Probing many-body interactions in an optical lattice clock
We present a unifying theoretical framework that describes recently observed many-body effects during the interrogation of an optical lattice clock operated with thousands of fermionic alkaline earth atoms. The framework is based on a many-body master equation that accounts for the interplay between elastic and inelastic -wave and -wave interactions, finite temperature effects and excitation inhomogeneity during the quantum dynamics of the interrogated atoms. Solutions of the master equation in different parameter regimes are presented and compared. It is shown that a general solution can be obtained by using the so called Truncated Wigner Approximation which is applied in our case in the context of an open quantum system. We use the developed framework to model the density shift and decay of the fringes observed during Ramsey spectroscopy in the JILA Sr and NIST Yb optical lattice clocks. The developed framework opens a suitable path for dealing with a variety of strongly-correlated and driven open-quantum spin systems.
One of the ultimate goals of modern physics is to understand and fully control quantum mechanical systems and to exploit them both, at the level of basic research and for numerous technological applications including navigation, communications, network management, etc. To accomplish these objectives, we aim at developing the most advanced and novel measurement techniques capable of probing quantum matter at the fundamental level.
Some years ago, the second – the international unit of time – was defined by the Earth’s rotation. However, with the discovery of quantum mechanics and the quantized nature of the atomic energy levels, it became clear that atomic clocks could be more accurate and more precise than any mechanical or celestial reference previously known to man. Thus, in 1967 the second was redefined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine energy levels of a caesium atom. Since then, the accuracy of atomic clocks has improved dramatically, by a factor of 10 or so every decade. The characterization of the unit of time plays a central role within the International System of Units (SI) because of its unprecedented high accuracy and because it is also used in the definitions of other units such as meter, volt and ampere.
Thanks to the development of laser trapping and cooling techniques Wieman et al. (1999); Phillips (1998), the best caesium standards have reached an accuracy of one part in . However, caesium clocks are limited by the fact that they are based on atomic transitions in the microwave domain. Because the quality factor of the clock is proportional to the frequency, optical clocks with frequencies that can be times higher than microwaves, offer an impressive potential gain over their microwave counterparts. Optical frequencies on the other hand are very difficult to measure, as the oscillations are orders of magnitude faster than what electronics can measure. The implementation of frequency comb technology Hall (2006) has provided a coherent link between the optical and microwave regions of the electromagnetic spectrum, greatly simplifying optical frequency measurements of high accuracy. After the development of frequency combs, the interest in optical clocks has grown rapidly. Now, optical clocks based on single trapped ions and neutral atoms are the new generation of frequency standards with a sensitivity and accuracy as high as one part in Chou et al. (2010); Rosenband et al. (2008); Bloom et al. (2013).
Optical clocks operated with fermionic neutral alkaline earth atoms (AEA), such as Sr or Yb, have matured considerably. Those employ an optical lattice to tightly confine the atoms so that Doppler and photon-recoil related effects on the transition frequency are eliminated. State-of-the-art neutral-atom-optical clocks have surpassed the accuracy of the Cs standard Ludlow et al. (2008) and just recently, thanks to advances in modern precision laser spectroscopy, are reaching and even surpassing the accuracy of single ion standards Bloom et al. (2013). The most stable of these clocks now operate near the quantum noise limit Nicholson et al. (2012); Hinkley et al. (2013). The stability arises from the intrinsic atomic physics of two-valence-electron atoms that possess extremely long lived singlet and triplet states (clock states), with intercombination lines nine orders of magnitude narrower than a typical dipole-allowed electronic transition.
The potential advantage of neutral-atom clocks over single trapped ion clocks is that, in the former, a large number of atoms is simultaneously interrogated. This could lead to a large signal-to-noise improvement; however, high atom numbers combined with tight confinement also lead to high atomic densities and the potential for non-zero collisional frequency shifts via contact atom- atom interactions. With atom-light coherence times reaching several seconds, even very weak interactions (e.g., fractional energy level shifts of order ) can dominate the dynamics of these systems.
To suppress these interactions, the use of ultracold, spin-polarized fermions was proposed. The idea was to exploit the Fermi suppression of -wave contact interactions while freezing out -wave and higher wave collisions at ultracold atomic temperatures. Indeed, at precision level of , the JILA Sr clock did not exhibit a density-dependent frequency shift Boyd et al. (2007), however as the measurement precision progressed, density-dependent frequency shifts were measured in spin polarized fermions, at the JILA Sr clock Campbell et al. (2009); S. Blatt et al. (2009); Swallows et al. (2011a) and at the NIST Yb clock Lemke et al. (2009).
When those density-dependent frequency shifts were first observed, they were attributed to -wave collisions allowed by inhomogeneous excitation Takamoto et al. (2006); Campbell et al. (2009); A. M. Rey and A. V. Gorshkov and Rubbo (2009); Gibble (2009); Yu and Pethick (2010), under the assumption that -wave interactions were suppressed at the operating temperatures (K) Campbell et al. (2009). The basic initial understanding, obtained from a mean-field treatment, was that excitation inhomogeneities induced by the optical probing laser made the initially indistinguishable fermionic atoms distinguishable, and thus allowed them to interact via -wave collisions.
However, studies of the cold collision shift in the NIST Yb optical lattice clock using Ramsey spectroscopy revealed that -wave interactions were the dominant elastic interactions in that system Lemke et al. (2011). Furthermore, evidence of inelastic -wave interactions was reported in both Yb and Sr atomic clocks Ludlow et al. (2011); Bishof et al. (2011a). Although the importance of many-body interactions in optical clocks has been recognized theoretically A. M. Rey and A. V. Gorshkov and Rubbo (2009); Gibble (2009); Yu and Pethick (2010), only recent measurements have revealed their many-body nature Martin et al. (2013); Martin (2012). In those measurements, the role of -wave collisions was further suppressed by operating the Sr clock with highly homogeneous atom-laser coupling. This results in dominant -wave interactions with a collective character, as we will explain below.
At this point it is important to emphasize that recent advances in modern precision laser spectroscopy, with record levels of stability and residual laser drift less than mHz/s Nicholson et al. (2012); Swallows et al. (2011b); Martin (2012) are the crucial developments that are allowing us to deal with clocks operated at a very different conditions than those ones dealt with just few years ago. The level of precision spectroscopy achievable in current atomic clocks is now providing the required spectral resolution to systematically resolve and study the complex excitation spectrum of an interacting many-body system. This was certainly not the case in prior clock experiments where interaction effects were subdominant and where a mean-field treatment was more than enough to describe the clock behavior. For example in 2006, a 2 Hz spectral resolution has achieved for the Sr atomic transition and no interaction effects were observable at the time Boyd et al. (2006).
In this paper we present a unifying theoretical framework that goes beyond a simple mean-field treatment and that is capable of describing the full many-body dynamics of nuclear spin-polarized alkaline earth atoms during clock interrogation. The two clock states are treated as an effective spin degree of freedom. Both, elastic and inelastic two-body collisions and single-particle losses are present during the dynamics, and thus a pure Hamiltonian formulation is not sufficient. Instead, we develop a master equation formulation which is capable of treating the quantum evolution of an open spin system. We provide analytic/exact solutions of the master equation dynamics in parameter regimes where exact treatments are possible. For the more generic situations we solve the dynamics relying on the so called Truncated Wigner Approximation (TWA) Blakie et al. (2008); Polkovnikov (2010). In contrast to previous theoretical treatments of the clock dynamics, which were limited to treating two-particles or many-particles but at the mean-field level or under the all-to-all approximation A. M. Rey and A. V. Gorshkov and Rubbo (2009); Gibble (2009); A. M. Rey and A. V. Gorshkov and Rubbo (2009); Gibble (2009); Yu and Pethick (2010); Swallows et al. (2011a), the TWA method allows us to include both elastic and inelastic collisions beyond the mean-field level, finite temperature effects and inhomogeneities generated by either the laser during the pulse interrogation or by many-body interactions. Those are shown to be crucial for properly modeling observed many-body dynamics, especially at K, at which excitation inhomogeneities can not be neglected. To our knowledge this is the first time that the TWA is applied to describe an open quantum system in the presence of inelastic losses.
Although this paper focuses on optical lattice clocks, the developed theoretical framework is generic for driven open-quantum systems and should be a useful platform for dealing with a variety of current experimentally relevant systems including trapped ions Britton et al. (2012); Kim et al. (2010), polar molecules Neyenhuis et al. (2012); Hazzard et al. (2013); Gorshkov et al. (2011a, b); Hazzard et al. (2011); Yan et al. (2013), nitrogen vacancy centers Prawer and Greentree (2008), and atoms in optical cavities Baumann et al. (2010); Gopalakrishnan et al. (2011) among others.
The remainder of the paper is organized as follows. In section II, we introduce the reader to the basic operation of an atomic clock and derive the many-body Hamiltonian that describes the dynamics of nuclear spin-polarized fermionic atoms during clock interrogation. We then proceed to derive a simplified effective spin model which assumes frozen motional-excitations during the dynamics. In section III, we solve for the dynamics under the assumption of collective spin interactions (all-to-all interactions) in a closed system. In Section IV, we show how to treat the observed two-body losses and introduce a master equation, which we solve under the collective-interactions approximation. We also show how to use the TWA to deal with the open quantum system dynamics. In section V, we relax the frozen-motional-degrees-of-freedom approximation and derive an improved spin Hamiltonian with cubic spin-spin interactions which account for the virtual occupation of excited motional modes. In section VI, we go beyond the all-to-all interactions approximation and present a more general prescription that can address both non-collective interactions and losses and single-particle inhomogeneities. In section VII, we apply the developed theoretical framework to model the dynamics during Ramsey spectroscopy observed in the JILASr and the NIST Yb optical lattice clocks, and finally, in section VIII, we present the conclusions. In Appendixes 1-5, we present some details omitted in the main text.
Ii Many-body physics during clock operation
ii.1 A simple overview of an optical lattice clock
The general design of an optical lattice clock is shown in Fig. 1. It consists of two components, a laboratory radiation source and an atomic system with a natural reference frequency determined by quantum mechanics to which the laboratory radiation source can be compared. Here, the laboratory radiation source is an ultra-stable continuous-wave laser. It acts as the local oscillator (or pendulum) for the clock and is used to probe an electromagnetic resonance in an atom. The atomic signal can then be used to determine the difference between the laser frequency and that of the reference atom, allowing laser frequency to be monitored and stabilized to the preferred value. A frequency comb Hall (2006) provides the gears of the clock, allowing measurements of the laser frequency relative to other high accuracy clocks in either the optical or microwave domains.
The two main quantities that characterize the performance of a clock are the accuracy and the precision. The accuracy is determined by how well the measured frequency matches that of the atom’s natural frequency. In general, the accuracy will depend on the atomic species used and how well it can be isolated from environmental effects during spectroscopy. The precision of the clock is more commonly referred to as the stability, which represents the repeatability of the measured clock frequency over a given averaging time . For quantum projection noise limited measurements, it is typically expressed as Bauch (2003)
Here, is the line quality factor of the clock transition for a linewidth and is associated with the signal-to noise-ratio achieved for interrogating atoms in the measurement cycle time .
Fermionic AEA such as Sr and Yb have unique properties that make them ideal candidates for the realization of atomic clocks Katori (2011). The clock states are the ground singlet state, , and a long lived triplet state , with intercombination lines both electric and magnetic dipole forbidden and as narrow as a few mHz, see Fig. 2. In the ground state (), and to leading order in the excited state (), the electronic degrees of freedom have neither spin nor orbital angular momentum Boyd et al. (2006). This means that the atoms in the clock states only interact with external magnetic fields through the nuclear spin degrees of freedom which have a -factor 1000 times smaller than the electronic orbital one. Therefore, AEA are much less sensitive to magnetic field fluctuations and/or to intensity and phase noise on the optical fields than conventional alkali atoms.
The clock transition is only allowed (i.e. laser light weakly couples to ) because in the excited state the hyperfine interaction leads to a small admixture of the higher-lying states Boyd et al. (2007). This small admixture strongly affects the magnetic moment and causes the -factor of the excited state to be significantly different from that of the ground state ( for Sr). The different -factor allows for the addressability of the various Zeeman levels in the presence of a bias magnetic field as demonstrated in Ref. Boyd et al. (2006).
Optical spectroscopy in atomic clocks is sensitive to atomic motion due to the Doppler effect. To overcome this limitation, atoms are confined in a tight optical lattice formed by a standing wave light pattern to eliminate broadening and frequency shifts due to atomic motion. Within a lattice site the atoms are tightly trapped in the so-called Lamb-Dicke regime where the length scale associated with their motion is much smaller than the wavelength of the laser probing the atoms. Moreover the optical lattice can be carefully designed to operate at the so-called magic wave length at which the light shifts on the clock states are equal and the clock frequency is not perturbed Ye et al. (2008).
In optical lattice clocks, one can simultaneously probe large samples of laser-cooled atoms which can potentially lead to high frequency stability. Nevertheless, this precision may come at the cost of systematic inaccuracy due to atomic interactions. As mentioned before, the use of ultracold spin-polarized fermions was thought to be the key to avoid interaction effects. However, it has been recently shown that this is not the case Campbell et al. (2009); Swallows et al. (2011a); Bishof et al. (2011b, a); Martin et al. (2013); Lemke et al. (2011); Ludlow et al. (2011).
ii.2 Many-body Hamiltonian for spin polarized fermionic atoms
In order to model the dynamics during clock interrogation, we will consider a nuclear spin-polarized ensemble of fermionic atoms with two accessible electronic degrees of freedom associated with the - electronic levels (see Fig. 2) which we denote as and respectively. stands for the ground state and for a excited state. We focus on the case where the atoms are trapped in an external potential which is the same for and (i.e. at the “magic wavelength” Ye et al. (2008)). If the atoms are illuminated by a linearly polarized laser beam with bare Rabi frequency , they are governed by the following many-body Hamiltonian Gorshkov et al. (2009, 2010); A. M. Rey and A. V. Gorshkov and Rubbo (2009); Yu and Pethick (2010); Hazzard et al. (2011); Martin et al. (2013)
Here is a fermionic field operator at position for atoms with mass in electronic state () or () while is the corresponding density operator. We consider two possible interaction channels: -wave and -wave (see Fig. 2). Since polarized fermions are in a symmetric nuclear state, their -wave interactions are characterized by only one scattering length , describing collisions between two atoms in the antisymmetric electronic state, . The -wave interactions can have three different scattering volumes , , and associated to the three possible electronic symmetric states (, , and respectively. takes into account the interrogation of the atoms by a laser that has frequency and wavevector and is detuned from the atomic transition frequency by .
ii.3 Effective spin model
We consider the situation of a deep 1D lattice, , along , which creates an array of two-dimensional discs or “pancakes” and induces a weak harmonic radial (transverse) confinement with an angular frequency . The lattice confines the atoms to the lowest axial vibrational mode. This analysis can be straightforwardly generalized to the case of a 2D lattice in which two directions are frozen and only one is thermally populated.
We expand the field operator in a non-interacting atom basis, , where is the ground longitudinal mode in a lattice site and are transverse harmonic oscillator eigenmodes. creates a fermion in mode and electronic state . In this basis and in the rotating frame of the laser, can be rewritten as Swallows et al. (2011a); Lemke et al. (2011); Martin et al. (2013):
Here is the atom number operator in mode and state , is a Kronecker delta function, is the radial harmonic oscillator length, and are single-particle energies in the trap. We have used a Gaussian approximation for , which is excellent for the deep lattice used in experiments. with the recoil energy, , is the lattice beams’ wave-number and is the lattice depth. The coefficients and characterize - and -wave matrix elements, respectively, which depend on the harmonic oscillator modes and satisfy and . Explicitly,
Here are Hermite polynomials.
In Fig. 3 we show the mode dependence of the functions and . Since those are computed in the harmonic oscillator mode basis, they are long-range in mode-space. While the function scales (for ) as , and grows with increasing energy, as expected from -wave interactions, the function scales (for ) as , and thus decreases with increasing energy. In Fig. 4, we also show the dependence of the mean and standard deviation of the -wave interaction parameters as a function of temperature (T). There one can see that is almost independent in this quasi-2D geometry. This is expected because while the actual -wave interactions for fixed density should increase linearly with Kanjilal and Blume (2004), the latter growth is compensated by the linear decrease with of the density in a 2D harmonic trap. Fig. 4 (bottom) shows a histogram of which is peaked about its average value. The histogram was computed at K but it is almost independent.
For the laser-atom interaction Hamiltonian we follow Refs. Campbell et al. (2009); S. Blatt et al. (2009); A. M. Rey and A. V. Gorshkov and Rubbo (2009); Gibble (2009); Yu and Pethick (2010) and assume that the probe is slightly misaligned with a small component along the -direction:
with the misalignment angle (see Fig 2). We also assume we are in a regime in which laser induced sideband transitions can be neglected, and define a mode-dependent effective Rabi frequency given by
where are the Lamb-Dicke parameters, and are Laguerre polynomials Wineland and Itano (1979). In Fig. 4 we show the ratio between the mean Rabi frequency, , and the standard deviation, , as a function of temperature. increases with T since the atomic cloud spreads as it heats up. In the Lamb-Dicke regime, , becomes:
Under typical operating conditions, Hz, and as will be discussed below, the system is in the regime where its mean interaction energy per particle is about two orders of magnitude weaker than the energy splitting between neighboring single-particle transverse vibrational modes. We refer to this regime as the vibrational-weakly-interacting regime. Thus, to leading order, only collision events that conserve the total single-particle energy need to be considered. Under these conditions the atom population is frozen in the initially populated modes and only the electronic degree of freedom vary during the clock interrogation. For an initial state with at most one atom per mode (-polarized state), it is possible under these conditions to reduce to a spin- model with the spin encoded in the states. For atoms and labeling the thermally populated harmonic oscillator modes as with , the spin model can be written as:
Here , with Pauli matrices and the identity matrix.
encapsulate the temperature dependence of the interactions.
Note, however, that in the case of a pure harmonic spectrum, mode changing collisions are energetically allowed even under weak interactions due to (i) the linearity of the harmonic oscillator spectrum and (ii) the separability of the harmonic oscillator potential along the and directions. Condition (i) allows two particles in modes and to collide and scatter into modes and without violating the energy conservation constraint. Condition (ii) allows the same two particles to scatter into modes and . Those issues, in principle, can impose important limitations on the validity of the spin model in a harmonic trap. In practice, however, the trapping potential is not fully harmonic. It comes from the Gaussian beam profile of the lasers and is given by with the beam waist. To leading order, the trapping potential is harmonic but for an atom in a mode there are higher order corrections of the energy beyond leading order: , with . At typical operating conditions of the Yb and Sr experiments: Hz, m, K, and a mean occupation mode numbers , the difference of for nearby modes is larger than Hz which is not negligible compared to typical interaction energy scales Hz. The first term in thus prevents processes (i), while the second term, which breaks the separability of the potential, prevents processes (ii). Based on this argument we first restrict our analysis to only processes that conserve the number of particles per mode.
A further simplification of Eq. (9) can be made when atoms are initially prepared in the totally symmetric Dicke manifold with Arecchi et al. (1972) at time . Here, is the eigenvalue of the collective operator and .
In this case there are two important physical mechanisms that prevent leakage of the population outside the symmetric Dicke manifold. (i) The weak dependence of the interaction matrix elements on the thermally populated modes so that the mode-dependent coupling constants , , and are peaked at their averages , and (See Figs. 3-4). (ii) The fact that . Here are the corresponding standard deviations. The latter is satisfied in part because is the only interaction term that has a contribution arising from -wave interactions. In general, those are expected to dominate over -wave interactions since -wave collisions are suppressed by the centrifugal barrier which is estimated to be greater than K Campbell et al. (2009); Lemke et al. (2011). It must be said, nevertheless, that the actual values of the -wave and -wave scattering parameters between two atoms and one and one are not known. (i) and (ii) impose a large energy gap in the Hamiltonian which suppresses transitions between manifolds with different total collective spin , caused by the inhomogeneities and Rey et al. (2008). Consequently, to a very good approximation it is expected that the dynamics can be projected into the manifold with an effective Hamiltonian given in terms of collective operators:
The term is a constant of motion and does not play any role in the collective dynamics.
In addition to the above conditions on the interactions, staying in the manifold requires that the probing laser generates negligible excitation inhomogeneity, i.e. . In this case we can write the atom-light Hamiltonian in terms of collective operators. The validity of this condition depends on the misalignment angle, [Eq. (6)], and the average vibrational mode population S. Blatt et al. (2009). At the current operating conditions, the JILA Sr clock can achieve Martin et al. (2013), and the collective mode approximation for the atom-light Hamiltonian is well satisfied. The latter is not necessarily the case for the Yb optical clock experiment at NIST due to the higher temperatures, as we will elaborate later. The restriction of the dynamics to the Dicke manifold is relaxed in Sec. VI where we also investigate the parameter regime in which it is valid.
Iii Ramsey interrogation: Collective case
In Ramsey spectroscopy (see Fig. 5), a well established tool in atomic physics, atoms are typically prepared in the same internal state, say (e.g. via optical pumping). Next, one applies a strong resonant linearly polarized light pulse for time . By strong, we mean that the Rabi frequency must be much larger than the atomic interaction energy scales but weaker than the harmonic oscillator frequency, , to avoid laser induced mode changing processes.
This first pulse rotates the spin state of the atom at mode to , here is the pulse area. Subsequently, the atoms are allowed to freely evolve for a dark time . Finally a second pulse is applied for a time and the population of the states measured.
In the following, we will discuss the case of a fixed number of atoms, , prepared in the modes , and only later we will worry about spatial and thermal averages. Note that under the collective-mode approximation .
iii.1 Analytic solution
Let us first start dealing with the situation in which there is no excitation inhomogeneity and the initial (second) pulses are just rotations of the collective Bloch vector by an angle , respectively. In this case, the state of the system before the free dynamics is
with being collective Dicke states. During the free evolution, the Hamiltonian reduces to which introduces just a phase, to each of the states. Here is an eigenvalue of and takes integer or half-integer values (depending on whether is even or odd) satisfying . Expressions for the evolution of the spin operators after a dark-time evolution can be exactly computed:
with and and given by
The normalized contrast, which is a measure of the amplitude of the Ramsey fringes, is defined as the magnitude of the projection of the collective Bloch vector on the x-y plane normalized by half of the total number of atoms:
from the above expression one obtains that
The contrast is extracted from measurements of the fraction of excited atoms, by varying the laser detunning, or by varying the phase of the second pulse along the x-y plane. For the former case:
Let’s now discuss the physics encapsulated in Eqs. (18-20). The term is the so called density-dependent frequency shift, which gives rise to a density-dependent measurement of the atomic transition frequency. The quantity determines the contrast of the Ramsey fringes. In the weakly interacting regime, , and for , . This means that interactions act as an effective magnetic field along the quantization axis with magnitude , which depends both on the total atom number and the population difference between excited and ground state. This is consistent with just a simple interpretation of the frequency shift as being the average energy difference experienced by an atom in state with respect to an atom in state due to the presence of other atoms. As we explain in Sec. III.2, this can be derived from a mean field analysis which factorizes the interaction term as . In this regime, the condition determines the pulse area at which the shift is canceled. This is the ideal operating pulse area for a clock Ludlow et al. (2011). Note that if is equal to zero, no density shift is expected at . This is consistent with the intuition that due to the equal population of both and states, the mean energy shift experienced by an atom in state due to others is exactly canceled by the opposite energy shift experienced by an atom in . In this weakly interacting regime, as expected in the case that interactions act just as a mere effective magnetic field, which causes the Bloch vector just to precess with no Ramsey fringe-contrast decay.
Outside the weakly interacting regime, two important corrections to this picture arise. One is the fact that the shift is no longer linear in [from Eq. (22)]. The second one is that the Ramsey fringe-contrast collapses and revives. The collapse is well approximated by a Gaussian decay, . The revivals take place at times with an integer. This behavior of the contrast is closely linked to the decay of coherence in a matter-wave due to the nonlinearities arising from the atom-atom interactions and subsequent revival due to the discreteness of the spectrum of the many-body system. The observation of collapses and revivals of a Bose-Einstein condensate (BEC) loaded in an optical lattice was first reported in Ref. Greiner et al. (2002).
iii.2 Mean-field solution
Even though the all-to-all interactions allows for an exact solution of the many-body dynamics in Ramsey spectroscopy, it is convenient to introduce an approximate mean-field treatment. The mean-field treatment will be very helpful for dealing with inelastic collisions, which are experimentally relevant. A simple and enlightening way to carry out a mean-field treatment is to use the Schwinger-boson representation, which maps spin operators to two-mode bosons subject to a constraint Auerbach (1994). It represents the spin operators as
with the constraint
with a bosonic annihilation operator of mode . The mean-field treatment, which gives rise to the so-called Gross-Pitaevskii Equation (GPE) Pethick and Smith (2002), replaces the field operators by c-numbers, . The latter approximation is justified in the weakly interacting regime when there is a macroscopic population of those modes and the state of the system can be regarded as a simple product state with no significant entanglement. Translated to the spin language, those conditions imply that the system can be well described as a spin coherent state. For purposes that will become clearer later, it is convenient to introduce the density matrix and . The latter satisfies the following equations of motion:
are the components of the Bloch vector.
For the Ramsey dynamics, the mean-field treatment is almost trivial, and time evolution corresponds to a precession of the Bloch vector induced by an effective magnetic field
This behavior can be clearly seen in Eq. (34) by noticing that during the dark time , thus is a constant of motion, and precesses at a rate