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Giant spin oscillations in an ultracold Fermi sea

Collective behavior in many-body systems is the origin of many fascinating phenomena in nature ranging from swarms of birds and modeling of human behavior to fundamental magnetic properties of solids. We report on the first observation of collective spin dynamics in an ultracold Fermi sea with large spin: We observe long-lived and large-amplitude coherent spin oscillations, driven by local spin interactions. At ultralow temperatures, Pauli blocking stabilizes the collective behavior and the Fermi sea behaves as a single entity in spin space. With increasing temperature, we observe a stronger damping associated with particle-hole excitations. As a striking feature, we find a high-density regime where excited spin configurations are collisionally stabilized.

collective effects involving the spin degree of freedom may lead to fascinating interaction-induced phenomena (12)(13)(14)(15)(16)(17)(18)(19). For bosonic atoms with a total spin of 1, 2,3 f = spin-dependent phenomena have been intensively studied (20,21 and references therein). In these experiments, the macroscopic occupation of a common single-particle wave function, which is inherent to Bose-Einstein condensation, strongly favors collective behavior. For fermions, in contrast, every singleparticle state is occupied with only one particle and hence macroscopically many of these states are occupied in a Fermi sea. This naturally raises the following questions: Can the whole Fermi sea exhibit collective and coherent dynamics of its internal spin degree of freedom (Fig. 1A)? Does this dynamics preserve the spatial structure or do emerging spin components populate new spatial modes? Recently, first experiments with high-spin fermions allowed to study ground-state and dynamical properties in the regime of deep optical lattices (22,23), few-body physics (24) and also the emergence of spatial spin waves (25), leaving, however, the questions raised above largely unanswered. Here, we report on the first observation of coherent spin dynamics in an ultracold We have observed these effects in a quantum degenerate gas of 40 K in an optical dipole trap (26), well isolated from its environment. The hyperfine ground state of 40 K has a total spin 9 / 2 f = , which gives rise to ten spin states with magnetic quantum numbers 9 / 2, ,9 / 2 m = − … . To record the population of these different spin components, we release the atoms from the trap and subsequently measure the spin occupations after expansion in a Stern-Gerlach field (26). To understand collective spin-changing dynamics, we first consider the underlying microscopic quadratic Zeeman energies and differential spin-dependent interaction energies for the different spin configurations determines whether spin oscillations can occur or not (20). At large magnetic fields, spin-changing collisions are suppressed energetically and the Fermi sea remains in its initial spin configuration. For low magnetic fields, the two energy scales become comparable and resonant spin oscillations are induced. As a key result, Fig. 1B demonstrates this kind of coherent spin oscillations for timescales as long as 2s for an ultracold atomic Fermi ensemble. In the following we show that this peculiar behavior can be theoretically described via a mean-field approach (25). Furthermore, all key observations are explained surprisingly very well within a single-spatial-mode approximation, i.e. the whole Fermi sea evolves in time as a single spin entity. The high-spin Fermi sea is described by the single-particle density matrix in Wigner representation ( ) , ij w x p , being a matrix in spin space depending on position and momentum. Its time evolution obeys the Boltzmann equation (26) Here, x p accounts for the spin-independent center-of-mass motion of the atoms in the trap and ( , ) ij I x p is the collisional integral accounting for particle-hole excitations. Spin-changing dynamics is driven by the commutator including the mean-field potential ( ) x p with the interaction tensor klji U and the quadratic Zeeman qS . The commutator drives the spin-changing dynamics in the presence of coherences given by finite off-diagonal matrix elements of ij w (26). In Fig. 1B, the system is prepared with very small initial coherences sufficient to initialize a spin instability followed by collective spin oscillations, a behavior also observed for Bose-Einstein condensates (27).

Fig. 2. Collective behavior at ultralow temperatures.
Frequency (main graphs) and amplitude (insets) of collective spin oscillations versus magnetic field (A) and density (B), both for an initial state as in Fig. 1C. Experimental parameters are are deduced from fits (26). Solid lines in insets correspond to the amplitude and the shading in the main graph to a Fourier spectrum, both deduced from calculations within a single-mode approximation without free parameters (26).
The interplay between the kinetic and interaction energy in equation (1) can lead to a fascinating effect: If the spatial dynamics of the trapped atoms is reasonably faster than the spin-changing dynamics, then the spatially-dependent interactions are averaged out leading to an effective longrange interaction potential (28). This induces collective spin-changing behavior of the whole Fermi ensemble despite the underlying complex spatial structure (26), justifying the above-mentioned single-mode approximation. Fig. 1C shows coherent spin oscillations at ultralow temperatures for an ensemble initially prepared in a particular coherent spin superposition (26). In contrast to the instability-induced oscillations shown in Fig. 1B, this state immediately exhibits spin oscillations and can thus be systematically compared to theoretical calculations. At high magnetic fields, we find low-amplitude and high-frequency oscillations in contrast to slow and giant spin oscillations at low magnetic fields as depicted in Fig. 2A. By varying the density and thereby the interaction energy within the Fermi sea ( Fig. 2B), we find an increase in amplitude and a rather constant frequency for increasing interaction. The system precisely follows the mean-field description within the single-mode approximation without free parameters. As one key result of this work, we therefore conclude that spin-changing collisions can induce collective behavior in a Fermi sea of trapped atoms preserving the spatial structure for long timescales, even for macroscopic systems with several 5 10 particles and a spatial extension larger than 100μm .
A fundamental question is how finite temperature affects the observed collective behavior. Fig. 3A shows the strong impact of increasing temperature on the spin-changing dynamics in our experiment, revealing long-lived oscillations at low temperatures and a nearly full suppression of coherent evolution at higher temperatures. Again we find good agreement for the observed frequencies and amplitudes with the single-mode approximation (see Fig. 3B). However, for higher temperatures we observe substantially stronger damping of the coherent oscillations, as depicted in Fig. 3C. We attribute this to an enhanced rate of particle-hole excitations, which are described by the collisional integral ( , ) ij I x p in equation (1). These incoherent collisions are characterized by a finite momentum exchange of the colliding particles. In contrast, the collisions inducing collective spin oscillations conserve the individual momenta and are captured by the mean-field potential. At ultralow temperatures, the Fermi statistics does not provide any free momentum states due to the Pauli exclusion principle, hence collisions involving momentum exchange are strongly suppressed (29). For increasing temperatures, however, free momentum states become available in the Fermi sea and allow for increased particle-hole excitations. In a relaxation approximation ansatz (30), we calculate the corresponding damping rate without free parameters and find good agreement with the observed damping for temperatures up to f 0.4T T = (see Fig.   3C). For even higher temperatures the coherent oscillations are fully suppressed, which underlines that ultralow temperatures are the key ingredient to enable collective spin-changing dynamics in a Fermi sea.
Finally, we investigate the occurrence of spin instabilities like the one presented in Fig. 1B as a function of the density and the magnetic field. As a particularly surprising feature we find a regime at high densities where magnetically excited initial states remain stable for seconds. We study this effect by investigating the emergence of spin instabilities, which is depicted in the stability the larger density. We attribute this stabilization to a self-induced spin-Zeno-effect: At large density, increased incoherent collisions "project" the spins back onto the initial state and thus suppress the coherent evolution observed at low densities. Note that this intriguing spin stabilization is a pure collisional effect, which is highly sensitive to slight changes of the experimental parameters and leads to a completely different macroscopic behavior of the Fermi sea. for the experiments depicted in Fig. 3 and Fig. 4. We estimate the uncertainty of the trapping frequencies to be 10% and of the magnetic field to be less than 3mG .

S1.2 Initialization of collective spin dynamics in a Fermi sea
A major challenge for observing collective spin dynamics in a fermionic many-body system is to satisfy two preconditions as worked out in section (S2): finite interactions to allow for spinchanging collisions in the sample and finite single-particle coherences. We employ two different approaches to fulfill these conditions and hence to initialize the collective spin dynamics: For the experiments depicted in Figs. 1B and 4, a very small seed in the initial state is sufficient to initialize self-induced collective dynamics. We checked that this instability-induced collective dynamics sets in even if the amount of seed atoms is below the detection limit. For the measurements depicted in Figs. 1C, 2 and 3, in contrast, we explicitly prepare initial coherences as described in the following. Consider the single-particle density matrix, which describes the Fermi sea in its groundstate hyperfine manifold with a total spin of 9 / 2 F = and ten spin states 9 / 2,...,9 / 2 m = − . The diagonal elements represent the spin occupations, whereas the non-diagonal elements represent the corresponding coherences. In the experiment, we evaporate a binary spin mixture of . The resulting single-particle density matrix 0 ρ hence consists of the corresponding two diagonal entries whereas all non-diagonal elements are zero as depicted in Fig. S1A.
Subsequently, we apply a short radio-frequency pulse of about 10µs at a low magnetic field, which creates non-diagonal elements and initializes spin-changing dynamics. At low magnetic field strengths, the radio-frequency coupling is dominated by the linear Zeeman effect, which results in a coupling of all ten spin components.
The resulting density matrix is given by Here, , x y S are the corresponding generators of rotation and ϑ corresponds to the rotation angle in spin space. Fig. S1A shows the impact of the radio-frequency manipulation on the density matrix of the high-spin Fermi sea, where one clearly observes occupations in new spin states as well as emerging coherences. Fig. S1B shows the experimental result of the radio-frequency pulse for different rotation angles corresponding to different radio-frequency pulse powers. We compare the observed spin occupations depending on the rotation angle ϑ to (S1) and find very good agreement with the Rabi frequency as the only free parameter.
For the experiments shown in Figs. 1C, 2 and 3 we apply a radio-frequency pulse corresponding to a rotation angle in spin space of 0.44 ϑ = . This state is depicted in Fig. S1A in (III) and exhibits a nearly equal occupation of the 1 / 2 ± and 3 / 2 ± spin components resulting in large-amplitude oscillations which are well suited for our investigations.

S1.3 Experimental detection analysis
Here, we describe the experimental determination of the spin populations, the total particle number and the temperature of our samples.
In our experiments, we employ time-of-flight imaging. After a variable evolution time, all optical potentials are suddenly switched off and an applied Stern-Gerlach field leads to the separation of the spin states within an expansion time of typically 18.5ms . We detect the atoms with resonant absorption imaging which allows us to measure the total particle number N as well as all relative spin populations simultaneously.
To determine the temperature of our samples, we release the two-component Fermi gas from the trap and employ a fugacity fit, which is independent of the particular trapping frequencies. We apply this fit to the sample after a time-of-flight of typically 21ms without a Stern-Gerlach field and without spin-changing dynamics. The temperature determination is performed separately to avoid deformations of the cloud due to the Stern-Gerlach field. Note that the observed temperatures reflect the initial temperatures of our samples. We checked that applying a short radio-frequency pulse as applied in Figs. 1C, 2 and 3 does not increase the temperature. A slight temperature increase is observed during the experiments which we mainly attribute to singlephoton scattering events.
With the measured experimental parameters (the trap frequencies, particle numbers and temperatures), we determine the density and also perform all numerical calculations. In the whole paper, density always refers to the total peak density p n of the system. We estimate the errors for the investigated parameters in the following way: Errors in the observed relative spin populations shown in Figs. 1B and C correspond to the standard deviation. The errors of the magnetic field depicted in Fig. 2A reflect the systematic uncertainty of the magnetic field and always lie within the data points. The error in the density shown in Fig. 2B mainly results from uncertainties of the trapping frequencies. Note that in the rather shallow trap configuration employed in this experiment, the influence of the Gaussian shape of the dipole trap is enhanced and might lead to a systematic overestimation of the determined density. Errors in temperature in Fig. 3 are taken from the fugacity fit, which is independent of the particular trapping frequencies, and mostly lie within the data points.

S1.4 Fitting procedure of the spin oscillations
In this section, we describe how we experimentally determine the frequency, amplitude and damping rate from the observed spin-changing dynamics evaluated in Figs. 2 and 3.

S2 Derivation of the collisionless Boltzmann equation
In this section, we derive the collisionless Boltzmann equation, which we employ in equation (1) in the main text for the description of collective dynamics of a spinor Fermi gas in the mean-field approximation.
In our system, the single-particle state of an atom of mass m is described by a spinor field operator ˆ( ) consisting of a single-particle and an interaction part. Here and in the following, sums over spin indices include all internal states. The first two terms of the single-particle part describe the motion of atoms in a trapping potential t r a p , whereas the third part accounts for the presence of an external homogeneous magnetic field along the spin quantization axis, generating a non-linear Zeeman splitting. Note that in our system the quadratic Zeeman energy is dominating. In the interaction part, the coupling constants ijkl U are composed of the scattering lengths S a for even total spin S of the scattering particles, . We describe the time evolution of the system by first introducing the single-particle density matrix where ˆ( ) t ρ denotes the statistical many-body density operator whose time evolution is given by . Therefore, the single-particle operator evolves in time according to †( When evaluating the right-hand-side of this expression, one encounters quartic expectation values in the interaction part, to which we apply a Hartree-Fock approximation and decompose them as such that we obtain a closed, but nonlinear equation for ( , ) mn ρ x y only, without retaining fewparticle density matrices . For a semiclassical treatment, we introduce the Wigner-transform where we have introduced the notation 2 0 for the free particle motion in the trap. We also have defined the Hartree-Fock mean-field potential In the next step, we apply a semiclassical approximation to the orbital degrees of freedom, retaining only the leading term in the gradient expansion above, truncating it at 1 α = . This is justified as long as the mean-field potential varies slowly compared to the single-particle wavelengths involved. In the general case, one has to include the external fields in this expansion as well, but for a harmonic potential as well as for the quadratic Zeeman effect induced by a homogeneous external magnetic field the truncation is exact. We obtain a semiclassical equation of motion taking the form of a collisionless Boltzmann equation

S3 Single-mode approximation
In this section, we derive a single-mode approximation for a fermionic many-body system, which is widely employed in the main text and describes the observed collective spin-changing behavior very well.

S3.1 Transformation to the rotating frame
We use a dimensionless description of the many-body system in units of the trapping potential, which is obtained by measuring lengths and momenta in direction i in units of / i mω  and i m ω  , respectively, and by using ω  , ω and 1 ω − as the units for energy, angular velocity, and time.
Here ω can be defined as the geometric mean of the trap frequencies. In these units, While the left-hand side of this equation describes the spin-independent classical motion in the harmonic trap, the right-hand side induces coherent spin dynamics. The latter happens on a much longer timescale than the former. Neglecting the right-hand side for short times, the spin configuration, which initially, at 0 t = , is described by the matrix w at a given point ( ) i.e.
This motivates a transformation to a reference frame in phase space that is co-moving with the classical dynamics, namely ( ) ( ) x p x x p p x p for a function g acting in phase space. In the new frame, the dynamics is described by Now the mean-field potential also depends on the new momentum variables and is given by an integral along d rotating axes Here ( ) ( )

S3.2 Dynamically-induced effective long-range interactions
If the dynamics generated by the commutator is slow compared to the dynamics in the trap, we can approximate the mean-field potential ' V by averaging over the rotational dynamics along the different trap directions, as it is captured by u and v . For this purpose, we artificially induce a second time τ describing the dynamics in the trap, while t captures the dynamics induced by the commutator. This allows us to average the mean-field potential with respect to the former. In the following, we distinguish two cases.

S3.2.1 Commensurate trap frequencies
If the trap frequencies are commensurate, we can average over a time span given by the least Thus each phase space point ( ) ', ' x p interacts via the mean-field potential with the or by the one-dimensional system. For the latter, we can write ( ) yielding the effective long-range interaction potential in phase space between two particles 1 and 2, introduced in (28).

S3.2.2 Incommensurate trap frequencies
Thus, we arrive at a mean-field potential corresponding to an anisotropic long-range interaction between two particles 1 and 2 in phase space.

S3.2.3 Description in single-mode approximation
the simulations in Fig. 2 and of 1Hz Γ = for the simulations in , which exhibits large-amplitude spin oscillations normalized in this graph, but clearly the emergence of any spatial deformations is suppressed. In this simulation, parameters have been used where the kinetic motion induced by the trap is faster than the interaction-driven spin-changing dynamics, as it is the case in our experiments.
Numerically, we can also access a regime, where the kinetic motion induced by the trap and the interaction-driven spin-changing dynamics occur on comparable timescales. The result of the corresponding normalized time evolution in this regime is exemplarily plotted in the lower graph in Fig. S2A. We find the intriguing formation of pronounced spatial structures, clearly revealing the breakdown of the single-mode approximation. Note that this regime is experimentally not accessible in our system. To ensure the validity of the single-mode approximations even further for the parameters, at which the experiments have been performed, we also compare the singlemode results with the full calculations for different magnetic fields where we choose the same parameters. Fig. S2B shows the extracted frequencies and amplitudes as a function of the magnetic field for both calculations revealing perfect agreement. We conclude that the singlemode approximation is valid within our parameter regime as already shown in the experiments in the main text.

S4 Incoherent collisions in the relaxation approximation
In this section, we deduce the collisional integral ( , ) ij I x p and the temperature-dependent damping rate within the relaxation approximation.
To estimate the damping rate we derive an explicit expression for the collision term. We follow the approach of Ref. (31), where a Boltzmann equation is derived from the T-matrix describing a single scattering event of two particles. This approach can be generalized to arbitrary spin by introducing a T-matrix for each channel defined by the total spin S of both particles: