This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 128.200.11.129 This content was downloaded on 23/07/2015 at 18:54 Please note that terms and conditions apply. Photon-exchange induces optical nonlinearities in harmonic systems View the table of contents for this issue, or go to the journal homepage for more 2015 J. Phys. B: At. Mol. Opt. Phys. 48 065401 (http://iopscience.iop.org/0953-4075/48/6/065401) Home Search Collections Journals About Contact us My IOPscience
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Photon-exchange induces optical nonlinearities in harmonic systems
View the table of contents for this issue, or go to the journal homepage for more
2015 J. Phys. B: At. Mol. Opt. Phys. 48 065401
(http://iopscience.iop.org/0953-4075/48/6/065401)
Home Search Collections Journals About Contact us My IOPscience
Received 21 July 2014, revised 15 October 2014Accepted for publication 2 December 2014Published 20 February 2015
AbstractThe response of classical or quantum harmonic oscillators coupled linearly to a classical field isstrictly linear; all nonlinear response functions vanish identically. We show that, if the oscillatorsinteract with quantum modes of the radiation field, they acquire nonlinear susceptibilities. Theeffective third order susceptibility χ (3) induced by four interactions with quantum modescontains collective resonances involving pairs of oscillators. All nonlinearities are missed by theconventional approximate treatment based on the quantum master equations.
Keywords: harmonic, vacuum, nonlinearities
(Some figures may appear in colour only in the online journal)
1. Introduction
When a harmonic oscillator is coupled linearly to a classicalfield via the interaction xE(t) its response is strictly linear [1].This can easily be seen from the Heisenberg equation ofmotion for a damped harmonic oscillator,
ω γ+ − =x x x e mE t¨ ˙ ( )02 . The Fourier transform of this
equation gives a linear response for any value of the field [2]ω χ ω ω=x E( ) ( ) ( )(1) , where
χ ωω ω γω
=− +
e m( )
i. (1)(1)
02 2
For this system all higher order susceptibilities vanish iden-tically as a result of quantum interference of different path-ways of the density matrix [2, 3]. A finite nonlinear responsemay be induced by two mechanisms: adding anharmonicitesto the potential or by incorporating a nonlinear couplingbetween the oscillator and the field e.g. E t x( ) 2. In multi-dimensional spectroscopy, these are known as mechanicaland electronic nonlinearities, respectively [4]. Nonlinearcoupling to a bath [5] can also cause a nonlinear response.
Multidimensional spectroscopy of molecular vibrationswith infrared pulses is widely used to study the secondarystructure of proteins, hydrogen bonding in liquid water, pro-tein folding, and chemical exchange [6–11]. These
applications use an exciton Hamiltonian. In optical spectro-scopy of semiconductors [12–16] and photosynthetic com-plexes [17], Coulomb interactions cause anharmonicitieswhereby Pauli exclusion affects the dipole coupling.
A Liouville-space superoperator formalism has beenapplied to all of the above techniques and has facilitated theanalysis of complicated signals by allowing the pre-selectionof relevant pathways (represented diagrammatically).
The vanishing of the nonlinear response of harmonicoscillators makes them a convenient, background-free refer-ence system for describing the response of more complexsystems. The oscillator picture is natural for intermolecular andintramolecular vibrational modes, where the oscillators repre-sent the actual coordinates of atoms. Multidimensional spec-troscopy of molecular liquids has been formulated using thismodel [4]. The same model applies to the optical response ofmany-body exciton systems such as molecular aggregates orsemiconductor nanostructures [18]. In these applications theoscillators are not the actual particles (electrons), but rathercollective, quasiparticle coordinates that represent electron–hole pairs. The harmonic oscillator model serves as a referencefor the quasiparticle representation of the many-body response.Although it is nearly impossible to visualize the wavefunctionof a many-body Fermion system, a few quasiparticles can beeasily managed and calculated.
Journal of Physics B: Atomic, Molecular and Optical Physics
Here, we show that coupling to vacuum modes of theradiation field induces nonlinearities in the response of asystem of harmonic oscillators, and can result in new, col-lective resonances. We demonstrate that coupling via theexchange of vacuum photons induces nonlinearities in theresponse of a system of harmonic oscillators. Photonexchange can occur either sequentially, where one oscillatoremits the photon and another absorbs it, or non-sequentially,where there are interactions with the classical fields in-between. Sequential photon exchange can be described usingthe quantum master equation (QME) approach [19, 20]. Thecoupling between oscillators via photon exchange of thequantum field is then described by effective dipole–dipole andspontaneous emission (superradiance) coupling terms. In thenear-field limit, the dipole–dipole coupling parameter variesas −rab
3 with the distance between oscillators, while in the far-field limit this dependence becomes −rab
1. We show that sys-tems described by the QME remain linear. This should beexpected because the Hamiltonian is quadratic in the bosoncreation and annihilation operators. However, a more generaldiagrammatic expansion of the signal generates terms withnon-sequential photon exchange with a field correlationfunction of the vacuum modes which is quartic in the bosoncreation and annihilation operators. This induces a finitenonlinear response. Four-wave mixing (FWM) of classicaland quantum fields coupled to two harmonic oscillators aand b, has both single-oscillator and collective resonances.Figure 1(a) shows two uncoupled harmonic oscillators and (b)shows the level-scheme with the single and collectivetransition frequencies involved in the induced nonlinearresponse.
The nonlinear response of a system of N non-interactingoscillators is N times the single oscillator response, whichvanishes in the harmonic case. To second order, coupling tothe vacuum modes of the electromagnetic field results in asusceptibility that is given by a sum of a product of individualsusceptibilities. For instance, in [21] we showed that thecoupling of the quantum modes to first-order for a systemcomposed of particles i and j can create a fifth-order
susceptibility χ (5) which is the product of two susceptibilitiesχ χ χ= i j
(5) (3) (3) . We showed that these non-additive con-
tributions can induce collective resonances that only arise in aquantum field framework and, in [22], such resonances werepredicted for two level chromophores. Corrections to har-monic systems come at higher order in the vacuum-modecoupling.
The first-order correction to the nonlinear response isproportional to two interactions with the quantum fields,which are initially in the vacuum state. This contribution isproportional to the semi-classical single-oscillator χ (3), whichvanishes for our model. It is well known that a Hamiltonianwhich is quadratic in the boson creation and annihilationoperators can be diagonalized and thus there should be nononlinear effects; these require an anharmonicity. For ourmodel, the lowest-order finite nonlinear response is fourthorder in the quantum fields. Using Wickʼs theorem this cor-relation function can be written as a product of two quadraticfield correlation functions. The effective χ (3), caused byinteraction with four quantum modes. χ (3) has two contribu-tions. The first is the product of χa
(3) from oscillator a and χb(3)
from oscillator b and represents a resonant energy transfer,where each oscillator interacts twice with the quantum modes.We find that the quartic quantum field correlation functionvanishes for this contribution. The second contribution to thesusceptibility is of the form χ χa b
(5) (1) , which involves both acascading process [21], where oscillator b emits a photon andoscillator a absorbs it, and a process where oscillator a emitsand absorbs the same photon. The latter is typically neglectedwhen calculating the resonant energy transfer betweenmolecules [27]. We show that this term may not be neglectedwhen considering harmonic oscillators, since it is responsiblefor the finite nonlinear response. The time-ordered χa
(5) cor-relation function contains two dipole-operators associatedwith emission and absorption of a single photon with oscil-lator a, which cannot be factorized out to renormalize thenonlinear response function.
Figure 1. (a) Two uncoupled harmonic oscillators a and b with transition frequencies ωa and ωb. (b) The relevant level scheme for the χ (3)
susceptibility. The quantum mode corrections induces an effective χ (3) that contains the the collective transition frequencies ω ω ω= ±± a b
(yellow) and ω ω ω= ±±˜ 2 a b (green); as well has the single oscillator frequencies (blue and red) in the two-particle eigenstates. Theexchange of oscillator a and b in the level scheme also occurs.
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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
Note that the oscillator picture can represent the col-lective excitations of a complicated system composed ofmany particles (e.g. electrons). The process where oscil-lator a emits and absorbs a photon is analogous to similarprocesses that occur on the level of the constituent parti-cles and are interpreted as renormalizing their bare prop-erties (mass and charge). These latter processes aretherefore implicitly included in the oscillator properties butdiffer from the renormalization of the collective excitationsof this many-particle system (modeled by the harmonicoscillator levels) that we consider in this paper. On amicroscopic level, this can represent the emission by oneconstituent particles and absorption by another. Such pro-cesses originate in intramolecular self-energy terms whichare normally neglected in molecular quantum electro-dynamics [27].
This paper is organized as follows. In section 2, wepresent the model of the system and its response in a compactformal expression that contains all orders in the fields andserves as a starting point for perturbative calculations of thecorrections to various signals due to quantum modes of theradiation field. In section 3, we calculate the correction to thelinear absorption spectrum from coupling to the quantum fieldmodes. In section 4, we demonstrate that this coupling leadsto a finite third-order nonlinear response and in section 5, wepresent the FWM signal obtained with continuous wave (cw)lasers. For comparison, the response calculated using theQME is presented in section 6. The results are discussed insection 7.
2. Model and the nonlinear response
We consider a system of uncoupled harmonic oscillatorswhich interact with the radiation field and described by theHamiltonian
= +H H H , (2)0 int
where
∑ω=α
α α αH B B (3)0†
is the oscillator Hamiltonian and αB† ( αB ) is the boson creation(annihilation) operator of the αth oscillator which satisfy thecommutation δ=α β αβB B[ , ]† .
The radiation field is given as
= +E t E t E tr r r( , ) ( , ) ( , ), (4)v0
where = +E t t tr r r( , ) ( , ) ( , )00 0
* is a classical incomingfield and the second term represents the quantum vacuummodes = +E t t tr r r( , ) ( , ) ( , )v
v v†
⎛⎝⎜
⎞⎠⎟ ∑ π ω
ϵ=λ
λλ
ω− +( )tc
ar k( , )2
e . (5)vv
vt
k
kk r
,
1 2( )
,i i ·
v
vv v
denotes the quantization volume, ϵ λ( ) is the polarizationvector for mode νk and λ is the polarization. λakv is the
Boson annihilation operator. The matter–field interaction,in the interaction picture, with respect to the Hamiltonian,H0 is
∑= − +α
α α( )H E t V t V tr( , ) ( ) ( ) . (6)int†
μ=α α αV t B t( ) ( ) is the dipole coupling and μα is the transitiondipole matrix element. To simplify the expression inequation (6) we do not invoke the rotating-wave-approximation.
The optical response will be calculated using super-operator algebra [23] for the density matrix in Liouville space.The frequency dispersed response to the electric field is givenby [2]
⎡⎣⎢
⎤⎦⎥
∫∫
ω ω= −
× ω
−∞
∞
−−∞
−
S t
V t
r( ) ( , ) d
( )e e , (7)LH T T t
0*
( )d it
iint
where denotes the imaginary part. Equation (7) will beperturbatively expanded in the field–matter interaction tocalculate the correction to the linear and nonlinear responsefunctions due to vacuum mode coupling. We define the linearcombinations of superoperators
= −−O O O (8)L R
and
= ++ ( )O O O1
2, (9)L R
where the two superoperators OL and OR are defined by theiractions from the left ↔O X OXL and from the right
↔O X XOR [23].
3. Photon exchange correction to the linearresponse χ ð1Þ
The diagrammatic expansion of equation (7) to first-order inHint, with the classical fields, is presented in figure 2(a). Inthese diagrams we work in the ± representation. Time pro-gresses as one moves up the diagrams and the vertical linesrepresent the density matrices of the different molecules a andb. This provides a formal book keeping tool for the variouscontributions to the signal [21].
The superoperator corresponding to the interactionHamiltonian is = +− − + + −EV E V E V( ) . Because the classicalfields are c-numbers, i.e. E− = 0 the dipole operators asso-ciated with classical fields are −V . The first-order expansion(figure 2(a)) for two uncoupled harmonic oscillators a and b,
3
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
gives the classical linear response [2]
ω ω χ ω ω= −S ( ) ( ) ( ; ), (10)(1)0
20(1)
where the susceptibility is given as
⎛
⎝⎜⎜
⎞
⎠⎟⎟χ ω ω
μ
ω ω Γ− = −
− ++ ⇔
a b( ; )
1
i, (11)
a
a a0(1)
2
2
where ⇔a b represents the interchange of oscillators a and b.Equation (10) describes the process where oscillator a absorbsand emits the classical field ω. See the corresponding levelscheme diagrams, figure 2(a).
The leading correction due to the vacuum modes issecond-order: emission followed by absorption. We expandequation (7) to third-order in figure 2(b), (two interactionswith the quantum modes and one with the classical mode).The last interaction, at time τ2, is with a classical field( ω r( , )0
* equation (7)) acting on oscillator a. Diagram 2(b)1depicts oscillator b absorbing a photon at τ1 from the classicalfield. That photon is converted into a single photon thatinteracts with the vacuum electromagnetic field at τv1 andpropagates to oscillator a where it is absorbed at τv2. Finally,oscillator a generates the detected signal at τ2. See the levelscheme diagram shown above figure 2(b)1. Because
ρ =−OTr [ ˆ ] 0, the last interaction with the quantum modemust be +E (i.e., ⟨ ⟩+ ±E E ). Similarly, the interaction at τv1 mustbe a quantum mode, since ⟨ ⟩− −V V b vanishes. As a result, thequantum field correlation function will be ⟨ ⟩+ −E E . Infigure 2(b)1 we label the dipole operator associated with eachinteraction with the field.
The effective coupling between the two oscillatorsdepends on their distance in space and their dipole moments.We assume that the angle of the dipole moments are fixed.Figure 3 illustrates the configuration of the dipole moments μa⃗and μb⃗. The emission and absorption of a single photon byoscillator a mediated by the vacuum is shown in figure 2(b)2.See the corresponding level scheme diagram above the ladderdiagram.
The signal is calculated in appendix A. Combining theabsorption spectrum with the second-order vacuum modecorrection gives the effective linear susceptibility
The susceptibility χ0(1) is given by equation (11) and the
correction due to the interaction with the quantum modesreads
⎛⎝⎜⎜
⎞⎠⎟⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟
χ ω ωμ μ
ω ω Γ
ω γ ωω ω Γ
μ
ω ω Γω
πϵ
ω ω Γ
− = −− +
×+
− +
−− +
×− +
+ ⇔
( )
( )
J r r
c
a b
( ; )1
i
( ; ) i ( ; )
ie
1
i 2
1
i. (13)
a b
a a
ab ab ab ab
b b
k r
a
a a
a a
1(1)
2
*
i
2
23
03
ab0
Here ωπϵ c2
3
03is proportional to the spontaneous decay, and
originates from diagram 2(b)2, The effective dipole–dipole
Figure 2. (a) Diagram for the linear absorption spectrum equation (10) (χ (1)), for a harmonic oscillator. (b) Diagrams for the nonlinearcorrection to the linear signal equation (10). Lines corresponding to interaction with a classical field are solid blue and with a quantum fieldare wavy red. The corresponding level schemes are shown above. For diagram rules see [21].
Figure 3. The dipole moments μa and μb in configuration for twooscillators separated by the distance rab.
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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
coupling between oscillators is
⎡⎣⎢⎢
⎡⎣⎢⎢
⎤⎦⎥⎥
⎤⎦⎥⎥
ωμ μ ω
πϵθ θ
θ θ
= −
− + +( )
J rc
y
y
y
y
y
y
( ; )4
sin sincos ( )
( )
cossin ( )
( )
cos ( )
( )(14)
ab aba b
a bab
ab
a bab
ab
ab
ab
* 3
03
2 3
and the effective cooperative emission [19] is
⎡⎣⎢⎢
⎡⎣⎢⎢
⎤⎦⎥⎥
⎤⎦⎥⎥
γ ωμ μ ω
πϵθ θ
θ θ
=
+ + −( )
rc
y
y
y
y
y
y
( ; )4
sin sinsin ( )
( )
coscos ( )
( )
sin ( )
( ), (15)
ab aba b
a bab
ab
a bab
ab
ab
ab
* 3
03
2 3
where ω≡y r c/ab ab .The dipole–dipole and spontaneous emission contribu-
tions originate from the process depicted in diagram 2(b)1.For the dipole–dipole coupling in the near zone, ≪y 1ab , thesecond term in the square brackets of equation (14) becomesdominant and the susceptibility scales as −rab
3, which isrecognizable as the static dipole–dipole coupling. For the farfield limit, ≫y 1ab , the first term in equation (14) becomesdominant and the susceptibility depends the separation dis-tance as −rab
1.The linear absorption spectrum, equation (10),
ω χ ω ω= − −S ( ) ( ; ), (16)0(1)
3 0(1)
3 3
where the effective susceptibility is given by equation (11).Equation (16) is plotted in figure 4(a) using a red-dashed line.
The linear signal that includes the correction from thequantum field is given by
ω χ ω ω= − −S ( ) ( ; ). (17)(1)3
(1)3 3
with χ ω ω−( ; )(1)3 3 given by equation (12). The correction
ωS ( )(1)3 is plotted in figure 4(a) using a solid-blue line. The
absorption spectrum (dashed-red) shows two absorptionpeaks at ω ω= a3 , ωb. The correction from the quantummodes (solid-blue line) in figure 4(a) changes the absorptionpeak at ω ω= b3 to have both absorption and emission fea-tures. As rab increases, ( πλ =r 2 0.2ab a in figure 4(b)), thecorrection from the quantum modes weakens and χ1
(1)
approaches χ0(1) .
4. χ 3ð Þ nonlinearity induced by photon-exchange
The third-order susceptibility vanishes for harmonic oscilla-tors when coupling to the quantum vacuum modes isneglected. The lowest-order possible correction is second-order in the quantum modes, which are initially in the vacuumstate. This contribution vanishes as well, as can be understoodfrom the diagrammatic expansion of equation (7) to fifth orderin Hint. See figure 5. This process is third-order in the classicalfields and second-order the quantum fields. In this process,oscillator b absorbs a photon from the classical field at τ1 and
emits a vacuum photon at τv1, which then propagates tooscillator a where it is absorbed at τv2. Oscillator a interactswith the classical field at τ2 and τ3 and generates the detectedsignal at τ4. These diagrams offer a compact representation ofthe signal generation. Because the susceptibility can writtenas χ χ χ= ,ab b a
(5) (1) (3) there will be eight possible quantumpathways for oscillator a and one for oscillator b. Overall, thethird-order process contains three classical field interactionsand be viewed as an effective χ (3). For the vacuum trace to befinite, the final interaction with a quantum mode on oscillatora must be + −E V and the quantum mode interaction onoscillator b must be − +E V . From figure 5, the matter corre-lation function for oscillator a will have form ⟨ ⟩+ − − −V V V V a
and oscillator b will have the form ⟨ ⟩+ −V V b. The semiclassicalresponse ⟨ ⟩+ − − −V V V V is zero for the harmonic system. Sincethe Hamiltonian equation (3) is quadratic in the boson crea-tion and annihilation operators, the nonlinear response van-ishes when coupling to the quantum modes is neglected.When the interaction Hamiltonian equation (6) is expanded to
Figure 4. The absorption spectrum for a cw laser (dashed-red) of twoharmonic oscillators a and b and the correction the linear responsefrom the quantum modes equation (12) (solid-blue) are plotted in theunits μ
| |a
1 22
. The distance between oscillators is (a) πλ =r 2 0.09ab a
and (b) πλ =r 2 0.2ab a . The oscillator frequencies used areω = −12 000 cma
1, ω = −14 000 cmb1 and the decay rates are
Γ Γ= n ˜a a and Γ Γ= m ˜b b, where n and m are the excitation level of theoscillator, Γ = −˜ 20 cma
1 and Γ = −˜ 10 cmb1. The dipole moments are
in the units μ μ= nna a a, , μ μ= m0.99b a.
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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
quadratic order in the quantum vacuum modes, the correctionvanishes and harmonic oscillator remains linear. Obtaining anon-vanishing correction will require us to expand the signalequation (7) to higher order in the quantum modes.
The next order correction to the nonlinear response isfourth-order. For two harmonic oscillators a and b, the signalcan be separated into two contributions. The first is whereoscillators a and b both interact twice with the quantummodes. This contribution to the signal is 7th order in Hint andshown in figure 6. The permutations of the quantum fieldmodes in the diagram of figure 6 will give a total of 14diagrams, which are not shown here. This can describe theprocess where oscillator b interacts with the classical fields atτ1, τ2, and then interacts with the vacuum electromagneticfield to emit a photon at τv1, and absorb a photon at τv2.Oscillator a absorbs a photon at τv3, and emits a photon at τv4
via the vacuum. Then oscillator a interacts with the classicalfield at τ3 and generates the detected field at τ4. Similar pro-cesses will also occur due to the permutations of the quantumand classical fields. The associated susceptibility χab
(5) can be
factorized as χ χ .a b(3) (3) Similar to the diagrams with two
quantum modes (figures 2, 5) the last quantum mode onoscillator a at time τv4 in the diagram 6 has a −V associatedwith it. The first quantum mode on oscillator a at time τv3 canbe ±V . However, since ⟨ ⟩+ − − −V V V V a vanishes for the har-monic oscillator, the interaction at time τv3 must be +V . Foroscillator b, the last quantum mode at τv2 is +V . Because⟨ ⟩+ − − −V V V V b vanishes, this requires the first quantum modeat τv1 to be +V . The diagram in figure 6, will acquire a quantumfield correlation function of form ⟨ ⟩+ − − −E E E E , which van-ishes for the bosonic fields. This means that there is noresonant energy transfer between two harmonic oscillators.
The second contribution yields a finite nonlinear signaland involves three quantum modes interacting with oscillatora and one quantum mode interacting with oscillator b and viseversa. See figure 7 for the diagrammatic expansion. Thiscontribution can describe the process where oscillator babsorbs a photon from the field at τ1. Oscillator b then emits aphoton at τv1 and that propagates to oscillator a where it isabsorbed at τv2. Oscillator a emits and absorbs a photon withthe vacuum electromagnetic field at τv3 and τv4. Finally,oscillator a interacts with the field at times τv2 and τv4 to
generate the signal. The nonlinear response will include allpermutations of the quantum and classical field modes, whichwill describe a similar processes. The resulting susceptibilityis χ χ χ=ab a b
(5) (5) (1) , which is an effective χ ,(3) since it is thirdorder in external fields.
Derivation of the heterodyne signal for the diagram infigure 7 is given in appendix B. Equation (B18) gives theeffective third-order susceptibility
Figure 5. The nonlinear response equation (7) for two uncoupled harmonic oscillators expanded to fifth order, two interactions with thequantum modes (wavy red) and four interactions with the classical modes (solid blue).
Figure 6. Diagram for the nonlinear response equation (7) expandedto seventh order, four interactions with the quantum modes (wavyred) and four interactions with the classical modes (solid blue) fortwo uncoupled harmonic oscillators. These diagrams can representresonant energy transfer between harmonic oscillators.
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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
where ωμ μ′˜ ( )aa v( , )
1 is given by equation (A6) and the dipole–dipole and spontaneous emission coupling term are givenby equations (14), (15), respectively. In equation (18), thesusceptibility χ ν ν ν
+−−+−−,2 2 1 appears twice to account for the
two permutations of the quantum mode coupling. Thesuperscript ν ν ν,i j k corresponds to the designation of thequantum modes at times τv4, τv3 and τv2, respectively. Forexample, the index ν ν ν,2 2 1 corresponds to ωv1 at time τv2
and ωv2 at τv3 and τv4. If ωv2 is positive at τv3 then it will benegative at time τv4, and vise versa. This is because itmust emit and absorb the the same quantum mode ωv2.Here
∫∫
χ ω ω ω ω ω ω
τ τ τ τ
τ τ τ τ θ τ τ
Δ τ τ τ τ τ τ α τ τ
=
× −
×
×
ν ν νν ν ν
ν ν ν ν
ωτ ω τ ω τ ω τ ω τ ω τ
+−−−−+
+−−−−+ +−
− + + + + −
( , , , , , )
d d d d
d d d d ( )
( , , , , , ) ( , )
e , (19)
v v v v v v
v v v v
,3 2
4 3 2 1
,4 3 2 1
i i i i i iv v v v v v
1 2 21 2 2
4 3 2 2 2 1
1 2 24 3 2
11
4 1 4 3 3 2 3 2 2 2 2
∫∫
χ ω ω ω ω ω ω
τ τ τ τ
τ τ τ τ θ τ τ
Δ τ τ τ τ τ τ α τ τ
=
× −
×
×
ν ν νν ν ν
ν ν ν ν
ωτ ω τ ω τ ω τ ω τ ω τ
+−−−+−
+−−−+− +−
− + + + + −
( , , , , , )
d d d d
d d d d ( )
( , , , , , ) ( , )
e , (20)
v v v v v v
v v v v
,3 2
4 3 2 1
,4 3 2 1
i i i i i iv v v v v v
1 2 21 2 2
4 3 2 2 2 1
1 2 24 3 2
11
4 1 4 3 3 2 3 2 2 2 2
∫∫
χ ω ω ω ω ω ω
τ τ τ τ
τ τ τ τ θ τ τ
Δ τ τ τ τ τ τ α τ τ
=
× −
×
×
ν ν νν ν ν
ν ν ν ν
ωτ ω τ ω τ ω τ ω τ ω τ
+−−+−−
+−−+−− +−
− + + + + −
( , , , , , )
d d d d
d d d d ( )
( , , , , , ) ( , )
e , (21)
v v v v v v
v v v v
,3 2
4 3 2 1
,4 3 2 1
i i i i i iv v v v v v
1 2 21 2 2
4 3 2 2 2 1
1 2 24 3 2
11
4 1 4 3 3 2 3 2 2 2 2
∫∫
χ ω ω ω ω ω ω
τ τ τ τ
τ τ τ τ θ τ τ
Δ τ τ τ τ τ τ α τ τ
=
× −
×
×
ν ν ν
ν ν ν ν
ωτ ω τ ω τ ω τ ω τ ω τ
+−+−−−
+−+−−− +−
− + + − + +
( , , , , , )
d d d d
d d d d ( )
( , , , , , ) ( , )
e . (22)
v v v
v v v v v v
v v v v
,3 2
4 3 2 1
,4 3 2 1
i i i i i iv v v v v v
2 2 12 2 1
4 3 2 2 2 1
2 2 14 3 2
11
4 2 4 3 3 2 3 2 2 1 2
The fifth-order susceptibility Δ for oscillator a is definedas
Δ τ τ τ τ τ τ
τ τ τ τ τ τ=
ζ ζ ζ ζν ν ν
νζ ζ
νζ ζ
ν
+−
−V V V V V V
( , , , , , )
( ) ( ) ( ) ( ) ( ) ( ) , (23)
v v v
L v v va
,4 3 2
4 3 2
1 2 3 4
4 3 24 3 2
44 1 2
33 3 4
22
where ζi can be ±. The first-order susceptibility α fromoscillator b is
α τ τ τ τ=ν ν+− + −V V( , ) ( ) ( ) . (24)v v
b1 1
11
11
The two frequency integrations in equation (18) over ωv1
and ωv2 may be done analytically using contour integra-tion. Note that the frequencies ω1, ω2 and ω3 in thesusceptibility (18) can be positive or negative. The inte-gration over τ4 in equations (19)–(22) will give a deltafunction, which originates from time translational invar-iance that can be used to evaluate the ω3 integration inequation (B18). This leaves the integrations over ω1 andω2 in equation (B18). The final diagrams correspondingthe susceptibility χa
(5) susceptibility are shown infigures 12, 13. The wavy green lines correspond to theexchange of vacuum photon between oscillators and thewavy red lines correspond the emission and absorption ofthe same quantum modes on oscillator a. The χ (5)
response can be read off the diagrams. For example,figure 12(I)1, has the form ⟨ ⟩+ − − − − − +V V V V V V Vv v v4 3 2 , where
− +V Vv v3 2 corresponds to the emission and absorption of aphoton with oscillator a via the vacuum and cannot befactorized out of the correlation function due to time-ordering. Diagrams that include interactions in-betweenthe emission and absorption of the vacuum photon withthe same oscillator vanish when performing the integra-tions in the susceptibilities (see appendix B).
The simulation of this signal with ω ω= a1 and ω ω= b2
is plotted in figures 8(b)–(m) as a function of ω3. Infigure 8(b), the spectrum shows a single oscillator resonanceω ω= a3 . The ω ω= b3 resonance is shown in figure 8(c). Thecollective resonances at ω ω ω= −b a3 , ω ω+b a are shownin figures 8(d), (e), respectively, and have dispersive features.
Figure 7. Diagrammatic expansion of equation (7) to seventh-orderin Hint. The diagram contains four interactions with the quantummodes and four interactions with the classical field modes. Thiscontribution gives rise to an effective third-order susceptibility.
7
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
The double oscillator resonances ω ω= 2 a3 , ω2 b are shown infigures 8(f), (g), respectively. The collective peaks atω ω ω= −2 a b3 , ω ω+2 a b, are shown in figures 8(h), (i) andhave absorption features. Interchanging ωa and ωb gives theω ω ω= −2 b a3 , ω ω+2 b a resonances. See figures 8(j), (k),respectively. The triple resonances at ω ω= 3 a3 , ω3 b areshown in figures 8(l), (m), respectively. The resonancesω ω ω= +2 b a3 , ω ω−2 b a, ω2 b and ω3 b occur with theinterchange of oscillator a and b in ω ω ωS ( , , )1 2 3 . Forcomparison, we include the linear absorption spectrumfigure 4(a) in figure 8(a) with the distance between oscillatorsas πλ =r 2 0.09ab a . All peak positions in figure 8 can be read
off the level scheme in figure 1. The peaks at ω ω= a3 , ωb arethe dominant peaks in the FWM spectrum.
In the susceptibility equation (25), we scan the frequencyω3 that corresponds to the last classical mode at time τ4 in thediagram of figure 7. A single-particle resonance ω ω= a3 ismeasured when the last two interactions are with the classicalmodes. When there are interactions with quantum modes in-between τ4 and τ3 the quantum modes mix the frequency ω1,ω2, ω3 with the matter frequency for oscillator a, creatingcollective resonances in the spectrum. This can be verified bydiagrammatically expanding the susceptibility and using thefrequency delta function.
Figure 8. (a) The linear absorption spectrum and the correction to the linear absorption spectrum from coupling to the quantum modesfigure 4(a) is shown for comparison. (b)–(m) The FWM spectrum equation (25) χ ω ω ω ω ω ω− − − − +( ; , , )(3)
3 1 2 1 2 3 is plotted, withω ω= a1 and ω ω= b2 . The spectrum is plotted in arbitrary units and different panels are rescaled.
8
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
6. The QME misses the optical nonlinearities
We have shown that the coupling to the quantum vacuumfield modes affects the response to the classical field 0.Conventionally, the coupling to the quantum vacuum fieldmodes is described by the QME for the reduced matter den-sity matrix ρ [19, 24].
⎜ ⎟
⎡⎣ ⎤⎦⎛⎝
⎞⎠∑
ρ ρ
γ ρ ρ ρ
= − + +
+ − −α β
αβ β α β α β α
H H H
B B B B B B
˙i
,
1
2
1
2, (26)
D0 int0
,
† † †
where βB † ( βB ) are the raising (lowering) operators for theharmonic oscillator
∑=α β
αβ α β≠
H J B B . (27)D†
αβJ is the dipole–dipole coupling due to the interaction withthe quantum modes and γαβ is the cooperative spontaneousemission rate, which represents the effective coupling medi-ated by the exchange of photons [25, 26]. αβJ and γαβ aregiven in equations (14) and (15). The interaction HamiltonianHint
0 in the rotating wave approximation reads
⎡⎣ ⎤⎦ ∑= − +α
α αH e t B t e t B tr r( , ) ( ) ( , ) ( ) . (28)int0
0†
0†
To calculate the optical response using the QME we firstcompute the expectation value of the displacement andmomentum of the βth harmonic oscillator. In second quanti-
zation, the dimensionless displacement ω= x x m˜ 1 and
momentum =ω
p̃ p
mof the βth oscillator is given by
= +β β β( )x B B˜1
2, (29)†
= − −β β β ( )p B B˜i
2. (30)†
The expectation value of the operators is
⎡⎣ ⎤⎦ρ∂
∂= +
ββ β( )
x
tB B
˜ 1
2Tr ˙ , (31)†
⎡⎣ ⎤⎦ρ∂
∂= − −
ββ β ( )
p
tB B
˜ i
2Tr ˙ . (32)†
Using the bosonic commutator relations we obtain
⎛⎝⎜
⎞⎠⎟
∑
∑
ω
γωω
∂
∂= +
+ −
ββ β
α βαβ α
ααβ
β β
α αα β
≠
( )
x
tp J p
m
mx e t r
˜˜
1˜
˜ 2 , , (33)0
∑ ∑ω γ∂
∂= − + +
−
ββ β
α βαβ α
ααβ α
β
≠
( )
p
tx J x p
e t r
˜˜
1˜ ˜
2 i , . (34)0
We have neglected the Lamb shift by assuming ω ω≃β β(0) .
Equations (33) and (34) are linear as expected for a systemwith a quadratic Hamiltonian. The dipole–dipole interac-tion αβJ couples the momentum of one oscillator to dis-placement coordinates of other. The cooperative rateintroduces inter oscillator momentum–momentum anddisplacement–displacement coupling. The cooperativeemission term with α β= represents super-radiancecontribution.
Equations (33) and (34) can be solved by transformationto the frequency domain. It can be also solved by finding thenormal modes (eigenvalues of the system and decoupling themomentum from coordinates). The system of equations islinear with dimensionality 2N, where N is the number ofharmonic oscillators. For e.g., two oscillators one needs tosolve four linear equations. This is a generalization ofequation (1).
The QME contains sequential photon-exchange, whichgenerates an effective (dipole–dipole) interaction between theoscillators, as well as coupling to vacuum modes, which leadsto spontaneous emission. The effective interaction betweenparticles is instantaneous whereas the underlying photonexchange process may be interrupted by other field–matterinteractions (this is shown in the latter two diagrams offigure 5 for example). Thus, this calculation shows that thenonlinear response is caused by nonconsecutive photonexchange and is neglected by the QME, since it only includesconsecutive processes.
7. Discussion
In summary, we have calculated corrections to the linear andnonlinear response of a harmonic system due to coupling tothe vacuum modes of the radiation field. The inducednonlinear response, which is fourth-order in the quantummodes contains two types of processes. The first is theresonant energy transfer between oscillators. It arises wheneach oscillator a and b interact two times with the quantummodes and this contribution is zero in the heterodynedetected signal for the harmonic system. However, this nottrue for the homodyne detected signal generated by acoherent spontaneous emission [2] from oscillator a and b.The signal originates from two interactions with the samefield mode which is initially in the vaccum state. The dia-grammatic expansion is shown in figure 9. In essence, thelast two modes acting on oscillator a and b, which arequantum modes, generate the measured signal. The otherfour quantum modes correspond to the resonant energytransfer process.
In the second process oscillator a interacts three times withthe quantum modes and oscillator b interacts once. Here,
9
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
oscillator b emits a photon into the vacuum and oscillator aabsorbs it, then oscillator a emits and absorbs the same photonwith the vacuum. This process generates the finite nonlinearresponse. We showed that the FWM spectrum for the hetero-dyne detected signal contains new collective resonances.
We have calculated the effective χ (3) generated by thecoupling between two harmonic oscillators. It is also possibleto envision a three-oscillator process which involves fourinteractions with the quantum modes and four interactionswith the classical modes. This process is shown dia-grammatically in figure 10. The last quantum mode onoscillator a at time τv4 has a −V . For oscillators b and c, thelast quantum mode at times τv2 and τv1 be +V . The interaction atτv3 can be ±V . For −V , ⟨ ⟩+ − − −V V V V vanishes and for +V ,⟨ ⟩+ − − −E E E E vanishes. Thus the nonlinear signal will becomposed of two oscillator processes.
Acknowledgments
The support of the National Science Foundation (Grant CHE-1361516), the Chemical Sciences, Geosciences, and Bios-ciences division, Office of Basic Energy Sciences, Office ofScience, US Department of Energy is gratefully acknowledged.RG would like to thank ME Raikh for useful discussions.
Appendix A. Photon-exchange contribution to thelinear response
From figure 2, we can immediately write the expression forthe photon exchange contribution to the linear absorptionspectrum equation (10) as
⎡⎣⎢ ∫ ∫∫ ∫
ω τ τ
τ τ θ τ τ
τ τ τ τ
=
× −
×
ωττ
τ τ
ν ν
−∞
∞
−∞
−∞ −∞
− + −
{
S
V V V V
( )2i
d e d
d d ( )
( ) ( ) ( ) ( )
v
v v v
L va
vb
(1)4 2
i
1
2 1
v v
22
2
2
1
2
2 1
22
11
⎤⎦
ω τ
τ τ
τ τ τ τ
ω τ
τ τ
×
×
+
×
×
ν
ν
ν ν
ν
ν
+
− +
− + −
+
− + }
r E r
E r E r
V V V V
r E r
E r E r
( , ) ( , )
( , ) ( , )
( ) ( ) ( ) ( )
( , ) ( , )
( , ) ( , ) . (A.1)
L b v b
v a a
L v va
L a v a
v a a
0†
01
2 1
0†
01
22
11
22
11
22
11
The classical fields can be pulled out of the correlationfunction. The signal reads
⎡⎣⎢⎢
⎧⎨⎩
⎤⎦
∫ ∫
∫ ∫
∑ω τ τ
τ τ θ τ τ
τ τ τ τ
ω τ τ τ
τ τ τ τ
ω τ
τ τ
=
× −
×
×
+
×
×
αβ
ωττ
τ τ
ν ν
ν ν
ν ν
ν ν
−∞
∞
−∞
−∞ −∞
− + −
+ + −
− + −
+
+ −
}( )
S
V V V V
r E r E r E r
V V V V
r E r
E r E r
( )2i
d e d
d d ( )
( ) ( ) ( ) ( )
( , ) ( , ) ( , ) ( , )
( ) ( ) ( ) ( )
, ( , )
( , ) ( , ) . (A.2)
*
*
v
v v v
L va
vb
L b a v b v a
L v va
L a a
v a v a
(1)4 2
i
1
2 * 1
0† 0
1†
2 * 1
0† 0
1
†
v v
22
2
2
1
2
2 1
22
11
22
11
22
11
22
11
The integrations over time can be done by castingequation (A.2) into the frequency domain and yields
⎪
⎪
⎧⎨⎩
∫∑ωωπ
ωπ
ωπ
δ ω ω ω ω
μ μ ω ω μ μ
=
× − − +
× ⃗ ⃗
αβ −∞
∞
+
{( )
S
r E r
( )2i d
2
d
2
d
2
( , ) ( , )
v v
v v
a b a b a b
(1)4
1
1
*0† 0
1 *
1 2
1 2
Figure 9. Diagram for the homodyne detected nonlinear responseequation (7) expanded to seventh order, six interactions with thequantum modes (wavy red) and two interactions with the classicalmodes (solid blue) from two uncoupled harmonic oscillators.
Figure 10.Diagrams for the nonlinear response equation (7) for threeuncoupled harmonic oscillators expanded to seventh order, fourinteractions with the quantum modes (wavy red) and threeinteractions with the classical modes (solid blue). This contributionto the signal vanishes.
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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
ω ω
ω ω ω ω
μ ω ω μ μ
ω ω
ω ω ω ω
×× −
+ ⃗ ⃗
×
× −
ν ν
ν ν
+ −
+
+ −
}
E r E r
G G G
r E r
E r E r
G G G
( , ), ( , )
( ) ( ) ( )
( , ) ( , )
( , ), ( , )
( ) ( ) ( ) , (A.3)
v a v b
b a g v
a a a a a
v a v a
a a g v
†
1 1
20† 0
1 *
†
1 1
22
11
1
22
11
1
where the Greenʼs function ωG ( )a is given as
ω =ω ω Γ− +G ( )a
1
ia a. Evaluation of the field correlation func-
tion can be done by inserting the mode expansion of the fieldsequation (5), giving
⎡⎣ ⎤⎦
⎡⎣⎤⎦
∫
∫
τ τ
ωπ
ω
ω ω
ω ω
δ ω ω δ ω ω
δ ω ω δ ω ω
=
× −
=
× − −
− + +
να
νβ
αβν ν
ω τ τ ω τ τ
νβ
να
αβν ν
+ −
− − −
( ) ( )
( ) ( )
E r E r
r r
( , ), ( , )
1
2
d
2( )
e e ,
, , ,
1
2d ( )
( ) ( )
, (A.4)
v v
v v
v v
v v
†
( , )
i ( ) i ( )
†
( , )
v v v v
22
11
1 2
1 2 1 2
22
11
2 1
1 2
1 2
where
ωπϵ
δω
= − +αβαβ
αβ
( )r c
r( )
2·
sin ( ), (A.5)i j
ij i j( , )
0
2
= −αβ α βr r r| | is the distance between oscillators and
ω ωπϵ
δ=αα
c( )
2(A.6)l m
l m( , )
3
03 ,
is the cooperative emission (superradiance). Insertingequations (A.5), (A.6), we can perform two integrations inequation (A.3), using the delta functions.
⎧⎨⎩
⎫⎬⎭
∫
∫
∑ωμ μπ
ω ω ω ω
ω
πμ μ ω ω ω
μ
πω ω ω ω
ω
πμ μ ω ω ω
=
× ⃗ ⃗ −
+
× ⃗ ⃗ −
αβ+
−∞
∞
+
−∞
∞
S r E r G G
G
r E r G G
G
( )i
2( , ) ( , ) ( ) ( )
d
2˜ ( ) ( )
i
2( , ) ( , ) ( ) ( )
d
2˜ ( ) ( ) . (A.7)
a bb a a b
v
a b ab v g v
aa a a a
v
a a aa v g v
(1)2 0
† 01
2*
2
2 0† 0
1
2*
1
1 1
1
1 1
The last integration is done using contour integration, whichwill select a particular phase in the sine-function ofequation (A.5), ω αβe r ci . Using the identity
⎡⎣⎢
⎛⎝⎜
⎞⎠⎟
⎤⎦⎥
δ
δ δ
− +
= − − + − − +
( )
( ) ( )k
rrr
krrr
k r k r
1·
eˆ ˆ
13ˆ ˆ
i 1e . (A.8)
ij i j
kr
ij i j ij i jkr
32
i
2 2 3 3i
Equation (A.9) becomes
⎧⎨⎩ ∑ωμ μπ
ω ω ω=αβ
+S r E r G( )
i
2( , ) ( , ) ( )a b
b a a(1)
2 0† 0
1
⎪
⎪
⎫⎬⎭
ω γ ω
ω ω Γ
μ
πω ω ω ω
μ μ ωπϵ ω ω Γ
×−
− +
+
×− +
αβ αβ αβ αβ
+
( )
( )
J r r
r E r G G
c
( ; ) i ( ; )
i
i
2( , ) ( , ) ( ) ( )
2
1
i. (A.9)
b b
aa a a a
a a
b b
2
2 0† 0
1
* 3
03
The effective dipole–dipole coupling ωJ r( ; )AB AB is given as
⎡⎣⎢
⎡⎣ ⎤⎦
⎡⎣ ⎤⎦⎛⎝⎜⎜
⎞⎠⎟⎟
⎤⎦⎥⎥
ω ωπϵ
μ μ μ μ
ωω
μ μ μ μ
ω
ω
ω
ω
= −
× ⃗ ⃗ − ⃗ ⃗
×
− ⃗ ⃗ − ⃗ ⃗
× +
αβ αβ
αβ αβ
αβ
αβ
αβ αβ
αβ
αβ
αβ
αβ
′ ′
′ ′
( ) ( )( )
( ) ( )( )
J rc
r r
r c
r c
r r
r c
r c
r c
r c
( ; )4
· · ˆ · ˆ
cos ( )
· 3 · ˆ · ˆ
cos ( )
( )
sin ( )
( ), (A.10)
g f f g g f f g
g f f g g f f g
3
03
˜ ˜ ˜ ˜
˜ ˜ ˜ ˜
3 2
and the effective corporative emission rate γαβ is given by
⎡⎣⎢
⎡⎣ ⎤⎦
⎡⎣ ⎤⎦⎛⎝⎜⎜
⎞⎠⎟⎟
⎤⎦⎥⎥
γ ω ωπ ϵ
μ μ μ μ
ωω
μ μ μ μ
ω
ω
ω
ω
=
× ⃗ ⃗ − ⃗ ⃗
×
− ⃗ ⃗ − ⃗ ⃗
× −
αβ αβ
αβ αβ
αβ
αβ
αβ αβ
αβ
αβ
αβ
αβ
′ ′
′ ′
( ) ( )( )
( ) ( )( )
rc
r r
r c
r c
r r
r c
r c
r c
r c
( ; )4
· · ˆ · ˆ
sin ( )
· 3 · ˆ · ˆ
sin ( )
( )
cos ( )
( ). (A.11)
g f f g g f f g
g f f g g f f g
3
03
˜ ˜ ˜ ˜
˜ ˜ ˜ ˜
3 2
The vector r̂ connects the two dipoles such that their polarangle defines the parallel and perpendicular components asμ μ θ=α α α∥ cos and μ μ θ=α α α⊥ sin and the same for μβ.
See figure 3. Equations (A.10) and (A.11) can be cast in theform of equations (14), (15). From equation (A.9) we canwrite the effective χ (1), equation (12).
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J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
Appendix B. The nonlinear response to quartic order
In the nonlinear response there are two finite quantum fieldcorrelation functions, ⟨ ⟩+ − + −E E E E and ⟨ ⟩+ + − −E E E E . Thefield correlation functions ⟨ ⟩+ − − ±E E E E vanish.
For bookkeeping purposes, it is convenient to separatethe signal according to the possible field correlation functions
The signal ω( )I(3) is given diagrammatically in figure 11.
The three other signals ω( )II(3) - ω( )IV
(3) can be deduced fromthe diagrams in figure 11.
Using the diagrammatic perturbation, we can immedi-ately write the expression corresponding to the diagrams infigure 11
⎜ ⎟⎛⎝
⎞⎠
∫ ∫
∫
ω τ τ τ τ
τ τ ω τ τ
τ τ τ
τ τ Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
= − −
× …
×
× −
×
+
+
ωτ
ν ν
ν ν ν
ν ν ν
ν ν ν
ν ν ν
−
+ +
+ + +
− − +−
+−−−−+
+−−−+−
+−−+−−
{
}
r E r E r
E r E r E r
E r E r
( )2 i
d e d d d
d d ( , ) ( , ) ( , )
( , ) ( , ) ( , )
( , ) ( , ) ( ) ( , )
( , , , , , )
( , , , , , )
( , , , , , ) . (B.2)
I
v v L a a a
b v a v a
v a v b v v v
v v v
v v v
v v v
(3)7
4i
3 2 1
0† 03
02
01
1
,4 3 2
,4 3 2
,4 3 2
4
4 1
44
33
22
11 2 1
11
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
The ω( )II(3) signal has similar diagrams to figure 11, with
τv3 and τv2 having a +V , −V , respectively. The signal isgiven by
⎜ ⎟⎛⎝
⎞⎠
∫ ∫
∫
ω τ τ τ τ
τ τ ω τ τ
τ τ τ
τ τ Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
= − −
× …
×
× −
×
+
+
ωτ
ν ν
ν ν ν
ν ν ν
ν ν ν
ν ν ν
−
+ +
+ + −
+ − +−
+−−+−−
+−−−+−
+−+−−−
( ) ( )( ) ( )
{
}
r E r E r
E r E r E r
E r E r
( )2 i
d e d d d
d d ( , ) ( , ) ( , )
( , ) , ,
, , ( ) ( , )
( , , , , , )
( , , , , , )
( , , , , , ) . (B.3)
II
v v L a a a
b v a v a
v a v b v v v
v v v
v v v
v v v
(3)7
4i
3 2 1
0† 03
02
01
1
,4 3 2
,4 3 2
,4 3 2
4
4 1
44
33
22
11 2 1
11
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
The ω( )III(3) signal has similar diagrams to figure 11, with the
quantum mode at time τv1 occurring after τv2. The corre-sponding signal is given as
⎜ ⎟⎛⎝
⎞⎠
∫ ∫
∫
ω τ τ τ τ
τ τ ω τ τ
= − −
× …
ωτ−
+ +
r E r E r
( )2 i
d e d d d
d d ( , ) ( , ) ( , )
III
v v L a a a
(3)7
4i
3 2 1
0† 03
02
4
4 1
τ τ
τ τ
τ Θ τ τ Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
×
×
× − −
×
+
ν
ν ν
ν ν
ν ν ν
ν ν ν
+ +
+ −
− +−
+−−+−−
+−−−+−
( )( )
{}
E r E r
E r E r
E r
( , ) ( , )
, ( , )
, ( ) ( ) ( , )
( , , , , , )
( , , , , , ) . (B.4)
b v a
v a v b
v a v v v v v
v v v
v v v
01
1
,4 3 2
,4 3 2
44
33
11
22 3 1 1 2
11
4 3 24 3 2
4 3 24 3 2
The ω( )IV(3) signal has similar diagrams to figure 11, with the
quantum mode at time τv1 occurring after time τv3. The cor-responding signal is given as
⎜ ⎟⎛⎝
⎞⎠
∫ ∫
∫
ω τ τ τ τ
τ τ ω τ τ
τ τ τ τ
τ Θ τ τ Θ τ τ
α τ τ Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
= − −
× …
×
× − −
×
+
ωτ
ν ν ν
ν
ν ν ν ν
ν ν ν
−
+ +
+ + − +
−
+− +−−+−−
+−−−+−
{}
r E r E r
E r E r E r E r
E r
( )2 i
d e d d d
d d ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
( , ) ( ) ( )
( , ) ( , , , , , )
( , , , , , ) . (B.5)
IV
v v L a a a
b v a v b v a
v a v v v v
v v v v
v v v
(3)7
4i
3 2 1
0† 03
02
01
1,
4 3 2
,4 3 2
4
4 1
44
11
33
22 4 1 1 3
11
4 3 24 3 2
4 3 24 3 2
The expansion of the field correlation functions is done inappendix C. Inserting the field correlation functionsequations (C.11)–(C.14) in equations (B.2)–(B.5), respec-tively, yields
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎤⎦⎥
⎫⎬⎭
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ
δ τ τ
δ τ τ δ τ τ
δ τ τ
Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
=
×
× …
×
× − +
× − − +
− − −
+ − − +
− − +
× −
×
+
+
μμωτ
νν ν ν
ν
ν ν ν
ν ν ν
ν ν ν
′−
+
+ +
′ ′
′′′
′′′
+−
+−−−−+
+−−−+−
+−−+−−
( )
( ){
}
c
r E r
E r E r
r
r
c
r
c
r
c
r
c
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )
1
( )
( )
( , )
( , , , , , )
( , , , , , )
( , , , , , ) , (B.6)
I
v v L a a
a b
ab
v v v vab
v vab
v v v vab
v vab
v v v
v v v
v v v
v v v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
1
,4 3 2
,4 3 2
,4 3 2
4
4 1
2 3 1 4
1 4
2 4 1 3
1 3
2 11
1
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
12
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
Figure 11.Diagrams for the nonlinear response equation (7) expanded to seventh order, four interactions with the quantum modes (wavy red)and four interactions with the classical modes (solid blue) from two uncoupled harmonic oscillators. The diagrams have the field correlationfunction τ τ τ τ⟨ ⟩ν ν ν ν
+ + − −E r E r E r E r( , ) ( , ) ( , ) ( , )v a v a v a v b4
43
32
21
1 .
13
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
⎜ ⎟
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ
δ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
α τ τ
=
×
× …
×
× − +
× − − +
− − −
×
+
+
×
μμωτ
νν ν ν
ν ν ν
ν ν ν
ν ν ν
ν
′−
+
+ +
′ ′
′′′
+−−+−−
+−−−+−
+−+−−−
+−
( )
( )
{
}
c
r E r
E r E r
r
r
c
r
c
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )
1
( , , , , , )
( , , , , , )
( , , , , , )
( , ), (B.7)
II
v v L a a
a b
ab
v v v vab
v vab
v v v
v v v
v v v
v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
,4 3 2
,4 3 2
,4 3 2
1
4
4 1
3 4 1 2
1 2
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
11
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎫⎬⎭
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ
δ τ τ
δ τ τ δ τ τ
δ τ τ Θ τ τ
Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
=
×
× …
×
× − +
× − − +
− − −
+ − − +
− − − −
× −
×
+
μμωτ
νν ν ν
ν
ν ν ν
ν ν ν
′−
+
+ +
′ ′
′′′
′′′
+−
+−−+−−
+−−−+−
( )
( ){
}
c
r E r
E r E r
r
r
c
r
c
r
c
r
c
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )
1
( )
( )
( )
( , )
( , , , , , )
( , , , , , ) , (B.8)
III
v v L a a
a b
ab
v v v vab
v vab
v v v vab
v vab
v v
v v v
v v v
v v v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
1
,4 3 2
,4 3 2
4
4 1
2 4 1 3
1 3
2 3 1 4
1 4 3 1
1 21
1
4 3 24 3 2
4 3 24 3 2
⎛⎝⎜
⎞⎠⎟
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
=
×
× …
×
μμωτ
′−
+
+ +
c
r E r
E r E r
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )
IV
v v L a a
a b
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
4
4 1
⎜ ⎟
⎜ ⎟
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
δ τ τ
δ τ τ
δ τ τ Θ τ τΘ τ τ Θ τ τα τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
× − +
× − +
− − −
× − −× − −×
×
+
νν ν ν
ν
ν ν ν
ν ν ν
′ ′
′′′
+−
+−−+−−
+−−−+−
( )
{}
r
r
c
r
c
1
( ) ( )
( ) ( )
( , )
( , , , , , )
( , , , , , ) . (B.9)
ab
v vab
v vab
v v v v
v v v v
v
v v v
v v v
2
1
,4 3 2
,4 3 2
1 4
1 4
2 3 4 1
1 3 1 2
11
4 3 24 3 2
4 3 24 3 2
In equation (B.9) we have inserted the theta functionΘ τ τ−( )v v1 2 . In the field correlation function of equation (B.9),τv1 is always greater than τv2 justifying the insertion of this term.The second delta function in equations (B.6)–(B.9) does notcontribute since >r 0ab , giving
⎜ ⎟
⎟
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎛⎝
⎞⎠
⎛⎝⎜
⎞⎠
⎫⎬⎭
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ
δ τ τ δ τ τ
Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
=
×
× …
× − +
× − − +
+ − − +
× −
×
+
+
μμωτ
νν ν ν
ν
ν ν ν
ν ν ν
ν ν ν
′−
+
+ + ′ ′
′′′
′′′
+−
+−−−−+
+−−−+−
+−−+−−
( )( )
}
( )
{
c
r E r
E r E rr
r
c
r
c
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )1
( )
( )
( ) ( , )
, , , , ,
, , , , ,
( , , , , , ) , (B.10)
I
v v L a a
a bab
v v v vab
v v v vab
v v v
v v v
v v v
v v v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
1
,4 3 2
,4 3 2
,4 3 2
4
4 1
2 3 1 4
2 4 1 3
2 11
1
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝
⎞⎠
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
α τ τ
=
×
× …
× − +
× − − +
×
+
+
×
μμωτ
νν ν ν
ν ν ν
ν ν ν
ν ν ν
ν
′−
+
+ + ′ ′
′′′
+−−+−−
+−−−+−
+−+−−−
+−
( )
{
}
c
r E r
E r E rr
r
c
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )1
( )
( , , , , , )
( , , , , , )
( , , , , , )
( , ), (B.11)
II
v v L a a
a bab
v v v vab
v v v
v v v
v v v
v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
,4 3 2
,4 3 2
,4 3 2
1
4
4 1
3 4 1 2
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
11
14
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
⎜ ⎟
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎫⎬⎭
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ
δ τ τ δ τ τ
Θ τ τ Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
=
×
× …
× − +
× − − +
+ − − +
× − −
×
+
μμωτ
νν ν ν
ν
ν ν ν
ν ν ν
′−
+
+ + ′ ′
′′′
′′′
+−
+−−+−−
+−−−+−
{}
( )
c
d
r E r
E r E r
r
r
c
r
c
( )1
8
1
2
d e d d
d d ( , ) ( , )
( , ) ( , )
1( )
( )
( ) ( ) ( , )
( , , , , , )
( , , , , , ) , (B.12)
III
v v L a a
a b
abv v v v
ab
v v v vab
v v v v v
v v v
v v v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
1
,4 3 2
,4 3 2
4
4 1
2 4 1 3
2 3 1 4
3 1 1 21
1
4 3 24 3 2
4 3 24 3 2
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎛⎝
⎞⎠
∫ ∫
∫
ωπ πϵ
δ τ τ τ τ
τ τ ω τ
τ τ
δ τ τ δ τ τ Θ τ τ
Θ τ τ Θ τ τ α τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
=
×
× …
× − +
× − + − −
× − −
×
+
μμωτ
νν ν ν
ν
ν ν ν
ν ν ν
′−
+
+ + ′ ′
′′′
+−
+−−+−−
+−−−+−
{}
( )
c
r E r
E r E rr
r
c
( )1
8
1
2
d e d d d
d d ( , ) ( , )
( , ) ( , )1
( ) ( )
( ) ( ) ( , )
( , , , , , )
( , , , , , ) . (B.13)
IV
v v L a a
a bab
v vab
v v v v
v v v v v
v v v
v v v
(3)3 6 3
0
2
4i
3 2 1
0† 03
02
01
2
1
,4 3 2
,4 3 2
4
4 1
1 4 2 3 4 1
1 3 1 21
1
4 3 24 3 2
4 3 24 3 2
Equations (B.12) and (B.13) can be added yielding
⎜ ⎟
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎫⎬⎭
∫ ∫
∫
ω ω
π πϵ
δ τ τ τ τ
τ τ ω
τ τ
τ Θ τ τ
δ τ τ δ τ τ
δ τ τ δ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
α τ τ
+
=
×
× …
×
× − − +
× − − +
+ − + −
×
+
×
μμωτ
νν ν ν
ν ν ν
ν ν ν
ν
′−
+ +
+ ′ ′
′′′
′′′
+−−+−−
+−−−+−
+−
( )
{}
c
r
E r E r
E rr
r
c
r
c
( ) ( )
1
8
1
2
d e d d d
d d ( , )
( , ) ( , )
( , ) ( )1
( )
( )
( , , , , , )
( , , , , , )
( , ). (B.14)
III IV
v v L a
a a
b v vab
v v v vab
v vab
v v
v v v
v v v
v
(3) (3)
3 6 30
2
4i
3 2 1
0†
03
02
01
2
,4 3 2
,4 3 2
1
4
4 1
1 2
2 4 1 3
1 4 2 3
4 3 24 3 2
4 3 24 3 2
11
The term Δ ν ν ν+−−−−+
,4 3 2 is absent for equation (B.14). This isbecause Δ ν ν ν
+−−−−+,4 3 2 corresponds to an interaction with quantum
fields before the classical fields. This term vanishes, since weassume that the quantum fields are initially in the ground state.We can freely insert Δ ν ν ν
+−−−−+,4 3 2 , since it is zero, into
equation (B.14) and add ω( )I , ω( )III and ω( )IV , yielding
⎜ ⎟
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎫⎬⎭
∫ ∫ ∫
ω ω ω
π πϵ
δ τ τ τ τ τ τ
ω τ τ
τ
δ τ τ δ τ τ
δ τ τ δ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
α τ τ
+ +
=
× …
×
× − +
× − − +
+ − + −
×
+
+
×
μμωτ
νν ν ν
ν ν ν
ν ν ν
ν ν ν
ν
′−
+ +
+ ′ ′
′′′
′′′
+−−−−+
+−−−+−
+−−+−−
+−
( )
{
}
c
r E r E r
E rr
r
c
r
c
( ) ( ) ( )
1
8
1
2
d e d d d d d
( , ) ( , ) ( , )
( , )1
( )
( )
( , , , , , )
( , , , , , )
( , , , , , )
( , ). (B.15)
I III IV
v v
L a a a
bab
v v v vab
v vab
v v
v v v
v v v
v v v
v
(3) (3) (3)
3 6 30
2
4i
3 2 1
0† 03
02
01
2
,4 3 2
,4 3 2
,4 3 2
1
44 1
2 3 1 4
1 3 2 4
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
11
The total signal can be found by adding equations (B.11) and(B.15), yielding
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛⎝⎜
⎞⎠⎟
⎧⎨⎩⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎡⎣
⎤⎦⎛⎝
⎞⎠
∫
∫ ∫
ωπ πϵ
τ
τ τ τ τ τ ω τ
τ τ
δ δ τ τ δ τ τ
δ τ τ δ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
δ τ τ δ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
α τ τ
=
× …
× − +
× − − +
+ − − +
×
+
+
× − + − +
×
+
+
×
ωτ
νν ν ν
μμ
ν ν ν
ν ν ν
ν ν ν
ν ν ν
ν ν ν
ν ν ν
ν
−
+
+ + ′ ′
′′′′
′′′
+−−−−+
+−−−+−
+−−+−−
′′′
+−−+−−
+−−−+−
+−+−−−
+−
( )
{
}
Sc
r E r
E r E r
r
r
c
r
c
r
c
( )1
8
1
2d e
d d d d d ( , ) ( , )
( , ) ( , )
1( )
( )
( , , , , , )
( , , , , , )
( , , , , , )
( )
( , , , , , )
( , , , , , )
( , , , , , )
( , ). (B.16)
v v L a a
a b
abv v v v
ab
v v v vab
v v v
v v v
v v v
v v v vab
v v v
v v v
v v v
v
(3)3 6 3
0
2
4i
3 2 10† 0
3
02
01
2
,4 3 2
,4 3 2
,4 3 2
,4 3 2
,4 3 2
,4 3 2
1
4
4 1
2 3 1 4
2 4 1 3
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
3 4 1 2
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
11
Using equations (C.9) and (C.10), equation (B.16) can be castinto the following form
15
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
⎜ ⎟⎛⎝
⎞⎠
⎡⎣ ⎤⎦
⎡⎣⎤⎦
⎡⎣
⎤⎦
⎡⎣
⎤⎦
∫∫ ∫
∫
∫
ω τ
τ τ τ τ τ
ω τ τ
τμ μ
ωπ
ω γ ω
ωπ
ω
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
Δ τ τ τ τ τ τ
α τ τ
= − −
× …
×
×
× −
×
×
+
×
+
+
+×
+
+
×
ωτ
μ μ
ω τ τ ω τ τ
ω τ τ ω τ τ
ν ν ν
ν ν ν
ν ν ν
ω τ τ ω τ τ
ν ν ν
ν ν ν
ν ν ν
ν
−
+ +
+
′
− + −
− + −
+−−−−+
+−−−+−
+−−+−−
− + −
+−−+−−
+−−−+−
+−+−−−
+−
{
}
( ) ( )
S
r E r E r
E r
J r r
( )2 i
d e
d d d d d
( , ) ( , ) ( , )
( , )d
2
; i ;
d
2˜ ( )
e
e
( , , , , , )
( , , , , , )
( , , , , , )
e
( , , , , , )
( , , , , , )
( , , , , , )
( , ), (B.17)
v v L
a a a
ba b
v
ab ab v ab ab v
vaa v
v v v
v v v
v v v
v v v
v v v
v v v
v
(3)7
4i
3 2 10†
03
02
01
( , )
i ( ) i ( )
i ( ) i ( )
,4 3 2
,4 3 2
,4 3 2
i ( ) i ( )
,4 3 2
,4 3 2
,4 3 2
1
v v v v v v
v v v v v v
v v v v v v
4
4 1
1
1 1
2
2
2 2 3 1 1 4
2 2 4 1 1 3
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
2 3 4 1 1 2
4 3 24 3 2
4 3 24 3 2
4 3 24 3 2
11
where ωJ r( ; )ab ab v1 and γ ωr( ; )ab ab v1 are given byequations (14) and (15), respectively. Next we use the Fouriertransform ∫ ω= ω
,i j k are given in equations (19)–(22), respectively.The integrations over ωv1 and ωv2 are done using contour
integration. The contour integration over ωv2 vanishes for thediagrams in figure 11 that have quantum and classical fieldinteractions in-between the emission and absorption of thesame photon with oscillator a. With this observation, we caneliminate the diagrams that contain interactions with the fieldin-between the emission and absorption of a photon byoscillator a via the vacuum. This is done in figures 12–13,where we draw all diagrams which contribute to the signal.The green wavy lines correspond to the emission of a quan-tum mode from oscillator b and the absorption by oscillator a.The red wavy lines correspond to the emission and absorptionof a photon by oscillator a via the vacuum.
In diagrams figures 12(1), (4), (6) it is possible to have τv4
and τv3 red and τv2 and τv1 green. However, this contributionvanishes in the expansion of the field correlation function,because it corresponds to ⟨ ⟩+ − − +E E E E , which vanishes.
The final expression of the susceptibility for diagrams infigures 12–13 is given as equation (18).
Appendix C. The field correlation function
The field correlation functions can be expanded using theexpression for the field equation (5) and Wickʼs theoremgiving
⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
× −
× −
+ −
× −
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
ω τ τ ω τ τ
ω τ τ ω τ τ
ω τ τ ω τ τ
+ + − −
′ ′
− −
− −
− −
− −
{
}
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e e
e e
e e
e e , (C.1)
v a v a v a v b
vab v
vaa v
( , ) ( , )
i ( ) i ( )
i ( ) i ( )
i ( ) i ( )
i ( ) i ( )
v v v v v v
v v v v v v
v v v v v v
v v v v v v
44
33
22
11
1
1
2
2
2 2 3 2 3 2
1 1 4 1 4 1
2 2 4 2 4 2
1 1 3 1 3 1
⎡⎣ ⎤⎦⎡⎣ ⎤⎦
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
× −
× −
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
ω τ τ ω τ τ
+ − + −
′ ′
− −
− −
{}
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e e
e e , (C.2)
v a v a v a v b
vab v
vaa v
( , ) ( , )
i ( ) i ( )
i ( ) i ( )
v v v v v v
v v v v v v
44
33
22
11
1
1
2
3
3 3 4 3 4 3
1 1 2 1 2 1
16
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
Figure 12. Similar to figure 11, the diagrams that vanish after the contour integration ωv2 in equation (B19) have been dropped. The red wavy
lines correspond to the emission and absorption of a photon with oscillator a. The green wavy lines to the emission of a photon fromoscillator b and the absorption the photon by oscillator a. The diagrams proportional to (I) τ τ τ τ⟨ ⟩ν ν ν ν
+ + − −E r E r E r E r( , ) ( , ) ( , ) ( , )v a v a v a v b4
43
32
21
1 .(II) τ τ τ τ⟨ ⟩ν ν ν ν
+ + − −E r E r E r E r( , ) ( , ) ( , ) ( , )v a v a v b v a4
43
31
12
2 .
17
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦⎡⎣ ⎤⎦
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
× −
× −
+ −
× −
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
ω τ τ ω τ τ
ω τ τ ω τ τ
ω τ τ ω τ τ
+ + − −
′ ′
− −
− −
− −
− −
{
}
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e e
e e
e e
e e , (C.3)
v a v a v b v a
vaa v
vab v
( , ) ( , )
i ( ) i ( )
i ( ) i ( )
i ( ) i ( )
i ( ) i ( )
v v v v v v
v v v v v v
v v v v v v
v v v v v v
44
33
11
22
2
2
1
1
1 1 3 1 3 1
2 2 4 2 4 2
1 1 4 1 4 1
2 2 3 2 3 2
⎡⎣ ⎤⎦⎡⎣ ⎤⎦
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
× −
× −
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
ω τ τ ω τ τ
+ − + −
′ ′
− −
− −
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e e
e e , (C.4)
v a v b v a v a
vaa v
vab v
( , ) ( , )
i ( ) i ( )
i ( ) i ( )
v v v v v v
v v v v v v
44
11
22
33
3
3
1
1
1 1 4 1 4 1
3 2 3 3 3 2
where ων ν′ ( )aa( , ) and ων ν′ ( )ab
( , ) are given by equations (A.6),
(A.5). Using the fact that ων ν′ ( )aa( , ) and ων ν′ ( )ab
( , ) are oddfunctions the field correlation functions can be cast in thefollowing form
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
ν ν ν ν
ν ν μ μ
+ + − −
′ ′
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
v a v a v a v b
vab v
vaa v
( , ) ( , )
44
33
22
11
1
1
2
2
⎡⎣⎤⎦
×
+
ω τ τ ω τ τ
ω τ τ ω τ τ
− + −
− + −
e
e , (C.5)
i ( ) i ( )
i ( ) i ( )
v v v v v v
v v v v v v
2 2 3 1 1 4
2 2 4 1 1 3
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
×
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
+ − + −
′ ′
− + −
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e , (C.6)
v a v a v a v b
vab v
vaa v
( , ) ( , )
i ( ) i ( )v v v v v v
44
33
22
11
1
1
2
3
3 3 4 1 1 2
⎡⎣⎤⎦
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
×
+
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
ω τ τ ω τ τ
+ + − −
′ ′
− + −
− + −
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e
e , (C.7)
v a v a v b v a
vaa v
vab v
( , ) ( , )
i ( ) i ( )
i ( ) i ( )
v v v v v v
v v v v v v
44
33
11
22
2
2
1
1
1 1 3 2 2 4
1 1 4 2 2 3
∫ ∫τ τ τ τ
ωπ
ωωπ
ω=
×
ν ν ν ν
ν ν μ μ
ω τ τ ω τ τ
+ − + −
′ ′
− + −
E r E r E r E r( , ) ( , ) ( , ) ( , )
d
2( )
d
2( )
e . (C.8)
v a v b v a v a
vaa v
vab v
( , ) ( , )
i ( ) i ( )v v v v v v
44
11
22
33
3
3
1
1
1 1 4 3 2 3
The integration over ω can be done using the identity
⎜ ⎟⎛⎝
⎞⎠∫ ω
ω ω τ τ−r
cd sin ev
v ab i ( )1
1 2
Figure 13. Similar to figure 12 with the field correlation function (III) τ τ τ τ⟨ ⟩ν ν ν ν+ + − −E r E r E r E r( , ) ( , ) ( , ) ( , )v a v a v b v a
44
33
11
22 (IV)
τ τ τ τ⟨ ⟩ν ν ν ν+ − + −E r E r E r E r( , ) ( , ) ( , ) ( , )v a v b v a v a
44
11
33
22 .
18
J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 065401 R Glenn et al
⎜ ⎟
⎜ ⎟
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
π δ τ τ
δ τ τ
= − − +
− − −
r
c
r
c
2
i
. (C.9)
ab
ab
1 2
1 2
The integration over ωv2 can be using the identity
∫ ω ωμ μ ω τ τ′ −d ( )ev aa v( , ) i ( )v v v
3 33 3 4
⎜ ⎟⎛⎝
⎞⎠πϵ π
δ δ τ τ= −μμ′′′′
c4
i
2( ), (C.10)v v
03
3
3 4
where δ′′′ is the third derivative of the delta-function.Inserting equations (C.9) and (C.10) into the field correlationequations (C.1)–(C.8) functions gives
⎜ ⎟⎛⎝
⎞⎠
τ τ τ τ
πϵ πϵ πδ= − − +
ν ν ν ν
μμ νν ν ν
+ + − −
′ ′ ′ ( )
E r E r E r E r
c
( , ) ( , ) ( , ) ( , )
2 4
i
2
v a v a v a v b
0 03
32
44
33
22
11
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎧⎨⎩⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎫⎬⎭
δ τ τ δ τ τ
δ τ τ
δ τ τ δ τ τ
δ τ τ
× − − +
− − −
+ − − +
− − −
′′′
′′′
r
r
c
r
c
r
c
r
c
1( )
( )
, (C.11)
abv v v v
ab
v vab
v v v vab
v vab
2 3 1 4
1 4
2 4 1 3
1 3
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛⎝
⎞⎠
⎧⎨⎩⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎫⎬⎭
τ τ τ τ
πϵ πϵ πδ
δ τ τ δ τ τ
δ τ τ
= − − +
× − − +
− − −
ν ν ν ν
μμ νν ν ν
+ − + −
′ ′ ′
′′′
( )
E r E r E r E r
c
r
r
c
r
c
( , ) ( , ) ( , ) ( , )
2 4
i
2
1( )
, (C.12)
v a v a v a v b
abv v v v
ab
v vab
0 03
32
44
33
22
11
3 4 1 2
1 2
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛⎝
⎞⎠
⎧⎨⎩⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎫⎬⎭
τ τ τ τ
πϵ πϵ π
δ
δ τ τ
δ τ τ δ τ τ
δ τ τ
δ τ τ δ τ τ
= −
× − +
× − +
− − − −
+ − +
− − − −
ν ν ν ν
μμ νν ν ν
+ + − −
′ ′ ′
′′′
′′′
( )
E r E r E r E r
c
r
r
c
r
c
r
c
r
c
( , ) ( , ) ( , ) ( , )
2 4
i
2
1
( )
( ) , (C.13)
v a v a v b v a
abv v
ab
v vab
v v
v vab
v vab
v v
0 03
3
2
44
33
11
22
1 3
1 3 2 4
1 4
1 4 2 3
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎛⎝
⎞⎠
⎡⎣⎢
⎛⎝
⎞⎠
⎛⎝
⎞⎠
⎤⎦⎥
⎫⎬⎭
τ τ τ τ
πϵ πϵ πδ
δ τ τ
δ τ τ δ τ τ
= − − +
× − +
− − − −
ν ν ν ν
μμ νν ν ν
+ − + −
′ ′ ′
′′′
( )
E r E r E r E r
c
r
r
c
r
c
( , ) ( , ) ( , ) ( , )
2 4
i
2
1
( ) . (C.14)
v a v b v a v a
abv v
ab
v vab
v v
0 03
32
44
11
22
33
1 4
1 4 2 3
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