Nonlinear polariton excitations in quantum dot arrays

Physics Letters A 165 (1992) 148-152 North-Holland PHYSICS LETTERS A

Nonlinear polariton excitations in quantum dot arrays

Pawel Hawrylak Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada KIA OR6

Marek Grabowski ’ Frank J. Seiler Research Laboratory, US Air Force Academy, Colorado Springs, CO 80840. USA

and

Jacek A. Tuszynski Department of Physics, University ofAlberta, Edmonton, Alberta, Canada T6G 2Jl

Received 3 February 1992; accepted for publication 3 March 1992 Communicated by A.A. Maradudin

We investigate the tunneling of photons in a quasi-one-dimensional array of polarizable quantum dots using Maxwell-Bloch equations. This allows a first principle study of nonlinear polaritons in saturable, dispersive and absorptive, periodic media. The nonlinear field equation for the steady state generates linearized excitations which form photonic bands and nonlinear excitations: gap solitons. The possibility of manipulating the photonic gaps and gap solitons by separately contacting the dots is examined.

The interaction of photons with electronic excitations in an artificially structured dielectric medium leads to many interesting and important phenomena; photonic gaps and gap solitons [ 1,2 ] providing a good example. The modulation of the dielectric medium is on the order of the wavelength of light, typically much greater than the length scale associated with both individual atoms and their separation. Hence a classical description of the medium in terms of its polarizability is sufficient. If one considers the interaction of individual atoms with the field, the quantum mechanical description of this interaction leads to many interesting phenomena such as Rabi oscillations, photon ethos, and self-induced transparency [ 31. The dielectric medium and individual atom phenomena can be combined by creating arrays of mesoscopic atoms [ 4 ] (quantum dots) frozen in a semiconductor matrix. A typical separation a between dots can be comparable to the wavelength corresponding to the lowest radiative transition of the dot at frequency CO,. Hence one must describe self-con- sistently spatial and temporal collective behaviour of the field and quantum dots. This behaviour can be mod- ified by selectively contacting individual dots and changing their equilibrium state by, for example, electrically injecting electron-hole pairs. To describe the essential features of such a complex system we shall adopt a model one-dimensional array of two-level dots. Each quantum dot is characterized by its complex polarization P(x, t ) and population inversion 8n (x, t). The time t is measured in units of the characteristic transition frequency w& ’ and the distance x in units of the separation of dots a. The collective state of quantum dots and the linearly polarized complex electromagnetic field E(x, t) are described by the Maxwell-Bloch equations [ 3,5]:

E,(x,t)-fE,,(x,t)=~~~~(x,t), P,=-i(l-iy)P+ii(E+E*)&r,

’ Permanent address: Department of Physics, University of Colorado, Colorado Springs, CO 80933, USA.

148 Elsevier Science Publishers B.V.

Volume 165, number 2 PHYSICS LETTERS A 11 May 1992

8n,=-i(E+E*)(P*-P)-r(8n-8n0). (1)

Here v is the effective velocity of light, y and rare the inverse of the relaxation rates of polarization and population inversion, respectively, and 8n0 is the equilibrium population inversion in the absence of the field. ano= 1 corresponds to the ground state being fully occupied, while 8n0= - 1 corresponds to a fully inverted state. A different 8n0 might be obtained by, e.g., separately contacting individual dots and electrically injecting electron-hole pairs. The polarization P(x, t) is coupled to the field E(x, t) via the spatially dependent coupling constant /? (proportional to the dipole moment of the dot) of the form

All parameters of the theory can be determined experimentally by studying the linear response of the system. The Maxwell-Bloch equations provide a full description of the spatial and temporal behaviour of the field

and the dots. In the spatially homogeneous medium (the Jaynes-Cummings model) the full solution of the Maxwell-Bloch equations exhibits chaotic behaviour [ 6 1. The spatio-temporal behaviour for a pulse width shorter than the relaxation time in a homogeneous medium exhibits self-induced transparency [ 71. Even in a homogeneous medium the nonlinear interactions lead to nontrivial selection rules for the solitary waves [ 8 1. We shall leave the temporal behaviour in the coherent regime to a separate presentation and concentrate here on the spatial behaviour of pulses long enough for the transient effects to be unimportant. In the rotating wave approximation the stationary solution P(x) of the Bloch equation for oscillating fields E( x, t ) = exp ( - iot )E( x) , P(x, t) =exp( -iwt)P(x) relates the polarization P(x) to the local field E(x) via the field dependent susceptibility K(E):

1 (1-o)+iy 8n0 K(E)= z ( l-o)2+y2+ (y/r)EE* .

(2)

This complex susceptibility reflects the transition from saturable dispersive to saturable absorptive nonlinearity as the frequency sweeps through the resonance in the unbiased dot. Note that depending on the value of equilibrium population inversion the susceptibility of the individual dot can change from absorption to gain. In the steady state the nonlinear field equation determines the spatial distribution of the field:

&.(x)+k*E(x) = -k2p2(x)K(E)E(x) . (3)

The wavevector k=w/v corresponds to the propagation of the field between the dots. It is convenient [ 9 ] to define the field degrees of freedom in terms of its amplitude A (x) and phase g(x): E(x) =A (x) exp [i@(x) I, and to identify them with coordinates of a fictitious particle. The spatial coordinate can now be identified with time. The corresponding momenta are defined as p=A, and j=A*&. The variable j is just the energy flux: k= - fi(E*EX -EEz). The equation of motion for the field now takes a very simple and intuitive form:

p,=-k2A+j2/A3-k2B2(x) Re[lc(A)]A, jX= -k2/12(x) Im[+f)]A2. (4)

The first equation describes a harmonic, isotropic oscillator in the presence of a time dependent nonlinear force proportional to the dispersive part of the susceptibility and in the presence of a time dependent “centrifugal force” (the term proportional to j2/A3). The variable j can be interpreted as the angular momentum of the oscillator. The second equation describes the evolution of the momentum j and is governed purely by the absorptive part of the susceptibility. Clearly from the form of the susceptibility, eq. (2), and eq. (4) it is obvious that the losses lead to a collapse while gain leads to a growth of the centrifugal barrier of the harmonic oscillator. In the absence of losses (gain) j=const and in the case of small amplitudes (nonsaturable nonlinearity), eq. (4) has been studied in detail in ref. [ 9 1. Following ref. [ 9 ] the solution of eq. (4 ) can be written in the form of a compact three-dimensional nonlinear map:

R’=R(Q2+J2/R2), Q’=2c-sk/3’ic’(R’) - (R/R’)Q, J’=J-~s/~~K”(R’)R’, (5)

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Volume 165, number 2 PHYSICS LETTERS A I1 May 1992

where R = A 2, Q = sp/kA + c, J= js/ k, s = sin ( k) , c = cos ( k) and K’ and K” stand for the real and imaginary parts of the susceptibility. Eq. (5) allows for the analysis of the global properties of the nonlinear wave equation.

Let us first consider the stability of a plane wave E(x) = T exp ( ikx) . Starting with the initial condition R = T 2,

Q=cos( k), J= T2 sin(k) we iterate the map given by eq. ( 5) N times. The orbits whose amplitude R remains within a fixed radius R,,, from the initial condition are classified as stable. Hence we can create a phase diagram ( T2, w) of stable orbits. In fig. 1 we show a phase diagram corresponding to the parameters N= 11, ~=0.2, p= 1.0, y=r=O.Ol, R,,,= 102. All dots in the absence of the field E(x) are in their ground state. The black crosses in fig. 1 correspond to stable orbits. Fig. 1 shows two wide photonic gaps on both sides of the resonant frequency. These gaps begin to fill up with stable states - gap solitons. The region in the vicinity of the resonant frequency is quite complex (disordered) for a low amplitude of the plane wave. As the amplitude T2 increases the absorptive losses saturate and plane waves with larger amplitudes become more stable. Eq. (5) and the phase diagram it generates can be used to study the transmission experiment. However, we shall adopt a more standard (but entirely equivalent) approach to the transmission problem in terms of transfer matrices.

We decompose the field E(x) between the (rr - 1 )st and nth dot in terms of forward (A,_, ) and backward

(B,_ , ) propagating components:

E(x)=A,_~ exp{ik[x-(n-l)]}+&_, exp{-ik[x-(n-l)]}.

Following ref. [ lo] we cast the problem of finding the solutions to eq. (3) in terms of the transfer matrix M, defined by

CL:)= (

[ 1 -iikk(En)p2] exp( -ik) -$ikrc(E,)P’exp( -ik) A,

ikK(En)p2 exp( +ik) [ 1 ++ikrc(E,)P’] exp( +ik) >( > B, ’ (6)

The linear optical properties are obtained from this transfer matrix using the linearized susceptibility. The polariton Bloch wavevector Q can be easily computed from the trace of the transfer matrix M 2 cos (Q) = tr (M). The effect of quantum dots is to renormalize the photon group velocity at the bottom of the polariton band: Q=k,/w, and to introduce gaps in the polariton spectrum. The lowest polariton gap falls in the frequency range vn/?‘/ [/3’+ 2 ( 1 - vn) ] < o < vn. This allows one to determine two parameters of the theory: v and /I. For the parameters corresponding to fig. 1 the lowest polariton band is in the frequency range (0.36, 0.63).

To calculate the transmission coefficient we start with a plane wave solution at the end of the sample in the form of a transmitted wave T exp (ikx). This solution is iterated backwards and matched across the first layer to the wave composed of incident, Z, and reflected, R, components: E(x) = Z exp( ikx) + R exp ( - ikx). The transfer matrix across the first layer is slightly different from the transfer matrix M. In this way we can de-

Fig. 1. Stability diagram (T’, w) of the plane wave.

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termine the incident wave intensity 1 Zl ’ as a function of the transmitted intensity I TI 2 for each frequency w and number of layers N. Several such functions for selected frequencies are shown in fig. 2. For frequencies in the photonic band of the linear theory the transmitted intensity is a linear function of the incident intensity. For higher incident intensity the transmitted intensity 1 TI 2 becomes a multivalued function of the incident intensity I II 2 in the usual way. For frequencies in the gap of the linear theory the incident intensity 1 I) ’ goes

toward a constant as the transmitted intensity I TI ’ vanishes. Hence a finite incident intensity is needed for a finite transmission to occur. The photonic gaps can be penetrated by a beam of sufficient intensity by prop-

agating a solitary wave. The situation in fig. 2 resembles bifurcation phenomena in equilibrium phase transitions: the multivalued-

ness in the band of linear theory is reminiscent of first order phase transitions, and the multivaluedness in the gap is reminiscent of second order phase transitions. Naturally the behaviour of the system will be different for increasing or decreasing incident intensity. Even more complex behaviour is expected in the vicinity of the

resonance (see fig. 1). Is it then possible to define a meaningful transmission coefficient 1 TI 2/ 1 II ’ as a function of frequency o? We propose to characterise the system by a threshold transmission coefficient. The threshold transmission coefficient corresponds to the ratio of the transmitted intensity to the minimum incident intensity

required for finite transmission. In short, for each frequency we construct the incident intensity as a function of the transmitted intensity and find a minimum. The threshold transmission coefficient corresponds to this minimum. We show the linear transmission coefficient (fig. 3a) and the threshold transmission coefficient (fig. 3b) as a function of frequency for the same set of parameters as in figs. 1 and 2. The threshold transmission coefficient illustrates nicely the effects of nonlinearity in terms of filling up the polariton gaps with nonlinear, solitonlike excitations. Fig. 3c illustrates briefly the effect of altering the equilibrium value of inversion of every other dot from 6n0= 1 to 6n0= 0. This essentially removes half of the dots and leads to changes in the transmission coefficient.

In summary, we studied Maxwell-Bloch equations for a periodic array of two-level quantum dots. The spatial distribution of the electromagnetic field “dressed up” by induced polarization of quantum dots is described by a nonlinear wave equation. Global solutions of the wave equation can be studied by means of a three-dimensional nonlinear map representing all irreducible variables. This nonlinear map admits stable linearized solutions as polariton bands. The nonlinear solutions corresponding to periodic orbits of the map describe solitons. Energy carrying solitons lead to transmission in the polar&on gaps and gaps in the polariton bands. Po-

-2

10

ro6 10 10 10 10 10 10 1

incident intensity

Fig. 2. Incident intensity III* versus transmitted intensity 1 TI * forasetoffrequenciesw=0.05,0.10 ,..., 0.70.

Fig. 3. Threshold transmission coeffkient as a function of fre-

quency o for (a) linear system, (b) nonlinear but unpolarized system, (c ) polarized nonlinear array.

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laritons and solitons can be manipulated by separately contacting individual dots leading to a tunable optical system.

The Coulomb effects associated with biexcitons in individual quantum dots [ 111 neglected here can be in- corporated into a more accurate susceptibility.

References

[ 1] W. Chen and D.L. Mills, Phys. Rev. Lett. 58 (1987) 160. [2] C.M. de Sterke and J.E. Sipe, Phys. Rev. A 43 ( 1991) 2467. [ 31 L. Allen and J.H. Eberly, Optical resonances and two-level atoms (Wiley, New York, 1975). [4] M.A. Reed, R.T. Bate, K. Bradshaw, W.M. Duncan, W.R. Frensley, J.W. Lee and H.D. Shih, J. Vat. Sci. Technol. 4 ( 1986) 358;

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P. Hawrylak, in: Proc. NATO Workshop on Light scattering in superlattices, Mt. Tremblant (Plenum, New York, 1992 ) [6] P.W. Milonni, M.L. Shih and J.R. Ackerhalt, in: World Scientific lecture notes in physics, Vol. 6. Chaos in laser-matter interactions

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Nonlinear polariton excitations in quantum dot arrays