High momentum splitting of matter-waves by an atom chip field gradientbeam-splitter

Shimon Machluf, Yonathan Japha and Ron Folman∗

Ben-Gurion University of the Negev(Dated: August 14, 2012)

The splitting of matter-waves into superposition states is a fundamental tool for studying the basictenets of quantum behavior, as well as a building block for numerous technological applications. Wereport on the first realization of a beam-splitter by a combination of magnetic field gradients and aradio-frequency technique. It may be used for freely propagating or trapped atoms in a Bose-Einsteincondensate or a thermal state. It has the advantageous feature of endowing its superposition statewith a large differential momentum in the direction parallel or transverse to the atoms’ motion,thereby, for example enabling to open large angles. As large space-time area of an interferometerincreases its sensitivity, this may be used for new kinds of interferometry experiments (e.g. largeangle Sagnac interferometry). Furthermore, it is also simple to use, fast, and does not require light.

PACS numbers: 37.10.Gh, 32.70.Cs, 05.40.-a, 67.85.-d

For the last two decades, matter-waves, in the formof ultra cold atoms [1], have been the source of numer-ous new experimental insights into the basic tenets ofquantum theory. One of the basic tools for such stud-ies including the creation and analysis of interferometry,dephasing, entanglement and squeezing, is the coherentspatial splitting of the wave function.

For freely propagating atoms, beam splitting has beenachieved some time ago (e.g. [2]). Such splitting mainlyutilizes light (e.g. [3]). For a Bose-Einstein Condensate(BEC), coherent splitting has been achieved with an op-tical trap [4] as well as optical, radio-frequency (RF) andmicro-wave interactions within magnetic traps [5–7].

Atom interferometers have been used for accurate andhigh precision measurements of gravity [8], the fine-structure constant [9–11], gravity gradients [12], Newtonsgravitational constant [13, 14], and one of the few terres-trial tests of general relativity that is competitive withastrophysics [15].

Of specific interest are beam-splitters (BS) whichtransfer a large transverse momentum to the atoms [16–19], enabling large angle interferometry which may en-able experiments of paramount importance. Aside fromobvious technological applications such as sensors for ac-celeration (e.g. our work [20]) and the gravitational field,large angle interferometry may enable new experimentalinsight on low-frequency gravitational waves, the Lense-Thirring effect, the equivalence principle, atom neutral-ity, or measurements of fundamental constants with sen-sitivity to supersymmetry (see for example [17] and ref-erences therein).

The method presented here, which may be named a”field gradient BS”, is very fast, has a large dynamicrange, and does not require lasers, stable optics, or micro-wave electronics. It may be employed for a beam of freelypropagating atoms (i.e. an atomic beam or a fountain)as well as a trapped BEC.

The general scheme of the BS and its output is demon-

strated in Fig. 1. It is based on a Ramsey-like sequenceof two π/2 rotations with a field gradient applied dur-ing the interval between them. Consider two-level atomswith internal states |1〉 and |2〉. We start with the atomsinitially in the state |2〉. The first π/2 rotation transfersthem into the superposition state 1√

2(|1〉+ |2〉). We then

apply a field gradient which constitutes a state-selectiveforce Fj (j = 1, 2) for a time T , which is typically shorterthan the time it takes the atoms to move in the force

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FIG. 1: (color online) Absorption images of the atoms: (a) inthe trap before the release, (b) after weak splitting of less than~k in 5µs interaction time and 14 ms time-of-flight (TOF),(c) after a strong splitting of 40 ~k in 1 ms interaction timeand 2 ms TOF, and (d) the four clouds separated by anotherstrong gradient (Stern-Gerlach) after the BS. Images (b) and(c) show the high dynamic range this method can reach with-out any complicated sequence. Image (d) shows that eachcloud in (b) and (c) is constructed from atoms in the |1〉 and|2〉 states.

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field. The state of the atoms after time T is givenby |ψ(x, T )〉 = 1√

2

(|1〉eiF1·xT/~ + |2〉eiF2·xT/~)ψ0(x),

where ψ0(x) is the spatial wave-function at t = 0. Eachlevel gains a phase φ(x, t) = −∆Ejt/~ = Fj ·xT/~, whichis equivalent to a momentum transfer pj = FjT . Thesecond π/2 rotation transfers the atoms into the super-position state

|ψ〉 =1

2(|1,p1〉+ |1,p2〉 − |2,p1〉+ |2,p2〉) (1)

such that each of the internal states |j〉 is in a super-position of two momentum states |p1〉 and |p2〉. Thiscompletes the operation of spatial splitting. One maythen use another pulse to separate between the |1〉 and|2〉 states to realize two parallel interferometers for noiserejection (e.g. Stern-Gerlach pulse for magnetic sublevels[Fig. 1(d)]), or keep the two interferometers in a stateof overlap (e.g. to have two clocks measuring differentproper times [21]). If one wishes to stay with just one ofthe two states, one may typically find a dedicated tran-sition to discard the redundant state.

This paper is structured as follows: First we explainthe specific aspects of our present realization of the newmethod, including the fact that differential momentumtransfer as high as 100~k and above is feasible (k taken forreference as the wave number of a 1 µm wave-length pho-ton). We then describe the details of our experiment. Asproof-of-principle, we present experimental results thatexhibit that this method is quick and versatile for bothun-trapped and trapped BECs. Finally, we present theobserved interference fringes and we discuss the criteriafor phase stability.

In our realization of the field gradient BS, we uti-lize Zeeman sub-levels. We begin with 87Rb atoms inthe |F,mF 〉 = |2, 2〉 state (F and mF are the hyperfinelevel and Zeeman sub-level quantum numbers), where the|1〉 ≡ |2, 1〉 and |2〉 ≡ |2, 2〉 states serve as our two-levelsystem. The atoms are subjected to an homogeneousmagnetic field such that there is an energy difference be-tween the |1〉 and |2〉 states ∆E = ~ω12, due to the Zee-man splitting. An important aspect of our scheme is thatthe magnetic field is strong enough so that the non-linearZeeman shift drives the transition to the |2, 0〉 level outof resonance so that we can manipulate our system as apure two-level system [22]. Next, one applies two π/2 RFpulses with a Rabi frequency Ω, which form a Ramsey-like interferometer [23]. The difference from a standardRamsey interferometer is due to the magnetic gradientwhich is applied between the two pulses (interaction timeT ). If the gradient is applied perpendicular to the mo-tion of the atoms, the differential momentum will be inthe transverse direction and a two dimensional splittingwill occur (allowing for an area enclosing interferometer).If the gradient is applied in the direction of motion, asis done in this paper, a one dimensional interferometerensues (namely, the two wave packets separate along the

same line of their motion). In our proof-of-principle ex-periment, we demonstrate the above described scheme byperforming all the sequence on a BEC in free-fall [24] aswell as in a trap.

In Fig. 1 we present a few sequences and results. Weutilize an atom chip [25–27] in order to easily create alarge magnetic gradient. Fig. 1(a) presents the trappedBEC before it is released. In Fig. 1(b,c) we presentsplitting events with a differential momentum of ~k and40~k respectively, exhibiting the large dynamic range ofthis method. In Fig. 1(d) we add a long pulse of fieldgradient before the imaging so as to verify the state ofthe atoms (Stern-Gerlach type separation).

Adopting the general explanation of Eq. (1) to ourspecific system, the operation of the field gradient BSmay be explained also by simple kinematics. During theinteraction time T a differential acceleration between thewave-packets is induced. Focusing on the |2〉 state, fol-lowing the Ramsey-like sequence we have two |2〉 wavepackets, one which was accelerated as a |2〉 state through-

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FIG. 2: (color online) (a) The resulting wave-packet veloc-ities for the |p1〉 (blue) and the |p2〉 (red) momentum statesafter the BS (in units of mm/s and ~k) following from thenumerical integration of Eq. (2) over the interaction time T .Here we have used our experimental parameters, namely 3A in a 200 µm wide wire, and an atom-surface distance of100 µm. Inset I: expanding the time axis to 1 ms, we ob-serve the limit of our method due to the large distance whichdevelops between the atoms and the wire. Inset II: Utilizingimproved experimental parameters (but just as feasible) wefind that differential momentum transfers of above 100~k arepossible in an interaction time of a few µs. The parametersare 2 A in a 10 µm wide wire (a thickness of 2 µm gives 107

A/cm2 which is more than sustainable for such short pulses),and an atom-surface distance of 10 µm. (b) For T = 5 µs and(c) T = 10 µs, we also present the atomic density (GP simula-tion) of the“near field” fringes formed just after the splitting.These represent the point of view explaining the operation ofthe field gradient BS in the context of a Fourier transform ofthe amplitude in the “near field” (see text).

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out time T and one which was accelerated as a |1〉 state.The momentum kick for a wave-packet of a certain mF

state at a distance z below the chip wire carrying currentI is given by

dp

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=mF gFµBµ0I

2πz21

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where µ0 is the magnetic permeability of free space, µB isthe Bohr magneton, and gF is the Lande factor for the hy-perfine state F . The second term is a finite-size geomet-ric factor which includes D, the width of the chip wire.Fig. 2 presents the resulting momentum kick, which isobtained by integrating the differential equation (2) overthe interaction time T . It is also shown that a differentialmomentum of 100~k and above is feasible.

In Fig. 2(b-c) we present a Gross-Pitaevskii (GP) sim-ulation of the atomic density just after the beam-splitter(“near field”). The density shows a sinusoidal patternwith a wavelength λ = 2π~/∆p, where ∆p is the mo-mentum transfer difference. In simple terms, this densitydistribution is caused by the fact that the magnetic gradi-ent gives rise to varying transition frequencies and conse-quently multiple Ramsey frequencies along the atom en-semble. This is similar to varying Rabi frequencies alongthe ensemble that have been observed when a drivingfield having an intensity gradient was applied [28–30]. Asa perfect sine function Fourier transforms into two iden-tical and counter propagating k components, this “nearfield” density distribution transforms after a long timeinto a pair of wave-packets with momentum difference∆p. It may also be noticed that the “near-field” densitypattern followed by momentum splitting is the inverseof a recombination process where two wave-packets withdifferent momenta merge coherently and form a periodicfringe pattern.

In more detail, we start our experiment with a BEC of∼ 104 87Rb atoms in the |2, 2〉 state. We utilize the sameset-up described in [22]. The atoms are subjected to amagnetic field such that the energy difference betweenthe |1〉 and |2〉 states, ∆E = ~ω12, due to the Zeemansplitting, is about 25 MHz. The non-linear Zeeman shiftof about 250 kHz, also allows us to eliminate remains ofthe mF = 1 state due to the evaporative cooling stage(on the order of 9%). Next, to form a Ramsey-like in-terferometer we apply two π/2 RF pulses with a Rabifrequency of Ω = 20 − 25 kHz. We verify the Zeemansub-level population by applying a strong Stern-Gerlachfield [22].

We first bring a trapped BEC to a distance of 100 µmfrom the chip surface. Once the atoms are released, theRF radiation as well as the magnetic gradient originatefrom the chip or its mounting. The gradient is inducedby a 2−3 A current in a 200 µm wide and 2 µm thick goldwire. The sequence is extremely simple and versatile. InFig. 3 we present the characterization of the BS (whilesplitting has been observed also with thermal atoms, here

we characterize atoms in a BEC state). A simple linearrelation is revealed. This is to be expected for short inter-action times during which the atoms move only slightlyand the acceleration in Eq. (2) is fairly constant. It isworth emphasizing the extremely short interaction times,in the range of 1− 100 µs, required for this field gradientBS to transfer significant momenta.

Next, we show that our scheme could also be imple-mented on a trapped BEC, thereby opening the road forguided“multi-pass” interferometry, such as the “multi-pass” Sagnac we have previously suggested [20]. We re-peat the above scheme by performing all the sequence ona cloud trapped in an Ioffe-Pritchard (IP) magnetic trap,where the main difference is that the magnetic gradientis fixed. Furthermore, the gradient exists also during theπ/2 pulses, but due to the wide spectrum of the shortpulse it is still quite effective in creating the superpo-sition (Ω = 5 − 10 kHz, and a π pulse transfers up to90% of the atoms, compared with more than 95% in freefall). The two different magnetic potentials give rise to adifferential acceleration in the z (gravity) direction, thusenabling the realization of the above scheme. The trapfrequencies ωmF

of the two states are ω2 = 2π × 100 Hzand ω1 = 2π × 100/

√2 with a minimum magnetic two-

level splitting of 18 MHz and a non-linear Zeeman shiftof 100 kHz in the transition to the |2, 0〉 state.

In Fig. 4(a) we present the two potentials by theirequipotential surfaces, as they are situated below theatom chip. In Fig. 4(b-d) we utilize a 1D energy versusposition (z) plot to described the evolution of the sys-tem during and after the BS sequence. Due to gravity,

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FIG. 3: (color online) Characterization of the field gradientBS: The differential velocity between the two observed peaksas a function of the interaction time T . The error bars aretaken from the difference between a few data sets. The the-oretical prediction includes errors in the atoms’ position of asingle pixel (±2.5 µm) and in the wire resistance (±0.2 Ohm).At short times, the data is a bit higher than the theoreticalexpectation, probably due to uncertainties regarding the mea-sured current. The linear dependence on T is to be expectedfrom the solution of Eq. (2) at short enough interaction timeswhen the change of the atom-surface distance is very small.Inset: Our current pulse shape. The “overshoot” at shorttimes is responsible for the larger acceleration for small T .

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the centers of the combined magnetic and gravitationaltrapping potentials for the two levels are shifted by ∆z =g/ω2

2 . It follows that when atoms initially at the level |2〉are transferred by the first π/2 pulse into the level |1〉,they experience an acceleration dvz/dt = ω2

1∆z = g/2.For interaction times T πω1 these atoms move onlysightly along the potential gradient such that, as in thefree fall scheme, the momentum splitting grows almostlinearly with T . This almost linear dependence is shownin Fig. 5. Here, the atom-atom collisional repulsion isresponsible for an additional contribution to the velocity,as shown in the experimental data and confirmed by ournumerical GP solution.

Finally, we touch upon the issue of phase stability. Be-yond the presentation and characterization of the tech-nique and kinematics of a novel and unique beam split-ter, which were the goals of this paper, we now presentas an outlook, a brief discussion of the requirements for

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FIG. 4: (color online) Splitting a trapped BEC: (a) An il-lustration of the two traps below the atom chip showing thevertical separation between the two traps, and the differencein size due to the different confinement. The energy differ-ence between the two potentials varies almost linearly alongthe extent of the cloud and hence one should expect two welldefined momentum components and a clear two peak sepa-ration after TOF as a function of the interaction time, asindeed presented in Fig. 5. (b-d) Trapping potential of thetwo states (blue for |1〉 and red for |2〉) in the direction ofgravity (the energy difference was minimized for visibility),and the position and momentum (size of arrow) of the fourparts of the wave-function during the splitting process. (b)The splitting just after the first π/2 pulse, where the positionof both clouds is the trap minimum of the |2〉 state. (c) Theclouds after the second π/2 pulse where only the |1〉 part wasaccelerated before the pulse. There is still spacial overlap (thedistance in exaggerated in the image for clarity). (d) The fourparts after some oscillation time in the trap, showing that onepart of the |2〉 state didn’t move, the other part of the |2〉 wasslowed by the trapping potential almost to a halt, while thetwo parts of |1〉 were accelerated.

stability.

In Fig. 6 we present typical images of interferencefringes as observed in our experiments. In the free-fall ex-periment we use an additional gradient pulse at a specifictime ∆t after the first one, when the two wave-packetsare already separated by a distance d = ∆p∆t/m, ∆pbeing the momentum transfer difference of the first kick.The second gradient pulse gives a stronger kick to thewave-packet with smaller momentum, which is at thistime closer to the chip wire relative to the wave-packetwith larger momentum. The duration of the secondmomentum kick is tuned such that after this kick thetwo wave-packets have the same momentum (with spa-tial separation d). They expand and overlap after aTOF to give rise to interference fringes with periodic-ity λ = 2π~t/md. The density at the center of theatomic cloud in Fig. 6(a,b) was fitted to the functionA exp[−(z− z0)2/2σ2](1 + v sin(2π(z− z0)/λ+φ), wherez0 and σ are the center and length of the joined wave-packet and v is the visibility.

The stability of the phase φ of the interference pattern,namely, its repeatability over many realizations, is an im-portant issue when analyzing the ease of use of a newBS. For lack of an interferometer scheme in our setup,our “poor man’s” recombination scheme described abovecould not enable us to prove that we have attained sta-bility in our proof-of-principle field gradient BS. In thefollowing we describe what we believe the criteria for sta-bility are.

The phase difference between the two momentum com-ponents may be mainly attributed to the differential time

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2sin(ωT )/ω+vr, where g is the gravitational accelera-

tion, ω is the trap frequency for mF = 1 and vr = 0.58mm/secis an additional velocity due to atom-atom repulsive interac-tion (no fitting parameters). The first term follows from anintegration of Eq. (2), while the collisional constant vr is ob-tained from a full GP simulation. The linearity of the graph isdue to the small interaction times such that sin(ωT )/ω ≈ T .

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integral of the magnetic energy [whose gradient is givenin Eq. (2)] during the interaction time, which is for our re-alization ∆φ = (gFµBµ0IT/π~D)arctan(D/2z). For theexperimental values T = 5 µs, I = 3 A and z = 100 µmthis phase difference is ∆φ ≈ 2π × 165. For phase sta-bility we require the variation of ∆φ to be much smallerthan π. For example, for a variation of 0.2π we needa time stability of δT ∼ 3 ns, and current stability ofδI ∼ 1 mA. It is interesting to note that taking the nar-row wire limit of ∆φ together with Eq. (2) yields the re-lation ∆φ ≈ ∆pz/~, and consequently, moving the atomscloser to the wire improves phase stability.

Fig. 6(c) shows an interference pattern generated witha BEC split in the trap. Here we have allowed the atomsto oscillate in the trap for a period of about 2 ms, whichis approximately a quarter of a period, so that the rel-ative velocity between the two wave-packets is almostzero. The trap is then released and the two wave pack-ets, positioned at z ≈ 0 and at z ≈ ∆p/mω ≡ d, expand,overlap, and form multiple interference fringes. In thiscase it turns out that the main limitation on phase sta-bility is shot-to-shot fluctuations of the magnetic field atthe trap bottom. In our experimental setup we have in-

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FIG. 6: (color online) Interference fringes: (a-b) Large(T = 5µs) and small (T = 10µs) periodicity fringes offreely falling atoms (BEC) observed after the field gradientBS (TOF = 14 ms), when recombining the two |2〉 wave-packets by utilizing a differential magnetic gradient. Thedifferent interaction time results in a momentum differencebetween the clouds, hence different distance before the re-combination, and ends with different fringe spacing, 33µmand 16µm, respectively. The fitted function is A exp(−(z −z0)2/2σ2)(1 + v sin(2π(z − z0)/λ + φ)) (see text for details).(c) Fringes observed after the field gradient BS is appliedto a trapped BEC (T = 700 µs, TOF = 10 ms), whenrecombining the two |2〉 wave-packets by allowing them tooscillate in the trap before releasing them (2 ms oscillationtime). The 8 µm periodicity fits well with the well knownformula for fringe separation ht/md. We fit as before butwith a Thomas-Fermi envelope, with a reduction of the max-imum due to saturation effects. Using the definition f(A) ≡Amax1 − (z − z0)2/w2, 03/2(1 + v sin(2π(z − z0)/λ + φ)),the fit function is f(A)/[1 + f(A/Asat)]. The low visibility isattributed mostly to our limited optical resolution (≈ 5 µm)and our less-than-perfect π/2 pulses.

dependently verified that trap bottom fluctuations are ofthe order of 10 kHz. This corresponds to a phase shift ofδφ ∼ 2π × 7 for a splitting time of T = 700 µs.

To conclude, we have presented a new scheme for afast beam-splitter based on magnetic field gradients andtransitions between Zeeman states. The scheme utilizesa Ramsey-like process which introduces a differential ac-celeration between two wave-packets. This field gradientbeam-splitter is shown to enable a very large splittingmomentum in very short times while also allowing for awide dynamic range. The new beam-splitter may be usedfor a trapped BEC as well as an atomic beam and maybe utilized for future interferometry experiments both forfundamental studies and technological applications suchas large angle (and therefore large area) Sagnac acceler-ation sensors, including in guided “multi-pass” schemes.

We are most grateful to David Groswasser, Zina Bin-shtok, Shuyu Zhou and the rest of the AtomChip groupfor their assistance.

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[24] Unfortunately, we currently do not have an atomic beamat our disposal.

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