Nonlinear two-level dynamics of quantum time crystals

Abstract

A time crystal is a macroscopic quantum system in periodic motion in its ground state. In our experiments, two coupled time crystals consisting of spin-wave quasiparticles (magnons) form a macroscopic two-level system. The two levels evolve in time as determined intrinsically by a nonlinear feedback, allowing us to construct spontaneous two-level dynamics. In the course of a level crossing, magnons move from the ground level to the excited level driven by the Landau-Zener effect, combined with Rabi population oscillations. We demonstrate that magnon time crystals allow access to every aspect and detail of quantum-coherent interactions in a single run of the experiment. Our work opens an outlook for the detection of surface-bound Majorana fermions in the underlying superfluid system, and invites technological exploitation of coherent magnon phenomena — potentially even at room temperature.

Introduction

Perpetual ground state motion in equilibrium defines a time crystal, but observing such motion is famously unfeasible. Experimental time crystal realisations thus bend either the equilibrium or the perpetuity requirement, reaching stability only if isolated from the environment and the observer. Consequentially, coupling separate time crystals while retaining sufficient isolation is challenging, and time crystals have yet not been studied in a dynamic environment. We arrange spontaneous two-level dynamics of interacting time crystals, each consisting of a million million magnons, in the superfluid B phase of helium-3. In this system, the observable time crystal life time can be extended up to a thousand seconds — a thousand million periods of motion — in the absence of a driving force, while the underlying superfluid system provides intrinsic feedback for engineering coherent dynamics.

Magnons in the B phase of superfluid helium-3 arise as the quanta of transverse spin waves, associated with magnetisation that precesses about the external magnetic field. At sufficient magnon density and low enough temperature, the precession synchronises spontaneously at uniform frequency and phase, forming a magnon Bose–Einstein condensate. The spontaneous synchronisation can be demonstrated by pumping magnons to a higher energy level in the confining trap from which they spontaneously fall to the ground state, or even by pumping incoherent magnons to the system using a noise drive. This shows that the magnon state in the condensate is decoupled from the drive. The transverse spin precession of the magnon condensate therefore manifests the characteristic spontaneous periodic motion of a time crystal.

The time crystal can be created using two different pumping techniques. Using a continuous drive yields a Floquet, that is discrete, time crystal. Here we use the pulsed technique where the drive is turned off before the time crystal evolution begins. This approach allows us to study uncontrived time crystal dynamics and interactions in the absence of external enforcement. The time crystal formation during the pumping pulse and its evolution thereafter is characterised by two timescales. The first timescale, about 0.1 seconds, describes the time crystal thermalisation — that is, how quickly the precession becomes coherent at the ground level in a trap, following the pumping of magnons. The second timescale is the time crystal lifetime. In an isolated sample container this lifetime grows exponentially towards infinity as temperature decreases. In practice there are also losses in the circuit that is coupled to the precessing spins for control and observation purposes. It is therefore necessary to allow for a finite lifetime. The time crystal remains well defined as long as the lifetime is much longer than the thermalisation time.

In superfluid helium-3, Cooper pairs possess orbital momentum whose average distribution, parametrised by the vector L, is axially symmetric in the sample container. The time crystal is trapped in the middle of the superfluid sample by that distribution owing to spin-orbit interaction. We fine-tune the trap by adding a magnetic field profile as detailed in the Methods. We also place a free surface of the superfluid above the bulk trap centre. The free surface distorts the distribution of L, resulting in a second local minimum, located 3 millimetres above the bulk trap minimum. Magnons can be trapped and form time crystals in either of the traps or both of them simultaneously. In this Article we concentrate on the lowest energy level in each trap. We denote the time crystals "bulk" and "surface" corresponding to the physical location. The location of a time crystal is identified from experimental records by its response to changes in the magnetic field profile.

Let us denote the bulk time crystal population — that is, the number of trapped magnons — as the bulk population, and the surface population likewise. The bulk and surface precession frequencies are determined by the profile of the confining trap, and the coupling between the crystals by the overlap of their wave functions, as detailed in the Methods. We will show that the dynamics of the coupled levels are described by a macroscopic two-level Hamiltonian, Equation 1 in the source: Planck's reduced constant multiplying a two-by-two matrix whose diagonal entries are the bulk precession frequency, itself a function of the instantaneous bulk population, and the surface precession frequency, and whose off-diagonal entries are both the coupling. Note that this two-level system is conveniently parametrised by a macroscopic Bloch sphere, with the relative precession phase between the time crystals corresponding to the azimuthal angle and level populations corresponding to projection on the vertical axis, in a direct analogy with the Bloch sphere description of microscopic two-level systems such as qubits. In what follows we measure all frequencies in the frame rotating at the Larmor frequency taken at the centre of the bulk trap; the gyromagnetic ratio of helium-3 is about minus two hundred million radians per second per tesla.

The essential difference of Equation 1 from a standard two-level Hamiltonian is the dependence of the bulk frequency on the bulk population, which arises due to a nonlinear feedback provided by the spin-orbit trap: a large local magnon density expands the trap, changing the distribution of L, thus decreasing the bulk frequency. Near the free surface the distribution of L is fixed perpendicular to the surface. Therefore the surface frequency is constant to a good approximation. As the time crystal populations decay, the bulk population decreases and the bulk frequency increases. We can thus make the energy levels in the double trap cross by selecting suitable initial populations. A rigorous description of the macroscopic time crystal wave functions, the trapping potential, and the feedback mechanism are derived in the Methods. We emphasise that all the relevant technical explanations can be found in the Methods also where not explicitly referenced.

We note that another characteristic feature of time crystals, the lack of heating under continuous drive, is also manifest in this system. Under continuous pumping, the number of magnons in the time crystal is determined by the chemical potential that corresponds to the pumping frequency. When this number would be exceeded, the time crystal spontaneously decouples from the drive, thus preventing overheating. Similarly, during the slow population decrease after a pumping pulse, the chemical potential — the precession frequency — is continuously adjusted to the changing magnon number. Hence, the precession period and coherence become incommensurate with and thus independent of the drive pulse even if the drive was originally resonant.

In this Article, we study the dynamics of the two-level time crystal system, carrying out two experiments. In the first experiment, where the level crossing takes place at small bulk population, we showcase crossing dynamics that follow the textbook description: the system is initially in the ground state, but at an avoided level crossing both levels are populated owing to Landau-Zener population transfer. This "superposition" state continues to be modified by Rabi population oscillations after the crossing. The second experiment starts from a "superposition"; the level crossing takes place at large bulk population where the feedback mechanism transforms into a dynamically changing coupling. Analysis of the second experiment shows that co-existing time crystals lay many-body interactions bare for the capable observer in a single run of the experiment. That is, in this case, the level crossing dynamics cannot be described analytically, but while coherent quantum phenomena are often hidden from direct inspection, time crystals have no such limitations.

Results

Basic two-level dynamics at small bulk population

The time crystal levels can be populated in desired proportion by a radio-frequency pulse via adjacent coils. To highlight the two-level dynamics, we populate only the bulk time crystal in the beginning of the experiment. After the pulse, the coherent precession of magnetisation induces an oscillating signal in the coils, which allows inferring the precession frequency, and the signal amplitude yields the magnon number. The pumping is followed by exponential decay of the bulk population with a time constant controlled by temperature as detailed in the Methods.

In this first experiment the ground level is initially located in the bulk trap, the bulk population decays at a rate determined by that time constant, and the bulk frequency increases slowly as the trap recovers a narrower shape. Meanwhile, the surface frequency remains constant. Hence, the bulk frequency eventually crosses the surface frequency before levelling out. In a coupled two-level system, a level crossing has specific consequences: the observed frequencies are the dressed eigenfrequencies of the Hamiltonian, which deviate from the undressed frequencies in the Rabi regime — that is, when the coupling exceeds the magnitude of the difference between the two undressed frequencies. Due to this hybridisation, the observed levels avoid crossing each other, and the global ground level smoothly switches from bulk to surface. Population transfer between the levels is also observed, as both of the levels are populated after the avoided crossing.

Traversing the avoided crossing adiabatically would allow the entire magnon population to follow the global ground state, but here some magnons move to the state with higher eigenenergy, that is, higher precession frequency. This process is generally known as Landau–Zener–Stueckelberg–Majorana tunnelling, or Landau–Zener tunnelling. The population transferred depends on the rate of level crossing at the point where the two undressed frequencies are equal. For usual coupled non-linear oscillators with damping, the crossing rate is determined directly by the damping. Here the decay time constant of 3.5 seconds corresponds to a crossing rate of about 24 hertz per second. Using this and the coupling directly extracted from the experiment as explained below yields a predicted Landau–Zener population transfer fraction of 0.9 per cent to the excited state, which is two orders of magnitude smaller than that observed in the experiment. In this case, only one of the two levels would be visible after the crossing. In similar experimental runs with a smaller crossing rate we have observed population transfer up to twenty orders of magnitude larger than the corresponding Landau-Zener prediction.

We can analyse this striking mismatch by simulating the time evolution of the two-level Hamiltonian numerically. We feed the experimentally determined bulk and surface decay rates, the corresponding initial populations, and the measured dependence of bulk frequency on bulk population into a numerical simulation of the two-level Hamiltonian. The coupling constant is used as a fitting parameter, yielding a coupling of 1.4 hertz.

The outcome of the simulation, plotted in the same way as the experimental signal, can be compared directly with the experiment by subtracting the simulated time-dependent Fourier spectrum from the experimental one. The simulation underestimates the change of the surface time crystal frequency near the crossing, which causes the largest deviation between the Fourier spectra. Otherwise the typical deviation between the two signals, as normalised by the total signal at the crossing, is less than 5 per cent. In particular, the simulation replicates the magnitude of the population transfer, that is, 60 per cent of magnons move to the excited state. We emphasise that repeated runs of the simulation with perturbed input parameters reveal that this qualitative level of population transfer is insensitive to the precise value of any of the input parameters.

To explain this observation, we extract the undressed frequencies from the simulation in the region near the avoided crossing. As a result of the feedback of population on bulk frequency, the bulk frequency is changing both owing to the slow decay of the bulk population and because of the population transfer from bulk to surface by Rabi oscillations. Thus, their combined effect increases the crossing rate. The magnitude of the Landau-Zener population transfer is determined within about 50 milliseconds of the crossing, and in this window the bulk frequency can be linearised. Inserting the accelerated crossing rate into the Landau-Zener formula yields the expected population transfer of 61 per cent, in good agreement with the simulated population transfer of 60 per cent. That is, the population transfer follows the Landau-Zener description with the crossing rate taken at the instant of level crossing. Note that the Landau-Zener description is thus valid even if the crossing rate is regulated by intrinsic feedback. We conclude that the observed population transfer strongly supports the two-level interpretation of the time-crystal dynamics.

Far from the avoided crossing, the two-level interaction is characterised by alternating-current Josephson population oscillations between the levels. Owing to the feedback in the bulk trap, the oscillations result in a side band that follows the bulk trace. The frequency of the population oscillations is set by the difference of the time crystal precession frequencies, equal to the difference of their chemical potentials. Thus, the side band is separated from the bulk trace by the magnitude of that frequency difference. The amplitude of the population oscillations is determined by the coupling, and the relative side band amplitude by the slope of the bulk frequency against bulk population. Thus, we can extract the coupling directly from the experimental data, yielding 1.7 plus or minus 0.4 hertz in the level crossing region, in good agreement with the simulation fitted value of 1.4 hertz.

Put together, the Josephson population oscillations, the Landau–Zener population transfer, and the agreement on the two-level coupling independently extracted from the different aspects of the population dynamics confirm that the two time crystals form a macroscopic two-level system.

Dynamical coupling regime at large bulk population

Magnon time crystal dynamics, enhanced by the nonlinear feedback, can be analysed also directly without resorting to a numerical simulation of the system. This is advantageous as it will allow untangling interactions involving multiple time crystals that go beyond the two-level description. As a simple demonstration of this capability, we introduce a level crossing in a region where the bulk population is an order of magnitude larger than above. The resulting trap modification affects not only the bulk frequency but also the constriction between the time crystals. This causes the coupling to change dynamically in the course of the crossing. Both levels are populated in the beginning of the experiment to allow following their dynamics directly.

In this experiment the coupling is changing, but qualitatively the dynamics follow a similar pattern as above: the ground state moves from bulk to surface when the undressed frequencies cross at about 3.8 seconds. The moment of the crossing is identified by a sharp increase in the ground state trace relaxation rate, from 0.06 per second to 0.53 per second. The increase is attributed to increased dissipation in the surface trap due to surface-mediated emission of other spin wave modes and potentially surface-bound Majorana states, but a detailed study is left for the future. We note that the states can be identified also by adjusting the magnetic field profile and rerunning the experiment; the relaxation rate is a convenient shortcut for distinguishing the two levels.

Josephson population oscillations between the two time crystals are seen as the side band that follows the bulk time crystal trace. As explained above, the side band is separated from the bulk trace by the magnitude of the difference between the bulk and surface frequencies. This separation is characteristic of the Josephson effect, and it changes in time because the bulk frequency changes. The surface time crystal is not followed by a similar side band, because the surface trap is rigid and hence population oscillations result in no side bands. A second bulk trace side band should be located symmetrically on the other side of the bulk trace, but it exactly coincides with the surface trace and is therefore not resolvable.

The side band amplitude allows us to extract the coupling between the time crystals. The extracted coupling is the largest in the beginning of the experiment and decreases when the bulk population decreases. That is, the constriction between the time crystals is affected by the bulk trap modification, which makes the coupling larger when the bulk population is large. This is qualitatively in line with the trap modification mechanism discussed in earlier work.

Near the avoided crossing, interference effects prohibit direct access to the population oscillation in the experiment. However, in a two-level system the coupling can also be extracted from the minimum frequency separation of the dressed frequencies of the two levels at the avoided crossing, which is equal to twice the coupling. This is done by interpolation. The result is in good agreement with the dependence extracted from the side bands. Note that near the avoided crossing the separation of the dressed, that is observed, frequencies equals the difference between the Josephson frequency and the Rabi frequency. That is, Rabi population oscillations smoothly replace the Josephson oscillations, increasing the population oscillation frequency as compared with the Josephson frequency, which goes to zero.

It is worth noting that the relaxation rate of the bulk time crystal depends on whether it is the global ground state or the global excited state. The bulk relaxation rate is 0.06 per second until the level crossing, when it is the ground state, and it increases to 0.2 per second after the level crossing, when it is the excited state. The same observation, correspondingly, applies to the surface time crystal relaxation rates. We emphasise that such change is never observed in the absence of the level crossing, for example if the bulk time crystal is the ground state throughout the experiment. That is, the excited state seems to slowly leak magnons to the ground state. This observation hints that there is an additional incoherent channel that allows magnons to move from the excited state to the ground state, showing independently that the two time crystals interact and that the level crossing has physical consequences that penetrate the dynamics in the two-level system.

The above analysis confirms that the two-level description is valid and robust against dynamic variation of its parameters, and that direct experimental observations provide continuous access to all relevant aspects of the interaction.

Discussion

To summarise, we have shown that the dynamics and interactions of the two adjacent magnon time crystals are quantitatively described by a two-level Hamiltonian. The levels are modified by a nonlinear feedback, arising owing to spin-orbit interaction in the underlying superfluid system. This allows engineering intrinsic time crystal dynamics in the absence of continuous external drive. We show that when the two-level eigenfrequencies approach one another, the coupling between the levels results in an avoided crossing with ensuing Landau-Zener population transfer from the global ground state to the excited state. Rabi population oscillations, combined with the feedback mechanism, increase the population transfer by orders of magnitude. This is quantified by comparing numerically simulated population dynamics with the experiments. We also show that all relevant observables and parameters including the eigenfrequencies and the coupling between the time crystals can be simultaneously extracted from the experiment. We emphasise that each measurement sequence shown in this Article corresponds to a single run of the experiment, but the phenomena are well reproducible.

We have shown that the spin-orbit interaction can be harnessed to create a nonlinear feedback for magnons in a coherent time crystal system. Nonlinear feedback is needed for spin-based versions of quintessential quantum devices such as the SQUID. It remains an interesting task to explore the time-crystal two-level system further by demonstrating parametric pumping of magnons and logic gate operations between the two levels. For example, parametric pumping can be arranged by modulating the magnetic part of the trap at the coupling frequency. Additionally, any number of co-existing time crystals can be accommodated in a magnetic landscape to increase the number of degrees of freedom, and the flexible trap can be turned off by adjusting the external magnetic field. These are important capabilities for realising magnon-based devices. To access phenomena such as quantum entanglement, few-magnon operations can be implemented using nano-fluidic confinement and ultra-sensitive nuclear magnetic resonance techniques. We emphasise that similar physical phenomena including quasiparticle Bose-Einstein condensation and the emergence of time crystals can be accessed in certain solid-state room-temperature systems, for example based on magnons in yttrium iron garnet films. This opens the outlook of quasiparticle-based coherent on-chip applications in ambient conditions, including coherent quantum information processing.

The three-dimensional topological superfluid is wrapped by a two-dimensional system of surface-bound quasiparticles, among them Majorana fermions. At the free surface there are no impurities, unlike at sample container walls, and surface-bound Majorana fermions are expected to manifest themselves as detectable zero-temperature magnetic dissipation. Majorana fermions have remained elusive despite a decade of searching in different condensed matter systems. The hybridised two-level state is in direct contact with the free surface of the superfluid, making an extremely sensitive probe for the bound Majorana states. We have provided preliminary evidence that the surface induces magnetic dissipation and that details of this signature can be explored using the time-crystal two-level system.

Figure captions

Figure 1. Schematic illustration of the experiment. The superfluid helium-3 sample is contained in a quartz glass cylinder. The magnon time crystal is trapped in the middle of the container by the combined effect of a minimum in the static magnetic field, created using a pinch coil, and by the spatial distribution of the superfluid orbital momentum L. The coherent precession of magnetisation in the time crystal is observed using transverse pick-up coils. The static magnetic field is oriented parallel to the axis of the cylinder. The ripple on the superfluid free surface is added for illustrational purposes.

Figure 2. Time crystal two-level system. (a) The distribution of L confines magnons in two local minima, hosting two adjacent time crystals: one in the bulk of the superfluid and the other one touching the free surface. In each time crystal, magnetisation is precessing coherently, which couples to measurement circuitry. (b) Magnons in the bulk modify the confining trap created by the distribution of L. When the bulk population is large, the textural trap is widened, which modifies also the surface time crystal's wave function. This increases the coupling between the states. Changes in the trap and the wave functions have been exaggerated for illustrational purposes. (c) The state of the two-level system can be illustrated using a Bloch sphere where radial distance corresponds to the total magnon number, the relative phase between the time crystals' precession corresponds to the azimuthal angle, and the polar angle describes the relative weights of the two-level basis states in the "superposition".

Figure 3. Two-level time crystal dynamics at small bulk population. (a) The signal from the pick-up coils, analysed with windowed Fourier transformation, shows the bulk time crystal as a moving sharp peak. Frequency is plotted in the rotating frame. The excitation pulse at time zero is framed out for clarity. Initially the bulk frequency is below the surface frequency, but as the population in the bulk trap decays, at 3.3 seconds the global ground state moves to the surface in an avoided crossing. The excited state, now located in the bulk, is simultaneously populated. Rabi, that is Josephson, population oscillations are seen as a side band. Coupling extracted from the side band extrapolates to 1.7 plus or minus 0.4 hertz at the crossing, in good agreement with the fitted simulation value of about 1.4 hertz. (b) The numerical simulation recreates the population transfer and the side band, confirming that the population transfer is due to a Landau-Zener transition. In the absence of measurement noise, the side band of the surface time trace is also weakly visible. (c) Subtracting the simulation from the experiment shows that the point-wise residuals remain smaller than 5 per cent. The relative difference is normalised by the total signal at the crossing. In this measurement the temperature was 180 microkelvin and the Larmor frequency 833 kilohertz.

Figure 4. Analysis of two-level dynamics at small bulk population. (a) The dressed ground level time crystal frequency in the simulation follows that extracted from the experiment. The experimental line is obtained by tracing the maximum in the Fourier spectrum of Figure 3, with oscillations filtered out by the long time window. The ground state frequency initially corresponds to the bulk frequency, and after the avoided crossing at 3.3 seconds it corresponds to the surface frequency. Simultaneously, the excited level frequency switches from the surface value to the bulk value. Oscillations of the frequencies after the crossing arise owing to the population oscillations. (b) We can hence use the simulation to extract the undressed frequencies, which cross at 3.3 seconds. The crossing rate is accelerated intrinsically by the feedback of population on bulk frequency. The Landau-Zener population transfer magnitude is determined by this speed-up. (c) The measured signal amplitude agrees with the simulated dressed populations. Both the total population from simulation and the measured signal are normalised to one at the crossing. (d) The undressed bulk and surface populations are extracted from the simulation. The total population is normalised to one at the crossing. After the crossing, the bulk population with its exponential decay compensated averages at 0.61, corresponding to the fraction of population transferred to the excited state by the Landau-Zener mechanism.

Figure 5. Two-level time crystal with dynamic coupling at large bulk population. (a) The time crystals are created at time zero. Frequencies are plotted in the rotating frame. (b) Initially the ground level is located in the bulk and the excited level at the surface. At about 3.8 seconds the ground level moves smoothly to the surface in an avoided crossing. (c) Most of the population follows the ground level movement from bulk to surface, identified by a sharp increase in the exponential relaxation rate. Total population is normalised to one at the crossing. (d) Population oscillations between the time crystals are seen as a side band of the bulk crystal trace at the side-band frequency. The side band's frequency separation from the bulk trace is equal to the frequency separation of the main traces. (e) The coupling can be extracted from the side band and main trace amplitudes, in good agreement with that estimated by linear interpolation from the separation of the main traces in panel b. In this measurement the temperature was 150 microkelvin and the Larmor frequency 624 kilohertz.

Methods

Experiment

The superfluid helium-3 sample is placed in a cylindrical quartz-glass container, 15 centimetres long and 6 millimetres in diameter, in a nuclear demagnetisation refrigerator. The lower end of the sample container connects to a volume of sintered silver powder surfaces, thermally linked to the nuclear refrigerant. This allows cooling the helium-3 down to 130 microkelvin. Temperature of the superfluid is measured using a quartz tuning fork, and pressure is equal to saturated vapour pressure, which is vanishingly small at these low temperatures. The superfluid transition temperature at saturated vapour pressure is about 0.9 millikelvin. The sample container is surrounded by two transverse nuclear-magnetic-resonance coils, which are part of a tank circuit resonator with a quality factor of about 150, and a pinch coil used to create an axial minimum of the magnetic field. The resonance frequency of the tank circuit can be tuned in eight equidistant steps between 550 kilohertz and 833 kilohertz, corresponding to external magnetic fields between 16.5 and 25 millitesla. The signal is amplified by a cold preamplifier and room temperature amplifiers.

The free surface is located 3 millimetres above the centre of the magnetic field minimum. The location of the free surface is adjusted by removing helium-3 slowly until the desired location is achieved, measuring the pressure of helium-3 gas in a calibrated volume that results from the removal of liquid from the originally fully filled sample container. The outcome is favourably compared with the observed magnon spectrum and a numerical model of the trap.

The time crystal wave function can be written as an amplitude multiplied by the exponential of minus the imaginary unit times the precession frequency times time, where the precession frequency is related to the chemical potential by Planck's reduced constant, the phase term is contained in the amplitude, and the number of magnons is the squared modulus of the amplitude. The tipping angle of the precessing magnetisation, measured from the magnetic field, parametrises the spatial profile of the wave function: the magnon number is the volume integral of the squared sine of half that tipping angle. The signal induced in the pick-up coils is sinusoidal, corresponding to the magnetisation along the axis of the coil — in other words, the real part of the rotating complex wave function. The measured signal amplitude is proportional to the amplitude of the time crystal wave function, Equation 2 in the source: the amplitude equals a constant times the square root of the magnon number, where the constant contains the so-called filling factor of the state within the coils, the amplification provided by the tank circuit resonator and other amplifiers in the measurement circuit, and physical constants.

A desired level in the trap can be populated by a radio-frequency pulse via the pick-up coils, followed by slow population decay owing to two mechanisms. The fermionic thermal excitations of the superfluid cause non-hydrodynamic spin diffusion; this contribution can be made exponentially small in the zero-temperature limit — a thousand-second lifetime has been achieved — or dominant at higher temperatures. Observing and controlling the quasi-perpetual time crystal motion inevitably causes also external dissipation, in our case radiation losses in the measurement circuitry. Both of these dissipation mechanisms cause exponential population decay in time. The time crystals are well defined provided the lifetime, which here is about 10 seconds, is much longer than the time it takes for the time crystal to form after the pulse, here about 0.1 seconds.

The two-level system, in the absence of coupling between the states, is described by the "superposition" wave function: a bulk amplitude times the exponential of the imaginary unit times the bulk frequency times time, plus a surface amplitude times the exponential of the imaginary unit times the surface frequency times time, where each amplitude is the square root of that level's population times a phase factor. Only the relative phase enters the dynamics of the system. Hence the bulk amplitude can be chosen to be real, and the pair of amplitudes is conveniently illustrated by a macroscopic Bloch sphere: the surface of the sphere corresponds to states with the total magnon number, and the interior to smaller magnon numbers reached during the population decay. The weights of the basis states in the superposition — the fraction of the total population in the bulk or surface state — are given by the polar angle, and the relative phase corresponds to the azimuthal angle in the equatorial plane of the sphere. Equation 3 in the source states that the relative phase evolves in time as the difference of the initial phases plus the time integral of the difference between the bulk and surface frequencies. We note that controlling the relative phase is beyond the scope of the present work and requires adjusting the coil geometry.

Level dynamics in a flexible trap

The B phase of helium-3 is a p-wave superfluid, hence the orbital momentum of the Cooper pairs is equal to one. In the sample container cylinder, the average orbital momentum L is distributed symmetrically — the "texture" — owing to the orienting effects of the magnetic field and the container walls. In addition, we create an axial minimum of the magnetic field using a pinch coil, which confines the magnons due to the Zeeman energy. The bulk trapping potential therefore has a magnetic part, Equation 4 in the source: Planck's reduced constant times the local Larmor frequency, which depends on position. It also has a component created by the distribution of L owing to the spin-orbit interaction, Equation 5 in the source: Planck's reduced constant times four fifths of the squared B-phase Leggett frequency divided by the Larmor frequency, multiplied by the squared sine of half the tipping angle of the orbital anisotropy axis, that angle being measured from the direction of the magnetic field along the cylinder axis.

Bringing the free surface above the trap centre distorts the order-parameter trap, since the orbital tipping angle is zero at the free surface, creating a local minimum at the surface. Note that we study the time crystals in a frame rotating at the Larmor frequency where the uniform magnetic field is absent. Where the Larmor frequency is written without an explicit reference to position, this means the Larmor frequency in the middle of the bulk trap, corresponding to the minimum of the harmonic trapping potential. The time crystals located in the two traps can be identified and their frequencies adjusted by changing the profile of the field minimum, aided by the different relaxation rates.

The harmonic bulk trap has a radial trapping frequency of about 200 hertz corresponding to the spin-orbit potential and an axial trapping frequency of about 20 hertz corresponding to the magnetic potential. The resulting precession frequency is the Larmor frequency plus the radial trapping frequency plus half the axial trapping frequency. Therefore the axial trap can be neglected in the analysis, and it is convenient to measure all frequencies in the frame rotating at the Larmor frequency.

The textural part of the trapping potential feels local magnon density due to spin-orbit interaction. The equilibrium texture minimises a range of free-energy contributions, including the orienting effects of the magnetic field and the sample container walls. An important additional contribution is the spin-orbit interaction energy, Equation 6 in the source: the squared modulus of the wave function multiplied by the textural trapping potential, where the wave function carries the spatial variation of magnon density which gives rise to the feedback effect. That is, the bulk trap profile and the shape of the time crystal wave function depend on the bulk population such that the derivative of the bulk frequency with respect to the bulk population is negative. In the limit of large magnon number the bulk trapping frequency follows Equation 7 in the source: the zero-magnon trapping frequency multiplied by one minus a constant times the bulk population, all raised to a power of about five sevenths, where the constant depends on the rigidity of the textural trap and the profile of the magnetic field minimum. We emphasise that although the bulk frequency changes during the decay of the magnon time crystal, the change is very slow as compared with the Larmor frequency of about one megahertz, and we can thus assume that the wave functions always correspond to the instantaneous trap shape. Note that the surface trap is rigidified by the adjacency of the free surface, and the surface frequency is therefore independent of the surface population to a good approximation.

It is possible to describe the self-trapping effect numerically in a self-consistent calculation of the order parameter texture, the resulting trap, the time crystal wave function, and the population decay. That is however not necessary for understanding the experiments presented in this Article, because finding a general form of Equation 7 can be circumvented by fitting and numerical differentiation of the experimental data where necessary, and all other effects can be measured independently. For simplicity, we refer to Equation 7 in the discussion below, but the reader should bear in mind that the general form of the nonlinearity is more complicated.

Josephson coupling

Let us study the observable consequences of the population oscillation. We use the language of the Josephson effect, analogous to the alternating-current Josephson effect, as the oscillation amplitude can only be reliably extracted from the experiment far from the avoided crossing. Near the avoided crossing one should use the more general Rabi oscillation picture.

Equation 8 in the source gives the amplitude of the Josephson population oscillation: the coupling times the square root of the product of the bulk and surface populations, divided by the magnitude of the difference between the bulk and surface frequencies. The bulk and surface oscillation amplitudes are equal and opposite. The Josephson frequency is the magnitude of that frequency difference. This oscillation modulates the bulk condensate frequency as follows from the self-trapping relation, Equation 7. The frequency modulation is sinusoidal to a good approximation, because the amplitude of the population oscillation is small as compared with the total population and Equation 7 can be linearised.

Equation 9 in the source writes the resulting instantaneous bulk time crystal frequency as the bulk frequency at the instantaneous population, plus a modulation amplitude times the cosine of the difference between the bulk and surface frequencies multiplied by time. Equation 10 in the source connects that modulation amplitude to the population oscillation amplitude: it is the population oscillation amplitude times the derivative of the bulk frequency with respect to the bulk population. Fourier decomposition of the resulting frequency-modulated signal yields Equation 11 in the source: the amplitude of the harmonic of order n is the Bessel function of the first kind of that order, evaluated at the ratio of the modulation amplitude to the frequency difference, multiplied by the modulus of the corresponding rotating phase factor. The bulk main trace corresponds to order zero. Combining the above expressions, and denoting the first side band amplitude, the coupling term can be linearised and expressed in quantities that can be directly measured, Equation 12 in the source: twice the side band amplitude times the squared frequency difference, divided by the product of the squared bulk amplitude, the surface amplitude and the derivative of the bulk frequency with respect to the squared bulk amplitude. Here we assumed that the filling factors of the bulk and surface states in Equation 2 are equal and constant. Where the time crystal shapes are changing due to changes in the trap profile, the coupling extracted using the above expression is therefore only approximate.

The side band of the bulk time crystal is seen in Fourier analysis of the experimental signal. The coupling extracted from this record using Equation 12 extrapolates to about 1.7 hertz at the crossing, in good agreement with the fitted simulation value of about 1.4 hertz. Note that there should be another side band symmetrically at lower frequency than the bulk trace, but it is covered by the surface trace at exactly the same frequency.

The surface trap is only weakly modified in similar fashion, yielding no visible side bands in the experiment. That is, the Josephson effect in a fully rigid trap results in no side bands owing to complex interference of the two wave functions. This can be confirmed by solving the dynamics of the rigid non-decaying coupled system analytically. We used this result to test the validity of the numerical simulation discussed below.

Near the avoided crossing one should use the more general Rabi oscillation picture. Solving for the eigenfrequencies of the Hamiltonian in the Rabi regime yields the Rabi frequency as the square root of the sum of the squared frequency difference and the square of twice the coupling. In the limit where the coupling is much smaller than the frequency difference this reduces to the Josephson frequency. The region where the two differ is not directly visible in the experiments due to interference effects.

Landau–Zener tunnelling

In the presence of exponential population dissipation, the time crystal population follows Equation 13 in the source: the initial population multiplied by the exponential of minus twice the time divided by the relaxation time constant of the measured signal, for each of the bulk and surface levels. For the surface time crystal this makes little difference other than that the population decays slowly. The bulk time crystal frequency depends on the bulk population according to Equation 7, and the frequency therefore increases during the decay. Hence, we have obtained the flexible two-level system described by the Hamiltonian of Equation 1.

Let us choose the zero-magnon bulk frequency to be above the surface frequency, and the initial bulk population such that the initial bulk frequency is below the surface frequency. Now the frequencies of the surface and bulk time crystals will cross in the eigenbasis where the coupling vanishes. If the coupling is positive and the bulk population decreases adiabatically, magnons in the bulk trap will smoothly move to the surface trap, remaining in the global ground state in an avoided crossing. The minimum frequency separation of the global ground state and the excited state at the avoided crossing is twice the coupling, as can be solved from the Hamiltonian.

If the avoided crossing is passed non-adiabatically, a part of the ground state magnon population moves to the excited state. This phenomenon is known as the Landau–Zener–Stueckelberg–Majorana effect. In our case this means that after the avoided crossing some population remains in the bulk trap, which corresponds to the new excited state in the system. Equation 14 in the source gives the fraction of population promoted to the excited state: the exponential of minus two pi times the squared coupling, divided by the magnitude of the time derivative of the difference between the bulk and surface frequencies. Note that while in the canonical Landau-Zener problem the time derivative is constant, in our case it keeps changing. However, the magnitude of the Landau-Zener population transfer is determined within a time window of order the inverse square root of that time derivative around the level crossing — at most about 100 milliseconds wide in our experiment. Therefore the bulk frequency can be linearised, or in other words, the time derivative taken at the avoided crossing gives the correct Landau-Zener population transfer.

Numerical simulation

The time crystal two-level Hamiltonian can be combined with the slow decay into a pair of coupled equations of motion for the bulk and surface wave functions, in which the imaginary unit times the time derivative of each wave function equals that level's frequency minus the imaginary unit times its relaxation rate, multiplying its own wave function, minus the coupling multiplying the other level's wave function.

Data availability

The data used in this study are available in the Zenodo database under the accession code https://doi.org/10.5281/zenodo.6510863.

Open Access

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S. Autti, P. J. Heikkinen, J. Nissinen, J. T. Mäkinen, G. E. Volovik, V. V. Zavyalov and V. B. Eltsov — Low Temperature Laboratory, Department of Applied Physics, Aalto University, Finland; Department of Physics, Lancaster University; Department of Physics, Royal Holloway University of London; and the L. D. Landau Institute for Theoretical Physics. Published as Nature Communications 13, 3090 (2022), doi.org/10.1038/s41467-022-30783-w, open access under CC BY 4.0.

(Running heads, page numbers, reference-number markers and the publisher’s footer have been dropped; display equations are rendered in words keyed to their source equation numbers and inequalities are written out; the five figures are not reproduced, though their captions are; the fifty-nine references are at the source.)

(On this site: Volovik’s book-length case for superfluid helium-3 as a laboratory model of the quantum vacuum is at /library/stm-6ee45bd8be, his work with Salomaa on quantised vortices in the same liquid at /library/stm-477c51c498, and the Aalto experiment that used neutron-irradiated helium-3 as an analogue of cosmological defect formation — with Eltsov and Volovik among the authors — at /library/stm-065e484325. The general programme of reading spacetime off a condensed-matter ground state is surveyed at /library/stm-9d9474c4b5 and /library/stm-a50c456d62, with Unruh’s founding argument at /library/stm-dcc76e413a. Superfluid treatments of the cosmic vacuum itself are at /library/stm-4a4bcf1126 and /library/stm-395f34ea17.)