Revisiting the impact of dissipation on time-reversed one-axis-twist quantum-sensing protocols

Spin squeezing can increase the sensitivity of interferometric measurements of small signals in large spin ensembles beyond the standard quantum limit. In many practical settings, the ideal metrological gain is limited by imperfect readout of the sensor. To overcome this issue, protocols based on time reversal of unitary one-axis-twist (OAT) spin-squeezing dynamics have been proposed. Such protocols mitigate readout noise and, when implemented using cavity feedback, have been argued to also be robust against dissipation as long as the collective cooperativity of the system is sufficiently large [Davis et al., PRL 116, 053601 (2016)]. Here, we perform a careful systematic study of dissipative effects on three different implementations of a OAT twist-untwist sensing scheme (based on symmetric as well as asymmetric cavity feedback and on a Tavis-Cummings interaction). Our full treatment shows that the three approaches have markedly different properties and resilience when subject to dissipation. Moreover, the metrological gain for an implementation using symmetric cavity feedback is more sensitive to undesired dissipation than was previously appreciated.

Even though all three of these schemes implement the same OAT Hamiltonian, they differ in the structure of undesired dissipative processes that arise as a byproduct of the OAT interaction.These differences may be crucial since dissipation competes with the unitary OAT dynamics and limits the attainable level of spin squeezing.Nevertheless, it has been shown that all three schemes generate spin squeezing in the presence of dissipation if the collective cooperativity is large [22,24,28].
Another interesting feature of the OAT Hamiltonian is 1.Metrological gain Gmet = 1/N (∆ϕ) 2 as a function of the collective cooperativity for a sensing protocol using timereversal of one-axis-twist (OAT) dynamics.We use N = 10 5 spins and include readout noise equal to the standard projection noise of a coherent-spin state (Ξ 2 det = 1.0).Thick solid curves correspond to different implementations of the OAT Hamiltonian [symmetric cavity feedback (SCF), asymmetric cavity feedback (ACF), and a Tavis-Cummings (TC) interaction with single-spin dephasing or single-spin relaxation].Solid black line: ideal metrological gain G ideal met that can be achieved at zero dissipation.The dashed gray curve shows the expression G est met , defined in Eq. (23), which has been derived in Ref. 26 using a heuristic model of dissipative effects and applies to the SCF and TC schemes.As shown by the thick SCF and TC curves, nonperturbative effects of dissipation can lead to strong deviations from this result even though the collective cooperativity N η is large.In particular, the collective cooperativity required to reach Gmet > 1 is underestimated by almost an order of magnitude (shaded red area).The dash-dotted gray curve indicates Gmet = 0.1N η.
that it can be used for amplification of small signals in a spin system [16,26] (sometimes also called "phase magnification" [11]), which is an important technique to render 2. Sketch of the three considered schemes to experimentally implement the one-axis-twist (OAT) Hamiltonian ĤOAT, defined in Eq. (1), in an ensemble of N spin-1/2 systems with levels {|↑⟩ i , |↓⟩ i }i=1,...,N .(a) ĤOAT can be implemented using a direct Tavis-Cummings (TC) coupling between the spin-1/2 systems and a detuned cavity mode with frequency ωc and detuning ∆.Undesired dissipative processes are single-spin relaxation (rate γ rel ), single-spin dephasing (rate γ ϕ ), and cavity decay (rate κ).Alternatively, the OAT Hamiltonian can be implemented using feedback from a driven cavity mode, with drive frequency ω dr and ampliude β in , either by (b) symmetric cavity feedback (SCF), where both |↑j⟩ and |↓j⟩ are coupled to the excited level |ej⟩ via the cavity mode with equal but opposite-in-sign detuning ∆, or by (c) asymmetric cavity feedback (ACF), where only |↑j⟩ is coupled to |ej⟩ via the cavity mode with detuning ∆.Undesired dissipative processes in the cavity-feedback schemes are decay of the excited level (rate Γ) and cavity decay.
quantum sensing protocols robust to imperfect readout [11,26,30,31].OAT-based spin amplification relies on two time-reversed applications of the OAT Hamiltonian (which is why it is sometimes called a "twist-untwist protocol") and, similar to the generation of spin squeezing, the question arises of how the amplification dynamics depends on the undesired dissipation and how this impacts the achievable metrological gain.In the seminal proposal of OAT amplification [26], an appealing intuitive argument suggested that, for a SCF-based implementation, a large collective cooperativity was enough to ensure metrological gain in the presence of dissipation.This heuristic argument assumed moderate readout noise and that the signal amplification was independent of dissipation.Surprisingly, to the best of our knowledge, this prediction has not been rigorously tested.Moreover, the robustness predicted in Ref. 26 for the SCF-based implementation is not generic to all implementations of OAT amplification: A recent analysis of a TC-based implementation of the OAT Hamiltonian [30] showed that undesired dissipative processes can strongly degrade the amplification, such that significant levels of gain are only possible in an experimentally very challenging regime of extremely large single-spin cooperativities.
It is clearly crucial to applications to understand when one can achieve large metrological gain in the presence of readout noise and dissipation.In this work, we address this question by performing a careful analysis of the requirements on cooperativity that are needed to obtain significant levels of metrological gain and signal amplification in TC, SCF, and ACF-based OAT amplification schemes.We use a combination of numerically exact solutions of the quantum master equation (QME) for small system sizes and a mean-field theory (MFT) analysis for larger system sizes.We find that, depending on the level of readout noise, the requirement for achieving large metrological gain is very different in each scheme.This is in contrast to the simpler task of generating spin squeezing alone, where all implementations yield similar constraints.Moreover, we find that the metrological gain in the SCF and TC schemes is more sensitive to undesired dissipation than previously thought.
This paper is organized as follows.In Sec.II, we introduce the three different schemes to implement a OAT Hamiltonian.For each of the schemes, we then derive an effective QME including only the spin degrees of freedom of interest in Sec.III.In Sec.IV, we review the twist-untwist sensing protocol proposed by Davis et al. [26] and their heuristic analysis for the scaling of the metrological in the presence of dissipation.We then analyze numerically which requirements need to be satisfied to achieve large amplification in the different schemes in Sec.V.For the cavity-feedback based schemes, we find an interesting dependence of the gain on a scaled single-spin cooperativity, which we interpret using an intuitive argument based on MFT in Sec.VI.Finally, in Sec.VII, we analyze the full metrological gain of the cavity-feedbackbased schemes.We conclude in Sec.VIII.

II. THE SYSTEM
We consider an ensemble of N atoms with two levels |↑ j ⟩ and |↓ j ⟩, j ∈ {1, . . ., N }, which form an effective two-level system that is used for sensing.The state |↑ j ⟩ (|↓ j ⟩) is the eigenstate of the Pauli operator σ(j) z with eigenvalue +1 (−1), and the lowering operator is σ The ensemble of two-level systems can be described by a collective spin vector with components Ŝx = ( Ŝ+ + Ŝ− )/2, Ŝy = ( Ŝ+ − Ŝ− )/2i, and Ŝz = where χ denotes the OAT strength and both its sign and magnitude can be controlled experimentally.A Hamiltonian of the form of Eq. (1) can be engineered by coupling the spins to a bosonic mode with frequency ω c , damping rate κ, and annihilation operator â.The system is de-scribed by a QME of the form (ℏ = 1) where Ĥ0 = ω c â † â + ω s Ŝz and the specific form of the Hamiltonian Ĥ1 as well as the dissipative processes L spin of the spins depend on the details of the spincavity coupling scheme.The superoperator D[ Ô]ρ = Ô ρ Ô † − { Ô † Ô, ρ}/2 denotes a standard Lindblad dissipator.
For the TC-based implementation of OAT, shown in Fig. 2(a), they are given by ĤTC where g is the spin-cavity coupling strength and γ rel (γ ϕ ) is the single-spin relaxation (dephasing) rate.To generate the OAT Hamiltonian (1) from the TC interaction, one considers the dispersive regime where the spins are highly detuned from the cavity mode, The SCF-based implementation of OAT, shown in Fig. 2(b), uses a third, auxiliary atomic level |e j ⟩ and is defined by The levels |↑ j ⟩ and |↓ j ⟩ are equally detuned by an amount ±∆ from a cavity-assisted transition to the auxiliary level |e j ⟩, i.e., ω s = 2∆, and the excited level can decay back into the levels |↑ j ⟩ and |↓ j ⟩ with equal rates Γ.This coupling scheme is used in microwave atomic clocks, where the levels |↑ j ⟩ and |↓ j ⟩ are typically long-lived states connected by a clock transition [11,26] such that relaxation and dephasing processes within the {|↑ j ⟩ , |↓ j ⟩} manifold (which were present in L TC spin ) can now be neglected.The OAT strength χ and its sign are controlled by the amplitude β in and the detuning δ = ω dr − ω c of a coherent drive of the optical cavity, which leads to a steady-state intracavity photon number Finally, for the ACF-based implementation, shown in Fig. 2(c), only the level |↑ j ⟩ is coupled to |e j ⟩ and we  (8) describing the dynamics in the subspace spanned by the states {|↑j⟩ , |↓j⟩}j=1,...,N for the different implementations of the OAT Hamiltonian shown in Fig. 2. The single-spin cooperativity η, the number of photons ncav in the cavity due to a coherent drive, and the amplitude βin(t) of the coherent drive are defined in Eqs. ( 9), (6), and (5), respectively.
where ω c = ∆+ω e −ω s /2.This coupling scheme is used in optical atomic clocks, where the levels |↑ j ⟩ and |↓ j ⟩ are again long-lived states connected by a clock transition.Since only |↑ j ⟩ is coupled to the auxiliary |e j ⟩ level, the level |e j ⟩ can only decay back to |↑ j ⟩ at a rate Γ.

III. EFFECTIVE QUANTUM MASTER EQUATION
As discussed in the Introduction, all three coupling schemes shown in Fig. 2 generate the desired OAT Hamiltonian (1), but they differ in the specific form of additional dissipative terms.To derive an effective QME describing the dynamics in the subspace spanned by the states {|↑ j ⟩ , |↓ j ⟩} j=1,...,N , we eliminate the auxiliary levels {|e j ⟩} j=1,...,N and the cavity mode â for each coupling scheme.As derived in Appendix A, they all lead to an effective QME of the form where the different rates are defined in Table I.Since we are only interested in the effect of the OAT term χ Ŝ2 z , we will work in a rotating frame at the frequency ωs such that the term ωs Ŝz can be ignored.
As shown by Table I, the three implementations of the OAT Hamiltonian differ significantly in the form of their dissipative processes.The TC scheme has a collective relaxation process, whose strength can be tuned relative to the OAT strength χ TC by changing the ratio κ/∆ TC , but the single-spin relaxation and dephasing rates are independent of κ/∆ TC .Both the SCF and the ACF scheme have a collective dephasing process, whose strength can be tuned relative to the OAT strength by changing the ratio κ/δ, but the single-spin dissipation rates now depend on the ratio κ/δ, too.Moreover, in the absence of the OAT Hamiltonian, the dissipative processes in the three schemes try to stabilize very different steady states: The dissipation in the TC scheme aims to relax each spin into its ground state, ⟨σ The ACF scheme dephases the system in the z basis, such that the steady state has ⟨σ z ⟩ is a constant of motion.The SCF scheme, finally, features a combination of single-spin dephasing as well as singlespin excitation and relaxation processes at equal rates, which evolve the system into a completely mixed state, ⟨σ These very different dissipative processes compete with the unitary OAT dynamics, which raises the question whether the schemes will have different requirements to achieve signal amplification and large metrological gain.An important figure of merit to compare the strength of the desired unitary OAT dynamics to the undesired dissipative processes is the single-spin cooperativity, which is defined as for the SCF and ACF schemes, and has the form for the TC scheme.As far as spin squeezing is concerned, it has already been shown that (despite the different dissipative processes) spin squeezing can be achieved in all three schemes if the collective cooperativities N η and N η ϕ,rel are large (see Refs. 22, 24, and 28 as well as Appendix B).

IV. OAT TWIST-UNTWIST SENSING PROTOCOL A. Coherent dynamics
We start by reviewing the ideal OAT twist-untwist protocol in the absence of any dissipation [11,26], with a specific focus on the underlying amplification process.The goal of the sensing protocol is to measure a small signal that changes the level-splitting frequency of all N ensemble spins [2, 3].Such a signal causes a small excess precession angle ϕ in a Ramsey-type interferometric protocol.In the absence of any other noise sources, the estimation error ∆ϕ, with which the angle ϕ can be inferred from the final measurement, depends only on the projection noise of the sensing state in the direction of the signal acquisition (which we take to be the Ŝy direction without loss of generality): where (∆S y ) 2 = ⟨ Ŝ2 y ⟩ − ⟨ Ŝy ⟩ 2 .Spin squeezing reduces this estimation error by reshaping the projection noise, and is quantified by the Wineland parameter [3,32] where the variance (∆S ⊥ ) 2 is minimized over all directions perpendicular to the polarization of the collective spin vector Ŝ ≡ ( Ŝx , Ŝy , Ŝz ) ⊤ .
A key figure of merit is the metrological gain G met , which quantifies how much the estimation error has been reduced compared to minimum error achievable using a product state: For a spin-squeezed state, and in the absence additional readout noise, one finds G met = 1/ξ 2 R .Hence, the Wineland parameter quantifies the sensitivity improvement of a spin-squeezed state (which has 1/N ≤ ξ 2 R < 1) over a coherent-spin state (which has ξ 2 R = 1) in an ideal Ramsey-type measurement.
In practice, the final readout process of the Ramsey interferometer will be imperfect, i.e., it will add (technical) detection noise.This detection noise can be quantified in terms of an equivalent amount of projection noise and is typically given in multiples Ξ 2 det of the projection noise N/4 of a coherent-spin state [33].Assuming that spin squeezing has not significantly reduced the polarization of the sensing state, |⟨ Ŝ⟩| ≈ N/2, the total estimation error in the presence of projection and detection noise is Thus, in the presence of nonzero detection noise, spin squeezing reduces ∆ϕ only if Ξ 2 det < ξ 2 R , i.e., even a small level of detection noise will ultimately limit the estimation error if spin squeezing becomes strong enough.
To benefit from spin squeezing even in cases where detection noise is appreciable, Davis et al. [26] proposed a twist-untwist sensing protocol that amplifies the signal encoded in the spin ensemble prior to readout.It allows one to achieve a sensitivity below the standard quantum limit even if Ξ 2 det ≈ 1.The basic principle of this amplification mechanism is sketched in Fig. 3: The spin system is initialized in a coherent-spin state where all spins are polarized in the x direction, Ŝx |ψ 0 ⟩ = N/2 |ψ 0 ⟩.In the first "twist" step, the OAT Hamiltonian ĤOAT defined in Eq. ( 1) is applied for a time t sqz to transform |ψ 0 ⟩ into a spin-squeezed state |ψ 1 ⟩ = e −i ĤOATtsqz |ψ 0 ⟩.This state is used as the initial state of a sensing protocol that effectively encodes a small signal ϕ ≪ 1 of interest by rotating the state about the y axis, The resulting nonzero Ŝz polarization ⟨ψ 2 (ϕ)| Ŝz |ψ 2 (ϕ)⟩ ∝ ϕ is amplified in the last "untwist" step, where |ψ 2 (ϕ)⟩ is evolved using − ĤOAT for a time t unsqz , |ψ 3 (ϕ)⟩ = e +i ĤOATtunsqz |ψ 2 (ϕ)⟩.Choosing t unsqz = t sqz ensures that |ψ 3 (ϕ)⟩ is close to a coherent-spin state but, due to the nonlinearity of the OAT Hamiltonian, the final Ŝy polarization is a gain factor G ≫ 1 larger than if one simply rotated the initial state |ψ 0 ⟩ by an angle ϕ about the z axis, The amplification dynamics generated by the second application of ĤOAT can be understood in terms of Heisenberg equations of motion of the system [30].The OAT Hamiltonian (1) generates a dynamics of the form Since Ŝz commutes with ĤOAT , ⟨ Ŝz ⟩ is a constant of motion and the dynamics has a quantum-nondemolition structure: As long as ⟨ Ŝx ⟩ ≈ N/2 is positive, the ⟨ Ŝy ⟩ polarization will grow linearly in time at rate that is ≈ N χ⟨ Ŝz ⟩.The curvature of the Bloch sphere leads to deviations of this linear growth because ⟨ Ŝx ⟩ decreases with increasing ⟨ Ŝy ⟩, such that ⟨ Ŝy ⟩ takes a maximum value after an evolution time t ideal amp ≈ 1/χ √ N for N ≫ 1.At this time, ⟨ Ŝy ⟩ is an amplified version of the original signal stored in ⟨ψ 2 (ϕ)| Ŝz |ψ 2 (ϕ)⟩ ∝ ϕ with a gain factor The gain G ideal max quantifies the amplification of the signal encoded in the spin ensemble and should not be confused with the metrological gain G met defined in Eq. ( 13).
Since the detection noise is unchanged by the amplification process, the second OAT step improves the signalto-noise ratio of the readout process such that the final estimation error is given by (∆ϕ Comparing this result with Eq. ( 14), one can interpret the impact of amplification in Eq. ( 18) as a suppression of the detection noise, Ξ 2 det → Ξ 2 det /(G ideal max ) 2 = eΞ 2 det /N , combined with an effective squeezing operation, ] is shown by the solid black line in Fig. 1.
Note that, instead of using a twist and an untwist step with different signs of the OAT interaction strength, one can also rotate the squeezed state after the first twist step, encode the signal by a rotation about the Ŝy axis, and then use another twist step with the same sign of the OAT interaction strength to amplify the signal and convert the fluctuations back to those of a coherent spin state.This strategy was implemented in Ref. [11].

B. Impact of dissipation on the sensing protocol
The dissipative terms in the effective QME (8) lead to important modifications of the ideal amplification protocol outlined in Sec.IV A: both collective and local dissipation cause dephasing of the sensor that reduces ⟨ Ŝx ⟩ and ⟨ Ŝy ⟩ during the amplification step and thus competes with the unitary OAT dynamics that attempts to increase ⟨ Ŝy ⟩.Moreover, local relaxation and excitation processes break the conservation of ⟨ Ŝz ⟩ and destroy the quantum-nondemolition structure of the amplification process shown in Eq. ( 16), where a static signal encoded in the ⟨ Ŝz ⟩ polarization is transduced into the ⟨ Ŝy ⟩ polarization.Both of these effects reduce the gain to Finally, the effects of decoherence during the first "twist" step will be amplified during the subsequent "untwist" step, such that the projection noise of the final state may be larger than that of a coherent-spin state: where σ 2 diss ≥ 0 captures the enhanced fluctuations of the final state, expressed in multiples of the fluctuations N/4 of a coherent-spin state.Note that Eq. ( 20) is an implicit definition of σ 2 diss , i.e., no additional assumptions on the form of σ 2 diss are necessary.One can extract G and σ 2 diss from a (numerical) solution of Eq. ( 8) for small but nonzero ϕ, which provides one with ⟨ Ŝy (t amp )⟩ and (∆S y ) 2 (t amp ).
Combining all of these effects, one obtains the following form of the estimation error in the presence of dissipation: This result shows that, in the limit of large readout noise, Ξ 2 det ≫ 1 + σ 2 diss , the metrological gain depends only on the ability to create large gain G in the amplification process, G met ≈ G 2 /Ξ 2 det .In the opposite regime of small readout noise, Ξ 2 det ≲ 1 + σ 2 diss , the maximum metrological gain depends both on the enhanced fluctuations σ 2 diss and on the gain G, G met ≈ G 2 /(1 + σ 2 diss ).Davis et al. [26] analyzed the performance of the SCF scheme in the latter regime using a phenomenologically motivated expression for σ 2 diss , Here, the first term describes growth of the projection noise of the squeezed state due to collective dephasing and the last term accounts for fluctuations of the Ŝz component due to spin-flip processes mediated by the excited state, where (∆S sc z ) 2 = N χt(2δ/κ + κ/2δ)/6η.A similar analysis has been performed by Chu et al. [27], which is based on analytical solutions for the special cases where only collective dephasing or only local dissipation is present and then combines these results to find an approximate solution of the full problem.A key assumption of these works is that the gain as a function of time is not reduced by dissipation, G ≈ G ideal max , such that the contributions to the estimation error stemming from projection noise and detection noise are negligible compared to σ 2 diss /G 2 .The detuning δ can then be chosen to minimize Eq. ( 22), leading to a scaling of the metrological gain with the collective cooperativity, This estimate suggests that achieving a large metrological gain simply requires having a sufficiently large collective cooperativity N η.In Fig. 1, we compare this result (dashed grey line) with a numerical optimization of G met based on MFT (thick green curve), see Sec.VI for details.We see that the result of Davis et al. underestimates the minimum collective cooperativity N η required to achieve G met > 1 by almost an order of magnitude (highlighted by the red shaded area in Fig. 1), even though it does capture the correct scaling of G met with N η (up to a prefactor of ≈ 1.5) at extremely large cooperativities.The discrepancy at modest N η suggests that dissipation impacts G met more severly than what is captured in the perturbative analysis of Ref. 26.Indeed, as we show in Sec.V below, dissipation can actually reduce the gain G even if N η ≫ 1, such that the first term in Eq. ( 22) is underestimated, which in turn affects the optimization of the detuning δ and the optimal twisting time.These all lead to the discrepancy between the full results and the simple estimate seen in Fig. 1.

V. GAIN IN THE PRESENCE OF DISSIPATION
We now start our more rigorous analysis of how dissipation impacts the different implementations of the OAT twist-untwist amplification protocol.In a first step, we analyze how the dissipative terms in the QME (8) reduce the achievable gain G.As discussed in Sec.IV, this gain G is the relevant quantity to be maximized in the limit of large readout noise, Ξ 2 det ≫ 1.In a second step, the impact of dissipation on the metrological gain G met will be analyzed in Sec.VII.
Since spin-squeezing is not improving the estimation error (∆ϕ) in the regime Ξ 2 det ≫ 1, we ignore the first squeezing step for now, i.e., we set t sqz = 0 and use a coherent spin state rotated about the Ŝy axis as our initial state, The restriction t sqz = 0 does not matter if there is only unitary dynamics generated by ĤOAT .If there is additional dissipative dynamics [as described by Eq. ( 8)], setting t sqz to zero makes a difference since dissipation acting during the first twist step can depolarize the state |ψ 2 (ϕ)⟩ and increase its fluctuations, which in turn affects the subsequent amplification dynamics during the untwist step.However, we found that the results for G do not change significantly if the initial squeezing step is omitted.More details can be found in Appendix C, where we analyze the gain G of the full twist-untwist protocol.
In analogy with the definitions in Eqs. ( 17) and ( 19), we define the maximum gain G(η) in the presence of dissipation with a given single-spin cooperativity η as Here, t amp is the time over which |ψ(ϕ)⟩ has been evoved under ĤOAT and λ is a detuning parameter that can be optimized experimentally: λ = ∆ TC /κ for the TC scheme and λ = δ/κ for the SCF and ACF schemes.It effectively controls the relative strength of the collective and single-spin dissipation (cf.Table I).

A. TC implementation of OAT dynamics
The impact of dissipative processes on the gain in a TC implementation of OAT dynamics has been analyzed in detail in Ref. 30: the collective and single-spin relaxation processes cause ⟨ Ŝz ⟩ to decay towards the joint ground state, which leads to a nonzero ⟨ Ŝy ⟩ polarization even if no signal has been applied, ϕ = 0. Therefore, the definition (25) of the gain needs to be modified by subtracting this background, where δ⟨ Ŝy (t amp )⟩ = ⟨ Ŝy (t amp , ϕ)⟩ − ⟨ Ŝy (t amp , 0)⟩.The need for a background subtraction makes the TC implementation of OAT dynamics practically useless in many sensing applications.Even with this background subtraction, it was found that a significant fraction of the ideal gain, e.g., G sub (η)/[lim η→∞ G sub (η)] = 1/2, can only be achieved for large single-spin cooperativities η ϕ ≫ √ N and η rel ≫ N 0.9 [30].This condition is much more stringent than a condition in terms of the collective cooperativity N η ϕ,rel , since it becomes harder to satisfy with increasing size of the spin ensemble.

B. SCF implementation of OAT dynamics
Comparing the dissipation rates of the TC scheme with those of the SCF scheme (see Table I), we find that the collective decay of the TC scheme has been replaced by collective dephasing, and that single-spin relaxation is counteracted by an equally strong single-spin excitation process.Therefore, if no signal is applied, ⟨ Ŝz ⟩ = 0 remains zero and there is no growth of ⟨ Ŝy ⟩.Consequently, no background subtraction is needed in the SCF scheme and we can define the gain according to Eq (25).However, the dynamics is still different from the purely unitary case given by Eq. ( 16) since ⟨ Ŝz ⟩ is no longer a constant of motion of L SCF spin and dephasing will reduce the ⟨ Ŝx ⟩ and ⟨ Ŝy ⟩ polarizations.Therefore, even though the amplification process for SCF-based OAT dynamics appears to be more robust than the one of the TC scheme, one still expects it to be compromised in the presence of collective and single-spin dissipation.To quantify this effect, we numerically calculate the gain G(η) by integrating the QME (8) for fixed χ SCF and different values of η as well as δ/κ, starting from the initial state |ψ(ϕ)⟩ defined in Eq. (24).We maximize the gain G(η) over t amp and δ/κ (see Appendix C for details).The results are shown by the markers in Fig. 4.
As shown in Fig. 4(a), the onset of gain dynamics is controlled by the collective cooperativity: When plotted as a function of N η, all curves of the bare gain G(η) obtained for different system sizes collapse for G(η) ≪ 1 and N η ≲ 10.In this regime, however, collective dephasing and single-spin dissipation dominate and reduce TABLE II.Overview of the conditions on the cooperativity to reach O(1/2) of the maximum possible gain Gmax = limη→∞ G(η) in different amplification schemes.
For larger collective cooperativities, the curves quickly fan out and ultimately converge to the respective ideal values G max = lim η→∞ G(η) ∝ √ N in the absence of dissipation.The relevant figure of merit to quantify the amplification performance is the fraction G(η)/G max of gain that can be achieved for a given single-spin cooperativity η compared to the maximally possible gain G max .In Fig. 4(b), we show that the rescaled gain curves collapse quite well for a broad range of ensemble sizes N when plotted as a function of the scaled cooperativity √ N η (which is smaller than the collective cooperativity by a large factor √ N ).This suggests that the condition to observe a significant fraction of the ideal gain G max (e.g., Thus, even if the collective cooperativity is large, N η ≫ 1, the actual gain G(η) can still be significantly smaller than G max .
The exponent α = 1/2 of the prefactor N α in Eq. ( 27) can be rigorously extracted from the numerical data by plotting G(η)/G max as a function of η, identifying the threshold values η thres (f ) where G(η)/G max = f equals a certain fraction 0 < f < 1, and fitting the corresponding values of η thres (f ) as a function of N .As shown in Fig. 4(c), the exponent for all considered ratios f < 0.8 is consistent with α = 1/2.

C. ACF implementation of OAT dynamics
One may wonder if the ACF scheme performs better than the SCF scheme analyzed so far because it does not suffer from single-spin excitation and relaxation processes (see Table I).To check if the absence of these processes changes the amplification dynamics, we repeat the same analysis for the ACF scheme.We find that omitting single-spin excitation and relaxation does not change the physics significantly, and that the threshold condition to achieve G(η)/G max ≈ 1/2 is still given by Eq. ( 27).More details can be found in Appendix D. (a) Gain G(η) of the SCF scheme in the presence of undesired collective and single-spin dissipation, defined in Eq. ( 25), as a function of the collective cooperativity N η.The solid lines have been obtained using a numerical optimization of the gain using a second-order mean-field theory (MFT), see Sec.VI.The data points of the corresponding color have been obtained from a numerically exact solution of the quantum master equation ( 8

D. Comparison of the schemes
A summary of the conditions that are required to reach O(1/2) of the maximum gain in the different amplification schemes is given in Table II.Compared to a TCbased implementation of OAT, both the SCF and the ACF schemes feature an experimentally much more favorable condition on the required cooperativity, since it becomes easier to satisfy Eq. ( 27) with increasing ensemble size N .However, Eq. ( 27) is still more challenging than a condition in terms of the collective cooperativity, N η ϕ,rel ≳ 1, which governs dissipative amplification protocols based on superradiant decay [30,31].In atomic platforms, the difference between a condition in terms of √ N η and one in terms of the collective cooperativity N η may in practice not be very relevant, since large singlespin cooperativities η ≈ 10 [16] can readily be achieved and ensemble sizes can be quite large, N ≳ 10 5 [11].In solid-state platforms, however, the single-spin cooperativities are typically much lower [34], such that reaching a collective cooperativity exceeding unity can already be a challenge.
Importantly, undesired dissipative processes can significantly reduce the gain G(η) in all of the three implementations of OAT dynamics.This raises the question if non-perturbative effects will modify the analysis of the metrological gain performed by Davis et al. [26], since it is based on the assumption of a constant gain G(η) ≈ G max .We will address this question in Sec.VII below.

VI. ANALYSIS AND INTERPRETATION USING MEAN-FIELD THEORY
To get a better analytical understanding for the unusual scaling relation of the gain, Eq. ( 27), which is in between a collective-cooperativity and a single-spincooperativity criterion, and to analyze larger ensemble sizes N that are out of reach for a numerical integration of the QME (8), we now analyze the amplification dynamics using second-order MFT [35][36][37].This approach allows us to approximate the quantum dynamics by a closed set of differential equations of motion for the first moments S α = ⟨ Ŝα ⟩ and the (co)variances C αβ = ⟨{ Ŝα , Ŝβ }⟩/2−⟨ Ŝα ⟩⟨ Ŝβ ⟩ with α, β ∈ {x, y, z}.The complete set of MFT equations of motion corresponding to the QME ( 8) is given in Appendix E. Here, we only list the results for the first moments for Γ rel = 0, Note that, in the absence of dissipation, Γ ϕ = γ + = γ − = γ z = 0, and ignoring the covariance C xz , Eq. (28b) reproduces the Heisenberg equation of motion given in Eq. ( 16).
The MFT results for the gain, shown by the solid curves in Fig. 4, are obtained as follows.The system is initialized in a coherent-spin state pointing along the x axis, S x = N/2, C yy = C zz = N/4, and all other moments and (co)variances are set to zero.Starting from this initial state, the MFT equations of motion listed in Appendix E are numerically integrated for fixed χ SCF and different values of η as well as δ/κ.For each value of η the gain G(η) defined in Eq. ( 25) is calculated and maximized over the evolution time t amp and the ratio δ/κ determining the relative strength between collective dephasing and single-spin dissipation.
Since our MFT is based on the assumption that the state of the system can be approximated by a Gaussian state, deviations from the exact dynamics based on the full QME will occur at long evolution times.However, they provide a reasonable approximation of the full dynamics until the time of maximum gain.Therefore, by solving the MFT equations of motion numerically, one can typically obtain accurate predictions of the exponents of scaling laws in N , but the numerical prefactors may be off.As shown in Figs.4(a) and (b), the MFT results agree very well with the results obtained for small N by solving Eq. ( 8) numerically, but there is a small offset in the prediction of the gain in the ideal limit η → ∞, G ideal max | MFT > G ideal max .Similarly, t ideal amp obtained from a MFT simulation differs from the analytical result t ideal amp ≈ 1/χ √ N .Therefore, we extract G ideal max (as well as the corresponding limits of all other quantities, e.g., t ideal amp ) from the numerical data at large η rather than using the analytical results for rescaling.
Given the good agreement between MFT results and the data based on a numerically exact integration of the QME (8), we can use the MFT equations of motion to gain analytical insight into the amplification physics by deriving an intuitive argument for the surprising condition shown in Eq. (27).For √ N η ≳ 1, the performance of the SCF scheme is limited by single-spin dephasing, which can be inferred from the optimal value of δ/κ ≈ 1/2 for η ≲ 1, shown in Appendix C. We will therefore ignore collective dephasing, Γ ϕ ≈ 0, and set γ − = γ + = 2γ z = χ/η.In order to be insensitive to the single-spin dissipation and to obtain the maximum possible gain, the OAT dynamics should amplify the signal much faster than the dissipative timescale η/χ, which immediately leads to the condition √ N η ≫ 1.The essence of this argument also emerges if we perform a more quantitative analysis that is based on an approximate solution of the MFT equations of motion.At short times, we can ignore the covariances C yz and C xz in Eqs.(28a) and (28b) and integrate the resulting equations of motion: where we introduced the auxiliary function Starting from an initial state S x (0) = N/2, S y (0) = 0, S z (0) = N ϕ/2 with ϕ ≪ 1, neglecting collective dephasing, Γ ϕ = 0, and setting γ − = γ + = 2γ z = χ/η, we can now evaluate Eq. ( 25) and obtain the following expression for the gain, Inserting the OAT amplification time of an ideal system, t ideal amp = 1/χ √ N , and expanding the expression for √ N η ≫ 1, we find the approximate expression which approaches the correct scaling relation for the ideal gain if the condition ( 27) holds, lim η→∞ G(1/χ √ N , η) = √ N /2, and clearly displays an exponential suppression if √ N η is small.

VII. METROLOGICAL GAIN IN THE PRESENCE OF DISSIPATION
As discussed in Sec.IV A, the ultimate figure of merit for the performance of a OAT twist-untwist sensing scheme is the metrological gain G met defined in Eq. ( 13).In the regime of large readout noise, Ξ 2 det ≫ 1 + σ 2 diss , the metrological gain G met is proportional to the squared gain G and, consequently, a large G met will be achieved if the conditions summarized in Table II are satisfied.
Given that the gain physics in the presence of dissipation is already quite intricate and leads to an unusual scaling with cooperativity, cf.Eq. ( 27), we now turn our attention to the regime of moderate readout noise, Ξ 2 det ≲ 1 + σ 2 diss .In this regime, Davis et al. [26] argued that the metrological gain of the SCF scheme should be controlled by the collective cooperativity, G met ∝ √ N η.
Since their argument neglects the reduction of the gain G(η) by dissipation, corrections to this result are to be expected.Indeed, Fig. 1 surprisingly shows that G met ≪ √ N η is possible even though both the singlespin and the collective cooperativity are large.We analyze this behavior more closely in Fig. 5(a), which shows the maximum metrological gain G met (η) for the SCF scheme as a function of the collective cooperativity N η.The data shown by the markers represent a simulation of the full OAT twist-untwist sensing protocol sketched in Fig. 3 and described in Sec.IV A, which has been obtained as follows: The system is initialized in a coherent-spin state |ψ 0 ⟩ polarized in the x direction and is evolved by integrating the QME (8) for a time t sqz with a OAT strength χ to obtain a spin-squeezed state ρ1 .Subsequently, the signal ϕ is encoded into the  38), as a function of the collective cooperativity N η for (a) the SCF scheme and (b) the ACF scheme and different spin numbers N .In both subplots, the dashed gray line indicates the estimate est met for the SCF scheme, defined in Eq. ( 23), which has been derived in Ref. 26.It overestimates the onset of metrological gain even though the collective cooperativities N η are large.The dash-dotted gray line in (b) indicates a Gmet = 0.1N η scaling.The solid lines have been generated using the MFT equations of motion and the markers have been calculated by numerically exact integration of the QME (8) for small ensemble sizes N .The level of readout noise is Ξ 2 det = 1.0.Note that the abrupt changes in the slope of Gmet for the MFT curves are artifacts of the small number of sample points.
Finally, the rotated state ρ2 (ϕ) is evolved by integrating Eq. ( 8) for a time t unsqz = t sqz using the OAT strength −χ to obtain the final state ρ3 (ϕ) at time t end = t sqz + t unsqz = 2t (un)sqz .This procedure is repeated for different values of η and λ = δ/κ.As described in Sec.IV B, the gain G and the additional fluctuations σ 2 diss can be obtained from the expectation value ⟨ Ŝy (t end )⟩ and the variance (∆ Ŝy ) 2 of ρ3 (ϕ) using the definition and Eq. ( 20), respectively.Using these results as well as Eqs.( 13) and ( 21), the optimum metrological gain G met (η) for a given single-spin cooperativity η can be calculated using The solid lines in Fig. 5 correspond to a MFT simulation of the dynamics generated by Eq. ( 8), which allows us to investigate larger ensemble sizes N .Additional details on the numerical optimization of the metrological gain can be found in Appendix F. We compare our simulations with the dashed gray curve, which represents the result G est met for the SCF scheme [cf.Eq. ( 23)], which has been derived in Ref. 26 using phenomenological arguments.While it does capture the scaling of the metrological gain correctly in the regime G ideal met ≫ G met (η) ≫ 1, it is overly optimistic in a broad regime of parameters where the collective cooperativity is large, N η ≫ 1, but the metrological gain is small, G met ≳ 1.In particular, the onset of metrological gain is highly overestimated: G est met suggests that a moderate metrological gain of 5 dB requires a collective cooperativity of N η ≈ 100 for N = 100 spins.However, we find that the collective cooperativity that is actually required to reach G met = 5 dB is almost an order of magnitude higher, N η ≈ 1000.Finally, for small N ≲ 10 4 , the metrological gain depends not only on the collective cooperativity N η, but also on the number N of spins in the ensemble.Only for the highest considered spin numbers, the curves of G met for different N converge to a common behavior that grows ∝ √ N η for G met ≫ 1 before fanning out and saturating to the spin-number-dependent maximum metrological gain G ideal met = N/[e(1 + Ξ 2 det )] at large collective cooperativity.
Interestingly, the different dissipative processes of the SCF and the ACF schemes manifest themselves in the behavior of the metrologial gain for Ξ 2 det ≲ 1 + σ 2 diss , as shown in Fig. 5(b): The metrological gain of the ACF exceeds that of the SCF scheme and scales approximately linearly with the collective cooperativity G met ∝ N η [given by the dash-dotted gray line in Fig. 5(b)] before saturating to G ideal met .This corrobates the results reported in Ref. 16.
In Fig. 1, we compare the TC scheme with the ACF and SCF schemes for a fixed system size.We find that the TC scheme follows a similar G met ∝ √ N η scaling as the SCF scheme if G ideal met ≫ G met (η) ≫ 1, i.e., in an intermediate regime where the metrological gain is large but not yet saturating to its ideal value G ideal met .However, as for the SCF scheme, the onset of metrological gain is highly overestimated.Moreover, the TC-based implementation of a OAT twist-untwist protocol requires a much higher collective cooperativity than the SCF and ACF schemes to reach similar levels of G met .This is a result of the collective decay term in the TC scheme (cf.Table I) which highly disrupts the amplification dynamics (as described in Sec.V A) and also led to a much more stringent requirement to achieve significant levels of gain (cf.Table II).

VIII. CONCLUSION
In conclusion, we analyzed three different implementations of the OAT Hamiltonian, shown in Fig. 2, and compared their ability to amplify a small signal encoded in a nonzero ⟨ Ŝz ⟩ polarization.The three schemes differ in the form of undesired dissipative terms that accompany the unitary OAT interaction, which are summarized in Table I.Despite the fact that these differences do not lead to different conditions to generate spin squeezing, they do manifest themselves in very different conditions that need to be satisfied in order to see significant levels of gain G, which are summarized in Table II.Interestingly, the SCF and ACF schemes provide large gain if the scaled cooperativity √ N η is large.We used second-order MFT to provide an intuitive explanation for this unusual scaling.While this condition is less challenging than the one required by the TC scheme, it is experimentally more challenging than reaching a large collective cooperativity, which is required by amplification schemes based on collective dissipation [30,31].
Motivated by these surprising results, we revisited the metrological gain G met of the SCF, ACF, and TC schemes.In the regime where readout noise dominates, Ξ 2 det ≫ 1 + σ 2 diss , all implementations of the OAT twistuntwist scheme have G met ∝ G 2 /Ξ 2 det , i.e., large metrological gain can be achieved whenever G is large.In the opposite regime, Ξ 2 det ≲ 1 + σ 2 diss , we found that the ACF scheme provides parametrically larger G met than the SCF and TC schemes.Moreover, we found that the prediction G met ∝ √ N η, which had been derived in Refs.26 and 27 using a phenomenologically motivated model and the approximation that the action of different dissipative processes can be analyzed separately, respectively, highly overestimates the performance of the SCF and TC schemes when G met ≳ 1: Even though the collective cooperativity may be large, MFT simulations and numerically exact integration of the quantum master equation ( 8) for the SCF and TC schemes show that the dissipative processes can interrupt the amplification dynamics and lead to a much smaller metrological gain than previously appreciated.
Our study highlights that the form of undesired dissipative processes can have a huge impact on the performance of a OAT twist-untwist sensing protocol and that a careful deliberation is needed when choosing the specific implementation of OAT dynamics on a given quantum sensing platform.AMO systems provide large single-spin cooperativities and low readout noise, such that the ACF scheme can be implemented.Solid-state sensing platforms, however, are typically plagued by much higher levels of readout noise and reaching a large single-spin cooperativity can be significantly harder than achieving a large collective cooperativity.Therefore, even though cavity-feedback schemes could in principle be implemented in solid-state systems [29], amplification schemes based on collective dissipation [30,31] are expected to be more effective on these platforms.
The twist-untwist sensing protocol can also be implemented using two-axis-twist dynamics [38,39] or twistand-turn dynamics [40].Even though the two-axis-twist Hamiltonian generates higher levels of spin squeezing than OAT [4], it leads to parametrically the same metrological gain in a unitary twist-untwist sensing protocol [38].In the future, it would be interesting to check if this remains true if undesired dissipative processes are taken into account, and to compare the levels of metrological gain that can be achieved in different implementations of the two-axis-twist Hamiltonian in the presence of dissipation.In contrast to the SCF and ACF schemes, the cavity is undriven and therefore in its ground state.This allows us to ignore the −2χ TC â † â Ŝz term in ĤTC (which would otherwise lead to dephasing of the spins due to photonnumber fluctuations in the cavity).Moreover, we have |χ TC | ≪ ω s , such that we can ignore the small renormalization of the spin frequency.After an adiabatic elimination of the cavity mode, we thus find the effective QME This result can be rewritten in the form of Eq. ( 8) using the frequencies and rates shown in Table I.

SCF coupling scheme
For the SCF coupling scheme defined by the QME (2) and Eq. ( 4), we can diagonalize the undriven Hamiltonian Ĥ0 + ĤSCF where we defined the intracavity photon number n cav = |a| 2 due to the drive, cf.Eq. (6).To ignore higher-order correction terms to the Schrieffer-Wolff transformation defined in Eq. (A6), the condition |a| 2 ≪ (∆/g) 2 must hold.This implies Ω SCF n cav ≪ 2∆ and thus allows us to ignore the renormalization of the spin transition frequency.Finally, we adiabatically eliminate the cavity fluctuations d by assuming that the cavity relaxes to its steady state fast compared to the dissipative dynamics of the spins, which yields the effective QME where we defined the OAT strength

6 20 40
FIG.1.Metrological gain Gmet = 1/N (∆ϕ) 2 as a function of the collective cooperativity for a sensing protocol using timereversal of one-axis-twist (OAT) dynamics.We use N = 10 5 spins and include readout noise equal to the standard projection noise of a coherent-spin state (Ξ 2 det = 1.0).Thick solid curves correspond to different implementations of the OAT Hamiltonian [symmetric cavity feedback (SCF), asymmetric cavity feedback (ACF), and a Tavis-Cummings (TC) interaction with single-spin dephasing or single-spin relaxation].Solid black line: ideal metrological gain G ideal met that can be achieved at zero dissipation.The dashed gray curve shows the expression G est met , defined in Eq. (23), which has been derived in Ref. 26 using a heuristic model of dissipative effects and applies to the SCF and TC schemes.As shown by the thick SCF and TC curves, nonperturbative effects of dissipation can lead to strong deviations from this result even though the collective cooperativity N η is large.In particular, the collective cooperativity required to reach Gmet > 1 is underestimated by almost an order of magnitude (shaded red area).The dash-dotted gray curve indicates Gmet = 0.1N η.
FIG.2.Sketch of the three considered schemes to experimentally implement the one-axis-twist (OAT) Hamiltonian ĤOAT, defined in Eq. (1), in an ensemble of N spin-1/2 systems with levels {|↑⟩ i , |↓⟩ i }i=1,...,N .(a) ĤOAT can be implemented using a direct Tavis-Cummings (TC) coupling between the spin-1/2 systems and a detuned cavity mode with frequency ωc and detuning ∆.Undesired dissipative processes are single-spin relaxation (rate γ rel ), single-spin dephasing (rate γ ϕ ), and cavity decay (rate κ).Alternatively, the OAT Hamiltonian can be implemented using feedback from a driven cavity mode, with drive frequency ω dr and ampliude β in , either by (b) symmetric cavity feedback (SCF), where both |↑j⟩ and |↓j⟩ are coupled to the excited level |ej⟩ via the cavity mode with equal but opposite-in-sign detuning ∆, or by (c) asymmetric cavity feedback (ACF), where only |↑j⟩ is coupled to |ej⟩ via the cavity mode with detuning ∆.Undesired dissipative processes in the cavity-feedback schemes are decay of the excited level (rate Γ) and cavity decay.

FIG. 3 .
FIG. 3. Sketch of spin amplification using unitary oneaxis-twist (OAT) dynamics.(a) The system is initialized in a coherent-spin state pointing along the Ŝx direction.(b) The initial state is evolved under the OAT Hamiltonian (1) for a time tsqz to transform it into a spin-squeezed state.(c) The signal of interest is encoded in the Ŝz component of the state using a rotation about the Ŝy axis.(d) Finally, the OAT Hamiltonian with opposite sign is applied for a time tunsqz, which returns the fluctuations of the state to the level of a coherent-spin state and generates a nonzero Ŝy polarization that is an amplified version of the Ŝz polarization encoding the signal.
FIG. 4.(a) Gain G(η) of the SCF scheme in the presence of undesired collective and single-spin dissipation, defined in Eq. (25), as a function of the collective cooperativity N η.The solid lines have been obtained using a numerical optimization of the gain using a second-order mean-field theory (MFT), see Sec.VI.The data points of the corresponding color have been obtained from a numerically exact solution of the quantum master equation (8) for small spin numbers N , see Sec.IV B and Appendix C. The collective cooperativity controls only the onset of gain at N η ≲ 1 and G(η) ≪ 1.A gain of unity, G = 1 is indicated by the dotted black line.(b) Same data for G(η), normalized to the ideal value Gmax = limη→∞ G(η) in the absence of dissipation, collapses when plotted as a function of the scaled single-spin cooperativity √ N η.(c) Fits of the threshold values η thres (f ), defined by G(η thres )/Gmax = f , to extract the exponent of N leading to optimal collapse of the curves in (b).The fits have been performed over the range 100 ≤ N ≤ 10 4 and are based on the MFT results.
FIG. 4.(a) Gain G(η) of the SCF scheme in the presence of undesired collective and single-spin dissipation, defined in Eq. (25), as a function of the collective cooperativity N η.The solid lines have been obtained using a numerical optimization of the gain using a second-order mean-field theory (MFT), see Sec.VI.The data points of the corresponding color have been obtained from a numerically exact solution of the quantum master equation (8) for small spin numbers N , see Sec.IV B and Appendix C. The collective cooperativity controls only the onset of gain at N η ≲ 1 and G(η) ≪ 1.A gain of unity, G = 1 is indicated by the dotted black line.(b) Same data for G(η), normalized to the ideal value Gmax = limη→∞ G(η) in the absence of dissipation, collapses when plotted as a function of the scaled single-spin cooperativity √ N η.(c) Fits of the threshold values η thres (f ), defined by G(η thres )/Gmax = f , to extract the exponent of N leading to optimal collapse of the curves in (b).The fits have been performed over the range 100 ≤ N ≤ 10 4 and are based on the MFT results.

5 FIG. 5 .
FIG.5.Metrological gain Gmet, defined in Eq.(38), as a function of the collective cooperativity N η for (a) the SCF scheme and (b) the ACF scheme and different spin numbers N .In both subplots, the dashed gray line indicates the estimate est met for the SCF scheme, defined in Eq. (23), which has been derived in Ref.26.It overestimates the onset of metrological gain even though the collective cooperativities N η are large.The dash-dotted gray line in (b) indicates a Gmet = 0.1N η scaling.The solid lines have been generated using the MFT equations of motion and the markers have been calculated by numerically exact integration of the QME (8) for small ensemble sizes N .The level of readout noise is Ξ 2 det = 1.0.Note that the abrupt changes in the slope of Gmet for the MFT curves are artifacts of the small number of sample points.

TABLE I .
Parameters of the effective QME