# Stability and sensitivity of interacting fermionic superfluids to quenched
disorder

Jennifer Koch  Department of Physics and Research Center OPTIMAS, RPTU
Kaiserslautern-Landau, 67663 Kaiserslautern, Germany  Sian Barbosa  Department
of Physics and Research Center OPTIMAS, RPTU Kaiserslautern-Landau, 67663
Kaiserslautern, Germany  Felix Lang  Department of Physics and Research Center
OPTIMAS, RPTU Kaiserslautern-Landau, 67663 Kaiserslautern, Germany  Artur
Widera  Department of Physics and Research Center OPTIMAS, RPTU
Kaiserslautern-Landau, 67663 Kaiserslautern, Germany

###### Abstract

We probe the response of ultracold, interacting Fermi gases of Lithium (Li)
atoms in the BEC-BCS crossover to strong perturbations in space and time via
rapidly switched optical disorder potentials with focus on the BEC side. We
measure the time evolution of long-range phase coherence quantified via the
ability of the gas to expand hydrodynamically. While in the presence of static
disorder, the effects on a Bardeen-Cooper-Schrieffer (BCS) type superfluid are
predicted to be much smaller compared to a molecular Bose-Einstein condensate
(BEC), we find that the quantum properties of a resonantly interacting,
unitary Fermi gas are more strongly suppressed in the presence of quenched
disorder than in a molecular BEC. For quenches from a disordered potential, we
find that the unitary Fermi gas never recovers quantum hydrodynamics for all
parameters studied, while a molecular BEC always recovers quantum
hydrodynamics, even when the quench leads to strong particle losses of up to
$70\text{\,}\%$. Temperature measurements indicate an additional heating
channel specific to gases close to resonant interactions, leading to strong
local dephasing or pair breaking. Our work illustrates the striking difference
between the nonequilibrium phase diagram of the time-dependent disordered BEC-
BCS crossover compared to that of the static-disorder case. Moreover, our
results suggest an important role of local dephasing of fermionic pairs by
time-dependent local perturbations, which could be a key contribution to the
destruction of macroscopic quantum properties for time-dependent
perturbations.

††preprint: APS/123-QED

When an interacting system of fermionic particles is cooled below a critical
temperature, bosonic pairs form, and the system becomes superfluid or, for
charged fermions, superconducting. For ultracold Fermi gases, magnetic
Feshbach resonances allow to modify the effective interaction strength [1] and
thereby the underlying pairs from bound molecules to Cooper-type fermion pairs
along the so-called crossover from a molecular Bose-Einstein condensate (BEC)
to a Bardeen-Cooper-Schrieffer (BCS) superfluid [2, 3]. Between these two
regimes, on resonance, a unitary Fermi gas (UFG) is realized exhibiting, for
example, the largest superfluid critical velocity along the crossover [4, 5].
On resonance, the microscopic details of the gas are not relevant for the
macroscopic properties [6], which has allowed to deduce universal properties
highly relevant for nuclear matter, see for example [2, 3, 7, 8, 9, 10]. The
microscopic pairing has been investigated theoretically [11, 12] and
experimentally using radio-frequency (rf) spectroscopy, for example, revealing
different equilibrium properties such as the pair size or binding energy [13].
This microscopic pairing has profound consequences on the macroscopic quantum
state. The excitation spectrum, for example, has been measured along the
crossover in three-dimensional gases using Bragg spectroscopy [14, 15] and rf
spectroscopy [16] and shows a clear change from a phononic-type branch in the
BEC regime to a spectrum indicating dominant pair breaking in the BCS regime.

Figure 1: Experimental setup and method. (a) (I) Schematic drawing of the
experimental setup. The atomic cloud (red ellipsoid) is trapped in a
combination of an optical dipole trap (blue cylinder) and a magnetic saddle
potential (gray surface) and is superposed by the disorder field (green
ellipsoids). (II) The microscopic pair size $\xi_{\text{pair}}$ of a mBEC is
much smaller than that of a UFG and the pair distance $l$ equals roughly the
pair size at resonance. (III) In the mBEC regime, the Feshbach molecules
occupy the ground state of the harmonic trap and form a macroscopic
wavefunction; by contrast in a UFG, the atoms occupy the levels according to
the Pauli exclusion principle up to the Fermi energy $E_{\text{F}}$. (IV) The
microscopically relevant length scale is the coherence (healing) length
$\xi_{\text{phase}}$ quantifying the length on which the system can respond as
a macroscopic many-body wavefunction to perturbations such as disorder
speckles. (b) Computed ideal time evolution of the aspect ratio for a
degenerate gas (solid line) and a thermal gas (dashed line) for hydrodynamic
expansion in our experimental traps without disorder, showing a divergence of
the aspect ratio for a quantum fluid. For details see Appendix. (c) Measured
aspect ratio $A$ of a mBEC at $763.6\text{\,}\mathrm{G}$ (blue circles) and a
UFG at $832.2\text{\,}\mathrm{G}$ (red diamonds) as a function of the
expansion time $d$ around the maximum achievable value, which is limited by
experimental details, see Ref. [17]. Each point is an average of four
repetitions and the corresponding error bars show their standard deviations.
The lines (light blue and light red) are guides to the eye. (d) The same
measurement as in (c) but for thermal gases. (e) Absorption pictures for the
measurement points indicated in panels (c) and (d), where the left most
pictures show the initial quantum gases before expansion dynamics. The scales
in the images mark a distance of $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
also in the following figures.

A prominent consequence of the pair nature in disordered potentials is
summarized by the Anderson theorem [18], stating that delocalized Cooper pairs
are only little affected by local perturbations, leading to only a moderate
decrease of pairs for disordered systems. Indeed, theoretical investigations
of the BEC-BCS crossover in static disorder [19, 20, 21, 22, 23] show only a
slight modification of the critical temperature in the BCS regime [19],
together with a relatively small reduction of the order parameter and
condensate fraction [20, 21], and an area of stability in the crossover region
close to resonant interactions. By contrast, in the BEC regime, the critical
temperature is more strongly suppressed by static disorder, together with
order parameter and condensate fraction. The specific properties of the pairs
are thus also expected to modify the response of the superfluid to time-
dependent disorder and determine how fast quantum properties decay for strong
perturbation, and how fast they are recovered once the perturbation is absent.

We experimentally study the response of an ultracold gas of fermionic
lithium-6 atoms prepared in the two lowest Zeeman substates (typically $3\cdot
10^{5}$ atoms per spin state at a temperature
$T\approx$200\text{\,}\mathrm{n}\mathrm{K}$$) to fast switching of a disorder
potential. To tune the interaction between these two states, a broad Feshbach
resonance at $832.2\text{\,}\mathrm{G}$ is used [24]. This allows entering the
different regimes of the BEC-BCS crossover, quantified via the interaction
parameter $1/k_{\text{F}}a$ , with the absolute value of the Fermi wave-vector
$k_{\text{F}}=\sqrt{2mE_{\text{F}}}/\hbar$ with the mass $m$ of a lithium
atom, the Fermi energy $E_{\text{F}}\approx k_{B}\times 670\,$nK, $k_{B}$ the
Boltzmann constant, and the $s$-wave scattering length $a$.

Our observable revealing the response of quantum properties is the ability of
the gas to undergo quantum hydrodynamic expansion. Expansion measurements
probing hydrodynamic behavior have been used, for example, to characterize the
transition between ballistic and hydrodynamic expansion [25] and to study the
viscosity in a UFG [26]. Here, we employ hydrodynamic expansion as a measure
of long-range phase coherence [17]. It allows us to time-resolve a quantum
system’s response to a strong perturbation in space and time, as revealing the
existence or reformation of a well-defined global phase. We can thus trace a
genuine quantum property of a strongly interacting, three-dimensional many-
body system subjected to strong and time-dependent perturbation. While initial
studies [17] have been performed deep in the molecular BEC (mBEC) regime, we
focus on the markedly different stability and sensitivity of a mBEC and a UFG,
respectively.
Hydrodynamics is initiated by suddenly switching off the optical dipole trap
(see Fig. 1(a)), releasing the gas into a magnetic trap, where the change of
aspect ratio can be determined from absorption images. Here, the aspect ratio
is defined as the ratio of the full width at half maximum (FWHM) of the 1D
integrated column density distributions in $x$ and $y$ directions (see Fig.
1(a) and Appendix). The ensuing dynamics are markedly different for an ideal
gas, a classical (thermal) interacting gas, or a quantum gas [27]. While an
ideal gas shows a moderate change of aspect ratio due to single-particle
dynamics in the trap, this change is enhanced by collisional hydrodynamics for
repulsively interacting classical gases (see Fig. 1(b,d,e)). Quantum gases,
however, show a remarkably large change in aspect ratio, as shown in Fig.
1(b,c,e). This strong increase beyond classical collisional hydrodynamics is
tightly connected to the existence of a well-defined global phase, i.e., long-
range phase coherence [17]. We note that a comparable quantum enhancement
cannot be observed for the BCS side of the resonance, i.e., negative
interaction parameters $1/k_{\text{F}}a<0$ even close to resonance, in
agreement with previous observations of the vanishing of hydrodynamics when
approaching the BCS regime [28]. We attribute this to breaking of the
underlying fermionic pairs due to the strongly reduced density and hence gap
during the expansion. Our analysis is therefore restricted to positive
interaction parameters $1/k_{\text{F}}a>0$ which will be the focus throughout
the remainder of this work. For more details of the experimental setup, see
Appendix and Ref. [29].

The perturbation of the system is controlled through a repulsive optical
speckle disorder-potential with mean potential strength
$\bar{V}_{\mathrm{dis}}$ and correlation lengths (i.e. typical grain sizes) of
$\sigma_{x,y}=$750\text{\,}\mathrm{n}\mathrm{m}$$ and
$\sigma_{z}=$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$, see Fig.
1(a). The mean potential strength is of the order of the superfluid’s chemical
potential $\mu$ ($\mu_{\mathrm{BEC}}=k_{B}\times 390\,$nK for a mBEC and
$\mu_{\mathrm{UFG}}=k_{B}\times 480\,$nK for a UFG) and the short correlation
length is larger but of the order of the quantum gases’ coherence or healing
length $\xi$ ($\xi_{\mathrm{heal}}\approx 230\,$nm for $1/k_{\text{F}}a\approx
1$). Thus, the static perturbation is strong. Since the polarizability of a
Feshbach molecule is twice the value for a single atom, the potential height
of the speckle potential is twice that of the molecules compared to the
unbound atoms for the same laser power [30, 31]. In tunneling experiments
along the BEC-BCS crossover, for example, this does not seem to play a role,
and the dynamics for constant optical potentials along the crossover could be
compared [32, 33]. Measurements for atom losses after disorder quenches (see
Appendix) indicate, however, that the loss curves collapse to a single curve,
when the disorder laser power for the UFG is twice the power for a mBEC,
supporting our assumption of different potentials $V^{\mathrm{(mBEC)}}\approx
2V^{\mathrm{(UFG)}}$ for the same laser power. In the following, the disorder
is given in laser power of the disorder, providing comparison of potential
heights differing by a factor of two. Moreover, the disorder potential can be
either ramped adiabatically with respect to $h/\mu$ or be quickly switched on
or off with switching times being shorter than $h/\mu$, so that the many-body
system cannot adiabatically follow the dynamics. The speckle realization is
changed after each measurement. Moreover, for our parameters, classical
trapping cannot occur, because the mobility edge is close to the classical
percolation threshold in our system [30, 34]. For a detailed description and
characterization of the disorder potential see Appendix and [35, 36, 37].

Figure 2: Decay of quantum hydrodynamics. (a) Decay of the relative maximum
aspect ratio $A_{{\text{max}}_{\text{rel}}}$ (normalized to the maximum aspect
ratio without disorder) as a function of the disorder-pulse duration
$\tau_{\text{on}}$ for a mBEC at $763.6\text{\,}\mathrm{G}$
($1/k_{\text{F}}a=1.04$, blue circles) and a UFG ($1/k_{\text{F}}a=0$, red
diamonds). The blue (red) dashed lines show independently measured aspect
ratios for classical hydrodynamics in thermal gases at
$763.6\text{\,}\mathrm{G}$ ($832.2\text{\,}\mathrm{G}$), i.e. including
collisional hydrodynamics, and its uncertainty as shaded areas (for more
details see Appendix). The red and blue lines are exponential-decay fits to
the data. Insets show the experimental sequence of disorder quench pulse
(green), and subsequent expansion time $d$ and imaging (red vertical line) as
well as absorption pictures of the two regimes for a short and long disorder
pulse. (b) Half-life period $\tau_{1/2}$ as a function of interaction
parameter in the crossover for $5\text{\,}\mathrm{W}$ (orange octagons) and
$10\text{\,}\mathrm{W}$ (pink pentagons) disorder laser power. In the
crossover, the ability for hydrodynamic expansion decays significantly faster
compared to interaction parameters $1/k_{\text{F}}a>1$ far from the crossover
[17]. The black line shows the calculated dephasing time
$t_{\mathrm{ph}}=\hbar/\Delta
E=\hbar\sigma_{x,y}/(\bar{V}_{\mathrm{dis}}\xi_{\mathrm{pair}})$ of the two
atoms with opposite spin for $2\bar{V}_{\mathrm{dis}}$ for molecules (disorder
laser power of $10\text{\,}\mathrm{W}$). At this time scale, two paired atoms
acquire a phase difference of $2\pi$ in the disorder potential gradient, see
inset sketch.  Figure 3: Revival of quantum hydrodynamics. (a) Relative
revival of the maximum aspect ratio (normalized to the maximum aspect ratio
without disorder) $A_{{\text{max}}_{\text{rel}}}$ as a function of the revival
time $\tau_{\text{off}}$ in the mBEC regime (blue circles) and in the UFG (red
diamonds). The red and blue lines are exponential fits. The inset shows the
timing sequence. For the unitary (mBEC) regime, a speckle laser power of
$10\text{\,}\mathrm{W}$ ($5\text{\,}\mathrm{W}$) is used to approach identical
disorder potentials. (b) Absorption pictures for the measurement points
indicated in panel (a). (c) Maximum absolute aspect ratio $A_{\text{max}}$
after quenching out of disorder as a function of the interaction parameter
$1/k_{\text{F}}a$. Dots are measurements for $5\text{\,}\mathrm{W}$ (orange
octagons), $10\text{\,}\mathrm{W}$ (purple pentagons), and for
$0\text{\,}\mathrm{W}$ (black hexagons) disorder laser power. Each data point
is an average of the three largest aspect ratios during the expansion, its
uncertainty is the standard deviation of these three points. The blue (mBEC)
and red (UFG) data points, marked with a rectangular box, indicate the
independently measured aspect ratios of a thermal gas. The gray line is a
guide to the eye between the two points including its uncertainty. (d-f)
Measured relative temperature increase for different disorder quench protocols
shown in the insets, as a function of disorder power when applied to a mBEC
(blue circles) or a UFG (red diamonds). Importantly, the UFG is heated more
strongly by the same disorder quench compared to the mBEC, indicating an
additional absorption channel in the crossover. The pulse duration is set to
$\tau_{\mathrm{on}}=$100\text{\,}\mathrm{m}\mathrm{s}$$. Each data point is an
average of five repetitions and their standard deviation is indicated as error
bar.

Decay of quantum hydrodynamics
We first study the impact of a suddenly applied disorder potential on a
superfluid in different interaction regimes and analyze the collapse of the
hydrodynamic expansion after a sudden quench into the disorder potential of
variable duration $\tau_{\mathrm{on}}$. In addition to studying the
hydrodynamic behavior, we have additionally examined the density response for
all measurements along the crossover shown in the following and report it in
the Appendix. In Fig. 2(a), we compare the decay of the maximum achievable
aspect ratio of hydrodynamic expansion of a mBEC ($1/k_{\text{F}}a=1.04$) and
a UFG at resonance ($1/k_{\text{F}}a=0$) below the critical temperature as a
function of $\tau_{\mathrm{on}}$. All data are normalized to the maximum
aspect ratio change experimentally obtained without disorder, see Fig. 1(c).
Here, for the mBEC regime, the speckle laser power is set to
$5\text{\,}\mathrm{W}$, corresponding to
$\bar{V}_{\text{dis}}^{\mathrm{mBEC}}/\mu_{\mathrm{\text{mBEC}}}\approx 1.6$
and $10\text{\,}\mathrm{W}$ for the UFG
$\bar{V}_{\text{dis}}^{\mathrm{UFG}}/\mu_{\mathrm{\text{mBEC}}}\approx 1.3$,
ensuring approximately the same, strong disorder potential height in the two
regimes. As seen in Fig. 2(a), the maximum aspect ratio decreases after
suddenly applying the disorder potential for a certain time $\tau_{\text{on}}$
approximately exponentially, indicating a collapse of quantum hydrodynamics
and hence long-range coherence. The minimum aspect ratio reached in steady
state corresponds to the respective aspect ratio of a thermal gas in the mBEC
regime or at resonance. We find that the UFG stabilizes at a higher value of
the aspect ratio compared to the mBEC, which we attribute to higher
interaction strength and hence stronger classical hydrodynamics in this
regime. As shown in the Appendix, the change of aspect ratio depends on the
trap frequencies rather than potential depth. Therefore, the difference in
classical hydrodynamics cannot be due to the different mass. The half-life
period $\tau_{1/2}$, i.e., the time at which the aspect ratio has collapsed to
half of the initial value, of the UFG is shorter with $7\pm
1.\text{\,}\mathrm{\SIUnitSymbolMicro s}$ than for the mBEC with $11\pm
2.\text{\,}\mathrm{\SIUnitSymbolMicro s}$.
From studies in the mBEC regime [17], a relatively constant $\tau_{1/2}$ value
was found for a broad range of interaction parameters outside the crossover
$1/k_{\text{F}}a>1$. The underlying mechanism explaining the time scales and
numerical simulations of the phase evolution suggested imprinting of a random
local phase onto the mBEC by the disorder. The resulting local phase evolution
destroys long-range phase coherence, and for a broad range of interaction
strengths, it is only dependent on the properties of the disorder potential.
In Fig. 2(b) we show the $\tau_{1/2}$ values when entering the crossover. The
clearly faster decay of long-range phase coherence indicates that an
additional mechanism suppressing long-range phase coherence is present.

Revival of quantum hydrodynamics
Complementary, we study the revival of the long-range phase coherence when the
system is quenched out of a disordered potential. Experimentally, we
adiabatically ramp up the speckle potential in a
$50\text{\,}\mathrm{m}\mathrm{s}$ linear ramp to avoid excitations in the
cloud. The potential is held for $100\text{\,}\mathrm{m}\mathrm{s}$ before
suddenly switching off the disorder, see inset in Fig. 3(a). After a varying
hold time $\tau_{\mathrm{off}}$ without disorder, the cloud is expanded for a
time $d$ into the magnetic saddle potential to allow for hydrodynamic
expansion before imaging. The two regimes of mBEC and UFG show a strikingly
different behavior, illustrated in Fig. 3(a,b). While the long-range phase
coherence for the mBEC recovers completely with an exponential time constant
$\tau_{1/2}^{\text{off}}=$20\pm 3.\text{\,}\mathrm{ms}$$, the UFG does not
fully revive. We show the maximum achievable value of the aspect ratio in Fig.
3(c) along the crossover without disorder and for two different disorder laser
powers. Without disorder, we find a maximum aspect ratio that slightly
increases for decreasing interaction parameter $1/k_{\text{F}}a$ in the
crossover, until it rapidly decays for negative scattering length, i.e., on
the BCS side of the crossover $1/k_{\text{F}}a<0$. For strongest disorder
applied, the gas quenched out of disorder behaves as a classical gas for all
interaction strengths. An interesting behavior is observed for intermediate
disorder laser powers. Here, the gas can fully revive in the mBEC regime, an
agreement with previous studies [17]. For decreasing interaction parameter
$1>1/(k_{F}a)>0.5$, the maximum aspect ratio decays and approaches the
classical limit until it increases again for $0.5>1/(k_{F}a)>0$. However, for
all interaction strengths in the crossover, quantum hydrodynamics does not
revive, in stark contrast to the mBEC regime outside the crossover.

Open-system disorder quenches
In order to characterize the many-body state after the quench, we additionally
measure the temperature increase for different disorder ramp procedures, see
Appendix. In Fig. 3(d-f), we show the relative temperature increase for the
limiting cases of a mBEC and a UFG for a fully adiabatic ramp, a disorder
pulse (as used in Fig. 2), and rapid quench out of disorder (as used in Fig.
3(a,c)). Clearly the UFG is heated more strongly than the mBEC, especially for
the quench protocol out of disorder Fig. 3(f). The maximum thermal energy
increase in Fig. 3(f) for the UFG is $\Delta E/E_{F}\approx 0.09$. Compared to
the energy needed for pair breaking $2\Delta_{\mathrm{gap}}=1.8E_{F}$ [38].
This maximum thermal energy increase hence amounts to $5\text{\,}\%$ of the
energy needed for pair breaking.

In order to check if the temperature increase is the main reason for collapse
of quantum hydrodynamics, we study disorder quenches in an open system, where
high-energetic particles can escape from the trap effectively reducing the
mean energy. Fig. 4 shows the maximum achievable aspect ratio as a function of
the particle losses that occurred. We find that, even for losses of more than
$60\text{\,}\%$, the mBEC recondenses and shows a close-to-full revival of
hydrodynamic expansion, and hence long-range coherence. By contrast, the UFG
does not show a significant increase in quantum hydrodynamics for any particle
loss.

Figure 4: Quench in an open system. Amount of revival of the maximum aspect
ratio $A_{\text{max}}$ as a function of atom losses (see Appendix), which are
controlled via the trap depth. A mBEC, blue circles, (UFG, red diamonds) at
$763.6\text{\,}\mathrm{G}$ ($832.2\text{\,}\mathrm{G}$) is perturbed with
$5\text{\,}\mathrm{W}$ ($10\text{\,}\mathrm{W}$) disorder laser power. The
aspect ratio after a long revival time ($\tau_{\mathrm{off}}>200\,$ms) is
taken as the maximum achievable quantum hydrodynamic in equilibrium taken from
a fit to eq. (Eq.7) as in panel Fig. 3(a) and normalized to the maximum value
without disorder, see Appendix. The insets show absorption images for the
indicated parameters.

Discussion
In summary, our data presented suggests that even for a UFG initially
adiabatically prepared to populate the ground state of a disorder potential, a
quench out of disorder permanently destroys long-range quantum coherence. This
is in stark contrast to the equilibrium expectation, where a UFG has so far
been found to form a superfluid showing largest critical velocity. A first
intuitive understanding can be obtained by considering the microscopic pairing
structure along the BEC-BCS crossover [11, 39, 40, 41]. In the Appendix, we
compare the relevant atomic energy and coherence-length scales along the
crossover to energy and length scales of the disorder potential. Deep in the
mBEC regime, the pair size is much smaller than any length scale of the
disorder potential. At the same time, the molecular binding energy is so large
that no energy scale of the disorder is comparable. Thus, the disorder
potential primarily affects the global wave function, which is perturbed at
the length scale of the healing length and the energy scale of the chemical
potential, leading to wave-like excitations of the many-body system. In the
crossover, however, the pair size increases and becomes of the order of the
healing length, being smaller but of the order of the correlation length of
the disorder. As shown in the inset of Fig. 2(b), local gradients of the
disorder hence become relevant on the length scale of the pair. At the same
time, the binding energy of the molecular state decreases when approaching the
resonance $1/k_{\text{F}}a\to 0$. Here, the energy scales of the time-
dependent disorder (inverse ramping time
$E_{\text{quench}}=\hbar/t_{\text{on}}$, correlation energy
$E_{\sigma}=\hbar^{2}/(m\sigma^{2})$ [42], mean potential strength
$\bar{V}_{\text{dis}}$) at strong enough interaction become compatible to the
binding energy $E_{\text{B}}=\hbar^{2}/(ma^{2})$ [28] (see Appendix). This
microscopic matching of pair properties to the length and energy scales of the
quenched disorder can explain the additional absorption channel for energy in
the UFG. The total thermal energy absorbed, however, is not large enough to
account for substantial pair breaking. For two-dimensional strongly disordered
superconductors it was shown in Ref. [43] that the strongly disordered many-
body system can form superconducting islands separated by an insulating sea,
where the islands are not phase coherent. While our system is three
dimensional, and our disorder is formed by a repulsive potential far from the
percolation threshold, so that the system is expected to be always fully
connected, our observation might point toward a local dephasing of the UFG by
strong disorder quenches, potentially at the level of individual pairs.

Our studies of the stability of interacting Fermi superfluids along the BEC-
BCS crossover in the presence of a strong time-dependent perturbation clearly
show that the static properties of the disordered BEC-BCS crossover are very
different from the observations obtained in this work for the time-dependent
nonequilibrium case. In the future, it will be interesting quantiatively map
out the phase diagram of the disordered BEC-BCS crossover away from
equilibrium. Additionally, reducing the dimensionality of the gas and
producing homogeneous gases in different dimensions will provide a deeper
insight into the connection between microscopic pairing mechanism and
macroscopic nonequilibrium dynamics.

###### Acknowledgements.

This work was supported by the German Research Foundation (DFG) via the
Collaborative Research Center Sonderforschungsbereich SFB/TR185 (Project
277625399). J.K. was supported by the Max Planck Graduate Center with the
Johannes Gutenberg-Universität Mainz. We thank Carlos A. R. Sá de Melo,
Giuliano Orso, Giacomo Roati, Benjamin Nagler and Alexander Guthmann for
fruitful discussions and Eloisa Cuestas for carefully reading the manuscript.
We acknowledge help by Valeriya Mikhaylova with the graphical implementation
of Fig. 1a).

## Appendix

## Quantum-gas preparation

We evaporate our laser-cooled samples at a magnetic field of
$763.6\text{\,}\mathrm{G}$ ($a=4509a_{0}$, with the Bohr radius $a_{0}$) in
the BEC regime for all observed interacting regimes. During evaporation, the
laser power of the optical-dipole trap (ODT) is lowered from
$140\text{\,}\mathrm{m}\mathrm{W}$ to $8\text{\,}\mathrm{m}\mathrm{W}$ by two
subsequent exponential power ramps during $4.38\text{\,}\mathrm{s}$. This
final laser power of $8\text{\,}\mathrm{m}\mathrm{W}$ is relatively low and
enables controlling atom losses through disorder quenches. After the
evaporation step, the laser power is held for
$250\text{\,}\mathrm{m}\mathrm{s}$ to equilibrate the temperature of the
cloud. Subsequently, the trap is compressed by increasing the optical-dipole-
trap (ODT) laser power up to $80\text{\,}\mathrm{m}\mathrm{W}$ during
$300\text{\,}\mathrm{m}\mathrm{s}$. By varying the level of compression, the
amount of losses due to the disorder field can be adjusted. No losses occur at
a power of $80\text{\,}\mathrm{m}\mathrm{W}$.
After the condensate is formed, the magnetic field is adiabatically ramped to
the desired field for interaction control during
$200\text{\,}\mathrm{m}\mathrm{s}$. The cloud is trapped in a combination of
the ODT, created with a focused laser beam of a wavelength of
$1070\text{\,}\mathrm{n}\mathrm{m}$ and a magnetic saddle potential. Further,
the magnetic field strength determines the scattering length and, therefore,
the interacting regime between the two spin states.
Before disturbing the cloud by quenching the disorder field, another holding
time of $30\text{\,}\mathrm{m}\mathrm{s}$ ensures that no excitations of the
cloud are present after changing the magnetic field strength. The trap
frequencies in the radial directions, $x$ and $z$, increase with the square
root of the laser power of the ODT. The axial trap frequency, $y$ direction,
is within the range used, independent of the laser power. Changing the
magnetic field has a negligible influence on the trap frequency compared to
the genuine frequency. The trap frequencies are
$\omega_{x},\omega_{y},\omega_{z}=2\pi\times(345,23,220)\,\text{Hz}$ for an
ODT laser power of $80\text{\,}\mathrm{m}\mathrm{W}$ and a magnetic field
strength of $763.6\text{\,}\mathrm{G}$.

## Disorder potential

A far-off resonant, blue-detuned laser with a wavelength of
$532\text{\,}\mathrm{n}\mathrm{m}$ generates the potential. The envelope of
the laser is a Gaussian with waists of $466\pm
25.\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and $414\pm
25.\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ along two orthogonal
directions [44]. Moreover, the optical speckle potential is formed by shining
the collimated laser beam through two successive speckle plates. An objective
focuses the random phase pattern on the position of the atoms. Hence,
anisotropic speckle grains with sizes
$\sigma_{x,y}=$750\text{\,}\mathrm{n}\mathrm{m}$$ and
$\sigma_{z}=$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ are formed.
The speckle plates are mounted in a motorized turntable; one of them is
rotated by a certain angle after each measurement, so that the interference
potential changes in each experimental realization. We characterize the
strength of the disorder potential $V_{\text{dis}}$ by the mean disorder
potential $\bar{V}_{\text{dis}}$, which is the overall spatial average. For
comparison, we may express the mean disorder potential in units of the
unperturbed chemical potential. For the mBEC, the chemical potential [45]

$\mu_{\mathrm{mBEC}}=\frac{\hbar\bar{\omega}}{2}\biggl{(}\frac{15\frac{N_{\uparrow\downarrow}}{2}a_{\mathrm{dd}}}{a_{\mathrm{ho}}}\biggr{)}^{2/5}$
(1)

is established through the Gross-Pitaevskii equation, where
$a_{\mathrm{dd}}=0.6a$ [46] is the $s$-wave scattering length for molecules,
$a_{\mathrm{ho}}=\sqrt{\hbar/(2m\bar{\omega})}$ the harmonic oscillator length
with the mass $m$ of the lithium atom. The chemical potential of the UFG [47]

$\mu_{\mathrm{UFG}}=\sqrt{1+\beta}E_{\text{F}}$ (2)

is proportional to the Fermi energy with the universal constant
$\sqrt{1+\beta}$.

## Extraction of the maximum aspect ratio

Figure 5: Comparison of different fit functions to in situ and expansion
density profiles. Measured line density (gray points) in dependence of the $x$
and $y$ direction, once in the mBEC regime at $763.6\text{\,}\mathrm{G}$ and
once at resonance at $832.2\text{\,}\mathrm{G}$. (a,b) mBEC, in situ and
expansion in dependence of the $x$ direction, fit to the data with a Thomas-
Fermi fit in the mBEC regime (solid blue line). (c & d) mBEC, in situ and
expansion in dependence of the $y$ direction, fitting to the data with a
bimodal fit (solid orange line). The bimodal fit consists of a Thomas-Fermi
fit (solid blue line) and a fit with a Gaussian profile (rose dashed-dotted
line). (e & f) At unitarity, in situ and expansion in dependence of the $x$
direction, fit to the data with a Thomas-Fermi fit in the mBEC regime (solid
blue line) and with one of a UFG (red dashed line). (g & h) At unitarity, in
situ and expansion in dependence of the $y$ direction, fit to the data with a
bimodal fit (solid orange line). The bimodal fit consists of a Thomas-Fermi
fit in the mBEC regime (solid blue line) and a fit with a Gauss profile (rose
dashed-dotted line). For comparison, a fit with a Thomas-Fermi fit at
resonance (red dashed line). d) & h) show the line densities in $y$ direction
by expansion of the cloud. The measured line density is only fitted in the
area marked with the green background and the whole cloud is inferred. Due to
systematic imaging errors, it is difficult to fit the whole range. At
resonance, both fit-methods fit well with the data.

The aspect ratio is the ratio of the width of the clouds in the axial and
radial directions, where the width is extracted from the absorption images. We
integrate the density distributions in the imaging plane separately along the
$x$ and $y$ directions to obtain two one-dimensional density profiles. The
width of the cloud is determined as the fitting parameter of the 1D density
profile for each direction. However, the density profiles of a mBEC and a UFG
are generally different, and the Thomas-Fermi density profiles of a mBEC and a
UFG do not have the same exponent. To avoid any method-depending systematics
and to obtain a homogeneous evaluation method when analyzing the density
profiles along the crossover, we employ the same fitting profile for all
density distributions measured at various interaction parameters. This is
supported by Fig. 5, where the different fitting functions are compared and
show only small differences. Thus, we take the profiles that are suitable for
a BEC as fit functions for the whole crossover. After expansion, the radial
direction is the long axis of the cloud through inversion of the aspect ratio.
In the $x$ direction, we apply a 1D Thomas-Fermi profile $n_{\text{TF}}$ [48]

$n_{\text{TF}}\propto\left(1-\frac{i^{2}}{r_{\text{TF}_{i}}^{2}}\right)^{2},$
(3)

with $r_{\text{TF}_{i}}$ the radius in $i$-direction. Besides that, in the $y$
direction, the density shows a sharp peak and a non-neglecting background,
which makes it difficult to fit a Thomas-Fermi profile. Therefore, this
direction is fitted with a 1D-bimodal fit
$n_{\text{bimodal}}=n_{\text{G}}+n_{\text{TF}}$. The background is fitted with
a Gaussian 1D-profile $n_{\text{G}}$ given by

$n_{\text{G}}\propto e^{-\frac{i^{2}}{2s_{i_{\text{G}}}^{2}}},$ (4)

with the width $s_{i_{\text{G}}}$. We extract the full width at half maximum
(FWHM) of the total fitted 1D-density profiles for both directions. The aspect
ratio is the ratio of the FWHM of the $x$ direction by the one of the $y$
direction. By varying the expansion time $d$, we measure a change in the
aspect ratio. The maximum value quoted in the main text is extracted as the
mean from the three highest ratios during the expansion evolution, and their
standard deviation is indicated by the error bars.
For comparison, on resonance, the 1D Thomas-Fermi density profile
$n_{\text{TFU}}$ is [49]

$n_{\text{TFU}}\propto\left(1-\frac{i^{2}}{r_{\text{TFU}_{i}}^{2}}\right)^{5/2},$
(5)

with the radius $r_{\text{TFU}_{i}}$ in $i$-direction. Figure 5 shows fits of
the measured density profiles and compares a bimodal fit (Eq. 3 & 4) with the
Thomas-Fermi fit on resonance (Eq. 5) for clouds at resonance. Both versions
fit the data well.

## Influence of the interaction parameter on the expansion dynamics

A mBEC and a UFG show quantum hydrodynamic expansion [28]. In the case of a
mBEC, a UFG, or a thermal gas, the in-situ aspect ratio is independent of the
mass of the particles. The radii for all three cases individually depend on
the mass, but it cancels when computing the aspect ratio. Instead, the aspect
ratio depends on the trap frequencies. During expansion, the aspect ratio
$\frac{R_{x}(t)}{R_{y}(t)}=\frac{b_{x}(t)}{b_{y}(t)}\frac{\omega_{y}}{\omega_{x}}$
changes according to scaling factors $b_{i}$

$\ddot{b}_{i}-\frac{\omega_{i}^{2}}{b_{i}(b_{x}b_{y}b_{z})^{\gamma}}=0,$ (6)

which can be derived from the Euler equation [28]. The scaling factors are
independent of the mass. The expansion of the mBEC and the UFG differ by the
exponent gamma (BEC $\gamma=1$, UFG $\gamma=2/3$ ), since the difference of
the chemical potential depends on the density in the two regimes
($\mu_{\mathrm{BEC}}\propto n$, $\mu_{\mathrm{UFG}}\propto n^{2/3}$) [50].
Moreover, the trap frequencies are identical for a Feshbach molecule or a
single atom, and hence the aspect ratio during expansion is independent of the
mass of the particles. A thermal gas shows a ballistic expansion with scaling
factors also independent of the particle mass.

Figure 6: Mapping losses. Mapping of remaining atoms
$N_{\text{quench}}/N_{\text{ref}}$ in the trap after quenching the disorder
field as a function of the power of the ODT laser for the compression.
$N_{\text{quench}}$ ($N_{\text{ref}}$) is the atom number with (without)
disorder field. For the unitary regime ($832.2\text{\,}\mathrm{G}$), a speckle
power of $10\text{\,}\mathrm{W}$ (red, diamonds) and for the mBEC regime
($763.6\text{\,}\mathrm{G}$) a speckle power of $5\text{\,}\mathrm{W}$ is
applied (blue, circles). The data points for the decay (dark) and the ones for
the revival (light) are shown. The error bars are calculated by the error
propagation of the standard deviation of five repetitions. The solid lines are
fits with an error function. For comparison, the power of the ODT laser is
mapped to the trap depth because this is different for molecules and atoms for
the same powers (see red and blue axis). The left (right) inset shows the
disorder pulse shape for the decay (revival) of the quantum properties. The
imaging (vertical red line in inset) takes place in situ after an expansion
time $d=0$.

## Measuring particle losses versus trap depth

We adjust the atom losses induced by the disorder field through different
compression values of the trap. A lower ODT laser power leads to a smaller
compression and a lower trap depth, which leads to larger atom losses in a
disorder quench and vice versa. For a sufficiently large compression of the
trap, we measure no atomic losses that are caused by the speckle field. Figure
6 shows the mapping between the power of the ODT laser after compression and
the remaining particles in the trap. The measurements are performed for the
mBEC regime at $763.6\text{\,}\mathrm{G}$ and on resonance at
$832.2\text{\,}\mathrm{G}$. As noted in the main text, the disorder strength
of the speckle potential is twice as much for molecules compared to single
atoms for the same laser power. Hence, for the mBEC regime, the speckle-laser
power is set to $5\text{\,}\mathrm{W}$ and for the unitary regime to
$10\text{\,}\mathrm{W}$ to get the same effective disorder potential. As a
reference, the atom number is also measured without a disorder field for both
regimes. For an identical trap depth, the remaining particles in the trap are
equal in both regimes. In addition, we have investigated the losses for two
different instantaneous disorder quenches, one when switching the disorder
field on and one when switching it off. The amount of atoms that leave the
trap is higher for a quench into the disorder field compared to a quench out
of the disorder field, where we introduce the speckle potential adiabatically
rather than instantaneously (see Fig. 6). Moreover, we have measured similar
loss curves for the other magnetic field strengths used. No losses occur for
all applied magnetic fields for an ODT laser power of
$80\text{\,}\mathrm{m}\mathrm{W}$.
For these measurements, as for the measurements presented in the main text,
the evaporation of the cloud takes place at $763.6\text{\,}\mathrm{G}$, and
the power of the ODT laser is lowered to $8\text{\,}\mathrm{m}\mathrm{W}$.
Subsequently, the cloud is held for $250\text{\,}\mathrm{m}\mathrm{s}$ before
the trap is compressed. The power of the ODT laser and hence the trap
compression is set to a selected value. As the next step, the magnetic field
is ramped to the desired value in a duration of
$200\text{\,}\mathrm{m}\mathrm{s}$. Subsequently, the cloud is held in the
trap for $30\text{\,}\mathrm{m}\mathrm{s}$ before switching on the disorder
speckle laser. Lastly, the cloud is imaged in situ by absorption imaging. The
atom number is determined by summing up the intensity of the picture pixel-
wise.

## Determination of the half-life period by expansion dynamics

For the decay and the revival of the long-range phase coherence (where the
cloud fully recovers), the half-life period $\tau_{1/2}$ is determined similar
to Ref. [17]. When plotting the aspect ratio as a function of the speckle-
pulse length $\tau_{on}$ or the revival time $\tau_{\text{off}}$, it shows an
exponential evolution to a steady state.
This behavior is fitted with an exponential decay function (see Fig. 2(a) and
3(a))

$h(\tau)=a\,e^{-\gamma\tau}+o,$ (7)

with $\gamma$ is the decay constant and, $o$ and $a$ are further fit values.
Here, the half-life period is calculated via

$\tau_{1/2}=\ln(2)/\gamma.$ (8)

The uncertainty of the half-life period is taken as the fit uncertainty.

## Hydrodynamics from quantum to classical regimes

Figure 7: Expansion of a thermal gas. Maximum aspect ratio $A_{\text{max}}$ as
a function of the power of the ODT laser at the end of the evaporation. Shown
are experimental data of a mBEC at $763.6\text{\,}\mathrm{G}$ (blue) and a UFG
at $832.2\text{\,}\mathrm{G}$ (red). The experimental data are fitted with an
exponential decay function (solid lines). The final state is fitted with a
constant (dashed line) by the last six measured values. The fit uncertainties
are the blue and red-filled areas around the dashed lines.

In order to compare the maximum aspect ratio of quantum gases after quenching
the disorder potential with that of a gas above the critical temperature, we
record the hydrodynamic expansion of a thermal gas without disorder. The
temperature of the gas is adjustable via the final laser power of the ODT
during evaporation. The power of the ODT laser at the end of the evaporation
is varied between $8\text{\,}\mathrm{m}\mathrm{W}$ and
$350\text{\,}\mathrm{m}\mathrm{W}$. Afterward, the trap is compressed by
increasing the power of the ODT laser. To avoid further evaporation and have
the same conditions for the different powers at the end of the evaporation,
the power for the compression is set to $400\text{\,}\mathrm{m}\mathrm{W}$ for
all varied powers. At an ODT laser power of $350\text{\,}\mathrm{m}\mathrm{W}$
at the end of evaporation, the atomic cloud has a temperature of about
$1200\text{\,}\mathrm{n}\mathrm{K}$. We measure the maximum aspect ratio of
the cloud by expansion for two magnetic field strengths,
$763.6\text{\,}\mathrm{G}$ and $832.2\text{\,}\mathrm{G}$ (see Fig. 7). In
both regimes, we see a decrease in the maximum aspect ratio with increasing
power, i.e., temperature until it reaches a final state, a thermal gas,
exhibiting purely classical hydrodynamics. At this final state, there is no
long-range phase coherence present. For the UFG, the aspect ratio of the
final, thermal state is higher than that of the mBEC. We attribute this to
stronger interactions, leading to stronger collisional hydrodynamics. Due to
the higher ODT power of $400\text{\,}\mathrm{m}\mathrm{W}$ (largest value in
Fig. 7) compared to $80\text{\,}\mathrm{m}\mathrm{W}$ (shown in Fig. 2(a)
without losses), the aspect ratios are not comparable since the power effects
the trap frequencies. Therefore, the aspect ratio of the thermal gas at
$400\text{\,}\mathrm{m}\mathrm{W}$ is adjusted to the one in Fig. 2a) at
$80\text{\,}\mathrm{m}\mathrm{W}$. It is scaled according to
$\sqrt{\omega_{$80\text{\,}\mathrm{m}\mathrm{W}$}/\omega_{$350\text{\,}\mathrm{m}\mathrm{W}$}}$,
where $\omega$ is the geometric mean of the trap frequencies in the radial
directions. This follows from the ideal-gas condensate, where the expansion
radius is proportional to the square-root of the trap frequency [51].

## Energy calculations

Figure 8(a) compares relevant energies of the quenched disorder potential with
energy scales of the many-body system, especially the molecular binding energy
$E_{\text{B}}=\hbar^{2}/(ma^{2})$ [28], in the BEC-BCS crossover. Calculations
are done using experimental parameters, specifically total atom number
$N_{\uparrow\downarrow}=5.13\times 10^{5}$ (average of experimental values)
and trap frequencies
$\omega_{x},\omega_{y},\omega_{z}=2\pi\times(345,23,220)\,\text{Hz}$. The
Fermi energy $E_{\text{F}}=\hbar\bar{\omega}(3N_{\uparrow\downarrow})^{1/3}$
is calculated via the geometric mean $\bar{\omega}$ of the trap frequencies
[3]. Besides, the mean disorder potential strength $\bar{V}_{\text{dis}}$ is
shown and the correlation energy $E_{\sigma}=\hbar^{2}/(m\sigma^{2})$ [42] is
calculated. These two quantities are different for atoms or molecules at the
same laser power due to the difference in mass. The energy scale of the quench
time of the speckle field is given by $E_{\text{quench}}=\hbar/t_{\text{on}}$,
where the time to switch on the speckle field instantaneously is measured with
$t_{\text{on}}=$2.26\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$$.
Fig. 8(b) shows the change of coherence lengths $\xi_{\mathrm{pair}}$ and
$\xi_{\mathrm{phase}}$ of the interacting many-body system along the BEC-BCS
crossover, see Refs. [11, 39, 40, 41]. Deep in the mBEC regime, both differ
significantly. The molecules are relatively small and the healing length
$\xi=\xi_{\mathrm{phase}}$ is much larger, increasing for larger interaction
parameters $1/(k_{F}a)$. Hence, $\xi_{\mathrm{phase}}>\xi_{\mathrm{pair}}$.
Approaching the resonance, the healing length decreases, while the molecules
become larger as the molecular bound state energy approaches the dissociation
threshold. Upon entering the crossover $1/(k_{F}a)\approx 1$, the two length
scales approximately coincide $\xi_{\mathrm{phase}}=\xi_{\mathrm{pair}}$. On
resonance, the pair coherence length is larger than the phase coherence
length. In the BCS regime, the two quantities scale the same and differ only
by some constant factor. The coherence lengths allow estimating the effect of
local potential changes onto the fermionic pair.

## Temperature measurements

We have measured the relative temperature increase in the quantum gases for a
mBEC at $763.6\text{\,}\mathrm{G}$ and a UFG at $832.2\text{\,}\mathrm{G}$
through the disorder potential for different disorder ramp procedures (see
Fig. 3(d-f)) applying up to $10\text{\,}\mathrm{W}$ laser power creating the
disorder field. As a reference, we measure the gas temperature $T_{0}$ without
the disorder field. Three disorder ramping procedures are investigated. First
(Fig. 3(d)), the disorder field is adiabatically introduced by a linear ramp
during a $50\text{\,}\mathrm{m}\mathrm{s}$, held for
$100\text{\,}\mathrm{m}\mathrm{s}$, and subsequently switched off in
$50\text{\,}\mathrm{m}\mathrm{s}$ by a linear ramp. Second (Fig. 3(e)), the
disorder is rapidly switched on (switching time $<5\,\mu$s) for a rectangular
pulse for a duration of duration
$\tau_{\text{on}}=$100\text{\,}\mathrm{m}\mathrm{s}$$. Lastly (Fig. 3(f)), the
disorder field is adiabatically introduced by a linear ramp during a
$50\text{\,}\mathrm{m}\mathrm{s}$, and subsequently the power is held for
$100\text{\,}\mathrm{m}\mathrm{s}$, before the field is suddenly switched off.
After a holding time of $100\text{\,}\mathrm{m}\mathrm{s}$, the magnetic field
is adiabatically swept to $680\text{\,}\mathrm{G}$. Subsequently, the cloud is
imaged in situ via absorption imaging. At this magnetic field strength, a
bimodal fit to the density profiles allows the extraction of the temperature
of the cloud.

## Density response to disorder quenches

Figure 8: Energy and length scales. (a) Comparison of relevant energy scales
along the crossover in units of temperature as a function of the magnetic
field values. Calculations are done with experimental parameters, see
Appendix. (b) Pair correlation length $\xi_{\text{pair}}$ (gray dashed line)
and phase coherence length $\xi_{\text{pair}}$ (yellow line) as a function of
the interaction parameter $1/k_{\text{F}}a$, calculated for the three-
dimensional case at zero temperature according to Refs.[11, 39, 40, 41].
Figure 9: Density response to disorder quenches. Decay (dark blue and dark
red) and revival (light blue and light red) of the fragmentation $\alpha$,
normalized to the respective maximum of the fit function
($\alpha_{\text{rel}}$), as a function of the disorder pulse duration
$\tau_{\text{on}}$ or the revival time $\tau_{\text{off}}$. Shown are the
experimental values of a mBEC at $763.6\text{\,}\mathrm{G}$ (blue points) and
a UFG at $832.2\text{\,}\mathrm{G}$ (red diamonds), where each data point is
averaged from five measurements, and its standard deviation corresponds to the
error bar. The solid lines are fits with a Gompertz function. Further, the
offset $o$, which is extracted from the fit, is subtracted from the data and
the fit to obtain the fragmentation, which is only caused by the disorder
field. Figure 10: Interaction dependence of the density response to quenches
into disorder. Half-life period $\tau_{1/2}$ as a function of the magnetic
field. As for the expansion dynamics, the half-life period decreases from the
mBEC side towards the resonance. The inset shows the pulse shape for the decay
measurement. The cloud is immediately imaged in situ after the disorder pulse
(vertical red line). Error bars are determined via the fit uncertainties (see
Eq.12). Figure 11: Interaction dependence of the density response to quenches
out of disorder. (a) Half-time period $\tau_{1/2}$ as a function of the
magnetic field. The half-time period of the density variation decreases with
decreasing the interaction parameter. The inset shows the disorder realization
of the revival measurement. The cloud is imaged in situ (vertical red line)
after the revival time $\tau_{\text{off}}$. (b) Amount of revival of the
fragmentation $\alpha$ in dependence of the magnetic field $B$. The
fragmentation fully recovers for $5\text{\,}\mathrm{W}$ (orange) and even for
$10\text{\,}\mathrm{W}$ (purple) power of the disorder laser. The inset shows
in situ absorption pictures when the fragmentation reaches its highest value
due to the disorder field.

The disorder potential leads to a spatial density variation of the cloud. For
every parameter set used to measure the hydrodynamic response of quantum
gases, we have also measured the density response. The difference between
probing the density or the hydrodynamic response is the expansion time $d$.
The density variation is probed by in-situ
($d=$0\text{\,}\mathrm{m}\mathrm{s}$$) absorption images of the column-
integrated density distribution $n(x,y)$ of the atomic cloud while for probing
the hydrodynamic response the expansion time is in the order of
$d=$10\text{\,}\mathrm{m}\mathrm{s}$$. The evaluation procedure of the
absorption images is taken from [17]. To extract the disorder-induced
perturbation of the cloud, we first fit a 2D Thomas-Fermi profile [51]

$n_{\text{2D}}\propto\left(1-\frac{(x-x_{0})^{2}}{r_{y}^{2}}-\frac{(y-y_{0})^{2}}{r_{x}^{2}}\right)^{3/2},$
(9)

which is valid in the mBEC regime. For a UFG, the profile should be
distinguished from a mBEC in its power law. However, as discussed in the case
of hydrodynamic expansion above, we apply the evaluation described for the
mBEC regime for all data in order to obtain a homogeneous analysis.
The fitted smooth 2D Thomas-Fermi profile is subtracted from the measured
data, leaving only the density variations around the mean. The density
perturbation is quantified via the fragmentation $\alpha$, see Ref. [17],
defined as

$\alpha=\sqrt{\langle\Delta n^{2}\rangle-\langle\Delta n\rangle^{2}},$ (10)

with $\Delta=n-n_{\text{fit}}$, where $n_{\text{fit}}$ is the fitted 2D
Thomas-Fermi profile of the measured density distribution $n$. Further, the
brackets denote averaging over all pixels, where $n_{\text{fit}}>0$. Due to
fluctuations already without any disorder field $\alpha\neq 0$ [17]. In
contrast to the hydrodynamic expansion, the fragmentation occurs and recovers
on similar time scales. Moreover, the density distribution becomes completely
smooth, i.e. fragmentation completely vanishes after disturbing the cloud (see
Fig. 9). This is independent of the interaction strength, the powers of the
laser ($5\text{\,}\mathrm{W}$ and $10\text{\,}\mathrm{W}$), which creates the
disorder potential, and even in an open system with significant atom/molecule
losses (see Fig. 11 (b) 12(c)). We interpret this as a consequence of local
transport processes occurring, where we cannot distinguish classical or
quantum contributions.
For extracting the half-life period, the fragmentation is fitted with a
Gompertz function $g$ [17, 52]

$g(t)=a\,e^{-e^{-t/c}/b}+o,$ (11)

with the fit parameters $a$, $b$, $c$ and $o$. The half-life period
$\tau_{1/2}$ is calculated by

$\tau_{1/2}=-c\,\ln(b\,\ln(2)).$ (12)

The amount of the revival of the fragmentation is calculated from the fit
parameters. The minimal fragmentation $\alpha_{\text{min}}$ of the revival
(quench out of) is set in relation to the minimal fragmentation of the decay
measurements (quench into). For the decay measurement, the Gompertz function
is calculated for $t=0$ ($\alpha_{\text{min}}=a\,e^{(-1/b)}+o$). For the
revival measurement, $t$ is considered in the limiting case towards infinity
($\alpha_{\text{min}}=a+o$).
In a direct comparison of the half-life periods between a mBEC and a UFG, we
measure that the new final state, for the decay measurement, is assumed faster
for a UFG (UFG: $\tau_{1/2}^{\text{on}}=$72\pm
17.\text{\,}\mathrm{\SIUnitSymbolMicro s}$$, mBEC:
$\tau_{1/2}^{\text{on}}=$112\pm 23.\text{\,}\mathrm{\SIUnitSymbolMicro s}$$).
We attribute this to a larger interaction strength and, thus, increased
collision rate. The same can also be seen with the revival (UFG:
$\tau_{1/2}^{\text{off}}=$140\pm 21.\text{\,}\mathrm{\SIUnitSymbolMicro s}$$,
mBEC: $\tau_{1/2}^{\text{off}}=$169\pm 36.\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$). Over the BEC-BCS crossover, we see a decrease in the half-life period
with increasing magnetic field (see Fig. 10 and 11(a)). This can be attributed
to the larger scattering length, where local particle scattering and transport
mechanisms may be enhanced.
By directly comparing the hydrodynamics and the density response, we see that
the decay of the hydrodynamics for the mBEC an the UFG is one order of
magnitude faster than the density response, as has been seen in Ref. [17].
This can be understood intuitively for the mBEC regime, where the quenched
disorder potential imprints a random local phase on the many-body
wavefunction. This leads to quick dephasing and decay of long-range coherence.
Once local phase differences have been established, this phase difference
drives local currents that lead to the emergence of the density variation of
fragmentation at a later time. Therefore, the density response is delayed
compared to the hydrodynamic decay.

Figure 12: Density response for disorder quenches in an open system. Particle
losses for the mBEC (UFG) at $763.6\text{\,}\mathrm{G}$
($832.2\text{\,}\mathrm{G}$) for a power of the disorder laser of
$5\text{\,}\mathrm{W}$ ($10\text{\,}\mathrm{W}$) (a) Amount of the revival of
the fragmentation. The minimum of fragmentation of the revival measurement is
set in ratio to the minimum of the fragmentation $\alpha_{\text{min}}$ of the
decay measurement. Values are extracted from the fit parameters of the
Gompertz-fit, and the error bars are calculated via error propagation of the
fit uncertainties. (b) Half-life period $\tau_{1/2}$ for the decay (dark) and
revival (bright) of density variation for the unitary (red) and mBEC (blue)
regime in dependence of provoked atom losses. Revival takes roughly a factor
of two longer than the decay of the density. The decay time for expansion and
fragmentation is unaffected by losses, but the revival time increases with
particle losses. (c) Half-life period $\tau_{1/2}$ for the decay (dark) and
revival (bright) of the aspect ratio for a UFG (red) and mBEC (blue) in
dependence of provoked particle losses. Only the mBEC is shown for the revival
because the UFG does not fully revive.

Furthermore, we have also studied the density response for dissipative
quenches (see Fig. 12). Even with losses, the fragmentation fully revives for
a mBEC and a UFG (see Fig. 12(a)). The losses influence the revival time,
which increases for both interaction regimes with higher particle losses (see
Fig. 12(b)). The half-life period for the decay seems to be unaffected by the
particle losses. This might be explained by a modified density after loss have
occurred. The local collisions that re-smooth the density occur at a reduced
rate when the density is reduced, resulting in a longer time to reach
equilibrium. Further, the same is observed for the half-life period by the
hydrodynamic response of the mBEC (see Fig. 12(c)). As in [17], the half-life
period for the revival of a mBEC takes some orders of magnitude longer than
the decay. For reduced density, the time for reaching an equilibrium quantum
gas is reduced. We attribute this to the fact that the local collision rate
sets the time for decay of Bogoliubov excitations.

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