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Spin State Read-Out by Quantum Jump Technique

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Spin State Read-Out by Quantum Jump Technique 1 Spin State Read-Out by Quantum Jump Technique: for the Purpose of Quantum Computing E. Pazy Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel T. Calarco and P. Zoller Institut fur¨ Theoretische Physik, Universitat¨ Innsbruck, A-6020 Innsbruck, Austria Abstract—Utilizing the Pauli-blocking mechanism we in an isolated atom, self-assembled QDs are embedded show that shining circular polarized light on a singly in an underlying lattice ,i.e., a three-dimensional crystal charged quantum dot induces spin dependent fluorescence. structure. Even though the number of atoms in such Employing the quantum-jump technique we demonstrate dots is small compared to lithographically defined QDs, that this resonance luminescence, due to a spin dependent optical excitation, serves as an excellent read out mech- still this underlying lattice in which the QD is formed anism for measuring the spin state of a single electron strongly affects the single particle states through its band confined to a quantum dot. structure. Due to their discrete density of states, QDs have been I. INTRODUCTION suggested as the major building blocks for numerous quantum computing implementation schemes. These im- Semiconductor self-assembled quantum dots (QDs) plementation schemes can roughly be divided into those form spontaneously during the epitaxial growth process, which utilize the charge [2], or respectively the spin with confinement provided in all three dimensions by a [3] of charge carriers confined within a QD. Accurate high bandgap in the surrounding material (for example, measurement of a single qubit is an essential requirement see [1]). The strong confinement in such structures for the implementation of quantum computation (QC), leads to a discrete atom-like density of states which therefore highly precise methods for the measurement makes these QDs especially attractive for novel device arXiv:cond-mat/0404494v1 [cond-mat.other] 21 Apr 2004 of the spin of a QD confined charge carrier need to be applications in fields such as quantum computing, optics, devised. Implementation of a highly efficient solid-state and optoelectronics. It is therefore tempting to try to measurement scheme designed to measure the spin or import successful methods used in quantum optics to the charge of single electron is an extremely difficult task field of QDs. As we shall show, one should be cautious [5]. An important requirement is that the measurement in considering the atom-QD analogy. Unlike electrons apparatus do not induce decoherence of qubits while no E.Pazy, acknowledges support through a Kreitman fellowship. measurement is taking place. Laser based measurement 2 schemes are ideal in this respect since there is no state it will remain dark, i.e. no photons will be emitted dephasing while the laser field is turned off. There have from the QD. We will proceed to describe the effects of been proposals for optical measurements of charge using heavy/light hole mixing on our proposed measurement laser pulses [6]. Monitoring the fluorescence from a implementation scheme, showing that although mixing single quantum dot (QD) has been suggested as a mean invalidates the assumption of perfect Pauli-blocking it is to measure single scattering events within QDs [7], as still possible to use our measurement scheme to read out well as a means for final read out of the spin state the spin state of an electron. We also consider the effects for the purpose of quantum computation [8]. Recently, of finite detection efficiency. it has been verified experimentally that the spin state of an electron residing in a QD can be read using II. MODEL circularly polarized light [9]. In this paper we start with In the quantum jump technique the dynamics of a the case with no heavy/light hole mixing, which has also quantum system is described by a non-unitary time been recently treated in [10], and then we proceed to evolution defined by an effective non-Hermitian Hamil- demonstrate how it is still possible to devise an optical tonian. We start by defining the coherent evolution of a read out scheme even in the presence of heavy/light hole singly occupied QD coupled to a laser field. mixing. The general Hamiltonian describing the dynamics of Starting with the Hamiltonian for a singly charged a single QD interacting with a classical light field can QD coupled to the laser field, we consider σ+-polarized be schematically written as [1] laser pulses in resonance with the lowest excitation. H = H0 + Hcl = (Hc + Hcc)+ Hcl , (1) Under these conditions we will show how via the Pauli blocking mechanism, experimentally verified in QDs where Hc accounts for the non-interacting confined [11], we are able to obtain spin conditional coupling of carriers, i.e., electrons and holes, Hcc describes the a QD-confined electron to the laser field. Restricting the Coulomb interaction between charge carriers and Hcl single QD Hamiltonian to a simplified (ideal) three-level describes the interaction with the classical light field. scheme and utilizing the quantum jump method we will show how fluorescence can be employed to measure the A. Single particle states spin of a single electron residing in the QD. The idea To demonstrate the validity of our proposed mea- behind this read-out technique is the following: When surement scheme we employ the traditional approach the polarized laser pulse is switched on, due to Pauli- to obtaining single particle states, utilizing the envelope blocking only one of the spin states of the confined function approximation [1], [12] combining it with the electron will optically excite a charged exciton state pro- k p approximation [13], in which the QD wave function · ducing coherent oscillations between the electron-photon are defined in terms of Γ-like bulk band edge states. and trion state, these being disrupted by spontaneous Utilizing the above method we classify the single particle emission of photons. Therefore, depending on the initial states according to the value of ( S+m ,S +m ), where | | z z spin state of the electron, in one case the luminescence Sz and mz are projections of the spin and internal band will exhibit bright periods whereas for the other spin angular momentum in the crystal growth direction. We 3 obtain the states (3/2, 3/2) – heavy-hole (HH) sub- label the spin states of the excess electron with ± band, the (3/2, 1/2) states – light-hole (LH) subband- ± 0 c0†, 1/2 vac , (2) and the (1/2, 1/2) – spin-orbit split-off subband-the | i ≡ − | i ± 1 c† vac , (3) latter of which we can safely ignore being energetically | i ≡ 0,1/2 | i far apart. It is convenient for later calculations of the where c0†,σ(c0,σ) denote creation (annihilation) oper- dipole matrix element to introduce the four Γ point Bloch ators for electrons in their single-particle ground states functions [14] which serve as a basis for the crystal with spin projections σ and vac stands for the electron- | i states with energies corresponding to the top of the hole vacuum, i.e., the crystal ground state. In terms occupied valence band or the bottom of the conduction of Bloch functions these states are given by: 1 | i ↔ band. These states are labeled: S , X , Y , Z . The ı S , 0 ı S These are eigenstates of the bare | i | i | i | i | ↑i | i ↔ | ↓i following are the expressions for the HH and LH states c e e Hamiltonian H , with eigenvalues ǫ0, 1/2 and ǫ0,1/2 − in terms of the Bloch functions and the spin of the respectively, and are not affected by the carrier-carrier functions: interaction Hcc. (3/2, 3/2) 1 (X + ıY ) HH : ↔ √2 | ↑i (3/2, 3/2) 1 (X ıY ) − ↔ √2 | − ↓i C. Pauli blocking (3/2, 1/2) 2 Z + 1 (X + ıY ) LH : ↔ − 3 | ↑i √6 | ↓i On shining, in the growth direction, a σ+ polarized (3/2, 1/2) q 2 Z 1 (X ıY ) − ↔ − 3 | ↓i− √6 | − ↑iclassical light laser pulse defined by an amplitude E(t) q Further more we employ the effective mass approx- and central frequency ωL on a QD with an excess imation and strain-induced effects are not treated ex- electron in the ground state, if the laser is tuned on plicitly, rather, we consider an effective confinement the lowest interband excitation energy, a ground-state potential. exciton can be obtained. The Pauli exclusion principle It is important to stress that our proposed measure- forbids double occupancy of any of the electronic states. In particular, if the excess electron occupies the state 0 , ment scheme does not relay on the above approach for | i obtaining the single particle states, we could just as well no further electron can be promoted from the valence employed an atomistic psuedo-potential approach[15]. band into that state, and thus creation of a charged + The necessary requirements which need to be satisfied in exciton by a σ -polarized laser pulse is inhibited (left order to employ our scheme are that single particle states part of Fig.1). On the other hand, if the excess electron was in 1 , nothing could prevent a second electron from are well separated, energetically, and have well defined | i angular momentum values. being excited to the spin down state, thereby creating the charged exciton state (right part of Fig.1). We consider the absolute value of the Rabi frequency, defined as B. Qubit states 1/2,+3/2 2µ− E(t) Ω(t) 00 , (4) The states we are interested in measuring via the ≡ ¯h quantum-jump technique are the spin states of an excess (where µ00 denotes the dipole matrix element between electron confined to a QD.
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