Orbital-driven melting of a bosonic Mott insulator in a shaken optical lattice

# Orbital-driven melting of a bosonic Mott insulator in a shaken optical lattice

Christoph Sträter    André Eckardt Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
July 12, 2019
###### Abstract

In order to study the interesting interplay between localized and dispersive orbital states in a system of strongly interacting ultracold atoms in an optical lattice, we investigate the possibility to coherently couple the lowest two Bloch bands by means of resonant periodic forcing. For bosons in one dimension we show that a strongly interacting Floquet system can be realized, where at every lattice site two (and only two) near-degenerate orbital states are relevant, whose tunneling matrix elements differ in sign and magnitude. By smoothly tuning both states into resonance, the system is predicted to undergo an orbital-driven Mott-insulator-to-superfluid transition. As a consequence of kinetic frustration, this transition can be either continuous or first-order, depending on parameters such as lattice depth and filling.

###### pacs:
37.10.Jk,03.75.Lm,67.85.-d,75.30.Mb

## I I. Introduction

Orbital degrees of freedom play an important role in solid-state systems. A prominent example is the intriguing physics of heavy-fermion compounds that emerges from the interplay between dispersive conduction-band orbitals on the one hand and strongly localized orbitals, with a large effective mass and strong Coulomb interactions, on the other Hewson (1997); Coleman (2007); Si and Steglich (2010); Gulácsi (2004). However, in systems of ultracold atoms in optical lattices Bloch et al. (2008); Lewenstein et al. (2012) orbital degrees of freedom, spanning Bloch bands above a large energy gap, are typically frozen out, at least in the interesting deep-lattice tight-binding regime where interactions are strong. Here, we investigate the possibility to coherently open on-site orbital degrees of freedom in a strongly interacting optical lattice system by means of near-resonant lattice shaking. We consider spinless bosons in one dimension (1D) and show how to realize a “dressed-lattice” system, where effectively at every lattice site the strongly localized ground-band orbital is nearly degenerate and coupled to the much more dispersive first-excited-band orbital. The tunneling matrix elements of the two orbitals differ strongly in magnitude and also in sign, with the latter leading to kinetic frustration. We predict an orbital-driven phase transition between a Mott insulator (MI) and a superfluid (SF) state when the population of the light orbitals is adiabatically increased by lowering the interorbital detuning. As a consequence of frustration and strong interorbital interactions, this transition is found to be either continuous or first-order, depending on parameters such as filling or lattice depth.

In contrast to the present proposal, in previous experiments atoms were transferred non-adiabatically to excited bands of optical lattices by different methods Gemelke et al. (2005); Müller et al. (2007); Sias et al. (2007); Wirth et al. (2011); Ölschläger et al. (2011); Bakr et al. (2011). Moreover, lattice shaking has recently been employed for band-coupling in the weakly interacting regime, where condensation into two possible momentum states led to domain formation Parker et al. (2013). Such band-coupling has been studied theoretically for non/weakly interacting particles and isolated sites Arlinghaus and Holthaus (2011, 2012); Sowiński (2012); Zhang and Zhou (2014); Zheng et al. (2014); Choudhury and Mueller (2014); M Di Liberto et al. (2014). Also orbtial coupling via magnetic resonances has been proposed Pietraszewicz et al. (2012) and there has been theoretical interest in the physics of excited orbitals not involving lower-lying states Isacsson and Girvin (2005); Wu et al. (2006); Wu (2008a, b); Li et al. (2012a, b); Pinheiro et al. (2013). Finally, the perturbative admixture of excited orbitals has been studied in theory Li et al. (2006); Lühmann et al. (2008); Schneider et al. (); Büchler (2010); Hazzard and Mueller (2010); Dutta et al. (2011); Łacki and Zakrzewski (2013); Dutta et al. (2014) and experiment Campbell et al. (2006); Best et al. (2009); Will et al. (2010); Mark et al. (2011); Heinze et al. (2011); Jürgensen et al. (2014).

## Ii II. Realizing the two-orbital model

Consider spinless bosonic atoms of mass in an optical lattice . The and directions are frozen out by a deep lattice, we will assume , such that an array of 1D tubes with a dimerized lattice [Fig. 1(a)] is created. The recoil energy is needed to localize a particle on a lattice constant . For Rb a typical wave length of nm gives kHz. Each 1D tube is described by the multi-band Bose-Hubbard Hamiltonian , where

 ^Hkin = −∑ℓ∑α(−1)αJα(^b†α(ℓ+1)^bαℓ+h.c.), (1) ^Hos = ∑ℓ[∑αϵα^nαℓ+∑{α}U{α}2^b†α1ℓ^b†α2ℓ^bα3ℓ^bα4ℓ]. (2)

Here and are the bosonic creation and number operator for a Wannier orbital of Bloch band , localized at *[Sinceweconsideraweakdimerizationonly; $V_1<V_0$; wedonotneedtoemploygeneralizedWannierorbitals; localizedintheleftandrightminimumofeachdoublewell[Fig.~\ref{fig:lattice}(a)]; asin][]VaucherEtAl07. The band-center energies and tunnel parameters fulfill and , respectively. The interaction strengths vanish for odd , since , and depend on the transverse localization length and the scattering length ( nm for Rb).

In the tight-binding regime, is typically much larger than the temperature and the chemical potential so that the orbital degree of freedom is frozen out. We wish to coherently open this freedom by means of time periodic forcing with near-resonant frequency . In particular, the lowest band () shall be coupled to the more dispersive first excited band (), without creating coupling to even higher-lying bands (). In order to achieve such a controlled situation—also in the regime where interactions are strong compared to tunneling—we combine two strategies: First we choose a driving scheme, namely sinusoidally shaking the lattice back and forth, that for weak forcing does not lead to multi-“photon” interband transitions at resonances with integer . Second, we engineer the band structure by varying such that transitions to band 2 remain off resonant: With increasing the bands organize in pairs (0,1), (2,3), …such that and as well as and decrease, while increases. For , already a slight dimerization ensures that is noticeably smaller than , rendering the transition off-resonant when . At the same time is small enough to keep a relatively large ratio , retaining the desired feature that particles are much less dispersive than particles [Fig. 1(b,d)].

By moving the lattice like in direction, an inertial force is created, described by

 ^Hdr(t)=Kcos(ωt)∑ℓ[∑αℓ^nαℓ+∑α′αηα′α^b†α′ℓ^bαℓ]. (3)

Here vanishes for even . We employ a time-periodic unitary transformation with integers , designed to shift all band energies to values that are as close as possible to . This gives by choice of and by choice of ; all other are scattered somehow between and . The periodic time dependence of the transformed Hamiltonian appears in the interband coupling parameters and . For weak forcing the driving frequency is large compared to the intraband terms as well as to the band coupling [Fig. 1(d)]. This allows to average the rapidly oscillating terms in the Hamiltonian over one driving period and to approximately describe the system by the effective time-independent Hamiltonian , reading

 ^Heff = ^Hkin+∑ℓ[∑αϵ′α^nαℓ+K∑α′αη′α′α^b†α′ℓ^bαℓ (4) +∑{α}U′{α}2^b†α1ℓ^b†α2ℓ^bα3ℓ^bα4ℓ],

with and . For a more systematic derivation of Eq. (4), is defined as the generator of the time evolution over one period Shirley (1965) and computed using degenerate perturbation theory in the extended Floquet Hilbert space Sambe (1973), similar like in Refs. Eckardt and Holthaus (2007, 2008). In leading order one recovers Eq. (4). The leading correction contains tiny second-order coupling to bands of order to be neglected, where is a typical interband coupling matrix element and .

It is a crucial property of lattice shaking (3) that in the interband coupling is a single-photon process, with , and that scattering is a zero-photon process, with . No multi-photon processes are found for weak driving. Thus, in above the bands 0 and 1 are coupled to band 2 only, via the processes sketched in Fig. 1(c). These processes are, however, off-resonant, since . The bands 0 and 1 are, therefore, to good approximation isolated and described by the two-band (2B) model

 ^H2B = ^Hkin+∑ℓ[δ^n1ℓ−γ(^b†1ℓ^b0ℓ+h.c.)+2U10^n0ℓ^n1ℓ (5) +U002^n0ℓ(^n0ℓ−1)+U112^n1ℓ(^n1ℓ−1)],

where , and . For , , and we obtain , , , , , and .

describes a highly tunable 1D ladder system [Fig. 2(a)] with interesting properties: The tunneling matrix elements along both legs (i.e. in both bands) differ in sign and magnitude. The former leads to maximal kinetic frustration with a flux of per plaquette Honerkamp (2003); Hotta and Furukawa (2006); Eckardt et al. (2010); Struck et al. (2011); Greschner et al. (2013); Tieleman et al. (2013); Dhar et al. (2013); Yudin et al. (2014). The latter makes leg 0 more prone to localization than leg 1. The hybridization of both legs is controlled by the energy separation and the coupling , which can be tuned via the frequency and strength of the driving, respectively. Finally, the system features strong interorbital interactions , with two-particle energies .

In order to investigate the 2B model (5), the following experimental protocol can be pursued. After the system is prepared in (or close to) the undriven ground state, populating band 0, the driving strength is ramped up smoothly to the desired value. During this step the detuning is still large enough to suppress any significant occupation of band 1. Then, the orbital freedom is opened by smoothly lowering .

## Iii III. Orbital-driven Mott transition

We study the ground state of versus . For large positive (negative) only leg 0 (leg 1) will be occupied; the system effectively reduces to a 1D Bose-Hubbard chain. For integer filling of particles per site, the ground state of such a chain with tunneling and on-site repulsion is a gapped (i.e. incompressible) MI with localized particles if , where Elstner and Monien (1999). Otherwise, it is a gapless SF with quasilong-range order. Thus, for the system is a MI for , since , and a SF for , since .

It is instructive to control via the chemical potential , introduced by adding to . In Fig. 2(b), we plot the ground-state compressibility in the - plane, computed by time-evolving block decimation (TEBD) in imaginary time Wall and Carr (2009); Vidal (2004). As expected Elstner and Monien (1999), for we find incompressible MI phases at integer filling , interrupted by SF phases where changes in direction, while for the system is a compressible SF. When is lowered, the filling () of leg 0 (1) decreases (increases). In response, an orbital-driven transition occurs, either between a MI and a SF or, for fractional filling, between different SFs. For the given parameters, these are first-order transitions, except at the tip of the Mott phase, where a continuous transition is found. The discontinuous SF-to-SF transition, where the ground state changes abruptly, happens when near a boson suddenly prefers to delocalize with quasimomentum in leg 1, rather than with quasimomentum 0 in leg 0. The discontinuous MI-to-SF transition, to be explained below, is more subtle.

A strong-coupling argument explains the orbital-driven MI-to-SF transition. Within the MI state, increases smoothly when is lowered, and the larger the larger is the reduction of kinetic energy (or ) that a particle (or a hole) acquires by delocalizing along leg 1 on the MI background. When the kinetic energy reduction of a particle-hole excitation exceed its interaction-energy cost , these excitations proliferate and the ground state becomes a SF as seen in Fig. 2(b). This transition can also be observed in Fig. 2(c,middle) where we plot and the ground-state correlations versus , for the same parameters and sharp filling 111Experimentally and and the Fourier transforms of and can be measured via band mapping Müller et al. (2007).. While decays exponentially with in the MI phase, in the SF regime the decay is only algebraic. Therefore, the transition is indicated by a significant increase of correlations on longer distances such as . It is found near , in fair agreement with the above estimate giving .

In Fig. 2(c, middle) we can identify three different types of MI states, characterized by respective signs of the short-range correlations along both legs. This is a consequence of kinetic frustration; while the tunneling matrix elements and favor and , the rung coupling favors . We use the label MI if both legs retain their favored correlations (), and MI if leg dominates the other one [ (-1) for (1)]; similar labels are used for SF states. Due to the strong interleg interactions the system does not feature the chiral time-reversal symmetry broken MI or SF ground states with complex predicted in Ref. Dhar et al. (2013). Treating both and the as perturbation the MI-to-MI transition is predicted to occur at Sup (). Experimentally, this transition is hardly observable, since it occurs at tiny . The MI-to-MI, happening when is lowered further, is of greater importance. A perturbative treatment of the tunneling matrix elements , neglecting and on interaction-dominated doubly occupied sites, predicts this transition to occur when Sup (), in reasonable agreement with the numerics. These transitions can be observed also in a deeper lattice, where the system remains a MI for small [Fig. 2(c,top)].

For a lower lattice depth of , the MI-to-SF transition occurs earlier and already within the MI regime [Fig. 2(c,bottom)]. This is explained by the above estimates that predict the MI-to-SF transition to occur when , well before the estimated value for the MI-to-MI transition is reached. As a consequence, the MI-to-SF transition is rendered discontinuous. The discontinuity results from an abrupt change in the structure of the short-range correlations along leg 0. Namely, the SF phase is of SF type, with , such that the short-range correlations along leg 0 have to undergo a finite jump at the MI-to-SF transition. The same argument also explains the first-order nature of the orbital-driven MI-to-SF transitions for at filling , visible as a sharp jump of the compressibility in Fig. 2(b). All in all, the fact that the orbital-driven MI-to-SF transition can be discontinuous results from the combination of kinetic frustration, tunneling imbalance , and strong interband interactions , all stemming from the spatial structure of the Wannier orbitals.

## Iv IV. Preparation dynamics and heating

When is lowered slow enough, the system is expected to approximately follow the ground state of the 2B model (5), unless the first-order transition is crossed. This desired dynamics is effectively adiabatic Eckardt and Holthaus (2008), i.e. adiabatic with respect to , but diabatic with respect to tiny coupling matrix elements neglected in . We have simulated the time evolution of the system [Fig. 3(a)] using TEBD Wall and Carr (2009); Vidal (2004). For parameters like in Fig. 2(c,middle), is ramped from to within a time ms. In order to probe the validity of the 2B model (5), we include the major “loss” processes depicted in Fig. 1(c) by employing Hamiltonian (4) with three bands (0,1,2). In Fig. 3(a), one can clearly see that the occupation remains very low and that the overlap with the instantaneous 2B ground state stays close to 1. Both clearly shows that the driving does not cause detrimental heating and justifies a description of the driven system in terms of the 2B model (5). It, moreover, indicates that an effectively adiabatic time evolution is possible, despite a noticeable dip of the overlap at the Mott transition (resembling the behavior of Landau-Zener sweeps Lim and Berry (1991)). Thus, the protocol allows for the preparation of stable low-entropy states in an excited Bloch band.

The overlap plotted in Fig. 3(b) versus and measures the effective adiabaticity. Too small and spoil the adiabatic dynamics within the 2B model and for too large the coupling to band 2 becomes relevant. Moreover, for too large and slow second-order loss processes (not included) can occur. Finally, Fig. 3(c) shows that for strong interactions a minimal dimerization of is crucial. Different from the weakly interacting case Parker et al. (2013), we find significant transfer to the second excited band 2 for the simple cosine lattice ().

## V V. Conclusion and outlook

We have shown that lattice shaking is a feasible tool to coherently open on-site orbital degrees of freedom in a strongly interacting optical lattice system and that the interplay between Wannier orbits of different structure gives rise to rich physics already for spinless bosons in 1D. Extending the scheme to spinful fermions, the interplay between strongly localized and dispersive orbital states should permit to mimic aspects of the intriguing heavy-fermion physics and to realize periodic-Anderson-like models Hewson (1997); Coleman (2007); Si and Steglich (2010); Gulácsi (2004). The extension to higher dimensional lattices should provide a feasible scheme for the preparation of low-entropy states in excited bands as they have been discussed before Isacsson and Girvin (2005); Wu et al. (2006); Wu (2008a, b); Li et al. (2012a, b); Pinheiro et al. (2013) and, moreover, to couple them to strongly localized lowest-band orbits. Finally, by employing sufficiently off resonant forcing, keeping large enough, one might enhance and control the perturbative admixtures of excited bands Li et al. (2006); Lühmann et al. (2008); Schneider et al. (); Büchler (2010); Hazzard and Mueller (2010); Dutta et al. (2011); Łacki and Zakrzewski (2013); Dutta et al. (2014); Campbell et al. (2006); Best et al. (2009); Will et al. (2010); Mark et al. (2011); Heinze et al. (2011); Jürgensen et al. (2014), e.g. in order to enhance superexchange processes by engineering density-dependent tunneling.

###### Acknowledgements.
We thank Miklós Gulácsi for discussion. CS is grateful for support by the Studienstiftung des deutschen Volkes.

## Vi Appendix A: Transition between MI0 and MI01

For large negative , one can treat both and the tunneling matrix elements as a perturbation. For unit filling the unperturbed ground state takes the simple form

 |ψ0⟩=∏ℓ^b†0ℓ|vac⟩ (6)

with the vacuum state . A finite correlation will appear in third order. Namely one has

 ⟨ψ|^b†10^b11|ψ⟩ ≃ ⟨ψ1|^b†10^b11|ψ2⟩+⟨ψ2|^b†10^b11|ψ1⟩ (7) = 2⟨ψ1|^b†10^b11|ψ2⟩

with denoting the state correction appearing in th order perturbation theory. Here the relevant term of reads

 −γ^b†10^b00−δ|ψ0⟩=γδ^b†10∏ℓ≠0^b†0ℓ|vac⟩, (8)

and the relevant term of takes the form

 (γ^b†11^b01J0^b†01^b00(2U01+δ)U00−J1^b†11^b10γ^b†10^b00(2U01+δ)δ)|ψ0⟩ =(2J0U00−J1δ)γ2U12+δ^b†11∏ℓ≠0^b†0ℓ|vac⟩. (9)

With that we arrive at

 ⟨^b†10^b†11⟩≃(2J0U00−J1δ)2γ2(2U12+δ)δ (10)

leading to a sign change when both terms in the round bracket cancel each other. The change from positive to negative sign corresponds to the transition from MI to MI that is thus expected to occur at

 δ=U00J12J0. (11)

## Vii Appendix B: Transition between MI01 and MI1

We assume sharp filling and treat the tunnel terms as a perturbation. The unperturbed on-site problem is then given by

 ^H0 = δδ^n1−γ(^b†1^b0+^b†0^b1)+2U01^n0^n1 (12) +U002^n0(^n0−1)+U112^n1(^n1−1)

where we dropped the site index . In the subspace of one particle on a site the unperturbed ground state reads

 |ψ(0)⟩=(a0^b†0+a1^b†1)|vac⟩, (13)

with energy per site and , with , giving in leading order perturbation theory

 n0≃a20andn1≃a21. (14)

In the course of the perturbation calculation we also need defect states with one particle less (a hole) and one extra particle. The hole state is simply given by the vacuum

 |ψ(h)⟩=|vac⟩, (15)

with energy . The subspace with two particles on a site contains three states. For simplicity, we neglect the hybridization coupling and approximate the eigenstates with an additional particle by states with sharp occupations of the orbitals ,

 |ψ(p20)⟩ = 1√2(^b†0)2|vac⟩, (16) |ψ(p11)⟩ = ^b†0^b†1|vac⟩, (17) |ψ(p02)⟩ = 1√2(^b†1)2|vac⟩, (18)

with unperturbed energies , , and .

Re-introducing the site index the unperturbed ground state reads

 |ψ0⟩=∏ℓ|ψ(0)ℓ⟩=∏ℓ(a0^b†0ℓ+a1^b†1ℓ)|vac⟩. (19)

The correlation function between the neighboring sites and obtains a finite value in the first order of the perturbation expansion with respect to tunneling

 ⟨ψ|^b†α0^bα1|ψ⟩ ≃ ⟨ψ0|^b†α0^bα1|ψ1⟩+⟨ψ1|^b†α0^bα1|ψ0⟩ (20) = 2⟨ψ0|^b†α0^bα1|ψ1⟩.

Here the relevant terms of the first-order state correction possess an extra particle in one of the three possible states on site 1 and a hole on site 0. These terms are related to the perturbation and read

 [a20J0U00−2ε0(^b†01)2−a21J1U11+2δ−2ε0(^b†11)2 −a0a1(J1−J0)U01+δ−2ε0^b†11^b†01]∏ℓ≠0,1(a0^b0+a1^b1)|vac⟩. (21)

With this expression we obtain from Eqs. (20) and (14) that

 ⟨^b†00^b01⟩ ≃ 2n0U00(2U01+δ−2ε0)[2n0J0(2U01+δ−2ε0) (22) −n1(J1−J0)(U00−2ε0)]

and

 ⟨^b†10^b11⟩ ≃ −2n1(U11+2δ−2ε0)(2U01+δ−2ε0) (23) ×[2n1J1(2U01+δ−2ε0) +n0(J1−J0)(U11+2δ−2ε0)].

The transition from MI to MI is related to becoming negative. Approximating , which is consistent with our previous approximation to neglect on doubly occupied sites, the transition is expected to occur when

 n1n0≈4J0U01(J1−J0)U00 (24)

or, equivalently, when

 n0−n1≈(J1−J0)U00−4J0U01(J1−J0)U00+4J0U01. (25)

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