ivchenko_bookreg

with three-dimensionally confined photons. In a cavity with the Bragg mirror stacks in the vertical direction and optical square-shaped lateral confinement, the optical mode spectrum splits into 0D levels given by

where L is the lateral width and νx,y = 1, 2, ... are the lateral quantum numbers. Tridimensional light confinement was reported in an in-plane microcavity surrounded by a circular Bragg reflector [7.62]. The structure consists of deep concentric trenches etched in a GaAs/AlxGa1−xAs waveguiding heterostructure. In a wire-shaped microcavity with the lateral confinement along one direction, say along the x axis, the photon dispersion is described by a set of 1D modes

where qy is the photon wave vector along the optical-wire axis. Exciton-light coupling in cylindrical microcavities containing quantum wires was theoretically analyzed in [7.63].

Confined optical modes in semiconductor microcavities has opened a way for studies of mixing between 0D and 1D photons and N D excitons (N = 2, 1, 0) [7.64–7.69]. Recently Constantin et al. [7.70] have investigated in one study the coupling between 1D excitons and 3D, 2D, 1D and 0D photon states in high-finesse photon-well, photon-wire and photon-dot Bragg-air microcavites with embedded V-groove quantum wires. The interaction of electromagnetic waves and excitons is enhanced with decreasing the dimensionality of the states involved in the mixing.

7.5.2 Four-Wave Mixing in Microcavities

Turning to the nonlinear optical response of a QW microcavity in pump-probe experiments we first of all mention the exciton bleaching in the presence of free carriers generated under optical pumping, see, e.g., the review [7.52] and references therein. Physically, the pumping can lead to broadening of the exciton resonances, due to scattering of excitons by the photocarriers, and/or to a reduction in the exciton oscillator strength (described by the parameter Γ0), due to the phase-filling and screening e ects. If the former mechanism dominates then, with increased excitation, the transmission peaks at ω± widen and reduce in height without a remarkable reduction in the splitting ω+ − ω− [7.71]. Houdr´e et al. [7.72] studied a transition from the strongto the weak-coupling regime due to the photoinduced loss of exciton oscillator strength. From carrier-density-dependent photoluminescence data, they conclude that the oscillator strength saturates as (1 + N/Nsat)−1/2 with Nsat = 4.3 × 1011 cm−2 at 100 K, where N is the density of electron-hole

352 7 Nonlinear Optics

pairs. Saba et al. [7.73] realized biexcitonic mechanism of the nonlinear optical response in a semiconductor microcavity. In pump-probe measurements they observed a crossover from exciton to biexciton polaritons: with increasing pump intensity the linear exciton-polariton doublet (7.61) evolves into a triplet polariton structure and finally into a shifted biexciton-polariton doublet.

The dynamics of exciton polaritons and the interplay between the coherent and incoherent processes in quantum microcavities has been revealed in a number of experiments on four-wave mixing carried out in cavities containing III-V or II-VI QWs [7.74–7.76]. Combining (7.17) with (7.56) we can switch the theoretical consideration of four-wave mixing from a single QW to a λ-cavity with a QW in its center and write the basic equations [7.77]

describing the nonlinear dynamics of exciton polaritons in the cavity.

We will analyze the degenerate four-wave mixing in a particular case where all three frequencies ω0, ω¯ and ω˜ coincide and the incident pulses are

where γ = (Γ + γ¯)/2, s = (¯γ −Γ )/(2Ω) and 2Ω is the Rabi splitting. For the

two-level-like nonlinearity, the nonlinear source in (7.62a) can be transformed to

FN L(t) = iθ(t)FN(0)L

Here we assume pulse 1 to arrive before pulse 2, t2 > t1, and put t2 = 0 in which case the interpulse time delay T = −t1 = |t1|. According to (7.64) the electric field of the self-di racted wave 2k2 − k1 can be presented as

where the functions f1(t), f2(t) are independent of T , and the time-integrated signal has the form [7.77]

Thus, the signal S(3)(T ) is a sum of monotonous and oscillating terms. The three parameters S, a, φ0 entering (7.66) determine, respectively, the value of the monotonous term at zero delay, and the relative amplitude and initial phase of the oscillating term. A simple analytical form for these parameters follows if we assume γ to be small as compared with the Rabi splitting which implies the strong-coupling regime. Neglecting terms of the second and higher

˜

orders in γ/Ω we obtain for the parameters of the time-integrated signal

ω˜ = ω0 = ω¯. The parameters used in the computation correspond to those of Al0.2Ga0.8As λ-cavity with the distributed Bragg reflectors comprising 24 Al0.4Ga0.6As/AlAs stacks on the cavity front side (air interface) and 33 stacks on the substrate side. The refractive indices are as follows: next,l = 1, next,r = 3.63, n1 = 3.17, n2 = 3.45, nb = 3.54. The calculated values of γ¯ and

¯ −1 −1

Γ are 0.51 ps and 117 ps . A single QW was assumed to be embedded in the center of the active layer and values of Γ = 1.0 ps−1 and Γ0 = 0.05 ps−1

were chosen for the exciton nonradiative and radiative damping rates. In this

˜ −1

case the Rabi splitting 2Ω equals to 4.8 ps . The signal in Fig. 7.5 exhibits quantum beats between the exciton and cavity modes, and a significant beat-

ity with Ω δbi, the time-resolved four-wave-mixing signal exhibits 2Ω-

oscillations modulated by the second period Tbi = 2π/δbi corresponding to

˜ the biexciton binding energy δbi. For comparable but incommensurable Ω

and δbi, the nonlinear signal reveals damped irregular oscillatory behavior. Kelkar et al. [7.76] spectrally resolved four-wave mixing signal as a func-

tion of time delay and observed well-defined temporal oscillations at both the cavitylike and heavy-hole-excitonlike resonances. The period of oscillations is measured to be 440 fs and matches well with the splitting of 9.5 meV between the two exciton-polariton modes.

7.5.3 Angle-Resonant Stimulated Polariton-Polariton Scattering

The further progress in the understanding of nonlinear optical properties of microcavities was stimulated by an observation of the previously overlooked scattering process shown in Fig. 7.6 [7.78–7.80]. In this process, two polaritons

354 7 Nonlinear Optics

Fig. 7.5. The calculated time-resolved 2k2 − k1 signal for di erent time delays between the light pulses. The real time t is referred to the peak of pulse 2. The pattern reveals the damped oscillatory behavior in both τ - and t-dependencies. From [7.77].

from the lower branch ω− with the same in-plane wave vector qp injected by the oblique-incidence pump are scattered within the branch ω− into one “signal” polariton with q = 0 and one “idler” polariton with q = 2qp. Energy and momentum conservation restricts the range of qp to those satisfying

For particles with a parabolic dispersion this scattering process is forbidden. As discussed in Sect. 7.5.1 the strong coupling between the excitons and photons in a micricavity produces two new dispersion branches of mixed quasiparticles. The lower branch ω−(q ) has radically nonparabolic shape and allows extra scatterings to take part, in particular the process in Fig. 7.6. Substituting ω−(q ) from (7.60) into (7.68) and considering a microcavity with the photon resonance ω¯ tuned to ω0, we can come to the transcendental equation

1 + 4x2 − 4 + x2 + 1 − x = 0 ,

for the dimensionless variable x = q2/(2mV¯ ). This equation has two solutions, x1 = 0 and x2 = 1. Therefore, the above polariton-polariton scattering

occurs for qp = 2V m/¯ and the critical, or “magic”, angle of incidence is given by

˜

θp = arcsin (cqp/ω0) = arcsin nb 2Ω/ω0 , (7.69)

because, if the detuning ω¯ − ω0 is zero and the damping rates are ignored,

˜

the Rabi splitting 2Ω equals 2V .

ω

phot

+

ω0=ω

0

−

0

0

Fig. 7.6. (a) Polariton dispersion relation vs. incident angle θ0 at zero detuning. Microcavity (exciton) frequencies ωphot (ω0) shown dashed. Probe polariton (open circle) stimulates the scattering of pump polaritons (filled circles). (b) Sample structure and experimental geometry probed at normal incidence and time delay τ , while changing the pump angle θ0. From [7.78].

The scattering under consideration was first observed by Savvidis et al. [7.78] in pump-probe experimental set-up shown in Fig. 7.6. The pump pulse excites resonantly the polaritons ω−(qp) at the magic angle θp. With a controlled delay a normally-incident probe pulse follows the pump one and the spectrally-resolved reflected signal is measured as a function of the pump intensity. The probe polaritons with q = 0 stimulate the scattering of pump polaritons. Fig. 7.7 shows the reflected probe spectra when a broad-band 100 fs probe pulse (< 0.3 mW) is focused along the growth axis to a 50 µm spot on the sample. The no-pump spectrum has two dips revealing the

˜ ≈

exciton-polariton modes with the Rabi splitting 2Ω 7 meV. This spec-

356 7 Nonlinear Optics

trum is radically modified when the pump pulse arrives, provoking narrowband gains around the lower polariton of up to 70. These enormous gains for the injected pulses are observed when the pump is tuned to the vicinity of the lower-branch inflexion point and its incident angle is 16.5◦. To confirm the gain is real, a narrow bandwidth probe centered at the lower polariton was used, producing the amplified spectrum shown in inset in Fig. 7.7. The observed angle-resolved stimulated amplification is a clear and impressive demonstration of the bosonic nature of cavity exciton polaritons.

Fig. 7.7. Reflected probe spectra at τ = 0 for pump o , co-, and cross-circularly polarized to the probe. Spectral oscillations are caused by interference between reflections from front and back of the sample. Pump spectrum on lower trace. Inset: Reflected narrow-band probe spectra at τ = 0, with pump pulse on/o , together with pump PL without probe pulse (filled circles). From [7.78].

In the absence of the probe pulse, the pump luminescence collected in the same direction is over 2 orders of magnitude weaker. In [7.79, 7.80], the pump photoluminescence was studied in more detail under conditions of cw excitation of the similar QW microcavities. At low input powers Ip, the θ = 0◦ “signal” output is a linear function of Ip. The linear output-input regime is followed by a dramatic superlinear increase above the threshold pump density Ip 15 W/cm2. The bottom of the lower branch ω− forms a “trap” for exciton polaritons. When driven by a cw laser at the magic angle, the polaritons are sucked into the trap, condensing into a macroscopic quantum state which e ciently emits light. Also clearly apparent in spectra is the “idler” mode (with the wave vector 2qp) observed at the angle ϕ0 = 35◦. It is more than an order of magnitude weaker than the signal.

A theory of the angle-resolved polariton amplifier has been proposed by Ciuti et al. [7.81]. They consider the excitation configuration of Fig. 7.6 with the probe and pump producing the polaritons with the wave vectors q = 0 and q = qp, respectively. Ciuti et al. present a close set of equations of motion for the polarizations P0 ≡ P (q = 0), P1 ≡ P (qp) and P2 ≡ P (2qp) of the probe, pump and idler polaritons. For the steady-state regime with the

˜ −

probe electric field Eprobe(t) = Eprobe exp ( iωt) and the pump electric field

˜ −

Epump(t) = Epump exp ( iωpt), one has

˜ −iωt ˜ −iω t ˜ −i(2ω −ω)t

P0 = P0e , P1 = P1e p , P2 = P2e p ,

where ωp = ω−(qp), and the equations for the polarization amplitudes read

Here Fprobe, Fpump are the probe and pump driving terms proportional to the

˜ ˜

amplitudes Eprobe and Epump, Ωint is the coupling energy of the polaritonpolariton interaction and, for simplicity, the damping rate is taken the same for the three modes. Note that the frequencies ω0, ω−(qp) and ω−(2qp) are slightly blueshifted with respect to the unperturbed lower polariton branch.

The blueshift occurs due to the polariton-polariton interaction and is pro-

| ˜ |2 portional to P1 .

Taking (7.70a) and complex conjugate of equation (7.70c) we have the

˜ ˜ linear inhomogeneous system that determines the quantities P0 and P2 as a

358 7 Nonlinear Optics with

The nonlinear response of the probe becomes singular when one of the poles is real. Such a condition is fulfilled when the energy conservation law (7.68)

this case ω(−) becomes real and equal ω−(0) while ω(+) = ω−(0) + 2iγ. If the energy conservation is not satisfied, particularly if the pump incidence angle di ers from the magic angle, the threshold density is higher and the amplification smaller. The singularity of the probe polarization at ω = ω−(0) is only formal and becomes finite if (7.70b) for the pump is consistently solved [7.81].

Similar angular-resolved probe-pump experiments have been performed as well in wire-shaped microcavities [7.82]. The experiments show that again, due to polariton-polariton scattering, a high optical gain is observed at q = 0 and at the idler state 2qp, but for photonic wires the polariton energies can be located on di erent subbranches from the pump.

7.5.4 Stimulated Spin Dynamics of Polaritons

According to Fig. 7.7, the scattering process introduced in [7.78] is highly polarization dependent. It is strong for co-circularly polarized pump and probe beams and completely suppressed for their opposite circular polarizations. It thus seemed possible to describe all the intermediate situations where both pump and probe are elliptically polarized by simply decomposing them into σ+ and σ− components and considering these components independently. However, this picture has been ruled out by the experiment of Lagoudakis et al. [7.83] who have reported unusual polarization properties of a microcavity excited resonantly at the magic angle. For example, in the case of a linearly-polarized pump pulse and circularly polarized probe pulse the observed signal was linearly polarized but in the polarization plane rotated by 45◦ with respect to the pump polarization. Figures 7.8a, b and c plot the intensity of polarization components of the emitted signal versus the helicity, Ppump,c, of the elliptically polarized pump. The main axis of the pump polarization ellipse is parallel to the incidence plane and the probe polarization is kept circular σ+. The detected components are I+, I− (circular polarization), I1, I2 (linear polarization in the axes 1, 2 being parallel and perpendicular to the plane of pump incidence) and I1 , I2 (linear polarization in the axes 1 , 2 ), see (3.228). The polarization of the idler emission shows a similar behavior but di ers in details from that of the signal. Kavokin et al. [7.84] have presented two models, phenomenological and pseudospin ones, in order to interpret the behavior of the polarized emission of resonantly excited microcavities. The intensities of the six polarization components calculated in the phenomenological model by using two fitting complex parameters are shown in Figs. 7.8d, e and f. The pseudospin model in [7.84] is based on representing the polariton polarized states in terms of the psedospin component

Fig. 7.8. Intensity of signal emission, decomposed into (a),(d) circular, (b),(e) linear, and (c),(f) linear diagonal polarizations, as a function of the pump circular polarization degree, where (a)-(c) show the experimental data and (d)-(f) show the theoretical predictions. The probe is σ+ polarized. From [7.84].

360 7 Nonlinear Optics

Sz for circular polarization and Sx, Sy for linear polarization in the axes 1, 2 and 1 , 2 , in the same way as it is done in Sect. 5.5.2 for heavy-hole excitons. It is assumed that, due to exchange interaction, the pump pulse induces the splitting of the probe polaritons described by the Hamiltonian

similar to the Hamiltonian (5.150) or (5.156). Here σ˜x, σ˜z are pseudo Pauli matrices, Ωx is the pump-induced splitting proportional to the degree of pump linear polarization in the axes 1, 2 and Ωz is proportional to Ppump,c. The probe psedospin rotates around the vector (Ωx, 0, Ωz ) with the angular

velocity Ω = Ωx2 + Ωz2 which leads to the observed polarization phenomena. Concluding this chapter, we underline that spin dynamics of polaritons in photon-well, photon-wire and photon-dot microcavities can be an area rich on new e ects and deserves the further study.