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We have developed a novel model for theoretical description of polarised optical pulse resonant coherent interactions with multi-level quantum systems used to map the spin-dependent optically allowed dipole transitions in a semiconductor nanostructure. The approach takes advantage of the real pseudospin vector representation in its little known ramification due to Feynman. We have originally proposed a system Hamiltonian, designed to drive coherently the system from one state to another employing circularly (elliptically) polarised optical excitations. The resulting equations of motion are coupled to the vectorial Maxwell’s equations in real space and directly integrated by Finite-Difference Time-Domain (FDTD) method without invoking any approximations such as the slowly-varying envelope approximation (SVEA) and rotating-wave approximation (RWA). The latter permits a rigorous description of pulse propagation and interactions on an ultrashort time scale including few-optical cycle pulses. Using the model selective excitation of electric dipole transitions with DJz=1 or DJz=-1 by predefined helicity of the optical pulse have been demonstrated. The developed methodology has been successfully applied to study coherent propagation phenomena, such as Self-Induced Transparency (SIT), numerically, demonstrating polarised SIT-soliton formation (Fig. 14 click on the figure below). 

Fig. 14. Polarised soliton propagating in absorbing medium. The 2p SIT left-circularly polarised pulse completely excites and de-excites the two-level system of quantum absorbers thereby performing a full Rabi flop of the population inversion (Ex and Ey are the E-vector components in a plane perpendicular to z-axis).

The time evolution of a p-pulse is shown in Fig.15 (click on the figure to see animation) . The pulse amplitude is increased while the pulse duration is shortened so that the pulse area is conserved (stable solution in an amplifying medium).

Fig. 15. Right-circularly polarized SIT p pulse completely inverts the medium returning the population back to its ground state.

The exciton QW/QD transitions, excited by circularly polarised optical wave are modelled by a pair of two-level systems (two coupled oscillators) corresponding to the s+ and s--heavy-hole transitions (Fig. 16). 

Fig. 16. Schematic diagram of the selection rules for the electric dipole transitions from the heavy-hole valence band to conduction band in a III-V unstrained semiconductor quantum well. The (J,Jz) refer to the total angular momentum and its component along the propagation (quantization) direction z; s+ and - refer to the transitions excited by each sense of circularly polarized light and correspond to DJz=±1, respectively.

The circularly polarised optical wave in 2D is modelled by two orthogonal linearly polarised optical waves time-delayed with respect to each other or by the TM1 guided mode in a planar parallel mirror waveguide in 2D. We employ our group-theoretical pseudospin formalism based on the commutator Lie algebra of SU(4) group for description of the time evolution of the spin population in this degenerate four-level quantum system resonantly excited by circularly polarised light. The time evolution of the degenerate four-level system is represented by the generalized rotations of a real 15-dimensional coherence vector. The relaxation processes and the radiative decay are treated within the framework of the relaxation time approximation introducing phenomenological decay times. These evolution equations are solved self-consistently with the full-wave vector Maxwell’s equations directly in the time domain, by using the FDTD method.

p-pulse excites (or de-excites) completely the two-level system. The simulation results for the ultrashort pulse propagation are shown in Fig.17, for both polarisations of the injected ultrashort pulse s - and s+. In Fig. 14(a,b) all the population is assumed initially in the ground states. In Fig.17(a) a left-circularly polarised pulse sis injected. The pulse excites the first two-level system bringing the population to the upper state and does not affect the second system (see Fig.16). Fig. 17(b) shows a right-circularly pulse (s+) injected into the same system, the first two-level system is not affected, while the population of the second system is driven into the upper level. In Fig. 17 (c,d) the initial population is assumed equally distributed between level |1> and level |4>. Fig. 17(c) depicts a s--pulse exciting the first two-level system into the upper level and de-exciting the second one into the ground state. Fig. 17(d) shows the failure of a s+-pulse to affect the population in both systems. The remaining two cases, namely: population equally distributed between level |2> and level |3> and all population distributed equally between the upper-lying levels are considered analogously. The results show that specific spin states can be excited by choosing the proper polarisation of the injected ultrashort optical pulse. Thus we can coherently control the spin population of specific states in the four-level quantum system

 
 

Fig. 17.  A snapshot of a circularly polarised pulse at the simulation time t=333 fs. The E-field components (Ex, Ey) are plotted along with the population inversion in the first (r22- r11) and in the second (r44- r33) two-level systems., indicating the boundaries of the active medium embedded between two free-space regions. Homogeneous refractive index profile along the structure is assumed (n=1) (see text for discussion).

     We have numerically demonstrated self-induced transparency in a four-level system, initially prepared in a state with equally distributed population between the lower (upper)-lying level in the first two-level system and the upper (lower)-lying level in the second system (Fig.18). The pulse area of the injected pulse is chosen to correspond to a 2p-sech pulse according to the Pulse Area Theorem, i.e. E0=4.2186´109 Vm-1 with pulse duration Tp=100 fs. The ultrashort polarised pulse travels unchanged (Fig.18(b)) in the degenerate four-level medium driving locally the population through full Rabi-flops in both two-level systems, shown on expanded spatial scale in Fig. 18 (a) (blue and magenta curves).

(a)
(b)

Fig. 18. (a) A snapshot of a left-circularly polarised -- pulse propagating in a 4-level medium which is initially prepared in a state with population residing in the lower-lying level in the first system and in the upper-lying level of the second two-level system (r11e=1/2, r44e=1/2) at the simulation time t=200 fs ; (b) Polarised soliton propagation at t=133, 266, 400 fs.

     The spin-dependent model has been extended to two spatial dimensions and implemented in planar nonlinear optical waveguides and semiconductor microcavities (Fig. 19).

Fig. 19. Electric field, polarisation vector components and the population difference in the first and the second two-level systems. A left-circularly polarized (- ) p- pulse with hyperbolic secant envelope is injected from the left boundary (z=0). The population of the first two-level system is initially in the ground state representing absorbing medium while the second two-level system is initially inverted (gain medium). Upon the pulse arrival the first system is excited and the population is residing on the upper level while the second is de-excited into its ground state.

 Injecting a right-circularly polarised pulse in an initially inverted first pair of levels and second system in its ground state causes the return of the population to its ground level in the first two-level system and excitation of the second two-level system driving the population in the upper excited state.

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