Bistability of Feshbach resonance in optical cavity

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1 Cent. Eur. J. Phys. 1( DOI: /s Central European Journal of Physics Bistability of Feshbach resonance in optical cavity Research Article Xiao-Qiang Xu 1, You-Quan Li 1 Department of Physics, Hangzhou Normal University, Hangzhou 31007, People s Republic of China Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 31007, People s Republic of China Received January 014; accepted March 014 Abstract: PACS (008: Mn, Kk Keywords: We consider Feshbach resonance in an optical cavity where photons interact with atoms and molecules dispersively. From mean-field theory we obtain multiple fixed-point solutions, which is strongly related to the phenomenon of bistability. Adiabatic evolutions demonstrate hysteretic behaviors by varying pump-cavity detuning from opposite directions. We also use the quantum model to check mean-field results which match perfectly. The analysis here may enrich the study of particle-photon interaction systems. cold atoms optical cavity Feshbach resonance bistability Versita sp. z o.o. 1. Introduction The interaction between two-level atoms and photons has been popularly modeled by Dicke or Jaynes-Cummings Hamiltonian without considering inter-particle collisions. This research has formed an important part of quantum optics. Despite its long history, new features may still be expected from these systems, such as the recently discovered Josephson oscillation of photons between two weakly linked cavities [1, superradiance in optical lattices [, etc. A large number of atoms may realize Bose-Einstein condensation (BEC when cooled to below the critical temperature. In the strong coupling regime, the collective behaviors of the ultracold atoms may bring new physics into the particle-photon interaction system. For example, xiaoqiang.hsu@gmail.com from the oscillation of the center-of-mass of the condensate, the optomechanics in ultracold bosonic atomic gases has been observed [3 1. Meanwhile, the same optomechanics phenomena may also be observable in degenerate fermionic gases [13. Similar discussion can be extended to the case of multi-component BECs, such as double-well condensates [14 and spinor BECs [15, 16 in cavity, etc. Among multi-component condensates, there exists a special case: the atom-molecule mixture [17 0. We are interested in investigating the influence of photons on such a conversion system. There are two basic ways to produce molecules from atoms: photoassociation [1 and Feshbach resonance [. For the former case, there have been discussions on the molecular production rate [3, 4 and information transfer [5 in the quantized photoassociation. Here we will focus on the latter, the Feshbach resonance [, 6, which magnetically converts atoms into molecules. We assume photons interact with atoms and 47

2 Bistability of Feshbach resonance in optical cavity molecules dispersively. This nonlinear interaction may bring new features to both Feshbach resonance and cavity physics. This paper is organized as follows. In Sec. we will give the system Hamiltonian and derive the mean-field model. From the fixed-point solutions and the energy spectrum, we make explicit the existence of bistability. In Sec. 3 we will compare the quantum model with the mean-field results and discuss the quantum ground state properties. In the last section we will give a brief summary and discussion.. Mean-field bistability As we already know, Feshbach resonance [ can combine two atoms into a bound molecular state. Thus we may have Bose-Einstein condensates of both atoms and molecules [6. Together with the inter-conversion, we may describe this process in its second quantized form as [ α H FR = (ˆψ m ˆψ a ˆψa + H.c. + ε m ˆψ m ˆψm + λ a ˆψ a ˆψ a ˆψ a ˆψa + λ m ˆψ m ˆψ m ˆψ m ˆψm +λ am ˆψ a ˆψa ˆψ m ˆψm dr, where field operators ˆψ a (r and ˆψ m (r annihilate an atom and a molecule in the r position, respectively, while ˆψ a (r and ˆψ m(r are the corresponding Hermitian conjugates. Note that we consider the cigar-shape condensate, where the axial direction coincides with that of the cavity. The position invariable r is spared in these operators. Parameters λ a and λ m refer to the atomic and molecular interactions, while λ am refers to the atom-molecule interaction. We assume the condensate is dilute enough so that the assumption of s-wave contact interaction is applicable. Here ε m denotes the energy of the molecular state, while we set the energy of the atomic state to the zero point of energy. α measures the Feshbach coupling strength between atoms and molecules. When considering the Feshbach resonance process in an optical cavity, we need to include the effects of cavity modes. In a frame rotating at the pump frequency ω p, the Hamiltonian reads [14 H = H FR + c c [U a ˆψ a ˆψa + U m ˆψ m ˆψm dr ĉ ĉ + η ( ĉ + ĉ, (1 where the operator c annihilates a photon in the cavity. Here we assumed that the atoms and molecules couple to only one cavity mode with frequency ω c. Other modes are ignored for simplicity. = ω p ω c is the pumpcavity detuning. We assume that both pump-atom (ω p ω a and pump-molecule (ω p ω m detunings are large enough that the atomic and molecular upper levels can be adiabatically eliminated, and consequently U a = g a/(ω p ω a and U m = g m/(ω p ω m describe the atom-photon and molecule-photon dispersive interactions, while g a and g m are the atom-photon and molecule-photon couplings. η denotes the strength of the external pumping laser. We may obtain the Heisenberg equations of motion for field operators: i ψ a = αψ m ψ a + U a c cψ a + ( λ a ψ a ψ a + λ am ψ mψ m ψa, i ψ m = αψ a ψ a + ε m ψ m + ( λ m ψ mψ m + λ am ψ a ψ a ψm +U m c cψ m, ( (Ua iċ = c + η + c ψ a ψ a + U m ψ mψ m dr iκc, and phenomenologically include the photon loss term κ. The relaxation time scale of the cavity mode is of the order of 1/κ. For large κ which corresponds to the bad cavity limit, the evolution behavior of photons follows that of atoms and molecules, i.e., i ĉ = 0, which gives ĉ = η/[ + iκ (U a ˆψ a ˆψa + U m ˆψ m ˆψm dr. As a result, the mean photon number can be expressed as ĉ η ĉ = [ ( U a ˆψ ˆψ. (3 a a + U m ˆψ m ˆψm dr + κ Thus, we may replace the photon operator with the atomic and molecular operators, and obtain a set of effective evolution equations with no photons. Before that, we may adopt the mean-field approximation which is valid for large particle number, and replace the operators with their expectation values, i.e., the complex quantities ˆψ a = n ρa exp[iφ a, ˆψ m = n ρ m exp[iφ m. n is the particle density. ρ i and φ i (i = a, m correspond to the density probability and phase for the atomic and molecular condensates, respectively. Due to particle number conservation, we have the constraint ρ a + ρ m = 1. By defining the phase difference φ = φ m φ a, we may reformulate Eqs. ( and obtain the classical evolution equations φ = α cos φ 6ρ m 1 ε m + ( λ a λ am ρm + (4 λ am 4 λ a λ m ρ m (Ũ a Ũ m η +, [ Ũ a (Ũm Ũa ρ m + 1 ρ m = α(1 ρ m ρ m sin φ, (4 48

3 Xiao-Qiang Xu, You-Quan Li where we have defined new dimensionless quantities as α = α n/κ, λa = λ a n/κ, λm = λ m n/κ, λam = λ am n/κ, ε m = ε m /κ, η = η/κ, Ũ a = U a N/κ, Ũ m = U m N/κ, and = /κ. N = n(rdr is the total number of atoms, including those bound in molecules. Note that now the time is rescaled by κ. Due to the canonical conjugation relation between ρ m and φ, ρ m = H c / φ, φ = H c / ρ m, the classical Hamiltonian can be expressed as H c = α ρ m (1 ρ m cos φ + ε m ρ m + ( λ a λ am + 1 (4 λ am 4 λ a λ m ρm [( η N arctan Ũa Ũm ρ m + Ũa for nonzero κ. Note that positive Ũa Ũm is assumed when we derive the above equation. We choose 87 Rb near the resonance point G as our test model where λ a = Jm 3 [7, and also assume λ m = 1.λ a and λ am = 0.6λ a [8. For Feshbach coupling, we set α = Jm 3/ [9. We choose n = m 3 and N = , and the cavity parameters κ = 10 6 Hz, while U a = 10 3 Hz [30. In this case, we have α = , λ a = , λ m = , λ am = , and Ũ a = 40. In order to avoid the decoupling of Feshbach resonance from the cavity system, here we choose deeply bound molecules, and set Ũm = Ũa for simplicity and ε m = 0 (near Feshbach resonance. First, we investigate the fixed-point solutions of the system. We set φ = 0 and ρ m = 0 from Eqs. (4. The second equation gives φ = 0 or π. Substituting the value of φ back into the first equation, we can obtain the corresponding solutions of ρ m. Note that, there also exists another fixed-point solution, i.e., ρ m = 0.5 without well defined value of φ. This one can be obtained directly from the evolution equations of field operators. As we will show later, the φ = π case has lower energy for which we plot the number of fixed-point solutions in Fig. 1(a. For the parameters we choose, the distribution of number of fixedpoint solutions for φ = 0 in the ( η, space has a similar diagram. In order to determine the stability of each solution, we expand ρ m and φ around the fixed-point values by introducing small fluctuations, i.e., ρ m = ρ 0 m + δρ m, φ = φ 0 + δφ, where ρ 0 m and φ 0 satisfy φ 0 = 0 and ρ 0 m = 0 from Eq. (4. By expanding the equations to the first order of δρ m and ρ m Figure 1. (Color online (a Numbers of physical fixed-point solutions (labeled by the digits in each region for φ = π. (b Mean photon number in the cavity for different values of η which are labeled on top of each curve. The dashed curves correspond to unstable fixed-point solutions. δφ, we have δ φ = Mδρ m, δ ρ m = Nδφ, where M = α cos φ 0 6ρm (4 λ ρ 0 m ρm 0 am 4 λ a λ m (Ũa Ũm η [(Ũa Ũm ρm 0 + Ũa { }, 1 + [(Ũ a Ũ m ρm 0 + Ũ a N = α ( 1 ρ 0 m ρ 0 m cos φ 0. By diagonizing the matrix of δρ m and δφ, we may obtain the excitation frequency ω = MN. If both frequencies are real, we can conclude that the corresponding fixed-point is dynamically stable. Otherwise, complex frequencies correspond to unstable solutions since they may enlarge any small perturbation, and cause collapse of the system. When there is no external pumping, i.e., η = 0.0, the photons in the cavity will dissipate quickly, exerting no influence on the process of Feshbach resonance. Under the parameters we choose here, there is only one fixedpoint solution for φ = π, i.e., ρ m 0.7, indicating that about half of the atoms are converted into molecules. As the value of η is increased, from Fig. 1(a, we find that the interaction between photons and atoms (molecules induces multiple solutions which may indicate the phenomenon of bistability. In order to make this explicit, we plot the mean photon number versus in Fig. 1(b. When the value of η reaches a critical point at about 0.7, photon bistability emerges. In experiments, when the value of is tuned from opposite directions, different evolution behaviors of mean photon number may be observed, which is similar to the hysteresis effect in magnetic materials. Due to the one-to-one correspondence between the photon number and the molecule (atom distribution, we can also observe similar bistability phenomena for atoms and molecules. In Fig. (a, we show the evolution behaviors 49

4 Bistability of Feshbach resonance in optical cavity Figure. (Color online Adiabatic evolution of (a molecule density probability in mean-field theory and (b molecule number in quantum regime. Solid (red and dotted (black lines correspond to positive and negative tuning of with arrows indicating the tuning directions. In (a, the gray curve corresponds to the fixed-point solutions of ρ m. of molecule density probability ρ m when is adiabatically tuned from two opposite directions, i.e., (t γt ( γ is much smaller than the other system characteristic frequencies, which confirms our previous analysis. For larger values of η, we may enter the regime with zero fixed-point solutions and two fixed-point solutions, as indicated in Fig. 1.(a. The former case corresponds to the situation with no stable stationary state existing in that parameter regime, while for the case of two fixed-point solutions, there exist one stable ground state and one unstable excited state. An example is shown in Fig. 1(b for η = 5.0. Figure 3 illustrates the mean-field energy levels of the system when η = 1.5. As we noted before, the state with φ = π (solid black curves bears lower energy. Most importantly, the swallowtail loop in the figure (formed by two solid and one dashed black curves indicates the bistability we discussed above. At the exact crossing point of the energy loop, the energy levels are degenerate, causing the failure of the adiabaticity. Suppose we prepare the system in the ground state. When passes the crossing point, due to the degeneracy, the system has a finite possibility of entering the upper branch of the energy loop (the first excited state. In other words, it cannot adiabatically follow the initial ground state anymore. Different directions of (t tunings (increasing or decreasing may lead Figure 3. (Color online Mean-field energy levels when η = 1.5. The solid black (stable and dashed (unstable curves correspond to φ = π, while the gray solid curves belong to φ = 0. The black dotted curve corresponds to the solution of ρ m = 0.5 without well-defined φ. Inset: energy contour when η = 1.5 and = 4.0. to different branches of the swallowtail loop, corresponding to the time evolution patterns in Fig.. In the inset, we plot the energy contour, where darker color indicates lower energy. We can find one saddle point and two energy minimums at φ = π indicating two stable fixed-point solutions, which is strongly related to the bistability. 3. Quantum model In order to test and compare the results of the mean-field approximation, we want to consider the whole system in the quantum regime. From Eq. (1, we use the single mode approximation [31, 3 by assuming the atomic and molecular condensates have the same form of density distribution which should be applicable before phase separation caused by large U am. We write ˆψ a = φ(râ, ˆψ m = φ(r ˆm, where φ(r = n(r/n is the renormalized density distribution for both atomic and molecular condensates. â and ˆm are the annihilation operators for atoms and molecules, respectively. The normalization requires φ (rdr = 1. This gives â â + ˆm ˆm = ˆN which corresponds to the constraint of particle number conservation as we discussed in the mean-field section. We also consider the bad cavity limit (large κ, and follow the same spirit of the mean-field treatment by replacing the photon field with the atomic and molecular ones. Based on the Heisenberg equations i â = [â, Ĥq, 430

5 Xiao-Qiang Xu, You-Quan Li i ˆm = [ ˆm, Ĥq, now the system Hamiltonian (rescaled may be retraced as NĤ q κ = α N ˆm ââ + α N ˆmâ â + N ε m ˆm ˆm + λ a â â ââ + λ m ˆm ˆm ˆm ˆm + λ am â â ˆm ˆm [ η N arctan Ũa N â â Ũm N ˆm ˆm. (5 Note that the parameters are defined as in the mean-field case. We may notice the similar structure to the classical Hamiltonian. Figure 4. (Color online Quantum eigen-energy levels with N = 80 when η = 0.1. Values of other parameters are the same as in the mean-field case. The yellow area outlines a similar characteristic loop as for the mean-field energy levels. In order to obtain the eigen information of this Hamiltonian, we expand the system in the Fock basis j = j, N j, where j (j = 0,,..., N/ denotes the number of molecules and N j the number of atoms. Here, even N is assumed for simplicity. Similar discussions apply for odd N. By diagonalizing the matrix H ij = i NĤ q /κ j, we can obtain the eigen-solutions of the quantum system. We choose the same parameter values as in the mean-field analysis. In Fig. 4 we plot the system eigenenergies from which we can find the obvious correspondence between the quantum and mean-field models. The points of avoidcrossing form a similar swallowtail loop to the mean-field energy spectrum. Suppose the system ground state Ψ g can be expanded as Ψ g = N/ j=0 c j j, where c j are complex coefficients satisfying the normalization condition N/ j=0 c j = 1. In order to verify that the quantum ground state has π relative phase between atoms and molecules, we can generalize the method for double species BECs [36 to our case and define the normalized phase state as 1 N/ θ r = e ijθr j, (6 N/ + 1 j=0 where θ r = πr/(n/ + 1, (r = 0, 1,..., N/. Then the distribution of ground state in the phase space is given by P(θ r = θ r Ψ g = N/ N/ cj + j=0 d=1 N/ d j=0 c j c j+d cos(dθ r (7 From our calculation, we find that P(θ r always has a gaussian profile whose peak locates at θ r = π, coinciding with our mapping between the mean-field and quantum models. However, near the avoid-crossing point, the distribution function P(θ r becomes almost flat, indicating the loss of phase coherence caused by quantum fluctuation. The avoid-crossing of the lowest energy levels always corresponds to the phenomenon of bistability. From Landau- Zener theory we know, during the time evolution process, no matter how adiabatically is tuned, when the avoidcrossing point is passed, the system always has finite probability of jumping into higher energy levels, entering the energy loop as shown in Fig. 4. Different directions of adiabatic tuning may cause the system to enter different sides of the loop. As in the mean-field case, we also plot the adiabatic time evolution starting from different ground states with opposite directions of tuning of (t. As shown in Fig. (b, around 0 we notice a window indicating the quantum bistability. 4. Summary and discussion In this paper, we considered the ground state properties of Feshbach resonance in a cavity. With the dispersive interaction between particles and photons (cavity mode, we may observe the phenomenon of bistability by adiabatically varying the pump-cavity detuning. We started with the mean-field approximation, where we found that the appearance of bistability is strongly related to multiple fixed-point solutions. Then we compared the meanfield results with the quantum model, which coincides with previous discussions. Physically speaking, the phenomena of bistability results from nonlinear interactions between particles, or between particles and photons. We need to mention that, for the parameters chosen in Sec., the pure Feshbach resonance model has no multiple fixed-point solutions for φ = 0 or φ = π when η = 0.0. However, since the particle interactions themselves can induce nonlinearity in the system, they may induce bistability (even multi-stability in the pure Feshbach resonance system. When the interaction with photons is included, the system may exhibit more complex solutions of fixed-points which deserve further study. 431

6 Bistability of Feshbach resonance in optical cavity We also need to mention that, since we assumed uniform spatial distribution and motions of center-of-mass of condensates are ignored, our model cannot support the bistability caused by the optomechanics effect [5, 6. Note that bistability in cavity a QED system [33 which relies on the coupling between two atomic states induced by cavity photons has been observed. Here we considered the dispersive interactions between atomic (molecular lower state and photons where the higher atomic (molecular states are eliminated adiabatically. The interplay between Feshbach resonance and dispersive interactions with photons results in the bistability we discussed here, which is different from the typical QED system. One additional note is that it has been suggested to cool molecules in cavity in analogy to optomechanics [34, 35. Here we may offer another strategy. Atoms are easier to cool than molecules due to their relatively simpler energy spectrum. We can use Feshbach resonance to convert cooled atoms into molecules which meets the same end. The work is supported by NSFC Grant No and the funds from Hangzhou City for supporting Hangzhou- City Quantum Information and Quantum Optics Innovation Research Team. References [1 A.-C. Ji, Q. Sun, X. C. Xie, W. M. Liu, Phys. Rev. Lett. 10, 0360 (009 [ M. J. Bhaseen, M. Hohenadler, A. O. Silver, B. D. Simons, Phys. Rev. Lett. 10, (009 [3 S. Slama, S. Bux, G. Krenz, C. Zimmermann, Ph. W. Courteille, Phys. Rev. Lett. 98, (007 [4 S. Gupta, K. L. Moore, K. W. Murch, Dan M. Stamper- Kurn, Phys. Rev. Lett. 99, (007 [5 K. W. Murch, K. L. Moore, S. Gupta, Dan M. Stamper- Kurn, Nature Phys. 4, 561 (008 [6 S. Ritter, F. Brennecke, K. Baumann, T. Donner, C. Guerlin, T. Esslinger, Appl. Phys. B 95, 13 (009 [7 M. G. Moore, P. Meystre, Phys. Rev. A 59, R1754 (1999 [8 M. G. Moore, O. Zobay, P. Meystre, Phys. Rev. A 60, 1491 (1999 [9 P. Horak, S. M. Barnett, H. Ritsch, Phys. Rev. A 61, (000 [10 P. Horak, H. Ritsch, Phys. Rev. A 63, (001 [11 J. Larson, B. Damski, G. Morigi, M. Lewenstein, Phys. Rev. Lett. 100, (008 [1 C. Maschler, H. Ritsch, Phys. Rev. Lett. 95, (005 [13 R. Kanamoto, P. Meystre, Phys. Rev. Lett. 104, (010 [14 J. M. Zhang, W. M. Liu, D. L. Zhou, Phys. Rev. A 78, (008 [15 Y. Dong, J. Ye, H. Pu, Phys. Rev. A 83, (011 [16 J. M. Zhang, S. Cui, H. Jing, D. L. Zhou, W. M. Liu, Phys. Rev. A 80, (009 [17 L.-H. Lu, Y.-Q. Li, Phys. Rev. A 76, (007 [18 L.-H. Lu, Y.-Q. Li, Phys. Rev. A 77, (008 [19 X.-Q. Xu, L.-H. Lu, Y.-Q. Li, Phys. Rev. A 79, (009 [0 X.-Q. Xu, L.-H. Lu, Y.-Q. Li, Phys. Rev. A 80, (009 [1 J. Javanainen, M. Mackie, Phys. Rev. A 59, 3186 (1999 [ H. Feshbach, Ann. Phys. (N.Y. 19, 87 (196 [3 M. Jääskeläinen, J. Jeong, C. P. Search, Phys. Rev. A 76, (007 [4 C. P. Search, J. M. Campuzano, M. Zivkovic, Phys. Rev. A 80, (009 [5 H. Jing, M. Zhan, Eur. Phys. J. D 4, 183 (007 [6 S. J. J. M. F. Kokkelmans, M. J. Holland, Phys. Rev. Lett. 89, (00 [7 R. Wynar, R. S. Freeland, D. J. Han, C. Ryu, D. J. Heinzen, Science 87, 1016 (000 [8 J. Li, D.-F. Ye, C. Ma, L.-B. Fu, J. Liu, Phys. Rev. A 79, 0560 (009 [9 D. J. Heinzen, R. Wynar, P. D. Drummond, K. V. Kheruntsyan, Phys. Rev. Lett. 84, 509 (000 [30 L. Zhou, H. Pu, H. Y. Ling, W. Zhang, Phys. Rev. Lett. 103, (009 [31 T.-L. Ho, S. K. Yip, Phys. Rev. Lett. 84, 4031 (000 [3 L.-M. Duan, J. I. Cirac, P. Zoller, Phys. Rev. A 65, (00 [33 G. Rempe, R. J. Thompson, R. J. Brecha, W. D. Lee, H. J. Kimble Phys. Rev. Lett. 67, 177 (1991 [34 G. Morigi, P. W. H. Pinkse, M. Kowalewski, R. de Vivie-Riedle, Phys. Rev. Lett. 99, (007 [35 R. J. Schulze, C. Genes, H. Ritsch, Phys. Rev. A 81, (010 [36 H. T. Ng, C. K. Law, P. T. Leung, Phys. Rev. A 68, (003 43