Ultranarrow Superradiant Laser via Dressed Dark State

TL;DR

This paper proposes a superradiant laser scheme using dressed dark states in even isotopes of alkaline-earth-metal-like atoms, achieving ultranarrow linewidths and enhanced stability for precision measurement applications.

Contribution

It introduces a novel dressed-state protocol that enables superradiant lasing from dark states in even isotopes, overcoming their forbidden transition limitations.

Findings

Achieves mHz linewidth in superradiant regime

Laser frequency more robust against cavity and magnetic field fluctuations

Demonstrates potential for ultrastable frequency sources

Abstract

Superradiant laser, which exploits the clock transition of alkaline-earth-metal-like atoms to generate ultrastable light in the bad-cavity limit, has garnered much attention in the past few decades. Unlike their odd counterpart, the even isotopes of alkaline-earth-metal-like atoms possess simpler structures and longer-lived transitions, which would relax the field control requirements and enhance the frequency stability of the output light. However, due to the absence of hyper-fine interaction in even isotopes, the transition from the state $^{3}\mathrm{P}_{0}$ to $^{1}$S$_{0}$ is strictly forbidden, leading to a vanishing coupling strength between the cavity mode and the atoms (the state $^{3}\mathrm{P}_{0}$ as a dark state). In this work, we suggest a superradiant laser scheme by dressing this dark state with a small bright component by virtue of a static magnetic field. In contrast to other proposals utilizing natural atomic transitions, our dressed-state protocol can work from the crossover regime (coherence in both atoms and photons) to the superradiant lasing regime (coherence solely in atoms). Specially, by operating deep into the superradiant lasing regime, our scheme witnesses a dramatic line-narrowing feature (mHz level) while maintaining its power. Furthermore, compared to the crossover regime, the laser frequency in the superradiant lasing regime is more robust against the fluctuations of the cavity length and magnetic field strength. Our proposal demonstrates the potential for extracting ultranarrow light from even isotopes of the alkaline-earth-metal-like atoms and may find its role in frequency stabilization and related precision measurement scenarios.

Equations

$$\hat{H}$$

$${\hat{H}}_a$$

$${\hat{H}}_c$$

$${\hat{H}}_{a-c}$$

$$\hat{H}$$

$${\hat{H}}_I$$

$$\frac{d\hat{\rho }}{dt}$$

$$⟨\hat{X}\hat{Y}\hat{Z}⟩$$

$$\frac{d}{dt}⟨{\hat{c}}^{†}\hat{c}⟩=-\kappa ⟨{\hat{c}}^{†}\hat{c}⟩+\frac{N{\Omega }_d}{2i}\left(⟨{\hat{c}}^{†}\hat{\sigma }⟩-⟨{\hat{\sigma }}^{†}\hat{c}⟩\right).$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_z⟩$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_1^{†}{\hat{\sigma }}_2⟩$$

$$\frac{d}{dt}⟨{\hat{c}}^{†}\hat{\sigma }⟩$$

$${⟨{\hat{c}}^{†}\hat{c}⟩}_{\mathrm{s}}^{(0)}$$

$${⟨{\hat{\sigma }}_z⟩}_{\mathrm{s}}^{(0)}$$

$${⟨{\hat{\sigma }}_1^{†}{\hat{\sigma }}_2⟩}_{\mathrm{s}}^{(0)}$$

$${⟨{\hat{c}}^{†}\hat{\sigma }⟩}_{\mathrm{s}}^{(0)}$$

$$\lambda \equiv \frac{⟨{\hat{c}}^{†}\hat{c}⟩⟨{\hat{\sigma }}_z⟩}{N⟨{\hat{\sigma }}_1^{†}{\hat{\sigma }}_2⟩}.$$

$$S(\omega )=2{\int }_0^{\infty }dt\mathrm{Re}\left[{⟨{\hat{c}}^{†}(t)\hat{c}(0)⟩}_{\mathrm{s}}{e}^{-i(\omega -{\omega }_d)t}\right].$$

$$\frac{d}{dt}V(t)=\mathbb{W}V(t),$$

$$\mathbb{W}=\frac{1}{2}\left(\begin{array}[]{cc}2i\delta_{c}-\kappa&iN\Omega_{d}\\ -i\Omega_{d}\left\langle\hat{\sigma}_{z}\right\rangle_{\mathrm{s}}&-\Gamma\end{array}\right).$$

$$V(t)$$

$$\Delta \nu$$

$$\nu$$

$$c_P^{(i)}\equiv \left|\frac{\partial ({\nu }^{\ast }-\nu )}{\partial {\delta }_i}\right|,i=c,B,$$

$$c_P^{(c)}$$

$$c_P^{(B)}$$

$$\frac{d}{dt}⟨{\hat{c}}^{†}\hat{c}⟩$$

$$\frac{d}{dt}⟨{\hat{c}}^{†}{\hat{\sigma }}_{g{b}_0}⟩$$

$$+i\frac{{\Omega }_d}{2}⟨{\hat{\sigma }}_{b_0{b}_0}⟩+i\frac{{\Omega }_d}{2}(N-1)⟨{\hat{\sigma }}_{b_{0}g}^1{\hat{\sigma }}_{g{b}_0}^2⟩,$$

$$\frac{d}{dt}⟨{\hat{c}}^{†}{\hat{\sigma }}_{g{d}_0}⟩$$

$$+i\frac{{\Omega }_d}{2}⟨{\hat{\sigma }}_{b_0{d}_0}⟩+i\frac{{\Omega }_d}{2}(N-1)⟨{\hat{\sigma }}_{b_{0}g}^1{\hat{\sigma }}_{g{d}_0}^2⟩,$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_{d_0{b}_0}⟩$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_{b_0{b}_0}⟩$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_{d_0{d}_0}⟩$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_{b_{0}g}^1{\hat{\sigma }}_{g{b}_0}^2⟩$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_{d_{0}g}^1{\hat{\sigma }}_{g{d}_0}^2⟩$$

$$\frac{d}{dt}⟨{\hat{\sigma }}_{b_{0}g}^1{\hat{\sigma }}_{g{d}_0}^2⟩$$

$$+i\frac{{\Omega }_{b,0}}{2}⟨{\hat{\sigma }}_{b_0{d}_0}⟩⟨{\hat{\sigma }}_{b_{0}g}\hat{c}⟩+i{\Omega }_B⟨{\hat{\sigma }}_{d_{0}g}^1{\hat{\sigma }}_{g{d}_0}^2⟩-i{\Omega }_B⟨{\hat{\sigma }}_{b_{0}g}^1{\hat{\sigma }}_{g{b}_0}^2⟩,$$

$$\frac{d}{dt}{⟨{\hat{c}}^{†}\hat{c}⟩}^{(1)}$$

$$\frac{d}{dt}{⟨{\hat{\sigma }}_z⟩}^{(1)}$$

$$\frac{d}{dt}{⟨{\hat{\sigma }}_1^{†}{\hat{\sigma }}_2⟩}^{(1)}$$

$$\frac{d}{dt}{⟨{\hat{c}}^{†}\hat{\sigma }⟩}^{(1)}$$

Full text

I Introduction

Since the first detection in 1960 (Maiman, 1960), lasers, especially frequency-stabilized representatives, have found applications in diverse areas, such as optical atomic clocks (Derevianko and Katori, 2011; Ludlow et al., 2015) and atom or ion cooling (Hänsch and Schawlow, 1975; Wineland and Dehmelt, 1975; Cohen-Tannoudji and Phillips, 1990). Typically, the laser frequency is stabilized using feedback from a high-finesse optical cavity whose length is ultimately limited by the thermal noise of the end mirrors (Numata et al., 2004). Consequently, significant efforts have been devoted to reducing the Brownian noise of the mirrors utilizing high-quality materials, for instance, ultralow expansion (ULE) glass (Numata et al., 2004; Yu et al., 2019), sapphire (Storz et al., 1998), and single-crystal silicon (Kessler et al., 2012). Recently, as an alternative route, a superradiant laser has gained considerable attention due to its potential to extract light directly from atomic clock transitions. Specifically, in the bad-cavity limit where the photon dissipation rate is several orders of magnitude larger than the atomic relaxation rate, the atoms behave as a macroscopic spin through interacting with the same cavity mode (Meiser et al., 2009; Meiser and Holland, 2010a, b; Liu et al., 2020). Therefore, above a certain pump, the atoms emit light collectively, resembling the transient superradiance, but in a steady-state manner. As the atomic phases are synchronized, the superradiant emission light is line-narrowed, even narrower than the natural linewidth of the atomic transition. Unlike the conventional laser where the coherence is mainly stored in photons, the coherence of the superradiant laser is significantly influenced by the macroscopic atomic spin, demonstrating strong atom-atom correlations. Particularly, the superradiant laser can operate in two regimes: the crossover regime, where both the stimulated emission photons and macroscopic spin contribute to the system’s coherence, and the superradiant lasing regime, where in contrast the coherence is solely in atoms (Norcia and Thompson, 2016; Tieri et al., 2017; Debnath et al., 2018).

The superradiant laser requires a long-lifetime atomic clock transition, rendering the intercombination transitions in the alkaline-earth-metal-like atoms (e.g., Sr, Ca, Mg, Yb) favorable candidates. For example, leveraging the $7.5$kHz spin-forbidden 3P1-1S0 transition of the 88Sr atoms, the Ref. (Norcia and Thompson, 2016) achieves quasi-steady-state emission with a linewidth of around 6kHz. For a system composed of the 87Sr atoms ($I=9/2$), the weakly allowed double-forbidden transition 3P0-1S0 (linewidth around mHz) yields a millihertz superradiant laser (Meiser et al., 2009). Further pursuit of the stabilized laser can hinge on atomic transitions with longer lifetimes, such as the 3P0-1S0 transition in even isotopes of the alkaline-earth-metal-like atoms ($I=0$, lifetime around several thousands of years) (Santra et al., 2004). Additionally, compared with their odd counterpart with hyperfine interaction, the simpler structure of even isotopes exhibits scalar polarizability, thereby easing the field control requirement in the lasing scheme (Santra et al., 2005; Barber et al., 2006). However, the absence of the nuclear spin also indicates a vanishing single-photon transition matrix element (3P0 acting as a dark state), resulting in a negligible photon number in the steady state. In this sense, a proposal which can effectively couple the dark-state transition of even alkaline-earth-metal-like atoms to the cavity mode is urgently desired for the ultranarrow superradiant laser.

One approach for achieving this effective interaction is utilizing an intermediate level via external fields (Santra et al., 2005; Dong et al., 2023; Barber et al., 2006; Taichenachev et al., 2006; Dubey et al., 2025). For instance, although the transition from the clock state 3P0 to the ground state 1S0 is strictly forbidden, it can be mediated by the broad state 1P1 with probe and dressing lasers (Santra et al., 2005). In our previous proposal of coherence-assisted superradiant laser, we couple the atomic ground state 1S0 with the clock state 3P0 via two Raman beams, where the frequencies of the beams demand judicious control (Dong et al., 2023). Inspired by the work of magnetic-induced excitation of forbidden transition (Barber et al., 2006; Taichenachev et al., 2006; Dubey et al., 2025), here we explore the dark state superradiant laser system with a static magnetic field. In our proposal, the dark state 3P0 is dressed with a small bright 3P1 component via the magnetic dipole (M1) transition, and hence photons will be emitted when the cavity mode is resonant with this dressed dark state. Originating from the M1 transition, the lasing properties of our protocol, such as the threshold, steady-state radiation or power, and the coherence feature of the laser, can be manipulated by the magnetic field. Therefore, in contrast to other proposals utilizing natural atomic transitions, our dressed-state scenario can operate from the crossover regime (coherence in both atoms and photons) to the superradiant lasing regime (coherence solely in atoms). Specially, by operating deep into the superradiant lasing regime, our superradiant laser scheme witnesses a dramatic line-narrowing feature (mHz level) while maintaining its power. Furthermore, compared to the crossover regime, the laser frequency in the superradiant lasing regime is more robust against the fluctuations of the cavity length and magnetic field strength. Our magnetic-field-tuned proposal provides an alternative route for the superradiant laser system utilizing even alkaline-earth-metal-like atoms, and may find potential applications in frequency stabilization and the related precision measurement scenarios.

II Model

II.1 System Setup

In our scenario, $N$ cold bosonic alkaline-earth-metal-like atoms are confined in an optical cavity whose axis is along the $y$-direction (see Fig. 1(a)). The cavity mode couples with the intercombination transition of the atoms, i.e., the ground state $\left|g\right\rangle\equiv\left|{}^{1}\mathrm{S}_{0}\right\rangle$ and the sublevel of the bright state $\left|b_{0}\right\rangle\equiv\left|{}^{3}\mathrm{P}_{1},m=0\right\rangle$ where $m$ is the magnetic quantum number. Due to the absence of the nuclear spin, the 3P0-1S0 transition in even isotopes is strictly forbidden, leaving the state $\left|d_{0}\right\rangle\equiv\left|{}^{3}\mathrm{P}_{0}\right\rangle$ as a dark state (no lasing). Actually, a $z$-directioned static magnetic field would couple the dark state with the bright one through the M1 transition and thus induce an intermediate coupling between the cavity mode and the dark transition. The Hamiltonian of the system reads ($\hbar=1$)

[TABLE]

where $\hat{c}$ $(\hat{c}^{\dagger})$ represents the annihilation (creation) operator of the cavity mode with frequency $\omega_{c}$. $\omega_{b,0}$ ($\omega_{d,0}$) is the frequency of the atomic bare bright (dark) state which couples with the cavity mode with the Rabi frequency $\Omega_{b,0}$ ($\Omega_{d,0}=0$). A $z$-directioned magnetic field $B$ interacts with the bright-dark transition via the M1 process with the Rabi frequency $\Omega_{B}$. For alkaline-earth-metal-like atoms, the magnetic Rabi frequency is $\Omega_{B}=2\pi\times 1.14B$MHz ($B$ given in Gaussian unit) (Taichenachev et al., 2006). The frequency gap $\Delta\equiv\omega_{b,0}-\omega_{d,0}$ between the atomic bare bright and dark states is around $2\pi\times 1$THz (Courtillot et al., 2005).

As shown in Fig. 1(b), the atoms are pumped from the ground state $\left|g\right\rangle$ to the bare-dark state $\left|d_{0}\right\rangle$ incoherently. This pumping process can be achieved by using several beams to coherently pump the atom from $\left|g\right\rangle$ to an auxiliary excited state which then decays rapidly to $\left|d_{0}\right\rangle$ via spontaneous emission. A typical example is illustrated in Fig. 1(c). An $x$-polarized beam 1 couples $\left|g\right\rangle$ with the sublevel of 3P1, i.e., $\left|x\right\rangle\equiv\left(\left|{}^{3}\mathrm{P}_{1},m=-1\right\rangle-\left|{}^{3}\mathrm{P}_{1},m=1\right\rangle\right)/\sqrt{2}$ while a $y$-polarized beam 2 couples $\left|x\right\rangle$ with a sublevel of 3S1 ($\left|\mathrm{S}\right\rangle\equiv\left|{}^{3}\mathrm{S}_{1},m=0\right\rangle$). Finally, atoms in $\left|\mathrm{S}\right\rangle$ decay to $\left|d_{0}\right\rangle$ at a rate around several MHz. Notice that the transition from $\left|\mathrm{S}\right\rangle$ to $\left|b_{0}\right\rangle$ is dipole forbidden, i.e., no pump to $\left|b_{0}\right\rangle$.

II.2 Lasing from the dressed dark state

Although the cavity mode couples with the bright transition of the atoms, the vanishing pump indicates no population in the 3P1 state and thus no lasing. In fact, since the atoms are continuously pumped to the 3P0 state, in the steady state, the atoms will all be trapped in this dark state.

In the presence of the static magnetic field, the bright state also interacts with the dark state, which effectively opens the 3P0- 1S0 dark transition. In fact, this magnetic-field-induced lasing process can be elucidated more evidently with the dressed-state basis. Actually, for a moderate magnetic field ($B$ no more than $1\mathrm{T}=10^{4}\mathrm{G}$), the magnetic Rabi frequency $\Omega_{B}$ is much smaller than the frequency gap of the bright-dark transition (THz level). Hence, by treating the M1 transition term as a perturbation, the Hamiltonian in the dressed-state basis can be recast as

[TABLE]

where $\left|b\right\rangle\simeq\left|b_{0}\right\rangle+\Omega_{B}/\Delta\left|d_{0}\right\rangle$ ($\left|d\right\rangle\simeq\left|d_{0}\right\rangle-\Omega_{B}/\Delta\left|b_{0}\right\rangle$) stands for the dressed-bright (dressed-dark) state involving a small dark (bright) component. The Zeeman-shifted frequencies of the two dressed states are $\omega_{b}=\omega_{b,0}+\varpi_{B}$ and $\omega_{d}=\omega_{d,0}-\varpi_{B}$ with $\varpi_{B}\equiv-\Delta/2+\sqrt{(\Delta/2)^{2}+\Omega_{B}^{2}}$. The dressed-bright (dressed-dark) state coherently interacts with the cavity mode with the Rabi frequency $\Omega_{b}\simeq\Omega_{b,0}$ ($\Omega_{d}\simeq\Omega_{b,0}\Omega_{B}/\Delta$). Moreover, as the dark component in $\left|b\right\rangle$ is rather small, the pump to this dressed-bright state is still negligible, i.e., the atomic population in $\left|b\right\rangle$ is vanishing (see the following discussions). Accordingly, one can simplify this scheme to a two-level model by ignoring the dressed-bright state in Eq. (5). The approximated two-level dressed-dark-state Hamiltonian can be rewritten as (in the interaction picture)

[TABLE]

where $\delta_{c}\equiv\omega_{c}-\omega_{d}$ denotes the cavity-atom detuning (see Fig. 1(b)). In consequence, with the magnetic field, the long-lived dark state is dressed with a small bright component and thus coherently couples with the cavity mode. Above a certain threshold, the atomic population inversion would be generated between the ground and dressed-dark states, leading to the output of lasing photons.

III Dynamical Equations and Analysis

III.1 Dynamical Equations

In the effective two-level model, with the incoherent pump and dissipation processes, the dynamics of the density matrix $\hat{\rho}\equiv\hat{\rho}(t)$ of the total system is governed by the following master equation

[TABLE]

where the Lindblad operator is defined as $\mathscr{L}[\hat{\mathscr{O}}]\equiv\hat{\mathscr{O}}\hat{\rho}\hat{\mathscr{O}}^{\dagger}-\hat{\mathscr{O}}^{\dagger}\hat{\mathscr{O}}\hat{\rho}/2-\hat{\rho}\hat{\mathscr{O}}^{\dagger}\hat{\mathscr{O}}/2$. $\kappa$ ($\eta$, $\gamma_{d}\equiv\gamma\Omega_{B}^{2}/\Delta^{2}$) denotes the dissipation rate of the photon (pump rate, decay rate of the atom). $\hat{\sigma}_{i}\equiv\left|g\right\rangle_{i}\left\langle d\right|$ stands for the transition operator of the $i$th atom.

Owing to the infinite dimension of the cavity mode and the exponential scaling of the dimension of gain medium with the atomic number, approximate methods for solving the above master equation are urgently desired. Although the first-order mean-field theory yields useful results, for instance, the steady-state photon number and atomic inversion, it violates the phase symmetry and overlooks the quantum correlations of the system which are crucial for characterizing the coherence of the system. Here, beyond the first-order method, we solve the dynamical equations of the system within the second-order mean-field theory where the correlation between any two operators is retained and the phase symmetry of the system is preserved (Meiser et al., 2009; Dong et al., 2023).

In the second-order mean-field theory, the correlation of three or more operators is ignored, i.e.,

[TABLE]

Here $\left\langle\hat{X}\right\rangle\equiv\left\langle\hat{X}(t)\right\rangle\equiv\mathrm{Tr}(\hat{X}\hat{\rho})$ represents the expectation value of the operator $\hat{X}$. It has been demonstrated that the density matrix $\hat{\rho}$ is symmetric under the phase transformation $U(\theta)\equiv\exp[i\theta(\hat{c}^{\dagger}\hat{c}+\sum_{j=1}^{N}\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{j})]$ for any $\theta$. As a result, the expectation values of operators which are not symmetric would vanish. Moreover, as all the atoms interact with the field homogeneously, we can simplify the expressions of expectation values as $\left\langle\hat{\sigma}_{z,i}\right\rangle=\left\langle\hat{\sigma}_{z}\right\rangle\equiv\left\langle\hat{\sigma}^{\dagger}\hat{\sigma}-\hat{\sigma}\hat{\sigma}^{\dagger}\right\rangle$, $\left\langle\hat{c}^{\dagger}\hat{\sigma}_{i}\right\rangle\equiv\left\langle\hat{c}^{\dagger}\hat{\sigma}\right\rangle$, and $\left\langle\hat{\sigma}_{i}^{\dagger}\hat{\sigma}_{j\neq i}\right\rangle=\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle$ for all $i$.

As a first step, the dynamical equation for the expectation value of the photon number reads

[TABLE]

Apparently, the dynamics of the photon number is related to the atom-field correlation $\left\langle\hat{c}^{\dagger}\hat{\sigma}\right\rangle$. To close the dynamical equation, we obtain the derivatives of the atomic inversion $\left\langle\hat{\sigma}_{z}\right\rangle$, the atom-atom correlation $\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle$, and the atom-field correlation $\left\langle\hat{c}^{\dagger}\hat{\sigma}\right\rangle$ as follows

[TABLE]

where we have defined the notations $\Gamma\equiv\eta+\gamma_{d}$ and $d_{0}\equiv(\eta-\gamma_{d})/\Gamma$. To demonstrate the validity of the two-level dressed-dark-state model, we also present the dynamical equations of the original three-level model (see Appendix A).

In the bad-cavity limit, it has been elucidated that the atoms are collectively correlated in phase in the lasing region (Meiser et al., 2009; Meiser and Holland, 2010a; Debnath et al., 2018). Thus for a large atomic number $N\gg 1$, the collective correlation $N\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle$ in Eq. (12) would be much larger than the single-atom population inversion $\left\langle\hat{\sigma}_{z}\right\rangle$. Hence, by neglecting the small term $i\Omega_{d}\left[\left(\left\langle\hat{\sigma}_{z}\right\rangle+1\right)/2-\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle\right]/2$ in Eq. (12), one can achieve the approximate analytical steady-state solutions of the above dynamical equations (for the resonant case $\delta_{c}=0$)

[TABLE]

where $C=N\Omega_{d}^{2}/\kappa\Gamma$ is the generalized collective cooperative parameter. Here the subscript “$\mathrm{s}$” stands for the steady state while the superscript “$(0)$” indicates that Eqs. (13)-(16) is the lowest-order approximate results. The high-order corrections of the steady-state solution can be retained by taking $i\Omega_{d}[(\left\langle\hat{\sigma}_{z}\right\rangle+1)/2-\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle]/2$ in Eq. (12) as a perturbation (see Appendix B).

It is worth to mention that Eqs. (9)-(12) demonstrate the significance of coherence in the laser system. As is indicated, the growth of the intracavity photon number and atomic inversion relies on the field-atom coherence $\left\langle\hat{c}^{\dagger}\hat{\sigma}\right\rangle$ which is inherited from $\left\langle\hat{c}^{\dagger}\hat{c}\right\rangle\left\langle\hat{\sigma}_{z}\right\rangle$ and $N\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle$. Roughly speaking, $\left\langle\hat{c}^{\dagger}\hat{c}\right\rangle\left\langle\hat{\sigma}_{z}\right\rangle$ characterizes the strength of the stimulation emission while $N\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle$ represents the atom-atom correlation in the collective spin. In this sense, one can quantify the relative contribution of photons and atoms to the coherence of the system with the coherence contribution parameter

[TABLE]

With the approximate analytical solution in Eqs. (13)-(15), the coherence contribution parameter can be simplified as the ratio of the effective dissipation rate of the atom $\Gamma$ over the photon decay rate $\kappa$, i.e., $\lambda\simeq\Gamma/\kappa$. Obviously, $\lambda\gg 1$ ($\lambda\ll 1$) implies the coherence mainly in photons (atoms) while $\lambda\sim 1$ corresponds to the compatible case where the system coherence is in both photons and atoms. In fact, the coherence feature of the laser varies with system parameters, such as the magnetic field strength and pump rate. For reasons that would be clear in the next section, we refer to the system as operating in the crossover regime when the coherence can be stored in both atoms and photons, i.e., the coherence contribution parameter varies from $\lambda\ll 1$ to $\lambda\gg 1$ in the lasing region, while the superradiant lasing regime when the coherence is kept solely in atoms, i.e., $\lambda\ll 1$ throughout the lasing region. In our scenario, the laser system can operate from the crossover regime to the superradiant lasing regime with the modulation of the magnetic field, where the laser properties are distinct from one another (see Tab. 1 and the following discussion for more details).

III.2 Laser spectrum

Beyond the photon number in the steady state, the laser linewidth is another significant feature of the output laser. As the full width at half maximum (FWHM) of the laser spectrum, the laser linewidth describes the spread of the laser around its central frequency and characterizes the spectral purity of the laser. According to the Wiener-Kintchine theorem (Wiener, 1930; Khintchine, 1934; Huang, 2009), the laser spectrum is related to the correlation function of the cavity mode as

[TABLE]

Within the frame of the second-order mean-field theory, the dynamical equation of the correlation function $\left\langle\hat{c}^{\dagger}(t)\hat{c}(0)\right\rangle_{\mathrm{s}}$ can be cast as

[TABLE]

where we have defined the vector $V(t)\equiv\left(\left\langle\hat{c}^{\dagger}(t)\hat{c}(0)\right\rangle_{\mathrm{s}},\left\langle\hat{\sigma}^{\dagger}(t)\hat{c}(0)\right\rangle_{\mathrm{s}}\right)^{T}$ and matrix

[TABLE]

The correlation function vector $V(t)$ can be expressed as (Dong et al., 2023)

[TABLE]

where $w_{j}$ is the $j$th eigenvalue of $\mathbb{W}$ with the right and left eigenvectors $\left|j\right\rangle$ and $\left\langle\widetilde{j}\right|$ for $i,j=1,2$. The initial condition $V(0)$ corresponds to the steady-state solutions of Eqs. (9) and (12). Therefore, the laser spectrum behaves as a weighted superposition of two Lorentzian lineshapes. Concretely, the $j$th lineshape is centered around the frequency $\omega_{d}+\mathrm{Im}(w_{j})$ with linewidth $2\left|\mathrm{Re}(w_{j})\right|$. As illustrated in the experimental and theoretical works (Meiser et al., 2009; Norcia and Thompson, 2016; Bohnet et al., 2012; Debnath et al., 2018; Dong et al., 2023), the linewidth of the bad-cavity laser is dramatically narrowed as a result of the strong atom-atom correlation. Hence the linewidth and the central frequency of the laser spectrum can be approximated utilizing one of the eigenvalues of $\mathbb{W}$ (the one with the smaller real part)

[TABLE]

It is worth to emphasize that the analytical expression of the linewidth is derived from the first-order correction of $\left\langle\hat{\sigma}_{z}\right\rangle_{\mathrm{s}}$ (see Appendix C).

IV Output Laser Properties

In the above section, we have examined the dynamics of the dressed-dark-state superradiant laser system analytically which characterizes the properties of the laser (steady-state radiation, laser spectrum, and so on). As shown in Eqs. (13)-(16) and (22)-(23), since the lasing state in our scheme, i.e., the dressed-dark state, originates from the M1 transition, the behaviour of the output laser would be manipulated by this magnetic field. As an representative example, in the following we explore our proposal utilizing an ensemble of 88Sr atoms to demonstrate the features of this magnetic-field-induced superradiant laser more intuitively. Here the energy gap between the bare bright and dark states is $\Delta=2\pi\times 5.6$THz. The decay rate of the 3P1-1S0 transition is $\gamma=2\pi\times 7.5$kHz. As achieved in experiments (Bohnet et al., 2012; Norcia and Thompson, 2016; Norcia et al., 2016), the atomic number in the cavity can be up to $N=10^{6}$ with the cavity dissipation rate $\kappa=2\pi\times 150$kHz. The Rabi frequency of the coupling between the 3P1-1S0 transition and the cavity mode is $\Omega_{c}=2\pi\times 21.2$kHz.

IV.1 Steady-state radiation

In the superradiant laser system, the atom-atom correlation $\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle_{\mathrm{s}}$ is remarkable in portraying the features of the steady-state radiation. In the lasing process, together with the atomic collective decay resulting from the adiabatic elimination of the cavity mode, the non-collective pump and decay of the atoms drive the atoms to some collective states $\left|J,M\right\rangle$ (Dicke state) (Dicke, 1954) where $J$ and $M$ are the quantum numbers satisfying $(\hat{J}_{x}^{2}+\hat{J}_{y}^{2}+\hat{J}_{z}^{2})\left|J,M\right\rangle=J(J+1)\left|J,M\right\rangle$ and $\hat{J}_{z}\left|J,M\right\rangle=M\left|J,M\right\rangle$. As the total radiation rate of the system in the steady state is related to the collective correlation $\left\langle\hat{J}_{+}\hat{J}_{-}\right\rangle_{\mathrm{s}}\simeq N^{2}\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle_{\mathrm{s}}$ (Meiser and Holland, 2010a), the steady-state photon number inside the cavity is proportional to the atomic correlation $\left\langle\hat{c}^{\dagger}\hat{c}\right\rangle_{\mathrm{s}}\propto N^{2}\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle_{\mathrm{s}}$. Subsequently, according to the input-output theory, the emission power of the photons through the cavity mirror equals the cavity dissipation rate times the photon energy, i.e., $P=\kappa\hbar\omega\left\langle\hat{c}^{\dagger}\hat{c}\right\rangle_{\mathrm{s}}$ (see Eqs. (13), (15) and Fig. 2) (Walls and Milburn, 2008).

To be more specific, as demonstrated in the approximate analytical results Eq. (13), for a small pump rate, the generalized collective cooperative parameter $C$ is much larger than unity. Therefore, when the pump rate exceeds the effective decay rate of the dressed-dark state $\gamma_{d}\equiv\gamma\Omega_{B}^{2}/\Delta^{2}$, the atom-atom correlation $\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle_{\mathrm{s}}$ grows from negative to positive, which marks $\gamma_{d}$ as the lower threshold of the laser. Above this threshold, due to the pump-induced dephasing, a competition between the establishment and destruction of the macroscopic spin appears, and thus the photon number first increases and then decreases with the growth of the pump. Ultimately, the radiation ceases at the upper threshold $N\Gamma_{d,c}\equiv N\Omega_{b,0}^{2}\Omega_{B}^{2}/(\kappa\Delta^{2})$ where $\Gamma_{d,c}$ denotes the collective decay rate of the dressed-dark state. Notably, beyond this upper threshold, the atomic population is inverted, and the correlation between atoms vanishes (Meiser and Holland, 2010a).

Figures 2(a) and 2(b) display the steady-state radiation versus the pump rate for the cases $\Omega_{B}/\Delta=10^{-1}$ and $\Omega_{B}/\Delta=10^{-3}$, respectively. It is worth noting that for the system we discussed here, the boundary between the aforementioned crossover and superradiant lasing regimes is $\Omega_{B}/\Delta\simeq 10^{-2.15}$ (see Tab. 1). That is, $\Omega_{B}/\Delta=10^{-1}$ ($\Omega_{B}/\Delta=10^{-3}$) corresponds to the crossover (superradiant lasing) regime. In Fig. 2, the red solid lines represent the approximate analytical result of the two-level dressed-dark-state model (Eq. (13)), while the triangles stand for the numerical calculation results of the original three-level model (see Appendix A). The gray dashed vertical lines denote two thresholds of the laser. The coincidence of these results elucidates the validity of the two-level dressed-dark-state model in our scheme, which can be attributed to the negligible population of the dressed-bright state (see the black dashed lines in Figs. 2(c) and 2(d)).

IV.2 Laser linewidth

The approximate analytical result (Eq. (22)) depicts the behaviour of the laser linewidth in details. Especially, two feature emerge as the rise of the pump: (1) a constant linewidth $\Delta\nu\simeq\Omega_{d}^{2}/\kappa$ for a comparatively small pump ($\gamma_{d}<\eta\ll\kappa,N\Gamma_{d,c}$), which has been demonstrated in the superradiant laser system embedded with 87Sr atoms (Meiser et al., 2009); (2) linewidth narrowed with the increase of the pump $\Delta\nu\simeq\Omega_{d}^{2}/\eta{}^{2}$ ($\kappa\ll\eta<N\Gamma_{d,c}$), resembling a Schawlow–Townes-like behaviour where the phase diffusion diminishes gradually with the growth of the power (Haken, 1983). It is worth highlighting that the performance of the linewidth in an individual case would vary with the magnetic field. For instance, for a relatively large magnetic field $\Omega_{B}/\Delta=10^{-1}$ (the crossover regime) where the thresholds of the system satisfy $\gamma_{d}=2\pi\times 75\mathrm{Hz}\ll\kappa\ll N\Gamma_{d,c}=2\pi\times 30\mathrm{MHz}$, the linewidth remains several tens of kHz for a small pump and exhibits a Schawlow–Townes-like feature when the pump goes beyond the cavity dissipation rate (see Fig. 3(a)). In contrast, a reduced magnetic field $\Omega_{B}/\Delta=10^{-3}$ (the superradiant lasing regime) will shift the laser working range to a lower pump zone ($\gamma_{d}=2\pi\times 7.5\mathrm{mHz}\ll N\Gamma_{d,c}=2\pi\times 3\mathrm{kHz}\ll\kappa$), resulting in a constant linewidth at the mHz level throughout the operation region (see Fig. 3(b)). In Fig. 4, we also illustrate the contour plot of the laser power and linewidth versus the pump rate and magnetic field, which unveils the laser properties more clearly.

The features of the linewidth are closely related to the coherence and the working regimes of the system. With the coherence contribution parameter defined in Eq. (17), the laser linewidth can be recast as $\Delta\nu\simeq\Omega_{d}^{2}/[\kappa(1+\lambda)^{2}]$. When the main noise of the system comes from the cavity ($\kappa\gg\eta$, $\lambda\ll 1$) and the cavity mode can be eliminated adiabatically, the phase information of the system is carried by the atoms (Bohnet et al., 2012), leaving the laser linewidth a constant $\Delta\nu\sim\Omega_{d}^{2}/\kappa$ (coherence in atoms). Meanwhile, for the case of a large pump ($\eta\gg\kappa$, $\lambda\gg 1$), the intracavity photon number grows drastically, which in turn boosts the stimulated emission process and inhibits the phase diffusion of the light field. Consequently, the laser is narrowed with the rise of the pump $\Delta\nu\sim\Omega_{d}^{2}/\eta{}^{2}$ (coherence mainly in photons). Therefore, the linewidth exhibits the constant and Schawlow–Townes-like features in the crossover regime (the coherence contribution parameter from $\lambda\ll 1$ to $\lambda\gg 1$) in turn, while remaining constant in the superradiant lasing regime ($\lambda\ll 1$). What is more, as a result of the shrinkage of laser thresholds, the laser system can operate from the crossover regime to the superradiant lasing regime with the reduction of the magnetic field, where the laser linewidth is remarkably reduced (see the horizontal black dot-dashed line in Fig. 4).

This line-narrowing phenomenon is more evident in Fig. 5. For a fixed pump $\eta=2\pi\times 0.5\mathrm{kHz}$, the laser is continuously narrowed with the decrease of $\Omega_{B}/\Delta$ while maintaining its power. In this sense, one can achieve a laser with a fairly small linewidth and large power at a small pump by operating the system in the superradiant lasing regime. Emerging from the growth of the atom-atom correlation (see Fig. 5(b)), this ultranarrow laser at a quite small pump mitigates the heating effect, and thus would be beneficial to the practical use (Dong et al., 2023). For clarity, we have marked the lasing region satisfying $P\geq 10^{-11}\mathrm{W}$, $\Delta\nu\leq 2\pi\times 10\mathrm{mHz}$, and $\eta\leq 2\pi\times 10^{3}\mathrm{Hz}$ in Figs. 2-4 (see the purple marked region in Figs. 2 and 3 and the area surrounded by the white dashed line in Fig. 4). As is demonstrated, the laser satisfying the above conditions operates in the superradiant lasing regime.

IV.3 Frequency and pulling coefficient

As exhibited in Sec. III.2, the laser frequency refers to the central frequency of the output spectrum and is a weighted expectation of the cavity and lasing state frequency, i.e., $\nu\simeq(\kappa\omega_{d}+\Gamma\omega_{c})/(\kappa+\Gamma)$. Apparently, the resonance of the cavity mode with respect to the lasing state indicates a laser frequency as that of the state (see Appendix C). However, due to the inevitable thermal fluctuation of the cavity mirror, the laser frequency may deviate from this resonance frequency. Meanwhile, as the lasing state in our scheme is not a bare atomic state but a dressed state whose frequency varies with the magnetic field ($\omega_{d}=\omega_{d,0}-\varpi_{B}$ with $\varpi_{B}\equiv-\Delta/2+\sqrt{(\Delta/2)^{2}+\Omega_{B}^{2}}$), the fluctuation of the magnetic field strength (or equivalently, the magnetic Rabi frequency) may also pull the laser frequency away. Here we investigate this frequency pulling effect induced by the cavity or magnetic field fluctuations and try to uncover the frequency robustness of the photons in the superradiant lasing regime. The pulling coefficient can be defined as

[TABLE]

where $\nu^{*}-\nu$ represents the laser frequency change as a result of the cavity frequency shift $\delta_{c}=\omega_{c}-\omega_{d}$ or the magnetic Rabi frequency shift $\delta_{B}=\Omega_{B}^{{}^{\prime}}-\Omega_{B}$.

The laser frequency pulling via the cavity and magnetic field can be well controlled in the superradiant lasing regime. Notably, as illustrated in Eq. (23), the cavity and magnetic-field pulling coefficients read

[TABLE]

In Fig. 6, we plot the two pulling coefficients with respect to the pump for different magnetic field cases. On one hand, owing to the shrinkage of the laser thresholds, $c_{P}^{(c)}$ is suppressed when the system tends to the superradiant lasing regime, i.e., $c_{P}^{(c)}\ll 1$ for $\Gamma\ll\kappa$ (see Fig. 6(a)). Particularly, for $\Omega_{B}/\Delta=10^{-1}$ (the crossover regime), $c_{P}^{(c)}$ grows from $10^{-3}$ to almost unity with the rise of the pump. As a contrast, it is no more than $0.02$ when the magnetic field is reduced to $\Omega_{B}/\Delta=10^{-3}$ (the superradiant lasing regime). On the other hand, the magnetic field-pulling coefficient $c_{P}^{(B)}$ is approximately proportional to the magnetic field, indicating a relatively small pulling for a rather weak magnetic field. In Fig. 6(b), it has been demonstrated that the blue dashed line ($\Omega_{B}/\Delta=10^{-3}$, the superradiant lasing regime) is much lower than red dashed line ($\Omega_{B}/\Delta=10^{-1}$, the crossover regime). As a consequence, compared to the case in the crossover regime, the laser frequency is more robust against both the cavity and magnetic field fluctuations in the superradiant lasing regime, which relaxes the cavity-length and field control requirement of the scheme and brings convenience for the implementation of this scheme.

V Discussions and Conclusions

In this work, we explore a magnetic-field-induced superradiant laser scheme where the lasing state is an atomic dark state dressed with a small bright component via the magnetic dipole transition. Hence the laser properties, for instance, the laser thresholds, power, and linewidth, could be manipulated by this external field. Specifically, with the variation of the magnetic field, the laser system can operate from the crossover (coherence in both atoms and photons) to the superradiant lasing regimes (coherence only in atoms). In contrast to the crossover regime, the laser in the superradiant lasing regime exhibits a significant line-narrowing feature (mHz level) while maintaining its power. Further research reveals that this effect emerges from the strong atom-atom correlation generated from the atomic macroscopic spin. Moreover, although the laser frequency would be shifted by the cavity length and magnetic field fluctuations, it has been demonstrated that both pulling coefficients can be well controlled as long as the system operates in the superradiant lasing regime. This flexibility relaxes the field and cavity-length control requirements for the experimental implementation.

It is interesting to compare the proposal here with our previous coherent-assisted superradiant laser scheme (Dong et al., 2023). In our previous work, stemming from the coherence between the dressed bright and dark states constructed by two Raman beams, a new local minimum of the linewidth emerges at the cost of the laser power, manifesting the possibility of realizing a laser with a comparatively small linewidth at a low pump. However, since the laser frequency is hypersensitive to the fluctuation of the frequency difference of the Raman beams, the scenario in (Dong et al., 2023) demands a judicious control of the Raman beams. In our present model, the lasing state is a dressed-dark state induced by a static magnetic field (the dressed-bright state is negligible due to the large detuning). The laser linewidth in the present work is reduced by the growth of the atom-atom correlation without the loss of its power. Meanwhile, the laser frequency pulling is dramatically mitigated. Therefore, one can achieve a laser with a rather small linewidth and large power by operating the system deep into the superradiant lasing regime, where the laser frequency is robust against both the fluctuations of the cavity length and magnetic field strength.

Emerging from the magnetic-field-induced dressed-dark state, our proposal can be performed with many of the alkaline-earth-metal-like atoms (e.g., Sr, Mg, Ca, and Yb) and may find extensive applications in frequency reference and precise metrology scenarios such as gravitational wave detection and measurements of fundamental constants (Abramovici et al., 1992; Harry et al., 2006; Graham et al., 2013; Fortier et al., 2007; Rosi et al., 2014).

Acknowledgements.

We thank An-An Yao for helpful discussions. G.D is supported by National Natural Science Foundation of China (NSFC) Grant No. 12205211. Y.Y. is supported by NSFC Grant No. 12175204.

Appendix A The Dynamical Equations for the

three-level model

The dynamical equations for the three-level laser model shown in Fig. 1(b) are

[TABLE]

where c.c. stands for the conjugate complex. We have defined the detunings $\delta\equiv\omega_{c}-\omega_{d,0}$, $\Delta\equiv\omega_{b,0}-\omega_{d,0}$ and the operator $\hat{\sigma}_{nm}\equiv\left|n\right\rangle\left\langle m\right|$ for $n,m=g,b_{0},d_{0}$.

Appendix B The approximate analytical solutions

of the two-level dynamical equations

When taking the quantity $i\Omega_{d}[(\left\langle\hat{\sigma}_{z}\right\rangle+1)/2-\left\langle\hat{\sigma}_{1}^{\dagger}\hat{\sigma}_{2}\right\rangle]/2$ into account, we obtain the high order corrections for the solutions of Eqs. (9)-(12). For example, the dynamical equations for the first-order correction can be cast as

[TABLE]

with the steady-state solution

[TABLE]

Appendix C quantum regression theorem

As we have shown in Sec. III.2, the linewidth and central frequency of the output laser can be characterized by the eigenvalue of the matrix with the smaller real part. The eigen function of the matrix $\mathbb{\mathbb{W}}$ reads (Dong et al., 2023)

[TABLE]

where $w_{j}$ denotes the $j$th eigenvalue of the matrix with the right (left) eigenvector $\left|j\right\rangle$ ($\left\langle\widetilde{j}\right|$). Moreover, the eigenvectors of the matrix $\mathbb{W}$ are orthogonal and normalized to each other as $\left\langle\widetilde{j}|i\right\rangle=\delta_{i,j}$ for $i,j=1,2$. The function for the eigenvalue $w_{j}$ is

[TABLE]

Thus the eigenvalue with the smaller real part is

[TABLE]

In the last step, we have assumed a small detuning.

The linewidth and central frequency of the output laser are

[TABLE]

which is Eq. (23) in the main text. In Fig. 7, we plot the frequency of the output laser which demonstrates the validity of our approximated analytical solutions.