Coherent properties and rabi oscillations in two-level donor systems

NANOSTRUCTURES

Coherent properties and rabi oscillations in two-level donor systems

A. Latgé^I; F. J. Ribeiro^II; A. Bruno-Alfonso^III; L. E. Oliveira^IV; H. S. Brandi^V

^IDepartamento de Física, Universidade Federal Fluminense,24210-340, Niterói-RJ, Brazil

^IIDepto. de Física, Univ. Estadual de Feira de Santana, 44031-460, Feira de Santana-BA, Brazil

^IIIDepto. de Matemática, Fac. de Ciências, Univ. Est. Paulista, 17033-360, Bauru-SP, Brazil

^IVInstituto de Física, Unicamp, CP 6165, 13083-970, Campinas-SP, Brazil

^VInstituto de Física, Univ. Federal do Rio de Janeiro, 21945-970, Rio de Janeiro-RJ, Brazil

ABSTRACT

Coherent properties and Rabi oscillations in two-level donor systems, under terahertz excitation, are theoretically investigated. Here we are concerned with donor states in bulk GaAs and GaAs - (Ga, Al)As quantum dots. We study confinement effects, in the presence of an applied magnetic field, on the electronic and on-center donor states in GaAs - (Ga, Al)As dots, as compared to the situation in bulk GaAs, and estimate some of the associated decay rate parameters. Using the optical Bloch equations with damping, we study the time evolution of the 1s and 2p₊ states in the presence of an applied magnetic field and of a terahertz laser. We also discuss the role played by the distinct dephasing rates on the photocurrent and calculate the electric dipole transition moment. Results indicate that the Rabi oscillations are more robust as the total dephasing rate diminishes, corresponding to a favorable coherence time.

Keywords: Coherent properties; Rabi oscillations; Two-level donor systems

One of the proposals concerning new solid-state quantum computers (QC) is the possibility of using quantum dots (QDs) as the basic architecture for their implementation [1]. In that case, discrete electronic charge or spin states are the qubits responsible for encoding quantum information [2-4]. One crucial point is that the model-qubit system operates under the conditions that decoherence processes are weak and single-qubit and two-qubit unitary operations are controlled. This implies that a QC would be effective only if the decoherence times are much longer than the time involved in the single- and two-qubit operations. The use of laser pulses in controlling the qubit operations may overcome this limitation. Coherent optical excitations in two-level donor systems in bulk GaAs [2], under applied magnetic fields, were converted into deterministic photocurrents. The 1s and 2p₊ donor states are the model qubits coherently manipulated by laser radiation. A more favorable situation concerning the coherence time may be obtained if the excited donor state lies below the continuum. Donor-doped QDs which exhibit pronounced confining effects are then natural candidates to both theoretical [5] and experimental investigations.

Here we investigate the confinement effects of a model spherical QD, under applied magnetic field, on the electronic and on-center donor states in GaAs - (Ga, Al)As QDs. We investigate the conditions in which one may obtain a bound 2p₊ state in contrast to the resonant one in the study by Cole et al [2], and using the optical Bloch equations with damping terms [6], we study the time evolution of the 1s and 2p₊ donor states under the action of a terahertz laser.

The on-center donor Hamiltonian for a spherical GaAs - (Ga, Al)As QD, in the effective-mass approximation, is given by

where l_z = , V_b(r) is the QD barrier potential, g = = B/R^* = (/l_B)² is the ratio of the magnetic and Coulomb energies (for donors in GaAs, g = 1 corresponds to an applied magnetic field of » 6.9 T), R^*» 5.9 meV is the GaAs donor effective Rydberg, l_B = (c/eB)^1/2 is the magnetic length (or cyclotron radius), and and are the effective Bohr radius and effective Bohr magneton, respectively. Using hydrogenic-like envelope wave functions [7], the 1s and 2p_± energies may then be variationally obtained as a function of the z-direction applied magnetic field.

The magnetic-field dependence of the energies of donor states 1s, 2p_-, 2p₊, and of e_c = e₀ + g, for a R = 400 Å GaAs - Ga_0.7Al_0.3As spherical QD are shown in Fig. 1. Notice that e₀ is the energy of the lowest confined non-occupied electronic state. The arrow shows the 1s-2p₊ transition energy corresponding to 2.52 THz, which is the free-electron laser frequency used in the experimental measurements by Cole et al [2]. The confinement effects due to the QD are such that a magnetic field of » 3.0 T tunes the THz radiation to the corresponding 1s-2p₊ transition, with the 2p₊ below the continuum states, and this source of decoherence is removed [2].

The x-component of the corresponding 1s-2p_± dipole matrix element, = á1s|x|2p_±ñ, and the Rabi frequency W_R = E_THz/, where E_THz is the amplitude of the terahertz electric field (in the x-direction), are then calculated. Fig. 2 shows the á1s|x|2p_±ñ matrix elements as a function of the applied magnetic field and of the dot radius. Notice that the matrix-elements results for a GaAs - Ga_0.7Al_0.3As spherical QD of radius R = 1000 Å are essentially the same as for bulk GaAs, as expected. In the bulk regime, for small values of applied magnetic fields, the á1s|x|2p_±ñ matrix elements increase with increasing magnetic fields which can be related to the magnetic-field confinement effects being stronger for the 2p_± state as compared to the 1s state. This leads to a larger overlap between 1s- and 2p_±-like wave functions and therefore to a larger value of the matrix-elements. One notices the existence of a maximum around 2 - 3T, which may be traced back to the fact that, with increasing values of the magnetic field, the Landau magnetic length and Bohr radius (i.e., magnetic and Coulomb energies) become comparable.

The time evolution of the elements of the density matrix within a two-level model for the donor-QD system are obtained via standard procedures [5,6], from the set of optical Bloch equations, i.e.,

where w_L is the THz laser frequency, and w₂₁ is the energy separation of the 1s and 2p₊ impurity levels. The parameters g₁, g₂, and g₃ are recombination rates as introduced phenomenologically in Cole et al [2].

To calculate the time evolution of the photosignal corresponding to the 1s-2p₊ transition, we first estimate the recombination rates. The parameter g₁, giving the rate of spontaneous emission of photons due to 2p₊ ® 1s transitions, may be obtained by

Figure 3 shows that the 1s-2p₊ recombination rate for the R = 400 Å GaAs - Ga_0.7Al_0.3As QD increases with the magnetic field. However, the calculated values are negligible in the THz range of the oscillation frequencies of the problem, and may be neglected. The dephasing rate g₂ at the lowest THz field is estimated from far-infrared measurements as g₂ = 6.0 × 10¹⁰ rad s^-1 [8], and the ionization rate g₃ is set as g₃ = 0, since the 2p₊ excited donor state lies below the first Landau level [2].

Calculated results are shown in Fig. 4 (a), at resonance, for a R = 400 Å GaAs - Ga_0.7Al_0.3As spherical QD (B » 3.0T and E_THz = 3 × 10⁴V/m). One clearly notices that the displayed Rabi oscillations are more robust, as compared with the experiment by Cole et al [2] in doped bulk GaAs, and therefore it suggests that a donor-doped QD leads to a favorable coherence time so that qubit operations may be efficiently controlled. The corresponding contour plot of the photocurrent for varying pulse duration and applied magnetic field is depicted in Fig. 4 (b).

In summary, we have discussed the possible conditions under which decoherence is weak and qubit operations are efficiently controlled in QDs. Using the optical Bloch equations with damping, we are able to investigate, in a phenomenological manner, the coherence effects on Rabi oscillations associated to donor states confined in GaAs - (Ga, Al)As QDs in the presence of an applied magnetic field and under a terahertz laser. The pronounced confining effects of semiconductor QDs are shown to provide better coherence-time conditions for the Rabi oscillations.

Acknowledgments

The authors would like to thank the Brazilian Agencies CNPq, FAPERJ, FUJB, FAPESP, and the Institutes of Millenium (MCT) for partial finantial support.

[1] A. Barenco, D. Deustch, A. Eckert, and R. Jozsa, Phys Rev. Lett. 74, 4083 (1995); G. D. Sanders, K. W. Kim,W. C. Holton, Phys. Rev. A 60, 4146 (1999); M. S. Sherwin, A. Imamoglu, and T. Montroy, Phys. Rev. A 60, 3508 (1999); X.-Q. Li and Y. Arakawa, Phys. Rev. A 63, 012302 (2000); X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000); X.-Q. Li and Y. Yan, Phys. Rev. B 65, 205301 (2002).

[2] B. E. Cole, J. B. Williams, B. T. King, M. S. Sherwin, and C. R. Stanley, Nature 410, 60 (2001).

[3] A. Zrenner, E. Beham, S. Stufler, F. Findels, M. Bichler and G. Abstreiter, Nature 418, 612 (2002); E. Beham, A. Zrenner, F. Findels, M. Bichler and G. Abstreiter, Phys. Stat. Sol. (b) 238, 366 (2003).

[4] X. Li, Y. Wu, D. G. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi and L. J. Sham, Science 301, 809 (2003).

[5] H. S. Brandi, A. Latgé, and L. E. Oliveira, Phys. Rev. B 64, 233315 (2001); ibid. 68, 233206 (2003).

[6] C. Cohen-Tannoudju, J. Dupont-Roc, and G. Grynberg, Processus d'Interaction entre Photons et Atomes (Editions du CNRS, Paris, 1988).

[7] L. H. M. Barbosa, A. Latgé, M. de Dios-Leyva, and L. E. Oliveira, Sol. State Commun. 98, 215 (1996).

[8] H. Kobori, M. Inoue, and T. Ohyama, Physica B 302, 17 (2001).

Received on 4 April, 2005

References

• [1] A. Barenco, D. Deustch, A. Eckert, and R. Jozsa, Phys Rev. Lett. 74, 4083 (1995);
• G. D. Sanders, K. W. Kim,W. C. Holton, Phys. Rev. A 60, 4146 (1999);
• M. S. Sherwin, A. Imamoglu, and T. Montroy, Phys. Rev. A 60, 3508 (1999);
• X.-Q. Li and Y. Arakawa, Phys. Rev. A 63, 012302 (2000);
• X. Hu and S. Das Sarma, Phys. Rev. A 61, 062301 (2000);
• X.-Q. Li and Y. Yan, Phys. Rev. B 65, 205301 (2002).
• [2] B. E. Cole, J. B. Williams, B. T. King, M. S. Sherwin, and C. R. Stanley, Nature 410, 60 (2001).
• [3] A. Zrenner, E. Beham, S. Stufler, F. Findels, M. Bichler and G. Abstreiter, Nature 418, 612 (2002);
• E. Beham, A. Zrenner, F. Findels, M. Bichler and G. Abstreiter, Phys. Stat. Sol. (b) 238, 366 (2003).
• [4] X. Li, Y. Wu, D. G. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi and L. J. Sham, Science 301, 809 (2003).
• [5] H. S. Brandi, A. Latgé, and L. E. Oliveira, Phys. Rev. B 64, 233315 (2001); ibid. 68, 233206 (2003).
• [6] C. Cohen-Tannoudju, J. Dupont-Roc, and G. Grynberg, Processus d'Interaction entre Photons et Atomes (Editions du CNRS, Paris, 1988).
• [7] L. H. M. Barbosa, A. Latgé, M. de Dios-Leyva, and L. E. Oliveira, Sol. State Commun. 98, 215 (1996).
• [8] H. Kobori, M. Inoue, and T. Ohyama, Physica B 302, 17 (2001).

Publication Dates

• Publication in this collection
  06 July 2006
• Date of issue
  June 2006

History

• Received
  04 Apr 2005