Transport Enhancement of Irregular Optical Lattices with Polychromatic Amplitude Modulation

We demonstrate that the transport characteristics of deep optical lattices with one or multiple off-resonant external energy offsets can be greatly-enhanced by modulating the lattice depth in an exotic way. We derive effective stationary models for our proposed modulation schemes in the strongly interacting limit, where only one particle can occupy any given site. Afterwards we discuss the modifications necessary to recover transport when more than one particle may occupy the lattice sites. For the specific five-site lattices discussed, we numerically predict transport gains for ranging from $4.7\times 10^6$ to $9.8\times 10^{8}$.


I. INTRODUCTION
Ultracold atoms in optical lattice potentials are very rich and diverse physical systems. Their coherent dynamics make them candidates for quantum logic gates [1,2]. Additionally, they have proven to be an excellent quantum simulators: not only has the mott-insulator to superfluid phase transition been realized [3], but modulation techniques such as 'lattice shaking' [4] have recently been used to realize the Haldane model [5] as well as the Meissner effect [6] in these systems.
It has also been proposed that atomtronics components [7][8][9] can be realized in optical lattices. The typical scheme involves customized lattices that exhibit certain transport characteristics. Holographic masking techniques [10] make it possible to realize such lattices, while the ability to image single atoms [10][11][12] makes signal detection feasible.
This article is concerned with optimizing the transport characteristics of arbitrary optical lattices. We examine the transport properties of these systems by driving their end sites with reservoirs of neutral, ultracold atoms. When a chemical potential difference exists between the reservoirs, atomic transport, or current, may be induced across the system. For example, when the reservoirs and the lattice are conditioned so that each system site can support at most one atom, the flat energy landscape depicted in Fig. 1(a) allows the atoms to freely explore the entire lattice. This configuration yields the optimal transport response. If the landscape has one or more external energy differences, as in Fig. 1(b), the current is greatly suppressed.
We demonstrate that the transport response of an arbitrary N-site lattice can be greatly enhanced by modulating the field intensity that generates the lattice structure. We analyze two different modulation schemes: one involving independent, site-by-site control of the lattice parameters, the other assuming a global modulation of the entire structure. In a regime where the external energy differences are large compared to the intrasystem tunneling, we demonstrate that these modulations enhance transport by mapping the stationary arbitrary system Hamiltonian that prohibits transport onto an effective stationary Hamiltonian that optimizes the transport (Color online) A one-dimensional optical lattice whose end sites are driven by ultracold atomic reservoirs. In the strongly-interacting case where the reservoirs are condition to put only one atom into the system, the energetically flat lattice depicted in (a) yields an optimal transport current response where the arbitrary lattice depicted in (b) yields virtually zero current.
response. In the single site occupancy case described above, the proposed schemes map a system such as the one illustrated in Fig. 1(b) onto an effective lattice of the type in Fig. 1(a).
The conditions for transport optimization turn out to depend on the relative strength of the reservoir drive to the on-site atomic repulsion. We will show that they change depending on the maximum number of allowed atoms on a given site. We examine the single occupancy case in depth, and follow with a discussion of current recovery in situations involving higher occupancy.
The results we present are general, and they can be applied to a variety of systems: Although we analyze the one-dimensional open bosonic case in this work, a variation of the presented schemes are valid for either bosons or fermions, trapped in open or closed quantum systems comprised in one-, two-, or three-dimensional optical lattices or possibly potentials generated in atom-chip experiments. They might find application in experiments where the control of a coherent atomic signal is desired. They may also be useful in the development of more arXiv:1501.01672v1 [quant-ph] 7 Jan 2015 complex atomtronics components [13], where atomic flow may be required across coupled lattice sites with energy mismatches. Finally, the modulation schemes might be useful as a spectroscopic tool: scanning through modulation frequencies of an arbitrary lattice can lead to spikes in the transport response, which would infer properties of the structure of the lattice.
The structure of this article goes as follows: in Sec. II we introduce the Hamiltonian and discuss our open quantum system theory, and define the observable that we refer to as 'transport current'. In Sec. III we analyze the single excitation case in depth. A unitary transformation is used to demonstrate the formal equivalence between stationary flat lattices and complex modulated arbitrary lattices. After we discuss an experimentally-realizable lattice modulation, we propose two transport-recovering modulation schemes: one assuming site-by-site control of the lattice parameters and the other assuming a global modulation of the lattice. By employing a secular approximation, we derive effective stationary models for both modulation methods. Numerical simulations are then provided to confirm and characterize the transport recovered by both schemes. In Sec. IV we discuss transport recovery when up to two atoms can occupy each lattice site. Extensions to higher site occupancies are straight-forward.

A. System Hamiltonian
The systems we consider in this article are ultracold bosonic atoms trapped in lattice potentials whose tunneling matrix element can be modulated. Working in the lowest Bloch band and in the tight-binding regime, the system dynamics are well-described by the Bose Hubbard model [14]. As long as the lattice is not modulated with frequencies that overlap with resonances that would promote the atoms to higher Bloch bands [15], the barriermodulated Bose Hubbard model can be written aŝ where ω j is the external site energy,â † j (â) creates (annihilates) a particle on site j, U is the on-site interaction energy and J j (t) is the time-dependent nearest neighbor tunneling matrix element between sites j − 1 and j.

B. Time-Dependent Born-Markov Master Equation
We study the transport properties of these onedimensional systems using the tools developed in [13], where the left and right ends of the lattice are coupled to reservoirs acting as sources and sinks of ultracold atoms as in Fig. 1. We use a quantum master equation approach [16,17] to model the dynamics of the Bose-Hubbard system coupled to reservoirs of ultracold neutral atoms. Working in the zero-temperature limit, if the on-site interaction U is large and repulsive, then the reservoir's level occupancy can be characterized by a chemical potential µ, such that all states below µ are occupied and all states above are vacant. A chemical potential difference between the left and right reservoirs, µ L > µ R , may induce an atomic current across the lattice from left to right. When U J(t), a realistic parameter regime that we assume throughout this article, the particle manifolds are well-separated. If the values of µ L,R do not resonantly overlap with any system eigenenergies, the care taken in Ref. [13] is not necessary, and we can employ the standard Born and Markov approximations [17][18][19]. Under these conditions, the value of µ L determines the maximum allowed particles on any given site. In our model, we assert the value of µ L by truncating the system's basis at the appropriate level.
If the modulation frequencies associated with J(t) are of the same order as the other frequencies associated with the system, the rapid decay of the memory kernel ensures that the time-dependence of the system has a negligible affect the Liouvilian. This generates a master equation of the standard Lindblad form [20]. For an N-site lattice where atoms are being pumped onto site 1 by a reservoir with chemical potential µ L , and removed from site N by a reservoir with chemical potential µ R = 0, our master equation model is where κ is the decay rate of the reservoirs, which we assume to be identical for convenience.

C. Transport Current
We define atomic transport, or current, in these systems to be the average number of particles leaving the system from the N th site:

III. SINGLE ATOM EXCITATIONS
We first consider cases where µ L is set to maintain an occupancy of one atom on the first site. In this situation, Eq. (1) becomeŝ and the creation and annihilation operators in Eq (2) also become Pauli spin-flip operators. To make the notation less cumbersome we adoptσ † j =σ (j)

A. Modulating the Optical Lattice
For these cases, the stationary optical lattice Hamiltonian that exhibits maximal transport,Ĥ max , corresponds to Eq. (4) with ω j = ω 0 . Performing a unitary transformation that rotates withĤ 0 = ω 0 Now consider a stationary N -site system defined by Eq. (4) with arbitrary external energies ω j . When energy differences between adjacent sites become large compared to their tunneling amplitudes, transport is greatly suppressed. Rotating this system witĥ yields where δ j = ω j − ω j−1 , is the energy difference between sites j and j − 1.
Notice that if each tunneling rate in Eq. (7) is modulated with then Eq. (7) is equivalent toĤ max definded by Eq. (5). The modulation assumed in Eq. (8) is ideal, since it fully recovers the maximal transport response from the arbitrary stationary lattice.

Experimentally-realizable modulation
Since the tunneling rate depends on the light intensity that makes up the lattice [21], a practical modulation could involve varying this intensity. These dynamics can be modeled with where ω j is the modulation frequency. It should be stated that altering the intensity of the light field also shifts the external energies, and on-site interaction energies of the sites. However, these shifts respectively are a linear and power power law dependent in the intensity whereas the variation of J is exponential [21,22]. Since only energy differences affect transport, these shifts can be neglected if their differences are small compared to J, or if they are global.

B. Optimization Schemes
We analyze the transport response obtained from optical lattices experiencing two different applications of Eq. (9). The first assumes one is able to modulate each tunneling rate J j , at a unique frequency independent of all of the others as in the ideal case above. Under these conditions, we show that a substantial amount of the optimal current can be recovered if each J j is modulated at the frequency δ j corresponding to the energy difference between the j and j − 1 sites. The Hamiltonian for this system iŝ (10) Due to the sub-micron spacing between adjacent optical lattice sites [23], this modulation might only find application in an atom-chip experiment, where the currents through nanowires which generate the magnetic field that trap atoms can be manipulated independently from one another.
The second scheme involves a global modulation of the lattice. If there are M ≤ N unique, nonzero values of δ j , we show that a great deal of the system current can be recovered if every tunneling rate is modulated with a superposition of the M unique frequencies. The corresponding Hamiltonian beinĝ

C. Effective Stationary Models
Assuming a natural separation of parameters where the energy gaps δ j are large compared to the tunneling rates J j , stationary effective flat lattice models can be generated that accurately reproduce the dynamics of Eqs. (10) and (11). These effective models are significantly less demanding to numerically evolve. They also shed light on the transport recovery mechanism for both cases.

Site-by-site modulation
Assuming uniform tunneling J for simplicity, Eq. (10) in a frame rotating with Eq. (6) becomes (12) where we have expressed the cosines as exponentials.
When two adjacent sites have the same energy, the effective tunneling matrix element between these sites is simply J. When δ j = 0, and J |δ j |, a secular approximation similar to the rotating wave approximation can be made: for timescales large compared to max{1/|δ j |}, the oscillating terms in Eq. (12) become negligible compared to their stationary counterparts. Neglecting these terms yields an effective stationary tunneling matrix element that is reduced by a factor of 4.
That is, the dynamics of Eq. (10) are well-described by the flat stationary Hamiltoniañ where

Global lattice modulation
Assuming the global modulation from Eq. (11), a rotation with Eq. (6) implies Applying the same argument above, when δ j = 0 this time, the only stationary term is J ×M/2. While for δ j = 0, the only stationary term is J/4 as in the previous case. Therefore, the outcome is a flat stationary Hamiltonian added to a host of rotational pieces that can be neglected under the condition J δ j , |δ j − δ k |. For timescales large compared to max{1/|δ j |, 1/|δ j − δ k |}, the stationary effective Hamil-tonian is once again Eq. (13), but now with

D. Numerical Results
We now provide a series numerical simulations that quantify and confirm transport recovery, as well as the accuracy of the effective models for both proposed modulation schemes.

Two site lattices
We begin our analysis by deriving the current response for an arbitrary stationary two-site lattice defined bŷ This result provides benchmarks for the modulated lattices that follow. When Eq. (17) inserted into the Pauli equivalent of Eq. (2), this model forms a closed coupled system of equations involving σ † 1σ 1 , σ † 2σ 2 , σ † 1σ 2 and σ † 2σ 1 . In steady-state, the current response as a function of the δ 2 becomes with a maximum value of For a lattice with J/κ = 10, an energy offset of δ 2 /κ = 10 3 reduces the transport to ∼ I max /2, 500. We now consider the modulated system defined bŷ inserted into Eq. (2). Using the numerical values for J, δ 2 and κ just defined, Fig. 2 presents a numerical simulation of the steady-state current response of this open quantum system as a function of ω. The blue solid curve corresponds to the modulated open quantum system, while the magenta dashed curve is Eq. (19) for comparison. The system only has an appreciable current response when it is modulated with a ω δ 2 . Furthermore, although the system is modulated with Eq. (9) instead of the ideal modulation of Eq. (8), the current response is almost at its maximum value for this particular parameter regime.
Next, we fix the modulation frequency to be resonant with the energy offset, ω = δ 2 , and simulate the steadystate current response of Eq. (20) in Eq. (2) as a function response of the modulated system. The red solid curve is the ideal response determined by Eq. (19). The magenta dashed curve is the current response of the effective stationary model Eq. (13). The current obtained in the modulated system is always less than the ideal current for the flat lattice, but is nearly equal to the current of the effective Hamiltonian or all values of J.
In an experiment, it might not be advantageous or possible to vary the tunneling from some maximal J to zero. Next we examine the current response as a function of the deviation ∆J from J. The modulated Hamiltonian we simulate iŝ which converges to Eq. (20) as ∆J → J. These simulations assume δ 2 /J = 10 2 , and J/κ = 10. of the maximal current is recovered when ∆J = J/4, and over 90% when ∆J = J/2. This shows that even small amplitude changes can lead to significant current gains.

Many Site Disordered Lattices
In this section, we numerically verify transport recovery due to both modulation schemes presented in Sec. III B, as well as demonstrate the accuracy of the effective stationary models from Sec. III C for a variety of five-site disordered lattices. Assuming uniform tunneling, the Hamiltonian for the arbitrary, unmodulated system isĤ Consider the system depicted in Fig. 5(a) whose lattice parameters are ω j /J = [1, 2, 0, 0, 4] × 10 2 . This lattice has three unique energy gaps and one nearest neighbor degeneracy. When coupled to reservoirs whose decay rates satisfy J/κ = 10, this lattice exhibits a steady-state current response that is reduced by a factor of 4 × 10 10 compared to the maximum response determined by its flat counterpartĤ The effective flat, stationary models for the dynamics of this lattice under site-by-site and global modulations are illustrated in Figs. 5(b) and 5(c) respectively. A nu- merical simulation of the time evolution of this gapped lattice under site-by-site and global modulations is provided in Fig. 6. The recovered current for both cases is normalized by the steady-state maximal current response determined by Eq. (23). In steady-state, the site-by-site modulation recovers 20% of the maximum possible current, whereas the global modulation recovers 10%. This corresponds to a respective current gain of roughly 9 × 10 9 and 4 × 10 9 . The effective stationary models are also plotted in this figure. For times large compared to timescales set by the energy differences of this system (t ∼ 1 in the plot), the unique dynamics of both modulations are accurately recovered.  Fig. 5(a) under the site-by-site (green solid) and global (orange solid) modulation schemes. The Numerical evolution of the stationary effective Hamiltonia for the site-by-site (red dashed) and global (blue dashed) are also provided. Both modulation schemes increase the lattice's current response greater than a factor of 10 9 . Although the dynamics for both modulation cases are Noticeably different, the stationary effective models very accurately capture the evolution for all times greater than t = 1.
The steady-state current gains and accuracy of the cor-responding effective models for a variety of five-site lattices are provided in table I. The characteristics of the Hamiltonian described above are presented in the third row of this table. In all four cases, the atomic current is largely recovered and the stationary models accurately predict the long-time dynamics of each system under both site-by-site and global modulations.

IV. MULTIPLE ATOM EXCITATIONS
The optimization condition changes when the left reservoir is set to maintain an occupancy greater than one particle on the left site. We close considering a twosite lattice with a finite energy offset δ 2 , where µ L is set to maintain an occupancy of two atoms on the first site. Assuming Eq. (9) as the modulating function, the system Hamiltonian we consider iŝ For the numerical simulations presented in this section we assume δ 2 /J = 5 × 10 2 , J/κ = 10 and U/J = 10 2 . The Numerical steady-state current response of Eq. (24) in Eq. (2) as a function of ω is presented in Fig. 7. In the previous section, the current was greatly enhanced , leads to a current response that is very small compared to the response one obtains when modulating the lattice at δ2 + U .
when the system was modulated at the frequency of the gap δ 2 . Here, when ω = δ 2 , the current merely increases by a factor of 21. However, when ω = δ 2 − U , a current gain of 1.7 × 10 4 is observed. The effective stationary Hamiltonian corresponding to Eq. (24) with ω = δ 2 − U The percent difference between current predicted by this effective model and the full modulated system is 0.63%.
The reason for the change in the optimization condition can be understood by examining how the energetic relationships of the unmodulated system's Fockstates effectively change when the system is modulated at frequencies δ 2 and δ 2 − U . Figure 8 depicts the bare Fock energies of the unmodulated, energy-offset lattice. In the absence of any modulation, there are no system degeneracies, and the action of the reservoirs evolve an arbitrary system almost completely into the |2, 0 state.  This establishes a resonance between |1, 0 and |0, 1 , as well as |2, 1 and |1, 2 . When restricted to the single particle excitation case as in Sec. III, this condition optimizes transport since the action of the reservoirs drives population cycles through the lowest four states in the diagram. However, when µ L is set to maintain an occupancy of two particles, the energy mismatch between |2, 0 and |1, 1 assures that, regardless of the system's initial condition, the action of the reservoirs evolve the vast majority of the system population into the |2, 0 state. Current may leave the system, but only via a second-order, off-resonant transition from |2, 0 → |0, 2 . Such a transition only leads to the relatively small current gain seen in Fig. (7) when ω 1.25(δ 2 + U ). δ2 +U . The blue arrows represent actions of the left reservoir, the red arrows represent actions of the right reservoir, while the green arrows represent intrasystem transitions. As seen in (a), modulations with δ2 create an effective resonance between |1, 0 , and |0, 1 . However, the energetic missmatch between |2, 0 and |1, 1 prohibits effective transport when the reservoir is driving the |2, 0 state. (b) Alternatively, if the system is modulated with δ2 +U , the resonance between |2, 0 and |1, 1 recovers the maximal transport response for this system. Figure 9(b) shows the energetic relationship of the Eq. (25), the effective Hamiltonian associated with modulating the system at δ 2 − U . This modulation makes the system in question equivalent to the forward-biased atomtronic diode introduced in Ref. [7], but with J reduced by a factor of 4. The resonance between the |2, 0 and |1, 1 states assures a large transport response via cycles between the |1, 0 , |2, 0 , |1, 1 and |2, 1 states.
This section confirms that transport in higher particle manifolds is possible, but care needs to be taken to ensure that the correct resonance conditions are met.

V. CONCLUSION
In this article, we have demonstrated that modulating the tunneling barriers of a lattice structure can greatly enhance the transport characteristics that are otherwise suppressed by energetic mismatches in the system. Through a secular approximation, we have shown that the modulation techniques work by taking a stationary lattice that prohibits transport, and maps it onto an effective stationary lattice that supports transport. The bulk of the presented results were concerned with verifying current recovery for the case where at most one atom was allowed on any given site. In this regime, we examined the current response as a function of the modulation intensity, and verified our proposed modulation schemes and effective models for a variety of two-and five-site lattices. Afterwards, we showed that current in higher particle manifolds can also be recovered. However, the conditions for optimizing the transport changes: the modulation frequencies need to be chosen to create the correct effective resonances across particular particle manifolds in the system.