SCIENCE ADVANCES | RESEARCH ARTICLEPHYSICS

Observation of a transition between dynamical phases
in a quantum degenerate Fermi gas
Scott Smale1*, Peiru He2,3*, Ben A. Olsen1, Kenneth G. Jackson1, Haille Sharum1, Stefan Trotzky1,
Jamir Marino2,3, Ana Maria Rey2,3†, Joseph H. Thywissen1†

INTRODUCTION

The challenge faced in understanding out-of-equilibrium systems isthat the powerful formalism of statistical physics, which has allowed
a classification of quantum phases of matter based on simple principles
such as minimization of free energy, does not apply. A diverse range of
nonequilibrium phenomena has been observed, including synchronization (1–5), self-organization (6–9), quantum chaos (10, 11), Loschmidt
echo singularities (12), and time crystals (13, 14). A proposed organizing principle is that transitions, reminiscent of those found between
thermodynamic ground states, can also be found between dynamicalphases (15–19).
In general terms, a nonequilibrium phase transition is characterized by the existence of a critical point separating phases with distinct
dynamical properties in many-body systems. The analog of thermodynamic order parameters is found in long-time average observables,
which have a nonanalytic dependence on system parameters. In driven
open systems, energy and particle number are not necessarily conserved, and nonequilibrium transitions are typically signaled by different steady states that occur upon varying system parameters suchas pump or loss rates (20–22), independently of initial conditions. In
closed systems, dynamics are often initiated by quenching control
parameters, with qualitatively distinct behaviors observed below, above,or at a critical point (15–18) that can depend on the initial state of thesystem. The label “dynamical phase transition” has been applied not
only to the boundary between two dynamical phases but also to thenonanalytic behavior in real-time dynamics of the return probability
amplitude (12, 23), which does not require an order parameter to be
defined (24). The phenomenon under investigation in our work is theformer case, which we will refer to as a transition between dynamicalphases (TDP) to avoid confusion. The theoretical study of these transitions has encompassed a broad range of platforms including collective

1
Department of Physics and Centre for Quantum Information and Quantum Control, University of Toronto, Ontario M5S 1A7, Canada. 2JILA, National Institute ofStandards and Technology, and Department of Physics, University of Colorado,
Boulder, CO 80309, USA. 3Center for Theory of Quantum Matter, University ofColorado, Boulder, CO 80309, USA.
*These authors contributed equally to this work.
†Corresponding author. Email: arey@jilau1.colorado.edu (A.M.R.); jht@physics.
utoronto.ca (J.H.T.)

Smale et al., Sci. Adv. 2019; 5 : eaax1568

2 August 2019spin models (25–27), nonequilibrium phases of superconductors (28–30),
interacting fermions and bosons on the lattice (15–17, 31, 32), andquantum field theories (18, 33); however, experimental investigations
have so far been restricted to self-trapping transitions in bosonic systems(34–38) and the transverse-field Ising model realized with trapped-ion
chains (19).
Here, we report the observation of a transition between two dynamical phases of a quantum degenerate Fermi gas. The sample under investigation consists of neutral potassium atoms (40K) confined in aharmonic optical trap and cooled to nanokelvin temperatures. The controllable interactions of this closed quantum system enable a broad
search for nonequilibrium phenomena that arise from the interplayof atomic contact interactions, quantum statistics, and motion. Using
collective magnetization as an order parameter, system dynamics areobserved directly and compared to theoretical models at various levelsof approximation.
We understand and analyze our system through a mapping of thesingle-particle eigenstates of the harmonic trap onto a lattice in modespace, as depicted in Fig. 1. By tuning the interaction strength tosuppress collisions that would change the occupancy of the modes,
the atoms become pinned on the conceptual lattice, enabling the description of our system with a spin model (39–41); here
^ =ħ ¼ ∑hi^sZi  ∑Jij^s i ⋅ ^s j
H
i

i;j

ð1Þ

^Yi ; s^Zi g are spin-1/2 operators acting on the iths Xi ; swhere ^s i ¼ 1=2f^
atom and X, Y, and Z denote orientations in Bloch space. This is thecollective Heisenberg model, a canonical model for magnetism (42)in which the nonlocal spin-spin couplings Jij compete with an inhomogeneous axial field, hi (39–41). Similar treatments of fermionicsystems using a spin model have been used successfully in opticallattice clocks (39, 43–46) in the microkelvin regime. However, in
those experiments, undesirable inelastic collisions limited the number, N ≲ 50, and prevented the study of transitions between welldefined dynamical phases. The stability of low-lying hyperfine states
and control over interactions in our experiment allow us to explore
many-body dynamics in macroscopic ultracold samples of N ≃ 3 ×
104 alkali atoms at nanokelvin temperatures and test the spin modelin this new regime.
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A proposed paradigm for out-of-equilibrium quantum systems is that an analog of quantum phase transitions
exists between parameter regimes of qualitatively distinct time-dependent behavior. Here, we present evidenceof such a transition between dynamical phases in a cold-atom quantum simulator of the collective Heisenberg
model. Our simulator encodes spin in the hyperfine states of ultracold fermionic potassium. Atoms are pinned
in a network of single-particle modes, whose spatial extent emulates the long-range interactions of traditional
quantum magnets. We find that below a critical interaction strength, magnetization of an initially polarized
fermionic gas decays quickly, while above the transition point, the magnetization becomes long-lived because ofan energy gap that protects against dephasing by the inhomogeneous axial field. Our quantum simulation reveals
a nonequilibrium transition predicted to exist but not yet directly observed in quenched s-wave superconductors.

Copyright © 2019
The Authors, somerights reserved;
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SCIENCE ADVANCES | RESEARCH ARTICLE
Dephasing〉 with ^S¼ ∑i ^sX;Y;Z
as collective spin operators.
S X;Y;Z ¼ 〈^SiZ
However, since S is a constant of motion in our simulation and set tozero by the first p/2 pulse, we can simply interpret the signal as 2S/N,
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
with S ¼ ∑p¼X;Y;Z ðS p Þ2 as the total magnetization.

Exchange

The optical confinement creates a potential that is approximately
harmonic, with frequencies w = {wx, wy, wz} = 2p × {395,1140,950}
Hz along three spatial directions. Spin-dependent curvature in the confinement potential produces a further shift ±Dw in the oscillator frequency between the ∣↓〉 and ∣↑〉 states. Since the resultant energy
shift depends linearly on theof the single-particle motional ei indexygenstates, labeled by ni ¼ nxi,ni ,nzi g, it constitutes an inhomogeneousyaxial field in mode space, hi ¼ 2ni ˙ Dw ¼ 2ðnxi Dwx þ ni Dwy þ nzi Dwz Þ.
The strength of the inhomogeneity is tuned in two ways: using the polarization of one of the laser beams forming the optical trap to change
Dw (see Materials and Methods) and using temperature to change the within the range of 20 to 30.
average mode index nInteractions are proportional to the s-wave scattering length a of thecolliding atoms, which are tuned by a magnetic Feshbachresonanceﬃnear
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
20 mT (50). We factor Jij as UJ ij, where U ¼ 4pa mwx wy wz =ℏ sets
an overall scale and J ij is a mode-dependent coupling factor proportional to the density-density overlap of the single-particle eigenmodesof the ith and jth particles (see the Supplementary Materials). Becauseof the extended nature of the motional wave functions, the J ij are long
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pp
ranged and e1= ∣ni  nj ∣ in each direction p = x, y, z. The TDP is

Mode changes

We find that below a critical interaction strength, the magnetizationof an initially polarized gas quickly decays, while above this criticalpoint, the magnetization becomes long-lived and is protected by anenergy gap against inhomogeneous field-induced dephasing. These observed dynamical phases are a manifestation of an emergent property ina many-body dynamical quantum system. The transition would be absent for small particle number since the emergence of a sufficiently
strong gap from weak two-body collisions relies upon a collective nonlinearity. We then implement a many-body echo sequence (4, 47, 48),
which is a direct test of reversibility. This allows us to identify the boundaries of the parameter regime in which the complex far-from-equilibrium
dynamics of interacting fermions is quantitatively described by the collective Heisenberg model. The successful mapping allows us to implement
quantum simulations of the nonequilibrium phases predicted to exist in
quenched s-wave superconductors by Richardson-Gaudin models (28–30)
but not yet directly observed, given the need for ultrafast probes (49).

RESULTS

The simulation cycle begins with a noninteracting sample fully polarized in the lower spin state ∣↓〉, which ensures that no site in the modelattice is doubly occupied. Nonequilibrium dynamics are initiated by afast radio-frequency (rf) pulse that rotates the collective magnetization
into the XY plane. The time evolution of transverse magnetization isprobed using a Ramsey sequence: Following the initial p/2 pulse, atoms
evolve for a variable time t, after which a second p/2 pulse is applied, andthe total populations in the ∣↓〉 and ∣↑〉 states are measured with aStern-Gerlach technique. Shot-to-shot field drifts on the microteslascale prevent a reproducible accumulated phase in the Ramsey
sequence. We estimate the magnitude of the transverse coherence
by repeating the sequence at least 10 times and using a maximumlikelihood estimator that assumes a randomized interferometric phase(see Materials and Methods for further details).
This procedure
meaqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ
sures the total transverse magnetization 2
Smale et al., Sci. Adv. 2019; 5 : eaax1568

ðS X Þ2 þ ðS Y Þ2 =N, where

2 August 2019

X;Y;Z

observed in a weak-scattering regime, where atoms remain frozen in
their initial modes and dynamics involve only spin degrees of freedom(3, 4, 51). The precise control of a required an improved determination of the zero-crossing field, Bzc, at which a = 0. As described inMaterials and Methods, we find Bzc = 20.907(1) mT.
Dynamical phase diagram
Figure 2 shows an exploration of the nonequilibrium phase diagram
using total magnetization at 100 ms, S(t = 100 ms), as the orderparameter. Simulations were run with scans of mean interactionscans of axial field instrength J = 〈∑i,j Jij/N2〉T (Fig. 2, A to C) or
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
2~
homogeneity h ¼∑i hi =N  ð∑i hi =NÞ (Fig. 2, E to G), whereT

the indices i and j run over N populated modes. The thermal average,〈⋯〉T, is performed by averaging over different realizations of populatedmodes drawn from a Fermi-Dirac distribution. Several distinct regions
appear: a dynamical ferromagnet with high persistent S at large positiveor negative NJ and a paramagnetic phase with low S at smaller ∣NJ∣.
Measurements are compared to a mean-field treatment of Eq. 1,
where the jth atom experiences an effective magnetic field, Bj , which
depends only on the local field, hj, and the average magnetization of
the other atoms in the ensemble^ mf =ħ ¼ ∑^s j ⋅ Bj
H
j

ð2Þ

where Bj ¼ hj Z  ∑i 2Jij 〈^s i 〉. Here, the indices i and j run over a set of
N populated modes drawn from a finite temperature Fermi-Dirac
distribution, and Z is a unit vector. The TDP is a consequence of the
opening of an interaction energy gap between the fully polarized
manifold and the remainder of the Hilbert space, as further discussed
below. Numerical solutions of the corresponding 3N nonlinear Bloch~equations show that above some critical interaction strength, Jc ðhÞ,2 of 8Downloaded from http://advances.sciencemag.org/ on September 16, 2020

Fig. 1. Simulation of the collective Heisenberg model with a local inhomogeneousaxial field using weakly interacting fermions in a mode space lattice. Eachsite in mode space (left) has an occupancy of 0 or 1 and experiences a local fieldhi that causes single-spin precession at a rate that depends on the mode (topright). Atoms experience long-range spin-exchange interactions Jij (middle
right). Mode-changing collisions would move atoms between sites in modespace (bottom right) but are not included in the spin model.

X;Y;ZSCIENCE ADVANCES | RESEARCH ARTICLE

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~ Blue circles are
Fig. 2. Nonequilibrium phase diagram. (A to C and E to G) Magnetization S at t = 100 ms versus interaction strength J and inhomogeneity h.
experimental data, and red lines are theory calculations scaled by e −G(a)t at t = 100 ms (see Fig. 4). These are cuts through a nonequilibrium phase diagram(D) calculated using magnetization at t = 100 as the order parameter. The dashed white line shows a cut at the inhomogeneity used in Fig. 3. The solid white~ Error bars on datalines show the steady-state (t → ∞) phase boundary for a/b = 0.27 determined by the critical point of the magnetization in cuts of constant h.
points are statistical. Bands in theory correspond to uncertainty in Bzc. All theory calculations except for the solid white line are finite-time thermally averaged
numerics, as described in the main text.

the dynamics become gapped and ferromagnetic order (meta)stabilizes,~ the gas partially demagnetizes. The numerical calwhile below Jc ðhÞ,culations take into account finite temperature effects by averaging
over solutions generated by different sets of populated modes sampledfrom a Fermi-Dirac distribution (see the Supplementary Materials). In
the ungapped phase, exchange interactions are not strong enough toprevent demagnetization, rendering the system prone to dephasing
induced by inhomogeneous hi. In contrast to thermodynamic ferromagnetism, which occurs only for J > 0, the dynamical ferromagnet
is a spin-locked state that can be stabilized by either repulsive or attractive exchange interactions.
The red lines in Fig. 2 show S(t = 100 ms) calculated by this modelusing ab initio determination of J, a set of field curvatures that match allobservations in this manuscript (see the Supplementary Materials), anda decay envelope [e−G(a)t] discussed in detail further below. The whitesolid line in Fig. 2D shows the TDP steady-state (t → ∞ ) phase bound~ While we are unable to probe the system at infinite
ary that sets Jc ðhÞ.
time and thereby reveal the strict steady-state limit, we find, as shownby the corroboration in Fig. 2, that t = 100 ms ≈ 40 (wx/2p)−1 is sufficiently long to capture the nonequilibrium phase diagram as a functionof J and ~h.
Transition between dynamical phases
To gain understanding of the scaling expected near the TDP, one canuse a simplified “all-to-all” model, in which coupling constants are
replaced by their mean value, Jij → J. In this limit, Eq. 2 becomes integrable and maps to a Richardson-Gaudin model, a model used to describe fermionic superconductors when the Bardeen Cooper Schrieffer
Hamiltonian is expressed in terms of Anderson pseudospins (52).
Borrowing the methodology developed for dealing with dynamical phases
in superconductors (30, 53, 54), one can obtain the frequency spectrum
ruling the nonequilibrium dynamics from the roots of L2(u), where L(u)is the Lax vector of the auxiliary variable u (see the Supplementary
Smale et al., Sci. Adv. 2019; 5 : eaax1568

2 August 2019

Materials). The roots can be found using the property that L2(u) is an integral of motion of the dynamics and can be evaluated for convenience attime t = 0. When the roots compress in the neighborhood of the real axis,
the long-time limit ofS(t) relaxes to a zero. This corresponds to the normal
phase or “phase I” in the language of superconductors (30). On the other
hand, the appearance of a pair of complex conjugate roots determines the
TDP critical point and the emergence of a phase characterized by a nonzero steady-state order parameter, S(∞) > 0, i.e., “phase II.” While theprecise scaling of the order parameter near the TDP can be complex, since
it is determined by the spectrum of the hi, we find that above the criticalpoint in our system, it can be approximated by the analytic expression
pﬃﬃﬃ
pﬃﬃﬃ !
~
~
3ah
3ah
cot
Sð∞Þ ≈
2Jeff
NJeff

ð3Þ

This formula is exact (with a = 1 and Jeff = J) for the case of a onedimensional (1D) system Dwy, z = 0 at zero temperature. To account fornoncollective interactions, higher dimensions, and finite temperature, we
introduce renormalization parameters
a and Jeff = bJ. The critical interpﬃﬃﬃ~ ¼ 2 3ah=ðbNpÞ.
~An alternate order parameter
action strength is Jc ðhÞin the nonequilibrium ferromagnetic phase is the gap frequencypﬃﬃﬃ~
W ¼ 2∣Jeff ∣Sð∞Þ ≈ 3a hcot

pﬃﬃﬃ !
~
3ah
NJeff

ð4Þ

In the ferromagnetic phase,S(t) exhibits transient oscillations at the gap
frequency W, which slowly damp as it reaches S(∞). The gap frequency~ in a nonanalytic manner. As discussed below, wegoes to zero at Jc ðhÞobserve each of these signatures in the quantum simulation.
The collective nature of these phenomena is emphasized by analternative interpretation of the gap. The initial p/2 pulse can be said
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−15 −10 −5

SCIENCE ADVANCES | RESEARCH ARTICLE
to create a superposition of ∣S = N/2, ms〉 Dicke states, where S and mS2p
are eigenvalues of the collective operators ^S ¼ ∑p¼X;Y;Z ð^S Þ2 andZ^S , respectively. All mS states have the same energy in the rotating
frame of the rf pulse. A finite energy gap ℏW inhibits the production ofspin waves (generated by the inhomogeneous hi) and keeps the dynamics within the collective Dicke manifold: Flipping a single spinwould reduce S to (N/2 − 1) and change the exchange energy, propor2
tional to J ^S , by NJ.
Figure 3 (A to E) shows the qualitative change in dynamical be~ Below the TDP, S(t) decays monohavior as J crosses Jc for fixed h.
tonically in time (Fig. 3A), but above the transition, S(t) oscillates
around a nonzero magnetization (Fig. 3, C to E). All observations canbe reproduced by the same theoretical model shown in Fig. 2 (redlines) if J ij are scaled by 0.8 from their ab initio values, perhaps because of an increased sampling of trap anharmonicity due to the higher~ or because of a retemperature used in this dataset to increase hnormalization of coupling constants due to resonant mode-changing
processes (55). The TDP is seen in three observables (Fig. 3, F to H):
first, by a departure from ungapped dynamics; second, by looking fora jump in S at 100 ms; and last, by a finite value of W.
The c2 measure in Fig. 3F compares both data and calculations to
S J=0(t), the calculated time evolution for J = 0. The sharp increase
near NJ/2p ≈ 10 Hz in both experiment and theory indicates thequalitative deviation of the dynamics from the paramagnetic phase.
The magnetization at t = 100 ms (in Fig. 3G) also shows an increase
near NJ/2p ≈ 10 Hz. Numerical solutions of the mean-field dynamics
with thermal averaging at finite time (red band) and without thermal
averaging in steady state (solid black line) agree well with the experimental data. The simplified all-to-all model (Eq. 3, dotted black line)
agrees only qualitatively. The gap frequency W in Fig. 3H is found from

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50 100 150 200
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| |/

~ ¼ 2p  18:1ð1Þ Hz. As interaction strength increases, data (blueFig. 3. Transition between dynamical phases. (A to E) Time-dependent magnetization S(t) for fixed hpoints, with statistical error bars) and theory (red lines, with error bands due to bias field uncertainty) deviate from noninteracting dynamics (dashed black curve) aftersome time. (F) Experiment and theory are compared via c2 distance to noninteracting dynamics, both significantly deviating for NJ/2p ≳ 10 Hz or a ≳ 3a0. a.u., arbitrary
units. (G) The interpolated magnetization at 100 ms agrees well with numerical solutions of the mean-field dynamics with thermal averaging at finite time (red band)
and without thermal averaging in steady state (solid black line) but only qualitatively with the analytic approximation of Eq. 3 (dotted black line). (H) The gap frequency
W shows good agreement between data and all three levels of theory. W is fit to the analytic formula of Eq. 4 to find NJc/2p = 7.8(1.1) Hz, indicated by the green band.
Insets in (G) and (H) compare the analytic approximation given by Eqs. 3 and 4 (dotted black line), with the exact all-to-all numerical solution using the Lax vector
approach (orange line) and with the corresponding mean-field numerical solution assuming all-to-all couplings (green circles). Apart from the insets, theory curves are
scaled by e−G(a)t to account for beyond spin model processes (see Fig. 4). Error bars on data points are statistical. The theory uncertainty (red band) is dominated by Bzc,
and the width of the green Jc band is ± two standard deviations.
Smale et al., Sci. Adv. 2019; 5 : eaax1568

2 August 20194 of 8SCIENCE ADVANCES | RESEARCH ARTICLE
by introducing an empirical dephasing rate G(a) = G0 + (a/a0)2g to thetransverse magnetization such that SX, Y(t) → SX, Y(t)e−G(a)t. Figure 4(B and C) compares data with calculations using a best-fit G1
0 ¼ 0:57 s−1= 600 s. Here, G0
without echo and G10 ¼ 0:25 s with echo and gaccounts for the single-particle mode-changing processes generated
by magnetic field gradients, which are enhanced by a spin reversal(1, 40). g parametrizes mode-changing collisions that take place at
a rate that increases quadratically with a and takes a value anticipatedby kinetic theory for our experimental density, temperature, and polarization (see the Supplementary Materials). At larger ∣a∣, coherence islost and a quantum picture becomes unnecessary, as shown by the successof a semiclassical picture to describe diffusive transverse demagnetization(3, 4, 56–60). Figure 4A shows an extended phase diagram, where it canbe seen that trapped fermions simulate the collective Heisenberg model
only in a restricted parameter window of many-body quantum coherence.

Effective time reversal
Figure 4 describes a further set of simulations that probe the limits ofvalidity in which our system is described by a spin-lattice model. Figure 4(A and B) shows that S(t = 100 ms) decreases at sufficiently large a
despite a larger gap. This is accompanied by a breakdown in microscopic reversibility, as seen by comparing to a sequence with a many-body^ Y → H
^ Y (4, 47, 48).
reversal of the spin model (Fig. 4C), in which H
Signatures of time reversibility within the window ∣a∣≲ 20a0 are seenfrom the nearly J-independent dynamics of S in both the gapped andungapped phases. The reversibility of demagnetization in our system inthis regime is a significant validation of Eq. 1 since the many-body echo
sequence does not reverse all terms (e.g., the spin-independent harmonic oscillator term) in the full Hamiltonian.
Two processes prevent full reversibility in our simulation: straymagnetic field gradients and collisional processes. These are quantified

DISCUSSION

Magnetization
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0.0

In summary, we demonstrate the existence of a TDP in a neutral Fermi
gas in a regime of reversible dynamics near the zero crossing of a Feshbachresonance. We outline a direct connection to nonequilibrium phases in
the Richardson-Gaudin models for superconductivity, thereby extending
experimental observations of TDP’s beyond prior manifestations in
Josephson- and Ising-type systems (19, 34–38). Moreover, the excellent
agreement between spin-model calculations and a two-axis explorationof the dynamical phase diagram with >104 spins provide experimental
evidence of the scaling behavior and universal character of TDPs.
The collective nature of the dynamics observed here could protectmany-body states in other systems of interest for applied quantum technologies. For example, gap protection would increase the coherencetime in optical lattice clocks operated in the quantum degenerate regime (61). Furthermore, using the effective time-reversal capability

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400

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Time (ms)Fig. 4. From reversibility to the breakdown of collective Heisenberg model simulation. (A) Spin-model mean-field dynamical phase diagram across a wider range
than Fig. 2D showing significant dephasing. (B and D) Magnetization versus interaction strength, with constant bias field. Here, S decreases for very small or very large
interactions. The large ∣NJ∣ behavior is captured by augmenting the spin model dynamics with a phenomenological dephasing term (red curves). (C and E) Magne^ at t/2. The bias field is held at B1 in the first half of the evolution time,
tization versus interaction strength, with a many-body echo sequence that reverses the sign of HY
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, spinreversal, and readout pulses are performed at Bzc. The magnetization can be recovered for ∣NJ∣/2p ≲ 100 Hz, a region shown in more detail by the inset in (C). A small
systematic error DBzc ~ 2 mT leads to a shift Da ~ 0.3a0 in azc, a1, and a2, included in the theory curves.
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a fitting function that uses a damped sinusoid for later times and SJ=0
for early times (see the Supplementary Materials). By fitting the gap
parameter to the analytic formula in Eq. 4, we extract a nonzero criticalinteraction strength NJc/2p = 7.8(1.1) Hz. Using the location of the stepin c2 (Fig. 3G), we exclude time sequences with∣a∣< 3a0, where theoscillation frequency diverges. The excellent agreement of all three measures with theory based on Eq. 1 confirms that our quantum simulator
probes the TDP in the collective Heisenberg model.
The significance of the observation is further clarified in Fig. 3(G and H) by comparison to various approximation levels. Finite-time
effects and thermal averaging play a minor role, validating our interpretation of S at sufficiently large t as the steady-state order parameter.
Inhomogeneous coupling (Jij ≠ J) plays a significant role for S but lessso for W. Comparisons to the exact Lax vector analysis [insets in Fig. 3(G and H)] show the close similarity of the observed TDP to the phase
I–to–phase II transition in dynamical superconductors (28–30).

SCIENCE ADVANCES | RESEARCH ARTICLE
demonstrated here, together with technical improvements to magnetic field stability and homogeneity, our system could provide a fruitfulplatform to measure out-of-time order correlations and scrambling ofquantum information (48) or test spectroscopic protocols that use time
reversal to relax the detection resolution required for spectroscopy beyond the standard quantum limit (62).

MATERIALS AND METHODS

Smale et al., Sci. Adv. 2019; 5 : eaax1568

2 August 2019



p dq′
ðAcosq′ þ O  f Þ2
pð f ; A; OÞ ¼ ∫0 pﬃﬃﬃﬃﬃ exp
2s2
2psp
which is a convolution of the noise-free probability distribution andGaussian noise with width s, calibrated to be s = 0.01. We constructeda log-likelihood function for a set of n fraction measurements,nℓðA; O; ffi gÞ ¼ n1 ∑i¼1 logðpðfi ; A; OÞÞ, from which we numericallycomputed the maximum-likelihood amplitude A and fringe offset
O, as well as confidence intervals. Where O falls outside the range of
0.50 ± 0.05, we discarded the data; otherwise, the peak-to-peak amplitude 2A was taken as the best estimate of 2S/N.
SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/
content/full/5/8/eaax1568/DC1
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 noninteracting particles.
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. Nonequilibrium scattering rate.
Table S1. Determination of 40K Feshbach resonance parameters.
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The neutral atomic sample was prepared using laser cooling, magnetic
and optical trapping, and evaporative and sympathetic cooling in an
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200 mT/m2 best-fit curvature in the model. The optical contribution
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modify the trap frequency along x by roughly ±0.06 Hz or ±0.015%.

~ allows the vertical exploration in the dynamicalThis control over hphase diagram shown in Fig. 2.
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reconstruct the offset O and amplitude A of a Ramsey fringe F = O +Acos (q′) from a set of fractions fi acquired in several experimental runs
with the same conditions, we assumed a probability distribution

SCIENCE ADVANCES | RESEARCH ARTICLE

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Acknowledgments: We thank A. Koller and C. Luciuk 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 supported by NSERC; the Air Force Office of Scientific Research grants FA9550-13-10063 and FA9550-18-1-0319 and its Multidisciplinary University Research Initiative grant(MURI); the Army Research Office grants W911NF-15-1-0603, W911NF-16-1-0576, andW911NF-19-1-0210; the Defense Advanced Research Projects Agency (DARPA); the NSF grant
PHY1820885; JILA-NSF grant PFC-173400; and the National Institute of Standards andTechnology. Author contributions: The work was conceived by A.M.R., J.H.T., and S.T.
Experiments were performed by S.S., B.A.O., H.S., K.G.J., and S.T. Data were analyzed by S.S.,
P.H., and B.A.O. Theoretical models and simulation were done by P.H., J.M., J.H.T., and A.M.R. All
authors contributed to manuscript preparation. Competing interests: The authors declarethat they have no competing interests. Data and materials availability: All data needed
to evaluate the conclusions in the paper are present in the paper and/or the Supplementary
Materials.The datasets generated and analyzed during the current study are available fromthe corresponding authors upon reasonable request.
Submitted 26 February 2019
Accepted 26 June 2019
Published 2 August 201910.1126/sciadv.aax1568
Citation: S. Smale, P. He, B. A. Olsen, K. G. Jackson, H. Sharum, S. Trotzky, J. Marino, A. M. Rey,J. H. Thywissen, Observation of a transition between dynamical phases in a quantum
degenerate Fermi gas. Sci. Adv. 5, eaax1568 (2019).

Downloaded from http://advances.sciencemag.org/ on September 16, 2020

Smale et al., Sci. Adv. 2019; 5 : eaax1568

2 August 20198 of 8Observation of a transition between dynamical phases in a quantum degenerate Fermi gas
Scott Smale, Peiru He, Ben A. Olsen, Kenneth G. Jackson, Haille Sharum, Stefan Trotzky, Jamir Marino, Ana Maria Rey andJoseph H. Thywissen

Sci Adv 5 (8), eaax1568.
DOI: 10.1126/sciadv.aax1568

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