Roton-Maxon Excitation Spectrum of Bose Condensates in a Shaken Optical Lattice

Li-Chung Ha1, Logan W. Clark1, Colin V. Parker1, Brandon M. Anderson1,2, Cheng Chin1

1James Franck Institute, Enrico Fermi Institute and Department of Physics,University of Chicago, Chicago, IL 60637, USA and

2Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA(Dated: February 10, 2015)

We present experimental evidence showing that an interacting Bose condensate in a shaken opticallattice develops a roton-maxon excitation spectrum, a feature normally associated with superfluidhelium. The roton-maxon feature originates from the double-well dispersion in the shaken lattice,and can be controlled by both the atomic interaction and the lattice modulation amplitude. Wedetermine the excitation spectrum using Bragg spectroscopy and measure the critical velocity bydragging a weak speckle potential through the condensate – both techniques are based on a digitalmicromirror device. Our dispersion measurements are in good agreement with a modified Bogoliubovmodel.

PACS numbers: 03.75.Kk, 05.30.Jp, 37.10.Jk, 67.85.-d

In his seminal papers in the 1940s [1, 2], L. D. Landauformulated the theory of superfluid helium-4 (He II) andshowed that the energy-momentum relation (dispersion)of He II supports two types of elementary excitations:acoustic phonons and gapped rotons. This dispersionunderpins our understanding of superfluidity in helium,and explains many experiments on heat capacity and su-perfluid critical velocity. What is now called the “roton-maxon” dispersion in He II has been precisely measuredin neutron scattering experiments [3, 4] and is generallyconsidered a hallmark of Bose superfluids in the stronginteraction regime.

The roton-maxon dispersion carries a number of in-triguing features that distinguish excitations in differentregimes. The low-lying excitations are acoustic phononswith energy E = pvs, where p is the momentum and vsis the sound speed. At higher momenta, the dispersionexhibits both a local maximum at p = pm with energyE = ∆m and a minimum at p = pr with energy E = ∆r.The elementary excitations associated with this maxi-mum and minimum are known as maxons and rotons,respectively. The roton excitations, in particular, areknown to reduce the superfluid critical velocity belowthe sound speed. This is best understood based on theLandau criterion for superfluidity in which the critical ve-locity set by the roton minimum vc ≈ ∆r/pr is lower thanthe sound speed vs. The roton minimum also suggeststhe emergence of density wave order [5] and dynamicalinstability [6].

To explore the properties of these unconventional ex-citations, many theoretical works have proposed schemesfor producing the roton-maxon dispersion outside of theHe II system. Many proposals have been devoted toatomic systems with long-range or enhanced interactions,e.g. dipolar gases [6–8], Rydberg-excited condensates [9],or resonantly interacting gases [10]. Other candidates are2D Bose gases [11, 12], spinor condensates [13, 14], andspin-orbit coupled condensates [15, 16]. Experimentally,mode softening resulting from cavity-induced interaction

has recently been reported [17], which provides strong ev-idence for an underlying rotonlike excitation spectrum.

In this Letter, we generate and characterize an asym-metric roton-maxon excitation spectrum based on aBose-Einstein condensate (BEC) in a one dimensional(1D) shaken optical lattice. We implement Bragg spec-troscopy and identify the local maximum and minimumin the dispersion associated with the maxon and roton ex-citations. Furthermore, by dragging a speckle potentialthrough the BEC we show a reduction of the superfluidcritical velocity in the presence of the roton dispersion.

We create the roton-maxon dispersion by loading a 3DBose condensate into a 1D shaken (i.e. periodically phasemodulated) optical lattice. The lattice shaking techniquehas been used previously to engineer novel band struc-tures [18, 19] and to simulate magnetism [20–22]. Here,we phase modulate the lattice to create a double-wellstructure in the single-particle dispersion ε0(q), for whichthe ground state has a twofold degeneracy; see Fig. 1(a)and Ref. [21]. The double-well dispersion results froma near resonant coupling between the ground and firstexcited band through lattice shaking [21], and is a con-sequence of the parametric instability of a driven anhar-monic oscillator [18]. The dispersion with quasimomen-tum q can be calculated based on a Floquet model [21].A similar double-well dispersion can also be realized in aspin-orbit coupled system [23–27].

The double-well dispersion is modified by atomic inter-actions. Assuming the BEC is loaded into one of the twodispersion minima at quasimomentum q = q∗, we intro-duce the canonical momentum p = q − q∗ in the referenceframe where the condensate has zero momentum and en-ergy. The new dispersion is ε0(p) = ε0(p + q∗) − ε0(q∗).One finds that the dispersion are no longer symmetricdue to the existence of the other unoccupied minimum;see Fig. 1(b). Based on a modified Bogoliubov calcula-tion (see Supplemental Material [28] and Refs. [29, 30]),we diagonalize the Hamiltonian to obtain the excitationspectrum:

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FIG. 1: (color online) Generation of roton-maxon dispersionin a shaken lattice. (a) For a single atom, the lattice mod-ulation creates a double-well structure above a critical mod-ulation amplitude (top three lines) [21]. In our experiment,the atoms are prepared at the minimum with zero or negativemomentum (q∗ ≤ 0, red dot); see text. (b) With atomic in-teractions, a roton minimum (circle) and a maxon maximum(square) in the excitation spectrum can form. The dashed lineindicates the critical velocity limited by the roton minimumaccording to the Landau criterion for superfluidity. Disper-sions are upward offset with increasing modulation amplitudefor clarity. The lattice reciprocal momentum is hkL = h/λwhere λ is the wavelength of the lattice beams and h = 2πh isthe Planck constant.

E(p) =√ε(p)2 + 2µε(p) +∆ε(p), (1)

where ε(p) = [ε0(p)+ε0(−p)]/2, ∆ε(p) = [ε0(p)−ε0(−p)]/2and µ is the chemical potential. For a system witha double-well structure in ε0(p), the theory predictsa roton-maxon structure with the roton minimum oc-curing near p = −2q∗; see Fig. 1(b). Creation of anartificial roton˝ in the dispersion minimum of an anal-ogous spin-orbit coupled system was theoretically pro-posed in Ref. [15].

Our experiment to detect this unusual dispersionstarts with an almost pure cesium condensate of N0 =30 000 atoms loaded into a crossed beam optical dipoletrap (wavelength λ = 1064 nm) with trap frequencies(ωx, ωy, ωz) = 2π × (9.3,27,104) Hz [21]. We turn onan additional 1D optical lattice by retroreflecting one ofthe dipole trap beams in the x − y plane at 40○ with re-spect to the x-axis. The lattice depth is approximatelyV = 7 ER, where ER = h × 1.325 kHz is the photon re-coil energy of the lattice beam. The lattice potential isphase modulated at 7.3 kHz which is 0.7 kHz blue de-tuned from the ground to first excited band transitionat q = 0. The phase modulation creates admixed bands,and the ground band develops two minima in its disper-sion [21]. We preferentially load the BEC into one ofthe minima by providing a momentum kick before phasemodulating the lattice [21]. We define the direction ofthe kick as negative, and thus the BEC has a negative

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FIG. 2: (color online) Excitation spectra. (a) We measurethe excitation spectra with N0 = 30 000 atoms in a harmonictrap (square) and in a stationary lattice (circle) with DMD-based Bragg spectroscopy. The inset illustrates the movingoptical potential with velocity v and periodicity d created bythe DMD on the BEC (tilted ellipse); see text. The solidlines correspond to the Bogoliubov model with chemical po-tentials equal to the trap-averaged values. (b) For a BEC withN0 = 9000 atoms loaded in a shaken optical lattice, we mea-sure the excitation spectrum along the lattice direction. Themodulation amplitude (peak to peak) is ∆x = 33 nm. Thesolid line is the best fit based on Eq. (1). The inset shows atypical atom loss spectrum taken at k = −0.38 kL. In bothpanels, the scattering length is a = 47 a0.

momentum q = q∗ < 0 and the roton minimum is expectedat p = 2∣q∗∣; see Fig. 1(b).

To probe the dispersion we perform Bragg spec-troscopy [31] by illuminating the atoms with a sinusoidalpotential moving along the direction of the shaken lattice.The potential is created from a programmable digital mi-cromirror device (DMD) and a 789 nm laser, which pro-vides a repulsive dipole force. The DMD potential withvelocity v and periodicity d [see Fig. 2(a) inset] inducesa Raman coupling between the condensate with p = 0and finite momentum states with p = h/d. When the Ra-man detuning E = pv matches the energy of the finite

3

momentum state E(p), a resonant transfer will removeatoms from the condensate. We illuminate the atomswith the moving potential for 40 ms and measure theresidual condensate particle number after a 30 ms timeof flight (TOF). The dispersion can be mapped out byfinding the energy which gives the strongest reduction ofatom number in the condensate for each momentum p.

To test this technique, we compare the dispersions ofthe BEC in a harmonic trap and that in a V = 7 ER un-shaken lattice to Bogoliubov calculations; see Fig. 2(a).The measurement agrees well with the Bogoliubov spec-trum using the measured trap-averaged chemical poten-tials µ = h×120 Hz without the lattice and µ = h×150 Hzwith the lattice.

We now consider the dispersion of a BEC in a shakenoptical lattice, where the roton feature is expected. Herewe observe a distinct difference between the excitationsat positive versus negative momentum. We work witha modulation amplitude (peak to peak) of ∆x = 33 nmwhich guarantees a strong double-well feature. Fig. 2(b)shows the dispersion measurement, which contains a clearroton-maxon feature at positive momentum (hereafter,the roton direction). In contrast, we do not see this fea-ture for negative momentum (hereafter the nonroton di-rection).

We compare the measured roton spectrum with themodel in Eq. (1). Constraining the model to the ex-perimental parameters only yields qualitative agreementlikely due to interaction effects [32] which effectively mod-ify the modulation amplitude ∆x and lattice depth V .Thus we fit the data with Eq.(1) and find the best fit tohave µ = h×58(4) Hz, V = 5.9(1) ER and ∆x = 49(3) nm.The low chemical potential is expected and comes fromthe lower condensate number as well as the weaker, mo-mentum dependent atomic interactions in the admixedband.

The roton energy is determined by atomic interactionsand can be controlled by tuning the scattering length.To demonstrate this we prepare samples with the usualprocedure but at a higher scattering length a = 70 a0followed by ramping the magnetic field to reach the de-sired scattering length [33]. We measure the excitationspectrum in the roton direction with p > 0 at six differentscattering lengths, shown in Fig. 3(a).

We adopt a global fit to the data in Fig. 3(a) based onEq. (1) to determine the roton energy ∆r and the maxonenergy ∆m. Our observation shows that we can exper-imentally tune the scattering length to vary the rotonenergy by a factor of 3. Furthermore, we can use scal-ing arguments to distinguish the behavior of rotons andmaxons from the more conventional phonons. For smallchemical potentials, the excitation energy for phonons iswell known to scale as µ1/2, while the roton and maxonenergies are expected to depend linearly on µ; see Supple-mental Material [28]. Furthermore, for an adiabatic rampof the scattering length, the chemical potential should

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FIG. 3: (color online) Roton or maxon energy vs scatteringlength. (a) We measure the excitation spectra at differentscattering lengths a/a0=5 (circle), 13 (triangle), 24 (square),40 (diamond), 55 (pentagon) and 70 (star). The condensatenumber is N0 = 9000. Solid curves are fits based on Eq. (1). Aglobal optimization procedure gives V = 6.7(2) ER and ∆x =43(3) nm. (b) Roton energies (circle) and maxon energies(square) extracted from the fits in panel (a) are shown atdifferent scattering lengths. Solid curves are fits based on∆r = A(a/a0)

2/5 and ∆m = B + C(a/a0)2/5, from which we

obtain A = h×9(1) Hz, B = h×37(9) Hz and C = h×8(1) Hz.

scale as µ = n0g ∝ a2/5 where g ∝ a is the interactionstrength, and the condensate density in the harmonictrap is n0 ∝ a−3/5 [2]. Therefore, we plot the extractedroton and maxon energies as a function of a2/5 as a proxyfor the chemical potential; see Fig. 3(b). The observedlinear dependence confirms the expected scaling for ro-tons and maxons.

One significant consequence of the roton dispersion isthe suppressed superfluid critical velocity vc. We mea-sure the critical velocity of the BEC loaded into theshaken lattice by projecting a moving speckle patternusing the DMD. Instead of using a single laser beam [35–37] or a lattice with a definite spatial frequency [38], ourspeckle pattern contains a broad spectrum of wavenum-

4

bers up to the resolution (k ≈ 0.55 kL) of our projectionsystem. Furthermore, the potential remains locally per-turbative (≈ h × 1.1 Hz) to prevent vortex proliferation[39–41]. When the velocity of the speckle pattern reachesor exceeds the critical velocity, atoms are excited fromthe condensate. To prevent excitation in the low densitytail [38], we digitally mask out the region of the specklepattern which could overlap with the edge of the cloud.

We observe a clear threshold in speckle velocity abovewhich the condensate number decreases; see Fig. 4(a).The experimental sequence is similar to that used forBragg spectroscopy: we illuminate the cloud with a mov-ing speckle pattern for 100 ms followed by a 30 ms TOF.To find the critical velocity, we fit the remaining conden-sate number with a constant value intersecting a lineardecay. The intersection point determines the critical ve-locity vc. Above a critical value, we observe the conden-sate fraction decreases linearly with the speckle velocity.This is consistent with a previous observation of the crit-ical velocity in a Bose superfluid [38], along with a recentcalculation [42].

In order to understand the emergence of the roton-maxon dispersion, we measure critical velocity in boththe roton direction p > 0 and the nonroton direction p < 0with increasing modulation amplitude ∆x; see Fig. 4(b).In order to maintain a comparable chemical potential,we prepare the samples with a large ∆x = 33 nm andslowly ramp ∆x to the desired value. For small final ∆x <12 nm, vc is the same in both directions and decreasesas we approach the critical value ∆xc ≈ 12 nm (phononmode softening). When the gas enters the ferromagneticphase (∆x > 12 nm) [21], vc increases immediately alongthe nonroton direction, while in the roton direction vcremains small.

We compare the measurement with the critical velocitybased on the Landau criterion vL = min∣E(p)/p∣. As theexperiment conditions closely resemble those in Fig. 2(b),we evaluate the critical velocity with µ = h × 58 Hz,V = 5.9 ER and ∆x scaled by 1.5, the parameters whichbest fit that dispersion measurement. The calculatedvL, shown as dashed lines in Fig. 4(b), displays a dis-parity between the roton and nonroton directions for∆x > 15 nm, in agreement with our observation. Ourcritical velocities, however, are significantly lower thanvL. In early BEC experiments [36, 37], low critical veloc-ities were observed and explained by the large obstaclesthat disrupt the superflow and spin off vortices [39–41].In our experiment with weak speckle potential, a likelyscenario is that the critical velocity is limited by exci-tations generated in the low density regions above andbelow the cloud along the DMD projection axis.

In conclusion, we observe a roton-maxon dispersion ofa BEC in a shaken 1D optical lattice based on threepieces of evidence: the many-body excitation spectrum,the dependence of the excitation energies on the atomicinteractions, and the superfluid critical velocity measure-

-400 -200 0 200

0.8

0.9

1.0nonroton roton

(a)

Con

dens

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num

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ract

ion

Speckle velocity v (µm/s)

0 1 0 2 0 3 00

1 0 0

2 0 0

3 0 0

4 0 0

r o t o n

n o n r o t o n

Critic

al ve

locity

v c (µm/

s)

M o d u l a t i o n a m p l i t u d e � x ( n m )

( b )

FIG. 4: (color online) Superfluid critical velocity. (a) We mea-sure the residual condensate number fraction after dragginga speckle pattern through the center of the cloud at differ-ent velocities v along the roton direction (p > 0, solid dots)and the nonroton direction (p < 0, solid squares). The solidlines are fits used to determine the critical velocity. The in-set illustrates the experimental scheme; see text. (b) Criticalvelocities as a function of modulation amplitude are shown.Above the critical modulation amplitude ∆x > 12 nm, thecritical velocity is significantly lower in the roton direction.Our measurement is compared with the critical velocity cal-culated from Eq. (1) using Landau criterion (dashed lines). Inboth panels, the scattering length is a = 47 a0 and the initialcondensate number is N0 = 9000.

ment. Our results agree well with the Bogoliubov calcula-tion and suggest the roton/maxon excitations are distinctfrom acoustic phonons. Our experiment demonstratesthat shaken optical lattices are a convenient platform togenerate new types of quasiparticles in a dilute atomicgas, allowing future study of their dynamics, stability,and interactions. For instance, knowing the quasiparti-cle dispersion should allow a future experiment to createmacroscopic numbers of rotons, leading to possible rotoncondensation [42, 43], and separation of the rotons intodomains. In situ imaging would allow direct observationof the temporal evolution of such states.

5

We thank K. Jimenez Garcıa for discussion and carefulreading of the article, and U. Eismann and E. Hazlettfor assistance in the early phase of the experiment. L.-C. H. is supported by the Grainger Fellowship and theTaiwan Government Scholarship. L. W. C. is supportedby the NDSEG Fellowship. This work was supported byNSF MRSEC Grant No. DMR-1420709, NSF Grant No.PHY-0747907 and ARO-MURI Grant No. W911NF-14-1-0003.

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6

SUPPLEMENTAL MATERIAL

Optical Setup and Digital Micromirror Device(DMD)

The optical setup that we use for creating and probingthe roton-maxon dispersion is shown in Fig. S1(a). The1D optical lattice is created by retroreflecting one of the1064 nm dipole trap beams and is phase modulated usinga pair of acousto-optic modulators, as described in ourprevious work [S1]. We phase modulate the lattice at7.3 kHz, 0.7 kHz blue detuned from the ground to firstexcited band transition at q = 0. This modulation couplesthe two bands and produces a roton-maxon dispersion.

We have implemented a digital micromirror device(DMD: Texas Instruments, DLP LightCrafter 3000) totailor dynamic optical potentials for probing the disper-sion. The DMD consists of a 608 × 684 array of 7.6 µmsquare mirrors. Each mirror flips individually to one oftwo angles, separated by 24○. A mirror at the “on” an-gle will reflect light towards the atom cloud, while the“off” angle reflects light into a beam dump. We reflecta blue-detuned 789 nm laser off of the DMD and use ahigh-resolution objective lens to project the real spacepattern of mirrors in the “on” state onto the plane of theatom cloud. We use additional lenses to demagnify thepattern by a factor of 36. The resolution of the result-ing patterns is limited by the objective lens to ∼1 µm,approximately 5 micromirrors across. By having manymicromirrors in each resolution sized area we can gener-ate intermediate intensities in static patterns even thoughthe state of each micromirror is binary. The programmedpattern of “on” mirrors can be updated up to 4000 timesper second, allowing us to create motion in the projectedpatterns.

Bragg Spectroscopy with the DMD

In order to create single-wavevector moving lattices forBragg spectroscopy, we add additional beam selectionmasks in front of the DMD as shown in Fig. S1(b). Anapproximately sinusoidal DMD pattern will diffract thelaser into the desired beams as well as many harmon-ics. By always transmitting only two diffracted beamsthrough a mask we ensure that the projected opticalpotential contains only the desired wavevector for prob-ing the atoms. Shifting the pattern of the “on” mirrorstranslates the projected potential by the same amountregardless of the blocking mask. The second importantadvantage of beam selection is that removing the 0thorder diffraction allows us to double the maximum pro-jected wavevector. The 0th order diffracted beam is al-ways more intense than the ±1st, so when all are presentthe projected potential is dominated by the interferencewavevector between the 0th and ±1st. That wavevec-

objective

objectiveeyepiece

absorptionimagingbeam1D optical

lattice beams

AOMs forphase modulation

digitalmicromirrordevice (DMD)

DMD

dichroicmirrorDMD

illuminationbeam

atoms

eyepieces

beamselectionmask

CCD(a)

(b)

atoms

1D opticallattice beams

z

FIG. S1: Optical setup (a) Our optical setup is based on asingle high resolution objective lens, which allows us both toperform absorption imaging and to project arbitrary patternsfrom the DMD onto the atoms (see supplemental text fordetails). The 1D optical lattice is formed by retroreflectingone of the 1064 nm dipole trap beams after passing it throughtwo acousto-optic modulators (AOMs), which can be usedto phase modulate the lattice [S1]. (b) In order to projectclean optical potentials for probing dispersion, beam selectionmasks transmit only the two desired beams diffracted fromthe DMD. In this way the interference pattern on the atomscontains no confounding wavevectors.

tor is limited by the objective lens to approximatelyk = 0.55 kL. Once the 0th order beam is blocked, theinterference between the 1st and -1st dominates, whichraises the maximum projected wavevector to approxi-mately k = 1.1 kL and is sufficient for our experiments.

For any wavevector of the projected potential, thequasiparticle excitation frequency corresponds to the rateat which the pattern’s phase shifts by 2π. We typicallyuse sets of 9 lattice patterns, so that each pattern switchcorresponds to a phase change of 2π/9. We scan the exci-tation frequency by changing the rate at which we triggerthe DMD to cycle through the set of patterns. To makethe movement smoother for probing the small excitationfrequencies in Fig. 3, we use sets of 20 patterns instead.

The dispersion relation corresponds to the points inwavevector and frequency space at which we observe res-onant heating of the atom cloud. We determine those

7

points by probing the atomic sample at a fixed wavevec-tor and scanning the DMD triggering frequency. We typ-ically apply the exciting optical potential to the cloudfor 40 ms, then perform 30 ms time of flight (TOF) todetermine the number of atoms remaining in the conden-sate. When the excitation is resonant, atoms are excitedout of the condensate, which we can observe as a deple-tion of the atom number in the momentum state of thecondensate after TOF. Fig. S2 shows an example losscurve. The fit is to a Gaussian whose center we taketo be the resonance frequency. The resonance frequencyis not sensitive to the particular fit function chosen: fit-ting to a Lorentzian instead of a Gaussian typically shiftsresonances by tenths of Hz.

The example images in Fig. S2 show the difference be-tween the full and depleted clouds. For this excitation,which at k = 0.44 kL is near the roton momentum, theatoms missing from the main condensate peak (insidethe solid white circles) appear at the right side of the im-age in a location corresponding to the roton momentumafter TOF (inside the dashed white circle). For appliedpotentials with wavevectors far from the roton minimum,the excited atoms do not always appear in a predictableplace. However, the depletion of the condensate remainsa consistent signal for all excitation measurements andtherefore we use it throughout this work.

0 10 20 30 40 50 60

5000

6000

7000

8000

mai

n pe

ak a

tom

num

ber

excitation frequency (Hz)

22 μm-2

0 μm-2

atom

num

ber i

n m

ain

peak

excitation frequency (Hz)

20 μm

FIG. S2: Determining a dispersion point This plot showsthe atom number detected in the main peak after applying anexcitation of varying frequency with k = 0.44 kL to an atomicsample with a = 13 a0. Example images (each the average of4 or 5 experimental trials) illustrate the TOF results. Diffrac-tion peaks from the lattice are outside of the field of view. Theatom number is determined by integrating the signal presentin the solid white circle. The central image, corresponding toa near-resonant frequency, has a clearly depleted main peak.The dashed white circle indicates the location where atomstransferred to the roton minimum appear after TOF. Thesolid curve in the plot is a Gaussian fit which yields the exci-tation frequency of 29(3) Hz.

(b)

T= 0 ∆t

T= 20 ∆t

T= 40 ∆t

T= 60 ∆t

(a)

DMD

atoms

v

1D opticallattice beams

objective

eyepiece

FIG. S3: Critical velocity measurement (a) For criticalvelocity measurements a digital speckle pattern (light gray)moves at a steady speed v across the atomic sample, but isdigitally cropped with the DMD to illuminate only the cen-tral region of the sample (dark blue). We show the patternat four different elapsed times T = N × ∆t after the DMDhas been triggered N times, where ∆t is the delay betweentriggers. The thin vertical lines show the spatial period of thepatterns. (b) Critical velocity measurements do not requirea beam selection mask. We directly project the real spacespeckle pattern onto the atomic sample.

Critical velocity measurement

To measure the critical velocity of the condensate, wemove a speckle pattern through the atomic sample asshown in Fig. S3 and determine the minimum velocityrequired to heat the cloud. Heating is detected usingthe method shown in Fig. S2. The speckle pattern isgenerated on the DMD and directly projected onto theatoms with no beam selection mask in between. However,we do use a digital mask to ensure that we apply thespeckle only to the region of high chemical potential, seeFig. S3(a). We create a speckle pattern by randomlyturning on or off 4 × 4 sections of micromirrors insteadof individual mirrors. Each section still corresponds toan area smaller than our resolution limit but wastes lesslaser power into large-angle diffracted peaks that cannotbe collected by our projection optics. In principle thisspeckle pattern should excite the gas at a broad rangeof wavevectors, limited on the low end by the finite sizeof the condensate to approximately kmin = 2π/20µm ≈0.05 kL and limited on the high end by the resolution toapproximately kmax ≈ 0.55 kL.

To move the speckle we trigger the DMD to switch toa different pattern with the same speckle but shifted by0.3 µm in the plane of the atomic sample. The trig-gering rate f = 1/∆t determines the speckle velocityv = f × 0.3 µm. Because the DMD can only store upto 96 patterns, the speckle pattern is forced to repeat af-ter 96 × 0.3 µm= 29µm which is larger than the width of

8

a sample.

Modified Bogoliubov Spectrum

The shaken lattice is described by a single particleHamiltonian

H0 = −h2

2m

d2

dx2+ V sin2 [kL (x − x0 (t))] (S1)

where m is the mass of a particle, V is the lattice depthand x0 (t) = (∆x/2) sin (ωt). In what follows we will as-sume an extended system and ignore the harmonic trap-ping potential. We use the Trotter expansion to numer-ically calculate the single particle spectrum ε0(q) of thetime-averaged Hamiltonian [S1], see Fig. 1(a) in the maintext. We will henceforth work in momentum space andproject into the single particle band that is adiabaticallyconnected to the s-band in the limit of no shaking.

We can describe the interacting Bose gas with theHamiltonian

H =∑p

[ε0 (p) − µ] a†pap +

g

2v∑

q,p1,p2

a†p1+qa

†p2−qap1 ap2 ,

(S2)where g is the interaction energy, v is the volume ofthe sample, µ is the chemical potential and we have ap-plied a gauge transformation to shift the dispersion toε0 (p) = ε0 (p + q∗)−ε0 (q∗). Since the single particle spec-trum is asymmetric around the condensate momentum(p = 0), the standard Bogoliubov formula does not apply.To calculate the excitation spectrum of the system we as-sume a condensate at p = 0 and replace the annihilationoperator with a0 →

√N0, where N0 is the condensate

number. The Bogoliubov Hamiltonian is found by ex-panding to second order in the fluctuations around themean-field a0:

HBog = ∑p≠0

[(ε0 (p) + µ) a†pap +

µ

2(a†

pa†−p + apa−p)] , (S3)

where µ = N0g/v and we have neglected an overall mean-field energy shift of the condensate.

To diagonalize HBog we define a new set of operators

bp and b†−p implicitly through the relations

ap = upbp + vpb†−p, (S4)

a†−p = u−pb

†−p + v−pbp,

where we assume up, vp to be real. We require that theBogoliubov Hamiltonian is diagonal when expressed interms of the new operators:

HBog = ∑p≠0

E (p) b†pbp, (S5)

and that the new operators additionally satisfy the stan-dard commutation relations: [bp, b†p′] = δpp′ , [bp, bp′] =

0. We then calculate the commutators [ap, HBog],[a†

−p, HBog], [bp, HBog], and [b†−p, HBog]. Imposing the

the definition of Eq. (S4), as well as the constraint thatcommutation relations are preserved, results in a gener-alized eigenvalue equation [S2]

(ε0 (p) + µ µµ ε0 (−p) + µ

)u (p) = E (p)(1 00 −1

)u (p) ,

(S6)

where u (p) = (up, vp)T . Solving the eigenvalue equa-tion gives the Bogoliubov dispersion shown in Eq. (1) ofthe main text. The finite momentum of the condensatebreaks the symmetry in momentum around p = 0.

The Bogoliubov transformation coefficients can befound from the generalized eigenvector:

u (p) = 1

N (p)(f (p) +

√f2 (p) − µ2

−µ ) , (S7)

where N2 (p) = 2√f2 (p) − µ2 (

√f2 (p) − µ2 + f (p)) and

f (p) = ε (p) + µ normalizes the eigenvector such that

uT (p)(1 00 −1

)u (p) = 1.

Phonon, maxon, and roton excitations

As discussed in the main text, and shown in Fig. 1-3, the Bogoliubov spectrum has a linear dispersion nearp/q∗ ≪ 1, followed by a maxon at p = pm and a roton atp = pr. To calculate the phonon velocity we expand theBogoliubov spectrum for small p. The dispersion nearp = 0 is

E (p) ≈√

p2

2m∗( p2

2m∗+ 2µ)→

õ

m∗∣p∣ (S8)

with the effective mass

m∗ =⎛⎝d2ε0 (p)dp2

∣p=0

⎞⎠

−1

(S9)

which implies that the phonon velocity is given by v2s =µ/m∗.

Away from p = 0, and for sufficiently small µ≪ ε0 (p),the spectrum can be approximated by expanding inµ/ε (p):

E (p) ≈ ε0 (p) + µ +O⎛⎝[ µ

ε (p)]2⎞⎠. (S10)

This implies that the roton and maxon occur near thesingle particle minimum and maximum respectively. Theroton and maxon quasi-momenta are therefore well ap-proximated by pr ≈ 2∣q∗∣ and pm ≈ ∣q∗∣, with their energies

∆r ≈ µ, (S11)

∆m ≈ ε0 (−q∗) + µ, (S12)

9

respectively. In both cases the spectrum is linear in the

chemical potential, or equivalently in (a/a0)2/5, as wasfound in Fig. 3(b) of the main text.

Critical velocity by the Landau criteria

The critical velocity for a superfluid can be foundby considering a superfluid moving with velocity v ina reference frame K, simultaneously moving at a ve-locity v. The energy of the condensate in frame K isE = E (p). We apply a Galilean transformation to areference frame K ′ for which the container of the super-fluid is at rest. In frame K ′, the energy of the conden-sate is E = E (p) + vp + 1

2mv2. We see that in the lab

frame, the superfluid is only capable of dissipating en-ergy if E (p)+vp < 0. Since E (p) is positive, this impliesthat vp must be negative with ∣vp∣ ≥ E (p). Furthermore,the critical velocity must occur for the first p that satisfies∣vcp∣ = E (p). These arguments result in the celebratedLandau criteria:

vc = minp

∣E (p)p

∣ , (S13)

where ∣E (p) /p∣ is the phase speed of the excitation.For weak shaking the single particle band structure has

a single symmetric minimum at momentum p = q∗ = 0.Since the spectrum is symmetric in quasimomentumaround q∗ = 0, the Bogoliubov dispersion has the stan-dard form of

E (p) =√ε0(p)2 + 2µε0(p). (S14)

The critical velocity in the symmetric case is found byminimizing the phase speed over all p. Since there is noroton, the critical velocity is set by the speed of sound atsmall momenta:

vc0 =√

µ

m∗. (S15)

As the shaking amplitude increases, the dispersion nearthis minimum becomes increasingly flat as characterizedby (m∗)−1 → 0. At the critical shaking above whichthe double well structure emerges, the quadratic part ofthis dispersion exactly vanishes and the single particlespectrum is quartic at small p. This implies that E (p)∝p2 so E (p) /p → 0 as p → 0, and therefore the criticalvelocity must vanish. This dependence explains the dipin the critical velocity near ∆x = 12nm in Fig. 4(b).

Above the critical shaking amplitude the condensateoccupies the minimum at q∗ < 0 and the symmetry of theBogoliubov spectrum is broken. This asymmetry resultsin the condensate having two distinct critical velocitiesin the non-roton and roton directions. The critical ve-locity in the non-roton direction is set by the phononvelocity, because there is no excitation in that directionwith smaller phase speed. This is found by minimizingthe phase speed only for negative momenta

vc− = vs =√

µ

m∗. (S16)

In the roton direction, on the other hand, rotons canhave a smaller phase speed than phonons. Once againthe critical velocity is found by minimizing the phasespeed, but now for only positive momenta. This functionis minimized numerically to produce the dashed red linein Fig. 4(b) above the critical shaking value. For a suffi-ciently small chemical potential, such that the spectrumaway from p = 0 is well approximated by Eq. (S10), weindeed expect the rotons to have a smaller phase speedthan the phonons. This implies the critical velocity inthe roton direction is well approximated by

vc+ ≈ ∣ ∆r

2q∗∣ . (S17)

We therefore see that the limit of small chemical poten-tial, the ratio of the two critical velocities is given by

∣vc+vc−

∣ ≈ ∆r/2∣q∗∣√µ/m∗

≈ 1√8

õ

q∗2/2m∗, (S18)

where we have used ∆r ≈ µ and q∗2/2m∗ ≫ µ for ex-perimentally relevant values. Therefore the critical ve-locity when moving in the roton direction is significantlysmaller than the critical velocity in the non-roton direc-tion. This was observed in Fig. 4(b).

[S1] C. V. Parker, L.-C. Ha and C. Chin, Nat. Phys. 8, 769(2013).

[S2] C. J. Pethick and H. Smith, Bose-Einstein Condensationin Dilute Gases (Second Edition, Cambridge UniversityPress, 2008).