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Eltsov, VB AC Josephson Effect Between Two This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Autti, S.; Heikkinen, P. J.; Mäkinen, J. T.; Volovik, G. E.; Zavjalov, V. V.; Eltsov, V. B. AC Josephson effect between two superfluid time crystals Published in: Nature Materials DOI: 10.1038/s41563-020-0780-y Published: 01/02/2021 Document Version Peer reviewed version Please cite the original version: Autti, S., Heikkinen, P. J., Mäkinen, J. T., Volovik, G. E., Zavjalov, V. V., & Eltsov, V. B. (2021). AC Josephson effect between two superfluid time crystals. Nature Materials, 20(2), 171-174. https://doi.org/10.1038/s41563- 020-0780-y This material is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Powered by TCPDF (www.tcpdf.org) AC Josephson effect between two superfluid time crystals S. Autti1;2∗, P.J. Heikkinen1;3, J.T. M¨akinen1;4;5, G.E. Volovik1;6, V.V. Zavjalov1;2, and V.B. Eltsov1y 1Low Temperature Laboratory, Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland. yvladimir.eltsov@aalto.fi 2Department of Physics, Lancaster University, Lancaster LA1 4YB, UK. *[email protected] 3Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK. 4Department of Physics, Yale University, New Haven, CT 06520, USA 5Yale Quantum Institute, Yale University, New Haven, CT 06520, USA 6L.D. Landau Institute for Theoretical Physics, Moscow, Russia Quantum time crystals are systems charac- experiments here. In general, Bose-Einstein condensates terised by spontaneously emerging periodic or- are an established platform for studying both DC and der in the time domain1. While originally a AC Josephson effects10{15. phase of broken time translation symmetry was The manifest feature of magnon condensation is the 2 a mere speculation , a wide range of time crys- emergence of coherence of the precession frequency and 3{5 tals has been reported . However, the dynam- phase16{23. Importantly, the coherence is spontaneous ics and interactions between such systems have and the precession period is not influenced by the mech- not been investigated experimentally. Here we anism that creates the magnons. This can be confirmed study two adjacent quantum time crystals re- by pumping magnons to a higher-frequency level in a con- alised by two magnon condensates in superfluid fining trap, from which magnons then fall to the ground 3 He-B. We observe an exchange of magnons be- level, thereby choosing a period independent of that of tween the time crystals leading to opposite-phase the drive8. This periodic, observable motion in a many- oscillations in their populations | a signature of body system constitutes the essence of a time crystal. It 6 the AC Josephson effect | while the defining is detected in our system using nuclear magnetic reso- periodic motion remains phase coherent through- nance techniques (NMR), based on coupling the precess- out the experiment. Our results demonstrate ing magnetisation to nearby pick-up coils (\NMR coils", that time crystals obey the general dynamics of see Fig. 1). quantum mechanics and offer a basis to further In NMR experiments, magnon time crystals are char- investigate the fundamental properties of these acterised by two timescales24. The first timescale τ ∼ phases, opening pathways for possible applica- E 0:1 s describes how quickly the precession in the conden- tions in, e.g., quantum information processing. sate becomes coherent, following the pumping of incoher- 1 The idea of a quantum time crystal was met with en- ent magnons. The second timescale τN is magnon life- thusiasm. Nevertheless, it was quickly realised that ob- time. The system reaches exact particle conservation in serving perpetual periodic motion in a quantum ground the limit τN ! 1, which in an isolated sample container state faces fundamental limitations7. Thus, early ex- is approached exponentially as temperature decreases. In perimental work on time crystals concentrated on find- practice there are losses in the pick-up coils that are cou- ing routes around such restrictions, either by limiting pled to the precessing spins in order to control and ob- 25 the spontaneous symmetry breaking to discrete time serve the condensate . As long as τN τE, the coher- translation symmetry3{5, or by allowing slowly decaying ently precessing condensate approximates a ground state systems8. Meaningful interactions between time crystals at a fixed magnon number, thus approaching a perfect have so far remained elusive. In this Letter we study ground-state time crystal1. On the other hand, if losses the controlled interaction of two spatially-separated time are completely eliminated, the time-dependent state be- crystals, composed of macroscopic number of magnon comes unobservable. In the experiments presented here quasiparticles, while the periodic motion and parameters τN ∼ 3 s. of the interaction can be directly observed. One can then either observe the phenomena that A magnon Bose-Einstein condensate (BEC) in super- emerge during this slow decay, or compensate for the fluid 3He-B is a macroscopic quantum state described losses by pumping the system continuously. Under con- by a simple wavefunction Ψ = jΨjei(−µt/h¯+'M), where tinuous pumping, the condensate spontaneously finds a µ is the chemical potential9. Magnons are quanta of magnon number that corresponds to the chemical po- transverse spin waves, corresponding to magnetisation M tential µ set by the pumping frequency26,27 (typically that precesses at frequency f = µ/(2π¯h) around the ex- N ≥ 1012). This is the traditional way of studying 3{5 ternal magnetic field H, starting from initial phase 'M. time crystals . On the other hand, free decay with the Each magnon carries −¯h of spin, yielding total number pumping turned off realises time crystals in a novel, un- 2 2 8 of magnons N / jΨj / βM, where βM is the deflection perturbed environment . This is particularly instrumen- angle of M from the equilibrium direction along H. Here tal for studying dynamics and interactions of magnon- we assumed that β is small, which is satisfied in all our BEC time crystals, providing a fundamental advantage as 2 compared with continuous pumping: it removes the need to distinguish potential artefacts of the external driving Sample container Surface BEC force. In return for allowing the system to decay, one z also gains direct access to all the observables relevant to the Josephson effect. These features make magnon con- NMR coils densates an ideal laboratory system for studying time M crystals, their dynamics, and related emergent phenom- ena such as the Josephson effect. βM Magnons are trapped in the middle of the sample con- tainer cylinder by the combined effect of the superfluid order parameter distribution (\texture"), and an axial x minimum in the magnetic field25,28. This yields an ap- proximately harmonic three-dimensional trap. When a free surface is introduced 3 mm above the field minimum, the textural trap forms two local minima and, hence, y splits the magnon BEC spectrum into two physical lo- cations: (1) The bulk trap remains approximately har- monic; (2) an additional surface trap emerges, see Fig. 1. States in the measured experimental spectrum are identi- Pinch coil Bulk BEC fied based on their dependence on the profile of the mag- netic trap, controlled by the current in the pinch coil. In what follows we focus on the lowest-energy state of each H of the two traps. Experimental techniques are detailed 6 mm in the Methods section. We populate both the bulk and the surface traps by FIG. 1. Experimental setup: Quartz-glass sample con- a radio-frequency (RF) pulse, delivered via the pick-up tainer cylinder is filled partially with superfluid 3He-B, leaving coils. The duration, frequency, and amplitude of the a free surface of the superfluid approximately 3 mm above the pulse are chosen so that the two traps are populated ap- centre of the surrounding coil system. The space above the proximately equally. The signal from the pick-up coils is free surface is vacuum due to the vanishing vapour pressure visualised using time-windowed Fourier analysis (Fig. 2). of 3He at sub-mK temperatures. Magnons can be trapped in When the pumping is turned off, the condensate popula- this configuration in two separate locations, in bulk (coloured tions decrease slowly due to dissipation, seen in the de- blue) and touching the free surface (coloured red). Trans- crease of signal amplitude from both condensates. The verse NMR coils are used both for RF pumping of magnons bulk condensate frequency increases during the decay by into the BECs, and for recording the induced signal from the 20 Hz. This is because the textural trap is expanded coherently precessing magnetisation M (light yellow arrow). The amplitude of the recorded signal is proportional to β , when the number of magnons is large owing to spin-orbit M 26,27,29 the tipping angle of M, and its frequency corresponds to the interaction, making the trap shallower . The sur- precession frequency of the condensate. The condensates are face trap is more rigid owing to the boundary condition trapped by the combined effect of the distribution of orbital set by the free surface. Hence, the frequency of the sur- anisotropy axis of the superfluid (green arrows) via spin-orbit face condensate only changes by 7 Hz. During decay, coupling, and a minimum of the external magnetic field cre- the period (frequency) of each condensate is indepen- ated using a pinch coil (magenta wire loop).
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