Observation of a Transition Between Dynamical Phases in a Quantum DegenerateFermi Gas

Scott Smale,1, ∗ Peiru He,2, 3, ∗ Ben A. Olsen,1, † Kenneth G. Jackson,1 Haille Sharum,1

Stefan Trotzky,1 Jamir Marino,2, 3, ‡ Ana Maria Rey,2, 3, § and Joseph H. Thywissen1, 4, §

1Department of Physics, University of Toronto, Ontario M5S 1A7, Canada2JILA, National Institute of Standards and Technology, and University of Colorado,

Department of Physics, University of Colorado, Boulder, CO 80309, USA3Center for Theory of Quantum Matter, University of Colorado, Boulder, Colorado 80309, USA

4Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

A proposed paradigm for out-of-equilibrium quantum systems is that an analogue of quantumphase transitions exists between parameter regimes of qualitatively distinct time-dependent behav-ior. Here, we present evidence of such a transition between dynamical phases in a cold-atom quantumsimulator of the collective Heisenberg model. Our simulator encodes spin in the hyperfine statesof ultracold fermionic potassium. Atoms are pinned in a network of single-particle modes, whosespatial extent emulates the long-range interactions of traditional quantum magnets. We find thatbelow a critical interaction strength, magnetization of an initially polarized fermionic gas decaysquickly, while above the transition point, the magnetization becomes long-lived, due to an energygap that protects against dephasing by the inhomogeneous axial field. Our quantum simulationreveals a non-equilibrium transition predicted to exist but not yet directly observed in quencheds-wave superconductors.

Introduction

The challenge faced in understanding out-of-equilibriumsystems is that the powerful formalism of statisticalphysics, which has allowed a classification of quantumphases of matter based on simple principles such as mini-mization of free energy, does not apply. A diverse range ofnon-equilibrium phenomena have been observed, includ-ing synchronization [1–4], self-organization [5–8], quan-tum chaos [9, 10], Loschmidt echo singularities [11], andtime crystals [12, 13]. A proposed organizing principleis that transitions, reminiscent of those found betweenthermodynamic ground states, can also be found betweendynamical phases [14–18].

In general terms, a non-equilibrium phase transition ischaracterized by the existence of a critical point separat-ing phases with distinct dynamical properties in many-body systems. The analogue of thermodynamic order pa-rameters is found in long-time-average observables, whichhave a non-analytic dependence on system parameters.In driven open systems, energy and particle number arenot necessarily conserved, and non-equilibrium transi-tions are typically signaled by different steady states thatoccur upon varying system parameters such as pump orloss rates [19–21], independently of initial conditions. Inclosed systems, dynamics are often initiated by quench-ing control parameters, with qualitatively distinct behav-iors observed below, above, or at a critical point [14–17]that can depend on the initial state of the system. Thelabel “dynamical phase transition” has been applied notonly to the boundary between two dynamical phases, butalso to the non-analytic behavior in real-time dynamicsof the return probability-amplitude [11, 22], which doesnot require an order parameter to be defined [23]. Thephenomenon under investigation in our work is the for-

mer case, which we will refer to as a transition betweendynamical phases (TDP) to avoid confusion. The the-oretical study of such transitions have encompassed abroad range of platforms including collective spin models[24–26], non-equilibrium phases of superconductors [27–29], interacting fermions and bosons on the lattice [14–16, 30, 31], and quantum field theories [17, 32]; however,experimental investigations have so far been restricted toself-trapping transitions in bosonic systems [33–37] andthe transverse-field Ising model realized with trapped-ionchains [18].

Here, we report the observation of a transition betweentwo dynamical phases of a quantum degenerate Fermigas. The sample under investigation consists of neutralpotassium atoms (40K) confined in a harmonic opticaltrap and cooled to nanokelvin temperatures. The con-trollable interactions of this closed quantum system en-able a broad search for non-equilibrium phenomena thatarise from the interplay of atomic contact interactions,quantum statistics, and motion. Using collective mag-netization as an order parameter, system dynamics areobserved directly, and compared to theoretical models atvarious levels of approximation.

We understand and analyze our system through a map-ping of the single-particle eigenstates of the harmonictrap onto a lattice in mode space, as depicted in Fig. 1.By tuning the interaction strength to suppress collisionsthat would change the occupancy of the modes, the atomsbecome pinned on the conceptual lattice, enabling thedescription of our system with a spin model [38–40], here

H/~ =∑i

hisZi −

∑i,j

Jij si · sj , (1)

arX

iv:1

806.

1104

4v3

[qu

ant-

ph]

5 M

ar 2

019

2

Dephasing

Exchange

Mode changes

FIG. 1. Simulation of the collective Heisenberg modelwith a local inhomogeneous axial field using weaklyinteracting fermions in a mode space lattice. Each sitein mode space (left) has an occupancy of 0 or 1 and expe-riences a local field hi that causes single spin precession ata rate that depends on the mode (top right). Atoms experi-ence long-range spin-exchange interactions Jij (middle right).Mode-changing collisions would move atoms between sites inmode space (bottom right), but are not included in the spinmodel.

where si = 1/2σXi , σYi , σZi are spin-1/2 operators act-ing on the ith atom, and X, Y , and Z denote orientationsin Bloch space. This is the collective Heisenberg model(CHM), a canonical model for magnetism [41], in whichthe non-local spin-spin couplings Jij compete with an in-homogeneous axial field, hi [38–40]. Similar treatmentsof fermionic systems using a spin model have been em-ployed successfully in optical lattice clocks [38, 42–45]in the microkelvin regime. However, in those experi-ments, undesirable inelastic collisions limited the num-ber, N . 50, and prevented the study of transitions be-tween well defined dynamical phases. The stability oflow-lying hyperfine states and control over interactionsin our experiment allows us to explore many-body dy-namics in macroscopic ultracold samples of N ' 3× 104

alkali atoms at nanokelvin temperatures and test the spinmodel in this new regime.

We find that below a critical interaction strength, themagnetization of an initially polarized gas quickly de-cays, while above this critical point, the magnetizationbecomes long-lived, and is protected by an energy gapagainst inhomogeneous field-induced dephasing. Theseobserved dynamical phases are a manifestation of anemergent property in a many-body dynamical quantumsystem. The transition would be absent for small parti-cle number, since the emergence of a sufficiently stronggap from weak two-body collisions relies upon a collectivenonlinearity. We then implement a many-body echo se-quence, which is a direct test of reversibility. This allowsus to identify the boundaries of the parameter regime

in which the complex far-from-equilibrium dynamics ofinteracting fermions is quantitatively described by theCHM. The successful mapping allows us to implementquantum simulations of the non-equilibrium phases pre-dicted to exist in quenched s-wave superconductors byRichardson-Gaudin models [27–29] but not yet directlyobserved, given the need for ultra-fast probes [46].

Results

The simulation cycle begins with a non-interacting sam-ple fully polarized in the lower spin state |↓〉, which en-sures that no site in the mode lattice is doubly occu-pied. Non-equilibrium dynamics are initiated by a fastradio-frequency (rf) pulse that rotates the collective mag-netization into the XY plane. The time evolution oftransverse magnetization is probed using a Ramsey se-quence: following the initial π/2 pulse, atoms evolve fora variable time t, after which a second π/2 pulse is ap-plied, and the total populations in the |↓〉 and |↑〉 statesare measured with a Stern-Gerlach technique. Shot-to-shot field drifts on the microtesla scale prevent a repro-ducible accumulated phase in the Ramsey sequence. Weestimate the magnitude of the transverse coherence byrepeating the sequence at least 10 times and using amaximum-likelihood estimator that assumes a random-ized interferometric phase (see Methods and Materialsfor further details). This procedure measures the totaltransverse magnetization 2

√(SX)2 + (SY )2/N , where

SX,Y,Z = 〈SX,Y,Z〉 with SX,Y,Z =∑i sX,Y,Zi as collec-

tive spin operators. However, since SZ is a constant ofmotion in our simulation, and set to zero by the first π/2pulse, we can simply interpret the signal as 2S/N , with

S =√∑

p=X,Y,Z (Sp)2 the total magnetization.

The optical confinement creates a potential thatis approximately harmonic, with frequencies ω =ωx, ωy, ωz = 2π × 395, 1140, 950Hz along three spa-tial directions. Spin-dependent curvature in the confine-ment potential produces a further shift ±∆ω in the os-cillator frequency between the |↓〉 and |↑〉 states. Sincethe resultant energy shift depends linearly on the in-dex of the single-particle motional eigenstates, labelledby ni = nxi , nyi , nzi , it constitutes an inhomogeneousaxial field in mode space, hi = 2ni ·∆ω = 2(nxi ∆ωx +nyi∆ωy + nzi∆ωz). The strength of the inhomogeneityis tuned in two ways: using the polarization of one ofthe laser beams forming the optical trap to change ∆ω(see Methods and Materials), and using temperature tochange the average mode index n within the range 20 to30.

Interactions are proportional to the s-wave scatteringlength a of the colliding atoms, which are tuned by amagnetic Feshbach resonance near 20 mT [47]. We factorJij as UJij , where U = 4πa

√mωxωyωz/~ sets an over-

all scale, and Jij is a mode-dependent coupling factorproportional to the density-density overlap of the single-particle eigenmodes of the ith and jth particles (see Sup-

3

plementary Materials). Due to the extended nature ofthe motional wave functions, the Jij are long-ranged,

∼ 1/√|npi − npj | in each direction p = x, y, z. The TDP

is observed in a weak-scattering regime, where atoms re-main frozen in their initial modes, and dynamics involveonly spin degrees of freedom [3, 48, 49]. For these ex-periments, we needed to improve by an order of mag-nitude the accuracy to which the zero-crossing field Bzc

(at which a = 0) was known; data used to determineBzc = 20.907(1) mT are presented in the SupplementaryMaterial.

Figure 2 shows an exploration of the non-equilibriumphase diagram using total magnetization at 100 ms,S(t = 100 ms), as the order parameter. Simulationswere run with scans of mean interaction strength J =∑i,j Jij/N

2 (Fig. 2A,B,C) or scans of axial field inho-

mogeneity h =√∑

i h2i /N − (

∑i hi/N)2 (Fig. 2E,F,G).

Several distinct regions appear: a dynamical ferromagnetwith high persistent S at large positive or negative NJ ,and a paramagnetic phase with low S at smaller |NJ |.

Measurements are compared to a mean-field treatmentof Eq. 1, where the jth atom experiences an effectivemagnetic field, Bj , which depends only on the local field,hj , and the average magnetization of the other atoms inthe ensemble,

Hmf/~ =∑j

sj ·Bj (2)

where Bj = hjZ −∑i 2Jij〈si〉. Here the indices i, j

run over a set of N populated modes drawn from afinite-temperature Fermi-Dirac distribution, and Z is aunit vector. The transition between dynamical phases isa consequence of the opening of an interaction-energygap between the fully polarized manifold and the re-mainder of the Hilbert space, as further discussed be-low. Numerical solutions of the corresponding 3N non-linear Bloch equations (see Supplementary Materials)

show that above some critical interaction strength, Jc(h),the dynamics become gapped and ferromagnetic order(meta)stabilizes, while below Jc(h), the gas partially de-magnetizes. In the ungapped phase, exchange interac-tions are not strong enough to prevent demagnetization,rendering the system prone to dephasing induced by in-homogeneous hi. In contrast to thermodynamic ferro-magnetism, which occurs only for J > 0, the dynamicalferromagnet is a spin-locked state that can be stabilizedby either repulsive or attractive exchange interactions.

The red lines in Fig. 2 show S(t = 100 ms) calcu-lated by this model, using ab-initio determination of J ,a set of field curvatures that match all observations inthis manuscript (see Supplementary Materials), and adecay envelope, e−Γ(a)t, discussed in detail further be-low. The white solid line in Fig. 2D shows the TDPsteady-state (t → ∞) phase boundary which sets Jc(h).

While we are unable to probe the system at infinite timeand thereby reveal the strict steady-state limit we find,as shown by the corroboration in Fig. 2, that t = 100 ms≈ 40 (ωx/2π)−1 is sufficiently long to capture the non-

equilibrium phase diagram as a function of J and h.

To gain understanding of the scaling expected near theTDP one can use a simplified “all-to-all” model, in whichcoupling constants are replaced by their mean value,Jij → J . In this limit, Eq. 2 becomes integrable andmaps to the Bardeen-Cooper-Schrieffer Hamiltonian forfermionic superconductors expressed in terms of Ander-son pseudo-spin [50]. Borrowing the methodology devel-oped for dealing with dynamical phases in superconduc-tors [29, 51, 52], one can obtain the frequency spectrumruling the non-equilibrium dynamics from the roots ofL2(u), where L(u) is the Lax vector of the auxiliary vari-able u (see Supplementary Materials). The roots canbe found using the property that L2(u) is an integral ofmotion of the dynamics, and can be evaluated for con-venience at time t = 0. When the roots compress in theneighborhood of the real axis, the long-time limit of S(t)relaxes to a zero. This corresponds to the normal phase,or “phase I” in the language of superconductors [29]. Onthe other hand, the appearance of a pair of complex con-jugate roots determines the TDP critical point and theemergence of a phase characterized by a non-zero steady-state order parameter, S(∞) > 0, i.e. “phase II”. Whilethe precise scaling of the order parameter near the TDPcan be complex, since it is determined by the spectrum ofthe hi, we find that above the critical point in our systemit can be approximated by the analytic expression

S(∞) ≈√

3α h

2Jeffcot

(√3α h

NJeff

). (3)

This formula is exact (with α = 1 and Jeff = J) forthe case of a one-dimensional system ∆ωy,z = 0 at zerotemperature. To account for non-collective interactions,higher dimensions, and finite temperature, we introducerenormalization parameters α and Jeff = βJ . The crit-ical interaction strength is Jc(h) = 2

√3αh/(βNπ). An

alternate order parameter in the non-equilibrium ferro-magnetic phase is the gap frequency:

Ω = 2|Jeff |S(∞) ≈√

3α h cot

(√3α h

NJeff

). (4)

In the ferromagnetic phase, S(t) exhibits transient oscil-lations at the gap frequency Ω, which slowly damp as itreaches S(∞). The gap frequency goes to zero at Jc(h) ina non-analytic manner. As discussed below, we observeeach of these signatures in the quantum simulation.

The collective nature of these phenomena is empha-sized by an alternative interpretation of the gap. Theinitial π/2 pulse can be said to create a superposition of

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4 6 8 10 12

0.00.20.40.60.81.0

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A

B

C

D E

F

G

FIG. 2. Non-equilibrium phase diagram. (A–C,E–G) show magnetization S at t = 100 ms versus interaction strength J

and inhomogeneity h. Blue circles are experimental data, and red lines are theory calculations scaled by e−Γ(a)t at t = 100 ms(see Fig. 4). These are cuts through a non-equilibrium phase diagram (D), calculated using magnetization at t = 100 as theorder parameter. The dashed white line shows a cut at the inhomogeneity used in Fig. 3. The solid white lines show thesteady-state (t→∞) phase boundary for α/β = 0.27, determined by the critical point of the magnetization in cuts of constant

h. Error bars on data points are statistical; bands in theory correspond to uncertainty in Bzc. All theory calculations exceptfor the solid white line are finite time thermally averaged numerics as described in the main text.

|S = N/2,mS〉 Dicke states, where S and mS are eigen-values of the collective operators S2 =

∑p=X,Y,Z(Sp)2

and SZ respectively. All mS states have the same en-ergy in the rotating frame of the rf pulse. A finite energygap ~Ω inhibits the production of spin waves (generatedby the inhomogeneous hi) and keeps the dynamics withinthe collective Dicke manifold: flipping a single spin wouldreduce S to (N/2− 1), and change the exchange energy,proportional to JS2, by NJ .

Figures 3A-E show the qualitative change in dynami-cal behavior as J crosses Jc for fixed h. Below the TDP,S(t) decays monotonically in time (Fig. 3A), but abovethe transition, S(t) oscillates around a non-zero magneti-zation (Fig. 3C,D,E). All observations can be reproducedby the same theoretical model shown in Fig. 2 (red lines),if Jij are scaled by 0.8 from their ab-initio values, per-haps due to an increased sampling of trap anharmonicitydue to the higher temperature used in this data set toincrease h, or due to a renormalization of coupling con-stants due to resonant mode-changing processes [39, 40].The TDP is seen in three observables (Figs. 3F,G,H):first, by a departure from ungapped dynamics; second,by looking for a jump in S at 100 ms; and third, by afinite value of Ω.

The χ2 measure in Fig. 3F compares both data andcalculations to SJ=0(t), the calculated time evolution

for J = 0. The sharp increase near NJ/2π ≈ 10 Hzin both experiment and theory indicates the qualita-tive deviation of the dynamics from the paramagneticphase. The magnetization at t = 100 ms (in Fig. 3G)also shows an increase near NJ/2π ≈ 10 Hz. Numer-ical solutions of the mean-field dynamics with thermalaveraging at finite time (red band) and without ther-mal averaging in steady state (solid black line) agreewell with the experimental data. The simplified all-to-all model (Eq. 3, dotted black line) agrees only qualita-tively. The gap frequency Ω in Fig. 3H is found froma fitting function that uses a damped sinusoid for latertimes, and SJ=0 for early times (see Supplementary Ma-terials). By fitting the gap parameter to the analyticformula Eq. 4, we extract a nonzero critical interactionstrength NJc/2π = 7.8(1.1) Hz. Using the location ofthe step in χ2 (Fig. 3G), we exclude time sequenceswith |a| < 3a0, where the oscillation frequency diverges.The excellent agreement of all three measures with the-ory based on Eq. 1 confirm that our quantum simulatorprobes the TDP in the collective Heisenberg model.

The significance of the observation is further clarifiedin Figs. 3G,H by comparison to various approximationlevels. Finite-time effects and thermal averaging play aminor role, validating our interpretation of S at suffi-ciently large t as the steady-state order parameter. Inho-mogeneous coupling (Jij 6= J) plays a significant role for

5

Dis

tanc

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m n

on-in

t. dy

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(a.u

.)

F G H

Hold time (ms) Hold time (ms)Hold time (ms)Hold time (ms) Hold time (ms)0 50 1000 50 100 150 200

0.0

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Absolute value of collective interaction strength | |/

Mag

netiz

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n

at10

0m

sM

agne

tizat

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0 5 10 15 20 250 5 10 15 20 250 3 6

0 10 0 20 40 600 20 40 60

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freq

uenc

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30Jc

FIG. 3. Transition between dynamical phases. (A-E) Time-dependent magnetization S(t) for fixed h = 2π × 18.1(1) Hz.As interaction strength increases, data (blue points, with statistical error bars) and theory (red lines, with error bands dueto bias field uncertainty) deviate from non-interacting dynamics (dashed black curve) after some time. (F) Experiment andtheory are compared via χ2 distance to non-interacting dynamics, both significantly deviating for NJ/2π & 10 Hz, or a & 3a0.(G) The interpolated magnetization at 100 ms agrees well with numerical solutions of the mean-field dynamics with thermalaveraging at finite time (red band) and without thermal averaging in steady state (solid black line), but only qualitatively withthe analytic approximation of Eq. 3 (dotted black line). (H) The gap frequency Ω shows good agreement between data andall three levels of theory. Ω is fit to the analytic formula of Eq. 4 to find NJc/2π = 7.8(1.1) Hz, indicated by the green band.Insets in G, H compare the analytic approximation of Eq. 3 and Eq. 4 (dotted black line) with the exact all-to-all solutionfrom the Lax vector approach (orange line) and from the corresponding mean-field numerical solution (green circles). Apart

from insets, theory curves are scaled by e−Γ(a)t to account for beyond-spin-model processes (see Fig. 4). Error bars on datapoints are statistical; the theory uncertainty (red band) is dominated by Bzc, and the width of the green Jc band is ± twostandard deviations.

S, but less so for Ω. Comparisons to the exact Lax vectoranalysis (insets to Fig. 3G,H) show the close similarity ofthe observed TDP to the phase-I-to-phase-II transitionin dynamical superconductors [27–29].

Figure 4 describes a further set of simulations thatprobe the limits of validity in which our system is de-scribed by a spin-lattice model. Figures 4A,B show thatS(t = 100 ms) decreases at sufficiently large a, despitea larger gap. This is accompanied by a breakdown inmicroscopic reversibility, as seen by comparing to a se-quence with a many-body reversal of the spin model

(Fig. 4C), in which HΨ → −HΨ [49, 53]. Signaturesof time reversibility within the window |a| . 20a0 areseen from the nearly J-independent dynamics of S inboth the gapped and ungapped phases. The reversibil-ity of demagnetization in our system in this regime isa significant validation of Eq. 1, since the many-bodyecho sequence does not reverse all terms (e.g., the spinindependent harmonic oscillator term) in the full Hamil-tonian.

Two processes prevent full reversibility in our simula-tion: stray magnetic field gradients and collisional pro-

6

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4002000-200-400

0 40 80-40-80

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Fig. 2D

Fig. 3

Magnetization

Scattering lengthIn

hom

ogen

eity

Collective interaction strength

Bias

fiel

d

0 50 100

Time (ms)0 50 100

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fiel

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0 100-1000.0

1.00-20 20

FIG. 4. From reversibility to the breakdown of CHM simulation. (A) Spin-model mean-field dynamical phasediagram across a wider range than Fig. 2D, showing significant dephasing. (B,D) Magnetization versus interaction strength,with constant bias field. Here, S decreases for very small or very large interactions. The large-|NJ | behavior is capturedby augmenting the spin model dynamics with a phenomenological dephasing term (red curves). (C,E) Magnetization versus

interaction strength, with a many-body echo sequence that reverses the sign of HΨ at t/2. The bias field is held at B1 in thefirst half of the evolution time, yielding scattering length a1 and collective interaction strength NJ1, while in the second half,B2 is chosen such that a2 = −a1, and NJ2 = −NJ1. The initialization, spin-reversal, and readout pulses are performed at Bzc.The magnetization can be recovered for |NJ |/2π . 100 Hz, a region shown in more detail by the Inset of C. A small systematicerror ∆Bzc ∼ 2 µT leads to a shift ∆a ∼ 0.3a0 in azc, a1, a2, included in the theory curves.

cesses. These are quantified by introducing an empiricaldephasing rate Γ(a) = Γ0 + (a/a0)2γ to the transversemagnetization, such that SX,Y (t) → SX,Y (t)e−Γ(a)t.Figures 4B,C compare data with calculations using abest-fit Γ−1

0 = 0.57 s without echo and Γ−10 = 0.25 s with

echo, and γ−1 = 600 s. Here Γ0 accounts for the single-particle mode-changing processes generated by magneticfield gradients, which are enhanced by a spin-reversal[1, 39]. γ parametrizes mode-changing collisions thattake place at a rate that increases quadratically witha, and takes a value anticipated by kinetic theory forour experimental density, temperature, and polarization(see Supplementary Materials). At larger |a|, coherenceis lost, and a quantum picture becomes unnecessary, asshown by the success of a semi-classical picture to de-scribe diffusive transverse demagnetization [3, 54–58].Figure 4A shows an extended phase diagram, where itcan be seen that trapped fermions simulate the CHMonly in a restricted parameter window of many-bodyquantum coherence.

Discussion

In sum, we demonstrate the existence of a TDP in aneutral Fermi gas in a regime of reversible dynamicsnear the zero-crossing of a Feshbach resonance. Weoutline a direct connection to non-equilibrium phasesin the Richardson-Gaudin models for superconductivity,thereby extending experimental observations of TDP’s

beyond prior manifestations in Josephson and Ising-typesystems [18, 33–37]. Moreover, the excellent agreementbetween spin-model calculations and a two-axis explo-ration of the dynamical phase diagram with > 104 spinsprovide experimental evidence of the scaling behaviorand universal character of TDPs.

The collective nature of the dynamics observed herecould protect many-body states in other systems of inter-est for applied quantum technologies. For example, gapprotection would increase the coherence time in opticallattice clocks operated in the quantum degenerate regime[59]. Furthermore, using the effective time-reversal ca-pability demonstrated here, together with technical im-provements to magnetic field stability and homogeneity,our system could provide a fruitful platform to measureout-of-time order correlations and scrambling of quantuminformation [60] or test spectroscopic protocols that usetime reversal to relax the detection resolution required forspectroscopy beyond the standard quantum limit [61].

Methods and Materials

The neutral atomic sample is prepared using laser cool-ing, magnetic and optical trapping, and evaporativeand sympathetic cooling, in an atom-chip apparatus de-scribed previously [62]. The hyperfine states used to en-code spin information are the F = 9/2, mF = −9/2(for |↓〉) and mF = −7/2 (for |↑〉) states. At the be-ginning of a simulation sequence, the fermionic ensemble

7

has a typical temperature of several hundred nanokelvin,with data taken in the range T ∼ 0.3–0.5EF /kB , whereEF = (6N)1/3~ω is the Fermi energy, and ω is the geo-metric mean trap frequency. There is no optical lattice:confinement is produced by a crossed-beam 1064-nm op-tical dipole trap, and the map in Fig. 1 is purely concep-tual.

The magnetic field and its gradients are controlled us-ing a combination of microfabricated wires on the atomchip ≈200 µm from the atoms, and macroscopic coils ex-ternal to the vacuum system. The field is calibratedthrough rf spectroscopy of the |↓〉-to-|↑〉 transition. Dur-ing a typical experimental run, drifts are .1 µT. Mag-netic field gradients, which lead to periodic oscillationof the spin clouds (see Supplementary Materials), aremeasured by displacing the trap centre and repeating rfspectroscopy. In the optimal configuration, gradients are13(1), 12(1), 2(5) µT/m. The differential displacements∆i resulting from these gradients are small compared tothe harmonic oscillator length a2

i = ~/mωi, as dimen-sionless displacements ∆i/ai = 2.1, 0.4, 0.1 × 10−2. Inthis ∆i ai regime, the more general XXZ spin model[40] reduces to the Heisenberg model used here.

The effective axial field hi in the Heisenberg Hamil-tonian is the potential-energy differential between the|↑〉 and |↓〉 states, as sampled by the occupied motionaleigenstates. Since one can always subtract a constantpotential term, dynamics depend only on the inhomo-geneity in hi, quantified through its rms spread h. Thereare both magnetic and optical contributions to hi. Theleading-order magnetic-field contribution is curvature inreal space. Direct spectroscopic data can bound this to. 500 µT/m2, which is compatible with the 200 µT/m2

best-fit curvature in the model. The optical contribu-tion to h is tuned using the polarization of the trappinglight. For far-detuned light, the differential fractionalvector light shift experienced by the atoms is approxi-mately PgF (δFS/3δ), where P is the polarization of thelight, gF is the g-factor of the hyperfine sublevels, δ is thefrequency detuning, and δFS is the fine-structure splittingof the electronic excited state. For a full range of polar-ization P ∈ [−1, 1] of the ODT beam propagating alongthe z-axis, the resulting vector light shift should mod-ify the trap frequency along x by roughly ±0.06 Hz, or±0.015%. This control over h allows the vertical explo-ration in the dynamical phase diagram shown in Fig. 2.

The magnetization 2S/N is determined using repeatedmeasurements and a maximum likelihood estimator, asfollows. Each experimental image measures N↑, N↓ atthe end of a Ramsey sequence. The fraction of spin-↑,f = N↑/N , relates to the collective spin as f = 1/2 +(SY /N) cos θ + (SX/N) sin θ, where θ is the phase lagof the second π/2 pulse. However, the typical evolutiontimes (100 ms) exceed the clock coherence time (1.5 ms),so that the accumulated phase φ randomizes the orienta-tion of the transverse spin, SY = S cosφ,SX = S sinφ.

This yields f = 1/2 + (S/N) cos(θ′), where θ′ = θ − φ isa random phase. To reconstruct the offset O and ampli-tude A of a Ramsey fringe F = O +A cos(θ′) from a setof fractions fi acquired in several experimental runs withthe same conditions, we assume a probability distribution

p(f ;A,O) =

∫ π

0

dθ′√2πσπ

exp

[−(A cos θ′ +O − f)2

2σ2

]which is a convolution of the noise-free probability dis-tribution and Gaussian noise with width σ, calibratedto be σ = 0.01. We construct a log-likelihood func-tion for a set of n fraction measurements, `(A,O; fi) =1n

∑ni=1 log (p(fi;A,O)), from which we numerically com-

pute the maximum-likelihood amplitude A and fringe off-set O, as well as confidence intervals. Where O falls out-side the range 0.50±0.05, we discard the data; otherwise,the peak-to-peak amplitude 2A is taken as the best esti-mate of 2S/N .

∗ These authors contributed equally to this work.† current address: Yale-NUS College, 138527 Singapore‡ current address: Harvard University, Dept. of Physics,

Cambridge, MA 02138, USA§ Corresponding author emails: arey@jilau1.colorado.edu,

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Acknowledgements: We thank A. Koller and C. Lu-ciuk for early work on this project, and V. Gurarie,B. Lev, J. Thompson, M. Foster, and D. Stamper-Kurn for discussions. Funding: This work is sup-ported by NSERC, by the Air Force Office of Scien-tific Research grants FA9550-13-1-0063, FA9550-18-1-0319 and its Multidisciplinary University Research Ini-tiative grant(MURI), by the Army Research Office grantARO W911NF-15-1-0603, the Defense Advanced Re-search Projects Agency (DARPA) and Army Research

10

Office grant W911NF-16-1-0576, the National ScienceFoundation grant PHY1820885, JILA-NSF grant PFC-173400, and the National Institute of Standards andTechnology. Author Contributions: The work was con-ceived by A.R., J.T. and S.T. Experiments were per-formed by S.S., B.O., H.S., K.J., and S.T. Data was an-alyzed by S.S., P.H., and B.O. Theoretical models andsimulation was done by P.H., J.M., J.T., and A.R. Allauthors contributed to manuscript preparation. Com-

peting Interests: The authors declare that they have nocompeting interests. Data and materials availability:

The datasets generated and analyzed during the currentstudy are available from the corresponding authors uponreasonable request.

Supplementary Materials:

Fig. S1. Order parameters predicted by Lax vector analysis

in a 1D system.

Fig. S2. Approximate form of the Lax vector in a 3D system.

Fig. S3. Effective mean field potential.

Fig. S4. Magnetization dynamics for non-interacting particles.

Table S1. Determinations of the 40K Feshbach resonance pa-

rameters.

Fig. S5. Determination of the Feshbach zero-crossing.Fig. S6. Spin-echo amplitude near the Feshbach zero-crossing.Fig. S7. Fits to time series.Fig. S8. Equilibrium scattering rate versus temperature.Fig. S9. Non-equilibrium scattering rate.Refs. 63-72