Resonance fluorescence with bichromatic excitation

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Resonance fluorescence with bichromatic excitation and nonresonant squeezed reservoir: iterative approach

Can. J. Phys. Downloaded from www.nrcresearchpress.com by CSP Staff on 05/14/12 For personal use only.

S.S. Hassan and R.A. Alharbey

Abstract: An iterative approach, not restricted to field strength nor to resonance conditions, is used to solve the nonautonomous Bloch equations modeling the interaction of a bichromatically driven two-level atom in the presence of an off-resonant broadband squeezed vacuum (SV) radiation reservoir. The system parameters are chosen such that the absolute difference between the iterative and (exact) numerical solutions is O(10–2) or less. For some limiting cases, range of validity of the iterative procedure is estimated analytically. The derived analytical solutions are used to calculate the transient fluorescent spectrum. Some previously obtained results (both experimental and theoretical) in the normal vacuum case are recovered. In the SV case, with strong resonant (off-resonant) bichromatic field strength, the transient spectrum shows many symmetrical (asymmetrical) resonances of weights and locations dependent on the SV detuning parameter. PACS Nos: 32.50.+d, 42.50.p, 42.50.ct Résumé : Nous utilisons une approche itérative, sans restriction sur l’intensité du champ ni les conditions de résonance, pour résoudre l’équation de Bloch non autonome qui décrit l’interaction d’un atome à deux niveaux stimulé bichromatiquement avec un réservoir de radiation du vide comprimé (SV) à large bande et hors résonance. Les paramètres du système sont choisis de telle sorte que la différence absolue entre les résultats itératifs et les résultats numériques exacts soit de l’ordre de O(10–2) ou moins. Pour quelques cas limites, nous estimons analytiquement le domaine de validité de la procédure itérative. Les solutions analytiques obtenues sont utilisées pour calculer le spectre de fluorescence transitoire. Nous retrouvons certains résultats déjà obtenus (expérimentalement et théoriquement) dans le cas du vide normal. Dans le cas du SV, avec une forte intensité de champ bichromatique résonant (hors résonance), le spectre transitoire montre plusieurs résonances symétriques (asymétriques) de poids et de position qui dépendent du paramètre de désyntonisation. [Traduit par la Rédaction]

1. Introduction Atomic energy-level splitting is a direct response of atomic systems to an exciting field. The spectral nature of the scattered radiation due to atom ⊕ field interaction is investigated through the fluorescent spectrum (commonly known as the resonance fluorescence (RF) spectrum), which is termed the Fourier transform of the averaged two-time correlation function of the atomic dipole operators (see ref. 1 and references therein). For the case of a dissipating single two-level atom excited by an idealized (strong) monochromatic laser field, the theoretical prediction of Mollow in 1969 [2] for the RF spectrum, namely, a 3-peak lorentzian structure together with its experimental verification in 1974–1976 [3] form a landmark in this area of research. The authors in ref. 4 have extended Mollow’s investigation [2] to the case where the atomic dissipating processes take place in the presence of a broadband squeezed vacuum (SV) radiation field reservoir. Because of the stimulated processes induced by the SV field, the unequal transverse decay rates for the quadrature compo-

nents of the atomic polarization [5] may induce a narrowing (subnatural linewidth) or broadening of the central peak of the Mollow spectrum, depending on the relative phase of the driving and SV fields [4]. The investigations in ref. 4 and other related work [6] have considered the case of a resonant SV field, that is, the circular frequency of the (monochromatic) laser field and the central frequency of the broadband SV field are equal. For zero atomic detuning (atomic frequency equal to laser frequency) and nonresonant SV field (i.e., nonzero SV detuning) the authors in ref. 7 showed that the long time-averaged first harmonic component of the RF spectrum is insensitive to the SV phase. Mathematically speaking, for nonzero SV detuning the model Bloch equations for the averaged atomic variables are nonautonomous and have no formal (exact) analytical solutions. For nonzero atomic and SV detunings the work in ref. 8 presented an investigation for the first harmonic components of absorption and intensity via the continued fraction approach. Further, analytical results supported by computational display were presented for the RF spectrum in the

Received 29 September 2011. Accepted 29 March 2012. Published at www.nrcresearchpress.com/cjp on 3 May 2012. S.S. Hassan. University of Bahrain, College of Science, Department of Mathematics, P.O. Box 32038, Kingdom of Bahrain. R.A. Alharbey. King Abdul-Aziz University, Faculty of Science, Mathematics Department, P.O. Box 42696 Jeddah 21551, Kingdom of Saudi Arabia. Corresponding author: R.A. Alharbey (e-mail: [email protected]). Can. J. Phys. 90: 449–460 (2012)

doi:10.1139/P2012-039

Published by NRC Research Press


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strong field limit [9, 10] and for the RF, absorption, and dispersion spectra in the weak field limit [11]. Broadly speaking, these analytical investigations [9–11] for the RF spectrum show that: (i) in the strong field limit [9, 10], the superposition of all harmonics of the SV detuning in addition to the SV phase parameter induce extra lorentzian–dispersive structure in the vicinity of the central peak, asymmetry, and broadening–narrowing effects while, (ii) in the weak field limit [11] the RF spectrum is quite sensitive to the SV phase and the combination of atomic and SV detuning parameters. Recently [12], we have adopted an iterative scheme to solve the nonautonomous system of Bloch equations in the case of a monochromatically driven two-level atom in the presence of a nonresonant SV field. In effect, this iterative scheme [12] acts to reduce the nonautonomous Bloch equations of time-dependent harmonic coefficients to a system of differential equations (DE) with constant coefficients and nonhomogeneous time-dependent harmonic terms, which then admits exact analytical solutions. The iterative scheme [12] is valid for large SV detuning parameters or for arbitrary field strength. The analytically calculated and computationally displayed transient RF spectra [12] show that the (large) SV detuning and phase parameters induce extra asymmetry compared with earlier results [9, 10] with a small SV detuning parameter. Particularly in the strong field limit [12] one of the Mollow side peaks is either merged with the central peak or has its weight reduced noticeably. The simplest deviation from the ideal monochromatic property of the laser field is the two-frequency case, that is, the bichromatic laser field of two different frequency components u1 and u2 and of equal constant amplitudes U : U(exp (iu1t) + exp(iu2t)). The resulting model Bloch differential equations are still nonautonomous (even in the case of a resonant SV field), and hence they have no formal (exact) analytical solutions. Different treatments for the Bloch equations in the presence of a SV field were given in the strongly driven case with small frequency detuning (u1 – u2) [13]. Also, the authors in ref. 14 have investigated the nonoscillatory fluorescent spectrum component numerically with zero atomic detuning. An alternative form of the bichromatic field is the fully amplitude-modulated field: 3o cos(Uot) cos(ult), (3o, amplitude modulation index; Uo, modulation frequency; and ul, circular frequency), which was investigated in the normal vacuum (NV) case by many authors [15–18]. Experimentally [19], the observed RF spectrum of two-level-like Ba atoms under intense bichromatic field excitation in the NV case shows a series of side bands (comb-like structure) separated by the frequency detuning 2d = (u1 – u2) with alternating linewidths. Analysis of this observed spectrum within the dressed state context is discussed in refs. 20, 21, and 22. In the present paper, we extend our iterative scheme [12] to the case of a two-level atom driven by a bichromatic field in the presence of a broad-band (SV) radiation reservoir. Range of validity of the iterated terms is estimated in some limiting cases, otherwise, assisted by making the absolute difference between the iterative and exact numerical solutions O(10–2) or less. In the present bichromatic field excitation with SV field, there are three cases to be considered for the SV field central frequency, usv, namely:

Can. J. Phys. Vol. 90, 2012

• resonance case, usv = us, where us = (u1 + u2)/2 is the average circular frequency of the bichromatic field; • usv is centered at the first odd harmonics of d = (u1 – u2)/ 2; and • usv is centered at the first even harmonics of d. The paper is presented as follows. Iterative analytical solutions of the Bloch equations are derived in Sect. 2, and then utilized to calculate analytically the transient RF spectrum with graphical presentation and discussion in Sect. 3. A summary is given in Sect. 4.

2. Model Bloch equations and iterative solutions The model Bloch equations for the averaged atomic variables describing the interaction of a single two-level atom with bichromatic laser field in the presence of an off-resonant SV field are of the normalized form with g = 1 [13], D E D E D E D E b S 2iU b S þ Me2ikdt b S z cosðdtÞ ð1aÞ S þ ¼ a b ¼

D E b S

ð1bÞ

D E D E D E D E 1 b S z iU b Sz ¼ b b Sþ b S cosðdtÞ 2

ð1cÞ

with E D E d Db b S ;z ðtÞ S ;z ¼ dt The quantities hb S ;z ðtÞi are the averaged atomic polarization components and the inversion, respectively, where b S ;z ðtÞ are the spin-1/2 Pauli operators with algebra h i h i b b S; b S ¼ 2b S z ¼ b ð2aÞ Sz S Sþ; b and 1 b Sz S ¼ b S b 2

2 b S ¼ 0

1 b S S zb S ¼ b 2

ð2bÞ

The other symbols in (1) are [13]: a = (1/2)(1 + 2N) + iD; b = a + a*; the parameter k = (usv – us)/d (d ≠ 0) is the measure of detuning the SV carrier (central) frequency usv from the average frequency us of the bichromatic driving field; U is the Rabi frequency; N and M are the SV parameters (average photon number and degree of squeezing) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with |M| = NðN þ 1Þ; D = us – uo; and kd = usv – us are the atomic and SV detuning parameters, respectively. The iterative method used to solve system (1) follows closely that used in ref. 12 for the monochromatic driven case. First, for d = 0, (1) reduce to the case of a monochromatic driving field and resonant SV field, where the exact (general) solution of the corresponding Bloch equations with constant coefficients is given in ref. 12 (we use the notation hb S ;z ðtÞio for this resonant case, d = 0) D E D E X b S þ ðtÞ ¼ Ao ðtÞ þ Aa ðtÞ b S a ð0Þ ð3aÞ o

a


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D E ¼ b S ðtÞ

ð3bÞ

o

D E D E X b S z ðtÞ ¼ Bo ðtÞ þ Ba ðtÞ b S a ð0Þ o

ð3cÞ

a


with a = (+, –, z). The functions Ab(t) and Bb(t); b = (+, –, z, o) are given in Appendix A. Now, we rearrange (1) as follows: D E D E D E D E b S þ ðtÞ M b S ðtÞ 2iU b S z ðtÞ S þ ðtÞ ¼ a b h E D Ei D S ðtÞ þ 2iUðcosðdtÞ 1Þ b S z ðtÞ M e2ikdt 1 b ð4aÞ ¼

E D b S ðtÞ

ð4bÞ

D E D E D E D E 1 b S z ðtÞ ¼ b b S z ðtÞ iU b S þ ðtÞ b S ðtÞ 2 hD E D Ei iUðcosðdtÞ 1Þ b S þ ðtÞ b S ðtÞ ð4cÞ The iterative solutions of (4) (for d ≠ 0) are obtained by replacing hb S ;z ðtÞi in the square brackets by the resonant (i.e., d = 0) forms hb S ;z ðtÞio , (3). Thus, (4) becomes of the form (we use hb S ;z ðtÞi1 to denote the first iterative solution of (4)) D E D E D E D E b S þ ðtÞ ¼ a b S þ ðtÞ M b S ðtÞ 2iU b S z ðtÞ 1

1

¼

1

1

Mfo ðtÞ 2iUf1 ðtÞ ð5aÞ

E D b S ðtÞ

ð5bÞ

1

D E D E E D E D 1 b S z ðtÞ iU b S þ ðtÞ b S ðtÞ S z ðtÞ ¼ b b 1 1 1 1 2 iUf2 ðtÞ ð5cÞ where

E D S ðtÞ fo ðtÞ ¼ e2ikdt 1 b D E S z ðtÞ f1 ðtÞ ¼ ðcos dt 1Þ b

o

with di linear expressions in N. For zero-atomic resonance and strong field (U ≫ 1, N, D = 0), the roots of the cubic Q(s) are approximately:

1 1 3 s2;3 ¼ þ 4N iU s1 ¼ 2 2 2 and

Eþ1 ¼ O

1 e1 U þ e2 d

with ei linear expressions in N. In general, and for arbitrary bichromatic driving field strength the adopted data for the system parameters are taken such that the order of the extra terms Eb and Fb of the iterative solution, (7), are O(10–2) or less via the following checks: a. the average inversion jhb S z ðtÞi1 j 1=2, and b. the absolute difference between the iterative, (7), and the (exact) numerical solutions of the original system, (1), is O(10–2) or less. Next, we proceed to calculate analytically the transient RF spectrum using the iterative solutions, (7).

3. Transient spectrum

SðD; TÞ ¼ ð6bÞ

o

o

ð6cÞ

Equation (5) have constant coefficients and their solutions are listed in the form D E D E D E X b S þ ðtÞ ¼ b S þ ðtÞ þ Eo ðtÞ þ Ea ðtÞ b S a ð0Þ ð7aÞ o

a

D E ¼ b S ðtÞ 1

D E D E D E X b S z ðtÞ þ Fo ðtÞ þ Fa ðtÞ b S a ð0Þ S z ðtÞ ¼ b o

ii.

For weak off-resonant field (D ≫ U, N) and arbitrary field (d), the roots of the cubic Q(s) (Appendix A) are approximately: s1 = –(1 + 2N), s2,3 = –(1/2) (1 + 2N) ± iD, and U2 Eþ1 ðtÞ ¼ O Dðd1 þ d2 D þ d3 dÞ

The transient fluorescent spectrum has the form [1]

o

1

i.

ð6aÞ

E D E D S þ ðtÞ b S ðtÞ f2 ðtÞ ¼ ðcosðdtÞ 1Þ b

1

with the functions Eb(t) and Fb(t) are given in Appendix B. A rough estimate of the extra terms Eb(t) and Fb(t) can be deduced in (7) in some limiting cases of field strength. For simplicity we take |M|2 = N and M real and for the term Eþ1 ðtÞ (Appendix B), for example, one can show that

a

2 T

ZT

Tt Z

dt 0

D E S þ ðt þ tÞb dteiDt b S ðtÞ

ð8Þ

0

where T is the integrating time of the detector, and D = ud – us is the detector’s detuning parameter. The auto-correlation function in (8) takes the form as in ref. 12, namely D E 1 b b b S þ ðt þ tÞS ðtÞ ¼ Cþ ðtÞ þ S z ðtÞ 2

D E 1 S ðtÞ ð9Þ þ Co ðtÞ Cz ðtÞ b 2

ð7bÞ

where the quantaties Co,+,z(t) are now given by

ð7cÞ

Co ðtÞ ¼ Ao ðtÞ þ Eo ðtÞ Cþ ðtÞ ¼ Aþ ðtÞ þ Eþ ðtÞ Cz ðtÞ ¼ Az ðtÞ þ Ez ðtÞ

ð10Þ Published by NRC Research Press

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Fig. 1. The atomic inversion against the normalized time (t) (in units of g = 1), for weak resonant field (U = 1, D = 0) in the NV case (N = M = 0) with (a) d = 0.3, and (b) d = 1. Full line represents the iterative solution hb S z ðtÞi1 and dashed line represents the (exact) numerical solution of (1) for hb S z ðtÞi.

Can. J. Phys. Vol. 90, 2012 Fig. 2. The RF spectrum S(D) against the normalized detuning (D) — with g = 1 — for detecting time T = 1 and for the same data as Fig. 1.

Fig. 3. (a) The atomic inversion with data as Fig. 1a but for U = 10. (b) The RF spectrum S(D) at T = 1 for U = 10, D = 0.

The full form of spectrum (8) with (9) and (10), is best displayed graphically against the detector’s detuning parameter (D) at fixed T with initial ground state atom (hb S ð0Þi = 0, hb S z ð0Þi = –1/2). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The system parameters ðU; D; kd; N; jMj ¼ NðN þ 1Þ; fÞ in the following figures are chosen such that the iterative scheme is valid, as noted in Sect. 2. We consider the following cases. 3.1. NV case (N = M = 0) For resonant weak (or moderate) field amplitude, (U = 1, D = 0) and for d = 0.3 and 1, both the iterative and the (exact) numerical solutions for the atomic inversion hb S z ðtÞi coincide for time t ≲ 1 (Fig. 1). The RF spectrum in this case at T = 1 is shown in Fig. 2 where the spectrum has a prominent single peak surrounded by weak groves. For a strong resonant field (U = 10, D = 0) and for d = 0.3 the iterative and the numerical solutions coincide exactly for t ≲ 1 (Fig. 3a) Published by NRC Research Press

Hassan and Alharbey


Fig. 4. (a) The RF spectrum S(D) for U = 15, d = 0.35, and D = 0, 5 at (a) T = 1, and (b) T = 3.

453 Fig. 5. The RF spectrum S(D) against (D) in the thermal field case (N = 5, M = 0) for weak off-resonant field (U = 0.2, D = 5) and d = 1 at T = 1, 10. Inset figure shows the atomic inversions against t for the same data.

Fig. 6. As Fig. 5 but for U = 1, D = 15, N = 0.1, and d = 0.5.

and the RF spectrum in this case at T = 1 shows the usual three-peak Mollow structure (of the monochromatic case) but with extra smaller resonances between the central and the side peaks (Fig. 3b). In fact, this spectrum resembles qualitatively the transient spectrum (T ∼1) — Fig. 3f in ref. 18 — calculated iteratively in the high power limit O(g/U). For stronger field (U = 15) at resonance (D = 0) additional resonances appear between the central and the main side peaks at T = 1 (Fig. 4a) with further resonance peaks around the main side peaks at larger T = 3 (Fig. 4b). In the off-resonance case (D = 5) the same behavior survives, but the spectrum becomes asymmetrical (dashed curves in Fig. 4). 3.2. Thermal field case (N ≠ 0, M = 0) For off-resonant weak field amplitude, (U = 0.2, D = 5) and for N = 5, d = 1, (Fig. 5) the single lorentzian spectrum centered at D = –D for T = 1 has a higher weight for T = 10. Note that the iterative and the numerical solutions of the inversion exactly coincide for all time (t) For moderate U =

1, D = 15, N = 0.1, and d = 0.5 (Fig. 6) the spectrum at T = 1 is asymmetric with its main peak at D = –D and a weaker right peak. For larger T = 5 the main peak is much higher and more spiky while the smaller right peak washes away. Note, iterative and numerical solutions of the inversion coincide at t = 1 and differ by 10–2 at T = 5 (see inset of Fig. 6). For strong resonant field (U = 10, D = 0) and N = 5 (Fig. 7) the atomic inversion exhibits an oscillatory pattern around the zero saturation value and of very small weight with larger d = 10. The difference of both iterative and numerical solutions is O(10–2) for t ≳ 1. The corresponding RF spectrum at T = 1 is symmetric and has a main peak Published by NRC Research Press


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Fig. 7. (a) As Fig. 1 but for U = 10, D = 0, N = 5, and (a) d = 0.5 and (b) d = 10.

Fig. 8. , The RF spectrum S(D) against (D) for the same data as (a) Fig. 7a and (b) Fig. 7b.

with merged side-bands for d = 0.5 (Fig. 8a), which develops to many resonances with larger d = 10 (Fig. 8b). In fact, this spectrum structure in Fig. 8b resembles qualitatively the experimental spectrum with observed asymmetry attributed, however, to other background atoms (Fig. 2e of ref. 19). For the nonresonant strong field case (U = 10, D = –15) and N = 5, d = 5 the atomic inversion shows a larger period of oscillation with a difference of O(10–2) for almost all (t) between the iterative and the numerical solutions (Fig. 9a). Figure 9b shows that the corresponding RF spectrum has pronounced asymmetry and many resonance peaks with main peak at D ≃ 20 for large T = 10 (similar qualitative structure is observed in Fig. 2f of ref. 19). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3.3. SV field case ðN 6¼ 0; jMj ¼ NðN þ 1ÞÞ For resonant weak field (U = 0.2, D = 0) and SV parameters N = 5, f = 0, and for d = 5 the RF spectrum at T = 1 (Fig. 10a) is symmetric with central lorentzian at D = 0 and two smaller side peaks at D ≃ 10. The two side peaks correspond to the elastic (coherent) scattering at the two excitation frequencies u1 and u2 (or ±2d = ±(u1 – u2) in a rotating frame). As for the central peak, it corresponds to the spec-

trum of the SV field. Both spectrum and atomic inversion are insensitive to the SV detuning parameter (k). For nonresonant weak field (U = 0.2, D = 5) and for SV phase f = p/2 (Fig. 10b), the two elastic side peaks wash away and the central peak at D = –5 is skewed. Both the nonzero value of D and f = p/2 contribute to this asymmetric structure as well as make the two solutions (iterative and numerical) coincide for all time (t). For a strong resonant field (U = 15, D = 0) and for d = 0.35, N = 5, and f = p (Fig. 11a), the agreement between the iterative and the numerical solutions of the inversion is excellent for t ≲ 4 with oscillations reduced because of the relatively large value of N = 5. The SV detuning parameter (k) has no effect on the inversion for this choice of parameters. The RF spectrum at T = 1, (Fig. 11b), shows the main three-peak Mollow structure with many small symmetrical resonances between the central and the side peaks. The SV detuning parameter (k = 0, 1, 2) affects the weight of the central peak and the location of the extra resonances. Changing to f = p/2 has the effect to reduce the weight of the central peak and washes away the extra resonances with both side peaks merge with the tail of the central peak. Published by NRC Research Press

Hassan and Alharbey


Fig. 9. (a) As Fig. 1 but for U = 10, D = –15, N = 5, and d = 5. (b) The RF spectrum S(D) against (D) for the same data in (a) at T = 1, 10.

For nonresonant strong field (U = 15, D = –5) and d = 1 the spectrum at T = 10 for f = 0, (Fig. 12a), shows a sharp spike at D = 0 with asymmetrical high right and lower left side peaks (for k = 2 the left side peak is merged). For f = p/2 (Fig. 12b), the two side peaks are still asymmetric but their height is reduced noticeably relative to the central peak. Changing k = 0, 1, 2 affects the width of central peak at its top part and merging of the side peaks to the central one. For a different set of parameters, U = 15, D = d = 5, and f = p/2. Figure 13 shows that the transient spectrum at T = 1 has many asymmetrical resonances but less merged, with the k parameter having its effect on the height of the main threepeaks and their merging. Note that both iterative and numerical solutions of the atomic inversion associated with the data of Figs. 12 and 13 are qualitatively similar to that of Fig. 11a.

4. Summary In this paper, we have extended our iterative scheme adopted in ref. 12 to the nonautonomous Bloch equations modelling the interaction of a single two-level atom with a bichromatic laser field in the presence of a resonant and (or) off-resonant SV field. Comparison of the iterative and (exact)

455 Fig. 10. (a)ffi The RF spectrum S(D) in the SV case (N = 5, |M| = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi NðN þ 1Þ; f ¼ 0) for resonant weak field, (U = 0.2, D = 0) and d = 5. Inset shows the inversion for the same data. (b) as (a) but with D = 5, f = p/2.

numerical solutions for the averaged atomic variables are made. A rough estimate of the extra iterated terms is shown in some limiting cases. Otherwise, the chosen system parameters for which the absolute difference between iterative and numerical solutions is O(10–2) or less are further checked by the physical condition, jhb S z ðtÞij 1=2. The iterative analytical formulas have been used to calculate the RF spectrum in all cases of NV, thermal, and SV fields with off-resonance conditions. Previous results for the RF spectrum in the strong bichromatic field in the SV case (valid for small frequency difference of the bichromatic field) [10, 13] are recovered. Some of our RF spectrum results in the NV case (Fig. 3b) for strong resonant field are analogues to the calculated transient spectrum in the high power limit [18], and in the thermal field case (Figs. 8b and 9b) for strong resonant and offresonant field, respectively, are analogues to the experimentally observed transient and steady spectra in ref. 19. Published by NRC Research Press

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Fig. 11. (a) As Fig. 1 butp for U = 15, D = 0, d = 0.35, and for SV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi parameters N ¼ 5; jMj ¼ NðN þ 1Þ; and f ¼ p. (b) The RF spectrum S(D) for the same data in (a) at T = 1 and for different k = 0, 1, and 2.

Can. J. Phys. Vol. 90, 2012 Fig. 12. The RF spectrum S(D) at T = 10, for U = 10, D = –5, d = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1, and for SV parameters N ¼ 5; jMj ¼ NðN þ 1Þ, and (a) f = 0 and (b) f = p/2 for different k = 0, 1, and 2.

Fig. 13. The RF spectrum S(D) at T p = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1, for U = 15, D = d = 5, and for SV parameters N ¼ 5; jMj ¼ NðN þ 1Þ; and f ¼ p=2 for different k = 0, 1, and 2.

In general, our iterative scheme (up to the check of the extra terms to be O(10–2) or less as noted before) is not restricted to field strength nor resonance conditions. It remains open to verify other spectral results presented here by extending the experiment with bichromatic field in ref. 19 to the SV case. It is worth adding that time-dependent fluorescence spectrum as a mean to monitor the state of atoms has been the subject of experimental [23] and theoretical [24] investigation in the NV case.

Acknowledgment We would like to express our gratitude to the Deanship of Scientific Research at King Abdulaziz University (Jeddah, KSA) for the financial support. Published by NRC Research Press

Hassan and Alharbey


References 1. J.H. Eberly and K. Wodkiewicz. J. Opt. Soc. Am. 67, 1252 (1977). doi:10.1364/JOSA.67.001252; J.H. Eberly, C.V. Kunasz, and K. Wodkiewicz. J. Phys. B, 13, 217 (1980). doi:10. 1088/0022-3700/13/2/011. 2. B.R. Mollow. Phys. Rev. 188, 1969 (1969). doi:10.1103/ PhysRev.188.1969. 3. F. Schuda and C.R. Stroud, Jr., andM. Hercher. J. Phys. B, 7, L198 (1974). doi:10.1088/0022-3700/7/7/002; F.Y. Wu, R.E. Grove, and S. Ezekiel. Phys. Rev. Lett. 35, 1426 (1975). doi:10.1103/PhysRevLett.35.1426; W. Hartig, W. Rasmussen, R. Schieder, and H. Walther. Z. Phys. A, 278, 205 (1976). doi:10.1007/BF01409169. 4. H.J. Carmichael, A.-S. Lane, and D.F. Walls. Phys. Rev. Lett. 58, 2539 (1987). doi:10.1103/PhysRevLett.58.2539. PMID: 10034778. 5. C.W. Gardiner. Phys. Rev. Lett. 56, 1917 (1986). doi:10.1103/ PhysRevLett.56.1917. PMID:10032810. 6. S. Smart and S. Swain. Phys. Rev. A, 48, R50 (1993). doi:10. 1103/PhysRevA.48.R50; PMID:9909688; P. Zhou, S. Swain, and Z. Ficek. Opt. Commun. 148, 159 (1998). doi:10.1016/ S0030-4018(97)00617-2. 7. Z. Ficek and B.C. Sanders. J. Phys. B, 27, 809 (1994). doi:10. 1088/0953-4075/27/4/017. 8. S.S. Hassan, O.M. Frege, and N. Nayak. J. Opt. Soc. Am. B, 12, 1177 (1995). doi:10.1364/JOSAB.12.001177. 9. A. Joshi and S.S. Hassan. J. Phys. B, 30, L557 (1997). doi:10. 1088/0953-4075/30/17/002. 10. S.S. Hassan, A. Joshi, and O.M. Frege. Nonlinear Opt. Quantum Opt. 30, 23 (2003). 11. S.S. Hassan, A. Joshi, and M.F. Ali. Eur. Phys. J. D, 26, 301 (2003). doi:10.1140/epjd/e2003-00264-8. 12. H.A. Batarfi, R.A. Al-Harbi, R. Saunders, and S.S. Hassan. Can. J. Phys. 88, 529 (2010). doi:10.1139/P10-035. 13. A. Joshi and S.S. Hassan. Phys. Rev. A, 58, 4239 (1998). doi:10.1103/PhysRevA.58.4239. 14. P. Zhou, S. Swain, and Z. Ficek. Phys. Rev. A, 55, 2340 (1997). doi:10.1103/PhysRevA.55.2340. 15. G.S. Agarwal and N. Nayak. J. Opt. Soc. Am. B, 1, 164 (1984). doi:10.1364/JOSAB.1.000164; G.S. Agarwal and N. Nayak. J. Phys. B, 19, 3385 (1986). doi:10.1088/0022-3700/ 19/20/022; G.S. Agarwal and N. Nayak. Phys. Rev. A, 33, 391 (1986). doi:10.1103/PhysRevA.33.391. PMID:9896623. 16. S. Chakmakjian, K. Koch, and C.R. Stroud, Jr. J. Opt. Soc. Am. B, 5, 2015 (1988). doi:10.1364/JOSAB.5.002015. 17. W.M. Ruyten. J. Opt. Soc. Am. B, 6, 1796 (1989). doi:10. 1364/JOSAB.6.001796. 18. S.P. Tewari and M.K. Kumari. Phys. Rev. A, 41, 5273 (1990). doi:10.1103/PhysRevA.41.5273. PMID:9903767. 19. Y. Zhu, Q. Wu, A. Lezama, D.J. Gauthier, and T.W. Mossberg. Phys. Rev. A, 41, 6574 (1990). doi:10.1103/PhysRevA.41. 6574. PMID:9903069. 20. H.S. Freedhoff and Z. Chen. Phys. Rev. A, 41, 6013 (1990). doi:10.1103/PhysRevA.41.6013; PMID:9903004; H.S. Freedhoff and Z. Chen. Phys. Rev. A, 46, 7328 (1992). doi:10. 1103/PhysRevA.46.7328; PMID:9908078; Z. Ficek and H.S. Freedhoff. Phys. Rev. A, 48, 3092 (1993). doi:10.1103/ PhysRevA.48.3092. PMID:9909962. 21. G.S. Agarwal, Y. Zhu, D.J. Gauthier, and T.W. Mossberg. J. Opt. Soc. Am. B, 8, 1163 (1991). doi:10.1364/JOSAB.8. 001163. 22. D.L. Aronstein, R.S. Bennink, R.W. Boyd, and C.R. Stroud, Jr. Phys. Rev. A, 65, 067401 (2002). doi:10.1103/PhysRevA.65. 067401.

457 23. Y.S. Bai, A.G. Yodh, and T.W. Mossberg. Phys. Rev. Lett. 55, 1277 (1985). doi:10.1103/PhysRevLett.55.1277. PMID: 10031775. 24. N. Lu, P.R. Berman, Y.S. Bai, J.E. Golub, and T.W. Mossberg. Phys. Rev. A, 34, 319 (1986). doi:10.1103/PhysRevA.34.319. PMID:9897254.

Appendix A The functions Ab(t) and Bb(t) (b = +, –, z, o) in (3) are listed as follows (all quantities in normalized units of g = 1): Aþ ðtÞ ¼

3 X a1 ðsj Þ j¼1

Az ðtÞ ¼

Q 0 ðs

jÞ

3 X a3 ðsj Þ j¼1

Q 0 ðsj Þ

es j t

A ðtÞ ¼

3 X a2 ðsj Þ

Q 0 ðsj Þ j¼1

es j t

es j t

3 1 a3 ð0Þ X a3 ðsj Þ sj t e þ A0 ðtÞ ¼ 0 ðs Þ 2 Qð0Þ Q j j¼1

!

Bþ ðtÞ ¼ Aþ ðtÞða1 ! g 1 Þ

B ðtÞ ¼ Aþ ðtÞ a1 ! g 1

Bz ðtÞ ¼ Aþ ðtÞða1 ! g 2 Þ

Bo ðtÞ ¼ Ao ðtÞða3 ! g 2 Þ

where a1 ðsÞ ¼ ðs þ 2G Þðs þ G iDÞ þ

U2 2

1 a2 ðsÞ ¼ Mðs þ 2G Þ U2 2 a3 ðsÞ ¼ Uðs þ G iD MÞ g 1 ðsÞ ¼

U M s G þ iD 2

g 2 ðsÞ ¼ ðs þ G Þ2 þ D2 jMj2 QðsÞ ¼ s3 þ a2 s2 þ a1 s þ ao

a2 ¼ 4G

Q0 ¼

dQ ds

1 G ¼ ð1 þ 2NÞ 2

a1 ¼ 5G 2 þ D2 þ U2 jMj2 1 ao ¼ U2 G M þ M þ 2G G 2 þ D2 jMj2 2 and sj (j = 1, 2, 3) are the roots of the cubic Q(s). Published by NRC Research Press

458


Appendix B

The functions Eb(t) and Fb(t) (b = +, –, z, o) in (7) are listed as follows: 10 10 X X Eb ðtÞ ¼ Ebi ðtÞ Fb ðtÞ ¼ Fbi ðtÞ i¼1

i¼1

where (we use the prime notation


Eþ1 ðtÞ ¼ U

P 03 i;j¼1

i;j¼1;i6¼j )

3 X ½a2 ðsj Þ a1 ðsj Þ g 1 ðsi Þ ðsi idÞt e es j t 0 0 Q ðsj ÞQ ðsi Þðsi sj idÞ i;j¼1

E10 ðtÞ ¼

M Eþ ða2 a1 ! a2 ; g 1 ! a2 Þ 2iU 3

1 Ez4 ðtÞ ¼ Eþ1 a2 a1 ! a3 ; g 1 ! a3 a3 2

( 3 X ½a2 ðsj Þ a1 ðsj Þ g 1 ðsi Þ si t te Eþ3 ðtÞ ¼ 2iU ½Q 0 ðsi Þ 2 i¼1

i;j¼1

M Eþ3 a2 a1 ! a1 ; g 1 ! a1 2iU

Ezi ðtÞ ¼ Eþi ðg 1 ! g 2 Þ; i ¼ 1; 2; 3

Eþ2 ðtÞ ¼ iU

3 X 0 ½a2 ðsj Þ a1 ðsj Þ g 1 ðsi Þ

Q 0 ðsj ÞQ 0 ðsi Þðsi sj Þ

M Eþ ða2 a1 ! a2 ; g 1 ! a2 ; d ! 2dkÞ iU 2

E9 ðtÞ ¼

P3

3 X ½a2 ðsj Þ a1 ðsj Þ g 1 ðsi Þ ðsi þidÞt sj t e e 0 ðs ÞQ 0 ðs Þðs s þ idÞ Q j i i j i;j¼1

þ

E8 ðtÞ ¼

1 Ez5 ðtÞ ¼ Eþ2 a2 a1 ! a3 ; g 1 ! a3 a3 2 e e sj t

sj t

)

1 Ez6 ðtÞ ¼ Eþ3 a2 a1 ! a3 ; g 1 ! a3 a3 2

1 Eþ4 ðtÞ ¼ Eþ1 a2 a1 ! a3 ; g 1 ! a1 a2 2

Ez7 ðtÞ ¼

M Eþ1 a2 a1 ! a1 ; g 1 ! a3 ; d ! 2dk iU

1 Eþ5 ðtÞ ¼ Eþ2 a2 a1 ! a3 ; g 1 ! a1 a2 2

Ez8 ðtÞ ¼

M Eþ ða2 a1 ! a2 ; g 1 ! a3 ; d ! 2dkÞ iU 2

1 Eþ6 ðtÞ ¼ Eþ3 a2 a1 ! a3 ; g 1 ! a1 a2 2

Ez9 ðtÞ ¼

M Eþ3 a2 a1 ! a1 ; g 1 ! a3 2iU

M Eþ7 ðtÞ ¼ Eþ1 a2 a1 ! a1 ; g 1 ! a2 ; d ! 2d iU

Ez10 ðtÞ ¼

M Eþ ða2 a1 ! a2 ; g 1 ! a3 Þ 2iU 3

M Eþ ða2 a1 ! a2 ; g 1 ! a1 ; d ! 2dkÞ Eþ8 ðtÞ ¼ iU 2

(

3 ½a2 ðsj Þ a1 ðsj Þ g 2 ðsi Þ eðsi þidÞt esj t iU X Eo1 ðtÞ ¼ 2 i;j¼1 si ðsi sj þ idÞQ 0 ðsi ÞQ 0 ðsj Þ 3 g ð0ÞX ½a2 ðsj Þ a1 ðsj Þ sj t e eidt þ 2 0 Qð0Þ j¼1 ðsj idÞQ ðsj Þ

a2 a1 ! a1 ; g 1 ! a2

Eþ9 ðtÞ ¼

M Eþ 2iU 3

Eþ10 ðtÞ ¼

M Eþ ða2 a1 ! a2 ; g 1 ! a1 Þ 2iU 3

(

3 ½a2 ðsj Þ a1 ðsj Þ g 2 ðsi Þ eðsi idÞt esj t iU X Eo2 ðtÞ ¼ si ðsi sj idÞQ 0 ðsi ÞQ 0 ðsj Þ 2 i;j¼1

Ei ðtÞ ¼ Eþi g 1 ! g 1 ; i ¼ 1; 2; 3

3 g ð0ÞX ½a2 ðsj Þ a1 ðsj Þ sj t e eidt þ 2 Qð0Þ j¼1 ðsj þ idÞQ 0 ðsj Þ

1 E4 ðtÞ ¼ Eþ1 a2 a1 ! a3 ; g 1 ! a2 a1 2

(

1 E5 ðtÞ ¼ Eþ2 a2 a1 ! a3 ; g 1 ! a2 a1 2

Eo3 ðtÞ ¼ iU

1 E6 ðtÞ ¼ Eþ3 a2 a1 ! a3 ; g 1 ! a2 a1 2 E7 ðtÞ ¼

M Eþ a2 a1 ! a1 ; g 1 ! a1 ; d ! 2dk iU 1

þ

)

)

3 X ½a2 ðsi Þ a1 ðsi Þ g 2 ðsi Þtesi t si ½Q 0 ðsi Þ 2 i¼1

3 g 2 ð0ÞX ½a2 ðsj Þ a1 ðsj Þ sj t e 1 0 Qð0Þ j¼1 sj Q ðsj Þ

st ) 3 st X 0 ½a2 ðsj Þ a1 ðsj Þ g 2 ðsi Þ e i e j þ si ðsi sj ÞQ 0 ðsj ÞQ 0 ðsi Þ i;j¼1 Published by NRC Research Press

Hassan and Alharbey

1 Eo4 ðtÞ ¼ Eo1 a2 a1 ! a3 ; g 2 ! a3 a3 2 1 Eo5 ðtÞ ¼ Eo2 a2 a1 ! a3 ; g 2 ! a3 a3 2


1 Eo6 ðtÞ ¼ Eo3 a2 a1 ! a3 ; g 2 ! a3 a3 2 Eo7 ðtÞ ¼

M Eo1 a2 a1 ! a1 ; g 2 ! a3 ; d ! 2dk iU

Eo8 ðtÞ ¼

M Eo ða2 a1 ! a2 ; g 2 ! a3 ; d ! 2dkÞ iU 2

Eo9 ðtÞ ¼

Eo10 ðtÞ ¼

M Eo a2 a1 ! a1 ; g 2 ! a3 iU 3 M Eo ða2 a1 ! a2 ; g 2 ! a3 Þ iU 3

3 X g 1 ðsj Þ g 1 ðsj Þ g 1 ðsi Þ ðsi þidÞt e es j t Fþ1 ðtÞ ¼ iU 0 ðs ÞQ 0 ðs Þðs s þ idÞ Q j i i j i;j¼1

3 X g 1 ðsj Þ g 1 ðsj Þ g 1 ðsi Þ ðsi idÞt e es j t Fþ2 ðtÞ ¼ iU 0 0 Q ðsj ÞQ ðsi Þðsi sj idÞ i;j¼1 ( 3 X g 1 ðsj Þ g 1 ðsj Þ g 1 ðsi Þ si t Fþ3 ðtÞ ¼ 2iU te ½Q 0 ðsi Þ 2 i¼1 )

3 X 0 g 1 ðsj Þ g 1 ðsj Þ g 1 ðsi Þ s t sj t i e e þ Q 0 ðsj ÞQ 0 ðsi Þðsi sj Þ i;j¼1 1 Fþ4 ðtÞ ¼ Fþ1 g 1 g 1 ! g 2 ; g 1 ! a1 a2 2 1 Fþ5 ðtÞ ¼ Fþ2 g 1 g 1 ! g 2 ; g 1 ! a1 a2 2 1 Fþ6 ðtÞ ¼ Fþ3 g 1 g 1 ! g 2 ; g 1 ! a1 a2 2

459

Fþ9 ðtÞ ¼

M Fþ3 g 1 g 1 ! g 1 ; g 1 ! a2 2iU

Fþ10 ðtÞ ¼

M Fþ3 g 1 g 1 ! g 1 ; g 1 ! a1 2iU

Fi ðtÞ ¼ Fþi ðg 1 !g 1 Þ ; i ¼ 1; 2; 3 1 F4 ðtÞ ¼ Fþ1 g 1 g 1 ! g 2 ; g 1 ! a2 a1 2 1 F5 ðtÞ ¼ Fþ2 g 1 g 1 ! g 2 ; g 1 ! a2 a1 2 1 F6 ðtÞ ¼ Fþ3 g 1 g 1 ! g 2 ; g 1 ! a2 a1 2 F7 ðtÞ ¼

M Fþ g g 1 ! g 1 ; g 1 ! a1 ; d ! 2dk iU 1 1

F8 ðtÞ ¼

M Fþ g g 1 ! g 1 ; g 1 ! a2 ; d ! 2dk iU 2 1

F9 ðtÞ ¼

M Fþ3 g 1 g 1 ! g 1 ; g 1 ! a1 2iU

F10 ðtÞ ¼

M Fþ3 g 1 g 1 ! g 1 ; g 1 ! a2 2iU

Fzi ðtÞ ¼ Fþi ðg 1 ! g 2 Þ; i ¼ 1; 2; 3 1 Fz4 ðtÞ ¼ Fþ1 g 1 g 1 ! g 2 ; g 1 ! a3 a3 2 1 Fz5 ðtÞ ¼ Fþ2 g 1 g 1 ! g 2 ; g 1 ! a3 a3 2 1 Fz6 ðtÞ ¼ Fþ3 g 1 g 1 ! g 2 ; g 1 ! a3 a3 2 Fz7 ðtÞ ¼

M Fþ1 g 1 g 1 ! g 1 ; g 1 ! a3 ; d ! 2dk iU

Fz8 ðtÞ ¼

M Fþ2 g 1 g 1 ! g 1 ; g 1 ! a3 ; d ! 2dk iU

Fþ7 ðtÞ ¼

M Fþ g g 1 ! g 1 ; g 1 ! a2 ; d ! 2dk iU 1 1

Fz9 ðtÞ ¼

M Fþ3 g 1 g 1 ! g 1 ; g 1 ! a3 2iU

Fþ8 ðtÞ ¼

M Fþ2 g 1 g 1 ! g 1 ; g 1 ! a1 ; d ! 2dk iU

Fz10 ðtÞ ¼

M Fþ3 g 1 g 1 ! g 1 ; g 1 ! a3 2iU Published by NRC Research Press

460


(

3 ðg 1 g 1 ! g 2 Þg 2 ðsi Þ eðsi þidÞt esj t iU X Fo1 ðtÞ ¼ 2 i;j¼1 si ðsi sj þ idÞQ 0 ðsi ÞQ 0 ðsj Þ

1 Fo4 ðtÞ ¼ Fo1 g 1 g 1 ! g 2 ; g 2 ! a3 a3 2

3 g ð0ÞX ða2 ðsj Þ a1 ðsj ÞÞ sj t e eidt þ 2 Qð0Þ j¼1 ðsj idÞQ 0 ðsj Þ


3 g ð0ÞX ½a2 ðsj Þ a1 ðsj Þ sj t e eidt þ 2 Qð0Þ j¼1 ðsj þ idÞQ 0 ðsj Þ

Fo3 ðtÞ ¼ iU þ

)

3 X ½a2 ðsi Þ a1 ðsi Þ g 2 ðsi Þtesi t

2 si Q 0 ðsi Þ i¼1

3 g 2 ð0ÞX ½a2 ðsj Þ a1 ðsj Þ sj t e 1 Qð0Þ j¼1 sj Q 0 ðsj Þ

þ

1 Fo5 ðtÞ ¼ Fo2 g 1 g 1 ! g 2 ; g 2 ! a3 a3 2 1 Fo6 ðtÞ ¼ Fo3 g 1 g 1 ! g 2 ; g 2 ! a3 a3 2

(

3 ½a2 ðsj Þ a1 ðsj Þ g 2 ðsi Þ eðsi idÞt esj t iU X Fo2 ðtÞ ¼ si ðsi sj idÞQ 0 ðsi ÞQ 0 ðsj Þ 2 i;j¼1

(

)

) st

3 st X 0 ½a2 ðsj Þ a1 ðsj Þ g 2 ðsi Þ e i e j si ðsi sj ÞQ 0 ðsj ÞQ 0 ðsi Þ i;j¼1

Fo7 ðtÞ ¼

M Fo1 g 1 g 1 ! g 1 ; g 2 ! a3 ; d ! 2dk iU

Fo8 ðtÞ ¼

M Fo2 g 1 g 1 ! g 1 ; g 2 ! a3 ; d ! 2dk iU

Fo9 ðtÞ ¼

M Fo3 g 1 g 1 ! g 1 ; g 2 ! a3 iU

Fo10 ðtÞ ¼

M Fo3 g 1 g 1 ! g 1 ; g 2 ! a3 iU
