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# Dynamics of spatial phase coherence in a dissipative Bose-Hubbard atomic
system

Rémy Vatré  Raphaël Bouganne  Manel Bosch Aguilera  Alexis Ghermaoui  Jérôme
Beugnon  Raphael Lopes  Fabrice Gerbier<EMAIL_ADDRESS>Laboratoire Kastler Brossel, CollÃ̀¨ge de France, CNRS, ENS-Université PSL,
Sorbonne Université, 11 place Marcelin Berthelot, F-75231 Paris, France

###### Abstract

We investigate the loss of spatial coherence of one-dimensional bosonic gases
in optical lattices illuminated by a near-resonant excitation laser. Because
the atoms recoil in a random direction after each spontaneous emission, the
atomic momentum distribution progressively broadens. Equivalently, the spatial
correlation function (the Fourier-conjugate quantity of the momentum
distribution) progressively narrows down as more photons are scattered. Here
we measure the correlation function of the matter field for fixed distances
corresponding to nearest-neighbor (n-n) and next-nearest-neighbor (n-n-n)
sites of the optical lattice as a function of time, hereafter called n-n and
n-n-n correlators. For strongly interacting lattice gases, we find that the
n-n correlator $C_{1}$ decays as a power-law at long times, $C_{1}\propto
1/t^{\alpha}$, in stark contrast with the exponential decay expected for
independent particles. The power-law decay reflects a non-trivial dissipative
many-body dynamics, where interactions change drastically the interplay
between fluorescence destroying spatial coherence, and coherent tunnelling
between neighboring sites restoring spatial coherence at short distances. The
observed decay exponent $\alpha\approx 0.54(6)$ is in good agreement with the
prediction $\alpha=1/2$ from a dissipative Bose-Hubbard model accounting for
the fluorescence-induced decoherence. Furthermore, we find that the n-n
correlator $C_{1}$ controls the n-n-n correlator $C_{2}$ through the relation
$C_{2}\approx C_{1}^{2}$, also in accordance with the dissipative Bose-Hubbard
model.

##  I Introduction

Interference is a central phenomenon in quantum mechanics [1]. In the early
days of its development, the realization that the mechanical properties of a
quantum particle can display interference phenomena led to the discovery of
wave-particle duality, often presented in introductory textbooks as the first
example of the counter-intuitive features of quantum theory. Today, matter
wave interferences are routinely observed experimentally and used in high-
precision gravity or rotation sensors.

The textbook picture of a perfectly coherent matter wave is usually far from
experimental realities. In a real system, phase coherence is usually
maintained only in a coherence volume $l_{\mathrm{c}}^{\mathcal{D}}$, with
$l_{\mathrm{c}}$ the characteristic coherence length and $\mathcal{D}$ the
dimensionality. More formally, the degree of spatial phase coherence of a
quantum system can be quantified by the first-order correlation function of
the quantized matter field $\hat{\Psi}$ (also called single-particle density
matrix–SPDM) [2],

$\displaystyle\rho^{(1)}(\bm{s})=\int
d^{\mathcal{D}}\bm{R}\,\langle\hat{\Psi}^{\dagger}(\bm{R}+\bm{s}/2)\hat{\Psi}(\bm{R}-\bm{s}/2)\rangle.$
(1)

Here and in the following, the brackets $\langle\cdot\rangle$ denote the
expectation value. Physically, the SPDM $\rho^{(1)}(\bm{s})$ quantifies the
contrast of a matter wave interferometer with an arm separation $\bm{s}$.
Besides interferometry, the SPDM is also key to the description of degenerate
quantum fluids. Long-range spatial phase coherence where $l_{\mathrm{c}}$ is
comparable to the size of the system is the hallmark of Bose-Einstein
condensation in an ultracold gas of bosonic atoms [2, 3, 4, 5, 6, 7, 8].

Position measurements erase the spatial phase coherence present before the
measurement, at least on spatial scales comparable or greater than the
measurement resolution. The dynamics of the associated decoherence process is
an important question that also emerged during the development of quantum
mechanics, e.g. in the celebrated Heisenberg microscope thought experiment
[1], and has been revisited many times (see, e.g., Refs. [9, 10, 11, 12] for
recent discussions and experiments). A simple but experimentally relevant
example is a single atom illuminated by near-resonant light of wavelength
$\lambda_{0}$. Here, light scattering from the incident mode to an initially
empty mode can be interpreted as a weak, continuous measurement of the
position of the atom [13, 14, 15, 16, 17]. Indeed, collecting the scattered
photons with a suitable imaging system allows one (at least in principle) to
infer the location of the source, i.e. the atom. As more photons are scattered
as a single realization of the experiment unravels, the position distribution
of the atom (initially completely uncertain if the atom starts in a momentum
eigenstate) progressively narrows down to a particular random location. Upon
averaging over many realizations of the experiment, one retrieves a uniform
average position distribution. In terms of the SPDM, the pinning of the
particle position erases the spatial coherence of the initial state, leading
to an exponential damping of $\rho^{(1)}(\bm{s})$ as long as
$|\bm{s}|\gg\lambda_{0}$. In this elementary example, the damping rate is
simply given by the rate of spontaneous emission $\gamma_{\mathrm{sp}}$ (or
equivalently, the photon scattering rate).

Instead of interferometry, the spatial coherence of a quantum system can also
be characterized by measuring its momentum distribution $n(\bm{p})$ [2], since
$n(\bm{p})$ is the Fourier transform of the SPDM and therefore carries the
same information [5, 2, 7]. A single photon scattering event corresponds to
the absorption of an incident photon with wavevector $\bm{k}_{\mathrm{L}}$ and
the subsequent emission of another photon with wavevector $k_{0}\bm{u}$, where
$k_{0}\simeq k_{\mathrm{L}}$ and where the unit vector $\bm{u}$ is random. Due
to momentum conservation of the combined atom-light system, the atomic
momentum changes from $\bm{p}_{0}$ to
$\bm{p}_{0}+\hbar(\bm{k}_{\mathrm{L}}-k_{0}\bm{u})$. When many photons are
scattered, the atomic state undergoes a random walk in momentum space, a
phenomenon often called momentum diffusion. The characteristic width of the
momentum distribution increases diffusively as $\Delta k\propto\sqrt{t}$ for
long times. Momentum diffusion plays an essential role in the theory of laser
cooling, where it limits the achievable temperatures [18, 19]. In this
context, the momentum diffusion of a single particle has been thoroughly
studied.

In a recent work [20], we have investigated the decay of spatial coherence of
bosonic gases in optical lattices when illuminated by a near-resonant
excitation laser. Our main observation is a power-law decay of the coherence
of strongly interacting systems at long times, rather than the exponential
decay expected for an ensemble of non-interacting atoms undergoing momentum
diffusion. Our observations are in quantitative agreement with the theoretical
predictions of Poletti et al. [21, 22] based on a minimal model describing
light scattering by a quantum gas, hereafter denoted by “dissipative Bose-
Hubbard model” (see Section IV for precise definitions). This model explains
the power-law decay of coherence as the signature of a non-trivial dissipative
many-body dynamics, where the interplay between fluorescence (destroying
spatial coherence), and coherent tunnelling between neighboring sites
(restoring spatial coherence at short distances) is drastically modified by
interactions. In the present article, we extend our previous work in two
respects. First, we focus on one-dimensional systems, as opposed to the two-
dimensional systems studied in [20]. Second, we measure the spatial coherence
between nearest-neighbor (n-n) sites of the optical lattice (as in [20]) but
also between next-nearest-neighbor (n-n-n) sites.

The article is organized as follows. In Section II, we describe the
experimental setup and protocols and review the essential features of a
theoretical description of light scattering by atoms trapped in optical
lattices, ignoring many-body effects for the time being. In Section III, we
present our main experimental results concerning the measurement of n-n and
n-n-n correlators in one-dimensional lattice gases. In Section IV, we review
the solution of the dissipative Bose-Hubbard model proposed by Poletti et al..
In Section V, we apply these results to the computation of n-n and n-n-n
correlators measured experimentally. Finally, we conclude in Section VI.

##  II Experimental system

Figure 1: a: Sketch of the experimental geometry. Here $\bm{k}_{\mathrm{L}}$
and $\bm{\varepsilon}_{\mathrm{L}}$ denote the wave and polarization vectors
characterizing the mode of the excitation laser. b: Simplified atomic level
structure of 174Yb highlighting the two relevant energy levels $g$ and $e$.
The level splitting is $\hbar\omega_{0}$, the natural linewidth of the excited
state is $\Gamma$, and the Rabi frequency controlling the strength of the
interaction between the atom and the excitation laser is
$\Omega_{\mathrm{L}}$. The laser detuning from the resonance is defined as
$\delta_{\mathrm{L}}=\omega_{\mathrm{L}}-\omega_{0}$.

###  II.1 Experimental setup

We start with a brief description of the experimental apparatus and methods
(more details are given in [23, 24]). We prepare a quantum gas of bosonic
174Yb atoms in a three-dimensional cubic optical lattice. We start from a
nearly pure (more than 80 % condensed fraction) Bose-Einstein condensate (BEC)
prepared in a crossed optical dipole trap [23]. We then ramp up the depth of
the optical lattice potential along the vertical direction to a fixed value
$V_{z}=27\,E_{\mathrm{R}}$. Here
$E_{\mathrm{R}}=h^{2}/(2m_{\mathrm{Yb}}\lambda_{\mathrm{lat}}^{2})\approx
h\times 1.8\,$kHz is the recoil energy at the lattice wavelength
$\lambda_{\mathrm{lat}}\simeq 759.8\,$nm and $d=\lambda_{\mathrm{lat}}/2$ is
the lattice spacing. Simultaneously, we ramp down the depth of the crossed
dipole trap to a negligible value and then switch it off once the ramp is
completed. This sequence creates a stack of two-dimensional bulk Bose gases
with negligible tunnelling between adjacent planes. In a second step, we
increase the depths $V_{x,y}$ of the two horizontal lattices to their final
values. In this work, we prepare arrays of independent one-dimensional (1d)
lattice gases, as illustrated in Fig. 1a. We thus choose $V_{x}\ll
V_{y}\approx V_{z}$ to enable tunnelling only in the $x$ direction.

The density profile of the initial condensate is not uniform [7] and the
Gaussian envelopes of the lattice lasers result in an auxiliary harmonic
confinement in the $x-y$ plane [8]. Both effects lead to an inhomogeneous
distribution of atoms across all individual 1d gases [25, 26]. Varying the
total atom number changes the peak density and the sizes of the initial
Thomas-Fermi condensate, thereby allowing us to tune the peak density of our
system and prepare a wide range of density profiles [24]. Experiments
presented in this article are performed with gases containing $N\approx
5-6\cdot 10^{4}$ atoms, where unit-, double- and triple-occupations coexist in
the regime of deep lattices [24].

###  II.2 Controlled spontaneous emission

Figure 2:  a: Sketch of several possible transition processes after an
absorption-spontaneous emission cycle. Elastic scattering where atoms end up
in the same band and site is denoted by (1). Same-site, band-changing
processes such as the one labelled (2) are less frequent but not necessarily
negligible for typical lattice depths ranging from a few $E_{\mathrm{R}}$ to a
few tens of $E_{\mathrm{R}}$. In contrast, processes labelled (1’),(2’) that
involve tunnelling to a different site are exponentially suppressed due to the
reduced overlap between adjacent Wannier states. b1-2: Illustration of a
single radiating atom (b1) and of collective radiation of an ensemble of nine
atoms forming a cartoon Mott insulator state with one atom per site (b2). c:
Illustration of energy shifts due to on-site ($U_{\mathrm{dd}}^{\bm{0}}$) and
nearest-neighbor ($U_{\mathrm{dd}}^{\bm{1}}$) dipole-dipole interactions.

In the absence of near-resonant laser light, the coherence of the gas is long-
lived. Spontaneous emission induced by the far-detuned lattice lasers [27]
corresponds to heating times of several seconds [28, 29], making spontaneous
emission hardly distinguishable from other heating processes, e.g. due to
intensity fluctuations of the lattice lasers. To study the impact of
spontaneous emission in a controlled way, we use an additional excitation
laser tuned near the ${}^{1}\mathrm{S}_{0}-^{3}$$\mathrm{P}_{1}$ resonance at
$\lambda_{\mathrm{0}}\approx 556\,$nm (see Fig. 1b). The actual detuning of
the laser from the atomic resonance,
$\delta_{\mathrm{L}}=\omega_{\mathrm{L}}-\omega_{0}\approx(2\pi)\times 2\,$MHz
with $\omega_{0}=ck_{0}$ the resonance frequency and $k_{0}=2\pi/\lambda_{0}$,
is one order of magnitude larger than the spontaneous linewidth
$\Gamma\approx(2\pi)\times 180\,$kHz. We thus work in the regime of weak
optical saturation, where the spontaneous emission rate is given by [30]

$\displaystyle\gamma_{\mathrm{sp}}\simeq\frac{\Gamma}{2}\frac{\Omega_{\mathrm{L}}^{2}}{2\delta_{\mathrm{L}}^{2}+\frac{\Gamma^{2}}{2}}\ll\Gamma.$
(2)

We choose the dipolar electric coupling strength $\Omega_{\mathrm{L}}$
(controlled by the excitation laser intensity) such that
$\gamma_{\mathrm{sp}}\approx 520\,\mathrm{s}^{-1}$ for the experiments
presented in this article.

We now discuss how the motional quantum state of a single atom (or,
equivalently, of a non-interacting gas of atoms) trapped in an optical lattice
is modified by spontaneous emission when the gas is exposed to near-resonant
laser light. The periodic optical lattice potential $V_{\mathrm{lat}}$ is
created by a set of far-off-resonance laser beams distinct from the excitation
laser [31, 8]. Because of the periodic potential, the motional levels cluster
in allowed energy bands labeled by a band index $\nu$, with Bloch energy
eigenstates within each band also labeled by their quasi-momentum $\bm{q}$
restricted to the first Brillouin zone (BZ). In typical quantum gases
experiments, only the fundamental energy band is occupied before exposing the
atoms to near-resonant light. Once dissipation is enabled, light scattering
leads to either intra-band or inter-band transitions depending on whether or
not the band index $\nu$ changes. After a fluorescence cycle, the quasi-
momentum is conserved up to an arbitrary reciprocal lattice vector $\bm{Q}$,
i.e. $\bm{q}\to\bm{q}+\bm{k}_{\mathrm{L}}-k_{0}\bm{u}+\bm{Q}$. With this minor
modification, the phenomenon of momentum diffusion as described in the
Introduction generalizes almost directly to the lattice case [29].

For deep lattices and in the presence of interactions, it is often useful to
use spatially localized Wannier states instead of (spatially delocalized)
Bloch energy eigenstates [8]. The various types of transitions between Wannier
states are illustrated in Fig. 2a. By far the dominant process is elastic
scattering denoted by (1), where atoms end up in the same band and site,
whereas same-site band-changing processes such as the one labelled (2) are
much less frequent. Let us consider for concreteness interband transitions
from the fundamental band to one of the first excited bands labeled by
$\alpha=x,y,z$ in a cubic lattice. The transition rates scale as
$\gamma_{\mathrm{sp}}\eta_{\alpha}^{2}$, where the so-called Lamb-Dicke
parameter $\eta_{\alpha}\approx k_{\mathrm{L}}\sigma_{\alpha}$ (with
$\sigma_{\alpha}$ the harmonic oscillator width approximating the exact
Wannier functions [8]) decreases slowly with increasing lattice depth. In
contrast, processes involving tunnelling to a different site – such as the
ones marked (1’),(2’) in 2a – are exponentially suppressed due to the reduced
overlap between adjacent Wannier states, and thus negligible in practice. A
single-band description neglecting all bands but the fundamental one
corresponds to the idealized limit where $\eta_{\alpha}\to 0$ (which occurs
for infinite lattice depths). Lamb-Dicke parameters in our experiments are in
the range $\eta_{y/z}\sim 0.1$ and $\eta_{x}\sim 0.2-0.4$. As a result, one
expects that interband transitions take place on a time scale $\sim
10\,\gamma_{\mathrm{sp}}^{-1}$, as confirmed by experiments [20].

###  II.3 Analysis of momentum profiles

Figure 3: Extraction of nearest- and next-nearest-neighbors correlators
$C_{1}$ and $C_{2}$ from the measured momentum distributions (here for a
lattice depth $V_{x}=7.3\,E_{\mathrm{R}}$ and $t=0.1,0.5,2\,$ms). The first
Brillouin zone corresponds to $[-k_{\mathrm{L}},+k_{\mathrm{L}}[$. The blue
solid lines in the left panels a1-3 show the experimental data for the
normalized momentum distribution $\pi(k_{x})$, the black solid lines the best
fit using Eq. (6), and the orange lines the fit residuals. In the middle
panels b1-3, the blue dots show the contribution of the fundamental band
$\pi_{0}=\pi(k_{x})-f_{\mathrm{exc}}$ obtained by subtracting from the
measured $\pi$ the content $f_{\mathrm{exc}}$ of the fit function describing
the excited bands [Eq. (6)]. The dashed lines in the left and middle panels
show the “incoherent” part of the fundamental band contribution, i.e. the
calculated Wannier envelope weighted by the band population
$p_{0}\mathcal{W}_{0}(k_{x})$. In the right panels c1-3, the blue dots show
$\pi_{0}-p_{0}\mathcal{W}_{0}(k_{x})$, the black dots (offset vertically for
clarity) show $\pi_{0}-p_{0}\mathcal{W}_{0}(k_{x})-2C_{1}\cos(k_{x}d)$, and
the red solid line is the fitted curve
$2\mathcal{W}_{0}(k_{x})\big{(}C_{1}\cos(k_{x}d)+C_{2}\cos(2k_{x}d)\big{)}$.
The left and middle panels use the same (arbitrary) vertical scales. The
vertical scales of the rightmost panels are enlarged by the factors indicated
in the plots to make the details more visible.

All measurements in this article are done using absorption imaging of the
atomic gas. We record the images after releasing the gas from the optical
lattice trap and letting it expand for 20 ms. Such absorption images allow one
to infer the atomic momentum distribution integrated along the probe line of
sight, $\int n(\bm{k},t)dk_{z}$. As in our previous work [20], we use a model
function to fit the experimental momentum distributions. We first recall, for
completeness and to motivate our fit model, a few key notions concerning the
momentum distribution of a lattice gas, before discussing our fitting model in
details.

#### Definitions of single-particle correlators and quasi-momentum
distribution:

For a one-dimensional quantum gas occupying the fundamental Bloch band $n=0$,
the discrete single-particle correlator [32, 33, 31]

$\displaystyle
C_{s}=\frac{1}{N}\sum_{i}\langle\hat{a}_{i+s}^{\dagger}\hat{a}_{i}\rangle$ (3)

plays the same role as the SPDM $\rho^{(1)}$ for a gas in the continuum. Here
$\hat{a}_{i}$ is the operator annihilating an atom at site $i$ with position
$x_{i}=i\cdot d$, $s$ is the relative distance in units of the lattice spacing
$d$, and $N$ is the total atom number. In the case of a uniform system with
$N_{s}$ sites and filling factor $\bar{n}=N/N_{s}$, the general definition (3)
simply amounts to a normalization such that $C_{0}=1$, i.e.
$C_{s}=\langle\hat{a}_{i+s}^{\dagger}\hat{a}_{i}\rangle/\bar{n}$. The
definition (3) is more general and applies in particular to systems with a
non-uniform density, as realized in experiments. The one-dimensional momentum
distribution [33, 31, 8]

$\displaystyle
n_{0}(k_{x})=\mathcal{W}_{0}(k_{x})\left(1+\sum_{s\in\mathbb{N}^{\ast}}2C_{s}\cos(sk_{x}d)\right)$
(4)

is proportional to the product of a “structure factor”, the discrete Fourier
transform of $C_{s}$, by an envelope function $\mathcal{W}_{0}(k_{x})$. The
latter is the square of the Fourier transform of the real-space Wannier
function in the fundamental band and reflects the momentum distribution of a
single atom confined at one lattice site. The structure factor can also be
interpreted as a normalized quasi-momentum distribution. When taking excited
bands into account, Eq. (4) is generalized as
$n(k_{x})=n_{0}(k_{x})+n_{\mathrm{exc}}(k_{x})$, with
$n_{\mathrm{exc}}(k_{x})=\sum_{\mathrm{exc.\,\,bands}\,\,n}\mathcal{W}_{n}(k_{x})\left(1+\cdots\right)$
the contribution of the excited bands.

#### Model function:

We integrate the atomic distributions $\int n(\bm{k},t)dk_{z}$ inferred from
the absorption images over the irrelevant direction $k_{y}$, and normalize
them to the total atom number,

$\displaystyle\pi(k_{x},t)=\frac{1}{N_{\mathrm{tot}}(t)}\int
n(\bm{k},t)dk_{y}dk_{z}.$ (5)

Experiments presented in this article are performed with one-dimensional
systems characterized by short-ranged spatial coherence even in the
equilibrium state, due to the central role played by fluctuations in one-
dimensional systems [34]. As a result, the Fourier expansion in Eq. (4) can be
truncated to the first few terms. We fit the distribution $\pi$ with
$f(k_{x},t)=f_{0}(k_{x},t)+f_{\mathrm{exc}}(k_{x},t)$, where the contribution
of atoms in the fundamental band and excited bands are respectively

$\displaystyle f_{0}(k_{x},t)$
$\displaystyle=\mathcal{W}_{0}(k_{x})\Big{(}p_{0}+2\sum_{s=1}^{3}C_{s}(t)\cos(sk_{x}d)\Big{)},$
(6) $\displaystyle f_{\mathrm{exc}}(k_{x},t)$
$\displaystyle=\sum_{n=1}^{n_{\mathrm{max}}}\mathcal{W}_{n}(k_{x})p_{n},$ (7)

with $p_{n}$ the fractional population of band $n$. The fundamental band
correlators $C_{s}$ are the same as in Eq. (3). The definitions are such that
$\int\pi(k_{x},t)dk_{x}=1$, $\sum_{n}p_{n}=1$. In the excited bands
contribution, we neglect all correlators and use a cutoff
$n_{\mathrm{max}}=3$. We calculate the Wannier envelope functions
$\mathcal{W}_{n}(k_{x})$ at the lattice depths of interest, and use them as
input to the fitting procedure. The free parameters in the fit are thus the
fundamental band correlators $\\{C_{s}\\}_{s=1,\cdots,3}$ and the band
populations $\\{p_{n}\\}_{s=0,\cdots,3}$.

We show in Fig. 3a1-3 three illustrative momentum profiles along with the best
fit and the fit residuals. Comparing the measured distribution with the lowest
band Wannier enveloppe (shown as dashed lines) at various times, the
broadening of the overall distribution reflects the gradual population of the
excited bands due to the excitation laser, as discussed in Section II.2. The
fit residuals are featureless and the amplitude of the residual deviations is
comparable to the detection noise. We conclude that the fit function is able
to capture the relevant features of the measured momentum distribution, and in
particular to substract efficiently the contribution of the higher bands. To
appreciate the importance of the various terms, we show the contribution $\pi-
f_{\mathrm{exc}}$ of the fundamental band in Fig. 3b1-3. This quantity can be
compared to the momentum distribution of a band with the same overall
population but no coherence, shown as dashed lines. Fig. 3c1-3 shows the
extracted interference pattern from atoms in the fundamental band. When
$C_{1}\neq 0$, the probability to find atoms near $k_{x}=0,\pm
2k_{\mathrm{L}}$ increases due to constructive interference, while the
probability near $k_{x}=k_{\mathrm{L}}$ decreases due to destructive
interference. Note that the Wannier enveloppe suppresses the visibility when
$k_{x}\neq 0$. Subtracting the n-n contribution from the interference patter,
we obtain the black dotted lines where the additional modulation with period
$k_{\mathrm{L}}$ comes from n-n-n coherence.

##  III Observation of anomalous decay of coherence in one dimension

###  III.1 Observation of algebraic decay at long times

Figure 4: Nearest- (circles) and next-nearest (squares) neighbor correlators
$C_{1}$ and $C_{2}$ as a function of time for several lattice depths indicated
in the individual panels a-f. The blue dotted and solid lines show
respectively an exponential fit to $C_{1}(t)$ and an improved fit using a
function [Eq. (9)] that first decays exponentially before crossing over to an
asymptotic power-law decay. The asymptotic power law itself is shown by the
dashed line. The theory based on the dissipative Bose-Hubbard model (Section
IV) predicts the asymptotic law $C_{2}\approx C_{1}^{2}$. We test this
prediction by taking the square of the improved fit to $C_{1}$ without any
additional change (red dot-dashed lines). We observe a reasonable agreement
with the experimentally measured $C_{2}$ (red squares) at sufficiently long
times. The hashed gray areas show the regions $C_{s}<0.02$ where detection
noise becomes dominant. These regions are excluded from the fits. The
experimental error bars are statistical. We record typically five independent
realizations per data point $(V_{x},t)$. We fit each realization independently
and report the mean and standard deviation of the fitted $C_{1/2}$.

Fig. 4 shows the evolution of the nearest- (circles) and next-nearest
(squares) neighbors correlators $C_{1}$ and $C_{2}$ as a function of the
dissipation time for several values of the lattice depth $V_{x}$, that decay
as expected. The hashed gray areas in Fig. 4 show the detection noise, i.e.
the imaging noise in absorption images, mostly due to the optical shot noise
of the probe laser. Values of $C_{s}$ below a threshold of roughly $10^{-2}$
are dominated by noise and therefore not indicative of atomic coherence. We
use a conservative threshold value $C_{s}\geq 0.02$, excluding data points
with lower coherence from the fits. The dashed gray areas in Fig. 4 show the
excluded regions. Even at short times, the fitted third-nearest neighbor
coherence $C_{3}$ are typically barely distinguishable from the detection
noise. We have thus omitted $C_{3}$ from the figures to improve their
readability. Furthermore, the detection noise restricts the observation times
to $\gamma_{\mathrm{sp}}t\lesssim 5$ to maintain a sufficient signal-to-noise
ratio for $C_{1/2}$ for all lattice depths.

With suitable approximations, the quantum optical theory of relaxation can be
used to predict the evolution of the momentum distribution [29, 21, 22, 35].
The simplest model neglects all correlations from interatomic interactions or
collective effects in the radiation of the atomic ensemble: Each atom
interacts with the excitation laser and the vacuum field independently from
the presence of others. Assuming that only the fundamental Bloch band is
relevant, this model predicts that the initial coherences decay exponentially
(see [29, 35] and Appendix A),

$\displaystyle C_{s}(t)\simeq C_{s}(0)\exp(-\gamma_{\mathrm{sp}}t).$ (8)

This prediction is in poor agreement with the experimental results. Although
the initial decay of the measured coherences is well described by an
exponential function (dotted lines in Fig. 4), the fitted decay rates are
substantially larger than the value $\gamma_{\mathrm{sp}}$ suggested by Eq.
(8), see Fig. 5c. Furthermore, the experimental curves eventually depart from
the exponential behavior and settle at long times to a regime where the decay
becomes slower than predicted by Eq. (8). In that regime, the experimental
data tend to align as a straight line in a double-logarithmic plot, indicating
that the long-term decay follows an algebraic law $\propto t^{-\alpha}$
instead of an exponential one.

###  III.2 Atom losses

In addition to the decay of coherence and to interband transfer, we also
observe an overall decay of the total atom number due to the excitation laser.
We attribute these losses to light-induced two-body collisions converting a
fraction of the internal energy into kinetic energy and leading to atom
losses111A contribution from nearby photoassociation resonances is also
possible.. This observation is qualitatively and quantitatively in line with
our previous work on two-dimensional systems [20], where it was analyzed in
details. The loss time constant is roughly one order of magnitude slower than
the decoherence dynamics. The main effect of inelastic losses –the reduction
of atom number– is normalized away by the definition (3) of the single-
particle correlators. However, there remains a more subtle effect on the
coherence that will be discussed later in Section V.

###  III.3 Determination of the decay exponent

Figure 5: Results of fitting Eq. (9) to the experimental data: a: Power law
exponent $\alpha$; The dashed line in a shows the decay exponent $\alpha=1/2$
expected from theory [section IV]; b: power law amplitude $A$; c: initial
decay rate $\gamma_{0}$ plotted as a function of lattice depth. The filled
circles correspond to measurements made in the interval $5\,E_{\mathrm{R}}\leq
V_{x}\leq 10\,E_{\mathrm{R}}$ where the exponent can be reliably extracted,
while open circles show measurements outside this interval. The average
exponent for $5\,E_{\mathrm{R}}\leq V_{x}\leq 10\,E_{\mathrm{R}}$ is
$\alpha\approx 0.54(6)$. The blue squares in c are the decay rate of a single
exponentially decaying function $C_{1}(0)\mathrm{e}^{-\gamma_{0}t}$ fitted to
the data. All error bars are one standard deviation confidence intervals
estimated from the fit.

We now analyze in more details the behavior of the single-particle coherences
$C_{1}$ and $C_{2}$ shown in Fig. 4. The theory presented in Section IV
explains the origin of the asymptotic power-law tail (shown as dashed line in
Fig. 4). However, the same theory only predicts the asymptotic behavior and
cannot describe the transient behavior at short times
$\gamma_{\mathrm{sp}}t\lesssim 1$, where the initial coherence are quickly
suppressed before the power-law regime emerges [21]. Experimentally, we are
able to explore a relatively narrow time window
$0\leq\gamma_{\mathrm{sp}}t\lesssim 5$ due to the parasistic loss processes
mentionned above. As a result, modelling this transient behavior is important
to extract a reliable value for the power-law exponent.

The solid curve in Fig. 4 shows a fit to the experimentally measured
$C_{1}(t)$ using the function

$\displaystyle\mathcal{C}_{1}(t)=$ $\displaystyle
C_{1}(0)\exp(-\gamma_{0}t)+A\frac{1-\exp(-\gamma_{0}t)}{(\gamma_{\mathrm{sp}}t)^{\alpha}}.$
(9)

Eq. (9) should be viewed as a heuristic function describing a smooth crossover
from an initially exponential decay to a long-time algebraic decay with
exponent $\alpha$. For simplicity, we have chosen to use the same exponential
function to describe the initial decay and the build up of power-law behavior.
We use the initial decay rate $\gamma_{0}$, the asymptotic amplitude $A$ and
the decay exponent $\alpha$ as free fitting parameters.

The results of the fit are presented in Fig. 5. The main result concerns the
exponent $\alpha$ of the power law, plotted in Fig. 5a. The limitations on
detection sensitivity and observation times restrict the range of lattice
depths where a power-law tail can be reliably extracted to intermediate
values, roughly $5\,E_{\mathrm{R}}\leq V_{x}\leq 10\,E_{\mathrm{R}}$. For
clarity of presentation, we have marked with open symbols in Fig. 5
measurements outside of this “interval of reliability”. For lattice depths
below $\sim 5\,E_{\mathrm{R}}$, the observed decay is essentially described by
the exponential part, although a slower tail at long times seems to develop in
all cases (even without a lattice). For such low lattice depths, the fitted
initial decay rate $\gamma_{0}$ (circles in Fig. 5c) is almost identical to
the decay rate of a single exponentially decaying curve fitted to the data
(blue squares). For lattice depths above $\sim 12\,E_{\mathrm{R}}$, the
lattice dynamics becomes slower than the spontaneous emission time,
$\hbar/J\geq\gamma_{\mathrm{sp}}^{-1}$. As a result, when the power law tail
eventually appears, its amplitude is close or below the noise floor (see Fig.
5b) and extracting reliably the exponent becomes difficult. This behavior is
reflected by increased uncertainties on the fitted power law exponent outside
the interval $5\,E_{\mathrm{R}}\leq V_{x}\leq 10\,E_{\mathrm{R}}$. Inside the
“interval of reliability”, we find an average value $\alpha\approx 0.54(6)$
consistent with $\alpha=0.5$.

Another observation in Fig. 4 concerns the next-nearest-neighbor correlator
$C_{2}$. We find that $C_{2}$ is well described by $\mathcal{C}_{1}^{2}$ for
$t\gg\gamma_{0}^{-1}$, where $\mathcal{C}_{1}$, given in Eq. (9), is fitted to
the nearest neighbor coherence without additional adjustment parameter. This
behavior is consistent with an exponential decay of the correlator $C_{s}\sim
e^{-s}$ for $s\neq 0$. In the next Section IV, we analyze a minimal model that
explains the two effects that we observe experimentally, namely the emergence
of a power law behavior $C_{1}\propto 1/t^{1/2}$ and the relation
$C_{2}\approx C_{1}^{2}$.

##  IV The dissipative Bose-Hubbard model

For the sake of simplicity in the following discussion, we restrict ourselves
to a quantum gas in the fundamental band $\nu_{0}$, ignoring the possibility
of interband transfer. Extending the theory given below to take multiple bands
into account is straightforward in principle, but leads to a substantial
increase of the complexity. Ref. [29] establishes the minimal model describing
the dynamics of a gas of bosonic atoms in an optical lattice illuminated by
near-resonant light. This model can be described by a Lindblad master equation
of the form

$\displaystyle\frac{d}{dt}\hat{\rho}$
$\displaystyle=\frac{1}{i\hbar}\big{[}\hat{H},\hat{\rho}\big{]}+\mathcal{K}\big{[}\hat{\rho}\big{]}.$
(10)

Here $\hat{H}$ denotes the Bose-Hubbard (BH) Hamiltonian,

$\displaystyle\hat{H}$ $\displaystyle=-J\sum_{\langle
i,j\rangle}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{U}{2}\sum_{i}\hat{n}_{i}(\hat{n}_{i}-1),$
(11)

with $J$ and $U$ the nearest-neighbor tunnelling and interaction energies. The
superoperator $\mathcal{K}$ (hereafter called “dissipator”) reads

$\displaystyle\mathcal{K}\big{[}\hat{\rho}\big{]}$
$\displaystyle=\gamma_{\mathrm{sp}}\sum_{i}\hat{n}_{i}\hat{\rho}\hat{n}_{i}-\frac{1}{2}\Big{\\{}\hat{n}_{i}^{2},\hat{\rho}\Big{\\}}.$
(12)

We refer to the model defined by Eq. (10), $\hat{H}$ and $\mathcal{K}$ as
“dissipative Bose-Hubbard model” in the following. In Appendix A, we discuss
$N-$particle corrections, either collective radiative effects where the
spontaneous emission rate of $n$ atoms differs from $n\gamma_{\mathrm{sp}}$
(illustrated in Fig. 2b), or long-range interactions between the induced
electric dipoles carried by each atom mediated by multiple photon scattering
(illustrated in Fig. 2c) [29]. We conclude that collective effects are
negligible for the experimental parameters used in this work.

The dissipative BH model has been analyzed by Poletti et al. [21, 22]. After
an initial stage for $\gamma_{\mathrm{sp}}t\leq 1$, where spatial coherences
eventually present in the initial state are exponentially damped, they
discovered an asymptotic regime where relaxation slows down dramatically. The
theory of [22] is summarized in the Supplementary Material. The key result is
a classical master equation for the on-site number distribution $p_{n}$, i.e.
the probability to find $n$ bosons at any given site,

$\displaystyle\frac{dp_{n}}{d\tau}$ $\displaystyle=\sum_{\sigma=\pm
1}W_{n+\sigma}\big{[}\\{p_{n}\\}\big{]}\Big{(}p_{n+\sigma}-p_{n}\Big{)},$ (13)

with $\tau=t/T^{\ast}$ a dimensionless time. The dimensionless transition
rates are given by
$W_{n+1}\big{[}\\{p_{n}\\}\big{]}=\sum_{m}g_{\varepsilon,\overline{n}}(m,n+1)p_{m-1}$,
$W_{n-1}\big{[}\\{p_{n}\\}\big{]}=\sum_{m}g_{\varepsilon,\overline{n}}(m+1,n)p_{m+1}$,
with the function
$g_{\varepsilon,\overline{n}}(m,n)=mn/[(m-n+1)^{2}+(\overline{n}\varepsilon)^{2}]$.
The dynamics is governed by the dimensionless dissipation parameter

$\displaystyle\varepsilon=\frac{\hbar\gamma_{\mathrm{sp}}}{U\overline{n}},$
(14)

and by an emergent time scale $T^{\ast}$ such that

$\displaystyle\frac{1}{T^{\ast}}$
$\displaystyle=2z\gamma_{\mathrm{sp}}\left(\frac{J}{U\overline{n}}\right)^{2}.$
(15)

Here $z=2$ is the number of nearest-neigbors in one dimension and $\bar{n}$
the filling factor (mean number of atoms per lattice site).

Figure 6: Numerical solution of the master equation (13) for the on-site
distribution $p_{n}$. a: bare distribution as a function of occupation number
$n$. b: rescaled distribution
$\tilde{p}(u)=p_{n}\times\bar{n}\tau^{\frac{1}{4}}$ as a function fo the
scaling variable $u=(n-\bar{n})/(\bar{n}\tau^{\frac{1}{4}})$. The dashed line
shows the scaling distribution $f_{\infty}$ and the solid line a normalized
Gaussian function. We used the parameters $U/J=20$, $\varepsilon=0.1$ and
$\bar{n}=3$.

A numerical solution of the master equation (13) is shown in Fig. 6a. The
initial state is a Fock state with $\overline{n}=3$. The master equation
conserves the norm $\sum_{n}p_{n}=1$ and the average population
$\sum_{n}np_{n}=\overline{n}$. The distribution function broadens with time as
expected, but in a very peculiar way. Fig. 6b illustrates that the
distribution function obeys approximately a scaling dynamics. The distribution
functions at widely different times overlap when plotted versus the scaling
variable [22],

$\displaystyle
u\left(n/\bar{n},\tau\right)=\frac{n/\bar{n}-1}{\tau^{\frac{1}{4}}}.$ (16)

The scaling property implies that the standard deviation of $n$ increases with
time roughly as a power law $\propto\tau^{1/4}$, much more slowly than for a
standard diffusive process $\propto\tau^{1/2}$.

To gain further insight on the origin of this scaling behavior, we follow
again Poletti et al. [22] and consider the limiting case of very large filling
$\overline{n}\gg 1$. In that limit, the discrete variable $n$ and distribution
$p_{n}$ can be replaced by their continuous counterparts,
$x=n/\overline{n}\in[0,\infty[$ and $p(x,\tau)=p_{n}(\tau)/\delta x$, with
$\delta x=1/\overline{n}$. The classical master equation (13) maps to a
diffusion equation [22],

$\displaystyle\frac{\partial
p(x,\tau)}{\partial\tau}\approx\frac{\partial}{\partial
x}\Big{(}D(x)\frac{\partial p(x,\tau)}{\partial x}\Big{)}.$ (17)

with a diffusion coefficient $D(x)=1/[\varepsilon^{2}+(x-1)^{2}]$.

The diffusion equation admits two limiting cases of interest, a dissipation-
dominated regime $\hbar\gamma_{\mathrm{sp}}\gg U$ (i.e. $\varepsilon\gg 1$),
with a diffusion coefficient $D(x)\approx 1/\varepsilon^{2}$, and an
interaction-dominated regime $U\gg\hbar\gamma_{\mathrm{sp}}$ (i.e.
$\varepsilon\ll 1$), with a diffusion coefficient $D(x)\approx(x-1)^{-2}$. In
both cases, the diffusion coefficient behaves as a power law $D(x)\sim
x^{-\eta}$ with $\eta=0$ ($\varepsilon\gg 1$) or $2$ ($\varepsilon\ll 1$). For
the initial condition $p(x,0)=\delta(x)$, the diffusion equation (17) with a
power law diffusion coefficient admits a self-similar solution [22]

$\displaystyle p(x,\tau)$
$\displaystyle=\tau^{-\beta}f_{\infty}\left[u\left(x,\tau\right)\right],$ (18)

with scaling exponent $\beta=1/(\eta+2$). For an arbitrary initial condition
$p_{0}(x)$, the time evolution can be obtained from $p(x,\tau)=\int
dx_{0}p_{0}(x_{0})G(x-x_{0},\tau)$ owing to the linearity in $p(x)$ of the
diffusion equation.

In the dissipation-dominated regime $\varepsilon\gg 1$, the scaling solution
$f_{\infty}$ is a Gaussian function corresponding to a scaling exponent
$\beta=1/2$. This corresponds to standard diffusive behavior with $\Delta
x\propto\sqrt{t}/\varepsilon$. The diffusion coefficient in real units
$D_{\varepsilon}\sim 1/(T^{\ast}\varepsilon^{2})\sim
J^{2}/(\hbar^{2}\gamma_{\mathrm{sp}})\ll J/\hbar$ describes a diffusive
dynamics much slower than the ballistic dynamics of atoms tunnelling in the
lattice. Such a slowed down, dissipative dynamics can be interpreted as
resulting from the inhibition of coherent tunnelling by the quantum Zeno
effect [37].

The interaction-dominated regime $\varepsilon\ll 1$ is characterized by a
different scaling function $f_{\infty}(u)=\mathrm{e}^{-16u^{4}}/\Gamma(1/4)$
($\Gamma$ is the Euler gamma function) and exponent $\beta=1/4$. Numerical
simulations as in Fig. 5 show that even for finite $\bar{n}\geq 1$, the
distribution function obeys approximately the scaling relation
$p(x,\tau)\approx\tau^{-\beta}f_{\mathrm{approx}}\left[u\left(x,\tau\right)\right]$
with exponent $\beta\approx 0.25$. This approximate scaling behavior for
finite $\bar{n}$ explains the sub-diffusive dynamics observed in the numerical
calculation. However, we find empirically that the approximate scaling
function for finite $\bar{n}$ differs from $f_{\infty}$. Numerically, we find
that it is closer to a normalized Gaussian function
$f_{\mathrm{approx}}(u)=\mathrm{e}^{-u^{2}/2}/\sqrt{2\pi}$ (see Fig. 6b).

##  V Single-particle correlation function

To connect the theory and the experimental results, we now turn to the
calculation of the single-particle correlators. We find that the nearest-
neighbor correlator depends on the populations as (see the Supplementary
Material for details of the calculations)

$\displaystyle C_{1}(t)$
$\displaystyle\simeq\frac{J}{U\bar{n}}\sum_{m,n=0}^{+\infty}\frac{m(n+1)(n-m+1)}{(n-m+1)^{2}+(\varepsilon\overline{n})^{2}}$
$\displaystyle\hskip 42.67912pt\times\Big{[}p_{m}p_{n}-p_{m-1}p_{n+1}\Big{]}.$
(19)

We have also computed the next-nearest-neighbors correlator $C_{2}(t)$ in
terms of the distribution function $p_{n}$ using the same method as for
$C_{1}$. We do not reproduce here the rather lengthy expression [see Eq. (15)
in the Supplementary Material].

When the distribution $p$ assumes a scaling form
$p(n,\tau)\approx\tau^{-\beta}f\left[u\right]$, the correlators $C_{1,2}$ obey
simple laws (see Supplementary Material),

$\displaystyle C_{1}(t)$
$\displaystyle\simeq\frac{a_{1}}{\sqrt{2z\gamma_{\mathrm{sp}}t}},$ (20)
$\displaystyle C_{2}(t)$
$\displaystyle\simeq\frac{a_{2}}{2z\gamma_{\mathrm{sp}}t}=\frac{a_{2}}{a_{1}^{2}}C_{1}^{2}(t).$
(21)

The numerical coefficients $a_{1},a_{2}$ depend on the particular form of the
distribution function $f$. For the approximate but experimentally relevant
Gaussian scaling distribution, we find $a_{1}=a_{2}=1$ and

$\displaystyle C_{2}(t)\underset{\bar{n}\sim 1}{\simeq}C_{1}^{2}(t).$ (22)

Thus, the scaling approach fully explains the experimental observations of
Section III.

As for the distribution function, the numerical evaluations of $C_{1/2}$ shown
in Fig. 7 confirm the relevance of the scaling solution even for relatively
small filling factors. Fig. 7a shows that the asymptotic laws in Eqs. (20,21)
describe well the algebraic tails of the correlators for $\bar{n}=3$. Fig. 7b
concentrates on $C_{1}$, varying the filling factor in the interval
$\bar{n}\in[0.1,3.5]$. When $\bar{n}\gtrsim 1$, the correlator $C_{1}$
approaches the scaling law (20) from $\gamma_{\mathrm{sp}}t\gtrsim 1$ up to a
filling-dependent time $T_{\mathrm{sc}}$. Thus, one can define a “scaling
window” $t\in[\gamma_{\mathrm{sp}}^{-1},T_{\mathrm{sc}}]$ where the correlator
is well described by the scaling theory. Outside this window, the distribution
has broadened sufficiently to “feel the $n=0$ boundary”, the distribution
deviates from the scaling form and the algebraic behavior disappears.

To conclude this analysis, we briefly discuss how the overall picture of
scaling dynamics is modified in presence of losses, as in the experimental
system. We restrict ourselves to the “adiabatic regime” (approximately
realized in the experiments), where the loss rate is much slower than the
spontaneous emission rate. The scaling solution is still relevant, but with a
slowly decaying $\bar{n}(t)$ due to losses. Fig. 7b then allows us to
understand qualitatively that losses shorten the scaling window. For very
small loss rates, the density barely changes in the entire scaling window, and
the scaling picture of the evolution of coherence is therefore not
significantly modified. As the loss rate is increased, the system follows as
time goes the set of curves shown in Fig. 7b, moving to the right due to
spontaneous emission but also downwards towards lower filling factors because
of losses. Eventually, the system exits the scaling regime not because of the
initial finite duration of the scaling window, but because $\bar{n}(t)$ has
dropped to a value $\sim 1$ where the scaling behavior disappears. Numerical
simulations [20] support the claim that our experiments are performed in this
intermediate, adiabatic regime, where losses are slow enough to use an
adiabatic description, but fast enough to impact the evolution of coherence.

Figure 7:  a: Nearest-and next-nearest-neighbor correlators $C_{1}$ (blue) and
$C_{2}$ (red) evaluated for $U/J=20$ and $\epsilon=0.1$ with a Fock state with
$\bar{n}=3$ as initial state. The numerical calculations (solid lines) are
compared to the analytic scaling forms in Eqs. (20,21) [dotted lines]. The
dashed line shows the exponential decay curve expected for independent atoms.
b: Calculation of $C_{1}$ for several filling factors $\bar{n}\in[0.1,3.5]$
and the same parameters as in a. The dashed line shows the asymptote predicted
in the scaling limit $\bar{n}\to+\infty$, $\epsilon\to 0$.

##  VI Conclusion

In this article, we have given a detailed experimental and theoretical account
of the loss of spatial coherence in one-dimensional bosonic quantum gases
illuminated by near-resonant light. We analyze the momentum distribution to
extract nearest- and next-nearest-neighbors correlators of the bosonic field.
We find experimentally that the n-n correlator decays as a power law $\propto
1/\sqrt{t}$, and also that the n-n correlator controls the n-n-n correlator
according to $C_{2}\approx C_{1}^{2}$ at long times. Our findings are in
contrast to the simple model where the atomic ensemble is treated as mutually
independent scatterers, and where the decay would be exponential with a time
scale given by the inverse of the spontaneous emission rate. The power law
decay at long times is correctly predicted by the theory of [21], which agrees
quantitatively with our measurements in the sense that it predicts an exponent
$\alpha=1/2$ and the same relation $C_{2}\approx C_{1}^{2}$ between n-n and
n-n-n correlators as observed experimentally. The long-term algebraic decay is
an emergent dissipative many-body phenomenon, where both the dissipation and
strong interactions play a crucial role. This dynamics is independent of the
details of the particular initial state, which are erased during the early
part of the decay before the power law behavior is established.
Experimentally, we observe that the time scale governing this early decay is
much shorter than $\gamma_{\mathrm{sp}}^{-1}$. An accelerated early dynamics
is qualitatively consistent with the mean-field numerical simulations of [29],
and intuitively expected in an interacting system. A recent publication [38]
suggests that the initial decay should be controlled by a “non-Hermitian
susceptibility” that characterizes the response of the initial system to
dissipation. In analogy with its equilibrium counterpart, the non-Hermitian
susceptibility could potentially be useful as a novel probe of many-body
properties, for instance of the momentum distribution. We leave a deeper
investigation of this question for future work.

###### Acknowledgements.

It is a pleasure to dedicate this article to Jean Dalibard. We are genuinely
grateful to be a part of the research group that he built at LKB. Each of us
owe a great deal to Jean, not only for his essential contributions in our
joint projects, but also for the continuous inspiration and education.

##  Appendix A Theory of spontaneous emission from a Bose-Hubbard quantum gas

In this Appendix, we discuss the quantum-optical theoretical framework that
allows one to describe the behavior of an ultracold gas of interacting atoms
driven by near-resonant light. This framework (sometimes called “generalized
optical Bloch equations”) generalizes the well-known optical Bloch equations
[30] to account for the quantized motion of atoms (here, in a periodic lattice
potential). Such a generalization has been studied in depth in the context of
laser cooling [39, 40, 41]. The generic equations of motion take the form of a
master equation for the atomic density operator
$\\{\hat{\rho}_{\alpha,\beta}\\}_{\\{\alpha,\beta=e,g\\}}$, where each
component $\hat{\rho}_{\alpha,\beta}$ is an operator with respect to the
motional degrees of freedom. Here our treatment also takes light-induced many-
body effects into account following Refs. [42, 43, 44, 45, 29].

In the regime of weak laser excitation, saturation effects can be neglected
and the internal degrees of freedom can be eliminated adiabatically [30]. This
approch leads to an effective Lindblad master equation (10) describing the
dynamics of the motional degrees of freedom in the electronic ground state
$g$. The Hamitonian is $\hat{H}=\hat{H}_{0}+\hat{V}_{\mathrm{dd}}$, with
$\hat{H}_{0}$ the bare many-body Hamiltonian in the absence of the excitation
laser, including interactions and the optical lattice potential, and with

$\displaystyle\hat{V}_{\mathrm{dd}}$
$\displaystyle=\hbar\gamma_{\mathrm{sp}}\int
d^{3}\bm{r}_{1}d^{3}\bm{r}_{2}\,G_{\mathrm{L}}(\bm{r}_{12})\mathrm{e}^{-\mathrm{i}\bm{k}_{\mathrm{L}}\cdot\bm{r}_{12}}$
$\displaystyle\hskip
56.9055pt\big{(}\hat{\Psi}(\bm{r}_{1})\hat{\Psi}(\bm{r}_{2})\big{)}^{\dagger}\hat{\Psi}(\bm{r}_{1})\hat{\Psi}(\bm{r}_{2})$
(23)

the interaction mediated by light scattering between the induced electric
dipoles carried by each atom. We note $\hat{\Psi}(\bm{r}_{\alpha})$ the field
operator annihilating an atom in the ground state at position
$\bm{r}_{\alpha}$ and $\bm{r}_{12}=\bm{r}_{1}-\bm{r}_{2}$ the relative
distance. The many-body dissipator in the Lindblad master equation (10) takes
the form

$\displaystyle\mathcal{K}_{\mathrm{m-b}}\big{[}\hat{\rho}\big{]}=$
$\displaystyle\gamma_{\mathrm{sp}}\int
d^{3}\bm{r}_{1}d^{3}\bm{r}_{2}\,F_{\mathrm{L}}(\bm{r}_{12})\mathrm{e}^{-\mathrm{i}\bm{k}_{\mathrm{L}}\cdot\bm{r}_{12}}$
(24) $\displaystyle\hskip
28.45274pt\left(\hat{n}(\bm{r}_{1})\hat{\rho}\hat{n}(\bm{r}_{2})-\frac{1}{2}\Big{\\{}\hat{n}(\bm{r}_{1})\hat{n}(\bm{r}_{2}),\hat{\rho}\Big{\\}}\right),$

with $\hat{n}(\bm{r})=\hat{\Psi}^{\dagger}(\bm{r})\hat{\Psi}(\bm{r})$ the
particle density operator. The function
$K_{\mathrm{L}}=F_{\mathrm{L}}+\mathrm{i}G_{\mathrm{L}}=\bm{\varepsilon}_{\mathrm{L}}\cdot\bar{\bar{K}}\cdot\bm{\varepsilon}_{\mathrm{L}}^{\ast}$
is determined by the laser polarization $\bm{\varepsilon}_{\mathrm{L}}$ and by
the so-called dyadic Green function $K_{m,m^{\prime}}(\bm{r}_{12})$, the
propagator of a photon from $\bm{r}_{1}$ to $\bm{r}_{2}$ with polarization
changing from axis $m$ to $m^{\prime}$ [42, 43]. The two functions
$F_{\mathrm{L}}$ and $G_{\mathrm{L}}$ are shown in Fig. 8a,b for our
particular experimental situation with
$\lambda_{\mathrm{lat}}/\lambda_{0}\approx 759/556\approx 1.37$ and
$\bm{\varepsilon}_{\mathrm{L}}=\bm{e}_{x}$.

Figure 8: Real part (a) and imaginary part (b) of
$K_{\mathrm{L}}(\bm{s})=F_{\mathrm{L}}(\bm{s})+\mathrm{i}G_{\mathrm{L}}(\bm{s})=\bm{\varepsilon}_{\mathrm{L}}\cdot\bar{\bar{K}}(\bm{s})\cdot\bm{\varepsilon}_{\mathrm{L}}^{\ast}$.
The plots are drawn with our experimental parameters,
$\lambda_{\mathrm{lat}}/\lambda_{0}\approx 759/556\approx 1.37$ and
$\bm{\varepsilon}_{\mathrm{L}}=\bm{e}_{x}$ and for two orthogonal directions
$\bm{s})$ along $\bm{e}_{x}$ and $\bm{e}_{z}$. c: on-site Hubbard interaction
energy. The calculation includes the contribution of van der Waals interaction
and of dipole-dipole interactions. The solid (respectively, dashed) curve
corresponds to $U>0$ (resp., $U<0$). The yellow dot marks the experiments
presented in the main text.

We expand the master equation in the Wannier basis. For simplicity and to
avoid notational overload, we assume that the Wannier functions
$w(\bm{r}-\bm{r}_{i})$ are tightly localized and use a tight-binding
approximation. For one-body terms in the Hamiltonian, we only retain the
dominant contributions involving on-site energies and nearest-neighbor
tunnelling. The most generic form for two-body terms involves four sites
labeled $i,j,k,l$. We only keep the dominant terms with $i=j$ and $k=l$. The
bare many-body Hamiltonian $\hat{H}_{0}$ then reduces to the well-known Bose-
Hubbard Hamiltonian,

$\displaystyle\hat{H}_{\mathrm{BH}}$ $\displaystyle=-J_{0}\sum_{\langle
i,j\rangle}\hat{a}_{i}^{\dagger}\hat{a}_{j}+\frac{U_{0}}{2}\sum_{i}\hat{n}_{i}(\hat{n}_{i}-1),$
(25)

with $J_{0},U_{0}$ the tunnelling and interaction energies for atoms in the
fundamental band and with $\hat{n}_{i}=\hat{a}_{i}^{\dagger}\hat{a}_{i}$ the
on-site particle number operator. The d-d interactions and dissipator take the
form

$\displaystyle\hat{V}_{\mathrm{dd}}\simeq$
$\displaystyle\sum_{i}U_{\mathrm{dd}}^{\bm{0}}\hat{n}_{i}(\hat{n}_{i}-1)+\sum_{i\neq
j}U_{\mathrm{dd}}^{\bm{r}_{ij}}\hat{n}_{i}\hat{n}_{j},$ (26)
$\displaystyle\mathcal{K}_{\mathrm{m-b}}\big{[}\hat{\rho}\big{]}$
$\displaystyle\approx\sum_{ij}\Lambda_{\bm{r}_{ij}}\Big{(}\hat{n}_{i}\hat{\rho}\hat{n}_{j}-\frac{1}{2}\left\\{\hat{n}_{i}\hat{n}_{j},\hat{\rho}\right\\}\Big{)},$
(27)

with matrix elements

$\displaystyle\frac{U_{\mathrm{dd}}^{\bm{r}_{ij}}}{\hbar\gamma_{\mathrm{sp}}}$
$\displaystyle=\int
d^{3}\bm{r}_{1}d^{3}\bm{r}_{2}\,G_{\mathrm{L}}(\bm{r}_{12}+\bm{r}_{ij})|w(\bm{r}_{1})|^{2}|w(\bm{r}_{2})|^{2},$
(28) $\displaystyle\frac{\Lambda_{\bm{r}_{ij}}}{\gamma_{\mathrm{sp}}}$
$\displaystyle=\int
d^{3}\bm{r}_{1}d^{3}\bm{r}_{2}\,F_{\mathrm{L}}(\bm{r}_{12}+\bm{r}_{ij})|w(\bm{r}_{1})|^{2}|w(\bm{r}_{2})|^{2}.$
(29)

For $\bm{r}_{ij}\neq\bm{0}$, the functions vary slowly on the scale of the
Wannier functions, and the d-d interaction energy can be approximated as

$\displaystyle U_{\mathrm{dd}}^{\bm{r}_{ij}\neq\bm{0}}$
$\displaystyle\simeq\hbar\gamma_{\textrm{sp}}G_{\mathrm{L}}(\bm{r}_{ij}).$
(30)

Similarly, transition rates for any $\bm{r}_{ij}$ are approximately given by

$\displaystyle\Lambda_{\bm{r}_{ij}}$
$\displaystyle\simeq\gamma_{\textrm{sp}}F_{\mathrm{L}}(\bm{r}_{ij}).$ (31)

In our experimental situation, the ratio $\lambda_{\mathrm{lat}}/\lambda_{0}$
is such that the function $F$ is small in all directions when $\bm{r}_{ij}\neq
0$. As a result, one can simplify the dissipative term and consider a “zero-
range model” with

$\displaystyle\Lambda_{\bm{r}_{ij}}$
$\displaystyle\simeq\gamma_{\textrm{sp}}\delta_{ij}.$ (32)

Off-site terms due to d-d interactions are similarly small and typically
negligible even in comparison with the tunnelling energy.

The on-site d-d interaction parameter $U_{\mathrm{dd}}^{\bm{0}}$ must be
treated differently. For this term, the approximation leading to Eq. (30) for
off-site terms is invalid, due to the $1/r^{3}$ divergence of d-d interactions
at short distances [46]. In three dimensions, this divergence is regulated by
angular integration in three dimensions [46]. Fig. 8c shows a numerical
evaluation of the on-site interaction matrix element
$U=U_{0}+U_{\mathrm{dd}}^{\bm{0}}$. For the parameters used in this work, the
d-d term is a small correction to the total interaction energy and can be
safely neglected.

For higher dissipation rates comparable to or greater than
$E_{\mathrm{R}}/\hbar$, the dipole-dipole interaction energy can be
substantial. In our experimental configuration, each site of the optical
lattice resembles a pancake-shaped trap (the confinement is always stronger in
the vertical direction), and the atomic electric dipoles are primarily along
$(\bm{e}_{x}+\bm{e}_{y})/\sqrt{2}$. In this configuration, dipole-dipole
interactions are attractive. As a result, the on-site interaction energy
vanishes when the d-d interaction energy exactly cancels the background value,
and becomes negative above. This feature could be used in future experiments
to manipulate interactions among 174Yb atoms, or similar atoms lacking
accessible Feshbach resonances.

##  Appendix B Decay of coherence for non-interacting atoms

With the same approximations as in Appendix A, the dissipator describing the
effect of spontaneous emission on the motional state of a single atom in the
fundamental band of an optical lattice reads

$\displaystyle\mathcal{D}\big{[}\hat{\rho}\big{]}$
$\displaystyle=\gamma_{\mathrm{sp}}\sum_{i,j}\Lambda_{x_{ij}}\left(\hat{n}_{i}\hat{\rho}\hat{n}_{j}-\frac{1}{2}\\{\hat{n}_{i}\hat{n}_{j},\hat{\rho}\\}\right),$
(33)

where $\Lambda_{x_{ij}}$ is defined in Eq. (29) and where
$x_{ij}=x_{j}-x_{j}$. The dissipator leads to an exact equation of motion for
the spatially averaged coherence $C_{s}$,

$\displaystyle\frac{dC_{s}}{dt}$
$\displaystyle=-\gamma_{\mathrm{sp}}\big{(}1-\Lambda_{s}\big{)}C_{s}.$ (34)

Therefore, $C_{s}$ decays exponentially with a rate constant
$\gamma_{\mathrm{sp}}(1-\Lambda_{s})$ varying from zero when $s=0$ to
$\approx\gamma_{\mathrm{sp}}$ when $s$ is much larger than the light
wavelength $\lambda_{0}$. If we take a “zero range” limit where $\lambda_{0}$
is much shorter than the lattice spacing, the rate constant becomes
$\gamma_{\mathrm{sp}}(1-\delta_{s,0})$. This leads to Eq. (8) in the main
text.

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