arXiv:0812.0532v2 [cond-mat.str-el] 26 Feb 2009 time n h pia atc.For lattice. optical the ing al,tefato fprilso obyocpe sites doubly-occupied on occupancy particles double of (or fraction the tally, a bevda around at observed was hopping ln) hsgnrto fduln epi h Mott the in deep interaction doublons the of where (dou- generation insulator, sites This doubly-occupied generates modula- blons). lattice The the lattice. of optical the tion forming been of beams has strength light lattice the the optical modulating an harmonically by in investigated fermions of regime related neatn einwsrpre 8 ] utemr,in Furthermore, J¨ordens 9]. by [8, experiment reported the strongly was the to compress- the region the weakly of the of interacting decrease from A study over crossing 7]. the when 6, ibility in 5, 4, results [3, model equally-exciting now Hubbard has new, atoms 2]. fermionic to with [1, led setup same bosons phase the lattice in quantum Progress interacting paradigmatic by displayed the transition into insights merous lto fteltiewt frequency mod- with harmonic lattice A the site. of each ulation on spin-direction random a h od r rsn n h rudsaeo h system ( the sites of singly-occupied state of ground composed the largely is and present are bonds the xeieti agrta h ue-xhnecoupling super-exchange the than J larger is experiment on ntestrto time saturation the on bound notclltiei h rmwr fteHbadmodel Hubbard the of framework the in lattice optical an togo-ierepulsion on-site strong thl ligadwt h interaction the with and filling half at egbr.Tegon tt ftesse ossso one hopping of the consists with system site the per of particle state ground The neighbors. h opn amplitude hopping the c σ iσ † H ecnie emoi tm,sc as such atoms, fermionic consider We odaosi pia atcspoie swt nu- with us provided lattices optical in atoms Cold 4 = = oeetpmigo otisltr em odnrl vers rule golden Fermi insulator: Mott a of pumping Coherent raea tmi n ftopsil yefiestates hyperfine possible two of one in atom an create ↑ = τ , t mod ↓ 0 2 − /U t tsite at t 0 0 2 ≈ stefcso u rsn ok Experimen- work. present our of focus the is , eateto odne atrPyis h ezanInsti Weizmann The Physics, Matter Condensed of Department h 8,n infiatqatmcreain over correlations quantum significant no [8], i,j X ASnmes 37.s 11.d 31.15.aq 71.10.Fd, 03.75.Ss, numbers: PACS incoheren are Cou particles oscillations the e driving. Rabi where the an the description renders onto alternative to freedom ampli mapped of due hopping degrees be oscillations remaining the can Rabi system of coherent many-body modulation performs the o periodic pumping phase a the insulating discuss by we Here, insulator inv equilibrium. Mott conveniently of are out systems system atom the cold in phases novel more, 50 i odaospoieauiu rn osuymn-oysystems many-body study to arena unique a provide atoms Cold ,σ h/U i c n h sum the and iσ † 5 cce” 8 hc esa upper an sets which [8] “cycles”) (50 c jσ D t 1 H + U 0 nttt o hoeia hsc,EHZrc,89 uih S Zurich, 8093 Zurich, ETH Physics, Theoretical for Institute a esrdatrmodulat- after measured was ) D ee h emoi operators fermionic the Here, . stetemperature the As . sat . tal. et c U/ . 0 fe modulation a after 30% = τ
sat 6 + h t t ,j i, 0 U 0 U ≤ 8,tesrnl cor- strongly the [8], upesddet the to due suppressed 13 = X i oiae vrthe over dominates τ i mod ω U usoe nearest over runs c ≈ oiaigover dominating . i † 40 ,a 6, ↑ .Hassler F. seFg 1(a)]. Fig. [see c U/ ,sbetto subject K, T i ↑ c ~ Dtd oebr6 2018) 6, November (Dated: ≪ saturation i † ↓ transfers c T U i ↓ with ) nthe in , 1 (1) n .D Huber D. S. and here, o h eeaino obeocpnyb ouaigthe modulating time by a occupancy during double sequenc lattice of experimental optical an generation of Sketch the (a) for online) (color 1: FIG. lsrsdn ndul-cuidstsstrtsa value a at saturates sites doubly-occupied in residing cles fe time a after obyocpe ie ihart Γ rate a with sites doubly-occupied H seFg ] a h ietectto ftebn via rule bond golden Fermi the the of by given excitation rate direct a with the pairs (a) doublon-hole incoherent 2]: pumping of Fig. themselves process [see lend the ideas different modeling account Two to into bath. bonds effective bond remaining an (typical) the as taking single and a a 1(b)], adopt modeling [Fig. we via space, Hilbert description huge simple a in problem many-body pair. doublon-hole a site, site empty doubly-occupied an particle a to a creating next promotes thereby bond, (possibly) the and across system the to energy than bond different a created. con- on was doublon-hole resides it all pair where of the which consist in freedom figurations of degrees bath The h est fsae ftedulnhl otnu,and continuum, doublon-hole the of h states of density the lto fteltie aigtemdlto strength modulation by the given Taking be lattice. to the of ulation f

eff D in % | ept h atta h ubr oe ersnsa represents model Hubbard the that fact the Despite 10 20 30 H l updit ra continuum. broad a into pumped tly ossigo n odebde na ra fsites. of array an in embedded bond one of consisting mod 100 0 E (a) apd ecmaeti ceet an to scheme this compare We damped. g siae ydnmclpoe pushing probes dynamical by estigated 080 20 | g obyocpe ie nafermionic a in sites doubly-occupied f ln ftetolvlsse othe to system two-level the of pling eoe h nryo h rudstate, ground the of energy the denotes Γ ueo cec,Rhvt 60,Israel 76100, Rehovot, Science, of tute i Γ τ mod FGR rniin notedulnhl continuum doublon-hole the into transitions ,1 2, stemti lmn rsn rmtemod- the from arising element matrix the is up ue eso htde nthe in deep that show We tude. sat τU/h ≈ b lutaino h ffcieHletspace Hilbert effective the of Illustration (b) . etv w-ee ytmwhich system two-level ffective a rmeulbim Further- equilibrium. from far τ = D δt sat sat 2 ~ n siaigtedniyo states of density the estimating and π |h f witzerland | H mod sRb oscillations Rabi us τ fe niiilbidpof buildup initial an After . | (b) g i| up 2 ρ doublon h rcino parti- of fraction the , ( E g + ~ ω ;(2) ); H hole eff D ρ sat is e 2

(a) (b) the doublon-hole continuum; see (a) above. It turns out continuum bath |dai that the time scale (τR) for the Rabi cycle is in rough FGR Γdown decay FGR Γ R agreement with the experimental value of the saturation Γ down Ω up time, whereas the treatment via a direct excitation into |gi |Si Γin(out) the doublon-hole continuum leads to a time scale (τFGR) arbitrary time exp which is an order of magnitude larger (τsat . τR τFGR). For a quantitative agreement, one has to go beyond≪ a sin- FIG. 2: Comparison of two possible descriptions: (a) A rate FGR gle bond in the description of the pumping process. Dif- equation description where an incoherent pump rate [Γup ] into a broad doublon-hole continuum and subsequent decay ferent bonds are coupled via the pumping field, leading decay FGR to a Dicke-like description [11], but, most likely, many- [Γdown] or incoherent stimulated emission [Γup ] lead to a saturation. (b) Coherent Rabi oscillations [with frequency body effects dominate the coupling of different bonds; a

ΩR] between the ground state and a single exited state where systematic study of larger clusters deserves future efforts. decay out of the effective Hilbert space is introduced via cou- In deriving the effective Hamiltonian governing the pling to a bath [Γin(out)]. dynamics in the Hilbert space of a single bond (the sites are labeled by L and R), we neglect the confin- ing potential and focus on a homogeneous three dimen- as ρ 1/t0, we obtain the scaling of the pump rate sional setup. The ground state at half-filling involves ΓFGR ∝ δt2/t as a function of the hopping amplitude one particle per site without any spin order, i.e., the mod ∝ 0 t0 and the strength of the modulation δt. (b) Coher- particles on the bond occupy the three triplet states 1 0 1 ent T = c† c† 0 , T = c† c† + c† c† 0 , Rabi oscillations in the isolated bond between the ± L ( ) R ( ) √2 L R L R | i ↑ ↓ ↑ ↓ | i | i ↑ ↓ ↑ ↓ | i 1 ground state of the bond and the doubly-occupied ex- † † † †
and the singlet state S = √ cL cR cL cR 0 . These cited state with a Rabi frequency Ω δt proportional | i 2 ↑ ↓ − ↑ ↓ | i R states constitute the four degenerate ground
states of a to the modulation strength. The Rabi∝ oscillations gen- single bond and are realized with equal probability of 1/4 erate doublon-hole pairs and lead to a periodic modula- each. A harmonic modulation of the lattice beam with tion (with frequency Ω ) of the double occupancy. Note R V = V + δV cos(ωτ) leads to a time-dependent hopping that the rates of the two (extreme) processes, incoherent 0 FGR Hamiltonian pumping into a band (a) (with rate Γmod) and coher- ~ ent excitation of doublon-hole pairs (b) (with rate ΩR), Hhop = δt cos(ωτ) cL†σcRσ + H.c. , (3) have a different dependence on the modulation strength Xσ
δt and can therefore be discriminated experimentally. with δt/t = 3/4 V /E δV/V and τ denoting Here, we propose a third way where the coherent 0 − 0 r 0 the time. This expressionp is valid
for δV/V0 1 and Rabi oscillations produce a doublon-hole pair and, sub- ≪ δV/V E /V [12]. For current experiments with sequently, hopping of the doublon (or the hole) to the 0 ≪ r 0 neighboring sites leads to an incoherent decay of the ex- fermions [8],p where δV/V0 =0.1 and V0 does not exceed citation into the doublon-hole continuum. The process is 10Er, both conditions are safely fulfilled. In the deriva- tion of Eq. (3), we have dropped the constant hopping governed by two time scales, the Rabi frequency ΩR δt which periodically modulates the double occupancy∝ and amplitude proportional to t0, as hopping is blocked in the the decay Γ t which leads to a saturation. Mott insulator at half-filling. The hopping amplitude t0 hop ∝ 0 A similar system, where a two-level atom is coupled will be reintroduced later as it is responsible for the de- simultaneously to a laser field and to vacuum modes of cay of the doublon-hole pair by coupling the bond to the the radiation field, is well-studied in quantum optics [10]. surrounding lattice. The modulation couples the ground states to the doublon-hole states d = c† c† 0 , The two-level atom corresponds to the two states on the L(R) L(R) L(R) | i ↑ ↓| i bond and the laser plays the role of the harmonic mod- with the doublon located on the left (right) site. These ulation of the lattice. However, in the atomic example, states exhibit an energy offset U with respect to the en- the spontaneously emitted photon takes the atom back ergy of the ground states. The total Hamiltonian in the into the ground state where it continues with the Rabi Hilbert space spanned by the four ground states (triplet cycles, while in the present case the doublon (or hole) and singlet) and the two doublon-hole pairs is given by itself exits the system. The present process then resem- H = U d d + H U d d bles more strongly that of optical pumping into a third eff | αih α| hop ≈ | αih α| nondecaying level, with the accumulation in the pumped αX=L,R αX=L,R iωτ/~ iωτ/~ level playing the role of the buildup in double occupancy. + δt e da S + e− S da , (4) In the following, we first describe the effective model in h | ih | | ih |i the single bond Hilbert space driven by a harmonic mod- where the first term describes the energy offset by U and ulation of the lattice and derive the frequency of the Rabi the second term the driving due to the lattice modula- oscillations. We then account for the bath consisting of tion (3). In going from the first to the second equation the surrounding sites via a master equation approach and in (4), the rotating wave approximation has been used analyze it in different limits. We compare this approach and we have introduced the active and passive doublon- to the situation where the bond is directly excited into hole states given by d = ( d d )/√2; these are | a(p)i | Li ± | Ri 3 related to the left (right) doublon-hole state via a basis 50 change. Driving the system on resonance with ~ω = U, the singlet state is coupled to the excited state via Rabi oscillations. Within the present description, off-resonant in25 % coupling can be described in the standard manner. As D can be seen from Eq. (4), the triplet states are unaffected 0 by the lattice modulations. Only the singlet state is cou- 012τδt/~ 56 pled to the active doublon-hole pairs d . Going over to | ai the interaction representation with the reference dynam- FIG. 3: Fraction of particles residing in doubly-occupied sites ics given by the first term in (4), the relevant Hamiltonian after modulation time τ. After a buildup time corresponding ~ in the effective two-level system eff = span da , S to approximately one π-pulse [τR = π/ΩR = π/2δt], the H {| i | i} ~ ≈ ~ can be expressed as a matrix decay, with (weak) strength Γout = δt/ 0.24t0/ and Γin = 0, leads to a saturation at Dsat = 58% corresponding to the 0 δt fact that all “active” bonds, singlets, are excited in the system H = , (5) δt 0  (solid line). A more realistic situation where the damping is increased by a factor of 2 renders the oscillations barely which describes the coherent oscillations in the two-level visible (dashed line). Taking into account a finite value of system with the Rabi frequency Γin leads to a decrease of the saturation value. Note that, irrespective of the saturation value, the time scale needed to 2δt pump the system into a doubly occupied state is given by the ΩR = ~ . (6) Rabi frequency. The triplet states and the passive doublon-hole pair re- main eigenstates of the time-dependent Hamiltonian and that atoms which left the effective Hilbert space still con- therefore particles in such states are not driven. tribute to the double occupancy. The bath can be pic- So far, we have neglected the effect of t0 on the excited tured as a gas of doublons and holes (with equal num- state da . Provided the bond is in a doublon-hole state, bers). Initially, the bath does not contain any particles | i we can lose the state from our single bond description such that Γin = 0 (this approximation remains valid as via the hopping of the doublon (or the hole) out of the long as the density of particle is low); furthermore, we bond; cf. Fig. 1(b) [13]. This effect is captured by a bath estimate the escape rate by Γout = Γhop t0/~. For ∼ sat representing the states with the doublon and hole sep- Γin = 0, Eq. (8) provides the saturation value ρD = 1, arated from each other. Note that the bath states still i.e., the singlet is certainly transferred to one of the contribute to the double occupancy (as there is still a doublon-hole states. Using the dilute gas approximation, doublon present) but do not participate in Rabi oscilla- Γin = 0, we can solve the equation of motion for the tions (as the doublon and the hole are not located on the density matrix same bond). We denote the rate at which the doublon- 1 Γout 2ρ ρ hole bond decays into a separated doublon and a hole ∂ ρ = [H,ρ] 11 12 . (9) τ ~ ρ 0 by Γout, and the reverse rate at which a doublon and a i − 2  21  hole collide to form a doublon-hole pair by Γin. Taking Assuming that initially the bond is in a singlet state the coupling to the bath into account and performing a standard analysis similar to [14] leads to an incoherent 0 0 ρ(τ =0)= , (10) term (besides the standard term proportional to [H,ρ]) 0 1 in the evolution of the reduced density matrix ρ in the Eq. (9) gives the time evolution of the density matrix of effective Hilbert space eff = span da , S of the form H {| i | i} the bond. Qualitatively, the weight is transferred to the ~ ∂ρ 1 2(Γ ρ Γ ρ ) Γ ρ exited state da at a rate ΩR and then the decay into = out 11 − in 0 out 12 . (7) | i ∂τ −2  Γoutρ21 0  other states occurs with a rate Γout (cf. Fig. 3). inc To calculate the fraction of particles in a doublon state, Different from a standard master equation [14], Eq. (7) we need to know how many bonds are initially in a singlet is not probability preserving, since with probability ρ0 = configuration. We estimate the probability that a specific d 1 ρ11 ρ22 the doublon and the hole have separated site is part of a singlet as P =1 (3/4) 0.58, here d = − − S − ≈ and thus have left the effective Hilbert space eff. In the 3 is the “dimension” of the lattice modulation (it counts H steady state situation with ∂τ ρ = 0, the probability that the number of bonds per site) and we have assumed that the system contributes to the double occupancy is given the singlets are independent of each other and occupied by with a probability 1/4. The fraction of particles in a doublon state is given by sat sat sat sat Γout +Γin ρD = ρ0 + ρ11 =1 ρ22 = , (8) − Γ + 2Γ = PS ρD(τ) (11) out in D the sum of the probabilities that the system is in the with ρ (τ)=1 ρ (τ) the probability that this singlet D − 22 doublon-hole pair state (ρ11) or in the bath (ρ0). Note is transferred to a doublon state. The system quickly 4 reaches a steady state with sat = PS (cf. Fig. 3). In sponding doublon-hole continuum acquires a width of ap- three dimensions, we reach aD maximal value = 58%. proximately 24t (in three dimensions). A crude estimate Dsat 0 Note that at this value of double occupancy the dilute gas for the density of states is then given by ρ 1/24t0. We ≈ ~ approximation Γin = 0 becomes invalid. We expect that then arrive at the two time scales τR = π/ΩR = π /2δt ~ 2 2 Γin increases for increasing double occupancy and, there- and τFGR = ( /2π)24t0/δt = (24t0/π δt)τR 10τR, as- fore, the double occupancy saturates at a lower value. sociated with a Rabi oscillation and the rate≈ equation, A theory that aims at a quantitative result for the sat- respectively. For the last step, we have used δt/t0 0.24 uration value should take such effects into account and corresponding to current experiments. Note that≈ the solve the system self-consistently. Assuming that for long time scale to reach saturation in the incoherent Fermi times the density of particles in the bath is high such golden rule approach is an order of magnitude larger that Γ Γ , we obtain the result that ρsat = 50% than the one predicted by the Rabi oscillation picture. In in ≫ out D [cf. Eq. (8)] and sat = 29%, comparable to experimen- Ref. [8] the saturation sets in before 50 “cycles”, which tal results [8]. TheD time scale needed to excite the sys- is even smaller than τ [τ = 50h/U = (200δt/U)τ R mod R ≈ tem is approximately given by the time τR = π/ΩR of 0.6τR for U/6t0 = 13.6 and δt/t0 0.24], promoting a π pulse, irrespective of the saturation value obtained. the picture of the coherent Rabi oscillations.≈ In an ex- Note that the finite line-width of the perturbing laser, periment, one can test one theory against the other by the scattering between doublons, and an inhomogeneous changing the coupling strength δt while keeping all other environment for a single bond due to the trap are other parameters, such as the bandwidth 1/24t0, constant. The 2 processes which lead to a dephasing of the coherent single time scale for the saturation scales as δt− in the Fermi bond dynamics. As the effect of t0 on the excited state golden rule picture, whereas Rabi oscillations produce a leads already to an over-damped dynamics, these other saturation time inversely proportional to δt. effects are neglected here. Another approach to describe the saturation in the In summary, we have discussed two alternative expla- double occupancy is based on a rate equation for the nations for the buildup and the saturation of the dou- doublon fraction. The rates at which excitations are gen- ble occupancy in the modulation experiments of Ref. [8] erated are taken into account via a rate equation for the deep in the Mott insulating phase. The comparison to doublon concentration experiment favors the description in terms of an effec- tive two-level system with coherent driving, leading to ~ ∂τ = (1 )Γup Γdown. (12) the characteristic time scale of τR π/ΩR = π/2δt for D − D − D half a Rabi cycle. Given the large≈ decay rate propor- Within this description two independent processes are tional to t > ~Ω δt, only one π-pulse is observable. 0 R ≈ assumed. First, the system is excited with a rate Γup. The predictions can be experimentally tested by studying Second, at a later uncorrelated stage, these excitations the dependence of the saturation time on various system are removed with a rate Γdown [cf. Fig. 2(a)]. In order to parameters as V0/Er, δV/V0, and t0/U. compare the two schemes, the rate equation and the Rabi oscillation, we extract the time scales of both processes. We acknowledge fruitful discussions with E. Altman, The Fermi golden rule matrix element is approximately G. Blatter, H. P. B¨uchler, the group of T. Esslinger, S. given by δt2 [6]. Additionally, we need an estimate for the F¨olling, L. Goren, and A. R¨uegg. This work was sup- density of states. Once excited, the doublon as well as ported by the Swiss National Foundation through the the hole can move freely through the lattice. The corre- NCCR MaNEP.

[1] M.P.A. Fisher, P.B. Weichman, G. Grinstein, and D.S. T.A. Costi, R.W. Helmes, D. Rasch, and A. Rosch, Sci- Fisher, Phys. Rev. B 40, 546 (1989). ence 322, 1520 (2008). [2] M. Greiner, O. Mandel, T. Esslinger, T.W. H¨ansch, and [10] C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, I. Bloch, Nature 415, 39 (2002). edited by R. Balian, S. Haroche, and S. Liberman, Pro- [3] V.W. Scarola, L. Pollet, J. Oitmaa, and M. Troyer, ceedings of the Les Houches Summer School of Theoret- arXiv:0809.3239 (2008). ical Physics, 1975 (North-Holland, Amsterdam, 1977). [4] L. De Leo, C. Kollath, A. Georges, M. Ferrero, and O. [11] R.H. Dicke, Phys. Rev. 93, 99 (1954). Parcollet, Phys. Rev. Lett. 101, 210403 (2008). [12] When deriving (3), care has to be taken in expanding the [5] R.W. Helmes, T.A. Costi, and A. Rosch, Phys. Rev. Lett. exponential part of the hopping amplitude [15]. 100, 056403 (2008). [13] We think about the singly-occupied lattice as being the [6] S.D. Huber and A. R¨uegg, Phys. Rev. Lett. 102, 065301 vacuum with excitations given by doublons and holes. (2009). [14] C.W. Gardiner and P. Zoller, Quantum Noise, chapter [7] C. Kollath, A. Iucci, I.P. McCulloch, and T. Giamarchi, 5.1 (Springer, Berlin, 2000). Phys. Rev. A 74, 041604(R) (2006). [15] W. Zwerger, J. Opt. B: Quantum Semiclassical Opt. 5, [8] R. J¨ordens, N. Strohmaier, K. G¨unter, H. Moritz, and T. S9 (2003). Esslinger, Nature 455, 204 (2008). [9] U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch,