Interferometric measurement of the current-phase relationship of a superfluid weak link
S. Eckel, F. Jendrzejewski, A. Kumar, C.J. Lobb, G.K. Campbell
IInterferometric measurement of the current-phase relationship of a superfluid weak link
S. Eckel, ∗ F. Jendrzejewski, A. Kumar, C.J. Lobb, and G.K. Campbell
Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 20899, USA
Weak connections between superconductors or superfluids can di ff er from classical links due to quantumcoherence, which allows flow without resistance. Transport properties through such weak links can be describedwith a single function, the current-phase relationship, which serves as the quantum analog of the current-voltagerelationship. Here, we present a technique for inteferometrically measuring the current-phase relationship ofsuperfluid weak links. We interferometrically measure the phase gradient around a ring-shaped superfluid Bose-Einstein condensate (BEC) containing a rotating weak link, allowing us to identify the current flowing aroundthe ring. While our BEC weak link operates in the hydrodynamic regime, this technique can be extended toall types of weak links (including tunnel junctions) in any phase-coherent quantum gas. Moreover, it can alsomeasure the current-phase relationships of excitations. Such measurements may open new avenues of researchin quantum transport. A variety of quantum phenomena, such as Josephson ef-fects [1] and quantum interference [2, 3], can be observedby weakly connecting two superconductors or superfluids.Such a weak connection can be, for example, a narrow chan-nel or a potential barrier that allows for quantum tunneling.For any weak link, there is a relationship between the cur-rent and the phase di ff erence between the two superconduc-tors or superfluids. This current-phase relationship is essen-tial for understanding quantum transport through the weaklink [4]. In superconductors, the current-phase relationshipof weak links is measured routinely, and such measurementscan indicate the presence of exotic quantum states, such asMajorana fermions [5, 6] or oscillations in the order parame-ter [7]. In superfluid liquid helium, this current-phase relation-ship has been measured, but only indirectly [8]. In degenerateatomic gases, the current-phase relationship has not yet beenmeasured (although many of the e ff ects associated with weaklinks, e.g., Josephson e ff ects [9, 10], have been observed).Here, we interferometrically measure the phase around a ring-shaped superfluid Bose-Einstein condensate (BEC). We usethis technique to determine both the magnitude and sign ofpersistent currents in the ring. In the presence of a rotatingconstriction that acts as a weak link, we show how to measureits current-phase relationship.In a superfluid, the velocity v is related to the gradient of thephase φ of the macroscopic wavefunction by v = ( (cid:126) / m ) ∇ φ ,where (cid:126) is Planck’s constant divided by 2 π and m is the massof an atom. Ignoring the transverse degrees of freedom, thenumber current is then I = ( (cid:126) n / m ) ∇ φ , where n is theequivalent 1D density of the fluid along the direction of flow.In a weak link, the superfluid density will vary as a functionof position and velocity [11], resulting in a potentially com-plicated current-phase relationship [12]. For example, in anidealized Josephson junction, which is typically realized witha tunnel barrier, the phase drop across the weak link γ is re-lated to the current through I = I c sin γ , where I c is its criticalcurrent. Because the ideal Josephson junction can be hardto achieve, the current-phase relationships of experimentallyrealizable weak links can exhibit higher order harmonics orbecome multivalued [13–15].In the present case, we generate a constriction that acts as a weak link in that its critical velocity is much less than thatof the rest of the system [4]. However, our weak link is largecompared to healing length of the BEC, leading to a linearcurrent-phase relationship similar to that of a bulk superfluid.Previous works [9, 10, 16] used weak links that operated inthe tunneling regime; however, none of these measured thecurrent-phase relationship.Weak links have enabled manipulation of ring-shapedBECs, by both controlling a persistent current [17–21] andinducing flow between reservoirs [16, 22]. Because the wave-function must be single valued, the integral of ∇ φ aroundany closed path must be a multiple of 2 π . In particular,for a ring with mean radius R , this leads to the constraint( m / (cid:126) ) (cid:72) v ( θ ) Rd θ = π(cid:96) , where θ is the azimuthal angle andthe integer (cid:96) is a topological invariant known as the windingnumber. Transitions between these quantized states can oc-cur when a weak link stirs the superfluid at a critical rotationrate [19, 21]. In previous experiments with ring-shaped con-densates, the detection method used could only measure themagnitude of resulting winding number (cid:96) . Here, we use aninterference technique to measure the phase and therefore thecurrent flow around a ring-shaped BEC. We demonstrate thatwhen the rotating weak link is present, there is already a cur-rent around the ring even if (cid:96) =
0. This implies that while thewinding number is quantized, neither the average current northe total angular momentum of the BEC are quantized.To measure the phase around the ring, we use two BECs of Na atoms held in an optical dipole trap, as shown in Fig 1(a).One is shaped like a disk and serves as a phase reference. Theother is a concentric ring, which can sustain a persistent cur-rent. To detect the phase of the wavefunction and thus the cur-rent in the condensate, we interfere the two separate conden-sates, which can be accomplished after time-of-flight (TOF)expansion. In fact, such interference experiments provided thefirst conclusive proof that a BEC is a single, phase-coherentobject [23]. Later experiments used similar interference tech-niques to detect quantized vortices [24], to investigate the co-herence properties of a superfluid Fermi gas [25], and to studythe physics of both two dimensional [26] and one-dimensionalBose gases [27]. A method similar to that presented here hasbeen independently developed to investigate the supercurrent a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t
20 μm
75 μm l = -12 l = -6 l = -1 l = 3 l = 8 O p t i c a l d e p t h (a) (b)(c)(d) FIG. 1. (a)
In-situ image of the ring and disk BECs with dimensionsshown. (b) Example interferogram after 15 ms time-of-flight (left)when there is no current in the ring, including traces of the azimuthalinterference fringes to guide the eye (right). (c) Interferograms forvarious winding numbers, where the arrow indicates the direction offlow. (d) Traces of the interference fringes to guide the eye and countthe number of spiral arms. The extracted winding number is shownbelow the traces. generated by a rapid quench through the BEC transition [28].Measuring the interference of our BECs after TOF expan-sion yields a measurement of ψ ∗ D ψ D + ψ ∗ D ψ R + ψ ∗ R ψ D + ψ ∗ R ψ R ,where ψ D is the wavefunction of the disk and ψ R is the wave-function of the ring. The first term P D = ψ ∗ D ψ D produces nofringes as the disk expands. The terms that are of most interesthere contain the ring and the disk, P RD = ψ ∗ D ψ R + ψ ∗ R ψ D , andthey interfere once ψ R and ψ D expand such that they overlap.The last term, P R = ψ ∗ R ψ R , can also produce an interferencepattern once the ring has expanded further, such that its char-acteristic width | σ ( t ) | becomes comparable to R . At this point,the opposite sides of the ring can interfere with each other.For simplicity, let us first consider the interference patternwhen there is no weak link present and both BECs are at restbefore being released from the trap [Fig. 1(b)]. Without flow,the phase is independent of angle in both the disk and ring.The interference term P RD results in concentric circles. Theradial position of these azimuthal interference fringes dependson the relative phases between the two condensates; the radialseparation between fringes corresponds to a phase di ff erenceof 2 π . The interference term P R = ψ ∗ R ψ R produces similar con-centric circles, but with a contrast that is below our detectionthreshold [20, 29].If there is no weak link present but there is a non-zero wind-ing number in the ring, the resulting interference patterns aremodified. In this case, the phase of the ring wavefunction willbe given by φ = (cid:96)θ , assuming the ring is su ffi ciently smooththat both n and v are independent of the azimuthal angle θ .Such a phase profile represents a quantized persistent current:the current takes on discrete values (cid:96) I , where I = n Ω R and Ω = (cid:126) / mR . As shown in Refs. [20, 29], the interference P R is modified in this case: a hole with quantized size appearsat long times. Previous experiments [19, 21, 30] demonstrated xy θ Ω v θφ ( θ ) n ( θ ) v ( θ ) π - π γ R (a) (b)(c)(d) FIG. 2. (a) Schematic of the atoms in the trap with a weak linkapplied. The coordinate system used throughout is shown; θ = x axis. (b) A close up of the weak link region. Whenthe weak link is rotated at Ω , atoms flow through the weak link asshown by the stream lines. Larger velocities along the stream linescorrespond to darker lines. (c) The resulting density n ( θ ), velocity v ( θ ), and phase φ ( θ ) as a function of angle, with the phase drop γ across the weak link shown. (d) Method of extracting the the phasefrom an interferogram (left). First, we trace the interference fringesaround the ring (center) and then fit the discontinuity across the re-gion where the barrier was (right). quantized persistent currents in a ring by releasing the BECfrom a ring-shaped trap (without another BEC present) andobserving the size of the resulting hole. While this methoddetermines the magnitude of the current, it does not determinethe direction.In addition to modifying the P R term, a persistent currentalso modifies the interference term P RD , turning the (cid:96) = (cid:96) (cid:44) P R .) The combination of the ini-tial azimuthal velocity of the ring atoms and the expansion ofthe clouds creates spirals in the interference pattern. One canuse such spirals to measure the accumulated phase around thering α by tracking a maximum (or a minimum) of an inter-ference fringe from θ = θ = π . The net radial fringedisplacement divided by the spacing between fringes yields α/ π . Because α = π(cid:96) in the present case, this procedure isequivalent to counting the number of spiral arms, which de-termines the magnitude of (cid:96) , and noting their chirality, whichdetermines its sign.Adding the weak link modifies the interference pattern be-yond the spirals described above. The weak link, as shown inFig. 2(a)–(b), is a density-depleted region in the ring. Oncethe cloud is released, atoms from either side of the density-depleted weak link expand toward each other and interfere,causing additional interference fringes to appear in the radialdirection, as shown in Fig. 2(d). Just as in the case wherethere was no weak link, we can still measure α by trackingthe azimuthal interference fringes around the ring, excludingthe weak link region. To measure their radial displacementafter going from θ = θ = π , one must extrapolate thosefringes back through the weak link region. Dividing the sizeof the extrapolated radial displacement of a single fringe bythe spacing between the fringes once again yields α/ π . Here, α is not necessarily a multiple of 2 π . This measurement of α allows us to extract the current-phase relationship of the weaklink, as shown below.Before discussing the results, we first describe the exper-imental techniques. The ring and the disk traps are formedby the combination of two crossed lasers. A red-detunedlaser shaped like a sheet creates vertical confinement, whilean intensity-masked blue-detuned laser separates the ring trapfrom the disk trap to form the two BECs. A blue-detunedlaser generates the weak link by creating a Gaussian-shapedrepulsive potential of height U and 1 / e full-width of ≈ µ m(for details on the weak link, see Ref. [22]). This potentialdepletes the density in a small portion of the ring, as shownin Fig. 2(a). On average, a total of ≈ × atoms reside inthe traps. The ring BEC has a mean radius of 22 . µ m andannular width (twice the Thomas-Fermi radius) of ≈ µ m. Itcontains ≈
75 % of the atoms and has an initial chemical po-tential µ / (cid:126) ≈ π × (3 kHz). The central disk contains ≈
25 %of the atoms and has a Thomas-Fermi radius of ≈ µ m. Whilethe disk is approximately hard-walled, the ring is closer to har-monic with a measured radial trapping frequency of ≈
390 Hz.The distance between the inner radius of the ring and the diskis ≈ . µ m.To prepare the system in a well defined quantized persis-tent current state with a chosen (cid:96) , we stir our weak link ata corresponding Ω . Such stirring lasts for 1 s, during whichthe rotation rate of the weak link is constant but the strengthof the weak link potential ramps on linearly in 300 ms, holdsconstant for 400 ms, and ramps o ff in another 300 ms. Tomeasure the resulting (cid:96) , we hold the BECs for an additional100 ms, then release them, and lastly, image the interferencepattern after 15 ms TOF expansion. This procedure producedthe data shown in Fig. 1(b)–(c).We extend these results by measuring α in the presence of aweak link as a function of U , the rotation rate Ω , and the initialwinding number (cid:96) . First, we stir to set the initial windingnumber (cid:96) = ±
1, as described above. To get the highestfidelity for setting (cid:96) (95(2) %), we empirically find that U ≈ . µ and Ω ≈ ± . Ω and U . During the first 300 ms of this secondstage of stirring, the weak link potential ramps from zero tothe chosen U , and afterwards remains constant. We adjust thestarting position such that the weak link is at θ = − − − − Ω /2 π (Hz) − I b u l k / I A − − (a)(b)(c)(d) FIG. 3. Plot of the normalized current around the bulk of the ring, I bulk / I = α/ π , vs. the rotation rate Ω of the weak link for fourdi ff erent weak link potential stengths U : (a) 0 . µ , (b) 0 . µ , (c)0 . µ , (d) 0 . µ . The solid lines are the prediction of our model(see text). The dashed, vertical lines show the predicted transitionsbetween the di ff erent winding number branches. The thin, gray, di-agonal lines represent the case where all the atoms move around thebulk of the ring with the weak link, i.e., I bulk = n R Ω . potential turn o ff , which releases the cloud. After 17 ms TOF(for slightly better resolution), we again image the cloud.The above procedures result in a measurement of the phaseaccumulated around the ring, α , which is related to the cur-rent around the bulk of the ring through I bulk = n ( m / (cid:126) ) ∇ φ = n ( m / (cid:126) )( α/ π R ). We measure I bulk , normalized to I = n Ω R , as a function of Ω for a variety of di ff erent U ; Fig. 3shows four examples. As shown, there are discrete jumps in I bulk at specific rotations rates. At these critical rotation rates,the system experiences a phase slip which changes (cid:96) . Thesecritical rotation rates are dependent on U , and can be hys-teretic. Fig. 3(a)–(b) show such hysteresis. The size of thehysteresis loop is consistent with previous measurements [21].In addition, we measure a non-zero, superfluid I bulk for rota-tion rates below the critical rotation rate, where presumablythere are no excitations and (cid:96) = I bulk can be understood in the following way: As the weaklink rotates around the ring, it must displace superfluid fromin front of its path and superfluid must fill in behind it. Thenumber of atoms that must flow per unit time is proportionalto the di ff erence in the density in the weak link and the bulkof the ring. If the flow was only confined to the weak linkand in the direction opposite of the rotation, as shown bythe solid stream lines in Fig. 2(b), it would violate the con-dition (cid:72) v ( θ ) Rd θ =
0. Thus, the atoms in the bulk of thering must have some velocity in the same direction as the ro-tation (dashed flow lines) in order to cancel the phase accumu-lated by the atoms moving through the weak link, as shown inFig. 2(c). This is analogous to fluxoid vs. flux quantization insuperconductors [31]: although (cid:96) must always be quantized,neither the current I bulk or the total angular momentum are(see Supplemental).Initially, I bulk vs. Ω is linear, and Fig. 4(a) shows its mea-sured derivative dI bulk / d Ω | Ω= as a function of U . As U → I bulk = Ω . For a given rotation,increasing U displaces more atoms, resulting in a larger cur-rent around the bulk of the ring. As expected, dI bulk / d Ω | Ω= continues to increase until U = µ , at which point no atomscan move through the weak link and they all must movearound the ring, i.e., I bulk = n R Ω . This limit corresponds tosolid-body rotation; Fig. 3 shows this limit as thin gray lines.In a reference frame that rotates with the weak link, there isno flow in this limit and thus I WL =
0, where I WL is the cur-rent in the weak link’s frame. The opposite limit of U → I WL = n R Ω (where we have taken I WL > I bulk / I vs. Ω curves of Fig. 3 can be predicted using amodel based on the local density approximation (LDA) to theGross-Pitaevksii equation (see Supplemental and Ref [12]),but assuming a critical velocity as measured in Ref. [21]. (AnLDA treatment can be used because the azimuthal length ofthe weak link of ≈ µ m is larger than the healing length ξ = (cid:112) (cid:126) / m µ ≈ . µ m.) All parameters are measured inde-pendently; none are adjustable. The predictions of this modelare shown in Figs. 3 and 4(a) as the solid curves.The current-phase relationship is best evaluated in the weaklink’s frame, where I WL = n R Ω − I bulk . The phase dropacross the weak link, γ , that corresponds to I WL is given by γ = − π ( I bulk / I ) [see Fig. 2(c)]. For a constant Ω , thecurrent-phase relationship determines how much current flowspast the weak link ( I WL , measured in the weak link’s frame)and how much flows past a fixed point in the bulk of the ring( I bulk ). Using these relationships, we can extract the currentphase-relationship from the data in Fig. 3, the results of whichare shown in Fig. 4(b)–(d).For our BEC system, our model predicts that the current-phase relationship is roughly linear. Non-linearities causedby changes in the superfluid density with γ occur when thevelocity through the weak link nears the speed of sound; how-ever, because our critical velocity is lower than the speed ofsound, these non-linearities are small. Thus, our weak linkis far from an ideal Josephson junction. We also note thatour simple model cannot predict the current-phase relation-ship in the region indicated by the dotted line in Fig. 4(b)–(d).(The dotted lines merely guide the eye between the predictedbranches.) In this branch, we expect dissipation to play a keyrole in the dynamics.Ideally, one would want to apply our method to a weak linkthat could be tuned from the hydrodynamic flow regime ob-served here to the Josephson or tunneling regime. For theideal Josephson junction with a sinusoidal current-phase re- − − π −π π π d I b u l k d Ω n D R U / μ − I W L / I −
101 0 0.2 0.4 0.6 0.8 100.51 (a)(b)(c)(d) γ (radians) FIG. 4. (a) Derivative of the initial bulk current dI bulk / d Ω vs. U ,normalized to the expected value in the limit where U /µ ≥ n R .The solid line shows the prediction of the LDA model. (b)–(d) Ex-tracted current-phase relationships from the data in Fig. 3, for threedi ff erent weak link potential strengths U : (b) 0 . µ , (c) 0 . µ , (d)0 . µ . γ is the phase across the weak link and I WL is the currentthrough it, normalized to I = n R Ω ≈ × atoms / s. The solidcurves represent the prediction of our theoretical model. The dashedlines merely guide the eye by connecting the multiple branches ofthe current-phase relationship. lationship, the I bulk vs. Ω lines in Fig 3 would be curved,a signature that has yet to be observed in degenerate atomicgases. To obtain such a signature, one would need a potentialbarrier whose width is comparable to the healing length of thecondensate to suppress hydrodynamic flow but allow quantummechanical tunneling.In conclusion, we have demonstrated a technique for mea-suring the current-phase relationship of a weak link in a dilute-gas superfluid BEC. We demonstrate that a rotating weak linkalways generates a superfluid current in the bulk of the ring,even when the rotation rate is less than any critical velocityin the system. The magnitude of that current is determinedby the current-phase relationship. Our new method will allowfor better characterization of weak links and, in the case of atunnel junction, should provide the signature of the existenceof idealized Josephson junctions in BEC systems. In addition,measurement of the current-phase relationship enables predic-tion of the hysteretic energy landscape of our system [21],which, like the energy landscape of a flux qubit, should bequantized [32]. More broadly, it is possible that this methodcan be extended to measure the current-phase relationships ofvarious excitations, such as solitonic-vortices [33]. Lastly, thispowerful tool may prove important for studying transport inother, exotic forms of quantum matter, such as unitary Fermigases [34], Tonks-Giradeau gases [35, 36], and quasi-2D con-densates near the BKT transition [26].The authors thank J.G. Lee for technical assistance, M. Ed-wards, E. Tiesinga, R. Mathew, and W. T. Hill III for usefuldiscussions. We thank W.D. Phillips for an extremely thor-ough reading of the manuscript. This work was partially sup-ported by ONR, the ARO atomtronics MURI, and the NSFthrough the PFC at the JQI. S. Eckel is supported by a Na-tional Research Council postdoctoral fellowship. ∗ [email protected][1] B. Josephson, “Coupled Superconductors,” Rev. Mod. 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