magnetization Embedded Image (Fig. 1). Initially, interactions between these identical ultracold fermions is inhibited by the Pauli exclusion principle. The states we use also block any local mechanism for spin relaxation, unlike the scenario typical in liquids or solids. However, the differential magnetic moment Embedded Image between the internal states allows a magnetic field gradient Embedded Image to twist the magnetization across the cloud into a spiral pattern, leading to a gradient in transverse magnetization. This gradient drives diffusive spin transport that erodes the coherence irreversibly. In contrast, for a weakly interacting Fermi gas, collisionless spin waves lead to reversible dynamics (17). For the strongly interacting Fermi gas probed here, the evolution of transverse magnetization Embedded Image is modeled with Embedded Image, neglecting trap effects, where Embedded Image (18). This equation is solved by Embedded Image such that Embedded Image gives the time scale of demagnetization. Because there is no spatial gradient in the magnitude of magnetization, the dynamics do not probe Embedded Image.

Fig. 1 Magnetization dynamics.

A π/2 pulse at t = 0 initializes the system with a homogeneous magnetization (My = 1 in the rotating frame) perpendicular to the magnetic field, which is along z. A spin spiral develops because of a magnetic field gradient and drives diffusive spin currents. The upper and lower sequences show evolution without and with a π pulse, respectively.

The effect of spin diffusion on magnetization is measured using the spin-echo technique described in Fig. 1. The spin-refocusing π pulse at tπ swaps the population of the states Embedded Image and Embedded Image, which causes the spin spiral to start untwisting. This partial rephasing also reduces the rate of diffusion. At Embedded Image, the model anticipates a spin echo withEmbedded Image (1)The final cloud-averaged Embedded Image is indicated by the contrast in Embedded Image and Embedded Image atom number after a final π/2 pulse with variable phase (16).

We observe that demagnetization occurs in several milliseconds (Fig. 2A, inset). Fitting Embedded Image with an exponential decay function Embedded Image, we find a range of Embedded Image, compatible with Embedded Image in Eq. 1. Constraining Embedded Image, we extract τM across a wide range of gradients (Fig. 2A) and fit it to find that the Embedded Image scaling of Eq. 1 holds even for the case of the trap-averaged magnetization. At an initial temperature Embedded Image, where TF is the Fermi temperature of the spin-polarized gas (16), a single-parameter fit of Embedded Image to the data yields Embedded Image, where the uncertainties are the statistical error from the fit and the systematic error from the gradient calibration, respectively. This is a direct measurement of the time- and trap-averaged diffusivity that does not rely on any calibration other than that of the gradient.

Fig. 2 Spin diffusion.

(A) Magnetization decay times τM at various gradients at an initial temperature of Embedded Image Horizontal and vertical error bars reflect the systematic error in the gradient calibration and the fit error, respectively. The solid line is a single-parameter fit of Embedded Image to the data, and the shaded area denotes the combined statistical and systematic uncertainty. The inset shows a sample of Embedded Image at B′ = 8.2(7) G/cm with a fit using Eq. 1. (B) Measured Embedded Image for various initial temperatures at a single gradient B′ = 18(1) G/cm. To achieve temperature control, the data were collected with N = (1 – 7) × 104. Horizontal error bars are statistical uncertainty; vertical error bars combine gradient-calibration and fit uncertainty. The insets show sample data for Embedded Image at two initial temperatures with a fit using Eq. 1.

In Fig. 2B, we choose a constant gradient and vary Embedded Image. Diffusivity is larger in hotter clouds, as both the typical velocity and the mean free path increase with temperature. At lower temperatures, we observe that Embedded Image does not continue to decrease but appears to saturate. Careful examination of the demagnetization dynamics at our lowest initial temperatures (see insets to Fig. 2B) also suggests an acceleration of demagnetization at later times. An apparently time-dependent Embedded Image could be due to its polarization dependence, as is predicted below the so-called anisotropy temperature, where Embedded Image differs from Embedded Image (13, 19). It might also arise from spin-rotation effects (20, 21). However, we find the deviations from Eq. 1 to be small, and we are unable to distinguish between these possibilities and other systematics. Within the probed range of temperature, the trap- and time-averaged Embedded Image is consistent with a quantum lower bound of Embedded Image.

Demagnetization transforms the system of N particles in a single spin state to a mixture of two spin states, each with N/2 particles. The final Fermi energy of the trapped system Embedded Image therefore is reduced by a factor of 21/3 compared with the initial EF,i. Furthermore, demagnetization releases attractive interaction energy. Together these effects increase temperature (16), so that each measurement of Embedded Image has to be understood as a time average over a range in temperatures. The intrinsic heating, together with the initial polarization of the cloud, ensures that the gas remains in the normal phase throughout the evolution (22).

The observation of suppressed spin transport indicates strong interparticle scattering but does not reveal how a thermodynamic interaction energy emerges. In a complementary set of measurements, we study the microscopic transformation of the gas by following the dynamical evolution of pair correlations that are enabled by demagnetization. Instead of measuring Embedded Image, we probe the gas with a pulse that couples Embedded Image to Embedded Image, an initially unoccupied internal state that interacts only weakly with Embedded Image and Embedded Image (16). The transfer rate to Embedded Image is measured as a function of the frequency detuning δ above the single-particle resonance. In a strongly interacting gas in equilibrium, the high-frequency tail of such a spectrum is known to be proportional to Tan’s contact parameter Cdr Embedded Image(r) times δ–3/2 (2330). The contact density Embedded Image is a local measure of the pair correlation, that is, the number of pairs of opposite spins at short distance, where g is the coupling constant and ψσ is the annihilation operator with spin σ. As is clear from its definition, Embedded Image is also proportional to the local interaction energy. Although contact has been shown to relate various thermodynamic and many-body properties of a short-range interacting gas, it has so far been studied only in equilibrium and only with an unmagnetized gas (25, 3033).

Figure 3A shows that after a short hold time, the spectrum exhibits only the single-particle peak, whereas after a longer hold time, the spectrum develops a high-frequency tail. Similar spectroscopic measurements starting from a polarized Fermi gas have shown the emergence of mean-field shifts after decoherence (34). Here, we study the high-frequency tail of the spectrum, finding that it has a δ–3/2 scaling at Embedded Image for each hold time t, which indicates that pair correlations can be described with a contact parameter throughout the dynamics (Fig. 3B).

Fig. 3 Time-resolved spectroscopy to measure Tan’s contact

at B′ = 8.2(7) G/cm and Embedded Image. (A) The normalized rate Embedded Imageof atom transfer (16) from Embedded Image to Embedded Image is shown after 10 μs (solid line) and 2.5 ms (dashed line), both without a spin-refocusing pulse. The red line in the right panel is a fit of a power-law Embedded Image to the high-frequency tail for t = 2.5 ms. (B) The transfer rate Embedded Image rescaled with Embedded Image versus hold time, where Embedded Image, revealing a plateau for large positive detunings Embedded Image. The inset compares contact values extracted from these full spectra (open circles) and those measured at a fixed detuning δ/2π = 125 kHz (filled circles). The latter method allows single-shot study of contact dynamics, and thus reduced statistical noise, as seen here. The contact is normalized by the final Fermi momentum in the trap and the total number of atoms. Error bars in (B) show one standard deviation of repeated measurements.

Figure 4A shows that, under various protocols, the contact starts at zero and grows in time toward a maximal value of Embedded Image, where Embedded Image is the Fermi momentum in the final state of the trapped gas and N is the total number of atoms. This is comparable to equilibrium values observed previously at Embedded Image in (33), which lies between the initial and final temperatures of these data (16). At longer times (t > 5 ms), Fig. 4A shows a slow reduction of contact, which is likely due to heating; however, in this work, we focus on the short-time dynamics. A fit using an empirical rise function Embedded Image to the short-time data yields an exponent of Embedded Image with a spin-reversal and Embedded Image without a spin-reversal, reminiscent of the magnetization loss function Embedded Image. Further connection between contact and magnetization is demonstrated by Fig. 4B, which traces the contact during a spin-reversal sequence: The rise of Embedded Image is slowed by the refocusing pulse and plateaus at the spin-echo time, around which transverse spin diffusion is suppressed.

Fig. 4 Linking contact and magnetization.

The time evolution of contact is studied using single-shot spectroscopy, as illustrated by the inset of Fig. 3B. (A) Contact versus hold time t with (in green) and without (in red) a spin-refocusing π pulse at t/2. Solid lines show a fit to Embedded Image. (B) Contact versus hold time t with a π pulse applied at tπ = 1 ms. At the echo time, the spin spiral has untwisted and no longer drives a diffusive spin current. Error bars in (A) and (B) show statistical error across repeated measurements, and both data sets were taken under the same conditions as Fig. 2A. (C) Contact growth time, τC, versus demagnetization time, τM [colors as in (A)], measured at various gradients. Shaded areas are bounded by the limits of fully paired and uncorrelated spins. (D) Normalized contact versus magnetization obtained by relating measurements of Embedded Image and C(t) for all sampled gradients B′. The solid line shows the behavior of uncorrelated spins. A typical statistical error bar is shown on a single point.

Figure 4C compares τM and τC, both with and without an echo. A linear relationship is found, which is surprising at first, since magnetization is a one-body vector observable and contact is a two-body scalar observable. The connection comes from the Pauli exclusion principle, which requires that if two particles are in the same location, as is required for a contact interaction, their spin state must be the antisymmetric spin-singlet state. For uncorrelated spin pairs, the probability to be in a spin-singlet state is Embedded Image. Combining this assumption with the diffusion model for magnetization predicts Embedded Image and Embedded Image with and without an echo, respectively. The maximum singlet probability for a given magnetization is Embedded Image and would instead give Embedded Image that is Embedded Image larger. Data in Fig. 4C show an approximately linear relation whose slope is between these two limits.

Comparing the full range of measured values for normalized C and Embedded Image at various times and gradients in Fig. 4D also shows a functional form between the uncorrelated Embedded Image and the fully paired Embedded Image. A calculation based on a large-Embedded Image expansion (3537) predicts that Embedded Image changes between these limiting behaviors as T goes from 2Tc to Tc, where Tc is the critical temperature for pair superfluidity (16). Because a singlet pair has no net spin, the observation of enhanced Embedded Image is also consistent with previous observations of reduced magnetic susceptibility due to strong attractive interactions in the normal state (38, 39).

Alternatively, an apparent reduction in Embedded Image might arise from a lag in the evolution of C behind Embedded Image. However, we find no statistically significant dependence on gradient, which is evidence for a local equilibration of C on a faster time scale than the system-wide demagnetization. A true steady-state transport measurement, on the other hand, would suffer from an inhomogeneous magnetization due to imbalanced chemical potentials in the trap. Our dynamic measurement avoids this problem, because longitudinal spin transport is strongly suppressed on the millisecond time scale (8, 9).

In conclusion, we have shown how a transversely spin-polarized Fermi gas decoheres and becomes strongly correlated at an interaction resonance. A diffusion constant of Embedded Image challenges a quasiparticle-based understanding of transport by implying the necessity of maximally incoherent quasiparticles. A similar limit to the quasiparticle lifetime would explain the ubiquitous T-linear resistivity in metals (40) and a quantum-limited shear viscosity (7).

Supplementary Materials

Materials and Methods

Figs. S1 and S2

References (4147)

References and Notes

  1. Supplementary materials are available on Science Online.
  2. Acknowledgments: We thank B. Braverman, I. Kivlichan, L. LeBlanc, and T. Pfau for experimental assistance; D. DeMille, T. Enss, L. Jiang, A. Leggett, and A. Paramekanti for discussions; A. Aspect for manuscript comments; and M. Ku and M. Zwierlein for sharing their unitary equation-of-state data. S.T. acknowledges support from the Centre for Quantum Information and Quantum Control. G.J.A.E. acknowledges support from Ontario. This work was supported by NSERC, CIFAR, the University of Hong Kong, and AFOSR under agreement no. FA9550-13-1-0063.
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