## Abstract

We propose and analyze a novel scheme to realize electromagnetically induced transparency (EIT) via robust electron spin coherence in semiconductor quantum wells. This scheme uses light hole transitions in a quantum well waveguide to induce electron spin coherence in the absence of an external magnetic field. For certain polarization configurations, the light hole transitions form a crossed double-V system. EIT in this system is strongly modified by a coherent wave mixing process induced by the electron spin coherence.

©2003 Optical Society of America

Electromagnetically induced transparency (EIT) exploits destructive quantum interference associated with a nonradiative coherence to make an otherwise opaque medium transparent [1–4]. Optical processes underlying EIT can also lead to remarkable phenomena including lasing without inversion, slow and stopped light, and stimulated Raman adiabatic passage [1–5]. The tremendous success of EIT studies in atomic and atomic-like systems has stimulated considerable interest in extending these studies to semiconductor systems, both for a better understanding of nonradiative quantum coherences in semiconductors and for possible applications ranging from classical to quantum information processing.

While EIT in interband optical transitions in semiconductors have been demonstrated via the use of a variety of nonradiative quantum coherences including exciton spin coherence, biexciton coherence, and intervalence band coherence [6–8], these coherences are extremely fragile against both carrier-phonon and carrier-carrier scattering. These studies have all been carried out with ultrafast pulses at low temperature. It is very difficult, if possible at all, to extend these approaches to elevated temperatures.

Recent studies of optical spin manipulations in semiconductor have shown that electron spin coherence, as a form of nonradiative quantum coherence, is exceptionally robust. Electron spin coherence can be preserved over remarkably long time and length scales and especially can persist to room temperature [9–10]. Destructive quantum interference associated with the robust electron spin coherence thus provides a promising avenue for realizing room temperature EIT in semiconductors.

Interband dipole optical transitions near the band edge of semiconductors such as a GaAs quantum well (QW) are characterized by transitions between the doubly degenerate conduction bands with *S*
_{z}=1/2 and -1/2 and the doubly degenerate heavy-hole (hh) and light hole (lh) valence bands with *J*
_{z}=±3/2 and ±1/2, respectively. The two hh transitions share no common states. No electron spin coherence can be induced via hh transitions in the absence of an external magnetic field. The spin coherence, however, can be induced in the presence of an external magnetic field in the Voigt configuration, as shown in earlier experimental studies [9–11]. A recent theoretical proposal on EIT via electron spin coherence in a QW has also required the presence of a strong external magnetic field [12].

In this paper we propose and analyze an EIT scheme that exploits polarization selection rules for the lh transition to induce electron spin coherence in the absence of an external magnetic field. As shown in Fig. 1(a), the two electron spin states can couple to a common lh valence band state. To induce electron spin coherence, the relevant dipole optical transitions need to be driven by two optical fields, one of which is polarized along the z-axis (the growth direction). The other field can be either σ+ or σ- polarized. Note that the field polarized along the z-axis has to propagate in the x-y plane (the plane of the QW), which requires the use of a waveguide geometry. We assume the lateral extension of the waveguide to be much greater than both the quantum well thickness and the optical wavelength. We also assume that the fundamental waveguide mode, which we consider in this paper, can approximately be represented as a plane wave.

To realize EIT via the electron spin coherence, we can use a probe beam polarized along the z-axis and a control beam polarized along the x-axis (see Fig. 1(b)). Both beams couple to respective lh transitions and propagate in the QW waveguide along the y-axis. Note that the nonradiative coherence between the two lh valence bands decays rapidly and features a decoherence time a few orders of magnitude shorter than that of the electron spin coherence. Effects of the hole spin coherence are thus negligible. In this limit, we can view the above EIT scheme as a crossed double-V system.

EIT in this crossed double-V system can differ significantly from ordinary V-type systems. For convenience, we label the four energy levels in the lh transition as |a>, |b>, |c>, and |d>, as shown in Fig. 1(a). The electron spin coherence, *ρ _{ad}*, induced through the |b>-to-|a> and |b>- to-|d> transitions can interact with a σ+ polarized control field via the |b>-to-|d> transition. The resulting

*ρ*leads to the usual EIT in a single-V system. This spin coherence, however, can also couple to a σ- polarized control field via the |c>-to-|a> transition. The resulting

_{ab}*ρ*oscillates at a frequency of 2ν-ν

_{dc}_{1}(ν and ν

_{1}are the frequency of the control and probe beams, respectively). For convenience, we refer to this second process as a coherent wave mixing process.

For a more detailed analysis, we model the coherent optical interactions using the atomic-like 4-level system shown in Fig. 1(a). This model is of course oversimplified. A realistic quantitative description would need to include the influence of the interactions between carriers and also that of scatterings of the carriers by phonons and disorders. Nevertheless, the present model can provide a qualitative illustration of the special nonlinear optical processes in the crossed double-V system.

The Hamiltonian for the crossed double-V system, *H*=*H*
_{0}+*V*, where *H _{0}* is the unperturbed Hamiltonian and

*V*is the Hamiltonian for the relevant dipole optical interaction within the rotating wave approximation, can be written in the form:

where |Ω_{1}| denotes the Rabi frequency of the probe field and |Ω_{+}| and |Ω_{-}| denote those of the coupling field: Ω∝*E* with *E* being the complex electric field amplitude. For the linear polarized control field, |Ω_{+}|=|Ω_{-}|. To include the coherent wave mixing process, we write the relevant density matrix elements, up to the first order in the probe field, as follows:

where the complex amplitudes are slowly-varying functions of time.

The coherent wave mixing process discussed above can also lead to population pulsation. To the second order of the probe field, populations, as represented by the diagonal matrix elements, can oscillate with a beat frequency of ±2(ν-ν_{1}). In contrast to the conventional population pulsation process, population pulsation in the crossed double-V system arises from electron spin coherence and does not occur in the first order of the probe field. Effects of population pulsation thus do not contribute up to the first order of the probe field.

The equations of motion for the relevant populations and complex amplitudes, up to the first order in the probe field but to all orders of the control field, are given as (decay rates are added phenomenologically):

where *γ _{ad}* is the decay rate for the spin coherence, Γ is the recombination rate for upper states |a> and |d>, and Δ

_{1}=ω-ν

_{1}with ω being the lh transition frequency. We take ω=ν although a generalization to nonzero detuning for the control field is straightforward. We have also assumed that all the dipole transitions involved have the same decoherence rate,

*γ*. Note that the symmetry of the optical transitions involved implies that

*A’*=

*A*,

*B’*=

*B*,

*C’*=

*C*,

*Y*=

*X**,

*ρ*=

_{bb}*ρ*, and

_{cc}*ρ*=

_{aa}*ρ*.

_{dd}With continuous-wave control and probe fields, we obtained the steady-state solution for the above equations of motion:

where *I*=|Ω_{+}|^{2}/*γ*Γ is the normalized saturation intensity. Note that *B*
^{(1)} depends on the phase difference Δϕ=2ϕ-ϕ_{1} where ϕ and ϕ_{1} are the phase of the control and probe field, respectively. As can be seen from Eqs. (3), (11) and (12), *A*
^{(1)} leads to a polarization at the same frequency and wave vector as the probe field. In comparison, *B*
^{(1)}, which arises from the coherent wave mixing process discussed earlier, leads to a polarization at frequency 2ν-ν_{1} and wave vector **2k-k**
_{1} where **k** and **k**
_{1} are the wave vector for the control and probe beams, respectively.

To describe the propagation of the probe in the crossed double-V system, we define two susceptibilities:

where *N* is an effective density averaged over the cross-section of the probe beam and *µ* is the dipole matrix element for the z-component of the lh transition. *χ*(*ν*
_{1}) is the susceptibility that contains contributions from absorption saturation and the usual EIT process. *$\overline{\chi}$*(*ν*
_{1}) is the susceptibility associated with the coherent wave mixing process.

We plot both real and imaginary parts of χ(ν_{1}) as a function of the probe detuning in Fig. 2 and those of *$\overline{\chi}$*(*ν*
_{1}) in Fig. 3. For a qualitative illustration of absorption and dispersion characteristics, we take γ=10^{12} Hz, Γ=10^{8} Hz, γ_{ad}=10Γ, Δϕ=0, and the Rabi frequency for the control field to be |Ω_{+}|=|Ω_{-}|=10^{11} Hz. Figure 2 shows a sharp absorption dip in Im[χ(ν_{1})] and a steep variation in Re[χ(ν_{1})] near zero detuning. Both of these features result directly from the long-lived electron spin coherence and are characteristic of EIT in ordinary V-type three-level systems. It can be shown that the double V-system modifies the details of χ(ν_{1}) and that it enhances the transparency governed by Imχ(ν_{1}=ν). The susceptibility, *$\overline{\chi}$*(*ν*
_{1}), plotted in Fig. 3 also displays sharp spectral features near zero detuning with a magnitude comparable to that for χ(ν_{1}), indicating that the coherent wave mixing process is as strong as the usual EIT process. This is expected since both processes arise from the electron spin coherence.

The transmission, group velocity, and other propagation characteristics of a probe pulse near zero control-probe detuning are strongly modified by the coherent wave mixing process. At zero control-probe detuning (ν=ν_{1}), the nonlinear polarization induced by the usual EIT process and that by the coherent wave mixing process have the same frequency (see Eq. (3)). The complete propagation characterics, however, are difficult to determine since at finite control-probe detuning, the polarizations induced by the usual EIT and the coherent wave mixing processes feature different frequencies and wave vectors. Solutions to Maxwell’s wave equations including contributions from both polarizations are needed and will be discussed in a future publication.

A drawback of the proposed EIT scheme is that the control beam becomes strongly attenuated as it propagates along the QW waveguide. To avoid strong attenuation of the control, one can use a control beam that is still x-polarized but propagates along the z-axis (orthogonal to the probe) instead of the y-axis. This orthogonal geometry, however, can also lead to a peculiar behavior: the optical field induced by the coherent wave mixing process can propagate along the direction of -**k**
_{1} (counter-propagating with respect to the probe). This is due to the fact that only the in-plane component of the wave vector needs to be conserved for phase matching since the motion of carriers are confined to the plane of the QW.

In summary, we have proposed and analyzed a novel scheme to realize EIT via electron spin coherence in semiconductor QWs. This scheme uses lh transitions in a QW waveguide to induce electron spin coherence in the absence of an external magnetic field. EIT in this scheme is strongly modified by a coherent wave mixing process induced by the electron spin coherence. The present theoretical model oversimplifies optical interactions in a QW. For a more quantitative description, a microscopic many-body theory and solutions to Maxwell’s wave equations are needed and will be treated in future work. Given the robustness of electron spin coherence even at room temperature, the proposed EIT scheme can potentially lead to room temperature EIT operation in a semiconductor, opening up new avenues for applications of nonradiative quantum coherence and especially EIT processes.

This work is supported by ONR/DARPA-SpinS, ARO, NSF-DMR, and JSOP.

## References and links

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