AC Josephson effect between two superfluid time crystals
Samuli Autti, Petri J. Heikkinen, Jere T. Mäkinen, Grigori E. Volovik, Vladislav V. Zavjalov, Vladimir B. Eltsov
AAC Josephson effect between two superfluid time crystals
S. Autti , ∗ , P.J. Heikkinen , , J.T. M¨akinen , , , G.E. Volovik , , V.V. Zavjalov , , and V.B. Eltsov † Low Temperature Laboratory, Department of Applied Physics, Aalto University,POB 15100, FI-00076 AALTO, Finland. † vladimir.eltsov@aalto.fi Department of Physics, Lancaster University, Lancaster LA1 4YB, UK. *[email protected] Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK. Department of Physics, Yale University, New Haven, CT 06520, USA Yale Quantum Institute, Yale University, New Haven, CT 06520, USA L.D. Landau Institute for Theoretical Physics, Moscow, Russia
Quantum time crystals are systems characterised by spontaneously emerging periodic order in thetime domain . A range of such phases has been reported (e.g. reviews ). The concept has even beendiscussed in popular literature , and deservedly so: while the first speculation on a phase of brokentime translation symmetry did not use the name “time crystal” , it was later adopted from 1980’spopular culture . For the physics community, however, the ultimate qualification of a new conceptis its ability to provide predictions and insight. Confirming that time crystals manifest the basicdynamics of quantum mechanics is a necessary step in that direction. We study two adjacent quantumtime crystals experimentally. The time crystals, realised by two magnon condensates in superfluid He-B, exchange magnons leading to opposite-phase oscillations in their populations - – AC Josephsoneffect — while the defining periodic motion remains phase coherent throughout the experiment. A magnon Bose-Einstein condensate (BEC) in superfluid He-B is a macroscopic quantum state described by asimple wavefunction Ψ = | Ψ | e − i ( µt/ ¯ h + ϕ M ) , where µ is the chemical potential . Magnons are quanta of transverse spinwaves, corresponding to magnetisation M that precesses at frequency f = µ/ (2 π ¯ h ) around the external magnetic field H , starting from initial phase ϕ M . Each magnon carries − ¯ h of spin, yielding total number of magnons N ∝ | Ψ | ∝ β ,where β is the deflection angle of M from the equilibrium direction along H . Here we assumed that β is small, whichis satisfied in all the experiments presented in this Letter. In general, Bose-Einstein condensates are an establishedplatform for studying both DC and AC Josephson effects .The manifest feature of quasiparticle condensation is the emergence of spontaneous coherence of the precessionfrequency and phase . In our system it also guarantees that the precession period forms uncontrived: for example,it is possible to pump magnons to a higher-frequency level in a confining trap, from which magnons then fall tothe ground level, thereby choosing a period independent of that of the drive . This periodic, observable motionconstitutes the essence of a time crystal. It is detected in our system using nuclear magnetic resonance techniques(NMR), based on coupling the precessing magnetisation to nearby pick-up coils (see Fig. 1).In NMR experiments, magnon time crystals in superfluid He-B are characterised by two timescales . The firsttimescale τ E ∼ . τ N is magnon lifetime. The system reaches exact particle conservation inthe limit τ N → ∞ , which in an isolated sample container is approached exponentially as temperature decreases. Inpractice there are losses in the pick-up coils that are coupled to the precessing spins in order to control and observe thecondensate . It is also impractical to carry out experiments at the lowest achievable temperature . It is thereforenecessary to allow for a finite τ N , which is acceptable as long as τ N (cid:29) τ E .One can then either observe the phenomena that emerge during this slow decay, or compensate for the losses bypumping the system continuously. Under continuous pumping, the condensate spontaneously finds a magnon numberthat corresponds to the chemical potential µ set by the pumping frequency . On the other hand, free decay withthe pumping turned off is particularly instrumental for studying dynamics and interactions of magnon-BEC timecrystals as it removes the need to distinguish potential artefacts of the external driving force. These features makemagnon condensates an ideal laboratory system for studying time crystals, their dynamics, and related emergentphenomena.Magnons are trapped in the middle of the sample container cylinder by the combined effect of the superfluid orderparameter distribution (“texture”), and an axial minimum in the magnetic field . This yields an approximatelyharmonic three-dimensional trap. Introducing a free surface, located 3 mm above the field minimum, modifies thetextural trap creating two local minima and, hence, splits the magnon BEC spectrum into two physical locations:(1) The bulk trap remains approximately harmonic. (2) An additional surface trap emerges, see Fig. 1. States inthe measured experimental spectrum can be identified based on their dependence on the profile of the magnetic trap,controlled by changing the current in the pinch coil. In what follows we concentrate on studying the lowest-energystate of each of the two traps.The measurement begins by populating both the bulk and the surface traps using a radio-frequency (RF) pulse. a r X i v : . [ c ond - m a t . o t h e r] M a r The duration, frequency, and amplitude of the pulse are chosen so that the two traps are populated approximatelyequally. The signal recorded from the pick-up coils is then visualised using time-windowed Fourier analysis (Fig. 2a).When the pumping is turned off, the condensate populations decrease slowly due to dissipation as seen in the decreaseof recorded signal amplitude from both condensates. The bulk condensate frequency is also increasing during thedecay by 20 Hz. This is because the textural trap feels the local pressure of magnons via spin-orbit interaction, andthereby becomes expanded when the number of magnons is large, making the trap shallower . The surface trapis rigidified by the textural boundary condition set by the free surface. Hence, the frequency of the surface condensateonly changes by 7 Hz. The period (frequency) of each condensate is independent of the drive pulse features duringthe decay. This justifies calling the observed states time crystals. In what follows we refer to the magnon-condensatetime crystals simply as “(time) crystals”.In addition to the traces corresponding to the time crystals, Fig 2 also features two side bands. The side bands areseparated from the main traces by the frequency difference between the two crystals, which changes slowly in time.We interpret the side bands as follows: The phase difference between the time crystals follows d( ϕ bulk − ϕ surf ) / d t = − ( µ bulk − µ surf ) / ¯ h , where the phase ϕ = − µt/ ¯ h + ϕ M . The changing phase difference therefore drives an alternatingJosephson supercurrent of constituent particles between the time crystals.This is seen as amplitude oscillations ofthe two signals, producing the side bands. The frequency of the oscillations is set by the difference of the timecrystal precession frequencies, equal to the difference of their chemical potentials (Fig. 2b). These observations arecharacteristic to AC Josephson effect . The remarkable advantage of the time crystal compared to superfluidsand superconductors is that all four variables in the canonical Josephson equation ( ϕ bulk , ϕ surf , µ bulk , µ surf ) are nowmeasured directly in the same experiment.We fit the measured signal, in short time windows, directly with two sine curves. This allows extracting thesignal amplitude from each time crystal separately (Fig. 2c). The result shows that the bulk crystal signal amplitude(population) oscillates at a frequency equal to the frequency difference of the bulk and the surface time crystals.The surface crystal signal shows similar oscillations with the opposite phase, as expected for AC Josephson effect.The surface crystal however also features in-phase oscillations. Below we demonstrate, using numerical simulations,that the bulk crystal oscillation modifies the textural trapping potential around it periodically, and the modificationpropagates along the texture to change the shape of the surface time crystal. This shape change modifies the effectivefilling factor of the surface crystal between the pick-up coils, hence changing the resulting signal, while the magnonnumber in the surface time crystal is not affected by this modification.It is worth emphasising that in the frame rotating with frequency µ/ ¯ h , the (initial) azimuthal angle of magnetisation ϕ M in each of the two time crystals remains stable over more than 10 periods of oscillation (Fig. 2e), despite thepopulation exchange, and the slow decay of overall magnon number. The azimuthal angle can be extracted by feedingthe signal recorded from the pick-up coils to a software lock-in amplifier, locked to the frequency extracted from FFTanalysis in Fig. 2a. The Josephson oscillations are filtered out by choosing a lock-in time constant longer than theJosephson frequency. The remaining drift is attributed to inaccuracy of the used reference frequency. The phasestability culminates the robustness of the time crystal, well-defined periodicity being the defining feature of brokentime translation symmetry.Let us confirm that the trapping potential connects the two time crystals indirectly by building a self-consistentnumerical simulation of the two underlying magnon condensates in the flexible trap (see Methods). The calculationqualitatively reproduces the remarkable features seen in the experiment (Fig. 2d): The signal from the surface timecrystal shows twice shorter period than the signal from the bulk crystal. This is caused by changes in the shape of thesurface trap, imposed by oscillations in the bulk crystal population. The calculation is quasi-static, meaning that thetrap is assumed to adjust to changes in the magnon distribution instantaneously. This explains why the signal fromthe surface crystal is aligned differently with the bulk crystal oscillations than observed in the experiment. Comparingthe amplitude of the simulated oscillations with the experiment also supports the view that the observed Josephsonoscillations in the two recorded experimental signals correspond to equal and opposite changes in the populations ofthe two time crystals. The oscillations of the trapping potential also directly change the precession frequencies of bothcondensates , thus changing the frequency difference in phase with the population changes. That should result indistortion of the sinusoidal population exchange, yielding additional side bands in Fig. 2. In practice this effect is tooweak to be distinguished in the experiment.In conclusion, we report an experimental realisation of two adjacent quantum time crystals that exchange constituentparticles via the AC Josephson effect. The time crystals are created in a flexible trap in superfluid He-B, emerging astwo spatially separate magnon BECs associated with coherent spin precession. The configuration of two interactingcondensates is stabilised in the proximity of a free surface of the superfluid. The Josephson oscillations in the number ofparticles each time crystal contains are seen as opposite-phase amplitude variations in the measured signal. Flexibilityof the trapping potential connects the two time crystals also indirectly providing additional interaction that results inan in-phase component of oscillation, as verified by numerical simulations. In the rotating frame, the azimuthal angleof each of the two time crystals remains stable and well defined in the course of all these perturbations. Notably,all the observables that characterise the AC Josephson effect, the time crystal precession phases and their chemicalpotentials, are directly measured in the same experiment. This relatively novel phase of matter therefore deserves itsplace in physicists’ vocabulary.It remains an interesting task for future to study more sophisticated time crystal interactions. For instance, onecould simulate the Hamiltonian of a Penrose-type “gravitationally” induced wave function collapse by allowing twotime crystals in their flexible traps to collide. On the other hand, long-lived coherent quantum systems with tunableinteractions, such as the robust time crystals studied here, provide a platform for building novel quantum devicesbased on spin-coherent phenomena . For example, the dependence of the chemical potential on the time crystalpopulations, coupled by the Josephson junction, could be used as a shunting “capacitor” for the junction, forming anequivalent to the transmon qubit. Such devices based on macroscopic spin coherence could perhaps be implementedeven at room temperature . ACKNOWLEDGEMENTS
This work has been supported by the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (Grant Agreement No. 694248). The experimental work was carried outin the Low Temperature Laboratory, which is a part of the OtaNano research infrastructure of Aalto University andof the European Microkelvin Platform. S. Autti acknowledges financial support from the Jenny and Antti Wihurifoundation, and P. J. Heikkinen that from the V¨ais¨al¨a foundation of the Finnish Academy of Science and Letters.
COMPETING INTERESTS
The authors declare no competing interests.
AUTHOR CONTRIBUTIONS
All authors contributed to writing the manuscript and discussing the results. Experiments were carried out andplanned by S.A., J.T.M, P.J.H, V.V.Z, and V.B.E. Theoretical work was done by S.A., G.E.V, and V.B.E.
METHODS
We cool superfluid helium-3 down to 130 µ K (0 . T c ) using a nuclear demagnetisation cryostat . Temperatureis measured using quartz tuning forks . The superfluid transition temperature at 0 bar pressure is T c = 930 µ K.The superfluid is contained in a quartz-glass cylinder (radius 3 mm), placed in an external magnetic field of about25 mT, aligned along the axis of the container. We emphasise that the results in the present Letter are not specificto this field or temperature: similar AC Josephson oscillations were observed down to 17mT and at temperatures upto 0 . T c .The sample container is surrounded by transverse coils, needed for creating and observing the magnon condensateusing NMR. First, the coils can be used to create a transverse radio-frequency (RF) field H rf , which tips magnetisationwithin the coils by a small amount. This allows pumping magnons into the sample. Second, the coils are used forrecording the resulting coherent precession of magnetisation that induces an electromotive force (EMF) into thepick-up coils (Fig. 1).The free surface is located 3 mm above the centre of the magnetic field minimum. The distance of the freesurface is determined by comparing the observed magnon spectrum with the numerical model described below, andconfirmed by measuring the pressure of He gas in a calibrated volume that results from the removal of liquid fromthe originally fully-filled sample container. The free surface distorts the textural trap and splits the magnon spectruminto two physical locations, as detailed in the main text. Analysis of the whole observable spectrum will be publishedseparately .We simulate the magnon condensates in a quasi-static approximation using a two-step model following the lines ofRefs. 23,25,33. The first step is to calculate the trapping potential in the absence of magnons, and solve the correspond-ing magnon spectrum. This is achieved by minimising the free energy functional of the equilibrium superfluid ,including the orienting effects of the magnetic field, sample container walls, and the free surface. The effect of thefree surface is assumed to be described by the same parameter values that apply to solid walls. The magnetic fieldis calculated based on the known geometry of the coil system. In the absence of the free surface this results inan approximately harmonic trap for magnons . In the presence of the free surface the two spatially-separatedcomponents of the calculated spectrum semi-quantitatively correspond to those observed experimentally , but allthe states touching the free surface are shifted upwards by roughly 150 Hz in the simulation as compared with theexperimental spectra. While this means that the surface condensate shape is not described perfectly, it provides amore-than-sufficient starting point for the purposes of the present work. Finding detailed quantitative agreementremains a task for future studies.The second step in the model construction is to enable non-zero magnon density. The textural part of the trappingpotential feels local magnon density due to spin-orbit interaction. For a fixed magnon distribution, this contributioncan be included in the textural free energy minimisation. A self-consistent solution for given magnon number can thenbe found by fixed point iteration, as described in Refs. 25,33. Signal from the condensates is calculated accordingto the EMF induced in the pick-up coils due to the coherently precessing magnetisation in each condensate. Wecalibrate the simulation signal amplitude using the measured frequency shift as a function of signal amplitude .Josephson oscillations are emulated in our model by adding opposite-phase equal-amplitude oscillations of magnonnumber between the two condensates.Magnons placed in the surface-touching condensate change the trap confining them less than those placed in thebulk condensate, as seen in Fig. 2a. This is because the boundary condition for the texture set by the free surfaceis orders of magnitude stronger than the effect of magnons. For simplicity, we therefore neglect the effect of thesurface condensate population altogether in the calculation of the trapping potential. In our experiments the bulkcondensate frequency is slightly higher than that of the surface-touching condensate. In the simulation we tune thecurrent in the pinch coil such that the bulk condensate has the lowest frequency in the system, 50 Hz below thesurface condensate frequency in the limit of zero magnons. This allows finding a self-consistent solution at all magnonnumbers straightforwardly, as the self-consistency step in the simulation targets the the bulk condensate only. Thissimplification does not change the textural connection between the bulk condensate and the surface condensate, orthe coupling of the condensates to the pick-up coils. Wilczek, F. Quantum time crystals.
Phys. Rev. Lett. , 160401 (2012). Sacha, K. & Zakrzewski, J. Time crystals: a review.
Reports on Progress in Physics , 016401 (2017). Else, D. V., Monroe, C., Nayak, C. & Yao, N. Y. Discrete time crystals. arXiv:1905.13232 (2019). Wilczek, F. The exquisite precision of time crystals.
Scientific American
November (2019). Gibney, E. The quest to crystallize time.
Nature , 164166 (2017). Ball, P. In search of time crystals.
Physics World , 29–33 (2018). Ball, P. Out of step with time.
Nature Materials , 569 (2018). Andreev, A. F. Bose condensation and spontaneous breaking of the uniformity of time.
Journal of Experimental andTheoretical Physics Letters , 1018–1025 (1996). Jospehson, B. Possible new effect in superconducting tunneling.
Phys. Lett. , 251–253 (1962). Bunkov, Y. M. & Volovik, G. E.
Novel Superfluids , vol. 1 (Oxford University Press, Oxford, 2013). Borovik-Romanov, A. S. et al.
Observation of a spin-current analog of the Josephson effect.
JETP Lett. , 1033–1037(1988). Levy, S., Lahoud, E., Shomroni, I. & Steinhauer, J. The a.c. and d.c. Josephson effects in a Bose-Einstein condensate.
Nature , 579–583 (2007). Valtolina, G. et al.
Josephson effect in fermionic superfluids across the BEC-BCS crossover.
Science , 1505–1508 (2015). Abbarchi, M. et al.
Macroscopic quantum self-trapping and Josephson oscillations of exciton polaritons.
Nature Physics ,275–279 (2013). Bunkov, Y. M., Fisher, S. N., Gu´enault, A. M. & Pickett, G. R. Persistent spin precession in He-B in the regime of vanishingquasiparticle density.
Phys. Rev. Lett. , 3092–3095 (1992). Fisher, S. et al.
Thirty-minute coherence in free induction decay signals in superfluid He-B.
J. Low Temp. Phys. ,303–308 (2000). Borovik-Romanov, A. S., Bun’kov, Y. M., Dmitriev, V. V. & Mukharskii, Y. M. Long-lived induction signal in superfluid He-B.
JETP Lett. , 1033–1037 (1984). Bunkov, Y. M. & Volovik, G. E. Magnon Bose–Einstein condensation and spin superfluidity.
Journal of Physics: CondensedMatter , 164210 (2010). Autti, S., Eltsov, V. B. & Volovik, G. E. Observation of a time quasicrystal and its transition to a superfluid time crystal.
Phys. Rev. Lett. , 215301 (2018). Volovik, G. E. On the broken time translation symmetry in macroscopic systems: Precessing states and off-diagonal long-range order.
JETP Letters , 491–495 (2013). Heikkinen, P. J. et al.
Relaxation of Bose-Einstein condensates of magnons in magneto-textural traps in superfluid He-B.
J. Low Temp. Phys. , 3–16 (2014). Bunkov, Y. M. & Volovik, G. E. Magnon condensation into a Q-ball in He-B.
Phys. Rev. Lett. , 265302 (2007). Autti, S. et al.
Self-trapping of magnon Bose-Einstein condensates in the ground state and on excited levels: From harmonicto box confinement.
Phys. Rev. Lett. , 145303 (2012). Thuneberg, E. V. Hydrostatic theory of superfluid He-B.
J. Low Temp. Phys. , 657–682 (2001). Autti, S., Heikkinen, P. J., Volovik, G. E., Zavjalov, V. V. & Eltsov, V. B. Propagation of self-localized Q -ball solitons inthe He universe.
Phys. Rev. B , 014518 (2018). Penrose, R. Gravitational collapse and space-time singularities.
Phys. Rev. Lett. , 57–59 (1965). Sato, Y. & Packard, R. E. Superfluid helium quantum interference devices: physics and applications.
Reports on Progressin Physics , 016401 (2011). Kreil, A. J. E. et al.
Josephson oscillations in a room-temperature Bose-Einstein magnon condensate. arXiv:1911.07802 (2019). Heikkinen, P. J., Autti, S., Eltsov, V. B., Haley, R. P. & Zavjalov, V. V. Microkelvin thermometry with Bose-Einsteincondensates of magnons and applications to studies of the AB interface in superfluid He.
J. Low Temp. Phys. , 681–705(2014). Blaauwgeers, R. et al.
Quartz tuning fork: Thermometer, pressure- and viscometer for helium liquids.
J. Low Temp. Phys. , 537–562 (2007). Blaˇzkov´a, M. et al.
Vibrating quartz fork: A tool for cryogenic helium research.
J. Low Temp. Phys. , 525–535 (2008). Heikkinen, P. J.
Magnon Bose-Einstein condensate as a probe of topological superfluid . Ph.D. thesis, Aalto University Schoolof Science (2016). https://aaltodoc.aalto.fi/handle/123456789/20580 . Autti, S.
Higgs bosons, half-quantum vortices, and Q-balls: an expedition in the He universe . Ph.D. thesis, Aalto UniversitySchool of Science (2017). https://aaltodoc.aalto.fi/handle/123456789/26282 . Kopu, J. Numerically calculated NMR response from different vortex distributions in superfluid He-B.
J. Low Temp. Phys. , 47–58 (2007).
NMR coilsPinch coilSample container xz β M Surface BECBulk BEC H FIG. 1.
Experimental setup:
Quartz-glass sample container cylinder is filled partially with superfluid He-B, leaving afree surface of the superfluid approximately 3 mm above the centre of the surrounding coil system. The space above the freesurface is vacuum due to the vanishing vapour pressure of He at sub-mK temperatures. Magnons can be trapped in thisconfiguration in two separate locations, in bulk (coloured blue) and touching the free surface (coloured red). Transverse NMRcoils are used both for RF pumping of magnons into the BECs, and for recording the induced signal from the coherentlyprecessing magnetisation M . The amplitude of the recorded signal is proportional to β , the tipping angle of M , and itsfrequency corresponds to the precession frequency of the condensate. The condensates are trapped by the combined effectof the distribution of orbital anisotropy axis of the superfluid (green arrows) via spin-orbit coupling, and a minimum of theexternal magnetic field created using a pinch coil. The external field H is oriented along the z axis of the sample container. time (s) a m p li t ude ( m V ) periods of oscillation a m p li t ude ( m V ) a cd time (s) f r equen cy ( H z ) Surface-BEC time crystalBulk-BEC time crystalSidebandSideband
FFT time (s) -2000200 0 2.25 . . Oscillation periods deg r . ) ̀ M ( eba FIG. 2.
Time crystal AC Josephson effect: ( a ) Two co-existing magnon-BEC time crystals, created with an RF drive pulseat t = 0, are seen as peaks in the Fourier spectrum of the signal recorded from the NMR coils. For clarity, the exciting pulse isleft just outside the time window shown here. The FFT amplitude refers to the voltage measured from the pick-up coils afterpre-amplification, and f L = 833 kHz is the Larmor frequency. The upper trace corresponds to the magnon-BEC time crystalin the bulk, and the lower trace to the time crystal touching the free surface. The bulk trap is the more flexible of the two, andthe bulk time crystal frequency hence increases during the decay more than that of the surface crystal. Population oscillationsbetween the time crystals result in amplitude oscillations of the two signals, seen as two side bands. ( b ) The changing frequencydifference of the two time crystals (cyan dash line) matches the frequency of the population oscillations between them, extractedfrom the bulk crystal side band (magenta line). ( c ) Direct fits to the recorded signal at the frequency of the bulk crystal (bluedash line) and the surface crystal (solid red line) reveal AC Josephson oscillations of population between the two crystals: theopposite-phase component of the amplitude oscillation is attributed to the AC Josephson effect, while the in-phase componentin the surface crystal signal is due to trapping potential changes imposed by the bulk crystal oscillations. ( d ) The quasi-staticnumerical simulation reveals that changes in the bulk crystal population (blue dash line) distort the trapping potential, addingan additional component to the calculated signal from the surface crystal (solid red line). ( e ) The azimuthal angle of precessingmagnetisation ϕ M in the rotating frame in the bulk crystal (blue dash line) and the surface crystal (solid red line) are extractedby feeding the raw signal to a software lock-in amplifier, locked to the corresponding frequency traces in panel aa