Hybrid Toffoli gate on photons and quantum spins

Quantum computation offers potential advantages in solving a number of interesting and difficult problems. Several controlled logic gates, the elemental building blocks of quantum computer, have been realized with various physical systems. A general technique was recently proposed that significantly reduces the realization complexity of multiple-control logic gates by harnessing multi-level information carriers. We present implementations of a key quantum circuit: the three-qubit Toffoli gate. By exploring the optical selection rules of one-sided optical microcavities, a Toffoli gate may be realized on all combinations of photon and quantum spins in the QD-cavity. The three general controlled-NOT gates are involved using an auxiliary photon with two degrees of freedom. Our results show that photons and quantum spins may be used alternatively in quantum information processing.

Quantum computing is an active area of research because of its ability to efficiently solve difficult problems without efficient classical algorithms [1][2][3][4] . The quantum computer, the elementary quantum element in quantum applications, is still difficult to realize with the methods of modern science. Based on the qubit system in two-dimensional Hilbert space, most quantum algorithms 1-4 require a large number of qubits to encode information [5][6][7] . These quantum algorithms may be realized by special quantum circuits consisting of basic gates corresponding to unitary matrices. In other words, the design of quantum algorithms is equivalent to the decomposition of a unitary matrix into a product of matrices chosen from a basic set 8,9 . From classical matrix decomposition, such as cosine-sine decomposition 9 , multiple controlling logic gates have been fundamental to the multiple-qubit evolution. Finding efficient ways to synthesize these controlling logic gates may allow large-scale quantum computing tasks to be performed on a shorter time-scale.
Because classical computing is designed around irreversible gates, it is impossible to directly translate this expertise into the quantum world. The Gottesman-Knill Theorem says that Clifford gates (CNOT, Hadamard, S) can be classically simulated efficiently, so they are probably not sufficiently universal for quantum computation. These gates, together with other one-qubit gates, not generated by the gates in the Clifford group, form a universal set of gates for quantum computation 10 . Based on classical reversible logic 11 , the Toffoli gate 8,9 has played a central role in this field; it is a controlled controlled-NOT acting on three bits. The Toffoli gate is also of interest in other quantum applications, for example, as a building block in phase estimation 12 , error correction 13 , and fault tolerant quantum circuits 14 . Much progress has been made, and various physical architectures have been used, including NMR systems 13 , ion traps 15,16 , linear optics 17 , superconductors 18 and atoms 19,20 . These experiments may create opportunities to investigate efficient quantum circuits for synthesizing quantum operations.
Qubit-based quantum applications require a two-level structure on atom, ion or photon systems that naturally have many accessible degrees of freedom (DOFs). These DOFs may be regarded as high-dimensional systems. In fact, high-dimensional systems may provide different quantum correlations and may be useful in quantum information processing [21][22][23][24][25][26][27][28][29] . High-dimensional systems are flexible in terms of improvements to the channel capacity 21,22 and communication security 24,25 . Moreover, they also provide an alternate way of scaling quantum computation. By extending a proposal 29 , Lanyon et al. 30 recently demonstrated a general technique that harnesses multi-level information carriers to significantly reduce the realization complexity of multiple-control logic gates. By making use of a multiple-level target system, they showed that the Toffoli gate and general two-qubit controlled-unitary gates may be realized with linear optics. Regrettably, their multiple-level target system is unscalable for large-scale applications such as Shor's algorithm. This flaw is then addressed by using multiple-level auxiliary states 31 , which may result in a high-dimensional quantum Fourier transformation.
Motivated by their scheme 23,[29][30][31] , in this paper, we propose modified proposals of the Toffoli gate by using auxiliary photons with two DOFs as an auxiliary four-dimensional quantum state. Previous results have shown that two DOFs of photons may be used to fuse hybrid quantum information 32 , reduce quantum resources [33][34][35] , and construct a universal ququart quantum computer 36 . Our application using two DOFs of photons is for the scalability of qubit-based quantum computations 23,30 and to avoid high-dimensional quantum Fourier transformations 31 . Moreover, from the strong field provided by a Fabry-Perot-type cavity, cavity QED may have a very strong effect even at the single photon level. This effect is very useful for large-scale quantum computation. In fact, by exploring the giant optical circular birefringence induced by quantum-dot spins in one-sided optical microcavities 32,33,[37][38][39][40][41][42][43][44][45] , a spin may be interacted with a linearly circularly polarized photon. Based on the cavity QED, the Toffoli gate can be deterministically implemented on all combinations of photons and spins using an auxiliary photon with the polarization DOF and the spatial mode DOF. Our schemes extend previous schemes [13][14][15][16][17]19,20,34,35 with six CNOT gates, recent proposals 29-31 with three CNOT gates and the multiple-level logic state. All of our input quantum systems are qubits. The multiple-dimensional system, i.e., one photon with two DOFs, is used as an auxiliary system to carry the control information 30 . With these constructions, the multiple DOFs will not cause confusion in quantum information processing due to different dimensions of encoded quantum systems 31 . The disentangling operations only involve single photon operations and detectors 31 . Furthermore, our Toffoli gate may be realized on all combinations of photons and quantum spins. Thus they may be very useful for hybrid quantum information processing from recent experiments [44][45][46][47][48][49][50][51][52][53][54] .

Results
The Toffoli gate is an important three-qubit entangling gate in quantum logic gates [11][12][13] . It will flip the target qubit conditional on the two control qubits. Combined with the one-qubit Hadamard, the Toffoli gate offers a simple universal quantum gate set in comparison to the CNOT gate and one-qubit rotations 10,55 . Generally, a Toffoli requires at least five two-qubit gates or six CNOT gates 11,54 . If an additional logic state is permitted for the target, a reduced decomposition requires only three two-qubit gates [29][30][31] . The enhanced decomposition is achieved by harnessing a third level of the target information carrier, i.e., a qutrit with logical states , 0 1 and 2 . Motivated by this idea [29][30][31] , two DOFs of one photon as a multiple-dimensional system will be used as the control information carrier but not the target information carrier. Four logic states , , , denote bases of the polarization DOF and spatial mode DOF of one photon respectively, where R and L denote right and left circularly polarizing photons, respectively, and d i denotes the spatial modes of one photon. In the following, we also denote = XY X Y : By exploring the interaction of quantum-dot spins and a circularly polarized photon 32,33,[37][38][39][40][41][42][43][44][45] , a Toffoli gate may be realized on the spins and photons regardless of the type of control and target qubits, using three general CNOT gates. These hybrid CNOT gates are typical controlling flip operations on the different DOFs of one photon or different types of quantum systems. These schemes show hybrid implementations of the Toffoli gate with photons and quantum spins using a reduced number of controlling qubit gates. QD-cavity system. Consider a singly charged GaAs/InAs quantum dot (QD) inside a micropillar cavity [37][38][39] , which consists of a λ-cavity between two GaAs/Al(Ga)As distributed Bragg reflectors. The QD is located in the center of the cavity to achieve maximal light-matter coupling. If the QD is neutral, optical excitation generates a neutral exciton. If the QD is singly charged, i.e., a single excess electron is injected, optical excitation can create a negatively-charged exciton (X − ), which consists of two electrons bound to one hole [37][38][39] . Due to Pauli's exclusion principle, for the spin state ↑ ≡ + 1 2 , X − in the state ↑ ↓ with the two electron spins antiparallel is created by resonantly absorbing a left circularly polarized photon L , where the heavy-hole spin state ≡ + 3 2 ; for the spin state ↓ ≡ − 1 2 , X − in the state ↓ ↑ with the two electron spins antiparallel is created by resonantly absorbing a right circularly polarization photon R , where heavy-hole spin state ≡ − 3 2 , as shown in Fig. 1. In the limit of a weak incoming field [40][41][42] , the spin cavity system behaves like a beam splitter. Based on the transmission and reflection rules of the cavity for an incident circular polarization photon conditioned on the QD-spin state, the dynamics of the interaction between the photon and spin in a QD-microcavity coupled system is described as below 32 under ideal conditions. In the following, this ideal spin-cavity unit is used to realize the Toffoli gate on photons and quantum-dot spins for efficient quantum information processing. Then, the experimental spin-cavity unit will be discussed in the last section.
Toffoli gate on a three-photon system. Consider three linearly circularly polarized photons A, B and C in the states Our goal is to realize the Toffoli gate with the following form where the photons A and B are the controlling qubits while the photon C is the target photon. The detailed circuit is shown in Fig. 2. This construction is completed with three auxiliary quantum electron spins e i in the state + = ( ↑ + ↓ )/ 2 and an auxiliary photon D in the state Rd 1 . The Toffoli gate T AB,C is completed with the following three controlled gates. First, from the subcircuit S 1 shown in Fig. 2(a), the photon A as an input pulse passes through the cPS 1 , cavity Cy 1 , cPS 2 , sequentially. Then W 1 is performed on the spin e 1 . Now, the pulse D from the spatial mode d 1 passes through the H 1 , cPS 3 , cavity Cy 1 , cPS 4 , H 2 , sequentially. After these operations, the joint system consisting of the photons A and D, and the spin e 1 is changed from φ + Rd after the measurement of the electron spin e 1 under the basis is performed on the photon A for the measurement outcome − e 1 . This circuit has realized the controlled-NOT gate , CNOT A D P on the input photon A and the polarization DOF of the auxiliary photon D, which is different from previous CNOT gate on the same type of input system. Second, from the subcircuit S 2 shown in Fig. 2(b), the photon B passes through the cPS 5 , cavity Cy 2 , cPS 6 , sequentially. Then W 2 is performed on the spin e 2 . Now, the photon D passes through the BS 1 , cPS 7 , X 1 , cavity Cy 2 , X 2 , cPS 8 , BS 2 , sequentially. After these operations, the joint system consisting of the photons A, B and D, and the spin e 2 is changed from φ Φ +  Figure 1. Schematic energy level and optical selection rules due to Pauli's exclusion principle. â in and â out are the input and output field operators of the waveguide, respectively. L and R represent the left circularly and right circularly polarized photons, respectively. ↑ and ↓ represent the spins of the excess electron. ↑ ↓ and ↓ ↑ represent the negatively charged exciton X −1 .
after the measurement of the electron spin e 2 under the basis | ± 〉 { }, where a Pauli phase flip σ Z p is performed on the photon B for the measurement outcome − e 2 . This circuit has realized the controlled-NOT gate , CNOT B D S on the input photon B and the spatial mode DOF of the auxiliary photon D. Third, from the subcircuit S 3 shown in Fig. 2(c), the pulse D from the spatial mode d 2 passes through the cPS 9 , cavity Cy 3 , cPS 10 , sequentially. Then W 3 is performed on the spin e 3 Now, the photon C passes through the H 3 , cPS 11 , cavity Cy 3 , cPS 12 , H 2 , sequentially. After these operations, the joint system consisting of the photons A, B, C and D, and the spin e 3 is changed from  . X i denote wave plates to perform the polarization flip transformation + R L L R . Z i denote waveplates to perform the phase flip transformation − R R L L. cPS i represent circularly polarizing beamsplitters that transmit R and reflect L . cBS i represent 50%50 circularly polarizing beamsplitters to perform the Hadamard operation 2 . Cy i denote the QD-cavity charged the electron spin e i . If there are two input lines of one cavity, the photon represented with red lines passes through the cavity firstly, and then the photon represented with black lines passes through the cavity.
after the measurement of the spin e 3 under the basis | ± 〉 { }, where a phase flip σ Z p is performed on the photon D from the spatial mode a 2 for the measurement outcome − e 1 . This circuit may be viewed as the controlled-NOT gate CNOT D,C performed on the auxiliary photon D and the input photon C as follows which is an essential three-qubit operation. Finally, by performing the single qubit measurements on the photon D under the basis In the experiment, this measurement may be completed with the 50%50 circularly polarizing beamsplitter cBS 3 , two circularly polarizing beamsplitters cPS 13  shown in equation (7) may collapse into Thus, the Toffoli gate T AB,C shown in equation (3)  Toffoli gate on a three-spin system. Consider three electron spins e i in the states This section is to realize the Toffoli gate where the electron spins e 1 and e 2 are the controlling qubits, while the electron spin e 3 is the target qubit.
The detailed circuit is shown in Fig. 3 by using an auxiliary photon D in the state Rd 1 . This Toffoli gate is realized with the following three controlled gates on electron spins. First, the auxiliary photon D from the spatial mode d 1 passes through the half waveplate H 1 to H 2 sequentially. The joint system consisting of the photon D and the electron spin e 1 changes from ψ Rd . Similarly, this measurement may be implemented in the experiment with the 50%50 circularly polarizing beamsplitter cBS 3 , two circularly polarizing beamsplitters cPS 7   Toffoli gate on hybrid three-qubit systems. The present Toffoli gate on a three-photon system shown in the Fig. 2 and a three-spin system shown in Fig. 3 may be combined to realize Toffoli gate on hybrid three-qubit systems. Thus, the three input qubits may be an arbitrary combination of photons and quantum spins. Because of the symmetry of two control qubits, four different cases are to be considered, as shown in Fig. 4. First, let two photons A and B jointly control an electron spin e; their initial states are φ A 1 , φ B 2 and ψ e 3 , respectively. The detailed circuit is shown in Fig. 4(a). From the , CNOT A D P realized with the subcircuit S 1 , the joint system consisting of three qubits and an auxiliary photon D changes from  Furthermore, from the CNOT gate realized with the subciruit S 6 , the joint system Ω 2 shown in the equation (17)  . In the experiment, this measurement may be implemented in experiments with a 50%50 circularly polarizing beamsplitter, two circularly polarizing beamsplitters, two half waveplates, and four single photon detectors, as shown in Fig. 2(c). The recovery operations are similar to these shown in Table 1. Thus, a Toffoli gate has been realized on the two photons and one spin using three CNOT gates.
Second, consider two electron spins e 1 and e 2 in the states ψ i e i that jointly control one photon A in the state φ The detailed circuit is shown in Fig. 4(b). From the CNOT gates realized with the subcircuit S 4 and S 5 in Fig. 3 Moreover, from the CNOT realized with the subcircuit S 3 in Fig. 2 . The detailed circuit is shown in Fig. 4(c). Similar to the subcircuits shown in Fig. 4(a,b), from the CNOT gates realized with the subcircuits S 1 in Fig. 2(a), S 5 in Fig. 3 and S 3 in Fig. 2(c), the joint system of the three input qubits and the auxiliary photon D changes from φ ψ φ Rd The detailed circuit is shown in Fig. 4(d). Similar to the subcircuit shown in Fig. 4(c), from the CNOT gates realized with the subcircuits S 1 in Fig. 2(a), S 5 in Fig. 3 and S 6 in Fig. 3 for the polarization DOF and spatial mode, respectively. The recovery operations are the same as those in Fig. 4(c). Thus, the spin qubit may be jointly controlled by one photon and one spin.

Discussion
The optical selection rules of a QD-cavity system shown in equation (1) play core roles in the present Toffoli gates. In the resonance conditions Δ ω x = Δ ω c = 0, if one neglects the cavity side leakage κ s ≈ 0, it easily follows that |r 0 | → 1 and |r| → 1 when the cooperativity parameter g 2 /(κγ) of cavity QED is large enough. Thus, our six Toffoli gates are deterministic and faithful. However, the side leakage from the cavity is unavoidable in the experiment 44,45,[47][48][49][50][51][52][53][54] . In the following, consider two kinds of transition channels for the cavity photon. The first is the cavity decay due to transmission through the cavity mirror, whose rate is κ. Every other unwanted photon loss, such as cavity absorption and scattering, are characterized by the overall loss rate κ s . Taking into account the coupling through the cavity decay channel and neglecting the spatial dependence, the relation of the input field operator â in and output operator â out may be approximated with an experimental reflection coefficient [37][38][39]  These complex coefficients indicate that the reflected light may experience a phase shift [32][33][34][35][36][44][45][46] . Under resonant conditions Δ ω c = Δ ω x = 0, the reflection coefficients |r| and |r 0 | are evaluated in Fig. 5, and the phase shifts θ and θ 0 are evaluated in Fig. 6 inrelation to the decay ratios of cavity κ s /κ and the cooperativity parameter C = g 2 /(κγ) of cavity QED 56,57 , which is a geometric parameter that characterizes the absorptive, emissive, or dispersive coupling of an atom to the cavity mode. Based on Fig. 5, the reflection coefficients will satisfy |r| ≈ 1 and |r 0 | ≈ 1 when C ≫ 10 and κ s /κ → 0, and these additional conditions are not required for relative phase shifts θ 0 = π and θ = π because r and r 0 are real under the resonant conditions Δ ω c = Δ ω x = 0. Hence, the real reflection coefficients r and r 0 will be considered under the resonant conditions. In fact, the ideal optical selection rules shown in equation (1) are changed into E P pp p j j , where P j is a successful reflection probability of the j-th photon from a micropillar cavity 37,50,54,57 , and  denotes the index set of photons involved in each scheme. Its efficiency is evaluated in Fig. 7(a). To detail the influence of the practical input-output process on the Scientific RepoRts | 5:16716 | DOI: 10.1038/srep16716 fidelity of the final joint system after this Toffoli gate, we take the case in which the detector D Rd 1 clicks as an example and obtain the average fidelity F pp,p , as evaluated in Fig. 8(a). Here, where the integral is evaluated over all possible input states, Ψ i and Ψ f are the ideal final state and the experimental final state with side leakages, respectively. For our second Toffoli gate on three electron spins shown in Fig. 3, three electron spins e 1 , e 2 and e 3 are involved, and one photon D is used; its success is determined by the photon D, which is detected at the detector , , D D D Rd Ld Rd 1 1 2 or D Ld 2 click. The practical efficiency E ss,s is evaluated in Fig. 7(b) whereas the average fidelity F pp,p is evaluated in Fig. 8(b) for the photon D detected at the detector D Rd 1 as an example. For the other four cases, one can obtain similar results.
Typically, the cavity side leakage may greatly affect the efficiency and fidelity of the Toffoli gate. As shown in the Figs 7 and 8, high efficiency and fidelity may be achieved even in the weakly coupling regime when κ κ  s . Otherwise, the strong coupling defined by g ≫ (κ, γ) is necessary [39][40][41][48][49][50][51][52][53][54] . The classical strong-coupling condition corresponds to the single-photon Rabi frequency 2g being larger than the geometric mean of the atomic and cavity line widths. In general, the system can be parameterized in terms of two dimensionless parameters, namely, the ratios g/κ and g/γ in the cavity QED description or, in the classical description, the cooperativity parameter C and the line width ratio κ/γ. The cavity QED strong-coupling condition 2g > (κ, γ) corresponds to a normal-mode splitting that is much larger than  the line widths of the normal modes. The cooperativity parameter of cavity QED is shown to play a central role and is given a geometrical interpretation. The cooperativity has been realized up to 27 58 . Under this cooperativity, the efficiencies E PP,P and E SS,S are greater than 91.24% for κ s /κ ≈ 0.2 48,54 ; the average fidelities F PP,P and F SS,S are greater than 93.47% for κ s /κ ≈ 0.2 48,54 . If one hopes to achieve a fault tolerance threshold of 7.5 × 10 −3 on a two-dimensional lattice of qubits 59 , the cavity leakage ratio should be κ s /κ < 0.04 and the cooperativity should be C > 28 for a photonic Toffoli gate, whereas the cavity leakage ratio should be κ s /κ < 0.03 and the cooperativity should be C > 34 for a Toffoli gate on a three-spin system. When the fault tolerance threshold is reduced to 1 × 10 −3 using controlled phase gates based on dipole-induced transparency 60 , the cavity leakage ratio should be reduced to 0.02, and the cooperativity should be improved to C > 38 for a photonic Toffoli gate, whereas the cavity leakage ratio should be reduced to 0.015 and the relative coupling strength should be improved to 4.2 for a Toffoli gate on a three-spin system. κ s /κ = 0.05 has been reported, which could be achieved by taking a pillar  microcavity with the quality factor of Q = 165000 demonstrated in ref. 54 and decreasing the reflection of the top mirror to reduce the quality factor to Q = 9000, which is still in the strong-coupling regime 48 .
If the experimental electron spin decoherence and trion dephasing 41,42 are considered, the real efficiency and fidelity are slightly decreased when the hole spin coherence time is longer than three orders of the cavity photon lifetime 44,50,51 . Moreover, by using the spin echo technique 57,61 and the nanosecond spin resonance microwave pulse 47 to protect the electron spin coherence, faithful Hadamard transformations may be implemented on the electron spin for our six Toffoli gates. The heavy-light hole mixing may be reduced by engineering the shape, size and type of the charged exciton 61 . The optical selection rule has been experimentally realized with the spin state of a single trapped atom and the polarization state 44,45 . To achieve weak excitation, some adiabatic conditions are used to ensure that the X − stays in the ground state for the most time. With a first-order approximation, we can adiabatically eliminate â from the third subequation of equation (31) by substituting the steady-state solution to the first two subequations of equation (31). Under the adiabatic condition 1 0 0 , the system may be unchanged between the ground state E 0 and excite state E 1 under the first-order approximation. Here, Δ E 10 = E 1 − E 0 . If the dephasing is considered for the atomic system, it may be modeled by introducing phenomenological decay terms or noise operators , ,ˆf g h into three subequations of equation (31). Because the output modes are initially in a vacuum, the = f 0. By substituting the steady-state solution to the third subequation of equation (31) 2 62 . Of course, the present Toffoli schemes are also conditional on the perfect overlap of the cavity mode with the two spatially separated optical beams, the phase stability of the interferometer composed of the cBS, and the perfect time overlap of two beams passing through several interferometers.
In conclusion, we have investigated the possibility of hybrid quantum computation assisted by the quantum spins and photons with two DOFs. Six deterministic Toffoli gates are realized on the joint system of all combinations of the photon or the quantum spin systems. Compared with previous Toffoli gates [13][14][15][16][17]19,20 , our Toffoli gates may be realized with three general control-NOT gates, which are similar to the schemes in ref. [29][30][31]. Unlike the multiple dimensional quantum target state of the photonic Toffoli gate 18,30 , all the input systems are qubit systems, whereas the additional multiple-dimension logic state is used as the auxiliary system. With the modification, one does not need to consider the different dimensional quantum systems to encode information in quantum applications. This method is similar to that in ref. 31. However, their disentangling operations are necessary and essential controlled operations or high-dimensional operations on the auxiliary system. If our photon with two DOFs is considered, their Fourier disentangling operations require two controlled operations. However, with our schemes, even if the photon with two DOFs is used as an auxiliary system, we do not need to implement controlled operations or high-dimensional operations on the auxiliary system. Our disentangling operations are only single-qubit operations. Moreover, the Toffoli gate may be realized on different quantum systems, which may be very useful depending on the specific requirements. Different from the Toffoli gate 34 on the three-atom system, our Toffoli gate may be implemented on a hybrid photon and spin system. Our optical cavity system is easier than the Toffoli gate 35 using the double-side cavity system. Compared with their six controlled qubit operations 34,35,63 , our circuits are also compact by as a result of the auxiliary high-dimensional system and cost only three controlled qubit operations. Our theoretical results show that photons and quantum spins may be used alternatively in quantum information processing. Of course, the optical selection rules may be affected by the cavity leakage and spin coherence in quantum dots or the exciton coherence in the experiment. With the recent experiments regarding QD-cavity system [47][48][49][50][51][52][53][54] and the quantum gate between a flying optical photon and a single trapped atom 32 , our results are expected to be applicable for large-scale quantum computation.

Method
Optical selection rules. A singly charged GaAs/InAs QD 32,33,37-45 has four relevant electronic levels ↑ , ↓ , ↑ ↓ , and ↓ ↑ . An exciton consisting of two electrons bound to one hole with negative charges can be created by the optical excitation of a photon and an electron spin. In theory, consider the interaction between a single cavity mode and a single two-level spin interacting with a single cavity mode at optical frequencies. By neglecting the spatial dependence 37,44,45 , taking into account the coupling through the cavity decay channel and neglecting the spatial dependence, the master equation of the whole system can be expressed by the Lindblad form 1 2 where H = H 1 + H 2 + H 3 . ρ is an arbitrary system operator. ω =ˆ † a a H 1 is the Hamiltonian of the input photon pulse. σ σ = ( + ) is the standard Jaynes-Cummings Hamiltonian for a two-level system interacting with a single electromagnetic mode by applying the rotating wave approximation and dropping the energy nonconserving terms. ( ) a t are cavity input operators with the standard commutation relations . σ − and σ + are the Pauli raising and lowering operators respectively.
is the system Hamiltonian for the dipole, ω c is the resonant frequency of the dipole, and σ z is the Pauli operator for the population inversion. κ is the decay rate of the cavity field due to ohmic losses in the metal.  ρ ρ ρ ρ = − ( + − ) κ κ +ˆˆˆˆˆ † † † a a a a a a 2 1 2 s accounts for the damping of the input photon pulse. κ s is the decay rate of the cavity side leakage mode due to scattering into free-space modes. The scattering rate κ s may be calculated classically from the Larmor formula. where r(ω) is defined in equation (27). If the quantum dot is uncoupled from the cavity (g = 0), r(ω) is reduced to r 0 (ω) as shown in equation (28). For the strong coupling regime g ≫ (κ, γ), one can get |r| ≈ 1 and |r 0 | ≈ 1 under resonant conditions by adjusting ω, ω x and ω c . Thus, if the excess electron spin lies in the spin state ↑ , the input light ( ) L R acquires a phase shift of θ = arg[r(ω)](θ 0 = arg[r 0 (ω)]) by passing through the cavity. Conversely, if the excess electron spin lies in the spin state ↓ , the input light ( ) R L acquires a phase shift of θ = arg[r(ω)](θ 0 = arg[r 0 (ω)]) by passing through the cavity. Thus, two phase shifts may be obtained as 37 When the side leakage and cavity loss are ignored, the optical selection rules shown in equation (1) are followed by adjusting frequencies to achieve the phase shifts θ 0 = π and θ = 0 32,33,44,45 . with one half waveplate, one circularly polarizing beamsplitter and two single photon detectors, the electron spin e can be faithfully disentangled. The experimental performances depend on the experimental optical selection rules shown in equation (1).