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12 Cavity QED

The term cavity QED stands for cavity quantum electrodynamics. It treats themodication of the emission properties of emitters (e.g. atoms) by the modicationof the quantum vacuum.

12.1 Spontaneous decay

In previous chapters we have already treated two methods to calculate the sponta-neous decay of a two-level atom in free space.

The starting point was usually a Hamiltonian of the form:

H = H atom + H vac u m + H int (588)12

0z +k

a+k ak + k

(gk+ ak + h.c.) (589)

Now we will use three different methods to derive the spontaneous emission rate:

12.1.1 Method 1: master equation

This method is the most general one.

It leads to an expression for the time evolution of a small system (e.g. an atom)coupled to a reservoir (e.g. a thermal bath of harmonic oscillators).

For the reduced density matrix S of the small system we found:

t

S = 1i

[H A , S ] 2

(nR + 1) + S (t) S (t)+

2

nR S (t) + + S (t) (590)

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with the rate

= 3

02

123 0 c3 (591)

The time evolution of the matrix element

e|S (t)|e = pe(t) = p0 exp(t) (592)which follows from this equation reveals the irreversible exponential decay.

12.1.2 Method 2: Wigner-Weisskopf theory

In this theory we started with the general state vector

|(t) = ce(t)e i 0 t |e, {0} +k

cgk (t)e i k t |g, {1k} (593)which after substitution into the Schroedinger equation gave the following coupledequations:

ce(t) = i

k

gke i(k 0 )t cgk (t) (594)

cgk (t) = igkei(k 0 )t ce(t) (595)

Integration and iteration resulted nally in the same exponential decay with thesame rate as given above.

12.1.3 Method 3: Fermis golden rule

It is intuitive to use this third method.

The approximation is to setce(t) = ce(0) = 1 (596)

in the equation above for cgk (t).

Then one nds easily:

|cgk (t)|2 = |gk|2sin2 [(k 0)t/ 2]

(k 0)2/ 4 (597)

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The probability for the atom to be in the upper state isP e = 1

k|cgk (t)|2 (598)

1 dk |gk|2 sin2 [(k 0)t/ 2](k 0)2/ 4 (599)By noting that gk = k/ 2 0V 12 and averaging over all possible orientationsbetween the atomic dipole 12 and the electric eld vector one can derive:P e = 1

1

602 c3 d 3

|12

|2 sin

2 [( 0)t/ 2]( 0)2/ 4

(600)

Here we used the mode density of free space in the integral. However, this is notnecessarily correct! The key point of cavity QED is:

The spontanoeus emission of an atom is not an intrinsic property, but depends on the atoms environment!

In the equation above for the decay in free space, however, for long enough times(but short enough to justify rst order perturbation theory, ce(t) = 1) it is:

limt

sin2 [( 0)t/ 2]( 0)2/ 4

= 2 ( 0)t (601)and thus

dP e(t)dt

= 3|12|230 c3

P e(t) = P e(t) (602)=

2 2 | 12E |2D free () (603)

The last row is a form of Fermis golden rule with the density of states (DOS)D free () and the interaction 12E .

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12.2 Spontaneous emission in cavities

A typical cavity QED situation is shown in the following gure.

Figure 83: A typical cavity QED system: a single atom is inside a Fabry-Perot cavity [from Meystrein Sponaneous emission an laser oscillation im microcavities, H. Yokoyama, K. Ujihara, eds.]

A rst approach to study the modication of spontaneous emission is to replace thedensity of states for free space Dfree () phenomenologically with a cavity modieddensity of states Dcav ():

Dcav () = 2V

1(/ 2)2 + ( cav

)2

(604)

with the cavity damping rate .

A useful denition for the quality of a cavity is the cavity Q-factor:

Q = cav / (605)

With this expression one nds:

Enhanced spontaneous emission for an atom in resonance with the cavity:

cav = free3Q42

30V with 0 = 2c/ (606)

Suppressed spontaneous emission for an atom off-resonance with the cavity:cav = free

3162Q

30V

with 0 = 2c/ (607)

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Enhancement and suppression have been demonstrated for the rst time in experi-ments by Kleppner [D. Kleppner, Phys. Rev. Lett. 47, 233 (1981)] and Haroche [P.Goy, et al., Phys. Rev. Lett. 50, 1903 (1983)].

A more rigorous result is derived via the master equation (at zero temperature):

t

= 1i

[H A , ] 2

+ (t) + (t)+ 2 (t)+

2 a+ a(t) + (t)a+ a 2a(t)a+ (608)

where is the coupling of the atom to a fraction of free space (not covered by thesolid angle of the cavity) and is the damping rate of the cavity (loss of photons toa reservoir).

It is useful to write this master equation as

t

= 1i

[H eff , ] + (t)+ + a (t)a+ (609)

In the subspace

{|e, 0 ,

|g, 1

}this equation of motion is equivalent to a Schroedinger

equation with an effective (non-Hermitian) Hamiltonian H eff

H ef f = H A i 2

+ i 2

a+ a (610)

acting on the (unnormailzed) one-quantum state

|(t) = ce(t)ei/ 2 |e, 0 + cg(t)e i/ 2 |g, 1 (611)with = 0: i t |(t) = H eff |(t) (612)With this one nds

dce(t)dt

= 2

ce(t) igcg(t) (613)dcg(t)

dt = (i + / 2)cg(t) igce(t) (614)

In the following we divide two regimes for the approximate solution of this equation.

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12.2.1 Weak coupling regime

In this regime we assume g , .Formally integrating of the equation above yields:

cg(t) = igt

0

dt ce(t )e (i+ / 2)( t t ) (615)

In the weak coupling regime ce(t ) evolves slowly and can be taken out of the integral.

Then:cg(t) = igi + / 2 ce(t) (616)

Substituting this yields:

dce(t)dt

= ( / 2) + g2 (/ 2 i )

2 + 2/ 4ce(t) (617)

Hence:

|ce(t)|2 = P e(t) = ce(0)exp(cav ) with (618)cav = +

2g2

1

1 + 2(2/ )2 (619)

Replacing g with the expression for the free space rate yields in resonance ( = 0):

(e)cav = 3Q42

30V

(620)

and off-resonance:

(s)cav = 3

162Q30V

(621)

These expressions are identical to the results derived above.

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12.2.2 Strong coupling regime

The general solution for ce(t) for arbitrary g, , is of the form

ce(t) = ce1e 1 t + ce2e 2 t (622)

with

1,2 = 12

2

+ 2

+ i 12

2

+ 2

+ i 2

4g21/ 2

(623)

with some coefficients ce1 and ce2.

If g

, it follows

1,2 = 122

+ 2

+ i ig (624)

This leads to a damped oscillation of |ce(t)|2 with the Rabi frequency = 2 g whitha damping constant ( + )/ 4.The following shows a plot of the excited state probability.

Figure 84: Excited state probabilty in the weak and strong coupling regime [from Meystre inSponaneous emission an laser oscillation im microcavities, H. Yokoyama, K. Ujihara, eds.]

The spectrum of the uorescence light consists of two Lorentzians split by = 2g.

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12.3 The spectrum in the strong coupling regime

The appearance of a doublet structure in the uorescence spectrum in the strongcoupling regime is easily interpreted in the dressed atom picture.

The dressed state energies of the combined atom-eld system are

E 2n = (n + 1 / 2)c Rn (625)E 1n = (n + 1 / 2)c + Rn (626)

with the Rabi frequency

Rn = 1

2 2 + 4 g2(n + 1) (627)

The according eigenstates are

|2n = sin n |en + cos n |gn + 1 (628)|1n = cos n |en + sin n |gn + 1 (629)

where

tan2n = 2g n + 1

(630)

The spontaneous decay occurs now from the manifold with one quantum ( |e, 0 , |g, 1 )to the manifold with zero quantum ( |g, 0 ). Obviously, if there is some coupling withnon-zero g, the one-quantum manifold splits, but not the zero-quantum manifold.In case of exact resonance both dressed states of the manifold have equal atomicand photonic weight.

However, if the detuning is large it is

|10 |e, 0 g |g, 1 (631)

|20 g |e, 0 |g, 1 (632)

|10 is thus predominantly atomic and |20 predominantly photonic.

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Figure 85: Manifolds of dressed state eigenstates [from Meystre in Sponaneous emission an laseroscillation im microcavities, H. Yokoyama, K. Ujihara, eds.]

The following gure shows a simulated dressed states spectrum in the strong couplingregime when the detuning is changed.

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Figure 86: Simulated dressed states spectrum in the strong coupling regime when changing thedetuning

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The next gure shows an experimental result from an atomic physics experiment

performed in the optical domain [R. J. Thompson et al., Phys. Rev. Lett. 68, 1132(1992)]. Cs atoms from an atomic beam were sent through a high-nesse cavity andthe transmission of a probe laser through the cavity was measured.

Figure 87: Normal mode splitting of a cavity-atom system for decreasing average number of atomsin the cavity [R. J. Thompson et al., Phys. Rev. Lett. 68, 1132 (1992)].

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12.4 Modication of spatial emission pattern

If an emitter is placed inside a cavity not only the total emission rate, but also theemission pattern is changed.We assume the following geometry of a dipole in a planar cavity formed by twomirrors of reectivity R1 and R2:

Figure 88: Geometry for the calculation of the electric eld inside a Fabry-Perot cavity [fromBjoerk in Sponaneous emission an laser oscillation im microcavities, H. Yokoyama, K. Ujihara,eds.]

The starting point for the analysis is Fermis golden rule:

=2

2 | 12E |2D() (633)

The modication of the rate by the cavity can be interpreted in wo ways:1. One assumes there is an electric vacuum eld of magnitude E 0 = 20 V ineach mode, but the density of states D() is changed.2. One assumes a density of states as in free space, but a modied electric eld

strength. The calculation of the modied eld is a classical calculation where

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one assumes that a vacuum eld of strength E 0 impinges on the cavity and is

enhanced or suppressed.

Since E 0 and D() always enter together, both interpretations are valid.

One can now write the spontaneous emission per unit angle and unit frequencyfor a y-dipole as:

(, ,) =2

2 | 12E (, ,)|2 2V 4c33

(634)

For s-polarized radiation (E-eld perdendicular to the plane of incidence):

(, ,) = 2

212()

20V 2V 4c33

cos2 = 30 cos2

8 (635)

For -polarized radiation (E-eld parallel to the plane of incidence):

(, ,) = 30 sin2 cos2

8 (636)

One can check that the total spontaneous emission of a y-dipole is:

= 308

2

0

d2

0

d sin (cos2 + sin 2 cos2 ) = 30

4 (1 +

13

) = 0 (637)

Now one only has to calculate the vacuum eld entering a cavity.

The eld at the position z 0 inside the cavity originating from E 0 from the direc-tion , (and traveling in + and

direction) is:

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E = E 0 T 2 exp(ikz(L z 0)) 1 + exp(2 i ) R1R2 exp(2ik z L) + ...= 1 R2 exp(ik(L z 0)cos )1 R1R2 exp(2ik cos)

E 0 (638)

E + = E 0 T 2R1 exp(ikz(L + z 0)) 1 + exp(2 i ) R1R2 exp(2ik z L) + ...= R1(1 R2)exp( i [k(L + z 0)cos + ])1 R1R2 exp(2ik cos) E 0 (639)

where kz = k cos = 2 cos/ .

It is now possible to calculate the parallel and z-component of the eld for s-polarization

|E || |2 = |E + + E |2 and |E z|2 = 0 (640)and for -polarization:

|E || |2 = |E + + E |2 cos2 and |E z|2 = |E + + E |2 sin2 (641)

Resonances in the expression occur at kz = k cos = m/ 2.The full width half maximum (FWHM) of these resonances in wavelength andangle is:

= 2(1 R1R2)

2L (R1R2)1/ 4 (642)

= 2(1 R1R2)2L (R1R2)1/ 4 (643)By replacing the electric eld eld per photon in the expressions for the sponta-neous emission with the modied elds an analytical expression for the decay rateof the dipole at an arbitrary position and orientation can be derived.

We will not give these rather lengthy expressions, but show the qualitative be-haviour in the following gure:

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Figure 89: Radiation in the x-z-plane from a z-dipole in free space (a), in a cavity of reectivityR = 0 .95 and length 0.5 (b) and 1.0 (b) [from Bjoerk in Sponaneous emission an laser oscillationim microcavities, H. Yokoyama, K. Ujihara, eds.]

Figure 90: Radiation in the x-z-plane from a y-dipole in a cavity of reectivity R = 0 .95 and length0.49 (a), 0.5 (b), and 0.51 (c) [from Bjoerk in Sponaneous emission an laser oscillation immicrocavities, H. Yokoyama, K. Ujihara, eds.]

Figure 91: Radiation in the x-z-plane from a y-dipole in a cavity of reectivity R = 0 .95 andlength 1.0 (a), 1.0 (b), and 1.5 (c). In (b) the dipole is off-centered at z = / 4 [from Bjoerk inSponaneous emission an laser oscillation im microcavities, H. Yokoyama, K. Ujihara, eds.]

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12.5 Modication of energy levels (vacuum shift)

In the discussion of the Wigner-Weisskof theory of spontaneous emission the timeevolution of the coefficient ce(t), i.e. a two-level atom in the excited state and nophoton present in the mode was derived:

ce0 =

1602 c3 d 312

t

0

dt e i(k 0 )( t t )ce0(t ) (644)

The evaluation of the last integral gives:

limt

t

0 dt e i(k 0 )( t t ) = ( 0) P i 0 (645)

It follows:ce0 =

2

+ i ce0(t) (646)

A non-vanishing causes a shift of the energy level. Experimentally, e.g. thefrequency of a transition between two levels of an atom in vacuum is measured. Suchan experiment already accounts for the shift. However, if the vacuum is modied,e.g. by placing the atom inside a resonant structure, then the not only the transitionrate , but also the shift is modied. It can be detected as a modication of thetransition frequency.First experiments to demonstrate the level shift have been demosntarted in the 80swith single atoms. We follow here the analysis by Hinds and Feld (Phys. Rev. Lett.59, 2623 (1987)). It starts from expressions for the spontaneous decay rate andthe level shift for a two-level atom with transition frequency 12 in a cavity withmode k .

= |12 | 2 2 kV (12 k)D(k , k)dkdk (647) = |12 | 2 2

kV

112 k

D(k , k)dkdk (648)

where 12 is the atomic dipole moment, describes the polarization of the eld, V the quantization volume, and D(k , k) is the number of modes per unit frequencyinterval per unit solid angle.

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If the atom is inside a cavity formed by a pair of mirrors, then the mode density

included by the cavitys solid angle is modied. For these k-vector the density is:D cav (k , k) = D free L(k) (649)

where

L(k) = (1 + F )1/ 2

1 + F sin2(kL/c ) (650)

with the cavity length L and the parameter F = 4R/ (1 R)2 being related to themirror reectivity R.Substituting this results in:

= free [1 + (L(k) 1) f ( cav ] (651)where f ( cav ) is the fraction of total free-space spontaneous emission ordinarly emitted into the solid angle cav .A substitution of this expression gives for the shift of the resonance transition:

= freef ( cav

4F sin(212L/c )

1 + F sin2(212L/c ) (652)

The following gures show setup and results from Hinds and Feld:

Figure 92: Top (a) and side view (b) of the experiment by Hinds and Feld [PRL 59, 2623 (1987)].

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Figure 93: Observed intensities, linewidths, and frequency shifts as a function of the cavity detuning(from bottom to top). The cavity length decreases from left to rigth

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12.6 Atoms in cavities

Cavity-QED effects have rst been studied in the microwave regime using Rydbergatoms. We have discussed various experiments in previous chapters.Rydberg atoms provide a very large dipole moment, and thus couple strongly to theelectromagnetic eld. It is also possible to make superconducting microwave cavitieswith very large Q-factors. Therefore, the strong coupling regime can be obtainedwith Rydberg atoms.With improved optical coating techniques for tiny mirrors cavity QED experimentswere performed in the visible (An al., PRL 73, 3375(1995); Rempe, Thompson,Kimble, PRL 68,1132 (1992)).

12.6.1 Generalization of the Jaynes-Cummings-Hamiltonian

In earlier atomic physics experiments in the visible several atoms were interactingwith high-Q cavities in ultra-high vacuum chambers. A generalization of the Jaynes-Cummings-Hamiltonian is obtained when several (identical!) atoms are allowed tointeract with a single cavity mode( Tavis-Cummings-Hamiltonian , Physical Review170 , 379 (1968)):

H JC = 12

0

N A

l=1 zl +

a

+

a +

N A

l=1 g( r l)(a+l + a

+

l ) (653)

The eigenvalues N p and eigenstates |N,p of this Hamiltonian can be derived ana-lytically (Tavis and Cummings, Physical Review 170 , 379 (1968)).N p = ( A 2 p) N 1

g4

324N (10 p(A p) (A 1)(A 2)) (654)

|N,p = |N, p + g2

82N ((A 2 p + 1)

p(A p + 1) |N, p 1

(A 2 p1) ( p + 1)( A p)|N, p )(655)

whereN = g N A/ 2 + 1 / 2 (656)is the collective Rabi frequency, A is the number of atoms and N is the total number

of excitation. The dressed states can be ordered according to the total number of excitations, i.e. the number of photons or the number of atoms in the system.

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g 0 g (2)01/2 g (A)0

1/2

g (2)01/2 g 2(N)0

1/2

g (N)01/2

g (N +1)01/2+-

+-

+- +-

+-+-+-

Number of Excitations

Number of Atoms

1 2 A

Figure 94: Structure of the eigenstates of the Tavis-Cummings Hamiltonian

12.6.2 Important parameters for Cavity-QED

The most important parameters for the cavity is its Q-factor Q (ratio of resonancefrequency and resonance width; determines the eld enhancement in the cavity)and its mode volume V mode (the electric eld per photon is 1/ V mode ; determinesatom-eld coupling strength).In order to observe cavity QED effects it is important to have both a large Q and asmall mode volume V mode or to have a large cavity nesse F

F = F SR

f cL

Qf

with the cavity length L.

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A typical setup for a cavity QED experiment is shown in the following gure. It

consists of a tunable (piezo control) high-nesse cavity and a beam of (cold) atomspassing through it.

Figure 95: A typical cavity QED experiment with important rates.

There are several rates/times which determine the dynamics of the system:

The atom-eld coupling constant g The interaction time T of atoms with the cavity (e.g. given the transition timein an atomic beam experiment)

The photon damping rate of the cavity

The spontaneous emission rate of the atom (by coupling to free eld modes)Two important parameters in cavity QED with atoms are:

critical photon number n0Number of photons required to observe non-linear effects

n0 2

2g2

with = max {, 1/T } critical atom number N 0Number of atoms in the cavity to observe switching of optical response

N 0 2

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If both numbers are large it does not matter when the photon number or atom

number changes by one. This is the classical limit. In the opposite case, e.g. in thestrong coupling regime, both numbers can be much smaller than one ( n0 10 4and N 0 10 2 have been realized).In recent experiments (http://www.mpq.mpg.de/qdynamics/index.html,http://www.its.caltech.edu/ qoptics/people.html) ultra-cold atoms from atomic trapsare sent into optical cavities in a controlled way. This provides a very long interac-tion time. Even a mechanical back-action on the cold atoms can be observed.

Figure 96: Picture of a high-ness optical cavity

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Figure 97: Scheme of a setup where atoms from an atom trap (MOT) are injected into an opticalcavity

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The following to gures show an experimental setup where the the mode structure of

a single neutral atom (Rb) inside an optical cavity is probed. The atom is trappedinside the cavity via a far-detuned trapping laser (dipole trap). The measuredtransmission of a weak probe laser shows a pronounced double-paek structure, aclear indication of strong coupling.

Figure 98: Experimental setup to measure the mode structure of a strongly coupled single atomin an optical cavity via transmission of a probe laser [Maunz et al., Phys. Rev. Lett. 94, 033002(2005)]

In the strong coupling regime a single atom is capable to block the transmission of a macroscopic pump beam completely!

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Figure 99: Experimental results from [Maunz et al., Phys. Rev. Lett. 94, 033002]: Transmissionof probe laser. The detuning between the cavity and the atom is adjusted by tuning the Stark

shift of the atom via the trapping-eld power expressed in terms of the transmitted power, P. Theaverage transmission shows well-resolved normal mode peaks. A Monte Carlo simulation (solidlines) describes the data well. (2005)]

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12.7 Articial atoms

Another approach is to use quantum dots (articial atoms) instead of real atoms.Quantum dots are small semiconductor crystals (several nanometers in size) whichare grown epitactically. The crystals form a three-dimensional potential for chargecarriers (electrons and holes). Due to the small size of quantum dots the energy levelsfor both electrons and holes are discrete. Radiative recombination of electrons andholes in quantum dots thus leads to discrete spectral lines similar as in atoms.

Figure 100: SEM picture of a sample of InAs quantum dots [Reithmeier et al. Nature 432, 197(2004)]

As the solid state environment provides inferior isolation (compared to UHV vacuumchambers in atomic physics experiments) it is more difficult to obtain the strongcoupling regime. However, it is possible to make optical cavities at the fundamentallimit of the cavity volume (( / 2)3).Recently, two groups in Tucson Arizona [T. Yoshie, et al. Nature 432, 200 (2004)]and Wurzburg [Reithmeier et al. Nature 432, 197 (2004)] have succeeded to demon-strate the strong coupling regime in all solid-state systems.

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Figure 101: Experimental results from [Reithmeier et al. Nature 432, 197 (2004)]. C denotes the

cavity resonance and X the emission line of a single quantum dot

12.8 Examples and applications of CQED-systems

Cavity QED has been driven by fundamental research. However, with the demon-stration of cavity QED effects in the visible and even in solid-state systems severalapplications become possible.

The enhanced and directed spontaneous emission leads to much more efficientLEDs and lasers. A requirement is the fabrication of a laser cavity on theorder of the size 3 which is possible with todays technology.

Faster modulation speed of optical devices is obtained by shortening the lifetimeof the active material. If the fraction of spontaneous emission in a certain mode, the so-called -factor,approaches unity thresholdless lasers are obtained

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In the strong coupling regime a coherent transfer of excitations, i.e. from lightto matter or vice versa, is possible. This is of paramount importance for co-herent interfaces between light and matter, needed for quantum informationprocessing.

In the strong coupling regime electro-optic devices based on single photons andsingle emitters are feasible as already the presence of a single photon or a singleatoms causes signicant non-linearity.

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