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
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 , quantum information and computation , 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 , 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 . 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 , 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.
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 s can be expressed in Wannier representation, s(x) = Σk b̂ (s) k w(x – xk), where s = 1,2 stands for the internal-state index, b̂ (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 . Considering only the nearest-neighbor hopping, we obtain the effective Hamiltonian
where n̂ (s) k = b̂ (s)† k b̂ (s) k is the atom-number operator, and ĵ (s) k = b̂ (s)† k b̂ (s) k+1 is the hopping operator. The atom-atom interaction term is Ĥaa = Σ2 s1,s2=1 (Uij/2) Σk b̂ (s1)† k b̂ (s2)† k b̂ (s2) k b̂ (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 = ωp − ωc 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(x − xk)[p 2/2m + V(x)]w(x − x k+j), uj = ∫ dxw(x − xk)g 2(x)/Δw(x − x k+j), ηk = ∫ dxw(x − xk)g(x)h(x)/Δw(x − xk), 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 = nλ, 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
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 , 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 n̂ (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
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 . 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)† b̂ (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.
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 x̂), which replaces b̂i by b̂ i+n.
The SF in state |s〉 is given by |SFs〉 = (MN N!)−1/2(ΣM k=1 b̂ (s)† k)N|0〉, and the deep MI state |MIs〉 = (n!)−M/2∏M k=1 b̂ (s)†n k|0〉. For demonstration purpose, we use a partially-coherent trial wave-function ∏(ΣL j=1 b̂ (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!)−1/2(Σk b̂ (1)† k + Σb̂ (2)† k)N|0〉, for MI state U|MI〉 = (2n n!)−M/2 ∏k(b̂ (1)† k + b̂ (2)† k+d)n|0〉, and for the partially coherent (PC) state we get ∏i[ΣL j=1(b̂ (1)† i+j + b̂ (2)† i+d+j)]N|0〉 = B̂ † 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,
and, for the MI state,
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
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 , 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 b̂ (1)† k b̂ (2) k〉/(Δ + iκ). We have aMI = 0 for the MI state, and aSF = MN/(Δ + iκ) for the SF state. For the PC state, we have aPC = 0 when d > L, and aPC = (L − d)N/(Δ + iκ) when d ≤ L. 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. . 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 . 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 . When the atoms are prepared in MI-MI, SF-MI, or MI-SF states, scattering light intensities are the same
when the atoms are prepared in the SF-SF state, the scattering light intensity is different
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 , the scattering intensity is Î = γη 2 0 Σq1,q2 b̂ (2)† q2 b̂ (2) q1 b̂ (1)† q1 b̂ (1) q2. In the SF states, only the zero-momentum state is occupied, 〈SF|b̂ (s)† q1 b̂ (s) q2|SF〉 = Mnδ q1,q2 δ q1,0; in the MI state, 〈MI|b̂ (s)† q1 b̂ (s) q2|MI〉 = nδ q1,q2. In all four case, we always have I = γη 2 0Σq1 n̂ (2) q1 n̂ (1) q1, where n̂ (s) q = b̂ (s)† q b̂ (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.
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.
We thank T. Vogt for critical reading our manuscript. This work is supported by NKBRSFC (2005CB724503, 2006CB921402 and 2006CB921401) and NSFC (10674007, 10874008, 10934010).
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