Magnetic Field-controlled Transmission and Ultra-narrow Superradiant Lasing by Dark Atom-light Dressed States
MMagnetic Field-controlled Transmission and Ultra-narrow Superradiant Lasing byDark Atom-light Dressed States
Yuan Zhang,
1, 2, ∗ Chong Xin Shan, and Klaus Mølmer † School of Physics and Microelectronics, Zhengzhou University, Daxue Road 75, Zhengzhou 450052 China Donostia International Physics Center, Paseo Manuel de Lardizabal 4,20018 Donostia-San Sebastian (Gipuzkoa), Spain Department of Physics and Astronomy, Aarhus University,Ny Munkegade 120, DK-8000 Aarhus C, Denmark
Optical lattice clock systems with ultra-cold strontium-88 atoms have been used to demonstratesuperradiant lasing and magnetic field-controlled optical transmission. We explain these phenomenatheoretically with a rigorous model for three-level atoms coupled to a single cavity mode. We identifya class of dark atom-light dressed states which become accessible due to mixing with bright dressedstates in the presence of a magnetic field. We predict that these states, under moderate incoherentpumping, lead to lasing with a linewidth of only tens of Hz, orders of magnitude smaller than thecavity linewidth and the atomic incoherent decay and pumping rates.
Introduction
Conventional lasers rely on opticalcoherence established by stimulated emission froma population-inverted medium and have a spectrallinewidth set by the Schawlow-Townes limit [1]. Incontrast, superradiant lasers rely on coherence in themedium, established by the collective atom-light interac-tion, and have a minimal linewidth given by the Purcellenhanced atomic decay rate Γ c [2–5]. Recent experiments[6] showed that the two lasing mechanisms may co-existin a superradiant crossover regime with, e.g, Sr alkalineearth atoms trapped in a one-dimensional optical latticeinside an optical cavity, see Fig. 1(a). Theoretical stud-ies [7, 8] revealed that lasing in this regime benefits fromboth the optical and atomic coherence and can achieve alinewidth even smaller than Γ c .The same system was employed recently [9] to demon-strate magnetic field-controlled transmission of light,which was explained with a model involving three cou-pled oscillators [9, 10]. In this Letter, we develop a morerigorous description in the spirit of the Jaynes-Cummingsmodel for two-level systems [11], to explain this phe-nomenon with dressed atom-light states, see Fig.1 (c)and (d). Our analysis further reveals that the interplayof bright and dark dressed states is responsible for a newlasing mechanism with ultra-narrow linewidth. Model
Fig. 1 (b) shows the singlet ground state S , the triplet excited state P and a further excitedstate used for incoherent excitation of the atomic system.The spin-forbidden S − P transition becomes allowedin Sr atoms due to the spin-orbit interaction inducedstate-mixing [12]. In the presence of a static magneticfield, the triplet excited state of the k th atom is splitdue to the Zeeman effect into three states denoted as | e ± ,k (cid:105) , | e ,k (cid:105) and the atom emits σ ± , π polarized light atdifferent frequencies. The transitions σ ± both couple tothe single fundamental cavity mode with polarization in ∗ [email protected] † [email protected] g k Single Atom B (a) (b) σ - σ + π | e - ,k > | e + ,k >| e >| g k > S P Δ g √ | >>| B +,0 B -,0 >| D g k Tranmissionor Superradiance | B k >| D k > | g k > Single Atom (c) (d) |0> | G > Weak Excitation Limit /2 Δ k /2 Δ |0>| D > |0>| B > |1>| G > Figure 1. Panel (a) illustrates thousands of Sr atomstrapped in a one-dimensional optical lattice inside an opti-cal cavity. Panel (b) shows the ground state ( S ) and thethree excited states ( P ), which are split due to the mag-netic field. The atomic excitation is pumped coherently toa higher excited state (black solid arrow) and decays rapidlyto the excited states of interest (dashed arrows) and returnsto the ground state by spontaneous emission (wavy arrows)and coherent coupling to the cavity mode (red thin lines) [6].Panel (c) shows the equivalent scheme with one bright andone dark excited state in a single atom. Panel (d) shows thatthree dressed atom-light states are formed due to the Zeemansplitting and the collective atom-light interaction, leading tothree peaks in the transmission spectrum and steady-statesuperradiance. the horizontal direction.To analyze the scenario detailed above, the N atoms are described by the Hamiltonian H a = (cid:80) Nk =1 H k with the single atom contribution H k = (cid:80) s = e + ,e − (cid:126) ω s | s k (cid:105) (cid:104) s k | , where ω e ± = ω a,k ± ∆ k / . Here,the energy of the ground state is set as zero, ω a,k = 2 πc/λ is the frequency of the S − P transition with the wave-length λ = 689 nm, the splitting ∆ k = 2 π × . B MHzis determined by the static magnetic field B in unit ofGauss [9]. The cavity mode is described by the Hamil-tonian H c = (cid:126) ω c a + a with the frequency ω c , and cre- a r X i v : . [ qu a n t - ph ] M a r Z ee m a n Sp li tt i n g M H z (a) B +,0 B -,0 D T r a n s m i ss i o n C a v i t y P h a s e × (b) D B -,0 B +,0 N o r m a li z e d T r a n m i ss i o n (c) Group 1
Group 2 B +,0 Group 3Group 3Group 2 D B -,0 N o r m a li z e d T r a n m i ss i o n (d) Group 3 B +,0 B -,0 Group 3
Figure 2. Magnetic field-controlled transmission spectrum(relative to ω c ) of systems with . × Sr atoms probedwith a weak laser (a,b) and a strong laser (c,d). Panel (a)shows the transmission for different values of Zeeman split-ting ∆ / π . Panel (b) shows the transmission (solid blackcurve) and phase (dashed blue curve) for a given Zeemansplitting ∆ / π = 2 MHz. Panels (c) and (d) show the trans-mission with and without Zeeman splitting ∆ / π = 2 MHz,respectively. The symbols D , B ± , and group 1,2,3 associatethe peaks with the transitions between the dressed states asshown in Fig. 1(d) and Fig. A1 in the Appendix A. Otherparameters are specified in the text. ation and annihilation operators a + and a . The inter-action between the atoms and the cavity mode is de-scribed by the Hamiltonian H a − c = (cid:80) k H a − c,k with H a − c,k = (cid:126) g k ( | e + ,k (cid:105) (cid:104) g k | + | e − ,k (cid:105) (cid:104) g k | ) a + h . c . in the ro-tating wave approximation, where we assume the samecoupling strength g k = 2 π × . kHz for both σ ± transi-tions. We ignore the excited state | e ,k (cid:105) since it does notcouple with the cavity mode of interest.To understand the interaction between a single atomand the cavity mode, we introduce two new ex-cited states | B k (cid:105) = ( | e + ,k (cid:105) + | e − ,k (cid:105) ) / √ , | D k (cid:105) =( | e + ,k (cid:105) − | e − ,k (cid:105) ) / √ . The single atom-cavity mode in-teraction Hamiltonian H a − c,k = (cid:126) √ g k | B k (cid:105) (cid:104) g k | a + h.c. indicates that the bright (dark) states | B k (cid:105) ( | D k (cid:105) )do (do not) couple with the cavity mode. But theyare coupled by the magnetic field-induced Zeeman-splitting via H a,k = (cid:126) ω a ( | B k (cid:105) (cid:104) B k | + | D k (cid:105) (cid:104) D k | ) + (cid:126) (∆ k / (cid:80) k ( | B k (cid:105) (cid:104) D k | + | D k (cid:105) (cid:104) B k | ) , see Fig. 1 (c).To describe the interaction between the entire atomicensemble and the cavity mode, we consider theweak excitation limit and introduce the ground state | G (cid:105) = (cid:81) k | g k (cid:105) and collective singly-excited bright | B (cid:105) = N − / (cid:80) k | B k (cid:105) (cid:81) j (cid:54) = k | g k (cid:105) and dark state | D (cid:105) = N − / (cid:80) k | D k (cid:105) (cid:81) j (cid:54) = k | g k (cid:105) . This allows us to approxi-mate the interaction and atomic Hamiltonian as H a − c ≈ (cid:126) √ N g | B (cid:105) (cid:104) G | a + h.c. , H a ≈ (cid:126) ω a ( | B (cid:105) (cid:104) B | + | D (cid:105) (cid:104) D | ) + (cid:126) (∆ /
2) ( | B (cid:105) (cid:104) D | + | D (cid:105) (cid:104) B | ) . Here, we assume identicalatoms, i.e. ω a = ω a,k , ∆ = ∆ a,k , g = g k for all k . In the spirit of the Jaynes-Cumming model we in-troduce the basis of product states | n (cid:105) | D (cid:105) , | n (cid:105) | B (cid:105) , | n + 1 (cid:105) | G (cid:105) with the photon number states | n (cid:105) ( n =0 , , ... ) and the atom-ensemble states, and decompose H s = H c + H a + H a − c ≈ ⊕ n H ( n ) in this basis with H ( n ) = (cid:126) ω a − ω c ∆ / / ω a − ω c g n g n + (cid:126) ( n + 1) ω c . (1)Here, g n = √ n + 1 √ N g is the coupling strength forgiven photon number n . The Hamiltonian H ( n ) can bediagonalized for all choices of the physical parameters, seethe Appendix A. Here, we focus on the particular casewith ω a = ω c and obtain the atom-light dressed states | D n (cid:105) = N n (2 g n | D (cid:105) | n (cid:105) − ∆ | G (cid:105) | n + 1 (cid:105) )) , (2) | B ± ,n (cid:105) = − ( N n / √ | D (cid:105) | n (cid:105) + 2 g n | G (cid:105) | n + 1 (cid:105) ) ∓ (1 / √ | B (cid:105) | n (cid:105) , (3)with the frequencies ω ,n = ( n + 1) ω c , ω ± ,n =( n + 1) ω c ± (cid:113) g n + (∆ / and the factor N n = [∆ +4 g n )] − / . Magnetic Field-controlled Transmission
We first con-sider the three dressed states | D (cid:105) , | B ± , (cid:105) with lowest fre-quencies, ω , = ω c , ω ± , = ω c ± (cid:112) N g + ∆ / , whichare obtained from Eqs. (2) and (3) by seting n = 0 . Wesee that the dressed states | D (cid:105) , | B ± , (cid:105) depend on theground state | G (cid:105) and the dark atomic state | D (cid:105) , respec-tively, due to the magnetic field-induced Zeeman split-ting ∆ , which can be understood as a mixing of the darkand bright excited states. In addtion, the excitation fre-quency of | D (cid:105) coincides with the bare cavity frequencywhile that of | B ± , (cid:105) differ from the cavity mode, seeFig. 1 (d). Thus, these dressed states can be resonantlyexcited and their photonic components cause the threetransmission peaks [9], see Fig. 2 (a). Further more, theintensities of the center and side peaks are proportionalto the populations ∆ / (∆ + 8 N g ) , N g / (∆ + 8 N g ) of the higher photonic components in the dressed states.Thus, as the Zeeman-splitting ∆ increases, the centerpeak becomes stronger while the side peaks separate andbecome weaker, see Fig. 2 (a). This is precisely what isobserved in the experiment [9]. In addition, when the fre-quency ω d of a probe laser is swept across the resonancefrequencies, ω , , ω ± , , the frequency difference betweenthe probe laser and the dressed states changes sign andthis leads to a phase change in the transmitted signalaround the resonant frequencies as shown in Fig. 2 (b).If we probe the system with a stronger laser, it is pos-sible to excite the dressed states | B ± ,n (cid:105) and | D n (cid:105) withhigher photon number n > , see Fig. A1 in the Ap-pendix A. As a result, we expect more peaks in the trans-mission spectrum, see Fig. 2 (c,d), associated with thenine resonant transitions as indicated in Fig. A1 in theAppendix A. These transitions are among the dark orbright dressed states (the first group), between the darkand bright dressed states (the second group), and be-tween the bright states of different sign (the third group).So far, we restrict the analysis to the states with at mosta single atomic excitation but will relax this restrictionin the following. Master Equation
In the above analysis, we disregardedthe dissipation in the system such as photon loss andspontaneous emission. To model these processes and alsothe atomic incoherent excitation, which is prerequisite forlasing, we establish a master equation (see the AppendixB): ∂∂t ρ = − i (cid:126) [ H a + H c + H a − c + H d , ρ ] − κ D [ a ] ρ − (cid:88) k
12 ( γ + ,k − γ − ,k ) (cid:0) D (cid:2) A kgD , A kBg (cid:3) ρ + D (cid:2) A kBg , A kgD (cid:3) ρ (cid:1) − (cid:88) k
12 ( γ + ,k + γ − ,k ) (cid:0) D (cid:2) A kgB (cid:3) ρ + D (cid:2) A kgD (cid:3) ρ (cid:1) − (cid:88) k
12 ( η + ,k − η − ,k ) (cid:0) D (cid:2) A kDg , A kgB (cid:3) ρ + D (cid:2) A kgB , A kDg (cid:3) ρ (cid:1) − (cid:88) k
12 ( η + ,k + η − ,k ) (cid:0) D (cid:2) A kBg (cid:3) ρ + D (cid:2) A kDg (cid:3) ρ (cid:1) . (4)Here, H d = √ κ (cid:126) Ω( e − iω d t a + + h . c . ) describes the cou-pling between the cavity mode and the probe laser witha strength Ω and a frequency ω d , and √ κ is the trans-mission coefficient of the left mirror. The superoper-ators are defined as D [ o ] ρ = { o + o, ρ } / − oρo + and D [ o, p ] ρ = { po, ρ } / − oρp (with any operator o, p ), de-scribing cavity loss with the rate κ = κ + κ = 2 π × kHz due to the left ( κ ) and right mirror ( κ ), andatomic spontaneous emission with rates γ = γ + ,k = γ − ,k = 2 π × . kHz. Eq.(4) includes also the inco-herent atomic excitation with rates η + ,k , η − ,k , e.g., ob-tained via excitation of higher short-lived excited states,see Fig.1(b). For simplicity, we have introduced the ab-breviations A kst = | s k (cid:105) (cid:104) t k | ( s, t = g, B, D ) and ignoredthe negligible pure atomic dephasing in the optical lat-tice clock system. Note that the dissipation introduces adissipative coupling between the dark and bright atomicstates as captured by terms of the form D [ o, p ] ρ .To simulate systems with thousands of atoms, we uti-lize second-order mean-field theory [8]. In this theory,we derive the equation ∂ (cid:104) o (cid:105) /∂t = tr { o∂ρ/∂t } with Eq.(4) for the expectation value (cid:104) o (cid:105) of any observable o ,see the Appendix B. The equation for the mean pho-ton number in the cavity (cid:104) a + a (cid:105) couples to the atom-photon correlations (cid:10) aA kst (cid:11) which in turn depend on theatom-atom correlations (cid:104) A kst A k (cid:48) s (cid:48) t (cid:48) (cid:105) ( k (cid:54) = k (cid:48) ) and third-order correlations, e.g. (cid:10) a + aA kst (cid:11) . To truncate the hier-archy of equations, we approximate third-order quanti-ties with products of lower-order terms, e.g. (cid:10) a + aA kst (cid:11) = (cid:104) a + a (cid:105) (cid:10) A kst (cid:11) + (cid:104) a + (cid:105) (cid:10) aA kst (cid:11) + (cid:10) a + A kst (cid:11) (cid:104) a (cid:105)− (cid:104) a + (cid:105) (cid:104) a (cid:105) (cid:10) A kst (cid:11) ,resulting in closed non-linear equations, which involvethese quantities and also the photon-photon correlation (cid:104) aa (cid:105) , the cavity field amplitude (cid:104) a (cid:105) , the atomic state pop- ulations (cid:10) A kss (cid:11) and the atomic polarization (cid:10) A kst (cid:11) ( s (cid:54) = t ).Through the atom-atom correlations, the equations cap-ture atomic collective effects and lasing involving multi-ply excited atomic states. Assuming identical propertiesof all atoms, (cid:10) A kst (cid:11) and (cid:10) aA kst (cid:11) are the same for all k , and (cid:104) A kst A k (cid:48) s (cid:48) t (cid:48) (cid:105) are identical for all pairs ( k, k (cid:48) ). This allowsus to reduce the number of independent elements to just and their equations can be readily solved.Fig. 2 shows the calculated transmission spectrumof a system with . × atoms. Here, we illu-minate the system by a Gaussian laser pulse Ω =Ω exp (cid:110) − ( t − τ ) / (cid:2) σ (cid:3)(cid:111) with maximum amplitude Ω , peak time τ and duration σ , and evaluate the trans-mission as the ratio between the Fourier transform of thetime-dependent input strength √ κ Ω and the output am-plitude √ κ |(cid:104) a (cid:105)| (here and in the following, we assume κ = κ ). The results agree with the experiment [9], withthe parameters σ = 26 . ns, τ = 264 . ns, Ω = 10 √ kHz (a,b) or Ω = 400 √ kHz (c,d). C a v i t y P h o t o n Δ / π = , . , . , , M H z S u b r a d i a n c e S u p e rr a d i a n c e S u p e rr a d i a n t L a s i n g (a) Γ c / Experiment N o r m a li z e d Sp e c t r u m (b) Filter Cavity Frequency k Hz - - - -
20 0 30 50 80 100 Δ =0 η / γ =5 Superradiant Lasing -10 -8 -5 -2 0 2 5 8 10Filter Cavity Frequency MHz
SubradianceSuperradiance N o r m a li z e dSp e c t r u m -6 -4 -2 0 2 4 6Filter Cavity Frequency MHz Filter Cavity Frequency H z - - - - (c) Δ /2 π =0.1 MHz η / γ =5 SubradianceSuperradiant LasingSuperradiance η / γ Log ( L i ne w i d t h / π H z ) (d) Δ =0 Δ /2 π =0.1 MHz Γ c /2 π =1.5 kHz Figure 3. Magnetic field-controlled subradiance, superradi-ance and superradiant lasing in systems with . × atoms.Panel (a) shows the intracavity photon number versus theincoherent pumping rate η for increasing Zeeman splitting ∆ . Panels (b) and (c) show the normalized steady-statespectra for different η , without ( ∆ = 0 ) and with Zeemansplitting ∆ / π = 0 . MHz [note that in the upper panel (c)the frequency scale is in Hz]. Panel (d) compares the spec-trum linewidth in the system without (upper curves) and with(lower curves) Zeeman splitting. The black dots and the redcurves are the accurate results and those given by the semi-analytical expression (5), respectively. The vertical and hori-zontal dotted line in (d) indicate the pumping reported in [6]and the Purcell enhanced rate Γ c , respectively. Other param-eters are specified in the text. Magnetic Field-controlled Superradiant Lasing
We nowconsider a cavity that is not driven by a laser but theatoms are incoherently pumped, e.g, through the co-herent excitation of the short-lived higher excited statein Fig.1(b). We consider the continuous pumping η = η + ,k = η − ,k and hence a continuous steady-state releaseof the atomic excitation energy via the cavity mode. Inthe absence of Zeeman-splitting ∆ = 0 , the three-levelatoms behave like two-level systems since the dark atomicexcited state is incoherently populated by the pump butcouples with neither the bright atomic excited state northe cavity mode. As discussed in Ref. [8, 13], this sys-tem undergoes transitions from subradiance to superra-diance and finally to superradiant lasing as the pumpingstrength η overcomes the Purcell enhanced decay rate Γ c = 4 g /κ = 2 π × . kHz and the atomic decay rate γ ,respectively, see the lower curve in Fig. 3 (a) for ∆ = 0 .To calculate the spectrum of the emitted radiation, wemimic the spectral measurement by coupling the cavityoutput to a narrow-linewidth filter cavity, and then calcu-late the photon number in the filter cavity as a functionof its frequency [8], see the Appendix C, D. Fig. 3(b)shows the spectrum for a system with . × atoms inthe absence of Zeeman splitting ∆ = 0 . As the systemis pumped more strongly it undergoes the lasing tran-sition. The two peaks in the spectrum for weak pump-ing can be attributed to transitions between the brightdressed states | B ± , (cid:105) and the ground state in Fig. 1 (d).For stronger pumping, the peaks approach each otherbecause the small incoherent pumping reduces the effec-tive collective coupling J ≤ N/ of the atomic ensemblewith the cavity mode [8, 14, 15], which is shown in Fig.4 (a). There, we visualize the collective atomic excita-tion by treating the bright B k → g k and dark D k → g k transitions as separate two-level transitions and intro-ducing corresponding pseudo Dicke states [17], see theAppendix F. We see that as the incoherent pumping η increases, the bright transition explores the lower rung ofsub-radiant Dicke states due to the incoherent pumpingbut terminates at the states of lowest symmetry due tothe balanced absorption and stimulated emission, whilethe dark state undergoes no stimulated emission processand hence it explores also states with J ≥ M > , seethe leftmost red and green dots in Fig. 4(a), respectively.In the presence of Zeeman splitting ∆ / π = 0 . MHz,the dark atomic excited states couple with the brightstates and thus influence the intra-cavity photon numberand the spectrum. For η > γ , a finite value of ∆ thusmakes the intra-cavity photon number increase by aboutten times, see Fig. 3 (a). In addition, for weak and inter-mediate pumping η , we find a new peak in the spectrumaround the cavity mode frequency, see Fig. 3 (c). Thispeak can be attributed to transitions between the darkdressed state | D (cid:105) and the ground state in Fig. 1 (d).As η increases, the side peaks become weaker and ulti-mately indiscernible while the center peak gets strongerand more narrow. For more results on the influence ofthe Zeeman-splitting on the spectrum, see Fig. A2(a) inthe Appendix E. In this case, the Dicke state evolutionis very similar, see Fig. 4 (a), except that M B < isnegative (note M B > is positive for ∆ = 0 and larger η ). In Fig. 3(d), we compare the spectrum linewidthfor the systems with (upper black dots) and withoutZeeman-splitting ∆ (lower black dots). For ∆ = 0 andweak pumping η < γ the linewidth corresponds to thefrequency difference between the two side peaks. Re-markably, the linewidth for ∆ (cid:54) = 0 decreases to about Hz for stronger pumping η , which is orders of magnitudesmaller than the linewidth of π × kHz for ∆ = 0 ,the atomic decay rate of π × . kHz, the pumping rate π × . kHz, the cavity loss rate of π × kHz andthe Purcell decay rate Γ c = 2 π × . kHz.To understand the ultra-narrow linewidth, we have de-rived a semi-analytical expression: Γ = κ − θ (cid:2) ( γ + 2 η ) (cid:0)(cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11)(cid:1) − ∆Im (cid:10) A kDB (cid:11)(cid:3) θ ( (cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11) ) , (5)with the parameter θ = 8 N g / [( γ + 2 η ) + ∆ ] , see theAppendix E. Fig. 3 (d) shows that Eq. (5) (red curves)reproduces the accurate results (black dots) very well.Eq. (5) indicates that the linewidth depends only on thepopulation inversion (cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11) > between thebright excited and the ground states in the absence ofZeeman-splitting ∆ = 0 , while it depends crucially onthe coherence (cid:10) A kDB (cid:11) between the bright and dark statesfor ∆ (cid:54) = 0 . For the optimal pumping rate η and Zeemansplitting ∆ , we observe an almost perfect cancellation ofthese contributions in the enumerator of Eq. (5), leadingto the ultra-narrow spectrum, see Fig. A2 (d) of theAppendix E. - - / N M / N (a) Bright Dark } } Δ /2 π =0.1 MHz Δ =0 - - - - - - - - Log [ P ho t on N u m be r ] (b) η / γ =0.1 η / γ =0.5 η / γ =5 Figure 4. (a) Evolution of the quantum numbers J s , M s ofpseudo Dicke states | J s , M s (cid:105) associated with the bright s = B (red dots) and dark s = D (green dots) atomic transition asthe pumping strength η increases (arrows) for the systemswithout (left) and with (right) Zeeman-splitting ∆ = 2 π × . MHz. The results are shifted for the sake of clarity andthe dashed lines indicate the boundaries of each set of Dickeladders. (b) Optical emission spectra on logarithmic scale forthe pumping η/γ = 0 . , . , in Fig. 3 (c). Other parametersare specified in the text. For weak pumping, the spectrum is explained by thethree dressed atom-light states formed by the first excitedstate of the atomic ensemble, see Fig. 1(d), and it is con-sistent that for strong pumping, the ultra-narrow spec-trum is due to the dressed atom-light states formed bythe highly excited states of the atomic ensemble. Thus,in principle, we can construct a similar energy diagramas Fig. A1 of the Appendix A with adjacent highly ex-cited states of the atomic ensemble, and expect (1) co-herence between the bright and dark atomic state due tothe Zeeman-splitting induced state-mixing, and (2) con-tinuation of the three spectral peaks for strong pump-ing. Indeed, by displaying the spectra in Fig. 3 (c) ona logarithmic scale, we do observe two side peaks withextremely small intensity outside the ultra-narrow andintense peak, see Fig. 4 (b).The lasing mechanism bears strong resemblance withthe lasing without inversion mechanism (LWI) [18], suchas the formation of dressed states and the absence of pop-ulation inversion. Conventional LWI involves three-levelsystems interacting with a strong driving field, leadingto coherence in the system and dressed states, and witha weak probe field, which is amplified due to transitionsamong the dressed states. In our system, the quantizedcavity field and the Zeeman coupling play roles analogous to the driving field and the probe field in LWI, the darkand bright dressed states are formed and evolve due toboth couplings, and the incoherent pumping drives thepopulation and coherence dynamics and hence the emis-sion by the system, see the Appendix G.
Conclusion
In summary, our theory explains themagnetic field-controlled optical transmission demon-strated in Ref. [9] and predicts more transmission peaksfor stronger probe fields associated with higher energyatom-light dressed states. With incoherently pumpedatoms, we predict steady-state superradiant lasing witha remarkable orders of magnitude reduction of the laserlinewidth in the presence of a magnetic field. A theoryof lasing without inversion incorporating collective andsuperradiant effects may provide further insight in ourresults. We anticipate that the super-narrow lasing mayfind applications in optical atomic clocks with strontiumatom [19, 20] and other atoms like calcium and ytter-bium.This work was supported by the Villum Foundation. [1] A. L. Schawlow and C. H. Townes, Phys. Rev. , 1940(1958).[2] D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, Phys.Rev. Lett. , 163601 (2009).[3] D. Meiser and M. J. Holland, Phys. Rev. A , 033847(2010).[4] M. A. Norcia, M. N. Winchester, J. R. K. Cline, J. K.Thompson, Sci. Adv. , 31601231 (2016)[5] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J.Holland, and J. K. Thompson, Nature , 78 (2012)[6] M. A. Norcia and J. K. Thompson, Phys. Rev. X ,011025 (2016)[7] D. A. Tieri, M. Xu, D. Meiser, J. Cooper, and M. J.Holland, arXiv preprint: 1702.04830 (2017)[8] K. Debnath, Y. Zhang and K. Mølmer, Phys. Rev. ,063837 (2018)[9] M. N. Winchester, M. A. Norcia, J. R. K. Cline, and J.K. Thompson, Phys. Rev. Lett. , 263601 (2017)[10] Z. X. Liu, B. Wang, C. Kong, H. Xiong, and Y. Wu,Appl. Phys. Lett. , 111109 (2018)[11] E.T. Jaynes and F.W. Cummings, Proc. IEEE. , 89(1963)[12] M. M. Boyd, Phd Thesis, High Precision Spectroscopy ofStrontium in an Optical Lattice: Towards a New Stan-dard for Frequency and Time, University of Washington,2007[13] M. Xu, D. A. Tieri, E. C. Fine, J. K. Thompson and M.J. Holland, Phys. Rev. Lett. , 154101 (2014)[14] Y. Zhang, Y. X. Zhang and K. Mølmer, New J. Phys. ,112001 (2018)[15] N. Shammah, S. Ahmed, N. Lambert, et. al., Phys. Rev.A , 063815 (2018)[16] The collective dynamics of an ensemble of N identicaltwo-level atoms can be described by Dicke states | J, M (cid:105) ,where the collective spin quantum number J ≤ N/ de-scribes the symmetry and collective coupling of the statesto the quantized field, and − J ≤ M ≤ J describes theexcitation of the system. As shown in [14, 15] and in Fig. 4(a), for η < γ the incoherent pumping mainly causesreduction of the Dicke state quantum number J whilefor η > γ , it mainly increases J and hence the effectivecoupling strength.[17] R. H. Dicke, Phys. Rev. 93, 99 (1954)[18] J. Mompart and R. Corbalá n, J. Opt. B: Quantum Semi-class. Opt. 2 R7-R24 (2000).[19] S. A. Schäffer, M. Tang, M. R. Henriksen, A. A.Jørgensen, B. T. R. Christensen, J. W. Thomsen,arXiv:1903.12593v1[20] C.-C. Chen, S. Bennetts, R. G. Escudero, B. Pasquiou,and F. Schreck Phys. Rev. Applied , 044014 (2019)[21] Y. Zhu, Phys, Rev. A , R6149 (1992)[22] G. Yang, Z. Tan, B Zou, Y. Zhu, Opt. Lett. , 6695-6698(2014) Appendix A: Hamiltonian Diagonalization
In this section, we diagonalize the system Hamil-tonian H s = H a + H c + H a − c in the basis ofdark and bright atomic states. To facilitate thederivation, we remind the reader about the atomicHamiltonian H a = (cid:126) ω a (cid:80) Nk =1 ( | B k (cid:105) (cid:104) B k | + | D k (cid:105) (cid:104) D k | ) + (cid:126) (∆ / (cid:80) k ( | B k (cid:105) (cid:104) D k | + | D k (cid:105) (cid:104) B k | ) and the atom-cavitymode interaction H a − c = (cid:126) √ g (cid:80) k | B k (cid:105) (cid:104) g k | a + h.c. .Here, we assume that all the atoms are identical.In the weak excitation limit, we can introduce theground state | G (cid:105) = (cid:81) k | g k (cid:105) and the bright and dark col-lective excited state | B (cid:105) = (1 / √ N ) (cid:80) k | B k (cid:105) (cid:81) j (cid:54) = k | g k (cid:105) , | D (cid:105) = (1 / √ N ) (cid:80) k | D k (cid:105) (cid:81) j (cid:54) = k | g k (cid:105) of the atomicensemble, and approximate the atomic ensembleHamiltonian as H a ≈ (cid:126) ω a ( | B (cid:105) (cid:104) B | + | D (cid:105) (cid:104) D | ) + (cid:126) (∆ /
2) ( | B (cid:105) (cid:104) D | + | D (cid:105) (cid:104) B | ) and the atom-cavity mode in-teraction Hamiltonian as H a − c ≈ (cid:126) √ N g | B (cid:105) (cid:104) G | a + h.c. .Following the spirit of the Jaynes-Cumming model fortwo-level atoms [11], we introduce the atom-photon prod-uct states | n (cid:105) | D (cid:105) , | n (cid:105) | B (cid:105) , | n + 1 (cid:105) | G (cid:105) and expand theapproximated Hamiltonian H s = ⊕ n H n in the basis ofthese states. For any given photon number n , H n isgiven by Eq. (1) in the main text. By diagonalizingthis sub-Hamiltonian, we obtain three atom-light (pho-ton) dressed states | D n (cid:105) = N n [2 (cid:0) g n + ω ac δ ,n − δ ,n (cid:1) | D (cid:105) | n (cid:105)− δ ,n ∆ | B (cid:105) | n (cid:105) − g n ∆ | G (cid:105) | n + 1 (cid:105) )] , (A1) | B ± ,n (cid:105) = N n [2 (cid:0) g n + ω ac δ ± ,n − δ ± ,n (cid:1) | D (cid:105) | n (cid:105)− δ ± ,n ∆ | B (cid:105) | n (cid:105) − g n ∆ | G (cid:105) | n + 1 (cid:105) ] , (A2)with the frequencies ω ,n = ( n + 1) ω c + δ , ω ± ,n =( n + 1) ω c + δ ± ,n . Here, δ + ,n , δ ,n , δ − ,n are the frequencyshifts relative to the frequency ( n + 1) ω c of the photonnumber states | n + 1 (cid:105) and the normalization factors aredefined as N n = [4 (cid:0) g n + ω ac δ ± ,n − δ ± ,n (cid:1) + ( δ ± ,n ∆) +( g n ∆) ] − / . Here, the label + , , − sorts the shifts ina descending order and we introduce the abbreviations ω ac = ω a − ω c and g n = √ n + 1 √ N g . These shifts arethe solutions of the equation g n ω ac − (cid:104) g n + (∆ / − ω ac (cid:105) δ − ω ac δ + 4 δ = 0 , (A3)and can be written as δ + ,n = 16 γ (cid:0) (cid:15) + 4 ω ac γ + γ (cid:1) , (A4) δ ,n = 112 (cid:18) ω ec − (cid:16) − √ i (cid:17) (cid:15)γ − (cid:16) √ i (cid:17) γ (cid:19) , (A5) δ − ,n = 112 (cid:18) ω ec − (cid:16) − √ i (cid:17) γ − (cid:16) √ i (cid:17) (cid:15)γ (cid:19) , (A6) with the abbreviations (cid:15) = 12 g n + 3∆ + 4 ω ac , γ = (cid:114) α + 3 (cid:113) − g n + ∆ ) + 24 βω ac − ω ac , β = − g n − g n ∆ + ∆ , α = − ω ac (cid:0) g n − + 4 ω ac (cid:1) .Although there are complex numbers in the above ex-pressions, the shifts are always real. For the particularcase with ω c = ω a , the above results are reduced to Eqs.(2) and (3) in the main text. In Fig. A1, we show theformation of the dressed states and possible transitionsbetween them. δ δ δ n δ n δ n + δ n + |0> | G > Strong Excitation: Group 1Weak Excitation g Group 3Group 2 √ n2N |0> | G > g √ Δ /2 Δ /2 √ (n+1)2N g Δ /2 | >>| B +,0 B -,0 >| D | >>| B +,n-1 B -,n-1 >| D n-1 | >>| B +,n B -,n >| D n | >>| B +,n B -,n >| D n | >>| B +,n B -,n >| D n | >>| B +,n-1 B -,n-1 >| D n-1 | >>| B +,n-1 B -,n-1 >| D n-1 |0>| D > |0>| B > |1>| G >| n − D > | n − B > | n >| G >| n >| D > | n >| B > | n + G > Figure A1. Energy diagram. The left part shows collectiveatom-photon product states and their interaction through theZeeman-splitting ∆ / and the collective atom-cavity modecoupling √ n Ng . The right part shows three groups of tran-sitions (vertical lines) among the atom-light dressed states. Appendix B: Second-order Mean-field Equations
In this section, we show how to solve the master equa-tion with second-order mean-field theory. Firstly, we re-capture the master equation ∂∂t ρ = − i (cid:126) [ H a + H c + H a − c + H d , ρ ] − κ D [ a ] ρ − (cid:88) k γ + ,k D (cid:2) A kg + (cid:3) ρ − (cid:88) k γ − ,k D (cid:2) A kg − (cid:3) ρ − (cid:88) k η + ,k D (cid:2) A k + g (cid:3) ρ − (cid:88) k η − ,k D (cid:2) A k − g (cid:3) ρ. (B1)The Hamiltonians H a , H c , H a − c , H d have been intro-duced in the main text. The Lindblad terms describe thecavity photon loss with the rate κ , the spontaneous emis-sion of the atomic excited states | e ± ,k (cid:105) with the rates γ ± ,k and the pumping of these states with the rates η ± ,k . Thesuperoperator is defined as D [ o ] ρ = { o + o, ρ } / − oρo + (with any operator o ). For simplicity, we introduce theabbreviations A kst = | s k (cid:105) (cid:104) t k | .Using the expansion | e + ,k (cid:105) = (1 / √
2) ( | B k (cid:105) + | D k (cid:105) ) , | e − ,k (cid:105) = (1 / √
2) ( | B k (cid:105) − | D k (cid:105) ) , we can transform themaster equation (4) to the equation (4) in the maintext. For simplicity, we introduce the abbreviations Γ ± ,k = ( γ + ,k ± γ − ,k ) , Λ ± ,k = (cid:0) η + ,k ± η − ,k (cid:1) .From the master equation (4) in the main text, wecan derive the equation ∂ (cid:104) o (cid:105) /∂t = tr { o∂ρ/∂t } for theexpectation value (cid:104) o (cid:105) of any observable o . Following this,we derive the equation for the photon number ∂∂t (cid:10) a + a (cid:11) = − κ (cid:10) a + a (cid:11) − √ κ Ω2Im e iω d t (cid:104) a (cid:105)− (cid:88) k √ g k Im (cid:10) aA kBg (cid:11) , (B2)which depends on the cavity mode amplitude (cid:104) a (cid:105) and theatom-photon correlation (cid:10) aA kBg (cid:11) . In the same way, wederive the equation ∂∂t (cid:104) a (cid:105) = − ( iω c + κ/ (cid:104) a (cid:105)− i √ κ Ω e − iω d t − i (cid:88) k √ g k (cid:10) A kgB (cid:11) , (B3)which depends on the polarization of the atoms (cid:10) A kgB (cid:11) .In general, the equation for (cid:10) A kst (cid:11) reads ∂∂t (cid:10) A kst (cid:11) = − iω ka (cid:88) r = B,D (cid:0) δ t,r (cid:10) A ksr (cid:11) − δ s,r (cid:10) A krt (cid:11)(cid:1) − i (∆ k / (cid:88) r (cid:0) δ t,r (cid:10) A ks ¯ r (cid:11) − δ s, ¯ r (cid:10) A krt (cid:11)(cid:1) − i √ g k (cid:16) δ t,B (cid:10) aA ksg (cid:11) − δ s,g (cid:10) aA kBt (cid:11) + δ t,g (cid:10) a + A ksB (cid:11) − δ s,B (cid:10) a + A kgt (cid:11)(cid:17) − Λ + ,k (cid:16) δ s,g (cid:10) A kgt (cid:11) + δ t,g (cid:10) A ksg (cid:11) − (cid:88) r δ s,r δ t,r (cid:10) A kgg (cid:11)(cid:17) + Λ − ,k (cid:88) r δ s,r δ t, ¯ r (cid:10) A kgg (cid:11) − Γ + ,k (cid:88) r (cid:104) (cid:0) δ s,r (cid:10) A krt (cid:11) + δ t,r (cid:10) A ksr (cid:11)(cid:1) − δ s,g δ t,g (cid:10) A krr (cid:11)(cid:105) − Γ − ,k (cid:88) r (cid:104) (cid:0) δ s ¯ r (cid:10) A krt (cid:11) + δ t,r (cid:10) A ks ¯ r (cid:11)(cid:1) − δ s,g δ t,g (cid:10) A kr ¯ r (cid:11)(cid:105) . (B4)In the above equation, ¯ r denotes B, D when r labels D, B ,respectively. In addition, we also need the equation for the atom-photon correlations ∂∂t (cid:10) aA kst (cid:11) = − (cid:18) iω c + 12 κ (cid:19) (cid:10) aA kst (cid:11) − i √ κ Ω e − iω d t (cid:10) A kst (cid:11) − iω ka (cid:88) r = B,D (cid:0) δ t,r (cid:10) aA ksr (cid:11) − δ s,r (cid:10) aA krt (cid:11)(cid:1) − i (∆ k / (cid:88) r (cid:0) δ t,r (cid:10) aA ks ¯ r (cid:11) − δ s, ¯ r (cid:10) aA krt (cid:11)(cid:1) − i √ g k (cid:104) δ t,B (cid:10) aaA ksg (cid:11) − δ s,g (cid:10) aaA kBt (cid:11) − δ s,B (cid:10) a + aA kgt (cid:11) + δ t,g (cid:0)(cid:10) a + aA ksB (cid:11) + (cid:10) A ksB (cid:11)(cid:1)(cid:105) − i (cid:88) k (cid:48) (cid:54) = k √ g k (cid:48) (cid:68) A k (cid:48) gB A kst (cid:69) − Λ + ,k (cid:16) δ s,g (cid:10) aA kgt (cid:11) + δ t,g (cid:10) aA ksg (cid:11) − (cid:88) r δ s,r δ t,r (cid:10) aA kgg (cid:11)(cid:17) + Λ − ,k (cid:88) r δ s,r δ t, ¯ r (cid:10) aA kgg (cid:11) − Γ + ,k (cid:88) r (cid:104) (cid:0) δ s,r (cid:10) aA krt (cid:11) + δ t,r (cid:10) aA ksr (cid:11)(cid:1) − δ s,g δ t,g (cid:10) aA krr (cid:11)(cid:105) − Γ − ,k (cid:88) r (cid:104) (cid:0) δ s ¯ r (cid:10) aA krt (cid:11) + δ t,r (cid:10) aA ks ¯ r (cid:11)(cid:1) − δ s,g δ t,g (cid:10) aA kr ¯ r (cid:11)(cid:105) . (B5)In the above equations, we encounter the expectationvalues of three operators, e.g. (cid:10) aaA kBt (cid:11) . If we derivethe equations for these quantities, we will encounter theexpectation values of four operators and so on, whichcreates a hierarchy of equations. To truncate this hier-archy, we apply the third order cumulant expansion toapproximate the expectation values of three operators,e.g. (cid:10) aaA kBt (cid:11) = (cid:104) a (cid:105) (cid:10) aA kBt (cid:11) + (cid:104) a (cid:105) (cid:10) aA kBt (cid:11) + (cid:10) A kBt (cid:11) (cid:104) aa (cid:105) − (cid:104) a (cid:105) (cid:104) a (cid:105) (cid:10) A kBt (cid:11) . By doing so, we also need the equationfor the photon-photon correlation (cid:104) aa (cid:105) : ∂∂t (cid:104) aa (cid:105) = − ( i ω c + κ ) (cid:104) aa (cid:105)− i √ κ Ω e − iω d t (cid:104) a (cid:105) − i (cid:88) k √ g k (cid:10) aA kgB (cid:11) . (B6)Eq.(11) also depends on the atom-atom correlations (cid:10) A kst A ks (cid:48) t (cid:48) (cid:11) . The equation for these correlations is givenby the rather lengthy Eq. (B7).As we assume identical atoms , (cid:10) A kst (cid:11) , (cid:10) aA kst (cid:11) are iden-tical for any atom k and (cid:104) A kst A k (cid:48) s (cid:48) t (cid:48) (cid:105) are identical for anyatom pair k, k (cid:48) . As a result, we can reduce the numberof independent variables to × ,where sums (cid:80) k , (cid:80) k (cid:48) (cid:54) = k are replaced with single termsmultiplied by N and N ( N − in the equations. Appendix C: Spectrum Computation with a FilterCavity
To calculate the lasing spectrum, we introduce a fil-ter cavity and couple it to the main system by supple-menting the master equation (4) in the main text with ∂∂t (cid:68) A kst A k (cid:48) s (cid:48) t (cid:48) (cid:69) = − iω ka (cid:88) r (cid:16) δ t,r (cid:68) A ksr A k (cid:48) s (cid:48) t (cid:48) (cid:69) − δ s,r (cid:68) A krt A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) − i (∆ k / (cid:88) r (cid:16) δ t,r (cid:68) A ks ¯ r A k (cid:48) s (cid:48) t (cid:48) (cid:69) − δ s, ¯ r (cid:68) A krt A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) − i √ g k (cid:16) δ tB (cid:68) aA ksg A k (cid:48) s (cid:48) t (cid:48) (cid:69) − δ sg (cid:68) aA kBt A k (cid:48) s (cid:48) t (cid:48) (cid:69) + δ t,g (cid:68) a + A ksB A k (cid:48) s (cid:48) t (cid:48) (cid:69) − δ s,B (cid:68) a + A kgt A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) − Λ + ,k (cid:16) δ s,g (cid:68) A kgt A k (cid:48) s (cid:48) t (cid:48) (cid:69) + δ t,g (cid:68) A ksg A k (cid:48) s (cid:48) t (cid:48) (cid:69) − (cid:88) r δ s,r δ t,r (cid:68) A kgg A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) + Λ − ,k ( δ s,B δ t,D + δ s,D δ tB ) (cid:68) A kgg A k (cid:48) s (cid:48) t (cid:48) (cid:69) − Γ + ,k (cid:88) r (cid:20) (cid:16) δ s,r (cid:68) A krt A k (cid:48) s (cid:48) t (cid:48) (cid:69) + δ t,r (cid:68) A ksr A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) − δ s,g δ t,g (cid:68) A krr A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:21) − Γ − ,k (cid:88) r (cid:20) (cid:16) δ s, ¯ r (cid:68) A krt A k (cid:48) s (cid:48) t (cid:48) (cid:69) + δ t,r (cid:68) A ks ¯ r A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) − δ s,g δ t,g (cid:68) A kr ¯ r A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:21) − iω k (cid:48) a (cid:88) r (cid:16) δ t (cid:48) ,r (cid:68) A kst A k (cid:48) s (cid:48) t (cid:48) (cid:69) − δ s (cid:48) ,r (cid:68) A kst A k (cid:48) s (cid:48) t (cid:48) (cid:69)(cid:17) − i (∆ k (cid:48) / (cid:88) r (cid:16) δ t (cid:48) ,r (cid:68) A kst A k (cid:48) s (cid:48) ¯ r (cid:69) − δ s (cid:48) , ¯ r (cid:68) A kst A k (cid:48) rt (cid:48) (cid:69)(cid:17) − i √ g k (cid:48) (cid:16) δ t (cid:48) ,B (cid:68) aA kst A k (cid:48) s (cid:48) g (cid:69) − δ s (cid:48) ,g (cid:68) aA kst A k (cid:48) Bt (cid:48) (cid:69) + δ t (cid:48) ,g (cid:68) a + A kst A k (cid:48) s (cid:48) B (cid:69) − δ s (cid:48) ,B (cid:68) a + A kst A k (cid:48) gt (cid:48) (cid:69)(cid:17) − Λ + ,k (cid:48) (cid:16) δ s (cid:48) ,g (cid:68) A kst A k (cid:48) gt (cid:48) (cid:69) + δ t (cid:48) ,g (cid:68) A kst A k (cid:48) s (cid:48) g (cid:69) − (cid:88) r δ s (cid:48) ,r δ t (cid:48) ,r (cid:68) A kst A k (cid:48) gg (cid:69)(cid:17) + Λ − ,k (cid:48) ( δ s (cid:48) ,B δ t (cid:48) ,D + δ s (cid:48) ,D δ t (cid:48) B ) (cid:68) A kst A k (cid:48) gg (cid:69) − Γ + ,k (cid:48) (cid:88) r (cid:20) (cid:16) δ s (cid:48) ,r (cid:68) A kst A k (cid:48) rt (cid:48) (cid:69) + δ t (cid:48)(cid:48) ,r (cid:68) A kst A k (cid:48) s (cid:48) r (cid:69)(cid:17) − δ s (cid:48) ,g δ t (cid:48) ,g (cid:68) A kst A k (cid:48) rr (cid:69)(cid:21) − Γ − ,k (cid:48) (cid:88) r (cid:20) (cid:16) δ s (cid:48) , ¯ r (cid:68) A kst A k (cid:48) rt (cid:48) (cid:69) + δ t (cid:48) ,r (cid:68) A kst A k (cid:48) s (cid:48) ¯ r (cid:69)(cid:17) − δ s (cid:48) ,g δ t (cid:48) ,g (cid:68) A kst A k (cid:48) r ¯ r (cid:69)(cid:21) . (B7) the terms (cid:0) ∂∂t ρ (cid:1) m = − ( i/ (cid:126) ) [ H f + H f − c , ρ ] − χ D [ b ] ρ [8].The filter cavity Hamiltonian H f = (cid:126) ω f b + b is specifiedby a frequency ω f , the creation b + and annihilation op-erator b of photons. The filter cavity-system interaction H f − c = (cid:126) β ( b + a + a + b ) is specified with the couplingstrength β , and the Lindblad term describes photon lossin the filter cavity with a rate χ .To calculate the spectrum, we consider the equationfor the mean photon number (cid:104) b + b (cid:105) in the filter cavity ∂∂t (cid:10) b + b (cid:11) = − χ (cid:10) b + b (cid:11) + β (cid:10) b + a (cid:11) , (C1)and the field amplitude (cid:104) b (cid:105) in the filter cavity ∂∂t (cid:104) b (cid:105) = − ( iω f + χ/ (cid:104) b (cid:105) − iβ (cid:104) a (cid:105) . (C2)The equation (C1) depends on the photon-photon cor-relation between the main and filter cavity (cid:104) b + a (cid:105) , whichfollows the equation ∂∂t (cid:10) b + a (cid:11) = [ i ( ω f − ω c ) − ( χ + κ ) / (cid:10) b + a (cid:11) − iβ (cid:0)(cid:10) b + b (cid:11) − (cid:10) a + a (cid:11)(cid:1) − i (cid:88) k √ g k (cid:10) b + A kgB (cid:11) . (C3)This equation depends on the photon-atom correlation (cid:10) b + A kgB (cid:11) , which follows the equation ∂∂t (cid:10) b + A kgr (cid:11) = − i √ g k (cid:0) δ r,B (cid:10) b + aA kgg (cid:11) − (cid:10) b + aA kBr (cid:11)(cid:1) + (cid:2) i (cid:0) ω f − ω ka (cid:1) − χ/ − Λ + ,k − Γ + ,k / (cid:3) (cid:10) b + A kgr (cid:11) + iβ (cid:10) a + A kgr (cid:11) − ( i ∆ k / − ,k / (cid:10) b + A kg ¯ r (cid:11) . (C4) In the above equation, r = B, D indicates the quanti-ties related to the bright and dark atomic excited state.These quantities couple with each other through the Zee-man splitting ∆ k and through the incoherent pumping Λ − ,k = ( η + ,k − η − ,k ) / . Notice that the latter termexists only if the incoherent pumping η ± ,k are differentfor the two extreme excited states | e ± ,k (cid:105) . In addition,the above equation depends on the expectation values ofthree operators, e.g. (cid:10) b + aA kgg (cid:11) , and we apply the thirdorder cumulant expansion to approximate these valueswith low-order quantities.To reduce the backaction of the filter cavity on themain system, we shall assume a small value for β . Weshall also assume that χ is smaller than the linewidth ofthe spectrum that we want to measure. Under these as-sumptions, we can first determine the steady-state expec-tation values of the system observables, and then obtainthe filter cavity correlations and mean values for differentfilter-cavity frequencies ω f . Appendix D: Systems without Atomic and FieldCoherence
In the previous sections, we outlined the equations forrather general systems. However, if there is no initialcoherence in the field or the atoms and the system is alsonot driven by the probe laser, the cavity field amplitude (cid:104) a (cid:105) , (cid:104) b (cid:105) and the polarization (cid:10) A kgr (cid:11) vanish at all times.In this case, we can neglect all these quantities in allthe equations to get the following small set of simplifiedequations.The equation for the photon number can be now writ-ten as ∂∂t (cid:10) a + a (cid:11) = − κ (cid:10) a + a (cid:11) − (cid:88) k √ g k Im (cid:10) aA kBg (cid:11) . (D1)The equation for the atom-photon correlation becomes ∂∂t (cid:10) aA krg (cid:11) = (cid:2) i (cid:0) ω ka − ω c (cid:1) − κ/ − Λ + ,k − Γ + ,k / (cid:3) (cid:10) aA krg (cid:11) + ( i ∆ k / − Γ − ,k / (cid:10) aA k ¯ rg (cid:11) − i (cid:88) k (cid:48) (cid:54) = k √ g k (cid:48) (cid:68) A k (cid:48) gB A krg (cid:69) + i √ g k (cid:2) δ r,B (cid:10) a + a (cid:11) (cid:10) A kgg (cid:11) + (cid:0)(cid:10) a + a (cid:11) + 1 (cid:1) (cid:10) A krB (cid:11)(cid:3) . (D2)The equation for the atom-atom correlation becomes ∂∂t (cid:68) A kgr A k (cid:48) r (cid:48) g (cid:69) = (cid:104) i (cid:16) ω k (cid:48) a − ω ka (cid:17) − Λ + ,k − Γ + ,k / − Λ + ,k (cid:48) − Λ + ,k (cid:48) / (cid:105) (cid:68) A kgr A k (cid:48) r (cid:48) g (cid:69) − ( i ∆ k / − ,k / (cid:68) A kg ¯ r A k (cid:48) r (cid:48) g (cid:69) + ( i ∆ k (cid:48) / − Γ − ,k (cid:48) / (cid:68) A kgr A k (cid:48) ¯ r (cid:48) g (cid:69) − i √ g k (cid:16) δ r,B (cid:10) A kgg (cid:11) − (cid:10) A kBr (cid:11)(cid:17) (cid:68) aA k (cid:48) r (cid:48) g (cid:69) − i √ g k (cid:48) (cid:10) aA krg (cid:11) ∗ (cid:16)(cid:68) A k (cid:48) r (cid:48) B (cid:69) − δ r (cid:48) ,B (cid:68) A k (cid:48) gg (cid:69)(cid:17) . (D3)The equation for the population and polarization ofatoms become ∂∂t (cid:10) A krr (cid:48) (cid:11) = − ( i ∆ k / − ,k / (cid:10) A kr ¯ r (cid:48) (cid:11) + ( i ∆ k / − Γ − ,k / (cid:10) A k ¯ rr (cid:48) (cid:11) − Γ + ,k (cid:10) A krr (cid:48) (cid:11) − i √ g k (cid:16) δ r (cid:48) ,B (cid:10) aA krg (cid:11) − δ r,B (cid:10) aA kr (cid:48) g (cid:11) ∗ (cid:17) + Λ + ,k δ r (cid:48) ,r (cid:10) A kgg (cid:11) + Λ − ,k δ r (cid:48) , ¯ r (cid:10) A kgg (cid:11) , (D4) ∂∂t (cid:10) A kgg (cid:11) = − √ g k Im (cid:10) aA kBg (cid:11) + 2Γ − ,k Re (cid:10) A kBD (cid:11) − + ,k (cid:10) A kgg (cid:11) + Γ + ,k (cid:0)(cid:10) A kBB (cid:11) + (cid:10) A kDD (cid:11)(cid:1) . (D5)To calculate the spectrum, we first solve the aboveequations in the steady-state and then utilize the resultsas the input parameters for the following simplified equa-tions for the filter cavity-related quantities: ∂∂t (cid:10) b + b (cid:11) = − χ (cid:10) b + b (cid:11) + β (cid:10) b + a (cid:11) , (D6) ∂∂t (cid:10) b + a (cid:11) = [ i ( ω f − ω c ) − ( χ + κ ) / (cid:10) b + a (cid:11) − iβ (cid:0)(cid:10) b + b (cid:11) − (cid:10) a + a (cid:11)(cid:1) − i (cid:88) k √ g k (cid:10) b + A kgB (cid:11) , (D7) ∂∂t (cid:10) b + A kgr (cid:11) = − i √ g k (cid:10) b + a (cid:11) (cid:0) δ r,B (cid:10) A kgg (cid:11) − (cid:10) A kBr (cid:11)(cid:1) + (cid:2) i (cid:0) ω f − ω ka (cid:1) − χ/ − Λ + ,k − Γ + ,k / (cid:3) (cid:10) b + A kgr (cid:11) + iβ (cid:10) a + A kgr (cid:11) − ( i ∆ k / − ,k / (cid:10) b + A kg ¯ r (cid:11) . (D8)In the last equation, we have utilized (cid:10) b + aA kBB (cid:11) = (cid:104) b + a (cid:105) (cid:10) A kBB (cid:11) and (cid:10) b + aA kgg (cid:11) = (cid:104) b + a (cid:105) (cid:10) A kgg (cid:11) .For N identical atoms, we expect that (cid:10) A kgg (cid:11) , (cid:10) aA krg (cid:11) , (cid:10) b + A kgr (cid:11) , (cid:10) A krr (cid:48) (cid:11) are identical for different atom k and (cid:68) A k (cid:48) gr (cid:48) A krg (cid:69) are identical for different atom pair k (cid:48) , k . Asa result, the number of independent elements is just . Appendix E: Semi-analytical Expression ofSpectrum Linewidth
In this section, we derive a semi-analytical expressionfor the steady-state spectrum linewidth. To proceed, weconsider the steady-state version of Eq. (D6) and (D8)with g = g k , ω a = ω ka , ∆ = ∆ k , Λ ± = Λ ± ,k , Γ ± = Γ ± ,k for all k : (cid:10) b + b (cid:11) = − i ( β/χ ) (cid:0)(cid:10) b + a (cid:11) − (cid:10) ba + (cid:11)(cid:1) , (E1) ξ (cid:10) b + A kgB (cid:11) − i
12 ( i ∆ + Γ − ) (cid:10) b + A kgD (cid:11) = β (cid:10) a + A kgB (cid:11) + √ g (cid:0)(cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11)(cid:1) (cid:10) b + a (cid:11) , (E2) ξ (cid:10) b + A kgD (cid:11) − i
12 ( i ∆ + Γ − ) (cid:10) b + A gB (cid:11) = β (cid:10) a + A kgD (cid:11) + √ g (cid:10) A kBD (cid:11) (cid:10) b + a (cid:11) . (E3)Here, we have introduced ξ = ω a − ω f − i ( χ/ + + Γ + / . We solve Eqs. (E2) and (E3) andobtain (cid:10) b + A kgB (cid:11) = βε + ε (cid:10) b + a (cid:11) (E4)with the abbreviations ε = ε [ ξ (cid:10) a + A kgB (cid:11) + i ( i ∆ + Γ − ) (cid:10) a + A kgD (cid:11) ] , ε = ε √ g [ ξ ( (cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11) ) + i ( i ∆ + Γ − ) (cid:10) A kBD (cid:11) ] and ε − = ξ + ( i ∆ + Γ − ) .Then, we consider the steady-state version of Eq. (D7)and its complex conjugate: i ( ω f − ω c ) − ( χ + κ ) / (cid:10) b + a (cid:11) − iβ (cid:0)(cid:10) b + b (cid:11) − (cid:10) a + a (cid:11)(cid:1) − iN √ g (cid:10) b + A kgB (cid:11) , (E5) − [ i ( ω f − ω c ) + ( χ + κ ) / (cid:10) ba + (cid:11) + iβ (cid:0)(cid:10) b + b (cid:11) − (cid:10) a + a (cid:11)(cid:1) + iN √ g (cid:10) b + A kgB (cid:11) ∗ . (E6)Inserting Eqs. (E1) and (E4) into the above equations,we obtain the coupled equations (cid:20) τ ∗ − β /χ − β /χ τ (cid:21) (cid:20) (cid:104) b + a (cid:105)(cid:104) ba + (cid:105) (cid:21) = iβ (cid:20) ν ∗ − ν (cid:21) (E7)0with the abbreviations τ = i ( ω f − ω c − N √ gε ∗ ) +( χ + κ ) / β /χ and ν = (cid:104) a + a (cid:105) − N √ gε ∗ . The so-lution is (cid:10) b + a (cid:11) = iβ (cid:0) ν ∗ τ − νβ /χ (cid:1) τ τ ∗ − β /χ , (E8) (cid:10) ba + (cid:11) = iβ (cid:0) ν ∗ β /χ − τ ∗ ν (cid:1) τ τ ∗ − β /χ , (E9)and Eq. (E1) yields (cid:10) b + b (cid:11) = β χ ν ∗ τ + τ ∗ ν − ( ν + ν ∗ ) β /χτ τ ∗ − β /χ . (E10)To compute the spectrum, we require that β is smalland thus the above expression can be simplified as (cid:10) b + b (cid:11) ≈ β χ ντ . (E11)This expression suggests that the linewidth of the spec-trum is mainly determined by τ . Ignoring the terms pro-portional to β again, we can approximate this term as τ ≈ i ( ω f − ω c − Z ) + κ/ (E12)with the abbreviation Z = N g ( ξ ∗ ) + ( i ∆ − Γ − ) / ξ ∗ (cid:0)(cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11)(cid:1) − i
12 ( − i ∆ + Λ − ) (cid:10) A kDB (cid:11) ] , (E13)and the parameter ξ ≈ ω a − ω f − i (Λ + + Γ + / .To proceed further, we consider the resonant condition ω f ≈ ω ka + i Γ / ≈ ω c + i Γ / (with the linewidth Γ to bedetermined) and Λ − = 0 (i.e. the balanced pumping η + ,k = η − ,k ), which allows us to simplify Eq. (E13) as Z = − i N g (Γ / + + Γ + / + (∆) / i (∆ / (cid:10) A kDB (cid:11) + (Γ / + + Γ + / (cid:0)(cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11)(cid:1) ] . (E14)Inserting the above expression to Eq. (E12), we establishthe relation Re τ ≈ − Γ / Z + κ/ . Equating thisexpression to Γ / , we obtain an equation for Γ , whichcan be solved numerically. If the spectrum linewidth issmall, i.e. Γ < Λ + , ∆ , we can simplify Eq. (E14) byignoring the dependence of the denominator on Γ . Inthis case, we can achieve the following semi-analyticalexpression for the linewidth Γ = (cid:2) θ ( (cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11) ) / (cid:3) − { κ/ − θ [(Λ + + Γ + / (cid:0)(cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11)(cid:1) − (∆ / (cid:10) A kDB (cid:11) ] } (E15)with the abbreviation θ = 2 N g / [(Λ + + Γ + / + ∆ / ,which can be rewritten as Eq. (5) in the main text. Theexpression (E15) indicates that the linewidth is deter-mined by not only the population inversion of the bright N o r m a li z e dSp e c t r u m -0.02 -0.01 0.00 0.01 0.02Filter Cavity Frequency kHz η / γ =5 Δ /2 π =0.1,0.3,0.5,0.7,0.9,1.1 MHz -4.00 -2.00 0.00 2.00 4.00Filter Cavity Frequency MHz η / γ =0.1 Δ /2 π = (a) η / γ P opu l a t i on (b) k kk - - - - - η / γ Log (| Λ + + Γ + / )(< A BB >-< A gg >|) ( c ) < > ∆ =0 k k k k kk - - - η / γ Log (-( Λ + + Γ + / )(< A BB >-< A gg >) Log (- Δ / I m < A D B >) (d) ∆ /2 π =0.1 MHz kk k Figure A2. Supplemental results for systems with . × atoms under incoherent pumping. Panel (a) shows the spec-tra with increasing Zeeman splitting ∆ for weak pumping η = 0 . γ (lower part) and strong pumping η = 5 γ (upperpart). Panel (b) shows the populations (cid:10) A kgg (cid:11) (black solidcurve), (cid:10) A kBB (cid:11) (red dashed curve) and (cid:10) A kDD (cid:11) (green dottedcurve), which are similar for the systems with and withoutZeeman splitting ∆ . Panel (c) shows the population differ-ence between the bright atomic excited state (cid:10) A kBB (cid:11) and theground state (cid:10) A kgg (cid:11) (multiplied with a factor), which is neg-ative for η/γ < . but becomes positive for η/γ > . , forthe system without Zeeman splitting. Panel (d) shows thepopulation difference as in (c) and the imaginary part of thecoherence between the bright and dark atomic excited state (cid:10) A kDB (cid:11) (multiplied with a factor) for the system with Zeemansplitting. Other parameters are the same as used in the maintext. state, i.e. (cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11) , but also the off-diagonal el-ement (cid:10) A kDB (cid:11) . It is through these elements the darkatomic excited states contribute to the lasing.Figure A2 supplements the results shown in Fig. 3 inthe main text for . × atoms in the absence and pres-ence of the magnetic field-induced Zeeman splitting ∆ .Fig. A2(a) shows the influence of the Zeeman-splitting onthe steady-state spectrum. For weak pumping η = 0 . γ the two side peaks become weaker while the center peakgets stronger with increasing ∆ due to the increased influ-ence of the dark atomic excited state. For strong pump-ing η = 5 γ , the single peak becomes even narrower when ∆ increases from . MHz to . MHz, but it widensagain when ∆ increases further. Fig. A2(b) shows thatthe population of the ground state (cid:10) A kgg (cid:11) and dark atomicexcited state (cid:10) A kDD (cid:11) decrease and increase monotonouslywith increased pumping η , while the population of thebright atomic excited state (cid:10) A kBB (cid:11) first increases for η smaller than the spontaneous emission rate γ and then1decreases, because the stimulated absorption balancesthe emission. The change of population is similar forthe systems with ∆ = 0 and ∆ (cid:54) = 0 , and the linewidthreduction cannot be simply explained by the populationdynamics.To proceed, we compare the population difference (cid:10) A kBB (cid:11) − (cid:10) A kgg (cid:11) between the bright atomic excited stateand the ground state with the coherence (cid:10) A kDB (cid:11) betweenthe bright and dark atomic excited states, see Fig. A2(c)and (d). Fig. A2(c) shows that, in the absence of Zee-man splitting ∆ = 0 , the coherence is always zero andthe population difference is negative for the pumping η smaller than . γ but becomes positive otherwise. Thus,in this case, the linewidth narrowing can be attributed tothe population inversion as in normal lasing. However,Fig. A2(d) shows that, in the presence of the Zeeman-splitting ∆ / π = 0 . MHz, the population difference isalways negative, while the imaginary part of the coher-ence becomes much larger than the population differ-ence for large pumping η . Associated with the expres-sion (E15), these results indicate that in the case with ∆ (cid:54) = 0 the negative population difference causes light ab-sorption and spectral broadening while the coherence isresponsible for the gain and the spectrum narrowing. Appendix F: Bright and Dark Pseudo-Dicke States
We have further examined how the higher excitedstates of the atomic ensemble are involved in the spectralfeatures. For two-level atoms, these states can be wellcharacterized by the Dicke states [17] | J, M (cid:105) , where thecollective spin quantum number J ≤ N/ characterizesthe symmetry of Dicke states and the collective couplingof these states to the quantized field, and − J ≤ M ≤ J describes the excitation of the atoms. According to ourprevious study [8], the mean of these numbers can be cal-culated with the quantities in the second-order mean fieldtheory: M = N ( (cid:10) A kee (cid:11) − (cid:10) A kgg (cid:11) ) and J = N + N ( N − (cid:104)(cid:68) A kge A k (cid:48) eg (cid:69) + (cid:68) ( A kee − A kgg )( A k (cid:48) ee − A k (cid:48) gg ) (cid:69)(cid:105) ( k (cid:54) = k (cid:48) )for identical atoms. Here, e, g indicates the upper andlower level of the two-level atoms.To apply the Dicke states to our three-level atoms, weconsider the two transitions D k → g k , B k → g k as shownin Fig. 1 (c) in the main text as two sub-two-level sys-tems and define the pseudo Dicke states | J s , M s (cid:105) withthese transitions. Then, the quantum numbers J s , M s can be computed by simply replacing e with D or B inthe above formulas. We refer to these states as pseudoDicke states since the two transitions are not indepen-dent but connected through conservation of population,i.e. (cid:10) A kDD (cid:11) + (cid:10) A kBB (cid:11) + (cid:10) A kgg (cid:11) = 1 .Fig. 4(a) in the main text shows the evolution ofthe Dicke quantum numbers in the Dicke ladders asthe pumping increases. The evolution starts from thebottom-right corner along the bottom boundary and ex-plores the sub-radiative states at the left corner for the Δ 𝑝 absorptiongain0 Ω 𝑑 Ω 𝑝 Δ 𝑝 Ω 𝑑 Ω 𝑝 (a) Δ 𝑝 absorptiongain 0 Ω 𝑑 Ω 𝑝 Δ 𝑝 Ω 𝑑 Ω 𝑝 (b) Figure A3. Lasing without inversion from systems withcascade (a) and V-type (b) energy scheme (left), where Ω d , Ω p denote the coupling with the driving and probe classical field,respectively, ∆ p is the frequency detuning of the probe field.The middle part of both panels shows the formation of thedressed states due to the coupling with the driving field. Theright part shows the gain and absorption for different detuning ∆ p . The spheres indicate the relative population of the levels.Figure redrawn from [18] with the permission of J. Mompart. bright transition while it continues further along the up-per boundary and explores the symmetric Dicke statesclose to the upper-right corner for the dark transition.The evolution is similar for the systems without and withZeeman splitting ∆ . However, the careful examinationshows M B > ( M B < ) for ∆ = 0 ( ∆ (cid:54) = 0 ).By assembling the information revealed by Fig. A2and Fig. 4(a) in the main text , we conclude that forweak pumping the spectrum can be associated with thedressed atom-light states formed by weakly excited statesof the atomic ensemble. For stronger pumping, we ex-pect that the super-narrow spectrum is associated withdressed atom-light states formed by highly excited statesof the atomic ensemble. It is difficult to give explicitexpressions for such dressed states but we expect thatsimilar energy diagrams as shown in Fig. A1 can beformed by replacing the ground and first excited stateof the atomic ensemble with the adjacent highly excitedstates of the atomic ensemble and thus expect still threepeaks in the spectrum for strong pumping. By reexamin-ing the spectra on the logarithmic scale, see Fig. 4(b) inthe main text , we do see two other side peaks beside thesuper-narrow center peak. Since these peaks are ordersof magnitude smaller than the center peak, we can notsee them in normal scale. Appendix G: Comparison with Lasing withoutInversion
The lasing mechanism revealed by our analysis bearsmuch resemblance with lasing without inversion (LWI)[18], which also relies on atomic coherence in the ab-sence of population inversion. Some LWI schemes rely onatoms interacting strongly with a driving field and weaklywith a probe field and do not necessarily require incoher-ent pumping. Fig. A3 shows cascade (a) and V-type (b)three-level systems for LWI [18], which are similar to oursystems, both in the bare state and the dressed eigen-state basis due to the coupling with the strong driving2field. The rightmost panels show the amplification andabsorption of the probe field as function of the detuning ∆ pp