PHYSICAL REVIEW A 84, 063613 (2011)

Dynamical phase interferometry of cold atoms in optical lattices

Uri London and Omri GatRacah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

(Received 12 October 2011; published 9 December 2011)

We study the propagation of cold-atom wave packets in an interferometer with a Mach-Zehnder topology basedon the dynamical phase of Bloch oscillation in a weakly forced optical lattice with a narrow potential barrierthat functions as a cold-atom wave-packet splitter. We calculate analytically the atomic wave function, and showthat the expected number of atoms in the two outputs of the interferometer oscillates rapidly as a function of theangle between the potential barrier and the forcing direction with period proportional to the external potentialdifference across a lattice spacing divided by the lattice band energy scale. The interferometer can be used as ahigh-precision force probe whose principle of operation is different from current interferometers based on theoverall position of Bloch oscillating wave packets.

DOI: 10.1103/PhysRevA.84.063613 PACS number(s): 03.75.Dg, 03.75.Kk, 67.85.De

I. INTRODUCTION

The dynamics of a quantum particle in a periodic environ-ment is in many ways similar to that of a free particle, but whensubject to a uniform force it does not accelerate indefinitelybut performs periodic motion—Bloch oscillations. This effectis often explained using the interference of Wannier-Starkresonances, but can also be understood in terms of classicaldynamics generated by an effective Hamiltonian, where thekinetic term is replaced by the band dispersion—Peierlssubstitution [1–6]. The band dispersion is periodic so thatthe direction of the velocity, that is, the gradient of theHamiltonian with respect to momentum, oscillates, and theclassical trajectories generated by the effective Hamiltonianare periodic.

The high controllability and weak coupling to the environ-ment of cold-atom systems make them ideal for observation ofBloch oscillation [7–12]. One of the experimental applicationsof cold-atom Bloch oscillations has been force measurements,in particular of the acceleration of gravity [13–18] that hasrecently achieved 10−7 accuracy. This application is basedon a precise knowledge of the periodic potential and a largenumber of oscillations that enable an accurate measurement ofthe period. The experiments are carried out with very weaklyinteracting Bose atoms to minimize dephasing.

A cold-atom wave packet undergoing Bloch oscillationsin an optical lattice with an external force accumulates adynamical phase during its propagation. The dynamical phaseaccumulates much faster than the oscillations phase, so that asingle Bloch oscillation yields a high-precision measurementof the external force. However, the measurement of a phaserequires an interferometric setup. For this purpose, we proposeto use a narrow one-dimensional (1D) potential barrier as acold-atom wave-packet splitter to form an interferometer witha Mach-Zehnder topology, shown schematically in Fig. 1. Theauxiliary mirrors of optical Mach-Zehnder interferometers arenot required here, since the trajectories in the two arms of thecold-atom interferometer are curved, and naturally recombine.

A tunnel barrier for cold atoms can be implemented bya narrow, blue-detuned laser beam. The coherent tunnelingof Bose-Einstein condensates through a barrier significantlynarrower than the spatial extent of the condensate has beenused in experiments to study condensate dynamics in a

double-well potential [25–27]. Here we propose to use thesame technique in an optical lattice instead of a trap. Thewidth of the barrier, equal to several lattice spacings, fits therequirements of the interferometer.

The proposed setup is therefore as follows: a cold-atomwave packet is loaded adiabatically into an optical lattice withlattice spacing much shorter than the (spatial) wave-packetwidth, so that only a single energy band is occupied. Weassume that the band is nondegenerate and does not crossother bands. The atoms are also subject to a weak externalforce along a lattice direction. The wave packet is acceleratedby the external force and impinges on a narrow potential barrierthat is nearly perpendicular to the force direction; it thensplits, and the wave-packet fragments propagate in the twoarms and recombine in a second collision with the potentialbarrier. The interferometer structure is summarized in Fig. 1.High sensitivity is derived from the fast oscillations of theinterferometer outputs as a function of the tilt angle of thetunnel barrier.

The interferometer system relies on three small parameters.The principal parameter is ε, which is the ratio of the potential-energy difference across a lattice unit divided by the energy-band energy scale. The inequality ε � 1 guarantees thevalidity of the semiclassical approximation and provides thenecessary scale separation between propagation and scatteringin the operation of the interferometer. It is also essential forthe interferometric sensitivity of the phase, since the accumu-lated phase difference in the two arms is proportional to ε−1.The second small parameter is δ, which is the ratio of the latticespacing and the spatial uncertainty of the cold-atom wavepacket. δ � 1 is necessary for the localization of the wavepacket in a single band, while momentum localization requiresthat ε � δ. The third small parameter is the lattice spacing a

divided by the width w of the tunnel barrier. w has to be muchshorter than the wave-packet width to prevent wave-packetdistortion during the scattering, and much longer than a toprevent band transitions. The setup conditions are, therefore,

ε � δ � a

w� 1. (1)

Good visibility of the interference between the two wavepackets in the outputs of the interferometer depends on phase-space overlap. Time-reversal symmetry guarantees that the

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URI LONDON AND OMRI GAT PHYSICAL REVIEW A 84, 063613 (2011)

FIG. 1. (Color online) A schematic diagram of the cold-atominterferometer, consisting of a square optical lattice whose axes areshown as the x and y axes, an external force �F in the x direction,and a tunnel barrier (red) tilted by a small angle α with respectto the y axis. The initial wave packet indicated at the bottom ofthe figure (blue) at phase-space point (�qi, �pi) splits at time ti byscattering into transmitted and reflected subwave packets [whoseclassical reference orbits (�q (T )

i , �p(T )i ) and (�q (T )

i , �p(T )i ) (respectively)

are indicated in blue and green (respectively)] that scatter oncemore from the tunnel barrier at times tT and tR (respectively) andrecombine. The interference of the transmitted-transmitted (TT) andreflected-reflected (RR) wave packets is sketched near the top of thefigure. Some examples of the expected fraction of the atoms in theTT-RR output are shown in Figs. 2 and 3.

transmitted and reflected wave packets recombine at the beamsplitter with full overlap after a complete Bloch oscillation ifthe barrier is perpendicular to the force direction, regardlessof the band dispersion; it follows by continuity that there issignificant overlap when the tilt angle α of the beam splitterwith respect to this direction is small enough. The populationin the outputs exhibits fast oscillations as a function of α

with a period of order ε, shown in Eq. (22) and Figs. 2 and3. The range of α with high visibility is of the order of ε

12

for the optimal choice, δ ∼ ε12 , so that the number of visible

oscillations is large, ∼ε− 12 . In the initial, constant visibility

range of α, the oscillations are purely sinusoidal with a perioddetermined by the initial momentum and the band dispersion,as shown in Eq. (24).

The wave-packet motion in the arms of the interferometeris planar, undergoing Bloch oscillations in one direction andfree motion in the perpendicular direction. A 1D optical

lattice is therefore required, with the lattice axis along thedirection of the external force. Nevertheless, the interferometeroperates equally well in a 2D lattice, and such operation offersadditional probes, as discussed below. Since the 1D-latticecase is obtained as a special case of the 2D one, we carry outthe analysis for a 2D lattice, and present the 1D special casefor the key results.

The recombination of the wave packet occurs after a singleBloch oscillation. However, the interferometer geometry im-plies that the wave packets propagating in the interferometerarms cross the barrier once also during the Bloch period.The wave packets in general do not recombine during thisintermediate crossing, so it is useless for the purpose ofinterferometry. A practical method of avoiding these spuriouscollisions and subsequent degradation of the interferencepattern is to make the barrier potential time dependent. Thebarrier potential needs to be maintained only during the initialand final collisions, and can be turned off during the inter-mediate crossing of the barrier, since these are well separatedin time. We will assume that this plan is carried out in ourcalculations and, for the sake of completeness, point out themodifications that arise if the barrier is time independent.

Bloch oscillations are a wave phenomenon that is notparticular to quantum mechanics. In addition to cold-atomssystems, they have been experimentally observed and studiedin guided light systems [19,20]. These systems are typicallyparaxial, and since the paraxial wave equation is identical to theSchrodinger equation with the propagation direction playingthe role of the time coordinate, the propagation of a quantumwave packet in a periodic potential and of a beam in a waveguide array are described by the same analysis. This resultalso holds for the present study; for concreteness, we expressall of our results in cold-atom terms, as it is the more naturalapplication.

In addition to the spatial interferometry considered here,waves propagating in a periodic medium can split and interferein the band degree of freedom [21–24]. Instead of a tunnelbarrier, this requires a coupling between bands, possibly byLandau-Zener tunneling. This type of interference does notdirectly measure the external force unless the periodic structureis controlled.

The analysis of the interferometer operation is based on twobuilding blocks: semiclassical wave-packet propagation [29]in the arms of the interferometer, which is presented in Sec. II,and quantum normal form theory [30] of the scattering of thewave packets off of the tunnel barrier, which is the subject ofSec. III. The results of the wave-packet propagation analysisand wave-packet scattering are combined in Sec. IV to yieldexpressions for the interferometer output as a function of the tiltangle. In Sec. V, we discuss applications of the interferometerforce measurement and as a probe of the optical latticestructure. It is shown that the interferometer is experimentallyrealizable for a standard setup with gravity as the externalforce.

II. WAVE-PACKET BLOCH OSCILLATIONS

The propagation of the wave packets in the arms of theinterferometer is governed by the combination of the opticallattice potential and the external force �F directed along a lattice

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DYNAMICAL PHASE INTERFEROMETRY OF COLD ATOMS . . . PHYSICAL REVIEW A 84, 063613 (2011)

0.002 0.004 0.006 0.008 0.010

0.2

0.4

0.6

0.8

1.0

Detectionprobability

0.0005 0.0010 0.0015 0.0020

0.2

0.4

0.6

0.8

1.0

Detectionprobability

(a) (b)

FIG. 2. (Color online) Probability of observing an atom in the TT-RR output of the interferometer as a function of the tilt angle α. Theatoms propagate in a one-dimensional tight-binding period-two potential superlattice with hopping energy 1

2 U , on-site potential ±U , and lattice

spacing a = 2πh

√U

m. The thick blue line was calculated with the full expression (22), and the thin violet line is the small angle approximation

(24). The initial momentum-space width δ = 14

√ε and the barrier width w = 1

2π

a√δ. Panels (a) and (b) show the interference patterns for

ε = 10−3 and ε = 10−4, respectively.

axis, which we choose as the x axis, so that the single-atomHamiltonian is

�p2

2m+ Vper(�q) − �q · �F, (2)

where �q and �p are the atom’s position and momentum,respectively, and Vper is the lattice potential. We assume that theinteratom interactions are negligible, and that the many-atomstate is a product of identical one-atom states.

The optical lattice is either one or two dimensional. In eithercase, we suppose that the wave packet is fully supported in oneof the energy bands of the (unforced) lattice potential whoseenergy dispersion isE(px,py), where �p is the quasimomentum,and measure energy in units of U , the energy-band width. If theoptical lattice is one dimensional, then E(px,py) = E1(px) +

12m

p2y , where E1 is the one-dimensional dispersion, and py

is the y component of the ordinary momentum. We assumethat the energy band is nondegenerate and separated from theother bands by a gap of order U . We also choose h

a, where a

is the lattice spacing, as the unit of momentum, so that E isperiodic with period one in px . The external force is weak,in the sense that ε ≡ aF

U� 1. In the following, we will write

simply momentum instead of quasimomentum when there isno risk of confusion.

The initial single-atom state |i〉 is a wave packet with meanposition �qi and momentum �pi , and momentum uncertaintyδ � 1. It follows that the position uncertainty is much largerthan a. In explicit calculations, we let |i〉 be a Gaussian wavepacket with wave function 〈 �p|i〉 = c exp[− ( �p− �pi )2

4δ2 − ih�qi · �p].

Under these conditions, an effective classical dynamics canbe applied as a systematic approximation for the propagationof the wave packet. An effective Hamiltonian Heff determinesthe in-band dynamics, derived from a semiclassical expansionin ε, whose leading term, Peierls substitution [1], is obtainedby replacing the kinetic term in the Hamiltonian with the banddispersion, so that

Heff = E( �p) − �q · �F, (3)

0.001 0.002 0.003 0.004 0.005

0.2

0.4

0.6

0.8

1.0

Detectionprobability

0.0002 0.0004 0.0006 0.0008 0.0010

0.2

0.4

0.6

0.8

1.0

Detectionprobability

(a) (b)

FIG. 3. (Color online) The interferometer output for propagation in a separable two-dimensional optical lattice that is a superposition of x

and y potential superlattices having the same properties as the one used in Fig. 2. The significance of the axes, the curves, the initial state, thewidth of the barrier, and the values of ε are the same as in Fig. 2.

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URI LONDON AND OMRI GAT PHYSICAL REVIEW A 84, 063613 (2011)

and letting �q and �p be canonically conjugate, [qn,pm] = ihδnm.Here, �q is a coarse-grained position observable with anuncertainty much larger than a. We assume that the opticallattice is space-reflection invariant in x and y, so that thesubleading term in Heff is O(ε2) and, therefore, negligible.

The semiclassical propagation of the wave packet generatedby Heff consists of three elements: a phase-space drift alongthe classical trajectory of the center of the wave packet(�qt , �pt ), a deformation (squeezing) of the wave packet bythe linearized flow at the wave-packet center, and an overallphase accumulation proportional to the classical action of thewave-packet center trajectory [28,29]. Each part is realized asa unitary operator, so that the wave packet at time tf can berepresented by

|f 〉 = eiγf i T (�qf , �pf )M(Sf i)T†(�qi, �pi)|i〉, (4)

where

γ = −1

h

∫ f

i

[12 ( �p · d �q − �q · d �p) − Heffdt

]is the dynamical phase,

T (�q0, �p0) = exp

[i

h(�q · �p0 − �p · �q0)

]

is the Heisenberg phase-space shift operator, and M(S) is thesqueezing (metaplectic) operator derived from the symplecticphase-space deformation matrix

Sf i =(

Sqq Sqp

Spq Spp

),

with block components Sqq = ∂qf,n/∂qi,m, Spq = ∂pf,n/

∂qi,m, Sqp = ∂qf,n/∂pi,m, and Spp = ∂pf,n/∂pi,m.The problem is therefore reduced to the calculation of

the classical trajectories of Heff . The effective Hamiltonianis integrable, and its flow can be calculated explicitly. TheHamilton equations are

∂t �q = �∇E( �p), (5)

∂t �p = �F . (6)

The momentum equation implies that �pt = �pi + �F t ; they position equation gives qt,y = qi,y + ∫ t

tidt ′vy(t ′), where

vy(t) = ∂yE( �pt ), and conservation of energy implies thatqt,x = qi,x + 1

F[E( �pt ) − E( �pi)]. The periodicity of E implies

that the wave packet performs Bloch oscillations in the x

direction with period 1F

, while moving freely in the y direction.It follows that the Sqq and Spp blocks of the deformation

matrix are unit matrices, the Spq block is the zero matrix, andthe elements of the Sqp block are

∂qf,x

∂pi,x

= 1

F[vx( �pf ) − vx( �pi)], (7)

∂qf,y

∂pi,x

= ∂qf,x

∂pi,y

= 1

F[vy( �pf ) − vy( �pi)], (8)

∂qf,y

∂pi,y

= 1

F

∫ f

i

dpx∂yvy(px,pi,y). (9)

The expression for the dynamical phase can be simplified to

γf i = 1

2h( �pf · �qf − �pi · �qi) − U

hF

∫ f

i

dpxE(px,pi,y). (10)

In our choice of units, the coefficient UhF

= 2πε

, so that theclassical limit, where h → 0 keeping all classical quantitiesfixed, is equivalent here to the limit ε → 0, and ε is the effectivePlanck constant. In this limit, the dynamical phase diverges,giving rise to the high interferometric sensitivity, as the smallwavelength does in classical optical interferometry.

III. WAVE-PACKET SPLITTING OFF A TUNNEL BARRIER

The atom wave packet is split by tunneling through a one-dimensional potential barrier or a potential trench, that is, amodulation of the optical potential of the form ub(�q · n), wheren is the unit normal to the beam splitter, and ub is localized ona scale w of magnitude a � w � a/δ.

For the purpose of analyzing the scattering of the wavepacket by the beam splitter, we ignore temporarily the externalforce, and consider a standard scattering problem of anincoming unit amplitude plane wave with quasimomentum�p in an energy band with dispersion E , that splits into atransmitted wave with quasimomentum �p with amplitudet and a reflected wave with quasimomentum �p(R) withamplitude r . The inequality a � w guarantees that thescattered wave remains in the same energy band. Latticetranslation symmetry implies that the tangential componentof the quasimomentum is conserved, pt = p

(R)t , and energy

conservation implies that E( �p) = E( �p(R)); we assume that theband dispersion is such that there is a single solution �p(R),other than �p, to these two conditions.

The initial quasimomentum and wave-packet splitter orien-tation define a classical band of allowed energies betweenE− = minpn

E( �p + pnn) and E+ = maxpnE( �p + pnn). The

condition a � w also implies that unless E( �p) − E− is closeto max ub, or E( �p) − E+ is close to min ub, then either |t | � 1or |r| � 1. For concreteness, we choose to study scatteringfrom a potential barrier and let |E( �p) − E− − max ub| � 1 sothat both t and r are appreciable to obtain good interferometricvisibility. We also assume that the initial momentum is suchthat E( �pi) is far from the edges E± of the interval of allowedenergies.

The typical barrier maximum and dispersion minimumare quadratic, and under the conditions laid out above, thescattering amplitudes are determined by the local behavior ofE and ub near their extrema [30]. Denoting by �pm and qm theabscissas of the extrema, we define the classical action

I = E( �p) − E( �pm) − ub(qm)√−∂2

nE( �pm)∂2qu(qm)

, (11)

and the standard theory of scattering from a parabolic barriergives [31]

t( �p) = ei Ih

(log Ih−1)

√2π

�

(1

2− i

I

h

)e

π2

Ih , (12)

r( �p) = −iei I

h(log I

h−1)

√2π

�

(1

2− i

I

h

)e− π

2Ih . (13)

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A scattering wave packet is a superposition of stationaryscattering states, |i〉 = ∫

dpψ( �p)| �p〉. After the center of theincoming wave packet collides with the barrier, it splits intoa transmitted wave packet, |i〉T = ∫

d �pt( �p)ψ( �p)| �p〉, and areflected wave packet, |i〉R = ∫

dpr(p)ψ(p)√| det J || �p(R)〉,

where J = ∂ �p∂ �p

(R)is the Jacobian matrix of the reflection trans-

formation. Since the incoming wave packet is concentratednear �pi , the transmitted wave packet is approximately

|i〉T =∫

dpt( �pi)ψ( �p)| �p〉 = t( �pi)|i〉, (14)

but the dependence of the scattering amplitudes on �p causesa position-space shift and distortion of the wave packet.Nevertheless, the shift and distortion are weak if the t and r donot change appreciably over the range of momenta that supportthe wave packet, or equivalently, if the barrier width w is muchsmaller than the position uncertainty a

δ[32], as we assume. In

this case, the position and momentum shifts incurred by thescattering are of O(w) and O(w

aδ2) (respectively), which are

negligible in comparison with the respective uncertainties; therelative deformation is even smaller, of O((wδ

a)2).

By the same reasoning, the reflected wave packet can bewritten as |i〉R = r(pi)

∫d �pψ( �p)

√| det J || �p(R)〉. Since themomentum integration is localized, J can be evaluated at �pi ,and the reflected momentum is approximately �p(R) = �p(R)

i +Ji( �p − �pi), where Ji = J ( �pi). The reflected wave packet istherefore

|i〉R = r(pi)T (�qi)T( �p(R)

i

)M

((J t

i )−1 0

0 Ji

)T †( �pi)T

†(�qi)|i〉.

(15)

Finally, we reconsider the effect of external force. Inthis case, the quasimomentum is no longer a good quantumnumber, so we label the stationary states by the value of �p atthe barrier. These states are propagating in a large interval ofO( a

ε) size around the barrier and can therefore be used as a

basis for a scattering theory for the wave packets consideredhere, with position uncertainty of O( a

δ) � a

ε. The preceding

arguments and Eqs. (12)–(15) are valid also with the weak,uniform external force up to small corrections.

IV. ANGLE INTERFEROMETRY

The wave-packet interferometry takes place in three steps.An initial wave packet |i〉 of mean position �qi and momentum�pi is first split by scattering from the wave-packet splitter into atransmitted wave packet |i〉T and a reflected wave packet |i〉R ,with mean position �qi and momentum �p(R)

i . The transmittedwave packet then undergoes Bloch oscillation in the x directionand propagates freely in the y direction before impingingon the beam splitter again at time tT and position �q(T )

c withmean momentum �p(T )

c . The wave packet |c〉T then splits againinto a transmitted-transmitted (TT) wave packet |c〉T T at �q(T T )

c

and �p(T T )c , and a transmitted-reflected (TR) wave packet |c〉T R

at �q(T R)c and �p(T R)

c . The states and variables related to thepropagation and splitting of the reflected wave packet aresimilarly defined; see Fig. 1.

The system has a Mach-Zehnder geometry with twooutputs, one interfering TT with RR wave packets, and the

other interfering TR with RT wave packets. The interferometeris sensitive to the difference between the dynamical phasesaccumulated by the transmitted and reflected wave packetsduring their propagation. This phase difference changes fastas a function of the tilt angle α between the beam splitter andthe y axis.

A. Wave-packet interferometry

We now calculate the interference pattern, concentratingfrom this point forward on the TT-RR output. The interferencetakes place after both the transmitted and the reflected wavepackets have collided again with the beam splitter. When thetilt angle α is positive, tR > tT , so that tf = tR . The final stateof the TT wave packet is therefore

|f 〉T T = t( �p(T )

c

)t( �pi)e

iγ(T T )f i T

(�q(T T )f , �p(T T )

f

)M

(S

(T T )f i

)× T †(�qi, �pi)|i〉, (16)

with momentum-space wave function

〈 �p|f 〉T T = t( �p(T )

c

)t( �pi)e

− 2πiε

∫ �p(T T )f,x

�pi,xE(px,pi,y )dpx

× 1√2πδ2

eih

(�q(T T )f · �p(T T )

f −�qi · �pi )e− ih�q(T T )f · �p

× e− 1

4δ2 ( �p− �p(T T )f )·C(T T )·( �p− �p(T T )

f ) (17)

for C(T T ) = 1 + i 2δ2

hS

(T T )f,pq , and the final state of the RR wave

packet is

|f 〉RR = |c〉RR = r( �p(R)

c

)r( �pi)e

iγ(R)ci T

(�q(R)c

)T

( �p(RR)c

)×M

([(J (R)

c

)t]−10

0 J (R)c

)T †( �p(R)

c

)T †(�q(R)

c

)× T

(�q(R)c , �p(R)

c

)M

(S

(R)ci

)T †(�qi, �p(R)

i

)T (�qi)T

( �p(R)i

)×M

((J t

i

)−10

0 Ji

)T †( �pi)T

†(�qi)|i〉, (18)

with wave function

〈 �p|c〉RR = r( �p(R)

c

)r( �pi)e

− 2πiε

∫ �p(R)c,x

�p(R)i,x

E(px,pi,y )dpx

× eih

(�q(R)c · �p(R)

c −�qi · �p(R)i )√

2πδ2 det(Ji) det(J (R)c )

e− ih�q(R)c · �p

× e− 1

4δ2 ( �p− �p(RR)c )·C(RR)·( �p− �p(RR)

c ) (19)

for

C(RR) = [(J (R)

c

)t]−1[(J t

i

)−1J−1

i + i 2δ2

hS(R)

c,pq

](J (R)

c

)−1.

Conservation of energy and transverse momentum imply thatin Cartesian coordinates,

Ji =(

v(R)i,x v

(R)i,y

−sin α cos α

)−1(vi,x vi,y

−sin α cos α

), (20)

J (R)c =

(v(RR)

c,x v(RR)c,y

−sin α cos α

)−1(v(R)

c,x v(R)c,y

−sin α cos α

). (21)

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The state in the TT-RR output is |f 〉T T + |c〉RR , so that theprobability that an atom is detected there is |t( �p(T )

c )t( �pi)|2 +|r( �p(R)

c )r( �pi)|2 + 2 Re T T 〈f |c〉RR . The interference term takesthe form

T T 〈f |c〉RR

= t( �p(T )

c

)∗t( �pi)

∗r( �p(R)

c

)r( �pi)

× e

2πiε

[∫ �p(T T )

f,x

�pi,xE(px,pi,y )dpx−

∫ �p(R)c,x

�p(R)i,x

E(px,p(R)i,y )dpx ]

× e− ih

(�q(T T )f · �p(T T )

f −�qi · �pi )eih

(�q(R)c · �p(R)

c −�qi · �p(R)i )√

det(Ji) det(J

(R)c

)det

{12 [(C(T T ))∗ + C(RR)]

}× e

− 14δ2 [ �p(T T )

f ·(C(T T ))∗· �p(T T )f + �p(RR)

c ·C(RR)· �p(RR)c ]

× e1

4δ2 v·[(C(T T ))∗+C(RR)]−1·v, (22)

where

v = 2iδ2

h

(�q(T T )f − �q(R)

c

) + (C(T T ))∗ · �p(T T )f + C(RR) · �p(RR)

c .

The phase of the interference term changes as a function ofα on a scale of ε, while the amplitude and the frequency ofphase oscillations change more slowly. These properties arevisible in the interference patterns shown by thick blue linesin Figs. 2 and 3 for one- and two-dimensional optical lattices,respectively. The interference patterns were calculated for twovalues of ε, 10−3 and 10−4.

The results presented in this section were derived underthe assumption that the T and R wave packets do not scatterbefore the recombination event. Since the trajectories of thewave packets in the interferometer arms cross the barrier line,this requires temporal control of the barrier amplitude so thatit vanishes at the intermediate crossing times of the T and Rwave packets, and is kept constant at the initial splitting timeand final recombination times. If the barrier amplitude is notcontrolled temporally, further splitting would make the finalstate a superposition of six (rather than four) wave packetsor more. Nevertheless, the interference in the TT-RR andTR-RT channels persist with the modification that the state|f 〉T T is multiplied by an additional transmission amplitudet( �p(T )

x ) compared with Eqs. (16) and (17), where �p(T )x is the

momentum of the T wave packet when it crosses the barrier,and, similarly, |c〉RR gains an additional factor of t( �p(R)

x ) withrespect to Eqs. (18) and (19). In the following, we assume thatthe barrier potential is controlled appropriately and thereforeomit these additional factors that reduce the amplitude of theinterferometric oscillations in Eq. (22), but otherwise do notaffect the results.

B. Interferometry for small angles

The interference is effective when the interference term(22) is large, and this requires that the wave packets |f 〉T T and|c〉RR overlap in position as well as in momentum. For mosttilt angles, �q(T T )

f = �q(RR)c and �p(T T )

f = �p(RR)c . However, if the

beam splitter is aligned with the y axis, then by symmetryp

(R)i,x = −pi,x , p

(R)i,y = pi,y , and vy(−px,py) = vy(px,py), so

that after one Bloch period, the two subwave packets meetat the beam splitter at q(T )

c,x = q(R)c,x = qi,x , q

(T )f,y = q

(R)f,y =

qi,y + 1F∂y E(pi,y), where E(pi,y) = ∫ 1

0 dpxE(px,pi,y). Thereis, therefore, full overlap between the TT and the RR wavepackets for α = 0, and high interferometric visibility for smallenough α.

Since α is small, it is possible to approximate the classicaldata in the interference term by its Taylor expansion. The O(α)approximation gives

T T 〈f |c〉RR = [t( �pi)∗r( �pi)]2[1 + α(h1 + h2)] (23)

× e− 2πiε

α∂αp(R)i,y ∂ E(pi,y ),

where

h1 = ∂αp(T )c,x

∂x t( �pi )∗t( �pi )∗

+ ∂αp(R)c,x

∂xr( �pi )r( �pi )

,

h2 = − 12∂α

[det(Ji) det

(J (R)

c

)det

{12 [(C(T T ))∗ + C(RR)]

}].

The second correction to the interference term αh2 is O(α)and therefore negligible for small α; similarly, αh1 = O(w

aα)

is also negligible throughout the region of validity of Eq. (23),where, therefore,

T T 〈f |c〉RR = (t∗r)2e4πiε

∂ E(pi,y )α, (24)

using the shorthand t = t( �pi), r = r( �pi). It follows that forsmall enough angles, the probability of observing the atomin the TT-RR output oscillates sinusoidally as a function of α

on a fast scale of O(ε) with large visibility. The thin violetlines in Figs. 2 and 3 show the interference pattern with theapproximation given by Eq. (24) that becomes increasinglyaccurate for smaller α. If the band dispersion is known, theseoscillations can be used to measure the force F with highprecision, as discussed further below.

The domain of validity of Eqs. (23) and (24) is limited bythe size of the neglected O(α2) terms. These terms arise inthe combinations α2

δ2 , α2

ε, and α2δ2

ε2 in Eq. (22). In particular,the first and third combinations determine the scale of decayof visibility as a result of momentum mismatch and positionmismatch between the TT and RR wave packets, respectively.It follows that the domain of validity is

α � min

(δ,ε

12 ,

ε

δ

). (25)

Evidently, the largest range is obtained when δ ∼ ε12 , that is,

when the momentum and position uncertainties are balanced,so that the domain of validity is α � ε

12 . In this case, since

the oscillation period is O(ε), the number of equal periodoscillations up to a fixed tolerance is O(ε− 1

2 ). When α iscomparable with ε

12 , the visibility of the interference pattern

decreases and the oscillation period changes, as shown inFigs. 2 and 3, which also demonstrate that the number ofhigh-visibility constant-period oscillations increases when ε

becomes smaller.

V. CONCLUSIONS

An experiment of angle interferometry would detect thefraction of atoms scattered in the TT-RR arm for the variable tiltangle of the tunnel barrier, keeping the rest of the parameters

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DYNAMICAL PHASE INTERFEROMETRY OF COLD ATOMS . . . PHYSICAL REVIEW A 84, 063613 (2011)

fixed. Writing Eq. (24) in physical units, this fraction wouldbe approximately equal to the probability

T T〈f |c〉RR = |t |4 + |r|4 + 2 Re((t∗r)2e2i α

hF∂ E(pi,y )

). (26)

With the optimal choice of initial uncertainty, this expressionis valid for values of α significantly smaller than ε− 1

2 . A time-of-flight measurement can yield the squared magnitude of thefull momentum-space wave function, the TT-RR part of whichis given by Eqs. (17) and (19).

The interferometer period can be further simplified whenthe optical lattice is separable, a case that includes one-dimensional lattices, obtaining

T T〈f |c〉RR = |t |4 + |r|4 + 2 Re((t∗r)2e2i

vi,y

hFα), (27)

where vi,y is the constant velocity in the direction perpendic-ular to the force.

If gravity is used as the external force acting on atoms withmass m in lattice with spacing a, then

ε = m2a3g

u(πh)2, (28)

where g is the acceleration of gravity and u is the bandwidth inunits of the recoil energy, (πh)2

a2m. It follows that the interferomet-

ric condition ε � 1 is experimentally accessible. For example,for sodium atoms in a 1

2 589 nm lattice, ε = 1u

3.3 × 10−2;a lattice depth of 0.25 recoil energy, that can be set upwith sub-mW lasers [33], gives u ≈ 0.4, and ε = 8.4 × 10−2.Spatial localization of 3 μm and barrier width of 1 μm can bechosen to satisfy the basic inequalities given by Eq. (1).

A natural application of the interferometer is force mea-surement, which can reach high-precision thanks to the high

sensitivity of the interference pattern, and for this application aone-dimensional lattice suffices, and the force can be deducedfrom the interferometer output using Eq. (26). The statisticalerror in a single measurement is of the order of the inversesquare root of the number of atoms, which can be 10−3 intypical experiments. If the measurement is repeated for 100values of α, then the statistical error is reduced by anotherfactor of 10. The induced measurement error in the force isreduced by the number of visible interferometric oscillationsthat is of the order of

√ε, giving an error estimate of

10−5. The interferometer measures the combination Fvi,y

ratherthan the force itself, so that high-precision measurement ofthe transverse speed is needed to reach this value of forcemeasurement sensitivity. This sensitivity is not as good as thatof existing interferometers that rely on the classical Blochoscillation phase, but the dynamical phase interferometerhas the advantage of requiring a single Bloch oscillation,facilitating much faster measurements.

The interferometer can also be used to characterize thedispersion of two-dimensional lattices with a given externalforce. This application can become especially interestingif the lattice breaks space-reflection symmetry so that theinterferometer can measure a nontrivial Berry phase [34]. Thisquestion, however, requires a more accurate analysis that isbeyond the scope of the present paper.

ACKNOWLEDGMENTS

We benefited from informative and helpful discussionswith Nir Davidson and Nadav Katz. This research wassupported by the German-Israeli Foundation under Grant No.980-184.14/2007.

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