PHYSICAL REVIEW LETTERS 122, 153602 (2019)

Conductivity Spectrum of Ultracold Atoms in an Optical Lattice
Rhys Anderson,1 Fudong Wang,1 Peihang Xu,1 Vijin Venu,1 Stefan Trotzky,1 Frédéric Chevy,2 and Joseph H. Thywissen1,3
1

2

Department of Physics, University of Toronto, Ontario M5S 1A7 Canada
Laboratoire Kastler Brossel, ENS-PSL Research University, CNRS, UPMC-Sorbonne Université,Collège de France, 24 Rue Lhomond, 75005 Paris, France
3
Canadian Institute for Advanced Research, Toronto, Ontario M5G 1M1 Canada(Received 27 May 2018; published 18 April 2019)

We measure the conductivity of neutral fermions in a cubic optical lattice. Using in situ fluorescencemicroscopy, we observe the alternating current resultant from a single-frequency uniform force applied bydisplacement of a weak harmonic trapping potential. In the linear response regime, a neutral-particle analog
of Ohm’s law gives the conductivity as the ratio of total current to force. For various lattice depths,
temperatures, interaction strengths, and fillings, we measure both real and imaginary conductivity, up to afrequency sufficient to capture the transport dynamics within the lowest band. The spectral width of the real
conductivity reveals the current dissipation rate in the lattice, and the integrated spectral weight is related tothermodynamic properties of the system through a sum rule. The global conductivity decreases withincreased band-averaged effective mass, which at high temperatures approaches a T-linear regime.
Relaxation of current is observed to require a finite lattice depth, which breaks Galilean invariance andenables damping through collisions between fermions.
DOI: 10.1103/PhysRevLett.122.153602

The resistance of a metal is normally dominated byphonons, impurities, and crystalline defects, with electronelectron scattering playing a minor role. For ultrapuresamples, it has been found that this situation can bereversed with collisional physics instead playing a majorrole in electrical properties [1–4]. Optical lattices provide
ultracold atoms with a crystalline environment of comparable purity, and also with an effectively infinite Debye
temperature [5]. These properties allow for the study oftransport in conditions inaccessible to typical materials: attemperatures comparable to the Fermi energy, yet where
phonons are absent and the crystal remains intact [6].
Moreover, the strength of particle-particle scattering,which is the sole remaining source of dissipation, can betuned using a Feshbach resonance or by adjusting thelattice depth.
In this Letter, we study the conductivity of ultracold
fermions in an optical lattice subject to weak harmonic
confinement. Nonequilibrium transport of ultracold fermions in periodic potentials has been investigated previously through step response [7–9], in a mesoscopic twoterminal geometry [10–12], in the context of disorderinduced localization [13,14], through quasimomentum
relaxation [15], by observing diffusion [16,17], and by
studying expansion dynamics [18–21]. Here we realize theproposal of Wu, Taylor, and Zaremba [22], closely related
to the proposal of Tokuno and Giamarchi [23], to measureconductivity σðωÞ directly through the ratio of the current
response JðωÞ to an alternating force FðωÞ. In this
proposal, the weak harmonic confinement of the system0031-9007=19=122(15)=153602(7)results in a low-frequency resonance in σðωÞ near the
harmonic trap frequency. The spectral width and weight ofthis resonance reveal the current dissipation rate and carrier
inertia, which are the key low-energy transport propertiesof a metal.
Our sample is a balanced spin mixture of the two lowest
hyperfine states of fermionic 40K, trapped in a cubic latticewith period aL ¼ 527 nm. Typically N ¼ 104 atoms areloaded into the lattice at a depth V ¼ 2.5ER and scattering
length as ¼ 180a0 between the spin states, where ER ¼
ℏ2 ðπ=aL Þ2 =2m is the recoil energy, m is the mass of a 40K
atom, and a0 is a Bohr radius. At this depth, t ¼ h × 570 Hz
is the nearest-neighbor tunneling strength. Temperatures T
are measured independently for each dataset [24], andtypically range from 1.2t to 3.3t (here kB ¼ 1), or 0.3T F
to 0.9T F , where T F is the Fermi temperature.
A periodic displacement of one or both of the laser
beams forming a crossed dipole trap (XDT) [Fig. 1(a)]creates the analog of the externally applied voltage in anelectronic conductivity measurement. The uniform force
Fβ is linear in the trap displacement dβ in direction β,
through an in-plane spring constant mω2XDT , where
ωXDT ¼ 2π × 32ð1Þ Hz. The amplitude of the periodic
force is increased linearly over 150 ms, and then held
constant for 50 ms. After a further variable time td , up to
two drive periods, the dynamics are frozen by increasing V
to 60ER in 0.1 ms, and the drive is turned off. The in situ
density distribution of the cloud is recorded at V ¼ 103 ER
via fluorescence in a quantum gas microscope [25–33]
apparatus described previously [34].

153602-1© 2019 American Physical Society

PHYSICAL REVIEW LETTERS 122, 153602 (2019)z lattice(a)(a)
y lattice

(d)

XDT2
XDT1
(c)

(µm)

(d)

2

2

(e)

x lattice

(b)

(e)

c

0

b

-2
0

25
drive time

50
(ms)

response

displacement (µm)

(b)

1

20 Hz
50 Hz
60 Hz

0
0
5
drive amplitude

(c)

10
µm)

FIG. 1. Measurement. (a) Atoms in a 3D optical lattice are
driven by periodic displacement of one or both trapping beams(XDT). (b),(c) In situ images are taken after various drive times td
and the center of mass is extracted. (d) The displacement of thecenter of mass is fit to a single-frequency response (solid line)
typically across two periods in trap displacement (dashed line).
Data is for a 40 Hz drive and V ¼ 2ER lattice. (e) The response
amplitude Ax is shown versus drive dx in typical conditions and atseveral ω. The linear response limit is found for Ax ≲ 1 μm, as
seen by comparison to the lines fit at low Ax.

From images of the central four planes [see Figs. 1(b),1(c)],we determine projections of a site-granulated
center-of-massP
position operator R̂α¼x;y ¼ N −1 i;s rα;i n̂i;s, where n̂i;s is thenumber operator for an atom of spin s on lattice site i locatedat rα;i . hR̂α ðtÞi is fit to Aα cos½ωtd − ϕα  [see Fig. 1(d)], whereω is the drive frequency (typically 2π × 10–200 Hz) and Aα
and ϕα are fit parameters. The steady-state bulk current is thenhĴα ðωÞi ¼ NdhR̂α ðωÞi=dt. As shown in Fig. 1(e), remaining
in the linear-response regime restricts hR̂α i to the micronscale, emphasizing the need for high-resolution measurement. In complex notation, the global conductivity σ αβ ðωÞcan be determined through the equivalent of Ohm’s law [22],XhĴ α ðωÞi ¼
σ αβ ðωÞFβ ðωÞ:ð1ÞβIn terms of fit variables and drive strength, σ αβ ðωÞ ¼−iNωAα ðωÞ exp½iϕα ðωÞ=Fβ ðωÞ. We write the conductivity
in dimensionless form σ=σ 0 , where σ 0 ≡ Na2L =ℏ sets thescale of the Mott-Ioffe-Regel limit [35].
Figure 2(a) shows an example of on-diagonal conductivity, σ xx . We observe both a peak in Re σ xx and a zerocrossing in Im σ xx at finite frequency, in contrast to the dc
peak expected in a conventional metal with Drude-like
response. This can be understood as a capacitive effect of
the harmonic trap, which shifts the peak response to its
oscillation frequency, renormalized by the effective mass ofparticles in the lattice [9].

(f)FIG. 2. Conductivity spectra. (a) Real and imaginary ondiagonal conductivity, and (b) difference in real off-diagonal
conductivities, versus drive frequency. Here, V ¼ 2.5ER , withbackground as . Lines show fits to Eq. (2). (c) For V ¼ 0, a
Fourier-limited response is observed, with mSxx =N ¼ 1.01ð8Þ
and Γ ¼ 18ð1Þ s−1 . (d),(e),(f) For increasing V, the response
broadens and spectral weight (shaded) decreases. By V ¼ 4ER,
mSxx =N ¼ 0.37ð2Þ, and Γ ¼ 370ð140Þ s−1 . The upwards shift infrequency is due to increased confinement from the lattice beams.

Figure 2(b) shows that in the same conditions, σ xy − σ yx
is a smaller and noisy signal. When integrated using a sum
rule for off-diagonal conductivity [36], we find a cyclotron
frequency of 2π × ð0  2Þ Hz. This is expected, since nogauge field is applied here, however the method could be
used to explore the finite-frequency anomalous conductivity of systems with broken time-reversal symmetry[37–40]. In the remainder of this Letter, we report only
on-diagonal response.
In a pure harmonic trap, Re σ xx is a measurement-timelimited peak at the bare trap frequency [41] [see Fig. 2(c)],but the addition of a lattice broadens the response byenabling current dissipation [Figs. 2(d)–2(f)]. The Kubo
relation [42,43] gives σðωÞ as the Fourier transform of theretarded current-current correlation function, and thus afinite current lifetime τ broadens the ac conductivity spectraby τ−1. The damping of the current is seen through thediminished quality of the resonance, similar to cavityperturbation techniques employed in microwave spectroscopy of conductors [44,45]. In the absence of phonons,
defects, and impurities, collisions between the fermions are
responsible for dissipation of current. However, low-energy
collisions in the parabolic sector of the dispersion relationϵðkÞ do not contribute, since velocity and quasimomenta
are proportional, as in free space [22,41,46–48]. The fullband structure breaks Galilean invariance at higher quasimomenta, and enables collisional current dissipation [49].
The broadening of the global response includes not
only irreversible decoherence due to collisions, but alsodephasing due to the nonparabolic dispersion relation.
To deconvolve these two effects, we developed an effectivemodel based on linear response theory using the exact
eigenmode structure of the single-band confined-lattice
Hamiltonian [50–52], ĤCL ¼ Ĥ0 þ ĤP . In this model, aPtight-binding (TB) kinetic energy Ĥ0 ¼ −t hi;ji;s ĉ†i;s ĉj;s,153602-2

PHYSICAL REVIEW LETTERS 122, 153602 (2019)
(b)

(a)

FIG. 3. Transport time. (a) The best-fit broadening Γ foundfrom σ xx ðωÞ spectra in a variety of conditions. Varying initial N
from 5 × 103 to 50 × 103 creates variable density-weighted
filling per spin state n↑ , which is measured in situ, and ranges
from 0.09(1) to 0.19(2). Varying scattering length as from−240 a0 to 470 a0 results in U=t that ranges from −0.9 to 1.8.
Depositing additional heat energy Q with a nonadiabatic latticepulse before loading results in a range of T=t from 1.5 to 3.0. The
spectra in Figs. 2(d)–2(f) correspond to the circled points in thevariable-V series. The red band shows the current damping rate
τ−1 calculated with a kinetic theory over the range of measured
temperatures and densities; ΓPol (dashed) is measured for anoninteracting gas. (b) The same data and theory plotted withaxes nondimensionalized to account for temperature scaling of
the current dissipation rate. Some data for smallest Γ is omitted
for clarity.

where ĉi;s is the annihilation operator and hi; ji are adjacentsites, P
is combined with parabolic confinement Ĥ P ¼
ðm=2Þ α;i;s ω20α r2α;i n̂i;s , with trap frequency ω0α in the α
direction. For noninteracting atoms, linear response theory
predicts that the global conductivity at ω is given byðCLÞσ xx ðω; ΓÞ ¼

Nω X ðf p − f p0 Þjhp0 jR̂x jpij2;iℏ p0 ≠p ω − ωpp0 þ iΓ=2ð2Þwhere f p is the occupation of the eigenstate jpi of ĤCL ,and ℏωpp0 is the energetic splitting between states. The
broadening Γ represents the adiabatic ramp rate of theperturbation, but here is extended to also model weakly
dissipative effects such as interaction-induced collisional
damping [53–55]. Conductivity spectra are fit to Eq. (2)with variable Γ, Maxwell-Boltzmann f p ðTÞ, and a small(≤ 2.5 Hz) trap frequency shift. Examples are shownin Fig. 2.
Figure 3 shows the best-fit Γ found for 21 different
σ xx ðωÞ spectra with N, as , or V varied, or a quantity of heat
energy Q added before lattice loading to change T. The
lowest Γ are clustered near some nonzero minimum, whichis comparable to the ΓPol ¼ 36ð4Þ s−1 found for a spinpolarized ensemble (dashed horizontal line) at V ¼ 2.5ER .
Contributors to ΓPol could include finite measurement time,nonquadratic terms in the confinement, and imperfectionsin the optical lattice. An independent study of the Fourier
limit, with only the XDT beams [24], gives Γ ≥ 18ð3Þ s−1
for the time sequence used here. A study of the effects ofcontrolled disorder on σðωÞ is beyond the scope of thiswork, but would be an interesting complement toRefs. [13,14,18,56–59].
For Γ ≫ ΓPol, Fig. 3(a) shows a roughly linear trend
versus the product n↑ U 2 =t, where n↑ is the densityaveraged filling, and U and t are calculated parameters
for the Fermi-Hubbard Hamiltonian ĤFH ¼ Ĥ0 þ Ĥ U , withP
on-site interaction ĤU ¼ U i n̂i↑ n̂i↓ . We interpret Γ in thislimit as a measure of the current dissipation rate τ−1 .
The linear scaling may be understood in terms of a
Boltzmann-like n↑ U2 proportionality to density and scattering cross section, which is reasonable for the low
densities and moderate interaction strengths explored here
[15,19,43]. These observed trends are compared to a threedimensional kinetic calculation of collisional current
dissipation in a uniform lattice [24,43], which in the
low-n↑ , U ≪ t limit, givesτ−1 ≈ n↑U2
CðT=tÞ;ℏtð3Þ

where CðT=tÞ varies between 0.11 and 0.36 in the range
1.2t ≲ T ≲ 3.3t considered for these data. The calculatedτ−1 (shaded region in Fig. 3) compares well with themeasured Γ once Γ > ΓPol .
Contained within CðT=tÞ is the effectiveness of scattering in dissipating current, for which the role of the lattice isessential: some collisions can exchange momentum withthe light field, enabling mass current to be damped. The
Tﬃ
pﬃﬃﬃ
dependence of CðT=tÞ is quite different from the T
scaling of the collision rate in a free gas. A vanishingT 2 e−Δuk =T Fermi-liquid signature is expected at low T andω, where Δuk is the umklapp gap [47], whereas saturationof the rate of current dissipation will occur in the hightemperature limit [6]. An inflection point exists between
these limits [24], leading to an approximately linear
dependence of τ−1 on T=t in the range explored here.
Plotted in Fig. 3(b) is the same data as in Fig. 3(a) but withdimensionless axes that reflect this temperature scaling.
The scaled ℏΓ=T data agree well with calculations ofℏτ−1 =T, further supporting the conclusion that we candetermine transport time from our conductivity spectra.
Note that since ℏΓ=T < 1 for all measurements in
Fig. 3(b), it is reasonable to classify our system as aconventional metal, in which only a single damping time isexpected [60].
A second quantity deduced from the conductivity spectrais the frequency integral of Re σ, or “f sum.” The exact sum
rule is [22,24]

153602-3

PHYSICAL REVIEW LETTERS 122, 153602 (2019)(a)sum is additionally a measure of kinetic energy through2
2
STB
xx ¼ −aL hĤ 0 i=3ℏ , and one finds

(b)

STB
xx ¼

FIG. 4. Spectral weight of data sets with various N, as , Q, andV, as described in the caption of Fig. 3. (a) The partial f sum
Sxx =N, scaled by the TB effective mass m0 , is shown versus
measured T=t. Data are compared to three treatments of theuniform-lattice HM: a single band populated using MB statistics(red band), STB(purple solid line), and the asymptoticxxm0 Sxx =N ¼ t=T (black dashed line). The width of the redband is due to the variation of Sxx with V, for fixed T=t. The
red line is for V ¼ 2.5 ER. (b) Measured Sxx robustly agrees
with the noninteracting single-band calculation, even up toas ∼ 1.2 × 103 a0 .

S∞
αβ ≡

2
π

Z ∞
0

dω Re σ αβ ðωÞ ¼

N
h½R̂ ; Ĵ i;iℏ α βð4Þwhere the angle brackets denote a thermal average. For any
Psystem described by Ĥ ¼ Ni¼1 p̂2i =2m þ Vðr̂1 ; …; r̂N Þ,
one can show S∞
αβ ¼ ðN=mÞδα;β, independent of temperature, interaction strength, or trapping environment. We
find that this sum rule is satisfied without the lattice, as
shown in Fig. 2(c), where Sxx ¼ 1.01ð8Þ N=m.
Figure 4 shows the spectral weight found by integrationðCLÞof the best-fit σ xx , across a wide range of conditions. We
see that within the sampled frequency band, the spectral
weight is generically less than N=m, with the remaining
spectral weight transferred to interband transitions [9,61]
that need not be associated with low-frequency transport.
An effective low-energy Hamiltonian Ĥ eff can capture this
response: here, the partial f sum is Sxx ¼ h½R̂α ; Ĵsb
β i, with apurely single-band current Ĵ sb
¼
½
Ĥ
;
R̂
,
and
its correeffββsponding conductivity. The intraband response has anincreased carrier inertia: Sxx ¼ N=mband , where mband ¼
h1=mxx ðkÞi−1 is the band mass [62]. Alternately, N=mband
determines the current impulse response to a force applied
quickly compared to τ−1 , but slowly compared to theinverse band gap [63,64].
The red band in Fig. 4(a) shows the calculated f sum
with the Fermi-Hubbard Hamiltonian, using a MaxwellBoltzmann thermal average of single-particle states. Acrossa wide range of conditions, the data agree well with thepredictions of this uniform-lattice theory with no freeparameters. In the TB limit of an isotropic lattice, the f

N I 1 ð2t=TÞ
;m0 I 0 ð2t=TÞð5Þwhere I n ðzÞ is a modified Bessel function, and m0 is the TB
effective mass at k ¼ 0 [24]. This result is shown as apurple line in Fig. 4(a), capturing the salient trend in Sxx .
For the highest-temperature data, m0 Sxx =N approaches
t=T [dashed black line in Fig. 4(a)], which is a regime
previously discussed for single-band Hubbard models in
the context of T-linear dc resistivity [6,65–67]. The 1=T
regime of spectral weight is accessible with atoms in an
optical lattice because even at T ≫ t, we do not leave the
Hubbard regime, nor are phonons introduced or the crystal
structure affected. This is to be distinguished from the Tlinear resistivity that occurs at lower temperature in bothincoherent and conventional metals, and which is often
attributed to the temperature dependence of τ−1 [15,60,68–72]. Eventually, Sxx will vanish for large T=t: both flat anduniformly filled bands are inert to transport.
Figure 4(b) shows that Sxx is relatively unaffected by as,
despite the strong variation of τ−1 with as seen in Fig. 3.
This illustrates a basic property of optical conductivity:
scattering cannot “destroy” conductivity, but may only
move it from one part of the Re σðωÞ spectrum toanother [43].
Discussion.—The joint significance of Sxx and τ is that
their product gives an upper bound on the low-frequency
conductivity. For example, in the weak-trap limit ω0 → 0 of
Eq. (2), with fixed Γ ¼ τ−1 , the peak conductivity would be
Sxx τ. The same product is also found in Drude response
σ D ¼ Sxx =ð−iω þ τ−1 Þ at ω ¼ 0 [73]. Associating conductivity with the product of a dynamical factor and athermodynamic quantity is also found in the NernstEinstein form of conductivity, as the product of diffusivity
and compressibility [6,16,17,74].
More generally, the significance of Sxx and Γ is that they
determine the leading orders of conductivity for large ω, iSxx ΓSxx
1σðωÞ →
ð6Þ
þ 2 þO 3 ;ωωω
up to a cutoff [6,75]. The first term can be shown on generalgrounds using Kramers-Kronig relations. The second term
is model specific, but is found in a Drude response, in our
phenomenological quantum model, and in kinetic theory.
Furthermore, the coefficients of these leading terms can befound from a spatial average of local responses, at least in
kinetic theory [24]. This means that the f sum and current
damping rate for a trapped system can be obtained byintegrating spatially the response of a uniform system. One
expects such a “local density” picture to become valid in
the high-frequency limit since the amplitude of the driven

153602-4

PHYSICAL REVIEW LETTERS 122, 153602 (2019)
motion is vanishingly small, and over one oscillation cycle,
atoms in each region of the cloud do not explore the fullsystem. Experimental evidence for this correspondence isgiven in Figs. 3 and 4 by the excellent agreement between
uniform-lattice calculations and the Sxx and τ measured
with atoms in a lattice with weak confinement.
In sum, our work demonstrates how ac conductivity of
trapped atoms can be determined, and how the spectra weobserve can be understood in terms of transport time and band
mass. In particular, we show that dissipation in the regime we
explore is due to the combination of interactions between
fermions and the breaking of Galilean invariance by thelattice. Direct extensions of this work could include measuring the conductivity spectra of strongly correlated insulators, non-Fermi-liquid metals, and resonant superfluids.
We thank A. Georges, T. Giamarchi, A. MacDonald, E.
Mueller, A. Paramekanti, A. Rosch, E. Taylor, Zhigang
Wu, and Shizhong Zhang for stimulating conversations.
This work is supported by NSERC, by AFOSR underFA9550-13-1-0063, and by ARO under W911NF-15-10603. F. C. acknowledges support from the European
Union (ERC Grants ThermoDynaMix and CritiSup2).
R. A. and F. W. contributed equally to this work.

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