1 state trapped in a crossed dipole trap, around
which we slowly turn on a blue-detuned 2D
optical lattice to a depth of 40ER, whereER=
ħk^2 /2mis the recoil energy,ħis the reduced
Planck constant,mis the Rb atom mass, and
k=2p/772 nm is the lattice wave vector
( 14 , 16 , 18 ).Theatomsenduptrappedina
2D array of nearly identical“tubes”with
negligible tunneling among them. The num-
ber of particles per tube varies from 26 to 0
( 18 ). The axial trapping frequency is approx-
imately the same in all the occupied tubes,
wz/2p= 18.1 ± 0.36 Hz. The Lieb-Liniger model
that describes these 1D gases is character-
ized by the dimensionless coupling strength
g( 3 ). For large values ofg, there are strong
correlations among the single-particle wave
functions because it is too energetically cost-
ly for them to appreciably overlap. In our
tubes,g= 4.44/n1D, wheren1Dis the local 1D
density inmm–^1 ( 19 ). With our initial trapping
parameters, the weighted averagegis 8.5, and
the smallestgis 4.2 at the center of the central
tube. Our theoretical analysis assumes the T-G
gas limit ofg→ 1.
To demonstrate dynamical fermionization,
we suddenly reduce the depth of the crossed
dipole trap att=0sothat,whencombined
with the weak axial anti-trap due to the blue-
detuned 2D lattice, there is an approximately
flat potential over an axial range of ~40mm
(fig. S1A). After a variabletev,weturnoffthe2D
lattice. We use a non-sudden turnoff (32ms),
which allows for a longer TOF expansion in
the axial direction before the atoms spread
transversely to a region where gravity is not
well canceled by our magnetic field gradient
(fig. S2A). This turnoff still removes interac-
tions fast enough that the TOF distribution is
barely distinguishable from what is obtained
after a sudden shutoff (fig. S2B). At timetdet=
70ms,wetakeabsorptionimagesoftheatoms
(Fig. 1C) and integrate over the transverse
direction to obtain the TOF 1D distributions
( 18 ). The results are shown in Fig. 2A. The
initially peaked“bosonic”TOF distribution
smoothly deforms and approaches a rounded
“fermionic”TOF distribution over the first
12 ms [see also fig. S1B, which shows the full
width at half maximum (FWHM) of the TOF
distributions]. In the 12 ms over which the dis-
tribution has mostly fermionized, the FWHM
of the axial spatial extent of the atoms (before
TOF) grows from 22mm to 42mm. Whentev>
15 ms, the atoms have expanded to where the
axial potential is insufficiently flat and starts to
affect the TOF distribution.
We theoretically simulated our experiment
using the continuum limit of a lattice hard-
core boson model ( 20 ), which incorporates all
the experimental details, including the initial
size, the evolution up totev, the TOF, the in-
strumental resolution (4.8mm), and the sum
over tubes ( 18 ). The results are shown in Fig.1462 27 MARCH 2020•VOL 367 ISSUE 6485 SCIENCE
Fig. 1. Timing and measurement.(A)Rapid removal of interaction energy. Atoms are initially confined in a 2D
optical lattice of 1D tubes (left). When the 2D lattice is shut off, rapid transverse expansion reduces the density, taking
away interaction energy and allowing a good momentum measurement after TOF. The drawing at right illustrates
the moment when the density has dropped by a factor of 20, before atoms in adjacent tubes start to overlap.
Absorption imaging is done along the line of sight. (B) Timing diagram (not to scale horizontally). (i) Lattice depth as
a function of time. (ii) Axial trap depth as a function of time for the dynamical fermionization measurement.
U 0 = 0.628ERis the initial trap depth andUflat=0.068ERis the depth required to yield an approximately flat potential.
Att= 0, the depth is suddenly lowered to cancel out the residual anti-trap due to the lattice beams. All traps are
shut off at a variabletev, and imaging occurs at a fixedtdet(relative tot= 0). (iii and iv) Axial trap depth as a function
of time for the Bose-Fermi oscillation experiments. The axial trap depth is suddenly changed att= 0, and the
atomsevolveinthenewtrapforavariabletev. The absorption image is taken att=tev+tTOF.(C) Absorption images
fortev=0(top)andtev= 15 ms (bottom) after quenching to a flat potential. The images are averages over 30 shots.
Sudden lattice shutoff makes the atoms expand rapidly transversely. The 1D TOF distributions (in thezdirection,
vertical in the images) are obtained by integrating the images transversely.
Fig. 2. Dynamical fermionization.(A)Normalized experimental axial TOF distributions for a range oftev
values. Each profile is an average of 30 implementations. By 15 ms, an asymptotic shape has been reached.zTOF
denotes spatial positions after the time of flight. (B) Theoretical simulation of the experiment in the T-G limit,
with no free parameters ( 18 ). (C) The corresponding theoretical momentum distributions (rescaled bytdet).
(D) Experimental distributions for four of thetevvalues(0,1,3,and9ms)shownin(A)(coloredcurves),
separately compared to the corresponding theoretical curves from (B) (dotted black lines). (E)Comparisonof
curves attev= 15 ms. The experimental (red) and theoretical (dotted black) TOF curves are rescaled bytdetso
that they can be compared to the theoretical T-G gas (orange) and NIF gas (green) momentum distributions.
The agreement between the two TOF curves shows that the experiment is well described by the T-G limit. That
the T-G gas momentum distribution overlaps with the TOF curves shows that finite size and resolution effects
cease to be appreciable. Overlap with the NIF gas momentum distribution shows thattevis long enough that the
momentum distribution has largely evolved into its asymptotic shape, which is the distribution of rapidities.
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