Superfluid Quantum Interference in Multiple-Turn Reciprocal Geometry

Supradeep Narayana and Yuki SatoThe Rowland Institute at Harvard, Harvard University, Cambridge, Massachusetts 02142, USA(Received 3 May 2011; revised manuscript received 23 May 2011; published 20 June 2011)

We report the observation of superfluid quantum interference in a compact, large-area matter-wave

interferometer consisting of a multiple-turn interfering path in reciprocal geometry. Utilizing the Sagnac

effect from Earth’s rotation in conjunction with a phase shifter made of superfluid heat current, we

demonstrate that such a scheme can be extended for sensitive rotation sensing as well as for general

interferometry.

DOI: 10.1103/PhysRevLett.106.255301 PACS numbers: 67.25.dg, 03.75.!b, 07.60.Ly, 85.25.Dq

In conventional interferometry devices, beams of light[1] or atoms [2] are split and recombined while enclosingan area A. Certain physical interactions give rise to a phaseshift between the two beams, with the size of the shiftproportional to A and the ‘‘mass’’ of the constituent parti-cles (m for atoms and @!=c2 for photons). Because of thelarger mass, a phase shift for atoms with the same inter-ferometer area is typically 1010–11 times larger than it is forvisible photons. This makes the matter-wave based inter-ferometer a promising candidate for providing unprece-dented sensitivities to external phase shifting influences.For example, when configured as a Sagnac-interferometry-based gyroscope [3], such a device could be used forrotation sensing with possible applications expected ingeodesy, seismology, inertial navigation, and potentiallytesting fundamental theories such as frame-draggingeffects [4] predicted by general relativity.

For the precision measurements identified above, inter-ferometer parameters become a compromise between thedesire for the best resolution achievable and the reality ofthe noisy environment. Although increasing the area wouldimprove their sensitivity, interferometers need to be ascompact as possible to suppress phase shifts due to envi-ronmental noise as well as those due to stray fields. Alarger apparatus would also limit the applications of thedevice. A natural solution to these rather contradictoryrequirements is to let the interfering path circulate arounda small physical area multiple times. One can even envi-sion counter-winding the path in reciprocal geometry. Sucha system would allow common rejection of phase shiftsdue to static perturbations, essentially making the device a1st-order gradiometer [5].

Some of these techniques have been explored with highfinesse mirrors in ring-laser interferometers [6] and withmultiple fiber turns in fiber-optic gyroscopes [7]. Althoughthis past decade has seen dramatic progress in researchers’ability to coherently manipulate the motion of atoms, thesemultiple-turn approaches have been challenging for atominterferometers in confined geometry due to large disper-sive coupling of motion between the guide and confiningdirections in curved atomic waveguides [8,9]. Here we

demonstrate a superfluid 4He matter-wave interferometerwith multiple turns in reciprocal geometry. We show thatthis type of configuration can be extended for sensitiverotation sensing as well as for general interferometry.Our results also reveal important differences betweensuperfluid-based interferometers and atom and optical in-terferometers as well as superconducting quantum inter-ference devices.A picture of our apparatus is shown in Fig. 1(a) and is

schematically represented in Fig. 1(b). A small volume isbounded on one side by two arrays of nanoscale channelsand the other side by a flexible diaphragm. Each arrayconsists of 5625 channels spaced on a 2 !m square latticein a 60 nm-thick silicon nitride membrane. Flow measure-ments above the superfluid transition temperature T"

FIG. 1 (color online). (a) Experimental apparatus.(b) Schematic representation. The X’s indicate the two channelarrays. A flexible metalized diaphragm (D) and a fixed electrode(E) are used to create pressure difference across the nanochan-nels. The diaphragm also forms the input element for a sensitivemicrophone based on superconducting electronics (not shown) todetect the mass current oscillation. A heater (R) and a thin Cusheet (S) connected to the helium bath outside serve as a heatsource and a temperature sink for one set of loops.!#h indicatesthe thermally driven phase shift. The interference path encloses

an area characterized by three area vectors, ~A1, ~A2, and ~A3. Theheight of the apparatus is "9 cm. (c) Simplified view of theinterference loop.

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indicate that the channels are 45# 7 nm in diameter. Theinside is filled with superfluid 4He, and we configure theapparatus to form a loop of superfluid interrupted by twoweak link junctions. See Fig. 1(c).

In an ideal weak coupling limit, the mass current acrossthe channel arrays is governed by the Josephson current-phase relation I $ I0 sin!#, where I0 is the critical currentand!# is the quantum phase difference across the junction[10,11]. In operation, we apply a chemical potential differ-ence!! (combining pressure and temperature differences)across the pair of channel arrays. Fluid within each arrayexhibits mass current oscillation [12] at a Josephson fre-quency !!=h. The combined oscillation amplitude fromtwo arrays exhibits interference depending on the externalphase shift !#ext injected into the system. In a generalcase where each array exhibits different oscillation ampli-tudes I1 and I2, the overall amplitude can be written

as %I1 & I2'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!cos2%!#ext=2' & $2sin2%!#ext=2'

p, where

$ ( %I1 ! I2'=%I1 & I2' is the asymmetry factor. We mea-sure this amplitude by taking a fast Fourier transform of theJosephson current oscillations.

Phase shift !#ext can be induced in variety of ways. Arotation-induced phase shift has been studied extensivelyusing light [1], neutrons [13], neutral atoms [3,14–16], andelectrons [17]. Such a phase shift based on the Sagnac

effect is given by !#s $ %2m4=@' ~" ) ~A, and through thisrelation, the interferometer constitutes a gyroscope thatresponds to the slightest deviation of its frame from aninertial frame.

Since superfluid velocity is fundamentally related to thequantum mechanical phase gradient by vs $ %@=m4'r#,one can also inject phase bias along an interferometer loopby locally creating a thermally driven superfluid current.Such a phase shift is given by !#h $ %m4=@'%l=%'*%&n=&&ssT' _Q, where &, &n, and &s are the total, normal,and superfluid densities, s is the entropy per unit mass, T isthe temperature, and % and l are the cross-sectional areaand the length of the heat current pipe [18].

We distinguish superfluid quantum interference devicesfrom other interferometers in which collective effects arenot significant. Single atom interferometers or even atomslaunched from magneto-optical traps are typically notdense enough to exhibit significant nonlinear effects. Onthe other hand, for a superfluid interferometer, all theatoms are in a single macroscopic entangled state, andthe device technology relies on the nonlinear dynamicsof Josephson weak links placed in a loop governed by amacroscopic wave function. A superfluid interferometer isa neutral analogue of a dc superconducting quantum inter-ference device (dc SQUID [19]), where the role of amagnetic field is now played by inertial as well as gauge-field rotations [20]. The neutral fluid aspect of the devicehas some interesting consequences, as will be shown be-low. We note that an area-enclosing geometry for propa-gating Bose-condensed atoms has been recently achievedwith a chip-limited sensing area of "1500 !m2 [21].

The interfering path of the device reported here consistsof a reciprocal set of two-turn loops (four turns in total).The overall length of the path is 53 cm and the totalenclosed area A+ (if unwound) is 225 cm2, "22 timeslarger than the typical areas used for single-loop devicesemployed in the past [22,23] and 3 orders of magnitudelarger than typical areas used in state-of-the-art atom gyro-scopes [14,24]. We have configured the device in recipro-cal geometry so that most of this area is canceled by the

counter-winding loops [with ~A1 & ~A2 , 0 in Fig. 1(b)],

leaving an effective area j ~A3j" 6:6 cm2 by design. Aheater (R) and a thermal sink (S) are installed in the deviceso that!#h can be induced in one-half of the reciprocal setof loops. This acts as an adjustable differential phaseshifter. The total phase shift in the device is !#ext $!#s & !#h & !#b, where !#b represents any otherconstant current bias in the loop.We first demonstrate the reciprocal principle by measur-

ing Earth’s rotation ~"E through the Sagnac phase shift!#s

with !#h $ 0. The area vectors ~A1, ~A2, and ~A3 all pointhorizontally in the laboratory. Thus we can vary the product~"E ) ~A by reorienting the interferometer about a verticalaxis in the lab frame. Figure 2 shows the result of thisreorientation for several different temperatures. We plotthe overall mass current oscillation amplitude vs !#s=2'.It is clear that interference persists even in this multiple-turn

geometry. Here ~"E ) ~A $ "EA cos(L sin#, where# is the

direction of ~A with respect to an east-west line, and (L ,42:3- is the latitude of Cambridge. Excellent fits (solid

lines) yield ~A $ %7:1 cm2'A3 , ~A3. The fact that we ob-serve the Sagnac effect with an associated area of A3 ratherthan the total area A+ shows that the phase shift in A1 isindeed canceled by that in A2, confirming a successfulimplementation of the reciprocal path configuration.For interferometry that does not require the Sagnac effect

(such as Berry’s phase investigations and gauge-field rota-tion of polarized Bose condensates [20,25]), the reciprocalconfiguration can be used to maintain the interfering pathlength while removing some of the unwanted area thatinevitably couples the device to the environmental noise.

FIG. 2 (color online). Measured mass current oscillationamplitude vs Sagnac phase shift induced by Earth’s rotation.T" ! T $ 6, 5, 4, 3, and 2 mK (from the top).

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In that configuration, one (or both) of the reciprocal sectionscan be used for probing particular physical interactions ofinterest. To test such a differential mode of operation, wehave driven superfluid current in one-half of the reciprocalset of the interfering path as depicted in Fig. 1(b). Theobserved oscillation amplitude vs !#h=2' is plotted inFig. 3(a). In this case, the Sagnac contribution !#s to theoverall phase shift can be viewed as an unwanted perturba-tion, which is significantly reduced by the canceled area as!#s / A. Fits (solid lines) yield l=% $ 5:7* 104 .1=m/.The design values of l $ 0:19 m and % $ 3:6* 10!6 m2

give l=% $ 5:3* 104 .1=m/. This result complements theSagnac result shown in Fig. 2 and provides further evidencethat the matter-based interferometry is possible with super-fluid in a multiple-turn reciprocal path.

Laser interferometers overcome their intrinsic sensitivitydisadvantage by the use of a large sensing area madepossible by better beam splitters and high finesse mirrors.Atom interferometers stay competitive with their intrinsi-cally high sensitivity, although their sensing areas remainrelatively small, currently limited by the lack of better atomoptics and, more importantly, dispersion from waveguideimperfections amplified by betatron oscillations [8,26].Although a lack of such dispersive effects in a superfluidsystem makes it tempting to wind more loops, we showthat the sensing area cannot be scaled up indefinitely. InFig. 3(c), we plot the measured oscillation amplitude vs!#h for a shorter loopwith two turns [See Fig. 3(d) vs 3(b)].The total path length for this apparatus is 28 cm, "50% ofthe four-turn configuration. The same arrays of channels areused and channel dimensions are verified with normal flowmeasurements above T". We find that the modulation depthis shallower for the devicewith a longer path. To explain thisbehavior, we turn to the effect of increasing hydrodynamicinductance associated with the interfering path.

In the field of superconducting magnetometry [19], aseparate pickup loop, having a much larger flux capturearea than the ‘‘bare’’ SQUID, is typically utilized andcoupled to the SQUID inductance via a flux transformer.Such a scheme, made possible by the mutual inductancebetween the primary and secondary coils, allows an appro-priate impedance matching required for the significantincrease of sensing area. Unlike the charged superconduct-ing counterpart, no coupling exists between two loops ofneutral superfluid placed nearby. Therefore, in the super-fluid quantum interference device, the pickup elementneeds to be the superfluid loop consisting of two weaklink junctions (equivalent to a bare SQUID). This un-coupled circuitry of superfluid interferometers makes thehydrodynamic impedance matching an issue as the inter-fering path is made longer. The configuration resemblesuncoupled SQUIDs often employed in SQUID-based mi-croscopes and gradiometric miniature susceptometers forthe study of small magnetic samples [19,27].Writing the fluid kinetic energy in terms of the mass

current E $ LI2=2, the superfluid hydrodynamic induc-tance can be identified as L $ %@=m4'%dI=d!#'!1 [28].For superfluid in a weak link junction, I $ I0 sin!# leadsto kinetic inductance of the weak link (called the Josephsoninductance) LJ $ @=%m4I0' at !# $ 0. In contrast, forsuperfluid flowing through the rest of the interfering pathmade of a simple tube of length l and cross-sectional area%, the inductance Lt $ l=%&s%'.We model our interferometry device as an electrical

circuit represented in Fig. 4(a). The total current passingthrough the device is IT $ I1 sin!#1 & I2 sin!#2, where!#1 and!#2 are the phase drops across the two junctions.The hydrodynamic inductance Lt associated with thelong interfering path also gives rise to a phase drop%m4=@'LtI2 sin!#2. The phase integral condition [18]H ~r# ) d~l $ 0 applied to this circuit yields %!#1 !!#2' ! ) sin!#2 &!#ext $ 0. Here ) ( Lt=LJ is theratio of the inductance associated with the interfering pathto the Josephson inductance. We note that ) is related tothe modulation parameter *L $ 2LI0=$0 (often used inthe analysis of SQUID dynamics) by ) $ '*L, with themagnetic flux quantum $0 replaced by the superfluid

FIG. 3 (color online). (a) Measured mass current oscillationamplitude vs !#h=2'. (b) Four-turn interfering path. (c) Samemeasurement as (a) for a 2-turn interferometer. (d) Two-turninterfering path. For (a) and (c), T" ! T $ 6, 5, 4, 3, and 2 mK(from the top).

FIG. 4 (color online). (a) Electrical analogue of the apparatus.(b) Simulated interference patterns vs!#ext=2' for different)’s.

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circulation quantum +4 $ h=m4 [18]. The interferometerbehavior is now described by IT%(' where ( ( %!#1 &!#2 & ) sin!#2'=2.

The Fourier transform of the current-phase relationcan be written as IT%('=Imax $ a0=2&

P1n$1.an cos%n('&

bn sin%n('/, where an $ %1=''R'!' IT%(0' cos%n(0'd(0 andbn $ %1=''R'!' IT%(0' sin%n(0'd(0. We define the normal-

ized amplitude of the first harmonic I1st=Imax $!!!!!!!!!!!!!!!!!a21 & b21

q

and plot it as a function of ) in Fig. 4(b). It is apparent thatincreasing the interfering path length (and hence )) leadsto a shallower modulation depth, as evidenced in thisexperiment. Configured as a gyroscope, although oneachieves significant rotational sensitivity gain from theincreased area as ," / ,#s=A, higher series inductancecompromises some of this gain.

For the 4-turn reciprocal device and the 2-turn devicediscussed here, we estimate )" 9 and 4, respectively.Increasing the cross-sectional area of the interfering pathshould decrease Lt and hence ), allowing one to harnessthe maximum sensitivity gain with large-area interferome-ters. Increased compressibility of the fluid arising fromlarger fluid volume should be addressed in such configu-rations. Alternatively, one may take advantage of the simi-larity to dc SQUIDs and employ the so-called multiloopdc-SQUID configuration where several large pickup loopsare connected in parallel across the same junctions toreduce the loop inductance while giving an effective areathat equals that of a single loop [19].

Josephson phenomena have been observed in Bose-Einstein condensates (BECs) with a weak link [29,30].The first closed-loop atom circuit has been recently real-ized [31], paving a way to a BEC version of a dc SQUID.As in the case of a superfluid device described above, aBEC quantum interference device is expected to producenontrivial gauge potentials by rotation, making it a prom-ising candidate for a sensitive inertial sensor. Multiple-turnand reciprocal concepts are applicable to such systems, andequivalent impedance matching should be considered.

For the device with counter-wound sensing loopsdiscussed here, the rotational sensitivity (suppressed bydesign) is found to be , 4* 10!7 rad= sec =

!!!!!!Hz

pat T" !

T $ 6 mK. If the interfering path is unwound or woundonly in one direction, the sensitivity would then increaseby a factor of A+=A3, giving a projected resolution of, 1* 10!8 rad= sec =

!!!!!!Hz

p.

In conclusion, we have demonstrated a proof-of-principle superfluid 4He quantum interference devicewith a multiple-turn reciprocal path. With considerationsto the effect of increasing hydrodynamic inductance asso-ciated with a longer interfering path, this approach shouldplay an important role in developing a new generation offorce sensors suited for various applications such as rota-tion sensing. Our findings also reveal several fascinatingsimilarities and dissimilarities between superfluid inter-ferometers and atom and optical equivalents as well assuperconducting quantum interference devices.

We thank D. Rogers for machining, S. Bevis for infra-structure support, and J. VanDelden for nanofabricationwork. Y. S. is grateful to R. Packard for introducing theidea of astatic arrangement and for continual encourage-ment and discussion. Y. S. also acknowledges stimulatingdiscussion with E. Hoskinson, A. Joshi, O. Naaman, D.Pekker, K. Penanen, D. Schuster, and I. Siddiqi. Thisresearch was supported by the Rowland Institute atHarvard University. Preliminary studies that made thisexperiment possible were carried out at UC Berkeley underthe ONR Grant No. N00014-06-1-1183 and the NSF GrantNo. DMR 0244882.

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