Steady time crystal coupled to a mechanical mode as a cavity-optomechanics-like platform

Experimental setup

The pattern container we use is a cylindrical quartz-glass tube (15 cm lengthy, 2R = 5.85 mm diameter, Fig. 1). The ³He positioned within the cylinder is cooled down by a nuclear demagnetisation fridge into the superfluid B part. The decrease finish of the pattern container connects to a quantity of sintered silver powder surfaces, thermally linked to the nuclear refrigerant. This permits cooling the ³He within the cylinder all the way down to 130 μK.

Temperature of the superfluid is measured utilizing a quartz tuning fork^29,56, immersed within the superfluid. In the low-temperature regime investigated on this manuscript, the fork’s resonance width relies upon linearly on the thermal excitation density, which in flip depends upon temperature as (propto exp (-Delta /{ok}_{{{rm{B}}}}T)), the place Δ is the superfluid hole and ok_B is the Boltzmann fixed. Additionally, we are able to utilise the relief fee of the majority time crystal as a thermometer³⁰—its leisure fee likewise relies upon exponentially on temperature, (kappa propto exp (-Delta /{ok}_{{{rm{B}}}}T)).

The stress of the superfluid pattern is the same as saturated vapour stress, which is vanishingly small at these low temperatures. The superfluid transition temperature at this stress is T_c ≈ 0.93 mK. The transverse nuclear magnetic resonance (NMR) pick-up coil, positioned across the pattern container, is a part of a tank circuit resonator with the standard issue of about 150. The setup contains additionally a pinch coil to create a minimal alongside the vertical axis of the in any other case homogeneous axial magnetic subject. The resonance frequency of the tank circuit is 833 okHz in all measurements offered on this paper, akin to an exterior magnetic subject of 25 mT. We use a chilly preamplifier⁵⁷ and room temperature amplifiers to amplify the voltage induced within the NMR coils.

The free floor is positioned 3 mm above the situation of the magnetic subject minimal. We modify the liquid degree by eradicating ³He ranging from the initially totally crammed pattern container whereas measuring the stress of ³He gasoline in a calibrated quantity.

The time crystals are created by a brief (~1 ms) excitation pulse through the NMR coils. The pulse is resonant with an excited state within the entice that confines the magnons (particulars of the entice are described under). In about 0.1 s, nicely after the tip of the pump pulse, the magnons spontaneously kind a time crystal on the bottom state of the entice in a Bose-Einstein condensate part transition. The time crystal frequency and magnon quantity may be inferred from the AC voltage induced within the NMR coils, originating from the coherent precession of magnetisation within the time crystal at ω_TC. Details of the time crystal wave features and sign readout are defined in ref. ¹¹ and the Methods sections of Refs. ^18,19.

Signal evaluation

The precession of magnetisation within the time crystal produces a voltage sign within the pick-up coils. This sign is fed to a lock-in amplifier, which is often locked 2–3 okHz above the time crystal’s precession frequency and is thus used for the frequency downconversion. The lock-in output is sampled at 48 okHz frequency. An instance of such frequency-shifted report is proven in Fig. 1b. We remodel the wave report to frequency area by a windowed quick Fourier transformation (FFT) with a 3 × 10⁴ level window measurement and a ten% shift between home windows.

We then hint the frequency of the central band within the FFT sign in time. In most circumstances that is the utmost of the FFT sign, however care must be taken at giant mechanical drive amplitudes, the place the sideband amplitude might exceed the central band amplitude. This hint is then utilized by the becoming algorithm.

The amplitude of the FFT spectrum is match individually in every window to the FFT of the mannequin sign (U(t)=Asin int_{0}^{t}{omega }_{{{rm{TC}}}}({t}^{{prime} }),d{t}^{{prime} }) with ω_TC from Eq. (1) and (theta (t)={theta }_{max }sin {omega }_{{{rm{exc}}}}t). The floor modulation frequency ω_exc is set from the FFT of the geophone report (see the following part), measured concurrently with the time crystal sign. There are 4 becoming parameters: the general amplitude A, mixtures (G=g{theta }_{max }^{2}) and (Theta={theta }_{0}{theta }_{max }^{-1}) describing coupling to the floor mode and the typical frequency 〈ω_TC〉. Note that it is necessary for the right match to permit ω₀ in Eq. (1) to rely upon time as magnons are decaying from the entice, because the widths of the spectral bands are affected by this time dependence, particularly close to the start of the sign. From a single becoming parameter 〈ω_TC〉 for a window, we mannequin ω₀(t) utilizing time dependence of the traced frequency described above.

Mechanical forcing and calibration

A simplified schematic of the pattern positioning and the mechanical assist is proven in Supplementary Fig. 1 (See Supplemental Information). The cryostat is levitated on high of 4 energetic air spring dampers with connected distance sensors. These dampers are usually used to isolate the cryostat from mechanical vibrations within the surrounding assist constructions. The sensors feed their output to a proportional-integral-derivative (PID) controller. To induce mechanical forcing, we modulate the set level of one of many air springs as ({A}_{{{rm{nom}}}}sin ({omega }_{{{rm{exc}}}}t)) leading to small deviations of the cryostat’s tilt angle with the drive frequency ω_exc. The air springs are positioned at 90 cm radius and roughly 1.65 metres above the liquid floor within the pattern cell. The assist body itself is inflexible, leading to practically horizontal oscillations of the pattern. Similar mechanical forcing has been utilized beforehand for floor wave research in superfluid ³He and ⁴He²⁸.

We calibrate the lean angle by attaching a laser pointer to the cryostat and making use of a static tilt by altering the setpoint of the airspring we use for modulation. We then monitor the place of the laser pointer at fastened distance and calculate the corresponding tilt angle. This approach, we’re in a position to apply tilt angles ≲1.2°. In dynamic measurements, the utilized setpoint shifts are saved nicely under this worth. For recording amplitude of the dynamic drive, the cryostat is supplied with the geophone. It is put in at room temperature on the extent of the air spring dampers, and thus conversion of the geophone sign to the floor oscillation amplitude requires calibration described under.

We join the nominal mechanical forcing amplitude A_nom to the movement of the cryostat utilizing a voltage V_gp produced by the geophone. The result’s proven in Supplementary Fig. 2 (See Supplemental Information) from the identical knowledge set as in Fig. 3 of the primary textual content. We discover that the mechanical movement amplitude is nicely described by a operate of the shape

which incorporates three becoming parameters: a scale issue C, free exponent ν, and fixed B to permit for non-zero base degree within the measured voltage.

Converting the mechanical forcing as measured by the geophone to excitation of the free floor movement may be carried out in three steps. If we drive the floor mode on resonance, the utilized drive energy is the same as dissipated energy. The dissipation is noticed as a warmth stream into the superfluid, originating from the free floor movement. The ensuing temperature gradient may be measured immediately by evaluating the temperature measured by the thermometer fork on the backside of the container and utilizing the time crystal as a thermometer on the high of the container³⁰, see Supplementary Fig. 3a (See Supplemental Information).

The temperature gradient may be transformed to energy utilizing the identified thermal conductivity of superfluid ³He-B in a cylindrical container³¹. The thermal resistance of ³He-B at related experimental situations (stress, temperature, magnetic subject) was measured to be (R’_{rm T} approx 0.15,mu)Okay pW⁻¹ for a 5-cm-long cylindrical container with 8 mm diameter. Our cylindrical container has a  ≈ 6 mm diameter and the magnon condensate is positioned some  ≈ 14 cm above the underside quantity containing the thermometer fork. Scaling the thermal resistance with the lengths and inverse sq. radii, we get an estimate R_T ≈ 0.75 μK pW⁻¹ between the magnon time crystal and the warmth exchanger quantity in our experimental geometry.

The energy may be transformed into a spread of movement as follows. The fraction of vitality misplaced per cycle in a damped harmonic oscillator is

the place Q is the standard issue of the oscillator. The high quality issue estimated because the temperature dependent a part of the dependence proven in Fig. 2 provides Q ≈ 375, whereas the overall high quality issue, together with the zero-temperature offset for the floor wave resonance width, is Q ≈ 65. The gravity potential vitality saved within the free floor movement on the most amplitude of the oscillation cycle is

the place ρ_He = 81.9 kg m⁻³ is the density of the superfluid⁵⁸ and g_g = 9.81 m s⁻² is the free-fall acceleration. Thus, the dissipation energy turns into

which yields (P/{theta }_{max }^{2}approx 8.1,) pW deg⁻² for the overall high quality issue Q ≈ 65.

We can now categorical the measured temperature distinction (Delta tilde{T}({A}_{{{rm{exc}}}})=Delta {T}_{{{rm{TC}}}}({A}_{{{rm{exc}}}})-Delta {T}_{{{rm{fork}}}}), the place A_exc = V_gp(A_nom)/V_gp(A_nom = 0.098) is the normalised amplitude, as

Thus, drive amplitude is set as ({theta }_{max }^{2}{{{rm{(deg)}}}}^{2}approx Delta tilde{T}/6.1,mu)Okay. To receive direct relation between ({theta }_{max }) and A_exc (and thus V_gp), we carry out a single parameter match proven in Supplementary Fig. 3b (See Supplemental Information), leading to ({theta }_{max }^{2}{{{rm{(deg)}}}}^{2}approx 2.62,{A}_{{{rm{exc}}}}). In evaluation and plotting of the information in the primary textual content as a operate of the drive amplitude ({theta }_{max }), we convert utilized excitation A_nom to V_gp after which to ({theta }_{max }) utilizing calibration expressions above.

In Figure 3d the higher fringe of the shaded areas correspond to the calibration obtained as defined above, utilizing the complete Q ≈ 65. This calibration is used for ({theta }_{max }) scale in different panels of Fig. 3. In basic, we might count on a few of the dissipated warmth to be carried away by a layer of floor sure states on the partitions of the container and the free floor^34,59. The temperature-independent a part of dissipation is generated on the surfaces. If all of this warmth is carried away alongside the floor, by no means coming into the majority, then Q ≈ 375 might be acceptable to calibrate the lean angles as an alternative. The decrease fringe of the shaded areas in Fig. 3d corresponds to this restrict. It is believable that solely part of the generated warmth escapes the majority, and these two excessive limits we take as uncertainty of the calibration, which determines uncertainty vary of the decided optomechanical coupling g.

If we postulate that for the floor time crystal static and dynamic coupling coincide, we are able to match the dynamic coupling to the static measurements to select up a selected warmth launch fraction from the uncertainty band. The match proven by the dashed line in Fig. 3d provides g_surf  ≈ 3.74 Hz deg⁻². This worth corresponds to 84% of the temperature-independent a part of the dissipated warmth being carried away by the floor sure states. From the overall dissipation on the lowest experimental temperature this makes 69%.

Gravity wave resonances

For inviscid, incompressible, and irrotational stream, the dispersion of gravity waves in a cylindrical container a lot deeper than its diameter is given by⁶⁰

the place ok_i is the wave quantity, and σ = 155 μN/m is the floor rigidity of ³He⁶¹.

The velocity subject u associated to a scalar potential ϕ may be calculated as

For an infinitely deep cylindrical container, the potential related to the planar elementary mode is⁶²

the place (A={omega }_{{{rm{m}}}}{theta }_{max }{ok}_{1}^{-2}) is the wave amplitude, J_i is the Bessel operate of the primary variety or order i, ok₁ is the wave variety of the elemental mode, z is the vertical coordinate (z = 0 at fluid floor, destructive values in direction of the fluid), and φ is the azimuthal angle.

The ensuing velocity subject is then

which moreover units the magnitude of ok₁ by requiring the radial stream to be zero on the container wall. In different phrases, the wave quantity ok₁ is the primary answer to equation

setting ok₁ ≈ 1.8412/R.

Taking under consideration the meniscus impact, the gravity wave dispersion relation is modified to⁶³

For the bottom mode with ok₁, we get ω_SWm/2π ≈ 12.4 Hz, in good settlement with experimental observations.

Surface wave oscillations close to the container axis

The floor top profile of the primary mode takes the shape⁶⁴

To derive the utmost tilt angle in the course of the time evolution of the floor wave, we set (sin varphi=1). We can estimate the lean angle θ from the radial by-product of Eq. (M12):

Solving for θ and taking the restrict r → 0 (the time crystal is positioned near r = 0), to main order we get

the place the final approximation outcomes from Aok₁/2 ≪ 1. Therefore, we arrive to

the place we have now set ({theta }_{max }equiv A{ok}_{1}/2). This is the time dependence utilized in the primary textual content.

Magnon trapping potential

The axial trapping potential for (optical) magnons within the superfluid is ready by the magnetic subject profile as

the place ω_L is the native Larmor frequency and γ is the gyromagnetic ratio of ³He. The magnetic subject profile is created with a solenoid making a subject with inhomogeneity on the extent of 10⁻⁴, and by a pinch coil which produces an area minimal within the magnetic subject alongside the vertical axis, proven in Fig. 1.

The radial trapping potential is ready by the textural configuration through the spin-orbit interplay

the place Ω_L is the B-phase Leggett frequency and β_L is the polar angle of the Cooper pair orbital angular momentum, measured from the course of the static magnetic subject. In the absence of gravity waves, the minimal vitality configuration corresponds to the so-called flare-out texture⁶⁵. For low magnon numbers, akin to on the finish of the time crystals’ decay in our experiments, the flare-out texture with β_L ∝ r close to the axis leads to an roughly harmonic trapping potential with a attribute measurement set by the magnetic therapeutic size ξ_H²⁶. However, for giant magnon numbers, the potential is closely modified and may even develop into self-bound^11,66 or box-like²⁶. The complete trapping potential is a mixture of the magnetic and textural components.

For zero or low pinch coil currents the magnetic a part of the potential turns into much less necessary than the textural half. The floor vitality orients the orbital angular vector perpendicular to the floor, setting β_L = π/2 at container partitions and β_L = 0 on the free floor. The spatial variation of β_L creates the floor entice, which might moreover be recognized by the quicker magnon leisure fee¹⁸.

Optomechanical Hamiltonian

We describe the mix of a time crystal within the superlfuid entice and the transferring free floor as an optomechanical system with the Hamiltonian

the place ({hat{a}}^{{dagger} }) and (hat{a}) are magnon creation and annihilation operators, ({hat{b}}^{{dagger} }) and (hat{b}) are mechanical mode quanta (right here ripplons) creation and annihilation operators, g₁ and g₂ are the linear and quadratic coupling constants, respectively, and ω_m corresponds to the resonance frequency of the mechanical mode. The first “cavity” time period on the right-hand-side originates from equivalence of the time crystal frequency and magnon chemical potential for magnon CTC. We observe strategy of Ref. ⁶⁷ however the laser-drive-like phrases are usually not related to our experiments and don’t seem in Eq. (M18).

Eq. (1) in the primary textual content may be derived from Eq. (M18) by the next substitutions: g₂ → g, g₁ → 2gθ₀, and ({tilde{omega }}_{{{rm{TC}}}}to {omega }_{{{rm{0}}}}+2pi g{theta }_{0}^{2}), the place θ₀ is an efficient static tilt of the floor within the absence of oscillations. Thus, tilting the free floor with respect to the gravity potential leads to the modulation of the time crystal frequency. Note that by altering geometry (expressed as θ₀), the coupling may be easily tuned from quadratic to predominantly linear.

For mechanical movement as given in Eq. (M15), we get

the place Δω_TC = ω_TC(t) − ω₀. From Eq. (M19), one can see that there are parts at each ω and a pair ofω. The common frequency reads

that’s, the imply frequency will increase as (propto frac{1}{2}g{theta }_{max }^{2}). The origin of g from free-energy issues within the superfluid is mentioned within the supplemental materials.