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

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References

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  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]
  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]
  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]
  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]
  7. E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, "Feshbach resonances in atomic Bose-Einstein condensate," Phys. Rep. 315, 199-230 (1999).
    [CrossRef]
  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]
  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]
  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]
  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]
  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]
  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]
  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 40, 2471-2478 (1989).
    [CrossRef] [PubMed]
  28. A. Albus, F. Illuminati, and J. Eisert, "Mixtures of bosonic and fermionic atoms in optical lattices," Phys. Rev. A 68, 023606 (2003).
    [CrossRef]
  29. P.W. Anderson, "Random-Phase Approximation in the Theory of Superconductivity," Phys. Rev. 112, 1900-1916 (1958).
    [CrossRef]
  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]
  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]
  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]
  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]
  34. P. Meystre, G. Rempe, and H. Walther, "Very-low temperature behaviour of a micromaser," Opt. Lett. 13, 1078 (1988).
    [CrossRef] [PubMed]
  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. 72 3506-3509 (1994).
    [CrossRef] [PubMed]
  38. B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686-1689 (1998).
    [CrossRef] [PubMed]

condmat (1)

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

Nature (3)

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]

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]

Opt. Lett. (1)

Phys. Rep. (1)

E. Timmermans, P. Tommasini, M. Hussein, and A. Kerman, "Feshbach resonances in atomic Bose-Einstein condensate," Phys. Rep. 315, 199-230 (1999).
[CrossRef]

Phys. Rev. (1)

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

Phys. Rev. A (7)

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]

P. Filipowicz, J. Javanainen, and P. Meystre, "Theory of a microscopic maser," Phys. Rev. A 34, 3077-3087 (1986).
[CrossRef] [PubMed]

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]

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]

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]

Phys. Rev. Lett. (20)

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]

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]

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]

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]

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]

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]

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]

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]

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. 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]

Science (3)

B. P. Anderson and M. A. Kasevich, "Macroscopic Quantum Interference from Atomic Tunnel Arrays," Science 282, 1686-1689 (1998).
[CrossRef] [PubMed]

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]

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]

Other (1)

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

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Figures (4)

Fig. 1.
Fig. 1.

Average number of molecules, 〈b 〉, versus Θ and β for Nex =25 and η=0

Fig. 2.
Fig. 2.

Maximum value of 〈b 〉 as a function of η and b for Nex =25.

Fig. 3.
Fig. 3.

Q versus Θ for Nex =25 and η=0 with (a) β 2=0 and (b) β 2=0.1. (c) shows the effect of thermal noise with n̄T=0.1 and all other parameters same as (a).

Fig. 4.
Fig. 4.

p̄n as a function of Θ for Nex =50 and β=η=0 and n̄T=0.01

Equations (21)

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H ̂ = i ( H ̂ 0 i + H ̂ I i ) + H ̂ T
H ̂ 0 i = h ¯ ( ω b + δ ) n ̂ b i + h ¯ ω f ( n ̂ 1 i + n ̂ 2 i ) + 1 2 h ¯ U b n ̂ b i ( n ̂ b i 1 ) + h ¯ U x n ̂ b i ( n ̂ 1 i + n ̂ 2 i ) + h ¯ U f n ̂ 1 i n ̂ 2 i ,
H ̂ I i = h ¯ χ ( t ) b i c 1 , i c 2 , i + H . c . ; H ̂ T = < i , j > [ J b b i b j + J f ( c 1 , i c 1 , j + c 2 , i c 2 , j ) + H . c . ]
U p q = 2 h ¯ a p q π μ p q ξ 1 ξ ( p ) 2 + ξ ( q ) 2
U b 4 J b 2 n ̂ b i ( t ) + 1 + ( 2 n ̂ b i ( t ) + 1 ) 2 1 .
σ = c 1 c 2 , σ + = σ = c 2 c 1
σ z = c 1 c 1 + c 2 c 2 1
h ̂ = h ¯ ( ω b + U x ) n ̂ b + h ¯ ( ω f + U x n ̂ b ) σ z
+ h ¯ ( χ ( t ) b σ + χ * ( t ) b σ + ) + h ¯ 2 U b n ̂ b ( n b 1 )
𝓡 n b = [ 2 ω f ω b + ( 2 U x U b ) n b ] 2 + 4 χ 2 ( n b + 1 ) .
ρ ( t ) t pump = Γ 2 σ = 1 , 2 ( n ¯ [ c σ c σ ρ 2 c σ ρ c σ + ρ c σ c σ ] + ( 1 n ¯ ) [ c σ c σ ρ 2 c σ ρ c σ + ρ c σ c σ ] )
ρ 0 , 0 ( σ ) ( t ) = exp ( Γ t ) ρ 0 , 0 ( σ ) ( 0 ) , ρ 1 , 1 ( σ ) ( t ) = 1 ρ 0 , 0 ( σ ) ( 0 ) exp ( Γ t ) ,
ρ ( t ) t decay = 1 2 γ 1 [ b b ρ 2 b ρ b + ρ b b ]
1 2 γ 3 [ B B ρ 2 B ρ B + ρ B B ] ,
ρ ( b ) ( t l + τ ) = Tr atoms [ e i h ̂ τ ρ ( t l ) e i h ̂ τ ] F ( τ ) [ ρ ( t l ) ] ,
ρ ( b ) ( t l + τ ) = n p n ( t l + τ ) | n n | ,
p n ( t l + τ ) = [ 1 C n + 1 ( τ ) ] p n ( t l ) + C n ( τ ) p n 1 ( t l ) .
ρ ( b ) ( t l + 1 ) = exp ( 𝓛 T ) F ( τ ) ρ ( b ) ( t l ) .
𝓛 [ ρ ¯ ( b ) ] = T 1 ( 1 F ( τ ) [ ρ ¯ ( b ) ] ) ,
p ¯ n = p ¯ 0 l = 1 n C l ( τ ) γ T l
d d t n ¯ = γ N e x ( n ¯ + 1 ) sin 2 ( χ τ [ η + β n ¯ ] 2 + ( n ¯ + 1 ) ) [ η + β n ¯ ] 2 + ( n ¯ + 1 ) γ n ¯ .

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