About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volume 2010 (2010), Article ID 342915, 31 pages
http://dx.doi.org/10.1155/2010/342915
Research Article

Dynamics of Entanglement between a Quantum Dot Spin Qubit and a Photon Qubit inside a Semiconductor High-Q Nanocavity

1CREOL College of Optics and Photonics, University of Central Florida, Orlando, FL 32826, USA
2NanoScience Technology Center, University of Central Florida, Orlando, FL 32826, USA
3Department of Physics, University of Central Florida, P. O. Box 162385, Orlando, FL 32816, USA

Received 15 September 2009; Accepted 24 November 2009

Academic Editor: Shao-Ming Fei

Copyright © 2010 Hubert Pascal Seigneur et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate in this paper the dynamics of entanglement between a QD spin qubit and a single photon qubit inside a quantum network node, as well as its robustness against various decoherence processes. First, the entanglement dynamics is considered without decoherence. In the small detuning regime (Δ=78𝜇eV), there are three different conditions for maximum entanglement, which occur after 71, 93, and 116 picoseconds of interaction time. In the large detuning regime (Δ=1.5meV), there is only one peak for maximum entanglement occurring at 625 picoseconds. Second, the entanglement dynamics is considered with decoherence by including the effects of spin-nucleus and hole-nucleus hyperfine interactions. In the small detuning regime, a decent amount of entanglement (35% entanglement) can only be obtained within 200 picoseconds of interaction. Afterward, all entanglement is lost. In the large detuning regime, a smaller amount of entanglement is realized, namely, 25%. And, it lasts only within the first 300 picoseconds.

1. Introduction

In order to continue satisfying Moore’s law and sustain technological growth, a quantum approach that takes advantage of the wave nature of particles needs to be considered. Currently, the main problem is no longer the physical realization of the qubit, but rather the engineering of a practical quantum computing architecture or quantum network. As a consequence of decisive factors such as on-chip implementation, efficient transfer of quantum information, scalability, CMOS compatibility, and cost, the general consensus is that various implementations of the qubit should be combined in order to obtain an efficient quantum technology. This calls for qubits that are good for storage such as atoms to be used at quantum networks nodes while qubits that have desirable properties for travel such as photons as well as a coherence quantum interface between these qubits allowing for the exchange of information.

To the end of realizing an efficient quantum computing architecture, this promising composite qubit approach to a quantum technology has been proposed for ion trap [1] and also for neutral atoms [2]. We on the other hand have proposed a similar approach in connection with semiconductor-based artificial atoms or quantum dots (QDs) [3]. What does our scheme consist of? It consists of engineering a photonic crystal chip hosting a quantum network made of QDs spin embedded in defect cavities (storage qubits), which constitute the nodes of the network. These storage qubits can interact with other storage qubits at other locations or nodes by means of single photons (traveling qubits), which are guided through waveguides. Interestingly, this coherent interface, which is responsible for the state of the storage qubits to be mapped onto the traveling qubits or the entanglement between them, is itself a qubit system, the cavity-QED qubit (exchange or interaction qubit). Figure 1 depicts a storage qubit (an electron spin in a QD) interacting with a traveling qubit (a single photon) inside a quantum network node.

342915.fig.001
Figure 1: Storage, exchange and traveling qubits inside a quantum network node.

Our approach to a quantum network offers unique benefits with respect to the other composite qubit schemes. For instance, realistic on-chip implementation using photonic crystal has been shown to be plausible [4]; which is not the case in both the ion trap and neutral atom qubits. In addition, even though low temperatures are desirable for minimizing decoherence for the QD spin qubit, it is no where near the extreme temperatures needed for the functioning of the superconducting qubit. They are also much more robust against the influence of temperature than ion trap qubits. In fact, spin lifetimes up to 20 milliseconds have been reported [5]. This technology is easily scalable as additional nodes for the quantum network are generated by just creating additional cavities with embedded QDs in the photonic chip. Besides, because it is a semiconductor-based quantum technology, it is anticipated to be CMOS compatible and cost effective. The use of single photons as traveling qubits as well as the wavelengths considered makes not only the on-chip transfer of quantum information but also the long distance quantum communication by means of optical fibers efficient.

In these many regards, combining subwavelength photonic structures and semiconductor quantum dots provides an unmatched environment for the implementation of storage, exchange, and travelling qubits in comparison to other composite qubit schemes. Scaling up such existing technology by means of computer models is critical in order to build a fully functional quantum network. Such model is expected to lead to a thorough understanding of not only the coherent interaction between a QD and a single photon but also the entanglement dynamics between such particles produced inside quantum networks in a controlled way, in particular the fidelity or amount of such entanglement. For that reason, computer models are a fundamental step in gaining control over quantum information. As a result, we investigate in this paper the dynamics of entanglement between a QD spin qubit and a single-photon qubit inside a quantum network node, as well as its robustness against various decoherence processes.

2. Theory of Quantum Network Nodes

Modeling quantum network nodes requires the ability to describe the following physical systems (1) a QD, (2) a single-photon field, and (3) the interaction between these two in a nanocavity. How though are descriptions of the physical systems making up quantum network nodes used to implement the various qubits? First, the storage qubit is implemented using the spin states of a single excess electron in the conduction band of the QD. What about the traveling qubit? The polarization states of the single-photon in the single mode cavity, whether in the linear or circular polarization eigenbasis, could be used to describe the states of the traveling qubit. It turns out that the implementation of the quantum network nodes using semiconductor QD will necessitate the representation of the traveling qubit to be in the circular polarization eigenbasis. Last, the exchange qubit takes on the form of the “dressed” states in the “dressed” atom picture.

2.1. Modified Jaynes-Cummings Model

Quantum network nodes can be effectively described by a model very similar to the Jaynes-Cummings Model. The main difference between the model for the quantum network nodes and the JC model is the construction of the two-level system, which it is 2-fold degenerate in the excited state and 4-fold degenerate in the ground state, shown in Figure 2. It is important to note that these four degenerate ground states are not connected specifically with light and heavy holes, although the selection rules are identical. We will refer to these degenerate states as |𝟑/𝟐 (2-fold degenerate valence band states with angular momentum projection ±3/2) and |𝟏/𝟐 (2-fold degenerate valence band states with angular momentum projection ±1/2) [6, 7]. It is assumed that the QD is spherical in shape resulting in the confining potential with symmetry approximately identical to that of the first Brillouin zone of the anticipated cubic lattice of the semiconductor crystal. Accordingly, only a shift in the energy levels occurs while the degeneracy between |𝟑/𝟐 and |𝟏/𝟐 states is conserved (A degeneracy lift would not prevent this scheme from working [8], it would only change the time and condition necessary to perform entanglement.) When idle, the quantum dot system has all its ground state levels occupied while only one electron occupies the 2-fold degenerate excited state resulting ideally in an “infinite” decay time. The ability for the excited state to retain this electron is critical to store and process quantum information in our scheme as we will see in the rest of this section. Last, unlike the Jaynes-Cummings Model, the model for the quantum network nodes will take account of various decoherence processes.

342915.fig.002
Figure 2: Two-level approximation of QD for the quantum network nodes.

Because the QD system as a pseudo two-level system can be expressed in the total angular momentum (orbital angular momentum and spin) eigenbasis, there are clearly defined optical transition rules for the electrical dipole interaction between ground and excited states, namely, that the orbital angular momentum quantum number 𝑙 changes by ±1, and the spin is conserved as well as the parity of the envelop function. Furthermore, it follows that in the Faraday geometry, which requires the quantization axis of the excess electron spin to be parallel to the direction of light propagation, the empty state in the conduction band is only populated by circular polarized light (either |𝜎+ or |𝜎 depending on the excess electron spin state). This is illustrated in Figure 3 for the case when the excess electron spin is initialized to |.

342915.fig.003
Figure 3: Dipole selection rules in the quantum network nodes.

Consequently, the total Hamiltonian for describing the quantum network node system is thus written as 𝐇𝐇=QD+𝐇Photon+𝐇QD-Photon,(2.1) where 𝐇QD=𝜔𝟑/𝟐𝝈𝟑/𝟐𝐯,𝟑/𝟐𝐯+𝜔𝟑/𝟐𝝈𝟑/𝟐𝐯,𝟑/𝟐𝐯+𝜔𝟏/𝟐𝝈𝟏/𝟐𝐯,𝟏/𝟐𝐯+𝜔𝟏/𝟐𝝈𝟏/𝟐𝐯,𝟏/𝟐𝐯+𝜔𝑒𝝈𝟏/𝟐𝐜,𝟏/𝟐𝐜+𝜔𝑒𝝈𝟏/𝟐𝐜,𝟏/𝟐𝐜,𝐇Photon̂𝐚=𝜈𝝈+̂𝐚𝝈+̂𝐚+𝜈𝝈̂𝐚𝝈,𝐇QD-Photon=𝑔𝟑/𝟐̂𝐚𝝈+𝝈𝟑/𝟐𝐯,𝟏/𝟐𝐜+𝑔𝟑/𝟐̂𝐚𝝈𝝈𝟑/𝟐𝐯,𝟏/𝟐𝐜+𝑔𝟑/𝟐𝝈𝟏/𝟐𝐜,𝟑/𝟐𝐯̂𝐚𝝈++𝑔𝟑/𝟐𝝈𝟏/𝟐𝐜,𝟑/𝟐𝐯̂𝐚𝝈+𝑔𝟏/𝟐̂𝐚𝝈+𝝈𝟏/𝟐𝐯,𝟏/𝟐𝐜+𝑔𝟏/𝟐̂𝐚𝝈𝝈𝟏/𝟐𝐯,𝟏/𝟐𝐜+𝑔𝟏/𝟐𝝈𝟏/𝟐𝐜,𝟏/𝟐𝐯̂𝐚𝝈++𝑔𝟏/𝟐𝝈𝟏/𝟐𝐜,𝟏/𝟐𝐯̂𝐚𝝈.(2.2) The various coupling strengths from valence band states with total angular momentum 𝑗=±3/2 to conduction band states with total angular momentum 𝑗=±1/2 are be denoted 𝑔𝟑/𝟐, and the various coupling strengths from valence band states with total angular momentum 𝑗=±1/2 to conduction band states with total angular momentum 𝑗=±1/2 are denoted 𝑔𝟏/𝟐.

2.2. Entanglement Process

From dipole selection rules, it can easily be shown that the coupling strengths associated with 𝑒𝟏/𝟐 and 𝑒𝟑/𝟐 excitons can be expressed as 𝑔𝟏/𝟐=(1/3)𝑔𝟑/𝟐. This unbalance in the transition strengths between the 𝑒𝟏/𝟐 and 𝑒𝟑/𝟐 excitons is what makes possible both the process of mapping out quantum information from the storage qubit onto the traveling qubit and creating entanglement between them. How? Because linearly polarized light is nothing but a balanced superposition of right- and left-hand circular polarized light such that |𝜓𝑝=|=(|𝝈+𝐳+|𝝈𝐳)/2, the right and left circular components of the polarization accumulate different phases due to the unbalance in transition strengths resulting in the rotation of the single-photon linear polarization. This is referred to as the single-photon Faraday Effect [9].

As a result, if the spin of the excess electron is initialized to |, then the single-photon polarization rotates in a right-hand circular motion since the right circular polarization component accumulates a larger phase while interacting with |𝟑/𝟐 valence states as depicted in Figure 3. However, if the spin of the excess electron is initialized to |, then the single photon polarization rotates a left-hand circular motion. This Pauli blocking mechanism resulting in the conditional rotation of the single photon linear polarization based on the state of the excess electron spin can effectively be used to encode or map quantum state from the storage qubit onto the traveling qubit or even for creating entanglement between them.

2.3. Creating Entanglement

Quantum entanglement is a well-established quantum property that has no counterpart in classical physics; it occurs when the total wave function of the mixed system cannot be written in any basis, as a direct product of independent substates (i.e., tensor product). As a result, the system of the two entangled qubits individually represented, for instance, by states |𝟎 and |𝟏 has four new computational basis states designated |𝟎𝟎, |𝟎𝟏, |𝟏𝟎, and |𝟏𝟏. In the quantum network node, in order to create entanglement, the spin state of a single excess electron in the conduction band of the QD must be initialized to |𝜓𝑒=(1/p2)(|+|); this is depicted in Figure 4.

342915.fig.004
Figure 4: Spin state initialization for entanglement.

Under these conditions, it is unclear in which direction does the polarization of the single photon rotates. The entanglement between the QD excess electron spin and the single photon polarization is predicted to be the greatest at what would correspond to a 45-degree rotation of the linear polarization of the single photon or 𝜑=𝜋/4 [3]. Transforming to a Bell state eigenbasis for the storage qubit (electron spin) and traveling qubit (photon), we thus have the following maximally entangled Bell state: ||𝜓𝑒𝑝=𝑒𝑖𝑠𝑜𝟑/𝟐||𝜓𝑋𝟑/𝟐+𝑒𝑖𝑠𝑜𝟏/𝟐||𝜓𝑋𝟏/𝟐=𝑒𝑖𝑠𝑜𝟑/𝟐12||𝝈+||+|𝝈||+𝑒𝑖𝑠𝑜𝟏/𝟐12|𝝈||||𝝈++||=𝑒𝑖𝜃||||𝑒𝑖𝜑+||||𝑒𝑖𝜑2.(2.3)

3. Modeling Quantum Network Nodes

The density matrix formalism [10, 11], which is an elegant formulation of quantum mechanics, is used for the modeling of our quantum network. Most importantly, using the density matrix formalism to describe a composite quantum system such as quantum network nodes is indispensable for the analysis of quantum entanglement. This is because the density matrix is able to describe correlations between observables in the various subsystems unlike the Schrodinger’s equation formalism. For instance, the entanglement of two qubits forming a composite system represented by the density matrix 𝝆 can be computed directly as either the Von Neumann entropy [12] or the normalized linear entropy [13]. So in order to study the dynamics of the entanglement between qubits inside a quantum network node, all that is needed is the time evolution of the density matrix describing the subsystems making up the quantum network node, which is obtained by setting up properly an equation of motion and then solving for it. This is often referred to as the Louiville or Von Neumann Equation of motion for the density matrix, and it is shown in 𝑑𝑖𝑑𝑡𝝆=𝝆1𝐇,2𝝆𝚪,.(3.1)

3.1. Master Equation

The Louiville or Von Neumann Equation is the most general form of a master equation for a system whose states are described in term of a density matrix 𝝆 and whose interactions are described according to the Hamiltonian matrix 𝐇. However, because relaxation processes are more complicated than just the anticommutator of a single relaxation matrix Γ with the density matrix 𝝆, we use the following master equation, which is more suitable for the modeling of our quantum network using two relaxation matrices 𝐖 and ̂𝜸, 𝑑𝜌𝑚𝑚𝑖𝑑𝑡=𝑘𝐻𝑚𝑘𝜌𝑘𝑚𝜌𝑚𝑘𝐻𝑘𝑚+𝛿𝑚𝑚𝑘𝑚𝜌𝑘𝑘𝑊𝑚𝑘𝛾𝑚𝑚𝜌𝑚𝑚,(3.2) where 𝑊𝑚𝑘 are transitions rates affecting the diagonal elements of the density matrix and 𝛾𝑚𝑚 are decoherence rates affecting both diagonal and off-diagonal elements of the density matrix. The matrix 𝝆 is an 𝑁×𝑁 matrix with 𝑚=1,2,,𝑁 and 𝑚=1,2,,𝑁.

3.2. Matrix Transformations

Our problem is currently expressed in terms of the subsequent matrix equation 𝝆=𝐇𝝆𝝆𝐇𝝆𝐖[̂𝜸]𝝆+𝛿,(3.3) where 𝛿[𝝆][𝐖]=𝛿𝑚𝑚𝑘𝑚𝝆𝑘𝑘𝑊𝑚𝑘, and [̂̂𝜸][𝝆]=(𝜸𝝆)𝑚𝑚=𝛾𝑚𝑚𝜌𝑚𝑚. However, the Runge-Kutta algorithm we are using to solve this system of 1st-order Ordinary Differential Equations numerically requires that the problem is expressed in the following form: 𝑦=[𝑦].(3.4)

Equation (3.3) must be rewritten in the form of (3.4). This means rearranging the density matrix 𝝆 into a column vector 𝑦, and the commutator [̂𝜸𝐇,]+𝐖 into a matrix such that 𝝆𝑇L𝑦[1],[𝑚𝑚],(3.5)𝐇,𝑚𝑚+̂𝜸𝐖𝑇L[𝑚𝑚],[𝑛𝑛].(3.6) These operations amount to a transformation to Louiville space where 𝐋[𝑚𝑚],[𝑛𝑛], is a superoperator (tetradic matrices) acting on that space with dimension 𝑁2. Equation (3.6) can be broken down into the following three matrix transformations 𝐇,𝑚𝑚𝑇L[𝑚𝑚],[𝑛𝑛]𝐖,(3.7)𝑚𝑚𝑇L𝚪1[𝑚𝑚],[𝑛𝑛]̂𝜸,(3.8)𝑚𝑚𝑇L𝚪2[𝑚𝑚],[𝑛𝑛],(3.9) such that [𝑚𝑚],[𝑛𝑛]=[𝑚𝑚],[𝑛𝑛]+𝚪1[𝑚𝑚],[𝑛𝑛]+𝚪2[𝑚𝑚],[𝑛𝑛].(3.10)

3.3. Algorithms

The first operation transforming 𝝆𝑚𝑚 to 𝑦[1],[𝑚𝑚] is quite straightforward for an arbitrary size square matrix. The essence of the algorithm is to create a column vector 𝑦 whose number of elements is 𝑁×𝑁 and then fill it up using one row of the density matrix at the time.

The second operation transforming an arbitrary size square matrix [𝐇,]𝑚𝑚 to [𝑚𝑚],[𝑛𝑛] is more complex. Before generating algorithms for problems of arbitrary size, it is useful to work out a simple example. Let assume that the density 𝝆 and Hamiltonian 𝐇 matrices are 2×2 matrices so that 𝑚=1,2 and 𝑚=1,2 and therefore 𝑁=2𝜌𝝆=11𝜌12𝜌21𝜌22,𝐻𝐇=11𝐻12𝐻21𝐻22.(3.11)

The commutator of the Hamiltonian and the density matrix in matrix form is 𝝆=𝐻𝐇,12𝜌21𝐻21𝜌12𝐻11𝜌12𝐻12𝜌22𝐻12𝜌11𝐻22𝜌12𝐻21𝜌11𝐻22𝜌21𝐻11𝜌21𝐻21𝜌22𝐻21𝜌12𝐻12𝜌21.(3.12)

Next, the flowing operation [𝐇,]2×2[]4×4 is performed on the Hamiltonian 𝐇 such that (𝑑/𝑑𝑡)[𝝆]𝑚𝑚=((𝑖/))[𝐇,𝝆]𝑚𝑚(𝑑/𝑑𝑡)𝑦[1],[𝑚𝑚]=[𝑚𝑚],[𝑛𝑛]𝑦[1],[𝑚𝑚],𝐇𝐻11𝐻12𝐻21𝐻220𝐻21𝐻120𝐻12𝐻11𝐻220𝐻12𝐻210𝐻22𝐻11𝐻210𝐻21𝐻120.(3.13) The algorithm for an arbitrary square matrix of size 𝑁2 by 𝑁2 is considered next. First, we need to define new indices 𝑖 and 𝑗 which are used to identify sub-block matrices. So (3.13) becomes

𝐇𝐻11𝐻12𝐻21𝐻220𝐻21𝐻12𝐻11𝐻22𝐢=𝟏,𝐣=𝟏𝐻12𝐻12𝐢=𝟏,𝐣=𝟐𝐻21𝐻21𝐢=𝟐,𝐣=𝟏𝐻22𝐻11𝐻21𝐻120𝐢=𝟐,𝐣=𝟐.(3.14)

The strategy adopted to fill this matrix up consists of dividing the matrix into smaller quadrants or matrices 𝐋𝑖𝑗 of size 𝑁×𝑁𝐋=11𝐋12𝐋1𝑚𝐋21𝐋𝑚1𝐋𝑚𝑚.(3.15) The off-diagonal matrices 𝐋𝑖𝑗(𝑖𝑗) identified with the indices 𝑖 and 𝑗 are filled with element from the original Hamiltonian [𝐇,]𝑚𝑚 using the same indices 𝑖 and 𝑗 such that matrices 𝐋𝑖𝑗 are themselves diagonal matrices filled with elements 𝐻𝑖𝑗 of 𝐇. Therefore 𝐋𝑖𝑗=𝐻𝑖𝑗0000𝐻𝑖𝑗00000000𝐻𝑖𝑗.(3.16) As for the diagonal matrices (𝐋𝑖𝑗(𝑖=𝑗)), they are constructed such that 𝐋𝑖=𝑗=𝐻𝑖𝑗𝐻1,1𝐻1,2𝐻1,3𝐻1,𝑚𝐻2,1𝐻𝑖𝑗𝐻2,2𝐻2,3𝐻2,𝑚𝐻3,1𝐻3,2𝐻𝑖𝑗𝐻3,3𝐻3,𝑚𝐻𝑚,1𝐻𝑚,2𝐻𝑚,3𝐻𝑖𝑗𝐻𝑚,𝑚.(3.17)

Finally, the relaxation matrices 𝐖 and ̂𝜸 also need to be transformed. In the original master equation, 𝐖 and ̂𝜸 are 𝑁×𝑁 matrices and are written as

𝐖𝑚𝑚=𝑊11𝑊12𝑊13𝑊1𝑚𝑊21𝑊22𝑊23𝑊2𝑚𝑊31𝑊32𝑊33𝑊(𝑚1)𝑚𝑊𝑚1𝑊𝑚2𝑊𝑚(𝑚1)𝑊𝑚𝑚,̂𝜸𝑚𝑚=𝛾11𝛾12𝛾13𝛾1𝑚𝛾21𝛾22𝛾23𝛾2𝑚𝛾31𝛾32𝛾33𝛾(𝑚1)𝑚𝛾𝑚1𝛾𝑚2𝛾𝑚(𝑚1)𝛾𝑚𝑚,(3.18) First, Γ1 matrix is considered. It is built both from relaxation matrices 𝐖 and ̂𝜸 in connection with the diagonal elements of the density matrix. When 𝑚=𝑚, the term relating to the transition rates in (3.2) in matrix form is written as 𝛿𝝆𝐖=𝛿𝑚𝑚𝑘𝑚𝜌𝑘𝑘𝑊𝑚𝑘=𝑘𝜌𝑘𝑘𝑊1𝑘000𝑘𝜌𝑘𝑘𝑊2𝑘0000𝑘𝜌𝑘𝑘𝑊𝑚𝑘,(3.19) whereas the term relating to the decoherence rates in (3.2) in matrix form is written [̂𝜸]𝝆=𝛾𝑚𝑚𝜌𝑚𝑚=𝜌𝑚𝑚𝑘𝑊𝑘𝑚=𝜌11𝑘𝑊k1000𝜌22𝑘𝑊𝑘20000𝜌𝑘𝑘𝑘𝑊𝑘𝑚,(3.20) since 𝛾𝑚𝑚=12𝑘𝑊𝑘𝑚+𝑊𝑘𝑚=12𝑘𝑊𝑘𝑚+𝑊𝑘𝑚=𝑘𝑚𝑊𝑘𝑚.(3.21)

Combining (3.19) and (3.20), we get the following new term where 𝐖 and ̂𝜸 are forming the matrix 𝐖,

𝝆𝐖𝝆𝐖[̂𝜸]𝝆=𝛿=𝛿𝑚𝑚𝑘𝑚𝜌𝑘𝑘𝑊𝑚𝑘𝜌𝑚𝑚𝑘𝑊𝑘𝑚=𝑘𝜌𝑘𝑘𝑊1𝑘𝜌11𝑘𝑊𝑘1000𝑘𝜌𝑘𝑘𝑊2𝑘𝜌22𝑘𝑊𝑘20000𝑘𝜌𝑘𝑘𝑊𝑚𝑘𝜌𝑘𝑘𝑘𝑊𝑘𝑚.(3.22)

Before coming with an algorithm that can transform 𝐖 to Γ1 for a problem of arbitrary size, it is useful to work out a simple example. It is assumed for the moment that 𝝆, 𝐖, and ̂𝜸 are 2×2 matrices such that 𝑚=1,2 and 𝑚=1,2 and therefore 𝑁=2𝜌𝝆=11𝜌12𝜌21𝜌22,𝑊𝐖=11𝑊12𝑊21𝑊22,̂𝛾𝜸=11𝛾12𝛾21𝛾22=𝑊2100𝑊12.(3.23) From the master equation, the terms related to transition and decoherence rates are therefore expressed as 𝛿𝝆𝐖[̂𝜸]𝝆=𝛿𝑚𝑚𝑘𝑚𝜌𝑘𝑘𝑊𝑚𝑘𝜌𝑚𝑚𝑘𝑊𝑘𝑚=𝜌22𝑊12𝜌11𝑊2100𝜌11𝑊21𝜌22𝑊12.(3.24) In Louiville space, the term 𝛿[̂𝝆][𝐖][𝜸][𝝆] is rewritten 𝑦T[1],[4]Γ1[4],[4] such that 𝑦T𝚪(1,𝐽)1𝜌(𝐼,𝐽)1100𝜌22𝑊2100𝑊21𝑊000000001200𝑊12.(3.25) Thus, in order to obtain Γ1 for an arbitrary size problem, the strategy is to create a square matrix of size 𝑁2×𝑁2 such that the term 𝛿[̂𝝆][𝐖][𝜸][𝝆] is rewritten 𝑦T[1],[𝑚𝑚]Γ1[𝑚𝑚],[𝑛𝑛], and then fill Γ1. The notation for matrix indices in state space is as follows 𝑖=1,2,,𝑁 and 𝑗=1,2,,𝑁. Their counterparts in Louiville space are 𝐼=1,2,,𝑁2 and 𝐽=1,2,,𝑁2. The key is to identify the columns (or 𝐽’s) in Γ1 that needed to be filled, and then fill out the appropriate rows (or 𝐼’s) with the appropriate rates. The columns or 𝐽’s to be filled are the ones whose indices are matching the 𝐽’s in 𝑦T(1,𝐽) that correspond to diagonal terms in the original density matrix 𝝆𝑚𝑚, namely, 𝜌𝑚=1,𝑚=𝟏𝑦1,𝐽=𝟏 and 𝜌𝑚=𝟐,𝑚=𝟐𝑦1,𝐽=𝟒 as shown in the example above. A general expression for the columns to be filled is 𝐽=(𝑗1)𝑁+𝑗,where𝑗=1,2,,𝑁.(3.26) It turns out that the set of indices for rows and columns to be filled are identical (𝐼=𝐽); therefore determining the indices for the columns to be filled also give the ones for the rows.

Therefore, the algorithm to generate elements of Γ1 for a problem of an arbitrary size is as follows 𝚪1̂𝜸(𝐼,𝐽)=𝐖(𝑖,𝑗)(for𝐼𝐽),(𝑖,𝑗)(for𝐼=𝐽),(3.27) where 𝐼=(𝑖1)𝑁+𝑖 and 𝐽=(𝑗1)𝑁+𝑗 with 𝑖=1,2,,𝑁 and 𝑗=1,2,,𝑁. In Louiville space, when considering only the diagonal terms of the density matrix, Γ1 can be written as 𝐖=𝑘1𝑊𝑘1𝑊21𝑊31𝑊𝑘1𝑊12𝑘2𝑊𝑘2𝑊32𝑊𝑘2𝑊13𝑊23𝑘3𝑊k3𝑊𝑘(𝑚1)𝑊1𝑚𝑊2𝑚𝑊(𝑘1)𝑚𝑘𝑚𝑊𝑘𝑚.(3.28)

Next, Γ2 is considered. Unlike Γ1, Γ2 is associated with the case when 𝑚𝑚 in the master equation see (3.3). This means that 𝛿[𝝆][𝐖]=0; therefore, only ̂[𝜸][𝝆] contributes to the master equation. Furthermore, Γ2 is generated from only the decoherence rates 𝛾𝑚𝑚, which are off-diagonal elements of ̂𝜸 such that ̂𝜸𝑚𝑚=12𝑘𝑊𝑚𝑘+𝑊𝑘𝑚.(3.29) These rates are included in the master equation by means of a Hadamard or Schur product between the relaxation matrix ̂𝜸 and the density matrix 𝝆 such that [̂𝜸]𝝆=𝛾𝑚𝑚𝜌𝑚𝑚=0𝛾12𝜌12𝛾13𝜌13𝛾1𝑚𝜌1𝑚𝛾21𝜌210𝛾23𝜌23𝛾2𝑚𝜌2𝑚𝛾31𝜌31𝛾32𝜌320𝛾(𝑚1)𝑚𝜌(𝑚1)𝑚𝛾𝑚1𝜌𝑚1𝛾𝑚2𝜌𝑚2𝛾𝑚(𝑚1)𝜌𝑚(𝑚1)0.(3.30)

Before coming with an algorithm that can transform ̂𝜸 to Γ1 for a problem of arbitrary size, it is useful to work out a simple example. It is assumed for the moment that 𝝆 and ̂𝜸 are 2×2 matrices such that 𝑚=1,2 and 𝑚=1,2 and therefore 𝑁=2𝜌𝝆=11𝜌12𝜌21𝜌22,̂𝜸𝑖𝑗=0𝛾12𝛾210=0𝑊12𝑊210.(3.31)

From the master equation, the terms related to decoherence rates are therefore expressed as

[̂𝜸]𝝆=𝛾𝑚𝑚𝜌𝑚𝑚=0𝜌12𝑊12𝜌21𝑊210.(3.32) In Louiville space, the term [̂𝜸][𝝆] is rewritten Γ2[4],[4]𝑦[1],[4] such that

𝚪2(𝐼,𝐽)𝑦𝛾(𝐼,1)00000𝛾120000𝛾210𝜌000011𝜌12𝜌21𝜌22.(3.33) Consequently, in order to obtain Γ2 for an arbitrary size problem, the strategy is to create first a vector 𝛾[𝑚𝑚],[1] from ̂𝜸𝑚𝑚 whose number of elements is 𝑁2 and then fill it up by concatenating the rows of the density matrix at the time. Next, Γ2, a square matrix of size 𝑁2×𝑁2, is generated by filling its diagonal elements with elements from the vector 𝛾[𝑚𝑚],[1]. Evidently, the matrix Γ2 always consists of a diagonal matrix. The master equation in (3.3) now takes on the following form:

𝑦=[𝑦][𝑦]T𝚪1𝚪2[𝑦].(3.34)

3.4. Solving for Entanglement

The entanglement of two qubits forming a composite system represented by the density matrix 𝝆𝐴𝐵 can be computed directly as either the Von Neumann entropy [12] or the normalized linear entropy [13] of the reduced density matrix of either of the two qubits. Equations (3.35) below show, respectively, the Von Neumann and normalized linear entropies

𝑆VN=TrATrB𝝆𝐴𝐵log2TrB𝝆𝐴𝐵,𝑆NL=21TrATrB𝝆𝐴𝐵2.(3.35) Whether it is calculated from the Von Neumann entropy or the normalized linear entropy, the entanglement between two qubits ranges from 0 to 1, where 1 means that they are maximally entangled.

Let us consider the “pure” state density matrix describing the combined spin-photon system within a single high-Q cavity shown in (3.36). It contains time-dependent variables 𝜌𝑖𝑗, which describe the probability of occupying each distinct set of states in the cavity system, which are |,𝝈𝐳, corresponding to the state when the excess electron spin being down with a left circularly polarized (LCP or 𝝈𝐳) photon in the cavity, |,𝝈+𝐳 to the excess electron spin being down with a right circularly polarized (RCP or 𝝈+𝐳) photon in the cavity, |,𝐗𝟏/𝟐 to the excess electron spin being down with a 𝑒𝟏/𝟐 exciton (trion), |,𝐗𝟑/𝟐 to the excess electron spin being down with a 𝑒𝟑/𝟐 exciton (trion), |,𝝈𝐳 to the excess electron spin being up with an LCP photon in the cavity, |,𝝈+𝐳 to the excess electron spin being up with a RCP photon in the cavity, |,𝐗𝟏/𝟐, to the excess electron spin being up with a 𝑒𝟏/𝟐 exciton (trion), and |,𝐗𝟑/𝟐 to the excess electron spin being up with a 𝑒𝟑/𝟐 exciton (trion)

||||=𝝆=𝝍(𝑡)𝝍(𝑡),𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||𝜌11𝜌21𝜌31𝜌41𝜌51𝜌61𝜌71𝜌81𝜌12𝜌22𝜌32𝜌42𝜌52𝜌62𝜌72𝜌82𝜌13𝜌23𝜌33𝜌43𝜌53𝜌63𝜌73𝜌83𝜌14𝜌24𝜌34𝜌44𝜌54𝜌64𝜌74𝜌84𝜌15𝜌25𝜌35𝜌45𝜌55𝜌65𝜌75𝜌85𝜌16𝜌26𝜌36𝜌46𝜌56𝜌66𝜌76𝜌86𝜌17𝜌27𝜌37𝜌47𝜌57𝜌67𝜌77𝜌87𝜌18𝜌28𝜌38𝜌48𝜌58𝜌68𝜌78𝜌88||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐(3.36) Next, in order to solve for the entanglement dynamics inside our quantum network node, we need first the Hamiltonian, which is obtained in the rotating frame, as well as the relaxation matrices 𝐖 and ̂𝜸

𝐇=,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||00𝑉𝜈3/200000𝑉𝜈1/2000000𝑉1/2𝐸𝑐00000𝑉3/200𝐸𝑐000000000𝑉𝜈1/20000000𝑉𝜈3/20000𝑉1/20𝐸𝑐000000𝑉𝟑/𝟐0𝐸𝑐||,𝝈𝐳||,𝝈+𝐳||,𝐗1/2||,𝐗3/2||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐(3.37) where 𝐸c is the energy of the excess electron, 𝜈 the photon frequency, 𝑉h𝟑/𝟐=𝑔3/2 the interaction energy associated with the 𝑒𝟑/𝟐 excitons, and 𝑉h𝟏/𝟐=𝑔𝟏/𝟐 the interaction energy associated with the 𝑒1/2 excitons

𝐖=,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||𝑊15000𝑊150000𝑊26000𝑊26000𝑊23𝑊23𝑊34𝑊37𝑊38𝑊3400𝑊37𝑊38𝑊140𝑊34𝑊14𝑊34𝑊47𝑊4800𝑊47𝑊48𝑊15000𝑊150000𝑊26000𝑊260000𝑊37𝑊47𝑊570𝑊57𝑊37𝑊47𝑊78𝑊7800𝑊38𝑊480𝑊68𝑊78𝑊68𝑊38𝑊48𝑊78||,𝝈𝐳||,𝝈+𝐳||,𝐗1/2||,𝐗3/2||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐(3.38) where 𝑊23, 𝑊14, 𝑊57, and 𝑊68 are nonreversible decay rates, respectively, from the trion state |,𝐗𝟏/𝟐 to the photonic state ,𝝈+𝐳|, from the trion state |,𝐗𝟑/𝟐 to the photonic state,𝝈𝐳|, from the trion state |,𝐗𝟏/𝟐 to the photonic state ,𝝈𝐳|, and from the trion state |,𝐗𝟑/𝟐 to the photonic state ,𝝈+𝐳|. Next, the Fermi contact terms are depicted by 𝑊15 representing a transition for the excess electron spin from |,𝝈𝐳 to ,𝝈𝐳| and 𝑊26 representing a transition for the excess electron spin from |,𝝈+𝐳 to ,𝝈+𝐳|. Last, the hole hyperfine interaction terms are depicted by 𝑊37, 𝑊48, 𝑊34, 𝑊78, 𝑊38, and 𝑊47. They, respectively, represent the transition from the following trion state |,𝐗𝟏/𝟐 to the trion state ,𝐗𝟏/𝟐|, the transition from the following trion state |,𝐗𝟑/𝟐 to the trion state ,𝐗𝟑/𝟐|, the transition from the following trion state |,𝐗𝟏/𝟐 to the trion state |,𝐗𝟑/𝟐, the transition from the following trion state |,𝐗𝟏/𝟐 to the trion state ,𝐗𝟑/𝟐|, the transition from the following trion state |,𝐗𝟏/𝟐 to the trion state ,𝐗𝟑/𝟐|, and the transition from the following trion state |,𝐗𝟑/𝟐 to the trion state |,𝐗𝟏/𝟐

,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||̂0𝛾𝜸=21𝛾31𝛾41𝛾51𝛾61𝛾71𝛾81𝛾120𝛾32𝛾42𝛾52𝛾62𝛾72𝛾82𝛾13𝛾230𝛾43𝛾53𝛾63𝛾73𝛾83𝛾14𝛾24𝛾340𝛾54𝛾64𝛾74𝛾84𝛾15𝛾25𝛾35𝛾450𝛾65𝛾75𝛾85𝛾16𝛾26𝛾36𝛾46𝛾560𝛾76𝛾86𝛾17𝛾27𝛾37𝛾47𝛾57𝛾670𝛾87𝛾18𝛾28𝛾38𝛾48𝛾58𝛾68𝛾780||,𝝈𝐳||,𝝈+𝐳||,𝐗1/2||,𝐗3/2||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐(3.39) where each element 𝛾ij is calculated according to (3.29) .

Then, the amount of entanglement between the excess electron spin and the single photon polarization can be worked out from the density matrix. The normalized linear entropy is considered

||||=𝝆=𝜓(𝑡)𝜓(𝑡),𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||𝝆𝝈𝝈𝝆𝝈𝐗𝝆𝝈𝝈𝝆𝝈𝐗𝝆𝐗𝝈𝝆𝐗𝐗𝝆𝐗𝝈𝝆𝐗𝐗𝝆𝝈𝝈𝝆𝝈𝐗𝝆𝝈𝝈𝝆𝝈𝐗𝝆𝐗𝝈𝝆𝐗𝐗𝝆𝐗𝝈𝝆𝐗𝐗||,𝝈𝐳||,𝝈+𝐳||,𝐗1/2||,𝐗3/2||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐(3.40) Tracing the density matrix over the cavity polariton results in the spin reduced density matrix such that TrPolariton𝝆𝝆=red,spin=TrPhoton𝝆+TrExciton𝝆=𝝈+𝐳||𝝆||𝝈+𝐳+𝝈𝐳||𝝆||𝝈𝐳+𝐗𝟑/𝟐||𝝆||𝐗𝟑/𝟐+𝐗𝟏/𝟐||𝝆||𝐗𝟏/𝟐=𝝆Tr𝝈𝝈𝝆Tr𝝈𝝈𝝆Tr𝝈𝝈𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗𝝆Tr𝐗𝐗𝝆Tr𝐗𝐗𝝆Tr𝐗𝐗=𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗=2×2,(3.41)

where 𝝆𝝈𝝈=𝜌𝝈+𝐳𝝈+𝐳𝜌𝝈+𝐳𝝈𝐳𝜌𝝈𝐳𝝈+𝐳𝜌𝝈𝐳𝝈𝐳𝝆𝝈𝝈=𝜌𝝈+𝐳𝝈+𝐳𝜌𝝈+𝐳𝝈𝐳𝜌𝝈𝐳𝝈+𝐳𝜌𝝈𝐳𝝈𝐳,𝝆𝝈𝝈=𝜌𝝈+𝐳𝝈+𝐳𝜌𝝈+𝐳𝝈𝐳𝜌𝝈𝐳𝝈+𝐳𝜌𝝈𝐳𝝈𝐳𝝆𝝈𝝈=𝜌𝝈+𝐳𝝈+𝐳𝜌𝝈+𝐳𝝈𝐳𝜌𝝈𝐳𝝈+𝐳𝜌𝝈𝐳𝝈𝐳,𝝆𝐗𝐗=𝜌𝐗𝟑/𝟐𝐗𝟑/𝟐𝜌𝐗𝟑/𝟐𝐗𝟏/𝟐𝜌𝐗𝟏/𝟐𝐗𝟑/𝟐𝜌𝐗𝟏/𝟐𝐗𝟏/𝟐𝝆𝐗𝐗=𝜌𝐗𝟑/𝟐𝐗𝟑/𝟐𝜌𝐗𝟑/𝟐𝐗𝟏/𝟐𝜌𝐗𝟏/𝟐𝐗𝟑/𝟐𝜌𝐗𝟏/𝟐𝐗𝟏/𝟐,𝝆𝐗𝐗=𝜌𝐗𝟑/𝟐𝐗𝟑/𝟐𝜌𝐗𝟑/𝟐𝐗𝟏/𝟐𝜌𝐗𝟏/𝟐𝐗𝟑/𝟐𝜌𝐗𝟏/𝟐𝐗𝟏/𝟐𝝆𝐗𝐗=𝜌𝐗𝟑/𝟐𝐗𝟑/𝟐𝜌𝐗𝟑/𝟐𝐗𝟏/𝟐𝜌𝐗𝟏/𝟐𝐗𝟑/𝟐𝜌𝐗𝟏/𝟐𝐗𝟏/𝟐.(3.42)

Next, the 2×2 matrix TrPolariton𝝆 needs to be squared. As a result, we obtain

TrPolariton𝝆2=𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗𝝆Tr𝝈𝝈𝝆+Tr𝐗𝐗2=[][],𝒜𝒜+𝒪][𝒜+𝒵𝒪𝒜+𝒵𝒪][𝒪+𝒵𝒵(3.43) where 𝒜denotes (Tr𝝆𝝈𝝈+Tr𝝆𝐗𝐗), denotes (Tr𝝆𝝈𝝈+Tr𝝆𝐗𝐗), 𝒪 denotes (Tr𝝆𝝈𝝈+Tr𝝆𝐗𝐗), and 𝒵 denotes (Tr𝝆𝝈𝝈+Tr𝝆𝐗𝐗).

Finally, we take the trace of this square matrix over the spin such that 𝜂=TrspinTrPolariton𝝆2=𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗2+𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗+𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗+𝝆Tr𝝈𝝈+𝝆Tr𝐗𝐗2.(3.44) The normalized linear entropy is then

𝑆NL[]=21𝜂.(3.45)

4. Results

It is assumed that a GaAs/InGaAs QD with emission wavelength 𝜆QD of 1.182𝜇m, which corresponds to the frequency 𝜔QD=1.5941015rad/𝑠, is placed inside a photonic crystal cavity. Next, assuming the dipole moment associated with the InGaAs QD for interactions involving |𝟑/𝟐 electrons is 𝝁𝐯𝐜=29𝐷, the frequency of the photon 𝜔=𝜔QDΔ=1.5941015rad/𝑠, and the cavity mode volume 𝑉0=0.039𝜇m3, the interaction frequency 𝑔𝟑/𝟐 involving the valence states |𝟑/𝟐 was found to be 132109rad/s leading to the interaction energy in the Hamiltonian 𝑉(3/2)=𝑔𝟑/𝟐 and the interaction frequency 𝑔𝟏/𝟐 involving the valence states |𝟏/𝟐 to be 𝑔𝟑/𝟐/3=76109rad/𝑠 corresponding to the interaction energy 𝑉𝟏/𝟐=𝑔𝟏/𝟐. First, the entanglement dynamics is considered without decoherence, and then the effects of various decoherence processes on the entanglement dynamics are investigated.

4.1. Entanglement Dynamics without Decoherence

There are two regimes of interest when studying the dynamics entanglement inside the cavity with or without decoherence, namely, the small and large energy detuning regimes with respect to the QD and the field. In both regimes, we consider interaction times needed to rotate the linear polarization of the photon by 90 degrees from one pure state to the other expecting the condition of maximum entanglement to occur somewhere in the range during which the photon is in a superposition of its polarization eigenstates.

4.1.1. Case  1: Small Δ, Spin Initialized to 𝛼|+𝛽|, Photon Initialized to |

The “small” detuning energy is selected to be Δ=78𝜇eV, which corresponds to the optimized value Δ=0.86𝑔𝟑/𝟐 so as to have a photon and not an exciton back into the cavity once the polarization has rotated by 90 degrees or 2𝜑=𝜋. This takes approximately 130picoseconds. The density matrix is initialized to (4.1), where 𝛼 and 𝛽 are the probability amplitudes associated with spin state | and spin state |, respectively,

1𝝆(t=0)=2,𝝈+𝐳||,𝝈𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐|||𝛼|2|𝛼|200𝛽𝛼𝛽𝛼00|𝛼|2|𝛼|200𝛽𝛼𝛽𝛼000000000000000000𝛼𝛽𝛼𝛽00||𝛽||2||𝛽||200𝛼𝛽𝛼𝛽00||𝛽||2||𝛽||2000000000000000000||,𝝈𝐳||,𝝈+𝐳||,𝐗1/2||,𝐗3/2||,𝝈𝐳||,𝝈+𝐳||,𝐗𝟏/𝟐||,𝐗𝟑/𝟐(4.1)

The results are shown in Figures 5, 6, 7, and 8. In the small Δ regime, there are three peaks or maxima in the calculated electron-photon system entropy, which means that there are three different conditions for maximum entanglement. This differs somehow from the prediction obtained from the standard parameterization approach [14]. These peaks occur after 71,93, and 116 picoseconds of interaction time. However, there is a trade-off since the probability of creating an exciton is nearly 0.4 or 40% for the first 2 peaks and 45% for the last peak. This is not desirable from an engineering point of view since the aim would be to have a high probability of releasing the photon back into the quantum network waveguides once maximally entangled. Entanglement is also confirmed by the fact that the Poincare sphere shows intermediate polarization states inside the unit sphere.

fig5
Figure 5: Excess electron spin dynamics on the bloch sphere for small Δ.
fig6
Figure 6: Photon polarization dynamics on the poincare sphere for small Δ.
342915.fig.007
Figure 7: Spin-photon entanglement dynamics for small Δ.
342915.fig.008
Figure 8: Right and left circular polarization prob. amplitudes for small Δ.
4.1.2. Case  2: Large Δ, Spin Initialized to 𝛼|+𝛽|, Photon Initialized to |

Here, the “large” detuning energy is selected to be Δ=1.5meV, and the 90 degrees rotation of the linear polarization takes place in approximately 1.225nanoseconds. The results are shown in Figures 9, 10, 11. and 12. In the large Δ regime, there is only one peak or maxima of magnitude 1 in the calculated electron-photon system entropy, as predicted in by the standard parameterization approach [14]. This peak occurs at 0.625 nanoseconds for all cases. Furthermore, the probability of creating an exciton is effectively always zero. This is great from an engineering point of view since there will always be a high probability of releasing a photon back into the quantum network waveguides once maximally entangled.

fig9
Figure 9: Excess electron spin dynamics on the bloch sphere for large Δ.
fig10
Figure 10: Photon polarization dynamics on the poincare sphere for large Δ.
342915.fig.0011
Figure 11: Spin-photon entanglement dynamics for large Δ.
342915.fig.0012
Figure 12: Right and left circular polarization prob. amplitudes for large Δ.
4.2. Entanglement Dynamics with Decoherence

Recently, the hyperfine coupling with the nuclear spins of the semiconductor host material (105 nuclei in a single quantum dot), which acts as a random magnetic field (up to a few Tesla in InAs quantum dots), has been identified as the ultimate limit, at low temperature, to the electron spin relaxation or decoherence in quantum dots. Consequently, assuming low temperature conditions, only the various decoherence processes associated with hyperfine interactions will be considered; where as, those associated with phonon interactions are ignored.

First, the Fermi contact term is considered. It relates to the direct interaction of the nuclear dipole with the spin dipoles and is only non-zero for states with a finite electron spin density at the position of the nucleus (those with unpaired electrons in the conduction band). Therefore, the Fermi contact hyperfine interaction does not affect the trion states, just the following states |,𝝈𝐳, |,𝝈+𝐳, |,𝝈𝐳, and |,𝝈+𝐳. A typical longitudinal relaxation time for semiconductors is on the order of 𝑇11ns [15].

Second, due to the p symmetry of the Bloch wavefunctions in the valence band, the coupling of the nuclei with holes can be generally neglected because the Fermi contact interaction vanishes. The coupling constants of the hole-nuclear interaction are significantly smaller than the coupling constants of the electron-nuclear interaction. One recent reference work determined the longitudinal relaxation time for hole-nuclear interaction to be 𝑇1=14ns [16]. Furthermore, all trion states are affected, namely, |,𝐗𝟏/𝟐, |,𝐗𝟑/𝟐, |,𝐗𝟏/𝟐, and |,𝐗𝟑/𝟐. It is assumed that 𝑇1 is the same for both |𝟏/𝟐-nucleus and |𝟑/𝟐-nucleus interactions. It is also interesting to note that there are not any dark states for trions.

Last, radiative recombination rates of trions are also included in our model of decoherence. In the case of InGaAs quantum dots with an emission wavelength of 𝜆=950nm, a radiative recombination lifetime of 𝜏𝑋=0.84ns for negatively charged excitons was reported [17] as well as 𝜏𝑋=1.24ns in the case of InAs quantum dots with an emission wavelength of 𝜆=850nm [18]. On the other hand, GaAs quantum dots spin relaxation as phonon-assisted Dresselhaus spin-orbit scattering was estimated 𝑇1=34𝜇s at 𝑇=4.5K. This illustrates the fact that these types of decoherence processes can be ignored at low temperature.

4.2.1. Case  1: Small Δ, Spin Initialized to 𝛼|+𝛽|, Photon Initialized to |

The “small” detuning energy is again selected to be Δ=78𝜇eV. The results are shown in figures 13, 14, 15, and 16. In the small Δ regime, for all case of the excess electron spin being in a superposition of its eigenstates, a decent amount of entanglement (35% entanglement) is obtained within only the first 200 picoseconds. Afterward, all entanglement is lost. Moreover, the dynamics of entanglement is interesting in that for the lower amounts of superposition in the excess spin state, maximum entanglement is reached faster, yet it decreases also faster.

fig13
Figure 13: Excess electron spin dynamics on the bloch sphere for small Δ.
fig14
Figure 14: Photon polarization dynamics on the poincare sphere for small Δ.
342915.fig.0015
Figure 15: Spin-photon entanglement dynamics for small Δ.
342915.fig.0016
Figure 16: Right and left circular polarization prob. amplitudes for small Δ.
fig17
Figure 17: Excess electron spin dynamics on the bloch sphere for large Δ.
fig18
Figure 18: Photon polarization dynamics on the poincare sphere for large Δ.
342915.fig.0019
Figure 19: Spin-photon entanglement dynamics for large Δ.
342915.fig.0020
Figure 20: Right and left circular polarization prob. amplitudes for large Δ.
4.2.2. Case  2: Large Δ, Spin Initialized to 𝛼|+𝛽|, Photon Initialized to |

Also, the “large” detuning energy is selected to be Δ=1.5meV. A smaller amount of entanglement is realized, namely, 25%. And, it lasts only within the first 300 picoseconds. These results are shown in Figures 17, 18, 19, and 20. Here again the dynamics of entanglement is unexpected as lower amounts of superposition in the excess electron spin states result in a larger maximum for the entanglement.

5. Conclusion

Some entanglement between an electron spin qubit within a quantum dot and a single-photon qubit interacting inside a high-Q nanocavity can be obtained even in the presence of severe decoherence processes. Whether or not such amount of entanglement for such short periods of time can be useful in a real physical system remains to be seen. For a certainty, the performance of such scheme will have to be improved. There are three areas one could look into.

First, performances could be improved by increasing interaction frequencies. This is done by making it a resonant process, or by reducing the volume of the electromagnetic mode, or by changing material system so as to obtain a strong dipole moment for the quantum dot or by increasing dramatically the quality factor of the cavity.

Second, performances could be increased by eliminating or reducing the effect of decoherence. Indium atoms has I9/2 spin and arsenic atom have I3/2 spin; which make InAs quantum dots bad candidate as far as decoherence is concern. Other semiconductors with lower or without nucleus spin could be used. The dephasing times of the II–VI compounds are 3–10 times larger than dephasing times for III–V compounds [19]; however, for wurzite-type semiconductors to have similar optical selection rules as in the case of zinc-blende-type semiconductors, propagation along the c axis is needed [20].

At last, improvements in performances could also be obtained performing manipulations on the nuclear system [21], or using hole spin in the valence band instead of the electron spin in the conduction band as storage qubit since it is less influenced by the nucleus spin.

Acknowledgment

The authors acknowledge the support from NSF ECCS-0725514.

References

  1. D. L. Moehring, M. J. Madsen, K. C. Younge, et al., “Quantum networking with photons and trapped atoms (invited),” Journal of the Optical Society of America B, vol. 24, no. 2, pp. 300–315, 2007. View at Publisher · View at Google Scholar · View at Scopus
  2. T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science, vol. 317, no. 5837, pp. 488–490, 2007. View at Publisher · View at Google Scholar · View at Scopus
  3. M. N. Leuenberger, M. E. Flatte, and D. D. Awschalom, “Teleportation of electronic many-qubit states encoded in the electron spin of quantum dots via single photons,” Physical Review Letters, vol. 94, no. 10, Article ID 107401, 4 pages, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. D. Englund, A. Faraon, B. Zhang, Y. Yamamoto, and J. Vučković, “Generation and transfer of single photons on a photonic crystal chip,” Optics Express, vol. 15, no. 9, pp. 5550–5558, 2007.
  5. M. Kroutvar, Y. Ducommun, D. Heiss, et al., “Optically programmable electron spin memory using semiconductor quantum dots,” Nature, vol. 432, no. 7013, pp. 81–84, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. Al. L. Efros, M. Rosen, M. Kuno, M. Nirmal, D. J. Norris, and M. Bawendi, “Band-edge exciton in quantum dots of semiconductors with a degenerate valence band: dark and bright exciton states,” Physical Review B, vol. 54, no. 7, pp. 4843–4856, 1996. View at Scopus
  7. A. I. Ekimov, F. Hache, M. C. Schanneklein, et al., “Absorption and intensity-dependent photoluminescence measurements on Cdse quantum dots—assignment of the 1st electronic-transitions,” Journal of the Optical Society of America B, vol. 10, pp. 100–107, 1993. View at Publisher · View at Google Scholar
  8. C. Y. Hu, A. Young, J. L. O'Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: applications to entangling remote spins via a single photon,” Physical Review B, vol. 78, no. 8, Article ID 085307, 5 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus
  9. H. P. Seigneur, M. N. Leuenberger, and W. V. Schoenfeld, “Single-photon Mach-Zehnder interferometer for quantum networks based on the single-photon Faraday effect,” Journal of Applied Physics, vol. 104, no. 1, Article ID 014307, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Reviews of Modern Physics, vol. 29, pp. 74–93, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, New York, NY, USA, 1997.
  12. C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Physical Review A, vol. 54, no. 5, pp. 3824–3851, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. Silberfarb and I. H. Deutsch, “Entanglement generated between a single atom and a laser pulse,” Physical Review A, vol. 69, no. 4, Article ID 042308, 8 pages, 2004. View at Publisher · View at Google Scholar · View at Scopus
  14. M. N. Leuenberger, “Fault-tolerant quantum computing with coded spins using the conditional Faraday rotation in quantum dots,” Physical Review B, vol. 73, no. 7, Article ID 075312, 8 pages, 2006. View at Publisher · View at Google Scholar · View at Scopus
  15. I. A. Merkulov, Al. L. Efros, and M. Rosen, “Electron spin relaxation by nuclei in semiconductor quantum dots,” Physical Review B, vol. 65, no. 20, Article ID 205309, 8 pages, 2002. View at Scopus
  16. B. Eble, C. Testelin, P. Desfonds, et al., “Hole-nuclear spin interaction in quantum dots,” Physical Review Letters, vol. 102, no. 14, Article ID 146601, 4 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. P. A. Dalgarno, J. M. Smith, J. McFarlane, et al., “Coulomb interactions in single charged self-assembled quantum dots: radiative lifetime and recombination energy,” Physical Review B, vol. 77, no. 24, Article ID 245311, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus
  18. G. Munoz-Matutano, J. Gomis, B. Alen, et al., “Exciton, biexciton and trion recombination dynamics in a single quantum dot under selective optical pumping,” Physica E, vol. 40, no. 6, pp. 2100–2103, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. C. Testelin, F. Bernardot, B. Eble, and M. Chamarro, “Hole-spin dephasing time associated with hyperfine interaction in quantum dots,” Physical Review B, vol. 79, no. 19, Article ID 195440, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. G. D. Scholes, “Selection rules for probing biexcitons and electron spin transitions in isotropic quantum dot ensembles,” Journal of Chemical Physics, vol. 121, no. 20, pp. 10104–10110, 2004. View at Publisher · View at Google Scholar · View at Scopus
  21. B. Eble, P.-F. Braun, O. Krebs, et al., “Spin dynamics and hyperfine interaction in InAs semiconductor quantum dots,” Physica Status Solidi B, vol. 243, no. 10, pp. 2266–2273, 2006. View at Publisher · View at Google Scholar · View at Scopus