## Lattice array of molecular micromasers

Optics Express, Vol. 12, Issue 1, pp. 30-41 (2004)

http://dx.doi.org/10.1364/OPEX.12.000030

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### 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

5. P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates,” Phys. Rev. A **58**, R797–R800 (1998). [CrossRef]

6. D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of Coupled Atomic and Molecular Bose-Einstein Condensates,” Phys. Rev. Lett. **84**, 5029–5032 (2000). [CrossRef] [PubMed]

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^{87}Rb BEC [8

8. R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, “Molecules in a Bose-Einstein Condensate,” Science **287**, 1016–1019 (2000). [CrossRef] [PubMed]

^{85}Rb BEC [9

9. E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E. Wieman, “Atom-molecule coherence in a Bose-Einstein condensate,” Nature **417**, 529–533 (2002). [CrossRef] [PubMed]

^{40}K [10

10. C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, “Creation of ultracold molecules from a Fermi gase of atoms,” Nature **424**, 47–50 (2003); M. Greiner, C. A. Regal, and D. S. Jin, “Emergence of a molecular Bose-Einstein condensate from a Fermi gas,” Nature 426, 537–540 (2003). [CrossRef] [PubMed]

^{6}Li [11

11. K. E. Strecker, G. B. Partridge, and R. G. Hulet, “Conversion of an Atomic Fermi Gas to a Long-Lived Molecular Bose Gas,” Phys. Rev. Lett. **91**, 080406 (2003). [CrossRef] [PubMed]

12. S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Denschlag, and R. Grimm, “Pure Gas of Optically Trapped Molecules Created from Fermionic Atoms,” Phys. Rev. Lett. **91**, 240402 (2003) [CrossRef] [PubMed]

13. J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, “Production of Long-Lived Ultracold Li Molecules from a Fermi gas,” Phys. Rev. Lett. **91**, 240402 (2003) [CrossRef]

14. M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, “Observation of Bose-Einstein Condensation of Molecules,” Phys. Rev. Lett. **91**, 250401 (2003) [CrossRef]

15. S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Denschlag, and R. Grimm, “Bose-Einstein Condensation of Molecules,” Science **302**2101 (2003). [CrossRef] [PubMed]

16. D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. **89**, 040402 (2002). [CrossRef] [PubMed]

17. B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S. Kotochigova, P. Julienne, and P. Zoller, “Creation of a Dipolar Superfluid in Optical Lattices,” Phys. Rev. Lett. **90**, 110401 (2003). [CrossRef] [PubMed]

18. M. G. Moore and H. R. Sadeghpour, “Controlling two-species Mott-insulator phases in an optical lattice to form an array of dipolar molecules,” Phys. Rev. A **67**, 041603(R) (2003). [CrossRef]

19. K. Mølmer, “Jaynes-Cummings Dynamics with a Matter Wave Oscillator,” Phys. Rev. Lett. **90**, 110403 (2003). [CrossRef] [PubMed]

20. T. Esslinger and K. Mølmer, “Atoms and Molecules in Lattices: Bose-Einstein Condensates Built on a Shared Vacuum,” Phys. Rev. Lett. **90**, 160406 (2003). [CrossRef] [PubMed]

*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

21. K. Goral, M. Gajda, and K. Rzazewski, “Multimode Dynamics of a Coupled Ultracold Atomic-Molecular System,” Phys. Rev. Lett. **86**, 1397–1400 (2001). [CrossRef] [PubMed]

*fermionic*atoms in an optical lattice. In case that only two fermions occupy each lattice site this problem [22

22. C. P. Search, W. Zhang, and P. Meystre, “Molecular Micromaser,” Phys. Rev. Lett. **91**, 190401 (2003). [CrossRef] [PubMed]

23. P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A **34**, 3077–3087 (1986). [CrossRef] [PubMed]

24. D. Meschede, H. Walther, and G. Müller, “One-Atom Maser,” Phys. Rev. Lett. **54**, 551–554 (1985). [CrossRef] [PubMed]

25. G. Rempe, F. SchmidtKaler, and H. Walther, “Observation of sub-Poissonian photon statistics in a micromaser,” Phys. Rev. Lett. **64**, 2783–2786 (1990). [CrossRef] [PubMed]

27. A. M. Guzman, P. Meystre, and E. M. Wright, “Semiclassical theory of a micromaser,” Phys. Rev. A **40**, 2471–2478 (1989). [CrossRef] [PubMed]

*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.

22. C. P. Search, W. Zhang, and P. Meystre, “Molecular Micromaser,” Phys. Rev. Lett. **91**, 190401 (2003). [CrossRef] [PubMed]

*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

5. P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates,” Phys. Rev. A **58**, R797–R800 (1998). [CrossRef]

6. D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of Coupled Atomic and Molecular Bose-Einstein Condensates,” Phys. Rev. Lett. **84**, 5029–5032 (2000). [CrossRef] [PubMed]

8. R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, “Molecules in a Bose-Einstein Condensate,” Science **287**, 1016–1019 (2000). [CrossRef] [PubMed]

*V*

_{f}seen by the atoms is

*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

16. D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. **89**, 040402 (2002). [CrossRef] [PubMed]

*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.

*ϕ*

_{0}(

*ζ*)=exp(-

*z*

^{2}/2)/

*π*

^{1/4}. Here,

*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

*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.

*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

8. R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, “Molecules in a Bose-Einstein Condensate,” Science **287**, 1016–1019 (2000). [CrossRef] [PubMed]

*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.

*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

4. D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum Phases in an optical lattice,” Phys. Rev. A **63**, 053601 (2001). [CrossRef]

28. A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A **68**, 023606 (2003). [CrossRef]

*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

2. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature **415**, 39–44 (2002). [CrossRef] [PubMed]

3. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. **81**, 3108–3111 (1998). [CrossRef]

*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

29. P.W. Anderson, “Random-Phase Approximation in the Theory of Superconductivity,” Phys. Rev. **112**, 1900–1916 (1958). [CrossRef]

*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

*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,

*ω*

_{b}+

*δ*→

*ω*

_{b}and

*ω*

_{f}+

*U*

_{f}/2→

*ω*

_{f}.

*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.

*bosonic*atoms into molecules in an optical lattice [19

19. K. Mølmer, “Jaynes-Cummings Dynamics with a Matter Wave Oscillator,” Phys. Rev. Lett. **90**, 110403 (2003). [CrossRef] [PubMed]

*T*and let dissipation take over. This sequence is then repeated to build up the molecular field.

*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

*e,n*

_{b}〉, the probabilities for the system to be in the two states of the manifold after a time

*τ*are

*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.

*T*when

*χ*=0. This can be achieved by using a state dependent optical lattice [1

1. G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. **82**, 1060–1063 (1999). [CrossRef]

31. O. Mandel, M. Greiner, A. Widera, T. Rom, T. Hänsch, and I. Bloch, “Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials,” Phys. Rev. Lett. **91**, 010407 (2003). [CrossRef] [PubMed]

*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

3. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. **81**, 3108–3111 (1998). [CrossRef]

_{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*.

*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

30. C. P. Search, S. Pötting, W. Zhang, and P. Meystre, “Input-output theory for fermions in an atom cavity,” Phys. Rev. A **66**, 043616 (2002). [CrossRef]

*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

*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.

*t*)

*t*)→1 for Γ

*t*≫1. Therefore for times

*t*≫Γ-1, the fermions are “pumped” into the state |

*e*〉=

*c*

^{†}2

*a*

_{f}is the atomic scattering length [32

32. P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, “Three-Body Recombination of Ultracold Atoms to a Weakly Bound s Level,” Phys. Rev. Lett. **77**, 2921–2924 (1996). [CrossRef] [PubMed]

*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

*B*=

*bc*

_{1}

*c*

_{2}=

*bσ*- [33

33. M. W. Jack, “Decoherence due to Three-Body Loss and its Effect on the State of a Bose-Einstein Condensate,” Phys. Rev. Lett. **89**, 140402 (2002). [CrossRef] [PubMed]

*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

*γ*=

*γ*

_{1}+

*γ*

_{3}. We note that for

*a*

_{f}~100

*a*0 and

*mf*~10 a.m.u. one has

*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

6. D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of Coupled Atomic and Molecular Bose-Einstein Condensates,” Phys. Rev. Lett. **84**, 5029–5032 (2000). [CrossRef] [PubMed]

*γI*≈10

^{3}s

^{-1}for a vibrational relaxation rate of 10

^{-10}cm

^{3}/s [35

35. P. Soldan, M. T. Cvita, J. M. Hutson, P. Honvault, and J. M. Launay“Quantum Dynamics of Ultracold Na+Na2 Collisions,” Phys. Rev. Lett.89, 153201 (2002);N. Balakrishnan, R. C. Forrey, and A. Dalgarno, “Threshold phenomena in ultracold atom-molecule collisions,” Chem. Phys. Lett.280, 1–4 (1997). [CrossRef] [PubMed]

*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].

*ĥ*. Since

*n*molecules, this implies that the photoassociation fields must satisfy

*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

*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.

*=*ρ ¯

^{(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

*T*

^{-1}with the

*average*rate at which atoms are injected into the cavity [23

23. P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A **34**, 3077–3087 (1986). [CrossRef] [PubMed]

*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.

*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.

*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}.

*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).

*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*,

34. P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,” Opt. Lett. **13**, 1078 (1988). [CrossRef] [PubMed]

*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.

*Q*parameter,

*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.

*N*

_{ex}→∞ are similar to first-order phase transitions [27

27. A. M. Guzman, P. Meystre, and E. M. Wright, “Semiclassical theory of a micromaser,” Phys. Rev. A **40**, 2471–2478 (1989). [CrossRef] [PubMed]

*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*,

*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*.

37. O. Benson, G. Raithel, and H. Walther, “Quantum jumps of the micromaser field: Dynamic behavior close to phase transition points,” Phys. Rev. Lett. **72**3506–3509 (1994). [CrossRef] [PubMed]

*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

37. O. Benson, G. Raithel, and H. Walther, “Quantum jumps of the micromaser field: Dynamic behavior close to phase transition points,” Phys. Rev. Lett. **72**3506–3509 (1994). [CrossRef] [PubMed]

*n*

_{b}〉 as Θ is increased [23

23. P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A **34**, 3077–3087 (1986). [CrossRef] [PubMed]

38. B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science **282**, 1686–1689 (1998). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, “Quantum Logic Gates in Optical Lattices,” Phys. Rev. Lett. |

2. | M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature |

3. | D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. |

4. | D. van Oosten, P. van der Straten, and H. T. C. Stoof, “Quantum Phases in an optical lattice,” Phys. Rev. A |

5. | P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates,” Phys. Rev. A |

6. | D. J. Heinzen, R. Wynar, P. D. Drummond, and K. V. Kheruntsyan, “Superchemistry: Dynamics of Coupled Atomic and Molecular Bose-Einstein Condensates,” Phys. Rev. Lett. |

7. | E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, “Feshbach resonances in atomic Bose-Einstein condensate,” Phys. Rep. |

8. | R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, “Molecules in a Bose-Einstein Condensate,” Science |

9. | E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E. Wieman, “Atom-molecule coherence in a Bose-Einstein condensate,” Nature |

10. | C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, “Creation of ultracold molecules from a Fermi gase of atoms,” Nature |

11. | K. E. Strecker, G. B. Partridge, and R. G. Hulet, “Conversion of an Atomic Fermi Gas to a Long-Lived Molecular Bose Gas,” Phys. Rev. Lett. |

12. | S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Denschlag, and R. Grimm, “Pure Gas of Optically Trapped Molecules Created from Fermionic Atoms,” Phys. Rev. Lett. |

13. | J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, “Production of Long-Lived Ultracold Li Molecules from a Fermi gas,” Phys. Rev. Lett. |

14. | M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, “Observation of Bose-Einstein Condensation of Molecules,” Phys. Rev. Lett. |

15. | S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Denschlag, and R. Grimm, “Bose-Einstein Condensation of Molecules,” Science |

16. | D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, “Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator,” Phys. Rev. Lett. |

17. | B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S. Kotochigova, P. Julienne, and P. Zoller, “Creation of a Dipolar Superfluid in Optical Lattices,” Phys. Rev. Lett. |

18. | M. G. Moore and H. R. Sadeghpour, “Controlling two-species Mott-insulator phases in an optical lattice to form an array of dipolar molecules,” Phys. Rev. A |

19. | K. Mølmer, “Jaynes-Cummings Dynamics with a Matter Wave Oscillator,” Phys. Rev. Lett. |

20. | T. Esslinger and K. Mølmer, “Atoms and Molecules in Lattices: Bose-Einstein Condensates Built on a Shared Vacuum,” Phys. Rev. Lett. |

21. | K. Goral, M. Gajda, and K. Rzazewski, “Multimode Dynamics of a Coupled Ultracold Atomic-Molecular System,” Phys. Rev. Lett. |

22. | C. P. Search, W. Zhang, and P. Meystre, “Molecular Micromaser,” Phys. Rev. Lett. |

23. | P. Filipowicz, J. Javanainen, and P. Meystre, “Theory of a microscopic maser,” Phys. Rev. A |

24. | D. Meschede, H. Walther, and G. Müller, “One-Atom Maser,” Phys. Rev. Lett. |

25. | G. Rempe, F. SchmidtKaler, and H. Walther, “Observation of sub-Poissonian photon statistics in a micromaser,” Phys. Rev. Lett. |

26. | M. O. Scully and M. S. Zubairy, “Quantum Optics” (Cambridge University Press, Cambridge, UK, 1997). |

27. | A. M. Guzman, P. Meystre, and E. M. Wright, “Semiclassical theory of a micromaser,” Phys. Rev. A |

28. | A. Albus, F. Illuminati, and J. Eisert, “Mixtures of bosonic and fermionic atoms in optical lattices,” Phys. Rev. A |

29. | P.W. Anderson, “Random-Phase Approximation in the Theory of Superconductivity,” Phys. Rev. |

30. | C. P. Search, S. Pötting, W. Zhang, and P. Meystre, “Input-output theory for fermions in an atom cavity,” Phys. Rev. A |

31. | O. Mandel, M. Greiner, A. Widera, T. Rom, T. Hänsch, and I. Bloch, “Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials,” Phys. Rev. Lett. |

32. | P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, “Three-Body Recombination of Ultracold Atoms to a Weakly Bound s Level,” Phys. Rev. Lett. |

33. | M. W. Jack, “Decoherence due to Three-Body Loss and its Effect on the State of a Bose-Einstein Condensate,” Phys. Rev. Lett. |

34. | P. Meystre, G. Rempe, and H. Walther, “Very-low temperature behaviour of a micromaser,” Opt. Lett. |

35. | P. Soldan, M. T. Cvita, J. M. Hutson, P. Honvault, and J. M. Launay“Quantum Dynamics of Ultracold Na+Na2 Collisions,” Phys. Rev. Lett.89, 153201 (2002);N. Balakrishnan, R. C. Forrey, and A. Dalgarno, “Threshold phenomena in ultracold atom-molecule collisions,” Chem. Phys. Lett.280, 1–4 (1997). [CrossRef] [PubMed] |

36. | D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, “Weakly bound dimers of fermionic atoms,” condmat/0309010. |

37. | O. Benson, G. Raithel, and H. Walther, “Quantum jumps of the micromaser field: Dynamic behavior close to phase transition points,” Phys. Rev. Lett. |

38. | B. P. Anderson and M. A. Kasevich, “Macroscopic Quantum Interference from Atomic Tunnel Arrays,” Science |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(270.3430) Quantum optics : Laser theory

**ToC Category:**

Focus Issue: Cold atomic gases in optical lattices

**History**

Original Manuscript: November 7, 2003

Revised Manuscript: December 22, 2003

Published: January 12, 2004

**Citation**

Chris Search, Takahiko Miyakawa, and Pierre Meystre, "Lattice array of molecular micromasers," Opt. Express **12**, 30-41 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-30

Sort: Journal | Reset

### References

- G. K. Brennen, C. M. Caves, P. S. Jessen, and I. H. Deutsch, "Quantum Logic Gates in Optical Lattices," Phys. Rev. Lett. 82, 1060-1063 (1999). [CrossRef]
- M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature 415, 39-44 (2002). [CrossRef] [PubMed]
- D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, "Cold Bosonic Atoms in Optical Lattices," Phys. Rev. Lett. 81, 3108-3111 (1998). [CrossRef]
- D. van Oosten, P. van der Straten, and H. T. C. Stoof, "Quantum Phases in an optical lattice," Phys. Rev. A 63, 053601 (2001). [CrossRef]
- P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, "Stimulated Raman molecule production in Bose-Einstein condensates," Phys. Rev. A 58, R797-R800 (1998). [CrossRef]
- D. J. Heinzen, R. Wynar, P. D. Drummond and K. V. Kheruntsyan, "Superchemistry: Dynamics of Coupled Atomic and Molecular Bose-Einstein Condensates," Phys. Rev. Lett. 84, 5029-5032 (2000). [CrossRef] [PubMed]
- E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, "Feshbach resonances in atomic Bose-Einstein condensate," Phys. Rep. 315, 199-230 (1999). [CrossRef]
- R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, and D. J. Heinzen, "Molecules in a Bose-Einstein Condensate," Science 287, 1016-1019 (2000). [CrossRef] [PubMed]
- E. A. Donley, N. R. Claussen, S. T. Thompson, and C. E.Wieman, "Atom-molecule coherence in a Bose-Einstein condensate," Nature 417, 529-533 (2002). [CrossRef] [PubMed]
- C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, "Creation of ultracold molecules from a Fermi gase of atoms," Nature 424, 47-50 (2003); M. Greiner, C. A. Regal, and D. S. Jin, "Emergence of a molecular Bose-Einstein condensate from a Fermi gas," Nature 426, 537-540 (2003). [CrossRef] [PubMed]
- K. E. Strecker, G. B. Partridge, and R. G. Hulet, "Conversion of an Atomic Fermi Gas to a Long-Lived Molecular Bose Gas," Phys. Rev. Lett. 91, 080406 (2003). [CrossRef] [PubMed]
- S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Denschlag, and R. Grimm, "Pure Gas of Optically Trapped Molecules Created from Fermionic Atoms," Phys. Rev. Lett. 91, 240402 (2003). [CrossRef] [PubMed]
- J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, "Production of Long-Lived Ultracold Li Molecules from a Fermi gas," Phys. Rev. Lett. 91, 240402 (2003). [CrossRef]
- M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle, "Observation of Bose-Einstein Condensation of Molecules," Phys. Rev. Lett. 91, 250401 (2003). [CrossRef]
- S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Denschlag, and R. Grimm, "Bose-Einstein Condensation of Molecules," Science 302, 2101 (2003). [CrossRef] [PubMed]
- D. Jaksch, V. Venturi, J. I. Cirac, C. J. Williams, and P. Zoller, "Creation of a Molecular Condensate by Dynamically Melting a Mott Insulator," Phys. Rev. Lett. 89, 040402 (2002). [CrossRef] [PubMed]
- B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S. Kotochigova, P. Julienne, and P. Zoller, "Creation of a Dipolar Superfluid in Optical Lattices," Phys. Rev. Lett. 90, 110401 (2003). [CrossRef] [PubMed]
- M. G. Moore and H. R. Sadeghpour, "Controlling two-species Mott-insulator phases in an optical lattice to form an array of dipolar molecules," Phys. Rev. A 67, 041603(R) (2003). [CrossRef]
- K. Mølmer, "Jaynes-Cummings Dynamics with a Matter Wave Oscillator," Phys. Rev. Lett. 90, 110403 (2003). [CrossRef] [PubMed]
- T. Esslinger and K. Mølmer, "Atoms and Molecules in Lattices: Bose-Einstein Condensates Built on a Shared Vacuum," Phys. Rev. Lett. 90, 160406 (2003). [CrossRef] [PubMed]
- K. Goral, M.Gajda, and K. Rzazewski, "Multimode Dynamics of a Coupled Ultracold Atomic-Molecular System," Phys. Rev. Lett. 86, 1397-1400 (2001). [CrossRef] [PubMed]
- C. P. Search, W. Zhang, and P. Meystre, "Molecular Micromaser," Phys. Rev. Lett. 91, 190401 (2003). [CrossRef] [PubMed]
- P. Filipowicz, J. Javanainen, and P. Meystre, "Theory of a microscopic maser," Phys. Rev. A 34, 3077-3087 (1986). [CrossRef] [PubMed]
- D. Meschede, H. Walther, and G. Müller, "One-Atom Maser," Phys. Rev. Lett. 54, 551-554 (1985). [CrossRef] [PubMed]
- G. Rempe, F. SchmidtKaler, and H.Walther, �??Observation of sub-Poissonian photon statistics in a micromaser,�?? Phys. Rev. Lett. 64, 2783-2786 (1990). [CrossRef] [PubMed]
- M. O. Scully and M. S. Zubairy, Quantum Optics, (Cambridge University Press, Cambridge, UK, 1997).
- A. M. Guzman, P. Meystre, and E. M. Wright, "Semiclassical theory of a micromaser," Phys. Rev. A 40, 2471-2478 (1989). [CrossRef] [PubMed]
- A. Albus, F. Illuminati, and J. Eisert, "Mixtures of bosonic and fermionic atoms in optical lattices," Phys. Rev. A 68, 023606 (2003). [CrossRef]
- P.W. Anderson, "Random-Phase Approximation in the Theory of Superconductivity," Phys. Rev. 112, 1900-1916 (1958). [CrossRef]
- C. P. Search, S. Pötting, W. Zhang, and P. Meystre, "Input-output theory for fermions in an atom cavity," Phys. Rev. A 66, 043616 (2002). [CrossRef]
- O. Mandel, M. Greiner, A. Widera, T. Rom, T. Hänsch, and I. Bloch, "Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials," Phys. Rev. Lett. 91, 010407 (2003). [CrossRef] [PubMed]
- P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, "Three-Body Recombination of Ultracold Atoms to a Weakly Bound s Level," Phys. Rev. Lett. 77, 2921-2924 (1996). [CrossRef] [PubMed]
- M. W. Jack, "Decoherence due to Three-Body Loss and its Effect on the State of a Bose-Einstein Condensate," Phys. Rev. Lett. 89, 140402 (2002). [CrossRef] [PubMed]
- P. Meystre, G. Rempe, and H. Walther, "Very-low temperature behaviour of a micromaser," Opt. Lett. 13, 1078 (1988). [CrossRef] [PubMed]
- P. Soldan, M. T. Cvita, J. M. Hutson, P. Honvault, and J. M. Launay, "Quantum Dynamics of Ultracold Na + Na2 Collisions," Phys. Rev. Lett. 89, 153201 (2002); N. Balakrishnan, R. C. Forrey, and A. Dalgarno, "Threshold phenomena in ultracold atom-molecule collisions," Chem. Phys. Lett. 280, 1-4 (1997). [CrossRef] [PubMed]
- D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, "Weakly bound dimers of fermionic atoms," condmat/ 0309010.
- O. Benson, G. Raithel, and H. Walther, "Quantum jumps of the micromaser field: Dynamic behavior close to phase transition points," Phys. Rev. Lett. 72 3506-3509 (1994). [CrossRef] [PubMed]
- B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686-1689 (1998). [CrossRef] [PubMed]

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