Observation of time quasicrystal and its transition to 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,POB 15100, FI-00076 AALTO, Finland. Email: samuli.autti@aalto.fi

2Department of Physics, Lancaster University, Lancaster LA1 4YB, UK3L.D. Landau Institute for Theoretical Physics, Moscow, Russia

(Dated: April 3, 2018)

We report experimental realization of a quantum time quasicrystal, and its transformation to aquantum time crystal. We study Bose-Einstein condensation of magnons, associated with coherentspin precession, created in a flexible trap in superfluid 3He-B. Under a periodic drive with anoscillating magnetic field, the coherent spin precession is stabilized at a frequency smaller thanthat of the drive, demonstrating spontaneous breaking of discrete time translation symmetry. Theinduced precession frequency is incommensurate with the drive, and hence the obtained state isa time quasicrystal. When the drive is turned off, the self-sustained coherent precession lives amacroscopically-long time, now representing a time crystal with broken symmetry with respect tocontinuous time translations. Additionally, the magnon condensate manifests spin superfluidity,justifying calling the obtained state a time supersolid or a time super-crystal.

Originally time crystals were suggested as class a quan-tum systems for which time translation symmetry isspontaneously broken in the ground state, so that thetime-periodic motion of the background constitutes itslowest energy state [1]. It was quickly shown that theoriginal idea cannot be realized with realistic assump-tions [2–6]. This no-go theorem forces us to search forspontaneous time-translation symmetry breaking in abroader sense (see e.g. review in Ref. [7]). One avail-able direction is systems with off-diagonal long range or-der, experienced by superfluids, Bose gases, and magnoncondensates[5, 8]. In the grand canonical formalism, theorder parameter of a Bose-Einstein condensate (BEC)— the macroscopic wave function Ψ, which also de-scribes conventional superfluidity — oscillates periodi-cally: Ψ = 〈a0〉 = |Ψ|e−iµt, where a0 is the particle an-nihilation operator and µ is chemical potential. Such aperiodic time evolution can be observed experimentallyprovided the condensate is coupled to another conden-sate. If the system is strictly isolated, i.e., when numberof atoms N is strictly conserved, there is no referenceframe with respect to which this time dependence can bedetected. That is why for the external observer, the BEClooks like a fully stationary ground state.

However, for example in the Grand Unification exten-sions of Standard Model there is no conservation of thenumber of atoms N due to proton decay [9]. There-fore, in principle, the oscillations of the macroscopicwave function of an atomic superfluid in its ground statecould be identified experimentally and the no-go theo-rem avoided if we had enough time for such experiment,about τN ∼ 1036 years. In general, any system with off-diagonal long range order can be characterized by tworelaxation times [5]. One is the life time τN of the corre-sponding particles (quasiparticles). The second one is thethermalization time, or energy relaxation time τE , dur-ing which the superfluid state of N particles is formed.

If τE � τN , the system relatively quickly relaxes to aminimal energy state with quasi-fixed N (the superfluidstate), and then slowly relaxes to the true equilibriumstate with µ = 0. In the intermediate time τE � t� τNthe system has finite µ, and thus becomes a time crystal.Note that in the limit of exact conservation of particlenumber mentioned above, τN →∞, the exchange of par-ticles between the system and the environment is lostand the time dependence of the condensate cannot beexperimentally resolved.

Bose-Einstein condensates of pumped quasiparticles,such as photons [10], are in general a good example of sys-tems with off-diagonal long-range order, where the con-dition τN � τE is fulfilled. Time-crystals can be conve-niently studied in experiments based on the magnon BECstates in superfluid phases of 3He, where the life timeof quasiparticles (magnons) can reach minutes. MagnonBEC in 3He was first observed in the fully gapped topo-logical B phase [11, 12], then in the chiral Weyl super-fluid A phase [13, 14], and recently in the polar phasewith Dirac lines [15]. Magnon number N is not con-served, but the decay time τN � τE , see Fig. 1. Fort < τN the magnon BEC corresponds to the minimumof energy at fixed N . The life time τN is long enoughto observe the Bose-condensation and effects related tothe spontaneously broken U(1) symmetry, such as acand dc Josephson effects, phase-slip processes, Nambu-Goldstone modes, etc [16]. Each magnon carries spin −~,and the number of magnons is thus N = (S − Sz)/~,where S is the total spin and the z axis is directedalong the applied magnetic field H. The state with|Sz| < S corresponds to the precessing macroscopic spinwith transverse magnitude S⊥ =

√S2 − S2

z and the bro-ken U(1) symmetry within the condensate is equivalentto broken SO(2) symmetry of spin rotation about the di-rection of H. The magnon BEC is manifested by sponta-neously formed coherent spin precession, and is described

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FIG. 1. (Color online) Time super-crystal in superfluid 3He-B emerging from spontaneous coherence in freely precessingmagnetization. (Top) The coherent precession of magnetiza-tion, Mx + iMy = γS⊥e

iωt, is established with time constantτE after the excitation pulse at t = 0. The signal is pickedup by the NMR coils (Fig. 2) and downconverted to lowerfrequency (with reference at 834 kHz). (Bottom) Since themagnetic relaxation in superfluid 3He is small, the numberof magnons N slowly decreases with time scale τN � τE .During relaxation the precession remains coherent and thestate represents Bose-Einstein condensate of magnons untilthe number of magnons drops below the critical value (whichis these experiments corresponds to signal below the noiselevel).

by the wave function 〈S+〉 =√

2S 〈a0〉 = S⊥eiωt, where

S+ = Sx + iSy. Here the role of the chemical potentialis played by the global frequency ω of the precession. Acharacteristic feature of coherent precession is that thisfrequency is constant in space even for a spatially in-homogeneous magnon condensate in an inhomogeneoustrapping potential.

The coherently precessing spins are observed in NMRexperiments via corresponding precession of magnetiza-tion M = γS, where γ is the gyromagnetic ratio (Fig. 2).Equilibrium non-zero magnetization M = χH is createdby an applied static magnetic field, χ being the magneticsusceptibility. The magnetic field defines the Larmor fre-quency ωL = |γ|H. Then a transverse radio-frequency(rf) pulse, Hrf = Hrf x e

iωdrivet, is applied to deflect thespins by angle β with resperct to H. This correspondsto pumping N = S(1− cosβ)/~ magnons to the sample.After the pulse, the signal picked up by the NMR coils

NMR coils

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FIG. 2. (Color online) Experimental setup: Sample containeris made from quartz glass and filled with superfluid 3He-B.NMR coils create pumping field Hrf and pick-up inductionsignal from coherently precessing magnetization M of themagnon BEC. The trapping potential for the condensate iscreated by a spatial distribution of the orbital anisotropy axisof 3He-B (small green arrows) and an axial minimum in thestatic magnetic field produced by a pinch coil.

rapidly decays due to dephasing of the precessing spinscaused by inhomogeneity of the trapping potential. Aftertime τE , collective synchronization of the precessing spinstakes place, leading to the formation of the magnon BECwith off-diagonal long range order. This process is theexperimental signature of the time crystal: the systemspontaneously chooses a coherent precession frequency,and one can directly observe the resulting periodic timeevolution (Fig. 1).

Note that the periodic phase-coherent precessionemerges in the interacting many-body magnon system,which experiences the spin superfluidity [16]. The spon-tanously broken U(1) symmetry and the interaction be-tween magnons in magnon BEC give rise to the Nambu-Goldstone modes, which can be identified with phononsof this time crystal. The frequency of the precession isdetermined by interactions (here of spin-orbit type, seebelow) and is robust to perturbations in the system. Allthis fits the presently adopted criterions of ”time crystal”,which exhibits typical hallmarks of spontaneous symme-try breaking, such as long-range order and soft modes[17]. The coherent precession can be also considered asmacroscopic realization of the time crystal behavior ofexcited eigenstates [18].

Also another direction to circumvent the no-go theo-rem has been suggested – Floquet time crystals emergingunder a drive [19–29]. As distinct from the breaking ofcontinuous time-translation [1–5], here the discrete timesymmetry t→ t+T is spontaneously broken, T being theperiod of the driving force. The system spontaneously

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acquires a larger period than that of the drive, nT , i.e.ωcoherent = ωdrive/n, where ωdrive = 2π/T . For example,the period may be doubled, n = 2 [30]. In Ref. [31], aparametric resonance was observed, in which the magnonBEC generates pairs of (longitudinal) Higgs modes withωHiggs = ωdrive/2, although no direct demonstration ofdoubling of the period of the response of the magnonBEC itself was provided. The breaking of discrete timesymmetry may also result in the formation of time qua-sicrystals [32–36], where the periodic drive gives rise tothe quasiperioic motion with, say, two incommensurateperiods. We now discuss an observation of a time qua-sicrystal in a magnon BEC obtained by applying a drive,and its evolution to a superfluid time crystal (time super-crystal) when the drive is switched off.

In our experiments, the magnon BEC is trapped in apotential well. The trap is formed by the combinationof the spatial distribution of the orbital anisotropy axisof the Cooper pairs, called texture, and by a magneticfield minimum, created with a pinch coil (Fig. 2). Thepotential well is harmonic, and the magnon condensatecan be excited on several different energy levels in it, notonly in the ground state. The ground state can be si-multaneously populated by relaxation from the chosenexited level, forming a system of two coexisting conden-sates [37]. Similar off-resonant excitation of the coherentspin precession was first observed in Ref. [39]. It requiresan excitation at a higher frequency than the frequency ofthe coherent precession in the ground state, in partialanalogy with lasers [40] in the sense that a coherent sig-nal emerges from incoherent pumping.

One important property of magnon condensates in thetextural trap — as compared with, say, atomic BECs inultracold gases [41] — is that the trap is flexible. Thetrap is modified by the magnon BEC, which owing tothe spin-orbit interaction repels the texture and extendsthe trap. As a result the energy levels in the trap de-pend on the magnon number N in the condensate. Thismechanism is similar to the formation of hadrons in theso-called MIT bag model in quantum chromodynamics(QCD), in which free quarks dig a box-like hole in theQCD vacuum [42, 43]. And indeed, in the limit N →∞the harmonic trap transforms to the box [37, 38]. Theflexible trap provides an effective interaction between themagnons. In atomic BECs interactions also lead to de-pendence of the chemical potential on the number of par-ticles, but the functional dependence on N is different.The eigenstates in the magnon trap determine possiblefrequencies of the coherent precession. The dependenceof the precession frequency onN is seen in Fig. 3 at t > 0:during decay of the magnon BEC its ground-state energylevel increases as the trap shrinks in size and eventuallyreaches the undisturbed harmonic shape.

In Fig. 3 the frequency ωdrive of the driving rf field cor-responds to that of the second radial axially symmetricexcited state in the harmonic trap (level (2,0)). The drive

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FIG. 3. (Color online) Time crystal and time quasicrystal.The measured signal is analyzed using time-windowed Fouriertransformation, revealing two distinct states of coherent pre-cession seen as sharp peaks. Driving field Hrf with the fre-quency ωdrive, applied at t < 0, excites magnons on the sec-ond radial excited level in the harmonic trap. The majorityof these magnons moves to the ground level, where they formthe magnon BEC. Their precession is seen as the signal atthe smaller frequency, ωcoherent < ωdrive. The total signal isquasiperiodic in time, see Fig. 4. At t = 0 the rf pumpingis switched off and the (2,0) level quickly depopulates. Onlythe periodic signal from the precession of the ground-levelmagnon BEC remains, like in Fig. 1. Its frequency is slowlyincreasing in time in the flexible trap, following the decay ofthe magnon number N . This emphasizes robust nature ofbreaking of continuous time translation symmetry and signi-fies the time crystal.

pumps magnons to this level forming magnon condensatethere. One can see additional oscillations spontaneouslygenerated at a lower frequency ωcoherent < ωdrive, whichcorresponds to the BEC forming at the ground state inthe trap by magnons coming from the exited state. Notethat the frequencies of the ground and exited states canbe tuned by changing the magnetic field magnitude and,independently, the spacing of the states by the magneticfield profile. The spacing further depends on the numberof magnons in the trap, and one can choose the frequen-cies to be incommensurate. This demonstrates that thediscrete time symmetry, t → t + T , of the drive is spon-taneously broken leaving a state composed of precesisonat two incommensurate frequencies ωcoherent and ωdrive.We emphasize that the two states coexist in the sametrapping potential and hence occupy the same volume.The observed state is therefore a time quasicrystal (theterm time quasicrystal was introduced in Ref. [44]). Themeasured signal with oscillations at two well-resolved fre-

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FIG. 4. (Color online) Quasi-periodic signal from time qua-sicrystal at t < 0 in Fig. 3 is shown in time domain. Thesection plotted here begins at t = −4.97 s. The amplified sig-nal from NMR coils is downconverted to lower frequency usingLarmor frequency as a reference and digitized at 44 kHz rate.From the total signal the drive (lower plot) is subtracted, ascalibrated in a separate measurement without magnon BEC.The quasi-periodic signal (middle blue line in the upper plot)is then revealed. It is further splitted using band-pass fil-ters to the signal from the ground-level magnon BEC (uppergreen line) at (ωcoherent−ωL)/2π ≈ 290Hz and signal from the(2,0) level at (ωdrive−ωL)/2π ≈ 980Hz (lower red line). Theselines have been shifted vertically for clarity. Error bars on theright denote measurement noise amplitude in the respectivefrequency bands. Oscillations at (2,0) level proceed at exactlythe drive frequency as shown by the dashed vertical lines. Thefrequency locking occurs automatically by self-tuning of thesteady-state number of magnons [37].

quencies is shown in Fig. 4.

After the pumping is stopped (at t = 0 in Fig. 3),the excited-state condensate rapidly decays. What isleft is the condensate in the ground state, whose fre-quency slowly increases with time following the decayof the magnon number N . This state is a time crystal,as discussed above. That is, the driven time quasicrystalwith broken discrete time translation symmetry trans-forms to the time crystal with broken continuous timesymmetry.

In conclusion, in a single experiment we have observedboth types of time crystals discussed in the literature.These are states with broken continuous and discretetime translation symmetries. They are found in coher-ent spin dynamics of superfluid 3He-B, interpreted usingthe language of magnon Bose-Einstein condensation in aflexible trap provided by the 3He-B order parameter dis-tribution. The discrete time translation symmetry break-ing takes place under an applied rf drive. The magnoncondensation is then manifested by coherent spin preces-sion at a frequency smaller than the drive. The inducedprecession frequency is incommensurate with the drive,

giving rise to a time quasicrystal with the discrete time-translation symmetry being broken. When the drive isturned off, the self-sustained coherent precession lives fora long time, while the number of magnons decays onlyslowly. This is a time crystal. Both the time crystaland the time quasicrystal are formed in the topologicalsuperfluid 3He-B [45, 46] and possess spin superfluidity.Therefore these states can be called time supersolids.

We thank Jaakko Nissinen for stimulating discussions.This work has been supported by the European ResearchCouncil (ERC) under the European Union’s Horizon2020 research and innovation programme (Grant Agree-ment No. 694248). The work was carried out in the LowTemperature Laboratory, which is part of the OtaNanoresearch infrastructure of Aalto University. S. Autti ac-knowledges financial support from the Jenny and AnttiWihuri foundation.

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