Quasi-thermalization of non-interacting particles in a quadrupole potential: Analog simulation ofWeyl fermions

Daniel Suchet,1 Mihail Rabinovic,1 Thomas Reimann,1 Norman Kretschmar,1 Franz Sievers,1

Christophe Salomon,1 Johnathan Lau,2 Olga Goulko,3 Carlos Lobo,2 and Frederic Chevy1

1Laboratoire Kastler Brossel, ENS-PSL Research University,CNRS, UPMC, College de France, 24, rue Lhomond, 75005 Paris

2Mathematical Sciences, University of Southampton, Highfield Southampton, SO17 1BJ,UK3Department of Physics, University of Massachusetts, Amherst, MA 01003, USA

(Dated: July 9, 2015)

We study theoretically, numerically, and experimentally the relaxation of a collisionless gas in a quadrupoletrap after a momentum kick. The non-separability of the potential enables a quasi thermalization of the singleparticle distribution function even in the absence of interactions. Suprinsingly, the dynamics features an effectivedecoupling between the strong trapping axis and the weak trapping plane. The energy delivered during the kickis redistributed according to the symmetries of the system and satisfies the Virial theorem, allowing for theprediction of the final temperatures. We show that this behaviour is formally equivalent to the relaxation ofmassless relativistic Weyl fermions after a sudden displacement from the center of a harmonic trap.

PACS numbers:

From the early days of kinetic theory, understanding howtime-irreversibility emerges at the macroscopic level out of atime-reversible microscopic system has always been the ob-ject of fierce debates among physicists [1] and is still one ofthe most outstanding problems in contemporary science andphilosophy [2]. Owing to the unprecedented control of theirphysical properties, ultracold atoms have risen as a unique andversatile platform for the study of fundamental problems andhave shed new light on the foundations of statistical physicsand thermodynamics. Using the tools of atomic physics, itis for instance possible to observe negative temperature foratomic center-of-mass motion [3], or clarify the role of inte-grability in the dynamics of one-dimensional systems [4]. Oneof the most intriguing questions is the existence of relaxationin integrable systems where non-trivial constants of motionconstrain strongly their dynamics. It has been proposed thatthese systems could be described using Generalized Gibbs En-sembles [5] that were recently observed experimentally [6].Ensembles of non-interacting particles provide another exam-ple of generalized Gibbs ensembles [7]. Indeed, since theenergies of each particle are independently conserved, theyprovide as many constants of motion and should in principlepreclude the relaxation of the system.

In this Letter, we study both experimentally and theoret-ically the relaxation of non-interacting particles trapped ina quadrupole potential after a momentum kick. Althoughthe absence of collisions precludes thermalization toward aBoltzman distribution, we demonstrate that the system doesinstead reach a quasi-thermal equilibrium, which we charac-terize through the width of its momentum distribution alongeach spatial direction. This behaviour stands in stark con-trast to the one found in a harmonic trap, where the systemwould oscillate forever as a trivial consequence of Kohn’s the-orem [8]. This discrepancy arises from the non-separability ofthe magnetic potential which in turn implies complex phase-space trajectories. We show that the final kinetic tempera-

tures are strongly constrained by energy conservation, virialtheorem and microreversibility. Surprisingly, although thenon-separable character of the potential should mix all direc-tions of motion, the dynamics along the symmetry axis of thetrap appear to be only very weakly coupled to the transverseplane. As a consequence, the steady state temperatures areobserved to be partially anisotropic. An equivalent way tobreak Kohn’s theorem is to assume non-separability in the ki-netic energy, as is the case for massless relativistic particlesin a harmonic potential. Using a canonical transformation,we show that our results directly emulates the relaxation ofWeyl (ie massless) fermions after a sudden displacement fromtrap center, extending the celebrated physics of electrons intwo-dimensional graphene [9] or in 3D-compounds such asHgCdTe or HgMnTe [10].

One of the most common techniques for atom trappingis the magnetic quadrupole trap [11]. Two coils in anti-Helmholtz configuration along the z-axis create a magnetic

field of modulus B0(r) = b√

∑i α2i x2

i , where b is the magneticgradient and αx = αy = 1, αz = −2 to satisfy the rotationalsymmetry as well as Maxwell’s equations. In this configu-ration the z-direction exhibits stronger confinement and thusgives rise to a pancake-shaped cloud. For spin-polarized alkaliatoms in the electronic ground-state, this inhomogeneous fieldacts as a trapping potential V0(r) = µBB0(r), where µB is theBohr magneton, leading to the single-particle Hamiltonian

h(r,p) =p2

2m+µBb

√x2 + y2 +4z2. (1)

This Hamiltonian is equivalent to that of spinless relativisticparticles confined by a harmonic potential [18] through themapping Xi = −cpi/µBbαi and Pi = µBbαixi/c with c as anarbitrary velocity scale. The Hamiltonian (1) reads

H = Pc+∑i

12

α2i

µ2Bb2

mc2 X2i . (2)

arX

iv:1

507.

0210

6v1

[co

nd-m

at.q

uant

-gas

] 8

Jul

201

5

2

The first term corresponds to the (non-separable) kinetic en-ergy of a massless particle moving at velocity c and the secondone is readily identified as a harmonic potential.

To study the dynamics of these equivalent systems, we ap-ply a momentum kick to a thermal cloud of trapped non-interacting fermions, giving additional momentum q to eachof them and thereby an energy ∆E = q2/2m. We then mea-sure the position and momentum distributions and focus onthe long time limit.

The initial phase-space density of the thermal cloud is givenby a Boltzmann distribution at temperature T0:

f0 (r,p) =1Z

exp(− 1

kBT0h), (3)

and its temporal evolution f (r,p; t) is described by the colli-sionless Liouville equation:

∂t f (r,p; t) = −L f (r,p; t) , (4)f (r,p; t = 0+) = f0(r,p−q). (5)

Here, the Liouville operator is defined as L = p.∂r +F0.∂pwith F0 = −∂rV0 being the force field emanating from thequadrupole potential. We define the kinetic temperature ofthe sample along the direction i as the second moment of themomentum distribution at time t after the kick, i.e. kBTi =〈p2

i 〉t/m with

⟨p2

i⟩

t =∫

d3rd3p f (r,p; t) p2i . (6)

For small values of q2/2mkBT0 one can expand this expres-sion up to second order [19] and express

⟨p2

i⟩

t as a quadraticmapping of the momentum kick q:

⟨p2

i⟩

t =⟨

p2i⟩

0− +12∑

jθi j(t)q2

j . (7)

Here we introduced the 3×3 matrix θ which characterizes theredistribution of the initial kick q along all three spatial direc-tions. The result (7) can be interpreted in terms of heating: thekinetic energy imparted onto the center of mass of the cloudgets converted into internal energy and thereby increases thewidth of the momentum distribution. It should be emphasizedthat this transfer does not rely on collisions at all. It is rathera single particle process due solely to the complexity of theavailable phase-space trajectories [12] which originates fromthe non-separability of the underlying potential (1). As men-tioned earlier, a stationary regime is reached after sufficientlylong time, a situation strikingly different from the harmonictrap case, where the system would continue to oscillate owingto Kohn’s Theorem [8].

We focus on the limit of the θ(t) matrix as t→∞. Using themicroreversibility of the Liouville operator, we show in [19]that θ (∞) is symmetric and constrained by energy conservationand Virial theorem. Due to these conditions, the θ (∞) matrix

0 20 40 100Time

60 80

1.0

0.9

Co

M p

osi

tio

n

1.2

Tem

per

atu

re

1.0

0

-1.0

1.1

FIG. 1: Time evolution of the kinetic temperature and center of massposition along x (blue solid line), y (gray dashed line) and z (red dot-ted line) after a momentum kick q = (q0,0,0) with q0 =

√mkBT0, as

given by numerical simulation over 10 000 particles. The oscillationsare damped to zero and temperature increases within the (x,y)-planebut not along the z-axis. Axes are scaled in reduced units (see text).

is fully parametrized by two independent numbers c‖ and c⊥:

θ(∞) =

16

6c⊥ 2−6c⊥+3c‖ 2−3c‖2−6c⊥+3c‖ 6c⊥ 2−3c‖

2−3c‖ 2−3c‖ 6c‖

(8)

To obtain the value of the remaining unknown parameters, wesimulate numerically the relaxation dynamics of the cloud af-ter the kick. Since the problem is collisionless we integrate theequations of motion for 10000 particles with initial conditionsselected at random according to the initial distribution (5). Forall simulations and graphs presented below, we work in a re-duced unit system where m = kBT0 = µBb = 1. The results fora kick of unit magnitude q0 =

√mkBT0 along the x axis, i.e.

within the weakly confined plane, are shown in Fig. 1. Oneobserves two striking features: first, the temperatures alongx and y converge to the same value within ' 100 time unitst0 = q0/µBb, displaying a quasi-thermalization even thoughno collisions are present. Second, the temperature along z isalmost unaffected by the kick along x. This is also surpris-ing since, for each atom, energy is continuously transferredbetween degrees of freedom along the trajectory due to thenonseparability of the potential. In the opposite case, a kickalong z increases the temperature along z but leaves Tx and Tynearly unchanged [19].

In terms of the components of θ (∞) these conditions areequivalent to θ

(∞)xy = θ

(∞)xx and θ

(∞)zx = 0, hence c‖ = 1/3 and

c⊥ = 2/3. The matrix thus has the following form:

θ(∞) =

13

1 1 01 1 00 0 2

. (9)

The absence of cross-thermalization implies that, even ifthe system does reach a stationary regime, its final state can-not be described by a Boltzmann distribution (3). This is

3

a direct consequence of entropy conservation. Indeed, ac-cording to Thermodynamics’ Second Law, Boltzmann’s dis-tribution maximizes the entropy of the system for a given en-ergy. Starting from such a distribution with total energy Eand entropy S, the kick delivers the additional energy per par-ticle of ∆E = q2/2m, but it does so without increasing thesystem’s entropy. The latter must be conserved throughoutthe evolution because the system is collisionless. The quasi-equilibrium state thus exhibits a larger energy E +∆E for thesame entropy S, in contradiction to the usual growth of the en-tropy expected for a collisional and fully ergodic system. Todescribe this equilibrium state, one can resort to generalizedGibbs ensembles, as described in [7]. The definition (6) thenidentifies temperature as the second moment of the general-ized Gibbs distribution.

Using the explicit form (9) of θ (∞), we can express the heat-ing of the cloud as a function of the kinetic energy deliveredduring the momentum kick. For a kick along the strong axisz we find for the steady state temperatures:

∆Tx = ∆Ty = 0 & ∆Tz =2

3kB∆E. (10)

By contrast, a kick along any direction within the weak planeleads to a temperature increase given by:

∆Tx = ∆Ty =1

3kB∆E & ∆Tz = 0 (11)

These analytical results are in very good agreement with nu-merical simulations [19].

These predictions were tested experimentally using a laser-cooled ensemble of spin-polarized Lithium atoms [13]. Theirpreparation starts with a dual species magneto-optical trap(MOT) which is loaded with fermionic 6Li and 40K. In asecond step the clouds are subjected to blue detuned D1 mo-lasses [14, 15], cooling them down to the ≈ 50µK regime.Subsequently the atoms are optically pumped into their low-field seeking stretched Zeeman states, |F = 3/2,mF = 3/2〉and |9/2,9/2〉, respectively. Finally, we ramp a magneticquadrupole field up to b = 82G/cm within 500 ms, capturing107 6Li and 109 40K atoms. Inter-species- as well as p-wavecollisions among 40K atoms [16] allow for the complete ther-malization of the two clouds at approximately T0 = 300 µK.This is a convenient working temperature since lower valueswould imply noticeable Majorana losses within the Li sam-ple. After thermalization, 40K atoms are removed from thetrap by shining resonant light, which leaves 6Li unaffected.The initial temperature of the latter is well below the p-wavethreshold and Pauli principle forbids any s-wave interactionsince the sample is fully polarized. The lithium cloud thusforms an ensemble of non interacting particles, as is requiredto test the aforementioned theoretical predictions. The typi-cal size of such a cloud is r0 = kBT/µBb' 0.5mm along thestrong axis z.

The kick is performed by quickly turning on a magneticbias field B for a short duration τ (typically between 0 anda few hundred µs). τ is chosen to be much smaller than the

z - quench qZ [q0]

Tem

per

atu

re in

crea

se ∆

T [T

0]

x - quench qX [q0]

0

1.5

0.5

2.0

0 1.50.5 1.0

0

0.5

1.0

1.5a)

a) b)

2.0 0 1.50.5 1.0 2.0

1.0

FIG. 2: Predicted temperature increase as a function of the strengthof the CoM momentum kick, assuming a trap center displacementof δ = 5. The dashed lines are given by equations (10) and (11)with ∆E = q2/2m while the solid lines represent (14) and (15). Datapoints shown here are obtained by numerically solving the classicalequations of motion along x (blue squares), y (gray triangle) and z(red circles). Axes are scaled in reduced units.

period of the ensuing oscillation of the cloud’s center-of-mass(CoM) with typical values of' 10ms. The additional field hasthe effect of displacing the center of the trapping potential by adistance δ =(Bx/b,By/b,Bz/2b), such that V (r)=V0(r−δ ).We achieve typical displacements of δz ' 4r0 and δx ' 7r0.

If the bias field is strong enough, the trap minimum is dis-placed far away from the center of the cloud, so that it expe-riences a uniform gradient µBb. In this situation we expectthe cloud to acquire a momentum q ∼ µBbτ , similar to freefall in gravity. The bias field B is then switched off again inless than 20 µs. Depending on the duration of the kick, thecloud as a whole will also start to move and cover a distanceof d ∼ (µBb/2m)τ2. Since this unwanted effect scales withthe second power of the kick duration, it can be neglected forshort times and consequently, the cloud mostly acquires anoverall CoM velocity during the kick.

However, at our operating temperature, the amount ∆E ofenergy transferred to each particle is not simply given byq2/2m. During the kick, even though the center of mass re-mains nearly still, the cloud expands as it is exposed to a lineargradient. Therefore it gains additional potential energy whenrecaptured by the initial trapping potential. The complete cal-culation up to second order in τ leads to the following expres-sions [19]:

∆E =1

2mτ

2∫

d3rd3p f0 (F −F0)2 , (12)

q2 = τ2(∫

d3rd3p f0 (F −F0)

)2

, (13)

where F =−∂rV is the force field of the biased magnetic trap

4

and F0 = −∂rV0 its stationary counterpart. Thus the energygiven to the cloud displays a dependence on the magnitude ofthe displacement δ as well. If no kick is applied, F = F0 andno energy is transferred. On the other hand, in the asymptoticlimit δ � r0 the force F is almost uniform and equations (10)and (11) read

∆Tx = ∆Ty = 0 & ∆Tz =12

q2/mkB (14)

and

∆Tx = ∆Ty =12

q2/mkB & ∆Tz = 0. (15)

These results have been tested successfully with numericalsimulations (fig. 2).

Experimentally, we performed kicks along z as well as xand measured both the steady state temperature after a relax-ation time of 500ms and the center of mass velocity right afterthe kick using time-of-flight imaging. Further details regard-ing the experimental procedures can be found in [19]. Theheating ∆T = T −T0 and the CoM momentum q induced bythe momentum kick are extracted from the difference betweenthe corresponding values at quasi-equilibrium and the onespresent right after the kick. Results are presented in Fig. 3 andclearly support the effective decoupling between directions.

More quantitatively, for kicks along x, the bestfits are ∆Tx/T0 = 0.58(5)stat(20)syst × q2/q2

0 and∆Tz/T0 = 0.13(3)stat(5)syst × q2/q2

0 along the pas-sive symmetry axis. For kicks along the strongaxis z, ∆Tz/T0 = 0.60(5)stat(20)syst × q2/q2

0 and∆Tx/T0 = −0.14(5)stat(8)syst × q2/q2

0 along the passivesymmetry plane. Statistical errors are reduced by averagingover 60 measurements, and systematic errors originate fromresidual Eddy currents after fast switch off of the magnetictrapping potential. A conservative estimate of systematicerrors is obtained from the standard deviation of the ∆T (q)fits (such as in Fig. 3) measured for 4 different magnetic fieldgradients.

Without any adjustable parameter, the fits are in good agree-ment with the theoretical predictions of 0.50×q2/q2

0 expectedin the asymptotic limit δ � r0. For kicks along x, this limitis not fully satisfied and eq. (12) predicts a slightly reducedcoefficient of ∆Tx/T0 ' 0.4× q2/q2

0, still maintaining a rea-sonable agreement with the measured value [19].

Using Eq.(2), we predict the relaxation dynamics of mass-less relativistic fermions dragged away from the center of aharmonic trap. Equations (10) and (11) show that the CoMoscillations will be damped and the position distribution willreach a steady state with anisotropic broadening. The linearityof the Hamiltonian (2) with respect to momentum drasticallycontrasts its behaviour with that of a massive non-relativisticgas, which would simply oscillate according to Kohn’s theo-rem [8].

In conclusion, we have studied the quasi-thermalization ofan ensemble of particles confined in a quadrupole potentialand explored the analogy with a gas of three-dimensional

0.4

0.2

0.6

1.00.2 0.60 1.50.5 1.0

0

a)

a) b)

-0.2

0

0.5

1.0

1.5

z - quench qZ [q0]

Tem

per

atu

re in

crea

se ∆

T [T

0]

x - quench qX [q0]

FIG. 3: Temperature increase ∆T = T − T0 along x (blue squares)and z (red circles) as a function of the CoM momentum acquiredduring the kick. Dashed lines represent the theoretical predictions(14) and (15). Solid lines are fits to the experimental data. Error barsrepresent the temperature statistical uncertainty and shaded zonesgive the 95% confidence level of the fits. a) x momentum kick atb = 70G/cm. b) z momentum kick at b = 55G/cm.

massless relativistic particles in a harmonic trap. We havedemonstrated that even in the absence of collisions the sys-tem reaches a non-thermal stationary state that we charac-terized as a function of the external excitation imparted ontothe system. This work opens up interesting prospects for fu-ture research. First, we will further investigate the origin ofthe weak coupling between the axial and transverse directionsto determine whether this behaviour is a signature of hiddensymmetry properties of the system. By using smaller atomsamples one could also investigate the puzzling revival phe-nomenon predicted by Poincare for hamiltonian systems [17].Finally, using a Ioffe-Pritchard trap in which a bias field givesrise to a non-zero magnetic field at the trap center, the field

takes the form B =√

B20 +b2 ∑i α2

i x2i . In this case, the analog

system would be described by the relativistic kinetic energyE =

√m2c4 + p2c2 where the mass can be tuned via B0.

The authors would like to thank J. Dalibard and F. Ger-bier for stimulating discussions. We are grateful to E. Dem-ler for pointing out to us the analogy to massless relativisticparticles. O. Goulko acknowledges support from the NSF un-der the Grant No. PHY-1314735. C. Salomon expresses hisgratitude to the A. von Humboldt foundation and to Prof. I.Bloch and T. W. Hansch for their kind hospitality at LMUand MPQ. This work was supported by Region Ile de France(Dim nanoK/IFRAF), Institut de France (Louis D. prize) andthe European Union (ERC grant ThermoDynaMix).

5

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[3] S. Braun, J. P. Ronzheimer, M. Schreiber, S. S Hodgman,T. Rom, I. Bloch, and U. Schneider. Negative absolute temper-ature for motional degrees of freedom. Science, 339(6115):52–55, 2013.

[4] T. Kinoshita, T. Wenger, and D. S Weiss. A quantum Newton’scradle. Nature, 440(7086):900–903, April 2006.

[5] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii. Relaxationin a completely integrable many-body quantum system: an abinitio study of the dynamics of the highly excited states of 1dlattice hard-core bosons. Phys. Rev. Lett., 98(5):050405, 2007.

[6] T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuh-nert, W. Rohringer, I. E Mazets, T. Gasenzer, and J. Schmied-mayer. Experimental observation of a generalized gibbs ensem-ble. arXiv preprint arXiv:1411.7185, 2014.

[7] Ph. Chomaz, F. Gulminelli, and O. Juillet. Generalized gibbsensembles for time-dependent processes. Annals of Physics,320(1):135–163, 2005.

[8] W. Kohn. Cyclotron resonance and de haas-van alphen oscilla-tions of an interacting electron gas. Phys. Rev., 123:1242–1244,Aug 1961.

[9] A. K Geim and K. S. Novoselov. The rise of graphene. Naturematerials, 6(3):183–191, 2007.

[10] M. S. Zholudev F. Teppe W. Knap V. I. Gavrilenko N. N.Mikhailov S. A. Dvoretskii P. Neugebauer C. Faugeras A-L.Barra G. Martinez M. Orlita, D. M. Basko and M. Potemski.Observation of three-dimensional massless Kane fermions in azinc-blende crystal. Nature Physics, 10:233–238, 2014.

[11] D. E. Pritchard. Cooling Neutral Atoms in a Magnetic Trap forPrecision Spectroscopy. Phys. Rev. Lett., 51:1336, 1983.

[12] J.T.M. Walraven E.L. Surkov and G.V. Shlyapnikov. Collision-less motion of neutral particles in magnetostatic traps. PRA,49(6):4778, 1994.

[13] A. Ridinger, S. Chaudhuri, T. Salez, U. Eismann, D.R. Fernan-des, K. Magalhaes, D. Wilkowski, C. Salomon, and F. Chevy.Large atom number dual-species magneto-optical trap forfermionic 6li and 40k atoms. Eur. Phys. J. D, 65(1):223, 2011.

[14] D. Fernandes, F. Sievers, N. Kretzschmar, S. Wu, C. Sa-lomon, and F. Chevy. Sub-doppler laser cooling of fermionic40k atoms in three-dimensional gray optical molasses. EPL,100(6):63001, 2012.

[15] F. Sievers, N. Kretzschmar, DR Fernandes, D. Suchet, M. Ra-binovic, S. Wu, C. V. Parker, L. Khaykovich, C. Salomon, andF. Chevy. Simultaneous sub-doppler laser cooling of fermionic6Li and 40K on the D1 line: Theory and experiment. Phys. Rev.A, 91:023426, Feb 2015.

[16] B. DeMarco, J. L. Bohn, J. P. Burke, M. Holland, and D. S.Jin. Measurement of p-wave threshold law using evaporativelycooled fermionic atoms. Phys. Rev. Lett., 82:4208–4211, May1999.

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[18] E Demler, private comm., 2015[19] For additional simulations, detailed calculations and experi-

mental sequence, see supplementary materials.

SUPPLEMENTARY INFORMATION

Properties of the θ matrix

Here we demonstrate explicitly the derivation of the θ ma-trix (eq. 8 from the main text) and show the numerical resultsmentioned in the article.

Dynamical Equations

We consider a gas of spin polarized fermions of massm initially at thermal equilibrium at a temperature T in aquadrupole trap described by a force-field F0(r) = −∇V0.The initial phase space density f0 is given by Boltzmann’sdistribution

f0(r,p) =1Z

e−β (p2/2m+V0(r)). (16)

At t = 0 we kick the atoms and give an additional momen-tum q to each of them. Right after the kick, the phase-spacedensity is thus f (r,p, t = 0+) = f0(r,p− q). For low q wecan expand the phase-space density as

f (r,p, t = 0+) = (17)

f0(r,p)

[1+β

p ·qm−β

q2

2m+β

2 (p ·q)2

2m2 + ...

].

(18)

We have to solve the collisionless Boltzmann Equation

∂t f =−L [ f ], (19)

with L = p · ∂r/m +F · ∂p. The Liouville operator Lobeys the following properties:

1. L [ f0] = 0, since Boltzmann’s distribution is a station-ary solution of Boltzmann’s equation.

2. L commutes with the parity operators Πi=x,y,z chang-ing the (xi, pi) phase-space coordinates into (−xi,−pi).This means that the symmetry of a function in a givendirection is conserved under the action of L .

3. By contrast, if we apply the parity operator only on ror p, then L →−L and the parity of the function isinverted.

4. L is antisymmetric for the scalar product

〈α|β 〉=∫

d3rd3p f0(r,p)α(r,p)β (r,p), (20)

i.e. 〈α|L [β ]〉=−〈L [α]|β 〉.

6

Setting f (r,p; t) = f0(r,p)α(r,p; t), according to the pre-vious properties, α is solution of

∂tα =−L [α], (21)

with the initial condition

α(0+) = 1+βp ·q

m−β

q2

2m+β

2 (p ·q)2

2m2 . (22)

Using this new function, the average value of any physicalquantity G(r,p) is calculated by

〈G〉=∫

d3rd3p f0(r,p)α(r,p)G(r,p) = 〈G|α〉. (23)

Eq. (21) is formally solved by αt = Ot [α(0+)], with the evo-lution operator Ot = exp(−tL ). Owing to the antisymmetryof L , Ot is orthogonal for the scalar product 〈.|.〉, particularlyO†

t = O−1t = O−t .

The momentum-width is given by

〈p2i 〉t = 〈p2

i |Ot [α(0+)]〉

= 〈p2i |1〉(1−

βq2

2m)+ 〈p2

i |Ot [βp ·q/m]〉

+β 2

2m2 〈p2i |Ot [(p ·q)2]〉.

(24)

Due to the symmetry properties of L , Ot [p · q] is an odd-function of the phase-space coordinates and the second termof the sum in eq. (24) vanishes. We consider first the last termof the sum. It can be rewritten as

β 2

2m2 〈p2i |Ot [(p ·q)2]〉= β 2

2m2 ∑j,k

q jqk〈p2i |Ot [p j pk]〉. (25)

For j 6= k, p j pk is odd in the j and k directions, and sois Ot [p j pk]. Just as for the previous term, the scalar productvanishes in this case only, simplyfing the last term of eq. (24)to

β 2

2m2 〈p2i |Ot [(p ·q)2]〉= 1

2 ∑j

q2j〈

β p2i

m|Ot [

β p2j

m]〉. (26)

We can finally write the momentum width of the cloud as

〈p2i 〉t = 〈p2

i |1〉+12 ∑

jq2

j〈β p2

im|Ot [

β p2j

m]−1〉. (27)

At t = 0−, the cloud is not excited and we have α = 1,hence 〈p2

i 〉0− = 〈p2i |1〉. At t = 0+, the kick increases the mo-

mentum of each fermion by q changing the momentum-widthto 〈p2

i 〉0+ = 〈p2i 〉0− + q2

i . Taking t = 0 in Eq. (27), we there-fore see that

q2i =

12 ∑

jq2

j〈β p2

im|β p2

j

m−1〉, (28)

hence

〈β p2i

m|β p2

j

m−1〉= 2δi j. (29)

Using this identity, we recast Eq. (27) to

〈p2i 〉t = 〈p2

i 〉0− +12 ∑

jq2

jθi j(t), (30)

with

θi j(t) = 〈β p2

im|(Ot −1)[

β p2j

m]〉+2δi j. (31)

θi j owns the following properties:

1. θi j(t) is an even function of t. Indeed, from the expan-sion of the exponential, we get

θi j(t) = 2δi j + ∑n≥1

(−t)n

n!〈β p2

im|L n[

β p2j

m]〉 (32)

Since L inverts the parity of the momentum depen-dence, L n[p2

j ] has a parity (−1)n in each momentumcoordinate. By calculating the integral over p to ob-tain the scalar product, the contribution of odd-n’s musttherefore vanish, leaving only the even terms of thesum.

2. θi j(t) = θ ji(t). Indeed, using the orthogonality

of Ot , we see that 〈β p2i

m |(Ot − 1)[β p2

jm ]〉 = 〈(O−t −

1)[β p2i

m ]|β p2j

m 〉, and hence the θ matrix is symmetricθi j(t) = θ ji(−t) = θ ji(t).

Stationary State

We discuss now the equilibrium properties of the gas at t→∞. At equilibrium, the system obeys the Virial theorem, whichis obtained by studying C(t) = 〈r ·p〉. We have

C = 〈r ·p|∂tα〉=−〈r ·p|L [α]〉. (33)

Using the antisymmetry of L , this expression can be recastas

C = 〈L [r ·p]|α〉. (34)

The action of L on r ·p is straightforward derived by

L [r ·p] = p2

m+r ·F =

p2

m−r ·∂rV. (35)

We consider now a homogeneous potential of degree n. Ac-cording to Euler’s theorem, r ·∂rV = nV , hence

C = 2Ek−nEp, (36)

7

z - quench qZ [q0]

Tem

per

atu

re in

crea

se ∆

T [T

0]

x - quench qX [q0]

0

1.5

0.5

0 1.50.5 1.0

0

0.25

0.5

0.75a)

a) b)

2.0 0 1.50.5 1.0 2.0

1.0

FIG. 4: Temperature increase ∆T (blue along x, gray along y, redalong z) after a kick along x (a) and along z (b), as a function ofthe kick strength. The excitation is ideal and the energy given to thecloud is ∆E = q2/2m, unlike the real case scenario picture in Figure2 of the main text. Solid lines are given by eq. (10) and (11) of themain text. Filled symbols are numerical simulation results. The ydata points are indistinguishable from the shown x points.

where Ep and Ek are the expectation values of potential andkinetic energies. In the stationary regime (C = 0) the Virialtheorem gives 2Ek = nEp. For a quadrupolar potential (n= 1),this implies Ep = 2Ek. In other words, if E is the total energy,〈E〉= 3Ek.

Directly after the kick, the total energy is calculated by

〈E〉0+ = 〈E〉0− +q2/2m = 3E0−k +q2/2m. (37)

Using energy conservation, we have for t = ∞,

〈E〉0+ = 〈E〉+∞ = 3E+∞

k = 3E0−k +

32 ∑

i jθi j(+∞)

q2j

2m. (38)

Comparing Eq. (37) and eq. (38), it becomes clear that theθ(∞)i j must satisfy the condition

∑j

θ(∞)i j = 2/3, (39)

with θ(∞)i j = θi j(+∞). Furthermore, assuming a rotational in-

variance of the potential around the z-axis in case of a station-ary state, the matrix θ

(∞)i j can be fully parametrized by the two

diagonal elements c‖ = θzz and c⊥ = θxx = θyy leading to

θ∞ =

16

6c⊥ 2−6c⊥+3c‖ 2−3c‖2−6c⊥+3c‖ 6c⊥ 2−3c‖

2−3c‖ 2−3c‖ 6c‖

(40)

As shown in the letter, this expression leads to eq. (9) and(10). We tested these properties by numerical simulations overan ensemble of 104 particles and found very good agreement,as shown in figure S4.

Realistic Excitation

The previous analysis was made for a pure momentum kick.In practice, the cloud gets slightly displaced and expands dur-ing our excitation; as a result, the energy ∆E transferred toeach atom is not simply 1

2 q2/m. To get a quantitative descrip-tion of the excitation, we investigate the time evolution of theenergy of the cloud.

Consider the evolution of a physical quantity G(rt ,pt) for aparticle initially characterized by the phase-space coordinates(r0,p0). According to Liouville’s equation, its evolution un-der the force field F is given by

dGdt

= {G,H}= L [G], (41)

with L defined as in eq. 19 and H being the Hamiltonian.If the excitation is short enough, we can expand the solutionover time:

G = G0 + tL [G0]+t2

2L 2[G0]+ ... (42)

Averaging over the initial coordinates of the particle leadsto

〈G〉t =∫

d3r0d3p0 f0(r0,p0)G(rt ,pt) (43)

' 〈G〉0 + t∫

f0L [G]0 +t2

2

∫f0L

2[G]0 + ... (44)

where for clarity we suppressed the infinitesimal phase-spaceelement in the integral. In the following steps we rewriteL = L0 + (F −F0)∇p with F and F0 being the excitingand stationary force fields. Since f0 is stationary in the forcefield F0, we have L0[ f0] = 0 and hence

∫f0L [G] =

∫f0(F −F0)∇p[G] (45)

So the mean value of G can be further simplified into

〈G〉t ' 〈G〉0 + t∫

f0(F −F0)∇pG

+t2

2

∫f0(F −F0)∇p

(L0 +(F −F0)∇p

)G

(46)We apply the resulting eq. (46) to the momentum and the

total energy of the system in the static trap up to the leadingorder, obtaining

q2 = 〈p〉2t ' (t∫

f0(F −F0))2 (47)

∆E = 〈H0〉t −〈H0〉0 't2

2m

∫f0(F −F0)

2, (48)

8

0 5 10 15

1.5

1.0

0.5

Ener

gy

tran

sfer

E·m

/q2

20 25

2.0

Trap displacement δ

FIG. 5: Energy increase of the cloud versus the normalized trap dis-placement δ during momentum kick along z (red circles) and x (bluesquares). Solid lines: prediction from eq. (48). Dashed line: naiveprediction ∆E = q2/2m. Filled symbols: numerical simulation.

where the energy conservation in the static trap (L0[H0] = 0for H0 = p2/2m+V0) was used.

We model the excitation as a displacement δ of the trapcenter, such that F (r) = F0(r− δ). Figure S5 displays theactual ratio between the energy ∆E transferred to the atomsand the quantity q2/m as a function of the displacement δ

along x or z. The result is indeed very different from the naiverelation ∆E = 1

2 q2/m. For a kick along z the ratio dependsonly weakly on the trap displacement and reaches quickly itsasymptotic value of ∆E = 3

4 q2/m. By contrast, the excessenergy along x is sensitive to the actual value of δ and theasymptotic value ∆E = 3

2 q2/m is reached for δ > 30r0. Forthe experimentally relevant value δ ' 7 heating is reduced byabout 20% with respect to this asymptotic value.

Experimental sequence

In our experiments temperatures and kick velocities aremeasured with a standard time-of-flight (TOF) technique: thetrapping potential is abruptly switched-off and the atomic

cloud evolves freely during few ms. It is then imaged on aCCD camera by a resonant laser beam. The center of massvelocity is given by tracking the center of the distribution dur-ing the TOF while the temperature is measured using the stan-dard deviation of the position distribution for sufficiently longTOF.

A potential source of errors comes from Eddy currentswhich appear when switching abruptly off the quadrupolemagnetic trap with gradients of the order of 100−150 G/cm.These currents last for∼ 2ms and are more pronounced alongthe vertical direction z where the optical table provides anasymmetry. The transient magnetic field creates a position-dependent Zeeman effect which deforms the atomic cloudprofile at short TOF durations. This results in an error in themeasurement of the center of mass momentum with or with-out kick. For instance, in the absence of a kick we observe asmall parasitic velocity v0 which is proportional to the mag-netic gradient b and reaches 30cm/s at b = 165G/cm. There-fore to evaluate the actual momentum delivered to the cloudby the kick solely, we subtract v0 measured after the thermal-ization time of 500ms from the velocity right after the kick.

Long TOF were difficult to achieve because of the rela-tively high temperature of the cloud and of the small massof Lithium atoms. Cooling further down the sample wouldlead to higher Majorana losses during the 500ms of thermal-ization, resulting in an additionnal heating of the cloud. Inorder to mitigate the influence of the inhomogeneous Zeemaneffect we therefore used a highly saturated probe beam witha saturation parameter of s = 24 in order to power-broadenthe absorption resonance to a width of Γ′ = Γ

√1+ s, where

Γ = 6MHz is the natural width of the D2 line used for theimaging transition.

The fit errors are displayed by the error bars on figure 3 andaccount for our statistical errors of typicaly 0.05/mkB. Per-forming the experiment with 4 different magnetic gradients,we estimate a systematic uncertainty of 0.2/mkB for the fittedamplitude of the parabolic dependence of the heating on themomentum kick in figure 3.