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Scheme for preparation of coherent hybrid atomic–molecular beams
Physics Letters A 324 (2004) 272–276 www.elsevier.com/locate/pla
Scheme for preparation of coherent hybrid atomic–molecular beams Hui Jing ∗ , Yu-Zhu Wang Laboratory for Quantum Optics, Shanghai Institute of Fine Machines and Optics, Chinese Academy of Sciences, Shanghai 201800, PR China Received 30 September 2003; accepted 19 January 2004 Communicated by P.R. Holland
Abstract We study a possible scheme to realize coherent hybrid atomic–molecular beams based on MIT output coupler for Bose– Einstein condensate just by adding a tunable narrow magnetic Feshbach resonance region immediately below the atomic trap. The quantum dynamics of a simplified model in short-time limits, combining the optical out-coupling and magnetically induced intra-mode tunnelling between output atoms and their dimers, reveals an interesting squeezing-free property for the output atomic mode if the input photons is in a coherent state, although an injected optical squeezed state can really lead to the atomic squeezing in the propagating mode. 2004 Elsevier B.V. All rights reserved. PACS: 03.75.Nt; 03.75.Mn; 05.30.Jp
The realization of Bose–Einstein condensate (BEC) in cold dilute atomic gases has become a rich playground to manipulate and demonstrate various properties of quantum degenerate gases, accompanying many impressive advances in both experimental and theoretical efforts [1]. Recently the appealing schemes for preparation of quantum degenerate molecular gases via a magnetic Feshbach resonance [2,3] or an optical photo-association (PA) [4,5] within an atomic BEC and the interesting properties of the formed atom–molecule mixtures have been studied extensively. Photo-association directly generates the resonant intra-mode coupling and subsequent molecular
* Corresponding author.
E-mail address: [email protected] (H. Jing). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.01.051
samples with temperatures in the µK range, but the practical efficiency of molecules generation is severely limited by radiative decay and collisions between particles [5]. The versatile technique of Feshbach resonance in which the spin of one of two colliding atoms flips in a magnetic field is generally exploited in an optical trap [2], and its advantage of the possibility of reducing the negative effect of collisions has led to many important applications besides creating a molecular BEC with near-unit efficiency [3], such as BEC collapse or implosion control and matter-wave bright soliton [6]. Through a combination of magnetic and optical methods, Mackie considered a static magnetic field interacting with BEC in which the role of meanfield shifts and vibrational relaxation is minimized for the production of a stable molecular condensate [7]. In a recent beautiful experiment by Herbig et al. [2],
H. Jing, Y.-Z. Wang / Physics Letters A 324 (2004) 272–276
a pure molecular quantum gas spatially separated from its atomic partner was observed just by adding a magnetic levitation field to the ordinary Feshbach resonance technique within a BEC of cesium atoms. Most recently, Zhang et al. in their novel paper [8] proposed a magnetic Feshbach technique applying to a travelling atomic beam instead of an optically trapped atomic condensate, and studied the interesting atomic filamentation and quantum pair correlation in the atom laser. The key point in their model is the change of atomic binary collisions in an effective focusing Kerr medium introduced by eliminating the molecular mode for the case of far from resonance. In this Letter, by considering the MIT output coupler which in 1997 [9] first realized a pulsed atom laser based on the rf out-coupling of magnetically trapped atoms, we address the question of whether the presence of a magnetic region tuned near a Feshbach resonance immediately below the magnetic trap can lead to some different quantum statistical behaviors for the output atoms. Through a simplified model taking into account of the magnetically induced intra-mode tunnelling dynamics in the optical out-coupling process, we propose that the quantum conversion property between the input photons and the output atoms, as revealed in our previous work [10], can still exist in the case of molecular formation in propagating mode, at least for short-time limits. This means that, unlike the usual Feshbach resonance case in which the trilinear intra-mode tunnelling terms lead to a squeezed atom– molecule condensate [3,5,11], the output atoms may exhibit an interesting squeezing-free property in such special situation. In addition, by tuning the external magnetic field and initial trapped condensate density [12] and then applying a magnetic levitation field [2], one may control and separate the molecular fraction in the hybrid beams, which we hope to be realized in our magnetic trap for rubidium atomic condensate [13]. Turning to the situation of Fig. 1, we assume for simplicity that large number of Bose-condensed atoms tightly confined in a magnetic trap have two states, |1 and |2, with the initial condensation occurring in the trapped state |1. State |2, which has different trapping properties and is typically unconfined by the magnetic trap, is coupled to |1 by a one-mode optical field tuned near the |1 → |2 transition. The interaction of the field may thus generate condensate in state |2, from an initial condensate which is entirely in
273
Fig. 1. Schematic illustration for coherent hybrid atomic–molecular beams. The amplitude of the short resonant rf pulse should be attenuated to warrant a unitary evolution with a factorized structure for the total atomic wave function [9,10]. The magnetic Feshbach pulses also should be short to avoid any strong loss of atoms from condensate in the proximity of a Feshbach resonance [14].
state |1. The Feshbach resonance occurs within a horizontal “sheet” region where the resonantly enhanced atom–molecule coupling, at a particular value of an external magnetic field, coherently convert [3] the output atoms into molecules in single-state |3. In the second quantized notation, boson annihilation operators for input photons, trapped atoms and output atoms and subsequent molecules are denoted, respectively, by a, b1 , b2 and g. The free or diagonalized part of the total Hamiltonian are written in terms of the optical and magnetic detunings ∆ and δ for the two coupling precesses, in which the molecular decaying into uninterested states may be included by a non-Hermitian term in the form of δ → δ − 12 iΓ [3], with Γ being proportional to the total molecular decay rate. Defining the optical (Rabi) and magnetic coupling frequencies as and γ , the four-mode Hamiltonian in the interacωR tion picture is then (h¯ = 1) Hint = −ωR ab1 b2† + a † b1† b2 − γ b2† 2 g + g † b22 , (1) which is different in nature from the two-color PA case [11]. The Kerr collisions between the particles occupying the same or different states have been ignored, which is the main simplification of our model. It is possible in principle for the atomic collisions to be rather larger, but in the near-resonance case it is reasonable to assume the dominant terms to be the photon–atom and atom–molecule interactions at least for very dilute ensemble, as considered in many works [5,11,15].
274
H. Jing, Y.-Z. Wang / Physics Letters A 324 (2004) 272–276
We suppose the initial state of the system is theoretically described as |ψ(0) = |α1 ⊗ |(0) ⊗ |0g with |(0) = |02 ⊗ |ξ . Here |α1 is a Glauber coherent state of the operator b1 characterizing the condensed atoms in the trapped state |1, namely, b1 |α = √ Nc e−iθ |α; |02 denotes the initial untrapped atomic and molecular states |2 and |3 as vacuum states since there is no occupying particles in them; and the initial state |ξ of the input photons is taken now as a coherent state and later as a squeezed state [16]. Besides, the initial phase θ of the trapped BEC will not affect the physical results [10]. In the Bogoliubov approximation [10,17], we can ignore the slow change of the large number Nc of the condensed atoms in the trap, which means that the√operators b1 , b1† can be replaced with a c-number Nc . Such a simple threemode model makes it possible for an analytical calculation. The general depleted case would be explored elsewhere. Now we consider the time evolution of the input photons, output atoms and subsequent molecules, which are determined by the Heisenberg equations of motion in the forms as a˙ = iωR b, b˙ = iωR a + 2iγ b†g and g˙ = iγ b2 , respectively. Here we have denoted b2 √ as b and ωR = ωR Nc . Taking into account of the fact that the loss of atoms from a condensed state occurs in impressively short time scales (up to two hundreds of µs) and as longer is the time spent near a Feshbach resonance as higher is the loss of atoms [14], we can focus on the short-time behaviors of the present dynamical system by readily deriving the solutions in second order of evolution time as
Note that the main feature of our present scheme is that the magnetic Feshbach resonance process starts not directly in the initial trapped atomic condensate, but in the “developing” atomic mode which is initially in a vacuum state. Comparatively, the formally similar model studied by Calsamiglia et al. [11] on two-color coherent PA within a Fock-state atomic BEC comprises a linear part for molecular bound–bound transition, and their three-mode Hamiltonian was reduced to two-mode case just by eliminating the intermediate molecular mode under large evolution frequency approximation. Using the solutions obtained above, one can easily study the interested quantum statistical properties for the output atoms and their dimers. As a concrete example, here we analyze the different quantum squeezing behaviors in this system and we would see that in the short-time limits, unlike the usual Feshbach resonance case, the output atoms can exhibit an interesting squeezing-free property even in the presence of the nonlinear atom–molecule coupling within the propagating beams; however, if the input photons is prepared in a squeezed state, the output atoms may also show the quadrature squeezed effects, indicating a possible control of quantum statistics of the output atoms in short-time limits by manipulating the quantum state of the input photons. As is well known [16], the field quadratures X1b and X2b can be defined as X1b = 12 (b + b†) and X2b = 1 † 2i (b − b ), and following Buzek et al. [18], we can introduce the squeezed coefficients as
1 2 2 t a(0) a(t) = a(0) + iωR tb(0) − ωR 2 − ωR γ t 2 b† (0)g(0),
Si =
1 2 | [X1 , X2 ]|
(i = 1, 2)
(3)
or equivalently as Si (t) = 2 Nb (t) ± 2C1 b2 (t) − (Ci b(t))2 , where C1 ≡ Re, C2 ≡ Im and Nb (t) refers to the particle occupation. The Xib component is squeezed at time t0 when Sib (t0 ) < 0. For the case of initial optical coherent state, namely, a|ξ = √ an −iφ Na e |ξ , the final results for the output atomic mode can be obtained as the following forms:
b(t) = b(0) + iωR ta(0) + 2iγ tb† (0)g(0) 1 2 2 − ωR t b(0) + ωR γ t 2 a † (0)g(0) 2 − γ 2 t 2 b† (0)b2 (0) + 2γ 2 t 2 b(0)g † (0)g(0), g(t) = g(0) + iγ tb2 (0) − ωR γ t 2 a(0)b(0) − γ 2 t 2 2b† (0)b(0) + 1 g(0).
(∆Xi )2 − 12 | [X1 , X2 ]|
(2)
Obviously, for the special cases of γ = 0 or reducing the γ -related terms into the effective Kerr interactions [8,11] between the output atoms, we just return to our previous models considered in Ref. [10].
2 2 S1b (t) = 3ωR t |ξ |2 sin2 φ > 0, 2 2 t |ξ |2 cos2 φ > 0, S2b (t) = 3ωR
(4)
which means that there would be no squeezed effect for the output atoms in this short-time limits. For the out-state photons we can easily obtained the final
H. Jing, Y.-Z. Wang / Physics Letters A 324 (2004) 272–276
φ = 2nπ , it will become (for r > 0):
results as
2 2 t |ξ |2 cos2 φ, S1a (t) = 3 1 − ωR 2 2 t |ξ |2 sin2 φ, S2a (t) = 3 1 − ωR which means that there will be no squeezing also for the photons. Similar results can be easily seen for the molecular mode. This result is quite different from the usual Feshbach resonance case in which the tunnelling dynamics due to the magnetic atom– molecule coupling interaction will inevitably lead to a squeezed atomic–molecular condensate [11]. Note that this results only hold for the case of short evolution time. Certainly, more accurate theoretical studies on the interplay of the optical and magnetic coupling terms are required in further mathematical treatments. If, for another case, the initial state for input photons is the squeezed state [16]: |ξ = S(ξ )|m, where the squeezed operator 1 ∗ 2 1 † 2 S(ξ ) = exp ξ a − ξ(a ) 2 2 with ξ = r exp(iφ), representing a unitary transformation on the coherent state |m, it turns out that there is still be no squeezing in the molecular mode (in fact, the molecular fraction is so small that it could be effectively omitted in the second-order of evolution time), but some different results really appear for the out-state atomic mode. In particular, for an injected vacuum-squeezed state (m = 0), the results can be written in the following simple form: 2 2 t sinh r(sinh r + cos φ cosh r), S1b (t) = 2ωR 2 2 S2b (t) = 2ωR t sinh r(sinh r − cos φ cosh r).
275
(5)
Obviously, for the initial state of the atomic field, Eq. (5) yields S1b (0) = S2b (0) = 0, which means there is no squeezing, as it should be. However, the different values of the squeezed parameters r and φ really can lead to the atomic squeezing in different quadrature components. For example, if one chooses the squeezed angle as cos φ = −1, or φ = (2n + 1)π (n = 0, 1, 2, . . .), we can get (for r > 0): 2 2 t 1 − e−2r < 0, S1b (t) = −ωR 2 2 2r t e − 1 > 0, S2b (t) = ωR which just means that the quadrature component X1b is squeezed; however, if one chooses cos φ = 1 or
2 2 2r t e − 1 > 0, S1b (t) = ωR 2 2 t 1 − e−2r < 0, S2b (t) = −ωR which means that the squeezed effect transfers to X2b component. Clearly, these results are closely related to the restrained effect of molecular formation in the present situation of our scheme. In summary, we have investigated an interesting scheme for preparation of coherent hybrid atomic– molecular beams just by adding a magnetic Feshbach resonance process for the output atoms, within a simplified three-mode model. If the input photons are in a coherent state, the output atoms will maintain the coherence at least for short-time limits; but for an injected optical squeezed state, the quadrature squeezed effect really can happen for the out-state atoms by controlling squeezed parameters of the input photons. Of course, these results hold only for the short evolution time and, more importantly, the molecular population is even smaller in the short time limits [3,12]. However, according to Yurovsky et al. [12], there exists the optimal conditions under which large fraction of the atomic population can be converted into an unstable molecular condensate which could persist long enough to allow its coherent transfer to a more stable state. If the molecular fraction could be obviously increased, our scheme might be regarded as the first step towards a molecular laser based on a magnetic trap for BEC since adding a magnetic levitation field as proposed by Herbig et al. [2] to the hybrid beams might completely separate the molecular component out of the modulated hybrid beams. In addition, our method can also be readily extended to analyze the interested phenomena of other quantum statistical effects, and even to systems with an initial pure molecular condensate dissociating into an atomic BEC which is coupled to another kind of atomic state, in which some new effects may be expected since it essentially corresponds to different form of initial states for the dynamical system. While much works are needed to clarify the effects of practical experimental circumstances like the atomic collisions, the environmental noise and the three-body recombination problems [19], our scheme here should readily lend itself to such studies.
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Acknowledgements The author gratefully acknowledges the support of K.C. Wong Education Foundation, Hong Kong. This work is also supported by China Postdoctoral Science Foundation and NSF of China under grant No. 10304020.
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Scheme for preparation of coherent hybrid atomic–molecular beams Hui Jing ∗ , Yu-Zhu Wang Laboratory for Quantum Optics, Shanghai Institute of Fine Machines and Optics, Chinese Academy of Sciences, Shanghai 201800, PR China Received 30 September 2003; accepted 19 January 2004 Communicated by P.R. Holland
Abstract We study a possible scheme to realize coherent hybrid atomic–molecular beams based on MIT output coupler for Bose– Einstein condensate just by adding a tunable narrow magnetic Feshbach resonance region immediately below the atomic trap. The quantum dynamics of a simplified model in short-time limits, combining the optical out-coupling and magnetically induced intra-mode tunnelling between output atoms and their dimers, reveals an interesting squeezing-free property for the output atomic mode if the input photons is in a coherent state, although an injected optical squeezed state can really lead to the atomic squeezing in the propagating mode. 2004 Elsevier B.V. All rights reserved. PACS: 03.75.Nt; 03.75.Mn; 05.30.Jp
The realization of Bose–Einstein condensate (BEC) in cold dilute atomic gases has become a rich playground to manipulate and demonstrate various properties of quantum degenerate gases, accompanying many impressive advances in both experimental and theoretical efforts [1]. Recently the appealing schemes for preparation of quantum degenerate molecular gases via a magnetic Feshbach resonance [2,3] or an optical photo-association (PA) [4,5] within an atomic BEC and the interesting properties of the formed atom–molecule mixtures have been studied extensively. Photo-association directly generates the resonant intra-mode coupling and subsequent molecular
* Corresponding author.
E-mail address: [email protected] (H. Jing). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.01.051
samples with temperatures in the µK range, but the practical efficiency of molecules generation is severely limited by radiative decay and collisions between particles [5]. The versatile technique of Feshbach resonance in which the spin of one of two colliding atoms flips in a magnetic field is generally exploited in an optical trap [2], and its advantage of the possibility of reducing the negative effect of collisions has led to many important applications besides creating a molecular BEC with near-unit efficiency [3], such as BEC collapse or implosion control and matter-wave bright soliton [6]. Through a combination of magnetic and optical methods, Mackie considered a static magnetic field interacting with BEC in which the role of meanfield shifts and vibrational relaxation is minimized for the production of a stable molecular condensate [7]. In a recent beautiful experiment by Herbig et al. [2],
H. Jing, Y.-Z. Wang / Physics Letters A 324 (2004) 272–276
a pure molecular quantum gas spatially separated from its atomic partner was observed just by adding a magnetic levitation field to the ordinary Feshbach resonance technique within a BEC of cesium atoms. Most recently, Zhang et al. in their novel paper [8] proposed a magnetic Feshbach technique applying to a travelling atomic beam instead of an optically trapped atomic condensate, and studied the interesting atomic filamentation and quantum pair correlation in the atom laser. The key point in their model is the change of atomic binary collisions in an effective focusing Kerr medium introduced by eliminating the molecular mode for the case of far from resonance. In this Letter, by considering the MIT output coupler which in 1997 [9] first realized a pulsed atom laser based on the rf out-coupling of magnetically trapped atoms, we address the question of whether the presence of a magnetic region tuned near a Feshbach resonance immediately below the magnetic trap can lead to some different quantum statistical behaviors for the output atoms. Through a simplified model taking into account of the magnetically induced intra-mode tunnelling dynamics in the optical out-coupling process, we propose that the quantum conversion property between the input photons and the output atoms, as revealed in our previous work [10], can still exist in the case of molecular formation in propagating mode, at least for short-time limits. This means that, unlike the usual Feshbach resonance case in which the trilinear intra-mode tunnelling terms lead to a squeezed atom– molecule condensate [3,5,11], the output atoms may exhibit an interesting squeezing-free property in such special situation. In addition, by tuning the external magnetic field and initial trapped condensate density [12] and then applying a magnetic levitation field [2], one may control and separate the molecular fraction in the hybrid beams, which we hope to be realized in our magnetic trap for rubidium atomic condensate [13]. Turning to the situation of Fig. 1, we assume for simplicity that large number of Bose-condensed atoms tightly confined in a magnetic trap have two states, |1 and |2, with the initial condensation occurring in the trapped state |1. State |2, which has different trapping properties and is typically unconfined by the magnetic trap, is coupled to |1 by a one-mode optical field tuned near the |1 → |2 transition. The interaction of the field may thus generate condensate in state |2, from an initial condensate which is entirely in
273
Fig. 1. Schematic illustration for coherent hybrid atomic–molecular beams. The amplitude of the short resonant rf pulse should be attenuated to warrant a unitary evolution with a factorized structure for the total atomic wave function [9,10]. The magnetic Feshbach pulses also should be short to avoid any strong loss of atoms from condensate in the proximity of a Feshbach resonance [14].
state |1. The Feshbach resonance occurs within a horizontal “sheet” region where the resonantly enhanced atom–molecule coupling, at a particular value of an external magnetic field, coherently convert [3] the output atoms into molecules in single-state |3. In the second quantized notation, boson annihilation operators for input photons, trapped atoms and output atoms and subsequent molecules are denoted, respectively, by a, b1 , b2 and g. The free or diagonalized part of the total Hamiltonian are written in terms of the optical and magnetic detunings ∆ and δ for the two coupling precesses, in which the molecular decaying into uninterested states may be included by a non-Hermitian term in the form of δ → δ − 12 iΓ [3], with Γ being proportional to the total molecular decay rate. Defining the optical (Rabi) and magnetic coupling frequencies as and γ , the four-mode Hamiltonian in the interacωR tion picture is then (h¯ = 1) Hint = −ωR ab1 b2† + a † b1† b2 − γ b2† 2 g + g † b22 , (1) which is different in nature from the two-color PA case [11]. The Kerr collisions between the particles occupying the same or different states have been ignored, which is the main simplification of our model. It is possible in principle for the atomic collisions to be rather larger, but in the near-resonance case it is reasonable to assume the dominant terms to be the photon–atom and atom–molecule interactions at least for very dilute ensemble, as considered in many works [5,11,15].
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We suppose the initial state of the system is theoretically described as |ψ(0) = |α1 ⊗ |(0) ⊗ |0g with |(0) = |02 ⊗ |ξ . Here |α1 is a Glauber coherent state of the operator b1 characterizing the condensed atoms in the trapped state |1, namely, b1 |α = √ Nc e−iθ |α; |02 denotes the initial untrapped atomic and molecular states |2 and |3 as vacuum states since there is no occupying particles in them; and the initial state |ξ of the input photons is taken now as a coherent state and later as a squeezed state [16]. Besides, the initial phase θ of the trapped BEC will not affect the physical results [10]. In the Bogoliubov approximation [10,17], we can ignore the slow change of the large number Nc of the condensed atoms in the trap, which means that the√operators b1 , b1† can be replaced with a c-number Nc . Such a simple threemode model makes it possible for an analytical calculation. The general depleted case would be explored elsewhere. Now we consider the time evolution of the input photons, output atoms and subsequent molecules, which are determined by the Heisenberg equations of motion in the forms as a˙ = iωR b, b˙ = iωR a + 2iγ b†g and g˙ = iγ b2 , respectively. Here we have denoted b2 √ as b and ωR = ωR Nc . Taking into account of the fact that the loss of atoms from a condensed state occurs in impressively short time scales (up to two hundreds of µs) and as longer is the time spent near a Feshbach resonance as higher is the loss of atoms [14], we can focus on the short-time behaviors of the present dynamical system by readily deriving the solutions in second order of evolution time as
Note that the main feature of our present scheme is that the magnetic Feshbach resonance process starts not directly in the initial trapped atomic condensate, but in the “developing” atomic mode which is initially in a vacuum state. Comparatively, the formally similar model studied by Calsamiglia et al. [11] on two-color coherent PA within a Fock-state atomic BEC comprises a linear part for molecular bound–bound transition, and their three-mode Hamiltonian was reduced to two-mode case just by eliminating the intermediate molecular mode under large evolution frequency approximation. Using the solutions obtained above, one can easily study the interested quantum statistical properties for the output atoms and their dimers. As a concrete example, here we analyze the different quantum squeezing behaviors in this system and we would see that in the short-time limits, unlike the usual Feshbach resonance case, the output atoms can exhibit an interesting squeezing-free property even in the presence of the nonlinear atom–molecule coupling within the propagating beams; however, if the input photons is prepared in a squeezed state, the output atoms may also show the quadrature squeezed effects, indicating a possible control of quantum statistics of the output atoms in short-time limits by manipulating the quantum state of the input photons. As is well known [16], the field quadratures X1b and X2b can be defined as X1b = 12 (b + b†) and X2b = 1 † 2i (b − b ), and following Buzek et al. [18], we can introduce the squeezed coefficients as
1 2 2 t a(0) a(t) = a(0) + iωR tb(0) − ωR 2 − ωR γ t 2 b† (0)g(0),
Si =
1 2 | [X1 , X2 ]|
(i = 1, 2)
(3)
or equivalently as Si (t) = 2 Nb (t) ± 2C1 b2 (t) − (Ci b(t))2 , where C1 ≡ Re, C2 ≡ Im and Nb (t) refers to the particle occupation. The Xib component is squeezed at time t0 when Sib (t0 ) < 0. For the case of initial optical coherent state, namely, a|ξ = √ an −iφ Na e |ξ , the final results for the output atomic mode can be obtained as the following forms:
b(t) = b(0) + iωR ta(0) + 2iγ tb† (0)g(0) 1 2 2 − ωR t b(0) + ωR γ t 2 a † (0)g(0) 2 − γ 2 t 2 b† (0)b2 (0) + 2γ 2 t 2 b(0)g † (0)g(0), g(t) = g(0) + iγ tb2 (0) − ωR γ t 2 a(0)b(0) − γ 2 t 2 2b† (0)b(0) + 1 g(0).
(∆Xi )2 − 12 | [X1 , X2 ]|
(2)
Obviously, for the special cases of γ = 0 or reducing the γ -related terms into the effective Kerr interactions [8,11] between the output atoms, we just return to our previous models considered in Ref. [10].
2 2 S1b (t) = 3ωR t |ξ |2 sin2 φ > 0, 2 2 t |ξ |2 cos2 φ > 0, S2b (t) = 3ωR
(4)
which means that there would be no squeezed effect for the output atoms in this short-time limits. For the out-state photons we can easily obtained the final
H. Jing, Y.-Z. Wang / Physics Letters A 324 (2004) 272–276
φ = 2nπ , it will become (for r > 0):
results as
2 2 t |ξ |2 cos2 φ, S1a (t) = 3 1 − ωR 2 2 t |ξ |2 sin2 φ, S2a (t) = 3 1 − ωR which means that there will be no squeezing also for the photons. Similar results can be easily seen for the molecular mode. This result is quite different from the usual Feshbach resonance case in which the tunnelling dynamics due to the magnetic atom– molecule coupling interaction will inevitably lead to a squeezed atomic–molecular condensate [11]. Note that this results only hold for the case of short evolution time. Certainly, more accurate theoretical studies on the interplay of the optical and magnetic coupling terms are required in further mathematical treatments. If, for another case, the initial state for input photons is the squeezed state [16]: |ξ = S(ξ )|m, where the squeezed operator 1 ∗ 2 1 † 2 S(ξ ) = exp ξ a − ξ(a ) 2 2 with ξ = r exp(iφ), representing a unitary transformation on the coherent state |m, it turns out that there is still be no squeezing in the molecular mode (in fact, the molecular fraction is so small that it could be effectively omitted in the second-order of evolution time), but some different results really appear for the out-state atomic mode. In particular, for an injected vacuum-squeezed state (m = 0), the results can be written in the following simple form: 2 2 t sinh r(sinh r + cos φ cosh r), S1b (t) = 2ωR 2 2 S2b (t) = 2ωR t sinh r(sinh r − cos φ cosh r).
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(5)
Obviously, for the initial state of the atomic field, Eq. (5) yields S1b (0) = S2b (0) = 0, which means there is no squeezing, as it should be. However, the different values of the squeezed parameters r and φ really can lead to the atomic squeezing in different quadrature components. For example, if one chooses the squeezed angle as cos φ = −1, or φ = (2n + 1)π (n = 0, 1, 2, . . .), we can get (for r > 0): 2 2 t 1 − e−2r < 0, S1b (t) = −ωR 2 2 2r t e − 1 > 0, S2b (t) = ωR which just means that the quadrature component X1b is squeezed; however, if one chooses cos φ = 1 or
2 2 2r t e − 1 > 0, S1b (t) = ωR 2 2 t 1 − e−2r < 0, S2b (t) = −ωR which means that the squeezed effect transfers to X2b component. Clearly, these results are closely related to the restrained effect of molecular formation in the present situation of our scheme. In summary, we have investigated an interesting scheme for preparation of coherent hybrid atomic– molecular beams just by adding a magnetic Feshbach resonance process for the output atoms, within a simplified three-mode model. If the input photons are in a coherent state, the output atoms will maintain the coherence at least for short-time limits; but for an injected optical squeezed state, the quadrature squeezed effect really can happen for the out-state atoms by controlling squeezed parameters of the input photons. Of course, these results hold only for the short evolution time and, more importantly, the molecular population is even smaller in the short time limits [3,12]. However, according to Yurovsky et al. [12], there exists the optimal conditions under which large fraction of the atomic population can be converted into an unstable molecular condensate which could persist long enough to allow its coherent transfer to a more stable state. If the molecular fraction could be obviously increased, our scheme might be regarded as the first step towards a molecular laser based on a magnetic trap for BEC since adding a magnetic levitation field as proposed by Herbig et al. [2] to the hybrid beams might completely separate the molecular component out of the modulated hybrid beams. In addition, our method can also be readily extended to analyze the interested phenomena of other quantum statistical effects, and even to systems with an initial pure molecular condensate dissociating into an atomic BEC which is coupled to another kind of atomic state, in which some new effects may be expected since it essentially corresponds to different form of initial states for the dynamical system. While much works are needed to clarify the effects of practical experimental circumstances like the atomic collisions, the environmental noise and the three-body recombination problems [19], our scheme here should readily lend itself to such studies.
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H. Jing, Y.-Z. Wang / Physics Letters A 324 (2004) 272–276
Acknowledgements The author gratefully acknowledges the support of K.C. Wong Education Foundation, Hong Kong. This work is also supported by China Postdoctoral Science Foundation and NSF of China under grant No. 10304020.
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