Atomic coherent state in Schwinger bosonic realization for optical Raman coherent effect
aa r X i v : . [ qu a n t - ph ] D ec Atomic coherent state in Schwinger bosonic realizationfor optical Raman coherent effect
Hong-yi Fan , Xue-xiang Xu , , and Li-yun Hu ∗ Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China; College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, 330022, China.
November 9, 2018
Abstract
For optical Raman coherent effect we introduce the atomic coherent state (or the angular mo-mentum coherent state with various angular momemtum values) in Schwinger bosonic realization,they are the eigenvectors of the Hamiltonian describing the Raman effect. Similar to the fact thatthe photon coherent state describes laser light, the atomic coherent state is related to Ramanprocess.
Atomic coherent states (or the angular momentum coherent state with various angular momemtumvalues) are sometimes referred to in the literature as spin coherent states or Bloch states [1–6].They have been successfully applied to many branches of physics [7–10]. For example, Arecchi etal. applied atomic coherent states to describe interactions between radiation field and an assemblyof two-level atoms [4]. Narducci, Bowden, Bluemel, Garrazana and Tuft [7] used atomic coher-ent state to study multitime correlation function for systems with observables satisfying an angularmomentum algebra, which suggested a convenient classical-quantum correspondence rule for an-gular momentum degrees of freedom. Takahashi and Shibata [9] transformed some equation ofmotion for density matrix of a damped spin system into that of a quasi-distribution. Gerry and Ben-moussa [10] have studied the generation of spin squeezing by the repeated action of the angularmomentum Dicke lowering operator on an atomic coherent state. In this work we shall introducethe atomic coherent state in Schwinger bosonic realization to study Raman coherent effect in thecontext of quantum optics.It is known that the Raman coherent effect, a monochromatic light wave incident on a Ramanactive medium gives rise to a parametric coupling between an optical vibrational mode and themode of the radiation field, the so-called Stocks mode. (In the case of Brillouin scattering, thereis a similar coupling, where the vibrations are at acoustical, rather than optical frequencies.) Thesimplest Hamiltonian model for describing Raman coherent effect is H = ω a † a + ω b † b − iλ (cid:0) a † b − ab † (cid:1) , (1)which is a two coupled oscillator model. In this work we shall show that the atomic coherentstate (some assembly of angular momentum states, so named angular momentum coherent state)expressed in terms of Schwinger bosonic realization of angular momentum [11] has its obviousphysical background, i.e., a set of energy eigenstates of two coupled bosonic oscillators with theHamiltonian can be classified as the atomic coherent state | τ i j according to the angular momentum ∗ Corresponding author. E-mail address: [email protected]. j , where τ is determined by the dynamic parameters ω , ω , λ . Thus the Raman coherenteffect is closely related to atomic coherent state theory, while the laser is described by the coherentstate theoretically. The atomic coherent state with angular momentum value j is defined as [4–7] | τ i = exp( µJ + − µ ∗ J − ) | j, − j i = (1 + | τ | ) − j e τJ + | j, − j i , (2)where J + is the raising operator of the angular momentum state | j, m i , | j, − j i is the lowest weightstate annihilated by J − , and µ = θ e − i ϕ , τ = e − i ϕ tan( θ . (3)In the j -subspace the completeness relation for | τ i is Z d Ω4 π | τ i h τ | = j X m = − j | j, m i h j, m | = 1 j , (4)where d Ω = sin θ d θ d ϕ , and h τ ′ | τ i = (1 + τ ′ τ ∗ ) j (1 + | τ | ) j (1 + | τ ′ | ) j . (5)Using [ J + , J − ] = 2 J z , [ J ± , J z ] = ∓ J ± , one can show that | τ i obeys the following eigenvector equa-tions, ( J − + τ J + ) | τ i = 2 jτ | τ i , ( J − + τ J z ) | τ i = jτ | τ i , (6) ( τ J + − J z ) | τ i = j | τ i . Employing the Schwinger Bose operator realization of angular momentum J + = a † b, J − = ab † , J z = 12 (cid:0) a † a − b † b (cid:1) , (7)where [ a, a † ] = 1 , [ b, b † ] = 1 and | j, m i is realized as | j, m i = a † j + m b † j − m p ( j + m )!( j − m )! | i = | j + m i ⊗ | j − m i , ( m = − j, · · · , j ) , (8)note that the last ket is written in two-mode Fock space, then | j, − j i = | i ⊗ | j i , and the atomiccoherent state | τ i is expressed as | τ i = e µJ + − µ ∗ J − | i ⊗ | j i = 1 p (2 j )! [ b † cos( θ a † e − i ϕ sin( θ j | i = 1 (cid:16) | τ | (cid:17) j j X l =0 s (2 j )! l !(2 j − l )! τ j − l | j − l i ⊗ | l i (9)2specially when j = 0 , | τ i = | i is just the two-mode vacuum state in Fock space. Using the normalordering form of the two-mode vacuum projector | i h | =: e − a † a − b † b : , we can use the techniqueof integration within an ordered product of operators [12, 13] to prove in j -subspace, Z d Ω4 π | τ i h τ | = 1(2 j )! Z π dθ sin θ Z π dφ : (cid:18) b † cos θ a † e − iφ sin θ (cid:19) j × (cid:18) b cos θ ae iφ sin θ (cid:19) j exp (cid:0) − a † a − b † b (cid:1) :=: (cid:0) a † a + b † b (cid:1) j (2 j + 1)! e − a † a − b † b : , (10)the completeness relation of | τ i in the whole two-mode Fock space can be obtained after summingover j : ∞ X j =0 (2 j + 1) Z d Ω4 π | τ i h τ | = ∞ X j =0 : (cid:0) a † a + b † b (cid:1) j (2 j )! e − a † a − b † b := 1 , (11)which means that atomic coherent states in Schwinger bosonic realization with all values of j formsa complete set. Now we inquire whether the atomic coherent state with a definite angular momentum value j is thesolution of the stationary Schrodinger equation H | τ i = E | τ i . (12)In order to solve Eq.(12) we directly use Eq.(9) and the relation a † | n i = √ n + 1 | n + 1 i , a | n i = √ n | n − i , (13)to calculate H | τ i = 1 (cid:16) | τ | (cid:17) j j X l =0 s (2 j )! l !(2 j − l )! [ ω (2 j − l ) + ω l ] τ j − l | j − l i ⊗ | l i− iλ (cid:16) | τ | (cid:17) j j X l =1 s (2 j )!( l − j − l + 1)! (2 j − l + 1) τ j − l | j − l + 1 i ⊗ | l − i + iλ (cid:16) | τ | (cid:17) j j − X l =0 s (2 j )!( l + 1)!(2 j − l − τ j − l ( l + 1) | j − l − i ⊗ | l + 1 i . (14)3et l ∓ → l in the second and third term of the r.h.s. of Eq.(14), respectively, we have H | τ i = 1 (cid:16) | τ | (cid:17) j j X l =0 s (2 j )! l !(2 j − l )! τ j − l (cid:26) [ ω (2 j − l ) + ω l ] − iλ (2 j − l ) 1 τ + iλτ l (cid:27) | j − l i ⊗ | l i = 1 (cid:16) | τ | (cid:17) j j X l =0 s (2 j )! l !(2 j − l )! τ j − l (cid:26) (cid:18) ω − i λτ (cid:19) j + (cid:20) ( ω − ω ) + iλ (cid:18) τ + 1 τ (cid:19)(cid:21) l (cid:27) | j − l i ⊗ | l i = 2 (cid:18) ω − i λτ (cid:19) j | τ i + 1 (cid:16) | τ | (cid:17) j j X l =0 s (2 j )! l !(2 j − l )! τ j − l (cid:20) ( ω − ω ) + iλ (cid:18) τ + 1 τ (cid:19)(cid:21) l | j − l i ⊗ | l i . (15)We see when the following condition is satisfied, iλτ + τ ( ω − ω ) + iλ = 0 ⇒ τ ± = ( ω − ω ) ± q ( ω − ω ) + 4 λ iλ . (16)then | τ ± i , expressed by Eq.(9), is the eigenstate of H with eigenvalue E = 2 (cid:18) ω − i λτ (cid:19) j = j (cid:20) ( ω + ω ) ± q ( ω − ω ) + 4 λ (cid:21) (17)Hence H ’s eigenvectors are classifiable according to the angular momentum value j . Especially,when ω = ω = ω , from Eqs.(16)-(17) we know τ ± = ∓ i, E ± = 2 j ( ω ± λ ) . We now investigate some fundamental atomic coherent states as H ’s eigenstates. In the case of j = 1 / , from Eq.(9) we know the eigenstate of H is | τ ± i j =1 / = 1 (cid:16) | τ ± | (cid:17) / ( τ ± | i ⊗ | i + | i ⊗ | i ) ω = ω → | i ± i j =1 / = 1 √ ∓ i | i ⊗ | i + | i ⊗ | i ) . (18)Indeed, one can check H | i + i j =1 / = ω + λ √ ( − i | , i + | , i ) . In the case of j = 1 , | τ ± i j =1 = 11 + | τ ± | X l =0 s (2)! l !(2 − l )! τ − l ± | − l i ⊗ | l i = 11 + | τ ± | (cid:16) τ ± | i ⊗ | i + √ τ ± | i ⊗ | i + | i ⊗ | i (cid:17) ω = ω → | i ± i = 12 (cid:16) − | i ⊗ | i ∓ i √ | i ⊗ | i + | i ⊗ | i (cid:17) . (19)In the case of j = 3 / , | τ ± i j =3 / = 1 (cid:16) | τ | (cid:17) / (cid:16) τ ± | i ⊗ | i + √ τ ± | i ⊗ | i + √ τ ± | i ⊗ | i + | i ⊗ | i (cid:17) ω = ω → | i ± i j =3 / = 12 / (cid:16) ± i | i ⊗ | i − √ | i ⊗ | i ∓ i √ | i ⊗ | i + | i ⊗ | i (cid:17) . (20)4n the case of j = 2 | τ ± i j =2 = 1 (cid:16) | τ | (cid:17) X l =0 s l !(4 − l )! τ − l ± | − l i ⊗ | l i = 1 (cid:16) | τ | (cid:17) (cid:16) τ ± | i ⊗ | i + 2 τ ± | i ⊗ | i + √ τ ± | i ⊗ | i + 2 τ ± | i ⊗ | i + | i ⊗ | i (cid:17) ω = ω → | i ± i j =2 = 14 (cid:16) | i ⊗ | i ± i | i ⊗ | i − √ | i ⊗ | i ∓ i | i ⊗ | i + | i ⊗ | i (cid:17) . (21)Thus we know how the eigenstate of H is composed of the Fock states. Knowing that H is diagonal in the basis of atomic coherent state | τ ± i , we can directly calculate itspartition function by virtue of its energy level. Z + ( β ) = Tr + (cid:0) e − βH (cid:1) = ∞ X j =0 j h τ + | e − βH | τ + i j = ∞ X j =0 e − βA j = 1 e η − | η = − βA = 1 e − βA − , (22)and Z − ( β ) = Tr − (cid:0) e − βH (cid:1) = ∞ X j =0 j h τ − | e − βH | τ − i j = 1 e − βB − (23)where A = ( ω + ω ) + q ( ω − ω ) + 4 λ ,B = ( ω + ω ) − q ( ω − ω ) + 4 λ . (24)satisfying H | τ + i = 2 Aj | τ + i , H | τ − i = 2 Bj | τ − i . Thus the total partition function is Z ( β ) = Z + ( β ) Z − ( β ) = (cid:18) e − βA − (cid:19) (cid:18) e − βB − (cid:19) , (25)and the internal energy of system is h H i e = − ∂∂β ln Z ( β )= − ∂∂β (cid:20) ln (cid:18) e − βA − (cid:19) + ln (cid:18) e − βB − (cid:19)(cid:21) = Ae Aβ − Be βB − . (26)5n summary, similar to the fact that the photon coherent state describes laser light, the atomiccoherent state is useful to classify the energy eigenstates of the Hamiltonian describing the Ramaneffect.This may be useful to further study stimulated Raman scattering since the scattered lightbehaves as laser light. ACKNOWLEDGEMENT:
We sincerely thank the referees for their constructive suggestion. Worksupported by the National Natural Science Foundation of China under grants: 10775097 and 10874174,and the Research Foundation of the Education Department of Jiangxi Province of China.
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