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Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal

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Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Autti, S.; Eltsov, V. B.; Volovik, G. E. Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal Published in: Physical Review Letters DOI: 10.1103/PhysRevLett.120.215301 Published: 25/05/2018 Document Version Publisher's PDF, also known as Version of record Please cite the original version: Autti, S., Eltsov, V. B., & Volovik, G. E. (2018). Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal. Physical Review Letters, 120(21), 1-5. [215301]. https://doi.org/10.1103/PhysRevLett.120.215301 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) PHYSICAL REVIEW LETTERS 120, 215301 (2018) Editors' Suggestion Observation of a Time Quasicrystal and Its Transition to a Superfluid Time Crystal S. Autti,1,2,* V. B. Eltsov,1 and G. E. Volovik1,3 1Low Temperature Laboratory, Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland 2Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom 3L.D. Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia (Received 5 February 2018; revised manuscript received 28 March 2018; published 25 May 2018) We report experimental realization of a quantum time quasicrystal and its transformation to a quantum time crystal. We study Bose-Einstein condensation of magnons, associated with coherent spin precession, created in a flexible trap in superfluid 3He-B. Under a periodic drive with an oscillating magnetic field, the coherent spin precession is stabilized at a frequency smaller than that of the drive, demonstrating spontaneous breaking of discrete time translation symmetry. The induced precession frequency is incommensurate with the drive, and hence, the obtained state is a time quasicrystal. When the drive is turned off, the self-sustained coherent precession lives a macroscopically long time, now representing a time crystal with broken symmetry with respect to continuous time translations. Additionally, the magnon condensate manifests spin superfluidity, justifying calling the obtained state a time supersolid or a time supercrystal. DOI: 10.1103/PhysRevLett.120.215301 Originally, time crystals were suggested as a class of times [5]. One is the lifetime τN of the corresponding quantum systems for which time translation symmetry is particles (quasiparticles). The second one is the thermal- spontaneously broken in the ground state, so that the time- ization time, or energy relaxation time τE, during which the periodic motion of the background constitutes its lowest superfluid state of N particles is formed. If τE ≪ τN, the energy state [1]. It was quickly shown that the original idea system relatively quickly relaxes to a minimal energy state cannot be realized with realistic assumptions [2–6]. This with quasifixed N (the superfluid state) and then slowly no-go theorem forces us to search for spontaneous time- relaxes to the true equilibrium state with μ ¼ 0. In the translation symmetry breaking in a broader sense (see, e.g., intermediate time τE ≪ t ≪ τN, the system has finite μ and the review in Ref. [7]). One available direction is a system thus becomes a time crystal. Note that in the limit of the with off-diagonal long-range order, experienced by super- exact conservation of the particle number mentioned above fluids, Bose gases, and magnon condensates [5,8]. In the τN → ∞, the exchange of particles between the system and grand canonical formalism, the order parameter of a Bose- the environment is lost, and the time dependence of the Einstein condensate (BEC)—the macroscopic wave func- condensate cannot be experimentally resolved. tion Ψ, which also describes conventional superfluidity— Bose-Einstein condensates of pumped quasiparticles, −iμt oscillates periodically: Ψ ¼haˆ 0i¼jΨje , where aˆ 0 is such as photons [10], are, in general, a good example of the particle annihilation operator and μ is a chemical systems with off-diagonal long-range order, where the potential. Such a periodic time evolution can be observed condition τN ≫ τE is fulfilled. Time crystals can be experimentally provided the condensate is coupled to conveniently studied in experiments based on the magnon another condensate. If the system is strictly isolated, i.e., BEC states in superfluid phases of 3He, where the lifetime when the number of atoms N is strictly conserved, there is of quasiparticles (magnons) can reach minutes. Magnon no reference frame with respect to which this time BEC in 3He was first observed in the fully gapped dependence can be detected. That is why for the external topological B phase [11,12], then in the chiral Weyl observer the BEC looks like a fully stationary ground state. superfluid A phase [13,14], and recently in the polar phase However, for example, in the Grand Unification exten- with Dirac lines [15]. The magnon number N is not sions of the Standard Model, there is no conservation of the conserved, but the decay time is τN ≫ τE, see Fig. 1. number of atoms N due to proton decay [9]. Therefore, in For t<τN, the magnon BEC corresponds to the minimum principle, the oscillations of the macroscopic wave function of energy at fixed N. The lifetime τN is long enough to of an atomic superfluid in its ground state could be observe the Bose condensation and effects related to the identified experimentally and the no-go theorem avoided spontaneously broken Uð1Þ symmetry, such as ac and dc if we had enough time for such experiment, about Josephson effects, phase-slip processes, Nambu-Goldstone 36 τN ∼ 10 years. In general, any system with off-diagonal modes, etc [16]. Each magnon carries spin −ℏ, and the long-range order can be characterized by two relaxation number of magnons is thus N ¼ðS − SzÞ=ℏ, where S is the 0031-9007=18=120(21)=215301(5) 215301-1 © 2018 American Physical Society PHYSICAL REVIEW LETTERS 120, 215301 (2018) Pumping of magnons Magnon BEC z Magnon BEC Sample container 0.2 E 0.15 0.1 NMR coils 0.05 0 (arb. un.) x M -0.05 x 0.2 -0.1 -0.15 0.15 2 ms -0.2 00.511.52 0.1 Time (s) 0.05 Pinch coil 0 6 mm (arb. un.) x -0.05 M FIG. 2. Experimental setup: Sample container is made from 3 -0.1 P = 0 bar quartz glass and filled with superfluid He-B. NMR coils create a T = 131µK N pumping field Hrf and pick up the induction signal from -0.15 M Decrease of magnon number coherently precessing magnetization of the magnon BEC. -0.2 The trapping potential for the condensate is created by a spatial 010203040distribution of the orbital anisotropy axis of 3He-B (small green time (s) arrows) and an axial minimum in the static magnetic field produced by a pinch coil. FIG. 1. Time supercrystal in superfluid 3He-B emerging from spontaneous coherence in freely precessing magnetization. (Top) iωt The coherent precession of magnetization, Mx þ iMy ¼ γS⊥e , susceptibility. The magnetic field defines the Larmor fre- τ is established with time constant E after the excitation pulse at quency ωL ¼jγjH. Then, a transverse radio-frequency (rf) t ¼ 0. The signal is picked up by the NMR coils (Fig. 2) and iω t pulse H ¼ H xˆe drive is applied to deflect the spins by down-converted to lower frequency (with reference at 834 kHz). rf rf 3 angle β with respect to H. This corresponds to pumping (Bottom) Since the magnetic relaxation in superfluid He is small, N ¼ Sð1 − βÞ=ℏ the number of magnons N slowly decreases with timescale cos magnons to the sample. After the τN ≫ τE. During relaxation the precession remains coherent, pulse, the signal picked up by the NMR coils rapidly decays and the state represents Bose-Einstein condensate of magnons due to dephasing of the precessing spins caused by inho- until the number of magnons drops below the critical value mogeneity of the trapping potential. After time τE, collective (which in these experiments corresponds to a signal below the synchronization of the precessing spins takes place, leading noise level). to the formation of the magnon BEC with an off-diagonal long-range order. This process is the experimental signature total spin and the zˆ axis is directed along the applied of the time crystal: The system spontaneously chooses a coherent precession frequency, and one can directly observe magnetic field H. The state with jSzj <Scorresponds to the resulting periodic time evolution (Fig. 1). the precessingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi macroscopic spin with transverse magnitude S ¼ S2 − S2 Uð1Þ Note that the periodic phase-coherent precession ⊥ z, and the broken symmetry within the emerges in the interacting many-body magnon system, SOð2Þ condensate is equivalent to broken symmetry of the which experiences the spin superfluidity [16]. The sponta- H spin rotation about the direction of . The magnon BEC is nously broken Uð1Þ symmetry and the interaction between manifested by spontaneously formed coherent spin pre- ˆ þ magnons in magnon BEC give rise to the Nambu- pcessionffiffiffiffiffiffi and is described by the wave function hS i¼ Goldstone modes, which can be identified with phonons iωt ˆ þ ˆ ˆ 2Shaˆ 0i¼S⊥e , where S ¼ Sx þ iSy. Here, the role of of this time crystal.
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