Abstract
Harnessing spins as information carriers has emerged as an elegant extension to the transport of electrical charges^{1}. The coherence of such spin transport in spintronic circuits is determined by the lifetime of spin excitations and by spin diffusion. Fermionic quantum gases allow the study of spin transport from first principles because interactions can be precisely tailored and the dynamics is on directly observable timescales^{2,3,4,5,6,7,8,9,10,11,12}. In particular, at unitarity, spin transport is dictated by diffusion and the spin diffusivity is expected to reach a universal, quantumlimited value on the order of the reduced Planck constant ħ divided by the mass m. Here, we study a twodimensional Fermi gas after a quench into a metastable, transversely polarized state. Using the spinecho technique^{13}, for strong interactions, we measure the lowest transverse spin diffusion constant^{14,15} so far 6.3 (8) × 10^{3} ħ/m. For weak interactions, we observe a collective transverse spinwave mode that exhibits mode softening when approaching the strongly interacting regime.
Main
Studying transport in lowdimensional nanostructures has a long and rich history because of its nontrivial features and its relevance for electronic devices. The most common case, charge transport, has great technological implications and determines the current–voltage characteristics of a device. With the development of the field of spintronics^{1}, however, spin transport has also moved into the focus of the research interest. Spin transport has unique properties, setting it aside from charge transport: first, the transport of spin polarization is not protected by momentum conservation and is greatly affected by scattering^{5,16}. Therefore, the question arises: what is the limiting case of the spin transport coefficients when interactions reach the maximum value allowed by quantum mechanics? Second, unlike charge currents (which lead to charge separation and the buildup of an electrical field, counteracting the current), spin accumulation does not induce a counteracting force.
The main mechanism to even out a nonequilibrium magnetization M(r,t) = M(r,t)p(r,t) is spin diffusion, which is an overall spinconserving process. Other, nonspinconserving processes are much slower in ultracold Fermi gas experiments and are neglected hereafter. The gradient of the nonequilibrium magnetization drives two distinct spin currents^{17}: The first term produces a longitudinal spin current and the second term induces a transverse spin current. These spin currents are proportional to the longitudinal and transverse spin diffusivity, respectively.
In general, the spin diffusivity in D dimensions behaves as , where v is the collision velocity, n is the density and σ is the elastic scattering cross section between atoms. For shortrange swave interactions at unitarity, the cross section attains its maximum value allowed by quantum mechanics σ∼λ_{dB}^{D−1}, where degenerate regime (of any dimensionality) the de Broglie wavelength λ_{dB} is of the order of 1/k_{F} and hence the meanfree path is l_{mfp} = 1/(n σ)≈1/k_{F}, where k_{F} is the Fermi wave vector of the gas and n∼k_{F}^{D} is the density. Hence, the spin diffusion constant is given by ℏ/m. This quantum limit can also be viewed as a result of the uncertainty principle by noticing that the meanfree path l_{mfp} = 1/n σ is limited by the mean interparticle spacing^{11}. The simple scaling argument, however, hides much of the rich underlying physics. In particular, it cannot explain the Leggett–Rice effect^{14,15,18}, the difference between longitudinal and transverse spin diffusivities^{19}, and the transition to weak interactions where the physics changes because the system evolves from collisiondominated to collisionless. The lowest spin diffusion constant for longitudinal spin currents has been measured to be in threedimensional (3D) degenerate Fermi gases at unitarity^{5}, approximately two orders of magnitude smaller than in semiconductor nanostructures^{16}.
Here, we study the coherence properties of a transversely polarized 2D Fermi gas under the influence of a magnetic field gradient. By employing the spinecho technique^{13} we gain access to the intriguing, but little understood, transverse spin dynamics in 2D Fermi systems^{17}. The experiment starts by producing a Fermi gas of ^{40}K atoms. The motion of the atoms is restricted to a 2D plane^{20,21} by an optical lattice of wavelength λ = 1,064 nm, with the radial confinement being harmonic with a trap frequency of ω_{r} = 2π×127 Hz. After production of a spinpolarized, 2D gas, we apply three resonant radiofrequency pulses to rotate the spin state by respective angles π/2−π−π/2 about the S_{x}axis in spin space. Consecutive pulses are separated by a time τ and we refer to the spin evolution time as t = 2τ. The initial π/2pulse creates transverse spin polarization in a coherent superposition between and (Fig. 1). An applied magnetic field gradient B′≡∂ B_{z}/∂ x gives rise to a transverse spin wave of wave vector Q = δ γ t B^{′}, where δ γ = 2π × 152 kHz G^{1} is the difference of the gyromagnetic ratio for and at 209.15 G and t is the evolution time (see Methods). On timescales much shorter than the trapping period, Q is independent of the interaction strength^{22}, and nearby atoms acquire a relative phase angle of Δϕ≈Q/k_{F}. This lifts the spin polarization of the Fermi gas and the spins can collide with each other as they move in the harmonic potential or diffuse. Trivial dephasing induced by the magnetic field gradient is reversed by the πpulse and the spin state refocuses after a time τ if no decoherence has occurred. The final π/2pulse maps the spin state from S_{y} onto S_{z}, which is measured by performing a Stern–Gerlach experiment in time of flight, and we record 〈M_{z}〉 = (N_{↑}−N_{↓})/(N_{↑}+N_{↓}), where N_{↓/↑} is the total number of spin down/up atoms.
Even though the interaction potential between atoms is not explicitly spin dependent, an effective spinexchange interaction is mediated by the required antisymmetrization of the scattering wave function because of Pauli’s exclusion principle. In binary collisions, this leads to the identical spinrotation effect^{23}: in a collision both spins rotate about the axis defined by the sum of their spin orientations. In the Fermi degenerate regime, the binary collision picture has to be replaced by Landau’s quasiparticle description. Here, quasiparticle excitations are restricted to near the Fermi surface and can be considered as being affected by a ‘molecular field’ resulting from the effective spinexchange interaction^{14,15,24,25}. As a result, the spin wave cants out of the S_{x}–S_{y}plane and forms a spin spiral which acquires a component along S_{z} (see Fig. 1c). The magnitude of the identical spinrotation effect is proportional to the meanfield interaction strength g = −2πℏ^{2}/[mln(k_{F}a_{2D})]of the gas^{26}. Here, k_{F} is the Fermi wave vector for the initially spinpolarized sample and a_{2D} is the 2D scattering length (see Methods). We perform the experiment near the Feshbach resonance at 202.1 G, which allows tuning the strength of the interaction.
In the strongly interacting regime, that is, −1<ln(k_{F}a_{2D})<3, spin transport is dominated by diffusion and the spinecho signal attains a characteristic nonexponential decay of the form^{13,14,15} . Fitting the envelope of the spinecho signal (see Fig. 2a) with this function allows us to deduce the transverse spin diffusion constant. (Our experimental timing is not phase stable with respect to the Larmor precession of ≈50 MHz. Hence, our analysis focusses on the envelope of the spin echo, from which the shown data points typically scatter less than 10%. The combined error of preparation and detection of the magnetization for a single data point is less than 1%.) We extract the transverse spin diffusion constant for various interaction strengths and find at a shallow minimum around ln(k_{F}a_{2D}) = 0 (Fig. 2b). From the arguments given above, we expect the diffusivity to be on the order of ℏ/m. Observing a smaller value than for longitudinal spin diffusion in three dimensions^{5} is possibly less linked to the dimensionality rather than to the phase space available for collisions necessary to drive spin diffusion^{19}: In the case of longitudinal spin currents, a gradient of the M_{z}(x) polarization along the xdirection can be considered as a spatial variation of the local Fermi surfaces k_{F,↑}(x) and k_{F,↓}(x) (Fig. 2c). Only spins in a small region near the Fermi surface can diffuse from x to x+dx, invoking the typical T^{−2} scaling for quasiparticles in the deeply degenerate regime. In contrast, in the case of transverse spin currents the Fermi surfaces at different positions are of the same size but have slightly different directions of magnetization (see Fig. 2d). Hence, a spin moving along x from anywhere between the Fermi surfaces for spinup and spindown must scatter to reach local equilibrium, which scales as n_{↑}−n_{↓} and provides a much larger phase space. The result is that for a degenerate system the transverse diffusivity is smaller than the longitudinal one and becomes independent of temperature^{19}.
In the weakly interacting attractive Fermi liquid regime^{27}, that is, ln(k_{F}a_{2D})>3, we observe a qualitatively different behaviour (Fig. 3a). The envelope of the spinecho signal decays exponentially and is modulated by a slow oscillation, and both the frequency and the damping constant of the oscillation depend on the interaction strength. We attribute this slow periodic modulation to transverse spin waves which are excited by the spinecho sequence. The spin current induced by the magnetic field gradient can only be inverted completely by the πpulse when applied in phase with the harmonic motion. This can be qualitatively (for zero interactions) understood in a phasespace picture by considering that the magnetic field gradient causes a spatial offset between the potentials for the and the spin states and, correspondingly, the phasespace trajectories of the two spin components are displaced^{9}. Indeed, we observe that the difference of the centreofmass momenta 〈k_{↑}〉−〈k_{↓}〉 of the two spin densities n_{↑}(k) and n_{↓}(k) oscillates outofphase with the contrast of the spinecho signal (Fig. 3a,b). The total density profile n(k) = n_{↑}(k)+n_{↓}(k) is stationary, which indicates a pure spin mode.
The measured envelope of the spinecho signal (Fig. 3a) is fitted with an empirical function of the form: A exp(−γ t)(1−Bsin(ω t/2)^{3})+C. Here, and C≪1 are amplitudes and a global offset of the spinecho signal, respectively. This fit function reproduces the nonsinusodial signal shape and allows us to extract the oscillation frequency ω and amplitude decay γ. The exact shape can be calculated analytically only in the case of vanishing interactions^{9}. The exponent is fixed to the value 3 for all interactions and we do not observe a systematic dependence of the deduced frequencies and damping rates on the exact choice of the exponent.
For zero interaction strength the mode is at half the trap frequency and we observe a softening of the spin mode when approaching the strongly interacting, collisiondominated regime (Fig. 4a,b). This shows that for strong interactions the spinwave dynamics is superseded by spin diffusion. Such a behaviour is reminiscent of the collisionless to collisiondominated transition of the spindipole oscillation^{5,28}. The decay rate γ of the oscillation (see Fig. 4c) is smallest for the noninteracting gas, where it is two orders of magnitude smaller than the dephasing rate 1/τ_{Ramsey} in the inhomogeneous magnetic field (see Methods). As the interaction strength is increased, the rate γ is growing. We fit this growth with a powerlaw behaviour γ = α/ln(k_{F}a_{2D})^{β}+γ_{0} and determine the exponent to be β = 1.3±0.15. Observing a scaling with an exponent close to unity signals that the loss of global coherence is dominated by the identical spinrotation effect, which scales proportionally to the meanfield interaction strength g. In contrast, the rate of elastic collisions scales as ∝1/ln(k_{F}a_{2D})^{2}, that is, β = 2. This suggest the following decoherence mechanism in a spin1/2 Fermi gas: after building up in the magnetic field gradient, the spin wave decays when neighbouring atoms have acquired a sufficiently large phase angle Δϕ between their spins such that in a spinconserving collision both spins rotate out of the S_{x}–S_{y}plane. Initially, this is a coherent process, which subsequently dephases due to the coupling of the spin rotation to the momentum of the atoms^{29}. Our spinecho measurements show that the dephasing time measured by the Ramsey technique underestimates the actual spin decoherence time in the weakly interacting regime by orders of magnitude.
In strongly interacting ultracold atomic gases and condensedmatter systems in which spinrelaxation processes are suppressed, spin diffusion is the dominant process to equilibrate a nonequilibrium spin polarization. The search for a quantummechanical bound on spin diffusivity has been ongoing in various systems^{5,16,17,19}, and here we have reported the lowest yet spin diffusion constant of 6.3(8) × 10^{3} ℏ/m. Our work could pave the way towards an alternative mechanism to form ferromagnetic states of strongly repulsively interacting Fermi gases^{22}.
Methods
Preparation of the 2D gases.
We prepare a quantum degenerate Fermi gas of ^{40}K atoms in a onedimensional optical lattice of wavelength λ = 1,064 nm populating a stack of approximately 40 individual 2D quantum gases^{21,30}. The radial confinement of the 2D gases is harmonic with a trap frequency of ω_{r} = 2π×127 Hzand the axial trap frequency is ω_{z} = 2π×75 kHz. Using a radiofrequency cleaning pulse, we ensure preparation of a spinpolarized gas in the ground state with 1.5×10^{5} atoms, corresponding to a densityaveraged Fermi energy of the spinpolarized gas E_{F} = h × 12.3(5) kHz at a temperature of k_{B}T/E_{F} = 0.24(3). Our experiments on the spin dynamics start by ramping the magnetic field from B = 209.15 G to the desired value B near the Feshbach resonance in 30 ms, and then waiting there for 200 ms. Subsequently, we apply the spinecho pulse sequence (see main text) of three radiofrequency pulses with frequencies of ∼ 50 MHz. The pulses have square envelopes and durations of t_{π/2}∼46 μs for a π/2pulse and 2×t_{π/2} for a πpulse. The pulses are separated by equal times τ, which gives a total spin evolution time t = 2τ.
Interactions in two dimensions.
For 2D scattering with relative momentum q, the amplitude of the outgoing cylindrical wave for lowenergy scattering is^{31} f(q) = 4π/[ln(1/q^{2}a_{2D}^{2})+iπ]. This defines the 2D scattering length a_{2D}, which is linked to the binding energy of the confinementinduced dimer by E_{B} = ℏ^{2}/m a_{2D}^{2}. The logarithmic dependence of the scattering amplitude on momentum shows that f(q) is never independent of energy. In the regime of manybody physics of the weakly interacting 2D Fermi gas, the coupling strength is parameterized by the coefficient^{26} g = −2πℏ^{2}/[m ln(k_{F}a_{2D})]. The regime of zero interaction corresponds to , which is qualitatively different from the threedimensional case, where an infinite scattering length is observed in the regime of strongest interaction. In contrast, in two dimensions the regime of strongest interaction arises for ln(k_{F}a_{2D}) = 0, as according to the optical theorem the twobody scattering cross section σ = −Im[f(q)]/q = 4/q attains the maximal possible value dictated by unitarity, which is essentially the deBroglie wavelength. In the strongly interacting regime the mean field expansion in powers of 1/ln(k_{F}a_{2D}) breaks down. Instead, the energy per particle, after subtracting the twobody bound state energy, approaches a universal value^{32} 0.204 E_{F}/2.
Ramsey measurements: dephasing in the magnetic field gradient.
We assess the timescale of simple dephasing of the spin in the magnetic field gradient by performing Ramsey spectroscopy on the noninteracting gas using the following sequence: the initial π/2pulse is followed by a second π/2pulse with a variable time delay τ. We observe a Ramsey coherence time of τ_{Ramsey} = (600±30)μs. As the Ramsey time is more than an order of magnitude shorter than the inverse trap frequency, we neglect the motional contribution to the dephasing. Instead, we relate the Ramsey time to the dephasing time of the (quasi stationary) spins across the whole cloud, and we find from this the magnetic field gradient. Assuming a Thomas–Fermi distribution of atoms in the 2D potential, we fit the data with P(t) = 8J_{2}(ϕ)/ϕ^{2}, where J_{2} is the Bessel function of the first kind. The phase shift between the centre and the edge of the cloud, that is, the Fermi radius R_{F}, is ϕ = δ γ B′R_{F}t. The difference of the gyromagnetic factors, δ γ(B), is a function of the magnetic field and defined as the difference between the derivative of the Zeeman energies for the two spin states, that is, ℏδ γ(B) = ∂ E_{−9/2}/∂ B_{B}−∂ E_{−7/2}/∂ B_{B}. From a nonlinear leastsquares fit we deduce B′ = 4.7(2) G cm^{−1} for R_{F} = 17.7(5) μm and δ γ(209.15 G) = 2π × 152 kHz G^{−1}.
Change history
04 June 2013
In the version of this Letter originally published online, in the third main paragraph, the formula for the maximum elastic scattering cross section between atoms should have read σ∼λ_{dB}^{D−1}. In the previous paragraph, penultimate sentence, the terms 'parallel to M' and 'orthogonal to M' should not have been included. In the section 'Interactions in two dimensions', the first equation should have included square brackets, as here: f(q) = 4π/[ln(1/q^{2}a_{2D}^{2})+iπ]. These corrections have been made in the HTML and PDF versions of the Letter.
23 December 2013
In the version of this Letter originally published, the stated value for the difference in the gyromagnetic ratio for the ↑> and ↓> state, δγ, was missing a factor of 2π. Consequently, the diffusion constants were overestimated by a factor of (2π)^{2}. In particular, the value at ln(k_{F}a_{2D}) = 0 stated in the text should be 6.3(8) × 10^{3}ħ/m instead of 0.25(3)ħ/m. Other quantities, such as the magnetic field gradient, which was independently calibrated, are not affected. All other statements and observations in the published version are correct and remain unaffected. These errors have now been corrected in the online versions of the Letter.
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Acknowledgements
We thank E. Altman, G. Conduit, E. Demler, U. Ebling, A. Eckardt, C. Kollath, M. Lewenstein, A. Recati and W. Zwerger for discussions and B. Fröhlich and M. Feld for contributions to the experimental apparatus. The work has been supported by EPSRC (EP/J01494X/1, EP/K003615/1), the Leverhulme Trust (M. Koschorreck), the Royal Society, the Wolfson Foundation, and the AlexandervonHumboldt Professorship.
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The measurements were conceived by M. Koschorreck and M. Köhl, data were taken by M. Koschorreck, D.P., and E.V., data analysis and writing of the manuscript was performed by M. Koschorreck and M. Köhl.
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Koschorreck, M., Pertot, D., Vogt, E. et al. Universal spin dynamics in twodimensional Fermi gases. Nature Phys 9, 405–409 (2013). https://doi.org/10.1038/nphys2637
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DOI: https://doi.org/10.1038/nphys2637
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