letters to nature 5.Haake,E,Kus,M.Scharf,R.Classical and quantum chaos for a kicked top.Z Phys B 65,381-395 (19871. 6.Sanders,B.C.Milburn,G.I.The effect of measurement on the quantum features of a chaotic 3tem.Z.PhxB77,497-510(1989). WA 7.Habib,S..Shizume.K.Zurek.W.H.Decoherence,chaos.and the correspondence principle Ph3.ReK.Left.80,4361-43651998). 8.Graham.R..Schlautmann.M.Zoller,P.Dynamical localization of atomic-beam deflection by a modulated standing light wave.Phys Rev.A 45,R19-R22(1992). 9.Moore,F.L.Robinson,LC..Bharucha,C.F,Sundaram,B.&Raizen,M.G.Atom optics realization of the quantum 8-kicked rotor.Plrys.Rev.Lett.75,4598-4601(1995). 10.Hensinger,W.K,Truscott,A.G..Upcroft,B.Heckenberg N.R&Rubinsztein-Dunlop,H.Atoms in 8 an amplitude-modulated standing wave-dynamics and pathways to quantum chaos.IOpt.B2,659- 667(20001. 0 10 20 30 40 11.Hensinger,W.K.et al Experimental study of the quantum driven pendulum and its cassical analog in Modulation periods atom optics.Phrys.Rev.A (in the press). 12.Arnold,V.1.Mathematical Methads of Classical Mechanics (Springer,New York,1979). 13.Kozuma,M.etal Coherent splitting of Bose-Finstein condensed atoms with optically induced Bragg Figure 5 Momentum distributions as a function of the number of modulation periods, diffraction.Phry's.Rev.Lett.82,871-875 (1999). showing the tunnelling oscillation between negative and positive momenta.We note that 14.Steck,D.A..Oskay,W.H.&Raizen,M.G.Observation of chaos-assisted tunneling between islands of the zero-momentum state remains mostly unpopulated,even when the mean momentum stabilitv.Science (in the press). is zero.The colour coding ranges from blue to red for atomic populations ranging from small to large Acknowledgements We thank C.Holmes for discussions.The NIST group was supported by the ONR,NASA and ARDA,and the University of Qucensland group was supported by the ARC.A.B. was partially supported by DGA(France).and H.H.was partially supported by the A.v.Humboldt Foundation.W.K.H.and B.U.thank NIST for hospitality during the ity is that the tails of the oscillating quantum wave packets may extend outside the region of regular motion,allowing the atoms to 'leak out'into the classically chaotic region.This possibility is Correspondence and requests for materials should be addressed to W.K.H. (e-mail:hensinge@physics.uq.cdu.au). currently under investigation.The contribution of multiple Floquet states could lead to complicated multi-frequency oscillations,and an envelope for the tunnel oscillations appearing as decay,as observed for some parameters in our simulations.The effects of spontaneous emission and atom-atom interactions should be small. Quantum theory predicts dynamical tunnelling to occur for Quantum interference various values of the scaled well depth K,the modulation parameter e and modulation frequency and also predicts a strong sensitivity of superfluid 3He of the tunnelling period and amplitude on these parameters.For e=0.23,k=1.75 and a/2=250kHz we measured a tunnelling R.W.Simmonds,A.Marchenkov,E.Hoskinson,J.C.Davis period of approximately 13 modulation periods.As shown in R.E.Packard Fig.4b,for e=0.30,K=1.82 and a/2=222kHz we find a tunnelling period of 6 modulation periods with a significantly Physics Department,University of California,Berkeley,California 94720,USA longer decay time than in Fig.4a.We have also experimentally 444 observed an increase in the tunnelling period when k is decreased Celebrated interference experiments have demonstrated the and when all other parameters are held constant.This is the wave nature of light'and electrons,quantum interference opposite of what one would expect for spatial barrier tunnelling. being the manifestation of wave-particle duality.More recently, Our observation of dynamical tunnelling of atoms in a modu- double-path interference experiments have also demonstrated lated standing wave opens the door to further studies in quantum the quantum-wave nature of beams of neutrons',atoms'and nonlinear dynamics.By varying the hamiltonian parameters and the Bose-Einstein condensates'.In condensed matter systems, initial conditions,we observe dynamical tunnelling for a variety of double-path quantum interference is observed in the d.c.super- mixed phase space configurations.This may be 'chaos-assisted conducting quantum interference device(d.c.SQUID).Here we tunnelling,and a tunnelling rate that varies wildly as system report a double-path quantum interference experiment involv- parameters are changed would be a signature of such tunnelling'. ing a liquid:superfluid 'He.Using a geometry analogous to the By introducing noise or spontaneous emission in a controlled superconducting d.c.SQUID,we control a quantum phase shift manner,we could systematically investigate the role of decoherence by using the rotation of the Earth,and find the classic inter- in tunnelling and explore the classical limit of chaotic systems.By ference pattern with periodicity determined by the He quantum carefully following the evolution of wave packets loaded into the of circulation. chaotic region from a Bose-Einstein condensate,we could probe Our basic interferometer topology is shown in Fig.la.Schemati- quantum chaos'with the unprecedented resolution afforded by cally,the device is a circular loop of radius R which includes two minimum uncertainty wave packets. superfluid He Josephson weak links's.These weak links each During the preparation of this Letter,we learned of an consist of a 65 x 65 array of 100-nm apertures etched in a 60-nm- experiment4 reporting tunnelling of a different motional state in thick silicon nitride membrane.Similar arrays have previously been a similar system. □ shown to be characterized by a current-phase relation givenby the Josephson formula: Received 10 May:accepted 12 June 2001. I=I sino (1) 1.Tomsowvic,S.Tunneling and chaos.Physica Scripta T90,162-165 (2001) 2.Caldeira,A.O.Leggett,A.J.Quantum tunneling in a dissipative system.Arn.Phrys.149,374-456 Here I is the mass current flowing through the array,is the (1983). quantum phase difference across the array,and I.is the critical 3.Davis,M.J.Heller,E.J.Quantum dynamical tunneling in bound states.J.Chem Phys.75,246-254 current characterizing the array. (1981). 4.Dyrting.S..Milburn,G.J.Holmes,C.A.Nonlinear quantum dynamics at a classical second order The interferometer is predicted to behave as a single weak link resonance.P%y.RaE48,969-978(1993). with an effective critical current,(see Methods).If the inter- NATURE|VOL 4125 JULY 2001 www.nature.com 然©2001 Macmillan Magazines Ltd 55
letters to nature NATURE | VOL 412 | 5 JULY 2001 | www.nature.com 55 ity is that the tails of the oscillating quantum wave packets may extend outside the region of regular motion, allowing the atoms to `leak out' into the classically chaotic region. This possibility is currently under investigation. The contribution of multiple Floquet states could lead to complicated multi-frequency oscillations, and an envelope for the tunnel oscillations appearing as decay, as observed for some parameters in our simulations. The effects of spontaneous emission and atom±atom interactions should be small. Quantum theory predicts dynamical tunnelling to occur for various values of the scaled well depth k, the modulation parameter e and modulation frequency q, and also predicts a strong sensitivity of the tunnelling period and amplitude on these parameters. For e 0:23, k 1:75 and q=2p 250 kHz we measured a tunnelling period of approximately 13 modulation periods. As shown in Fig. 4b, for e 0:30, k 1:82 and q=2p 222 kHz we ®nd a tunnelling period of 6 modulation periods with a signi®cantly longer decay time than in Fig. 4a. We have also experimentally observed an increase in the tunnelling period when k is decreased and when all other parameters are held constant. This is the opposite of what one would expect for spatial barrier tunnelling. Our observation of dynamical tunnelling of atoms in a modulated standing wave opens the door to further studies in quantum nonlinear dynamics. By varying the hamiltonian parameters and the initial conditions, we observe dynamical tunnelling for a variety of mixed phase space con®gurations. This may be `chaos-assisted tunnelling', and a tunnelling rate that varies wildly as system parameters are changed would be a signature of such tunnelling1 . By introducing noise or spontaneous emission in a controlled manner, we could systematically investigate the role of decoherence in tunnelling and explore the classical limit of chaotic systems. By carefully following the evolution of wave packets loaded into the chaotic region from a Bose±Einstein condensate, we could probe `quantum chaos' with the unprecedented resolution afforded by minimum uncertainty wave packets. During the preparation of this Letter, we learned of an experiment14 reporting tunnelling of a different motional state in a similar system. M Received 10 May; accepted 12 June 2001. 1. Tomsovic, S. Tunneling and chaos. Physica Scripta T 90, 162±165 (2001). 2. Caldeira, A. O. & Leggett, A. J. Quantum tunneling in a dissipative system. Ann. Phys. 149, 374±456 (1983). 3. Davis, M. J. & Heller, E. J. Quantum dynamical tunneling in bound states. J. Chem Phys. 75, 246±254 (1981). 4. Dyrting, S., Milburn, G. J. & Holmes, C. A. Nonlinear quantum dynamics at a classical second order resonance. Phys. Rev. E 48, 969±978 (1993). 5. Haake, F., Kus, M. & Scharf, R. Classical and quantum chaos for a kicked top. Z. Phys. B 65, 381±395 (1987). 6. Sanders, B. C. & Milburn, G. J. The effect of measurement on the quantum features of a chaotic system. Z. Phys. B 77, 497±510 (1989). 7. Habib, S., Shizume, K. & Zurek, W. H. Decoherence, chaos, and the correspondence principle. Phys. Rev. Lett. 80, 4361±4365 (1998). 8. Graham, R., Schlautmann, M. & Zoller, P. Dynamical localization of atomic-beam de¯ection by a modulated standing light wave. Phys. Rev. A 45, R19±R22 (1992). 9. Moore, F. L., Robinson, J. C., Bharucha, C. F., Sundaram, B. & Raizen, M. G. Atom optics realization of the quantum d-kicked rotor. Phys. Rev. Lett. 75, 4598±4601 (1995). 10. Hensinger, W. K., Truscott, A. G., Upcroft, B., Heckenberg, N. R. & Rubinsztein-Dunlop, H. Atoms in an amplitude-modulated standing wave±dynamics and pathways to quantum chaos. J. Opt. B 2, 659± 667 (2000). 11. Hensinger, W. K.et al. Experimental study of the quantum driven pendulum and its classical analog in atom optics. Phys. Rev. A (in the press). 12. Arnold, V. I. Mathematical Methods of Classical Mechanics (Springer, New York, 1979). 13. Kozuma, M. et al. Coherent splitting of Bose-Einstein condensed atoms with optically induced Bragg diffraction. Phys. Rev. Lett. 82, 871±875 (1999). 14. Steck, D. A., Oskay, W. H. & Raizen, M. G. Observation of chaos-assisted tunneling between islands of stability. Science (in the press). Acknowledgements We thank C. Holmes for discussions. The NIST group was supported by the ONR, NASA and ARDA, and the University of Queensland group was supported by the ARC. A.B. was partially supported by DGA (France), and H. H. was partially supported by the A. v. Humboldt Foundation. W.K.H. and B.U. thank NIST for hospitality during the experiments. Correspondence and requests for materials should be addressed to W.K.H. (e-mail: hensinge@physics.uq.edu.au). 20 Modulation periods Momentum (ùk) 10 30 40 4 0 –4 –8 8 0 Figure 5 Momentum distributions as a function of the number of modulation periods, showing the tunnelling oscillation between negative and positive momenta. We note that the zero-momentum state remains mostly unpopulated, even when the mean momentum is zero. The colour coding ranges from blue to red for atomic populations ranging from small to large. ................................................................. Quantum interference of super¯uid 3 He R. W. Simmonds, A. Marchenkov, E. Hoskinson, J. C. Davis & R. E. Packard Physics Department, University of California, Berkeley, California 94720, USA .............................................................................................................................................. Celebrated interference experiments have demonstrated the wave nature of light1 and electrons2 , quantum interference being the manifestation of wave±particle duality. More recently, double-path interference experiments have also demonstrated the quantum-wave nature of beams of neutrons3 , atoms4 and Bose±Einstein condensates5 . In condensed matter systems, double-path quantum interference is observed in the d.c. superconducting quantum interference device6 (d.c. SQUID). Here we report a double-path quantum interference experiment involving a liquid: super¯uid 3 He. Using a geometry analogous to the superconducting d.c. SQUID, we control a quantum phase shift by using the rotation of the Earth, and ®nd the classic interference pattern with periodicity determined by the 3 He quantum of circulation. Our basic interferometer topology is shown in Fig. 1a. Schematically, the device is a circular loop of radius R which includes two super¯uid 3 He Josephson weak links7,8. These weak links each consist of a 65 ´ 65 array of 100-nm apertures etched in a 60-nmthick silicon nitride membrane. Similar arrays have previously been shown to be characterized by a current±phase relation given9 by the Josephson formula: I Ic sinf 1 Here I is the mass current ¯owing through the array, f is the quantum phase difference across the array, and Ic is the critical current characterizing the array. The interferometer is predicted to behave as a single weak link with an effective critical current, I p c (see Methods). If the inter- © 2001 Macmillan Magazines Ltd
letters to nature ferometer is rotated at angular velocity I is modulated by an interferometer should evolve according to the equation3: interference term0 =(》 中=- [2ms P(t)dt (3) (2) oh wherep is the density of the liquid and P is the pressure differential where A is the area vector of the loop and K3=h/(2m)is the He across the ends of the device.So if the interferometer behaves as a quantum of circulation.Thus the effective critical current is pre- single Josephson weak link,a static pressure Po applied across the dicted to be modulated with a period determined by the rotation interferometer will cause the quantum phase to increase linearly flux through the interferometer,analogous to the manner in which in time,leading to mass current oscillations at a Josephson magnetic flux modulates the critical current of a d.c.SQUID. frequency: The interference occurs because the phase of the superfluid wavefunction receives equal and opposite shifts in the two arms 2P0 = (4) of the interferometer.The phase shift arising from the rotation flux hp is identical in form to that in other neutral-matter quantum As described in Fig.1 legend,the deflection of the flexible rotation interferometers3,as well as in the optical Sagnac membrane reveals both the pressure across the interferometer and interferometer2 if the photon effective mass is taken to be hole2. the mass current through it.We have developed a feedback method The goal ofour experiment is to demonstrate two ideas.(1)That that permits us to drive the system at constant pressure by applying the superfluid interferometer is indeed characterized as a single a time-varying voltage to the membrane.We select a pressure for weak link with a well defined critical current I,and (2)that by which the Josephson frequency lies near 270 Hz,a spectral region changing the orientation of the plane of the loop with respect to the away from parasitic acoustic noise lines in the displacement Earth's rotation vector,the current I'can be modulated according to transducer. the interference term in equation(2).I:should exhibit periodicity Figure 2 shows a Fourier transform of the mass current through of(2-A)/K3. the interferometer that results from the constant-pressure drive A geometrically more accurate sketch of our interferometer being applied for about 6 seconds.A sharp peak at 273Hz is is shown in Fig.1b.The quantum phase difference across the clearly visible.This corresponds to the Josephson oscillation.This =sinΦ Weak links x⑨ Electrode SQUID Membrane sense coil Figure 1 Two views of our superfluid quantum interferometer.a,Schematic diagram of trapped persistent curent which is partially shunted into the input coi of a d.c.SQUD if the basic interferometer loop.As shown in the Methods section,the total current through the membrane moves?.Knowledge of the deflection of the membrane,(),reveals the the two sides of the loop (is predicted to depend on the total phase difference across the pressure head across the interferometer.The rate of change of the deflection reveals the loop,,as given by equation (1)with replaced by (given by equation (2).b,The mass current through the interferometer.The noise in the displacement sensor is interferometer geometry used in this experiment.Although the two weak links are moved 2x10-5mHz-We use a feedback system that applies a voltage to the electrode closer together,the topology is unchanged and/is still given by equation(2).The such that the pressure across the interferometer is fixed at some predetemined value, nominal 'loop'area,A,is ~6 cm2,and the tube cross-sectional diameter is ~0.3 cm. typically near 1.5 mPa.Temperature is monitored with two thermometers,one using Pressure differences can be applied to the flexible metallized membrane(and hence platinum nuclear magnetic resonance and a second using lanthanum cerium magnesium across the two weak links)by applying voltages between it and the adjacent rigid nitrate (LCMN).The experiment is performed at zero ambient pressure. electrode.The membrane is coated with a superconducting film.The sense coil contains a 56 然©2001 Macmillan Magazines Ltd NATURE|VOL 412|5 JULY 2001 www.nature.com
letters to nature 56 NATURE | VOL 412 | 5 JULY 2001 | www.nature.com ferometer is rotated at angular velocity Q, I p c is modulated by an interference term10 I p c 2Ic cos p 2Q×A k3 2 where A is the area vector of the loop and k3 [ h= 2m3 is the 3 He quantum of circulation. Thus the effective critical current is predicted to be modulated with a period determined by the rotation ¯ux through the interferometer, analogous to the manner in which magnetic ¯ux modulates the critical current of a d.c. SQUID. The interference occurs because the phase of the super¯uid wavefunction receives equal and opposite shifts in the two arms of the interferometer. The phase shift arising from the rotation ¯ux is identical in form to that in other neutral-matter quantum rotation interferometers3,11, as well as in the optical Sagnac interferometer12 if the photon effective mass is taken to be ~q=c 2 . The goal of our experiment is to demonstrate two ideas. (1) That the super¯uid interferometer is indeed characterized as a single weak link with a well de®ned critical current I p c , and (2) that by changing the orientation of the plane of the loop with respect to the Earth's rotation vector, the current I p c can be modulated according to the interference term in equation (2). I p c should exhibit periodicity of 2Q×A=k3. A geometrically more accurate sketch of our interferometer is shown in Fig. 1b. The quantum phase difference across the interferometer should evolve according to the equation13: © 2 # 2m3 r~ P tdt 3 where r is the density of the liquid and P is the pressure differential across the ends of the device. So if the interferometer behaves as a single Josephson weak link, a static pressure P0 applied across the interferometer will cause the quantum phase to increase linearly in time, leading to mass current oscillations at a Josephson frequency8 : qj 2m3 ~r P0 4 As described in Fig. 1 legend, the de¯ection of the ¯exible membrane reveals both the pressure across the interferometer and the mass current through it. We have developed a feedback method that permits us to drive the system at constant pressure by applying a time-varying voltage to the membrane. We select a pressure for which the Josephson frequency lies near 270 Hz, a spectral region away from parasitic acoustic noise lines in the displacement transducer. Figure 2 shows a Fourier transform of the mass current through the interferometer that results from the constant-pressure drive being applied for about 6 seconds. A sharp peak at 273 Hz is clearly visible. This corresponds to the Josephson oscillation. This Electrode SQUID sense coil Membrane x(t) A a b Weak links ⇒ I = Icsin A b a c d I2 = Icsinφ2 * I1 = Icsinφ1 It Ω Φ Φ Ω It Φ Figure 1 Two views of our super¯uid quantum interferometer. a, Schematic diagram of the basic interferometer loop. As shown in the Methods section, the total current through the two sides of the loop (It ) is predicted to depend on the total phase difference across the loop, ©, as given by equation (1) with Ic replaced by I p c (given by equation (2)). b, The interferometer geometry used in this experiment. Although the two weak links are moved closer together, the topology is unchanged and I p c is still given by equation (2). The nominal `loop' area, A, is ,6 cm2 , and the tube cross-sectional diameter is ,0.3 cm. Pressure differences can be applied to the ¯exible metallized membrane (and hence across the two weak links) by applying voltages between it and the adjacent rigid electrode. The membrane is coated with a superconducting ®lm. The sense coil contains a trapped persistent current which is partially shunted into the input coil of a d.c. SQUID if the membrane moves26. Knowledge of the de¯ection of the membrane, x(t ), reveals the pressure head across the interferometer. The rate of change of the de¯ection reveals the mass current through the interferometer. The noise in the displacement sensor is 2 3 10 2 15 m Hz 2 1=2 . We use a feedback system that applies a voltage to the electrode such that the pressure across the interferometer is ®xed at some predetermined value, typically near 1.5 mPa. Temperature is monitored with two thermometers, one using platinum nuclear magnetic resonance and a second using lanthanum cerium magnesium nitrate (LCMN). The experiment is performed at zero ambient pressure. © 2001 Macmillan Magazines Ltd
letters to nature frequency agrees with equation (4)to within the systematic uncer- reorientation of a superfluid He loop to change the frequency in a tainty (~10%)in the electrostatically derived calibration of the nonlinear Helmholtz oscillators containing a single Josephson pressure scale.We have determined that the oscillation frequency weak link. scales linearly with Po at least up to 1 kHz,and that there is no Figure 3 shows the result of our reorientation experiment.We higher-harmonic signal detectable with the present signal-to-noise plot the normalized measured current magnitude I as a function ratio.This implies that the overall current-phase relation is sine- of (20-A)/K.The data reveal the double-path quantum inter- like,and that the two separated arrays are phase coherent.The area ference pattern.The solid curve is the interference pattern under the Josephson oscillation peak in Fig.2 is proportional to e, expected from equation(2).The periodicity of the observed inter- the critical current for the interferometer. ference pattern is found to be as predicted:(20-A)/K3,within the This oscillation signal results from the quantum coherence experimental precision (which is limited by knowledge of the loop among all the apertures in each array,as well as between both area). arrays.Thus,not only are both weak links themselves coherent,but The interference pattern predicted by equation(2)and shown in the ~102 atoms within the torus are also quantum phase coherent Fig.3 is the central feature of this experiment.The device displays a with them.This demonstrates the first point,that the interferometer remarkable phenomenon:two-path quantum interference in a behaves as a single Josephson weak link with a well defined critical liquid.The pattern shows that the interferometer is indeed the current,I. superfluid equivalent of a d.c.SOUID. Furthermore,weak-link arrays have previously been shown'4 to It is natural to ask if the superfluid quantum interference device exhibit a characteristic rigid-pendulum oscillation mode whose could be developed into a sensitive rotation sensor,perhaps to small-amplitude frequency p is proportional to I.We have perform meaningful geodesy measurements or experiments on observed this normal mode motion in our interferometer and find general relativity.Following the analysis2 of superconducting d.c. I:,again confirming that the device behaves as a single weak SQUIDs,the intrinsic noise in this type of device arises from link characterized by equations(1)and(3). Nyquist noise in the various dissipative processes associated with The second goal of our experiment is to see if phase gradients, the weak links themselves2.The potential intrinsic sensitivity can generated by absolute rotation of the loop,create an interference only be reached if other extrinsic noise sources-such as tempera- pattern to modulate I,as predicted in equation(2).We chose the ture drifts,environmental noise and electronic SQUID readout loop area A to be 6 cm'so that the rotation of the Earth,D,can itself noise-are reduced by orders of magnitude from the values in the lead to several periods of the modulation pattern.The plane of the present experiment. loop is vertical-that is,the vector A points horizontally in the We have demonstrated a double-path superfluid interferometer laboratory.Thus we can vary the rotation flux in the loop,A, which is the analogue of a d.c.SQUID.The entire interferometer by reorienting the interferometer about a vertical axis in the exhibits dynamic behaviour (Josephson oscillations and pendulum laboratory frame.This reorientation technique was used previously motion)similar to that of a single weak link.Owing to quantum in superfluid He to show that rotation flux could vary the phase- interference,the effective critical current is modulated by rotation slip critical velocity in an apertureOther experiments employed flux through the enclosed area.Our double-path interference Josephson current 1.0 oscillation peak 0.8 0.6 (-s 6u)quajino ssew 以 0.4 0.2 1 0.0 -1.0 -0.5 0.0 0.5 1.0 20-A 0 K3 240 250 260270280 290 Josephson frequency(Hz) Figure 3 The interference pattem of a superfluid quantum interference gyroscope.The figure shows a plot of the effective critical current as a function of (20-A)/.where Figure 2 The spectrum of the mass current during a 2-s interval of the data stream from 3 is the quantum of circulation(see equation (2)).The error bars,which are the standard the SQUID position transducer.A typical data stream is 6s long,limited by the magnitude error of typically 100 FFTs,are smaller than the size of the plotted points.The temperature of the voltage we can apply to the diaphragm.Due to imperfect pressure regulation,the was approximately 0.8T,where T is the critical temperature of the superfluid.We Josephson peak drifts slightly during the 6-s data stream.So we break the transient into have applied a temperature correction (no greater than 4%)to account for systematic 2-s intervals,and use a fast Fourier transform (FFT)routine to produce a figure like errors from small temperature drifts recorded by the LCMN during the course of the that above for each interval.The peak is due to Josephson oscillations at 273Hz.We measurement.The rotation flux was varied by reorienting the normal to the loop's plane compute a number proportional to the area under the peak (shaded in the figure).At a through an angle of/2 with respect to the east-west direction.Then the rotation fixed orientation we record multiple data streams (typically 17).and average the flux R-A is equal to Acose,sin,where is the direction of A with respect to resultant areas.This average is taken as the measure of/,the critical current of the an east-west line,and=38 is the latitude of Berkeley.The solid line is a plot of interferometer. equation(②. NATURE|VOL 4125 JULY 2001 www.nature.com 然©2001 Macmillan Magazines Ltd 57
letters to nature NATURE | VOL 412 | 5 JULY 2001 | www.nature.com 57 frequency agrees with equation (4) to within the systematic uncertainty (,10%) in the electrostatically derived calibration of the pressure scale. We have determined that the oscillation frequency scales linearly with P0 at least up to 1 kHz, and that there is no higher-harmonic signal detectable with the present signal-to-noise ratio. This implies that the overall current±phase relation is sinelike, and that the two separated arrays are phase coherent. The area under the Josephson oscillation peak in Fig. 2 is proportional to I p c , the critical current for the interferometer. This oscillation signal results from the quantum coherence among all the apertures in each array, as well as between both arrays. Thus, not only are both weak links themselves coherent, but the ,1022 atoms within the torus are also quantum phase coherent with them. This demonstrates the ®rst point, that the interferometer behaves as a single Josephson weak link with a well de®ned critical current, I p c . Furthermore, weak-link arrays have previously been shown14 to exhibit a characteristic rigid-pendulum oscillation mode whose small-amplitude frequency qp is proportional to Ic p . We have observed this normal mode motion in our interferometer and ®nd qp ~ Ip c p , again con®rming that the device behaves as a single weak link characterized by equations (1) and (3). The second goal of our experiment is to see if phase gradients, generated by absolute rotation of the loop, create an interference pattern to modulate I p c , as predicted in equation (2). We chose the loop area Ato be 6 cm2 so that the rotation of the Earth, QE, can itself lead to several periods of the modulation pattern. The plane of the loop is verticalÐthat is, the vector A points horizontally in the laboratory. Thus we can vary the rotation ¯ux in the loop, QE×A, by reorienting the interferometer about a vertical axis in the laboratory frame. This reorientation technique was used previously in super¯uid 4 He to show that rotation ¯ux could vary the phaseslip critical velocity in an aperture15,16. Other experiments employed reorientation of a super¯uid 3 He loop to change the frequency in a nonlinear Helmholtz oscillator17,18 containing a single Josephson weak link. Figure 3 shows the result of our reorientation experiment. We plot the normalized measured current magnitude I p c as a function of 2Q×A=k3. The data reveal the double-path quantum interference pattern. The solid curve is the interference pattern expected from equation (2). The periodicity of the observed interference pattern is found to be as predicted: 2Q×A=k3, within the experimental precision (which is limited by knowledge of the loop area). The interference pattern predicted by equation (2) and shown in Fig. 3 is the central feature of this experiment. The device displays a remarkable phenomenon: two-path quantum interference in a liquid. The pattern shows that the interferometer is indeed the super¯uid equivalent of a d.c. SQUID. It is natural to ask if the super¯uid quantum interference device could be developed into a sensitive rotation sensor19±21, perhaps to perform meaningful geodesy measurements or experiments on general relativity. Following the analysis22 of superconducting d.c. SQUIDs, the intrinsic noise in this type of device arises from Nyquist noise in the various dissipative processes associated with the weak links themselves23. The potential intrinsic sensitivity can only be reached if other extrinsic noise sourcesÐsuch as temperature drifts, environmental noise and electronic SQUID readout noiseÐare reduced by orders of magnitude from the values in the present experiment. We have demonstrated a double-path super¯uid interferometer which is the analogue of a d.c. SQUID. The entire interferometer exhibits dynamic behaviour (Josephson oscillations and pendulum motion) similar to that of a single weak link. Owing to quantum interference, the effective critical current is modulated by rotation ¯ux through the enclosed area. Our double-path interference 240 250 260 270 280 290 0 1 2 3 4 5 Mass current (ng s–1) Josephson frequency (Hz) Josephson current oscillation peak Figure 2 The spectrum of the mass current during a 2-s interval of the data stream from the SQUID position transducer. A typical data stream is 6 s long, limited by the magnitude of the voltage we can apply to the diaphragm. Due to imperfect pressure regulation, the Josephson peak drifts slightly during the 6-s data stream. So we break the transient into 2-s intervals, and use a fast Fourier transform (FFT) routine to produce a ®gure like that above for each interval. The peak is due to Josephson oscillations at 273 Hz. We compute a number proportional to the area under the peak (shaded in the ®gure). At a ®xed orientation we record multiple data streams (typically 17), and average the resultant areas. This average is taken as the measure of I p c , the critical current of the interferometer. 2Ω⋅A κ3 –1.0 –0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Ic I*c max * Figure 3 The interference pattern of a super¯uid quantum interference gyroscope. The ®gure shows a plot of the effective critical current I p c as a function of 2Q×A=k3, where k3 is the quantum of circulation (see equation (2)). The error bars, which are the standard error of typically 100 FFTs, are smaller than the size of the plotted points. The temperature was approximately 0.8Tc, where Tc is the critical temperature of the super¯uid. We have applied a temperature correction (no greater than 4%) to account for systematic errors from small temperature drifts recorded by the LCMN during the course of the measurement. The rotation ¯ux was varied by reorienting the normal to the loop's plane through an angle of 6p/2 with respect to the east±west direction. Then the rotation ¯ux Q×A is equal to E A cosvL sin£, where £ is the direction of A with respect to an east±west line, and vL < 388 is the latitude of Berkeley. The solid line is a plot of equation (2). © 2001 Macmillan Magazines Ltd
letters to nature experiment advances the close analogy between the macroscopic 21.Herring.T.A.The rotation of the Earth.Rev.Geophys.Suppl 29,172-175 (1991). quantum physics of superconductivity and superfluidity.Both 22.Clarke,1.in SQUID Sensors:Fundamentls,Fabricntion and Applications (ed.Weinstock,H.)(Kluwer Acdemic.1996). systems exhibit persistent currents,quantized circulation (fluid 23.Simmonds,R.W.Marchnkov,A..Vitale,S.,Davis,J.C.Packard,R.E.New flow dissipation and magnetic),sino weak links and now,double-path quantum mechanisms in superfluid He.Phys.Rev.Lett.84,6062-6065(2000). interference. Q 24.Feynmann,R.P..Leighton,R.B.Sands,M.The Feynman Lectures in Physics Vol.3.Ch.21 (Addison Wesley,Reading.Massachusetts,1963). 25.Tilley,D.R.&Tilley,I.Superfhtidlity and Superconductivity 3rd edn 171 (Hilger,Bristol,1990). Methods 26.Paik,H.I.Superconducting tunable-diaphragm transducer for sensitive acceleration measurements. To derive equation(2)we follow a heuristic method similar to that which Feynman2 LApl.Ph5.47,1168-1178(1976). applied to the analogous superconducting case.Consider the interferometer in Fig.la to be rotating at angular velocity The total external current passing through the interferometer may be written as: Acknowledgements 4-lna+na=2aor色色）m(色色) We thank S.Vitale and K.Penanen for discussions;Y.Sato for assistance;A.Loshak for (5) making the aperture arrays;and E.Crump,D.Mathews and C.Ku for assistance in improving noise conditions in our building.This work was supported in part by NASA. We assume that the fluid in the torus is characterized at every point by a quantum phase the Office of Naval Research,the National Science Foundation,and the Miller Institute for factor which is macroscopically coherent.The rotation of the torus entrains the superflow Basic Research (J.C.D.). which is then almost that of a solid body,v,=OR.Around a closed path in the interferometer,as indicated by the dotted line in Fig.I,the phase can change only in Correspondence and requests for materials should be addressed to R.P. multiples of 2. (e-mail:packard@socrates.berkeley.edu). T中-dl=2xn= 票a++-=-0+ (6) The limit points,a-d,in the integrals are shown in Fig.la.In equation(6)we have used the concept that the phase gradient is proportional to superfluid velocity, Vo =(2m,v,/h.Equation (6)may be solved for which,when inserted into equation (5)gives Coexistence of superconductivity I=I sin (7 and ferromagnetism where=(,)/2 is the phase difference across the ends of the interferometer.We see that the rotating interferometer behaves as a single Josephson weak link,whose maximum in the d-band metal ZrZn2 current is given by the quantum interference term: C.Pfleiderer,M.Uhlarz*,S.M.Haydent,R.Vollmer",H.v.Lohneysen, =4(》 (8) N.R.Bernhoefts G.G.Lonzarichll Here K,=h/(2m,)is the quantum of circulation,A is the interferometer area vector, Physikalisches Institut,Universitat Karlsruhe,D-76128 Karlsruhe,Germany normal to the plane,and we have included a scalar product describing the more general H.H.Wills Physics Laboratory,University of Bristol,Bristol BS8 ITL,UK case when the rotation axis is not parallel to A. Forschungszentrum Karlsruhe,Institut fur Festkorperphysik, Received 5 March:accepted 14 May 2001. D-76021 Karlsruhe,Germany DRFMC-SPSMS,CEA Grenoble,F-38054 Grenoble,Cedex 9,France 1.Young.T.A Course of Lectures in Natural Philosophy and the Mechanical Arts Vol.1.364 (London. Cavendish Laboratory,University of Cambridge,Madingley Road, 1845);also as facsimile edition (New York,1971). 2.Davisson,C.L Are Electrons Waves?Franklin Inst.I.205,597 (1928). Cambridge CB3 0HE,UK 3.Werner,S.A.Studenmann,&Colella,R.Effect of Earth's rotation on the quantum mechanical phase of the neutron.Phys Rev.Lett.42,1103-1106 (1979). Keith,D.W.,Ekstrom,C.RTurchette,Q.A&Pritchard,D.E.An interferometer for atoms.Phys. It has generally been believed that,within the context of the Rax.Let66,2693-2696(1991). Bardeen-Cooper-Schrieffer (BCS)theory of superconductivity, 5.Andrews M.R.ctal.Observation of interference between two Bose condensates.Science 275,637-641 the conduction electrons in a metal cannot be both ferromag- (1997). 6.Barone,A.Paterno.G.Physic and Application of the Josephson Effect (Wiley.New York. netically ordered and superconducting'.Even when the super- 19821. conductivity has been interpreted as arising from magnetic 7.Varoquaux,E.&Avenel,O.Josephson effect and quantum phase slippage in superfluids Phrys.Rev. mediation of the paired electrons,it was thought that the super- Let.60.416-4191988). 8.Pereversev.S.V..Loshak,A.,Backhaus,S.,Davis,IC.Packard,R.E.Quantum oscillations between conducting state occurs in the paramagnetic phase.Here we two weakly coupled reservoirs of superfluid 'He.Nature 388,449-451(1997). report the observation of superconductivity in the ferromagneti- 9.Marchenkow,A.et al.Bi-state superfluid He weak links and the stability of Josephson states.Phys. cally ordered phase of the d-electron compound ZrZn2.The a.L1t.83.3860-3863(1999). specific heat anomaly associated with the superconducting transi- 10.Packard.R.E&Vitale,S.Principle of superfluid helium gyroscopes.Phys.Rev.B46,3540-3549 (192). tion in this material appears to be absent,and the superconducting 11.Gustavson,T.L.Bouyer,P.Kasevitch.M.A.Physical rotation measurements with an atom state is very sensitive to defects,occurring only in very pure interferometer gyro ope.Phys.Rev..Lt.78,2046-2049(I997). samples.Under hydrostatic pressure superconductivity and 12.Stedman,G.E.Ring-laser tests of fundamental physics and geophysics.Rep.Prog.Phys 60,615-687 ferromagnetism disappear at the same pressure,so the ferromag- 1997 13.Anderson,P.W.in Progress in Low Temperature Physics(ed Gorter.C.L)1-44 (North Holland. netic state appears to be a prerequisite for superconductivity.When Amsterdam,1967). combined with the recent observation of superconductivity in 14.Marchenkov,A..Simmonds,R.W,Davis,IC.Packard,R E.Observation of the Josephson plasma UGe2(ref.4),our results suggest that metallic ferromagnets may mode for a superfluid 'He weak link.Phys.Rev.B61,4196-4199(2000). 15.Schwab,K,Bruckner,N.Packard,R.E.Detection of the Earth's rotations using superfluid phase universally become superconducting when the magnetization is coherence.Nature 386,585-587(1997) small. 16.Avenel,O.,Hakonen.P.Varoquaux,E Detection of the rotation of the Earth with a superfluid The compound ZrZnz was first investigated by Matthias and gyrometer.P%s.R.LeH.78,3602-3605(1997). Bozorth'in the 1950s,who discovered that it was ferromagnetic 17.Mukharsky,Yu.,Varoquaux,E.Avenel,O.Current-phase relationship measurements in the flow of superfluid'He through a single orifice.Physica B 280,130-131 (2000). despite being made from nonmagnetic,superconducting constitu- 18.Mukharsky,Yu.,Avenel,O.Varoquaux,E.Rotation mea rements with a superfluid He gyrometer. ents.ZrZnz crystallizes in the C15 cubic Laves structure,as shown in P%ysica B284-288287-288(2000). the inset of Fig.1,with lattice constant a 7.393 A.The Zr atoms 19.Rowe.C.H.et al Design and operation ofa very large ring laser gyroscope.Appl Opt.38,2516-2523 form a tetrahedrally coordinated diamond structure and the mag- (1999. 20.Gustavson,T.L.Landragin,A.Kasevich,M.A.Rotation sensing with a dual atom interferometer netic properties of the compound derive from the Zr 4d orbitals, Sagnac gyroscope.Class.Quant Gravity 17,2385-2398 (2000) which have a significant direct overlap.Ferromagnetism develops 58 2001 Macmillan Magazines Ltd NATURE|VOL 412|5 JULY 2001 www.nature.com
letters to nature 58 NATURE | VOL 412 | 5 JULY 2001 | www.nature.com experiment advances the close analogy between the macroscopic quantum physics of superconductivity and super¯uidity. Both systems exhibit persistent currents, quantized circulation (¯uid and magnetic), sinf weak links and now, double-path quantum interference. M Methods To derive equation (2) we follow a heuristic method similar to that which Feynman24 applied to the analogous superconducting case. Consider the interferometer in Fig. 1a to be rotating at angular velocity . The total external current passing through the interferometer may be written as: It Ic sinf1 Ic sinf2 2Ic cos f1 2 f2 2 sin f1 f2 2 5 We assume that the ¯uid in the torus is characterized at every point by a quantum phase factor which is macroscopically coherent. The rotation of the torus entrains the super¯ow which is then almost that of a solid body, ns R. Around a closed path in the interferometer, as indicated by the dotted line in Fig. 1, the phase can change only in multiples of 2p. r=f×dl 2pn # b a 2m3 ~ ns×dl f1 # d c 2m3 ~ ns×dl 2 f2 2 2m3 ~ 2pR2 f1 2 f2 6 The limit points, a±d, in the integrals are shown in Fig. 1a. In equation (6) we have used the concept25 that the phase gradient is proportional to super¯uid velocity, =f 2msns=~. Equation (6) may be solved for f1 2 f2, which, when inserted into equation (5) gives It I p c sin© 7 where © [ f1 f2 =2 is the phase difference across the ends of the interferometer. We see that the rotating interferometer behaves as a single Josephson weak link, whose maximum current is given by the quantum interference term: I p c 2Ic cos p 2Q×A k3 8 Here k3 [ ~= 2m3 is the quantum of circulation, A is the interferometer area vector, normal to the plane, and we have included a scalar product describing the more general case when the rotation axis is not parallel to A. Received 5 March; accepted 14 May 2001. 1. Young, T. A Course of Lectures in Natural Philosophy and the Mechanical Arts Vol. 1, 364 (London, 1845); also as facsimile edition (New York, 1971). 2. Davisson, C. J. Are Electrons Waves? Franklin Inst. J. 205, 597 (1928). 3. Werner, S. A., Studenmann, J. L. & Colella, R. Effect of Earth's rotation on the quantum mechanical phase of the neutron. Phys. Rev. Lett. 42, 1103±1106 (1979). 4. Keith, D. W., Ekstrom, C. R., Turchette, Q. A. & Pritchard, D. E. An interferometer for atoms. Phys. Rev. Lett. 66, 2693±2696 (1991). 5. Andrews, M. R.et al. Observation of interference between two Bose condensates. Science 275, 637±641 (1997). 6. Barone, A. & Paterno, G. Physics and Application of the Josephson Effect (Wiley, New York, 1982). 7. Varoquaux, E. & Avenel, O. Josephson effect and quantum phase slippage in super¯uids. Phys. Rev. Lett. 60, 416±419 (1988). 8. Pereversev, S. V., Loshak, A., Backhaus, S., Davis, J. C. & Packard, R. E. Quantum oscillations between two weakly coupled reservoirs of super¯uid 3 He. Nature 388, 449±451 (1997). 9. Marchenkov, A. et al. Bi-state super¯uid 3 He weak links and the stability of Josephson p states. Phys. Rev. Lett. 83, 3860±3863 (1999). 10. Packard, R. E. & Vitale, S. Principle of super¯uid-helium gyroscopes. Phys. Rev. B 46, 3540±3549 (1992). 11. Gustavson, T. L., Bouyer, P. & Kasevitch, M. A. Physical rotation measurements with an atom interferometer gyroscope. Phys. Rev. Lett. 78, 2046±2049 (1997). 12. Stedman, G. E. Ring-laser tests of fundamental physics and geophysics. Rep. Prog. Phys. 60, 615±687 (1997). 13. Anderson, P. W. in Progress in Low Temperature Physics (ed. Gorter, C. J.) 1±44 (North Holland, Amsterdam, 1967). 14. Marchenkov, A., Simmonds, R. W., Davis, J. C. & Packard, R. E. Observation of the Josephson plasma mode for a super¯uid 3 He weak link. Phys. Rev. B 61, 4196±4199 (2000). 15. Schwab, K., Bruckner, N. & Packard, R. E. Detection of the Earth's rotations using super¯uid phase coherence. Nature 386, 585±587 (1997). 16. Avenel, O., Hakonen, P. & Varoquaux, E. Detection of the rotation of the Earth with a super¯uid gyrometer. Phys. Rev. Lett. 78, 3602±3605 (1997). 17. Mukharsky, Yu., Varoquaux, E. & Avenel, O. Current-phase relationship measurements in the ¯ow of super¯uid 3 He through a single ori®ce. Physica B 280, 130±131 (2000). 18. Mukharsky, Yu., Avenel, O. & Varoquaux, E. Rotation measurements with a super¯uid 3 He gyrometer. Physica B 284±288, 287±288 (2000). 19. Rowe, C. H. et al. Design and operation of a very large ring laser gyroscope. Appl. Opt. 38, 2516±2523 (1999). 20. Gustavson, T. L., Landragin, A. & Kasevich, M. A. Rotation sensing with a dual atom interferometer Sagnac gyroscope. Class. Quant. Gravity 17, 2385±2398 (2000). 21. Herring, T. A. The rotation of the Earth. Rev. Geophys. Suppl. 29, 172±175 (1991). 22. Clarke, J. in SQUID Sensors: Fundamentals, Fabrication and Applications (ed. Weinstock, H.) (Kluwer Academic, 1996). 23. Simmonds, R. W., Marchnkov, A., Vitale, S., Davis, J. C. & Packard, R. E. New ¯ow dissipation mechanisms in super¯uid 3 He. Phys. Rev. Lett. 84, 6062±6065 (2000). 24. Feynmann, R. P., Leighton, R. B. & Sands, M. The Feynman Lectures in Physics Vol. 3, Ch. 21 (Addison Wesley, Reading, Massachusetts, 1963). 25. Tilley, D. R. & Tilley, J. Super¯uidity and Superconductivity 3rd edn 171 (Hilger, Bristol, 1990). 26. Paik, H. J. Superconducting tunable-diaphragm transducer for sensitive acceleration measurements. J. Appl. Phys. 47, 1168±1178 (1976). Acknowledgements We thank S. Vitale and K. Penanen for discussions; Y. Sato for assistance; A. Loshak for making the aperture arrays; and E. Crump, D. Mathews and C. Ku for assistance in improving noise conditions in our building. This work was supported in part by NASA, the Of®ce of Naval Research, the National Science Foundation, and the Miller Institute for Basic Research (J.C.D.). Correspondence and requests for materials should be addressed to R.P. (e-mail: packard@socrates.berkeley.edu). ................................................................. Coexistence of superconductivity and ferromagnetism in the d-band metal ZrZn2 C. P¯eiderer*, M. Uhlarz*, S. M. Hayden², R. Vollmer*, H. v. LoÈhneysen*³, N. R. Bernhoeft§ & G. G. Lonzarichk * Physikalisches Institut, UniversitaÈt Karlsruhe, D-76128 Karlsruhe, Germany ² H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK ³ Forschungszentrum Karlsruhe, Institut fuÈr FestkoÈrperphysik, D-76021 Karlsruhe, Germany § DRFMC-SPSMS, CEA Grenoble, F-38054 Grenoble, Cedex 9, France k Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK .............................................................................................................................................. It has generally been believed that, within the context of the Bardeen±Cooper±Schrieffer (BCS) theory of superconductivity, the conduction electrons in a metal cannot be both ferromagnetically ordered and superconducting1,2. Even when the superconductivity has been interpreted as arising from magnetic mediation of the paired electrons, it was thought that the superconducting state occurs in the paramagnetic phase3,4. Here we report the observation of superconductivity in the ferromagnetically ordered phase of the d-electron compound ZrZn2. The speci®c heat anomaly associated with the superconducting transition in this material appears to be absent, and the superconducting state is very sensitive to defects, occurring only in very pure samples. Under hydrostatic pressure superconductivity and ferromagnetism disappear at the same pressure, so the ferromagnetic state appears to be a prerequisite for superconductivity. When combined with the recent observation of superconductivity in UGe2 (ref. 4), our results suggest that metallic ferromagnets may universally become superconducting when the magnetization is small. The compound ZrZn2 was ®rst investigated by Matthias and Bozorth5 in the 1950s, who discovered that it was ferromagnetic despite being made from nonmagnetic, superconducting constituents. ZrZn2 crystallizes in the C15 cubic Laves structure, as shown in the inset of Fig. 1, with lattice constant a 7:393 A. The Zr atoms Ê form a tetrahedrally coordinated diamond structure and the magnetic properties of the compound derive from the Zr 4d orbitals, which have a signi®cant direct overlap6 . Ferromagnetism develops © 2001 Macmillan Magazines Ltd