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What is discrete time crystal?
Discrete time-crystals (DTCs) are emergent non-equilibrium phases of periodically driven many-body systems, with potential applications ranging from quantum computing to sensing and metrology. To define a non-equilibrium steady state as genuine time crystalline phase, there are three accepted criteria
(i)The system should essentially be an interacting many-body system(can constitute of many interaction bodies, particles or modes distributed in space) exhibiting collective behaviour and break the discrete time translational symmetry of the periodically driven Hamiltonian beyond a threshold strength.
(ii) The collective phase should be robust and rigid to external perturbations at finite temperature.
(iii)The robust collective phase should have exponentially long (ideally infinite) autocorrelation time.
Having realized in quantum systems with ions, atoms and defects in solids, multiple questions regarding DTCs, for example, stability mechanism of DTCs against drive heating and fluctuations, possibility of realization of DTCs in a classical system and existence of other DTC phases with complex symmetry types beyond subharmonic entrainment hitherto remain unanswered.
first experimental realization of truly many-body classical DTCs
Here, we answer some of these above-mentioned questions by observing a range of new DTC phases as well as transitions between them in a classical nanoelectromechanical system (NEMS) based on coupled graphene and silicon nitride membranes. We confirm the time-crystalline nature of these symmetry-broken phases by establishing their many-body characters, long-range time and spatial order, and rigidity against parameter fluctuation or noise. Furthermore, we employ controlled mechanical strain to drive the transitions between phases with different symmetries, thereby mapping the emergent time-crystalline phase diagram. Overall, the rich phase diagram with various distinct DTC phases takes a step towards establishing time-crystals as a system with complexity rivaling that of solid state crystals.
Our system consists of many interacting Graphene and SiNx resonator modes
Fundamental mode of both the graphene drums can have resonance coupling with many high quality factor SiNx modes but not coupled to each other directly. Similarly, coupling among SiNx modes exist through graphene modes mediated interaction.
Cartoon of manybody system where many SiNx modes (red peaks) are interacting with fundamental modes of graphene resonators (navy blue and cyan Lorentzian peaks) on resonance. SiNx membrane have higher density of modes around 2-3MHz (where the fundamental frequency of tunable graphene resonator modes lies).
Experimentally observed phases satisfy criteria of genuine DTCs
(A) Displacement spectra measured at X1 for a particular DC gate voltage shows transition from non-crystalline phase to subharmonic phase with increasing amplitude of a parametric drive at twice of the frequency ω of the center of the clusters of hybrid modes (inset), thereby breaking discrete time translation symmetry. Region I exhibits the isolated cluster of hybrid modes corresponding to G1, region II shows a sharp transition to a correlated liquid-like phase while in region III, the system undergoes a transition to a DTC phase. (B, C, D) Subharmonic, anharmonic and biharmonic DTC phases are observed at different gate voltage. Lower panels of B-D show corresponding thermal peaks. All measurements carried out at the point X1 (blue) and X2 (green). (E, F) Quadrature plots (X1 - Y1 and X2 - Y2) corresponding to the two peaks of the anharmonic phase (Fig. C) show thermal-like fluctuations. (G, H) Cross quadratures (X1 - Y2 and X2 - Y1) show strong anti-correlation, thereby show evidence for emergence of long-range order in space. (I) We model the system by replacing zoo of many coupled SiNx modes with a direct coupling between the two otherwise decoupled nonlinear graphene modes with a common parametric drive and open to a thermal bath with damping and fluctuations. Fixed point analysis in parameter space (α vs δ) for particular drive strength shows different possible phases of the mean-field model. Red, blue, green and yellow regions are representing noncrystalline, anharmonic, biharmonic and subharmonic responses respectively. Blue and black dots (inset figures corresponding to each region) are stable and unstable fixed points respectively. Elliptical paths around an unstable fixed point represent limit cycles. (J) We check the rigidity of the observed phases with respect to noise by feeding in a white noise signal through capacitive arrangement of our experimental setup (discussed in the previous report) and record the resulting DTC spectra with increasing noise amplitude. We observe the subharmonic phase remains stable till the relative noise becomes comparable to the drive and subsequently melts to a broad noisy spectrum. Displacement spectrum density for minimum (blue) and maximum (red) noise strength is shown here. (K) Subharmonic peak amplitudes (blue and black curves represent experimental data and numerical result respectively) are plotted against relative noise spectrum density. Experimentally we observe sharp phase boundary at a specific noise threshold, reminiscent of a first-order thermodynamic phase transition, while numerical simulation of mean field model shows no signature of sharp transition . This stark difference is interpreted as a direct evidence for the many-body nature of the emergent DTC phases.
(A) Displacement spectra measured at X1 for a particular DC gate voltage shows transition from non-crystalline phase to subharmonic phase with increasing amplitude of a parametric drive at twice of the frequency ω of the center of the clusters of hybrid modes (inset), thereby breaking discrete time translation symmetry. Region I exhibits the isolated cluster of hybrid modes corresponding to G1, region II shows a sharp transition to a correlated liquid-like phase while in region III, the system undergoes a transition to a DTC phase. (B, C, D) Subharmonic, anharmonic and biharmonic DTC phases are observed at different gate voltage. Lower panels of B-D show corresponding thermal peaks. All measurements carried out at the point X1 (blue) and X2 (green). (E, F) Quadrature plots (X1 - Y1 and X2 - Y2) corresponding to the two peaks of the anharmonic phase (Fig. C) show thermal-like fluctuations. (G, H) Cross quadratures (X1 - Y2 and X2 - Y1) show strong anti-correlation, thereby show evidence for emergence of long-range order in space. (I) We model the system by replacing zoo of many coupled SiNx modes with a direct coupling between the two otherwise decoupled nonlinear graphene modes with a common parametric drive and open to a thermal bath with damping and fluctuations. Fixed point analysis in parameter space (α vs δ) for particular drive strength shows different possible phases of the mean-field model. Red, blue, green and yellow regions are representing noncrystalline, anharmonic, biharmonic and subharmonic responses respectively. Blue and black dots (inset figures corresponding to each region) are stable and unstable fixed points respectively. Elliptical paths around an unstable fixed point represent limit cycles. (J) We check the rigidity of the observed phases with respect to noise by feeding in a white noise signal through capacitive arrangement of our experimental setup (discussed in the previous report) and record the resulting DTC spectra with increasing noise amplitude. We observe the subharmonic phase remains stable till the relative noise becomes comparable to the drive and subsequently melts to a broad noisy spectrum. Displacement spectrum density for minimum (blue) and maximum (red) noise strength is shown here. (K) Subharmonic peak amplitudes (blue and black curves represent experimental data and numerical result respectively) are plotted against relative noise spectrum density. Experimentally we observe sharp phase boundary at a specific noise threshold, reminiscent of a first-order thermodynamic phase transition, while numerical simulation of mean field model shows no signature of sharp transition . This stark difference is interpreted as a direct evidence for the many-body nature of the emergent DTC phases.
Applications and significance
This is the first experimental realization of truly many-body classical DTCs stabilized by dissipation in by far the simplest system that satisfies all accepted criteria of time crystals. Observation of rich phase diagram along with sharp phase boundaries beyond mean-field paradigm are found to be emergent many-body features.
Correspondence of these DTC phases with their quantum counterparts can lead to insights into interacting quantum systems.
For NEMS, multiple graphene resonators in spatially ordered geometries can be extended to new space-time-crystalline phases.
Exceptional spectral rigidity of DTCs can be applied as unique frequency markers in information technology while tunability to phase boundaries can be used in sensors and metrology.
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