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Abstract
Narrow linewidth and high intensity of the fluorescence spectra are two important elements to improve the accuracy and efficiency of related practical measurements, but so far they have not been achievable simultaneously. We propose a new approach to sharpen the linewidth and to heighten the intensity simultaneously. Rather than double coherent mechanisms, it uses the spontaneous emission of inverted atoms in a cavity below the threshold as a resource for feedback control. The spontaneous feedback is simpler in principle and easier to realize technologically, and represents a kind of new way of controlling spontaneous emission.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Control and use of spontaneous emission are one of the most important subjects in quantum optics and nonlinear optics [1–7] and have a great deal of timely renewed interest (e.g., see [8–42]). Among others, linewidth narrowing and intensity elevation of fluorescence spectra turn out to be related important aspects to enhance the accuracy and efficiency in spectra-based high-precision measurements. Intuitively, slowing down the spontaneous decay of an atom gives the spectral line narrowing, and the total fluorescence power squashes in the narrow line. So far, existing schemes generally make the spectral lines narrow at a cost of decreasing the power [29–31]. Those schemes are mainly based on doubly coherent mechanisms, one of which splits the spectrum into Mollow triplet [8,9], and the other sharpens the spectral line but also suppresses the intensity. The fall in the intensity is due to the extra spectrum splitting [29] or the coherent population trapping [30,31].
In this article we propose a fundamentally different approach, with no use of coherence. It uses spontaneous emission as a resource for its own control. This spontaneous feedback is simpler but more efficient for the double purpose of sharpening the linewidth and boosting the intensity. Its advantages are listed as follows.
First, the spectral line becomes narrow and high continuously as the atom-cavity system approaches the threshold but remains below threshold. The ultimate limits are due to quantum fluctuations. The spontaneous photons from an ensemble of inverted atoms emit into a cavity mode, which remains amplified below lasing threshold and is coupled back to the atoms. The amplification below threshold makes extremely narrow the otherwise wide spectrum. Neither of incoherent pumping for the inversion and the cavity mode below lasing threshold causes the level splitting. The coupling back to the atoms acts as spontaneous feedback mechanism, which erects an ultrasharp peak over the natural spectrum. At least a half of the total power squashes into the sharp spectral line and thus boosts the spectral intensity.
Second, the spontaneous feedback holds also for the sidebands of resonant fluorescence. When the population inversion is established between dressed states and a cavity is tuned to resonant with the inverted dressed transition, the spontaneous emission from the inverted sideband can be used as a resource for feedback control. Similarly, the cavity mode remains amplified below threshold. The coupling of the cavity mode back to the atoms simultaneously sharpens and raises the fluorescence spectrum at the sidebands.
Thirdly, spontaneous emission as a resource for its own control is much easier to realize in principle and technologically than the doubly coherent schemes. Since the cavity mode is only amplified below lasing threshold, a bad cavity is efficient for the feedback control. In the absence of the coherent population trapping, one no longer needs the nearly parallel dipole moments, which are not existent in nature. Even for the substitutions by the dressed states, the quantum interference will be washed out by the decoherence of the dressed transitions [43,44]. Any added incoherent pump in the doubly coherent mechanisms increases the linewidth [31]. In addition to the cavity, there is no need to use specific reservoirs like the photonic band gap materials [24–28].
Although Haken [45] gave a brief calculation of the cavity linewidth below the threshold in the early days of laser theory, the linewidth holds only in a limited range. First, close to threshold unlimitedly, the linewidth given approaches zero unboundedly. Obviously, this is unphysical. The physical limitation will be given below. Second, far below threshold, the spectrum of the atom-cavity coupling system should be the total contributions of two Lorentz terms and their interference term, as shown in Eq. (7), rather than one single Lorentz spectrum. Only when the narrowing term dominates over other terms, is the linewidth given in [45] appropriate. So far, these two aspects have not yet been included in the existing references [1–7, 45–50]. For the above reason, a more complete and elaborate description is needed in the whole region below threshold. With this, it is convenient for us to apply this linewidth to related schemes, such as the resonance fluorescence of the atoms or molecules in a cavity. Related to this work, we note that great progresses have been made on collective emission and cavity QED [34–42]. For example, using subradiance and selective radiance in atomic arrays makes exponential improvement in photon storage fidelities [34]; using cavity-assisted Raman transitions manipulate spin dynamics and establishes squeezing in a spinor gas [35]; exploring single-photon time-dependent spectra in coupled cavity arrays [36]; detailed calculation and description of Fano-Agarwal couplings and non-rotating wave approximation in single-photon timed Dicke subradiance [37].
The remaining part of this article is organized as follows. In Sec. 2, we first give an analytical expression of the spontaneous emission spectrum of bare atom, and then discuss the limitation of the linewidth and the height of the spectral line. In Sec. 3, we show the applications of the spontaneous feedback mechanism in resonance fluorescence. The three subsections present the narrowing and heightening of the spectral lines of two-level dressed atom, three-level dressed atom and molecule system, respectively. Finally, conclusion is given in Sec. 4.
2. Spontaneous feedback by cavity for a bare atom
The analytical expression of the spontaneous emission is given in Eq. (7). The limitation of the linewidth due to the fluctuation of the population is also given explicitly in Eq. (9). Plotted in Fig. 1(a) is the spontaneous emission spectrum of inverted two-level atoms in a cavity below threshold. The spectrum displays an ultrasharp and ultrahigh peak regardless of large natural linewidths of the atoms and the cavity. The higher the spectral peak the narrower the linewidth. The ultrasharp linewidth is determined by the smallest one of decay rates of the atom-cavity coupled system. As the system approaches its threshold the smallest rate falls close to zero, as shown in Fig. 1(b). The linewidth and height are ultimately limited by the quantum fluctuations. It is clearly shown that the spontaneous feedback is a much easier and technologically less challenging way for obtaining narrow and high spectral lines than previous doubly coherent scheme [29–31]. This mechanism underlying the above effects is presented as follows.
Fig. 1 (a) Spontaneous spectrum S(ω) of inverted two-level atoms in a cavity for κ = γ, Λ = 1.2γ, γp = 0.5γ, and C = 2.7225. Note that the vertical axis is in logarithmic coordinate. (b) The smallest decay rate λ1 versus the cooperative parameter C for κ = 0.2γ (solid), γ (dashed), and 5γ (dotted).
2.1. Narrow and high spontaneous emission spectrum
Here we consider an ensemble of independent atoms as in [1–6, 49–54]. This should be distinguished from the Dicke model [55], in which identical atoms form a single quantum system, as studied by Tavis and Cummings [56, 57]. The Hamiltonian for the resonant interaction of N independent atoms with the cavity field is written in the dipole approximation and in an interaction picture as
where ħ is the Planck constant and g is the atom-field coupling strength. a and a† are the annihilation and creation operators for the cavity field, respectively. (; l, k = 1, 2) are the collective spin-flip (k ≠ l) and projection (k = l) operators. The independent atoms have independent phases for (). The master equation for the density operator ρ of the atom-field system is derived as [3] , with all relaxations included in , where we have used the common form for o = a, , , . κ and γ are the cavity and atomic decay rates, respectively, γp is the additional atomic phase damping rate, and Λ is the incoherent pumping rate from the ground state to an additional excited state from which the atoms decay rapidly to the excited state. It is the very way of creating population inversion for lasing action [1–5].Following the standard technique [2–5] we can study the quantum correlation spectra from the master equation. By arranging operators in the normal order (a†, σ21, σ22, σ12, a) and by defining the corresponding c-numbers (α*, v*, z2, v, α), we derive the set of Heisenberg-Langevin equations
together with those for (α*, v*) and the closure relation z1 + z2 = N (z1 is the c-number correspondence of σ11). The two g terms in Eqs. (2) and (3) describe clearly the feedback by the cavity to the collective atoms. Here we have defined Γ = γ + Λ + 2γp. F’s are the noise terms and have vanishing means and noise correlations 〈Fo(t)Fo′ (t′)〉 = Doo′ δ(t − t′), where the nonzero diffusion coefficients read as Dv*v = Dvv* = Γ〈z2〉 and Dz2z2 = 2γ〈z2〉.Here we focus on the case below threshold, as shown later. As in the usual case for independent [1–6,49–54], individual atoms have weak couplings with the field g ≪ γ, and thus the atom-field correlations have a negligible effect on the steady state values. For the entire ensemble, it has a total coupling constant with the cavity field. While the coupling constant for a single atom is small, the cooperativity parameter can be large if N is large. It is for this reason, many textbooks in quantum optics and laser physics [1–6, 49] separate the collective atomic operators from the field operators and then solve for the steady state. We follow the standard techniques. Our case is completely different from the one-atom case, where the individual atom has strong coupling with the cavity [29,58–60]. We can obtain the steady state solutions and the time evolutions by neglecting the noises temporarily. The steady state solutions are 〈α〉 = 〈v〉 = 0 and 〈z1,2〉 = NP1,2, where and are the mean populations of each atom. It should be noted that the population fluctuations δzl = zl − 〈zl〉 (l = 1, 2) will not be coupled to the quantities α and v because of 〈α〉 = 〈v〉 = 0. Then we can substitute 〈z1,2〉 for zl in Eq. (3). It is easy to find from Eqs. (2,3) that α and v evolve with time according to e−λ1,2t, where λ1,2 are the new decay rates of the coupled system
with the population difference P = P2 − P1. The stability requires that Reλ1,2 > 0. We see from Eq. (5) that λ1 > 0 is satisfied for 4CP < 1 and λ2 is always positive. However, once λ1 becomes negative, α and v become amplified. It is the case for 4CP > 1. Therefore there exists a threshold λ1 = 0, i.e., 4CP = 1. Our focus is on the case below and near the threshold, λ1 → 0+, i.e., 4CP → 1−, which requires population inversion P2 > P1. Then linearizing the atomic populations around the steady state, we obtain where the decay rate λ3 = γ + Λ is always positive and so the δz2 decay e−λ3t is stable (λ3 > 2γ for Λ > γ).The incoherent fluorescent field from the N independent atoms is determined by the fluctuating collective spin-flip operator . The fluorescence spectrum [1–5, 51–53] is derived in the explicit form
where ω̄ = ω − ω21, ηl > 0, and , (l = 1, 2). It should be noted that the calculation given in [45] excludes two latter terms, which can only describe the spectrum appropriately in a limited region. The first two Lorentz lineshapes in Eq. (7) describe the spontaneous emission at different rates λ1,2, while the third term originates from the constructive interference of the two emission processes. However, this interference is weak due to the opposite disparity of λ1,2 and η1,2 near threshold: λ1 ≪ λ2 and η1 ≪ η2. As a comparison, we recover the standard spectrum [1–5] from Eq. (7) for the case without the cavity coupling, where κ → ∞ and 4CP → 0 lead to λ1 = Γ/2 and λ2 = κ/2 → ∞. Shown in Fig. 1(a) is the spectrum S(ω) for the chosen parameters in the figure caption. These parameters give 4CP = 0.99 and describe the operation below and near threshold. The spectral linewidth is 7.6 × 10−3γ, which is about two thousandth of the natural width Γ = 3.2γ. The peak height is S(ω21) = 6.7 × 103, which is about 1.0 × 104 times as much as the natural peak height 4P2/Γ ∼ 0.625 (for P2 ≳ 0.5). Shown in Fig. 1(b) is the smallest decay rate λ1. As the system is very close to threshold, we have , , , and . The spectral linewidth is about d ≐ 2λ1 and the peak height is . Thus the spectrum is extremely narrow and high even though the involved natural linewidths κ and Γ are large. The spectrum integration gives the total intensity , which is extremely enhanced. Clearly it is attributed to the amplification of the cavity field below threshold and the feedback to the atoms.2.2. Linewidth limitation by quantum fluctuation
Although the expression of the spectrum in Eq. (7) is different from that in early studies [45–50], one common problem still remains. The intensity of the spontaneous emission goes to infinity as the linewidth approaches zero unboundedly (). This is obviously unphysical. However, it should be emphasized that the above narrowing and heightening of the spectrum are not unbounded, but ultimately limited by quantum fluctuations [3–5]. It is the population fluctuations that set a nonzero limit λ1min even for λ1 → 0+. From Eq. (6) we calculate the z2 variance , and then derive the ultimate limit
This limits the spectral width and height to respectively, which are proportional to and , respectively. As an example, for Λ ≳ γ, , κ ∼ Γ, C ∼ 1, and N ∼ 108, we have the ultimate limits dmin/Γ ∼ 10−4 and hmax/S0(ω21) ∼ 104.3. Spontaneous feedback in resonance fluorescence
3.1. Dressed two-level atom in a single-mode cavity
We first show that the narrowing and elevation mechanism holds for resonant fluorescence of two-level atoms. In this case, the ensemble of N-atoms is driven by a classical field with frequency ω0. The total Hamiltonian of the coupled system in the rotating frame of the driving field reads as H = H0 + HI, where H0 = ħΔσ22 + ħΩ(σ12 + σ21) describes the interaction between the atoms and one strong classical field, and HI = ħΔca†a + ħg(aσ21 + σ21a†) denotes the interaction of cavity field with the atoms. Of the new parameters, Ω is half Rabi frequency and Δ = ω21 − ω0 and Δc = ωc − ω0 are respectively the detunings of the atomic and cavity frequencies ω21 and ωc from the dressing field frequency ω0. It is most convenient to transform to the dressed state picture to show the essential mechanism. We assume the generalized Rabi frequency to be large, . After diagonalizing H0, we obtain the dressed states as [9]
where we have assumed tan(2θ) = 2Ω/Δ, 0 < θ < π/2. The dressed states have well separated eigenvalues ħ(Δ ∓ Ω̃)/2 since Ω̃ ≫ (γ, κ). The fluorescence radiation occurs at ω = ω0, ω0 ± Ω̃, respectively. In the secular approximation, we can rewrite the atomic damping terms into the same form as for the bare atoms, except for substitutions of (γ̃ = γ cos4 θ, Λ̃ = γ sin4 θ, ) for (γ, Λ, γp), respectively. With such substitutions, the dressed state populations P̃1,2, the decay rate Γ̃, the eigenvalues − λ̃1−3, the parameters η̃1,2, and the cooperativity parameter C̃ contained in λ̃1,2 have the same meanings and forms as above. The explicit expressions for P̃1,2 are and P̃2 = 1 − P̃1. For Δ < 0 we have P̃2 > P̃1. We tune the cavity field Δc = Ω̃ such that the atoms spontaneously emit photons into the cavity mode below threshold. Similarly, for Δ > 0 we tune the cavity field Δc = − Ω̃. Here we exemplify the former case. Performing the rotating transformation associated with the cavity detuning, we can describe explicitly the resonant interaction of the dressed atoms with the cavity field.The interaction Hamiltonian is derived as
where () represents the collective spin-flip and projection operators for the dressed atoms, g̃ = g cos2 θ is the effective interaction strength. It is seen that the interaction Hamiltonian is of the same form as Hamiltonian (1) except for the dependence of g̃, σ̃12 and σ̃21 on the normalized detuning Δ/Ω. It is such a parameter dependence that manipulates the dressed state populations and thus makes the fluorescence spectra controllable. Using the same techniques as above for the dressed transitions, we can derive the total fluorescence spectrum as where ω̄ = ω − ω0. S1,2 originate respectively from the dressed transitions at ω0 ∓ Ω̃ and have the same forms as (7) except for the substitutions of the tilde parameters and the respective central frequencies. stems from dressed transitions at ω0. The linewidths of the sidebands at ω0 ∓ Ω̃ are 2λ̃1 when λ̃1 → 0+. The height ratio of the sidebands is which is in agreement with the theoretical prediction [61,62] and experimental observation [63] although no sharp lines are involved there. For the numerical calculation we rewrite the cooperation parameter as , in which is independent of the normalized detuning Δ/Ω. Figure 2 shows the extremely narrow and high sidebands of the fluorescence spectrum below and near threshold. The nearest three-peaked spectrum locates at Δ/Ω = −2.68531, and its sidebands have their common width 2λ̃1 ≐ 1.4 × 10−2γ and their respective heights 5.6 × 103 and 33.Fig. 2 Fluorescence spectrum S(ω) of dressed two-level atoms in a cavity for κ = γ and C0 = 30. We take Ω̃ = 5γ to fix the sidebands for clear show.
3.2. Dressed three-level atom in a two-mode cavity
It is straightforward to generalize to the fluorescence spectra of three-level atoms. The atom-field interactions are shown in the insets of Figs. 3(a) and 3(b). The V-type atoms are excited from one common ground state |3〉 to two excited states |1, 2〉, from which they decay back to |3〉. The Λ-type atoms are excited from two ground states |1, 2〉 to one common excited state |3〉, from which they decay back to |1, 2〉. Exemplifying the V-type atoms we perform the numerical verification. The system Hamiltonian reads as
with where a1,2 and are the annihilation and creation operators for the cavity fields, respectively. The two Rabi frequencies are assumed to be equal, Ω1,2 = Ω. We have defined detunings Δl = ω3l −ωl and Δcl = ωcl − ωl (l = 1, 2). We assume that Δ1 = −Δ2 = Δ and . Diagonalizing H0 gives the dressed states [64] which have equally spaced eigenvalues 0, ±Ω̃, respectively. Thus, the fluorescence spectra have the five peaks, which locate symmetrically at ω = ω1, ω1 ± Ω̃, ω1 ± 2Ω̃, respectively. The cavity fields are tuned resonant with the dressed transitions at inner sidebands, respectively, i.e., Δc1 = −Δc2 = Ω̃. The spontaneous photons from the inverted dressed transitions enter the cavity modes below threshold for . The Λ-type atoms have similar behaviour for . Figure 3 displays the sharpening and boosting of the inner sidebands of fluorescence spectra S(ω) for the |1〉 ↔ |3〉 transitions of the two systems. We have assumed equal parameters g1,2 = g, κ1,2 = κ, and γ1,2 = γ, take κ = γ and C0 = 30, and fix Ω̃ = 5γ. In Fig. 3(a), for the nearest five-peaked spectrum at Δ/Ω = −1.1355, its inner sidebands have their common width 4.6 × 10−3γ and their respective heights 7.8 × 103 and 4.1 × 102. In Fig. 3(b), for the nearest five-peaked spectrum at Δ/Ω = 1.1915, its inner sidebands have their common width 4.0 × 10−3γ and their respective heights 9.2 × 103 and 4.6 × 102. Although we show in Fig. 3 the spectra from the |1〉 ↔ |3〉 transitions of the three-level atoms, the same features appear for the spectra from |2〉 ↔ |3〉 transitions (but not shown here) because of the symmetry of the parameters.The cascade configuration of three-level atoms is also suitable for the present mechanism. In this system, more often than not, we can have one inverted transition between dressed states. So long as conditions are satisfied, the narrowing and elevation of the spontaneous lines are predictable. However, the system has no longer the symmetry as for the V-type or Λ-type cases. Thus, in more cases, we only have narrow lines for one transition channel. In fact, so is the case if we break the symmetry between the two transitions in V-type or Λ-type atoms by not choosing symmetric Rabi frequencies or symmetric detunings.
Fig. 3 Fluorescence spectra from the |1〉 ↔ |3〉 transitions of dressed three-level atoms in (a) V and (b) Λ configurations.
3.3. Molecule system
Finally, it is possible to apply the above spontaneous feedback to narrow and heighten fluorescence spectra of molecules. Typically, within electronic states there are a number of vibrational levels, which are successively labeled by v = 0, 1, 2, · · · in ascending order. Thus molecular electronic spectra comprise a number of vibrational sub-bands. Non-radiative vibration relaxation, which takes place within the excited electronic state (v′ = 0, 1, 2, · · ·), is fast compared to the rate of emission. It is the case for large molecules in solution such as the dyes, often used in fluorescence spectroscopy [65]. The so-called relaxed fluorescence originates from the v′ = 0 excited vibrational level. Meanwhile, those levels above the v″ = 0 level within the vibrational ground state tend not to be occupied at room temperature for not too low vibrational frequencies. The incoherent pump starts from v″ = 0. It is advantageous for creating the population inversion for the electronic transitions v′ = 0 ↔ v″ > 0. When the cavity bandwidth κ is much less than the adjacent vibrational frequencies within the ground electronic state, the cavity will selectively couple one fluorescence transition v′ = 0 → v″ > 0 and thus leads to fluorescence spectrum narrowing and heightening.
4. Conclusion
In conclusion, we have proposed the spontaneous feedback mechanism for narrowing and heightening the spontaneous spectra. Spontaneous emission itself is used as the control resource. Ultimately, both the linewidth and the peak height are limited by the quantum fluctuations. The mechanism is valid for the single spectral line of the bare atoms, and also for the Mollow-like sidebands of the dressed atoms and the relaxed fluorescence spectra of molecules. In addition to the cavity, there is no aid of any extra added coherent resource. The spontaneous feedback control has loose conditions for experimental realization and practical applications. As our proposal, the dressed two-level atoms have more advantages for the experimental realization compared with the bare two-level atoms and the three-level atoms. For the former, varying the driving field amplitude and/or frequency is a controllable way to create population inversion on one dressed transition and select the cavity frequency [66,67]. The three-level schemes are relatively complicated because two driving fields and two cavity fields are employed.
Funding
National Natural Science Foundation of China (Grants No. 11474118 and No. 61178021); National Basic Research Program of China (Grant No. 2012CB921604).
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