Light scattering and dissipative dynamics of many fermionic atoms in an optical lattice

Transcription

1 Light scattering an issipative ynamics of many fermionic atoms in an optical lattice S. Sarkar, S. Langer, J. Schachenmayer,, an A. J. Daley, 3 Department of Physics an Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 560, USA JILA, NIST, Department of Physics, University of Colorao, 440 UCB, Bouler, CO 80309, USA 3 Department of Physics an SUPA, University of Strathclye, Glasgow G4 0NG, Scotlan, U. K. We investigate the many-boy issipative ynamics of fermionic atoms in an optical lattice in the presence of incoherent light scattering. Deriving an solving a master equation to escribe this process microscopically for many particles, we observe contrasting behaviour in terms of the robustness against this type of heating for ifferent many-boy states. In particular, we fin that the magnetic correlations exhibite by a two-component gas in the Mott insulating phase shoul be particularly robust against ecoherence from light scattering, because the ecoherence in the lowest ban is suppresse by a larger factor than the timescales for effective superexchange interactions that rive coherent ynamics. Furthermore, the erive formalism naturally generalizes to analogous states with SUN symmetry. In contrast, for typical atomic an laser parameters, two-particle correlation functions escribing boun imers for strong attractive interactions exhibit superraiant effects ue to the inistinguishability of off-resonant photons scattere by atoms in ifferent internal states. This leas to rapi ecay of correlations escribing off-iagonal long-range orer for these states. Our preictions shoul be irectly measurable in ongoing experiments, proviing a basis for characterising an controlling heating processes in quantum simulation with fermions. PACS numbers: 37.0.Jk, p, Hj I. INTRODUCTION In recent years, there has been remarkable progress towars quantitative applications of quantum simulators 4 to the stuy of many-boy physics in strongly interacting systems 5,6. In the case of fermionic atoms in optical lattices, the level of microscopic unerstaning an control achieve opens the oor towars the stuy of physics associate with the Hubbar moel 7, incluing magnetic correlations in two-component fermi gases in an optical lattice 8, an going beyon this to the stuy of many-boy physics with high egrees of SUN symmetry in group-ii-like atoms 9 3. At the same time, these experiments provie an ieal environment for probing out-of-equilibrium physics 4,5, both in terms of quench ynamics 6, an also issipative ynamics 3 6. Controlle issipative ynamics 7 coul be use in these systems to observe the emergence of pairing 5 or counter-intuitive long-range correlations 6, as well as Pauli-blocking an quantum Zeno effects in issipative processes 3,4. A key challenge for state-of-the art experiments with multiple spin species of fermions, is to realize entropies per particle low enough to observe quantum magnetism or other physics on small energy scales. This is particularly true of strongly interacting regimes U J, where J is the tunnelling amplitue to neighbouring sites, an U is the on-site interaction energy, in which the ominant orer is often riven by small terms 8 35 of the orer of J /U. Characterization an control over heating therefore takes on a special importance. From a theoretical point of view such ynamics are complicate, as they involve unerstaning the interplay between few-particle atomic physics, an out-of-equilibrium many-boy ynamics. However, the atomic physics of these systems is sufficiently well unerstoo that we can look to erive microscopic moels escribing heating of many-particle systems, either with technical noise such as fluctuations of an optical lattice potential 36,37, or with incoherent light scattering for one atom 38,39, up to many bosons For particles in optical lattices, incoherent light scattering gives a funamental limit to coherence times of manyboy states, an therefore minimal requirements for our unerstaning of out-of-equilibrium many-boy ynamics in quantum simulators 8,9,45. Here we stuy in etail the issipative ynamics of many fermionic atoms in a far-etune optical lattice ue to incoherent light scattering, fining effective ecoherence rates for the many-boy states that strongly epen on the etails of the many-boy state. In particular, we fin contrasting results for magnetic orer with strong repulsive interactions where the states are very robust, an for strong attractive interactions where the coherence of a gas of boun imers is rapily reuce. In orer to obtain these results, we consier the atomic physics of group-i an group-ii atoms, an erive a many-boy master equation that provies a microscopic escription, inepenent of the lattice geometry or imensionality. Using analytical techniques an by combining time-epenent ensity matrix renormalisation group methos with quantum trajectory techniques 55 59, we solve the master equation an etermine the ecoherence ynamics of a range of initial many-boy states. The resulting formalism can also be straight-forwarly generalise to heating of group-ii atoms with SUN magnetic orer, making the results we obtain here irectly observable in an relevant for a large variety of ongoing experiments. A complimentary article iscussing two postulate variants of the master equation we erive an solutions within a slave spin approach can

2 be foun in Ref. 60. II. OVERVIEW A. Summary of results for atoms in the lowest Bloch ban When we begin with atoms in the lowest Bloch ban of the optical lattice, an consier the ecoherence of manyboy states, we observe strikingly contrasting results in ifferent regimes. This is especially true for the case of strong interactions U J. While the rate of spontaneous emissions γ in typical current experiments can be of the orer of J, the physics of magnetically orere states with U > 0 an boun imers for U < 0 is riven by terms that arise in secon orer perturbation theory as J /U. Conservatively taking U 0J in this regime, we have γ J /U. Therefore, there is a anger that these states may be particularly susceptible to ecoherence at a rate that is relatively fast compare to the relevant ynamical timescales. We fin this concern to be well foune in the case of attractive interactions U < 0 with equal filling of two spin species. In that regime a gas of boun imers exists, in which imers tunnel in perturbation theory with amplitue J /U, an forms a superflui with long-range orer at low energy. However, this orer is strongly susceptible to spontaneous emissions: correlations escribing off-iagonal orer of imers ecay at a rate not only given by γ the rate of scattering for two inepenent particles, but instea 4γ. This arises from aitional superraiant enhancement ue to the inistinguishability of off-resonant photons scattere by atoms in ifferent internal states. This is in strong contrast to the case of magnetic orering for repulsive interactions U > 0, where minimising interaction energy favours insulating states with a single atom per site, an magnetic orering riven by a superexchange interaction of amplitue J /U. We show that these states can be particularly robust, which can be intuitively unerstoo as follows: for the typical experimental case, irect spin ecoherence oes not occur because the lattice lasers are far etune an photons scattere by atoms in ifferent internal states are inistinguishable. As a result, spontaneous emission only istinguishes between ifferent on-site particle number states, an not ifferent spin states. This suppresses ecoherence by a factor relate to the probability of oublyoccupie or unoccupie lattice sites assuming we begin with unit filling, which in higher imensions is of orer J /U. As a result, the ominant process involves transfer of particles to higher Bloch bans, which itself is suppresse by the Lamb-Dicke factor η 0.. Hence, these magnetically orere states shoul exhibit a particular robustness against ecoherence ue to spontaneous emissions. The general formalism erive here applies to higher imensions an can also be straight-forwarly generalise to larger numbers of internal states, incluing nuclear spins in group-ii atoms. In particular, the results we obtain in perturbation theory showing the robustness of magnetic orer can be irectly generalise to the heating of group-ii atoms with SUN magnetic orer. B. Outline of this article This paper is organize as follows: In Sec. III we summarise the atomic physics of a single group-i atom or group-ii atom an justify the microscopic assumptions we use as basis for escribing the many-boy ynamics. In Sec. IV we outline the erivation of the many-boy master equation for light scattering by fermionic atoms. Sec. V presents the intuitive regime of atoms in a oublewell potential, in preparation for Section VI, where we stuy the full many-boy ynamics on a lattice. We present a summary an outlook in Sec. VII. More technical etails of the calculations in Sec. III an Sec. IV are organize in Appenix A an B respectively. III. ATOMIC PHYSICS To provie the framework for the erivation of the master equation, we summarise the relevant atomic physics for the atomic species preominantly use in experiments with ultra-col atoms. Group-I alkali-metal atoms have been use wiely, an recently group-ii alkaline earthmetal atoms have been establishe for the realisation of systems with SUN symmetry,. We consier spontaneous emissions when an atom is trappe in an optical lattice create by far-etune laser fiel an show that the atom returns to the same state it starte from with very high probability. In aition the scattere photons are also inistinguishable, resulting in low irect spin ecoherence. In the following we provie two prototypical examples, 7 Yb group-ii an 6 Li group-i. A. Group-II atoms First we look at the case of group-ii atoms, specifically 7 Yb Fig. a. These atoms with two valence electrons have a groun state which is a spin singlet, with zero total electronic angular momentum. Hence, the groun states iffer only in the z-component of the nuclear spin, I = /, an we have two states in the lowest manifol. The electric fiel of the laser only couples irectly to the orbital motion of the electron, an we can efine a etuning from the most closely couple excite level, e.g., P using spectroscopic notation, as the ifference between the laser frequency an atomic transition frequency. If the fiel is far etune, i.e., is large compare with the hyperfine structure energy splitting δ hfs, then the iniviual hyperfine states cannot be

3 3 6 P I = e 3,# i e,# i e," i e 3," i hfs e,# i e," i! # e # i! " e " i P 3/ P / fs hfs S 0 g # i g " i a g # i g " i FIG.. a Atomic structure of 7 Yb nuclear spin I = /. We use the spectroscopic notation for the sublevels an show hyperfine structure splittings of the lowest singlet levels energies are not rawn to scale. The groun states have total electron spin of zero an states in this manifol essentially only iffer in the nuclear spin component m I. We write the two groun states as spin own an spin up states for m F =, respectively. b Reuction of this hyperfine structure to an effective four-level system where, for very large etuning δ hfs, ω ω, we can neglect the possibility of a spin flip an can take the photons scattere from each spin system to be ientical. resolve, an the hyperfine coupling cannot be use to rotate the nuclear spin state uring spontaneous emissions. Phrase in a ifferent way, we can note that a particular choice of groun state is always couple to a superposition of excite hyperfine states, which epens on the etuning. For large etuning this superposition is such that when the atoms return to the groun state, ecay channels corresponing to a spin flip interfere estructively an its relative rate is of the orer δ hfs / see Appenix A for an explicit calculation. In typical experimental setups where δ hfs 34 MHz 6,6, an far-off-resonance lattices can be etune by tens or hunres of nanometers 0 4 Hz, this rate of spin-flips is extremely small. In such a limit, the group- II atomic system can be regare as an assembly of two ecouple two-level systems for the two ifferent nuclear spin states Fig. b. Relative shifts of the transition frequencies between the levels are small, but to account for any small ifference, we efine transition frequencies ω an ω, as shown in Fig. b. For ω ω, the relative frequencies of the scattere photons cannot be resolve 63, resulting in suppression of irect spin ecoherence. We explore the ifferences between ientical an non-ientical photon scattering in more etail in Sec. IV. B. Group-I atoms In aition to the consierations in the group-ii case, group-i atoms such as 6 Li have nonzero electron spin in the groun state. We then nee to consier the role of fine structure coupling an inclue excite levels in P / an P 3/ as shown in Fig.. These have a fine structure energy ifference δ fs between them an hyperfine b I = S / C D E F A B FIG.. A iagram of the atomic structure of 6 Li nuclear spin I = showing the lowest hyperfine manifols energies not rawn to scale. The groun states in the S sublevel are labele by A, B,..., F an the excite states in P sublevels are labele by,,..., 8. These names will be use in the text in iscussing transitions between ifferent levels. structure energy splittings δ hfs,p/ an δ hfs,p3/ within each manifol of states. Analogously to the group-ii case, spin-flip processes that must change the nuclear spin are suppresse if the etuning is much larger than the hyperfine structure splitting, an also spin-flip processes changing the electronic spin are suppresse when the etuning is much larger than δ fs. An example of a spin flip between two groun states that have ifferent electron spins is g D g E in Fig.. The relative rate of spin flip processes is δ fs /. An example of the flip of a nuclear spin is g D g A, where the relative spin flip rate from a laser polarize along z-axis is δ hfs,p/ / δ hfs,p3/ /, with constants that can be compute from the ifferent ipole matrix elements. For a hyperfine structure splitting of δ hfs = 6. MHz for P / an 4.5 MHz for P 3/, an δ fs = 0.05 GHz for the fine structure splitting 64, we can again assume that the spin-flip processes are very strongly suppresse in far-etune lattices an are negligible on experimentally relevant timescales. More etails of these calculations can be foun in Appenix A. This conclusion also hols well when we consier the role of an external magnetic fiel, incluing in the Paschen-Back regime. For both 7 Yb an 6 Li, the spin flip rates stay negligibly small even when an external magnetic fiel is introuce see Appenix A. For very high magnetic fiels though the basic assumption ω ω has to be carefully revisite as the frequencies of the spontaneously emitte photons from ifferent spin states are ifferent now. IV. MASTER EQUATION FOR FERMIONIC MANY BODY SYSTEMS We now erive the master equation escribing fermionic atoms with two internal states, trappe in an optical lattice create by a far-etune laser fiel an

4 4 unergoing spontaneous emissions. We begin from the collective coupling of many atoms to the external raiation fiel which we consier as the reservoir or bath, an obtain the equation of motion for the reuce atomic ensity operator for the motion of the atoms, ρ trace over the bath in the form ρ = i[h, ρ] + Lρ. t Here the Hamiltonian H escribes the coherent ynamics whereas the Liouvillian Lρ correspons to the issipative ynamics ue to spontaneous emission events 46. Note that this erivation is analogous to the case of a single species of bosonic atoms treate in Ref. 4. Despite the ifferent particle statistics an in the presence of an aitional internal egree of freeom we remarkably obtain qualitatively equivalent terms an a very similar overall structure to the master equation. However, our generalization now takes into account the effects on the internal state ynamics for multiple electronic groun states, an below we will use this to investigate in etail the interplay between the motional ynamics an correlations in spin-orere states. The master equation we erive is, however, very general, an can be applie irectly to escribe fermions in a rich variety of regimes in an optical lattice 65. Furthermore, while we focus here on the twospecies case, we see from the structure of our calculation that both the master equation an the conclusions for spin-orere states can be straightforwarly generalize to SUN spin systems 0,3. In our treatment we take an ensemble of atoms, each with a mass m, an with four accessible internal states, electronic groun an excite states g an e for each of two spin manifols, giving rise to the four-level systems epicte in Fig. b. Initially, we will consier the limit where ω = ω = ω eg, enoting the corresponing transition wavenumber by k eg, so that the photons emitte are inistinguishable between the ifferent states. However, we will come back to check this assumption at the en of this section. The system is riven by a laser with frequency ω L corresponing to a wavenumber k L, far etune from the transition frequency by an amount = ω L ω eg. Therefore the interactions between the atom an the laser light involve a spatially epenent Rabi frequency Ωx which is proportional to the laser fiel strength an to the ipole moment the atom, eg. To write own the master equation in secon quantization we efine the spin s epenent fiel operators ψ s x an they obey fermionic anti-commutation relations {ψ s x, ψ s y} = δ s,s δx y. In orer to properly account for interactions, as well as losses from short-range contributions, we use stanar arguments to separate the ominant contribution to the ynamics at large istances from the short-range physics 66,67. This gives rise to interaction terms which for a ilute gas at low scattering energies can be completely characterise by the s-wave scattering length, an for which losses, e.g., ue to laser-assiste collisions at short istances can be accounte for via a small imaginary part of this length 68. The far etune laser rive allows aiabatic elimination of the atoms in the excite states 69 an working in a frame rotating with the laser frequency we obtain a master equation of the form see Appenix B H t ρ = i eff ρ ρh eff + J ρ. Here the non-hermitian effective Hamiltonian is: H eff = H 0 + H light eff + H int eff. 3 This effective Hamiltonian escribes in aition to the coherent ynamics an the collisional processes also the scattering processes that transfer away the groun state population therefore not trace preserving. The first term, H 0 is the Hamiltonian for non-interacting atoms in an optical lattice potential originating from the ac-stark shift 70 inuce by a staning wave of laser light: H 0 = s 3 xψ sx m + Ωx ψ s x. 4 4 To moel spontaneous emissions we couple the atoms to a raiation bath, namely the vacuum moes of the laser fiel. The effective Hamiltonian escribing the atom-light interaction is given by: H light eff = Γ s,s i Γ 3 x 3 ygk eg r ΩyΩ x 4 ψ sxψ s yψ s yψ sx 3 x Ωx 4 ψ sxψ s x i Γ s s,s 3 x 3 y ΩyΩ x 4 F k eg rψ sxψ s yψ s yψ sx, 5 where functions F an G are efine as { F ξ = 3 [ ˆξ ˆ eg ] sin ξ ξ + [ 3ˆξ ˆ cos ξ eg ] ξ sin ξ } ξ 3, 6

5 5 { Gξ = 3 [ ˆξ 4 ˆ eg ] cos ξ ξ + [ 3ˆξ ˆ sin ξ eg ] ξ + cos ξ } ξ 3, 7 an Γ is the Wigner-Weisskopf spontaneous ecay rate. The first term in H light eff gives the ipole-ipole create by photon exchange interaction energy. The secon term contains single-atom processes which absorb an then emit laser photons. The thir term escribes a collective two-atom excitation an e-excitation that can give rise to superraiance or subraiance in appropriate limits 69,7. Now as G ecays as a function of interatomic istance we can focus only on interaction on a small scale set by the laser wavelength. At very short istances k eg r 0 it is possible to absorb the ipoleipole interaction as a small moification to the collisional interactions, Heff int = 3 x gxψ xψ xψ xψ x. 8 This term contains short range low-energy two-boy scattering processes in the atomic system, characterize by a single parameter, the scattering length a s. The same scattering length can be obtaine using a pseuopotential in Heff int which is a contact potential66,67 with g = 4π a s /m. Now in the presence of laser light we also nee to take into account light assiste collisional interactions. A re-etune laser can give rise to optical Feshbach resonance resulting in moification of the scattering length which will epen on the laser intensity 68,7. This spatial epenence is reflecte in gx an away from the resonance we woul get back g 4π a s /m. Loss of atoms ue to light assiste collisions can be containe in an intensity epenent an thus spatially epenent imaginary part to the scattering length. As the rate of such losses are much less than the scattering rate, we can work in a regime where such loss processes have not occurre an can therefore leave out the imaginary part. Higher orer corrections such as three-boy collisions have also not been consiere in this Hamiltonian as we work with ilute gases. The last term in the master equation is the recycling term: J ρ = Γ 3 x 3 y ΩyΩ x 4 F k eg r ψ syψ s y ρ ψ sxψ s x s s, 9 which contains Linbla operators in the form of atomic ensities s ψ sxψ s x. As the function F k eg r falls off on the length scale of laser wavelength, a spontaneous emission process will ten to localize a particle within this length scale, ecohering the many-boy state. J ρ together with H eff also preserves the trace of the ensity operator. We can obtain a multi-ban Fermi-Hubbar moel for the coherent part of the evolution in the master equation by expaning the fiel operators in a Wannier basis 73, ψ s x = w n x x i c n,i,s, uner the assumptions of n,i nearest neighbor tunneling an local interaction in a eep lattice. Here, for the i-th site of the n-th Bloch ban, w n x x i is the Wannier function an c n,i,s is the fermionic annihilation operator for spin s. In an isotropic 3D cubic lattice we get, ρ = i[h, ρ] + Lρ, 0 t with the Fermi-Hubbar Hamiltonian, H = J n i,j,s cn i,s cn n,<i,j>,s + i,k,l,m,n j,s + n,i,s ɛ n i,s cn i,s cn i,s U k,l,m,n c k i,s cl i,s cm i,s cn i,s. Here J n i,j,s is the next neighbor tunneling rate corresponing to the kinetic energy, U k,l,m,n is onsite interaction energy coming mainly from collisional interaction with small moification from ipole interactions an ɛ n i,s is the onsite energy offset. The Linbla term escribing the scattering of laser photons: Lρ = i,j,k,l,m,n,s,s γ k,l,m,n i,j,s,s [ c k i,s cl i,s, [ c m j,s cn ]] j,s, ρ, an the matrix elements for ifferent scattering processes: γ k,l,m,n i,j =Γ 3 x 3 y F k egx y 4 Ω xωy w k x x i w l x x i w m x x j w n x x j. 3 Note that the notation use in Eq. 0 is the result of a simple regrouping of the terms in Eq. an chosen to make the following arguments towars the use of a single ban Fermi-Hubbar moel more transparent. In Lamb- Dicke regime i.e. Lamb-Dicke parameter, η = k L a 0 with a 0 as the extension of the Wannier functions in the lowest ban, for a re etune lattice spontaneous emissions ominantly return the atoms into the lowest Bloch Ban 4 as the relative probability for the atom to return to the first excite ban scales as η. Therefore we focus on the physics that arises from the treatment confine only to the lowest ban an write own the corresponing master equation,

6 6 t ρ = i[h F H, ρ] + L ρ. 4 We now only have a single ban Fermi-Hubbar Hamiltonian H F H = J <i,j>,s c i,s c j,s + U i n i, n i,, 5 where we have omitte the ban inices for the fermionic operators an the Liouvillian term is, L ρ = γ n i ρn i n i n i ρ ρn i n i. 6 i Here γ is the effective scattering rate obtaine by keeping only the onsite elements in Eq. 3 an the Linbla operators n i are number operators at each site n i = n i, + n i, = c i, c i, + c i, c i,. It is clear at this point that the issipative processes o not iscriminate between the ifferent spin orientations an can only ecohere the many-boy state by treating the lattice sites with ifferent total particle numbers ifferently. In a system where particle numbers for each species are conserve iniviually, the term in the Hamiltonian corresponing to an energy offset is just a constant an thus can be neglecte. Even though we have erive the master equation with two component systems in min, the generalization to any number of internal states is straightforwar an we can hanle SUN magnetism with the same formalism. For simplicity, we will mainly focus on the physics of systems with two internal states for the rest of this article. We now come back to the role of large etuning in avoiing irect spin ecoherence when the transition frequencies iffer for the two two level systems that represent the ifferent spin states, i.e., ω ω. We can moify the above erivation of the master equation at the expense of generating aitional terms an look at the ynamics in this more general case. In the following we illustrate the effect for a single particle fixe in space at x 0 having only two internal egrees of freeom. The corresponing master equation is given by t ρt = i s + s [ɛ s n s, ρ] γ s,s n sρn s n s n s ρ ρn s n s + s s γ s,s n sρn s n s n s ρ ρn s n s, 7 where the spin epenent scattering rates are efine as follows γ s,s = Γ s 3 x Ω x 0 Ωx 0 w 0 x x s s OJ /U OJ /U U>0 U<0 FIG. 3. Decoherence of fermions in a ouble-well. In the limit of strong interactions for U > 0, the groun state of Fermi- Hubbar Hamiltonian with strong repulsive interactions is primarily a spin singlet therefore symmetric spatially across the ouble-well. The population of oubly occupie sites is small OJ /U. Now for U < 0, the initial groun state is a coherent superposition of states with oubly occupie sites with OJ /U population in the spin singlet state. Spontaneous emission events over a significant perio of time lea to ecoherence of virtual ouble-occupations, an populate states in which the final steay-state population is evenly istribute in the state with single occupancy an those with oubly occupie sites. The general solution for the atomic ensity matrix can be obtaine analytically an is given by ρ ρt =, 0 ρ, 0e i ɛ+γ efft ρ, 0e i ɛ γ, efft ρ, 0 where γ eff = γ, + γ, γ, γ, / an ɛ = ɛ ɛ an the associate ecay rates Γ s can iffer between spin states. The off-iagonal elements of the ensity matrix ecay in magnitue exponentially with an effective rate γ eff. This irect ecoherence of the wave function is an effect of the spontaneous emission processes. Now in the limit of large etuning i.e. ω ω / 0 one can show, by taking a Taylor expansion of the function Γ s / s s aroun any of the spin values, that the ecay rate γ eff scales as ω ω /. Therefore, for large etuning the master equation contains cross-terms of equal magnitue to the iagonal terms γ eff 0, an there is no irect ecoherence in the system ue to spontaneous emissions. On the technical level this means the Liouvillian part in Eq. 7 reuces to a single particle an single-site version of Eq. 6. This case of ientical photon scattering is the stanar case for fermionic atoms both from group-i an group-ii in far etune optical lattices. V. DECOHERENCE IN A DOUBLE WELL We now procee to stuy the effects of spontaneous emissions as escribe by the master equation erive in the previous section, focussing on the resulting manyboy ynamics. We primarily take examples from strongly interacting regimes so that the spatial ecoherence in the many boy wave function ue to localization of the spin particle following a spontaneous emission event is minimal 4. We want to investigate the robustness of anti-ferromagnetic spin orer of two species

7 7 S r 0,t a U=8J time tj b U=-8J time tj FIG. 4. Decoherent ynamics starting from the groun state of one particle of each spin species interacting strongly in a ouble well M = for γ = 0.J, compute in both, our perturbative approach re ots an exact iagonalization soli line: a Decay of the rescale spin correlations between the sites [Eq. 6] for U = 8J. The ashe line inicates the steay state expectation value. b Time evolution of rescale oublon correlation [Eq. 7] for U = 8J which vanishes in the final steay state. fermions in the repulsive case an of the correlation function of the composite bosons forme in the case of strong attractive interactions. Before we present our results for larger lattice systems, we give an intuitive example iscussing the ecoherence in a ouble well. For bosons, the ynamics of a relate case is iscusse in Ref. 43. Here we particularly focus on the ynamics of the spin egree of freeom, which we treat first by consiering the case of an initial groun state with U > 0, U J. We then return to the case of elocalise oublons for strong attractive interactions U < 0. A. Repulsive interactions We consier an optical lattice chain with a length of two, containing one spin up particle an one spin own particle. Now, in the limit of strong interaction U J the groun state woul be a spin singlet with an amixture of states having both spins in one of the sites. It is instructive to work in a particular basis forme by combination of Fock states given by =, +, /, =,, /, 3 =, 0, 4 = 0,. We first calculate the groun state of the two site Fermi- Hubbar Hamiltonian which is nearly a spin singlet with OJ /U population in the manifol with ouble occupation at one of the sites Fig. 3. Evolving this initial state uner the master equation shows that the wave function of the system ecoheres ue to spontaneous emission until it reaches a steay state ρ = 0 where population is equally istribute in all three basis states that were populate at the initial time Fig. 3. The rate at which the spin correlation ecays is proportional to that of the increase in the population of oubly occupie states for a oubly occupie site S z = 0. We calculate this rate in perturbation theory in J/U where the coherences between the manifols are eliminate aiabatically to give the ecay rate of the spin orer. We begin by calculating the ecay rate of population in state which is being transferre to the oublet manifol spanne by 3 an 4. Now from the master equation, D r,t t ρ, = k=3,4 JRe iρ,k. 9 The coherences between state an the oublet manifol obey t ρ,3 = i J ρ 3,3 + ρ 4,3 ρ, + iu γρ,3, 0 an ρ,4 follows an analogue equation. Now the coherence within the oublet manifol given by t ρ 3,4 = i J ρ,4 ρ 3, 4γρ 3,4. Now in the limit U J, γ we can eliminate the coherences between state an the oublet manifol first an that leas us to an 4J t ρ γ, = U + γ ρ, ρ 3,3 ρ 4,4 + 4J ρ3,4 + ρ 4,3 Re, γ iu t ρ 3,4 = 4γρ 3,4 + 4J γ U + γ ρ 4,4 ρ, ρ 3,4. 3 Now we can eliminate the coherence in Eq. 3 as it contains a term proportional to γ whereas all the other terms are suppresse by a factor of OJ /U. We rewrite Eq. as 4J t ρ γ, U + γ ρ, ρ 3,3 ρ 4,4, 4 which gives a ecay rate proportional to β = 4J γ/u + γ. The result obtaine by evolving the master equation using exact iagonalization is in agreement to this anaytical value as illustrate in Fig. 4a. The spatial average of the spin correlation function is efine as, S x, t = Si z ts z M i+ xt. 5 i Here, Si z is the z component of spin at lattice site i, efine as Si z = n i, n i, /. By their nature, the spin correlations can also be interprete as a measure of ensity fluctuations. All spin components are equivalent ue to the SU symmetry of the lattice Hamiltonian an the issipative terms. Therefore we focus on the

8 8 z component of spin an for plotting purposes we also consier the rescale spatial average: S r x, t = S x, t S x, t = 0. 6 The physical process giving rise to the observe ecay can be outline as follows: A spontaneous emission event oes not ifferentiate between the ifferent spin states, but as we saw before, what it effectively etects is the occupation number at the site involve as inicate by the Linbla operators in Eq. 6. In this sense, it istinguishes states with oubly-occupie sites from states with singly occupie sites, ecohering virtual population of oubly occupie states. This rives the system away from the initial spin orere groun state which has mostly singly occupie sites with very small oubly occupie population OJ /U an transfers population from states with singly occupie sites to states with oubly occupie sites. The resulting state is no longer an eigenstate an the Hamiltonian therefore starts reistributing population coherently whereas spontaneous emission events continue to isrupt rebuiling of coherence. This interplay between the Hamiltonian an the issipative ynamics gives rise to the resulting ecoherence an change in spin correlation. The rate of ecoherence epens on the effective scattering rate γ as well as on the relative population in the oublet manifol which grows proportionally with its initial value. This is the reason we have a term OJ /U in the expression of β, an this reflects the ability of the Hamiltonian to populate oubly-occupie sites via tunneling in the presence of an energy gap. It also shows that the spin orer ecays much slower than the scattering rate an can be quite robust against spontaneous ecay for strongly interacting systems. Alternatively one coul argue that the initial state is the groun state which is an eigenstate of the Hamiltonian, an hence the first orer of time-epenent perturbation theory is given solely by the action of the issipative part on the initial state. For the correlation function of interest, we fin a vanishing first orer term as we on t have irect spin ecoherence. Hence the leaing orer term in the ecay rate has to be OJ /U. B. Attractive interactions We now consier the case of attractive interactions U < 0, again for one atom of each spin in a oublewell, where we observe markely ifferent ynamics for strong interactions. The groun state of the Hamiltonian now consists of states with ouble occupation, because the attractive interactions favour the formation of a imer. The key physical property is that the imer is elocalise over the two sites, i.e., the groun initial state is essentially a coherent superposition /. Again, there is a small amixture of the singlet singlyoccupie state, i.e., a population OJ /U in. The final steay state of the master equation at long times is the same with equal population in all these three basis states. However, the initial ynamics towars the steay state begin by rapily removing the coherence between 3 an 4, markely changing the state when we consier the ynamics of imers. We can calculate the ecay rate of the oublon correlation functions, e.g., in perturbation theory like before. Defining a spatially average an rescale oublon correlation function analogous to the spin case D r x, t = M i i t i+ xt i t = 0 i+ xt = 0, 7 we check the agreement between results obtaine in exact iagonalization an perturbation theory Fig. 4b. The ecay rate for the oublon correlations turns out to be 4γ an is approximately inepenent of the system size an filling factor, as we show up to first orer in time-epenent perturbation theory in Sec. VI C. In that section, we also iscuss the enhancement factor, which arises from a combination of having two atoms in a given site, an also having superraiant enhancement because of the scattering of ientical photons. An instructive way to check this enhancement factor is to look at the optical Bloch equations for a system of ientical two-level atoms fixe on lattice sites an solve for an effective ecay rate which is equivalent to calculating the rate of change in groun state population when the excite states can be aiabatically eliminate in the limit of large etuning of the riving laser fiel. For N atoms the atomic ensity operator ρ a obeys the following equation where the non-hermitian effective Hamiltonian H, written in terms of Pauli matrices, with H = N k= t ρ a = i[h, ρ a ] + Γ kl σ k ρ aσ + l, 8 k,l σk z Ω k σ+ k + σ k i Γ kl σ + k σ l. k,l 9 Here Ω k is the Rabi frequency for the k-th atom which we will take to be position inepenent an Γ kl = ΓF k eg r kl where Γ is just the spontaneous ecay rate of the excite state of an atom an the function F Eq. 6 introuces a localising effect on the scattering element between k-th an l-th site at a istance r kl on a scale set by the atomic transition wavelength k eg. We can etermine an effective ecay rate in the groun state population which initially when the system is the groun state ψ 0 is the rate of ecrease in the norm for evolution uner the effective Hamiltonian, namely,

9 9 Γ eff = δt [ ψ 0 e ih δt e ih δt ψ 0 ] ψ 0 k,l Γ kl σ k σ+ l ψ In the single atom case, using secon orer timeepenent perturbation theory the ipole coupling with the laser fiel is the perturbative part of the Hamiltonian, it is easy to calculate this effective ecay rate Γ single = ΓΩ /4. Now for two atoms we look at two limiting cases. When the atoms are separate by a istance much larger than the atomic transition wavelength, the scattering elements turn into on-site terms Γ kl Γδ k,l an the effective rate is Γ single. This is what one woul expect for the total ecay rate of two inepenent entities. In the opposite case, where the atomic istance is much smaller than the transition wavelength, all the scattering elements become inepenent of the istance between the atoms Γ kl Γ an we inee obtain an effective ecay rate of 4Γ single. VI. DYNAMICS FOR MANY ATOMS In orer to further quantify the impact of spontaneous emissions on many-boy correlations we iscuss the full many-boy problem using approximate analytic an numerically exact solutions to the master equation erive in Sec. IV. First we iscuss the effects of spontaneous emissions on the momentum istribution of noninteracting fermions. Secon we analyze the ecoherence of antiferromagnetic spin orer in the case of strong repulsive interactions, comparing time-epenent perturbation theory for J/U 0 to numerical ata. Depening on the system size we use either exact iagonalization or combine aaptive time-epenent DMRG with the quantum trajectory approach 55, to capture the ecoherence an time-epenence of first orer correlation functions in etail. While the first approach is exact, time-epenent DMRG is well establishe as a convenient means to moel the real-time ynamics inuce by stochastic processes in one-imensional systems that remain close to equilibrium. Our main results on the repulsive case are that the spin correlation functions are robust on experimentally relevant timescales an that the spin-ecoherence is governe by a single ecay rate which is suppresse by the number of ouble occupancies in the initial state. In the limit of strong attractive interactions, both the perturbation theory approach an the numerical simulations unveil ecay rates of the oublon correlation function enhance by a factor of four, which can be unerstoo as a consequence of superraiance 7,77,78. While the results shown in this section are obtaine from one-imensional optical lattices, they are irect consequences of the erive master equation an general conclusions as the robust magnetic orer an the impact of superraiance are therefore expecte to carry over to both higher spin-egrees of freeom an higher imensions. As a first general result we calculate the rate of energy increase inuce by the spontaneous emissions for N atoms. This can be obtaine analytically from the master equation Eq., as was one for bosons in Ref. 4, evaluating t H = TrL ρh. 3 The final result strongly resembles the result for bosons 4 an is not only inepenent of the interaction strength but also completely etermine by single particle physics 40 : t H = ΓΩ 0 kl 4 m N Ω = Ω 0 cos k L x. 3 However, as in the case of bosons, this result oes not properly characterize the heating inuce by spontaneous emissions as the energy increase preominantly results from excitations to higher bans which will in general not thermalize on experimental time-scales 4. For Bosons this has been quantifie in Ref. 44. Hence, even a qualitative analysis requires at least an analysis of first orer correlation functions such as spin correlations, momentum istribution functions or the single particle ensity matrix. In the following we perform such an analysis, first for free Fermions, then for repulsive interactions an finally for attractive interactions. A. Free Fermions The case of free fermions is another instructive example that can be ealt with exactly. We here focus on the time epenence of the momentum istribution for N fermions on M lattice sites n k = M c k,s c k,s, with c k,s = e ikl c l,s. 33 s M l= For U = 0 the Hamiltonian is iagonal in momentum space an hence the time-evolution of n k t is solely given by the action of the issipative part which results in t n k = Trn k L ρ = γ [n r, [n r, n k,s ]] r,s = γ n k + N M γ. 34 Therefore the steay state momentum istribution function n k N/M for t, i.e., the momentum istribution corresponing to all particles being localize in space by spontaneous emissions. The ynamics leaing to this state occur graually, as particles are sprea throughout the Brillouin zone via spontaneous emissions.

10 0 B. Repulsive interactions We now move to the richer case of repulsively interacting fermions with magnetic orering. In particular, we numerically stuy the ecay of spin correlations in the D Hubbar moel with repulsive interactions, for which unerstaning an characterizing the impact of ifferent heating mechanisms on the characteristic correlation functions is an important step on the way to experimentally realize quantum magnetism. In one spatial imension strong correlation effects give rise to interesting many-boy effects such as the absence of longrange orer, which can be utilize to benchmark experiments with ultra-col atoms against exact solutions 79, an powerful numerical methos 5. The one-imensional Heisenberg moel, one of the paraigm moels of quantum magnetism 80, can be obtaine from Fermi-Hubbar moel using perturbation theory 8 in J/U, which highlights the characteristic energy scale to observe quantum magnetism Furthermore the antiferromagnetic correlations persist to finite J/U an can be measure in experiments with ultra-col fermions 85. Here, investigate the ecay of the spatially average spin correlation functions efine in Eq. 5 for larger lattices. First, the perturbative result for the ecay rate obtaine for the ouble well in the previous section can be generalize for a chain of length M. By etermining the initial relative population in the oublet manifol given by N D = i N i 35 where i = c i, c i, is the oublon annihilation operator, in egenerate perturbation theory 86 in J/U, where the Fermi-Hubbar Hamiltonian reuces to a Heisenberg Hamiltonian 8 an systematic aiabatic elimination of coherences between the initial groun state in zeroth orer an the oublet manifol gives exactly the same transfer rate. Furthermore, within the same perturbative approach we can irectly relate our expression for the ecay rate of the spin correlation, namely β γn D. Secon, we will confirm the scaling preicte here numerically for repulsive D systems up to M = 3 lattice sites. Fig 5 a shows the ecay of the on-site contribution S x = 0, t at an interaction strength of U = 8J an γ = 0.J for ifferent system sizes M = 4, 8, 3, a timestep of tj = 0.0 an a DMRG matrix imension of bon imension of 8M to keep the iscare weight below 0 5 for the largest system at the largest time consiere tj = 0. We fin that for the average quantities finite size effects are small for M 6. Fig. 5c shows the main result, the ecay rate β extracte from a numerical fit of a e βt + const. [see Fig. 5b for an explicit example] to the curves as shown in Panel a for combinations of U/J = 4, 6, 8, 0, an γ/j = 0., 0., 0.05, 0.05 as a function of the effective ecay rate obtaine from perturbation theory, N D γ. Within the error bars obtaine i from the fits, the ecay rates obtaine from the numerical ata scale linearly with respect to N D γ an the system size epenence is mainly given by system size effects of N D. This corroborates our previously perturbative result, that the effective ecay rate is suppresse as U/J increases since it is proportional to the number of ouble occupancies in the initial state. Finally, Fig. 5 shows ata for S r δx, t, for istances x = 0,, 4 at M = 3, U = 8J an γ = 0.J. We fin that rescaling the ata accoring to Eq. 6 unveils that the correlation function ecays in a similar fashion inepenent of the istance. Therefore the alternating sign of S x, 0 a necessary conition for antiferromagnetic correlations is preserve uring the ynamics. To summarize, our numerical stuy of the ecay of spin correlation functions for the repulsive Fermi Hubbar moel unergoing spontaneous emissions shows that changes in antiferromagnetic correlations are inhibite because the rate is controlle by the number of ouble occupancies that can be forme. The energy gap plays an important role in suppressing the coherent processes that form virtually oubly-occupie sites, an leas to a suppression of the ecay of magnetic correlations somewhat analogous to the inhibition of iffusion seen for Bosons in Refs. 4 an 43. The rate of oubly occupie sites is an experimentally controllable parameter 8,87, an the time-epenence of the spin correlations shoul be irectly measurable in experiments, either using quantum gas microscopes 84,88, or other techniques such as moulation spectroscopy or Bragg scattering to etect local or longer-range spin fluctuations 8,89. Note that this robustness shifts the typical rate of ecay from γ to N D γ J /U γ. This compares favourably with the energy scale J /U of typical ynamics in this regime. Note that ue to the suppression, the new ominant effect of spontaneous emissions for large enough U will be transfer of particles to higher Bloch bans, on timescales given by /η γ. C. Attractive interactions This inhibition of the ecay of spin correlation functions is in strong contrast to the effects we observe for attractive interactions, as we saw in the case of the oublewell above. Here we analyse the characteristic correlation functions for many bosons with strong attractive interactions. Taking U < 0 an focusing on strong interactions at moerate to low ensities, we see that the groun state of the Fermi-Hubbar moel consists of boun imers that behave as composite bosons, an conense to allow conensation, an off-iagonal long-range orer of imers. In the strongly interacting regime we expect pairs to preominantly form in real space an hence, for a sufficiently low ensity, the groun state of our lattice moel has a large contribution of oubly occupie sites. To see this immeiately we can again use egenerate perturbation theory in J/U, as was one in Ref. 9, to fin an

11 S x,t β time tj 0.5 c a γ=0.j U=8J N D γ M=4 M=8 M=3 b M=3,γ=0.J,U=8J time tj M=3,γ=0.J,U=8J x=0 x= x= time tj S x,t S r x,t D r x,t D r x=5,t a M=8 x= b M=3 x= c M=3 γ=0.05j M=3 γ=0.j time tj x= U=-8J U=-6J time tj D r x,t D r x=5,t FIG. 5. Comparison of the ecay of spin correlations [see Eq. 5] average over the chain obtaine from exact iagonalization using the EXPOKIT package 90 for M = 4, 8 an tdmrg with D = 8 for M = 3. a Decay of the on-site contribution S x = 0, t for ifferent system sizes M = 4, 8, 3 at U = 8J an γ = 0.J using 500 trajectories for M = 4, 8 an 50 trajectories for M = 3. b Example fit to the ata shown in panel a for M = 3. c The ecay rates β extracte from numerical fits as shown in panel b as a function of N Dγ. The ashe lines in b an c are linear fits to the ata for ifferent system sizes [M=4 black squares an M=3 re triangles, which exhibit the scaling β N Dγ preicte by perturbation theory. Panel shows the rescale S r x, t [Eq. 6] for M = 3, U = 8J an ifferent x which shows only a weak istance epenence, especially at larger times. effective Hamiltonian H D that escribes the ynamics of boun pairs, H D = J U <i,j> i j n D i n D j, 36 which contains a oublon tunneling term as well as a nearest neighbour interaction term with n D i = i i being the on-site number operator for oublons. This moel favors pair formation on alternative sites as the system can ecrease its energy via virtual tunneling of oublons U < 0. Since these pairs can be approximately treate as bosons an Pauli-exclusion prohibits multiple pairs, the perturbative Hamiltonian is the one of harcore bosons with next-nearest neighbour interactions. At low ensities we expect a superflui of pairs for the groun state with an algebraic ecaying oublon correlation function Eq. 7. Here we stuy the ecay of those correlations uring the issipative ynamics. Given that the initial state is the groun state which is an eigenstate of the Hamiltonian, the first orer of timeepenent perturbation theory is given solely by the action of the issipative part on the initial state. We there- FIG. 6. Comparison of oublon correlation obtaine numerically an in perturbation theory ashe line in the strong attractive interaction limit, average over chain an rescale by the initial value [Eq. 7]. a Time epenence of the oublon correlation at x =, 5 for system size M = 8 at U = 8J an γ = 0.05J. We see that the quantum trajectory results iamons from tdmrg with D = 64 are in goo agreement with the result obtaine by oing exact iagonalization squares using the EXPOKIT package 90 an that perturbation theory oes not take into account the rebuiling of correlations estroye by spontaneous emissions an hence unerestimates the ecay at short istances, but overestimates the ecay at large istances. Using same line symbols in panel b we show the quantum trajectory results for spatial epenence for M = 3 at U = 8J an γ = 0.05J qualitatively similar to M = 8. c Effects of ifferent ecay rates for M = 3 at U = 8J. Depenence on interaction strength for M = 3, γ = 0.05J. The time for which our perturbation theory is reliable scales with U. b to show tdmrg ata using a bon imension D=8 an the number of trajectories use in all of the calculations here is 58. fore calculate: t i j = Tr i jl ρ = γ n k i jn k n k n k i j i jn k n k, 37 where k We first calculate n k = n k, + n k,. 38 n k i j i jn k = c i, c i, c j, c k, + c i, c i, c k, c j, δ k,j + c k, c i, c j, c j, + c i, c k, c j, c j, δ k,i, 39