Spin two-axis-twisting via coherent population trapping based cavity QED

We present a spin two-axis-twisting mechanism via coherent population trapping (CPT) based atom–photon interactions. CPT happens and the atoms are trapped in the dark state (coherent superposition of two ground states) when the ground states are resonantly coupled to a common excited state. Close to CPT, the atoms behave as two dark-state based spins, which interact with the common cavity vacuum fields. The otherwise nonexistent interaction is created between them and is identified to be responsible for the two-axis-twisting of the ground state spin. The essential difference from the previous schemes is the compatibility of the twisting spin squeezing with the resonant atom-light interaction. The CPT resonant unit serves as a kind of new ingredients for the quantum networks.

The mechanisms are mainly confined to the far off resonances or dispersive interactions for spins with small or almost vanishing coherence. In the common cases, on resonance or close to resonance, atoms are excited and their spontaneous emission will destroy any possible squeezing of the ground-state spin. In order to overcome the spontaneous emission, one has to be confined to the far-off-resonance regimes, where Ω k /|∆ k | ≪ 1 (Ω k are small, real Rabi frequencies associated with the kth driving fields and ∆ k are the large detunings of the atomic frequencies from the driving field frequencies). In the far-off-resonance regimes, the dispersion is dominant over the absorption although both of them are sufficiently weak. Since the atoms are hardly excited, spontaneous emission is almost avoided. Some cavity QED schemes have been proposed for the unitary one-and two-axis twisting [33][34][35][36][37][38][39][40][41][42][43][44][45]. Among them, Sørensen and Mølmer proposed to realize an effective one-axis twisting Hamiltonian like H eff =hβJ 2 z via double far-off-resonance stimulated Raman scattering [38] in Λ configuration. This method was extended to generate an effective two-axis twisting Hamiltonian in a similar cavity setup [39,40]. With the aid of phase-lock atom-photon coupling, Zhou et al [41] presented the two-axis countertwisting spin-squeezing Hamiltonian like H eff =hβ(J 2 z − J 2 y ) inside an optical cavity. However, those previous mechanisms based on the far off resonances or dispersive interactions have two kinds of limitations. One kind is only for small coherence or zero mean coherence, for which the atoms stay dominantly in a single state. The other kind is the second-order small quantities O(Ω k /∆ k ) O(Ω l /∆ l ) for the twisting strengths, where Ω k /|∆ k | ≪ 1 and Ω l /|∆ l | ≪ 1. This means that we are faced with a challenge to increase the coherence and the strengths. On one hand, the quantum networks needs the large coherence nodes. On the other, once the twisting strengths are not enough to overcome the damping rates, the possible squeezing will be washed out.
It seems that the twisting spin squeezing is not compatible with the resonant atom-light interaction. In this article we show coherent population trapping (CPT) [46] can be compatible with the twisting spin squeezing. So far it has been well known that CPT is one of the most remarkable resonant coherence effects in the atom-photon interactions. When two optical fields interact resonantly with three-level atoms, the atoms are pumped into a superposition of ground states and no longer excited but oscillate between two long-lived ground states. This oscillation is usually called 'dark resonance' [46][47][48][49][50]. The atoms enter one of the coherent superposition states of the two ground states, when dark resonance appears. This specific superposition state is called the 'dark state' . In fact, the dark resonance or dark state is the essential mechanism underlying CPT. As a most advantage close to the dark resonance, the fields experience large dispersion but negligible absorption, and the atoms have a long-lived coherence between the two ground states. A specific case is electromagnetically induced transparency [47][48][49][50], in which the probe field Rabi frequency is much weaker than the coupling field Rabi frequency, and the dark state is approximately the ground state coupled by the weak probe field. One of the greatest interests is in effects of giant nonlinearities on quantum correlations [51][52][53][54][55][56][57]. The CPT systems with equal or comparable Rabi frequencies have nonlinearities of infinite orders of the Rabi frequencies. Generally, the characteristic mechanisms responsible for quantum correlations hides deeply behind the dark resonance [58][59][60]. Dantan et al [58] predicted the existence of the ground-state spin squeezing and light entanglement close to CPT. To our knowledge, however, the hidden mechanism has not yet been clear so far.
In this article we present the CPT-based cavity QED mechanism for the spin two-axis-twisting. The advantages over the previous ones is the compatibility of the twisting spin squeezing with the resonant atom-light interaction. Because of the dark resonance, the atoms are hardly excited, but oscillate between two long-lived ground states and behave as two dark-state based spins. This gives the maximal coherence, which leads to giant dispersion nonlinearity versus negligible absorption. The nonlinearity makes the dark-state based spins interact with each other through the cavity vacuum fields. The induced interaction corresponds to the spin two-axis-twisting with almost maximal coherence. At the same time, the CPT-based scheme is robust against spontaneous emission decoherence processes because the atoms stay predominantly in the long lived ground states.
The remainder of this article is organized as follows. In section 2 we describe spin interaction of the CPT atoms due to the cavity vacuum fields. In section 3 we show the spin two-axis-twisting of the ground states. Finally, a conclusion is given in section 4.

CPT for cavity induced spin interaction
Shown in figure 2 is a setup we will consider, in which an ensemble of N independent three-level atoms with two ground states |1, 2⟩ and an excited state |3⟩ is placed in a two-mode cavity. The two electronic dipole-allowed transitions |1⟩ ↔ |3⟩ and |2⟩ ↔ |3⟩ in Λ configuration are coupled by two strong dressing fields of frequencies ω 1,2 , respectively. |1⟩ ↔ |2⟩ is electronic-dipole forbidden. Two cavity fields are applied to either wing (e.g. |1⟩ ↔ |3⟩) of the Λ configuration. We transform into the first rotating frame, in which the transitions |1⟩ ↔ |3⟩ and |2⟩ ↔ |3⟩ rotate with ω 1 and ω 2 , respectively. The master equation for the density operator ρ of the atom-field system is derived in the dipole approximation and in the ω 1,2 rotating frame as [61][62][63] with where H 0 describes the Hamiltonian for the interaction of the atoms with the dressing fields, while H I represents the interaction of the atoms with the applied cavity fields a 1,2 . In the above formulae,h is the Planck constant. J kl = ∑ N µ=1 J µ kl (J µ kl = |k µ ⟩ ⟨l µ |; k, l = 1, 2, 3) stands for the collective spin operator of the atomic ensemble. Ω l = µ l3 E l /(2h) is used as half Rabi frequencies, where µ l3 is the electric dipole moment and E l is the electric amplitude of the driving fields along the dipole moment. a l and a † l denote the annihilation and creation operators for the cavity fields, and g l is the atom-cavity coupling strength. ∆ l = ω l3 − ω l expresses the detuning of atomic transition frequency ω l3 from the dressing field frequency ω l , ∆ c l = ω c l − ω 1 gives the detuning of the cavity resonance frequency ω c l from ω 1 . The damping term in equation (1) consists of two parts, of which the first part L 1 ρ describes the spontaneous emission of the atoms from the excited state |3⟩ to the ground states |1, 2⟩ with rates Γ 1,2 , and the second part L 2 ρ depicts the cavity loss with rates κ 1,2 . For the present system, the strong classical fields E 1,2 and the cavity fields a 1,2 play a different role in manipulating the quantum properties of the atomic ensemble. The former not only creates coherence between the ground states but also gives new frequency components, while the latter interacts selectively with some degenerate transitions.
In the following we describe the dark state and giant nonlinearities, the dressed atomic states that are convenient for the CPT-based atom-photon interaction, and the cavity induced spin interaction. Schematic sketch for the proposed scheme. An ensemble of N atoms is placed at the center of the optical cavity and dressed by two strong classical fields with Rabi frequencies Ω1,2. As shown in the inset, the atom has two ground states |1, 2⟩ and an excited state |3⟩. The |1, 2⟩ ↔ |3⟩ transitions in Λ configuration are electric dipole allowed and coupled (orange lines) to two classical fields of frequencies ω1,2, while the |1⟩ ↔ |2⟩ transition is electronic-dipole forbidden. Two cavity fields a1,2 are applied (red and blue lines) to the common |1⟩ ↔ |3⟩ transition. ∆1,2 are the detunings of the atomic |1, 2⟩ ↔ |3⟩ transitions respectively from the classical field frequencies ω1,2, and ∆c 1,2 are the detunings of the cavity fields a1,2 from the common field frequency ω1.

Dark state and giant nonlinearities
In order to describe the physical mechanism, we first describe the interaction of the atoms with the classical dressing fields, but do not include the cavity fields temporarily. First we see the resonance case, ∆ 1 = ∆ 2 = 0 and take Ω 1,2 to be real. This can easily be seen by substituting the superposition states for the bare states |1, 2⟩. With this substitution, we can define new spin operators

D, 3). This substitution rewrites the Hamiltonian (3) in the form
Which indicates that only the superposition state |B⟩ is coupled to the applied fields, while the other superposition state |D⟩ is no longer involved in the interaction with the fields. Population transfer happens only through the optical pumping |B⟩ → |3⟩ and the successive decay |3⟩ → |D⟩. As a consequence, population leaves out of |B⟩ and |3⟩ and then deposits in |D⟩. After several radiative lifetimes, the atoms are trapped in the superposition state |D⟩, i.e. we have the populations where from now on we useJ kl ≡ ⟨J kl ⟩ to express the mean spins at steady state. For the above reason, the states |D⟩ and |B⟩ are usually called the dark and bright states, and the coherent effect is usually referred to as the dark resonance, which underlies CPT. In terms of the bare ground states |1⟩ and |2⟩, we have the atomic populationsJ and the atomic coherenceJ For equal Rabi frequencies, Ω 1 = Ω 2 , we have the maximal coherenceJ 12 = −N/2. What is essentially important is that the coherence determines the gain nonlinearity close to the dark resonance, ∆ 1,2 ̸ = 0, |∆ 1,2 | ≪ Ω 1,2 . From the atomic spin componentJ j3 we can define the susceptibilities as the response of the atoms to the dressing field E l . For general parameters, the nonlinear susceptibilities χ 1,2 , as defined in (13), are derived in appendix A. For ∆ 1 = −∆ 2 = ∆ and Γ 1 = Γ 2 = Γ, they are obtained from equation (A2) in the form , which show explicitly the nonlinear dependences on the Rabi frequencies Ω 1,2 . For Ω 1 = Ω 2 = Ω we have The imaginary part Imχ 1 for absorption and the real part Reχ 1 for dispersion are plotted in figure 3(a) in unit of |µ 13 | 2 /(ε 0h ) relative to the detuning ∆/Ω for given dressing field intensities Ω = 25.5Γ. When the dressing fields are tuned from the exact dark-state resonance (∆ = 0) by the detunings ∆ ̸ = 0, we have two transparency windows at ω 1 = ω 31 and ω 2 = ω 32 , respectively. The width of ∆ at half height for the transparency window is determined by Ω. Within the window, we have dominant dispersion nonlinearity over absorption. The ratios of the dispersion to absorption η j = Reχ j /Imχ j (j = 1, 2) are obtained as We plot the ratio of the dispersion to the absorption in figure 3(b) versus the normalized two-photon detuning ∆/Ω. We see the ratio takes a remarkably large value, η 1 ∼ 10 3 close to the center. The giant dispersion nonlinearity is just the consequence of the large coherenceJ 12 . It was shown that the giant dispersion nonlinearity plays a subtle role in the quantum correlations [58][59][60]. Here our purpose is to reveal that it will lead to the two-axis-twisting spin squeezing.

Dressed atomic states
It is convenient to merge the strong dressing fields into the atoms and to treat the atoms in terms of dressed states [64][65][66]. With the bare atomic states |1, 2, 3⟩ and the photon number states |n 1,2 ⟩ of the dressing fields E 1,2 ,the atom-field composite system are in an infinite ladder of triplets of near-degenerate eigen energy states |1, n 1 , n 2 − 1⟩, |2, n 1 − 1, n 2 ⟩, and |3, n 1 − 1, n 2 − 1⟩. When the interaction energy H 0 in equation (3) is merged into the interacting system, the triple near-degenerate states combine into a set of the superposition states, which are called the dressed states. Here we focus on the case when the dressing fields are strong and coherent. This means that the two dressing fields have their narrow distributions of photon numbers centered at large values, respectively. Within the narrow distribution around the large mean numbers of photons, we can treat the dressing fields as classical fields, which are related with Rabi frequencies Ω 1,2 .
For clarity we focus on equal Rabi frequencies, and opposite but equal and small detunings, In the rotating-wave frame and without including the photon numbers, the dressed states can be derived by diagonalizing the Hamiltonian (3) as where we have used the dark and bright states and the parameters In terms of the dressed states we can define atomic operators as J kl = ∑ N µ=1 |k µ ⟩⟨l µ | (k, l =0,1,2). Then we have the free Hamiltonian where |0⟩ has zero eigen energy.
Transforming the relaxation terms of the atoms to the dressed-state representation (Γ 1 = Γ 2 = Γ), we obtain the steady-state populations where we have used the closure relationJ00 +J11 +J22 = N. As a particular case, on the exact resonance ∆ = 0 (i.e. cos θ = 1, sin θ = 0), the dressed state |0⟩ and the dark state |D⟩ are exactly the same as each other. The populations are trapped in the dark statē which corresponds exactly to the maximal coherenceJ 12 = −N/2 between the ground states. Now we will focus on the regime close to the dark resonance |∆| ≪ Ω, i.e., cos θ .
It is seen that |0⟩ is almost the same as the dark state |D⟩, and that |1⟩ and |2⟩ consist primely of the states |B⟩ and |3⟩. From equation (21) we can obtain their populations to the second order small quantity (∆/Ω) 2
Because of |∆| ≪ Ω, we haveJ00 . = N,J11 =J22 . = 0, which indicates that the atoms almost stay in the dark state |0⟩ . = |D⟩. The dressed-state spins J01 and J02 of the CPT atoms can be looked upon as two dark-state based spins JD1 and JD2, i.e.

Dispersive cavity induced spin interaction
We explore the off-resonant interaction of two dressed spins J01 and J02 with the cavity fields. In terms of the dressed atomic states, the entire Hamiltonian in equations (2)-(4) are rewritten in a free part where ∆ c1 = −∆ c2 = ∆ c is assumed for simplification. Here we focus on the nonlinear spin interaction that happens when the cavity vacuum fields are far off one-photon resonances with the far separated dressed transitions Going into the interaction picture and taking a rotating approximation we have the time dependent interacting Hamiltonian from equation (4) H where H.c. is the Hermitian conjugate of the term before it, and the coupling constants are listed as with g 1,2 = g (real) used for simplification. The interaction of the degenerate dressed spins with the cavity fields are pictorially described in figure 4. For the dispersive interaction, we derive the effective Hamiltonian in appendix B. Close to the dark resonance (see equation (23)), we have the effective Hamiltonian from equation (B8) for the normalized where the cross coupling strength (equation (B6)) is With the mediation of the cavity vacuum, the dressed-state spins J01 and J02 are coupled to each other. For |∆| ≪ Ω, we haveΩ . = √ 2Ω and sin θ . = ∆/( √ 2Ω), and then we have which shows clearly the nonlinear dependence on the Rabi frequency Ω. The Rabi frequency Ω determines the detuning √ 2Ω − ∆ c for the dispersive shift g 2 N 2( √ 2Ω−∆c) of the dark state, and also manipulates the coherent perturbation |∆| √ 2Ω ≪ 1 (i.e. the deviation from the dark resonance) for the effective coupling strength β.

Two-axis-twisting spin squeezing
Now we can discuss the quantum nonlinear behavior of the ground state spin J 12 from the J01 and J02 coupling.

Ground state spin interaction
In this subsection we show that J DB as ground state spin and J D3 as excited-state involved spin have separable interactions of two-axis-twisting close to the dark resonance (|∆| ≪ Ω). In this case, we can see from equation (32) that the cross coupling strength is in the first order of ∆/Ω. For this reason, it is enough now to keep the dressed states to the zeroth order of ∆/Ω [i.e. to drop the sin θ terms in equation (17)],
The dressed-state spins J01 and J02 in the Hamiltonian (B8) are related to the ground state spin are independent of each other because of the independent commutation relations [J Dk , . It is seen that the ground state spin J DB and the excited state involved spin J D3 are both in a parametric interaction form. There is an essential difference between J DB and J D3 . The former involves only the superposition of the long lived ground states |D⟩ and |B⟩ but not the rapidly decaying excited state |3⟩, while the latter contains the rapidly decaying excited state |3⟩. This means that J DB has a long enough coherence time while J D3 has a much shorter coherence time. Therefore for J DB , it is easy for the cavity induced interaction to overcome the decoherence and to generate spin squeezing.

Two-axis-twisting
Let us focus on the subspace of only the ground states |1⟩ and |2⟩, where the spin two-axis-twisting interaction described by the Hamiltonian H 1 for the ground state spin J DB . For such a subspace, we define the Cartesian components J = (J x , J y , J z ), of which follow the commutation relation [ J y , J z ] = 2iJ x . Close to the dark resonance (cos θ . = 1, | sin θ| ≪ 1), we have their mean valuesJ x . = N andJ y =J z = 0 at steady state. Noting the relations we write the Hamiltonian H 1 in equation (35) in the form which shows the two-axis twisting for the (y, z) components of the spin J between the bare ground states. It is clear that the dark state based spin J DB and the bare ground state spin J 12 are equivalent to each other except for a rotation of the Bloch sphere. So far we have performed a series of steps for describing the interaction of the CPT atoms with the cavity fields. For clarity we summarize them as three main parts as in figure 5. The upper part describes the interactions of the bare atoms with the dressing fields and of the dressed atoms with the cavity fields, respectively. The middle part shows the induced, otherwise existent interaction between the dressed-state spins J01 and J02. The bottom part gives the ground state spin two-axis-twisting interaction, which is isolated from the middle part.
Including atomic and cavity relaxations we calculate the spin uncertainties in appendix C (equations (C14) and (C15)) as where γ is the total decay rate of J DB including the contributions of atomic phase damping γ p , atomic spontaneous decay Γ and the cavity loss κ 1,2 = κ (see equation (C7)), β is the cavity induced spin interaction strength containing the cavity decay (see equation (C8)), and D ± are the diffusion coefficients (see equation (C13)). Usually we have (γ, D ± ) ≪ |β|, as shown in appendix C, since the excited involved spin J D3 is negligibly weakly coupled to the ground spin J DB and since the cavity fields are largely detuned from the dressed spins, κ ≪ (Γ, |Ω − ∆ c |). Then the spin uncertainties (41) reduce to We have ξ 2 π/4 < 1 for β < 0 or ξ 2 −π/4 < 1 for β > 0. In principle, almost perfect squeezing is obtainable (ξ 2 π/4 → 0) when the evolution time is long enough, i.e. γ → 0, κ → 0. For a realistic case, good squeezing appears when the evolution time is much smaller than the relaxation time t ≪ γ −1 , and the interaction strength is relatively strong, 2|β|t ≫ 1. This corresponds to the evolution time t ∼ 3/(2|β|). For g ∼ 50 kHz, N ∼ 10 4 , sin θ ∼ 0.1, κ ∼ 100 kHz, andΩ − ∆ c ∼ 5 MHz, we have t ∼ 3 µs. In figure 6 we plot the spin uncertainty ξ 2 π/4 as the cooperatively parameter C = g 2 N/Γ for different cavity decay rate κ. The decay rates and frequencies are in units of the atomic decay rate Γ. As κ decreases and C increases, the spin uncertainty is greatly reduced. More than 80% squeezing is obtainable. Now we can summarize the main results as follows. (i) The CPT atoms behave as the dark-state based spins. The dark resonance is the first element of the present scheme. Through the dark resonance, the atoms are trapped in the dark state. The spontaneous noise is almost avoided. The good squeezing is based on the fact that almost no spontaneous noise is fed to the ground state spin. The physical mechanism is well described in terms of the dressed states. One of the three dressed states is just the dark state, while the other two dressed states are almost empty. Thus the CPT atoms act as two dressed-state spins J01 and J02, which have equal but opposite frequency differences from the cavity fields. In spite of the resonant interaction, the atoms are not excited and the spontaneous emission is avoided. Close to the dark resonance, the ratio of the dispersion to absorption is extremely large. The giant dispersion nonlinearities guarantee the absorptless interaction close to the resonance. As a consequence, the two dressed spins can be coupled to each cavity fields. Their simultaneous coupling with common fields leads to the interaction between them, as shown in equation (30).
(ii) The dispersive cavity fields induce the spin interaction. The dispersive cavity coupling is the requisite ingredient of the present scheme. Each of the two cavity vacuum fields a 1,2 couples each of the two dressed spins J01 and J02. The cavity effects include three aspects: cross coupling, enhanced decay, and frequency shift. In the dispersive cavity case, the cross coupling is much stronger than the other two. More importantly, the coupling of the dressed spins J01 and J02 does not cause cross coupling of the ground state spin J DB and the excited involved spin J D3 . This guarantees that the decoherence of J D3 is hardly fed to the decoherence of the ground states. So far, cold atoms are good candidates for a weak phase damping. As usual, the cavity losses are the main limiting factors to the squeezing. Figure 6 shows the effects of the cavity decay. As the cavity decay rate decreases, the curves of the uncertainty descent. When the cavity loss rate is remarkably weaker than the atomic decay rates (κ ≪ Γ), a good squeezing (more than 80%) is achievable for a relatively large cooperatively parameter. As the cavity decay rate rises the squeezing is significantly weakened. However, a squeezing of about 50% is achievable even though the cavity decay rate is comparable to the atomic decay (κ ∼ Γ).
(iii) The essential difference from the previous schemes is the compatibility of the resonant atom-light interaction with the twisting spin squeezing. Previous schemes are mainly based on the far off-resonant interactions. Researchers used the double far-off-resonance stimulated Raman scattering [38][39][40] or the far-off-resonance phase-lock [41]. It seems that the twisting spin squeezing is not compatible with resonant interactions. In sharp contrast, our scheme includes the dark resonance as the key element although the cavity fields are far off-resonant. Even in the presence of the resonant CPT unit, the atoms are not excited and spontaneous emission is hardly involved in the ground state spin interaction. The nonlinear spin interaction is created close to the dark resonance, where the dispersive nonlinearity is extremely stronger than the absorption.
Experimentally cold atoms confined in a magneto-optical trap [67] can be used for the present scheme. There are a great number of atomic structures that can be used as candidates, for example, the Rubidium 85 D 1 (795 nm) transition hyperfine structure. The two lower levels are the ground state hyperfine levels |1⟩ = |5 2 S 1/2 , F = 1⟩ and |2⟩ = |5 2 S 1/2 , F = 2⟩, separated from each other by 3 GHz, while the upper state is the excited state hyperfine level |3⟩ = |5 2 P 1/2 , F = 2⟩. The other excited state hyperfine level |5 2 P 1/2 , F = 1⟩ is 362 MHz below |3⟩ and has a negligible influence. The cavity parameters of [68] can be used for our purpose, including the beam waist w ∼ 35 µm, homogeneous laser beams of width d ∼ 50 µm, and an interaction volume of 10 −7 cm 3 . A density of 10 11 cm 3 is small enough to prevent coherence losses due to collisions, and corresponds to the number of atoms N ∼ 10 4 . For g ∼ 50 kHz, |∆|/( √ 2Ω) ∼ 0.1 (i.e. sin θ ∼ 0.1), andΩ − ∆ c ∼ ±5 MHz, we have β ∼ 500 kHz. For the atomic phase damping rate γ p ∼ 1.0 kHz and the cavity decay rate κ ∼ 100 kHz, this guarantees that the evolution time is much shorter than the relaxation times, |β| −1 ≪ κ −1 , γ −1 p .

Conclusion
In conclusion, we have presented the CPT based cavity QED mechanism for the spin two-axis-twisting of the two ground states. The ground states, when coupled coherently to a common excited state, are superposed as the dark state, where the atoms are trapped. Close to CPT, the atoms act as the two dark-state based spins. The otherwise nonexistent interactions happen between them through common cavity vacuum fields. The cavity induced interaction between the dark-state based spins underlies the mechanism for the two-axis-twisting. The essentially different feature of the proposed scheme form the previous ones is the compatibility of the two-axis-twisting spin squeezing with the resonant atom-light interactions. The inclusion of CPT resonant ingredients provides a new kind of nodes for the quantum networks. The great advantage lies in that, even in the presence of the resonance units, there is almost not spontaneous emission noise fed to the nodes and the squeezed spin states are prepared efficiently.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). The data that support the findings of this study are available upon reasonable request from the authors.
together the complex conjugates (J 21 , J 31 , J 23 ) and the closure relation J 11 + J 22 + J 33 = N. We have used the complex damping parameters γ 12 = i(∆ 1 − ∆ 2 ), γ 13 = 1 2 (Γ 1 + Γ 2 ) + i∆ 1 , and γ 32 = 1 2 (Γ 1 + Γ 2 ) − i∆ 2 . Here we have used the atomic operators for their corresponding c numbers without confusion, and therefore we have not necessarily written out the noise terms. The polarization elements (J 12 , J 13 , J 32 ) constitute a closed set of three equations when we treat (J 11 , J 22 ) as constants temporarily. Solving for the polarization elements first we then substitute them into the population equations. Solving for the populations we substitute them back to the polarization equations. By performing such steps and substituting the steady state solutions J 13 and J 23 into equation (13), we obtain the susceptibilities as where we have defined q l = |Ω l | 2 (l = 1, 2) dependent quantities q l = 2Re(q l /r), r = q 1 γ 12 + q 2 γ 32 + γ 12 γ 13 γ 32 ,

Appendix B. Dispersive cavity induced spin interaction
In this appendix we derive the effective Hamiltonian for the dispersive cavity induced spin interaction. The equation for the density matrix ρ isρ where the interaction HamiltonianH I (t) has fast oscillating terms. We have its formal solution Substituting the solution back we obtaiṅ ] . (B2) The first term is fast oscillating compared with the second one and is negligible to a good approximation. Then we can employ a Markovian approximation for the latter. Approximately, the evolution of the density operator isρ with the effective HamiltonianH where the indefinite integral is evaluated at time t without integral constant [69]. These arguments can be put on the more rigorous footing when we consider the time-averaged dynamics over a much longer period than the period of any of the oscillations present in the effective Hamiltonian. We assume that the cavity fields are kept in their vacuum states.
where the first term describes the cross spin interaction with the interaction coefficient while the second term represents the dynamical frequency shift the frequency shift quantity The cavity induced shift δ, which is much smaller than the atom-field detuningΩ − ∆ c , can be removed by adjusting the rotating frequency ∆ c → ∆ c + δ, as included in appendix C. After this, the Hamiltonian (B5) reduces simply to the two-spin cross interaction

Appendix C. Spin uncertainties including atomic and cavity decoherence
Now we describe the cavity induced interaction and calculate the spin uncertainties by including the atomic and cavity decoherence. We work close to the dark resonance, as shown by equation (23). Transforming to an appropriate rotating frame and using the reverse transform the ground state spin J DB = 1 √ 2 (J01 + J02) and the excited-state involved spin J D3 = 1 √ 2 (J01 − J02) of equation (34), we derive the Heisenberg-Langevin equations from the Hamiltonian (28) aṡ where we have included the phase damping rate γ p for J DB and the spontaneous decoherence rate Γ for J D3 and have taken κ 1 = κ 2 = κ. To balance the cavity induced shifts, as shown below, we have also reserved a small detuning −δ for J01 and J02 (i.e. for J DB and J D3 ), respectively, (|δ| ≪ |Ω − ∆ c |). We take the reserved detuning δ in equation (C1) as the cavity induced the frequency shift As a consequence, the reserved detuning balances the cavity induced shift. Note that the expression for δ is the same as that in equation (B7) for κ → 0. The white noise ingredients F JDB (t), F JD3 (t), F a1 (t), and F a2 (t) have their nonzero correlations ⟨F JDB (t)F J † DB (t ′ )⟩ = 2γ p Nδ (t − t ′ ), ⟨F JD3 (t)F J †

D3
(t ′ )⟩ = 2ΓNδ (t − t ′ ) and ⟨F a j (t) F a † j (t ′ )⟩ = κδ (t − t ′ ) (j = 1, 2). After eliminating the cavity fields we derive the Heisenberg-Langevin equationṡ Usually the former is negligibly weak compared with the latter, γ p ≪ Γ. The cavity induced parameters are written as and the noise terms are listed as ) . (C5) Eliminating adiabatically the fast decaying quantity J D3 , we obtain the equation forJ DB = J DB e iπ /4 aṡ where the total decay rate, the total cross strength, and the total noise are written respectively Note that the expression for β is the same as that in equation (32) for κ → 0, i.e. γ 1 → 0. We can calculate the noise correlations of F (t) as We use the phase quadraturesJ + =J DB +J † DB , andJ − = −i(J DB −J † DB ), and derive equations for their variances with the diffusion coefficients D ± = DJJ † + DJ †J ± DJJ ± DJ †J †.
The resulting variance are derived as After the reverse transform J DB =J DB e −iπ/4 , we obtain immediately the uncertainties