Effects of counter-rotating-wave terms of the driving field on the spectrum of resonance fluorescence

Transcription

1 PHYSICAL REVIEW A 88, ) Effects of counter-rotating-wave terms of the riving fiel on the spectrum of resonance fluorescence Yiying Yan, Zhiguo Lü, an Hang Zheng Key Laboratory of Artificial Structures an Quantum Control Ministry of Eucation), Department of Physics an Astronomy, Shanghai Jiao Tong University, Shanghai 4, China Receive 3 August 13; publishe 15 November 13) We investigate the fluorescence spectrum of a two-level system riven by a monochromatic classical fiel by the Born-Marovian master equation base on a unitary transformation. The main purpose is to unerstan the effects of counter-rotating-wave terms of the riving on spectral features of the fluorescence. We have erive an analytical expression for the fluorescence spectrum, which is ifferent from Mollow s theory, while Mollow s result on resonance is the limiting case of ours in moerately wea riving regimes. Our results emonstrate precisely that the counter-rotating-wave terms of the riving play an important role in the fluorescence spectrum for intense riving: i) the counter-rotating coupling suppresses the re sieban in the Mollow triplet an it enhances the blue one in explicitly contrast to the well-nown equal intensity of the sieban in Mollow s theory, ii) the higher-orer Mollow triplets appear as a characteristic spectral feature arising from counter-rotating-wave terms of the riving, an iii) a significant frequency shift of the siebans is observe, which epens on both the etuning an riving strength. DOI: 1.113/PhysRevA PACS numbers): 4.5.Ct, 4.5.Hz, 3.5.+, 3.8. t I. INTRODUCTION Mollow initially erive the resonance fluorescence spectrum of a riven two-level system TLS) [1]. The preicte resonance fluorescence spectrum has been verifie in traitional quantum optical experiments []. In Mollow s theory, the line shape of the resonance fluorescence spectrum epens on both the TLS spontaneous ecay rate an the riving strength. Specifically speaing, the ecay rate influences the full with at half maximum FWHM) of peas in the spectrum, an the riving strength etermines the splitting. For wea riving strengths, the resonance fluorescence spectrum is mae up of a single Lorentzian spontaneous emission line. For moerately strong riving strengths, instea of the single Lorentzian line, it consists of three split peas nown as the Mollow triplet. Recently, there has been increasing experimental investigation of the fluorescence spectrum in artificial atoms, for instance, semiconuctor quantum ots [3 9], single molecules [1], an superconucting circuits [11]. The experiments are not only to test funamental theory but also to evelop singlequantum emitters for quantum light spectroscopy an quantum information applications. It is reporte in Ref. [11] that the observe fluorescence spectrum is in quantitative agreement with the preictions from Mollow s theory. Accoring to Ref. [9], an iniviual Mollow sieban channel of the resonance fluorescence from a single quantum ot can act as an efficient single-photon source. However, so far most resonance fluorescence observe in experiments is uner conitions such that the riving strength is of the magnitue of the spontaneous ecay rate, which is far less than the bare transition frequency of the TLS. Thus, it is not surprising that the theory is consistent with the ata from experiments such as that in Ref. [11]. Apart from the experimental research on the fluorescence spectrum, theoretical investigations, which are attractive an significant, can be roughly groupe into two categories. One is stuying the influence of the various types of riving an environments in which TLSs exist on the fluorescence spectrum within the rotating-wave approximation RWA) of riving, such as the wors concerning the resonance spectrum of a quantum ot excite by a strong optical pulse [1] an the influence from a soli-state environment [13]. The other is extening the theory out of the framewor of the RWA of riving, such as the wor carrie out in Ref. [14], the main iea of which is treating the counter-rotating CR) wave terms of the riving as perturbation in the so-calle resse basis. In intense-riving regimes an on resonance, base on perturbation calculation, the authors foun unequally intense siebans, generation of higher-orer Mollow triplets with intensities comparable to those of the first, an an analytical expression for the shift of siebans in frequency [14]. Although the RWA-base theory succees in explaining classical fluorescence experiments, we believe that it is necessary to tae into account the effects of CR-wave terms of riving on the same footing as the rotating-wave terms for the following reasons. On the one han, the RWA loses its valiity when the riving strength is comparable to the magnitue of the bare transition frequency of the TLS an/or the riving frequency is etune from the resonance. The experiments in strongly riven TLS reporte in Refs. [15,16] reveal that theories beyon the RWA are neee to explain the observe phenomena. In this wor, we show clearly that for a finite-etuning case the spectrum without the RWA iffers from the one with the RWA even in wea-riving regimes. On the other han, the CR-wave terms have important effects on the coherently strongly riven ynamics of the TLS [17,18], such as the intriguing phenomenon nown as coherent estruction of tunneling CDT) [19]. Thus, one can expect that the CR-wave terms shoul introuce some characteristic spectral features of the scattere light from TLSs in intense-riving regimes. In this paper, we consier the effects of the CR-wave terms of the riving on fluorescence from a TLS riven by a monochromatic classical fiel. We tae into account the CR-wave terms of riving by means of the same unitary transformation as that of Ref. [], which has been use to stuy the coherently riven ynamics of the TLS. Using the unitary transformation, we obtain an effective Hamiltonian /13/885)/53811) American Physical Society

2 YIYING YAN, ZHIGUO LÜ, AND HANG ZHENG PHYSICAL REVIEW A 88, ) with RWA form, but the corresponing parameters become renormalize, which inclues the effects of the CR-wave terms of the riving. From the effective Hamiltonian, it is convenient to calculate the fluorescence spectrum by Mollow s metho Sec. III). In this sense our metho is as simple as the RWA metho. On the other han, our previous wor shows that the metho properly taes into account the CR-wave terms []. Using the effective Rabi frequency given in terms of the renormalize parameters, we have calculate the well-nown Bloch-Siegert shift up to fourth orer of the riving strength, which is exactly the same as the preiction of Floquet theory [1]. In this wor we emonstrate the spectral features arising from the CR-wave terms of the riving in three aspects: i) the asymmetry of the siebans with respect to the central pea, ii) the generation of the higher-orer Mollow triplets, an iii) frequency shifts of the siebans. We iscuss these characters an the ifference from other wors in the Secs. III A an III B. II. UNITARY TRANSFORMATION AND MASTER EQUATION The Hamiltonian escribing a TLS riven by a classical monochromatic fiel with the frequency in a vacuum electromagnetic fiel reas h = 1) H t) = 1 ω σ z + cos t)σ x + ω b b + 1 g b + b )σ x, 1) where σ x an σ z are Pauli matrices escribing the TLS. ω is the bare transition frequency between the two levels, is the riving strength, b b ) is the annihilation creation) operator of the th-moe electromagnetic fiel with frequency ω, an g is the coupling between the TLS an the th-moe electromagnetic fiel. In orer to tae into account the CR-wave terms of the riving, we perform a unitary transformation propose in Ref. []. The generator of the unitary transformation is given by St) = i ζ sin t)σ x, ) where we introuce a parameter ζ to be etermine later. The transformation H t) = e St) H t)e St) ie St) t e St) can be one to the en an yiels H t) = 1 [ ] ω cos ζ sin t) σ z + 1 [ ] ω sin ζ sin t) σ y + 1 ζ ) cos t)σ x + ω b b + 1 g b + b )σ x. 3) Using the ientity given in Ref. [], expizsin θ) = J n z)expinθ), 4) n= where J n z) is a Bessel function of the first in, we ivie the transforme Hamiltonian into two parts, H t) = H t) + H 1 t), 5) H t) = 1 ) ) ω J ζ σ z + ω J 1 ζ sin t)σ y + 1 ζ ) cos t)σ x + ω b b + 1 g b + b )σ x, 6) ) H 1 t) = ω J n+1 ζ sin [n + 1) t] σ y ω n=1 L ) + ω J n ζ cosn t)σ z. 7) ω n=1 L Up to now, the treatment is exact without any approximation. To procee, we first introuce an approximation where we rop H 1 t) because the higher-orer harmonic terms n,n ) are negligible accoring to Ref. []. Since our interest is the spectral features of the fluorescence arising from the CR-wave terms of the riving, we assume an RWA for the TLS-environment interaction as Mollow i [1]. This is the secon approximation we introuce in the following. To erive the master equation, we further assume wea TLS-environment coupling an the Born-Marovian approximation as usual [3]. These are all the approximations we use for eriving the master equation in this wor. If the parameter ζ is etermine as 1 ζ ) = ω J 1 ζ ), 8) we obtain our reformulate rotating-wave Hamiltonian, H t) = 1 ) ω J ζ σ z + eit σ + e iωlt σ + ) + ω b b + 1 g b σ + b σ + ), 9) where σ ± = 1 σ x ± iσ y ). We perform the unitary rotating transformation given by [ 1 Rt) = exp i t σ z + )] b b, 1) then obtain a time-inepenent Hamiltonian, H = Rt)H t)r t) + i Rt) R t) t = 1 δσ z + σ x ) + ω )b b + 1 g b σ + b σ + ) H S + H B + H 1, 11) where the effective etuning δ = ω J ζ ), H S is the resse TLS-riving Hamiltonian, is the environment H B 5381-

3 EFFECTS OF COUNTER-ROTATING-WAVE TERMS OF THE... PHYSICAL REVIEW A 88, ) term, an H 1 is the TLS-environment coupling. Provie the Born-Marovian approximation is use in the erivation of the master equation in the rotating basis, the master equation for the reuce ensity matrix of the TLS, ρ S t) = Tr B [ ρ SB t)], taes the same form as Mollow s: t ρ St) = i[ H S, ρ S t)] κ [σ +σ ρ S t) + ρ S t)σ + σ σ ρ S t)σ + ], 1) where the subscript S enotes the TLS an κ is the spontaneous ecay rate of the TLS. The erivation of the master equation is given in Appenix A. Accoring to Eq. 1), it is easy to obtain the Bloch equations, which tae the form σ +t) κ σ t) + i δ i = t κ σ z t) i δ i i i κ σ +t) σ t) +. 13) σ z t) κ Here, σμ t) = Tr[ ρ S t)σ μ ], μ =+,,z). The equations have the steay-state solutions σ + s = σ s = δ iκ) + 4 δ + κ, 14) σ z s = 4 δ κ + 4 δ + κ. 15) These results show that the solutions epen on the effective etuning an moifie riving strength. If we o not apply the unitary transformation but assume the RWA for the riving an follow the same proceure from Eq. 1) to 1), we obtain the same ifferential equation without moifie riving strength an etuning. In other wors, the parameters in the above equations are replace as follows:, δ δ = ω, 16) an they become the results of Mollow s master equation [1]. III. FLUORESCENCE SPECTRUM In general, the power spectrum of the scattere light is obtaine from the time-integrate Fourier transformation of the first-orer correlation function [4], Sω) = 1 π lim 1 T T t t g 1) t,t )e iωt t ), 17) T T where the correlation function g 1) t,t ) = U t)σ + Ut)U t )σ Ut ), with Ut) being the evolution operator for the original Hamiltonian. We analytically erive the fluorescence spectrum in Appenix B. The fluorescence spectrum consists of the so-calle coherent Rayleigh scattering) an incoherent inelastic scattering) components. The coherent one arising from the term σ μ τ )σ ν ) is the δ function, S c ω) = 1 I n, δω n ), 18) 4π n o with I n, = jn + j n+) σ+ s + j n j n+ σ s + σ + ) s + j n j n+1 + j n+1 j n+ ) σ z s σ s + σ + s ) + j n+1 σ z s, 19) where the summation is taen over all positive o integers an coefficients j n an j n have been efine in Appenix B. We enote the roots of the thir-egree polynomial B1), f p), by γ 1 an γ ± i. Then the incoherent part of the spectrum is written as S inc ω) = 1 4π Re [ γ 1 + in ω) I n,1 γ n o 1 + ω n) + I n, γ + in ω) γ + ω n + ) γ + in + ω) + I n,3 γ + ω n ) + I n,1 γ 1 in + ω) γ1 + ω + n) + I n, γ in + + ω) γ + ω + n + ) + I n,3 γ in ] + ω) γ + ω + nω, ) L ) with I n,l = jn R ) +,l + j nj n+ R ),l + j nj n+1 R ) z,l + j n j n+ R+) +,l + j n+ R+),l + j n+1 j n+ R+) z,l + j n j n+1 Rz) +,l + j n+1 j n+ Rz),l + j n+1 Rz) z,l, 1) I n,l = j n+ R ) +,l + j n j n+r ),l + j n+1j n+ R ) z,l + j n j n+r +) +,l +j n R+),l + j n j n+1r +) z,l + j n+1 j n+ R z) +,l + j n j n+1r z),l + j n+1 Rz) z,l. ) The quantities in Eqs. 1) an ) are efine an evaluate in Appenix B. Up to now, we have obtaine the moifie spectrum, which taes a more complex form than Mollow s. The expression inicates that there are higher-orer Mollow triplets, the central peas of which locate at o multiples of the riving frequency n. These higher-orer triplets fail to be capture with the RWA for the riving an are consistent with the preictions from the previous wors base on ifferent methos [14,5]. In Eq. ), the last three terms in the square bracets are the aitional terms arising from the CR-wave terms. A. Spectral features an effects of the CR-wave terms In the following, we illustrate precisely the spectral features inuce by the CR-wave terms. We set the bare transition frequency ω as the unit. At the same time, we only show the incoherent part of the fluorescence in the plots. To begin with, we examine whether the fluorescence spectrum on resonance obtaine from our metho coincies with Mollow s when the riving strength is moerately wea. We show the spectrum on resonance for riving strength

4 YIYING YAN, ZHIGUO LÜ, AND HANG ZHENG PHYSICAL REVIEW A 88, ) a b Mollow's Mollow's ω units of ω ω units of ω FIG. 1. Color online) The resonance fluorescence spectrum as a function of frequency ω for = 5κ an a) /ω =.1 an b) /ω =.1. The soli line represents our spectrum, an the ashe line shows Mollow s spectrum. = 5κ with /ω =.1,.1 in Figs. 1a) an 1b), respectively. It turns out that uner the moerately wea riving strength an resonance conition, the spectrum calculate by our metho is in goo agreement with Mollow s. However, when /ω =.1, our calculate spectrum is slightly ifferent from Mollow s because it has a higher blue sieban an a lower re sieban compare with Mollow s. These results inicate that the effects of CR-wave terms are negligible uner the wea riving strength an resonance conition. The effects of riving CR terms become clearer with increasing riving strength. Therefore, the RWA-base theory is aequate an succees in explaining classical fluorescence experiments. We now illustrate the influence of the CR-wave terms on the line shape of the fluorescence spectrum when increasing riving strength to the magnitue comparable to the bare transition frequency. In Figs. a) c), we plot the spectrum as a function of frequency ω for = 1κ an /ω =.6 with the three etuning values δ/ω =.,,., respectively. The asymmetry of the two siebans with respect to the central pea for /ω =.6 becomes noticeable. Comparing our results with Mollow s results in Figs. a) c), we notice the common character that the re sieban of our calculate spectrum is suppresse while the blue sieban is enhance. Furthermore, Fig. 3 shows that this effect can be strengthene by increasing the riving strength. Moreover, our theory preicts that a significant shift of the two siebans in frequency can be observe with proper etuning an riving strength. In Fig. 4, we show the secon Mollow triplet, whose a b Mollow's.1.5 Mollow's ω units of ω c ω units of ω.8.6 Mollow's ω units of ω FIG.. Color online) The fluorescence spectrum as a function of frequency ω for = 1κ, /ω =.6, an a) =.8ω,b) = ω, an c) = 1.ω. The soli line represents our spectrum, an the ashe line shows Mollow s spectrum

5 EFFECTS OF COUNTER-ROTATING-WAVE TERMS OF THE... PHYSICAL REVIEW A 88, ) p units of ω FIG. 3. Color online) The imensionless height κsω p )asa function of riving strength with ω p =,, corresponing to the central pea soli line), re sieban ashe line), an blue sieban ot-ashe line) in the first triplet. The riving frequency is = ω, an the ecay rate κ = 1 3 ω. All thic lines represent our preictions, an the thin ones are Mollow s preictions. central pea is at the frequency ω = 3 on resonance an /ω =.9. Even for such strong riving, the intensity of the secon Mollow triplet is much less than that of the first. This feature of the higher-orer triplet is istinguishe from that in previous wor [14]. Our metho emonstrates that the higher-orer triplet clearly comes from the counter-rotating term of the riving fiel which is epicte in Eq. ), an its intensity is etermine by a coefficient with higher-orer Bessel functions. Therefore, the experimental stuy of the higher-orer triplet is an interesting challenge. In Figs., 3, an 4, we show the spectral features arising from the CR-wave terms in three aspects: i) the asymmetry of the siebans with respect to the central pea, ii) the generation of the higher-orer Mollow triplets, an iii) shifts of the siebans. Although some characters have been reporte in previous wors [14,5]. we ientify that there are ramatic ifferences between our results an those in Ref. [14]. First of all, we clearly show that the CR-wave terms enhance the emission of the re sieban while suppressing Mollow' s ω units of ω FIG. 4. Color online) The fluorescence spectrum obtaine by our metho soli line) an Mollow s metho ashe line) on a logarithmic scale for = κ, /ω =.9, an = ω. The inset shows the spectrum obtaine in Ref. [14] with the same parameters. that of the blue one compare with Mollow s spectra. This feature eterministically comes from the CR-wave terms an is ifferent from the result claime in Ref. [14], i.e., the unequally intense Mollow siebans. Since the metho in Ref. [14] has taen into account the effects of the three ifferent ecay rates, which also cause the unequally intense Mollow siebans, we believe that the unequally intense Mollow siebans claime in Ref. [14] arise from the combine effects of both ifferent ecay rates an the CR-wave terms. Secon, our calculations clearly show the higher-orer Mollow triplets, which come from the effects of the CR-wave terms. Moreover, the etaile character is istinct from those of Ref. [14]. We fin that a higher-orer Mollow triplet with very wea intensities relative to the first one, while in Ref. [14] the secon Mollow triplet becomes comparable in intensity to the first triplet. We show numerical results with the same parameters as those in Fig. 6 of Ref. [14]. In Fig. 4, it is clearly seen that the magnitues of the three peas of the secon Mollow triplet are far less than the magnitue of the central line of the first one. The ratio of intensities between the emission lines of the secon Mollow triplet an central line of the first triplet is about 1/1 for /ω =.9, which shows the ramatic ifference between the results obtaine by the two methos see the inset in Fig. 4). In contrast, the results of Ref. [14] show that the height of the re sieban in the secon Mollow triplet is of the same magnitue as the central line in the first triplet for /ω =.9. However, the same sieban in our calculate spectrum almost isappears for this case. Finally, we show a comparison of the shifts of siebans relative to Mollow s spectrum preicte by our metho an Ref. [14]. In the wea-riving regime, both shifts are almost the same. However, as the riving strength increases from /ω =.7 to 1 [see Fig. 5b)], the shift obtaine in Ref. [14] sharply increases. This probably results from the seconorer energy corrections containing a ivergent term. As a comparison, the vali parameter space of our metho is clearly shown in Fig. 6 in our previous wor []. Further, the metho wors very well from the wea riving strength to intermeiate strong riving strength in the resonance an near-resonance regimes. Moreover, since the Bloch-Siegert shift obtaine by our metho is exact up to fourth orer, it strongly proves that our metho properly taes into account the effects of the CR-wave terms of the riving. Thus, we believe that the shift obtaine by our metho is reliable. B. CR-wave terms moulate frequency shift In the following, we focus on the frequency shift of the siebans cause by CR-wave terms of the riving which may be etectable in experiment. In a way similar to Mollow s theory, the thir-egree polynomial equation [Eq. B1) in Appenix B], which taes the same form obtaine by Mollow but with the corresponing renormalize coefficients, completely etermines the FWHM of the fluorescence spectrum, i.e., γ 1 an γ, an the splitting between the central pea an its satellite siebans, i.e.,. In general, the quantity etermine in Eq. B1) is ifferent from Mollow s ue to the effects of the CR-wave terms. As a result, the sieban s shift in frequency occurs. In orer to etermine the influence

6 YIYING YAN, ZHIGUO LÜ, AND HANG ZHENG PHYSICAL REVIEW A 88, ).6.ω a.8 b units of ω.4..ω.ω units of ω.6.4 Ref units of ω units of ω FIG. 5. Color online) a) The shift of siebans as a function of riving strength for κ = 1 3 ω an various riving frequencies: =.8ω soli line), = ω ashe line), an = 1.ω ot-ashe line). b) The comparison between the shift obtaine in Ref. [14] an ours for the exact resonance an κ = 1 3 ω. of the CR-wave terms on this frequency shift, we can efine the ifference = M as a significant frequency shift, where M enotes the splitting between the central pea an its siebans in Mollow s theory. It is nown that in the intense-riving limit κ), the splitting M is well approximate by the Rabi frequency R = δ + in Mollow s theory. Thus, the moulate frequency shift ue to CR-wave terms can be approximate to the ifference R R since the effective Rabi frequency R = δ + in our metho has taen into account the effects of the CR-wave terms. It is worth noting that the shift obtaine by our metho is beyon the secon-orer corrections []. As a comparison, the shift obtaine in Ref. [14] results from the secon-orer energy corrections. Figure 5 shows how the riving strength affects the shift. < means that the siebans of our preiction shift outwars to the center, > means that the siebans of our preiction shift inwars to the center, an = means that the frequencies of the siebans preicte by the two methos are equal. Since the shift is slight on resonance for wea riving, the RWA is a sufficiently goo approximation uner such conitions. However, a significant amount of shift emerges when the etuning δ =±.ω even for a small units of ω δ units of ω.1ω.1ω.9ω FIG. 6. Color online) The shift of siebans as a function of etuning δ for = κ an various ratios: /ω =.1 soli line), /ω =.1 ashe line), an /ω =.9 ot-ashe line). value of. This inicates that it is necessary to tae into account the effects of the CR-wave terms in the off-resonance case where the RWA breas own even for wea riving strength. When δ =.ω, the absolute value of the effective etuning δ changes nonmonotonously as increases. As a result, the shift increases slowly at first an then ecreases with increasing. Figure 5b) shows the ifference between the shift obtaine in Ref. [14] an our result on resonance. In the wea-riving regime, the shifts are almost the same. They rise with increasing. However, when the riving strength increases from /ω =.7 to 1, the shift obtaine in Ref. [14] sharply increases, which probably results from the ivergent term in the secon-orer energy correction. Figure 6 shows the epenence of the shift on the etuning δ for various riving strengths. For ω,theshiftistoo slight to be observe in the line shape of the spectrum. When is comparable to ω, we notice that a negative etune riving, i.e., >ω, is favorable for obtaining a larger shift with a given riving strength. In the intensive-riving regime, one has = R R.Asδ/ω increases from zero to. in the case /ω =.9, one can verify that the ifference between R an R increases monotonically. Thus, the shift increases to a maximum value of.6ω at δ =.ω. This helps us etect the CR-moifie frequency shift in experiment. IV. CONCLUSIONS In summary, we have evelope an analytical approach base on a unitary transformation to investigate the spectral features of the scattere light from a two-level system riven by a monochromatic classical fiel. The metho has taen into account the effects of the CR-wave terms of the riving but still retains the simple RWA form of the riving with the CR-moifie parameters, which is istinct from the traitional perturbative metho. We have erive an analytical expression for the fluorescence spectrum. Our calculate results clearly show the spectral features arising from the CR-wave terms in three aspects: i) the asymmetry of the siebans with respect to the central pea, ii) the generation of the higher-orer Mollow triplets, an iii) shifts of the siebans. First, since the re sieban is suppresse while the blue sieban is enhance

7 EFFECTS OF COUNTER-ROTATING-WAVE TERMS OF THE... PHYSICAL REVIEW A 88, ) in comparison with Mollow s triplet, our calculation shows that the CR-wave term of the riving inuces the asymmetry of the siebans. Secon, our theory also preicts the generation of the higher-orer Mollow triplets, which is qualitatively consistent with the results of other methos in previous wor [14,5]. However, we have foun that the intensities of the higher-orer Mollow triplets are far less than those of the first triplet. This feature is istinct from that of the previous wor. Thir, we reaily observe that the siebans in the first triplet shift from the positions of the corresponing siebans given by Mollow s theory in intense-riving regimes. Moreover, the shift epens on both the riving strength an the etuning. A negative etuning of the riving is favorable for generating a larger shift for a fixe, moerately intense riving strength. We expect that all the effects of the CR-wave terms woul be experimentally observable with elaborately esigne etuning an riving strength. Our results illustrate that the CR-wave terms of the riving have significant effects on the spectral features of fluorescence. ACKNOWLEDGMENTS This wor was supporte by the National Natural Science Founation of China Grants No , No , an No. 9111) an the National Basic Research Program of China Grant No. 11CB9). APPENDIX A: MASTER EQUATION UNDER BORN-MARKOVIAN APPROXIMATION In the interaction picture of the resse Hamiltonian, the ensity matrix satisfies the equation of motion, t ρi SB t) = i[ H I t), ρ I SB t)], A1) where ρ SB I t) = ei H S + H B )t ρ SB t)e i H S + H B )t an H I t) = e i H S + H B )t H 1 e i H S + H B )t = e i H S H 1 t)e i H S. Equation A1) can be integrate formally an yiels t ρ SB I t) = ρi SB ) i τ [ H I τ), ρ SB I τ)]. A) A3) Substituting expression A3) bac into Eq. A1) an taing the trace over the egree of freeom of the environment, we obtain the following equation: t ρi S t) = Tr B t τ [ H I t), [ H I τ), ρ I S τ)ρ B]], A4) where we have use the Born approximation ρ SB I τ) ρ S I τ)ρ B since in the wea-coupling regime the influence of the system on the environment is negligible. We further introuce Marovian approximation, which assumes ρ S I τ) ρi S t) in Eq. A4), an transform the equation of motion bac into Shröinger picture, t ρ St) = i[ H S, ρ S t)] t Tr B [ H 1 t),[e i H S t H 1 t t )e i H S t, ρ S t)ρ B ]]. A5) Here, we have replace the variable τ by t t an let the upper limit of the integral go to infinity, which is reliable since the integran vanishes sufficiently quicly as time increases [3]. At zero temperature, the master equation in Eq. A5) can be rewritten as t ρ St) = i[ H S, ρ S t)] g { e iω )t σ + e i H S t σ e i H S t ρ S t) e iω )t e i H S t σ e i H S t ρ S t)σ + 4 t e iω )t σ ρ S t)e i H S t σ + e i H S t + e iω )t ρ S t)e i H S t σ + e i H S } t σ A6) since Trb b ρ B ) =. It is not ifficult to erive following equations: e i H S t σ ± e i H S t = [ + 1 R R 1 + δ ) e i R t δ ) ] e R 4 ±i R t σ ± R 1 δ ) ] e ±i R t σ z + R [ δ 1 + δ ) e R i R t + R Substituting Eqs. A7) into Eq. A6), integrating with respect to t, an assuming 4 e i R t e i R t )σ. A7) R τ g 4 e±iω ± R )τ τ g 4 e±iω )τ κ, A8) namely, assuming that the ecay rate of a riven two-level system remains constant, we finally obtain the form of the master equation presente in Eq. 1)

8 YIYING YAN, ZHIGUO LÜ, AND HANG ZHENG PHYSICAL REVIEW A 88, ) APPENDIX B: THE DERIVATION OF THE FLUORESCENCE SPECTRUM For the fluorescence spectrum, it is sufficient to evaluate the integral 17) in the steay-state limit. In this limit, ue to the time-epenent unitary transformations ) an 1), we have the correlation function g 1) t,t ) = Tr[R t) e St) σ + e St) R t) Ũ τ) Rt )e St ) σ e St ) R t ) ρ ss ρ B Ũ τ)], B1) where ρ ss is the steay-state solution for master equation 1), Ũ τ) = exp ihτ), an τ = t t. The transform R t) e St) σ ± e St) R t) in the correlation function can be expresse in terms of σ ± an σ z with time-epenent coefficients, Rt)e St) σ ± e St) R t) = 1 + cos [ ζ sin t) ] σ ± e ±iωlt + 1 cos [ ζ sin t) ] σ e iωlt i sin [ ζ sin t) ] σ z = 1 [j n e ±inωlt + j n+ e inωlt )σ ± + j n e int + j n+ e±int )σ + j n+1 e inωlt + j n+1 e±int )σ z ], n o B) where the summation is taen over all positive o integers an the series are efine as j n = 1 + J ζ J n 1 ζ ), n = 1, ), n 1, B3) an j n = 1 J ζ J n 1 ζ ), n = 1, ), n 1. B4) When calculating the spectrum, accoring to Ref. [4], we approximate the integral by 1 T Iω) = lim T T 1 = lim T T 1 = lim T T 1 = lim T T T T T t t T T t e int im t σ μ τ)σ ν ) s e iωt t ) t e inτ im n) t σ μ τ)σ ν ) s e iωτ T t t e im n)t τ σ μ τ)σ ν ) s e in ω)τ t + t e im n)t τ σ μ τ)σ ν ) s e in ω)τ + = δ m,n τ σ μ τ)σ ν ) s e inωl ω)τ, B5) where σ μ τ)σ ν ) s = Tr[σ μ τ)σ ν ) ρ ss ρ B ], μ,ν =+,,z). Using this approximation, one can easily show that the spectrum is given by Sω) = 1 4π Re n o τe iωτ[ j n einτ + j n+ e inτ ) σ + τ)σ ) s + j n j n+e inτ + j n j n+ einτ ) σ τ)σ ) s + j n+1 j n+ e inτ + j n j n+1 einτ ) σ z τ)σ ) s + j n j n+ einτ + j n j n+e inτ ) σ + τ)σ + ) s + j n e inτ + j n+ einτ ) σ τ)σ + ) s + j n j n+1e inτ + j n+1 j n+ einτ ) σ z τ)σ + ) s + j n j n+1 einτ + j n+1 j n+ e inτ ) σ + τ)σ z ) s + j n j n+1e inτ + j n+1 j n+ einτ ) σ τ)σ z ) s + j n+1 e inτ + j n+1 einτ ) σ z τ)σ z ) s ], B6) where Rez) gives the real part of complex number z. To obtain the explicit expression for the spectrum, it is necessary to evaluate the correlation function σ μ τ)σ ν ) s, which is evaluate base on quantum regression theorem [6]. Notice that the ynamics of the quantity σ μ τ) = σ μ τ) σ μ s is etermine by the homogeneous part of Bloch equations 13). Using quantum regression theorem, we obtain a close set of

9 EFFECTS OF COUNTER-ROTATING-WAVE TERMS OF THE... PHYSICAL REVIEW A 88, ) ifferential equations for the quantity σ μ τ)σ ν ) = σ μ τ)σ ν ) s σ μ s σ ν s, which is τ σ μτ)σ ν ) = M μλ σ λ τ)σ ν ), λ B7) κ + i δ i M = κ i δ i. i i κ One can obtain the following Laplace transforms without ifficulty: where an g +ν p) = τe pτ σ + τ)σ ν ) = x ν + y ν ) i z ν p + i δ + κ ) + xν p + κ)p + i δ + κ), B9) f p) g ν p) = τe pτ σ τ)σ ν ) = ) x ν + y ν ) + i z ν p i δ + κ + yν p + κ)p i δ + κ), B1) f p) g zν p) = τe pτ σ z τ)σ ν ) = x ν + y ν ) δ + z ν [ δ + ) p + κ ] i ) p + κ xν y ν ), f p) B11) f p) = p 3 + κp + + δ + 5 ) ) 4 κ p + κ + δ + κ 4 B1) x ν = σ + )σ ν ), B8) B13) y ν = σ )σ ν ), z ν = σ z )σ ν ) B14) B15) are the initial conitions. These results lea to the solutions σ μ τ)σ ν ) = 3 l=1 R ν) μ,l es lτ. Here, R ν) μ,l = lim p s l p s l )g μv p), an s l enotes the three roots for f p) =. Consequently, the fluorescence spectrum can be completely etermine. B16) [1]B.R.Mollow,Phys. Rev. 188, ). [] R. E. Grove, F. Y. Wu, ans. Ezeiel, Phys. Rev. A15, ). [3]X.Xu,B.Sun,P.R.Berman,D.G.Steel,A.S.Bracer, D. Gammon, an L. J. Sham, Science 317, 99 7). [4] A. Muller, E. B. Flagg, P. Bianucci, X. Y. Wang, D. G. Deppe, W. Ma, J. Zhang, G. J. Salamo, M. Xiao, an C. K. Shih, Phys. Rev. Lett. 99, ). [5] S. Ates, S. M. Ulrich, S. Reitzenstein, A. Löffler, A. Forchel, an P. Michler, Phys.Rev.Lett.13, ). [6] E. B. Flagg, A. Muller, J. W. Robertson, S. Founta, D. G. Deppe, M. Xiao, W. Ma, G. J. Salamo, an C. K. Shih, Nat. Phys. 5, 3 9). [7] A. N. Vamivaas, Y. Zhao, C. Y. Lu, an M. Atatüre, Nat. Phys. 5, 198 9). [8] S. M. Ulrich, S. Ates, S. Reitzenstein, A. Löffler, A. Forchel, an P. Michler, Phys.Rev.Lett.16, ). [9] A. Ulhaq, S. Weiler, S. M. Ulrich, R. Roßach, M. Jetter, an P. Michler, Nat. Photonics 6, 38 1). [1] G. Wrigge, I. Gerhart, J. Hwang, G. Zumofen, an V. Sanoghar, Nat. Phys. 4, 6 8). [11] O. Astafiev, A. M. Zagosin, A. A. Abumaliov, Yu. A. Pashin, T. Yamamoto, K. Inomata, Y. Naamura, an J. S. Tsai, Science 37, 84 1). [1] A. Moelbjerg, P. Kaer, M. Lore, an J. Mor, Phys.Rev.Lett. 18, )

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