theory of fano effect in cavity quantum electrodynamics
TRANSCRIPT
PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
Theory of Fano effect in cavity quantum electrodynamics
Makoto Yamaguchi ,1,* Alexey Lyasota ,2,3 and Tatsuro Yuge 4
1Department of Physics, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan2Laboratory of Physics of Nanostructures, Institute of Physics, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
3Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney,New South Wales 2052, Australia
4Department of Physics, Shizuoka University, Shizuoka 422-8529, Japan
(Received 7 November 2020; accepted 25 December 2020; published 13 January 2021)
We propose a Markovian quantum master equation that can describe the Fano effect directly, by assuminga standard cavity quantum electrodynamics system. The framework allows us to generalize the Fano formula,applicable over the weak- and strong-coupling regimes with pure dephasing. A formulation of its emissionspectrum is also given in a consistent manner. We then find that the interference responsible for the Fano effect isrobust against pure dephasing. This is counterintuitive because the impact of interference is, in general, severelyreduced by decoherence processes. Our approach thus provides a basis for theoretical treatments of the Fanoeffect and insights into the quantum interference in open quantum systems.
DOI: 10.1103/PhysRevResearch.3.013037
I. INTRODUCTION
The Fano effect provides one of the key insights for thestudy of resonance physics [1–4]. The mechanism of thiseffect requires only a few ingredients, the generality of itsconcept is significant, and the predictions obtained are simplebut critical to understand a transition rate from an arbitrary ini-tial state that has two dissipation channels with interference;one is a direct channel to a continuum band of states and theother is an indirect channel via a discrete state into the samecontinuum (Fig. 1). As a result, the transition rate from theinitial state given by the Fano formula,
WF = W0(q + ε)2
1 + ε2, (1)
has a wide range of applications, such as atomic physics [5],Raman scattering [6,7], lasing without inversion [8,9], gainspectra in semiconductors [10], and photonic systems [11,12],especially when focused on the scattering (gain or absorption)problems for externally introduced field. The asymmetric res-onance profile of WF as a function of the reduced energyε ≡ 2(E − ER)/�R is now known as the characteristic signof the Fano effect, where the degree of the asymmetry isdetermined by the Fano parameter q; see also Fig. 1 for thedefinitions of relevant variables.
In general, however, the Fano effect is inherently not lim-ited to such scattering (gain or absorption) problems. Recent
Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.
studies have pointed out that the interference effect, indeed,plays important roles even in the simplest situations of cavityquantum electrodynamics (QED) [13–15], where an initiallyexcited two-level system (TLS) spontaneously emits a photoninto a continuum of radiation modes, directly and indirectlyvia a single mode cavity. Nevertheless, there are few theoret-ical studies to tackle the Fano effect, based on the moderntheories of open quantum systems [16–22]; the interferencebetween the dissipative channels is outside the scope of thesestudies. As a result, for example, it is difficult to discuss theinfluence of the strong coupling and/or the pure dephasing onthe Fano effect. Furthermore, the emission spectra can alsobe modified as a result of the interference [14,15,23]. In thiscontext, a more flexible and systematic theory has increasingimportance for understanding the Fano effect in open quantumsystems [24–26].
In this paper, we propose a Markovian quantum masterequation (QME) that provides a simple approach for the de-scription of the Fano effect, by assuming an ideal cavity QEDsystem. We find that the interference effect can be directlyimplemented into a Liouville superoperator. As a result, ananalytical expression of the transition rate, W , is obtained overthe weak- and strong-coupling regimes with pure dephasing,as a generalization of the Fano formula [Eq. (1)]. Furthermore,a formulation of its emission spectrum is also given in a con-sistent manner with our treatment of the QME. Then, despitethe smeared transition rate by the pure dephasing, we find thatthe interference effect is itself rather insensitive to the puredephasing, based on the calculated spectra. As a result, thedestructive interference can eliminate the emission line at theTLS transition energy with the help of pure dephasing. This isin contrast to a naive intuition that the impact of interferenceis severely reduced by decoherence processes in general. Theunderlying physics is elucidated by studying the fundamentalmechanism of the Fano effect from the viewpoint of spectra.
2643-1564/2021/3(1)/013037(14) 013037-1 Published by the American Physical Society
YAMAGUCHI, LYASOTA, AND YUGE PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
Continuum of states
ER
Interference
W0
Energy
E
Discrete state
Initial state
ΓR
FIG. 1. Schematic illustration of the interference for the Fano ef-fect. The initial state has two dissipation channels into the continuumof states, which can interfere with each other. E is the energy of theinitial state, while ER and �R are the resonance energy and width ofthe discrete state, respectively. W0 is the transition rate only due tothe direct channel.
Our scheme thus proposes a Markovian QME approach forthe Fano effect, achieves a generalization of the Fano formula,and gives insights into the quantum interference of the Fanoeffect. In consequence, our results can provide a basis fortheoretical treatments of the Fano effect in open quantumsystems.
The rest of the paper is organized as follows. In Sec. II,we describe our setup of the cavity QED system, where theTLS is coupled with the single mode cavity. At this stage,several assumptions are introduced to discuss the Fano ef-fect. In Sec. III A, we explain the Liouville superoperatorthat describes the interference effect in the QME. We thenderive the transition rate, W , by assuming that the TLS isinitially excited. In Sec. IV B, we formulate the emissionspectra in a consistent way with our treatment of QME. Basedon numerical calculations, then, we discuss the Fano effecton the emission spectra. In Sec. V, we summarize our results.Throughout the paper, we set h = 1 for simplicity.
II. SETUP
Our theoretical stage is a standard cavity QED systempositioned at the origin of spatial coordinates, where a TLSwith transition energy ω21 can interact with a single modecavity with resonant energy ωc by a coupling constant g. Weassume that the TLS and the cavity mode have decay ratesof γ and κ , respectively, as a result of the interaction with acontinuum of radiation modes in the environment.
In order to formulate the QME, we describe the totalHamiltonian H as
H = HS + HB + HSB, (2)
in the Schrödinger picture, where HS is the Hamiltonian ofthe system, HB is the Hamiltonian of the baths, and HSB is theinteraction Hamiltonian between the system and the baths. Weconsider the TLS and the cavity mode as the system, and thecontinuum of radiation modes as one of the baths. Hence, thesystem Hamiltonian, HS, is given by
HS = 12ω21σz + ωca†
c ac + (gσ+ac + g∗a†c σ−), (3)
where a†c and ac are the bosonic creation and annihilation
operators of the cavity photons, σ+ and σ− are the raising andlowering operators of the TLS, and σi (i = x, y, z) is the Paulioperator of the TLS [16]. Here, g is described by a complexnumber, g = |g|eiφ with φ = π/2. In a similar manner, theHamiltonian of the baths, HB, is described as
HB =∑
ωb†b + · · ·, (4)
where b† and b are the bosonic creation and annihilation
operators of a radiation mode with its energy ω. Here,
denotes the wave vector and the polarization, (k, λ), in thecontinuum. Contributions of other baths, responsible for thepure dephasing, are not shown in Eq. (4) to avoid digressingfrom the main subject. For the interaction Hamiltonian, HSB,we have
HSB = H (1)SB + H (2)
SB + · · ·, (5)
with
H (1)SB =
∑
(ξσ+b + ξ ∗ b†
σ−), (6)
H (2)SB =
∑
(ζa†c b + ζ ∗
b†ac), (7)
where ξ (ζ) is the coupling constant between the TLS (thecavity mode) and the th radiation mode. We describe ξ =|ξ|eiθ21 and ζ = |ζ|eiθc as complex numbers, where θ21 =π/2 in the same manner as g and θc is unknown in general.Again, other interaction Hamiltonians are not shown in Eq. (5)for brevity.
In Sec. III A, we will discuss the QME, based on theseHamiltonians. Before preceding further, however, we makethe following three assumptions:
(1) The absolute value of the detuning, ωc,21 ≡ ωc − ω21,under consideration is much smaller than ω21 and ωc.
(2) We assume that the coupling constants, ξ and ζ, can besimplified as a function of ω, i.e., ξ � ξ (ω) and ζ � ζ (ω).
(3) ξ (ω), ζ (ω), and D(ω) ≡ ∑ δ(ω − ω) do not depend
strongly on the energy ω. Here, D(ω) is the density of statesof the continuum.
As a result, the decay rates of the TLS and the cavity modeare given by
γ = 2π |ξ (ω21)|2D(ω21), κ = 2π |ζ (ωc)|2D(ωc), (8)
respectively. In fact, similar assumptions are implicitly usedin Fano’s original work [1]. However, we note that the secondassumption is drastic because, in general, ξ and ζ depend onthe direction of the wave vector and the polarization of theth mode as well as its energy. Nevertheless, we employ thisassumption to extract the essential features. ξ (ω) and ζ (ω)correspond to the coupling constants averaged over all direc-tions. In addition, based on the first and third assumptions,we ignore the dependence of γ and κ on ω21 and ωc, in thefollowing.
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III. THE QME APPROACH TO THE FANO EFFECT
A. A formulation of the QME
Now, we formulate the QME. For this purpose, we intro-duce the interaction picture with respect to H0 ≡ 1
2ω21σz +ωca†
c ac + HB:
O(t ) ≡ eiH0t Oe−iH0t ,
where O is an arbitrary operator. Under the Born-Markovapproximation, then, the time evolution of the reduced densityoperator of the system, ρS ≡ TrB[ρ], is given by
d
dtρS(t ) = − i[VS(t ), ρS(t )]
−∫ ∞
0dτTrB[HSB(t ), [HSB(t − τ ), ρS(t ) ⊗ ρB]],
(9)
where VS(t ) is the interaction picture of VS ≡ gσ+ac +g∗a†
c σ−, ρB is the density operator of the baths, and we haveused TrB[HSBρB] = 0 by assuming the continuum of radiationmodes is in vacuum. Here, we note that the reference Hamil-tonian of the interaction picture is H0, instead of HS + HB,and therefore, Eq. (9) is slightly different from the standardapproach [18]; see also Appendix A. As we see below, thisenables us to directly use the system operators of HSB, such asac, a†
c , and σ±, in the final form of dissipators because HSB(t ′)can be described by ac(t ′) = ace−iωct ′
, σ−(t ′) = σ−e−iω21t ′,
b(t ′) = be−iωt ′, and their Hermitian conjugates.
To derive the dissipators, we substitute Eq. (5) into thedouble commutator in Eq. (9):[
HSB(t ),[HSB(t − τ ), •]] =[
H (1)SB (t ),
[H (1)
SB (t − τ ), •]]+ [
H (2)SB (t ),
[H (2)
SB (t − τ ), •]]+ [
H (1)SB (t ),
[H (2)
SB (t − τ ), •]]+ [
H (2)SB (t ),
[H (1)
SB (t − τ ), •]]+ · · · . (10)
This equation means that the dissipators will be generated notonly by the direct terms (the first and second terms) but alsothe cross terms (the third and fourth terms) of the individualinteraction Hamiltonians [Eqs. (6) and (7)]. In fact, in Eq. (9),the direct terms give the well-known dissipators that describethe dissipative effect,
D21ρS = γ
2(2σ−ρSσ+ − σ+σ−ρS − ρSσ+σ−), (11)
DcρS = κ
2(2acρSa†
c − a†c acρS − ρSa†
c ac), (12)
by following the standard procedures to obtain the QME[16,17]. For simplicity, we neglect the terms corresponding tothe Lamb shift in our discussion. In contrast, the cross termsyield
DFρS =γF
2e−iωc,21t (2acρSσ+ − σ+acρS − ρSσ+ac)
+ γ ∗F
2eiωc,21t (2σ−ρSa†
c − a†c σ−ρS − ρSa†
c σ−), (13)
where γF is a complex number given by
γF = ei(θ21−θc )√ηγ κ. (14)
In the derivation, we have used the assumptions describedin Sec. II and introduced a phenomenological parameter η
(0 � η � 1) that describes the degree of overlap between thespatial radiation patterns of the TLS and the cavity mode;see also Appendix B. Here, η = 1 for the identical radiationpatterns, while η = 0 for the orthogonal ones. The super-operator DF is usually neglected by implicitly assuming theorthogonal radiation patterns (η = 0). Here, in contrast to γ
and κ , we note that γF is a complex number, the phase ofwhich is determined by θ21 and θc. Therefore, it is importantto treat the coupling constants as complex numbers in theoriginal interaction Hamiltonians [Eqs. (6) and (7)], whilesuch a treatment is not required for the description of D21
[Eq. (11)] and Dc [Eq. (12)]. To our knowledge, this is the firsttime such a dissipator is presented despite its simple form. Wenote that extension of our formulation to a finite temperaturecase is straightforward although the continuum was assumedin vacuum.
In addition to the dissipators shown above, in the follow-ing, we also use a dissipator that describes the pure dephasingeffect:
DphρS = γph
2(σzρSσz − ρS), (15)
where γph is the pure dephasing rate of the TLS [17]. As aresult, the QME in our paper is finally given by
d
dtρS = −i[VS, ρS] + (D21 + Dc + DF + Dph )ρS, (16)
in the interaction picture. At this stage, however, it is difficultto understand the effect of the dissipator, DF [Eq. (13)]. In thenext section, therefore, we study the transition rate, based onthe QME described here.
B. The transition rate
To clarify the effect of the dissipator, DF, we now discussthe transition rate of the initially excited TLS. For this pur-pose, we start with the time evolution of relevant expectationvalues, 〈O〉t ≡ Tr[Oρ(t )]. After transforming back into theSchrödinger picture, the QME yields the equations of motionfor the population of the excited state, ne(t ) ≡ 〈σ+σ−〉t , thephotons inside the cavity, nc(t ) ≡ 〈a†
c ac〉t , and the polariza-tion, p(t ) ≡ 〈σ+ac〉t :
nc = 2Re(ig+ p) − κnc, (17)
ne = 2Re(−ig− p) − γ ne, (18)
p = −ig∗+ne + ig∗
−nc − (iωc,21 + �tot )p, (19)
where g± and �tot are, respectively, defined by
g± ≡ g ± iγF
2(20)
and
�tot ≡ γ + κ
2+ γph. (21)
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Here, an approximation 〈σza†c ac〉t � −〈a†
c ac〉t has been intro-duced in Eq. (19) because, in our situation, the TLS is alwaysin the ground state when a photon is inside the cavity [16,22].In these equations, we can notice that the TLS can interactwith the cavity mode through the continuum of states becauseγF plays a similar role to the coupling constant g. However,such an interaction cannot reduce to a simple renormalizationof g. We need g+ and g− to describe the different ways ofcoupling between p and nc, and that between p and ne, as seenin Eqs. (17)–(19).
Here, it is not easy to obtain the analytical solutions forEqs. (17)–(19) although these equations can give the timeevolution over the weak- and strong-coupling regimes. Forour purpose to obtain the transition rate, W , however, there isno need to exactly solve the problem. In the strong-couplingregime, in particular, the transition rate can be measured fromthe decay of the envelope of the Rabi oscillations [22]. In thiscontext, to extract the transition rate, we further introduce anapproximation, p � 0, in Eq. (19):
p � −ig∗+ne + ig∗
−nc
iωc,21 + �tot, (22)
which averages out the Rabi oscillations if the system is in thestrong-coupling regime [27]. The coarse-grained time evolu-tion is then described by
nc = −(R+,− + κ )nc + R+,+ne, (23)
ne = −(R−,+ + γ )ne + R−,−nc, (24)
where
Rα,β ≡ Re
(2gαg∗
β
iωc,21 + �tot
). (25)
As a result, Eqs. (23) and (24) with the initial conditions,nc(0) = 0 and ne(0) = 1, can yield the analytic solutions:
ne(t ) = �+ + R+,− + κ
�+ − �− e�+t − �− + R+,− + κ
�+ − �− e�−t , (26)
nc(t ) = R+,+�+ − �− (e�+t − e�−t ), (27)
where �± denotes the eigenvalue of the coefficient matrix ofEqs. (23) and (24):
�± = − γ + κ + R+,− + R−,+2
±√(κ − γ + R+,− − R−,+
2
)2
+ R+,+R−,−.
By assuming κ � γ , the first term in Eq. (26) dominates theevolution of ne(t ). As a result, the transition rate, W , of theinitially excited TLS can be obtained as
W � − �+
= γ + κ + R+,− + R−,+2
−√(κ − γ + R+,− − R−,+
2
)2
+ R+,+R−,−. (28)
We note that W is described by −�−, instead, for κ � γ .However, we restrict our discussion to the case for κ � γ , forsimplicity, in accordance with Fano’s approach [1]. Equation(28) is one of our main results.
In order to discuss the meaning of our analysis and theobtained result [Eq. (28)], we show typical numerical re-sults with no pure dephasing (γph = 0) in Fig. 2. In ourcalculations, other parameters are assumed κ = 50 μeV, γ =0.05 μeV, |g| = 100 μeV, and θc = 0 [15,28,29], unless oth-erwise stated. In Fig. 2(a), we can see that the transition rate,W , shows the symmetric profile as a function of the reduceddetuning ε ≡ 2(ω21 − ωc)/κ when the spatial radiation pat-terns of the TLS and the cavity are orthogonal, η = 0 (thedotted line). This profile means that the transition rate issimply enhanced when the TLS comes into resonance with thecavity, as expected. In contrast, the profile becomes asymmet-ric with increasing the value of η, i.e., the degree of overlapbetween the spatial radiation patterns. This is a characteristicsignature of the Fano effect.
For the identical radiation patterns, η = 1, we also showthe time evolutions obtained by Eqs. (17)–(19) under the on-resonant and off-resonant conditions in Figs. 2(b) and 2(c),respectively. In Fig. 2(b), we can find an oscillating behaviorin the population of the excited state, ne(t ). This correspondsto the Rabi oscillation, indicating that the system is in thestrong-coupling regime. This is consistent with our settingof the parameters, |g| � κ + γ . We note that this parameterregime is beyond the applicable range of the Fano formula[Eq. (1)], despite the asymmetric profile of the transition rate.
To check the validity of our analysis, therefore, we alsoshow the time evolutions by Eqs. (26) and (27) (the dottedline) and by exp(−W t ) with Eq. (28) (the dashed line) inFig. 2(b). Here, we can see that the dotted line is roughlyalong the center line of the oscillations (the coarse-grainedevolution), and as a result, the dashed line shows good agree-ment with the decay of the envelope. These results suggestthat our approach works well because, in the strong-couplingregime, the transition rate is measured by the decay of itsenvelope by using the coarse-grained evolution, as mentionedabove. In Fig. 2(c), we can also verify that the asymmetrictransition rate in Fig. 2(a) indeed gives a difference in the timeevolutions between the positive and negative detuning. Wenote that the results by Eqs. (26) and (27) and by exp(−W t )are not shown for the sake of visibility because these arealmost exactly overlapped with the presented lines obtainedby Eqs. (17)–(19).
We have thus obtained the formula of the transition rateW [Eq. (28)] and numerically shown that the asymmetricprofile can be still found even in the strong-coupling regime.However, it is important to show that our result can exactlyrecover the Fano formula. For this purpose, we now focus onthe weak-coupling regime with κ � γ in accordance with theperturbation approach by Fano [1]. In this case, the magnitudeof the coupling constant |g| is much smaller than γ + κ � κ .We can then neglect the term, R+,+R−,−, in the square root ofEq. (28) and obtain
W � γ + R−,+. (29)
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FIG. 2. Numerical results for γph = 0. (a) The transition rate, W , as a function of the reduced detuning. The dotted and solid lines,respectively, indicate the orthogonal (η = 0) and identical (η = 1) spatial radiation patterns of the TLS and the cavity. The dashed line isthe result for the intermediate degree of overlap (η = 0.5). (b) The time evolution of the population, ne(t ), under the on-resonance conditionε = 0 with η = 1. The solid line is obtained from Eqs. (17)–(19). The oscillating behavior means that the system is in the strong-couplingregime. For comparison, the dotted line shows Eq. (26), while the dashed line gives exp(−W t ) with Eq. (28). (c) The same as (b) but under theoff-resonance conditions, ε = −100 (the red line) and ε = +100 (the blue line). In this panel, the results by Eq. (26) and exp(−W t ) are notshown for the sake of visibility because these are almost exactly overlapped with the presented lines. The open circles in panel (a) correspondto the time evolutions in panels (b) and (c). The parameters are assumed κ = 50 μeV, γ = 0.05 μeV, |g| = 100 μeV, and θc = 0.
It is interesting to note that, for η = 0 (g± = g), Eq. (29) re-covers the well-known Purcell effect W � γ + 2|g|2 �tot
ω2c,21+�2
tot
[22,30]. In contrast, by assuming η = 1 and γph = 0, Eq. (29)yields
W = γ + γ Re
[(q − i)(q∗ − i)
1 − iε
]= γ
|q + ε|2ε2 + 1
, (30)
where ε = 2(ω21 − ωc)/κ is the reduced detuning and q isdefined by
q ≡ 2|g|√γ κ
ei(φ+θc−θ21 ). (31)
We note that Eq. (30) has the same form as the Fano formula[Eq. (1)], except that the parameter q [Eq. (31)] is defined as acomplex number in general. This is consistent with the resultsdiscussed in charge transport experiments [31]. In Eq. (31),φ + θc − θ21 denotes the phase difference between the directchannel, θ21, and the indirect channel via the discrete state (thecavity mode), φ + θc. As a result, by assuming that the param-eter q is real, the Fano formula [Eq. (1)] can be reproduced.From these discussions, we can conclude that the presenteddissipator, DF [Eq. (13)], indeed, can describe the Fano effect,and that the transition rate, W [Eq. (28)], is a generalization ofthe Fano formula [Eq. (1)].
Finally, we show the effect of pure dephasing for |q| = 0and 3 in Figs. 3(a) and 3(b), respectively. In Fig. 3(a), we canfind that the transition rate has a strong dip when the TLS ison resonance with the cavity for γph = 0 μeV (the black solidline). This is consistent with the Fano formula. In the contextof cavity QED, however, this phenomena is interesting be-cause |q| = 0 means that there is no direct coupling betweenthe TLS and the cavity mode, i.e., g = 0 due to Eq. (31).As a result, at least in principle, an “antiresonance” [3] of
FIG. 3. The effect of pure dephasing on the transition rate: (a)|q| = 0 (|g| = 0 μeV) and (b) |q| = 3 (|g| = 2.37 μeV). The puredephasing rates, γph, are 0 μeV (black line), 3 μeV (red line), 30 μeV(blue line), and 300 μeV (gray line). The spatial radiation patternsare assumed identical between the TLS and the cavity (η = 1). Otherparameters are the same as Fig. 2.
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the transition rate is possible by the destructive interferencewhen g = 0, while its resonance effect is often discussedas the Purcell effect. By increasing the pure dephasing rate,however, we can see that the antiresonance effect is smearedout. Nevertheless, the effect of pure dephasing is weak up to�3 μeV. This feature is generally understood from Eqs. (17)–(19); the pure dephasing does not strongly influence on thetime evolution when γph � (γ + κ )/2 because γph appearsonly in �tot [Eq. (21)]. Hence, a similar behavior can be seenfor |g| = 2.37 μeV in Fig. 3(b), where the suppression of Wis again weakened by the pure dephasing, although the asym-metric profile can be still seen even if γph becomes comparableto (γ + κ )/2. The presented formulation thus allows us toquantitatively discuss the transition rate. However, one mayconsider that the smeared profiles of W are qualitatively trivialbecause, in general, the impact of interference is severelyreduced by the pure dephasing; it is natural that the suppres-sion of W by the destructive interference goes away by thepure dephasing, for example. Nevertheless, as we shall see inSec. IV C, our study on the spectra shows that the interferenceresponsible for the Fano effect is itself tolerant to the puredephasing and that there is another reason for the incompletesuppression of the transition rate.
IV. THE FANO EFFECT ON THE SPECTRA
In the previous section, we have described a dissipatorin the presence of the two direct and indirect dissipationchannels, which allows us to study the physical quantities ofthe system with the Fano effect. However, in reality, thesequantities are usually measured via environmental degrees offreedom, i.e., the radiation modes in the continuum. In thiscontext, further analysis is required to discuss the observablequantities in addition to the simple modification of the QME.In this section, we explain a formulation to obtain the emissionspectra in a consistent manner with our treatment of the QME.For this purpose, we first discuss the intensity detection in thepresence of the Fano effect. We then extend the idea to thespectroscopy. Finally, we numerically show the Fano effecton the emission spectra.
A. The intensity detection
We here formulate the intensity, S(t ), evaluated as the totalnumber of photons emitted from the system into all direc-tions per unit time, as a preliminary step toward the emissionspectra. In general, for the intensity detection, the optical fieldfrom the system is introduced into a photon-counting detectorafter propagating a certain distance in the continuum. Here,the incident energy per unit area and unit time on the detector,I (r, t ), is given by
I (r, t ) = 2nε0c0E−
(r, t ) · E+
(r, t ), (32)
where r is the position of the detector and n, ε0, and c0,respectively, denote the refractive index in the continuum, thevacuum dielectric constant, and the speed of light in vacuum.O(t ) ≡ eiHt Oe−iHt indicates the Heisenberg picture for anarbitrary operator O and the electromagnetic field operator,
E+
(r, t ), in the continuum is described by
E+
(r, t ) =∑k,λ
ek,λEkbk,λ(t )eik·r, (33)
with
Ek ≡ i
√ωk
2n2ε0V. (34)
Here, ek,λ is a unit vector along the polarization of the ra-diation mode (k, λ) and V is the quantization volume forthe continuum. We note that E
−(r, t ) = [E
+(r, t )]†. Since
I (t ) ≡ ∫ π
0 dθ∫ 2π
0 dφr2 sin θ I (r, t ) gives the power radiatedfrom the system into all directions, S(t ) is given by S(t ) �〈I (t )〉0/ω21 � 〈I (t )〉0/ωc. From the Heisenberg equation ofmotion, Eqs. (2)–(7) yield
d
dtbk,λ = −iωkbk,λ − iξ ∗
k,λσ− − iζ ∗k,λac, (35)
and its formal solution is given by
bk,λ(t ) = bk,λe−iωkt − iξ ∗k,λ
∫ t
0dt ′σ−(t ′)e−iωk (t−t ′ )
− iζ ∗k,λ
∫ t
0dt ′ac(t ′)e−iωk (t−t ′ ). (36)
This equation means that the operator of the radiation modes,bk,λ, is related to the two system operators, σ− and ac. Bysubstituting Eq. (36) into Eq. (33), E
+(r, t ) can be written as
E+
(r, t ) = E+free(r, t ) + E
+S (r, t ), (37)
where
E+free(r, t ) ≡
∑k,λ
ek,λEk,λbk,λei(k·r−ωkt ) (38)
denotes the free evolution of the radiation modes in the con-tinuum and
E+S (r, t ) ≡ E
+21(r, t ) + E
+c (r, t ) (39)
describes the electromagnetic field radiated from the system:
E+21(r, t ) ≡ e−iω21t
∫ t
0dt ′G21(t, t ′)σ−(t ′), (40)
E+c (r, t ) ≡ e−iωct
∫ t
0dt ′Gc(t, t ′)ac(t ′). (41)
Here, σ (t ′) ≡ σ (t ′)eiω21t ′and ac(t ′) ≡ ac(t ′)eiωct ′
are theslowly varying operators [16] and
G21(t, t ′) ≡ −i∑k,λ
ek,λEkξ∗k,λeik·r−iωk,21(t−t ′ ), (42)
Gc(t, t ′) ≡ −i∑k,λ
ek,λEkζ∗k,λeik·r−iωk,c(t−t ′ ) (43)
can be considered as a kind of propagator. Therefore, bysubstituting Eqs. (37) and (39) into Eq. (32), the expectationvalue of I (r, t ) is given by
I (r, t ) = 2nε0c0〈E−S (r) · E
+S (r)〉t
= 2nε0c0〈E−21(r) · E
+21(r)〉t
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THEORY OF FANO EFFECT IN CAVITY QUANTUM … PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
+ 2nε0c0〈E−c (r) · E
+c (r)〉t
+ 4nε0c0Re[〈E−21(r) · E
+c (r)〉t ], (44)
where 〈· · · E±free(r) · · · 〉t = 0 has been used because the con-
tinuum is assumed in vacuum in our paper. In the right-handside of the second equality, the first and second terms resultfrom the radiation from the TLS and the cavity, respectively,whereas the last term indicates their interference.
In order to study S(t ), therefore, further analysis onE
+21(r, t ) and E
+c (r, t ) is required. For this purpose, we intro-
duce the following approximations for Eqs. (42) and (43):
G21(t, t ′) � −i√
2ξ ∗e21(r)∑
k
Ekeik·r−iωk,21(t−t ′ ), (45)
Gc(t, t ′) � −i√
2ζ ∗ec(r)∑
k
Ekeik·r−iωk,c(t−t ′ ), (46)
based on the assumptions in Sec. II. Here, e21(r) and ec(r) arethe unit vectors along the polarization of the radiation patternsfrom the TLS and the cavity, respectively, in the far-fieldregion (kr � 1). A value of
√2 results from the sum of the
two orthogonal unit vectors along the polarization. Then, wecan calculate∑
k
Ekeik·r−iωk,α (t−t ′ )
= V
(2π )3
∫ ∞
0dk
∫ 2π
0dφk
∫ π
0dθkk2 sin θk
× Ekeikr cos θk−iωk,α (t−t ′ )
� −iπc0Eα
2nrωα
D(ωα )
× {δ(t − t ′ − td )eiωαtd − δ(t − t ′ + td )e−iωαtd }, (47)
where td ≡ nr/c0 is the delay time due to the propagation oflight from the origin, D(ω) = n3ω2V
π2c30
is the density of states inthe free space, and α ∈ {21, c}. Therefore, we have
E+21(r, t ) � −e21(r)
πE21c0√2nrω21
ξ ∗D(ω21)σ−(t − td ), (48)
E+c (r, t ) � −ec(r)
πEcc0√2nrωc
ζ ∗D(ωc)ac(t − td ). (49)
By substituting these equations into Eq. (44) with neglectingthe delay time, td, for simplicity, we can obtain
S(t ) �γ 〈σ+σ−〉t + κ〈a†c ac〉t + 2Re[γF〈σ+ac〉t ], (50)
where, based on the spirit of the phenomenological parameterη in Eq. (14), we have estimated
ei(θ21−θc )√γ κ
∫ π
0dθ
∫ 2π
0dφ
e21(r) · ec(r) sin θ
4π� γF,
because the integration means the average over the entire solidangle. It is interesting to note that, in Eq. (50), the individualcoefficients agree well with the dissipators in Sec. III A; thelast term is the direct consequence of the Fano effect. Thus,the effect of interference should be considered in the formu-lation of observable quantities as well as in the dissipators ofthe QME.
B. The emission spectra
For spectral detection, in general, the emitted photons areintroduced into a spectral apparatus, in which the photons aredispersed by a spectrometer before the number of the photonsis counted by a detector. In this context, the spectrometer canbe considered as a kind of a spectral filter for the detector.Here, we assume that the characteristic function of this filter-ing is Lorentzian with a central frequency ν and a width δs:
F (ω) ≡ A
(δs/2) − i(ω − ν), (51)
where A is a constant determined by the performance of thespectrometer. Since the number of the photons incident on thedetector depends on the central frequency ν and the width δs,the average photon flux density on the detector at positionr can be described as S(r, t, ν, δs ). The emission spectra arethen given by the profile of S(r, t, ν, δs ) as a function of ν,in which δs corresponds to the spectral resolution [32]. In thefollowing, we discuss the spectra integrated over time and di-rections, S(ν, δs ) ≡ ∫ ∞
−∞ dt∫ π
0 dθ∫ 2π
0 dφr2 sin θS(r, t, ν, δs ),for simplicity.
For this purpose, we describe the Fourier transform of theelectromagnetic field by
E+
(r, ω) ≡∫ ∞
−∞dt ′E+
(r, t ′)eiωt ′. (52)
Since E+
(r, ω) is composed of the free evolution of the radi-ation modes and the filtered field from the system,
E+
(r, ω) = E+free(r, ω) + F (ω)E
+S (r, ω), (53)
the inverse Fourier transform yields
E+
(r, t ) = E+free(r, t ) +
∫ ∞
−∞dt ′F (t − t ′)E+
S (r, t ′), (54)
where F (t ) = Ae−(δs/2+iν)t�(t ) and �(t ) is the step function.Hence, in a similar manner to Eqs. (32) and (44), the expec-tation value of the incident energy per unit area and unit timeon the detector is given by
I (r, t, ν, δs ) =2nε0c0
∫ ∞
−∞dt1
∫ ∞
−∞dt2
× F ∗(t − t1)F (t − t2)〈E−S (r, t1) · E
+S (r, t2)〉0
=2π |A|2∫ ∞
0dt ′J (r, t − t ′, ν, δs )e−δst ′
, (55)
where we have defined
J (r, t, ν, δs ) ≡2nε0c0
π
×Re∫ ∞
0dτ 〈E−
S (r, t−τ ) · E+S (r, t )〉0e(iν−δs/2)τ .
In the second equality of Eq. (55), we have used∫ t−∞ dt1
∫ t−∞ dt2[· · · ]
= ∫ ∞0 dt ′
1
∫ ∞0 dt ′
2[· · · ] + ∫ 0−∞ dt ′
1
∫ ∞−τ
dt ′2[· · · ],
by a transformation of variables, t ′1 = t1 − t2 and t ′
2 = t − t1.Since I (r, t, ν, δs ) depends on the coefficient A, we here set a
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YAMAGUCHI, LYASOTA, AND YUGE PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
condition of normalization:∫ ∞
−∞dt
∫ ∞
−∞dν
I (r, t, ν, δs )
2nε0c0=
∫ ∞
−∞dt〈E−
S (r) · E+S (r)〉t .
This equation means that the integration of I (r, t, ν, δs ) overfrequency ν and time t is identical to the total energy detectedat position r. We can then obtain 2π |A|2 = δs from Parseval’stheorem [32]. Hence, by integrating Eq. (55) over all direc-tions and time, S(ν, δs ) � I (ν, δs )/ω21 � I (ν, δs )/ωc yields
S(ν, δs ) = S21(ν, δs ) + Sc(ν, δs ) + SF(ν, δs ), (56)
with
S21(ν, δs ) = γ
πRe
∫ ∞
−∞dt
∫ ∞
0dτ 〈σ+(t − τ )σ−(t )〉0e(iν− δs
2 )τ ,
Sc(ν, δs ) = κ
πRe
∫ ∞
−∞dt
∫ ∞
0dτ 〈a†
c (t − τ )ac(t )〉0e(iν− δs2 )τ ,
SF(ν, δs ) = 1
πRe
∫ ∞
−∞dt
∫ ∞
0dτ (γF〈σ+(t − τ )ac(t )〉0
+ γ ∗F 〈a†
c (t − τ )σ−(t )〉0)e(iν− δs
2 )τ ,
where we have used Eqs. (39), (48), and (49). S21(ν, δs )and Sc(ν, δs ) are the time-integrated spectra of the TLS andthe cavity, respectively. In contrast, SF(ν, δs ) is the contri-bution due to their interference. Again, the proportionalityfactors in these expressions agree well with the dissipators inSec. III A.
For the application of this result, we note that the cor-relation functions of the form 〈Oi(t − τ )O j (t )〉0 have to beevaluated. However, the standard quantum regression theo-rem [16,17] cannot be directly applied because its form issimply outside the range of application. To circumvent thisdifficulty, we first consider 〈Oi(t )O j (t + τ )〉0. The quantumregression theorem, then, allows us to express the correlationfunction in the form 〈Oi(t )O j (t + τ )〉0 = ∑
k Cjk (τ )〈OiOk〉t
when τ � 0. By substituting t → t − τ , therefore, we canobtain 〈Oi(t − τ )O j (t )〉0 = ∑
k Cjk (τ )〈OiOk〉t−τ . As a result,we can estimate 〈Oi(t − τ )O j (t )〉0 by applying the quantumregression theorem, in which the dissipator, DF, again plays akey role to evaluate the correlation functions. In this sense, wenote that the consistency between the treatments of the QMEand the spectra has great importance.
By assuming the initially excited TLS, then, we obtain
Sα (ν, δs ) = Re
[1
γ+ − γ−
(fα (γ+)
iν + γ+ − δs/2− fα (γ−)
iν + γ− − δs/2
)]for α ∈ {21, c, F}, (57)
where we have defined
f21(γ±) ≡ γ
π
{ig−Ip −
(γ± + iωc + κ
2
)Ie
}, fc(γ±) ≡ κ
π
{ig∗
+I∗p −
(γ± + iω21 + γph + γ
2
)Ic
},
fF(γ±) ≡ 1
π
{ig−γ ∗
F Ic + ig∗+γFIe −
(γ± + iωc + κ
2
)γ ∗
F I∗p −
(γ± + iω21 + γph + γ
2
)γFIp
},
with Ie = ∫ ∞−∞ ne(t )dt , Ic = ∫ ∞
−∞ nc(t )dt , and Ip = ∫ ∞−∞ p(t )dt . In addition, γ± is given by
γ± ≡ −1
2(�tot + i(ω21 + ωc)) ± 1
2
√(κ − γ
2− γph + iωc,21
)2
− 4g∗+g−. (58)
These expressions are the second of our main results inour theoretical treatments. Here, we note that the obtainedspectrum, S(ν, δs ), has an explicit physical meaning, i.e., thenumber of photons counted per unit frequency, by definition.Therefore, the integration of the total spectrum over the fre-quency,
∫ ∞−∞ dνS(ν, δs ), indicates the total number of photons
finally emitted from the system, which must be 1 because weconsider the initially excited TLS and its relaxation within thelinear optical process. In order to verify this prediction, weanalytically integrate S(ν, δs ) and obtain∫ ∞
−∞dνS(ν, δs ) = γIe + κIc + 2Re[γFIp], (59)
where ∫ ∞
−∞dν
fα (γ±)
iν + γ± − δs/2= −π fα (γ±)
has been used. We note that Eq. (59) is consistent withthe time integration of Eq. (50). Furthermore, by assuminglimt→∞ nc(t ) = limt→∞ ne(t ) = 0, the time integrations of
Eqs. (17) and (18) together yield
nc(0) + ne(0) =∫ ∞
0dt[γ ne + κnc + 2Re(γF p)].
As a result, for nc(0) = 0 and ne(0) = 1, we can find∫ ∞
−∞dνS(ν, δs ) = 1, (60)
where nc(t ) = ne(t ) = p(t ) = 0 for t < 0 has been used.Equation (60) is consistent with the prediction that only onephoton is finally emitted from the system and this result en-sures the validity of our treatments, presented above. It isobvious that the simultaneous consideration of SF(ν, δs ) andDF is essential to achieve this property.
For our analysis on the spectra [Eq. (57)], in the nextsection (Sec. IV C), Ie, Ic, and Ip are analytically evaluatedby the coarse-grained evolution [Eqs. (22), (26), and (27)], forsimplicity. We note that the validity of this approach can bechecked by calculating Eq. (59) to be 1.
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THEORY OF FANO EFFECT IN CAVITY QUANTUM … PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
FIG. 4. Spectra for γph = 0 when the spatial radiation patterns of the TLS and the cavity are identical (η = 1). The resolution of thespectrometer is set to be δs = 20 μeV. Other parameters are the same as in Fig. 2. (a) The dependence of the spectral intensity, S(ν, δs ), on thedetuning. (b) The transition rate, W/γ , and the integrated value,
∫dνS(ν, δs ), as a function of the detuning. We note that W/γ is identical to
the solid line in Fig. 2(a).∫
dνS(ν, δs ) is calculated by Eq. (59) to check the validity of our numerical results. (c)–(n) The decomposition ofS(ν, δs ) [panels (c), (g), (k)] into S21(ν, δs ) [panels (d), (h), (l)], Sc(ν, δs ) [panels (e), (i), (m)], and SF(ν, δs ) [panels (f), (j), (n)]. The detuningsare −3.16 meV for the left column [panels (c)–(f)], +3.16 meV for the middle column [panels (g)–(j)], and 0.0 meV for the right column[panels (k)–(n)]. These detunings are indicated by the dashed lines in panel (a). (o) A schematic illustration of the dissipation channels forω21 < ωc. We note that the frequency of the field escaped via the cavity is dominated by ω21, instead of ωc. This process is achieved via thevirtual photon excitation inside the cavity and is the same as the classical forced oscillation.
C. Numerical results
Figure 4 shows numerical results for γph = 0 with η = 1,where the resolution of the spectrometer is set to be δs =20 μeV. Other parameters are the same as in Fig. 2. InFig. 4(a), we can see that the spectra, S(ν, δs ), are almost sym-metric with respect to ν − ω21 with changing the sign of thedetuning ω21 − ωc. This is in contrast to the asymmetric pro-file of the transition rate, W , shown in Fig. 4(b). The validityof these numerical results is supported by
∫ ∞−∞ dνS(ν, δs ) � 1
over the entire range of the calculations, as shown in Fig. 4(b).Furthermore, the color map is nearly the same as the spectrafor η = 0.0 although we do not show the results here. Never-theless, as we explain below, the Fano effect plays an essentialrole for a consistent understanding of these results.
To elucidate these points, at positive and negative detun-ings, ω21 − ωc = ±3.16 meV, the spectra are decomposedinto S21(ν, δs ), Sc(ν, δs ), and SF(ν, δs ), as shown in Figs. 4(c)–4(j). In the total spectra, S(ν, δs ), for both detunings [Figs. 4(c)and 4(g)], the main peaks appear at the TLS transition energy,ω21, with comparable spectral intensities. However, S21(ν, δs )for −3.16 meV detuning [Fig. 4(d)] is a factor of 107–108
larger than that for +3.16 meV detuning [Fig. 4(h)]. A similarbehavior can also be seen in Sc(ν, δs ) [Figs. 4(e) and 4(i)].This means that the spectra are strongly asymmetric if we just
consider DF in the QME but without the contribution ofSF(ν, δs ). However, the spectral intensities are largely re-duced by a destructive interference for −3.16 meV detuning[Fig. 4(f)], while enhanced by a constructive interference for+3.16 meV detuning [Fig. 4(j)]. As a result, S(ν, δs ) gives asimilar degree of spectral intensities for ±3.16 meV detun-ings. In contrast, at resonance condition (ω21 − ωc = 0 meV),an asymmetric vacuum Rabi splitting can be obtained, as aconsequence of the interference [Figs. 4(k)–4(n)]. This resultis in agreement with Ref. [15].
The Fano effect is thus essential for a consistent under-standing of the emission spectra. However, as stated above,there is a remarkable similarity between the total spectra forη = 1.0 and 0.0 (not shown) although the transition rate, W ,depends significantly on the value of η [Fig. 2(a)]. This isbecause the TLS has no other choice but to finally emit aphoton at the transition energy, ω21, after an infinite time, evenin the presence of the interference. This situation is schemat-ically illustrated in Fig. 4(o) by assuming ω21 < ωc, wherewe note that the frequency of the field escaped via the cavityis dominated by ω21, instead of ωc. This feature is evidencedby the dominant contribution of the emission line at ν = ω21
in Sc(ν, δs ) [Fig. 4(e)]. Such a transition process is mediatedvia the virtual photon excitation inside the cavity and can be
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YAMAGUCHI, LYASOTA, AND YUGE PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
FIG. 5. Spectra for γph = 30.0 μeV. Other parameters are the same as in Fig. 4. (a) The dependence of the spectral intensity, S(ν, δs ), onthe detuning. (b) The transition rate, W/γ , and the integrated value,
∫dνS(ν, δs ) [Eq. (59)], as a function of the detuning. For comparison,
the transition rate for γph = 0.0 μeV is indicated by the dotted line, which is identical to the solid line in Fig. 2(a). (c)–(n) The decompositionof S(ν, δs ) into S21(ν, δs ), Sc(ν, δs ), and SF(ν, δs ), in a similar manner to Figs. 4(c)–4(n). The detunings are −3.16 meV for the left column[panels (c)–(f)], −1.00 meV for the middle column [panels (g)–(j)], and 0.0 meV for the right column [panels (k)–(n)]. These detunings areagain indicated by the dashed lines in panel (a). (o) A schematic illustration of the dissipation channels for ω21 < ωc. The AZE channel isadditionally opened by the pure dephasing.
interpreted in the same manner as the classical forced oscil-lation. Therefore, the direct and indirect dissipation channelsboth produce fields with frequency ω21, as seen in Fig. 4(o).In consequence, the TLS has to finally emit a photon at itstransition energy, ω21, regardless of the interference. Althoughthe spectral width can be changed due to the modification ofthe transition rate, the difference is much below the resolution(δs = 20 μeV). As a result, the total spectra cannot be signif-icantly modified by the Fano effect alone by considering thatthe ν integral of S(ν, δs ) is fixed to 1 in the presented situation.
However, the situation is drastically changed when thepure dephasing is additionally introduced, as shown in Fig. 5,where the parameters are the same as in Fig. 4 except thatγph = 30.0 μeV. In Fig. 5(a), the spectra, S(ν, δs ), show anasymmetric behavior in contrast to Fig. 4(a). For positivedetuning (ω21 > ωc), the intensity at the cavity resonanceis enhanced in comparison with Fig. 4(a). This effect canbe understood from the viewpoint of the quantum anti-Zenoeffect (AZE) [22,33], known as one of the mechanisms re-sponsible for the off-resonant cavity feeding [22,33–36]. Fornegative detuning (ω21 < ωc), in contrast, we can find a pe-culiar spectral behavior. At the detuning of −3.16 meV, inparticular, a strong intensity reduction at ν = ω21 can be seenwith a further intensity enhancement at the cavity resonance(ν = ωc). This value of detuning corresponds to the originaldetuning position achieving the minimum of the transition rate
for γph = 0, instead of the shifted position for γph = 30 μeV[Fig. 5(b)]. Especially for negative detuning, thus, the Fanoeffect largely influences on the spectra with the help of puredephasing, even if the transition rate is significantly washedout (γph > (γ + κ )/2). This is seemingly counterintuitive ifwe consider that the pure dephasing spoils the interference be-tween the two dissipation channels. Again,
∫ ∞−∞ dνS(ν, δs ) �
1 in Fig. 5(b) ensures the validity of these results.To clarify the underlying physics, at the detuning of
−3.16 meV, the total spectrum is again decomposed intoS21(ν, δs ), Sc(ν, δs ), and SF(ν, δs ), in Figs. 5(c)–5(f). As seenin Fig. 5(a), S(ν, δs ) has a single main peak at the cavityresonance, ωc, in Fig. 5(c), which is in contrast to the resultfor γph = 0 in Fig. 4(c). Here, in Fig. 5(d), we can see thatS21(ν, δs ) has a main peak at ν = ω21 in a similar manner toFig. 4(d). This is reasonable because S21(ν, δs ) is the spectrumdirectly emitted from the TLS. On the other hand, in Fig. 5(e),an emission peak appears at ν = ωc in addition to the peak atν = ω21, in contrast to Fig. 4(e). The emission peak at ν = ωc
is a consequence of the AZE, as mentioned above. Then, bythe destructive interference shown in Fig. 5(f), the two emis-sion peaks at ν = ω21 are canceled out almost completely. Asa result, the emission peak at ν = ωc is highlighted in the totalspectra [Fig. 5(c)]. Here, the almost complete canceling of theemission peaks indicates that the interference is not spoiled bythe pure dephasing.
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THEORY OF FANO EFFECT IN CAVITY QUANTUM … PHYSICAL REVIEW RESEARCH 3, 013037 (2021)
Hence, the mechanism can be schematically illustrated inFig. 5(o), where the AZE dissipation channel is additionallyopened by the pure dephasing, in comparison with Fig. 4(o).This AZE channel provides an alternative route of radiationat the frequency of ωc. Its relative influence is maximizedespecially when the two dissipation channels are canceled outby the destructive interference. As a result, at the detuningof −3.16 meV, the emission line at the TLS transition energyis strongly reduced with a simultaneous intensity enhance-ment at the cavity resonance. This understanding is consistentwith our numerical results that such a phenomenon can bestill observed, even if γph is much smaller than (γ + κ )/2(Appendix C). Here, in this scenario, we should notice thatthe quality of interference for the Fano effect is not lostby the pure dephasing. This is because the two dissipationchannels are subjected to identical phase fluctuations by thepure dephasing, and therefore, the phase difference betweenthe two channels is not affected. At this stage, we can furthernotice that the insufficient suppression of the transition rate[Fig. 5(b)] is not due to the loss of the quality of interferencebut in fact due to the appearance of the AZE dissipationchannel. We can thus conclude that the interference effect isitself tolerant to the pure dephasing though the transition rate,W , is smeared out by the pure dephasing.
In contrast, such an effect rapidly diminishes by chang-ing the value of detuning. For example, at ω21 − ωc =−1.00 meV, the emission peak at ν = ω21 again becomesbright, as shown in Fig. 5(g). This is because the interferencerequires comparable strength of the two relevant field ampli-tudes. In fact, for the detuning of −3.16 meV, we can find thatS21(ν, δs ) is well matched to Sc(ν, δs ) in strength at ν = ω21,as seen in Figs. 5(d) and 5(e). However, this balance is lost bymaking ωc closer to ω21 because the amplitude via the cavity isenhanced in ratio by the resonance effect, as seen in Figs. 5(h)and 5(i). In consequence, the destructive interference becomesimperfect [Figs. 5(j) and 5(g)]. At zero detuning, then, thedifference caused by the pure dephasing becomes small inthis calculation [Figs. 4(k)–4(n) and 5(k)–5(n)]. For positivedetuning (ω21 > ωc), finally, the impact of the AZE channelis suppressed by the constructive interference, in comparisonwith negative detuning (ω21 < ωc).
We have thus clarified the Fano effect on the emission spec-tra, especially with and without the pure dephasing. Basedon the above discussion, however, we finally make three re-marks. First, the strong reduction of the spectral intensity canalso be expected by an appropriate amount of a nonradia-tive dissipation of the TLS, instead of the pure dephasing.In this case, the nonradiative dissipation channel plays analternative role for the AZE channel and the restriction of∫
dνS(ν, δs ) = 1 is eliminated. Furthermore, the nonradiativedissipation does not disturb the interference for the Fanoeffect. As a result, the destructive interference can highlightthe influence of the nonradiative dissipation and the spectralintensity can be reduced strongly without the simultaneousintensity enhancement at the cavity resonance. Second, weexpect that the interference of the two channels is also tolerantto the pure dephasing of the cavity, i.e., the fluctuations ofthe cavity resonance, ωc. This may be somewhat paradox-ical at first glance by considering the mechanism for therobustness against the pure dephasing of the TLS. However,
the relevant field escaped through the cavity is driven at thefrequency of ω21, mediated via the virtual photon excitation.This can be interpreted as the classical forced oscillation, asexplained above. Hence, the phase of this field is less sensitiveto the fluctuations of the cavity resonance under the detunedcondition. Finally, we expect that experimental demonstrationof the Fano effect is possible within current cavity QED se-tups. However, it would be still challenging to achieve the highdegree of the overlap parameter, η.
V. CONCLUSIONS AND OUTLOOK
We have presented a detailed analysis on the Fano effectin the cavity QED system, where the TLS is simply coupledwith the single mode cavity. Although such a system has beendiscussed by many authors in the past, the Fano effect hasbeen implicitly neglected in most cases. As a result, little wasknown about the Fano effect in this system. In our view, onereason is the absence of a flexible and systematic approach,based on the modern theories of open quantum systems.Therefore, in the early part of the present paper, we havefirst formulated the Markovian QME, based on the typicalinteraction Hamiltonians of the cavity QED system. It wasthen shown that the cross terms of the individual interactionHamiltonians yield a simple but unfamiliar type of Liouvillesuperoperator. Based on this treatment, we have found that theFano effect can be successfully described and the Fano for-mula can be generalized over the weak- and strong-couplingregimes with the pure dephasing effect. We have thus clearlyshown that the Markovian QME approach is advantageous forthe description of the Fano effect.
In the later part, on the other hand, we have focused on theemission spectra. Based on the same interaction Hamiltoni-ans, we have formulated the emission spectra, starting fromthe discussion on the simple intensity detection. As a result,the emission spectra were expressed in a consistent mannerwith the QME. Furthermore, it was numerically shown thatthe emission line at the TLS transition energy undergoesa strong intensity reduction by the destructive interferencein collaboration with the pure dephasing effect. This phe-nomenon can be observed even if the suppression of thetransition rate is largely washed out by the pure dephasing.By studying the decomposed spectra, then, we have clarifiedthe underlying mechanism, and finally concluded that theinterference between the two dissipation channels is itselftolerant to the pure dephasing, in contrast to the expectationthat the impact of interference is sensitive to decoherenceprocesses. This is because the two dissipation channels aresubjected to identical phase fluctuations, and therefore, thephase difference between the two channels is not affected. Theinsufficient suppression of the transition rate can be attributedto the appearance of the AZE dissipation channel.
The results described in this paper provide a fundamentaland prototypical methodology to treat the Fano effect in var-ious contexts, although the present paper was devoted to theproblem of the initially excited TLS in the cavity QED system.One direction for future research is the lasing action by includ-ing the effect of excitation because the interference betweenthe two channels is robust against the dephasing process.Another interesting direction is a consideration of multiple
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emitters inside the cavity. Superradiance and/or subradiancemay be affected by the Fano effect. It would be also interestingto study other relevant systems, such as circuit QED systems,optomechanical systems, and plasmon systems. We believethat our approach can provide insights on the Fano effect ina wide range of fields.
ACKNOWLEDGMENTS
M.Y. greatly appreciates fruitful discussion with Prof.Susumu Noda and Dr. Takashi Asano at Kyoto University,Japan, where part of this work was done during the author’sdoctoral studies (2007–2010). A.L. thanks Dr. Benjamin Dwirand Prof. Eli Kapon for valuable discussion of the observedphenomenon at Ecole Polytechnique Fédérale de Lausanne,Switzerland. This work was supported by Japan Society forthe Promotion of Science KAKENHI Grant No. JP18K03454.
APPENDIX A: DERIVATION OF EQ. (9)
Here, we derive Eq. (9). In the interaction picture withrespect to H0, the von Neumann equation for the total densityoperator ρ(t ) can be written as
d
dtρ(t ) = −i[VS(t ) + HSB(t ), ρ(t )]. (A1)
Therefore, by inserting its integral form
ρ(t ) = ρ(0) − i∫ t
0ds[VS(s) + HSB(s), ρ(s)], (A2)
into Eq. (A1), we obtain
d
dtρ(t )
= −i[VS(t ) + HSB(t ), ρ(0)]
−∫ t
0ds{[VS(t ), [VS(s), ρ(s)]] + [VS(t ), [HSB(s), ρ(s)]]}
−∫ t
0ds{[HSB(t ), [VS(s), ρ(s)]]
+ [HSB(t ), [HSB(s), ρ(s)]]}.By applying the Born approximation, ρ(t ) � ρS(t ) ⊗ ρB, withTrB[HSBρB] = 0, the trace over the bath eliminates the termswith an odd number of HSB:
d
dtρS(t ) = − i[VS(t ), ρS(0)] −
∫ t
0ds[VS(t ), [VS(s), ρS(s)]]
−∫ t
0dsTrB[HSB(t ), [HSB(s), ρS(s) ⊗ ρB]].
(A3)
Now, from Eq. (A2), we have
ρS(t ) = ρS(0) − i∫ t
0ds[VS(s), ρS(s)],
and therefore, Eq. (A3) yields
d
dtρS(t )
= −i[VS(t ), ρS(t )]
−∫ t
0dτTrB[HSB(t ), [HSB(t − τ ), ρS(t − τ ) ⊗ ρB]],
where the integration variable is changed to τ = t − s in thesecond term. By applying the Markovian approximation, theupper limit of the integral goes to infinity with ρS(t − τ ) �ρS(t ). We can thus obtain Eq. (9). In the degree of accuracy,this approach is true up to second order in VS + HSB, instead ofHSB alone. This means that the higher-order terms neglectedin Eq. (9) still have terms in second order of HSB. In thissense, this approach is different from the standard one [18].However, Eq. (9) is advantageous because the system oper-ators of HSB directly appear in the final form of Liouvillesuperoperators; there is no need to decompose the systemoperators into the eigenoperators of HS.
APPENDIX B: THE DISSIPATOR DUETO THE FANO INTERFERENCE
Here, we derive the dissipator [Eq. (13)] due to the Fanointerference. By substituting Eq. (10) into Eq. (9), the crossterms give
−∫ ∞
0dτTrB[H (1)
SB (t ), [H (2)SB (t − τ ), ρS(t ) ⊗ ρB]]
= γF(ωc)
2e−iωc,21t (acρS(t )σ+ − σ+acρS(t )) + H.c. (B1)
and
−∫ ∞
0dτTrB[H (2)
SB (t ), [H (1)SB (t − τ ), ρS(t ) ⊗ ρB]]
= γF(ω21)
2e−iωc,21t (acρS(t )σ+ − ρS(t )σ+ac) + H.c.,
(B2)
where γF(ω) is
γF(ω) ≡ 2∑
ξζ∗
∫ ∞
0dτei(ω−ωτ ),
� 2π∑
ξζ∗ δ(ω − ω). (B3)
In the second line of Eq. (B3), for simplicity, we have ne-glected the principal value contribution in the formula∫ ∞
0dτei(ω−ω ) = πδ(ω − ω) + iP 1
ω − ω
.
Here, γF(ω) is zero if the spatial radiation patterns between theTLS and the cavity mode are orthogonal because ξζ
∗ = 0. In
contrast, γF(ω) plays an important role when the two radiationpatterns are close with each other. Hence, in addition to thesecond assumption in Sec. II, i.e., ξ � ξ (ω) and ζ � ζ (ω),we further introduce a phenomenological parameter η (0 �
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η � 1) that describes the degree of the overlap between thetwo radiation patterns. Eq. (B3) is then given by
γF(ω) � 2πei(θ21−θc )√η∑
|ξ (ω)||ζ (ω)|δ(ω − ω)
= 2πei(θ21−θc )√η∑
√γ
2πD(ω)
√κ
2πD(ω)δ(ω − ω)
= ei(θ21−θc )√ηγ κ, (B4)
where Eq. (8) has been used. As a result of Eqs. (B1), (B2),and (B4), we can obtain Eq. (13).
APPENDIX C: NUMERICAL RESULTS FOR γph = 3 μeV
We here show numerical results for γph = 3 μeV in Fig. 6.In comparison with Fig. 4(a), the spectra [Fig. 6(a)] are largelychanged although the modification of W by the pure dephas-ing is weak [γph � (γ + κ )/2, Fig. 6(b)]. In this situation,the amount of pure dephasing is so small that the cavityfeeding effect is limited, as can be seen for positive detuning(ω21 > ωc). In contrast, we can find a strong intensity reduc-tion at ν = ω21 with a simultaneous intensity enhancement atthe cavity resonance (ν = ωc) when ω21 − ωc ∼ −3.16 meV.This is essentially the same phenomenon as we have seen inFig. 5(a). Hence, we can notice that the key point for thisobservation is not the magnitude of the pure dephasing butthe presence of the pure dephasing. This is consistent with ourscenario that the AZE channel additionally opened by the puredephasing is highlighted by the destructive interference of the
FIG. 6. Spectra for γph = 3 μeV. Other parameters are the sameas in Figs. 4 and 5. (a) The dependence of the spectral intensity,S(ν, δs ), on the detuning. (b) The transition rate, W/γ , and the in-tegrated value,
∫dνS(ν, δs ) [Eq. (59)], as a function of the detuning.
For comparison, the transition rate for γph = 0.0 μeV is indicated bythe dotted line, which is identical to the solid line in Fig. 2(a).
two dissipation channels. This result also supports our under-standing that the quality of interference for the Fano effect isnot lost by the pure dephasing, as explained in the main text.
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