# Coherent control of atomic transport in spinor optical lattices

**ABSTRACT** Coherent transport of atoms trapped in an optical lattice can be controlled

by microwave-induced spin flips that correlate with site-to-site hopping. We

study the controllability of homogeneous one-dimensional systems of

noninteracting atoms in the absence of site addressability. Given these

restrictions, we construct a deterministic protocol to map an initially

localized Wannier state to a wave packet that that is coherently distributed

over n sites. This is extended to analytic solutions for arbitrary unitary maps

given homogenous systems and in the presence of time-dependent uniform forces.

Such control is important for applications in quantum information processing

such as quantum computing and quantum simulations of condensed matter

phenomena.

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**ABSTRACT:**Resource tradeoffs can often be established by solving an appropriate robust optimization problem for a variety of scenarios involving constraints on optimization variables and uncertainties. Using an approach based on sequential convex programming, we demonstrate that a substantial fidelity robustness is obtainable against uncertainties while simultaneously using limited resources of control amplitude and bandwidth. What is required is a specific knowledge of the range and character of the uncertainties, a process referred to in the control theory literature as "uncertainty modeling." Using a general one-qubit model for illustrative simulations of a controlled qubit, we generate robust controls for a universal gate set. Our results demonstrate that, even for this simple model, there exist a rich variety of control design possibilities.Physical Review A 01/2013; · 3.04 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We present theoretical and experimental studies of the decoherence of hyperfine ground-state superpositions due to elastic Rayleigh scattering of light off resonant with higher lying excited states. We demonstrate that under appropriate conditions, elastic Rayleigh scattering can be the dominant source of decoherence, contrary to previous discussions in the literature. We show that the elastic-scattering decoherence rate of a two-level system is given by the square of the difference between the elastic-scattering amplitudes for the two levels, and that for certain detunings of the light, the amplitudes can interfere constructively even when the elastic-scattering rates from the two levels are equal. We confirm this prediction through calculations and measurements of the total decoherence rate for a superposition of the valence electron spin levels in the ground state of 9Be+ in a 4.5 T magnetic field.Physical Review Letters 11/2010; 105(20):200401. · 7.73 Impact Factor - SourceAvailable from: Andrea Alberti[Show abstract] [Hide abstract]

**ABSTRACT:**Spin-dependent optical potentials allow us to use microwave radiation to manipulate the motional state of trapped neutral atoms (F\"orster et al. 2009 Phys. Rev. Lett. 103, 233001). Here, we discuss this method in greater detail, comparing it to the widely-employed Raman sideband coupling method. We provide a simplified model for sideband cooling in a spin-dependent potential, and we discuss it in terms of the generalized Lamb-Dicke parameter. Using a master equation formalism, we present a quantitative analysis of the cooling performance for our experiment, which can be generalized to other experimental settings. We additionally use microwave sideband transitions to engineer motional Fock states and coherent states, and we devise a technique for measuring the population distribution of the prepared states.Journal of Physics B Atomic Molecular and Optical Physics 02/2013; 46(10). · 2.03 Impact Factor

Page 1

Coherent control of atomic transport in spinor optical lattices

Brian Mischuck∗and Ivan H. Deutsch

University of New Mexico

Poul S. Jessen

University of Arizona

(Dated: May 7, 2009)

Abstract

Coherent transport of atoms trapped in an optical lattice can be controlled by microwave-induced

spin flips that correlate with site-to-site hopping. We study the controllability of homogeneous

one-dimensional systems of noninteracting atoms in the absence of site addressability. Given these

restrictions, we construct a deterministic protocol to map an initially localized Wannier state to a

wave packet that that is coherently distributed over n sites. This is extended to analytic solutions

for arbitrary unitary maps given homogenous systems and in the presence of time-dependent uni-

form forces. Such control is important for applications in quantum information processing such as

quantum computing and quantum simulations of condensed matter phenomena.

1

arXiv:0905.1094v1 [quant-ph] 7 May 2009

Page 2

I. INTRODUCTION

Neutral atoms trapped in optical lattices have emerged as a rich platform for exploring

a wide variety of phenomena and devices based on coherent quantum dynamics. Examples

include quantum computers [1, 2, 3, 4], quantum simulators of condensed matter [5, 6], topo-

logical quantum field theory [7, 8], and quantum chaotic dynamics [9, 10, 11]. An essential

ingredient in these systems is the coherent control of atomic transport in the lattice. Such

transport is driven by time-dependent variations in the lattice potential and the application

of external fields. In its most basic form, the atoms’ ballistic tunneling between sites in

a sinusoidal potential can be controlled through time-dependent modulations of the lattice

depth and phase. The latter can be used to impart a time-dependent acceleration to the

lattice, thereby simulating the effects of an applied electric field for electrons in a crystal that

give rise to the fundamental paradigms of coherent transport in solid-state physics. Bloch

oscillations [12], Wannier-Stark ladders [13], Landau-Zener tunneling [14], and dynamical

localization [15] have all been demonstrated in optical lattices and explored as mechanisms

for coherent control.

More complex lattice geometries introduce additional features. For example, in a lattice

of double-wells, one can drive transport between sites in a pairwise manner, assuming a

sufficient barrier to ignore tunneling between different double-wells [16, 17, 18, 19]. In

this case, the control problem is substantially simplified, as the relevant Hilbert space in

a given time interval is restricted to a small discrete set of energy levels, as opposed to

the infinite chain of levels in a sinusoidal lattice. Control across the entire lattice can be

implemented by modifying the geometry so that the wells are alternatively coupled to all

nearest neighbors (left or right in 1D). Designer double-well lattices have been explored for

quantum information processing tasks such as quantum computing [20] and simulations of

condensed-matter phenomena [21].

Still richer control is possible for spinor lattices where the optical potential depends on

the atom’s internal spin state [22]. The lattice’s morphology can now be modified through

variation of the laser polarization as well as intensity, lattice phase, etc. The earliest pro-

posals for quantum logic in optical lattices via controlled collisions involved transport of the

atoms via time-dependent rotation of the direction of a laser beam’s polarization [1, 2, 23].

An alternative and perhaps more robust route to coherent control of atomic transport is to

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use external fields to drive spin-changing transitions that are correlated with atomic motion,

similar to the scheme proposed by Foot et al. [24]. Such protocols can make use of the tools

for robust control of spins [25, 26, 27], as developed in NMR, to the control of atomic motion

in the lattice.

In this article we explore methods for coherent control of atomic transport with

microwave-induced spin rotations between hyperfine levels and polarization-gradient lat-

tices. Our main focus is on controllability – how the Hamiltonian that governs the dynamics

restricts the possible unitary maps that one can implement, and how to design specific wave-

forms to carry out a given task. We will consider here the simplest problem of noninteracting

atoms in one dimension. While extensions to the interacting case are nontrivial, the current

work is an important stepping-stone in that direction.

The remainder of the paper is organized as follows. In Sec. II we establish the formalism

necessary to describe spinor lattices and their interaction with external fields. We apply

this to study the conditions for wave function control (the preparation of a desired spinor

wave function starting from a known localized Wannier state) and prescribe a constructive

algorithm for carrying out this task in Sec. III. We then generalize this in Sec. IV to the

case of more general unitary maps for unknown initial states. Finally, we summarize and

give an outlook towards future research in this area in Sec. V.

II. MICROWAVE-DRIVEN SPINOR LATTICES

Spinor lattices arise from the tensor nature of atom-photon interaction. In a monochro-

matic laser field Re?E(x)ei(φ(x)−ωLt)?, the light-shift potential takes the form,

VLS(x) = −1

4αijE∗

i(x)Ej(x),(1)

where αij is the atomic dynamic polarizability at frequency ωL, for atoms in a particular

ground-state manifold. We consider a 1D geometry consisting of counterpropagating laser

beams with linear polarizations, forming a relative angle θ (the “lin-θ-lin” geometry). Re-

stricting our attention to alkali atoms, when the laser field detuning is large compared to

the excited-state hyperfine splitting but not large compared to the fine-structure splitting,

ellipticity in the laser field leads to a fictitious magnetic field that varies periodically in

space, and results in a spin-dependent light shift [28]. Taking the quantization axis along

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one of the laser beams’ wave vectors, the spinor light-shift potential can then be written

VLS=

?

degenerate

F,mF

[EF− VF,mFcos(2kLz + δF,mF)]|F,mF??F,mF|, (2)

where

?

tice depth for lin-?-lin and gF is the Land´ e g-factor of the hyperfine manifold under

consideration.

EF

isthe energyof a hyperfinemanifold,VF,mF

=

V0

cos2θ + (gFmF/2)2sin2θ, and δF,mF= tan−1(gFmFtanθ/2).Here, V0 is the lat-

The addition of a static bias magnetic field breaks the degeneracy between Zeeman sub-

levels within a manifold and allows us to spectrally isolate different microwave transitions

between the manifolds. For concreteness, we consider133Cs, and choose a two-state sub-

space |F = 4,m = 3? ≡ |↑?,|F = 3,m = 3? ≡ |↓? to define a pseudo-spin-1/2 particle. Re-

stricting to this subspace, and adding a near-resonant microwave field with magnetic field

Bµwcos(ωµwt − φ) that couples these spin states, the Hamiltonian in the rotating frame

takes the form H = Hlatt+ Hµw, where

Hlatt=

p2

2m− V0cos(2kLz + δ0)| ↑??↑ | − V0cos(2kLz − δ0)| ↓??↓ |

Hµw= −∆µw

22

(3a)

σz−Ωµw

(cosφµwσx+ sinφµwσy).(3b)

Here V0= V4,3≈ V3,3, δ0= δ4,3≈ −δ3,3, Ωµw= ?↑| ˆ µ|↓?Bµw/¯ h is the microwave resonant

Rabi frequency, ∆µwis the microwave detuning from a hyperfine resonance defined by the

untrapped atoms, φµwis the phase of the microwave oscillator, and the Pauli-σ operators

are defined relative to the pseudo-spin.

Neglecting the kinetic energy and diagonalizing the Hamiltonian leads to adiabatic or

microwave-dressed potentials ,

V±(z) = −V0cosδ0cos(2kz) ±1

2

?

(2V0sinδ0sin(2kz) − ∆µw)2+ Ω2

µw.(4)

At ∆µw= 0, in the lin⊥lin (θ = π/2) geometry, the adiabatic potentials yield a period λ/4

“subwavelength” lattice [29, 30, 31]. In the context of a Hubbard Hamiltonian describing

interacting particles moving on a lattice [32], this configuration gives us greater freedom to

independently control the site-to-site tunneling rate J and the onsite interaction strength U

[33]. By employing both optical and microwave fields, the lattice depth dominates the control

of U while the the applied microwave dominates control of J. Moreover, the tunneling matrix

4

Page 5

element is complex, set by the microwave phase, allowing for time-reversible tunneling and

further control [34].

For θ ?= nπ/2, the adiabatic potentials take the form of a lattice of double-well potentials

arising from the asymmetry for transport to the left vs. the right. The parameters char-

acterizing the double well, including barrier height, tunneling matrix element, and energy

asymmetry (“tilt”), can be controlled through variations of lattice intensity/polarization,

microwave power, and detuning. The richness of this system should enable us to control

wave function coherence for spinors over multiple sites. Our early work on this subject

demonstrated spinor double-well coherence driven by Larmor precision in a quasistatic mag-

netic field [22]. The current approach, based on applied microwave fields should be much

more robust and controllable.

While the dressed-lattice adiabatic potentials guide intuition about the transport, quan-

titative predictions are more accurately made by considering the band structure of the

Hamiltonian, Hlatt, in Eq. (3). Associated with the spin s =↑ and s =↓ lattices are Bloch

states for band-n and quasimomentum-q, |ψ(s)

site-l, |φ(s)

?1/2

n,q?, and Wannier states for that band and lattice

n,l?, related by the usual Fourier transform over the first Brillouin zone,

|φ(s)

n,l? =

−1/2

ei2πlq|ψ(s)

n,q?dq.(5)

Here and throughout, lengths are measured in units of the lattice period L = λL/2 and wave

numbers in units of the reciprocal lattice vector K = 4π/λL. For sufficiently deep lattices

and atoms in the lowest lying bands, tunneling between sites is completely negligible over

the timescales of interest. In that case, the lattice Hamiltonian is diagonal both in the Bloch

and Wannier bases, with no energy variation over the q or l index.

Transport dynamics are driven by the microwaves tuned to cause transitions between

the ground bands associated with the spin-up and spin-down lattices. We assume that the

detuning and Rabi frequency are sufficiently small so that single-band/lattice tight-binding

(TB) model is a good approximation. Henceforth we drop the band index and set n = 0. In

the Wannier basis, the total Hamiltonian in the TB approximation is

HTB=

∞

?

l=−∞

−∆µw

2

σl

z−1

2

?

eiφµw?

ΩRσl,R

+ + ΩLσl,L

+

?

+ h.c.

?

,(6)

5

Page 6

where,

σl

z≡ |l,↑??l,↑| − |l,↓??l,↓|,

σl,R

(7a)

+ ≡ |l,↑??l,↓|,

σl,L

(7b)

+ ≡ |l − 1,↑??l,↓|

(7c)

are the Pauli operators for two-level transitions that pairwise couple spin-down Wannier

states to their neighbors on the right, |l,↓? → |l,↑?, and on the left |l,↓? → |l − 1,↑?.

Note, we have chosen an arbitrary labeling of the Wannier state indices by convention so

that a spin-down state and spin-up state to its right are both associated with the same

lattice period label, l. Because the microwaves transfer negligible momentum to the atoms,

translation of the atomic wavepacket is possible only when the probability amplitude of an

atom overlaps between neighboring sites. The Rabi frequencies for transitions to the left or

right are thus weighted by Franck-Condon factors, ΩR= ?φ↑

For the ground bands in the TB approximation, a large asymmetry in right-left transport

l|φ↓

l?Ωµw, ΩL= ?φ↑

l−1|φ↓

l?Ωµw.

and isolation of double wells arises from small asymmetry in right-left displacement of the

lattice due to the Gaussian overlap of the wavepackets (see Fig. 1).

The combination of spinor optical lattices and microwave-driven spin dynamics provides

a wide variety of parameters that can be modulated in real time during an experiment to

coherently control atomic transport. In the next section we study the formal controllability

of this system and develop constructive protocols to implement desired unitary maps.

III. WAVEPACKET CONTROL

Given the time-dependent Hamiltonian at hand, a first question to address is “controlla-

bility”, i.e., which class of unitary transformations can be generated by the arbitrary design

of the waveforms that parameterize that Hamiltonian. We consider first the problem of

preparing an arbitrary wavepacket that is coherently distributed over multiple sites of the

lattice, starting from an initially localized Wannier state. Unless specially designed to allow

for individual site addressability [4, 35], control in a typical lattice is limited by translational

invariance of the operations. Although a linear gradient breaks the translational symmetry,

the controllability of the system is still limited, as we show below. We thus restrict our

attention to strictly periodic lattices and allow for an additional constant force F on the

6

Page 7

z

z

€

ψ↑,↓(z)

2

€

U↓z ( )

€

U↓z ( )

z

z

€

ψ↑,↓(z)

2

€

U↓z ( )

€

U↓z ( )

FIG. 1:Controlled transport in lin-θ−lin. Spin-dependent lattices have a relative phase shift

due to polarization gradient. In the lin⊥lin configuration, θ = 90◦, there is no asymmetry in

right-left transport, and atoms can ballistically tunnel in the dressed potential. For a change of

just 10◦degrees away from lin⊥lin, in a lattice with an oscillation frequency of 20ER, the ratio of

the effective tunelling rates is ΩL/ΩR≈ 3500. The result is lattice of double wells with pair-wise

tunneling couplings.

atoms, as in the case of lattices held vertically in gravity, or when the overall lattice is

accelerated through time-dependent changes of the standing wave pattern.

We consider Hamiltonians which are composed of a translationally invariant part, H0,

with period L, and an applied force, F,

H(t) = H0(t) + F(t)x(8)

A particular example is H0(t) = HTB(t), as given in Eq. (6), with time-dependent variations

in microwave power and/or phase. The time evolution of such a system may be written in

the form of the time-ordered exponential

U(t) = T exp

?

−i

?t

0

(H0(t?) + F(t?)x)dt?

?

,(9)

Such a map has the property that if we translate the entire system by j times the period of

H0, then U → e−iRt

0F(t?)dt?jLU. If the initial state |φ? is a localized Wannier state, it satisfies

?φ|Tj|φ? = δj,0,(10)

7

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where Tjtranslates the system by jL. If state |φ? maps to |φ?? under a unitary evolution of

this form, then the evolved state satisfies this same condition, as follows from the identity

?φ?|Tj|φ?? = ?φ|U†TjU|φ? = eiRt

0F(t?)dt?jL?φ|Tj|φ? = δj,0.(11)

Thus any wavepacket prepared by these controls must be orthogonal to itself after a trans-

lation by an arbitrary number of periods. Furthermore, since the force F has dropped out,

the linear gradient does not impact the range of states that my be reached.

Are all states that satisfy this constraint reachable through some choice of the control

waveforms that parameterize HTBin Eq. (6)? To show that this is the case, we employ a

protocol for constructing a desired state-to-state mapping as defined by Eberly and Law in

the context of Jaynes-Cummings ladder [36]. First note that if we can map the state |φ? to

another state |φ??, and the control Hamiltonian allows the unitary map to be time-reversible,

then we can map |φ?? to |φ?. Thus, in order to show that we can get from a localized state

to any state satisfying Eq. (11), we consider the time-reversed problem of mapping such a

state to the initial Wannier state.

To construct the desired map, we employ a series of SU(2) rotations on resolvable sub-

spaces of the total Hilbert space. Such a collection of disjoint two-level systems can be

addressed in a lin-θ-lin spinor lattice in either an asymmetric configuration (θ ?= π/2), or

in a lin⊥lin configuration in the presence a sufficiently strong uniform force so that isolated

pairs of states are spectroscopically addressable (see Fig. 2). Note that in the latter case,

because of the presence of an external force, if we translate the system by an integer mul-

tiple of λ/2, the state picks up an extra phase due to the linear gradient. As we will see

below, our construction does not require these phases, and since the construction is capable

of synthesizing all reachable states, the phases are redundant. We can ignore the phases if

we choose the time over which the two level unitaries operate to be an integer multiple of

2π/FL, and we will assume this to be the case for the rest of this section.

We will restrict our attention to states with support strictly on a finite set of lattice sites

and zero probability amplitude outside some range. Such spinors can be represented as

|ψ? =

lmax

?

l=lmin

(cl↓|l ↓? + cl↑|l ↑?).(12)

The finite extent of the wave function, together with the condition expressed in Eq. (11),

places a constraint on the two-level subspaces. If we translate the entire state by lmax−lmin

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a.

U(z)

U(z)

c.

U(z)

U(z)

d.

U(z)

U(z)

b.

U(z)

U(z)

z

z

z

z

FIG. 2: Two different methods for isolating unitary maps on two-level subspaces. In (a) and (b),

the choice of polarization angles in a lin-θ-lin lattice isolate different sets of two-level systems with

transport either to the right or to the left. In (c) and (d), two-level systems in a lattice with a

linear gradient are spectrally addressed through their distinct microwave transition frequencies,

which differ by ±FL/2¯ h

then,

?ψ|Tlmin−lmax|ψ? = c∗

lmin↑clmax↑+ c∗

lmin↓clmax↓= 0.(13)

Thus, in order to satisfy Eq. (11), the two outermost two-level subspaces must be orthogonal.

Moreover, because of translational invariance, the unitary transformations that we apply are

equivalent at each period of the lattice. In particular, by unitarity, if we apply a rotation

operator that maps the subspace on the left end of the atomic distribution to pure spin up,

the subspace at the right end of the distribution must be rotated to pure spin down.

These observations are the core of our construction (see Fig. 3). A sequence of two-level

rotations can be used to map a coherent superposition delocalized across the lattice to one

localized at a single site in a single spin state. In the first step, a rotation is applied to

map all population at the left-most two-level system (l = lmin) to spin-up according to the

microwave-driven SU(2) transformation,

S =

1

√N

c∗

lmin,↑

clmin,↓ −clmin,↑

c∗

lmin,↓

,(14)

where N = |clmin,↑|2+ |clmin,↓|2. By the translation symmetry this simultaneously maps

9

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Step

1

Ini*al

State

Step

2

Final

State:

Step

3

Step

4

U(z)

U(z)

z

U(z)

U(z)

z

U(z)

U(z)

z

U(z)

U(z)

U(z)

U(z)

z

z

U(z)

U(z)

z

U(z)

U(z)

z

FIG. 3:An example of state preparation through sequential SU(2) rotations. The polarization

of the two counterpropagating beams is chosen to isolate sets of two-level systems. In step 1,

population in the leftmost two-level system is mapped to entirely spin up, forcing population in

the right most two level system to be entirely spin down by Eq. (13). In step 2, the polarization

is rotated so that a different set of two level systems are coupled. In step 3, the population in the

leftmost level is then mapped to entirely spin down. In step 4 the polarization is set to its original

configuration and steps 1-4 are repeated until the state is localized. For this particular case, we

have chosen to end the sequence spin down. A different choice for the final pulse would have ended

the state spin up.

all population at the right-most two-level system (l = lmax) to spin-down, resulting in the

spinor state

|ψ?? = c?

lmin↑|lmin↑? +

lmax−1

?

l=lmin+1

?c?

l↓|l ↓? + c?

l↓|l ↓??+ c?

lmax↓|lmax↓?,(15)

thereby shrinking the extent of the wave function by λ/2. The lattice is then reconfigured

10

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(either through polarization rotation, a change of microwave frequency, or a change in ac-

celeration) so that the opposite neighbors are coupled (spin-down now coupled to spin-up

neighbor on left). An appropriate SU(2) rotation is then applied to map all population on

the left most edge to spin down (simultaneously moving all population on right-most edge

to spin-up). Repeating, we form a sequence of rotations that take the outer edges of the

distribution and map them inwards in steps of λ/4 until all the population is localized at

one Wannier state. Reversing the order of the sequence provides the desired protocol for

constructing any single particle wave function in the given band and with support on a finite

number of lattice sites, subject to the constraint Eq. (11).

IV. IMPLEMENTING GENERAL UNITARY TRANSFORMATIONS

In the previous section we studied the construction of a particular unitary transformation

— mapping an initially localized Wannier state to a spinor wave function delocalized over a

finite number of lattice sites, under the constraint of Eq. (9). In this section we consider the

most general unitary map that we can implement under these constraints. We begin with

the case of perfect translational invariance of the lattice. In addition, since the microwave

photons possess negligible momentum, only Bloch states with the same quasimomentum are

coupled. Because of these symmetries, any unitarity transformation that we can synthesize

will be block diagonal in the Bloch basis, U = ⊕qUq, where each block is a U(2) matrix

that connects spin-up and spin-down states with the same quasimomentum q. In the basis

{|q,↑?,|q,↓?},

Uq= eiγ(q)

α(q) β∗(q)

β(q) −α∗(q)

, (16)

where |α(q)|2+ |β(q)|2=1. If the two lattices are sufficiently deep, tunneling is suppressed

so γ is independent of q and can be factored out of the problem, leading to SU(2) rotations

in each block.

The decomposition of a translationally invariant unitary transformation into a direct sum

of SU(2) matrices has important implications for the design of arbitrary maps. Generally the

design of a time-dependent waveform that generates an arbitrary unitary map is substantially

more complex than a protocol for state-to-state mapping on an initially known state [37].

Intuitively, this is because state-to-state maps only constrain one column of a unitary matrix,

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whereas the evolution of the orthogonal complement is not fully specified. The exception is

for a spin-1/2 system. By unitarity, specifying one column of an SU(2) matrix necessarily

constrains the other. Since our spinor lattice is described by a collection of noninteracting

spin-1/2 subspaces labeled by quasimomentum q, if we specify a state-to-state mapping of a

spinor Bloch state, we specify the SU(2) matrix on this block. We can achieve this using the

state-mapping protocol defined in Sec. III that takes an initially localized Wannier state to a

state distributed over a finite number of lattice sites. Such a map specifies a transformation

on each Bloch state according to the Fourier relationship between the probability amplitudes

in the Wannier and Bloch bases. Based on this relationship, we can use our state-to-state

map to design a more general class of unitary maps on the wave function.

To see this explicitly, consider the unitary evolution of an initial spin-up Wannier state

(take l = 0 without loss of generality),

U |0,↓? =

?1/2

∞

?

−1/2

dq (α(q)|q,↑? + β(q)|q,↓?)

=

l=−∞

(cl,↑|l,↑? + cl,↓|l,↓?). (17)

The quasimomentum functions α(q) and β(q) in Eq. (16) are the Fourier sums of probability

amplitudes in Wannier space,

α(q) =

∞

?

l=−∞

cl,↑e−i2πlq, β(q) =

∞

?

l=−∞

cl,↓e−i2πlq.(18)

As long as the Fourier transform of α(q) and β(q) have support only over a finite extent in

l, we can generate these functions by applying the state-mapping protocol of the previous

section to synthesize the probability amplitudes clin Eq. (18). For unitary maps defined

by α(q) and β(q) whose Fourier expansion in Wannier states does not have a strictly finite

support, more general control methods are required.

We can easily generalize our result to include control through applied spatially uniform

(possibly time-dependent) forces. In the TB approximation, expressed in the Wannier basis,

the Hamiltonian for a linear gradient potential in dimensionless units takes the form

Hgrad(t) =

∞

?

l=−∞

F(t)L[l|l ↓??l ↓ | + (l + δl(t))|l ↑??l ↑ |],(19)

where δl(t) arises due to the off-set between spin-up and down lattices. We allow for mod-

ulations of the overall force as studied in “shaken lattices” [38, 39, 40, 41, 42], and the

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Page 13

possibility of time-dependent variations in the relative positions of the two spin states, as

could be implemented through modulations in the laser beams’ polarization direction. The

combination of this Hamiltonian, together with microwave-driven control described by HTB

in Eq. (6), gives rise to a general unitary transformation that can be written using the

interaction picture in the form,

U(t) = D(t)UI(t), (20)

where,

D(t) = e−iRt

0δl(t?)F(t?)Ldt?, η =?t

0Hgrad(t?)dt?= e−iχ/2

?1/2

−1/2

dq|q − η??q| ⊗ e−iχσz/2

(21)

with χ =?t

0F(t?)dt?, and

id

dtUI(t) = HI(t)UI(t),(22)

where HI(t) = D†(t)HTB(t)D(t). The exact form of HIis rather complicated, but all that

really matters for our argument is that it is translationally invariant with the period of

the lattice, T†

jHI(t)Tj = HI(t). As a result, the solution, UI(t) will be block diagonal in

quasimomentum space, with the blocks consisting of SU(2) rotations. Thus by Eq. (20),

the general unitary evolution will have the form,

U(t) = e−iχ/2

?1/2

−1/2

dq|q − η??q| ⊗ Uq.(23)

A control sequence of microwave driven rotations in a uniform lattice followed by a time

dependent linear gradient can reach any unitary map of this form.

We contrast this control with that achievable in a one dimensional sinusoidal optical

lattice with time-dependent uniform forces and modulation of the lattice depth in the TB

approximation, without symmetry breaking for right vs. left transport. Haroutyunyan and

Nienhuis derived a general expression for the propagator in such a situation.[43]. In the

quasimomentum basis, the map is

U(t)|q? = e−ia(t)cos(2πq−b(t))|q − η?

(24)

where

a(t)eib(t)=

?t

0

dt?Ω(t?)eiη(t?)

(25)

and Ω(t) is the hopping rate between sites (symmetric to the left or to the right), and η(t)

is the same as above. The effect of the propagator is solely to induce a phase that varies

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Page 14

as the first order Fourier coefficient, in addition to shifting all of the quasimomentum by an

amount η. In contrast, Eq. (23) allows a broader class of unitaries to be synthesized.

V. SUMMARY AND OUTLOOK

We have described the control of transport of atoms through a microwave-dressed spinor

optical lattice. Asymmetric lattices and spectral isolation provides a means to break the

system up into a series of two-level systems, which aided the design of control routines. We

restricted our attention to translationally invariant systems, with no local addressing, but

the possibility of a uniform applied time-dependent force. Under these conditions, we can

determine the constraints of reachable states and more general reachable unitary maps that

take a localized atom at one site to a state coherently extended over n sites. Based on these

constraints, we propose a constructive protocol for carrying out these control tasks through

a sequence of SU(2) rotations acting in the two-level subspaces.

An important consideration for practical implementation of our protocol is robustness of

the control sequences to imperfections in the system. Because single particle transport is

driven by a series of SU(2) maps, certain errors may be fixed by borrowing techniques from

robust control of NMR systems[25, 26, 44]. In particular, spatial variations or miscalibrations

in the microwave field strength and real or fictitious (i.e. light-shift induced) magnetic

fields can be corrected with such techniques. On the other hand, the inevitable spatial

inhomogeneities in the lattice potential can lead to spatial variations in the energy common

to both of the levels, which causes an overall phase on the two-level systems that varies

in space. Because this phase error is not an SU(2) map, it cannot be removed with the

standard NMR composite pulse protocols nor their generalizations, and we must develop

new methods to correct this in order to make our protocol robust to lattice inhomogeneity.

This will be a topic of future investigation.

There are a number of ways in which our method can be extended to more general con-

trol of the transport of individual atoms. In the current work we restricted our attention

to the single band, tight-binding approximation, in uniform lattices with two spin levels.

More general protocols that do not restrict the bands can be used to study the control of

coupled spin and spatial degrees of freedom in a broader context. Moreover, it has recently

been shown [45] that the entire hyperfine manifold of magnetic sublevels is controllable with

14

Page 15

applied rf and microwave waveforms, leaving open the possibility of combining the control of

high dimensional spin and spatial degrees of freedom. In addition, breaking the translational

invariance with quadratic or higher order potentials should allow the controllability of the

system to be significantly enhanced. Other modifications of the spin-dependent potentials,

such as the spatial variation in microwave transition frequency that arises in a strong mag-

netic field gradient, can in principle allow spectral addressing of individual two-level systems

and extend the controllability of the system.

Finally, the techniques proposed here may also be extended to control the dynamics of

many-body systems. For instance, it might be possible to use the microwave drive to synthe-

size more arbitrary interactions between atoms than are dictated by the static Hamiltonian.

Once interactions are included in the model, many-body unitary maps can by built from

maps acting on restricted subspaces, as we have done here for single particles. Such tools can

play an essential role for quantum simulations of many-body Hamiltonians, both for stud-

ies of equilibrium properties such as the many-body phase diagram and non-equilibrium

phenomena such as Lodschmidt echos [34] and the dynamics of phase transitions.

Acknowledgments. We thank Seth Merkel for helpful discussions. This research was

supported by NSF Grants No. PHY-0555573 and No. PHY-0555673, and IARPA/NIST

Grant No. 70NANB4H1096.

∗Electronic address: bmischuc@unm.edu

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lattices, Physical Review Letters 82, 1060 (1999), doi:10.1103/PhysRevLett.82.1060, URL

http://link.aps.org/abstract/PRL/v82/p1060.

[2] D. Jaksch, H. Briegel,J. I. Cirac,C. W. Gardiner, and P. Zoller, Entangle-

ment ofatoms viacoldcontrolled collisions, PhysicalReviewLetters

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v82/p1975.

[3] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hansch, and I. Bloch, Controlled collisions

for multi-particle entanglement of optically trapped atoms, Nature 425, 937 (2003), ISSN 0028-

0836, doi:10.1038/nature02008, URL http://dx.doi.org/10.1038/nature02008.

15

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