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Emission spectrum of a Λ-type three-level atom driven by a two-mode field
29 July 2002
Physics Letters A 300 (2002) 147–156 www.elsevier.com/locate/pla
Emission spectrum of a Λ-type three-level atom driven by a two-mode field ✩ Hong Guo ∗ , Gao-xiang Li, Jin-sheng Peng Department of Physics, Huazhong Normal University, Wuhan 430079, PR China Received 23 January 2002; accepted 5 June 2002 Communicated by P.R. Holland
Abstract An analytical expression of emission spectrum for Λ-type three-level atom interacted with two-mode cavity field is given. The characters of the emission spectrum for the field in photon number state, coherent state and SU(2) coherent state are exhibited. The effects of the atomic initial state, the field property on the time-dependent physical spectrum are analyzed. 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50
1. Introduction Since the study of the emission spectrum of the cavity-bound atom has provided much fundamental insight into the subject of the radiation-matter interaction [1–6], it has received considerable attention over recent years. Gea-Banacloche et al. studied the emission spectrum for a two-level atom interacting with a single-mode cavity field. They pointed out that if the cavity is initially in a squeezed vacuum field, the atomic emission spectrum is insensitive to the relative phase between the atomic dipole and the field, due to the lack of coherent coupling between the atom and the squeezed field through one-photon transition process [1]. Zaheer and Zubairy found that the atomic emission spectrum is sensitive to the relative phase between the atomic dipole and the coherent field. For a certain choice of the relative phase, an asymmetric two-peaked spectrum substitutes for the usual symmetrically three peaks [4]. Ashraf investigated the emission spectra of a Λ-type three-level atom with the two nearly degenerate lower levels coupled to a single-mode field in an ideal cavity [7], and revealed the non-classical effects including the vacuum Rabi splitting and quantum beats. He also pointed out that the four-peaked spectrum evolves into the three-peaked spectrum when the photon number increases to a certain amount. Recently we studied the emission spectrum of a Λ-type three-level atom interacting with the gray-body radiation field, and showed that ✩ Project supported by the National Natural Science Foundation of China (60078016) and the Key Research Program of National Ministry of Education, PR China. * Corresponding author. E-mail address: [email protected] (H. Guo).
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 7 8 3 - 1
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the influences of the cavity absorptivity and the temperature on the atomic emission spectrum are of significance [8]. However, the emission spectrum of a Λ-type three-level atom interacting with a two-mode field in the cavity is not discussed in detail yet. As we know, the modes are anticorrelated when the two-mode field is in the SU(2) coherent state. The atomic transition sequence in a Λ-type three-level atom can be used to detect such correlations. Lai et al. have studied the dynamics of a Λ-type three-level atom interacting with a two-mode SU(2) coherent state. They found that the atomic occupation probabilities exhibit purely periodic behavior for equal atom-field coupling constants [9]. We ever studied the atomic coherent population trapping in a Λ-configuration three-level atom interacting with a twomode quantized field intermediated by degenerate two-photon processes when the atom is initially in a coherent superposition of its two lower states, and pointed out that these trapping field states are the Schrödinger cat states of the two-mode SU(2) coherent states whose probability amplitudes depend on the initial atomic preparation and the atom-field coupling constants [10]. So it is significance to study emission spectrum of a Λ-type three-level atom driven by the two-mode SU(2) coherent field. Our Letter is organized as follows. In Section 2, we derive an exact expression of emission spectrum for Λ-type three-level atom coupled to a two-mode cavity field, when the atom is in a coherent superposition of its three states and the two-mode field is in an arbitrary quantum state. In Section 3, we show the character of Λ-type three-level atomic emission spectrum for the cavity field in photon number state, coherent state and SU(2) coherent state and discuss the effect of the atomic initial state on emission spectra. Finally, we present a conclusion.
2. Model The system under consideration is a Λ configuration three-level atom (one upper and two lower levels) interacting with a two-mode field, as shown in Fig. 1. In the rotating-wave approximation the Hamiltonian of the system can be given as [9] H = H1 + H2 , H1 = ωa a + a + ωb b+ b +
(1a) 3
ωi |ii|,
(1b)
i=1
H2 = g1 aσ1+ + a + σ1 + g2 bσ2+ + b+ σ2 .
(1c)
Here a(b) and a + (b+ ) are the annihilation and creation operators for the cavity field of frequency ωa (ωb ), |i (i = 1, 2, 3) is the eigenstate of the atom with eigenfrequency ωi , and gi (i = 1, 2) is atom-field coupling constant. σ1 = |13|, σ2 = |23|. For simplicity, we only discuss the resonant case ωa = ω3 − ω1 , ωb = ω3 − ω2 .
Fig. 1. Diagram of the atom-field coupling system.
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Since H1 and H2 satisfy the commutation relation [H1 , H2 ] = 0
(2)
the time-dependent unitary transformation operator may be written as U (t, 0) = U1 (t, 0)U2 (t, 0),
(3a)
U1 (t, 0) = exp(−iH1 t)
exp(−iω1 t) + + × exp −i ωa a a + ωb b b t 0 0
U2 (t, 0) = exp(−iH2 t) g 2 a + aA + g 2 bb+ 1 1 2 2 C 1 g g ab+ [A − 1] 1 2 1 = C12 ig1 aB1 − C1 where
g1 g2 a + b[A2 − 1] C22 2 + g1 aa + g22 b+ bA2 C22 ig2 bB2 −
0 exp(−iω2 t) 0 ig1 a + B3 C3 ig2 b+ B3 − C3
−
C2
0 , 0 exp(−iω3 t)
(3b)
,
(3c)
A3
A1 = cos(C1 t),
A2 = cos(C2 t),
A3 = cos(C3 t),
B1 = sin(C1 t), C1 = g12 a + a + g22 bb+ ,
B2 = sin(C2 t), C2 = g12 aa + + g22 b+ b,
B3 = sin(C3 t), C3 = g12 aa + + g22 bb+ .
Supposing that the three-level atom is initially in the superposition of its eigenstates |ψa = c1 |1 + c2 |2 + c3 |3 and the field is |ψf = Fna ,nb |na , nb
(4)
(5)
na ,nb
then the initial state of the atom-field coupling system can be written as |ψ = |ψa ⊗ |ψf = (c1 |1 + c2 |2 + c3 |3)Fna ,nb |na , nb .
(6)
na ,nb
According to (3) and (6), we can obtain the two-time correlation function as g2 + g2 + σ1 (t1 ) + σ2 (t1 ) σ1 (t2 ) + σ2 (t2 ) g1 g1 ∗ = cos[βna ,nb (t1 − t2 )]D (t1 )D(t2 ) + E ∗ (t1 )E(t2 ) ,
(7a)
na ,nb
where βna ,nb =
g12 na + g22 nb ,
(7b)
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√ c1 na sin(βna ,nb +1 t) −ωb t sin(βna +1,nb t) −iωa t 2 2√ D(t) = e + Fna ,nb g2 nb e Fna +1,nb −1 g1 na + 1 βna ,nb βna +1,nb βna ,nb +1 √ c2 g2 nb sin(βna +1,nb t) −iωa t sin(βna ,nb +1 t) −ωb t 2√ 2 Fna ,nb g1 na + e + Fna −1,nb +1 g2 nb + 1 e g1 βna ,nb βna +1,nb βna ,nb +1 √ √ ic3 Fna ,nb −1 g12 na cos(βna +1,nb t)e−iωa t + Fna −1,nb g22 nb cos(βna ,nb +1 t)e−iωb t , + g1 βna ,nb g1 g2 c1 sin(βna +1,nb t) −iωa t sin(βna ,nb +1 t) −ωb t E(t) = e − Fna ,nb na e Fna +1,nb −1 (na + 1)nb βna ,nb βna +1,nb βna ,nb +1 2 c2 g2 sin(βna +1,nb t) −iωa t sin(βna ,nb +1 t) −iωb t + Fna ,nb nb e − Fna −1,nb +1 na (nb + 1) e βna ,nb βna +1,nb βna ,nb +1 ig2 c3 √ √ Fna ,nb −1 nb cos(βna +1,nb t)e−iωa t − Fna −1,nb na cos(βna ,nb +1 t)e−iωb t . + βna ,nb
(7c)
(7d)
The physical spectrum S(υ) of radiation field emitted by a cavity-bound atom is given by the expression [11] T S(υ) = Γ
T dt1
0
dt2 exp −(Γ − iυ)(T − t1 ) − (Γ + iυ)(T − t2 )
0
g2 g2 × σ1+ (t1 ) + σ2+ (t1 ) σ1 (t2 ) + σ2 (t2 ) , g1 g1
(8)
where T is the interaction time and Γ is the bandwidth of the filter. Substituting (7) into (8) and carrying out the integration over the interaction time, we obtain the following expression for the emission spectrum of a Λ-type three-level atom as: √ Γ g12 na S(υ) = G+ (na , nb )F (ωa , βna ,nb , −βna +1,nb ) − G− (na , nb )F (ωa , βna ,nb , βna +1,nb ) 8 n ,n βna +1,nb a b √ g 2 nb G+ (na − 1, nb + 1)F (ωb , βna ,nb , −βna ,nb +1 ) + 2 βna ,nb +1 2 − G− (na − 1, nb + 1)F (ωb , βna ,nb , βna ,nb +1 )
2√ g na G+ (na , nb )F (ωa , −βna ,nb , −βna +1,nb ) − G− (na , nb )F (ωa , −βna ,nb , βna +1,nb ) + 1 βna +1,nb √ g 2 nb G+ (na − 1, nb + 1)F (ωb , −βna ,nb , −βna ,nb +1 ) + 2 βna ,nb +1 2 − G− (na − 1, nb + 1)F (ωb , −βna ,nb , βna ,nb +1 ) √ g1 g2 na + 2 G+ (na − 1, nb + 1)F (ωb , 0, −βna ,nb +1 ) − G− (na − 1, nb + 1)F (ωb , 0, βna ,nb +1 ) βna ,nb +1 √ g1 g2 nb G+ (na , nb )F (ωa , 0, −βna +1,nb ) − βna +1,nb 2 2 2 − G− (na , nb )F (ωa , 0, βna +1,nb ) (9a) g1 βna ,nb
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with
√ G± (na , nb ) = c1 g1 na + 1 Fna +1,nb −1 + c2 g2 nb Fna ,nb ± c3 βna +1,nb Fna ,nb −1 , (9b) exp[i(υ − x + y + z)T ] − exp(−Γ T ) F (x, y, z) = (9c) . Γ + i(υ − x + y + z) In the next section we shall investigate the character of the emission spectrum for Λ-type three-level atom based on Eqs. (9).
3. Results and discussions 3.1. The atom in the upper state The Λ-type three-level atomic emission spectrum is dependent on the initial state of the atom-field coupling system. For the atom initially in the upper state, i.e. c1 = c2 = 0, c3 = 1, the emission spectrum S(υ) of a Λ-type three-level atom is reduced to Γ 2 √ S(υ) = g1 na Fna ,nb −1 F (ωa , βna ,nb , −βna +1,nb ) + F (ωa , βna ,nb , βna +1,nb ) 8 n ,n a b 2 √ + g22 nb Fna −1,nb F (ωb , βna ,nb , −βna ,nb +1 ) + F (ωb , βna ,nb , βna ,nb +1 ) √ + g12 na Fna ,nb −1 F (ωa , −βna ,nb , −βna +1,nb ) + F (ωa , −βna ,nb , βna +1,nb ) 2 √ + g22 nb Fna −1,nb F (ωb , −βna ,nb , −βna ,nb +1 ) + F (ωb , −βna ,nb , βna ,nb +1 ) √ + 2g1 g2 na Fna −1,nb F (ωb , 0, −βna ,nb +1 ) + F (ωb , 0, βna ,nb +1 ) 2 2 2 √ g1 βna ,nb . − g1 g2 nb Fna ,nb −1 F (ωa , 0, −βna +1,nb ) + F (ωa , 0, βna +1,nb ) (10) In general the emission spectrum for a Λ-type three-level atom interacting with the two-mode field in photon number state has twelve-peak structure. The peak height is related to the photon number na(b) and atom-field coupling constant g1(2) . The peak position is associated with not only the photon number na(b) and atom-field coupling constant g1(2) , but also the frequency ωa (ωb ) of the two-mode cavity field. The peaks are divided into two groups which are symmetrically located around ωa and ωb , respectively, as shown in Fig. 2. In the case of na = 0 (nb = 0), the four peaks located at ωa ±βna ,nb +1 ±βna +1,nb +1 (ωb ±βna +1,nb ±βna +1,nb +1 ) disappear, so in Fig. 2a there are only four peaks which is just the vacuum Rabi splitting. With the increase in the photon number na(b) of the two-mode cavity field, the two peaks located at ωa − βna ,nb +1 + βna +1,nb +1 (ωb − βna +1,nb + βna +1,nb +1 ) and ωa + βna ,nb +1 − βna +1,nb +1 (ωb + βna +1,nb − βna +1,nb +1 ) coalesce into each other, then give a single peak at ωa (ωb ). So we see only ten peaks in Fig. 2d. √ In the case of an uncorrelated two-mode coherent field, the photon distribution ρi is peaked at ni with a width ni (i = a or b). For uncorrelated two-mode field the variance of the two-mode photon number sum is equal to the sum of each mode photon number variance, so the peaks are broadened obviously even if g1 = g2 . As long as the two-mode cavity field intensity is very small, the contribution to the spectrum S(υ) of the vacuum state plays a dominant role due to the larger relative weight of the vacuum state in the photon number distribution of the extremely weak coherent field. With the increase of the mean photon number the vacuum state progressively loses its dominant influence. For the two-mode cavity field intensity is sufficiently large, a ten-peak spectrum emerges. When the average photon number increases the four sidebands around ωa move away each other as the other four sidebands around ωb , the two peaks located at ωa and ωb become higher as shown in Fig. 3.
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Fig. 2. The atomic emission spectrum S(υ) for the two-mode cavity field in photon number state, c1 = c2 = 0, c3 = 1, g1 = g2 , Γ = 0.2/g1 , T = 20g1 , (a) na = nb = 0; (b) na = nb = 1.0; (c) na = nb = 3.0; (d) na = nb = 5.0.
Fig. 3. Same as Fig. 2 but the field in two-mode coherent state, (a) na = nb = 0.5; (b) na = nb = 1.5; (c) na = nb = 5.0; (d) na = nb = 12.0.
H. Guo et al. / Physics Letters A 300 (2002) 147–156
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Fig. 4. Same as Fig. 2 but the field in two-mode SU(2) coherent state, (a) na = 0.5, nb = 0.5; (b) na = nb = 1.5; (c) na = nb = 5.0; (d) na = nb = 12.0.
The situation is quite different for the anticorrelated two-mode SU(2) coherent state. Because the two-mode SU(2) coherent state is the eigenstate of two-mode photon number sum operator a + a + b+b, the emission spectrum is very similar to that for a photon number state on the condition of g1 = g2 , as shown in Fig. 4. 3.2. The atom in the coherent superposition of its two lower states In order to study quantum interference effects exhibited by the system we suppose the atom is initially in the coherent superposition of its two lower states θ iφ θ |1 + sin e |2, 0 θ π, −π φ < π. |ψa = cos (11) 2 2 When the cavity field is initially in pure number state, from (9) we know that the emission spectrum S(υ) is related to the atomic populations and the photon number na(b) as shown in Fig. 5. In this case we can know from which lower level the atom jumps to the upper level according to the change of the photon number, so the interference between the two different one-photon transition processes vanish, the atomic emission spectrum S(υ) is insensitive to the phase of the atomic dipole. The result is quite different from the result in Ref. [8]. The phase of the atomic dipole plays an important role when a Λ-type three-level atom in coherent superposition of its two lower states interacts with a single-mode field in photon number state. For the case of the cavity field initially in uncorrelated two-mode coherent state or anticorrelated two-mode SU(2) coherent state, applying (9) we obtain 2 nb i(ϕ+η) θ θ + g2 e sin , S(υ) ∝ g1 cos 2 na 2
(12)
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H. Guo et al. / Physics Letters A 300 (2002) 147–156
Fig. 5. Same as Fig. 2 but atom initially in the coherent superposition of its two lower states, na = nb = 4, (a) θ = 0; (b) θ = π/6; (c) θ = π/4; (d) θ = π/2.
here ni is the mean photon number of the ith mode, η is the difference of the two-mode phase. Obviously the atomic emission spectrum S(υ) is associated with not only the field intensity and the atomic populations, but also the phase of the atomic dipole and the difference of the two-mode phase as illustrated in Fig. 6–9. That is to say, the shape of the spectrum may be controlled via the change of ϕ and η. If the parameters of the system initial state satisfy g1 θ = 2 arctan g2
na nb
and ϕ + η = π
(13)
the atomic emission spectrum S(υ) is equal to zero. Regardless of uncorrelated two-mode coherent state or anticorrelated two-mode SU(2) coherent state, the system initial state whose parameters satisfy (13) can be expressed as
nb [g12 (na + 1) + g22 (nb + 1)]
|Ena ,nb , (nb + 1)(g12 na + g22 nb ) 1 |Ena ,nb = g2 nb + 1|1, na + 1, nb − g1 na + 1|2, na , nb + 1 , g12 (na + 1) + g22 (nb + 1)
|Ψ =
Fna +1,nb
(14a)
na ,nb
(14b)
|Ena ,nb is the eigenstate of the Hamiltonian H , the eigenvalue corresponding to the eigenstate |Ena ,nb is the free energy of the atom and the field, so |Ena ,nb is the decoupled stationary state of the system. When the system is initially in the coherent superposition of |Ena ,nb , the atom-field coupling system is decoupled, so all the peaks of the atomic emission spectrum disappear. Meanwhile the atomic populations do not vary with the time evolution, that is the atomic population coherent trapping takes place. This is consistent with the results in Refs. [12,13].
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Fig. 6. Same as Fig. 5 but the field in two-mode coherent state, na = nb = 6.0, ϕ = η = 0, (a) θ = 0; (b) θ = π/12, S(υ) + 0.05; (c) θ = π/6, S(υ) + 0.1; (d) θ = π/4, S(υ) + 0.15.
Fig. 7. Same as Fig. 6 but the field in two-mode SU(2) coherent state, (a) θ = 0; (b) θ = π/12, S(υ) + 0.2; (c) θ = π/6, S(υ) + 0.4; (d) θ = π/4, S(υ) + 0.6.
Fig. 8. Same as Fig. 6 but θ = π/4, η = 0, (a) ϕ = 3π/5; (b) ϕ = π/2, S(υ) + 0.1; (c) ϕ = π/4, S(υ) + 0.2; (d) ϕ = 0, S(υ) + 0.3.
Fig. 9. Same as Fig. 7 but θ = π/4, η = 0, (a) ϕ = 3π/5; (b) ϕ = π/2, S(υ) + 0.1; (c) ϕ = π/4, S(υ) + 0.2; (d) ϕ = 0, S(υ) + 0.3.
4. Conclusion In summary, we have studied the influences of the system initial state on the emission spectrum of Λ-type threelevel atom driven by two-mode field. The emission spectrum of Λ-type three-level atom initially in the upper state is independent of the phase of field. When the cavity field is initially in photon number state, the atomic emission spectrum has nothing to do with the phase of the atomic dipole. If the Λ-type three-level atom is initially in the coherent superposition of its two lower states and the cavity field is in the coherent superposition of photon number state, the two different one-photon transition processes such as |1 → |3 and |2 → |3 interfere each other, so the emission spectrum exhibits a phase sensitivity [14]. Whether the cavity field is initially in uncorrelated two-mode coherent state or anticorrelated two-mode SU(2) coherent state, as long as the parameters of the system initial state satisfy (13), the interference between the transition processes |1 → |3 and |2 → |3 makes counter balance,
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which induces the atom-field coupling system to be decoupled. In this case all the peaks of the atomic emission spectrum disappear, and the atomic population coherent trapping takes place.
Acknowledgements This work is supported by the National Natural Science Foundation of China and the Key Program for Science Research from National Education Ministry of China.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
J. Gea-Banacloche, R.R. Schlicher, M.S. Zubairy, Phys. Rev. A 38 (1988) 3514. L. Xu, Z.-M. Zhang, Z.-F. Luo, J. Phys. B 25 (1992) 3075. J.J. Sanchez-Mondragon, N.B. Narozhny, J.H. Eberly, Phys. Rev. Lett. 51 (1983) 550. K. Zaheer, M.S. Zubairy, Phys. Rev. A 39 (1989) 2000. P. Zhou, S. Swain, G.-X. Li, J.-S. Peng, Opt. Commun. 134 (1997) 455. H. Guo, G.-X. Li, J.-S. Peng, Acta Phys. Sin. 49 (2000) 887. M.M. Ashraf, Phys. Rev. A 50 (1994) 741. H. Guo, J.-S. Peng, J. Mod. Opt. 48 (2001) 1255. W.K. Lai, V. Buzek, P.L. Knight, Phys. Rev. A 44 (1991) 2003. G.-X. Li, J.-S. Peng, Phys. Lett. A 219 (1996) 41. J.H. Eberly, K. Wodkiewicz, J. Opt. Soc. Am. 67 (1977) 1252. G.-X. Li, J.-S. Peng, Phys. Rev. A 52 (1995) 465. H. Guo, J.-S. Peng, Chin. J. Quantum Electronics 18 (2001) 255. J.-S. Peng, G.-X. Li, in: Introduction to Modern Quantum Optics, World Scientific, Singapore, 1998, pp. 160–177.
Physics Letters A 300 (2002) 147–156 www.elsevier.com/locate/pla
Emission spectrum of a Λ-type three-level atom driven by a two-mode field ✩ Hong Guo ∗ , Gao-xiang Li, Jin-sheng Peng Department of Physics, Huazhong Normal University, Wuhan 430079, PR China Received 23 January 2002; accepted 5 June 2002 Communicated by P.R. Holland
Abstract An analytical expression of emission spectrum for Λ-type three-level atom interacted with two-mode cavity field is given. The characters of the emission spectrum for the field in photon number state, coherent state and SU(2) coherent state are exhibited. The effects of the atomic initial state, the field property on the time-dependent physical spectrum are analyzed. 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50
1. Introduction Since the study of the emission spectrum of the cavity-bound atom has provided much fundamental insight into the subject of the radiation-matter interaction [1–6], it has received considerable attention over recent years. Gea-Banacloche et al. studied the emission spectrum for a two-level atom interacting with a single-mode cavity field. They pointed out that if the cavity is initially in a squeezed vacuum field, the atomic emission spectrum is insensitive to the relative phase between the atomic dipole and the field, due to the lack of coherent coupling between the atom and the squeezed field through one-photon transition process [1]. Zaheer and Zubairy found that the atomic emission spectrum is sensitive to the relative phase between the atomic dipole and the coherent field. For a certain choice of the relative phase, an asymmetric two-peaked spectrum substitutes for the usual symmetrically three peaks [4]. Ashraf investigated the emission spectra of a Λ-type three-level atom with the two nearly degenerate lower levels coupled to a single-mode field in an ideal cavity [7], and revealed the non-classical effects including the vacuum Rabi splitting and quantum beats. He also pointed out that the four-peaked spectrum evolves into the three-peaked spectrum when the photon number increases to a certain amount. Recently we studied the emission spectrum of a Λ-type three-level atom interacting with the gray-body radiation field, and showed that ✩ Project supported by the National Natural Science Foundation of China (60078016) and the Key Research Program of National Ministry of Education, PR China. * Corresponding author. E-mail address: [email protected] (H. Guo).
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 7 8 3 - 1
148
H. Guo et al. / Physics Letters A 300 (2002) 147–156
the influences of the cavity absorptivity and the temperature on the atomic emission spectrum are of significance [8]. However, the emission spectrum of a Λ-type three-level atom interacting with a two-mode field in the cavity is not discussed in detail yet. As we know, the modes are anticorrelated when the two-mode field is in the SU(2) coherent state. The atomic transition sequence in a Λ-type three-level atom can be used to detect such correlations. Lai et al. have studied the dynamics of a Λ-type three-level atom interacting with a two-mode SU(2) coherent state. They found that the atomic occupation probabilities exhibit purely periodic behavior for equal atom-field coupling constants [9]. We ever studied the atomic coherent population trapping in a Λ-configuration three-level atom interacting with a twomode quantized field intermediated by degenerate two-photon processes when the atom is initially in a coherent superposition of its two lower states, and pointed out that these trapping field states are the Schrödinger cat states of the two-mode SU(2) coherent states whose probability amplitudes depend on the initial atomic preparation and the atom-field coupling constants [10]. So it is significance to study emission spectrum of a Λ-type three-level atom driven by the two-mode SU(2) coherent field. Our Letter is organized as follows. In Section 2, we derive an exact expression of emission spectrum for Λ-type three-level atom coupled to a two-mode cavity field, when the atom is in a coherent superposition of its three states and the two-mode field is in an arbitrary quantum state. In Section 3, we show the character of Λ-type three-level atomic emission spectrum for the cavity field in photon number state, coherent state and SU(2) coherent state and discuss the effect of the atomic initial state on emission spectra. Finally, we present a conclusion.
2. Model The system under consideration is a Λ configuration three-level atom (one upper and two lower levels) interacting with a two-mode field, as shown in Fig. 1. In the rotating-wave approximation the Hamiltonian of the system can be given as [9] H = H1 + H2 , H1 = ωa a + a + ωb b+ b +
(1a) 3
ωi |ii|,
(1b)
i=1
H2 = g1 aσ1+ + a + σ1 + g2 bσ2+ + b+ σ2 .
(1c)
Here a(b) and a + (b+ ) are the annihilation and creation operators for the cavity field of frequency ωa (ωb ), |i (i = 1, 2, 3) is the eigenstate of the atom with eigenfrequency ωi , and gi (i = 1, 2) is atom-field coupling constant. σ1 = |13|, σ2 = |23|. For simplicity, we only discuss the resonant case ωa = ω3 − ω1 , ωb = ω3 − ω2 .
Fig. 1. Diagram of the atom-field coupling system.
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Since H1 and H2 satisfy the commutation relation [H1 , H2 ] = 0
(2)
the time-dependent unitary transformation operator may be written as U (t, 0) = U1 (t, 0)U2 (t, 0),
(3a)
U1 (t, 0) = exp(−iH1 t)
exp(−iω1 t) + + × exp −i ωa a a + ωb b b t 0 0
U2 (t, 0) = exp(−iH2 t) g 2 a + aA + g 2 bb+ 1 1 2 2 C 1 g g ab+ [A − 1] 1 2 1 = C12 ig1 aB1 − C1 where
g1 g2 a + b[A2 − 1] C22 2 + g1 aa + g22 b+ bA2 C22 ig2 bB2 −
0 exp(−iω2 t) 0 ig1 a + B3 C3 ig2 b+ B3 − C3
−
C2
0 , 0 exp(−iω3 t)
(3b)
,
(3c)
A3
A1 = cos(C1 t),
A2 = cos(C2 t),
A3 = cos(C3 t),
B1 = sin(C1 t), C1 = g12 a + a + g22 bb+ ,
B2 = sin(C2 t), C2 = g12 aa + + g22 b+ b,
B3 = sin(C3 t), C3 = g12 aa + + g22 bb+ .
Supposing that the three-level atom is initially in the superposition of its eigenstates |ψa = c1 |1 + c2 |2 + c3 |3 and the field is |ψf = Fna ,nb |na , nb
(4)
(5)
na ,nb
then the initial state of the atom-field coupling system can be written as |ψ = |ψa ⊗ |ψf = (c1 |1 + c2 |2 + c3 |3)Fna ,nb |na , nb .
(6)
na ,nb
According to (3) and (6), we can obtain the two-time correlation function as g2 + g2 + σ1 (t1 ) + σ2 (t1 ) σ1 (t2 ) + σ2 (t2 ) g1 g1 ∗ = cos[βna ,nb (t1 − t2 )]D (t1 )D(t2 ) + E ∗ (t1 )E(t2 ) ,
(7a)
na ,nb
where βna ,nb =
g12 na + g22 nb ,
(7b)
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√ c1 na sin(βna ,nb +1 t) −ωb t sin(βna +1,nb t) −iωa t 2 2√ D(t) = e + Fna ,nb g2 nb e Fna +1,nb −1 g1 na + 1 βna ,nb βna +1,nb βna ,nb +1 √ c2 g2 nb sin(βna +1,nb t) −iωa t sin(βna ,nb +1 t) −ωb t 2√ 2 Fna ,nb g1 na + e + Fna −1,nb +1 g2 nb + 1 e g1 βna ,nb βna +1,nb βna ,nb +1 √ √ ic3 Fna ,nb −1 g12 na cos(βna +1,nb t)e−iωa t + Fna −1,nb g22 nb cos(βna ,nb +1 t)e−iωb t , + g1 βna ,nb g1 g2 c1 sin(βna +1,nb t) −iωa t sin(βna ,nb +1 t) −ωb t E(t) = e − Fna ,nb na e Fna +1,nb −1 (na + 1)nb βna ,nb βna +1,nb βna ,nb +1 2 c2 g2 sin(βna +1,nb t) −iωa t sin(βna ,nb +1 t) −iωb t + Fna ,nb nb e − Fna −1,nb +1 na (nb + 1) e βna ,nb βna +1,nb βna ,nb +1 ig2 c3 √ √ Fna ,nb −1 nb cos(βna +1,nb t)e−iωa t − Fna −1,nb na cos(βna ,nb +1 t)e−iωb t . + βna ,nb
(7c)
(7d)
The physical spectrum S(υ) of radiation field emitted by a cavity-bound atom is given by the expression [11] T S(υ) = Γ
T dt1
0
dt2 exp −(Γ − iυ)(T − t1 ) − (Γ + iυ)(T − t2 )
0
g2 g2 × σ1+ (t1 ) + σ2+ (t1 ) σ1 (t2 ) + σ2 (t2 ) , g1 g1
(8)
where T is the interaction time and Γ is the bandwidth of the filter. Substituting (7) into (8) and carrying out the integration over the interaction time, we obtain the following expression for the emission spectrum of a Λ-type three-level atom as: √ Γ g12 na S(υ) = G+ (na , nb )F (ωa , βna ,nb , −βna +1,nb ) − G− (na , nb )F (ωa , βna ,nb , βna +1,nb ) 8 n ,n βna +1,nb a b √ g 2 nb G+ (na − 1, nb + 1)F (ωb , βna ,nb , −βna ,nb +1 ) + 2 βna ,nb +1 2 − G− (na − 1, nb + 1)F (ωb , βna ,nb , βna ,nb +1 )
2√ g na G+ (na , nb )F (ωa , −βna ,nb , −βna +1,nb ) − G− (na , nb )F (ωa , −βna ,nb , βna +1,nb ) + 1 βna +1,nb √ g 2 nb G+ (na − 1, nb + 1)F (ωb , −βna ,nb , −βna ,nb +1 ) + 2 βna ,nb +1 2 − G− (na − 1, nb + 1)F (ωb , −βna ,nb , βna ,nb +1 ) √ g1 g2 na + 2 G+ (na − 1, nb + 1)F (ωb , 0, −βna ,nb +1 ) − G− (na − 1, nb + 1)F (ωb , 0, βna ,nb +1 ) βna ,nb +1 √ g1 g2 nb G+ (na , nb )F (ωa , 0, −βna +1,nb ) − βna +1,nb 2 2 2 − G− (na , nb )F (ωa , 0, βna +1,nb ) (9a) g1 βna ,nb
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with
√ G± (na , nb ) = c1 g1 na + 1 Fna +1,nb −1 + c2 g2 nb Fna ,nb ± c3 βna +1,nb Fna ,nb −1 , (9b) exp[i(υ − x + y + z)T ] − exp(−Γ T ) F (x, y, z) = (9c) . Γ + i(υ − x + y + z) In the next section we shall investigate the character of the emission spectrum for Λ-type three-level atom based on Eqs. (9).
3. Results and discussions 3.1. The atom in the upper state The Λ-type three-level atomic emission spectrum is dependent on the initial state of the atom-field coupling system. For the atom initially in the upper state, i.e. c1 = c2 = 0, c3 = 1, the emission spectrum S(υ) of a Λ-type three-level atom is reduced to Γ 2 √ S(υ) = g1 na Fna ,nb −1 F (ωa , βna ,nb , −βna +1,nb ) + F (ωa , βna ,nb , βna +1,nb ) 8 n ,n a b 2 √ + g22 nb Fna −1,nb F (ωb , βna ,nb , −βna ,nb +1 ) + F (ωb , βna ,nb , βna ,nb +1 ) √ + g12 na Fna ,nb −1 F (ωa , −βna ,nb , −βna +1,nb ) + F (ωa , −βna ,nb , βna +1,nb ) 2 √ + g22 nb Fna −1,nb F (ωb , −βna ,nb , −βna ,nb +1 ) + F (ωb , −βna ,nb , βna ,nb +1 ) √ + 2g1 g2 na Fna −1,nb F (ωb , 0, −βna ,nb +1 ) + F (ωb , 0, βna ,nb +1 ) 2 2 2 √ g1 βna ,nb . − g1 g2 nb Fna ,nb −1 F (ωa , 0, −βna +1,nb ) + F (ωa , 0, βna +1,nb ) (10) In general the emission spectrum for a Λ-type three-level atom interacting with the two-mode field in photon number state has twelve-peak structure. The peak height is related to the photon number na(b) and atom-field coupling constant g1(2) . The peak position is associated with not only the photon number na(b) and atom-field coupling constant g1(2) , but also the frequency ωa (ωb ) of the two-mode cavity field. The peaks are divided into two groups which are symmetrically located around ωa and ωb , respectively, as shown in Fig. 2. In the case of na = 0 (nb = 0), the four peaks located at ωa ±βna ,nb +1 ±βna +1,nb +1 (ωb ±βna +1,nb ±βna +1,nb +1 ) disappear, so in Fig. 2a there are only four peaks which is just the vacuum Rabi splitting. With the increase in the photon number na(b) of the two-mode cavity field, the two peaks located at ωa − βna ,nb +1 + βna +1,nb +1 (ωb − βna +1,nb + βna +1,nb +1 ) and ωa + βna ,nb +1 − βna +1,nb +1 (ωb + βna +1,nb − βna +1,nb +1 ) coalesce into each other, then give a single peak at ωa (ωb ). So we see only ten peaks in Fig. 2d. √ In the case of an uncorrelated two-mode coherent field, the photon distribution ρi is peaked at ni with a width ni (i = a or b). For uncorrelated two-mode field the variance of the two-mode photon number sum is equal to the sum of each mode photon number variance, so the peaks are broadened obviously even if g1 = g2 . As long as the two-mode cavity field intensity is very small, the contribution to the spectrum S(υ) of the vacuum state plays a dominant role due to the larger relative weight of the vacuum state in the photon number distribution of the extremely weak coherent field. With the increase of the mean photon number the vacuum state progressively loses its dominant influence. For the two-mode cavity field intensity is sufficiently large, a ten-peak spectrum emerges. When the average photon number increases the four sidebands around ωa move away each other as the other four sidebands around ωb , the two peaks located at ωa and ωb become higher as shown in Fig. 3.
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Fig. 2. The atomic emission spectrum S(υ) for the two-mode cavity field in photon number state, c1 = c2 = 0, c3 = 1, g1 = g2 , Γ = 0.2/g1 , T = 20g1 , (a) na = nb = 0; (b) na = nb = 1.0; (c) na = nb = 3.0; (d) na = nb = 5.0.
Fig. 3. Same as Fig. 2 but the field in two-mode coherent state, (a) na = nb = 0.5; (b) na = nb = 1.5; (c) na = nb = 5.0; (d) na = nb = 12.0.
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Fig. 4. Same as Fig. 2 but the field in two-mode SU(2) coherent state, (a) na = 0.5, nb = 0.5; (b) na = nb = 1.5; (c) na = nb = 5.0; (d) na = nb = 12.0.
The situation is quite different for the anticorrelated two-mode SU(2) coherent state. Because the two-mode SU(2) coherent state is the eigenstate of two-mode photon number sum operator a + a + b+b, the emission spectrum is very similar to that for a photon number state on the condition of g1 = g2 , as shown in Fig. 4. 3.2. The atom in the coherent superposition of its two lower states In order to study quantum interference effects exhibited by the system we suppose the atom is initially in the coherent superposition of its two lower states θ iφ θ |1 + sin e |2, 0 θ π, −π φ < π. |ψa = cos (11) 2 2 When the cavity field is initially in pure number state, from (9) we know that the emission spectrum S(υ) is related to the atomic populations and the photon number na(b) as shown in Fig. 5. In this case we can know from which lower level the atom jumps to the upper level according to the change of the photon number, so the interference between the two different one-photon transition processes vanish, the atomic emission spectrum S(υ) is insensitive to the phase of the atomic dipole. The result is quite different from the result in Ref. [8]. The phase of the atomic dipole plays an important role when a Λ-type three-level atom in coherent superposition of its two lower states interacts with a single-mode field in photon number state. For the case of the cavity field initially in uncorrelated two-mode coherent state or anticorrelated two-mode SU(2) coherent state, applying (9) we obtain 2 nb i(ϕ+η) θ θ + g2 e sin , S(υ) ∝ g1 cos 2 na 2
(12)
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Fig. 5. Same as Fig. 2 but atom initially in the coherent superposition of its two lower states, na = nb = 4, (a) θ = 0; (b) θ = π/6; (c) θ = π/4; (d) θ = π/2.
here ni is the mean photon number of the ith mode, η is the difference of the two-mode phase. Obviously the atomic emission spectrum S(υ) is associated with not only the field intensity and the atomic populations, but also the phase of the atomic dipole and the difference of the two-mode phase as illustrated in Fig. 6–9. That is to say, the shape of the spectrum may be controlled via the change of ϕ and η. If the parameters of the system initial state satisfy g1 θ = 2 arctan g2
na nb
and ϕ + η = π
(13)
the atomic emission spectrum S(υ) is equal to zero. Regardless of uncorrelated two-mode coherent state or anticorrelated two-mode SU(2) coherent state, the system initial state whose parameters satisfy (13) can be expressed as
nb [g12 (na + 1) + g22 (nb + 1)]
|Ena ,nb , (nb + 1)(g12 na + g22 nb ) 1 |Ena ,nb = g2 nb + 1|1, na + 1, nb − g1 na + 1|2, na , nb + 1 , g12 (na + 1) + g22 (nb + 1)
|Ψ =
Fna +1,nb
(14a)
na ,nb
(14b)
|Ena ,nb is the eigenstate of the Hamiltonian H , the eigenvalue corresponding to the eigenstate |Ena ,nb is the free energy of the atom and the field, so |Ena ,nb is the decoupled stationary state of the system. When the system is initially in the coherent superposition of |Ena ,nb , the atom-field coupling system is decoupled, so all the peaks of the atomic emission spectrum disappear. Meanwhile the atomic populations do not vary with the time evolution, that is the atomic population coherent trapping takes place. This is consistent with the results in Refs. [12,13].
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Fig. 6. Same as Fig. 5 but the field in two-mode coherent state, na = nb = 6.0, ϕ = η = 0, (a) θ = 0; (b) θ = π/12, S(υ) + 0.05; (c) θ = π/6, S(υ) + 0.1; (d) θ = π/4, S(υ) + 0.15.
Fig. 7. Same as Fig. 6 but the field in two-mode SU(2) coherent state, (a) θ = 0; (b) θ = π/12, S(υ) + 0.2; (c) θ = π/6, S(υ) + 0.4; (d) θ = π/4, S(υ) + 0.6.
Fig. 8. Same as Fig. 6 but θ = π/4, η = 0, (a) ϕ = 3π/5; (b) ϕ = π/2, S(υ) + 0.1; (c) ϕ = π/4, S(υ) + 0.2; (d) ϕ = 0, S(υ) + 0.3.
Fig. 9. Same as Fig. 7 but θ = π/4, η = 0, (a) ϕ = 3π/5; (b) ϕ = π/2, S(υ) + 0.1; (c) ϕ = π/4, S(υ) + 0.2; (d) ϕ = 0, S(υ) + 0.3.
4. Conclusion In summary, we have studied the influences of the system initial state on the emission spectrum of Λ-type threelevel atom driven by two-mode field. The emission spectrum of Λ-type three-level atom initially in the upper state is independent of the phase of field. When the cavity field is initially in photon number state, the atomic emission spectrum has nothing to do with the phase of the atomic dipole. If the Λ-type three-level atom is initially in the coherent superposition of its two lower states and the cavity field is in the coherent superposition of photon number state, the two different one-photon transition processes such as |1 → |3 and |2 → |3 interfere each other, so the emission spectrum exhibits a phase sensitivity [14]. Whether the cavity field is initially in uncorrelated two-mode coherent state or anticorrelated two-mode SU(2) coherent state, as long as the parameters of the system initial state satisfy (13), the interference between the transition processes |1 → |3 and |2 → |3 makes counter balance,
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which induces the atom-field coupling system to be decoupled. In this case all the peaks of the atomic emission spectrum disappear, and the atomic population coherent trapping takes place.
Acknowledgements This work is supported by the National Natural Science Foundation of China and the Key Program for Science Research from National Education Ministry of China.
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