Engineering spin squeezing in a 3D optical lattice with interacting spin-orbit-coupled fermions
One of the most important tasks in modern quantum science is to coherently control and entangle many-body systems, and to subsequently use these systems to realize powerful quantum technologies such as quantum-enhanced sensors. However, many-body entangled states are difficult to prepare and preserve since internal dynamics and external noise rapidly degrade any useful entanglement. Here, we introduce a protocol that counterintuitively exploits inhomogeneities, a typical source of dephasing in a many-body system, in combination with interactions to generate metrologically useful and robust many-body entangled states. Motivated by current limitations in state-of-the-art three-dimensional (3D) optical lattice clocks (OLCs) operating at quantum degeneracy, we use local interactions in a Hubbard model with spin-orbit coupling to achieve a spin-locking effect. In addition to prolonging inter-particle spin coherence, spin-locking transforms the dephasing effect of spin-orbit coupling into a collective spin-squeezing process that can be further enhanced by applying a modulated drive. Our protocol is fully compatible with state-of-the-art 3D OLC interrogation schemes and may be used to improve their sensitivity, which is currently limited by the intrinsic quantum noise of independent atoms. We demonstrate that even with realistic experimental imperfections, our protocol may generate – dB of spin squeezing in second with – atoms. This capability exemplifies a new paradigm of using driven non-equilibrium systems to overcome current limitations in quantum metrology, allowing OLCs to enter a new regime of enhanced sensing with correlated quantum states.
A major frontier of contemporary physics is the understanding of non-equilibrium behaviors of many-body quantum systems, and the application of these behaviors toward the development of novel quantum technologies with untapped capabilitiesEisert et al. (2015). To this end, ultracold atomic, molecular, and optical systems are ideal platforms for studying unexplored regimes of many-body physics due to their clean preparation and readout, high controllability, and long coherence timesBloch et al. (2008); Gross and Bloch (2017). The exquisite capabilities of these systems have pushed the frontiers of metrology, quantum simulation, and quantum information science.
Optical lattice clocks in particular have seen some of the most impressive developments in recent years, reaching record levels of precision ()Campbell et al. (2017); Marti et al. (2018) and accuracy ()Bloom et al. (2014); McGrew et al. (2018). These advancements required important breakthroughs, including the capability to cool and trap fermionic alkaline-earth atoms in spin-insensitive potentialsTakamoto and Katori (2003); Barber et al. (2006); Ye et al. (2008); the development of ultra-coherent lasersKessler et al. (2012); Cole et al. (2013); Matei et al. (2017) to fully exploit an ultra-narrow clock transitionLudlow et al. (2015); and, more recently, the preparation of a quantum degenerate gas in a three-dimensional (3D) optical latticeCampbell et al. (2017); Marti et al. (2018); Goban et al. (2018). Nonetheless, all improvements in sensing capabilities to date have been based on single-particle control of internal atomic degrees of freedom. Such strategies will eventually have diminishing returns due to practical difficulties in (i) suppressing decoherence from external (motional) degrees of freedom, and (ii) interrogating more particles without additional systematic errors from interactionsMartin et al. (2013); Ludlow et al. (2015); Marti et al. (2018).
Pushing beyond the current independent-particle paradigm requires leveraging many-body quantum correlations. Entangled states such as spin-squeezed statesKitagawa and Ueda (1993); Wineland et al. (1992); Ma et al. (2011) can enhance measurement sensitivity, i.e. the uncertainty in the estimation of a parameter , below the standard quantum limit for uncorrelated particlesItano et al. (1993); Degen et al. (2017). The major challenge for progress in this direction is that generating entanglement requires interactions, which are generally undesirable because they degrade atomic coherence, thereby limiting clock performanceSwallows et al. (2011); Martin et al. (2013); Rey et al. (2014); Ludlow et al. (2011); Lemke et al. (2011); Ludlow et al. (2015). In fact, the most precise and accurate optical lattice clocks were designed to operate with fermionic atoms in identical nuclear and electronic states in order to suppress collisional decoherenceCampbell et al. (2009); Swallows et al. (2011); Campbell et al. (2017), as identical fermions cannot interact via the otherwise dominant (-wave) collisions at ultracold temperatures. However, an initially spin-polarized Fermi gas still exhibits interactions at later times due to spin-orbit coupling (SOC) that is induced by the laser that drives the clock transition (i.e. the “clock laser”)Wall et al. (2016); Kolkowitz et al. (2016); Livi et al. (2016); Bromley et al. (2018). Specifically, the momentum kick imparted by this laser imprints a position-dependent phase that induces inhomogenous spin precession and generates spin dephasing, thereby making atoms distinguishable and vulnerable to collisions. While a deep lattice can suppress SOC, it also intensifies the light scattering which currently limits the coherence time of the clockDörscher et al. (2018); Goban et al. (2018); Hutson et al. (2019).
In this work, we describe a scheme that can potentially lead to metrological advances in state-of-the-art optical lattice clocks through direct use of quantum entanglement by harnessing the interplay between nominally undesirable collisions and SOC. This scheme is made possible in the weak SOC regime by the formation of an interaction-energy gap that suppresses the SOC-induced population transfer from the exchange-symmetric Dicke manifold (spanned by spin-polarized and thus non-interacting states) to the remainder of Hilbert space. In this regime, the non-equilibrium dynamical processes induced by SOC result in all-to-all collective spin interactions that naturally generate spin-squeezed states in a manner robust to realistic experimental imperfections. To generate spin squeezing, our protocol only requires control over (i) the orientation of the clock laser and (ii) the optical lattice depth. These controls are straightforward to incorporate into current 3D clock interrogation sequences without sacrificing atom numbers or coherence times. Additionally, we show that by applying a modulated drive from the clock laser, one can further prepare states that saturate the Heisenberg limit for phase sensitivityKitagawa and Ueda (1993); Ma et al. (2011); Degen et al. (2017).
Ii From the Fermi-Hubbard model to one-axis and two-axis twisting
We consider fermionic atoms with two spin states (labeled and ) trapped in a 3D optical lattice. In this discussion, the spin states are associated with the two electronic states of a nuclear-spin-polarized gas. At sufficiently low temperatures, atoms occupy the lowest Bloch band of the lattice and interact only through -wave collisions. A schematic of this system is provided in Fig. 1(a), where tight confinement prevents motion along the vertical direction (), effectively forming a stack of independent 2D lattices. For simplicity and without loss of generality, however, we first consider the case when tunneling can only occur along one direction, , and thus model the system as living in one dimension.
An external laser with Rabi frequency and wavenumber along the tunneling axis resonantly couples atoms’ internal states through the Hamiltonian , where is a fermionic annihilation operator for an atom on site with internal state and is the position of site . This laser imprints a position-dependent phase that equates to a momentum kick when an atom changes internal states by absorbing or emitting a photon, thereby generating spin-orbit couplingWall et al. (2016); Livi et al. (2016). After absorbing the position dependence of the laser Hamiltonian into fermionic operators through the gauge transformation , which makes spatially homogeneous, the atoms are well-described in the tight-binding limit by the Fermi-Hubbard HamiltonianEsslinger (2010)
where is the nearest-neighbor tunneling rate; the SOC phase is the phase gained by spin-up atoms upon tunneling from site to site (in the gauge-transformed frame) with lattice spacing ; is the on-site interaction energy of two atoms; and is a number operator.
The Fermi-Hubbard Hamiltonian can be re-written in the quasi-momentum basis with annihiliation operators , where is a quasi-momentum and is the total number of lattice sites. In this basis, the single-particle Hamiltonian exhibits shifted dispersion relations that signify spin-orbit coupling [see Fig. 1(b)]:
When , interaction energies are too weak for collisions to change the occupancies of single-particle quasi-momentum modes. Atoms are then pinned to these modes, which form a lattice in quasi-momentum space [see Fig. 1(c)]Bromley et al. (2018). In this strong-tunneling limit, the Fermi-Hubbard Hamiltonian [Eqn. (1)] can be mapped to a spin- system with a collective ferromagnetic Heisenberg interaction and an inhomogeneous axial field, given byMartin et al. (2013); Rey et al. (2014); Bromley et al. (2018)
where is a collective spin operator; is a spin-1/2 operator for mode with components defined in terms of the Pauli matrices ; the sums over run over all occupied quasi-momentum modes; and is the SOC-induced axial field.
On its own, the collective Heisenberg term () in Eqn. (3) opens an energy gap , with the filling fraction of spatial modes, between the collective Dicke states and the remainder of Hilbert spaceRey et al. (2008); Martin et al. (2013); Norcia et al. (2018) with . Here and respectively label the eigenvalues of the collective spin operators and , with eigenvalues for non-negative and . The axial field generally couples states within the Dicke manifold to states outside it. In the weak SOC limit (i.e. ), however, the interaction energy gap suppresses population transfer between states with different total spin [see Fig. 1(d)]. In this regime, the virtual occupation of states outside the Dicke manifold can be accounted for perturbatively. The symmetries of SOC as expressed in Eqn. (3) dictate that this treatment should yield powers of when projected onto the collective Dicke manifold at higher orders in perturbation theory. At second order in perturbation theory (see Appendix A), we thus find that SOC effectively yields a one-axis twisting (OAT) model widely known to generate squeezing dynamicsKitagawa and Ueda (1993); Ma et al. (2011):
where is the mean and the variance of the axial field. The effect of the term is to generate a relative phase between states with different total spin and thus has no effect on dynamics restricted to a fixed . Note also that the collective spin rotation from can be eliminated by going into a rotating frame or by using a spin echo.
The entire protocol for preparing a squeezed state via OAT, sketched out in Fig. 1(e), reduces to a standard Ramsey protocol with a spin echo: after initially preparing a spin-down (i.e. ) polarized sample of ultracold atoms populating the lowest Bloch band of a lattice, a fast pulse is applied with the external interrogation laser to rotate all spin vectors into . The atoms then freely evolve for a variable time (possibly with spin-echo -pulses), after which the amount of metrologically useful spin squeezing, measured by the Ramsey squeezing parameter
can be determined experimentally from global spin measurements. Here is the mean collective-spin vector and is the variance of spin measurements along an axis orthogonal to , parameterized by the angle . Note that the above protocol concerns only the preparation of a spin-squeezed state, which ideally would be used as an input state for a follow-up clock interrogation protocol. SOC can be turned off during the latter protocol by properly re-aligning the clock laser.
The validity of the OAT model in Eqn. (3) relies on two key conditions concerning experimental parameter regimes. First, the on-site interaction energy should not be much larger in magnitude than the tunneling rate (clarified below); otherwise, one cannot assume frozen motional degrees of freedom and map the Fermi-Hubbard model onto a spin model. Second, the SOC-induced fields should be considerably smaller in magnitude than the interaction energy gap , as otherwise one cannot perturbatively transform SOC into OAT. These two conditions can be satisfied by appropriate choices of and the SOC phase , which are respectively controlled by tuning the lattice depth and changing the angle between the interrogation laser and shallow lattice axes [see Fig. 1(a)].
We demonstrate the importance of these conditions in Fig. 2, where we show numerical results from exact simulations of a 1D system with sites. Therein, optimal squeezing achievable under unitary dynamics is provided in dB, i.e. , while the time at which this squeezing occurs is provided in units of the nearest-neighbor tunneling time . At atom per lattice site, i.e. half filling of all atomic states in the lowest Bloch band, the spin model [Eqn. (3)] agrees almost exactly with the Fermi-Hubbard (FH) model [Eqn. (1)] up through . The agreement at half filling () is assisted by Pauli blocking of mode-changing collisions. Below half filling (), these two (FH and spin) models show good agreement at , while at mode-changing collisions start to become relevant and invalidate assumptions of the spin model. Note that we chose filling to demonstrate that our protocol should work even in this highly dopped case; in practice, optimized experiments are capable of achieving fillings closer to Brown et al. (2017). Interestingly, even with mode-changing collisions the Fermi-Hubbard model exhibits comparable amounts of squeezing to the spin model, and achieves this squeezing in less time. The spin and OAT models agree in the regime of weak SOC with , and exhibit different squeezing behaviors outside this regime as single-particle spin dephasing can no longer be treated as a weak perturbation to the spin-locking interactions.
The above scheme for OAT achieves optimal spin squeezing that scales as with minimal intervention, i.e. a standard Ramsey protocol. Further improvements upon this scheme can be made by introducing a time-dependent driving field that transforms the OAT Hamiltonian into a two-axis twisting (TAT) one. While the OAT model initially generates squeezing faster than the TAT model, the squeezing generation rate of OAT (measured in dB per second) falls off with time, while the squeezing generation rate for TAT remains approximately constant until reaching Heisenberg-limited amount of spin squeezing with sup (). Following the prescription in Ref. , we use the interrogation laser to apply an amplitude-modulated drive . If the modulation frequency satisfies and , where is the OAT squeezing strength in Eqn. (4) and is the zero-order Bessel function of the first kind, then up to (i) an term that contributes only overall phase factors, and (ii) an term that can be eliminated with a simple dynamical decoupling pulse sequence (see Appendix C), the effective Hamiltonian becomes or (see Appendix B), which squeezes an initial state polarized along the or axis, respectively.
Iii Experimental preparation of a squeezed state and practical considerations
Here we discuss specific implementations of the above protocols in the state-of-the-art 3D Sr optical lattice clock (OLC). This system has demonstrated the capability to load a quantum degenerate gas into a 3D lattice at the “magic wavelength” ( nm) for which both the ground () and first excited () electronic states of the atoms experience the same optical potentialCampbell et al. (2017). Furthermore, the 3D Sr OLC currently operates at sufficiently low temperatures to ensure vanishing population above the lowest Bloch band, such that its dynamics are governed by the Fermi-Hubbard Hamiltonian [Eqn. (1)]Esslinger (2010).
An external clock laser with wavelength nm resonantly interrogates the and states of the atoms and generates spin-orbit coupling (SOC)Wall et al. (2016). While the relative wavelengths of the lattice and clock lasers do not allow for weak SOC along all three lattice axes, weak SOC along two axes can be implemented by, for example, (i) fixing a large lattice depth along the axis, effectively freezing atomic motion along , and then (ii) making the clock laser nearly collinear with the axis, with only a small projection of its wavenumber onto the - plane [see Figure 1(a)]. The entire 3D OLC then factorizes into an array of independent 2D systems with atoms each, where is the number of lattice sites along each axis of the lattice. As in the 1D case, atoms within the 2D system experience all-to-all interactions, as well as spin-orbit coupling along two directions characterized by SOC phases . Generally speaking, higher-dimensional systems (e.g. 2D vs. 1D) are more desirable because they allow packing more interacting atoms into a fixed system volume, thereby increasing the maximally attainable amount of spin squeezing.
Figure 3 shows, for both OAT and TAT protocols, the maximally attainable amount of spin squeezing and the shortest time at which it occurs as a function of the lattice depth and linear lattice size in a single half-filled 2D layer (i.e. ) of the 3D OLC. Atoms are confined along the direction transverse to the 2D layer by a lattice of depth 60 , where is the atomic lattice recoil energy. The maximally attainable amount of spin squeezing by each protocol in Fig. 3 depends only on the atom number , while the shortest attainable time is determined by choosing the largest SOC phases which saturate . We impose this constraint on to ensure validity of the OAT Hamiltonian perturbatively derived in Appendix Asup ().
Currently, light scattering from the lattice beams induces decoherence of the clock on a time scale of secondsGoban et al. (2018); Hutson et al. (2019), which is much shorter than the natural lifetime of seconds (see Appendix D). This limitation imposes significant constraints on achievable spin squeezing, as shown in Figure 4 where the maximal squeezing with spin decay in the OAT case was determined using exact expressions for spin correlators derived in Ref. , while in the TAT case these correlators were determined by solving Heisenberg equations of motion for collective spin operators (see Appendix E). Due to the fast growth of Heisenberg operators in systems with all-to-all interactions, the latter method is not always capable of simulating up to the optimal squeezing time, and thus only provides a lower bound on the maximal squeezing theoretically obtainable via TAT. The results in Fig. 4 show that squeezing via OAT saturates with system size around (), while TAT allows for continued squeezing gains through (). Even with decoherence, our protocol may realistically generate – dB of spin squeezing in second with – atoms in a 2D section of the lattice, which is compatible with the atom numbers and interrogation times of state-of-the-art optical lattice clocksCampbell et al. (2017); Marti et al. (2018).
The source of decoherence considered above is not fundamental, and can be avoided by, for example, using two nuclear spin levels as spin-1/2 degrees of freedom that are interrogated by far-detuned Raman transitions instead of a direct optical transitionMancini et al. (2015). The strength of SOC for Raman transitions is tunable and, moreover, the lifetimes of ground nuclear spin levels are longer than 100 seconds in the latticeGoban et al. (2018). In this case, our protocol for preparing a squeezed state would additionally end with a coherent state transfer from nuclear to electronic degrees of freedom in order to retain metrological utility for the atomic clock.
We have proposed a new protocol to generate spin squeezing in a fermionic 3D optical lattice clock by combining nominally undesirable atomic collisions with spin-orbit coupling. To our knowledge, this is the first proposal to use atomic collisions for entanglement generation in state-of-the art atomic clocks, opening a path for the practical improvement of world-leading quantum sensors using correlated many-body fermionic states. Such capability could allow for major improvements in clock sensitivity and bandwidth, enhancing not only traditional timekeeping applications such as measurement standards, navigation (GPS), and telecommunications, but also geodesy and gravitational wave detection, precision tests of fundamental physics, and the search for new physics beyond the standard modelSafronova et al. (2018).
We acknowledge helpful discussions with M. Norcia, C. Sanner, and M. Mamaev. This work is supported by the Air Force Office of Scientific Research (AFOSR) grant FA9550-18-1-0319; the AFOSR Multidisciplinary University Research Initiative (MURI) grant; the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) grant W911NF-16-1-0576; the National Science Foundation (NSF) grant PHY-1820885; JILA-NSF grant PFC-173400; and the National Institute of Standards and Technology (NIST).
Appendix A Derivation of the effective one-axis-twisting model
Suppose we have a Hamiltonian of the form ()
and we consider -particle states initially in the ground-state manifold of , which have total spin . If the largest eigenvalue of is smaller in magnitude than half of the collective spin gap , i.e. the energy gap under between and its orthogonal complement , then we can formally develop a perturbative treatment for the action of on . Such a treatment yields an effective Hamiltonian on of the form , where is order in . Letting () be a projector onto (), we define the super-operators and by
where . The first few terms in the expansion of the effective Hamiltonian are then, as derived in Ref. ,
with . The zero-order effective Hamiltonian within the ground-state manifold. To calculate , we note that the ground-state manifold is spanned by the Dicke states
in terms of which we can expand the collective spin-z operator as . We can likewise expand the collective spin-x operator in terms of -oriented Dicke states as . The ground-state projector onto can be expanded in either basis as . Defining the mean and residual fields
we can then write
and in turn
where we used the fact that within the ground-state manifold. By construction, the residual fields are mean-zero, i.e. . Using the particle-exchange symmetry of the Dicke states, we can therefore expand
To calculate the second-order effective Hamiltonian , we let denote an eigenbasis of for the excited subspace , and set the ground-state energy to 0. We then define the operator
which sums over projections onto excited states with corresponding energetic suppression factors, in terms of which we can write
which is simply an operator-level version of the textbook expression for second-order perturbation theory. The only part of which is off-diagonal with respect to the ground- and excited-state manifolds and is , and the individual spin operators in this remainder can only change the total spin by at most 1. It is therefore sufficient to expand in a basis for states which span the image of under all within the manifold. Such a basis is provided by the spin-wave states
for Swallows et al. (2011). Using the fact that all spin- operators preserve the projection of total spin onto the axis, we then have that
where the relevant matrix elements between the Dicke states and the spin-wave states areSwallows et al. (2011)
Using the fact that , we can expand
where the sum over vanishes for and equals when , so
We therefore have that
where the term contributes a global energy shift which we can neglect, while the term is proportional to . In total, the effective Hamiltonian through second order in perturbation theory is thus
We benchmark the validity of this effective Hamiltonian via exact simulations of the spin [Eqn. (3)] and OAT [Eqn. (4)] Hamiltonians in a system of 20 spins, finding that the relative error in maximal squeezing (in dB) of the OAT model is less than 3% when sup ().
Appendix B Two-axis twisting, decoherence and the residual axial field
The protocol we use to transform one-axis twisting (OAT) into two-axis twisting (TAT) is as previously proposed in Ref. ; we provide a summary of this protocol here, in addition to some brief discussion of its implications for decoherence and the residual and terms of our OAT protocol. The TAT protocol begins with the OAT Hamiltonian with a time-dependent transverse field,
where is the modulation index of the driving field and the drive frequency , with the total number of spins. Moving into the rotating frame of subtracts this term from the Hamiltonian, and transforms operators as
In particular, the operators (i.e. the raising and lowering operators in the basis) transform simply as
For any operator and drive frequency , where is the operator norm of (i.e. the magnitude of the largest eigenvalue of ), we can generally make a secular approximation to say
where is the -th order Bessel function of the first kind. Expanding , one can thus work out that the effective Hamiltonian in the rotating frame of the drive is
Driving with a modulation index for which then gives us the effective two-axis twisting Hamiltonians
where denotes equality up to the addition of a term proportional to , which is irrelevant in the absence of coherent coupling between states with different net spin. In a similar spirit, one can work out that single-spin operators transverse to the -axis transform as
which implies that shifting into the rotating frame of the time-dependent drive takes
In practice, our protocols realize the OAT Hamiltonian in (30) with additional and terms [Eqn. (28)]. The effect of the term is to generate a relative phase between states with different total spin (with e.g. within the Dicke manifold). In the absence of coherent coupling between states with different total spin, therefore, the term has no effect on system dynamics. The term, meanwhile, is important; the magnitude of this term (as measured by the operator norm) is generally comparable to that of the squeezing term . Unlike in the case of OAT, does not commute with the TAT Hamiltonians, so its effects cannot be eliminated by a single spin-echo -pulse half way through the squeezing protocol. Nonetheless, we find that for () atoms, (10) -pulses in a CPMG (Carr-Purcell-Meiboom-Gill) sequenceCarr and Purcell (1954); Meiboom and Gill (1958) suffice to mitigate the effects of the term in the TAT protocol (see Appendix C). Phase control over these pulses, specifically choices of whether to apply or in any given -pulse, can be used to construct XY- pulse sequencesMaudsley (1986); Gullion et al. (1990) that are robust to pulse errors.
Appendix C Dynamical decoupling in the TAT protocol
The effective Hamiltonian resulting from a perturbative treatment of SOC is (see Appendix A)
where is a two-atom on-site interaction strength; is the number of lattice sites; is a residual axial field determined by the occupied quai-momentum modes (with atoms total); is the magnitude of a driving field; and is an effective OAT squeezing strength. The effect of the term is to generate a relative phase between states with different total spin (where within the Dicke manifold). In the absence of coherent coupling between states with different total spin, therefore, the term has no effect on system dynamics, and we are safe to neglect it entirely.
In the parameter regimes relevant to our discussions in the main text, the operator norms of and in (41) will typically be comparable in magnitude. The OAT protocol sets , and eliminates the effect of with a spin-echo -pulse applied half way through the squeezing protocol. The TAT protocol, meanwhile, effectively takes (as defined in Appendix B) and , where is the zero-order Bessel function of the first kind and is the modulation index of the amplitude-modulated driving field , satisfying . Unlike in the case of OAT, does not commute with the TAT Hamiltonian, so its effect cannot be eliminated with a spin-echo. Nonetheless, this term can be eliminated with a dynamical decoupling pulse sequence that periodically inverts the sign of while preserving .
Fig. C1 shows the maximal squeezing generated by and atoms via OAT, TAT, and TAT in the presence of the mean field as a function of the number of -pulses performed prior to the optimal TAT squeezing time. These pulses are applied in a CPMG sequence , where denotes Hamiltonian evolution for a time , denotes the application of an instantaneous -pulse , and is the number of pulses, such that the optimal TAT squeezing time is . The label TAT in Fig. C1 denotes squeezing through the Hamiltonian , where is the root-mean-square average of over choices of occupied spacial modes at fixed filling of all spatial modes in the lowest Bloch band of a periodic 2D lattice. While the modulation index is uniquely defined by , there are two choices of for which ; we use that which minimizes . Fig. C1 assumes an SOC angle (although results are independent of for ), a reduced field variance , and a filling . Note that as the filling , the residual axial field vanishes (), so TAT TAT.
Appendix D Decoherence in the 3D Sr optical lattice clock
Currently, light scattering from lattice beams in the 3D Sr optical lattice clock induces decoherence on a time scale of 10 secondsGoban et al. (2018); Hutson et al. (2019). This decoherence acts identically on all atoms in an uncorrelated manner, and can be understood by considering the density operator for a single atom, with effective spin states and respectively corresponding to the and electronic states. Empirically, the effect of decoherence after a time within the subspace of a single atom is to take for and
where are respectively decay rates for and . This form of decoherence can be effectively modeled by decay and dephasing of individual spins (respectively denoted and in Ref. ) at rates . In the language of the section that follows, we would say that this decoherence is captured by the sets of jump operators and with corresponding decoherence rates .
Appendix E Solving Heisenberg equations of motion for collective spin systems
In order to compute squeezing of a collective spin system, we need to compute expectation values of collective-spin operators. Choosing the basis for these operators, where with , we can expand all collective-spin Hamiltonians in the form
The evolution of a general operator under a Hamiltonian of the form in (43) is then given by
where ; is a set of jump operators with corresponding decoherence rate ; the decoherence operator is defined by
and is a matrix element of the time derivative operator . These matrix elements can be calculated analytically using product and commutation rules for collective spin operators. We can then expand Heisenberg operators in a Taylor series about to write
where are matrix elements of the -th time derivative. Expectation values of collective spin operators can thus be computed via the expansion in Eqn. (46), which at short times can be truncated at some finite order beyond which all terms have negligible contribution to .
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