Floquet maser

We report on a (Floquet) maser that delivers promising performance for applications in applied and fundamental science.

Abstract

The invention of the maser stimulated revolutionary technologies such as lasers and atomic clocks. Yet, realizations of masers are still limited; in particular, the physics of masers remains unexplored in periodically driven (Floquet) systems, which are generally defined by time-periodic Hamiltonians and enable observation of many exotic phenomena such as time crystals. Here, we investigate the Floquet system of periodically driven ¹²⁹Xe gas under damping feedback and unexpectedly observe a multimode maser that oscillates at frequencies of transitions between Floquet states. Our findings extend maser techniques to Floquet systems and open avenues to probe Floquet phenomena unaffected by decoherence, enabling a previously unexplored class of maser sensors. As a first application, our maser offers the capability of measuring low-frequency (1 to 100 mHz) magnetic fields with subpicotesla-level sensitivity, which is substantially better than state-of-the-art magnetometers and can be applied to, for example, ultralight dark matter searches.

INTRODUCTION

The masers (1–4) have become ubiquitous and resulted in innovations (5–9) ranging from lasers, atomic clocks, ultrasensitive magnetic resonance spectroscopy, and low-noise amplifiers to deep-space communications. Because the frequency of radiowaves produced by masers is highly stable, these devices enable exquisitely sensitive measurement of their frequency shifts caused by the interactions with external electromagnetic fields. This opens up exciting possibilities for developing precise metrology in applied and fundamental physics, such as magnetometry (9–11), temperature sensors (12), tests of Lorentz and charge-parity-time reversal symmetry violation (13), and searches for topological dark matter (14). The masers have been demonstrated in a variety of systems, such as ammonia molecules (1), hydrogen atoms (6), noble gas (15, 16), pentacene molecules (2), and silicon- and nitrogen-vacancy defect materials (3, 4, 10). However, the demonstrations of masers have remained unexplored in periodically driven (Floquet) systems (17), limiting the broad applications in sensing, spectroscopy, and fundamental physics. The generalization of masers to periodically driven systems would pave the way for many new applications, such as ultralow-frequency magnetic field sensing (18, 19) and searching for oscillating electric dipole moments (EDMs) [see review in (20, 21)].

To reach the above goal, a proper periodically driven maser gain medium should be considered. The recently developed notion of Floquet system [see review in (17)], which is only invariant under discrete time translations by a period, has spawned intriguing prospects, such as time crystals (22) and Floquet topological insulators (23). A variety of Floquet systems have been realized through periodic driving, ranging from periodically driven trapped ions (22) and atomic ensembles (23, 24) to nitrogen-vacancy centers (25). For the goal of developing periodically driven masers, Floquet systems should be an outstanding candidate. We note that the combination of maser techniques and Floquet systems can overcome the decoherence effect and thus permits a fresh look at many phenomena. For example, successful realization of maser using Floquet systems may open new opportunities for observing long-range temporal dynamics (22, 23) and spectroscopy with submillihertz resolution, with important implications in quantum metrology (26–28), for example, in searches for gravitational waves in evolved Laser Interferometer Space Antenna (eLISA) with the bandwidth of 1 to 100 mHz (18), measurement of the worldwide magnetic-background noise (including attempts at earthquake prediction) (19), and axion dark matter searches (29–31). Despite these appealing features, a demonstration of masers based on Floquet systems was heretofore lacking.

Here, we report the first theoretical and experimental demonstration of a Floquet-based maser composed of periodically driven ¹²⁹Xe spins in a vapor cell. Unlike the common masers that exploit inherent transitions (1–7), our maser is based on the synthetic dimensions supported by Floquet states of the system. We name the observed maser “Floquet maser,” which oscillates at the frequencies of transitions between Floquet states.

Using our maser technique, we observe ultrahigh-resolution spectra of the Floquet system with two orders of magnitude better resolution compared to that limited by decoherence. As the spectral resolution is greatly increased, a different regime emerges where high-order Floquet sidebands become notable and complex spectra are expected, enabling accurate measurement of physical parameters, e.g., atomic scalar and tensor polarizabilities (28), magnetic fields (18), ultralight bosonic exotic fields (31), and multiphoton coherences (32). As a first application, our maser constitutes a new quantum technology for measuring ultralow-frequency (1 to 100 mHz) magnetic fields with subpicotesla-level sensitivity, which is notably better than state-of-the-art magnetometers (18, 33–35). Moreover, we show that the present maser technique allows us to achieve a search sensitivity for the coupling of axion dark matter to masing spins well beyond the most stringent existing constraints (30, 31).

RESULTS

Setup and Floquet system

We use noble gas atoms ¹²⁹Xe with nuclear spin I = 1/2 in a setup depicted in Fig. 1A. A 0.5-cm³ cubic vapor cell made from pyrex glass contains 5-torr ¹²⁹Xe, 250-torr N₂, and a droplet (several milligrams) of enriched ⁸⁷Rb. ¹²⁹Xe spins are polarized by spin-exchange collisions with optically pumped ⁸⁷Rb atoms in a bias magnetic field B₀ (≈750 nT) along the polarized direction (z axis) (36). Similar to a microwave cavity in conventional masers (1, 2, 4), the ¹²⁹Xe spins in our experiments are embedded in a feedback circuit (16) (see Fig. 1A), which uses an atomic magnetometer (34) as a sensitive detector of ¹²⁹Xe spins and simultaneously supplies the real-time output audio-frequency signal of the magnetometer to the spins (see Materials and Methods). The ⁸⁷Rb atoms in the vapor cell act as a magnetometer for measuring the ¹²⁹Xe spin polarization P_x along the x direction; a corresponding feedback field B_f(t) = χP_x(t) is applied to the ¹²⁹Xe spins with a set of y coils around the vapor cell. Here, the proportionality constant χ (feedback gain) encapsulates the conversion factor of the atomic magnetometer and the connection polarity of the feedback coils. The amplitude of feedback gain χ can be adjusted with a rheostat in series with the feedback coils. The sign of χ can be controlled by the directions of the bias field (+z or −z) and feedback coils (+y or −y) (more details are presented in Materials and Methods). Similar to a resonant cavity (37), the self-induced feedback field B_f(t) carries the information about the spins and then acts back on the spins, leading to the well-known phenomenon of damping (9, 38) that is important in our maser scheme.

Fig. 1. Schematic of experimental setup and damping feedback mechanism.

(A) Schematic of the Floquet ¹²⁹Xe nuclear-spin maser. The ¹²⁹Xe atoms are polarized and detected by spin-exchange collisions with optically pumped ⁸⁷Rb (see Materials and Methods). Under a bias field and an oscillating magnetic field along z, the ¹²⁹Xe spins are magnetically coupled to a feedback circuit, which feeds back real-time B_f(t) along y and induces the damping of ¹²⁹Xe spins. DAQ, data acquisition. (B) Measured free decay ¹²⁹Xe signals for three different feedback gains (corresponding to different T_d). Here, the spin population P_z and the feedback gain χ are initially set as P_z > 0, χ<0 or P_z < 0, χ>0. T_d is well determined by corresponding decay time T₂ with $\sim \frac{\mathrm{\pi }}{15}$ excitation angle (see the main text). (C) Measured decay time T₂ as a function of damping time (black symbols). The red line is a fit with ${(T_{2,0}^{-1}+T_{\mathrm{d}}^{-1})}^{-1}$, where T_2,0 represents the intrinsic decoherence time without feedback. (D) Transient maser operations for two different damping times after flipping ∼π angle, inducing the inversion of ¹²⁹Xe spins population. The decay signal can be fitted with a hyperbolic secant function shown in the inset (see Materials and Methods).

We consider a Floquet system, where an oscillating magnetic field B_ac cos (2πν_act) (along z) periodically drives Zeeman energy levels. Unfortunately, the textbook approach (1–7, 10, 12) used to analyze masers is not well-suited for periodically driven systems. As noted in the pioneering work (39), a system with a time-periodic Hamiltonian can be equivalently represented as a time-independent Hamiltonian but with an infinite number of static energy levels. Within this framework, we can borrow some essential notions from the conventional time-independent masers and this, in turn, leads to the entirely new concept of Floquet maser. Specifically, a Floquet system has eigenstates (Floquet states) ∣ ± ⟩_n = Σ_n^′J_{n − n^′}( ± γB_ac/2ν_ac) ∣ ± , n^′⟩ (see Materials and Methods, Eq. 6) and energies E_{±, n}/2π = ± ν₀/2 + nν_ac (39, 40) (see Materials and Methods, Eq. 4). Here, ∣±, n^′⟩ denotes that the spin is in the spin-up (∣ + ⟩) state or in the spin-down (∣ − ⟩) state and the periodic driving field has the photon number n^′ (39, 40). 𝒥_{n − n^′} is the Bessel function of the first kind of order n − n^′. Under periodic driving of the oscillating field, the two-level (∣ + ⟩, ∣ − ⟩) spin system is extended to an infinite number of synthetic energy levels ∣ ± ⟩_n (see Fig. 2A), and these energy levels are time independent.

Fig. 2. Demonstration of Floquet maser.

(A) Floquet states of a periodically driven two-level system (Floquet system). The energy gap between the upper and lower Floquet states ∣ + ⟩_n and ∣ − ⟩_m is E_n,m/2π = (n − m)ν_ac + ν₀. The spin population P_z and the feedback gain χ are set as P_z > 0, χ > 0 or P_z < 0, χ < 0. (B) Signal of ¹²⁹Xe Floquet maser. The insets are zoom-in plots for the signal and the simulated spin population (P_z). (C and D) The corresponding amplitude spectra of the maser signal after eliminating the transient [ν_ac = 0 for (C); 0.9 Hz for (D)].

Masing effect on Floquet system

We first measure the feedback-induced damping of ¹²⁹Xe spins, because damping plays an important role in realizing masers (9, 16, 38). For simplicity, the periodic driving field is turned off when we measure damping. The measurement process is shown in Fig. 1B. The ¹²⁹Xe spins are initially polarized along +z and the feedback gain is set as χ<0, or the spins are initially polarized along −z and the feedback gain is set as χ>0. The spins are tilted by a small angle ${\mathrm{\theta }}_0\approx \frac{\mathrm{\pi }}{15}$ with a magnetic field pulse along the x axis, and then the free decay of ¹²⁹Xe signals is measured under self-induced feedback. In this case, the free decay signal can be fitted with a single-exponential decay with a decay rate given by $T_{2,0}^{-1}+T_{\mathrm{d}}^{-1}$ (see Materials and Methods), where the intrinsic decoherence time T_2,0 ≈ 13.65(1) s and T_d (the damping time) depends on feedback gain χ. By fitting the experimental data, we can find the decay rate and then calculate the corresponding T_d under different feedback gains χ, as shown in Fig. 1B. The results show that, by coupling nuclear spins to the feedback circuit, a regime can be reached in which damping constitutes the dominant mechanism of spin relaxation, e.g., T_d = 1.08(1)s ≪ T_2,0, and spin relaxation can be controlled by adjusting the feedback gain (Fig. 1C). This also suggests a method for active fast reset of long-lived spins [for example, ³He noble gas (41)] to their equilibrium state, improving the repetition rate of an experiment.

We next measure the spin dynamics when the ¹²⁹Xe spin population is suddenly inverted, corresponding to the case of θ₀ ≈ π. Figure 1D gives the observed free decay signals with designed T_d ≈ 3.18 s and 0.94 s, respectively. Unlike the exponential decay, the observed ¹²⁹Xe spin signal first increases to a maximum value at a certain time and then decays to zero, which can be described by a hyperbolic secant function (see Materials and Methods, Eq. 8). As first reported in (38), this is a transient maser when the threshold of the damping time T_d/T_2,0 ≪ 1 is fulfilled. However, the demonstrated maser cannot oscillate continuously because the population inversion is transient. To generate stationary maser dynamics (1, 4, 12), we can reverse the circular polarization of pump laser or alternatively reverse the sign of the feedback gain χ (see Materials and Methods) and simultaneously set the damping time smaller than the intrinsic decoherence time (i.e., T_d/T_2,0 < 1). Under these conditions, coupling of the spins to the damping feedback circuit can produce a self-sustained masing signal (9, 16).

We now consider the spin dynamics of the Floquet ¹²⁹Xe system under the damping feedback field. As discussed above, the Floquet system can be treated as a time-independent one with an infinite set of energy levels, shown in Fig. 2A. The key to a maser based on Floquet systems is the preparation of spin population between those Floquet states. In our experiments, population between Floquet states (∣ + ⟩_n and ∣ − ⟩_m) of the Floquet ¹²⁹Xe spins can be continuously prepared through ¹²⁹Xe−⁸⁷Rb spin-exchange collisions. Moreover, building on our demonstration of damping, the damping time is set to T_d ≈ 6.25 s, satisfying the threshold of T_d/T_2,0 < 1. When the feedback circuit is suddenly on, a feedback B_f(t) is induced by the Floquet system itself and oscillates with the frequencies of Floquet sidebands. The feedback field produces a torque on the spins that changes spin polarization (9, 16). This self-coupling can lead to stimulated Rabi oscillations between the Floquet states ∣ + ⟩_n and ∣ − ⟩_m. A steady-state maser oscillation is expected to build up. For different Floquet states pair n, m, the maser oscillation frequency is E_{n, m}/2π = (n − m)ν_ac + ν₀.

As a first illustration of the Floquet maser, Fig. 2B shows a time trace of ¹²⁹Xe spins under 0.900-Hz driving field and B_ac = 56.15 nT. The ¹²⁹Xe transverse polarization (P_x) shows characteristic initial transients, which subsequently level into a stationary oscillation. Because spin population (P_z) cannot be measured directly in our experiment, we simulate the maser with nonlinear Bloch equations (see Materials and Methods, Eq. 7) to understand the spin dynamics. During the quick collapse of P_x (see the top inset of Fig. 2B), the negative spin population (P_z < 0) reverses to positive (P_z > 0) in a short time (∼2 s ≪T_1,0 ≈ 21.5 s); this is the phenomenon of nuclear spin super-radiance (42). In the stationary window, the spin population P_z remains negative (see the bottom inset of Fig. 2B). To extract the oscillation frequencies of the stationary maser, we quantitatively investigate its spectrum. We take a time trace from the stationary window after eliminating the initial transients and apply Fourier transform. Unlike the common maser spectrum (without periodic driving) that has a single peak at ν₀ ≈ 8.915 Hz (Fig. 2C), four evident sidebands appear at ν_±1 = ν₀ ± 0.900 and ν_±2 = ν₀ ± 1.800 Hz (Fig. 2D), exactly at the frequencies of the transitions between Floquet states. All lines are at regular intervals equal to the periodic driving frequency ν_ac. In addition, the spectra of the Floquet maser for the case of ν_ac ≥ ν₀ are presented in the Supplementary Materials (section S1). Our result confirms that a self-organized oscillation between Floquet states can build up. In addition, a 4000-s continuous oscillation gives a full width at half maximum (FWHM) of 0.3 mHz, which is two orders of magnitude narrower than the decoherence-limited resolution (≈30 mHz).

For a given Floquet system, the coupling to the periodic driving is characterized by the modulation index γB_ac/ν_ac. The Floquet maser discussed above was studied in the weak-coupling regime of γB_ac/ν_ac ≪ 1. Strong-coupling regime (γB_ac/ν_ac ≫ 1) is indispensable in various applications of nonlinear atomic spectroscopy (32) and strong-field physics (43). To reach this regime, previous work usually required a large B_ac and ν_ac larger than the decoherence-limited linewidth. However, a large magnetic field also affects the atomic magnetometer and deteriorates its performance (44).

We provide an alternative way to reach the strong-coupling regime by greatly decreasing the driving frequency ν_ac as low as submillihertz, much below the decoherence-limited resolution. Thus, the modulation index could be substantially larger than 1. For example, if a 0.050-Hz field with a magnitude of 56.15 nT drives ¹²⁹Xe, γB_ac/ν_ac ≈ 13.2. In Fig. 3A, the sideband spectrum based on the Floquet maser (red line) exhibits at least 25 evident comb-like symmetric lines centered at the Larmor-frequency line (at ν₀). Similarly, frequencies ν_ac lower than T_2,0-limited linewidth, e.g., ν_ac=10 mHz with amplitude 56.15 nT, correspond to modulation index γB_ac/ν_ac ≈ 66. In this case, we observe at least 134 maser sidebands (Fig. 3B), obtained from 4000-s continuous oscillation. Compared with the maser in the weak-coupling regime, a great number of Floquet transitions can build up stationary maser oscillations in the strong-coupling regime, enabling exact reconstruction of the Floquet energy levels.

Fig. 3. Comb-like ultrahigh-resolution spectroscopy of Floquet maser.

(A) Comparison between conventional Floquet spectrum based on ¹²⁹Xe free-decay signal (blue lines) and a spectrum based on Floquet maser (red lines). The driving frequency is ν_ac = 0.050 Hz, and its amplitude is B_ac = 56.15 nT. a.u., arbitrary units. (B) Spectrum based on Floquet maser, where ν_ac = 0.010 Hz and B_ac = 56.15 nT.

We emphasize the difference between our maser and existing masers. First, our maser is based on transitions between Floquet states, whereas existing masers usually exploit inherent transitions (1–7, 10, 12). Our maser generates sidebands that are easily tunable by changing the frequency and amplitude of the periodic driving. As shown in this work, the Floquet maser is well suited for sensing oscillating driving field with a frequency resolution, independent of the masing spin decoherence and limited only by the stability of the maser (see section S5). Second, unlike the conventional maser that uses a microwave cavity (1, 2, 4, 6, 12), our maser makes use of a feedback circuit based on the signal of an atomic magnetometer to provide damping feedback, enabling the maser frequency down to the audio-frequency range. Even though the oscillation frequency is much smaller than that of earlier established microwave masers (1, 2, 4, 6, 12), we show below that our demonstrated maser could be particularly useful for precision measurements, such as ultralow-frequency magnetic field sensing (18, 19) and searching for ultralight new particles (29–31).

Applications of Floquet maser in magnetometry

Recently, experimental investigations have been reported toward achieving high sensitivity of measuring magnetic fields in low-frequency regime (1 to 100 mHz), which is of importance in applied (19) and fundamental physics (18, 29–31). However, it remains challenging for state-of-the-art magnetometers to reach femtotesla-level sensitivity owing to substantial 1/f noise. Combining our demonstrated Floquet maser technique with magnetometry, we realize a sensitive magnetometer that is well suited for operation in the ultralow-frequency regime. As we show below, our maser-based magnetometer can achieve subpicotesla-level sensitivity in the millihertz range. It would be possible to reach femtotesla-level sensitivity when the technical sources of maser instability are eliminated. The idea of our magnetometer is that the measured oscillating field applied to the spins can be seen as a periodic driving that generates sidebands around the maser oscillation frequency (ν₀ ≫ 1 Hz). Thus, the maser up-converts the low-frequency field to a higher frequency, where the 1/f noise arising from the Rb magnetometer is spectrally separated from the maser signal.

We first calibrate the magnetic response of the maser by applying an oscillating magnetic field with known amplitude and frequency. We focus on the prominent first-order sidebands occurring at ±ν_ac about the central frequency, whose amplitude is proportional to 𝒥₁(γB_ac/ν_ac) ≈ γB_ac/2ν_ac (here, we assume that the oscillating fields are small enough, satisfying γB_ac/ν_ac ≪ 1). As shown in Fig. 4A, a 2.25-nT magnetic field is applied along the z direction, and its frequency ν_ac is swept from 1 to 22 Hz. We Fourier transform the individual measurement traces (with 60-s acquisition time) and record the corresponding sideband-peak amplitudes. The experimental amplitudes of first-order sidebands are fitted to a 1/ν_ac function, which is in agreement with theory.

Fig. 4. Maser-based magnetometry on the first-order Floquet sideband of ¹²⁹Xe.

(A) The first-order Floquet sideband amplitude follows a 1/ν_ac dependence with the driving frequency ν_ac. The result of fit is ξ = 0.017/ν_ac; B_ac = 2.25 nT. (B) The first-order Floquet sideband amplitude follows a linear dependence with the driving amplitude B_ac. The fit result is ξ = 0.0055 B_ac + 0.0096, where ν_ac = 1.000 Hz. (C) Measured magnetic sensitivity of the maser-based magnetometer (note that the vertical and horizontal axes have a logarithmic scale).

Similarly, we measure the amplitude response by applying a ν_ac = 1.000 Hz magnetic field with varying amplitude, as shown in Fig. 4B. The response of the maser to the applied oscillating field is measured to be ξ_maser[V] ≈ 5.5 × 10⁻³B_ac[nT]/ν_ac[Hz].

To show the noise performance of the maser-based magnetometer, we take a 4000-s maser real-time data without periodic driving and evaluate the corresponding power spectral density outside the spectral peak at ν₀. The background noise is different in different frequency ranges (see fig. S7): frequency dependent for frequencies below ∼60 mHz and white (i.e., δξ_maser ≈ 4 × 10⁻⁵ V/Hz^1/2) for frequencies from ∼60 mHz to 10 Hz. Combining with the above-calibrated response, the magnetic field sensitivity of maser-based magnetometer is shown in Fig. 4C. In the frequency above 60 mHz, the magnetic sensitivity is estimated to be δB_ac ≈ 7.2ν_ac pT/Hz^1/2, where ν_ac is in hertz.

Below 60 mHz, the magnetic sensitivity reaches ∼700 fT/Hz^1/2, which is mainly limited by the effective magnetic field fluctuations generated by ⁸⁷Rb magnetization (see sections S4 and S5). Such fluctuations arise from instability in both pump laser and vapor cell temperature, which can be overcome by using existing methods (15, 45). By eliminating these sources of maser technical noise, it should be possible to reach a femtotesla-level sensitivity (see the red line in Fig. 4C). For example, after the elimination of technical noise, our maser-based magnetometer would achieve a sensitivity of about 7.2 fT/Hz^1/2 at 1 mHz, which is expected to be notably better than that in earlier work (18, 33–35). The present device, therefore, is complementary to the state-of-the-art superconducting quantum interference devices (33) and atomic (34) magnetometers that have high sensitivity above 1 Hz. Our device is particularly sensitive between 1 mHz and 1 Hz. Moreover, our magnetometer can operate in a nonzero magnetic field (e.g., in Earth’s field), in contrast with spin-exchange relaxation-free atomic magnetometers that operate at fields below ∼100 nT (44).

DISCUSSION

We have reported a novel maser based on time-periodic Floquet systems. Unlike conventional masers, our maser oscillates at frequencies of transitions between Floquet states. The generalization of the notion maser to periodically driven systems opens a new avenue to explore Floquet physics unaffected by decoherence effects. As we show in this work, the connection of maser technique and Floquet physics allows observing ultrahigh-resolution spectra of Floquet systems with submillihertz widths and high-order sidebands effect. This can greatly improve the accuracy of measuring system energies, magnetic fields, atomic scalar and tensor polarizabilities (28), ultralight bosonic exotic fields (31), nonlinear multiphoton coherences (32), etc. Although demonstrated for ¹²⁹Xe spins, our scheme of Floquet maser can be transferred to other types of experimental systems. For example, recent advances in cold atoms (23, 24) and dipolar spin ensembles (25) have led to progress in such areas of Floquet physics as time crystals and masers [for example, diamond maser (4, 12)]. We suggest future theoretical and experimental studies of masers based on a variety of Floquet systems, taking a fresh look at many phenomena, such as Floquet Raman transitions (25), Mollow-triplet sidebands (27), Autler-Townes splitting (46), and even time crystals (22, 23, 25).

Our Floquet technique is generic and can be easily applied to well-established masers, such as hydrogen masers (6), diamond masers (4, 12), and one-atom Rydberg maser (47), all promising as a new class of maser-based quantum sensors. For example, the application to hydrogen and diamond masers, whose gyromagnetic ratios are about three orders of magnitude larger than those of nuclei, yields improvements over currently achievable magnetic field sensitivity. As we show in this work, such sensors outperform state-of-the-art magnetometers (18, 33–35) with subpicotesla sensitivity at millihertz frequencies and can be immediately applied, for example, in the searches of gravitational wave observatory eLISA (1 to 100 mHz) (18). Moreover, we show that couplings between masing spins (such as protons or electrons) and oscillating exotic fields beyond the standard model (20) may enable their direct detection via Floquet maser magnetometry: The exotic fields are predicted to couple with standard model particles (masing spins in this case) and behave as an oscillating magnetic field, generating sidebands (31) on masing spins that can be measured with our maser. The maser can be applied to search for some exotic fields, such as ultralight axions (29–31) and other exotic spin-dependent interactions (20). Compared with existing approaches, our work has a unique advantage of measuring exotic fields with ultralow frequency, 1 to 100 mHz (corresponding to particle mass ∼10⁻¹⁸ to 10⁻¹⁵ eV). As discussed in Materials and Methods, our work is promising for improving the search sensitivity of axions by approximately two and three orders of magnitude compared to the values obtained in recent ultralow-field nuclear magnetic resonance experiments (30, 31).

MATERIALS AND METHODS

Experimental apparatus

A cubic vapor cell containing 5-torr ¹²⁹Xe, 250-torr N₂ buffer gas, and a droplet of enriched ⁸⁷Rb is placed inside a five-layer cylindrical mu-metal shield and is resistively heated to 140^∘C. The longitudinal and transverse relaxation times of ¹²⁹Xe spins are measured to be T_1,0 ≈ 21.5(2) s and T_2,0 ≈ 13.65(1) s in a z bias field (≈750 nT). As shown in Fig. 1A, ¹²⁹Xe atoms are polarized by spin-exchange collisions with ⁸⁷Rb atoms (36), which are pumped with circularly polarized laser light (≈ 10 mW) propagating along +z. The pump laser frequency is tuned to the center of the buffer gas broadened and shifted D1 line. The ⁸⁷Rb atoms also act as a sensitive magnetometer for measuring the nuclear magnetization of the ¹²⁹Xe spins via optical rotation of linearly polarized probe laser light (≈1 mW) propagating along x. The frequency of the probe laser is detuned from the D2 transition by about 100 GHz. In the presence of a z-bias field, the ⁸⁷Rb magnetometer is primarily sensitive to the ¹²⁹Xe x-magnetization producing an effective magnetic field $B_{\text{eff}}=\frac{8\mathrm{\pi }}{3}{\mathrm{\kappa }}_0{M}_x$ (36), where the enhancement factor κ₀ is about 500, and x-magnetization is M_x = μ_Xen_XeP_x with μ_Xe, n_Xe, and P_x being the ¹²⁹Xe nuclear magnetic moment, atomic density, and x-polarization, respectively. To suppress the influence of low-frequency noise, the polarization of the probe laser beam is modulated at 50 kHz with a photoelastic modulator, and the signal is demodulated with a lock-in amplifier.

The real-time output signal of the magnetometer generates the feedback field B_f(t) = χP_x(t) on ¹²⁹Xe spins along y. The sign of feedback gain χ is simultaneously determined by the direction of the bias magnetic field (+z or −z) and the connection polarity of the feedback coils (+y or −y), i.e., (+z, −y), (−z, +y) for χ > 0 and (−z, −y), (+z, +y) for χ < 0. Here, we have considered that the atomic magnetometer is sensitive to the x magnetic field (or x-magnetization of ¹²⁹Xe) but with a negative sign; i.e., the magnetometer’s output signal is proportional to “−B_xB_z” (36); thus, (±z, ∓y) corresponds to χ > 0 and similarly for the case of χ< 0.

Energies and states of Floquet systems

A two-level spin system driven with a radio-frequency (rf) field can be treated as a dressed-spin system (39, 40), in which rf photons with creation and annihilation operators ${\widehat{a}}^{†}$ and $\widehat{a}$ are introduced by the second quantization of the rf field. Here, the rf field B_ac cos (2πν_act) is applied parallel to the static magnetic field $B_0\widehat{z}$. The second-quantized Hamiltonian for the dressed-spin system can be written as (40).

$$\widehat{H}/2\mathrm{\pi }={\nu }_0{I}_z+{\nu }_{\text{ac}}{\widehat{a}}^{†}\widehat{a}+\mathrm{\lambda }{I}_z({\widehat{a}}^{†}+\widehat{a})$$ (1)

where ν₀ = ∣ γB₀∣ and γ ≈ − 1.18 × 10⁷ Hz/T is the gyromagnetic ratio of ¹²⁹Xe spins. The first term in Eq. 1 is the Zeeman interaction of the spin with B₀. The second term is the energy of the quantized rf field. The final term describes the coupling between the spin and the quantized rf field with strength $\mathrm{\lambda }=\mathrm{\gamma }{B}_{\text{ac}}/(2\sqrt{\overline{N}})$ (40), where $\overline{N}\gg 1$ is the average number of rf photons in the mode. We introduce basis states ∣ ± , n⟩, where ∣ ± , n⟩ = ∣ ± ⟩ ⊗ ∣ n⟩, n signifies the rf photon number of the driving field, and ∣ ± ⟩ denotes the eigenstate of σ_z, i.e., ∣ + ⟩ = (1,0)^T and ∣ − ⟩ = (0,1)^T. The Hamiltonian $\widehat{H}$ commutes with σ_z, and its diagonalization is reduced to that of Ĥ_ϵ in each of the two subspaces of σ_z

$${\widehat{H}}_{\mathrm{ϵ}}/2\mathrm{\pi }=\frac{\mathrm{ϵ}}{2}{\nu }_0+{\nu }_{\text{ac}}{\widehat{a}}^{†}\widehat{a}+\frac{\mathrm{ϵ}}{2}\mathrm{\lambda }({\widehat{a}}^{†}+\widehat{a})$$ (2)

where ϵ = ± 1. A displacement operator is defined as $\mathrm{D}(\mathrm{\xi })=e^{\mathrm{\xi }{\widehat{a}}^{†}-{\mathrm{\xi }}^{*}\widehat{a}}$, which has the following properties: D(ξ)D^†(ξ) = 1, $\mathrm{D}(\mathrm{\xi }){\widehat{a}}^{†}{\mathrm{D}}^{†}(\mathrm{\xi })={\widehat{a}}^{†}-\mathrm{\xi }$, and $\mathrm{D}(\mathrm{\xi })\widehat{a}{\mathrm{D}}^{†}(\mathrm{\xi })=\widehat{a}-\mathrm{\xi }$. Then, ${\widehat{H}}_{\mathrm{ϵ}}$ in Eq. 2 can be written as follows

$${\widehat{H}}_{\mathrm{ϵ}}/2\mathrm{\pi }=\mathrm{D}(-\frac{\mathrm{ϵ}\mathrm{\lambda }}{2{\nu }_{\text{ac}}})(\frac{\mathrm{ϵ}}{2}{\nu }_0+{\nu }_{\text{ac}}{\widehat{a}}^{†}\widehat{a}-\frac{{\mathrm{\lambda }}^2}{4{\nu }_{\text{ac}}}){\mathrm{D}}^{†}(-\frac{\mathrm{ϵ}\mathrm{\lambda }}{2{\nu }_{\text{ac}}})$$ (3)

This Hamiltonian ${\widehat{H}}_{\mathrm{ϵ}}$ has eigenstates ${\mid \mathrm{ϵ}⟩}_n={\mathrm{D}}^{†}(-\frac{\mathrm{ϵ}\mathrm{\lambda }}{2{\nu }_{\text{ac}}})\mid \mathrm{ϵ},n⟩$, i.e., the Floquet states. The energy of ∣ϵ⟩_n is

$$E_{\mathrm{ϵ},n}/2\mathrm{\pi }=\mathrm{ϵ}{\nu }_0/2+n{\nu }_{\text{ac}}$$ (4)

We now derive the explicit form of ∣ϵ⟩_n for $n\approx \overline{N}\gg 1$ on the basis of ∣ϵ, n⟩. Let $\mathrm{\xi }=-\frac{\mathrm{ϵ}\mathrm{\lambda }}{2{\nu }_{\text{ac}}}$. By using the Glauber formula: e^{A + B} = e^Ae^Be^{[A, B]/2}, where the two operators A and B both commute with their commutator, we have

$$⟨n-m\mid e^{\mathrm{\xi }{\hat{a}}^{\mathrm{†}}-{\mathrm{\xi }}^{*}\hat{a}}\mid n⟩\approx e^{-\frac{{\mathrm{\xi }}^{*}\mathrm{\xi }}{2}}{\mathcal{J}}_{-m}(2\mathrm{\xi }\sqrt{\overline{N}})$$ (5)

As $\mid \mathrm{\xi }\mid =\frac{\mathrm{\lambda }}{2{\nu }_{\text{ac}}}=\frac{\mathrm{\gamma }{B}_{\text{ac}}}{{\nu }_{\text{ac}}}\frac{1}{4\sqrt{\overline{N}}}\ll 1$, $e^{-\frac{{\mathrm{\xi }}^{*}\mathrm{\xi }}{2}}\approx 1$. Also, $2\mathrm{\xi }\sqrt{\overline{N}}=-\frac{\mathrm{ϵ}\mathrm{\gamma }{B}_{\text{ac}}}{2{\nu }_{\text{ac}}}$. Last, we obtain $⟨n-m\mid e^{\mathrm{\xi }{\hat{a}}^{\mathrm{†}}-{\mathrm{\xi }}^{*}\hat{a}}\mid n⟩={\mathcal{J}}_m\left(\frac{\mathrm{ϵ}\mathrm{\gamma }{B}_{\text{ac}}}{2{\mathrm{\nu }}_{\text{ac}}}\right)$, and the Floquet states are

$$\begin{array}{cc}\hfill {\mid \mathrm{ϵ}⟩}_n& =\mid \mathrm{ϵ}⟩{\mathrm{\Sigma }}_m\mid n-m⟩⟨n-m\mid e^{\mathrm{\xi }{\hat{a}}^{\mathrm{†}}-{\mathrm{\xi }}^{*}\hat{a}}\mid n⟩\hfill \\ & ={\mathrm{\Sigma }}_{n\prime }{\mathcal{J}}_{n-n\prime }\left(\frac{\mathrm{ϵ}\mathrm{\gamma }{B}_{\text{ac}}}{2{\mathrm{\nu }}_{\text{ac}}}\right)\mid \mathrm{ϵ},n\prime ⟩\hfill \end{array}$$ (6)

As a result of the periodic driving, the two-level (∣ + ⟩, ∣ − ⟩) spin system is extended to an infinite number of synthetic energy levels ∣ϵ⟩_n, as shown in Fig. 2A.

Maser mechanism with damping feedback

The feedback circuit uses a rubidium magnetometer to measure the nuclear polarization that acts back on the spins. The dynamics of the ¹²⁹Xe spin polarization $\overrightarrow{\mathbf{P}}=[P_x,P_y,P_z]$ is described by the nonlinear Bloch equations (9, 16)

$$\{\begin{array}{l}\frac{d{P}_x}{\mathit{dt}}=P_y{\mathrm{\omega }}_z(t)-P_z{\mathrm{\omega }}_y(t)-\frac{1}{T_{2,0}}{P}_x\\ \frac{d{P}_y}{\mathit{dt}}=P_z{\mathrm{\omega }}_x(t)-P_x{\mathrm{\omega }}_z(t)-\frac{1}{T_{2,0}}{P}_y\\ \frac{d{P}_z}{\mathit{dt}}=P_x{\mathrm{\omega }}_y(t)-P_y{\mathrm{\omega }}_x(t)-\frac{1}{T_{1,0}}{P}_z+{\gamma }_{\text{se}}(P_{\text{Rb}}-P_z)\end{array}$$ (7)

where $\overrightarrow{\mathrm{\omega }}=2\mathrm{\pi }\mathrm{\gamma }\overrightarrow{\mathbf{B}}=2\mathrm{\pi }\mathrm{\gamma }[B_0+B_{\text{ac}}\text{cos}(2\mathrm{\pi }{\nu }_{\text{ac}}t)]\widehat{z}+2\mathrm{\pi }\mathrm{\gamma }{B}_{\mathrm{f}}(t)\widehat{y}$. As the thermal polarization of ¹²⁹Xe is much smaller than that from spin-exchange collisions, we can neglect the thermal polarization in Eq. 7. The feedback field is B_f(t) = χP_x(t), where χ is the feedback gain determined by the feedback circuit. The gain is χ = 0 when the feedback circuit is disabled. P_Rb is the polarization of the rubidium atoms, which depends on the optical pumping and spin-relaxation rates (36). γ_se is the spin-exchange rate between the ¹²⁹Xe and ⁸⁷Rb. γ_se(P_Rb − P_z) represents the spin-exchange pumping of ¹²⁹Xe spins. We now consider the spin dynamics in two cases.

1) We consider the condition: P_z > 0, χ < 0 or P_z < 0, χ > 0, corresponding to the experiments shown in Fig. 1 (B and D). In this case, maser phenomena cannot be generated. Here, we assume B_ac = 0. In the beginning, the ¹²⁹Xe spin polarization reaches an equilibrium state. After that, a magnetic field pulse along x tilts the spins away from the z to y axis by an angle θ₀, and the evolution of the spin polarization $\overrightarrow{\mathbf{P}}$ becomes (9, 16)

$$\begin{array}{c}{P}_{+}(t)=P_0{T}_{\mathrm{d}}\frac{q}{T_{2,0}}\text{sech}[\frac{q}{T_{2,0}}(t-t_0)]e^{i(2\mathrm{\pi }\mathrm{\nu }\mathrm{t}-\mathrm{\pi }/2)}\\ P_z(t)=P_0{T}_{\mathrm{d}}/T_{2,0}\{q\text{tanh}[\frac{q}{T_{2,0}}(t-t_0)]-1\}\end{array}$$ (8)

where P₊ = P_x + iP_y, ν is the oscillation frequency, and

$$\begin{array}{c}q={[1+{(T_{2,0}/T_{\mathrm{d}})}^2+2\text{cos} {\mathrm{\theta }}_0(T_{2,0}/T_{\mathrm{d}})]}^{1/2}\\ t_0=-\frac{T_{2,0}}{q}{\text{tanh}}^{-1}\left[\frac{1}{q}\right(\frac{T_{2,0}}{T_d}\text{cos} {\mathrm{\theta }}_0+1\left)\right]\end{array}$$ (9)

As demonstrated in the Supplementary Materials (section S2), the oscillation frequency has a small shift from the Larmor frequency ν₀, i.e., ν = ν₀ + Δν, arising from an effect known as frequency pulling (16). The shift Δν linearly depends on the damping rate, i.e., Δν = α · 1/T_d, where α = 0.235 in our experiments. Note that longitudinal relaxation is neglected here. ∣P₊(t)∣ reaches its maximum value at t = t₀. We discuss two cases for the initial angle θ₀:

i) When the transverse excitation of nuclear spins is small (∣θ₀ ∣ ≪ 1), q ≈ (1 + T_2,0/T_d) and t₀ → − ∞. Because of t₀ < 0, ∣P_±(t)∣ should be monotonically decreasing. Last, we have P₊(t) ≈ P₀ sin θ₀e^{−(1/T_2,0 + 1/T_d)t}e^{i(2πνt − π/2)}, which is a single-exponential decay (see Fig. 1B);

ii) When the initial angle is θ₀ ≈ π, cos θ₀ ≈ − 1 and q ≈ ∣ 1 − T_2,0/T_d∣. Although the form of Eq. 8 is complex, we can still gain information from Eq. 8. On the basis of the definition of t₀ as described above, when T_d/T_2,0 ≥ 1, we have t₀ ≤ 0 and thus ∣P₊(t)∣ should be monotonically decreasing; when T_d/T_2,0 < 1, we have t₀ > 0 and ∣P₊(t)∣ increases to be maximum at t = t₀ and then decreases to be zero (see Fig. 1D).

2) We consider the condition: P_z > 0, χ > 0 or P_z < 0, χ < 0 and T_d/T_2,0 < 1, corresponding to the experiments shown in Fig. 2. In this case, the damping-induced torque provides a sufficient strength of feedback field for sustaining the maser oscillation. On the basis of numerical simulations, we show that the small transverse polarization component caused by misalignment or quantum fluctuation is sufficient for activating the Floquet maser (see section S3).

Estimation of search sensitivity for axion dark matter

Here, we discuss how to search for axions and axion-like particles (these are well-motivated dark matter candidates; we call these “axions” for brevity) via our maser technique and estimate the search sensitivity. As the nuclear spins of the maser on Earth move through the galactic dark matter halo, they couple to axion dark matter producing an effective oscillating magnetic field (29, 31), generating axion-driven maser sidebands.

The effective magnetic field B_axion

$$\{\begin{array}{l}{\nu }_{\text{axion}}=m_{\text{axion}}{c}^2/h\\ B_{\text{axion}}[\mathrm{T}]\approx 6·10^{-8}{\mathrm{g}}_{\text{aNN}}[{\text{Gev}}^{-1}]/{\mathrm{g}}_n\end{array}$$ (10)

where m_axion is axion mass, c is the velocity of light, h is the Planck constant, g_n ≈ − 3/2 is the nuclear Land$\stackrel{\prime }{\mathrm{e}}$ g-factor for ¹²⁹Xe, and g_aNN is the coupling constant to be measured that represents the coupling strength of neutrons (from ¹²⁹Xe nucleus) and axions. For example, the axion mass of 10⁻¹⁸ to 10⁻¹⁵ eV corresponds to the frequency ν_axion of ∼1 to 100 mHz. The experimental sensitivity (see Fig. 4C) of the maser-based magnetometer to real magnetic fields can directly translate to the sensitivity to the coupling constant g_aNN. As a result, we achieve the search sensitivity of axions ∣g_aNN ∣ ≈ 1.75 × 10⁻⁵ Gev⁻¹/Hz^1/2 for 1 to 60 mHz and ∣g_aNN ∣ ≈ 1.8 × 10⁻⁴ν_axion Gev⁻¹/Hz^1/2 from 60 mHz to 10 Hz. Benefitting from the narrow linewidth, the maser allows the detection of axion-driven frequency as low as millihertz (corresponding to axion mass ∼10⁻¹⁸ eV). Current experiments, with measurement time T_m = 10⁴ s and ν_axion = 1 mHz, yield a potential of limit on ∣g_aNN∣ ≈ 1.75 × 10⁻⁷Gev⁻¹, well beyond the most stringent existing constraints (30, 31).

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/7/8/eabe0719/DC1