Absence of heating in a uniform Fermi gas created by periodic driving
Constantine Shkedrov, Meny Menashes, Gal Ness, Anastasiya Vainbaum, Yoav Sagi
AAbsence of heating in a uniform Fermi gas created by periodic driving
Constantine Shkedrov, Meny Menashes, Gal Ness, Anastasiya Vainbaum, and Yoav Sagi ∗ Physics Department and Solid State Institute, Technion - Israel Institute of Technology, Haifa 32000, Israel (Dated: February 19, 2021)Ultracold atoms are a powerful resource for quantum technologies. As such, they are usuallyconfined in an external potential that often depends on the atomic spin, which may lead to inhomo-geneous broadening, phase separation and decoherence. Dynamical decoupling provides an approachto mitigate these effects by applying an external field that induces rapid spin rotations. However, acontinuous periodic driving of a generic interacting many-body system eventually heats it up. Thequestion is whether dynamical decoupling can be applied at intermediate times without altering theunderlying physics. Here we answer this question affirmatively for a strongly interacting degenerateFermi gas held in a flat box-like potential. We counteract most of the gravitational force by applyingan external magnetic field with an appropriate gradient. Since the magnetic force, and consequently,the whole potential, is spin-dependent, we employ rf to induce a rapid spin rotation. The drivingcauses atoms in both spin states to experience the same time-average flat potential, leading to auniform cloud. Most importantly, we find that when the driving frequency is high enough, there isno heating on experimentally relevant timescales, and physical observables are similar to those of astationary gas. In particular, we measure the pair-condensation fraction of a fermionic superfluidat unitarity and the contact parameter in the BEC-BCS crossover. The condensate fraction ex-hibits a non-monotonic dependence on the drive frequency and reaches a value higher than its valuewithout driving. The contact agrees with recent theories and calculations for a uniform stationarygas. Our results establish that a strongly-interacting quantum gas can be dynamically decoupledfrom a spin-dependent potential for long periods of time without modifying its intrinsic many-bodybehavior.
I. INTRODUCTION
Ultracold atoms can hold quantum information withintheir internal spin states for long periods of time [1–3].A common cause of decoherence is spatially inhomoge-neous spin-dependent potentials. As an example, the en-ergy difference between two internal states of opticallytrapped atoms usually vary in position due to differen-tial light shift. In a classical ensemble, each atom can betreated independently with the rest of the ensemble act-ing as a fluctuating bath [4, 5]. These fluctuations leadto decoherence of the qubit stored in this atom. Dynam-ical decoupling (DD) [6, 7], a generalization of the cel-ebrated Hahn echo technique [8], can substantially slowthis relaxation process by the application of multiple spinrotations [9].Dynamical decoupling has been applied successfullyin NMR [10–12], photonic systems [13], trapped ions[14, 15], electron spin in solids [16–19], ultracold atoms[9, 20], and Bose-Einstein condensates (BEC) [21, 22]. Inall cases, the decoupled system is weakly interacting andcan be treated in a mean-field approach. DD was notapplied to a strongly-interacting ensemble with the aimof protecting its many-body nature. The inherent prob-lem is that a generic interacting many-body system thatfollows the eigenstate thermalization hypothesis absorbsenergy and heats up when placed under periodic driving[23–25]. Nonetheless, recent theoretical works suggestthat the absorbed energy may be extremely small for a ∗ Electronic address: [email protected] very long intermediate duration [26–31]. The question iswhether these periods of metastability are long enoughto allow the formation and study of quantum collectivephases [32].In this work we address this question with a stronglyinteracting Fermi gas in the BEC-BCS crossover – aquintessential system that exhibits high- T c fermionic su-perfluidity [33]. In a non uniform external potential theexperimental conditions vary across the cloud. This situ-ation complicates the extraction of useful physical quan-tities, and in some cases, may lead to phase separation[34]. Although in situ measurements [35–38] and spatialselection [39–44] can give access to quasi-homogeneousobservables, it is better to create a uniform gas from theoutset. Indeed, in recent years, uniform Bose [45, 46]and Fermi [47, 48] gases have been created in flat opticaltraps. These traps are formed by several shaped laserbeams that create sharp repelling walls enclosing a darkvolume. The change in the gravitational energy acrossthe trap is substantial compared to the typical energy ofthe gas. Hence, to create a uniform gas, gravity mustbe counteracted. In previous experiments, this was ac-complished by applying a magnetic field with an appro-priate gradient. However, magnetic levitation works onlywhen all particles have approximately the same magneticdipole moment. This is not the case with generic hetero-geneous mixtures. A possible solution is to use opticallevitation [49], but generating such a potential, smoothon a nano-Kelvin scale, is a formidable task.Magnetic levitation can still cancel part of the grav-itational potential, if the magnetic moments of the twospins are not entirely opposite. This is our starting point:a spin-balanced K Fermi gas held in a combination of a a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b (b) total potential rf
Creating a uniform Fermi gas by periodic driv-ing. (a)
The gas, composed of two spin states (marked byred and blue colors and by opposite arrows), is trapped in abox-like optical potential. The two spins have different mag-netic dipole moments. As a result, it is only possible to par-tially counteract the gravitational potential with a magneticfield gradient, B z , set according to Eq.(4). The total externalpotential V ext ,s depends on the spin, s ∈ {↑ , ↓} , and conse-quently, the density distribution of each spin is different andnot uniform (color gradient in the top left figure). (b) Byadding a resonant rf field that drives rapid spin rotations,we create an effective spin-independent potential, given byEq.(3), in which the gas becomes homogeneous. Importantly,the intrinsic many-body behavior of the gas is unchanged bythis driving. flat optical potential and a magnetic field gradient corre-sponding to the average value of the spins (Fig. 1a). Inthis situation, the total potential is spin-dependent andthe density is not uniform. Here we show, theoreticallyand experimentally, that by applying an rf field that in-duces a rapid spin rotation, the time-averaged potentialexperienced by each of the spins is the same (Fig. 1b).By employing this DD procedure, we create a uniformthree-dimensional degenerate spin-balanced gas of Katoms. More importantly, the interaction Hamiltonian isinvariant under this rotation, and therefore the underly-ing many-body behavior is unchanged. We establish thisproperty by measuring the pair-condensate fraction atunitarity while applying a continuous driving. The con-densate fraction (CF) shows an intriguing non-monotonicdependence on the driving frequency. At low frequencies,driving impairs the gas conditions and reduces the CF.As the frequency increases further, the CF recovers andeven surpasses its value without the driving. At highdriving frequencies, we do not detect heating or loss ofatoms during the experiment which can be attributed tothe drive. Finally, we perform rf spectroscopy with a uni-form gas in the BEC-BCS crossover regime, and extractthe homogeneous contact parameter as a function of theinteraction strength.The structure of this paper is as follows. In sec-tion II we analyze theoretically the time-dependent prob-lem including the external driving field. We show that the interaction Hamiltonian is invariant under rf driv-ing. We derive expressions for the time-dependent single-body Hamiltonians, and discuss the high frequency limit,where the two spins experience the same diagonal exter-nal potential. In section III, we describe the experimentalsetup and measurement sequence. The results are pre-sented in section IV. We study with time-dependent insitu imaging the relaxation dynamics following the appli-cation of the driving field. The temperature of the uni-form gas is probed by Raman spectroscopy. The many-body behavior of the uniform driven gas is studied with apair-projection technique and rf spectroscopy. We studyboth the frequency dependence and the long time be-havior of the driven gas. Section V concludes with adiscussion and outlook.
II. THEORY
We consider fermions in two possible spin states, de-noted by ↓ and ↑ , where the energy of the latter islarger by (cid:126) ω . The two particles are placed in an ex-ternal potential and coupled by an rf field with a fre-quency ω rf . The Hamiltonian is a sum of three termsˆ H = ˆ H + ˆ H int + ˆ H rf that account for the single-particlekinetic and potential energy ( ˆ H ), the interaction energy( ˆ H int ) and the coupling to the external rf field ( ˆ H rf ). Inthe frame rotating with the ↑ spin, they are given by[50, 51]ˆ H = (cid:88) s = {↑ , ↓} (cid:90) d r ˆΨ † s ( r ) (cid:18) − (cid:126) ∇ m + V ext ,s ( r ) (cid:19) ˆΨ s ( r )(1a)ˆ H int = (cid:90) (cid:90) d r (cid:48) d r V ( r , r (cid:48) ) ˆΨ †↑ ( r ) ˆΨ †↓ ( r (cid:48) ) ˆΨ ↓ ( r (cid:48) ) ˆΨ ↑ ( r ) (1b)ˆ H rf = (cid:126) (cid:90) d r Ω e iω t (cid:0) e iω rf t + e − iω rf t (cid:1) ˆΨ †↑ ( r ) ˆΨ ↓ ( r ) + h.c. , (1c)where Ω is the Rabi frequency, V ( r , r (cid:48) ) is the two-bodyinteraction potential, V ext ,s ( r ) is the external potentialfor spin s , and ˆΨ s ( r ) are fermionic field operators obey-ing the anti-commutation relation { ˆΨ s ( r ) , ˆΨ † s (cid:48) ( r (cid:48) ) } = δ s (cid:48) s δ ( r − r (cid:48) ), with δ s (cid:48) s and δ ( x ) being the Kronecker andDirac delta functions, respectively.In our experiment, the rf field is resonant with the bareenergy difference ω rf = ω , and ω (cid:29) Ω. We thereforeemploy the rotating wave approximation and considerˆ H rf = (cid:126) (cid:82) d r Ω ˆΨ †↑ ( r ) ˆΨ ↓ ( r ) + h.c. . To clearly see the effectof the external rf field, we eliminate ˆ H rf by performinga unitary transformation, ˆ U = e i (cid:126) ˆ H rf t , into a referenceframe that rotates with the spins. Under this transfor-mation, the field operators become,ˆ U ˆΨ †↑ ( r ) ˆ U † = cos (cid:18) Ω t (cid:19) ˆΨ †↑ ( r ) + i Ω ∗ | Ω | sin (cid:18) Ω t (cid:19) ˆΨ †↓ ( r )(2)ˆ U ˆΨ †↓ ( r ) ˆ U † = i Ω | Ω | sin (cid:18) Ω t (cid:19) ˆΨ †↑ ( r ) + cos (cid:18) Ω t (cid:19) ˆΨ †↓ ( r ) . It is straightforward to verify that the interaction Hamil-tonian of Eq.(1b) is invariant under this transforma-tion, given that the interaction potential is symmetric V ( r , r (cid:48) ) = V ( r (cid:48) , r ). This is the case with contact interac-tions, V ( r , r (cid:48) ) = gδ ( r − r (cid:48) ) ∂∂r r [50], and with long-rangedipole-dipole interactions. As first noted by Zwierlein etal. [52], the invariance of contact interactions under rfrotations is the reason for the absence of a spectroscopicshift in the transition frequency between the spins.Next, we examine how ˆ H transforms under ˆ U . Theterms that do not depend on spin, in particular thekinetic terms, are unchanged by the transformation ofEq.(2). The spin-dependent terms (i.e., external poten-tial), on the other hand, give rise to both diagonal (spin-preserving) and off-diagonal (spin-changing) terms whichdepend on Ω t . Importantly, the time-average of the off-diagonal terms vanishes, while that of the diagonal termsbecomes independent of the spin, (cid:104) ˆ U ˆ H ext ˆ U † (cid:105) τ = (3) (cid:88) s = ↑ , ↓ (cid:90) d r ˆΨ † s ( r ) V ext , ↑ ( r ) + V ext , ↓ ( r )2 ˆΨ s ( r ) , where ˆ H ext and (cid:104)·(cid:105) τ denote the contribution of the ex-ternal potential to ˆ H and the time average over a longduration τ (cid:29) π/ Ω, respectively. We therefore obtainthat, to first order in the Magnus expansion [53], the rfdriving generates an effective Hamiltonian with a spin-independent external potential and all other terms thesame as in the non-driven Hamiltonian. Hence, we ex-pect to get the same many-body behavior as that of astationary system in the spin-averaged external poten-tial.In our experiment, the external potential is given by V trap + mgz + µ s B z z , where the first term is the flat opticalpotential, the second term is the gravitational potential,and the last term describes the interaction of a spin witha magnetic moment µ s with the external magnetic field,which is linear in position and oriented parallel to thegravitational force. To obtain a true flat total potential,the second and third terms are required to cancel eachother, but this is not possible if µ ↑ (cid:54) = µ ↓ (Fig. 1a). How-ever, by virtue of Eq.(3), the addition of the rf field allowsto achieve uniformity for both spins with (Fig. 1b): B z = − mgµ ↑ + µ ↓ . (4) III. EXPERIMENT
Our experiments are performed with a quantum de-generate gas of K atoms, prepared in an incoherent,spin-balanced mixture of the two lowest energy states, |↓(cid:105) = | / , − / (cid:105) and |↑(cid:105) = | / , − / (cid:105) , with the nota-tion | F, m F (cid:105) . The flat trap, V trap , is created by threelaser beams with a wavelength of 532nm [45, 54] (seeFig. 1); a ‘tube’ beam is created by a wide Gaussianbeam (125 µ m waist radius) that has a circular hole at itscenter, created by a digital mirror device [55]. The othertwo ‘end-cap’ beams are created by two highly ellipticalGaussian beams with waist radii of 5 . µ m and 180 µ m.Together, they generate a dark cylindrical volume withan approximate height of 48 µ m and a radius of 32 µ m,defined by the full width at half maximum of the atomicdensity. The cylinder symmetry axis is parallel to thegravitational force.The experimental sequence starts by cooling the gasto quantum degeneracy in a crossed optical dipole trap[56]. To improve the loading efficiency into the flat trap,we have added a second crossing beam to the opticaltrap described in Ref. [56]. This yields a harmonic trapwith trapping frequencies of w r ≈ π × w z ≈ π × N ≈ × atoms at T /T F ≈ .
24 in this trap, where N istotal atom number in both spin states and T F the Fermitemperature.To load the flat trap, the tube beam is ramped to30mW already at the beginning of the evaporation in theharmonic trap. The magnetic field gradient that coun-teracts gravity is ramped to its final value, as given inEq.(4), in 0 . dB z dy ≈ . .
8s before performing a measure-ment. The final typical conditions in the flat trap withthe rf field are N ≈ × atoms with T /T F ≈ . E F ≈ in situ density measurement ofthe gas (see Appendix A). The magnetic field is tunedaround the Feshbach resonance at 202 . a . It is ramped adiabati-cally to its final value in 10ms, 410ms before the end ofthe measurement. The strength of interactions is param-eterized by 1 /k F a , where k F is the Fermi wave-vector. Inthe last 200ms of the experimental sequence, the rf pulse0 50 100 150 200 − . . (a) (b) (c) Pulse duration (ms) O D s l o p e O p t i c a l d e n s i t y FIG. 2.
Relaxation to a uniform density.
The gas is pre-pared in a spin-polarized |↓(cid:105) state without the rf field. Themagnetic field is set according to Eq.(4), overcompensatinggravity for this state by 5 . t = 0, the rf field withΩ / π ≈ . in situ density of gas is recorded from the sideof the cylindrical trap after variable rf pulse duration (insetsshowing 3ms, 17 . g ( t ) = ae − t/τ cos (2 πft + φ ) + b (solid line),and obtain τ = 16(1)ms and f = 33 . ∼ |↓(cid:105) and |↑(cid:105) is a ≈ − a ( a is Bohrradius). is turned on with a typical Rabi frequency of aroundΩ / π ≈ . IV. RESULTS
Relaxation dynamics.—
Prior to turning the rf fieldon, the atomic densities of the two spins are not uni-form (Fig. 2a), because the magnetic field gradient givenby Eq.(4) over (under) compensates gravity for state |↓(cid:105) ( |↑(cid:105) ) by 5 . |↓(cid:105) atoms, and imaging them from the side of the cylinderafter different waiting times (Fig. 2a-c). To quantify thenon-uniformity of the gas, we plot the slope of the opticaldepth (OD) at the center of the trap, normalized by itsmaximal value at t = 0 (main part of Fig. 2). The gas re-laxes to a uniform density with damped oscillations. Theoscillation frequency is roughly given by the time it takesan atom with a Fermi velocity to traverse the trap heightback and forth. The density reaches a steady-state for rfpulse duration longer than 100ms.Since spin-polarized fermions do not interact via s- 0 1 20123 Momentum, | k z | /k F D i s t r i bu t i o n , n ( k z / k F ) harmonicuniform . . T / T F |||| FIG. 3.
The one-dimensional momentum distributionof a uniform periodically-driven Fermi gas.
The distri-bution, measured by Raman spectroscopy [60], is fitted withthe homogeneous Fermi-Dirac distribution given by Eq.(5)(black solid line). For comparison, we also fit a harmonictrap model (red dashed line) with the same average k F . Thehomogeneous model fit is clearly better. The measurementwas performed at a magnetic field of B = 209 . T /T F , as a function of the rfpulse duration. We observe no increase in T /T F up to 500ms,(this also holds for T alone). The Rabi frequency here isΩ / π ≈ . wave scattering, a question that may arise is how therelaxation process actually occurs. While the rf rotationcreates a |↑(cid:105) component on a short time-scale of the in-verse Rabi frequency, the gas remains spin polarized, al-beit with the spin oriented in a different direction. How-ever, small spatial inhomogeneities in the rf and magneticfields lead to spin dephasing, and eventually, throughatomic diffusion, also to decoherence. Therefore, thespin-polarized gas becomes a balanced spin mixture on arelatively short timescale of 10ms [57]. Thanks to the in-variance of the interaction Hamiltonian, the rf resonancefrequency does not change as the gas transforms frombeing non-interacting to strongly-interacting [52]. Momentum distribution of the uniform gas.—
An important issue to consider is heating, which mayoccur during the initial relaxation phase and during thecontinuous operation of the rf pulse. We obtain the tem-perature of the gas by measuring its momentum distribu-tion. Ordinarily, this is done by letting the gas expandballistically either in free space or in a harmonic trap[58]. Due to the relatively large initial size of the cloud,the free expansion requires particularly long expansiontimes, which are not always feasible. Expansion in a har-monic trap, on the other hand, is done for a quarter ofthe trap period, but is sensitive to anharmonicity of thetrap [47, 48, 58, 59].Here, we take a different approach and use Ramanspectroscopy, which has the advantage that it can beapplied to a trapped gas [60, 61]. The technique relieson a linear relation between the two-photon Raman de-tuning and the velocity of the atoms which are trans-ferred from state |↑(cid:105) to the initially unoccupied state | / , − / (cid:105) . In the experiment, the two Raman beamsare pulsed after the application of a 200ms-long DD rfpulse. By scanning the relative frequency between thebeams, we obtain a spectrum that is directly propor-tional to the one-dimensional momentum distribution[60]. A typical result with dynamically-driven uniformFermi gas is shown in Fig. 3. The number of atomsin state | / , − / (cid:105) is measured by selectively captur-ing them in a magneto-optical trap (MOT) and record-ing their fluorescence [56, 60]. To improve the detec-tion, we separated the wavelength of the MOT, whichis close to the D transition, from that of the collectedscattered photons [62]. To this end, we added a dedicatedprobe beam, tuned to the D transition, and filtered therecorded image with an ultra-narrow, 1nm, optical band-pass filter [63]. The intensities of the two Raman beamsare actively stabilized and programmed to follow a 1ms-long Blackman pulse [64]. The one-photon Raman de-tuning is around 46 . transition. Toreduce unwanted single-photon scattering, which consti-tutes most of the background signal, we incorporated atemperature stabilized etalon after the Raman laser tofilter the broadband amplified spontaneous emission.To determine the temperature, we fit the momentumdistribution with a doubly-integrated three-dimensionalhomogeneous Fermi-Dirac model, n ( k z ) = π TT F ln (cid:18) ζe − k z/k FT/TF (cid:19) , (5)where k z is in the direction of the two-photon momen-tum transfer [60], and ζ is the fugacity with the implicitform: Li ( − ζ ) = − √ π (cid:16) TT F (cid:17) − / , with Li n ( z ) being thePolylogarithm function. Notice that due to the double-integration, there is no sharp Fermi surface in this func-tional even at T = 0. The fit (solid black line in Fig. 3)yields a reduced temperature of T /T F = 0 . T = 7 . . k F (dashedred line). As expected, the homogeneous model fits thedata much better than the harmonic one ( χ = 1 . χ = 3 . (cid:29) E F / (cid:126) , the heating rate is too small to be detected. Pair condensation.—
We now turn to probe themany-body properties of a dynamically driven Fermi gas. When a spin-balanced Fermi gas is cooled below the crit-ical temperature, T c , atoms with opposite spins pair andcondense, forming a fermionic superfluid [33, 65–67]. Thevalue of T c depends on the interaction strength. The sur-vival of superfluidity is a stringent test to our dynamicaldecoupling scheme, since this phase is extremely sensitiveto heating and differential forces acting on the spins.In these experiments, we cool the gas below the su-perfluid transition at unitarity (1 /k F a ≈ .
5G (weak interactions) to unitarity.There, it is held for 5ms, during which the atoms pair-upand condense. The magnetic field gradient and rf pulseare present during the last 200ms, long enough to ensureequilibrium. Since during this time the magnetic fieldis changing, we program the rf frequency to track theresonance transition. We quantify the superfluid stateby measuring the condensate fraction, using the pair-projection technique [66, 68]. To this end, the trap isabruptly turned off and at the same time the magneticfield is ramped rapidly (40 µ s) to the BEC side of theresonance (199 . . / π ≈ E F /h ≈ h is Planck’s constant), while the atom numberstarts to increase around Ω / π ≈ . . C o nd e n s a t e f r a c t i o n · . . · Rabi frequency, Ω / (2 π ) (Hz) A t o m nu m b e r FIG. 4.
Condensate fraction and total atom numberat unitarity as a function of the driving frequency.
Atvery low frequencies (yellow shading), the spin oscillation isslow enough such that both observables are the same as with-out the driving, which are marked by horizontal lines withshading representing the uncertainty. At higher frequencies(blue shading), the driving has an adverse effect on the su-perfluid. At even higher frequencies (red shading), the atomnumber returns to its initial value, and the condensate frac-tion reaches an even higher value than for a stationary gas(inset). The conditions are measured after 5ms at the Fes-hbach resonance magnetic field, and the Rabi frequency isvaried by changing the rf pulse power. The observables ex-traction procedure is discussed in Appendix B.
In Fig. 5, we plot the total number of atoms and conden-sate fraction versus the waiting time at unitarity (blackcircles). We employ an rf field with a relatively highRabi frequency (Ω / π ≈ . dndt = − K n − K n − K n , (6)where n is the total atomic density. K = 1 / . − isthe single-body loss rate, determined by the rate of colli-sions with the residual gas in the vacuum chamber, andmeasured independently. K and K are the two- andthree-body loss rate coefficients. Previously, these pa-rameters were measured only with harmonically trappedgases, which complicated the analysis due to the non-linear density dependence in this model. Here, we bene- 10 − − − · Waiting duration (s) A t o m nu m b e r Ω / (2 π ) ≈ . − − − . . C o nd e n s a t e f r a c t i o n FIG. 5.
Time dependence of the atom number andcondensate fraction at unitarity.
Data is taken after dif-ferent waiting durations both with (black circles) and without(red squares) the rf driving. In both cases, the loss has a sim-ilar trend, which is well-fitted by the model in Eq.(6) (solidlines). Inset: The condensate fraction (same marks) is plot-ted together with the weighted average (solid lines), and itsstandard deviation (shades), shows no decrease. fit directly from the uniformity of the gas and from thefact its shape and volume are almost unchanged as theatom number diminishes. Fitting the data taken with therf pulse with both coefficients as free parameters (blacksolid line) yields K = 9(1) × − cm s − and K = 0.This shows that the loss is mainly due to three-bodyrecombination. If we set by hand K = 0, we obtain K = 4 . × − cm s − . Since in the data withoutthe rf pulse the density is not homogeneous, and in fact,differs between the two spin components, we do not useit to extract loss coefficients. Qualitatively, however, itis still fitted well by the model of Eq.(6) (red solid line).Our value for K is 10 times higher than the one mea-sured in a harmonic trap and at a significantly highertemperature in Ref. [70]. We note, however, that theirmaximal value of K , which was measured on the BECside of the resonance, agrees with our measurement atunitarity. The contact parameter of a uniform gas.—
Wenow turn to a measurement of the contact parameter inthe BEC-BCS crossover regime. This parameter is cen-tral to a set of universal thermodynamic and energeticrelations [72–77], many of which have been tested exper-imentally [78–81]. Previous works determined the valueof the contact with harmonically trapped gas at differ-ent temperatures and interaction strengths [78, 82, 83].Local measurements resolved the contact of a quasi-homogeneous sample [41, 43, 44, 84–87]. Until now, thecontact of a truly uniform gas was measured only at uni-tarity [88].We determine the contact from the power-law tail of rfline-shapes taken with the uniform gas [41, 56, 78, 88]. In − k F a ) − C o n t a c t , C ( N k F ) − − − Detuning, ν ( E F /h ) S i g n a l, P ( ν ) FIG. 6.
The contact of a uniform Fermi gas in theBEC-BCS crossover.
The contact is extracted from thetail of rf line-shapes taken at different interaction parameters.As an example, the inset shows the line-shape at ( k F a ) − =0 . T = T c improved by Popov (dash-dotted line) and at T = 0 (solidline) [89], a Luttinger-Ward calculation (dashed line) [90] anda GPF calculation (dotted line) [91, 92]. The FNDMC line[86, 93] is indistinguishable from the GPF line on this scale.Error bars stand for 1 σ confidence interval of the fit. contrast to the condensate fraction experiments, wherethe condensate was formed after ∼ / π ≈ . . |↑(cid:105) → | / , − / (cid:105) transition. The atomnumber in state | / , − / (cid:105) is again detected with flu-orescence imaging [56]. A typical rf lineshape is shownon a logarithmic scale in the inset of Fig. 6. A univer-sal power-law tail over two decades is clearly visible. Toextract the contact, we work in natural Fermi units andnormalize the spectrum to 1 /
2. The tail is then fittedwith C/ (2 / π ν / ) (black line in the inset), where C is the contact parameter in units of N k F . Owing to thehigh sensitivity of our fluorescence detection scheme, wekeep the rf power constant for all detunings, while themaximal transferred fraction is no more than 8 percent.In Fig. 6 we plot the contact of a uniform Fermigas at various interaction strengths in the BEC-BCScrossover. Starting from the BCS side ( a < C BEC = 4 π/k F a [90]. We find that al-ready above 1 /k F a ≈ .
8, our data are very close to C BEC . In contrast, the weak-coupling BCS limit of thecontact, C BCS = 4 ( k F a ) / /k F a = −
1. We compare our data to several theoriesand previous measurements. On the BEC side, there isa pronounced difference between the T = 0 and T = T c predictions [89]. Our data, which were taken slightlybelow T c , agrees with the T = 0 t-matrix calculation.We also find a good agreement with the Gaussian pairfluctuations (GPF) calculation [91, 92] and fixed-nodediffusion Monte Carlo simulation (FNDMC) [86, 93], es-pecially in the BEC region. A Luttinger-Ward calcu-lation [90] is slightly below our data on the BEC side.Close to unitarity (1 /k F a = − . C = 2 . C = 3 . C = 2 . C =3 . C = 3 . C = 3 . in situ thermody-namic measurements. Similar data taken in the BEC-BCS crossover, albeit with a quasi-homogeneous gas andabove T c , is in agreement with ours, to within the exper-imental accuracy [43]. V. DISCUSSION
In this work we have demonstrated that dynamical de-coupling can eliminate the effect of spin-dependent po-tentials without affecting the intrinsic many-body behav-ior. Our experiments are done with a spin-flip frequencythat is much higher than all other relevant experimen-tal scales. In this regime, we have found no detectableheating during the experiment due to the periodic driv-ing. Measurements of the condensate fraction and con-tact parameter show that the gas behaves the same asa stationary uniform Fermi gas. Our dynamical levita-tion scheme can be used to generate a uniform density ofother spin mixtures.The full Hamiltonian of Eq.(1) depends explicitly ontime and therefore does not conserve energy. In contrastto many-body localized systems [94–97], our gas is er-godic and thus it is not protected from heating [23–25].The fact we do not observe heating during the experimentmay lie in the intricate dynamical behavior of our systemon intermediate timescales. Following a quench, interact-ing many-body systems tend to rapidly attain a steadystate described by a generalized Gibbs ensemble, a phe-nomenon called prethermalization [98–102]. Prethermal-ization may also occur for periodically driven many-bodysystems [27, 28, 103, 104]. Moreover, Floquet prethermalstates absorb energy very slowly, and therefore are pre-dicted to persist for surprisingly long durations that scaleexponentially with the drive frequency [26–30]. It is plau-sible that following the application of the rf driving in ourexperiment, the ensemble quickly relaxes to a prethermalstate and absorbs energy so inefficiently that in practiceit does not heat. It is important to note, however, thatprevious theoretical works considered only discrete mod-els with bounded energy per site. Our experiment, onthe other hand, is done with an unbounded continuoussystem. It is still unknown if the rate of energy absorp-tion under periodic driving in our case is bounded andhow this bounds scales with the driving frequency. Itwill be interesting to investigate these questions and ex-plore the thermodynamic properties of the driven steadystate, in future experiments. Furthermore, driving canbe done with more sophisticated DD sequences in orderto generate specific local and global symmetries [31].
ACKNOWLEDGMENTS
We thank Amir Stern, Ari Turner, Netanel Lindner,Keiji Saito and David Huse for helpful comments. Thisresearch was supported by the Israel Science Founda-tion (ISF), grants No. 1779/19 and No. 218/19, and bythe United States - Israel Binational Science Foundation(BSF), grant No. 2018264.
Appendix A: Determination of the Fermi energy
We developed a systematic approach to calibrate theeffective volume occupied by the gas in the flat trap andextract the Fermi energy. To this end, we simulate thedensity in the box trap using a model potential, and fit itto the in situ integrated density, measured by absorptionimaging. For this calibration, we create a spin-polarizedgas at the same conditions as in the experiments pre-sented in this paper. This is done by first preparing thegas in the flat trap in a spin-balanced configuration asdescribed in the main text. Then we apply an adiabaticrapid passage selectively from state |↑(cid:105) that drives theatoms from this state to a final | / , +9 / (cid:105) state, leavingstate |↓(cid:105) untouched. The force created by the magneticgradient, initially working opposite to the gravity for the |↑(cid:105) state, flips its sign due to the change in the magneticnumber m F and starts working in the direction of gravity,ripping the atoms from the flat trap through the lowercap wall. This procedure removes all of the atoms thatwere initially in state |↑(cid:105) while loosing less than 10% fromstate |↓(cid:105) . The magnetic field gradient is set to perfectlycancel gravity for state |↓(cid:105) , making the density homoge-neous.The next step is taking in situ absorption images of thespin-polarized gas (see Fig. 7). The OD of the gas is toohigh to image directly. To reduce the OD for imaging, weapply a sequence of two rf pulses. The first pulse transfers ≈
90% of the atoms from state |↓(cid:105) to state |↑(cid:105) . The sec-ond pulse transfers all of the atoms from state |↑(cid:105) to state | / , − / (cid:105) , which is detuned by 92MHz from the opticaltransition. The last pulse ensures that no artifacts areintroduced to the imaging due to large atom number inoff-resonant states. The two pulses are completed withinless than 70 µ s, ensuring the density is unchanged duringthis procedure.The spin-polarized gas is essentially non-interactingand can be described by a Fermi-Dirac distribution. To µ m Density( µ m − ) Integrateddensity( µ m − )FIG. 7. Determination of the Fermi energy.
In situ atomic density measured by absorption imaging from the side(the cylinder symmetry axis is vertical). Top and right panelsshow the integrals along the corresponding sides of the trap(circles) and the fit which we use to calibrate the flat-potentialmodel (solid line). Note that only the vertical axis (rightpanel) should exhibit the characteristic flat distribution, asindeed is visible. The presented image is an average of 10experimental repetitions. fit the two-dimensional integrated density image, we cal-culate the density in a local density approximation [105]: n ( r, z ) = − λ − dB Li / [ − exp ( β [ µ − U ( r, z )])] , (A1)where λ dB is the de Broglie wavelength, β = 1 / ( k B T )with k B being the Boltzmann constant, U ( r, z ) is thetrapping potential and µ is the chemical potential, whichis set by the total number of atoms. The model potentialof the tube beam is parametrized by a power law func-tion, while the potential of the cap beams is taken as aGaussian function: U ( r, z ) = U r ( r/σ r ) p (A2)+ U z (cid:32) exp (cid:34) − z − z ) σ z (cid:35) + exp (cid:34) − z + z ) σ z (cid:35)(cid:33) . Here r is the radial coordinate relative to the symme-try axis of the tube (denoted by z ), and U r is the po-tential barrier of the cylindrical wall. The tube radius σ r = 32 µ m and the power-law exponent p = 13 . U z is the potential bar-rier of the cap beams, σ z = 7 . µ m is their waist radiiin the z direction, also measured directly. z = 24 µ mis half the separation between the two cap beams, mea-sured by imprinting the caps profile on a dilute expandedcloud of atoms [62]. The temperature, T , is found self-consistently together with momentum distribution mea-surements. The only free parameters in the fit are U r and U z . An example of a typical calibration is shown in Fig.7). Once the model potential is calibrated, we calculatethe Fermi energy using the local density approximation. µ m D e n s i t y ( a . u . ) Ω ≈ . ≈ µ m) A z i m u t h a ll y - a v e r ag e d d e n s i t y ( a . u . ) Ω ≈ ≈ µ m)FIG. 8. Extraction of the condensate fraction.
Upperpanels are absorption images of high (left) and low (right) con-densate fraction averaged over 10 experimental repetitions.Lower panels present the corresponding azimuthally-averagedsignals (blue lines). We fit a Gaussian (red line) to the thermalwings at radii > R mask (dashed black line). The condensatefraction is defined by the integrated signal above the Gaus-sian (yellow shading) over the total integrated signal (pinkplus yellow shadings). To make a fair comparison, each of thetwo distributions is normalized by its total number of atoms,61 . × and 68 . × in the right and left examples, re-spectively. Appendix B: Extracting the condensate fraction
To separate between the thermal wings and the cen-tral peak, we image the cloud after a relatively longexpansion. As a result, the absorption signal is weak.To improve the signal-to-noise ratio, we employ a deep- learning approach to filter out the background noise inthe images [106]. We have verified that this noise removalprocedure does not change significantly the reported CFvalues and only reduces the uncertainty. The recordeddensity can be roughly classified into three populations(see Fig. 8): unpaired atoms and dissociated pairs thathad either non-zero or zero center-of-mass momentum.The latter are the condensed pairs which constitute thecentral peak of the image. Each of the three populationsexpands differently, according to their respective momen-tum distribution. The total atom number is extracted bya direct integration of the OD image, and the error barsindicate statistical standard error only.Following the time of flight, the non-condensed partof the gas is characterized by a wider expansion with re-spect to the trap dimensions, while the condensed partspreads only slightly beyond its original size, set by thetube beam diameter [47]. To separate between the con-densed and non-condensed parts, we mask out the centralregion of the image (dashed line in Fig. 8) and fit onlythe tail of the azimuthally-averaged signal with a Gaus-sian. We extract the condensed population from the sig-nal that lies above the Gaussian fit (yellow shading inFig. 8). The mask radius, R mask , should be large enoughto leave only the thermal wings for fitting. When weanalyze data taken with no condensate, we observe thatthe width of the fitted Gaussian is almost independent ofthe mask radius up to around R mask ≈ µ m. For largerradii, the signal in the remaining thermal wings is tooweak, and the fit exhibits a systematic deviation. There-fore, we set the mask radius to R mask = 75 µ m. The CFerror bars in the figures indicate statistical error over 10repetitions. Finally, we note that the CF values we ob-tain are close to those reported in Ref. [88], taking intoaccount our measured reduced temperature. [1] U. Schnorrberger, J. D. Thompson, S. Trotzky, R. Pu-gatch, N. Davidson, S. Kuhr, and I. Bloch, Electromag-netically induced transparency and light storage in anatomic mott insulator, Phys. Rev. Lett. , 033003(2009).[2] Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C.Hung, C.-H. Lee, Y.-H. Chen, Y.-F. Chen, I. A. Yu,and Y.-C. Chen, Highly efficient coherent optical mem-ory based on electromagnetically induced transparency,Phys. Rev. Lett. , 183602 (2018).[3] Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao,S. Du, H. Yan, and S.-L. Zhu, Efficient quantum mem-ory for single-photon polarization qubits, Nat. Photon-ics , 346 (2019).[4] Y. Sagi, R. Pugatch, I. Almog, and N. Davidson, Spec-trum of two-level systems with discrete frequency fluc-tuations, Phys. Rev. Lett. , 253003 (2010).[5] Y. Sagi, I. Almog, and N. Davidson, Universal Scaling ofCollisional Spectral Narrowing in an Ensemble of Cold Atoms, Phys. Rev. Lett. , 093001 (2010).[6] L. Viola and S. Lloyd, Dynamical suppression of deco-herence in two-state quantum systems, Phys. Rev. A , 2733 (1998).[7] L. Viola, E. Knill, and S. Lloyd, Dynamical decouplingof open quantum systems, Phys. Rev. Lett. , 2417(1999).[8] E. L. Hahn, Spin echoes, Phys. Rev. , 580 (1950).[9] Y. Sagi, I. Almog, and N. Davidson, Process tomog-raphy of dynamical decoupling in a dense cold atomicensemble, Phys. Rev. Lett. , 053201 (2010).[10] H. Y. Carr and E. M. Purcell, Effects of Diffusion onFree Precession in Nuclear Magnetic Resonance Exper-iments, Phys. Rev. , 630 (1954).[11] S. Meiboom and D. Gill, Modified spin-echo method formeasuring nuclear relaxation times, Rev. Sci. Instrum. , 688 (1958).[12] U. Haeberlen, High resolution NMR in solids : selectiveaveraging (Academic Press, New York, 1976). [13] S. Damodarakurup, M. Lucamarini, G. D. Giuseppe,D. Vitali, and P. Tombesi, Experimental inhibition ofdecoherence on flying qubits via “bang-bang” control,Phys. Rev. Lett. , 040502 (2009).[14] M. J. Biercuk, H. Uys, A. P. VanDevender, N. Shiga,W. M. Itano, and J. J. Bollinger, Optimized dynamicaldecoupling in a model quantum memory, Nature ,996 (2009).[15] S. Kotler, N. Akerman, Y. Glickman, and R. Ozeri, Non-linear single-spin spectrum analyzer, Phys. Rev. Lett. , 110503 (2013).[16] J. Du, X. Rong, N. Zhao, Y. Wang, J. Yang, and R. B.Liu, Preserving electron spin coherence in solids by op-timal dynamical decoupling, Nature , 1265 (2009).[17] G. de Lange, Z. H. Wang, D. Riste, V. V. Dobrovitski,and R. Hanson, Universal dynamical decoupling of asingle solid-state spin from a spin bath, Science , 60(2010).[18] B. Naydenov, F. Dolde, L. T. Hall, C. Shin, H. Fed-der, L. C. L. Hollenberg, F. Jelezko, and J. Wrachtrup,Dynamical decoupling of a single-electron spin at roomtemperature, Phys. Rev. B , 081201 (2011).[19] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, andR. L. Walsworth, Solid-state electronic spin coherencetime approaching one second, Nat. Commun. , 1743(2013).[20] I. Almog, Y. Sagi, G. Gordon, G. Bensky, G. Kur-izki, and N. Davidson, Direct measurement of the sys-tem–environment coupling as a tool for understandingdecoherence and dynamical decoupling, J. Phys. B: At.,Mol. Opt. Phys. , 154006 (2011).[21] D. Trypogeorgos, A. Vald´es-Curiel, N. Lundblad, andI. B. Spielman, Synthetic clock transitions via contin-uous dynamical decoupling, Phys. Rev. A , 013407(2018).[22] H. Edri, B. Raz, G. Fleurov, R. Ozeri, and N. David-son, Observation of nonlinear spin dynamics andsqueezing in a bec using dynamic decoupling (2020),arXiv:2003.13101.[23] A. Lazarides, A. Das, and R. Moessner, Equilibriumstates of generic quantum systems subject to periodicdriving, Phys. Rev. E , 012110 (2014).[24] L. D’Alessio and M. Rigol, Long-time behavior of iso-lated periodically driven interacting lattice systems,Phys. Rev. X , 041048 (2014).[25] P. Ponte, A. Chandran, Z. Papi´c, and D. A. Abanin, Pe-riodically driven ergodic and many-body localized quan-tum systems, Ann. Phys. , 196 (2015).[26] T. Mori, T. Kuwahara, and K. Saito, Rigorous Boundon Energy Absorption and Generic Relaxation in Pe-riodically Driven Quantum Systems, Phys. Rev. Lett. , 120401 (2016).[27] T. Kuwahara, T. Mori, and K. Saito, Floquet–magnustheory and generic transient dynamics in periodicallydriven many-body quantum systems, Ann. Phys. ,96 (2016).[28] D. A. Abanin, W. D. Roeck, W. W. Ho, and F. Hu-veneers, Effective hamiltonians, prethermalization, andslow energy absorption in periodically driven many-body systems, Phys. Rev. B , 014112 (2017).[29] D. V. Else, B. Bauer, and C. Nayak, Prethermal Phasesof Matter Protected by Time-Translation Symmetry,Phys. Rev. X , 011026 (2017).[30] F. Machado, G. D. Kahanamoku-Meyer, D. V. Else, C. Nayak, and N. Y. Yao, Exponentially slow heating inshort and long-range interacting floquet systems, Phys.Rev. Research , 033202 (2019).[31] K. Agarwal and I. Martin, Dynamical enhancement ofsymmetries in many-body systems, Phys. Rev. Lett. , 080602 (2020).[32] I. Bloch, J. Dalibard, and W. Zwerger, Many-bodyphysics with ultracold gases, Rev. Mod. Phys. , 885(2008).[33] W. Zwerger, ed., The BCS-BEC Crossover and the Uni-tary Fermi Gas (Springer Berlin Heidelberg, 2012).[34] Y. Shin, M. W. Zwierlein, C. H. Schunck, A. Schirotzek,and W. Ketterle, Observation of phase separation in astrongly interacting imbalanced fermi gas, Phys. Rev.Lett. , 030401 (2006).[35] A. Schirotzek, Y. il Shin, C. H. Schunck, and W. Ket-terle, Determination of the superfluid gap in atomicfermi gases by quasiparticle spectroscopy, Phys. Rev.Lett. , 140403 (2008).[36] S. Nascimb`ene, N. Navon, K. J. Jiang, F. Chevy, andC. Salomon, Exploring the thermodynamics of a univer-sal fermi gas, Nature , 1057 (2010).[37] M. Horikoshi, S. Nakajima, M. Ueda, andT. Mukaiyama, Measurement of Universal Ther-modynamic Functions for a Unitary Fermi Gas, Science , 442 (2010).[38] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W.Zwierlein, Revealing the superfluid lambda transitionin the universal thermodynamics of a unitary fermi gas,Science , 563 (2012).[39] D. E. Miller, J. K. Chin, C. A. Stan, Y. Liu, W. Seti-awan, C. Sanner, and W. Ketterle, Critical velocity forsuperfluid flow across the BEC-BCS crossover, Phys.Rev. Lett. , 070402 (2007).[40] T. E. Drake, Y. Sagi, R. Paudel, J. T. Stewart, J. P.Gaebler, and D. S. Jin, Direct observation of the fermisurface in an ultracold atomic gas, Phys. Rev. A ,031601 (2012).[41] Y. Sagi, T. E. Drake, R. Paudel, and D. S. Jin, Mea-surement of the homogeneous contact of a unitary fermigas, Phys. Rev. Lett. , 220402 (2012).[42] Y. Sagi, T. E. Drake, R. Paudel, R. Chapurin, and D. S.Jin, Probing local quantities in a strongly interactingfermi gas, J. Phys. Conf. Ser. , 012010 (2013).[43] Y. Sagi, T. E. Drake, R. Paudel, R. Chapurin, andD. S. Jin, Breakdown of the fermi liquid descriptionfor strongly interacting fermions, Phys. Rev. Lett. ,075301 (2015).[44] C. Carcy, S. Hoinka, M. G. Lingham, P. Dyke, C. C. N.Kuhn, H. Hu, and C. J. Vale, Contact and Sum Rulesin a Near-Uniform Fermi Gas at Unitarity, Phys. Rev.Lett. , 203401 (2019).[45] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P.Smith, and Z. Hadzibabic, Bose-einstein condensationof atoms in a uniform potential, Phys. Rev. Lett. ,200406 (2013).[46] L. Chomaz, L. Corman, T. Bienaim´e, R. Desbuquois,C. Weitenberg, S. Nascimb`ene, J. Beugnon, and J. Dal-ibard, Emergence of coherence via transverse condensa-tion in a uniform quasi-two-dimensional bose gas, Nat.Commun. , 6162 (2015).[47] B. Mukherjee, Z. Yan, P. B. Patel, Z. Hadzibabic,T. Yefsah, J. Struck, and M. W. Zwierlein, Homo-geneous Atomic Fermi Gases, Phys. Rev. Lett. , , 060402 (2018).[49] K. Shibata, H. Ikeda, R. Suzuki, and T. Hirano, Com-pensation of gravity on cold atoms by a linear opticalpotential, Phys. Rev. Research , 013068 (2020).[50] S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory ofultracold atomic fermi gases, Rev. Mod. Phys. , 1215(2008).[51] P. Torma, Spectroscopies — theory, in Cold Atoms (Im-perial College Press, 2014) pp. 199–250.[52] M. W. Zwierlein, Z. Hadzibabic, S. Gupta, and W. Ket-terle, Spectroscopic insensitivity to cold collisions ina two-state mixture of fermions, Phys. Rev. Lett. ,250404 (2003).[53] S. Blanes, F. Casas, J. A. Oteo, and J. Ros, The magnusexpansion and some of its applications, Phys. Rep. ,151 (2009).[54] J. L. Ville, T. Bienaim´e, R. Saint-Jalm, L. Corman,M. Aidelsburger, L. Chomaz, K. Kleinlein, D. Perconte,S. Nascimb`ene, J. Dalibard, and J. Beugnon, Loadingand compression of a single two-dimensional Bose gas inan optical accordion, Phys. Rev. A , 013632 (2017).[55] K. Hueck, A. Mazurenko, N. Luick, T. Lompe, andH. Moritz, Note: Suppression of kHz-frequency switch-ing noise in digital micro-mirror devices, Rev. Sci. In-strum. , 016103 (2017).[56] C. Shkedrov, Y. Florshaim, G. Ness, A. Gandman, andY. Sagi, High-sensitivity rf spectroscopy of a stronglyinteracting fermi gas, Phys. Rev. Lett. , 093402(2018).[57] S. Gupta, Radio-frequency spectroscopy of ultracoldfermions, Science , 1723 (2003).[58] S. Tung, G. Lamporesi, D. Lobser, L. Xia, and E. A.Cornell, Observation of the Presuperfluid Regime ina Two-Dimensional Bose Gas, Phys. Rev. Lett. ,230408 (2010).[59] P. A. Murthy, D. Kedar, T. Lompe, M. Neidig, M. G.Ries, A. N. Wenz, G. Z¨urn, and S. Jochim, Matter-wave Fourier optics with a strongly interacting two-dimensional Fermi gas, Phys. Rev. A , 043611 (2014).[60] C. Shkedrov, G. Ness, Y. Florshaim, and Y. Sagi, Insitu momentum-distribution measurement of a quantumdegenerate Fermi gas using Raman spectroscopy, Phys.Rev. A , 013609 (2020).[61] G. Ness, C. Shkedrov, Y. Florshaim, O. K. Diessel,J. von Milczewski, R. Schmidt, and Y. Sagi, Observationof a smooth polaron-molecule transition in a degeneratefermi gas, Phys. Rev. X , 041019 (2020).[62] C. Shkedrov, High-sensitivity rf and Raman spec-troscopy of a quantum degenerate Fermi gas , Ph.d. the-sis, Technion - Israel Institue of Technology (2020).[63] A similar approach was recently reported by MartinSchlederer et al. , Single atom counting in a two-colormagneto-optical trap (2020), arXiv:2011.10081.[64] R. B. Blackman,
The measurement of power spectrafrom the point of view of communications engineering (Dover Publications, New York, 1959).[65] M. Greiner, C. A. Regal, and D. S. Jin, Emergence ofa molecular bose–einstein condensate from a fermi gas,Nature , 537 (2003).[66] C. A. Regal, M. Greiner, and D. S. Jin, Observationof Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett. , 4 (2004).[67] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.Schunck, and W. Ketterle, Vortices and superfluidityin a strongly interacting fermi gas, Nature , 1047(2005).[68] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F.Raupach, A. J. Kerman, and W. Ketterle, Condensationof Pairs of Fermionic Atoms near a Feshbach Resonance,Phys. Rev. Lett. , 120403 (2004).[69] J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E.Wieman, Magnetic field dependence of ultracold inelas-tic collisions near a feshbach resonance, Phys. Rev. Lett. , 728 (2000).[70] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin,Tuning p -Wave Interactions in an Ultracold Fermi Gasof Atoms, Phys. Rev. Lett. , 053201 (2003).[71] X. Du, Y. Zhang, and J. E. Thomas, Inelastic collisionsof a fermi gas in the BEC-BCS crossover, Phys. Rev.Lett. , 250402 (2009).[72] S. Tan, Energetics of a strongly correlated fermi gas,Ann. Phys. , 2952 (2008).[73] S. Tan, Generalized virial theorem and pressure relationfor a strongly correlated fermi gas, Ann. Phys. , 2987(2008).[74] S. Tan, Large momentum part of a strongly correlatedfermi gas, Ann. Phys. , 2971 (2008).[75] E. Braaten, D. Kang, and L. Platter, Exact relationsfor a strongly-interacting fermi gas from the opera-tore product expansion, Phys. Rev. Lett. , 205301(2008).[76] S. Zhang and A. J. Leggett, Universal properties of theultracold fermi gas, Phys. Rev. A , 023601 (2009).[77] E. Braaten, Universal relations for fermions with largescattering length, in The BCS-BEC Crossover and theUnitary Fermi Gas , Lecture Notes in Physics, Vol. 836,edited by W. Zwerger (Springer Berlin / Heidelberg,2012) pp. 193–231.[78] J. T. Stewart, J. P. Gaebler, T. E. Drake, and D. S. Jin,Verification of universal relations in a strongly interact-ing fermi gas, Phys. Rev. Lett. , 235301 (2010).[79] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D.Drummond, P. Hannaford, and C. J. Vale, Universalbehavior of pair correlations in a strongly interactingfermi gas, Phys. Rev. Lett. , 070402 (2010).[80] G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W.Jack, and R. G. Hulet, Molecular Probe of Pairing inthe BEC-BCS Crossover, Phys. Rev. Lett. , 020404(2005).[81] F. Werner, L. Tarruell, and Y. Castin, Number of closed-channel molecules in the bec-bcs crossover, Eur. Phys.J. B , 401 (2009).[82] E. D. Kuhnle, S. Hoinka, P. Dyke, H. Hu, P. Hannaford,and C. J. Vale, Temperature Dependence of the Univer-sal Contact Parameter in a Unitary Fermi Gas, Phys.Rev. Lett. , 170402 (2011).[83] M. G. Lingham, K. Fenech, T. Peppler, S. Hoinka,P. Dyke, P. Hannaford, and C. J. Vale, Bragg spec-troscopy of strongly interacting Fermi gases, J. Mod.Opt. , 1783 (2016).[84] N. Navon, S. Nascimb`ene, F. Chevy, and C. Salomon,The equation of state of a low-temperature fermi gaswith tunable interactions, Science , 729 (2010).[85] S. Hoinka, M. Lingham, K. Fenech, H. Hu, C. J. Vale,J. E. Drut, and S. Gandolfi, Precise Determination of the Structure Factor and Contact in a Unitary FermiGas, Phys. Rev. Lett. , 055305 (2013).[86] M. Horikoshi, M. Koashi, H. Tajima, Y. Ohashi, andM. Kuwata-Gonokami, Ground-State ThermodynamicQuantities of Homogeneous Spin-1 / ,041004 (2017).[87] S. Laurent, M. Pierce, M. Delehaye, T. Yefsah,F. Chevy, and C. Salomon, Connecting Few-Body In-elastic Decay to Quantum Correlations in a Many-BodySystem: A Weakly Coupled Impurity in a ResonantFermi Gas, Phys. Rev. Lett. , 103403 (2017).[88] B. Mukherjee, P. B. Patel, Z. Yan, R. J. Fletcher,J. Struck, and M. W. Zwierlein, Spectral Response andContact of the Unitary Fermi Gas, Phys. Rev. Lett. ,203402 (2019).[89] F. Palestini, A. Perali, P. Pieri, and G. C. Strinati, Tem-perature and coupling dependence of the universal con-tact intensity for an ultracold Fermi gas, Phys. Rev. A , 021605 (2010).[90] R. Haussmann, M. Punk, and W. Zwerger, Spectralfunctions and rf response of ultracold fermionic atoms,Phys. Rev. A , 063612 (2009).[91] H. Hu, X. J. Liu, and P. D. Drummond, Equationof state of a superfluid Fermi gas in the BCS-BECcrossover, Europhysics Letters (EPL) , 574 (2006).[92] E. D. Kuhnle, H. Hu, X.-J. Liu, P. Dyke, M. Mark, P. D.Drummond, P. Hannaford, and C. J. Vale, UniversalBehavior of Pair Correlations in a Strongly InteractingFermi Gas, Phys. Rev. Lett. , 070402 (2010).[93] S. Gandolfi, K. E. Schmidt, and J. Carlson, BEC-BCScrossover and universal relations in unitary Fermi gases,Phys. Rev. A , 041601 (2011).[94] P. Ponte, Z. Papi´c, F. Huveneers, and D. A. Abanin,Many-body localization in periodically driven systems,Phys. Rev. Lett. , 140401 (2015).[95] A. Lazarides, A. Das, and R. Moessner, Fate of many-body localization under periodic driving, Phys. Rev.Lett. , 030402 (2015). [96] V. Khemani, A. Lazarides, R. Moessner, and S. Sondhi,Phase structure of driven quantum systems, Phys. Rev.Lett. , 250401 (2016).[97] P. Bordia, H. Luschen, U. Schneider, M. Knap, andI. Bloch, Periodically driving a many-body localizedquantum system, Nat. Phys. , 460 (2017).[98] J. Berges, S. Bors´anyi, and C. Wetterich, Prethermal-ization, Phys. Rev. Lett. , 142002 (2004).[99] M. Moeckel and S. Kehrein, Interaction quench in thehubbard model, Phys. Rev. Lett. , 175702 (2008).[100] M. Eckstein, M. Kollar, and P. Werner, Thermalizationafter an interaction quench in the hubbard model, Phys.Rev. Lett. , 056403 (2009).[101] Michael Moeckel and Stefan Kehrein, Crossover fromadiabatic to sudden interaction quenches in the hubbardmodel: prethermalization and non-equilibrium dynam-ics, New J. Phys. , 055016 (2010).[102] M. Gring, M. Kuhnert, T. Langen, T. Kitagawa,B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Dem-ler, and J. Schmiedmayer, Relaxation and prethermal-ization in an isolated quantum system, Science ,1318 (2012).[103] S. A. Weidinger and M. Knap, Floquet prethermaliza-tion and regimes of heating in a periodically driven, in-teracting quantum system, Sci. Rep. , 45382 (2017).[104] D. Abanin, W. D. Roeck, W. W. Ho, and F. Huveneers,A rigorous theory of many-body prethermalization forperiodically driven and closed quantum systems, Com-mun. Math. Phys. , 809 (2017).[105] W. Ketterle and M. W. Zwierlein, Making, probingand understanding ultracold Fermi gases, in Proceedingsof the International School of Physics ”Enrico Fermi”,Course CLXIV , edited by M. Inguscio, W. Ketterle, andC. Salomon (IOS Press, Amsterdam, 2008).[106] G. Ness, A. Vainbaum, C. Shkedrov, Y. Florshaim, andY. Sagi, Single-exposure absorption imaging of ultracoldatoms using deep learning, Physical Review Applied14