An analytical approach for evaluating the optical spectrum emitted from a strongly coupled single...

An Analytical Approach for Evaluating the Optical Spectrum Emitted from a Strongly-Coupled Single Quantum-Dot Photonic-Crystal

Cavity System

Elahe Ahmadi, Hamid Reza Chalabi, Mehdi Miri, Mina Bayat, and Sina Khorasani

School of Electrical Engineering, Sharif University of Technology, Tehran, Iran Fax: +98-21-6602-3261, Email: khorasani@sharif.edu

ABSTRACT

A theory is presented for the quantum radiation emitted from a single exciton in a quantum dot. We assume that the quantum dot is in strong coupling to a slab photonic crystal cavity. A dielectric function of spatial coordinates is used to explain the effects of the macroscopic medium. It has been proved that the electric field in such a medium can be described using the so-called K-function. We derive a formula for obtaining the frequency spectrum, and present an analytical result for the optical spectrum, which is dependent on the K-function. We also have considered a slab photonic crystal configuration with hexagonal structure containing a cavity to evaluate the frequency spectrum in such a medium. FDTD method has been used to calculate the generalized-transverse green function and the K-function everywhere in the medium.

Keywords: Quantum Dots, Cavity Quantum Electrodynamincs, Photonic Crystals, K-function, FDTD

1. INTRODUCTION It has been known for a long time [1], that the spontaneous emission rate is dependent on the medium containing the light emitter. As a result, some characteristics of the emission process, such as emission rate and far field pattern, are defined by the surrounding enviornment. It is well known that the spontaneous decay of an excited atom can be strongly modified when it is placed inside microcavity.

In the strong coupling regime of cavity quantum electrodynamics (QED) the coherent coupling rate g between an excitonic transition and the cavity-mode exceeds the decay rates of the exciton and the cavity. This regime is of great interest for quantum information processing since it provides a coherent interface between stationary and flying qubits [2,3,4]. In contrast to atom-based systems, solid-state cavity-QED exhibits novel features that defy a simple explanation. In the particular case of quantum dot (QD) based systems, a prominent example that surfaced in earlier experiments operating both in the weak and the strong coupling regime is the effect of off-resonant cavity emission. Experiments performed with QDs, quite surprisingly show strong cavity fluorescence and potentially even lasing over a wide range of detunings [5].

In this paper, we extend previous analytical works on single quantum dot emission into a domain more suited to the planar PC medium. We concentrate exclusively on first-order quantum correlations effects, and derive an exact spectrum that is valid for any inhomogeneous dielectric. As we focus on the effects of the inhomogeneous dielectric, the atom is modeled simply as quantum harmonic oscillator in its ground states or first excited state, with fixed dipole orientation.

2. THEORY

Classical electromagnetic properties of a medium can be described by the photon Green function ( ), ;G r r ω′ , which is

the field response at r to a polarization dipole located at r′ , as a function of frequency. Green function is defined as a tensor (dyadic) as the solution of the following equation:

Photonic and Phononic Crystal Materials and Devices X, edited by Ali Adibi, Shawn-Yu Lin, Axel Scherer,Proc. of SPIE Vol. 7609, 76090F · © 2010 SPIE · CCC code: 0277-786X/10/$18 · doi: 10.1117/12.839540

Proc. of SPIE Vol. 7609 76090F-1

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.207.50.192. Terms of Use: http://spiedl.org/terms

( ) ( ) ( )2, , , , ( )G r r r G r r r r Ic

2ω′ ′ ′−∇×∇× ω + ε ω = δ −t tr r tr r r r r r r

However, for a lossless inhomogeneous dielectric, it is also useful to introduce a generalized-transverse Green function, which can be defined by using Maxwell’s equations, as the following:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )( )( )

( ) ( )( )( )

1 1, , , , ,/ /

f r f rK r r c

f r f rc c f r f r

G r r r r G r r r rr c r c

λ λ λ

λ λλ λ

λ λλ

′ ω′ ω ≡ω −ω ω

′′= − ω

ω −ω

′ ′ ′ ′= ω − δ = ω − δ −ε ω ε ω

∑ ∑

r rr rt r r

r rr r r rr r

t tr r r r r r r rr r

in which, the mode functions are obtained from:

( ) ( ) ( )2 0f r r f rc

λ λω

∇×∇× − ε =r rr r r r r

Therefore, the transverse delta function can be defined using these mode functions as:

( ) ( ) ( ) ( )*,T r r f r f r rε λ λλ

′ ′ ′δ = ε∑r rr r r r r

which has the following characteristic:

( ) ( ) ( )1 1 1,T T Tdr r r X r X rεδ ⋅ =∫r r r r r

In addition, longitude and transverse Green functions, ( ),LG r r ′t r r

and ( ), ;TG r r ′ ωt r r

, used in K-function are defined as:

( )( )( )

( ) ( )( )( )

( )2 21 1, , ,

/ /L T LG r r r r I r r r r

r c r cε ε⎡ ⎤′ ′ ′ ′= δ − − δ = δ⎣ ⎦ε ω ε ω

t tr r r r r r r rr r (7)

( ) ( ) ( )*2

2 2, ;T f r f rG r r c λ λ

′′ ω =

ω −ω∑r rr rt r r

Moreover, to describe the quantum mechanics of light-matter coupling, we adopt a canonical Hamiltonian approach and use a “two- level atom” model for the QD. The resulting Hamiltonian of the system includes one QD exciton, a sum over the light modes, and the coupling between the exciton and the light through the electric-dipole approximation:

( )( )† * †H a a i g a g aλ λ λ λ λ λ λλ λ

ωσ σ ω σ σ+ − − += + − + −∑ ∑h h h (9)

Then, the Electric Field is given by:

( ) ( ) ( )

1/ 2† *

E r t E r t E r t

E r t i a t f r

⎛ ⎞= ⎜ ⎟

⎝ ⎠

⎛ ⎞= − ⎜ ⎟

⎝ ⎠

r r rr r r

rr hr r

The following equations govern the time evolution of related operators:

[ ] ( )

† † †

where2 d

a a H i a gi

λ λ λ λ λ

λλ λ

ω σ σ

ω με

= = − + +

⎡ ⎤= = + +⎣ ⎦

⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

After taking Fourier transform:

( ) ( ) ( )( )

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

1/ 2 1/ 2 *

1/ 2 1/ 2*

f rE r i i f r

f ri i f r

λλ λλ

ω σ ω σ ωω ω

μω ωω σ ω σ ωε ε ω ω

μω ω σ ω σ ωε ε ω ω

= +−

⋅⎛ ⎞ ⎛ ⎞= × +⎜ ⎟ ⎜ ⎟ −⎝ ⎠ ⎝ ⎠

⋅⎛ ⎞ ⎛ ⎞− × +⎜ ⎟ ⎜ ⎟ +⎝ ⎠ ⎝ ⎠

rr r rr hr r

hrr r rh r

Using K Function, We have:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

E r f r f r

K r rc

λλ λ

ωω μ σ ω σ ωε ω ω

ωω μ σ ω σ ωε

⎡ ⎤= − ⋅ +⎣ ⎦−

⎡ ⎤= − ⋅ +⎣ ⎦

∑r rr r r r r

t r r r (13)

To get confidence in our approach, here we apply our result to a simplified case (Homogeneous media) and compare it with the classical result. Since the Homogeneous medium is translational invariant, for every function related the following relation holds:

( , , .) ( , .)F r r etc F r r etc′ ′= −r r r r

In this media, the equations for transverse and longitudinal delta functions reduce to:

( ) ( ) ( )

2 1 ˆ ˆ33 41 1 ˆ ˆ33 4

r r I I r rr

δ δπ

= − − ⊗

= + − ⊗

t tr r

t tr r (15)

Therefore, the longitudinal and transverse Green functions become:

( )( )

( ) ( )

( )( )

hom 2 3

hom 2 23

ˆ ˆ3 ˆ ˆ, / /44 /

ˆ ˆ3,4 / 3 /

in r cT

r r eG r P in r c I Q in r c r rrn c r

P z z z

Q z z z

rr rG r In c r n c

ω ω ωππ ω

π ω ω

− −

Ι − ⊗ ⎡ ⎤= − − + ⊗⎣ ⎦

= − +

= − + −

Ι − ⊗= +

t rt tr

In addition, the K function becomes:

( ) ( ) ( )( )

hom hom hom 2

1, , ,/

2ˆ ˆ/ /4 3 /

in r c

K r G r G r r In c

re P in r c I Q in r c r r Ir n c

ω ω ω δω

δω ω

= + −

⎡ ⎤= − + ⊗ −⎣ ⎦

t tt tr r r r

Therefore, the electric field is given by:

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

hom 20

1, , .

ˆ ˆ/ / .4

in r c

E r K rc

e P in r c Q in r c r rr c

ωω ω μ σ ω σ ωε

ω ω μ ω μ σ ω σ ωπε

= − +

⎡ ⎤= + +⎣ ⎦

r tr r r

r r (18)

The /in r ce ω term represents the retardation effect that is a known effect for the time varying fields. By introducing pr

defined as ( )σ σ μ+ −+r

, we arrive at the familiar classical result:

( ) ( ) ( ) ( )/ / /

22 2 2 3

ˆ ˆ ˆ ˆ ˆ ˆ, ( . ) 3 ( . ) 3 ( . )4 4 4

in r c in r c in r ce e eE r r r p p r r p p i r r p pc r ncr n r

ω ω ω

ω ω ωπε πε πε

= − + + − + + −r r r r r r r r

This in the time domain becomes:

( ) ( ) ( ) ( )2 2 2 30 0 0 /

1 1 1ˆ ˆ ˆ ˆ ˆ ˆ, ( . ) 3 ( . ) 3 ( . )4 4 4

evaluated at t t nr c

E r t r r p p r r p p r r p pc r cnr n rπε πε πε

′= −

⎡ ⎤= − + − + −⎢ ⎥⎣ ⎦

r r r r r r r r&& && & & (20)

The first, second and third terms are named as the Radiated field, Intermediate field and Static field, respectively.

Now, we revert to our main discussion. The optical spectrum is calculated from double time integration over the first order quantum correlation function:

( ) ( ) ( ) ( ) ( ) ( )2 12 1 2 1

, , ,i t tS r dt dt e E r t E r tωω∞ ∞

− − − += ⋅∫ ∫r rr r r

By substituting in the electric-field operator, we obtain the exact spectrum:

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( )( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

2 2 1 10 0

1 , ; , ;

i t i t

S r E r t e dt E r t e dt

E r E r

K r r K r r

ω ωω

ω μ ω μ σ ω σ ωε

ω μ σ ω σ ωε

∞ ∞− +−

− −

= − ⋅

= − ⋅ ⋅ − ⋅

= ⋅ ⋅

∫ ∫r rr r r

r rr r

t tr r r r r r

t r r r

Then, for evaluating the Spectral density, we need to calculate the exact form of ( )−σ ω . From Heisenberg equations, and after taking Fourier transform, we obtain the following equations:

( ) ( ) ( ) ( )( )* †0

i i g a g a−

−λ λ λ λ

σσ ω = − ω − ω

ω−ω ω−ω ∑ (23)

( ) ( ) ( ) ( )( )* †0

i i g a g a+

+λ λ λ λ

σσ ω = + ω − ω

ω+ω ω+ω ∑ (24)

( ) ( ) ( ) ( )( )

††

ia iga

ia iga λ

λ + −λλ

+ −λ

ω = + σ ω +σ ωω−ω ω−ω

ω = + σ ω +σ ωω+ω ω+ω

After some algebraic manipulations:

( ) ( ) ( ) ( )

( ) ( ) ( )( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

ia iai g g

λ λ+ +λ λ

λ λ λ

λ λ− −λ λ

λ λ λ

⎡ ⎤⎛ ⎞σ ω = σ + −⎢ ⎥⎜ ⎟⎜ ⎟ω+ω ω−ω ω+ω⎢ ⎥⎝ ⎠⎣ ⎦

− ω σ ω +σ ωω+ω

⎡ ⎤⎛ ⎞σ ω = σ − −⎢ ⎥⎜ ⎟⎜ ⎟ω−ω ω−ω ω+ω⎢ ⎥⎝ ⎠⎣ ⎦

− ω σ ω +σ ωω−ω

∑ (26)

where:

( ) 22 2

2C gλλ

ωω ≡

ω −ω∑ (27)

Therefore:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0 01 02

ia iaiC g g

ia iai C g g

λ λ− +λ λ

λ λ λ

λ λ−λ λ

λ λ λ

⎧ ⎡ ⎤⎛ ⎞⎪σ ω = ω σ + −⎢ ⎥⎜ ⎟⎨ ⎜ ⎟ω −ω − ω ω ω−ω ω+ω⎢ ⎥⎝ ⎠⎪ ⎣ ⎦⎩⎫⎡ ⎤⎛ ⎞ ⎪⎡ ⎤− ω+ω + ω σ − −⎢ ⎥⎜ ⎟ ⎬⎣ ⎦ ⎜ ⎟ω−ω ω+ω⎢ ⎥⎝ ⎠ ⎪⎣ ⎦⎭

∑ (28)

According to the initial state, we can calculate the exact spectral density for our system.

If we assume 0, gα = then:

( )( ) ( )( )

0,1 ;0; 2

giC e C g

λλ− λ

ω + ω+ω + ωω+ω

σ ω α =ω −ω − ω ω

∑ (29)

Therefore:

( ) ( ) ( )( ) ( )

( )( )( )

( ) ( )( )( )

2 22 2 2

1, , ;

S r K r r

K r r gC C

− −

ω = ω ⋅μ σ ω ⋅σ ωε

⎛ ⎞ω ⋅μ⎜ ⎟= ω + ω+ω + ω⎜ ⎟ω+ωε ω −ω − ω ω ⎝ ⎠

tr r r r

t r r r (30)

Otherwise, by assuming 0,eα = we have:

( ) ( ) ( )( ) ( )

( )( )( )

( ) ( )( )( )

2 22 2 2

1, , ;

, ;( + + ) 2

S r K r r

K r r gC C

− −

ω = ω ⋅μ σ ω ⋅σ ωε

⎛ ⎞ω ⋅μ⎜ ⎟= ω ω ω + ω+ω + ω⎜ ⎟ω+ωε ω −ω − ω ω ⎝ ⎠

tr r r r

t r r r (31)

3. NUMERICAL RESULTS

In this section, we apply the present theory to the structure shown in figure (1). We have used the FDTD method to obtain ( )kf r for evaluating the K- function. By using this method, we found out that our structure has one mode

( )1.491c eVω = deep inside the photonic band gap. Then, we calculated the mode’s profile for our structure. Therefore, we obtained K- function by equation (2).

Note that for evaluating the effect of cavity decay rate ( CΓ ) in our formulation we have replaced the K function by the

( ) ( ) ( )* 22

2 2 2, ,C

f r f rK r r c

iλ λ λ

′ ω′ ω ≡ω −ω + Γ ω ω∑r rr r

t r r (32)

For evaluation of the cavity decay rate, we again have used the FDTD simulation. For this purpose, we have calculated the energy stored in sphere in the vicinity of the exciton and have determined the rate with which it reduces in the logarithmic scale. Note that the cavity is excited with initially and then the energy stored in it decays.

The calculated cavity decay rate obtained by this method was 40μeV.

For including the exciton decay rate in our formalism, we attributed to it a complex frequency, which the imaginary part denotes to its decay rate.

Fig. 1. Schematic of a planar PC which is used for our simulation

Fig. 2. Profile amplitude and phase

According to the amount, the exciton and the cavity resonance frequency differ from each other, the detected spectra changes its shape, which is completely consistent with our qualitative prediction. When the detuning is large, the overall magnitude of the spectra is small. By decreasing the detuning, the amount of the spectra is increased. Note also the shift of the two maximums of the spectra, which are located in the cω and xω toward each other, which shows the existence of dressed states.

Fig. 3. Detected spectra for various cavity-exciton detunings. (a) exciton resonance frequency is the same as cavity mode frequency. (b) a cavity mode off-resonance (from the exciton) by 3 meV. (c) spectral density for different detunings, showing that by increasing

detuning the amplitude of spectra decreases

REFERENCES

[1] Purcell, E. M., Torrey, H. C., and Pound, R. V., “Resonance absorption by nuclear magnetic moments in a solid,” Phys. Rev. 69, 37-38 (1946).

[2] Monroe, C., “Quantum information processing with atoms and photons,” Nature 416, 238-246 (2002). [3] Cirac, J. I., Zoller, P., Kimble, H. J., and Mabuchi, H., “Quantum state transfer and entanglement distribution among

distant nodes in a quantum network,” Phys. Rev. Let. 78, 3221-3224 (1997). [4] Imamoglu, A., Awschalom, D. D., Burkard, G., Divincenzo, D. P., Loss, D., Sherwin, M., and Small, A., “Quantum

information processing using quantum dot spins and cavity QED,” Phys. Rev. Lett. 83, 4204-4207 (1999) [5] Strauf, S., Hennessy, K. , Rakher, M. T., Choi, Y. S., Badolato, A., Andreani, L. C., Hu, E. L., Petroff, P. M., and

Boumeester, D., “Self-tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. 96, 127404 (2006). [6] Hughes, S. and Yao, P., “Theory of quantum light emission from a strongly-coupled single quantum dot photonic-

crystal cavity,” Opt. Soc. Of Amer. 17, 3322-3330 (2009)

[7] Dung, H. T., Knoll, L. and Welsch, D-G., “Spontaneous decay in the presence of dispersing and absorbing bodies: General theory and application to a spherical cavity,” Phys. Rev. A 62, 053804 (2000).

[8] Hughes, S., “Coupled-cavity QED using planar photonic crystals,” Phys. Rev. Lett. 98, 083603 (2007). [9] Wubs, M., Suttorp, L. G. and Lagendijk, A. “Multiple-scattering approach to interatomic interactions and

superradiance in homogeneous dielectric,” Phys. Rev. A 70, 53823 (2004).

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