Realization of an inherent time crystal in a dissipative many-body system

Time crystals are many-body states that spontaneously break translation symmetry in time the way that ordinary crystals do in space. While experimental observations have confirmed the existence of discrete or continuous time crystals, these realizations have relied on the utilization of periodic forces or effective modulation through cavity feedback. The original proposal for time crystals is that they would represent self-sustained motions without any external periodicity, but realizing such purely self-generated behavior has not yet been achieved. Here, we provide theoretical and experimental evidence that many-body interactions can give rise to an inherent time crystalline phase. Following a calculation that shows an ensemble of pumped four-level atoms can spontaneously break continuous time translation symmetry, we observe periodic motions in an erbium-doped solid. The inherent time crystal produced by our experiment is self-protected by many-body interactions and has a measured coherence time beyond that of individual erbium ions.


Introduction：
Similar to ordinary crystals where atoms take periodic positions in space, time crystals are many-body states that spontaneously recur in time. However, a system spontaneously repeating its pattern implies the breaking of time translation invariance, which contradicts the time-independence of most states in conventional theory. The realization of time crystals was first addressed by Wilczek 1,2 , but subsequent no-go theorems indicated that they could not exist in thermal equilibrium states 3--5 .
Nevertheless, recent advancements have led to the realization of driven discrete time crystals [6][7][8] in close quantum systems, characterized by oscillations at twice the driving periods [9][10][11][12][13][14][15][16][17] . Yet, the heating associated with driving in these closed systems prevents the persistence of time crystal. Theoretical research suggests that dissipation may overcome the heating issues [18][19][20][21][22][23][24] , leading to the observations of dissipative discrete time crystals 25,26 . Moreover, if the driving becomes non-periodic and time-invariant, the studied system acquires continuous time translation symmetry. Although the potential heating problem of a continuous driving can be worse compared to that of a periodic force, continuous time translation symmetry can also be spontaneously broken in dissipative systems [18][19][20][21][22][23][24]27 . Recent experimental observation has confirmed the existence of continuous time crystal in an atom-cavity system 28 . Like periodically driven systems, there remains a built-in frequency stemming from the optical cavity 22,23 , which introduces an effective periodic modulation.
The aforementioned discrete and continuous time crystals depend on the imposition of external periodic constrains, such as periodic forces or cavities, to break time translation symmetry. However, according to the spirit of the original proposal, time crystals represent the spontaneous emergence of time-periodic motions within timeinvariant systems. These motions are inherently self-triggered and self-sustained without the need of introducing external periodic inputs. However, creating such an inherent phase is still a pending challenge.
Here we report in both theory and experiment that inherent time crystals can be realized in a dissipative quantum system, which represents built-in phases of manybody systems that do not rely on recurring forces or cavities. By exploiting the dipoledipole interactions within an erbium-ion ensemble, we not only observe the spontaneous breaking of continuous time translation, but also the emergence of temporal order. The resultant self-sustained motion exhibits a period determined by the system's parameters and is protected by many-body interactions from inner degrees of freedom. Furthermore, the persistence of the time crystal's oscillations reveals longrange time correlation beyond the coherence time of individual ions.

Results： Theoretical Model
We investigate a system consisting of a collection of four-level atoms that are driven by a continuous-wave (CW) laser, as depicted in Fig. 1a  Hamiltonian shown by Eq. (1) can describe various quantum systems [29][30][31][32] , and is particularly relevant to erbium-doped crystals. In these crystals, the erbium ions exhibit an optical transition of 1.5 m, and the effective spins for both their optical ground and excited states are = 1/2. Due to the presence of an average magnetic field produced by all other erbium ions in the crystal, the spin states experience a slight splitting. As a result, the erbium ions can be considered as a nearly-degenerate four-level system, as shown in Fig. 1b. When there is no light to drive the erbium ions, all ions remain in their ground states, and the optical resonant frequencies of neighboring ions are of no difference (as shown at the top of Fig. 1a). However, this situation changes if some ions are optically excited 33 . Generally, the magnetic dipole moments of ions in their optical excited states differ from that in the ground. Therefore, optically exciting an ion can instantaneously change the local magnetic field seen by its neighboring ions. The optical frequencies of nearby ions are thus modified as a result of the Zeeman effect 34,35 , as illustrated at the bottom of Fig. 1a, which in turn affects the absorption of light.
Moreover, the presence of optical spontaneous emission, spin relaxation, and decoherence makes the erbium ions inherently an open quantum system. The combined effect of the openness of the quantum system and the aforementioned excitationinduced effect can give rise to rich non-equilibrium dynamical phases 27 .
To explore the physical properties of the four-level erbium system, we use the To provide further clarification on the physics of the persistent oscillation, we examine the relations between different ( ) in Fig. 1d Fig. 1e suggests that the changes of 11 and 22 are not synchronized under the same optical driving. Instead they undergo a competing process.
It is the competition between different optical transitions that prevent the system from reaching a stationary state. For a pure two-level system, where there is only one possible optical transition, it is impossible to generate a competing process by inner degrees of freedom 18,27 . To observe an inherent time crystalline phase, the energy structure of interacting atoms needs to have more than two levels. This increased complexity allows the many-body interactions to act as intrinsic nonlinear interactions and provide positive feedback to the competition between different optical transitions (see Supplementary Note 2 and 3 for more details). The interplay between these complex processes leads to the formation of a temporal order. Unlike the time crystals so far demonstrated, the persistent oscillation of ( Fig. 1c) in such a CW-driven four-level system is purely self-organized, and its recurring frequency is only determined by the coupling parameters of the system itself, suggesting that the time crystalline order is inherent.

Time crystalline order
In the following, we discuss how to realize such an inherent time crystal in experiments. We   If we further increase in = 5.2 mW, similar oscillating ( ) can be observed, as shown in Fig, 2c. More importantly, when a segment of ( ) in Fig. 2c is zoomed in, a hidden periodic order can be identified, which in contrast is absent in the magnified view of Fig. 2b. This suggests that a temporally recurring motion, i.e., a time crystalline order, arises if in is increased from 4.7 to 5. The periodic oscillation in ( ) is significantly different from the well-known selfpulsing effect in erbium doped fiber lasers [39][40][41][42][43] . Self-pulsing is due to the dynamic interplay or the competition between the gain and the losses within a laser cavity. As a result, the net gain inside the cavity is periodically modulated and generate fluctuations in the laser output. However, our experiment did not adopt such a cavity configuration as one side of the sample was coated with an anti-reflective layer of less than 0.8 % reflectivity. Consequently, our observations could not have been attributed to the selfpulsing effect in lasers. As aforementioned, the measured ( ) is closely related to ( ). Thus, the periodicity in ( ) under a in = 5.2 mW optical driving indicates a repeating temporal order of ( ), which agrees well with the characteristics of a time crystalline order. Together with the spontaneous breaking of time translation symmetry shown in Fig. 2b and 2c, and the lack of imposed periodicity from a driving force or a cavity, the revealed periodicity in Fig. 2c and 2d suggests the self-formation of an inherent time crystalline phase.

Phase transitions
The emergence of a time crystalline order as shown in Fig This can cause an imprecise determination of the frequency position. To observe a welldefined temporal order, that is, a time crystal with a long-range temporal correlation or a long coherence time, it is critical to set a proper optical driving field. Phase noise of the inherent time crystal. The cross-correlation ( ) shows that the periodicity of ( ) remains even for 70 ms. Time points with phase discontinuity are marked by circles.
Noting that the spectrum of in = 9.5 mW in Fig. 3a has the narrowest linewidth, we use it to determine the coherence time of the time crystal. The full width at halfmaximum given by a Lorentz fitting is = 82 ± 30 Hz (Fig. 3a), corresponding to a coherence time of 2 = 1/ = 4 ± 2 ms, which is beyond that of individual erbium ions (for reference, the coherence time of erbium ions in a 50 ppm Er:Y2SiO5 at 4.2 K is on the order of 10 s 44 ). Ideally, periodic oscillations of a time crystal maintain phase coherence over indefinitely long periods of time 45,46 . Because the long-range temporal order of a time crystal is protected by many-body interactions and its coherence is robust to the dephasing effect from the environment.
Unlike an ideal one, our time crystal shows a finite coherence time in the longtime limit (Fig. 3b), suggesting that its periodic oscillations are consecutive and dephase here pave the way to create states of matter in a strongly correlated system. In particular, it should be possible to use many-body interactions to create robust quantum superposition states with long coherence times. Such phases can also be extended to microwave-driven systems for potential applications such as quantum metrology and quantum memories.

Methods:
Theoretical method: Due to the complex nature of many-body systems, the parameters of sys in Eq. (1), such as , and , , vary from atom to atom. It is impractical to calculate the response of such a system in a full quantum way. To obtain a basic understanding of the phases of the non-equilibrium erbium ensemble, we first disregard the inhomogeneities by setting , ≡ , , ≡ , Ω (r ) ≡ Ω , that is to say, the detunings and the Rabi frequencies for all erbium ions are the same. We then apply the mean-field approximation to our system, which means that the system density matrix can be factorized sys = ⨂ , and the reduced density matrix of the th atom is given Using this Hamiltonian, we can compute the Lindblad equation ̇= and obtain the time response of a collection of erbium ions (more details in Supplementary Note 1). One side of the sample was coated to achieve a reflectivity of 98.8 %, while the other end was coated with an anti-reflective layer of less than 0.8 % reflectivity. The sample is cooled to 4.0 K by a cryostat. The intensity instability of the of our laser (E15, NTK) is 2% in a 10s scale, and the laser linewidth is documented as less than 100 Hz.

Data analysis using cross-correlation function:
A reference pulse function with an oscillating frequency of is defined as ( ) = cos(

Supplementary Note 1. System Hamiltonian and mean-field approximation
We consider a collection of four-level atoms whose energy structure (2) In principle, the parameters in , such as , and , , varies from atom to atom.
It is impractical to calculate in a full quantum way the response of such a system.
Here we consider that the atoms are homogeneous, i.e., where ℎ contains the detuning terms and the optical driving terms, and For the first term in the above equation, we have Using again that Tr( − ) = 0, we obtain for the third term in Eq. (7) that With this Hamiltonian, we can now compute the Lindblad equation ̇= by using the Runge-Kutta method (Notre 2 or more detailed in our previous work).
In our experiment, the measured light intensity ( ) is related to ( ) through the macroscopic polarization ( , ) . We consider a laser field ℰ( , ) = ] , (25) represents the many-body interactions. ] , (26) one can compute the time evolution of ( ) through the Lindblad equation  (28) where is the average photon number of the thermal bath at the frequency between level and .
The parameters used in obtaining the results in Fig. 1 Figure S1 shows the calculated 33 ( ) for different . When there are no ion-ion interactions ( = 0 MHz), Rabi oscillations on the order of MHz can be observed shortly after the driving field is switched on (inset of Fig. S1(a)). However

II. Time crystalline order
When the time translation invariance of ( ) is broken in the long time limit due to the inclusion of many-body interactions, Rabi oscillations induced by the optical driving can still be observed throughout the entire time range. This suggests that the system continues to undergo coherent population oscillations between the optical ground and excited states. More importantly, under appropriate parameters, the unstable ( ) exhibits self-sustained oscillations at frequencies different from the Rabi oscillation frequencies. These regular oscillations indicate the formation of temporal order. For example, in Fig. S2(a), the Fourier spectrum of the calculated ( ) (Fig. 1C in the main text) shows a peak at 46.4 kHz, confirming the presence of temporal order.
Within the large parameter space of the four-level systems, it is found that the characteristic frequency of the inherent time crystal is sensitive to the optical transition overlaps. Specifically, we set the optical coupling strengths of |1⟩ to |3⟩ and that of |1⟩ to |4⟩ to be 1 and 2 , respectively, as shown in Fig.S2b. Here depends on the pump laser, and 1 and 2 are coupling coefficients.
Similarly, the optical coupling strengths of |2⟩ to |3⟩ and that of |2⟩ to |4⟩ are 2 and 1 , respectively. After obtaining the time response of ( ), we then carry out a Fourier transform to ( ) in the long time scale. The time-crystal frequency as a function of the ratio of 1 / 2 is plotted in Fig. S2c. For a range of 1 / 2 varying from 1.08 to 1.20, the crystal frequency changes from 8.3 to 20.7kHz. The result suggests a strong dependence of the crystal frequency on the ratio of 1 / 2 .
The physics of such dependence can be understood from Fig. 1D and 1E in the main text. As discussed, it is essential to have enough complexity in the energy structure to enable competition between different optical transitions. Such competition is necessary to bring an ensemble of atoms into a dynamically unstable phase. For pure two-level systems, the lack of complexity prevents such competition between different optical transitions and thus rules out a temporalorder phase. For four-level systems, the ratio 1 / 2 indicates the overlaps between the different transitions. Therefore, it is an important index of the competing processes and can significantly affect the time crystal frequency.
In contrast, our calculated results suggest the lifetime or the decoherence time of individual erbium ions has no significant impact on the time scale of the crystalline oscillation. The dissipation terms of erbium ions are more important in determining whether a time crystal phase can be form. While a too-low damping rate might not be enough to efficiently expel the heating due to the dynamic instability, a high rate can cause the rapid dephasing of the oscillations.
It is worth noting that the emergence of time crystalline order in our four-level system is not simply a result of combining different Rabi oscillations. As shown in Figure S1, without the presence of many-body interactions, these Rabi oscillations would only last for a short period of time and eventually fade out, indicating that the time crystalline behaviour cannot be solely explained by Rabi oscillations.
Furthermore, if the time crystal were solely the effect of Rabi oscillations, we would expect the crystal frequency to be affected by both the Rabi frequency and the ratio of transitions, which contradicts our calculations and experimental results.

Supplementary Note 3. Feedback loop of the four-level system
We use the calculated results in Fig. 1 in the main text to demonstrate the feedback loop of the four-level system. Starting for the steady solution corresponding to ( ) = 0, we assume that there is a small positive perturbation to 11 , as marked by in Fig. S3a. According to Fig. S3b, such an increase in 11 will lead to a decrease of 22 , as marked by . Then the decrease of 22 will result in a growth of 44 , as marked by in Fig. S3a. Finally, the increased 44 further enhances the increase of 11 as marked by in Fig. S3b. These processes form intrinsic positive feedback inside the four-level system. The positive feedback, together with the optical transitions that compete with each other, leads the system to a dynamical instability and a self-organized periodic temporal pattern.
The limit cycles as shown in S3 is a typical sign of the breaking of time translation symmetry. In the atom-cavity time crystal , if the coupling between the cavity field and the atomic polarization is linear, the system relaxes to a stationary state at the long-time limit. The limit-cycle behaviours therein are the result of the nonlinear interaction between the cavity field and the atomic polarization. In our system, the nonlinear interactions between different electronic transitions offer the possibility of breaking time translation symmetry. The two systems share similarity in math representations, as both the cavity field and the electronic transition can be represented by an oscillator in math.

Supplementary Note 4. Dipole-dipole interactions of erbium ions
Our Er:Y2SiO5 crystal has a concentration of 1000 ppm, corresponding to an average distance of 4nm between nearby erbium ions. The magnetic dipole-dipole interaction of erbium ions separated by 4nm is (10MHz). Note that erbium ions also possess electric dipole moments. It is estimated that the electric dipole In our experiment, the oscillating ( ) lasts for ever as long as the pump laser is on. This is in contrast to the continuous time crystal recently demonstrated, which has a lifetime limited by the atom loss. The persistence of ( ) in our experiment is explicitly shown in Fig. S4, where the ( ) after the laser ∼100ms after the switching of the laser is shown. As long as the pump laser is on its cw mode, such oscillations can be observed. This property is crucial as it enables us to monitor the spectra of ( ) in the long-time limit (or cw mode), as shown in Fig. 3 in the main text. Figure S5 shows the phase diagram of our system. Normally the output ( ) of our measurement is dynamically stable for different combination of pump power in and laser frequency . Starting from a cw state, the output of our system under increasing in first becomes dynamically unstable, then temporally periodic and finally dynamically irregular again.

Supplementary Note 6. Phase diagram
With increasing pump, the ( ) first becomes dynamically unstable, indicating the breaking of continuous time translation symmetry. The spectral range of ( ) is broad with a cut-off frequency ∼ 50MHz , which means that ( ) oscillates at the time scale of 10ns. However, no periodic signal can be identified from the correlation ⟨ ( ) ( + )⟩ or the spectra of ( ). In other words, even though the continuous time translation symmetry of our system is broken, there is lack of a temporal order. This regime is named as the phase of broke time translation symmetry I, as shown by the orange area in Fig. S5.
Depending on the , if the in is further increased, the many-body system can reach a phase with temporal order. For example, for = 0.00 GHz, if in is increased to 5.3mW, a periodic oscillation of 8.7KHz can be identified in the already-unstable ( ) . For = 0.50 GHz and = 1.12 GHz, a pump of in = 9.3mW and 6.0mW are needed, respectively (dark-red cross points in Fig. S5).
Note that under this circumstance, the ( ) is still dynamically unstable and the Here we show more data of the phase transitions at = 1.17GHz.
When the pump power is low = 5mW, there is no oscillating signal in Fig. S6a and S6b. If the pump laser is increased to 6.0mW, we can see that ( ) becomes dynamically unstable, indicating that the time translation symmetry is spontaneously broken by the erbium dipole-dipole interactions, as shown in Fig.   S6c. Its corresponding Fourier spectrum shows a periodic time pattern of 8.7kHz, as shown in Fig. S6d, indicating the forming of temporal order. Further increasing the pump power to 7.0mW, the 8.7kHz peak remains, and the changes in its amplitude and width are not obvious. However, oscillations at other frequencies become much more significant, as shown in Fig. S6f. Specifically, the spectral intensities at non-zero frequencies (apart from 8.7kHz) are approximately -45dB in Fig. S6d and -40dB in Fig. S6f. The 8.7kHz peak becomes less obvious as a result of the growing amplitudes of other frequencies. This may finally cause the time crystal to be unidentified. Note that such a transition is different from that of Fig.   3a in the main text, where the broadening of the 8.7kHz peak at strong in is caused by the high-order effects rather than growing background noise.

Supplementary Note 8. Intrinsic optical instability without time crystalline order.
The measured ( ) for in = 8mW and = 0.50GHz corresponding to Fig. 3a in the manuscript is shown in Fig. S7. After the laser is switched on at = 0ms, the system takes approximately 14ms to reach a phase of intrinsic optical instability.
However, there is no temporal periodicity in ( ) under this circumstance, as evidenced by the lack of peak in its spectrum measured in the long-time limit, which is shown as the orange line in Fig. 3A in the main text.

Supplementary Note 9. Data analysis using cross-correlation function
Here we present the method to obtain the phase information of ( ), as shown in • The time crystal frequency 0 = 8.7 kHz, which is obtained from the autocorrelation and the Fourier spectra of ( ), as shown in Fig. 2 in the main text.
• The intrinsic instability ( ) is considered as a rapidly-varying function of time. The typical frequency of ( ) is on the order of tens of MHz, as detailed in the literature.
Typically, we choose ∼ 0.1 ms. As aforementioned, ( ) varies rapidly at frequencies (10MHz). Thus any terms involved the fast oscillating ( ) is averaged to near zero when calculating the integral. The same conclusion also holds for the term with cos( 0 + ) . Using these two approximations, we then have Since ( ) is a slowly-varying function of time on the order of 1ms, and the integration time ∼ 0.1 ms, we can replace ( ) with ( ) in the above equation such that At the limit that ( 0 − ) → 0, we obtain that Neglecting a constant phase of 0 /2, we finally have Comparing Eq (35) and Eq (29), we can reproduce the period and the phase information of ( ) by choosing different (with a time resolution determined by ). To confirm the method's validity, we calculate the cross-correlation between the ( ) from 117ms to 121ms in Fig. S8a and a reference function with = 8.7kHz and = 0.125ms. The result is plotted as an orange line on top of Fig. S8a. It is obvious that the cross-correlation ( ) reproduces the period and the phase information of ( ) from a background of high frequency noise.
Note also that the analysis method here is not sensitive to the reference frequency . This property can be seen from Eq. (34), which does not depend on . That is to say, one does not have to precisely pre-measure 0 and set = 0 during the data analysis. As long as ( − 0 ) → 0, Eq. (34) holds. We present ( ) of three different in Fig. S8b. While the result of = 8.7kHz indicates a periodic oscillation of ( ), the period and the phase information of ( ) can also be revealed, almost identically, by the result of = 10.7kHz. However, if is further increased to 15.7kHz, the ( ) becomes improper and is no longer useful in extracting the phase information, as shown by the gray curve in Fig. S8b.

Supplementary Note 10. Periodicity at different delays
Using ( ) detailed in Note 8, one can reveal the periodicity and the phase information of ( ) more clearly than the raw data ( ) itself. Exemplified in Fig.   S9 are the cross correlation ( ) for different delay times of the same ( ) in The phase change consistently occurs when the 8.7 kHz signal slowly envelopes to a small amplitude (it never occurs when the 8.7 kHz oscillations are of large amplitude), indicating a continuous phase change rather than a sudden jump.
However, characterizing the laser's phase noise at 100 Hz scales is challenging, and we lack the necessary equipment for a conclusive determination of its nature.

Supplementary Note 11. Phase discontinuities in the theoretical model
To understand the effect of the phase noise in driving to the inherent time crystal, we have calculated the response of ( ) for a laser pump with phase shift at some specific moments. The results are shown in Fig. S10. We can see that the system takes approximately 4ms to self-organize to temporal order, as shown by the left inset of Fig. S10. At = 5.6ms and = 11.2ms, phase shifts of are imposed on the optical driving field, as marked by the red dashed lines in Fig. S10. These phase perturbations break the balance of the system. The ( ) thus losses its temporal order and takes approximately 4ms to self-organize to another one, as shown by the right inset of of Fig. S10. The period of this new temporal order is the same as the previous one. Still, there is a phase discontinuity between them. The calculated results of imposed phase discontinuities in the driving field agree well with our experiment: although the oscillating ( ) persists, the phase noise of the laser breaks ( ) into segments on the order of milliseconds.

Supplementary Note 12. Differences between self-pulsing and time crystal
It is well-known that self-pulsing effect can occur in erbium-doped fiber lasers , in which the laser output power fluctuates in a periodic or quasi-periodic manner without any external modulation. The fluctuations in the output power are generally caused by the competition between the gain of the laser medium and the losses due to the cavity's various components. As a result, the net gain inside the cavity is periodically modulated and generate fluctuations in the laser output. Although our time crystal and self-pulsing both exhibit periodic modulations in their outputs, they are fundamentally different and have different underlying physical mechanisms. The most important differences are: 1. The experimental setups are different. Optical cavity is a necessary for the well-know self-pulsing effects in erbium-doped fibre lasers. Our experimental setup differs significantly. One end of our sample is antireflection coated with a reflectivity of less than 0.8%, preventing the formation of a similar cavity. In addition, our sample is of 12mm long. In contrast, the fibre laser systems that manifest self-pulsing effect typically have a length on the order of meters, that is, two-orders-of-magnitude difference. Thus, the achievable optical gain in our crystal, given that the erbium concentrations are similar, is much less than that in fibre laser systems due to these factors. These differences distinguish our experimental setup from traditional fiber laser systems and should be considered in any comparisons or analyses. Additionally, the up to ∼50MHz frequency response observed in our time crystal phase exceeds the energy transfer rate between the erbium ions in the self-pulsing model. Therefore, the temporal order observed in our experiments likely arises from another mechanism instead of the well-known self-pulsing effect.