## Abstract

We consider the photoassociation of fermions trapped in a two-dimensional optical lattice into bosonic molecules, in the limit that intersite tunnelling is negligible. For the case of two fermions in different hyperfine states this process can be mapped into a generalized version of the Jaynes-Cummings Hamiltonian from quantum optics. We make use of this equivalence to show how to build a micromaser for the molecular field at each lattice site.

© 2004 Optical Society of America

In recent years there has been growing interest in the properties of ultracold atoms confined in optical lattices, the optical potentials created by detuned standing-wave laser beams. Much of this work has been motivated by the desire to build a quantum computer in which the atoms at each lattice site would represent an individual qubit [1]. An additional motivation is provided by the possibility to realize and investigate in detail strongly correlated lattice systems of condensed matter physics. Optical lattices and ultracold atoms offer a unique experimental environment for studying many-body theories because of the ability to control virtually every aspect of the system, from the structure of the lattice potential to the magnitude and sign of the interatomic interactions. This was epitomized by the recent experiment of Greiner *et al* [2] that produced a Mott-insulator transition [3, 4] in ultracold ^{87}Rb confined in an optical lattice.

Parallel to these developments, a major current thrust in the study of ultracold atomic systems is the possibility of forming molecular Bose-Einstein condensate (BEC) via either photoassociation [5, 6] or a Feshbach resonance [7]. To date, experiments have demonstrated the formation of large numbers of molecules via photoassociation of a ^{87}Rb BEC [8], as well as by using a Feshbach resonance applied to a ^{85}Rb BEC [9] and also to degenerate Fermi gases of ^{40}K [10] and ^{6}Li [11, 12, 13, 14, 15]. These two lines of research have recently converged in theoretical proposals to photoassociate bosonic atoms in the Mott-insulator state into molecules [16, 17, 18, 19, 20].

Photoassociation in optical lattices offers distinct advantages over quasi-homogenous systems. In particular, one can start from a degenerate gas of bosons or fermions (*N*~10^{5}) and obtain filling factors (occupation numbers per lattice site) of order unity. These low occupation numbers minimize the losses due to inelastic collisions between atoms and molecules in vibrational excited states. Also, the large energy separations that are possible between the lowest and the second Bloch band of the lattice allow one to restrict the center of mass states of the atoms and molecules to the lowest Wannier state of each lattice site, thereby avoiding many of the difficulties associated with multimode problems [21].

In this paper, we consider the photoassociation of *fermionic* atoms in an optical lattice. In case that only two fermions occupy each lattice site this problem [22] reduces mathematically to a situation almost identical to that of the micromaser [23, 24], a direct consequence the Pauli Exclusion Principle. One important difference is the appearance of a nonlinear frequency shift in the atom-molecule transition frequency due to the two-body elastic interactions between atoms and molecules, a term absent in the case of electromagnetic fields.

It is well established that the micromaser field exhibits a number of quantum mechanical features absent in normal lasers and masers. For example, they exhibit strong sub-Poissonian statistics [25] and the ability to dynamically generate Fock states [26]. They can also undergo multiple phase transitions characterized by sharp changes in the photon statistics from sub-Poissonian to super-Poissonian [27]. The close analogy between the situation at hand and the micromaser therefore opens up exciting avenues to generate non-classical states of molecular fieldsF and to study new types of non-equilibrium quantum phase transitions in matter-wave systems. More generally, it permits the extension of the numerous, well-established applications of cavity QED, to matter-wave optics. Of particular interest in this context is the fact that optical lattices automatically lead to the realization of an *array* of individually addressable molecular micromasers. From a theoretical point-of-view, this system is particularly interesting in that it permits one to analyze the dynamics of a matter-wave field exactly, without having to rely to a mean-field approach or to other approximate factorization schemes.

The goal of this paper is to extend and further discuss the results of Ref. [22]. We give a broad overview of fermionic photoassociation in optical lattices, review under which conditions under which it can be mapped to the micromaser problem, and present some of the important physical implications, in particular on the counting statistics of the molecular field.

We consider a 2-D optical lattice that confines fermionic atoms of mass *m*_{f}
in two hyperfine states |1〉 and |2〉 in the *xy*-plane. The atoms are coupled to a molecular boson of mass *m*_{b}
=2*m*_{f}
via a two-photon stimulated Raman transition [5, 6, 8]. The optical lattice potential *V*_{f}
seen by the atoms is

${V}_{f}\left(\mathbf{r}\right)={\displaystyle \sum _{\xi =x,y}}{V}_{\xi}^{\left(f\right)}{\mathrm{cos}}^{2}{k}_{\xi}\xi +\frac{1}{2}{\kappa}_{z}^{\left(f\right)}{z}^{2}$

where ${V}_{\mathrm{\xi}}^{\mathit{\left(}f\mathit{\right)}}$x>0 and we have included a transverse confining potential along the *z*-axis. A similar potential, with *f*→*b*, confines the molecules. We assume that the system is at zero temperature and that the filling factor — the number of fermions of each type that occupy every lattice site — is *n*_{F}
≤1 at all times. Hence the fermions only occupy the lowest Bloch band of the lattice. For deep lattices, for which the energy separation between the first and second Bloch bands is much greater than the atom-molecule interaction energy, the molecular state formed by photoassociation has likewise a center-of-mass wave function in the first Bloch band [16].

When restricted to the first Bloch band of the lattice, the Hamiltonian for the atom-molecule system is

where,

Here, *b*_{i}
is the bosonic annihilation operator for molecules in the Wannier state of the lowest Bloch band centered at the lattice site *i*, *w*_{b}
(**r**-**r**
_{i}
), and similarly, *c*_{σ,i}
is the fermionic annihilation operator for state |*σ*〉 with the Wannier wave function *w*_{f}
(**r**-**r**
_{i}
). The corresponding number operators *n̂*_{bi}
=*b*†_{i}
*b*_{i}
and *n〡*_{σi}
=*c†*_{σ,i}*c*_{σ,i}
have eigenvalues *n*_{bi}
and *n*_{si}
, respectively.

For deep lattices, the Wannier states of the first Bloch band can be approximated by using a harmonic expansion for the lattice potential at each of the nodes, in which case they are given by

${w}_{b,f}\left(\mathbf{r}\right)={\displaystyle \prod _{\xi =x,y,z}}\frac{{\varphi}_{0}(\xi \u2044{\ell}_{\xi}^{(b,f)})}{\sqrt{{\ell}_{\xi}^{(b,f)}}},$,

where *ϕ*
_{0}(*ζ*)=exp(-*z*
^{2}/2)/*π*
^{1/4}. Here,

${\ell}_{\xi}^{(b,f)}={\left(\frac{{\overline{h}}^{2}}{{m}_{b,f}{\kappa}_{\xi}^{(b,f)}}\right)}^{\frac{1}{4}}$

are the harmonic oscillator lengths and ${\kappa}_{x,y}^{(b,f)}$ =2${V}_{x\mathit{,}y}^{(b\mathit{,}f)}$${k}_{x\mathit{,}y}^{\mathit{2}}$ . The ground state frequencies are

${\omega}_{b,f}={\displaystyle \sum _{\xi}}{\omega}_{\xi}^{(b,f)}={\displaystyle \sum _{\xi}}\frac{\overline{h}}{2{m}_{b,f}{\ell}_{\xi}^{(b,f)2}}.$.

The terms proportional to *U*_{b}
,*U*_{x}
, and *U*_{f}
in Eq. (1) describe the on-site two-body interactions between molecules, atoms and molecules, and atoms, respectively. The coupling constants for the two-body interactions can be expressed in terms of the harmonic oscillator lengths as

where *U*_{pq}
=*U*_{p}
for *p*=*q* and *U*_{x}
otherwise. Here *µ*_{pq}
=*m*_{p}*m*_{q}
/(*m*_{p}
+*m*_{q}
) are the reduced masses and *a*_{pq}
are the s-wave scattering lengths for collisions between molecules, atoms in different states, and molecules and atoms.

The interaction Hamiltonian, *Ĥ*_{Ii}
, describes the conversion of atoms into ground-state molecules via two-photon stimulated Raman photoassociation. *χ*(*t*) is proportional to the far off-resonant two-photon Rabi frequency associated with two nearly co-propagating lasers with frequencies *w*
_{1} and *w*
_{2} [8]. Note also that the matter fields have been written in an interaction representation in which the molecular field is rotating at the frequency difference of the two lasers, so that *δ*=*ν*_{m}
-(ν_{1}+ν_{2})-(*ω*
_{1}-*ω*
_{2}) where *ℏ*_{νσ}
and *ℏν*_{m}
are the internal energies of the atoms and molecules, respectively.

As already mentioned earlier, the validity of the single-band approximation is based on the assumption that the various interaction energies are much smaller than the energy separation between bands,

$\overline{h}{\omega}_{b,f}\gg {U}_{b}{\hat{n}}_{bi}\left({\hat{n}}_{bi}-1\right),{U}_{f},{U}_{x}{\hat{n}}_{bi},\chi \sqrt{{\hat{n}}_{bi}}.$.

The last contribution to the Hamiltonian *Ĥ, Ĥ*_{T}
describes the tunnelling between lattice sites. In its expression the sum is over all nearest neighbors and *J*_{b}
and *J*_{f}
are the nearest neighbor hopping elements for the molecules and atoms, respectively. As is discussed in detail later on, the molecular micromaser operates by switching on the photoassociation laser in a train of short pulses of duration τ. In the present paper we assume that the optical lattice is sufficiently deep that we can ignore the intersite tunnelling of the fermions in these intervals, i.e., *τ*≪*ℏ*/*Jf*. In addition, if one wishes the individual micromasers in the array to remain independent of each other, it is necessary to require that molecular intersite tunnelling remains negligible for a time much longer than τ. The dynamics of a coupled array of molecular micromasers will be discussed in a future paper. In this paper, though, we assume that the molecules are in the Mott-Insulator state [4, 28],

We emphasize that this condition does not imply that the molecules are constrained to be in a Fock state at each site. Indeed, we show below that the *dynamics* of the open driven system that we consider normally generate states of the molecular field that differ significantly from a Fock state. We also note that within current experimental limitations, the condition (3) limits the number of molecules to a maximum of around 10 while *ℏ*/*J*_{f}
>1 ms [2, 3].

In the absence of intersite coupling, we need only consider the Hamiltonian *ĥ*=*Ĥ*_{0i}
+*Ĥ*_{Ii}
at a single lattice site. In what follows, we drop the lattice site label for notational clarity. We then proceed by introducing the mapping [29],

$${\sigma}_{z}={c}_{1}^{\u2020}{c}_{1}+{c}_{2}^{\u2020}{c}_{2}-1$$

where *s*
_{+}=|*e*〉〈*g*| and *σ*-=|*g*〉〈*e*| are the raising and lowering operators for a fictitious two-state system, and while *σ*_{z}
=|*e*〉〈*e*|-|*g*〉〈*g*| is the population difference between its upper and lower states. Here, |*e*〉=*c*
^{†}2${c}_{1}^{\u2020}$|0〉 and |*g*〉=|0〉. We note that this mapping only holds if *c*
_{1} and *c*
_{2} are fermionic operators, hence, our subsequent discussion does not hold for bosonic atoms. The mapping (4) allows us to reexpress *ĥ* exactly as,

$$+\overline{h}\left(\chi \left(t\right){b}^{\u2020}{\sigma}_{-}+{\chi}^{*}\left(t\right)b{\sigma}_{+}\right)+\frac{\overline{h}}{2}{U}_{b}{\hat{n}}_{b}\left({n}_{b}-1\right)$$

where we have dropped constant terms and made the redefinitions *ω*_{b}
+*δ*→*ω*_{b}
and *ω*_{f}
+*U*_{f}
/2→*ω*_{f}
.

In the absence of two-body collisions, *U*_{b}
,*U*_{x}
→0, this Hamiltonian reduces to the Jaynes-Cummings model of interaction between a quantized, single-mode electromagnetic field and a two-level atom. This model is a cornerstone of quantum optics [26]. Because it is exactly solvable, it permits the understanding of detailed aspects of the dynamics of light-matter interaction.

It is important at this point to contrast this model to a recent paper where K. Mølmer proposed the use of a Jaynes-Cummings Hamiltonian to describe the photoassociation of two different species of *bosonic* atoms into molecules in an optical lattice [19]. Because it deals with bosonic rather than with fermionic atoms, that model is limited to situations where the number of molecules at each site is ≤1. Our fermionic model, on the other hand, does not suffer from this limitation and allows the generation of states of the molecular field with occupation numbers greater than 1 at each lattice site.

We recall that the dynamics of the micromaser is governed by three mechanisms: the injection of a sequence of individual atoms from a very dilute atomic beam inside the microwave cavity, the Jaynes-Cummings interaction between these atoms and the cavity mode, and cavity dissipation, which can normally be neglected while an atom transits through the cavity, but that dominates the field dynamics in the intervals between atoms. A similar situation can be achieved in the present system: all that is required is to inject a sequence of pairs of fermionic atoms inside the lattice well, turn on the photoassociation lasers for some time interval τ to introduce a Jaynes-Cummings-like coupling between the fermions and the molecular field, and finally turn off these fields for a time *T *and let dissipation take over. This sequence is then repeated to build up the molecular field.

Consider first the Jaynes-Cummings-like part of the molecular micromaser dynamics. For *χ*=*const*, equation (5) can be solved just like the Jaynes-Cummings Hamiltonian within the two-state manifolds {|*e,n*_{b}
〉, |*g,n*_{b}
+1〉}. Within each manifold, the resulting dynamics is then in the form of quantized Rabi-like oscillations at the frequency

In particular, if the system is initially prepared in the state |*e,n*_{b}
〉, the probabilities for the system to be in the two states of the manifold after a time *τ* are ${\mid {c}_{e,{n}_{b}}\left(\tau \right)\mid}^{2}=1-{C}_{{n}_{b}+1}\left(\tau \right)$ and ${\mid {c}_{g,{n}_{b}+1}\left(\tau \right)\mid}^{2}={C}_{{n}_{b}+1}\left(\tau \right)$, where

${C}_{n}\left(\tau \right)=\frac{{4\mid \chi \mid}^{2}n}{{\U0001d4de}_{n-1}^{2}}{\mathrm{sin}}^{2}\left(\frac{1}{2}{\U0001d4de}_{n-1}\tau \right).$.

Note that unless *U*_{b}
=2*U*_{x}
, the detuning in *𝓡*_{n}
depends on the number of molecules present and thus resonant Rabi oscillations are only possible for a single manifold, an important difference with the optical case.

Let us now turn to the pump mechanism: it consists of continuously injecting pairs of fermionic atoms into each lattice site during time intervals *T* when *χ*=0. This can be achieved by using a state dependent optical lattice [1, 31] along with a beam of fermions in the internal states |*f*_{σ}
〉 that are incident on the lattice along the *z*-axis. The states |*f*_{σ}
〉 have polarizabilities opposite to those of |σ=1,2〉. Instead of a confining transverse potential, these atoms see therefore a repulsive potential centered at *z*=0. Provided that their incident energies are much less than the barrier height, tunneling is negligible and the atoms in states |*f*_{s}
〉 cannot enter the lattice. At the same time, we assume that the depth of the confining potential in the *z* direction for atoms in the states |σ=1,2〉 is sufficiently deep that the atoms are strongly confined around *z*=0 and tunneling out of the lattice is negligible. Under these conditions, two lasers directed along the lattice can be used to stimulate Raman transitions from the untrapped to the trapped states, |*f*
_{1}〉→|1〉 and |*f*
_{2}〉→|2〉[3]. The coupling constants are given by

$i{\kappa}_{k}=\frac{1}{2\sqrt{V}}\int {d}^{3}\mathbf{r}{e}^{ikz}{\Omega}_{2}\left(\mathbf{r}\right){w}_{f}^{*}\left(\mathbf{r}\right)$

where Ω_{2} is the two-photon Rabi frequency and we have approximated the incident beam of atoms in the states |*f*_{s}
〉 as being plane waves for the half space *z*<0 with momentum *ℏk* and normalization volume *V*.

For a broadband continuum of incident plane wave states, one can derive a Born-Markov master equation for the density matrix of the trapped fermions,

where *n̄*=*〈a*^{†}*σ*(*ωf*)*aσ*(*ω*_{f}
)〉 are the number of fermions in the incident beam with energy *ℏk*^{2}*/2mf*=*ωf* (assumed to be the same for both states |σ〉) and Γ=2*π*|κ(*ω*_{f}
)|2*D*(*ωf*) is the pumping rate with *D*(*ω*) being the density of states for the continuum [30]. In order to pump the lattice site with fermions we take *n̄*=1. Since the two fermionic species evolve independently in Eq. (7), we can express their density matrix as the tensor product of their density operators, *ρ*
^{(1)}(*t*)⊗*ρ*
^{(2)}(*t*). Eq. (7) then has the solution

where *ρ*
^{(σ)}_{0,0}=〈|*ρ*
^{(σ)}|0〉 and *ρ*
^{(σ)}1,1=〈0|*cσρ*
^{(σ)}
*c*^{†}*σ|*0〉. Since *ρ*
^{(s)}1,1 is the probability of having a fermion in the state |σ〉 occupying the lowest Wannier state of each lattice site, we see that the probability of having a pair of fermions in the states |1〉 and |2〉 at each site approaches 1 exponentially, i.e. ${\rho}_{1,1}^{\left(2\right)}$(*t*)${\rho}_{1,1}^{\left(1\right)}$(*t*)→1 for Γ*t* ≫1. Therefore for times *t* ≫Γ-1, the fermions are “pumped” into the state |*e*〉=*c*^{†}
2${c}_{1}^{\u2020}$|0〉 with unit probability.

Finally, the damping of the molecular mode is described via a phenomenological master equation that includes the dominant loss mechanisms. They include in particular three-body inelastic collisions between a molecule and two atoms in which the atoms form a dimer with the resultant loss of the atoms and molecule. This occurs at a rate

${\gamma}_{3}\sim \frac{\overline{h}{a}_{f}^{4}}{{m}_{f}{\left({\ell}_{x}^{\left(f\right)}{\ell}_{y}^{\left(f\right)}{\ell}_{z}^{\left(f\right)}\right)}^{2}},$,

where *a*_{f}
is the atomic scattering length [32]. Another loss mechanism is spontaneous emission from the molecular excited state from which the optical lattice and photoassociation beams are detuned. When *χ*(*t*)=0, these losses are due solely to scattering of lattice photons. This gives a rate *γ*
_{1}≈*γeω*^{(b)}
/4|Δ| for a blue detuned lattice and using a harmonic approximation for the optical potential at the lattice nodes. Here *γe* is the excited state linewidth and Δ the detuning from the excited state. The contribution to the master equation due to both loss mechanisms is

$$-\frac{1}{2}{\gamma}_{3}\left[{B}^{\u2020}B\rho -2B\rho {B}^{\u2020}+\rho {B}^{\u2020}B\right],$$

where *B*=*bc*_{1}*c*_{2}=*bσ*- [33].

In the limit of strong pumping, Γ≫*γ*
_{1,3}, the atoms reach their steady state before the state of the molecules has noticeably changed. We can then replace Eq. (8) with a coarse-grained master equation valid for time intervals much larger than Γ^{-1}. To do this we substitute the steady-state values of Eq. (7) into the density operator, *ρ*(*t*)=*ρ*
^{(b)}(*t*)⊗|*e*〉〈*e*| where *ρ*
^{(b)} is the molecular density operator, and then trace over the states of the atoms in Eq. (8). This yields finally

$\frac{\partial {\rho}^{\left(b\right)}\left(t\right)}{\partial t}=-\frac{1}{2}\gamma \left[{b}^{\u2020}b{\rho}^{\left(b\right)}-2b{\rho}^{\left(b\right)}{b}^{\u2020}+{\rho}^{\left(b\right)}{b}^{\u2020}b\right]\equiv \U0001d4db\left[{\rho}^{\left(b\right)}\right]$

where *γ*=*γ*
_{1}+*γ*
_{3}. We note that for *a*_{f}
~100*a*0 and *mf*~10 a.m.u. one has ${\gamma}_{3}^{-1}$~10s while ${\gamma}_{1}^{-1}$~1s for a detuning of 10000 linewidths and *ω*
^{(b)}
~2*π*×10^{4} s^{-1}. For molecules created in their rotational-vibrational ground state, we can ignore losses due to inelastic collisions between pairs of molecules, or between an atom and a molecule [6]. If the molecules are not created in their vibrational ground state, inelastic collisions give rise to an additional contribution to g with a decay rate of *γI*≈10^{3}s^{-1} for a vibrational relaxation rate of 10^{-10} cm^{3}/s [35]. This relaxation can be significantly suppressed in case the molecules are formed by fermionic atoms with a scattering length *a* large compared to the characteristic size *Re* of the interatomic potential. In that case, the Fermi statistics of the atoms lead to an order-of-magnitude reduction in collisional relaxation [36].

Just like in micromaser theory, the weak damping of the molecular mode allows us to assume that during the intervals t of photoassociation time evolution is unitary and given by *ĥ*. Since $\tau \sim 1\u2044\sqrt{n+1}\mid \chi \mid $ for complete Rabi oscillations to occur in the presence of *n* molecules, this implies that the photoassociation fields must satisfy $\sqrt{n+1}\mid \chi \mid \gg {\gamma}_{1,3}$. The resulting molecular density operator is then

where *t*_{l}
=*l*(*T*+t) for *l*=0,1,2,…. In order to determine *F*(*τ*) explicitly, we recall that at the end of each interval *T* every lattice site is occupied with unit probability by a pair of fermions for Γ*T*≫1. Since the fermions are uncorrelated with the state of the molecules already present, the reduced density matrix at *t*_{l}
+τ is then

where

Here *p*_{n}
=〈*n*|*ρ*
^{(b)}|*n*〉 and we have assumed that *r*^{(b)}
is initially diagonal in the number state basis. The formal solution for the molecular density operator is then

We note that *T* is controlled by the timing of the photoassociation beams, in contrast to conventional micromasers where *T* corresponds to the arrival times of the atoms and is therefore a random variable. It is know from the theory of the micromaser that the photon statistics of the generated field is a sensitive function of the pump mechanism. As such, the fact that the pump can now be easily manipulated provides us with an additional parameter to tailor the state of the molecular field, for instance in a closed loop configuration. Such possibilities will be investigated in detail in future work.

The steady-state molecular field is given by the return map condition *$\overline{\rho}$ ^{(b)}*=

*ρ*

^{(b)}(

*t*

_{l}+1)=

*ρ*

^{(b)}(

*t*l). If the interval

*T*is such that Γ

^{-1}≪

*T*≪

*g*

^{-1}, we can expand the exponential in the formal solution to lowest order in

*𝓛*. The steady-state condition is then

which has the same form as the steady-state solution of the micromaser if one identifies *T*
^{-1} with the *average* rate at which atoms are injected into the cavity [23]. The steady state solution in the number state basis is,

where *p̄*_{n}
is the probability of having *n* molecules and *p̄0* is determined by the conservation of probability, ∑*n*
*̄p*_{n}
=1. Eq. (12) has the same form as the zero-temperature micromaser photon statistics, except that *C*_{l}
(τ) contains an *l*-dependent detuning in *𝓡*_{l}
resulting from the two-body interactions.

The important features of the molecule statistics can be examined as a function of the number of pumping cycles per lifetime of the molecules *N*_{ex}
=1/*γT*, the micromaser pump parameter Θ=(*N*_{ex}
)^{1/2}
*χt*, the linear detuning η≡(2*ω*_{f}
-*ω*_{b}
)/2|*χ*|, and the dimensionless coefficient for the nonlinear detuning *β*=(2*U*_{x}
-*U*_{b}
)/2|*χ*|. The nonlinear detuning is unique to the molecular micromaser and represents a new degree of freedom that can be experimentally manipulated in order to adjust the molecular statistics. From Eq. (2) it follows that *β* could be controlled by either using a Feshbach resonance to adjust the s-wave scattering length or by changing the lattice potential.

Figure 1 shows the average number of molecules at each site, 〈*n̂*_{b}
〉, as a function of Θ and *β* for the linear resonance condition *η*=0. We see that the lasing threshold is not affected by the nonlinear detuning and occurs at Θ≈1. However, above threshold 〈*n̂b*〉 is strongly suppressed as *b* is increased. This is because the effective detuning between the fermion atom pairs and molecules increases with increasing molecule number due to the self-phase modulation of the molecules, *U*_{b}*n̂*_{b}
(*n̂*_{b}
-1), and the AC Stark shift of the atoms, *U*_{x}*n̂*_{b}*σ*_{z}
.

In contrast to the conventional micromaser, which is not affected by collisions, a finite linear detuning, *η*≠0, can be used to increase the number of molecules by partially compensating for the inherent nonlinear self-phase modulation, η+*βn*=0. Figure 2 shows the maximum value of 〈*n̂*_{b}
〉 as a function of *η* and *β*. It illustrates how the effect of a finite *β*>0(<0) can be compensated for by choosing a finite negative (positive) detuning, *η*<0(>0).

The sharp resonance-like dips in 〈*n̂*_{b}
〉 correspond to trapping states. They correspond to number states |*n*_{b}
〉 of the molecular field such that the photoassociation process undergoes an integer number *q* of Rabi oscillations from atoms to molecules and back during the time *τ*, $\tau {\U0001d4e1}_{{n}_{b}}=2\pi q$ [34]. From Eq. (12) it then follows that *p̄*_{n}
′=0 for all *n*′>*n* and the molecular field is unable to evolve beyond the state with *n*_{b}
molecules. However a slight change in the interaction time (or equivalently Θ) will result in an incomplete number of full Rabi oscillations and hence some probability of the fermionic pair having been converted to a molecule.

It is known that in the usual micromaser the presence of thermal microwave photons makes the observation of trapping states very challenging, since thermal fluctuations allow the photon number to increase beyond the trapping state value. Similarly, trapping states of the molecular micromaser will be destroyed by the presence of thermal molecules. However, while thermal cavity photons are present for all nonzero temperatures, thermal molecules are absent as a consequence of atom number conservation. Hence the trapping states should be more readily observable in the molecular micromaser.

Just like its microwave counterpart, the molecular micromaser normally exhibits highly non-classical statistics. Figure 3 shows the Mandel *Q* parameter,

$Q=\frac{\u3008{\hat{n}}_{b}^{2}\u3009-{\u3008{\hat{n}}_{b}\u3009}^{2}}{\u3008{\hat{n}}_{b}\u3009}-1$

for *β*
^{2}=0, which corresponds to the case of the usual micromaser, and for *β*
^{2}=0.1. For completeness, we have also added results in the presence of thermal noise, simulated by coupling the molecular density operator to a thermal bath with a standard master equation. Just above threshold, the molecule distribution is strongly super-Poissonian (*Q*>0) and then become sub-Poissonian (*Q*<0) until Θ≈2*π* after which *Q* shows very sharp oscillations. We note that the number fluctuations decrease with increasing |*β*| and show smaller super-Poissonian peaks. As with 〈*n*_{b}
〉, the very sharp resonances in *Q* are attributable to trapping states, which sharply truncate the width of the probability distribution leading to reduced fluctuations.

Besides the lasing threshold, the micromaser exhibits a number of additional transitions that in the limit *N*_{ex}
→∞ are similar to first-order phase transitions [27]. The first such transition occurs at Θ≈2*π* in Figs. 1 and 3. These transitions can be understood using a semiclassical equation of motion for 〈*n̂*_{b}
〉=*n̄* that describes the competition between the gain in the molecular field due to photoassociation and its decay at the rate *γ*,

To obtain this approximate equation we have made the factorization ansatz 〈${\widehat{n}}_{b}^{r}$
〉=〈*n̂*_{b}
〉
^{r}
. The threshold condition can be obtained by setting the linearized gain for *n̄*→0 equal to the losses, which yields Θ=1 independently of η and *β*.

In this “classical” description, there is in general more than one stable steady-state solution of Eq. (??). Of course, in a proper quantum-mechanical description all but one of them are actually metastable, and quantum fluctuations eventually lead the system to a unique steady state. Nonetheless, for a large enough excitation of the molecular field the lifetime of the metastable states is expected to be long enough as to be observable in a transient experiment, as has been the case in the conventional micromaser [37]. These (meta)stable steady-state solutions roughly correspond to the maxima in the probability distribution *p̄n* shown in Fig. 4 as a function of Θ, the correspondence becoming better the larger *N*_{ex}
. The peaks of the binomial or multinomial molecular distribution observed for some values of Θ correspond to bi- or multistable classical regimes. The small “islands” correspond to the presence of trapping states. (We have introduced a small amount of thermal noise in the plot to reduce their effect.) At those values of Θ for which there are two stable solutions the dynamics of the molecular state exhibit spontaneous quantum jumps and bistability in the molecule number [37]. The bimodal nature of the probability distribution gives rise to large number fluctuations and jumps in the mean molecule number 〈*n*_{b}
〉 as Θ is increased [23]. The study of the dynamics of these jumps, both theoretically and experimentally, will provide direct information on quantum noise-induced transitions in mesoscopic systems.

One considerable attraction of the present system is that it provides one with not just one, but a full array of molecular micromasers. Phase-locked laser arrays are of considerable interest as they permit e.g. the realization of high-power light sources. We are currently extending our analysis to include intersite molecular tunnelling. This will allow us to extend our system to much larger occupation numbers. At the same time we expect that the inclusion of tunnelling effects will provide valuable insights into the relative phases of the individual molecular micromasers and possibly lead to a phase locked micromaser array [38]. However, intersite tunnelling should also eliminate the effect of the trapping states on the molecule statistics in the same manner as thermal fluctuations, since inter-well molecular tunnelling will allow the molecule number to evolve past the limit imposed by the trapping state.

## Acknowledgments

This work is supported in part by the US Office of Naval Research, by the National Science Foundation, by the US Army Research Office, by the National Aeronautics and Space Administration, and by the Joint Services Optics Program. We acknowledge helpful discussions with B. P. Anderson.

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