* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Optically triggered spin entanglement of electrons
Particle in a box wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum dot cellular automaton wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Hidden variable theory wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum dot wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Quantum state wikipedia , lookup
Atomic theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Nitrogen-vacancy center wikipedia , lookup
Ferromagnetism wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Auger electron spectroscopy wikipedia , lookup
Scanning tunneling spectroscopy wikipedia , lookup
Bell's theorem wikipedia , lookup
EPR paradox wikipedia , lookup
Spin (physics) wikipedia , lookup
Atomic orbital wikipedia , lookup
Hydrogen atom wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Electron-beam lithography wikipedia , lookup
Optically triggered spin entanglement of electrons in semiconductors Claudia Sifel§ and Ulrich Hohenester Institut für Theoretische Physik, Karl–Franzens–Universität Graz, Universitätsplatz 5, 8010 Graz, Austria Abstract. We propose a single-electron doped quantum dot in a field-effect structure as an optically triggered turnstile for spin-entangled electrons. A short laser pulse excites a charged exciton, whose quantum properties are transferred through tunneling and relaxation to the spin entanglement between electrons in the dot and contact. PACS numbers: 73.21.La,03.67.-a,71.35.-y § To whom correspondence should be addressed ([email protected]) Optically triggered spin entanglement of electrons in semiconductors 2 Devices operating at the single-quantum level are a prerequisite for future quantum-computation and quantum-communication applications [1]. Within the field of semiconductors quantum dots, sometimes referred to as artificial atoms, are ideal candidates for such challenging future applications, in particular in view of their high compatibility with existing semiconductor technology: in recent years spectacular examples, such as single-photon [2] or single-electron [3] turnstile devices, have been demonstrated. The electron turnstile of Zrenner et al. [3] consists of a single quantum dot embedded in a field-effect structure: otpical excitation creates a single electron-hole pair (exciton), which decays through tunneling into a separate electron and hole in the contacts detectable as a photocurrent. Here we propose to extend this scheme for a turnstile electron-spin entangler [4]. The proposed structure (Fig. 1a) is identical to the one used by Zrenner et al., [3] with the only exception that the dot is initially populated by a single surplus electron; this can be achieved by applying an external bias voltage such that an electron is transferred from a nearby n-type reservoir to the dot, [5, 6] where further charging is prohibited because of the Coulomb blockade. Optical excitation of this structure then results in the excitation of a charged exciton, i.e., a complex consisting of two electrons and a single hole [5, 6]. Since within the field-effect structure the charged exciton is not a stable configuration, in a consequent step one electron and hole will tunnel out from the dot to the nearby contacts; here, the system can follow two pathways, where either the electron in the dot has spin-up and the one in the reservoir spin-down orientation or vice versa. According to the laws of quantum mechanics, the total state of the system thus becomes a superposition of these two configurations; as will be proven below, in this state the electron spins are maximally entangled. In our theoretical approach the dot is characterized by (Fig. 1b) the spin-degenerate electron ground states σ and the negatively charged exciton state 3, i.e., a Coulomb renormalized complex consisting of two electrons and a hole. As for the reservoirs we assume fermionic field operators Cσ† (ω) and D † (ω) [4], which, respectively, describe the creation of an electron (spin orientation σ) or hole with energy ω. Since we are dealing with an open system [7], i.e., a system interacting with its environment, we have to adopt a density-operator formalism. This is accomplished by a usual master-equation approach within Lindblad form [7]: X † ρ̇ ∼ Li ρL†i , (1) = −i(Heff ρ − ρHeff ) + i with Heff = Ho − 2i i L†i Li accounting for the dot states, Ho , and outscatterings; Li are the Lindblad operators which account for the different scattering channels [4], i.e., tunneling and scatterings in the reservoirs (see Ref. [4] for details). Eq. (1) is conveniently solved through unraveling [4, 7]: here the time evolution of ρ is split into a P Schrödinger-like and quantum jump part due to Heff and i Li ρL†i , respectively, which, within the spirit of Boltzmann’s equation, can be considered as generalized out- and in-scatterings. At each instant of time ρ can then be written as a mixture of two parts, P Optically triggered spin entanglement of electrons in semiconductors 3 one with probability p associated to the unscattered state, and one with probability (1 − p) associated to the scattered state. For the tunneling process we assume that the tunneling rates are the same for 3 → σ, and that the hole enters with a high excess energy into the contact where it immediately suffers an inelastic scattering, which guarantees that tunneling is an irreversible process (in the following we shall trace out the hole degrees of freedom). Since tunneling is spin-unselective the state Li ρL†i after scattering becomes a superposition state [4] P Rω ∝ σ Fc dω Cσ† |σ̄i, where F is the Fermi energy in the n-type reservoir, ωc a cutoff energy, and σ̄ a direction antiparallel to σ. Most importantly, within this state the spins of the electrons in the dot and reservoir are maximally entangled. We next address the question of how such spin entanglement is affected by scatterings: as for the electron in the reservoir we assume that spin-unselective scattering processes take place on a much shorter time-scale (∼ps) than spin-selective scatterings (∼ns [8]) which allows us to separately describe the two processes. In fact, the whole cascade process of (1) tunneling, (2) dephasing, and (3) spin disentanglement can be described through an unraveling of the master equation, and we obtain [4]: ρ(t) ∼ = p(0) (t)|3ih3| + const × (1) (2) XZ σσ 0 0 ωc dωdω 0 Cσ† (ω)|σ̄ihσ̄ 0 |Cσ (ω 0 ) F × p (t) + p (t) δ(ω − ω ) + p(3) (t) δ(ω − ω 0 )δσσ0 . (2) Here, p(i) (t) account for the probabilities that the system: (0) has remained unscattered; (1) one electron and hole have tunneled out from the dot, and that the electron in the reservoir has suffered a (2) spin-unselective, and finally (3) spin-selective scattering. The various contributions are depicted in Figure 2. Most importantly, we observe in Eq. (2) that spin-unselective scatterings destroy the phase coherence (i.e., dephasing) but do not affect the degree of spin entanglement. Thus, the decay of an optically excited charged-exciton indeed generates a robust spin entanglement between the electron in the dot and reservoir. Only spin-selective scatterings, which are expected to occur on a much longer time scale [8], will eventually destroy the spin entanglement. In conclusion, we have proposed a scheme for an optically triggered spin entanglement of electrons in semiconductors. It consists of a single-electron doped quantum dot embedded in a field-effect structure. Optical excitation of an additional electron-hole pair (charged exciton) is transferred through tunneling to a photocurrent, where the spins of the electrons in the dot and reservoir are maximally entangled. This entanglement is robust against dephasing and relaxation processes which are not spin-selective, and thus benefits from the long spin lifetimes in semiconductors. The proposed device might be useful in future quantum information applications to establish entanglement between spatially separated sites. Optically triggered spin entanglement of electrons in semiconductors 4 Acknowledgments Work supported in part by the Austrian science fund FWF under project P15752–N08. References [1] The Physics of Quantum Information, edited by D. Bouwmeester, A. Ekert, and A. Zeilinger (Springer, Berlin, 2000). [2] J. M. Gérard and B. Gayral, J. Lightwave Technol. 17, 2089 (1999); P. Michler et al., Science 290, 2282 (2000); C. Santori et al., Phys. Rev. Lett. 86, 1502 (2001). [3] A. Zrenner, E. Beham, S. Stufler, F. Findeis, M. Bichler, and G. Abstreiter, Nature (London) 418, 612 (2002). [4] C. Sifel and U. Hohenester, Appl. Phys. Lett. 83, 153 (2003). [5] R. Warburton et al., Nature (London) 405, 926 (2000). [6] F. Findeis, M. Baier, A. Zrenner, M. Bichler, G. Abstreiter, U. Hohenester, and E. Molinari, Phys. Rev. B 63, 121309(R) (2001). [7] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge, University Press, Cambridge, UK, 1997). [8] J. M. Kikkawa and D. D. Awschalom, Phys. Rev. Lett. 80, 4313 (1998). Optically triggered spin entanglement of electrons in semiconductors 5 Figure captions |3i (b) | ↑i | ↓i i-type F dot hν n-type (a) Figure 1. (a) Schematic band diagram of the proposed structure. (b) Level scheme of the spin-degenerate electron states |σ =↑, ↓i and the charged-exciton state |3i in the dot. 6 Optically triggered spin entanglement of electrons in semiconductors DCE7 '&(*),+-%./"#01 ! "#$%" & '2),+-%3045!" 6 : 78 9 %01;=<?>A@ B Figure 2. Schematic representation of the various contributions p(i) (t) of Eq. (2) as discussed in the text. Times are measured in units of the inverse tunneling rate −3 Γt for the spinΓ−1 t ; we assume Γd = 2 Γt for the spin-unselective and Γs = 3 × 10 (1) selective scatterings. The black dashed line shows the sum of p (t) and p(2) (t) which corresponds to the degree of spin entanglement.