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STUDY ON THE ACHIEVING TO STRONG COUPLING REGIME FOR InAs/GaAs QUANTUM DOT EMBEDDED IN THE NANOCAVITY
M.R. Mohebbifar, R.Kh. Gainutdinov, M.A. Khamadeev
Optics arid Nariophotoriics Department, Institute of Physics, Kazan Federal University, Kremlevskaya St., Kazan, 420008, Russia, e-mail: [email protected]
Abstract. In order to the generation of single photon and the production of dressed states between photons and electrons in nanophotonic structures, achieving strong coupling regime is necessary. One of the best ways achieving to strong coupling, is quantum dots embedded in nanocavitv of photonic crystals. In this paper, Hamiltonian of the interaction of nanoparticle - photon, based on semielassieal approach is obtained. Then total Hamiltonian of this system, by Rotating Wave Approximation (RWA) is calculated. By solving this Hamiltonian for Schrodinger equation, eigen values of this system, level shift factor and the strong coupling regime condition for InAs/GaAs quantum dots embedded in the nanocavitv was calculated. For three cavities by decay rates of 200^eV, 600^eV and lOOO^eV, the graph (AE — Yc — g) presents that when decreasing decay rate of nanocavitv, both coupling constant and levels shift are increased. This means that, only for high quality factor cavities, Strong coupling will occur. Indeed slope of coupling constant graph more than slope of decay rate of cavity. This means that coupling constant plays more important role for achieving to strong coupling regime.
Key words: InAs/GaAs quantum dot, strong coupling regime, nanocavitv, dressed state, cavity decay rate.
1. Introduction. A self-assembled single quantum dot (QD) is a nanocrvstal made of semiconductor materials that are small enough to exhibit quantum mechanical properties. The electronic properties of these materials are intermediate between those of bulk semiconductors and of discrete molecules. Due to their relatively higher efficiency compared to bulk, they have also found applications as optoelectronic devices [1-5], biotechnology, solar cell [5-12], single photon emitters f 13] or qubits for quantum computers [14], quantum interference, Rabi oscillations, Quantum coherence, photon anti-bunehing, etc. [15-23]. In zero-dimension structures, the free carriers are confined to a small region by a so called confinement potential providing the quantization of electronic energy states based on the size of the dots. Atom-like discrete energy-levels are occurred when confining the carriers in a nanoregion. The formation of self-assembled InAs QDs on GaAs substrates has attracted much interest, due to their promising applications in nanoscale devices. Various shapes of QDs can be grown by Stranski-Krastanow method [24-26]. The size and shape of such QDs depend on the growth conditions and the used techniques [27-29]. The optical properties of an ensemble of quantum dots are affected by the size distribution and the geometry of the dots [30]. The Stranski-Krastanow method is essentially a self-organized hetero-epitaxial growth during molecular beam epitaxy (MBE) [10]. In this technique, after a number of lattice-mismatched atomic layers deposited on a substrate, accumulated strain energy forces transition from layer to island growth. This happens when a so called wetting layer reaches to a critical 3 thickness of 3-4 nm [31-33]. Semiconductor nanocavitv systems with a QD have been investigated because of their unique physics based on cavity quantum electrodynamics and their potentials in future applications such as quantum information processing. In semiconductor microcavity systems [34], vacuum Rabi splitting in the strong coupling regime [35-38] and highly efficient lasing in the weak-coupling regime [39-47] have been observed. Here,
by solving this Hamiltonian for Sehrodinger equation, eigenvalues of InAs/GaAs quantum dots embedded in the nanoeavitv, level shift factor and the strong coupling regime condition for this nanosvstem for three cavities by decay rates of 200^eV, 600^eV and lOOO^eV was studied.
2. Results and Discussion. To simplify the problem, we start the study with the two-level atom system. In quantum mechanics, a two-level system is a system which can exist in any quantum superposition of two independent (physically distinguishable) quantum states. Figure 1 shows the schema of two-level atom system.
Ee
■g
Ю
ад
0
li>
Fig. 1. Schema of two-level atom system. The Hamiltonian of two-level atom system described by equation (1).
H = Eg\g)(g\ + Ee |e)(e| (1)
By using \g) = ^^ and \e) = ^^ and Pauli's matrix az = ^0 equal to equation (2).
Ha = . (2)
By semielassieal approach which in this approach the atoms are classical and the photons are quantum are studied, and by using "rotating wave approximation" (RWA) for this atom equal to equation (3).
Htnt = qr ■ E. (3)
where q is electric charge (—e), r is position of electron in the nucleus and E is electric field. E can be considered as an operator and:
qr = Pee\e)(e\ + Pgg\g)(g\ + Peg\e)(g\ + Pge\g)(e\, (4)
Pah = e(a\E\b) = j er ^a^*bd3r. (5)
This equation for a = b equal to zero and for a = b we have = Pab = Pbba with assumption A+ and A-
A+ = \e)(g\ = (0 1) , (6)
Л" = jgXej = (0 ^ . (7)
10
And dipolc operator describes by cquation(8)
eE = PA+ + PbA- . (8)
Interaction Hamiltonian becomes as follows:
Hmt = %A+ + gbA-)(a - at) (9)
where in these equations g is coupling constant and is annihilation(creation) operator. By applying operator to ground state atom goes to excited state and by to excited state atom goes to ground state. By using RWA interaction Hamiltonian becomes as follows:
Hint = %A+a + gbA-at). (10)
Finally, total Hamiltonian equal to equation (11),
Hiatal = + g*A~)(a, - af) + frwaa+ + . (11)
By using this Hamiltonian in the time-independent Schrodinger equation Energy Eigenvalues of this system:
= y2'A)2. (12)
Ec + Eg .7c±Ja 2 f7c-7a-2iA\2 E2 = ^^ - ---r - \-4-) (13)
where in these equations Ea{Ec) is the atom(cavitv) energy and Ya (7c) is the atom(cavitv) decay At resonance state which E0 = Ea = Ec
(14)
Here AE = E\ — E2 which represents the difference between two peaks called Splitting Index:
= (16)
The equation (16) expresses coupling regime of system. If g > \(yc — Ya)/4\ there is strong coupling for atom-cavitv system. If g > \(yc — Ya)/4\ there is weak coupling for atom-cavitv system.
Diagram of "Energy Splitting" - "Coupling Constant"
Coupling Constant(micro eV)
Fig. 2. Graph of coupling constant-energy splitting for three cavities by decay rates of 200^eV, 600^eV and lOOO^eV of InAs/GaAs quantum dots embedded in the nanocavitv.
Diagram " Enetgy Splrtlmg " - "Coupling Constant" - "Cawty Relaxation"
Fig. 3. Graph of coupling constant-cavity relaxation-energy splitting for InAs/GaAs quantum dots
embedded in the nanocavitv.
Figures 2 and 3 show results of calculation for InAs/GaAs quantum dots embedded in the nanocavitv for three cavities by decay rates of 200^eV, 600^eV and lOOO^eV. These figures present that when decreasing decay rate of cavity, both coupling constant and levels shift are increased. This means that, only for high quality factor cavities, Strong coupling will occur. Indeed slope of coupling constant graph more than slope of decay rate of cavity. This means that coupling constant plays more important role for achieving to strong coupling regime.
Conclusion. In this paper by solving the Schrodinger equation, eigenvalues of InAs/GaAs quantum dots embedded in the nanocavitv, level shift factor and the strong coupling regime condition for this nanosvstem for three decay rates cavities was studied. Results show that when decreasing decay rate of cavity, both coupling constant and levels shift are increased. This means that, only for high quality factor cavities, Strong coupling will occur. Indeed, slope of coupling constant graph more than slope of decay rate of cavity. This means that coupling constant plays more important role for achieving to strong coupling regime.
Acknowledgments. We would like to gratefully thank Prof. R.Kh. Gainutdinov for his guidance and the helpful discussions. Also we would like to thank the Department of Optics and Nanophotonics at Kazan Federal University for providing computer support for this project.
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