Theory of single molecule emission spectroscopy...elaborate detection schemes have been developed to...

Theory of single molecule emission spectroscopy

Golan Bel1, 2, 3, ∗ and Frank L. H. Brown3, †

1Department of Solar Energy and Environmental Physics,

Jacob Blaustein Institutes for Desert Research,

Ben-Gurion University of the Negev, Midreshet Ben-Gurion, 84990, ISRAEL2Center for Nonlinear Studies, Los Alamos National Laboratory,

Los Alamos, New Mexico 87545, USA3Department of Chemistry and Biochemistry and Department of Physics,

University of California, Santa Barbara, California 93106, USA

(Dated: April 8, 2015)

Abstract

A general theory and calculation framework for the prediction of frequency-resolved single molecule

photon counting statistics is presented. Expressions for the generating function of photon counts are de-

rived, both for the case of naive “detection” based solely on photon emission from the molecule and also

for experimentally realizable detection of emitted photons, and are used to explicitly calculate low-order

photon-counting moments. The two cases of naive detection versus physical detection are compared to

one another and it is demonstrated that the physical detection scheme resolves certain inconsistencies pre-

dicted via the naive detection approach. Applications to two different models for molecular dynamics are

considered: a simple two-level system and a two-level absorber subject to spectral diffusion.

PACS numbers: 82.37.-j, 05.10.Gg, 33.80.-b, 42.50.Ar

∗Electronic address: [email protected]†Electronic address: [email protected]

1

I. INTRODUCTION

Single molecule spectroscopy (SMS) is a widely-used and versatile experimental technique

in physics, chemistry and biology [1–12]. Often, SMS studies (experimental and theoretical)

[9, 12–50] focus on number statistics for the photons emitted from a molecule under laser irra-

diation. These statistics may literally count individual photons, or may examine fluctuations in

time-dependent intensities. In either case, significantly more information is obtained than would

be possible from a simple absorption experiment on a bulk sample.

In the majority of SMS studies, photons are detected indiscriminately, without frequency (or

polarization or propagation direction) resolution. However, SMS experiments that resolve emitted

photons by color have the potential to reveal even more information from spectroscopically active

systems than do their broadband counterparts. Perhaps the best known example of multi-color

detection schemes in SMS involves the exploitation of resonance energy transfer as a molecular

ruler [8, 26, 27, 51], where two differently colored dyes are attached to a molecule or super molec-

ular assembly to monitor system dynamics via the simultaneous detection of photons emitted from

both dyes. Similar studies can also be performed with semiconductor quantum dots [52] and more

elaborate detection schemes have been developed to extend two-channel photon detection to mul-

tiple channels [53] with an eye toward the study of complex condensed phase and biologically

relevant systems. Yet, even the simplest physical systems can yield surprisingly complex structure

in their emission spectrum, indicating the utility of achieving frequency resolution in the emission

spectroscopy for any SMS system. For example, the emission spectrum from a simple two-level

absorber was predicted by Mollow [54], and subsequently experimentally observed in various

systems [55–60], to display a triplet structure when driven sufficiently strongly or off-resonance.

Theoretical approaches building upon the pioneering work of Mollow have extended predic-

tions for the emission spectrum of a two-level absorber to the calculation of frequency dependent

intensity fluctuations and correlations [61–64]. However, these studies have focused primarily on

analytical approaches that are confined to certain limiting regimes and are applicable only to spe-

cific observables. Much of this work has been motivated by the beautiful experimental work of

Aspect et al. [58], where strong temporal correlations were observed between opposite sidebands

in the off-resonance excitation of strontium atoms. As important as this pioneering work is, the

experiment was designed to allow for the simplest possible theoretical description - based upon

a two-level emitter and widely separated emission peaks that can be unambiguously assigned to

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individual detectors. It is not immediately clear how the theoretical tools developed for this ideal-

ized situation could be carried over to more complex condensed phase systems that are typically

studied via SMS. In Ref. [37], we suggested an alternate approach to calculating photon emission

statistics that is not limited to two level systems. However, that approach is limited to situations

where all emitted photons are associated with individual spectral transitions that are sufficiently

well-separated to be unambiguously associated with corresponding detector channels. The ap-

proach can be viewed as an extension of generating function methods [65] developed to interpret

single-pair fluorescence resonance energy transfer (FRET) measurements [26, 27], but with the

ability to include quantum dynamics of the system and multiple detection channels.

The methods discussed above are not applicable to situations involving poorly resolved spectral

lines and a general treatment of photon emission statistics should be able treat such a case. In a

preliminary report [66], we presented a somewhat general scheme for the calculation of frequency

resolved photon counting statistics that is capable of dealing with poorly resolved spectral lines

and multi-state dynamics. Indeed, the only approximations of this approach are the assumption

of Markovian dynamics for the driven chromophore system and the assumption that moments of

the photon number operator for various field modes correspond to observable physical quantities.

In this work, we will refer to the methodology of ref. [66] as involving the “emission based”

approach to photon statistics. The purpose of this paper is three-fold. First, we present details

and an extended discussion of our emission based treatment introduced in ref. [66]. Second,

we demonstrate that the emission based approach can lead to negative photon intensities at short

times, which is nonsensical in the context of experimental measurements. Third, we generalize the

emission based approach to explicitly account for the response of physically realizable detectors,

thereby removing the problem of negative intensities and providing concrete predictions for direct

comparison to experiments.

The paper is organized as follows. In Section II, we introduce the generating function for pho-

tons emitted into discrete frequency bins and the formal relationship between this construct and

various photon counting moments. In section III, we introduce a Hamiltonian for the combined

chromophore system, semi-classical continuous wave (CW) exciting field and quantum radiation

field describing the emitted photons. The dynamics of this composite system are solved to present

the generating function and moments of Sec. II in terms of time-ordered integrals of correlation

functions of the chromophore system operators. Section IV presents an explicit scheme for the

calculation of these correlation functions under the Markov approximation for chromophore dy-

3

namics. Though the correlation functions themselves are relatively simple to express in terms of

simple matrix operations, the time-ordering operations inherent to the moments are tedious and

the final expressions are derived and presented in Appendix A. The generalization of the forego-

ing results to account for photon detection (as opposed to photon emission) is presented in Sec.

V; these results are closely related to the emission based results and involve the same correlation

functions of chromophore operators. However, the required time-integrations are distinct from the

emission based scheme and the final expressions are presented in Appendix B. Sections VI and

VII present applications of the theory to model two-level and four-level chromophore systems,

respectively. In Sec. VIII we conclude.

II. THE GENERATING FUNCTION

Our starting point is the Heisenberg representation of the number operator for photons with

wave vector−→k and polarization −→ε , which is defined using the creation and annihilation operators

as

Nk,ε (t) = â†k,ε (t) âk,ε (t) . (1)

Here, â†k,ε, âk,ε are the creation and annihilation operators, respectively, for photons with wave

vector−→k and polarization −→ε and t is the time.

The moment generating function is defined in a manner analogous to the usual classical defini-

tion, namely

G (s⃗, t) ≡

⟨∏k,ε

sNk,ε(t)

k,ε

⟩. (2)

The auxiliary variables sk,ε are introduced to facilitate extraction of photon statistics and the vec-

tor s⃗ ≡ {sk,ε}, on the left hand side of Eq. (2), is simply a shorthand notation to indicate the

collective set of auxiliary variables associated with all possible photon wave vectors and polariza-

tions. The averaging operation denoted by the angular brackets indicates a quantum mechanical

average over the initial density matrix of the combined molecule / radiation field system that will

be further specified in the following section, i.e. ⟨...⟩ ≡ Trace {...ρ (0)}. From the definition of

the generating function, it follows that the factorial moments of the number operator Nk,ε (t) are

obtainable by differentiating the generating function and setting s⃗ = 1. To be more explicit, the

4

nth (n is a positive integer) factorial moment is given by

⟨N

(n)k,ε (t)

⟩≡ ⟨Nk,ε (t) (Nk,ε (t)− 1) ... (Nk,ε (t)− n+ 1)⟩ =

∂nG (s⃗, t)

∂snk,ε

∣∣∣∣∣s⃗=1

. (3)

Similarly, the multivariate factorial moments are given by

⟨N

(n)k,ε (t)N

(m)k′ε′ (t)

⟩=∂n+mG (s⃗, t)

∂snk,ε∂smk′ε′

∣∣∣∣∣s⃗=1

, (4)

(n and m are positive integers) with extension to moments involving more than two photon modes

following immediately by introducing more derivatives. Making use of the identity xy = ey lnx,

Eq. 2 can be recast as

G (s⃗, t) =

⟨exp

(∑k,ε

â†k,ε(t)âk,ε(t) ln (sk,ε)

)⟩

=

⟨N exp

(∑k,ε

â†k,ε(t)âk,ε(t) (sk,ε − 1)

)⟩(5)

The second equality in Eq. (5) follows from a standard operator identity [67] and employs the

normal ordering operator, N . The normal ordering operator acts on all creation and annihilation

operators appearing to its right and rearranges them by placing the creation operators to the left

of the annihilation operators. When the generating function is written in this form, the derivatives

indicated in Eq. (4) immediately lead to the rather simple expression for the factorial moments⟨N

(n)k,ε (t)N

(m)k′ε′ (t)

⟩=⟨(â†k,ε (t)

)n (â†k′ε′ (t)

)m(âk′ε′ (t))

m (âk,ε (t))n⟩ . (6)The formalism presented in this section is completely general and applies to all photons in the

system, regardless of their origin (e.g. photons from vacuum fluctuations, photons emitted from

chromophores, photons from an applied field, etc.). Our primary concern in this work is with

photons emitted from a driven chromophore; section III details the model used to study these

photons and the manipulations necessary to cast the equations of this section into a form suitable

for practical calculations.

III. MODEL SYSTEM AND ITS DYNAMICS

The content of Eqs. 7 - 20, below, should be viewed as the natural extension of well-known

results in quantum optics for two-level chromophores to multi-level chromophore systems. These

5

results are presented in concise fashion, primarily to introduce notation and precisely define our

system; more detailed discussions (in the context of two-level chromophores) are available in Refs.

[62, 68, 69]. The Hamiltonian, Ĥ(t), of the systems considered in this work is of the form

Ĥ(t) = Ĥsys + ĤIe(t) + ĤI + ĤR. (7)

Ĥsys is the unperturbed Hamiltonian of the molecule-environment system that is being irradiated

by a laser source and is thus driven into emitting photons. ĤIe(t) stands for the interaction between

the system and the applied laser field; this interaction is treated semi-classically leading to an ex-

plicit time dependence within the Hamiltonian. ĤI describes the interaction of the atom/molecule

with the quantized radiation field. ĤR is the Hamiltonian of the quantized radiation field.

We will focus on model systems consisting of two electronic levels (ground |g⟩ and excited |e⟩

), so that

Ĥsys = |g⟩Hg ⟨g|+ |e⟩He ⟨e|+ Ĥb + V̂ ≡ ĤCH + Ĥb + V̂ (8)

Hg and He are the Hamiltonians for nuclear motion within the ground and excited states of the

chromophore (CH), respectively, with eigenfunctions and eigenvalues specified by

Hg |ng⟩ = ℏωng |ng⟩

He |me⟩ = ℏωme |me⟩ (9)

for ng = 1, ...Nmaxg, me = 1, ...Nmaxe. We will be concerned only with a finite number of

states Nmaxg and Nmaxe. Ĥb is the Hamiltonian of the thermal bath (environment), and V̂

describes the interaction of the chromophore with the environment. Note that we use the identifier

“sys” to specify that part of the total Hamiltonian associated with the “system”, as opposed to

the quantum radiation field and applied perturbing field. The designation “CH” reflects that sub-

portion of the system (the chromophore itself) that is actually responsible for interaction with

radiation. Those portions of the system not associated with the chromophore itself constitute a

reservoir/bath/environment that may be capable of inducing relaxation and thermal fluctuation

among the various chromophore states.

The system is driven by a CW plane-polarized external laser field characterized by wavevector−→k L and frequency ωL = c|

−→k L| such that

−→E e (−→r , t) = −→E 0ei

−→k L·−→r cos (ωLt) . (10)

6

Assuming that the molecule is located at −→r = 0 and treating the interaction within the dipole

approximation, we are led to

ĤIe = −−̂→µ ·−→E e (−→r = 0, t) = −ℏΩ0

(D̃+ + D̃−

)cos (ωLt) . (11)

On the right hand side of the above equation, we have used the dipole moment operator, calculated

in the Condon approximation, such that

−̂→µ = −→µ 0(D̃+ + D̃−

)(12)

where D̃+/D̃− are the raising/lowering operators between chromophore excited and ground states,

with matrix elements

D̃+me;ng = ⟨me |ng⟩

D̃−ng ;me = ⟨ng |me⟩ . (13)

We stress that the D̃± operators live entirely within the chromophore (CH) portion of the total

system Hilbert space. We also introduced the Rabi frequency

Ω0 =

−→E 0 · −→µ 0

ℏ. (14)

We stress that the semi-classical treatment for the applied field described above should not be

viewed as an approximation. Eq. 11 can be derived from a fully quantum-mechanical treatment

of the incident field without approximation [70, 71]. The semi-classical representation of the

interaction is very convenient for the analysis that follows.

The system is also coupled to the quantum radiation field. This interaction is written as

ĤI = −−̂→µ ·−→E q (0) = −i

(D̃+ + D̃−

)∑k,ε

Ek

(âk,ε − â

†k,ε

)(−→µ 0 · −→ε ) . (15)

In the above Ek =√

ℏωk2ε0V0

where ε0 is the vacuum permittivity and V0 is the volume of the cubic

box used to quantize the field (in what follows, the limit V0 →∞ will be taken, such that V0 does

not appear in any final results).

The radiation Hamiltonian is

ĤR =∑k,ε

ℏωk(â†k,εâk,ε +

1

2

). (16)

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A. The dynamics of the creation and annihilation operators and their relation to the system

dipole moment operators

In what follows, it is convenient to work with the creation and annihilation operators in the

interaction picture,

ak,ε (t) ≡ âk,ε (t) eiωkt

a†k,ε (t) ≡ â†k,ε (t) e

−iωkt, (17)

where ωk = c∣∣∣−→k ∣∣∣, c is the speed of light and the operators âk,ε (t) and â†k,ε (t) are the Heisenberg

operators associated with âk,ε and â†k,ε, respectively. The number operator of Eq. 1 conveniently

remains Nk,ε (t) = a†k,ε (t) ak,ε (t) in the interaction representation. Similarly, any operator in-

volving only pairs of â†k,ε and âk,ε (e.g. Eq. 6) retains its functional form when moving to the

interaction representation due the cancellation of the positive and negative complex exponentials.

Further, it is convenient to introduce the slowly varying rotating frame operators for the system

operators

D± ≡ e∓iωLtD̃±. (18)

In section II, we expressed the generating function for the number of photons in terms of

photon creation and annihilation operators. Our goal is to express all photon counting moments

in terms of the system operators alone. For this purpose, we derive the dynamics of the field

operators in relation to the dipole moment operators of the system. The dynamics of the creation

and annihilation operators are given bydak,ε (t)

dt=

1

iℏ

[ak,ε (t) , Ĥ (t)

]+ iωkak,ε (t) =

Ek (−→µ 0 · −→ε )ℏ

(D+ei(ωk+ωL)t +D−ei(ωk−ωL)t

),

da†k,ε (t)

dt=

1

iℏ

[a†k,ε (t) , Ĥ (t)

]− iωka†k,ε (t) =

Ek (−→µ 0 · −→ε )ℏ

(D+ei(−ωk+ωL)t +D−e−i(ωk+ωL)t

),

(19)

which follow immediately by applying the Heisenberg equations of motion to Eq. 17.

Applying the RWA to the equations above (i.e., neglecting terms oscillating with the laser

frequency, or higher) yields the formal solutions

ak,ε (t) = ak,ε (0) +Ek (−→µ 0 · −→ε )ℏ

t∫0

D− (τ) eiωkLτdτ

a†k,ε (t) = a†k,ε (0) +

Ek (−→µ 0 · −→ε )ℏ

t∫0

D+ (τ) e−iωkLτdτ. (20)

8

In the above equations ωkL ≡ ωk − ωL.

B. The generating function in terms of system operators

In principle, the results of Eq. 20 could be substituted directly into the expressions of sec.

II to derive expressions for photon counting statistics. Unfortunately, the field operators at time

zero do not commute with the system operators at positive times and this naive substitution results

in cumbersome theoretical expressions involving both field operators at time zero and system

operators over all positive times entangled together. We can simplify matters considerably by

recognizing that Eq. 5 can be rewritten

G (s⃗, t) = exp

[∑k,ε

(sk,ε − 1)∂

∂pk,ε

∂

∂rk,ε

] ⟨e∑

k,ε pk,εa†k,ε(t)e

∑k,ε rk,εak,ε(t)

⟩∣∣∣−→p =−→r =0 (21)where the variables pk,ε and rk,ε have been introduced as a mathematical device to extract the

required number of creation and annihilation operators via differentiation; they do not appear in

the generating function as they are set to zero following the indicated differentiations. Writing the

generating function in this way enforces the normal ordering operation explicitly. Looking at Eq.

20, we see that the time derivatives of ak,ε(t) commute with ak,ε(t) since the derivative involves

only the system operator D−(t) and no field components; similarly for the creation operators and

associated time derivative. This implies that we may write

∂

∂te∑

k,ε pk,εa†k,ε(t) = e

∑k,ε pk,εa

†k,ε(t)

(∑k,ε

pk,εEk (−→µ 0 · −→ε )ℏ

D+(t)e−iωkLt

)∂

∂te∑

k,ε rk,εak,ε(t) =

(∑k,ε

rk,εEk (−→µ 0 · −→ε )ℏ

D−(t)e+iωkLt

)e∑

k,ε rk,εak,ε(t)

which can be solved in terms of time-ordered exponentials of system operators

e∑

k,ε pk,εa†k,ε(t) = e

∑k,ε pk,εa

†k,ε(0)−→T exp

(∫ t0

dt1∑k,ε

pk,εEk (−→µ 0 · −→ε )ℏ

D+(t1)e−iωkLt1

)

e∑

k,ε rk,εak,ε(t) =←−T exp

(∫ t0

dt1∑k,ε

rk,εEk (−→µ 0 · −→ε )ℏ

D−(t1)e+iωkLt1

)e∑

k,ε rk,εak,ε(0). (22)

The operator←−T serves to rearrange the operators appearing to its right in standard time order, with

later times appearing to the left of earlier times. The operator−→T rearranges the operators in the

reversed time order, with later times appearing to the right of earlier times. Assuming that the

9

composite system / field density matrix factorizes at t = 0 and that the quantum field contains

zero photons at this time (i.e. ρ(0) = σsys(0)⊗|0⟩⟨0| with σsys(0) the reduced system density

matrix), the disentangled expressions of Eq. 22 lead immediately to⟨e∑

k,ε pk,εa†k,ε(t)e

∑k,ε rk,εak,ε(t)

⟩(23)

=

⟨ {−→T exp(∫ t0dt1∑

k,ε pk,εEk(−→µ 0·−→ε )

ℏ D+(t1)e

−iωkLt1

)}×{←−T exp

(∫ t0dt1∑

k,ε rk,εEk(−→µ 0·−→ε )

ℏ D−(t1)e

+iωkLt1

)} ⟩ .Substituting this expression in Eq. 21 yields our final expression for the generating function

G (s⃗, t) =

⟨TN exp

∑k,ε

E2k (−→µ 0 · −→ε )

2

ℏ2(skε − 1)

t∫0

t∫0

D+ (u)D− (v) e−iωkL(u−v)dudv

⟩ .(24)

The operator TN acts on all operators to the right of it by first arranging all the “+” operators to

the left of all the “−” operators and subsequently placing all “−” operators in standard time order

(latest times at the left) and all “+” operators in reversed time order (latest times at the right).

Eq. 24 is fully general and opens up the possibility to calculate a seemingly limitless array of

statistics related to photon emission, once the dynamics for the radiating system are established.

In the interest of both concreteness and simplicity, we focus further attention on statistical quan-

tities related to the number of photons emitted within a given frequency window, irrespective of

propagation direction and polarization state. The number operator for photons emitted within such

a window is defined as

N(ω,∆) ≡∑

k,ε:(ω−∆2 )≤ωk≤(ω+∆2 )

â†k,εâk,ε. (25)

where the indicated sum over k, ε is understood to include all photon modes consistent with the

indicated frequency window centered at ω with width ∆. Our previously introduced generating

functions (Eqs. 2, 5 and 24) are readily converted to this frequency resolved picture by consoli-

dating all photon modes associated with (ω,∆) together under the common auxiliary variable sω.

10

This leads to

G (s⃗, t) =

⟨∏ω

sN(ω,∆)(t)ω

⟩(26)

=

⟨N exp

(∑ω

N(ω,∆)(t) (sω − 1)

)⟩

=

⟨TN exp

Γ02π

∑ω

(sω − 1)∫ ω+∆/2ω−∆/2

dω′t∫

0

t∫0

D+ (u)D− (v) e−i[ω′−ωL](u−v)dudv

⟩ .In the final expression we have carried out the sum over wave vector directions and polarizations

by moving to the continuum limit of V0 → ∞ where the sums become integrals [68]. Here,

Γ0 ≡ω3eg |µ⃗0|2

3πε0ℏc3 and ωeg is the frequency corresponding to the energy gap between the excited and

ground electronic states of the molecule, i.e. ωeg = ωme − ωng where it is assumed that the elec-

tronic energy scale overwhelms the nuclear energetics and ωme − ωng is effectively constant for

all values of ng and me. This result approximates a factor of ω′3 within the frequency integral

(sum) by the constant factor ω3eg, which is analogous to the same approximation introduced in the

Wigner-Weisskopf treatment of spontaneous emission [68]. The constant Γ0 is, in fact, the sponta-

neous emission rate for a two-level absorber with frequency splitting ωeg. It is critical to recognize

that the sum (product) over frequencies appearing in Eq. 26, with windows defined by the sum-

mation in Eq. 25, assumes that the chosen frequency windows are completely non-overlapping.

Overlapping windows would invalidate the second and third lines of Eq. 26 as these results rely

upon the operator identity introduced in Eq. 5, which assumes that the creation/annihilation oper-

ators associated with different auxiliary variables commute with one another. We will also assume

for convenience, though it is not essential, that every frequency window considered shares the

same width, ∆. We also stress, that the vector −→s in Eq. 26 now represents the set of frequency

associated variables {sω}, in contrast to the usage introduced previously. This should not lead to

any confusion below. The case of broadband statistics, where all emitted photons are consolidated

under a single auxiliary variable s, is realized by taking the limit ∆→∞ in Eq. 26. The resulting

frequency integral reduces to a delta function in this limit, leaving

G(s, t) =

⟨TN exp

[(s− 1)Γ0

∫ t0

dt′D+(t′)D−(t′)

]⟩. (27)

This expression is equivalent to other formulations of broadband photon emission statistics in res-

onance fluorescence [34, 72]. In particular, broadband emission statistics calculated with Eq. 27

reproduce the predictions of simplified “generating function” calculations which count all photons

11

indiscriminately via tracking instantaneous spontaneous emission events from the chromophore

[33, 34, 37, 65]. However, it is worth emphasizing that the simplified approach used in the cal-

culation of broadband statistics can not be extended to the general case. Decay of an electronic

excitation into a particular field mode or narrow band of modes is a non-Markovian process and

there is no simple protocol to relate spontaneous emission events to photon emission with fre-

quency resolution.

C. Expressions for the moments in terms of system operators

Performing differentiations on Eq. 26 with respect to the sω variables results in factorial mo-

ments analogous to the mode resolved expressions previously discussed. For example,⟨N

(n)(ω,∆) (t)N

(m)(ω′,∆) (t)

⟩= (28)

(Γ02π

)n+m⟨ TN ω+

∆2∫

ω−∆2

dω1

t∫0

t∫0

D+ (u)D− (v) e−iω1L(u−v)dudv

n

×

ω′+∆

2∫ω′−∆

2

dω2

t∫0

t∫0

D+ (u)D− (v) e−iω2L(u−v)dudv

m

⟩,

which is general enough to cover all expressions we explicitly compute below.

In what follows, we will focus on a few quantities of special interest. The average number of

photons radiated into a given frequency window is provided by

⟨N(ω,∆) (t)

⟩=

Γ02π

ω+∆2∫

ω−∆2

dω1

t∫0

t∫0

⟨D+ (u)D− (v)

⟩e−iω1L(u−v)dudv. (29)

Another quantity of interest is the second multivariate moment, i.e.,

C2(ω,ω′,∆) (t) ≡⟨N(ω,∆) (t)N(ω′,∆) (t)

⟩− δω,ω′

⟨N(ω,∆) (t)

⟩=

Γ204π2

ω+∆2∫

ω−∆2

dω1

ω′+∆2∫

ω′−∆2

dω2

t∫0

t∫0

t∫0

t∫0

×

⟨TND+ (u)D+ (v)D− (w)D− (y)

⟩eiω1L(u−w)eiω2L(v−y)dudvdwdy. (30)

12

Carrying out both the normal and time ordering, the expression becomes

C2(ω,ω′,∆) (t) =Γ204π2

ω+∆2∫

ω−∆2

dω1

ω′+∆2∫

ω′−∆2

dω2

t∫0

t∫0

t∫0

t∫0

× (31)

⟨D+ (u)D+ (v)D− (w)D− (y)

⟩

eiω1L(u−w)eiω2L(v−y)

+eiω1L(u−y)eiω2L(v−w)

+eiω1L(v−y)eiω2L(u−w)

+eiω1L(v−w)eiω2L(u−y)

Θ(v − u)Θ (w − y) dudvdwdy,

where Θ(t) is the Heaviside theta function.

From Eqs. (29) and (30), we introduce a normalized correlation function as

C∆ (ω, ω′, t) =

⟨N(ω,∆) (t)N(ω′,∆) (t)

⟩−⟨N(ω,∆) (t)

⟩ ⟨N(ω′,∆) (t)

⟩√⟨N(ω,∆) (t)

⟩ ⟨N(ω′,∆) (t)

⟩ − δω,ω′ . (32)The chosen normalization ensures that, in the specific case of ω = ω′, the correlation function

reduces to Mandel’s Q parameter for the number of photons emitted within a given frequency

range, defined as

Q∆ (ω, t) = C∆ (ω, ω, t) =

⟨N(ω,∆) (t)N(ω,∆) (t)

⟩−⟨N(ω,∆) (t)

⟩2⟨N(ω,∆) (t)

⟩ − 1. (33)IV. CALCULATION OF THE MOMENTS FOR THE MODEL SYSTEM OF SEC. III

A. Evaluating correlation functions of chromophore operators

To make practical use of the expressions appearing in sec. III C, one must specify how the

multipoint correlation functions, central to these formulae, are to be evaluated. To this end, we in-

troduce a Markov approximation for evolution of chromophore dynamics; the relevant correlation

functions involve only D± operators acting in the chromophore (CH) space of the total system

Hilbert space. The Markov approximation involves the traditional set of assumptions routinely

introduced in the derivation of the optical Bloch equations / optical master equation and Redfield

theory; namely 1) a weak interaction between the chromophore and both the quantum radiation

field and system bath (i.e., that the ĤI and V̂ appearing in Eqs. 7 and 8 are “small”); 2) much

faster correlation times within the bath and quantum field than the time-scales associated with

13

chromophore dynamics; and 3) negligible influence of the chromophore upon the bath and quan-

tum field such that the density matrices associated with bath and quantum field remain thermalized

over all times for the purposes of calculating the dynamics of the chromophore. It is well known

that the above assumptions will lead to an elementary dynamics for the reduced chromophore

density matrix in the rotating frame σCH(t) ≡ Traceb,R{ρ(t)}

σ̇CH(t) = LσCH(t)

σCH(t) = eL(t−t1)σCH(t1). (34)

Explicit expressions for the matrix representation of the time-independent rotating frame reduced

Liouville super-operator L are presented in Sec. IV A. For present purposes, we simply note

that σCH(t) contains (Nmaxg + Nmaxe)2 elements; the matrix representation of L has dimensions

(Nmaxg +Nmaxe)2 × (Nmaxg +Nmaxe)2.

The TN operator ensures that all of the correlation functions related to photon counting observ-

ables are of the form

⟨D+ (u1)D+ (u2) ...D+ (um)D− (vm)D− (vm−1) ...D− (v1)⟩ =

Trace {D− (vm)D− (vm−1) ...D− (v1) ρ(0)D+ (u1)D+ (u2) ...D+ (um)} (35)

with um ≥ um−1 ≥ ... ≥ u1 and vm ≥ vm−1 ≥ ... ≥ v1. Multipoint correlation functions with

the time-ordering characteristics displayed in Eq. 35 can be calculated through a straightforward

generalization of the considerations leading to Eq. 34 [73]. The result requires the introduction of

some new notation. First, one has to order all the times ui, vi into a single set of times τ1 . . . τ2m

such that τ2m ≥ τ2m−1.... ≥ τ1. Then, the super-operator D(i) is defined by its action upon an

arbitrary quantum mechanical operatorA depending upon wether time τi was originally associated

with a D+(uj) or a D−(vk) operator:

D(i)A = D−A = D−A if τi ⇐⇒ vk

D(i)A = D+A = AD+ if τi ⇐⇒ uj. (36)

(The operators D− and D+ appearing above are Schrödinger picture operators (in the rotating

frame) and carry no time dependence.) Then [73],⟨D+ (u1)D

+ (u2) ...D+ (um)D

− (vm)D− (vm−1) ...D

− (v1)⟩

(37)

= TraceCH{D(2m)eL(τ2m−τ2m−1)D(2m−1)eL(τ2m−1−τ2m−2)...D(2)eL(τ2−τ1)D(1)eLτ1σCH (0)

}.

14

For explicit calculations it is most convenient to reformulate Eq. 37 within the Liouville space

formalism [74]. Recall that Liouville space is the vector space composed of all linear operators

acting upon the state vectors of a quantum mechanical Hilbert space; this includes both Hermi-

tian operators and non-Hermitian operators. Adopting the bra-ket-like notation of Blum [74], a

standard quantum mechanical operator in the Schrödinger representation A is associated with an

element |A) in Liouville space. The inner product of Liouville space is defined as

(A|B) = Trace{A†B

}(38)

and the super-operators introduced above, which act on quantum operators to yield new quantum

operators, are just simple operators within Liouville space acting on Liouville space elements to

yield new Liouvile space elements. For example, we have various equivalent ways to express the

non-equilibrium expectation value of the operator A

⟨A(t)⟩ = Trace {Aρ(t)} = Trace{AeLtρ(0)

}= Trace

{([eLt]†A†

)†ρ(0)

}=(A†|ρ(t)

)=(A†|eLtρ(0)

)=(A†∣∣ eLt |ρ(0)) = ([eLt]†A†|ρ(0)) . (39)

Using this formalism, Eq. 37 may be written⟨D+ (u1)D

+ (u2) ...D+ (um)D

− (vm)D− (vm−1) ...D

− (v1)⟩

(40)

= (I|D(2m)eL(τ2m−τ2m−1)D(2m−1)eL(τ2m−1−τ2m−2)...D(2)eL(τ2−τ1)D(1)eLτ1 |σCH (0))

where I is the identity operator. We now consider Liouville state elements that are eigenfunctions

of L. These normalized elements and their associated eigenvalues are defined by

L |α) = λα |α)

(α|L = (α|λα

(α|β) = δα,β, (41)

and obey the completeness relation ∑α

|α)(α| = I (42)

with identity super-operator I. It is important to recognize that L is not Hermitian and that the left

and right eigenvectors are not expected to be simply related to one another. However, each distinct

eigenvalue is associated with a pair of left and right eigenvectors and these eigenvectors obey the

orhonormality condition indicated above. The zero eigenvalue λ0 = 0 is of particular importance;

15

it is associated with the steady state chromophore density matrix as a right eigenvector |0) = |ρss)

and the identity operator as a left eigenvector (0| = (I|. The remaining eigenvalues are expected

to have negative real parts indicating a decay to the steady state. By repeated use of Eq. 42, Eq.

40 becomes

⟨D+ (u1)D

+ (u2) ...D+ (um)D

− (vm)D− (vm−1) ...D

− (v1)⟩= (43)

∑α1···α2m

D(2m)α1,α2meλα2m (τ2m−τ2m−1)D(2m−1)α2m,α2m−1eλα2m−1(τ2m−1−τ2m) · · ·×D(2)α3,α2eλα2 (τ2−τ1)D

(1)α2,α1e

λα1τ1

(α1|σCH(0)).=

∑α2···α2m

D(2m)0,α2meλα2m (τ2m−τ2m−1)D(2m−1)α2m,α2m−1eλα2m−1 (τ2m−1−τ2m) · · ·×D(2)α3,α2eλα2 (τ2−τ1)D

(1)α2,0

(if σCH(0) = σss).

Dα,β ≡ (α|D|β)

As indicated, the third line holds true if the system begins in steady state at t = 0, which will

be assumed for the cases studied in this work. Equation 43 presents an explicit expression for

the multipoint chromophore correlation functions presented in this work. The primary burdens

associated with evaluating this expression involve diagonalization of L, transforming the D(i)’s to

the basis of L eigenfunctions and carefully accounting for the interleaving of u, v times to insure

that the individual D(i)’s are associated with the proper D± character depending upon the set of

time arguments.

Before proceeding to specific cases in the following two sections, we briefly comment on an

apparent paradox associated with the Markov approximation invoked in this section. A key as-

sumption of the Markov approximation is that the quantum field remain in thermal equilibrium

at all times, for the purposes of calculating chromophore dynamics (see above). For the optical

frequencies of interest in this work and typical laboratory temperatures, this corresponds to a field

effectively devoid of photons. However, the entire point of expression 43 is to use it to count

photons emitted into the quantum radiation field. The resolution to this apparent paradox is to

recognize that, although the chromophore emits photons which are measurable, these photons are

few in number and are not expected to significantly influence the dynamics of the chromophore

system. Photon emission from the chrmophore is driven entirely by the applied laser field. Stated

differently, the Markov approximation assumes that photons emitted from the chromophore have

no effect on future chromophore dynamics and future emissions, although the emission of these

photons into the field is explicitly tracked and quantified by Eq. 26 and derived quantities. If

16

we assume that all the emitted photons are immediately detected, and thereby removed from the

system, this assumption is quite reasonable.

B. Calculation of the first moment

Equation (29) can be written in a manifestly time-ordered form

⟨N(ω,∆) (t)

⟩=

Γ02π

2Re

ω+∆2∫

ω−∆2

dω1

t∫0

t∫0

⟨D+ (u)D− (v)

⟩e−iω1L(u−v)Θ(u− v) dudv. (44)

Applying Eq. 43 to the correlation function in the expression above enables us to write it as⟨D+ (u)D− (v)

⟩Θ(u− v) =

∑α

D+0,αD−α,0eλα(u−v)Θ(u− v) , (45)

where we have assumed σCH(0) = σss. It then follows that⟨N(ω,∆) (t)

⟩=

Γ02π

2Re

[∑α

D+0,αD−α,0nα (t, ω,∆)

], (46)

where

nα (t, ω,∆) ≡

ω+∆2∫

ω−∆2

dω1

t∫0

t∫0

e(λα−iω1L)(u−v)Θ(u− v) dudv

=

ω+∆2∫

ω−∆2

dω1et(λα−iω1L) − 1− t(λα − iω1L)

(λα − iω1L)2. (47)

The first moment of the number of emitted photons is proportional to t in the long time limit

and the time derivative of average emitted photon number saturates to constant at long times. We

thus define the intensity of emitted photons as

I(ω,∆) (t) ≡⟨dN(ω,∆) (t)

dt

⟩, (48)

which becomes

I(ω,∆) (t) =Γ02π

2Re

[∑α

D+0,αD−α,0xα (t, ω,∆)

], (49)

with xα (t, ω,∆) given by

xα (t, ω,∆) =

ω+∆2∫

ω−∆2

dω1

t∫0

e(λα−iω1L)(t−v)dv =

ω+∆2∫

ω−∆2

dω11− e(λα−iω1L)t

iω1L − λα. (50)

17

The long time limit is conventionally decomposed into a coherent intensity associated with elastic

scattering and an incoherent part via I(ω,∆) = I(ω,∆)coh + I(ω,∆)incoh with

I(ω,∆)coh = Γ0⟨D+⟩ ⟨D−⟩δω,ωL = Γ0D

+0,0D−0,0δω,ωL , (51)

and

I(ω,∆)incoh =Γ02π

2Re

∑α

′D+0,αD−α,0

ω+∆2∫

ω−∆2

dω11

iω1L − λα

, (52)where the prime denotes the fact that we omit the 0 eigenvalue from the summation. It is to be

understood that the Kronecker delta function appearing in Eq. 51 assumes the value of one when

ωL lies within the ∆-wide frequency window centered at ω and is otherwise zero.

The formalism detailed in this section allows the calculation of the emission spectrum for any

detuning frequency (within the validity regime of the RWA) and for any system of the form de-

scribed in section III (as evaluated within the Markov approximation detailed in sec. IV A). It

is important to note that the spectrum defined by Eqs. 48 - 50 is the quantum equivalent to the

Page [75] and Lampard [76] spectra. Thus, for short times, it can yield apparently non-physical

results [77]. In particular, I(ω,∆) (t) can assume negative values at a given time, which would cor-

respond to the disappearance of previously emitted photons from the field. While such a process

(reabsorption of photons previously emitted by the chromophore) is perfectly allowable within

quantum mechanics, the possibility of negative “intensities” raises serious questions about how

one intends to actually detect photons experimentally - certainly photons that have been registered

on a measuring apparatus can not subsequently be unmeasured. The question of the experimentally

measured spectrum and higher order statistics will be discussed in sec. V.

C. Calculation of the second multivariate moment

The second multivariate moment is described in equations (30,31). The nature of the multi-

variate second moment requires that it be symmetric under ω ⇌ ω′ exchange. In order to use thissymmetry, we write it as

C2(ω,ω′,∆) (t) =Γ204π2

ω+∆2∫

ω−∆2

dω1

ω′+∆2∫

ω′−∆2

dω2 [f (t, ω1, ω2) + f (t, ω2, ω1)] , (53)

18

where

f (t, ω1, ω2) =

t∫0

t∫0

t∫0

t∫0

dudvdwdy× (54)

⟨D+ (u)D+ (v)D− (w)D− (y)

⟩ eiω1L(u−w)eiω2L(v−y)+eiω1L(u−y)eiω2L(v−w)

Θ(v − u)Θ (w − y) .In order to calculate the correlation function of Eq. (32), we express

⟨N(ω,∆) (t)

⟩ ⟨N(ω′,∆) (t)

⟩in

a way that will enable us to subtract it from the second moment. Using Eq. (29), we write it as

⟨N(ω,∆) (t)

⟩ ⟨N(ω′,∆) (t)

⟩=

Γ204π2

ω+∆2∫

ω−∆2

dω1

ω′+∆2∫

ω′−∆2

dω2 [h (t, ω1, ω2) + h (t, ω2, ω1)] , (55)

where

h (t, ω1, ω2) =1

2

t∫0

t∫0

t∫0

t∫0

⟨D+ (u)D− (v)

⟩ ⟨D+ (w)D− (y)

⟩e−iω1L(u−v)e−iω2L(w−y)dudvdwdy.

(56)

Subtracting the expression of Eq. (55) from the expression of Eq. (53), we can formally write the

numerator of the correlation function C∆ (ω, ω′, t) (Eq. (32)) as

C2(ω,ω′,∆) (t)−⟨N(ω,∆) (t)

⟩ ⟨N(ω′,∆) (t)

⟩=

Γ204π2

ω+∆2∫

ω−∆2

dω1

ω′+∆2∫

ω′−∆2

dω2 [M (t, ω1, ω2) +M (t, ω2, ω1)]

(57)

with M (t, ω1, ω2) = f (t, ω1, ω2) − h (t, ω1, ω2). As written, this expression is of little practical

use. To obtain numerical values for the correlation function, M (t, ω1, ω2) must be expressed in

terms of elementary pieces that are simple to calculate. This tedious procedure is detailed in

Appendix A. The function M (t, ω1, ω2), together with the expression for the first moment in Eq.

(46), yields the correlation function C∆ (ω, ω′, t) (Eq. (32)) as a combination of terms that are

explicitly evaluated in Appendix A. Numerical values of the correlation function (Eq. 57) are

obtained via numerical integration over the frequencies ω1 and ω2 .

19

V. CALCULATION OF THE MOMENTS WITH EXPLICIT CONSIDERATION DETECTOR

EFFECTS

A. Correspondence between the results of the previous section and the standard theory of broad-

band photon detection

In the standard theory of photon detection [68, 69, 78], one typically focuses interest on corre-

lation functions of the quantized electric field. These quantities are closely related to, but different

from, those introduced in the preceding section. To establish the relationship between these pic-

tures, we first review the case of broadband photon detection.

In contrast to the previous section, we imagine counting photons that are actually measured by

a detector, as opposed to counting photons that have been emitted from the irradiated system. Pho-

todetection typically relies on technology that directly measures the intensity of the measured light

incident upon the detector, e.g. as in a photomultiplier. We imagine an idealized photodetector

with the following properties: 1) The detector completely covers a spherical shell surrounding the

system, which is located at the center of the shell; 2) The distance from the system to the detector,

r, is in the far field of the emitting system; 3) The detector displays no frequency or polarization

dependence and has a quantum efficiency of one; detector response is instantaneous and there is

no dead time. 4) The presence of the detector does not affect the plane-wave modes of the cavity

used to quantize the field; i.e., the photons are unaffected by the detector until the time at which

they impinge upon it and are absorbed.

The Heisenberg electric field operators for the quantum radiation field are written [62, 68, 69]

−→E (r, t) =

−→E +(r, t) +

−→E −(r, t) (58)

−→E +(r, t) = i

∑k,ε

−→ε Ekâk,ε(t)eik·r

−→E −(r, t) = −i

∑k,ε

−→ε Ekâ†k,ε(t)e−ik·r.

The creation and annihilation operators appearing above can be written as time integrals over the

system operators as shown in Eqs. 17 and 20. Making this substitution and performing the sums

20

over k, ε, by taking the V0 →∞ limit and evaluating as an integral, leads to

−→E +(r, t) = (59)[(

1− rrr2

)· −→µ 0

] c8π2ε0r

e−iωLt∫ ∞0

dkk2(eikr − e−ikr

) ∫ t0

dt1D−(t1)e

iωkL(t1−t) +−→E +f (r, t)

=[(

1− rrr2

)· −→µ 0

] ω2eg4πε0c2r

e−iωL(t−r/c)D−(t− r/c) +−→E +f (r, t) ≡

−→E +S (r, t) +

−→E +f (r, t),

where the second equality is obtained by extending the lower limit of the k integral to −∞ and

approximating k ∼ ωeg/c as discussed under Eq. 26. The remaining k (equivalently, ω) integral

reduces to a Dirac delta function, which explains the remarkable simplification seen in this step.−→E +f (r, t) represents the “free” contributions to the field that would be present even in the absence

of the radiating system (i.e. the contribution from the a(0) pieces in Eq. 20) and−→E +S (r, t) the

contribution due to radiation from the system. The explicit form provided for−→E +S (r, t) presumes

that the detector is in the far field; contributions of order 1/r2 and smaller have been intentionally

discarded. The similar calculation for−→E −(r, t) yields

−→E −(r, t) =

[(1− rr

r2

)· −→µ 0

] ω2eg4πε0c2r

eiωL(t−r/c)D+(t−r/c)+−→E −f (r, t) ≡

−→E −S (r, t)+

−→E −f (r, t).

(60)

The intensity operator for light incident on a given point of our far-field detector is [69]

I(r⃗, t) = 2ε0c−→E −S (r, t) ·

−→E +S (r, t) + terms including

−→E ±f . (61)

Dividing this quantity by the energy per photon, ℏωeg provides an operator corresponding to the

local photon flux into the detector at position r. We integrate this quantity over the detector

surface to provide the total rate of photon detection operator (all photons hitting the detector are

registered due to the assumed perfect efficiency) and further integrate over time to yield an operator

corresponding to the total number of detected photons in the interval [0, t]

ddN(t)

dt=

2ε0cr2

ℏωeg

∫dΩ−→E −(r, t) · −→E +(r, t) + terms including −→E ±f (62)

dN(t) =2ε0cr

2

ℏωeg

∫ t0

dt′∫dΩ−→E −(r, t′) ·

−→E +(r, t′) + terms including

−→E ±f (63)

with the Ω integral extending over the full 4π solid angle of the detector. Carrying out the angular

integration explicitly yields

dN(t) = Γ0

∫ t0

dt′D+(t′ − r/c)D−(t′ − r/c) + terms including−→E ±f . (64)

21

The probability of actually observing n photons at the detector over the time interval [0, t] is

a classic problem in quantum optics [72, 78, 79], which we shall not re-derive here. Using the

present notation, the result is

p(n, t) =

⟨T:[dN(t)]ne−

dN(t)

n!

⟩(65)

where the T: ordering operator arranges all E− fields (irrespective of S or f character) to the left

of the E+ fields and time-orders the (−) fields with latest times to the right and time-orders the

(+) fields with latest times to the left. Inserting this into the expression for the generating function

for detected photons dG(s, t) =∑pn(t)s

n yields

dG(s, t) =⟨T: exp

[(s− 1)dN(t)

]⟩. (66)

Although not obvious on first inspection, it can be shown [80, 81] that the terms involving−→E ±f

make no contribution to Eq. 66, which is why we have not been careful in explicitly tracking

these contributions above. This is due to the particular time ordering associated with T: and the

fact that causality dictates that the free field at time τ must commute with all system operators at

times t < τ . Any term involving a free field contribution can be written such that the free field

contribution(s) appear at the leftmost and rightmost edges of the expression to be averaged. The

averaging operation then renders these terms zero due to the assumed absence of photons in the

field at t = 0. The only terms that remain in the generating function are those comprised solely of

system operators and we may write

dG(s, t) =

⟨TN exp

[(s− 1)Γ0

∫ t0

dt′D+(t′ − r/c)D−(t′ − r/c)]⟩

. (67)

This expression is nearly identical to Eq. 27, which corresponds to the monitoring of all emitted

photons, irrespective of frequency, propagation direction and polarization. The only difference

appearing between the two expressions is the translation in time of the system operators in the

current expression by −r/c, reflecting the finite time it takes for the emitted photons to reach the

detector. With this single (trivial) exception, the two methods for calculating photon statistics are

identical as applied to broadband monitoring/detection schemes.

We emphasize that the correspondence between Eqs. 26 and 67 follows from our completely

idealized treatment of the detector, which ensures the detection of all emitted photons. Certain

aspects of a non-ideal detector cause only minor changes to this expression. If the detector is

assumed to occupy only some fraction of the spherical surface surrounding the emitting system,

22

this affects only the constant pre-factor multiplying the time integral in Eq. 67 due to an incomplete

integration over Ω in Eq. 63. Similarly, a detector with quantum efficiency η < 1 requires further

scaling of dN(t) by this factor. A slightly more general version of Eq. 67 can therefore be written

dG(s, t) =

⟨TN exp

[(s− 1)αdΓ0

∫ t0

dt′D+(t′)D−(t′)

]⟩. (68)

where the constant αd < 1 accounts for the inability of the detector to capture all emitted photons.

We have also translated the origin of time by r/c in this expression relative to Eq. 67 in order to

make the correspondence with Eq. 27 manifest.

B. Detection of frequency filtered photons

In section IV B, we briefly commented that frequency-resolved statistics based on photon emis-

sion, as opposed to photon detection, are problematic at short times and can give rise to unphysical

results. Some deficiencies of the emission approach will be explicitly demonstrated in the calcu-

lations of the following section. To remedy these shortcomings, it is necessary to consider the

detection process explicitly in the theoretical description. Unlike the case of broadband statistics,

there is no clear way to establish correspondence between experimental photon counting measure-

ments at finite times and the theoretical approach of sec. II; the approach of sec. V A must be

extended to account for the physical process of spectral filtering as realized experimentally.

The procedure we will adopt to model the effect of spectral filtering on photon detection was

first discussed by Eberly and Wodkiewicz [77] in the context of classical fields. They suggested

that considering the response function of the physical device used to filter the photons prior to

detection is necessary to compute the spectrum at finite times and that such an approach removes

the unphysical artifacts associated with schemes related to that presented in sec. IV B. Subsequent

work [63, 82] has confirmed that the procedure suggested by Eberly and Wodkiewicz [77] can be

rigorously justified within the context of a quantized radiation field. Both the classical approach

and practical implementations of the quantum approach assume that the presence of spectral filters

and the detectors themselves do not affect the dynamics of the emitting system - i.e., any radiation

scattered off the filters and detectors is ignored for the purpose of calculating the evolution of

the system, which is assumed to evolve (as discussed above) in a quantum radiation field driven

by a semiclassical CW excitation at frequency ωL. We assume a filtering / detection scheme as

diagrammed in figure 1.

23

Fluoresced light emitted from the system is intercepted by a filter immediately prior to imping-

ing upon the detector. The filter is introduced because of its spectral filtering properties on the

light incident upon it. For our purposes, this filtering is most conveniently expressed by the effect

on the electric field in the time domain [77]

−→E +D(r, t) =

∫ t−∞

dt′H(t− t′)−→E +(r, t′)

−→E −D(r, t) =

∫ t−∞

dt′H∗(t− t′)−→E −(r, t′). (69)

Here,−→E ±D(r, t) represents the operators associated with the electric field that reaches the detector

after passage through the filter, as compared to the bare fields−→E ±(r, t), from the previous section,

that would would be present in absence of the filter. The response function H(t) reflects the

influence of the filtering device. For practical purposes, we assume a functional form for H(t)

motivated by the action of a lossless and highly reflective Fabry-Perot interferometer with central

frequency ω1 and bandwidth ∆

H(t) = ∆e−(∆+iω1)t. (70)

This yields detectable fields of

−→E +D(r, t) =

[(1− rr

r2

)· −→µ 0

] ∆ω2eg4πε0c2r

∫ t−∞

dt′e−(∆+iω1)(t−t′)e−iωLt

′D−(t′) +

−→E +f (r, t)

−→E −D(r, t) =

[(1− rr

r2

)· −→µ 0

] ∆ω2eg4πε0c2r

∫ t−∞

dt′e−(∆−iω1)(t−t′)e+iωLt

′D+(t′) +

−→E −f (r, t),(71)

where we have invoked the same translation in time by r/c here as was introduced in Eq. 68.

Though this filter response can be generalized in principle, the computational scheme we adopt

below largely depends on the exponential form of H(t). Substituting these results for−→E ±D(r, t)

in place of−→E ±(r, t) in the equations of the preceding section (i.e. Eqs. 63 and 66) leads to an

expression for the generating function of spectrally filtered photons

dG (s,∆, t) = (72)⟨TN exp

∆2αdΓ0 (s− 1) t∫0

t1∫−∞

t1∫−∞

D+ (u)D− (v) e−iω1L(u−v)e∆(u+v−2t1)dudvdt1

⟩ ,where we have again introduced the αd factor to account for the fraction of solid angle covered by

the detector and the detector quantum yield. Considering multiple detectors arranged around the

24

sample, each with a different central frequency, but identical bandwidths and αd factors, this result

generalizes to

dG (s⃗,∆, t) = (73)⟨TN exp

∑ω1

∆2αdΓ0 (sω1 − 1)t∫

0

t1∫−∞

t1∫−∞

D+ (u)D− (v) e−iω1L(u−v)e∆(u+v−2t1)dudvdt1

⟩ .where each sω1 is the auxiliary variable associated with the counts at a particular detector. In what

follows, we will always make the assumption that

αd = 1, (74)

in order to make direct comparison between the results of two theoretical schemes we have dis-

cussed (based on emitted photons and detected photons, respectively). However, it should be

stressed that this situation is not physically realizable, even in an idealized gedanken experiment.

It is not possible to place multiple αd = 1 filtering detectors around the sample since each of these

perfect detectors requires a full 4π solid angle around emitter and the action of one filter would

necessarily interfere with other filters placed behind it. Notwithstanding this point, the αd factors

are simple constants that contribute to the theoretical predictions for the photon counting moments

in a transparent way. In the following subsections, we will focus on the first two moments of de-

tected photon number. These expressions are readily generalized to account for arbitrary values of

αd simply by replacing Γ0 → Γ0αd wherever factors of Γ0 appear.

C. Calculation of the first moment

The first moment is given by:

⟨dN (ω1,∆, t)⟩ = ∆2Γ0

t∫0

dt1e−2∆t1

t1∫−∞

du

t1∫−∞

dv⟨D+ (u)D− (v)⟩e−iω1L(u−v)e∆(u+v)

= 2∆2Γ0Re

∑α

D+0,αD−α,0

t∫0

dt1e−2∆t1

t1∫−∞

du

u∫−∞

dve(λα−iω1L)(u−v)e∆(u+v)

= 2t∆2Γ0Re

[∑α

D+0,αD−α,01

2∆(∆− λα + iω1L)

]. (75)

25

The second line follows from similar considerations as introduced in Sec. IV B and the third has

simply evaluated the integrals. The intensity follows immediately and is stationary in time

dI (ω1,∆) = 2∆2Γ0Re

[∑α

D+0,αD−α,02∆(∆− λα + iω1L)

]. (76)

D. Calculation of the second multivariate moment

The correlation function is defined as

dC∆ (ω1, ω2, t) ≡⟨dN(ω1,∆) (t)

dN(ω2,∆) (t)⟩−⟨dN(ω1,∆) (t)

⟩ ⟨dN(ω2,∆) (t)

⟩√⟨dN(ω1,∆) (t)

⟩ ⟨dN(ω2,∆) (t)

⟩ − δω1,ω2 , (77)with the generalized Q parameter following as

dQ∆ (ω, t) ≡ dC∆ (ω, ω, t) =

⟨dN2(ω,∆) (t)

⟩−⟨dN(ω,∆) (t)

⟩2⟨dN(ω,∆) (t)

⟩ − 1. (78)Using similar motivation to the approach outlined in Sec. IV C, the second multivariate moment

follows from Eq. 73 as

dC2(ω1,ω2,∆) (t) ≡⟨dN(ω1,∆) (t)

dN(ω2,∆) (t)⟩− δω1,ω2

⟨dN(ω1,∆) (t)

⟩= Γ20∆

4× (79)t∫

0

e−2∆t2dt2

t∫0

e−2∆t1dt1

t2∫−∞

du

t2∫−∞

dw

t1∫−∞

dv

t1∫−∞

dy⟨TND+ (u)D+ (v)D− (w)D− (y)

⟩× eiω1L(u−w)eiω2L(v−y)e∆(u+v+w+y).

To calculate the numerator of the correlation function, the product of first moments:

⟨dN (ω1,∆, t)⟩⟨dN (ω2,∆, t)⟩ = ∆4Γ40

t∫0

dt1

t∫0

dt2e−2∆(t1+t2)× (80)

t1∫−∞

du

t1∫−∞

dv

t2∫−∞

dw

t2∫−∞

dy⟨D+ (u)D− (v)⟩⟨D+ (w)D− (y)⟩e−iω1L(u−v)−iω2L(w−y)e∆(u+v+w+y)

must be subtracted. In Appendix B we show how the correlation functions of eqs. (79) and (80)

can be expressed as sums of simple terms. The explicit forms of the individual terms are provided

in Appendix B, allowing calculation of the detected correlation functions.

26

VI. RESULTS FOR A TWO-LEVEL CHROMOPHORE

The two-level system serves as the prototypical example of an irradiating chromophore in quan-

tum optics and, indeed, most theoretical studies to date concentrate focus solely on this simplest

possible emitting system. Adopting the notation of sec. III, the system and chromophore of the

two-level-system are the same; there is no bath coupled to the optically active chromophore. That

is, the terms V̂ and Ĥb are absent from Eq. (8) and

Ĥsys = ĤCH =

1

2ℏω0 (|e⟩⟨e| − |g⟩⟨g|) . (81)

The dipole moment operator is given simply by

D ≡ −→µ 0(D+ +D−

); D+ = |e⟩⟨g| ; D− = |g⟩⟨e| . (82)

The reduced chromophore density matrix σCH(t) is 2 × 2 in dimension and corresponds exactly

to the density matrix associated with the optical Bloch equations. The Liouville super-operator L

from sec. IV A is well known for this case with the matrix representation

L =

−Γ0 0 −iΩ02 +i

Ω02

Γ0 0 +iΩ02

−iΩ02

−iΩ02

+iΩ02−iδL − Γ02 0

+iΩ02−iΩ0

20 iδL − Γ02

, (83)

using our previously introduced notations for the spontaneous emission rate, Γ0 and Rabi fre-

quency, Ω0. δL = ωL − ω0 is the detuning frequency of the applied field. This matrix is assumed

to act on a vector representation of the density matrix ordered as σCH = (σee, σgg, σge, σeg)†. In

this same basis, the super-operators associated with the dipole operator are

D+ =

0 0 0 0

0 0 1 0

0 0 0 0

1 0 0 0

D− =

0 0 0 0

0 0 0 1

1 0 0 0

0 0 0 0

. (84)

27

The three matrices in Eqs. 83 and 84, when supplemented with the parameter ∆ quantifying

tracked/detected bandwidth, completely specify the photon counting statistics for both emission

and detection schemes presented in sections IV and V. Numerical diagonalization of L allows

the L, D+ and D− matrices to be transformed to the basis of L eigenstates, which then allows

for immediate (if tedious) calculations via the formulae derived and presented in appendices A

and B. Thus, for the two-level chromophore, four physical parameters completely determine the

photon counting statistics: the Rabi frequency Ω0, the spontaneous emission rate Γ0, the detuning

frequency δL and the observation bandwidth ∆. For brevity of notation, we will henceforth set

Γ0 ≡ Γ and Ω0 ≡ Ω when discussing the two level chromophore; there are only a single possible

spontaneous emission rate and a single possible Rabi frequency in the absence of nuclear motions

coupled to the electronic transition.

A. Statistics in the long observation time limit

Mollow [54] first showed that, in the case of strong laser field (Ω > Γ) or off-resonant excitation

(δL > Γ,Ω), the emission spectrum of a driven two-level chromophore shows a triplet shape, while

for low intensity near resonant excitation, the spectrum has a single peak. We consider both of

these regimes and compare the emission statistics of sec. IV and detection statistics of sec. V. In

this section, we consider only the limit of long observation times and explicitly avoid the more

problematic artifacts of the emission based scheme. Short time observations will be discussed in

the following section. The case of long-time emission statistics was considered in our preliminary

report [66] and the primary focus of this section is to contrast the results obtained via the emission

and detection approaches.

In order to demonstrate the effect of the detector, we show in Fig. 2 single peak results (Ω =

Γ/√2) for both the emission and detection schemes; both the spectrum itself (intensity) and Q

parameter spectrum are displayed. In the left panels, we used a detector of width ∆ = Γ/10,

comparable to the width of the peak. In the right panels, we used ∆ = Γ/104, which is orders

of magnitude narrower than the peak. In Fig. 3, we present similar plots for the case of a triplet

spectrum, corresponding to Ω = 5Γ/√2.

Figures 2 and 3 display clear differences between the spectra as obtained via the two different

prescriptions (emission vs. detection). The primary effect at play here is signal “leakage” between

adjacent detection intervals. The emission scheme of sec. IV precisely assigns photons to a single

28

frequency bin, however, the Lorentzian response function associated with physical detection is

inherently less discriminate. A physical detector can register photons falling directly within its

∆-wide frequency window and also photons with frequencies considerably more distant from the

window center. This convolution between properties of the physical field and response of the

detector leads to an “over-counting” in the detected intensity spectrum as multiple windows can

all lay claim to the same photon. Hence, the detection spectrum is shifted to substantially greater

magnitude than the corresponding emission spectra.

The case of the Q parameter spectrum is more subtle, although the disparity between emission

and detection cases derives from the same origin involving convolution between the field and

detector response. As discussed in our original report [66], binning the Q spectrum can result in

complete reversal of sign, proceeding from broadband measurement with only a single frequency

bin to multiple narrow frequency bins. The Q parameter is a non-linear function of the physical

field and there is no requirement that the detector response lead to a simple “overcounting” type

effect. Indeed, the physically detectable Q parameter spectrum is seen to include both regions

that are higher than and lower than the emission Q spectrum based solely on emission. Although

one could argue that the emission and detection plots in figs. 2 and 3 are qualitatively similar,

they contain obvious differences and speak to the importance of using the proper theory including

detector response in the modeling of any future experiments.

Fig. 4 explicitly shows that the signal expected from detected photons is as much a function of

the detector width as of the underlying physics of the physical field itself. Here, the two physical

regimes from figs. 2 and 3 are considered on the same plot. Only the middle frequency bin centered

on ω0 is considered and the signals are plotted as a function of detector width δ. The singlet and

triplet signals show a reversal in dominance at intermediate bin sizes; this is true for both the

intensity and Q parameter. So, while the standard broadband Q parameter of the singlet regime

is considerably more negative than the corresponding triplet regime quantity, at small bin sizes

there is exactly the opposite behavior. Note that the plateau seen at large ∆ reflects convergence

to broadband regime. As discussed in the previous sections, both emission and detection schemes

converge to the same results in this particular limit.

In Fig. 5, we show the correlations between photons emitted at different frequencies. The

correlations were calculated using the naive emission-based approach (Eq. 32) and the long time

limit was taken analytically using the results of Appendix A. The two panels correspond to the

singlet (Ω = Γ/√2) and the triplet (Ω = 5Γ/

√2) parameters. The observation width was set to

29

∆ = Γ/10, which is narrow enough to resolve frequencies within the individual emission peaks.

Both singlet and triplet cases display positive correlations along the ω,−ω diagonal reflecting

the importance of energy conservation [58, 66]; fluctuations with an enhanced number of +ω

emissions are correlated with an enhanced number of −ω emissions. As we have previously

noted [66], the narrow band Q parameter centered at the excitation frequency, corresponding to

the ω1L = ω2L pixel at the origin of the two panels (n. b. the definition in Eq. 33), is strongly

positive and reflects bunching statistics. Anti-correlations between the numbers of photons at other

frequencies are the origin of the anti-bunching observed in broad band photon counting statistics

[72].

In Fig. 6, we plot the correlations between photons detected at different frequencies accounting

for the detectors’ response functions (see Eq. 77). These plots parallel the analogous calculations

presented in Fig. 5, which neglect the detectors. Although the general features of these plots are

similar to those determined via consideration of photon emission, there are obvious differences.

Most apparent are the differences in the vicinity of the ω1L = 0 and ω2L = 0 axes. Unlike the

emission scheme of Fig. 5, the detector response allows the large contribution from the elastic or

“coherent” [54, 62] delta-fuction at ω1L = 0 to bleed over into nearby frequency regions. The

mixture of coherent and incoherent contributions within all detector windows extends the positive

and negative correlations to the whole range of the spectrum. The appreciable tails of the detector

response also weaken the correlations along the off-diagonal as the energy conservation condition

is smeared out relative to the emission-based statistics. We note that, while statistics based on

emission are not directly experimentally observable, the physical picture afforded by the emission

calculations is somewhat more transparent than that obtained via direct calculation of detected

statistics. The convolution of the instrument response can only obscure the behavior of the radi-

ated field. The emission and detection based schemes are complementary to one another in the

long-time limit. Calculations based on photon detection provide a direct means to calculate ex-

perimentally observable phenomena, while the emission scheme may provide the clearest window

into the underlying physics.

B. Statistics at short times

In section IV, a possible problem with the naive emission-based approach to calculating pho-

ton statistics was anticipated. The formalism presented there for the time derivative of the first

30

moment of photon number (i.e. the intensity) has the appearance of a quantum generalization of

the time-dependent spectrum introduced by Page and Lampard [75, 76] and it is thus expected that

the expression for I∆(ω, t) is not restricted to be always positive [77]. It is readily verified that

the emission-based intensity does take on negative values at short times. In Fig. 7, we plot the

time evolution of both the singlet spectrum and the triplet spectrum from the preceding section

under resonant excitation conditions. Although the spectra converge to the fully positive results

of the preceding section at sufficiently long times, the early time results are highly oscillatory as

a function of detuning frequency and include negative regions. This behavior is restricted to the

emission-based calculation. The detected intensity, dI∆(ω, t) is independent of time and is always

non-negative.

Whether one views negative values of I∆(ω, t) as being entirely unphysical, or simply unmea-

surable, is really a matter of interpretation. If one insists on associating I∆(ω, t) with an intensity

of the electric field, then it is true that this quantity should be non-negative. However, return-

ing to our derivation of sec. IV, we see that I∆(ω, t) was never associated with a field intensity,

but rather was derived as the time derivative of the number of photons emitted into the field by the

chromophore. Insofar as some emitted photons may be subsequently reabsorbed, it is not expected

that I∆(ω, t) must remain strictly positive. The real problem with I∆(ω, t) is that there would not

appear to be any means to directly measure it. Physical detectors measure true intensities, which

are non-negative (registered photons are removed from the system by the detector). The scheme

presented in sec. V for calculation of the detected intensity, dI∆(ω, t) is thus to be preferred for

comparison to experiment, not only for the intensity itself, but for all higher order statistics as well.

Therefore, in what follows we only present the results of the scheme accounting for the response

function of the detector.

Fig. 8 displays the time evolution of the Q parameter for a very broad detector, ∆ = 1000Γ. A

detector of this width yields (with insignificant discrepancies) the traditional Mandel Q parameter,

in which all photons are counted; the results presented here are indistinguishable from the tradi-

tional calculations for broad band detection [72] and serve as a validation of the computational

approach. For the singlet parameters the Q parameter monotonically converges to its long time

limit while for the triplet parameters the convergence is not monotonic and the Q parameter oscil-

lates before it converges. For the very broad detector used in this figure the oscillations reflects the

system parameters showing a frequency of ≈ Γ. The inset displays the short-time behavior of the

main figure to emphasize the oscillations.

31

The time evolution of the generalized Q parameter, as would be measured by detectors of

width ∆ = Γ/10, is shown in Fig. 9. At short times, the generalized Q is small at all frequencies

reflecting nearly Poissonian statistics for small average photon number. At later times, Q converges

to to the long-time limit seen in Fig. 3 The figure also suggests that convergence requires times

t >> 1/∆ = 10/Γ, rather than the shorter time scale introduced by the system (∼ 1/Γ). The

fact that observation times must exceed both 1/∆ and intrinsic system time scales to achieve

convergence is seen more clearly in Fig. 10. Observation times shorter than (or comparable to)

either 1/Γ or 1/∆ lead to statistics significantly different from the long-time results (as indicated

by t = 1000/Γ in the figure). When the observation time is shorter than system time scales,

the statistics are different from long observation times regardless of detector width. For times

exceeding system time scales, convergence to the long-time asymtotics can be achieved, but only

if the time exceeds 1/∆. The figure also shows that, for a narrow band detector the Q parameter is

positive due to the correlations imposed by the detector, while for broad band detector it’s negative

reflecting the sub-Poissonian statistics of the photons from a single emitter.

Fig. 11, displays the time evolution of the correlation function (Eq. 77) for the case of the triplet

spectrum. As was the case for the single frequency observables discussed above, the correlations

grow into the long-time results of Fig. 6 but require observation times longer than 1/∆ to achieve

convergence.

C. Off-resonance excitation

The results presented above have all assumed resonant excitation by the laser field with ωL =

ω0. However, the methods presented in this work are not limited to this case and can be also

used to calculate the photon statistics for off-resonance excitation. To demonstrate the case of

off-resonant excitation, we chose the system parameters similar to the pioneering experiment of

Aspect et al. [58]. In this experiment, atoms were excited far from resonance such that the detuning

frequency was much larger than all system inverse time scales. As mentioned earlier, in this case

the spectrum shows a triplet–three peaks corresponding to the laser frequency and two side bands

at frequencies ωL ± ωL0. It was found that there is a strong correlation between the photons

emitted at the two side bands. In our calculations, the Rabi frequency is set to Ω = 400Γ, and

the detuning frequency is given by ωL0 = 20000Γ. The limit t → ∞ was taken analytically. In

Fig. 12, we show the intensity and Q parameters for detectors of width ∆ = 50Γ and ∆ = 500Γ.

32

These widths were chosen to ensure that all the photons emitted from a side band are detected by a

detector centered at the side band frequency. For these parameters, the coherent peak is 104 times

larger than side bands’ peaks. Therefore, in the top panels, showing the intensity, the y-axes were

truncated such that the two side bands are visible. The insets of these panels show the full triplet

spectra. The bottom panels show the corresponding Q parameter which is very small but exhibits

significant peaks at the side bands. Effectively, the narrow-band photon statistics is Poissonian at

all frequencies.

In Fig. 13, the correlation function for this system is displayed. The strong temporal corre-

lations between emitted photons in the opposing side bands [58] are reflected in the correlation

function, while there is no correlation between the photons in the central peak and those at the

side-bands. These results agree with the measurements of Aspect et al. [58] and reflect the energy

conservation between adsorbed and emitted photons; if a photon comes out of the atom with sig-

nificantly higher energy than the incoming photons, that emission will be accompanied by another

emission with a photon at lower energy to enforce energy conservation. The effect can be seen

if the detectors are simply in the vicinity of the two side-bands and show some overlap, but is

strongest if the detectors are centered exactly at ω = ωL ± ωL0.

VII. RESULTS FOR A FOUR-STATE CHROMOPHORE

To demonstrate the applicability of our method to more complex systems, we consider the case

of a chromophore coupled to a two-level system (TLS). This model is of both theoretical and

practical interest. A two-state chromophore coupled to a single TLS yields a four-state quantum

system, which is considerably more complex than a simple two-level chromophore, but remains

simple enough to study numerically without difficulty. Physically, TLS’s are known to play an im-

portant role in both the thermal physics and spectroscopy of low temperature glassy systems [83–

85] and the model we present can be viewed as representing the behavior of a single dye molecule

embedded in a glassy matrix, which we emphasized in our original report [66]. The mathematical

model we adopt here for a coupled two-level chromophore and TLS has been repeatedly described

in some detail in the literature [37, 85, 86]. For this reason, we will present only the equations

necessary to specify the pertinent super-operators L, D+ and D− necessary to calculate photon

statistics. Further details behind the derivation of these equations can be found in ref. [37], where

we considered exactly the same model in the context of broadband emission statistics and a much

33

simplified and approximate treatment of spectrally resolved emission statistics.

Physically, the model generalizes that of the two-level chromophore considered in the previous

section by introducing two nuclear states within both the excited and ground state manifolds of

the chromophore (see Fig. 14) and coupling these four states to a thermal bath representing the

long-wavelength phonons of the solid matrix. The states |a⟩ and |b⟩ are associated with the ground

electronic manifold and the states |c⟩ and |d⟩ with the excited electronic manifold. In the notation

of sec. III, a and b are associated with the ng index for ground nuclear states, whereas c and d

are associated with the ne index for excited electronic states. Within a traditional stochastic line-

shape model [87, 88], this picture would correspond to a chromophore undergoing two-state hop

spectral diffusion as driven by phonon-assisted tunneling of the TLS. The full quantum mechanical

treatment used here is somewhat more general than this due to the mixing between chromophore

and TLS states within the Hamiltonian, but it is convenient to think of this model as a proper

quantum treatment of spectral diffusion. The chromophore density matrix for this system, σCH(t)

is 4× 4, corresponding to super-operators of dimension 16× 16.

The super-operator L governing dynamics of the chromophore density matrix (see sec. IV A)

is comprised of several additive pieces:

L = W+ R+ LE + LΓ. (85)

The operator W is associated with the unperturbed dynamics of the “chromophore” component of

the total system, which in this case is comprised of the coupled bare chromophore and TLS pieces.

Using the language of sec. III, ĤCH is fully specified by [85, 86]

Hg = −ℏω02

+

(A

2− α

4r3

)σTLSz +

J

2σTLSx , (86)

He = +ℏω02

+

(A

2+

α

4r3

)σTLSz +

J

2σTLSx

Here, A and J are the asymmetry and tunneling matrix elements, respectively, for the TLS, and

σTLSz ,σTLSx are Pauli matrices in the basis of the TLS’s localized “minima” states. ω0 is the chro-

mophore electronic transition frequency in the absence of interactions. α is a chromophore-TLS

coupling constant and the strain-field mediated interaction between the two is of a dipolar nature -

leading to the displayed 1/r3 dependence in the physical distance between the two. Diagonalizing

ĤCH leads to the four eigenstates |g⟩|a⟩, |g⟩|b⟩, |e⟩|c⟩ and |e⟩|d⟩, increasing in energy alphabeti-

34

cally. In this basis, Eq. (86) can be written as:

Hg = ℏωa|a⟩⟨a|+ ℏωb|b⟩⟨b|, (87)

He = ℏωc|c⟩⟨c|+ ℏωd|d⟩⟨d|

with the frequencies

ωa = −1

2ω0 −

1

2ℏ√J2 + (A− P )2, (88)

ωb = −1

2ω0 +

1

2ℏ√J2 + (A− P )2,

ωc = +1

2ω0 −

1

2ℏ√J2 + (A+ P )2,

ωd = +1

2ω0 +

1

2ℏ√J2 + (A+ P )2

and we have set P ≡ α2r3

. Written in the |a⟩, |b⟩, |c⟩, |d⟩ basis (in super-operator elements specified

below indices with subscript g take the values a or b and indices with subscript e take the values c

or d), W is diagonal, with elements

Wngmg ;ngmg = −iωngmg

Wneme;neme = −iωneme

Wnemg ;nemg = −i(ωnemg − ωL)

Wngme;ngme = −i(ωngme + ωL). (89)

As introduced in sections III and IV A, σCH(t) is a rotating frame density matrix within the RWA

approximation. This is why ωL enters into W.

The dynamics due to the driving laser field follow immediately from the form of HIe in sec. III

to yield LE , with nonzero elements given by

LEngmg ;kelg = −LEkelg ;ngmg =i

2Ω0 ⟨ng | ke⟩ δmglg

LEngmg ;kgle = −LEkgle;ngmg = −i

2Ω0 ⟨le | mg⟩ δngkg

LEneme;kelg = −LEkelg ;neme = −i

2Ω0 ⟨lg | me⟩ δneke

LEneme;kgle = −LEkgle;neme =i

2Ω0 ⟨ne | kg⟩ δmele . (90)

R and LΓ contain the non-Hamiltonian dissipative contributions to the dynamics of σCH(t)

obtained by integrating out the acoustic phonons of the bath and photons of the quantum radia-

tion field, respectively. The elements of R are calculated using the full Redfield formalism [74]

35

described previously in refs. [37] and [86]. For example, a sampling of non-zero elements is

provided by

Rcc;dd = eβℏωeRdd;cc = CωeJ21

1− e−βℏωe(91)

Raa;bb = eβℏωgRbb;aa = CωgJ21

1− e−βℏωg

Rca;db =1

2

[ωeωg

Rcc;dd +ωgωe

Raa;bb]

Rdb;ca =1

2

[ωeωg

Rdd;cc +ωgωe

Rbb;aa]

ωg ≡ ωb − ωa

ωe ≡ ωd − ωc,

however, there are many more non-zero contributions, which we do not explicitly list here. Rcc;ddand Raa;bb are the thermal transition rates from d to c and from b to a, respectively. The rates of the

reverse transitions are seen to obey detailed balance. C is a collection of constants incorporating

the coupling strength between the TLS and the bath, which is typically taken as a parameter used

to fit an experiment rather than estimated from first principles [89]. Of course, the top two lines

simply express the phonon-assisted transition rates from state d to c and b to a, as expected. Other

elements follow similarly.

LΓ involves state-to-state spontaneous emission rates from states ne = c or d to states mg =

a or b

Γnemg = Γ0 |⟨ne | mg⟩|2 (92)

with

Γ0 ≡ω3eg |−→µ 0|

2

3πε0ℏc3(93)

as previously defined. The non-zero elements of LΓ are given by

LΓnene;nene = −∑mg

Γnemg

LΓmgmg ;nene = Γnemg

LΓij;ij = −1

2

(∑k ̸=i

Γik +∑k ̸=j

Γjk

)(94)

with all other elements being zero. The i, j, k indices in the third line run over all states irrespective

of ground or excited affiliation, however Γij = 0 unless the i and j indices have ne and mg

character, respectively.

36

The elements of D± follow immediately from the formalism specified in sec. IV A. The non-

vanishing elements are given by

D+ing ;ime = ⟨me | ng⟩

D−ngi;mei = ⟨ng | me⟩, (95)

where the index i runs over states of both excited and ground character.

As a first example, we consider the case of a slow TLS modulation, and strong coupling be-

tween the TLS and chromophore. (The case of weak TLS-chromophore coupling is closely related

to a purely stochastic description and is somewhat less interest

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