Expansion of matter waves in static and driven periodic potentials
C.E. Creffield, F. Sols, D. Ciampini, O. Morsch, E. Arimondo
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Expansion of matter waves in static and driven periodic potentials
C.E. Creffield, F. Sols
Departamento de F´ısica de Materiales, Universidad Complutense de Madrid, E-28040, Madrid, Spain
D. Ciampini , , O. Morsch , and E. Arimondo , CNISM-Pisa, Dipartimento di Fisica, Universit`a di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy and INO-CNR, Dipartimento di Fisica, Universit`a di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy (Dated: November 2, 2018)We study the non-equilibrium dynamics of cold atoms held in an optical lattice subjected to aperiodic driving potential. The expansion of an initially confined atom cloud occurs in two phases:an initial quadratic expansion followed by a ballistic behaviour at long times. Accounting for thisgives a good description of recent experimental results, and provides a robust method to extract theeffective intersite tunneling from time-of-flight measurements.
PACS numbers: 03.75.Lm, 03.65.Xp
I. INTRODUCTION
Experimental progress in confining ultracold atoms inoptical lattices has undergone spectacular progress in re-cent years. Optical lattice potentials are extremely cleanand controllable, and the excellent coherence propertiesof atomic condensates hold out the prospect of controllingtheir dynamics using quantum coherent methods. Thisis both interesting from the point of view of fundamentalphysics, and has many potential applications to quantuminformation processing, where it is vital that the coher-ence of the system is preserved during its time evolution.One such scheme is to use a time-periodic driving poten-tial to induce the effect termed “dynamical localization”[1] or “coherent destruction of tunneling” [2]. This is aquantum interference effect in which a particle acquiresa phase from its interaction with the driving potential,which leads to a renormalization of the single-particletunneling probability. For specific values of the driv-ing parameters the effective tunneling probability can behighly suppressed, providing a sensitive means of coher-ently controlling the localization of the atoms [3]. Thisrenormalization has recently been directly observed incold atom experiments [4–6].A convenient way of measuring the effective tunnel-ing is to observe the rate of expansion of a conden-sate once a harmonic potential trapping the atoms alongthe direction of the optical lattice has been switched off[4, 5]. Although the results of these experiments agreedwell with the theoretically expected scaling of the renor-malized tunneling probability with the Bessel functionof the driving strength, for particular initial conditionsthe scaling seemed to be quadratic rather than linear inthe Bessel function. A number of explanations of thisphenomenon have since been put forward, including apossible crossover from coherent to sequential tunneling[5, 7] induced by phase scrambling arising from dynam-ical instabilities, a time-averaging effect produced by fi-nite time-resolution of the measurement [8], and driving-induced atom-pairing [9]. A recent theoretical stabilityanalysis [10] has shown, however, that phase scrambling is unlikely to occur for the experimental parameters ofRefs. [4, 5], and the pairing mechanism would requirerather stronger interactions than were present in thoseexperiments. In this paper we propose an alternative ex-planation for the observed quadratic scaling of the renor-malized tunneling probability based on the exact form ofthe expansion of the condensate, which is linear in thelong time limit but quadratic for short times [11]. Usingthis complete time dependence in order to extract thetunneling probability from the experimental data givesquantitatively accurate agreement with theory, with noadjustable parameters.
II. MODEL AND ANALYSIS
The Bose-Hubbard model is described by the Hamil-tonian H BH = − J X h i,j i h a † i a j + H.c. i + U X j n j ( n j − X j V ( r j ) n j , (1)where a j /a † j are annihilation / creation operators for aboson on lattice site j , J (taken to be positive) describesthe hopping amplitude between nearest neighbor sites h i, j i , and U is the repulsive energy between two bosonsoccupying the same site. The operator n j = a † j a j is thestandard number operator, and V ( r ) is the external trappotential, which is usually considered to be parabolic, V ( r ) = mω T r /
2, where ω T is the trap frequency. Al-though simple in appearance, the Bose-Hubbard modelcan provide an excellent description [12] of ultracoldatoms held in optical lattice potentials.Adding a static and a sinusoidally varying force to thesystem leads to the general time-dependent potential H ( t ) = H BH + X j n j j (∆ + K cos ωt ) , (2)where ∆ is the static tilt applied to the lattice, and K and ω are the amplitude and frequency, respectively, of the os-cillating component. Experimentally, the two forces areintroduced into the rest frame of the optical lattice by ap-plying appropriate frequency differences to the acousto-optic modulators creating the lattice beams, as describedin detail in [4]. A time-periodic system of this type canbe analyzed using Floquet theory, revealing [13] that theeffect of the driving can be described by the static Hamil-tonian (1) with a renormalized tunneling J eff . For an un-tilted lattice (∆ = 0) this renormalization is given bythe zeroth-order Bessel function J eff = J J ( K ), wherefor convenience we define K ≡ K/ ¯ hω . Thus at the val-ues K = 2 . , . , . . . , at which the Bessel functionvanishes, the effective tunneling is suppressed. This effectthus provides a means to coherently control the dynamicsof trapped atoms, without altering any of the parametersof the optical lattice.When the trap potential along the lattice direction isremoved the atom cloud will expand in time, at a ratedetermined by J eff . To quantify this process we calculatethe spread of the wavefunction σ ( t ) = p h x i − h x i , (3)aligning the optical lattice with the x -axis. We begin bysetting the Hubbard interaction, U , to zero. In the con-tinuum approximation, valid when the kinetic energy ofthe condensate is much less than the width of the firstBloch band, the ground state of a parabolic trap is simplygiven by a Gaussian, ψ ( x ) = N exp[ − x / (2 a )], where N is the normalization such that R ∞−∞ | ψ ( x ) | dx = 1, and a = p ¯ h/ ( mω T ) is the harmonic trap-length. If the trappotential is now removed, this initial state will expandwhile remaining Gaussian. This expansion can be calcu-lated analytically [11] yielding the result σ ( t ) = σ q Jt/ ¯ h ) ( d L /a ) , (4)where d L is the spacing of the optical lattice, and σ = a/ √
2. Similar, but more complicated, expressionswere obtained by Korsch [14] using a lattice represen-tation instead of the continuum approximation, whichcoincide with this result. The expansion clearly occursin two different phases, separated by a crossover time t c = (¯ h/ | J | ) ( a/d L ) . For long expansion times, t ≫ t c ,the wavefunction spreads linearly with time, σ ( t ) ∝ | J | t ,reproducing the expected ballistic expansion of a releasedwavepacket. For short times, however, t < t c , the expan-sion is instead quadratic , σ ( t ) ∝ J t . It is importantto note that the expansion depends on both the magni-tude of the tunneling, J , as well as on the size of theinitial wavepacket a . In particular, a tightly-confinedwavepacket will have a higher spread in momentum, andso will enter the regime of linear expansion more quickly.The extreme case where only a single lattice site is filledwas considered in Ref. 1, where it was found that theexpansion was always linear with time. This result is ex- actly reproduced by Eq. (4) by appropriately taking thelimit a → III. RESULTS
Undriven lattice –
We first consider the case of astatic lattice, with ∆ , K = 0. In Fig. 1 we show thetime-dependence of the expansion of an initial Gaussianwavepacket, numerically evolved in time under the time-dependent Hamiltonian (2). Using J = 0 . E rec , wherethe recoil energy E rec = ¯ h π / md L , we see that the an-alytic expression (4) accords exactly with the numericalresult, indicating the validity of the continuum approxi-mation. The transition from the initial quadratic expan-sion to the ballistic regime is clearly visible. Halving thevalue of J used produces the expected result of reducingthe rate of expansion, and moving the crossover from thequadratic to the ballistic regime to a later time. Untilted driven lattice –
We now consider the ef-fect of including the time-dependent driving potential V ( t ) = K cos ωt . We choose a high driving frequency,¯ hω = 4 J , and tune the amplitude of the driving sothat J eff = 0 . J . In accordance with the predictionsof the Floquet analysis, we see that the expansion of thewavepacket in the driven system with a bare tunneling of J = 0 . E rec closely follows the result for J = 0 . E rec ,indicating that the driving field indeed renormalizes thetunneling as expected. Looking in detail at the expan-sion (inset of Fig. 1) we see that although on average theexpansion of the driven condensate closely matches thatof the static case with J = 0 . E rec , the driven resultcontains small amplitude oscillations with the same fre-quency as the driving. These oscillations arise from theintrinsic time dependence of the Floquet states them-selves. Their amplitude reduces as the driving frequencybecomes larger, indicating that the approximation ofmodelling the driven system with a renormalized staticHamiltonian becomes increasingly good. Finally we alsoshow in Fig. 1 the most dramatic effect of the renor-malization of tunneling. Since J eff = J J ( K ), tun-ing K to a zero of the Bessel function should result inthe complete suppression of tunneling (neglecting next-nearest neighbor tunneling [15]). We indeed see that set-ting K = 2 .
404 results in the condensate not expandingwith time, due to the vanishing of J eff . Similarly to the K = 1 .
22 case, this curve again displays small oscilla-tions, which become larger at low values of ω . This lowfrequency behaviour would correspond to the “dynami-cal localization” regime [1], where the wavepacket peri-odically returns to its initial state at stroboscopic times t = nT = n π/ω , but between these times can exhibitlarge excursions.In Ref. 4 the effective tunneling was deduced by mea-suring the expansion rate of the condensate at a fixedtime, and assuming this rate was directly proportional to
10 12 14Time (T)1.011.015 σ / σ σ / σ FIG. 1. (Color online) Above: Expansion of an initial Gaus-sian wavepacket in a flat lattice potential, obtained by the nu-merical propagation under Hamiltonian (2). When no drivingpotential is applied ( K = 0) σ ( t ) increases following Eq. (4).For J = 0 . E rec (black solid line) the expansion is clearlyquadratic initially, and becomes linear at long times. Setting J = 0 . E rec (red solid line) reduces the expansion rate asexpected. By applying a periodic driving potential the tun-neling can be renormalized to an effective value J eff . Tuning K = 1 .
22 reduces J eff so that the expansion of the con-densate (dashed red line) reproduces the J = 0 . E rec result.Setting K = 2 .
404 – the first zero of the Bessel function – pro-duces coherent destruction of tunneling (CDT), and the con-densate no longer expands with time (blue dash-dotted line).
Inset -
Detail of the periodically-driven result ( K = 1 . on average reproduces the J = 0 . E rec result, but shows small oscillations with the same frequencyof the driving. The amplitude of these oscillations decreaseswith increasing driving frequency. Below: Experimental com-parison of the free expansion of a condensate ( K = 0) with acondensate experiencing CDT ( K = 2 . J eff . Accordingly the ratio between the tunneling param-eters in the static and the driven lattice was calculatedas | J eff /J | = ( σ ( t ) − σ ) / ( σ stat − σ ), where σ stat is thesize of the condensate after expansion in the static lat-tice. For the experimental parameters ( d L = 426 nm and J/h = 270 Hz), we calculate a crossover time of t c ≃ . | J eff /J | ≥ .
1. Inorder to get better agreement with theory we now useEq. (4) containing the full expansion dynamics, giving | J eff /J | = s σ ( t ) − σ σ stat − σ . (5) | J e ff / J | FIG. 2. Dynamical suppression of tunneling in a periodicallydriven lattice. The effective tunneling parameter was calcu-lated assuming a simple linear expansion (open symbols) andby taking into account the exact expansion dynamics (filledsymbols). In the latter case the agreement with the theo-retically expected scaling (solid line) is clearly better. Theexperimental parameters were
J/h = 240 Hz, ω/ π = 4 kHzand the expansion time 150 ms. Figure 2 shows that, as expected, using Eq. (5) to cal-culate the renormalized tunneling gives better agreementwith theory.It is interesting to note that although J eff is stronglysuppressed near K = 2 .
4, it does not actually reach zerowhen the Bessel function vanishes. This is due to theeffect of higher-order hopping terms present in the sys-tem’s dynamics. Although they too are renormalized bythe driving potential, for sinusoidal driving they will notvanish at the same driving parameters as for the nearestneighbor hopping. The residual value of J eff , visible inFig. 2, is in reasonable agreement with the value of thenext-to-nearest neighbor hopping (around 5%) calculatedfor a similar system in Ref.[15]. Tilted lattice, resonant driving –
Applying Eq. (5) tothe experimental data on photon-assisted tunneling [5]leads to an even more striking improvement in the agree-ment between theory and experiment. In those experi-ments a tilt was applied to the lattice through a constantacceleration, leading to a suppression of tunneling byWannier-Stark localization. Periodic driving of the lat-tice at a frequency ω matching the energy offset betweentwo adjacent lattice wells then led to partial restorationof the tunneling probability, with the effective tunnelingprobability given by | J eff /J | = J ( K ), i.e. one expectsa scaling with the first-order Bessel function. As shownin the inset of Fig. 3, assuming linear expansion in or-der to extract J eff /J led to a scaling that interpolated | J e ff / J | | J e ff / J | a)b) FIG. 3. a) Effective tunneling for resonant driving in a tiltedlattice (photon-assisted tunneling). The effective tunnelingparameters were calculated for two different initial condensatesizes (around 15 µ m (open symbols) and around 17 µ m (filledsymbols)) using Eq. (4). For comparison, b) shows the sameexperimental data with the renormalized tunneling parametercalculated assuming linear expansion. One clearly sees thatin this case the experimental data interpolate between a lin-ear Bessel scaling (solid line) and a quadratic scaling (dashedline). between a linear and a quadratic dependence on J ( K ),depending on the initial size of the condensate (which inthe experiment was varied through the nonlinearity bychanging the atom number). If the full expansion dy-namics is taken into account through Eq. (5), however, both data sets give the same dependence on K which isvery close to the theoretical prediction. IV. CONCLUSIONS
We have shown that in order to extract the effec-tive tunneling from expansion measurements it is im-portant to account for the detailed time-dependence ofthe condensate expansion. When the effective tunnelingis small, or the initial width of the condensate is large,the crossover to ballistic expansion will not be reacheduntil very long times. Measurements made at earliertimes will thus underestimate the effective tunneling rate,which gives a quantitatively accurate interpretation ofthe “squared Bessel function” behaviour noted in Refs.4 and 5. Although we have not included the effects of in-teractions, this is a reasonable approximation for the sys-tems studied in Refs. 4 and 5 where the interactions werefairly small (
U/h ≃
10 Hz), and their effect rapidly be-came negligible as the condensate expanded and becamemore dilute. Using the correct expansion formula, Eq.(4), not only provides an accurate means of deducing thevalue of the effective tunneling, but is essential to studysubtle effects, such as the transition to diffusive tunnelingand the influence of higher-order tunneling terms, whichwould otherwise be masked by this behaviour when theeffective tunneling is small.
ACKNOWLEDGMENTS
This research was supported by the Acc´ıon Integrada /Azioni Integrate scheme (Spain-Italy). The authors alsoacknowledge support from the Spanish MICINN throughGrant No. FIS-2007-65723 and the Ram´on y Cajal pro-gram (CEC). We thank H. Lignier, C. Sias, Y. Singh,and A. Zenesini for assistance with the experiments. [1] D.H. Dunlap and V.M. Kenkre, Phys. Rev. B , 3625(1986).[2] F. Grossmann, T. Dittrich, P. Jung, and P. H¨anggi, Phys.Rev. Lett. , 516 (1991).[3] C.E. Creffield, Phys. Rev. Lett. , 110501 (2007).[4] H. Lignier, C. Sias, D. Ciampini, Y. Singh, A. Zen-esini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. ,220403 (2007).[5] C. Sias, H. Lignier, Y.P. Singh, A. Zenesini, D. Ciampini,O. Morsch, and E. Arimondo, Phys. Rev. Lett. ,040404 (2008).[6] E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic,and M.K. Oberthaler, Phys. Rev. Lett. , 190405(2008).[7] A.R. Kolovsky and H.J. Korsch, arXiv:0912.2587.[8] C. E. Creffield and F. Sols, Phys. Rev. Lett. , 250402 (2008).[9] C. Weiss and H.-P. Breuer, Phys. Rev. A , 023608(2009).[10] C.E. Creffield, Phys. Rev. A , 063612 (2009).[11] A. Galindo and P. Pascual, Quantum Mechanics I (The-oretical and Mathematical Physics) , Ch. 3 (Springer-Verlag, 1990).[12] D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, andP. Zoller, Phys. Rev. Lett. , 3108 (1998).[13] M. Holthaus, Phys. Rev. Lett. , 351 (1992).[14] H.J. Korsch and S. Mossmann, Phys. Lett. A , 54(2003); A. Klumpp, D. Witthaut, and H.J. Korsch J.Phys. A: Math. Theor. , 2299 (2007).[15] A. Eckardt et al. , Phys. Rev. A79