Classical many-body time crystals
Toni L. Heugel, Matthias Oscity, Alexander Eichler, Oded Zilberberg, R. Chitra
CClassical many-body time crystals
Time crystals are readily obtained in the steady state of many-body classical systems thatundergo period-doubling bifurcations.
Toni L. Heugel , ∗ , Matthias Oscity , , ∗ , Alexander Eichler , Oded Zilberberg ,and R. Chitra Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Straße 27, 8093 Zürich, Switzerland. Fachhochschule Nordwestschweiz FHNW, Klosterzelgstrasse 2, CH-5210 Windisch, Switzerland. Institute for Solid State Physics, ETH Zürich, Wolfgang-Pauli-Straße 27, 8093 Zürich, Switzerland. ∗ These authors contributed equally.
Discrete time crystals are a many-body state of matter where the extensive sys-tem’s dynamics are slower than the forces acting on it. Nowadays, there is agrowing debate regarding the specific properties required to demonstrate sucha many-body state, alongside several experimental realizations. In this work,we provide a simple and pedagogical framework by which to obtain many-body time crystals using parametrically coupled resonators. In our analysis,we use classical period-doubling bifurcation theory and present a clear distinc-tion between single-mode time-translation symmetry breaking and a situationwhere an extensive number of degrees of freedom undergo the transition. Weexperimentally demonstrate this paradigm using coupled mechanical oscilla-tors, thus providing a clear route for time crystals realizations in real materi-als.
In periodically modulated nonlinear systems, discrete time-translation symmetry can be1 a r X i v : . [ phy s i c s . c l a ss - ph ] M a r pontaneously broken, leading to inherently slower dynamics than that of the drive ( ).A rapidly expanding community is principally focused on such a phenomenon in periodically-driven closed quantum systems, where disorder and interactions are considered to be essen-tial for so-called discrete time crystals (
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ). Atime-crystalline phase of matter stabilized by many-body localization was first observed in aone-dimensional trapped-ion system ( ). Surprisingly, time crystals were also seen in three-dimensional ensembles of NV-centers ( ) and in spin- / nuclei in phosphate materials ( )where disorder-induced localization effects are absent. The latter results indicate a wider classof time-crystalline behavior, including classical counterparts ( ).A natural arena for realizing time crystals is provided by parametric resonators. A parametrically-pumped resonator mode plays an important role in many areas of science and technology. Inits best-known form, parametric pumping describes the modulation of a resonator’s potentialat twice its natural frequency (
1, 2, 3, 4 ). When the modulation depth exceeds an instabilitythreshold, the resonator undergoes a period-doubling bifurcation to a new regime stabilizedby nonlinearities ( ). This time-translation symmetry breaking (TTSB) leads to two stable parametric phase states that have equal amplitude, opposite phase, and half the oscillation fre-quency of the parametric drive (
23, 24, 25, 17 ). Interestingly, these states can be associated withtwo states of a classical bit (
26, 27, 28, 29, 30 ) or with an Ising spin (
31, 32, 33, 34, 35, 36, 37,38, 39, 40, 41, 23, 24, 25 ). Network of such coupled resonators have been proposed as simu-lation platforms for complex Ising-like models that are very hard to solve with conventionalcomputers (
17, 31, 32, 33, 34, 35, 36, 37, 38 ).In this work, we show that a many-body TTSB can be easily realized in a classical networkof dissipative parametric resonators. We present a general theoretical analysis and derive condi-tions for the manifestation of many-body
TTSB in this system. This is complemented by a sim-ple tabletop experimental demonstration using two coupled resonators. Our setup allows us to2une the coupling strength, and we find a regime where the modes of the system jointly undergoTTSB into well-defined parametric phase state configurations. Our experiment thus realizes thesimplest building block that highlights the plethora of accessible TTSB solutions. At the sametime, we test our understanding of the general many-body model against a well-controlled andaccessible experimental implementation. Our work lifts the ambiguity surrounding the conceptof time crystals by establishing sufficient conditions for their generation.We consider a classical network of N coupled nonlinear parametric oscillators, whose dy-namics is governed by N equations of motion ¨ x i + ω i [1 − λ cos(Ω p t )] x i + γ i ˙ x i + α i x i + η i x i ˙ x i − (cid:88) i (cid:54) = j β ij x j = 0 , (1)where dots mark differentiations with respect to time t , x i is the displacement, ω i is the eigen-frequency, γ i the dissipation, α i the quartic nonlinearity, and η i the nonlinear damping of the i th mode. The system is excited by a single parametric pump of modulation depth λ and frequency Ω p . Each mode i couples to other modes j (cid:54) = i in the form of a driving force in proportion to x i and with a coupling coefficient β ij .We can perturbatively solve the system using the slow-flow method ( ): we rewrite Eq. (1)as N first-order differential equations and perform a van der Pol transformation with frequency ω = Ω p / , followed by time-averaging, to obtain the slow-flow equation ˙ X = A ( X ) X , (2)where X = ( u , v , u , v , . . . , u N , v N ) T , with u i and v i the slowly varying phase-space quadra-tures of the individual resonators. This equation is valid if the dimensionless quantities − ( ωω i ) , λ , γ i /ω i , η i ω i x i , β ij ω i , and α i ω i x i are of order (cid:15) , where < (cid:15) (cid:28) ( ). These conditions are easily3atisfied for a network of nearly identical oscillators. The matrix A can be written as A = a ( X ) b · · · b N b , a ( X ) . . . ...... . . . . . . b ( N − N b N · · · b ( N − N a N ( X ) , (3)where the a i and b ij are given by, a i ( X ) = − ω (cid:18) a i, a i, a i, a i, (cid:19) , b ij = (cid:32) β ij ω − β ij ω (cid:33) , with i (cid:54) = j and i, j = 1 , , . . . , N , and using the definitions a i, = a i, = 4 γ i ω + η i ωX i , a i, =2 ( λω i + 2( ω i − ω )) + 3 α i X i , a i, = 2 ( λω i − ω i − ω )) − α i X i , and X i = u i + v i . Ingeneral, the number of steady-state solutions, both stable and unstable, to this N -body problemvaries from to N depending on the parameter regimes ( ).In the absence of nonlinearities, α i = η i = 0 , the natural description of the resonatornetwork is given by N normal modes with eigenfrequencies ν k , k = 1 , . . . , N . The dynamicsof the normal modes is determined by the eigenvalues and the eigenvectors of A . The N eigenvectors define the positions and momenta of the N normal modes. The time evolutionof the k -th normal mode is given by e Λ k t , with Λ k the respective eigenvalue. The motion willbe bounded for negative Re { Λ k } and manifest parametric instability, i.e., unbounded dynamicswhen Re { Λ k } > . Each normal mode exhibits a corresponding parametric stability phasediagram known as ‘Arnold tongues’, delineating regions where dissipation stabilizes the motionand regions where the linear system shows unbounded dynamics, see Fig. 1(a). In the following,we will focus on the dominant instability lobe occurring around twice the natural frequency ofthe normal mode k , Ω p ∼ ν k , when the parametric drive exceeds a threshold λ ≥ λ k th ( ).In general, it is not dissipation but the underlying nonlinearities ( α i , η i ) that stabilize thenormal-mode oscillations against unbounded growth ( ). At the boundary of its main instabilitylobe, each normal mode undergoes a period-doubling bifurcation alongside a spontaneous Z
26, 27, 28, 29, 30 ). It is important tonote that although a single normal mode can involve an extensive number of resonators of thenetwork, it does not give rise to a many-body TTSB because it does not involve an extensivenumber of independent degrees of freedom.A many-body TTSB phase is realized in the resonator network in a region where an ex-tensive number of normal modes undergo the aforementioned period-doubling transition. Asimple recipe to realize a many-body TTSB consists of finding the parametric pumping ampli-tude λ MBth ( ω P ) = min λ { λ > λ k th ( ω P ) , ∀ k } at which all normal modes are driven above theirrespective instability thresholds, see Fig. 1(c). There, each normal mode finds itself in a para-metric phase state, see Fig. 1(d). Note that the many-body threshold holds in the limit of weaknonlinearities and does not include corrections stemming from nonlinear inter-normal modecoupling. In the mean-field limit of N identical resonators, i.e., ω i ≡ ω and γ i = γ , withall-to-all coupling β ij = β/ √ N , apart from the symmetric mode, all other instability lobes co-incide with that of the antisymmetric ( a ) mode. The respective instability thresholds ( λ > λ th ) are given by ( ): λ s/a th = 4 ωω (cid:115) γ (cid:18) ω − ω ω + (cid:26) ( N − N , s − N , a (cid:27) β ω (cid:19) . (4)The overlap region of λ s/a th defines λ ≥ λ MBth (Ω p ) .In the following we discuss two limits, ‘strong’ and ‘weak’ coupling, that are defined relativeto the parametric modulation strength, λ . For weak β ij , the normal modes closely resemblethe underlying constituent resonators. However, as β ij increase, the normal modes becomecollective in nature. In the many-body TTSB phase the system can choose one of N to N configurations: in the weak coupling regime, these correspond to the possible configurations of5he N individual resonators (
17, 45 ), while in the strong coupling regime, they correspond tothe configurations of collective normal modes. In both cases, all these configurations manifestTTSB and the chosen configuration will depend on initial conditions, noise and the strength ofthe nonlinearities. To summarize, we predict that an array of coupled dissipative parametricresonators realizes a stable TTSB phase in its steady state. This phase endures in a wide regionof parameter space and is robust to fluctuations.We now report on an experimental demonstration of many-body TTSB in a system of twocoupled mechanical modes. Our setup is based on the lowest transverse vibrational modes oftwo macroscopic strings. The strings are clamped onto a fixed frame at one end, while theother end is attached to a stiff plate that has two purposes; firstly, the plate can be driven intovibrations parallel to the string axes by an electric motor. These vibrations modulate the tensioninside the strings and generate parametric pumping of both string modes. Secondly, the platetransmits vibrations between the strings, which leads to weak intrinsic coupling between themodes. In some experiments, we introduce strong mode coupling by way of a mechanicalconnection close to the mode antinodes, see Fig. 2(a).The motion of each string is independently measured with a dedicated piezo detector embed-ded into one clamping point. We use a lock-in amplifier (Zurich Instruments HF2 LI) to actuateplate vibrations and to read out the electrical signals from the two piezo detectors, which areproportional to the strings’ displacements. All measurements in this work were carried outin the form of frequency sweeps, where the actuation frequency Ω p = 2 ω and the detectionfrequency ω were swept slowly to capture the steady-state response of the modes.We use weak external driving for calibration of the modes, similarly to the procedure out-lined in Ref. ( ). In these experiments, the vibration amplitude is kept low and the influenceof the intrinsic coupling is negligible. From the Lorentzian response of each mode, we extracttypical values for ω , / π = 155 ± Hz (depending on ambient temperature) and Q , ∼ ,6hile we calculate the effective mass m = 4 . × − kg from the geometry of the strings.By fitting to the large-amplitude response under strong parametric pumping, we obtain the co-efficients of the nonlinear potential term, α = 11 . mV − s − and α = 6 . mV − s − , aswell as those of the nonlinear damping, η = 7 . µ V − s − and η = 3 . µ V − s − (in thestrong coupling case, we find η = 3 . µ V − s − and η = 1 . µ V − s − ) ( ). Finally,in the presence of strong coupling, we use the normal mode frequency splitting to estimate β = 36 . ± . Hz.
Strong coupling [Fig. 2(a)]: we first explore the regime where the two instability lobes cor-responding to the symmetric and antisymmetric normal modes are well separated, see Fig. 2(b).In Figs. 2(c) and (d), we show the measured amplitudes and phases of both strings under a com-mon parametric modulation as a function of frequency ω/ π , respectively. As the frequency isslowly swept upwards, both resonators oscillate with the same phase from Hz up to
Hz.As the frequency is ramped further, the resonators are in opposing phase states from
Hz upto . Hz. The modes exhibit identical symmetries (s/a) when the frequency is swept down-wards. These qualitative observations were consistent over many sweeps. The small peaksaround ω/ π = 153 Hz correspond to an unidentified eigenmode in the experimental setup thatdoes not appear to affect the modes of interest.We model the system with Eq. (2) for N = 2 using the parameters extracted from the ex-periment. The results of our calculations provide a simple understanding of the experimentalobservations: as the frequency is swept, either the symmetric or antisymmetric normal modesundergo TTSB at their respective instability thresholds, recreating the effective single-modeTTSB discussed earlier. The coupling between the normal modes induced by nonlinearities isirrelevant in this regime as one mode is strongly off-resonant with the other. The experimentalresults are well described by the phase-space bifurcation diagrams for each resonator plottedin Figs. 2(e) and (f). Despite the fact that both resonators participate in the TTSB of the sym-7etric or antisymmetric modes, many-body TTSB is not observed in this strong-coupling limitas the two instability lobes do not overlap for experimentally accessible parametric excitationstrengths. Weak coupling : next, we remove the connection between the strings and rely on the driv-ing plate to provide weak coupling between the string modes [Fig. 3(a)-(b)]. The experimentaldata look very different in this regime [Fig. 3(c)-(d)]. Both strings have nearly identical nat-ural frequencies (within mHz from each other) and exhibit hysteresis when sweeping thefrequency upwards and downwards. The frequencies where the oscillation drops to zero (dur-ing upsweeps) or jumps to a finite amplitude (during downsweeps) are precisely the same forboth resonators. The strings oscillate in phase during the upsweep and out of phase during thedownsweep. All of these features were reproduced over many sweeps.The theoretical model corresponds to normal modes that are split by a very small coupling β , such that their instability lobes overlap strongly [Fig. 3(b)]. Since both normal modes exhibitTTSB and are weakly coupled by nonlinearities, we witness the realization of two-body TTSB.As before, the experimental results for the amplitude and phase are consistently explained bythe weak coupling bifurcation diagram for both strings shown in Figs. 3(e)-(f). In comparisonwith the strong coupling scenario of Figs. 2(e)-(f), the weakly coupled system exhibits richerbehavior. The selection of symmetric and antisymmetric solutions as a function of the sweepingdirection may be explained in terms of the phase response of a linear resonator to a periodicexternal force. Below its natural frequency, a harmonic resonator oscillates with almost nophase lag in response to an external force. As the two string modes drive each other, theyprefer to move in phase. In contrast, since the harmonic resonator response has a phase lagof ∼ π above the natural frequency, the string modes preferably oscillate out of phase duringthe downsweep. This many-body TTSB state is stable against small detunings ω (cid:54) = ω androbust to noise (as seen in the experiment). Increasing noise levels are expected to preserve the8nderlying TTSB, but to induce transitions between the different stable solutions.Coupled parametric resonators provide the simplest platform to realize macroscopic stateswith robust discrete time-translation symmetry breaking. Period-doubling bifurcations in stablesteady-states provide a rich space of solutions that manifest such many-body phenomena. Thiscan be readily generalized to the quantum realm, where the bifurcations physics is replaced bydissipative first- and second-order phase transitions (
46, 47, 48, 49 ). Furthermore, higher-periodTTSB can also be realized in these systems through a judicious choice of modulated nonlin-earities ( ). In the weak coupling limit, the classical network can be viewed as an Ising ma-chine that simulates complex problems, where the system parameters can be tuned to engineerdesired ‘spin configurations’ of the Ising-like phase states. The analogous quantum networkcomprising dissipative Kerr parametric resonators is expected to manifest an equivalent TTSBphase (
51, 52, 49, 46, 47, 48 ). Such networks have been proposed as quantum annealers ( ), andfollowing this work can now be used as quantum simulators of many-body time crystals. References
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4, 43 ), cf. Eq. (1). Green tongue shapes indicate regions where the linear resonatorbecomes unstable. (b) Beyond the instability threshold in the first lobe, Ω p ∼ ω , the nonlin-ear resonator undergoes a period-doubling bifurcation and oscillates with frequency Ω p / . Dueto the period doubling, there exist two possible phase states with equal amplitude but oppositephase. In some frequency ranges, there is an additional zero-displacement stable solution. (c)Zoom of the stability diagram around the first instability lobe for N coupled resonators. Here, N nondegenerate normal modes (marked by different color and symbols) generically arise,cf. Eqs. (1). The red area indicates the region of many-body TTSB. (d) Inside the many-bodyTTSB, each of the normal modes resides in one of the phase states. The resulting multi-stateconfiguration depends on the coupling coefficients β ij , the nonlinearities, and noise fluctuations.16 x l (a)(b) l III
145 150 15502040 w/2p (Hz) V ( m V ) R M S
145 150 15502 p f (r ad ) (c)(d) (e) (f) u uv v (mV) (mV) I II
I II string 1 string 2
III I II w W p w/2p (Hz) w/2p (Hz) w/2p (Hz) Figure 2: Strongly coupled oscillators: (a) Schematic setup representing two parametrically-driven strings coupled via an additional mechanical connection. (b) Calculated normal-modestability diagram of the symmetric and antisymmetric eigenmodes of the coupled system. (c)Measured amplitude and (d) phase of strings 1 and 2 for the upsweep (orange and brown) anddownsweep (light and dark blue) where both oscillators are parametrically driven at frequency ω . (e)-(f) Simulated steady-state solutions of oscillators 1 and 2 in the rotating frame phasespace ( u , v ) calculated from the slow-flow equations, cf. Eq. (2) as a function of ω . The thick(thin) tubes are stable (unstable) solutions and white spheres indicate the positions of bifurca-tions. The stable branches corresponding to the experiment are highlighted in matching colorsfor up- and down-sweeps. 17 l (a)(b) l V ( m V ) R M S p f (r ad ) (c)(d) (e) (f) u uv v (mV) (mV)string 1 string 2 b
160 162 164160 162 164 w W p w/2p (Hz) w/2p (Hz) w/2p (Hz) w/2p (Hz) Figure 3: Weakly coupled oscillators: (a) Schematic setup representing two parametrically-driven strings weakly coupled via the driving plate. (b) Normal mode stability diagram. (c)Amplitude and (b) phase of strings 1 and 2 for the up-sweep (orange and brown) and down-sweep (light and dark blue) where both oscillators are parametrically-driven at frequency ω .(c) and (d) show the simulated steady-state solutions of oscillators 1 and 2 in the rotating-framephase space ( u , v ) calculated from the slow-flow equations, cf. Eq. (2) as a function of ω . Thethin tubes are unstable solutions and all other colored tubes represent stable solutions. Thewhite spheres in these plots denote bifurcations.18 upplemental Material for Classical many-body time crystals
Toni L. Heugel , ∗ , Matthias Oscity , , ∗ , Alexander Eichler , Oded Zilberberg , and R. Chitra Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Straße 27, 8093 Zürich, Switzerland. Fachhochschule Nordwestschweiz FHNW, Klosterzelgstrasse 2, CH-5210 Windisch, Switzerland. Institute for Solid State Physics, ETH Zürich, Wolfgang-Pauli-Straße 27, 8093 Zürich, Switzerland. ∗ These authors contributed equally.
I Derivation of λ th We present here the derivation of the many-body parametric driving threshold amplitude for N resonators that are equally coupled to one another. For the prupose of this calculation, it sufficesto consider N coupled linear resonators. The slow-flow equation describing this system (cf.Eq.(2) in the paper with coupling β ij = β for all i (cid:54) = j ) is given by: ˙ X = A X , (SI.1)where the matrix A is given by: A = a b · · · bb a . . . ...... . . . . . . bb · · · b a . (SI.2)and the individual matrix entries a and b are given by: a = (cid:18) − γ − ω ( λω + 2( ω − ω )) − ω ( λω − ω − ω )) − γ (cid:19) , (SI.3) b = (cid:18) β ω − β ω (cid:19) . (SI.4)The dynamics of the linear system can be deduced by decomposing the initial state X intothe eigenvectors of A . The time evolution of each eigenvector is then determined by e Λ t , where1 is the corresponding eigenvalue. ImΛ (cid:54) = 0 imposes an oscillatory behavior whose envelopedecreases exponentially for
ReΛ < and increases exponentially for ReΛ > . To evaluatethese eigenvalues and eigenvectors, it is useful to rewrite the matrix A as: A = Id N ⊗ a + · · · . . . . . . ...... . . . . . . · · · ⊗ b = Id N ⊗ a + M N ⊗ b . (SI.5)Based on the structure of A , the eigenvectors obey the ansatz w m = r m ⊗ s m , where r m are the N dimensional eigenvectors of M N with eigenvalue ρ m and s m are the 2-dimensional eigenvectorsof a + ρ m b with eigenvalues σ m . Since M N has N eigenvectors, r m , with corresponding s m ,this ansatz describes all the N eigenvectors and eigenvalues of the matrix A . The 2-vector s m describes the amplitude and momentum ( u m , v m ) in a particular mode’s phase-space and r m generically describes the relative amplitudes and the phase configuration, e.g., r m = (1 , − means that the two oscillators have opposite phases. We can readily show that this ansatz isindeed an eigenvector of A : Aw m = Id N r m ⊗ as m + M N r m ⊗ bs m (SI.6) = r m ⊗ ( a + ρ m b ) s m = σ m w m . Since M N has a simple structure, we see that the eigenvectors { r m } take the form r =(1 , , · · · ) with eigenvalue ρ = N − and r m = (0 , · · · , , − , · · · ) T , where the +1 is the m th entry, are eigenvectors of M N with eigenvalues ρ m = − ( m ∈ N , < m ≤ N − ). Notethat the eigenvectors r m effectively determine the normal mode transformations of the problem.Next, we evaluate the eigenvectors s m and eigenvalues σ m of a + ρ m b = (cid:18) a a − a a + a a (cid:19) , (SI.7)2here a = − γ , a = − λω ω and a = ( ω − ω ) ω − ρ m β ω . These are given by, σ m, ± = a ± (cid:113) a − a , (SI.8) s m, ± = (cid:18) ±√ a − a √ a + a (cid:19) . (SI.9)To summarize, the N eigenvectors of the matrix A are given by w m, ± = r m ⊗ s m, ± , (SI.10)with corresponding eigenvalues σ m, ± and ≤ m ≤ N − .If Re σ m, ± > , the corresponding w ± ,m grows exponentially indicating a parametric in-stability. We obtain the parametric driving threshold λ th,m for this instability by imposing thecondition: σ m, + = a + (cid:113) a − a = 0 . (SI.11)Note that we have a < , whereas (cid:112) a − a can be either real-valued and positive or complex-valued. Solving Eq. SI.11, we obtain λ th,m = 4 ωω (cid:113) a + a = 4 ωω (cid:115) γ (cid:18) ω − ω ω + ρ m β ω (cid:19) . (SI.12)For identical oscillators, we see that there are primarily two instability thresholds correspondingto (i) the instability of the symmetric normal mode, w , + , and (ii) to the instability of all othernormal modes: w m, + including the antisymmetric mode. II Calibration measurements
In Fig. S1, we present test measurements that we have performed to ensure that the weaklycoupled strings were degenerate in frequency. On timescales of hours, thermal drift sometimescaused detuning between the strings, which we balanced by adjusting the tension of the strings3eparately. In Fig. S2, we show the fits used to extract the nonlinear coefficients of the twoweakly coupled strings. Please refer to Ref. [23] of the main text for details regarding themodel of a nonlinear parametric oscillator. (a) (b)
158 160 162 1640123 V R M S ( m V ) w/2p (Hz)
158 160 162 1640123456 f (r ad ) w/2p (Hz) Supplementary Material Figure S1: (a) Amplitude and (b) phase response of the two res-onators in the linear regime. We use weak external driving, no parametric drive, and weakcoupling to observe a Lorentzian response. Light and dark blue correspond to resonator 1 and2, respectively. These measurements are taken immediately before the nonlinear parametricmeasurements shown in Fig. 3 of the main text to ensure that the two modes are degenerate infrequency. 4
58 160 162 164020406080 V R M S ( m V ) w/2p (Hz)
158 160 162 164020406080 V R M S ( m V ) w/2p (Hz) (a) (b) Supplementary Material Figure S2: (a) Amplitude response of resonator 1 and (b) resonator2 with weak coupling and strong parametric driving. Blue and magenta lines correspond tosweeps with increasing and decreasing frequency, respectively. These are the same data asshown in Fig. 3c of the main text. Solid and dashed black lines are stable and unstable theorysolutions, respectively. From fitting these solutions to the measured data, we retrieve the valuesof α , and η ,2