Controlled localization of interacting bosons in a disordered optical lattice

Quantum control of many-body states is essential for new quantum technology applications. Ultracold atoms trapped in optical lattices provide an excellent system to probe the coherent dynamical control of multiparticle states. A paradigm of the coherent control of cold atoms in optical lattices was provided by the superfluid (SF) to Mott-insulator (MI) transition via adiabatic ramping of the lattice potential SFMI . More recently, a theoretical proposal Holthaus06 to drive a SF-MI transition by using a fast oscillatory force has been experimentally demonstrated Arimondo09 . In this work we analyze the role played by disorder for interacting atoms in a lattice in the presence of a fast non-resonant oscillatory potential.

The coherent dynamics of tight-binding systems in the presence of external fields is a very timely subject and includes phenomena such as Bloch oscillations, Wannier-Stark localization, caused by static fields, or dynamical localization due to the interaction with ac fields HolHone96 ; Korsch . The energy eigenvectors of a quantum particle in a periodic lattice are Bloch waves. The corresponding eigenvalues form energy bands, and a single particle is delocalized over the lattice. Dunlap and Kenkre Dunlap88 showed that an external ac field localizes the particle when the ratio of the field magnitude and the field frequency is a root of the ordinary Bessel function of order zero. The use of Floquet theory gives rise to a great conceptual simplification HolHone96 ; the structure of the quasienergy bands depends on the parameters of the time-dependent external perturbation and therefore by fine tuning the amplitude and frequency of the field, tunneling and the consequent localization phenomena can be manipulated. The dynamical suppression of tunneling persists in the presence of interactions Holthaus06 and it was proposed to induce localization for a single particle in the presence of disorder Holthaus95 . Here, we combine the fast oscillatory force, the interactions between atoms, and disorder for a one-dimensional tight-binding system.

The interplay between disorder-induced localization and interactions is one of the main open issues in condensed matter physics. In the context of bosonic atoms trapped in optical lattices it has risen much interest in recent years DeMarco09 ; Inguscio and the exact ground state phase diagram of the disordered Bose-Hubbard (BH) model is still an open question Fisher89 ; Damski03 ; Burnett03 ; Kruger09 ; ReviewMaciej . For commensurate fillings and strong interactions in the absence of disorder, there exist insulating MI states characterized by an integer fixed number of bosons per lattice site and a gap in the excitation spectrum. In the presence of disorder an additional insulating phase appears, the Bose-Glass (BG) phase characterized by a gapless excitation spectrum and finite compressibility. For weak interactions and strong disorder an Anderson-Glass (AG) may appear where one or more condensates localize in Anderson states.

It has been shown previously that adiabatic ramping of a fast oscillatory force can drive an ordered system with commensurate filling from a superfluid regime into the localized regime as it effectively reduces and even suppresses the tunneling between neighboring sites Holthaus06 . This force periodic in time but constant in space can be produced in the laboratory through an effective acceleration induced by a linear-in-time dephasing of the lasers forming the optical lattice Raizen98 ; Arimondo09 . By changing appropriately the parameters of this acceleration, its amplitude and frequency, it becomes possible to control the transitions between different states of the BH Hamiltonian. We show here that adiabatic quantum evolution towards a localized state is still possible in the presence of disorder both for commensurate and non-commensurate fillings. The structure of the quasienergy levels becomes more complex when disorder is introduced. The breakdown of degeneracies eliminates spectral gaps giving rise to a denser spectrum, and in principle adiabatic evolution could be spoiled at least for reasonable time scales.

In section II we introduce the Hamiltonian and the Floquet theory used to describe the dynamics of the system. Section III discusses the time scales needed for inducing a transition into localized states for different ratios of the interaction and disorder strength. For commensurate filling we find that in spite of the breaking of symmetries induced by disorder, the time scale needed for adiabatic evolution is similar for an ordered and a disordered system. Localization is not possible for an ordered system with non-commensurate filling via the adiabatic ramping of the fast oscillatory potential HolthausEPL . We show that the breaking of symmetries induced by the disorder potential makes localization possible even at similar time scales than for the commensurate system. We also show how the adiabatic time scale shortens for increasing disorder strengths. Finally, we discuss the implications of our exact small system calculations for systems of experimentally relevant sizes. In Section IV we summarize our main conclusions.

We consider the dynamics of $N$ atoms trapped in the lowest Bloch band of an optical lattice at sufficiently low temperatures such that it can be described by the BH model. The corresponding Hamiltonian is (taking $\hbar=1$ throughout) SFMI

where ${a}_{j}$ is the bosonic destruction operator for an atom localized in lattice site $j=-M/2,..,M/2$ and ${n}_{j}={a}^{\dagger}_{j}{a}_{j}$. The parameter $J$ is the tunneling matrix element, $U$ is the on-site repulsive interaction, while $\epsilon_{j}\in[-1,1]$ denotes a site-dependent random external potential of strength $W$ and $C$ accounts for the external parabolic trap. Depending on the ratios $W/U$ and $U/J$ the ground states of $H_{0}$ show very different properties and, for large enough $N$, different parameter values span the phase-diagram of the multiparticle system.

For $W=0$, one can span the SF-MI quantum phase transition by changing the ratio $U/J$. In the limit $U/J\rightarrow 0$ the tunneling dominates and the system is SF, while for $U/J\gg 1$ the repulsive interactions dominate and the system is an insulator, with a gapped spectrum. For $W\neq 0$ and $U/J\gg 1$ the system is an insulator, but depending on the ratio $W/U$ and the mean occupation number $N/L$ ($L=M+1$ is the total length of the system, always odd in our calculations), the ground state is either a BG or a MI. Both states are localized, in the sense that there is no transport through the lattice, and thus the density matrix has exponentially decaying off-diagonal terms. What distinguishes both phases Inguscio ; Burnett03 is the structure of the spectrum, which is gapped for a MI while it is not for a BG. When disorder dominates $W/U>1$ the ground state is a BG, while for $W/U<1$ different theoretical methods predict a MI or a BG ground insulator state depending on the mean occupation number Fisher89 ; Damski03 ; Burnett03 ; Kruger09 . For small disorder, in the regime $U/J\rightarrow 0$, the system is SF with long-range coherence while in the region $U/W\ll 1$, the system condenses to an Anderson state Anderson localized in the regions of strong disorder. These properties do not smear out in the presence of a shallow trapping potential.

We impose a linear potential $V(t)$ across the lattice which oscillates in time Raizen98 ; Arimondo09 and the total Hamiltonian

is then periodic with period $T=2\pi/\omega$. The Floquet theorem establishes that, due to the periodicity of the Hamiltonian, the solutions to the Schrödinger equation $i\frac{\partial}{\partial t}\left|\,\Psi(t)\right\rangle=H(t)\left|\,\Psi(t)\right\rangle$ are linear combinations of functions of the form

where $\left|\,\phi^{e}(t)\right\rangle=\left|\,\phi^{e}(t+T)\right\rangle$. Inserting Eq. (4) into the Schrödinger equation one obtains the eigenvalue equation

where the Floquet Hamiltonian is defined as

and $e$ is the quasienergy of the system. As pointed out by Sambe Sambe , Eq. (5) can be solved using the standard techniques developed for time-independent Hamiltonians, provided we extend the Hilbert space to include the space of time-periodic functions. A suitable basis for this extended Hilbert space $\mathcal{R=H\bigotimes T}$ is $\{\left|\,\alpha\right\rangle\otimes\left|\,n\right\rangle\}$, where $\{\left|\,\alpha\right\rangle\}$ is a basis for the Hilbert space $\mathcal{H}$ of the system, and we define $\left\langle t\,\right|n\rangle=e^{in\omega t}$ with $n$ integer, as the basis functions in the vector space $\mathcal{T}$ of time periodic functions. In this basis, Eq. (5) becomes a matrix eigenvalue equation of infinite dimension with an infinite number of eigenvalues. It can be proven that if $e_{\alpha}$ is an eigenvalue with corresponding eigenvector $\left|\,\phi^{e_{\alpha}}(t)\right\rangle$, then $e_{\alpha}+p\omega$ with $p$ integer is also an eigenvalue with corresponding eigenvector $\left|\,\phi^{e_{\alpha}+p\omega}(t)\right\rangle$= $e^{ip\omega t}\left|\,\phi^{e_{\alpha}}(t)\right\rangle$. Accordingly, the eigenstates $\left|\,\phi^{e_{\alpha}+p\omega}(t)\right\rangle$ and $\left|\,\phi^{e_{\alpha}}(t)\right\rangle$ describe the same state in $\mathcal{H}$ apart from a global phase. Thus, in order to describe the time evolution in the Hilbert space we need only to solve the Floquet eigenvalue equation for $\varepsilon_{\alpha}\equiv e_{\alpha}\in(-\frac{1}{2}\omega,\frac{1}{2}\omega]$ called first Brillouin zone in analogy to the theory of periodic crystals. It can be shown that there are $D$ distinct quasienergy states in the first Brillouin zone, if the Hilbert space $\mathcal{H}$ is $D$-dimensional. The set of cyclic wave packets $\left|\,\Psi^{\varepsilon_{\alpha}}(t)\right\rangle$ in Eq. (4) obtained from the Floquet eigenstates provides a complete orthogonal basis in $\mathcal{H}$ at each $t$. The solution of the Schrödinger equation is then

where

From now on the term cyclic state refers to $\left|\,\Psi^{\varepsilon_{\alpha}}(t_{0})\right\rangle$ with $t_{0}$ being the beginning of a period.

We are interested in a regime in which $\omega$ is smaller than the energy separation to higher energy bands, so that our single-band BH model of Eq.(1) is still valid, but it is larger than the rest of the relevant energy scales of our system, i.e. $\omega\gg U,J,W$. In this high-frequency regime, one can show than the effect of the time-periodic external potential in Eq. (2) is to renormalize the tunneling term. To restore the lattice symmetry broken by the time-dependent potential it is useful to introduce a vector potential $A(t)=\int_{t_{0}}^{t}V(t^{\prime})dt^{\prime}$ and transform the solutions of the time-dependent Schrödinger equation as $\tilde{\Psi}(t)=\exp{\left[-iA(t)x\right]}\Psi(t)$. Note that the tunneling terms, combination of creation and destruction operators on neighboring sites, acquire a non-trivial phase. On the other hand, the space dependent phase induced by the vector potential cancels out in the interaction and disorder terms, as they only depend on creation and destruction operators on the same site. In the limit $\omega\rightarrow\infty$ one can average the phase over one period and show that the time-periodic external potential in Eq. (3) leads for $t_{0}=0$ to an effective time-independent Hamiltonian similar to $H_{0}$ with an effective tunneling Dunlap88 ; Dittrich91 ; Holthaus92 ; Holthaus06 ; Creffield08

where $\mathcal{J}_{0}(V/\omega)$ is the Bessel function of zero order. For finite but high-frequencies $\omega\gg U,J,W$, the Hamiltonian $H_{0}$ with the approximate effective tunneling rate of Eq. (9) still describes with high accuracy the cyclic states of the time periodic Hamiltonian (2) at $t_{0}$. For the zeroes of the Bessel function, the tunneling rate, although not exactly zero, is very much suppressed. Note that in this work we do exact time dependent calculations and do not use the effective Hamiltonian approximation. As shown below this approximation is very accurate and the main effect of the time-dependent potential Eq. (3) is to produce a renormalization of the effective tunneling in the lattice.

For an ordered lattice ($W=0$) the Hamiltonian Eq. (2) drives a transition Holthaus06 ; Arimondo09 from an initial SF state into a MI state by adiabatically ramping the potential $V$ to a value $V/\omega\sim 2.4$, which corresponds to the first zero of the Bessel function, where $J_{\rm eff}=0$. In a disordered one-dimensional lattice with no interactions this oscillatory force can be used for increasing the localization of particles in the Anderson state in the regime of fast oscillations Holthaus95 , while for slow oscillations, it can be used to decrease the localization Molina06a ; Molina06b .

Here, we combine the oscillatory driving force, the random potential, and the interatomic interactions to explore the possibility of localizing interacting particles not only in a MI state, but also in other insulating states. We explore in a controlled fashion the full parametric dependence of the ground state of interacting bosons in the presence of disorder. As we can effectively control the value of the effective tunneling matrix elements between the lattice sites we can move between the different states in the diagonal lines of the corresponding phase diagram $U/J$ vs. $W/J$ (i.e. we keep $U/W$ constant while changing the effective $U/J$). For this control through external periodic forces to be of any use, one should take into account the velocity of the ramping needed for the transition between states to take place. If one increases the amplitude of the external force adiabatically the system follows the eigenstates (cyclic states) of the Floquet Hamiltonian. We show that adiabatic behavior can be achieved for realistic parameters of our system.

We start at $t=0$ from the ground state of $H_{0}$ and switch on the time-dependent potential Eq. (3) assuming a discrete ramp $V_{m}=\Delta Vm$ with $m=1,\dots,N_{s}$. For each $V_{m}$ we solve the Floquet eigenvalue equation and obtain the corresponding quasienergies and cyclic states. Assuming that for each $V_{m}$ we let the system evolve during a whole period $T$ the evolved state reads

where $U$ is given by Eq. (8). We have checked that in the adiabatic limit we obtain similar results for a continuous ramping of the external potential. According to the generalized adiabatic principle for Floquet Hamiltonians of finite size, we consider that the evolution is adiabatic when the state $\left|\,\Psi(t)\right\rangle$ follows the same cyclic state during the ramping

for a particular $\alpha$. As we will see below, the variation of the quasienergies with potential amplitude is such that, in general, the levels become closer when $V/\omega\sim 2.4$ and the effective tunneling is coherently suppressed. In our calculations we have used only one ramping speed such that adiabatic evolution is assured in this region. Note, however, that the ramping could be faster initially for low values of $V$ and slowed down at the end. Calculations have been performed using the full spatial basis of $H_{0}$ and a truncated Fourier basis. The size of the Fourier basis is adjusted to assure convergence. We use the Arnoldi-Lanczos algorithm Lanczos to keep only the central eigenvectors with quasienergies $\varepsilon_{\alpha}$.

To keep the problem computationally tractable we restrict ourselves to systems of size $L=5$. The results shown below are typical examples that demonstrate that the coherent control of our system is feasible. We discuss the expected behavior for increasing system sizes and conclude that adiabatic following is expected to occur also for realistic experimental sizes. Note that for small system sizes, some possible phases of the disordered BH model may not appear Kruger09 . Also, we want to be cautious when identifying phases which occur in many-particle systems Fisher89 ; ReviewMaciej with the ground states of the Hamiltonian $H_{0}$ for small system sizes Burnett03 . Our small system ground states are just precursor of the phases that appear in a large particle system. In this work we are interested in the time-dependent phenomena and our aim is to show that one can move adiabatically between the ground states of the effective BH Hamiltonian for varying values of the parameters $U/J$ and $W/J$ with $U/W$ fixed. This would correspond to an adiabatic transition along the diagonal lines of the phase diagram $U/J$ vs. $W/J$ for a many-particle system of larger size.

All the results shown here correspond to parameters $\delta=0$, and $C=0.002$. To illustrate the effect of disorder in the system we present results for one realization of the random potential $\epsilon_{i}=\{0.6804,-0.4182,-0.8932,0.3686,-0.3248\}$. Any realization of disorder that completely breaks the symmetry of the system would yield similar adiabatic time scales although the precise value of the ramping time might depend on the exact form of the random potential. Specifically the behavior of the time scale with occupation number and disorder ratio $W/U$ do not depend on the actual realization of the random potential.

To characterize the state $\left|\,\Psi(t)\right\rangle$ we use the one-particle density matrix and the momentum distribution

where $Lk/(2\pi)=-M/2,..,M/2$. Both quantities evolve in time, as the fast oscillatory potential is ramped up.

For an ordered system ($W=0$) with commensurate filling, it has been shown theoretically Holthaus06 and experimentally Arimondo09 that the adiabatic ramping of the fast oscillatory potential in Eq.(3) can drive a SF-MI transition. In the presence of disorder, the reflection invariance of the system is broken, the properties of the states are different and there is no guarantee that the adiabatic time scales are within experimental reach. We show in fig. 1 the increasing complexity of the quasienergy levels with disorder strength. The breaking of symmetries makes the quasienergy spectrum more dense but strengthens the repulsion between energy levels.

We have shown that the adiabatic ramping of a constant force rapidly oscillating in time allows for the complete scan of the parametric dependence of the ground state of a disordered BH model. As shown previously Holthaus06 , the inclusion of an ac field in the Bose-Hubbard Hamiltonian is equivalent to a renormalization of the effective tunneling for the time independent spatial Hamiltonian. A succession of oscillatory fields where the amplitude is slowly increased gives rise in the adiabatic limit to a succession of effective Hamiltonians from some initial $J$ to the final $J_{\rm{eff}}=0$ and thus allows for a complete scan of the parameters $U/J$ and $W/J$ with $W/U$ constant.

We have found that in spite of the increasing complexity of the quasienergy diagram with disorder, adiabatic following of the ground state is feasible with a similar time scale as in a system without disorder. Adiabatic following of the ground state enables the transition from states with long-range coherence into localized states both for commensurate and non-commensurate fillings of the lattice potential. For strong disorder we have shown that the transition between insulating states localized in the most distorted regions of the lattice is possible. We have found that the addition of disorder shortens the time scale needed for adiabatic ramping if the disorder is weak or strong compared to the interaction strength. When the disorder is comparable to the interaction strength, adiabatic behavior is still feasible.

Our results indicate that this type of transitions between different regions of the phase diagram can be realized adiabatically for experimentally feasible time scales in systems with large number of particles. The adiabatic ramping of a fast oscillatory force could be used not only to control localization of disordered bosons but also to scan richer phase diagrams predicted for bosonic or fermionic systems in optical lattices.

This work was supported by the Spanish Government through projects FIS 2007-616866 and FIS2006-12783-C03-01. M R acknowledges support through the Ramon y Cajal programme. RAM’s contract is financed through the I3P by CSIC and the European Commission.