Abstract

We propose a new method of detecting quantum coherence of a Bose gas trapped in a one-dimensional optical lattice by measuring the light intensity from Raman scattering in cavity. After pump and displacement process, the intensity or amplitude of scattering light is different for different quantum states of a Bose gas, such as superfluid and Mott-Insulator states. This method can also be useful to detect quantum states of atoms with two components in an optical lattice.

© 2010 Optical Society of America

1. Introduction

Ultracold atomic gases in optical lattices created by pairs of off-resonance counter-propagating laser beams, offer new opportunities to investigate quantum information processing and strongly-correlated quantum matter [1, 2]. A fundamental and interesting question is how to precisely measure correlated characteristics of quantum gases in optical lattices. The methods to detect the quantum phase so far include spectroscopic method [3, 4], interference pattern by the absorption imaging [5, 6], and Hanbury-Brown-Twiss method [7–9]. Since coherence is so essential and crucial in quantum manipulation [3], quantum information and computation [10], the precise measurement of coherence length in optical lattices is very important.

Light scattering from a Bose-Einstein condensate (BEC), such as superradiant light scattering [11–14], scattering spectroscopy in optical lattice [15, 16], and coherent imaging [17], is a useful tool for probing quantum characteristics of BEC. Furthermore, by combining ultracold gases with cavity quantum electrodynamics (cQED) [18, 19], the light scattering spectroscopy described by the cQED [20–22] has been used to probe different quantum states of atoms in optical lattices [23]. Compared with the normally-used destructive absorption imaging [5,6] where superfluidity cannot always be identified from sharp interference patterns [24, 25], the scattering method provides a nondestructive way to measure the superfluid (SF) to Mott-Insulator (MI) quantum transition.

In this work, we propose a new method to detect quantum coherence of atoms trapped in an optical lattice in a cavity by measuring Raman scattering light intensity [26], where two optical lattices act distinctly on each state and two atomic internal states interfere after the pumping and displacement process. The atomic configuration with two internal states offers a huge advantage in precise measurement of the coherent length. The coherence length may provide a quantitative tool to distinguish between SF and MI phases, which can be useful to locate the quantum phase transition point.

 

Fig. 1. (a) Sketch of the system. Atoms with two spin components are held in a 1D optical lattice along the x axis. A pump laser incidents along the y axis, and the cavity mode is along the x axis. (b) Atomic levels picture. Atoms with two internal states |1〉, |2〉 and one excited state |e〉. The laser with frequency ωp pumps atoms transversely, and couples states |e〉 and |1〉. The cavity mode with frequency ωc is resonant to the transition between |e〉 and |2〉.

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2. Model

We consider N atoms trapped in a 1D optical lattice, with two lower internal states |1〉, |2〉, and one excited state |e〉 with energies h̅ω 1, h̅ω 2, h̅ωe, respectively, as shown in Fig. 1. The two components refer to two hyperfine states in the spin-dependent lattice. The atoms are initially prepared in state |1〉, or in both |1〉 and |2〉 states. The number of lattice sites is M, and the number of atoms per site including both components is n = N/M. A pumping beam far detuned from |1〉 → |e〉 transition is placed perpendicular to the cavity modes. The cavity modes, coupling states |2〉 and |e〉, are propagating along the axis direction. The scattering light leaked from the cavity modes can be detected.

When pump beam detuning is large enough, the excited population can be eliminated adiabatically. The atom-field operator ψ^s can be expressed in Wannier representation, ψ^s(x) = Σk (s) k w(xxk), where s = 1,2 stands for the internal-state index, (s) k is the atomic annihilation operator at site k, w(x) is the Wannier function of the lowest band, and xk is the position of site k. Here, we assume that the recoil energy is much smaller comparing with energy gap of optical lattice, so that most atoms are populated at the lowest energy band and the lattice excitation is neglected [20]. Considering only the nearest-neighbor hopping, we obtain the effective Hamiltonian

Ĥ=ΣkΣs=12[ε0n̂k(s)+ε1ĵk(s)+ε1*ĵk(s)]+ĤaaΔââ
+Σk[(u0n̂k(1)+u1ĵk(1)+u1*ĵk(1))ââ
+ηk(b̂k(1)âb̂k(2)+b̂k(1)âb̂k(2))],

where (s) k = (s)† k (s) k is the atom-number operator, and ĵ (s) k = (s)† k (s) k+1 is the hopping operator. The atom-atom interaction term is Ĥaa = Σ2 s1,s2=1 (Uij/2) Σk (s1)† k (s2)† k (s2) k (s1) k, where U 11 and U 22 are the on-site interaction energy between the same state, and U 12 = U 21 is the on-site interaction energy between different states. The annihilation operator of the scattering light with frequency ωc is â, and Δ = ωpωc is the detuning from the pumping field frequency ωp. The two terms in the last sum describe the atom-cavity-photon and atom-pumping-laser interaction. The constants εj, uj (j = 0, 1), and ηk are given by εj = ∫ dxw(xxk)[p 2/2m + V(x)]w(xx k+j), uj = ∫ dxw(xxk)g 2(x)/Δw(xx k+j), ηk = ∫ dxw(xxk)g(x)h(x)/Δw(xxk), where h(x) and g(x) are wave-functions of the pump beam and cavity mode respectively in the form of cos-functions of x. When the pump beam propagates along y-axis and a 0 = , where λ is the scattered light wavelength and a 0 is the lattice constant, the cavity mode changes by a phase factor 2π between neighboring site, and ηk is the same for every site, ηk = η 0; when a 0 = (n + 1/2)λ, ηk = (−1)k η 0. In the following, we just consider the situation of ηk = η 0 for simplification.

From the Heisenberg equation in the rotating-frame approximation, the scattered photon field in the steady state is given by

â=iΣkηkb̂k(1)b̂k(2)i[ΔΣk(u0n̂k(1)+u1ĵk(1)+u1*ĵk(1))]κ.

This steady state should be valid when the cavity loss rate κ is larger than the tunneling rate, and small enough to guarantee a single optical mode output [20], and in the steady state, the scattered light could simply represent the probed atomic state. Since the far-detuned light to the transition of atoms, Δ ≫ Σk(u 0 (1) k + u 1 ĵ (1) k + u * 1 ĵ (1)† k), the term with Σk in the denominator can be neglected, and the denominator is approximately a constant. The scattering intensity Î = â â is approximately given by

Îγ[Σkkηk*ηkb̂k(1)b̂k(2)b̂k(2)b̂k(1)+Σkηk2b̂k(2)b̂k(2)],

where γ = (Δ2 + κ2)−1 is related to the cavity.

3. Measurement of coherence length and quantum phase

When we place the two components BEC into an optical lattice, one advantage is that one component can be displaced without affecting the other [27]. Here the one-dimensional optical standing wave laser field is formed by two counterpropagating laser beams with linear polarizations, so the trapping laser could be decomposed to the same amplitude of σ + and σ polarized lights which form two lattices. The polarization angle of the returning laser beam can be adjusted by a quarter wave plate and an electro-optical modulator (EOM), hence the phase of σ ± light can be independently changed, and the corresponding polarized optical lattice is displaced while the other one stays still. As for the atomic levels, we choose the transition between |1〉 and |e〉 to be σ transition, i.e. Δm 1e = −1, and the transition between |2〉 and |e〉 is σ + transition, i.e. Δm 2e = 1. Thus, when we vary the phase of σ + trapping light, only the |2〉 state atoms will be displaced. This property offers a new way to measure the coherent length of Bose gas in lattice.

In the following, we propose a scheme to measure the coherent length of a Bose gas in the optical lattice prepared in one component with the help of the other component. In order to test the validation of this proposal, we use the extreme SF and MI state in homogeneous case as an example. We also use the partially-coherent state which is an intermediate state between SF and MI as a simple trial state to show the validation of our scheme. At present, states in the real inhomogeneous case are beyond our computation ability.

In our scheme, first, all atoms are initially prepared in state |1〉, then a π/2 pulse is applied to transfer half of the atoms to the other internal state |2〉, as shown in Fig. 2(a). The operator describing the π/2 pulse is Up = exp(−ib̂ (2)† (1)), meaning that the quantum state is coherently transferred from state |1〉 to state |2〉 in a Raman process. This process by applying the two beams in the same direction will not change the external state of the atom, for the atoms absorb and release photon in the same direction.

 

Fig. 2. Steps to measure the coherence length. (a) First, all the atoms are prepared in |1〉 state, and then a π/2 pulse is used to transfer half atoms to |2〉. (b) Secondly, we vary the phase of one optical lattice and perform the displacement action to the atoms in state |2〉. (c) Thirdly, the probe laser is used to detect the intensity of scattering light leaking from the cavity mode.

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Secondly, we adiabatically apply a d-lattice displacement to one component, as shown in Fig. 2(b). The displacement operator is given by Ud = exp(−i p · d · a 0 ), which replaces i by i+n.

The SF in state |s〉 is given by |SFs〉 = (MN N!)−1/2M k=1 (s)† k)N|0〉, and the deep MI state |MIs〉 = (n!)−M/2M k=1 (s)†n k|0〉. For demonstration purpose, we use a partially-coherent trial wave-function ∏(ΣL j=1 (s)† i+j)N|0〉 with coherence length La 0 between the MI and SF limits. Thus after the above two steps, U = UpUd, we have for SF state, U|SF〉 = (2N MNN!)−1/2k (1)† k + Σ (2)† k)N|0〉, for MI state U|MI〉 = (2n n!)−M/2k( (1)† k + (2)† k+d)n|0〉, and for the partially coherent (PC) state we get ∏iL j=1( (1)† i+j + (2)† i+d+j)]N|0〉 = PC|0〉.

Thirdly, we incident the probing pulse and detect the leaked light intensity, as shown in Fig. 2(c). We have, for the SF state,

ISF=γη02N(N1)γη02M2n2,

and, for the MI state,

IMI=γη02Mn2.

Hence, the ratio of the scattering light intensity in the SF to the intensity in the MI state is approximately M, which can be observed for big enough M. The scattering light intensity in the PC state is

 

Fig. 3. The scattering light intensity of cavity mode versus the scaled displacement (d/a 0) for the ηk = η 0 case, with M = 100 and n = 100. These atoms are initially prepared in the SF state (dashed line), MI state (dotted line), or the PC state with coherence length L = 60 sites (solid line).

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IPC{γη02(Ld)Mn2,d<L,γη02Mn2,dL.

When d is shorter than the coherence length L, IPC decreases monotonously with the displacement d until it reaches its low value at d = L. We plot the light scattering intensities in different states, as shown in Fig. 3. Different from the spectroscopic measurement [20], the scattering light intensity by Raman transition is measured and calibrated with respect to that from the SF state.

The phase of the scattering light field can also be detected by homodyne detection method, where we can directly measure the optical field, 〈a〉 = 〈Σk ηk (1)† k (2) k〉/(Δ + ). We have aMI = 0 for the MI state, and aSF = MN/(Δ + ) for the SF state. For the PC state, we have aPC = 0 when d > L, and aPC = (Ld)N/(Δ + ) when dL. Thus from this optical field we can detect the coherent length precisely.

Compared with the light-intensity measurement, the homodyne detection has several advantages. First, since the amplitude is zero for the MI state and proportional to MN for the SF state, it does not require calibration. Secondly, the optical field coefficient changes from 0 to (L−d)N in the transition from MI to SF states, which can be more precisely detected than the light intensity which coefficient varies between (L−d)Mn 2 and Mn 2. Lastly, it can directly provide information about the coherence length.

In our scheme, the coherent transport and splitting of atomic wave packets in spin-dependent optical lattice potentials can be controlled by adjusting the polarization angle of the two counter-propagating laser beams, as demonstrated in Ref. [27]. The coherence length can be measured with a precision up to the lattice spacing. This precision is much higher than those from existing methods usually on the scale of the condensate length. Improvement in the coherence-length measurement can help us with better understanding of quantum states.

4. Detection of quantum phase in two components

Our proposed method can also be applied to detect the coherence of a BEC prepared in both components. More complex quantum phases can be achieved in it than in a one-component gas, with the possibility of magnetic ordering in the MI state [28]. The two components can be prepared in SF or MI states respectively. Thus, in a two-component BEC, there exist at least SF-SF, MI-SF, SF-MI, and MI-MI phases [29]. When the atoms are prepared in MI-MI, SF-MI, or MI-SF states, scattering light intensities are the same

 

Fig. 4. Occupation number (s) q (a) or product (1) q (2) q (b) in momentum space versus the scaled momentum for atoms with one (a) or two (b) internal states, M=10, assuming the Wannier function is approximated as Gaussian. (a)With one internal state, occupation number for the SF state (solid line), and MI state (dotted line). (b) With two internal states, the product of two occupation numbers in the SF-SF state (dotted line), SF-MI or MI-SF states (dashed line), and MI-MI state (solid line), and the inset is enlarged picture for SF-MI or MI-SF states, and MI-MI state.

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IMIMI=IMISF=ISFMI=γη02Mn2;

when the atoms are prepared in the SF-SF state, the scattering light intensity is different

ISFSF=γη02M2n2.

Since the site number M is normally a big number, the scattering intensity of the SF-SF state is much larger than those of other states. Hence we can distinguish between the SF-SF and other states.

The result of Eqs. (7) and (8) can be understood in terms of the atom occupation number in momentum space which is shown in Fig. 4, in which the Wannier function is approximated as Gaussian. Using the annihilation operator of atoms in momentum space bq(s)=Σkeiqka0bk(s)M , the scattering intensity is Î = γη 2 0 Σq1,q2 (2)† q2 (2) q1 (1)† q1 (1) q2. In the SF states, only the zero-momentum state is occupied, 〈SF| (s)† q1 (s) q2|SF〉 = Mnδ q1,q2 δ q1,0; in the MI state, 〈MI| (s)† q1 (s) q2|MI〉 = q1,q2. In all four case, we always have I = γη 2 0Σq1 (2) q1 (1) q1, where (s) q = (s)† q (s) q is the occupation number of atoms in internal state i with wave-vector q. In the one-component case, due to atom number conservation, the occupation number in MI state is M-times wider than in SF state, and the maximum value in SF state is M-times higher than that in MI state, as shown in Fig. 4(a). In the two-component case, the product of two occupation numbers is shown in Fig. 4(b). In the SF-SF state, the height and width of the product are M 2 and 1 respectively, therefore the scattering light intensity is proportional to M 2. In SF-MI or MI-SF states, the height and width of product are M and 1, and the intensity is proportional to M. In the MI-MI states, the height and width of product are 1 and M, and the intensity is proportional to M.

5. Conclusion

In conclusion, the coherence of quantum states of Bose gases in optical lattices, including the SF and MI states, can be probed by the scattering intensity or amplitude of the scattering light. When the cavity modes interact with every sites of lattices in the same phase, the scattering light intensity ratio between SF and MI states can be as big as the lattice number, and the amplitude of scattering light is zero in the MI state and non-zero in the other states. In this method, the coherence length can be measured with high precision, which is helpful to locate the transition between SF and MI states. Furthermore, it can also be used in the coherence measurement for a two-component Bose gas. Our method offers new opportunities to probe atomic many-body states in optical lattices, which can be implemented with present techniques. Although computing the scattering result for the intermediate case with real experimental conditions is our of our reach now, if the state is known, the result can be obtained and used in extracting the coherent length. Such calculation for the real experimental case, together with other complicated issues such as the interaction in the shift stage in our scheme, will be explored in our future work.

Acknowledgments

We thank T. Vogt for critical reading our manuscript. This work is supported by NKBRSFC (2005CB724503, 2006CB921402 and 2006CB921401) and NSFC (10674007, 10874008, 10934010).

References and links

1. 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 atom,” Nature 415, 39–44 (2002). [CrossRef]   [PubMed]  

2. I. Bloch, “Quantum coherence and entanglement with ultracold atoms in optical lattices,” Nature 453, 1016–1022 (2008). [CrossRef]   [PubMed]  

3. E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999). [CrossRef]  

4. T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004). [CrossRef]   [PubMed]  

5. I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000). [CrossRef]   [PubMed]  

6. E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004). [CrossRef]  

7. M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005). [CrossRef]   [PubMed]  

8. S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005). [CrossRef]   [PubMed]  

9. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007). [CrossRef]   [PubMed]  

10. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

11. S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999). [CrossRef]   [PubMed]  

12. F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008). [CrossRef]  

13. Y. Wu and X. Yang, “Fully quantized theory of four-wave mixing with bosonic matter waves,” Opt. Lett. 30, 311–313 (2005). [CrossRef]   [PubMed]  

14. J. Cheng and Y.-J. Yan, “Quantum dynamics of a molecular matter-wave amplifier,” Phys. Rev. A 75, 033614 (2007). [CrossRef]  

15. D. Clément, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009). [CrossRef]   [PubMed]  

16. X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009). [CrossRef]  

17. L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007). [CrossRef]   [PubMed]  

18. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007). [CrossRef]   [PubMed]  

19. Y. Wu and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001). [CrossRef]  

20. I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007). [CrossRef]  

21. I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007). [CrossRef]   [PubMed]  

22. H. Zoubi and H. Ritsch, “Quantum phases of bosonic atoms with two levels coupled by a cavity field in an optical lattice,” Phys. Rev. A 80, 053608 (2009). [CrossRef]  

23. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007). [CrossRef]   [PubMed]  

24. R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007). [CrossRef]   [PubMed]  

25. Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008). [CrossRef]  

26. Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999). [CrossRef]  

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

28. L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003). [CrossRef]   [PubMed]  

29. G. H. Chen and Y. S. Wu, “Quantum phase transition in a multicomponent Bose-Einstein condensate in optical lattices,” Phys. Rev. A 67, 013606 (2003). [CrossRef]  

References

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  1. 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 atom,” Nature 415, 39–44 (2002).
    [CrossRef] [PubMed]
  2. I. Bloch, “Quantum coherence and entanglement with ultracold atoms in optical lattices,” Nature 453, 1016–1022 (2008).
    [CrossRef] [PubMed]
  3. E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
    [CrossRef]
  4. T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
    [CrossRef] [PubMed]
  5. I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
    [CrossRef] [PubMed]
  6. E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004).
    [CrossRef]
  7. M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
    [CrossRef] [PubMed]
  8. S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
    [CrossRef] [PubMed]
  9. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
    [CrossRef] [PubMed]
  10. M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  11. S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
    [CrossRef] [PubMed]
  12. F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
    [CrossRef]
  13. Y. Wu, and X. Yang, “Fully quantized theory of four-wave mixing with bosonic matter waves,” Opt. Lett. 30, 311–313 (2005).
    [CrossRef] [PubMed]
  14. J. Cheng, and Y.-J. Yan, “Quantum dynamics of a molecular matter-wave amplifier,” Phys. Rev. A 75, 033614 (2007).
    [CrossRef]
  15. D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
    [CrossRef] [PubMed]
  16. X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009).
    [CrossRef]
  17. L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
    [CrossRef] [PubMed]
  18. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
    [CrossRef] [PubMed]
  19. Y. Wu, and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001).
    [CrossRef]
  20. I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007).
    [CrossRef]
  21. I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007).
    [CrossRef] [PubMed]
  22. H. Zoubi, and H. Ritsch, “Quantum phases of bosonic atoms with two levels coupled by a cavity field in an optical lattice,” Phys. Rev. A 80, 053608 (2009).
    [CrossRef]
  23. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
    [CrossRef] [PubMed]
  24. R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
    [CrossRef] [PubMed]
  25. Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
    [CrossRef]
  26. Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999).
    [CrossRef]
  27. O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, “Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials,” Phys. Rev. Lett. 91, 010407 (2003).
    [CrossRef] [PubMed]
  28. L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003).
    [CrossRef] [PubMed]
  29. G. H. Chen, and Y. S. Wu, “Quantum phase transition in a multicomponent Bose-Einstein condensate in optical lattices,” Phys. Rev. A 67, 013606 (2003).
    [CrossRef]

2009

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009).
[CrossRef]

H. Zoubi, and H. Ritsch, “Quantum phases of bosonic atoms with two levels coupled by a cavity field in an optical lattice,” Phys. Rev. A 80, 053608 (2009).
[CrossRef]

2008

Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
[CrossRef]

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

I. Bloch, “Quantum coherence and entanglement with ultracold atoms in optical lattices,” Nature 453, 1016–1022 (2008).
[CrossRef] [PubMed]

2007

J. Cheng, and Y.-J. Yan, “Quantum dynamics of a molecular matter-wave amplifier,” Phys. Rev. A 75, 033614 (2007).
[CrossRef]

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
[CrossRef] [PubMed]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007).
[CrossRef]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007).
[CrossRef] [PubMed]

2005

Y. Wu, and X. Yang, “Fully quantized theory of four-wave mixing with bosonic matter waves,” Opt. Lett. 30, 311–313 (2005).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

2004

E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004).
[CrossRef]

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

2003

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

L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

G. H. Chen, and Y. S. Wu, “Quantum phase transition in a multicomponent Bose-Einstein condensate in optical lattices,” Phys. Rev. A 67, 013606 (2003).
[CrossRef]

2002

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

2001

Y. Wu, and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001).
[CrossRef]

2000

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef] [PubMed]

1999

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999).
[CrossRef]

Altman, E.

E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004).
[CrossRef]

Aspect, A.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Band, Y. B.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Bloch, I.

I. Bloch, “Quantum coherence and entanglement with ultracold atoms in optical lattices,” Nature 453, 1016–1022 (2008).
[CrossRef] [PubMed]

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

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

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef] [PubMed]

Boiron, D.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Bourdel, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Brennecke, F.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Chang, H.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Chen, G. H.

G. H. Chen, and Y. S. Wu, “Quantum phase transition in a multicomponent Bose-Einstein condensate in optical lattices,” Phys. Rev. A 67, 013606 (2003).
[CrossRef]

Chen, X.

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

Chen, X. Z.

X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009).
[CrossRef]

Chen, Y.

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

Cheng, J.

J. Cheng, and Y.-J. Yan, “Quantum dynamics of a molecular matter-wave amplifier,” Phys. Rev. A 75, 033614 (2007).
[CrossRef]

Chikkatur, A. P.

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Cl’ement, D.

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

Colombe, Y.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Demler, E.

E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004).
[CrossRef]

L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

Deng, L.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Diener, R. B.

R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
[CrossRef] [PubMed]

Doery, M.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Donner, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Duan, L.-M.

L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

Dubois, G.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Edwards, M.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Esslinger, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef] [PubMed]

Fabbri, N.

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

Fallani, L.

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

Fölling, S.

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

Fort, C.

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

Gerbier, F.

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

Gericke, T.

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

Greiner, M.

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

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Hagley, E. W.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Hänsch, T. W.

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

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef] [PubMed]

Helmerson, K.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Higbie, J. M.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

Ho, T.-L.

R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
[CrossRef] [PubMed]

Hogervorst, W.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Hoppeler, R.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Hunger, D.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Inguscio, M.

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

Inouye, S.

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Jeltes, T.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Julienne, P. S.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Kato, Y.

Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
[CrossRef]

Kawashima, N.

Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
[CrossRef]

Ketterle, W.

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Köhl, M.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

Kozuma, M.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Krachmalnicoff, V.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Leslie, S. R.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

Leung, P. T.

Li, J.

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

Linke, F.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Lukin, M.

E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004).
[CrossRef]

Lukin, M. D.

L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

Mandel, O.

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

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

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

Maschler, C.

I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007).
[CrossRef] [PubMed]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007).
[CrossRef]

McNamara, J. M.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Mekhov, I. B.

I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007).
[CrossRef]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007).
[CrossRef] [PubMed]

Moritz, H.

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

Perrin, A.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Phillips, W. D.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Pritchard, D. E.

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Reichel, J.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Ritsch, H.

H. Zoubi, and H. Ritsch, “Quantum phases of bosonic atoms with two levels coupled by a cavity field in an optical lattice,” Phys. Rev. A 80, 053608 (2009).
[CrossRef]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007).
[CrossRef] [PubMed]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007).
[CrossRef]

Ritter, S.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

Rolston, S. L.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Rom, T.

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

Sadler, L. E.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

Schellekens, M.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Schori, C.

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

Stamper-Kurn, D. M.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Steinmetz, T.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

Stenger, J.

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

Stöferle, T.

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

Trippenbach, M.

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

Trived, N.

Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
[CrossRef]

Vassen, W.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Vengalattore, M.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

Viana Gomes, J.

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Westbrook, C. I.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Widera, A.

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

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

Wu, Y.

Wu, Y. S.

G. H. Chen, and Y. S. Wu, “Quantum phase transition in a multicomponent Bose-Einstein condensate in optical lattices,” Phys. Rev. A 67, 013606 (2003).
[CrossRef]

Xia, L.

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

Xu, X.

X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009).
[CrossRef]

Yan, Y.-J.

J. Cheng, and Y.-J. Yan, “Quantum dynamics of a molecular matter-wave amplifier,” Phys. Rev. A 75, 033614 (2007).
[CrossRef]

Yang, F.

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

Yang, X.

Zhai, H.

R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
[CrossRef] [PubMed]

Zhou, Q.

Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
[CrossRef]

R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
[CrossRef] [PubMed]

Zhou, X.

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

Zhou, X. J.

X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009).
[CrossRef]

Zoubi, H.

H. Zoubi, and H. Ritsch, “Quantum phases of bosonic atoms with two levels coupled by a cavity field in an optical lattice,” Phys. Rev. A 80, 053608 (2009).
[CrossRef]

Nat. Phys.

I. B. Mekhov, C. Maschler, and H. Ritsch, “Probing quantum phases of ultracold atoms in optical lattices by transmission spectra in cavity quantum electrodynamics,” Nat. Phys. 3, 319–323 (2007).
[CrossRef]

Y. Kato, Q. Zhou, N. Kawashima, and N. Trived, “Sharp peaks in the momentum distribution of bosons in optical lattices in the normal state,” Nat. Phys. 4, 617–621 (2008).
[CrossRef]

Nature

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef] [PubMed]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature 450, 268–271 (2007).
[CrossRef] [PubMed]

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 atom,” Nature 415, 39–44 (2002).
[CrossRef] [PubMed]

I. Bloch, “Quantum coherence and entanglement with ultracold atoms in optical lattices,” Nature 453, 1016–1022 (2008).
[CrossRef] [PubMed]

I. Bloch, T. W. Hänsch, and T. Esslinger, “Measurement of the spatial coherence of a trapped Bose gas at the phase transition,” Nature 403, 166–170 (2000).
[CrossRef] [PubMed]

S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, “Spatial quantum noise interferometry in expanding ultracold atom clouds,” Nature 434, 481–484 (2005).
[CrossRef] [PubMed]

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

H. Zoubi, and H. Ritsch, “Quantum phases of bosonic atoms with two levels coupled by a cavity field in an optical lattice,” Phys. Rev. A 80, 053608 (2009).
[CrossRef]

G. H. Chen, and Y. S. Wu, “Quantum phase transition in a multicomponent Bose-Einstein condensate in optical lattices,” Phys. Rev. A 67, 013606 (2003).
[CrossRef]

E. Altman, E. Demler, and M. Lukin, “Probing many-body states of ultracold atoms via noise correlations,” Phys. Rev. A 70, 013603 (2004).
[CrossRef]

Y. Wu, and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816 (2001).
[CrossRef]

F. Yang, X. Zhou, J. Li, Y. Chen, L. Xia, and X. Chen, “Resonant sequential scattering in two-frequency-pumping superradiance from a Bose-Einstein condensate,” Phys. Rev. A 78, 043611 (2008).
[CrossRef]

J. Cheng, and Y.-J. Yan, “Quantum dynamics of a molecular matter-wave amplifier,” Phys. Rev. A 75, 033614 (2007).
[CrossRef]

X. Xu, X. J. Zhou, and X. Z. Chen, “Spectroscopy of superradiant scattering from an array of Bose-Einstein condensates,” Phys. Rev. A 79, 033605 (2009).
[CrossRef]

Phys. Rev. Lett.

L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, “Coherence-Enhanced Imaging of a Degenerate Bose-Einstein Gas,” Phys. Rev. Lett. 98, 110401 (2007).
[CrossRef] [PubMed]

D. Cl’ement, N. Fabbri, L. Fallani, C. Fort, and M. Inguscio, “Exploring Correlated 1D Bose Gases from the Superfluid to the Mott-Insulator State by Inelastic Light Scattering,” Phys. Rev. Lett. 102, 155301 (2009).
[CrossRef] [PubMed]

I. B. Mekhov, C. Maschler, and H. Ritsch, “Cavity-Enhanced Light Scattering in Optical Lattices to Probe Atomic Quantum Statistics,” Phys. Rev. Lett. 98, 100402 (2007).
[CrossRef] [PubMed]

E. W. Hagley, L. Deng, M. Kozuma, M. Trippenbach, Y. B. Band, M. Edwards, M. Doery, P. S. Julienne, K. Helmerson, S. L. Rolston, and W. D. Phillips, “Measurement of the Coherence of a Bose-Einstein Condensate,” Phys. Rev. Lett. 83, 3112 (1999).
[CrossRef]

T. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, “Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator,” Phys. Rev. Lett. 92, 130403 (2004).
[CrossRef] [PubMed]

R. B. Diener, Q. Zhou, H. Zhai, and T.-L. Ho, “Criterion for Bosonic Superfluidity in an Optical Lattice,” Phys. Rev. Lett. 98, 180404 (2007).
[CrossRef] [PubMed]

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

L.-M. Duan, E. Demler, and M. D. Lukin, “Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices,” Phys. Rev. Lett. 91, 090402 (2003).
[CrossRef] [PubMed]

Science

S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, J. Stenger, D. E. Pritchard, and W. Ketterle, “Superradiant Rayleigh Scattering from a Bose-Einstein Condensate,” Science 285, 571–574 (1999).
[CrossRef] [PubMed]

M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss Effect for Ultracold Quantum Gases,” Science 310, 648–651 (2005).
[CrossRef] [PubMed]

Other

M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

Cited By

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Figures (4)

Fig. 1.
Fig. 1.

(a) Sketch of the system. Atoms with two spin components are held in a 1D optical lattice along the x axis. A pump laser incidents along the y axis, and the cavity mode is along the x axis. (b) Atomic levels picture. Atoms with two internal states |1〉, |2〉 and one excited state |e〉. The laser with frequency ωp pumps atoms transversely, and couples states |e〉 and |1〉. The cavity mode with frequency ωc is resonant to the transition between |e〉 and |2〉.

Fig. 2.
Fig. 2.

Steps to measure the coherence length. (a) First, all the atoms are prepared in |1〉 state, and then a π/2 pulse is used to transfer half atoms to |2〉. (b) Secondly, we vary the phase of one optical lattice and perform the displacement action to the atoms in state |2〉. (c) Thirdly, the probe laser is used to detect the intensity of scattering light leaking from the cavity mode.

Fig. 3.
Fig. 3.

The scattering light intensity of cavity mode versus the scaled displacement (d/a 0) for the ηk = η 0 case, with M = 100 and n = 100. These atoms are initially prepared in the SF state (dashed line), MI state (dotted line), or the PC state with coherence length L = 60 sites (solid line).

Fig. 4.
Fig. 4.

Occupation number (s) q (a) or product (1) q (2) q (b) in momentum space versus the scaled momentum for atoms with one (a) or two (b) internal states, M=10, assuming the Wannier function is approximated as Gaussian. (a)With one internal state, occupation number for the SF state (solid line), and MI state (dotted line). (b) With two internal states, the product of two occupation numbers in the SF-SF state (dotted line), SF-MI or MI-SF states (dashed line), and MI-MI state (solid line), and the inset is enlarged picture for SF-MI or MI-SF states, and MI-MI state.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

H ̂ = Σ k Σ s = 1 2 [ ε 0 n ̂ k ( s ) + ε 1 j ̂ k ( s ) + ε 1 * j ̂ k ( s ) ] + H ̂ aa Δ a ̂ a ̂
+ Σ k [ ( u 0 n ̂ k ( 1 ) + u 1 j ̂ k ( 1 ) + u 1 * j ̂ k ( 1 ) ) a ̂ a ̂
+ η k ( b ̂ k ( 1 ) a ̂ b ̂ k ( 2 ) + b ̂ k ( 1 ) a ̂ b ̂ k ( 2 ) ) ] ,
a ̂ = i Σ k η k b ̂ k ( 1 ) b ̂ k ( 2 ) i [ Δ Σ k ( u 0 n ̂ k ( 1 ) + u 1 j ̂ k ( 1 ) + u 1 * j ̂ k ( 1 ) ) ] κ .
I ̂ γ [ Σ k k η k * η k b ̂ k ( 1 ) b ̂ k ( 2 ) b ̂ k ( 2 ) b ̂ k ( 1 ) + Σ k η k 2 b ̂ k ( 2 ) b ̂ k ( 2 ) ] ,
I SF = γ η 0 2 N ( N 1 ) γ η 0 2 M 2 n 2 ,
I MI = γ η 0 2 Mn 2 .
I PC { γ η 0 2 ( L d ) Mn 2 , d < L , γ η 0 2 Mn 2 , d L .
I MI MI = I MI SF = I SF MI = γ η 0 2 Mn 2 ;
I SF SF = γ η 0 2 M 2 n 2 .

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