Photon storage and routing in quantum dots with spin-orbit coupling

This supplementary text contains the energy eigenstates and electric-dipole matrix elements of the quantum dots (QDs), the expressions of the Heisenberg-Langevin equations, the solution of Eq. (4) and expressions of some coefﬁcients in the main text, the equations of motion for the single-particle wavefunctions derived from the Heisenberg-Langevin-Maxwell (HLM) equations, and the physical mechanism of suppressing the dephasing.


ENERGY EIGENSTATES AND ELECTRIC-DIPOLE MATRIX ELEMENTS OF THE QDS
The Hamiltonian of an electron in an anisotropic QD without SOC readŝ where A = 1 2 B(−y, x, 0) is the vector potential chosen to be in symmetric gauge, m is the effective mass of the electron, ω x (ω y ) is the trapping frequency in x (y) direction, g is Landé g-factor, µ B is Bohr magneton, B = (0, 0, B) is the external (static) magnetic field; the last term is Zeeman energy, withσ i the ith-component of Pauli matrices. It is easy to show that the above Hamiltonian can be written into the formĤ where Ω j = ω 2 j + e 2 B 2 /(4m 2 ) (j = x, y) are effective trapping frequencies, andL z = xp y − yp x is the z component of the orbit angular momentum of the electron.
By defining the creation and annihilation operatorsâ x = 1 (y + 2 y ∂ y ), andâ † y = 1 √ 2 y (y − 2 y ∂ y ), with j = h/(mΩ j ), the HamiltonianĤ 0 can be expressed asĤ describing a 2D harmonic oscillator under the action of the external magnetic field B.
When the QD has a significant Rashba SOC, its Hamiltonian is given byĤ =Ĥ +Ĥ SOC . Herê H SOC = g 1 σ xPy −σ yPx is contributed by the SOC, withP =p + eA the kinetic momentum and g 1 the strength of the SOC. The eigen energies E j and eigen states ofĤ can be obtained by an exact diagonalization in the basis of the 2D harmonic oscillator described byĤ 0 given above. These eigen states satisfy the equationĤ and can be expressed as where ξ j n x ,n y ,s are expansion coefficients for the jth eigen state. When the diagonalization is implemented, one can obtain ξ j n x ,n y ,s and E j (j = 1, 2, 3, ...). Using the relations x = x √ 2 â † x +â x and y = y √ 2 â † y +â y , we can obtain the electric-dipole matrix element related to the eigen states ψ j and ψ k , i.e. where with I m x ,m y ; n x ,n y = δ n y ,m y x ( √ m x + 1δ n x ,m x +1 + √ m x δ n x ,m x −1 ) + δ n x ,m x y ( m y + 1δ n y ,m y +1 + √ m y δ n y ,m y −1 ).

EXPRESSIONS OF THE HEISENBERG-LANGEVIN EQUATION
Here we give the explicit expressions of the Heisenberg-Langevin equation (3a) in the manuscript, with the microwave field coupling the two lower levels |1 and |2 taken into account. Note that in this case the Hamiltonian (2) in the manuscript is modified to beĤ H →Ĥ H +Ĥ M , wherê where Ω M = (e m · µ 21 )B m /h is the half Rabi frequency of the microwave field, with µ 21 the magnetic dipole matrix element associated with the lower levels |2 and |1 , and B m is the magnetic induction of the microwave field. In the following (also in the main text), we have assumed a zero total relative phase for simplicity, i.e., and φ m are the initial phase of two probe, two control, and microwave fields, respectively) [1]. Then the expressions of the Heisenberg-Langevin equations (3a) in the manuscript reads [2] i for diagonal elements, and the spontaneous emission decay rate, and γ dep αβ the dephasing rate between the levels |α and |β );F αβ are δ-correlated Langevin noise operators, with the two-time correlation function given by where D αβ,α β is the diffusion coefficient, which can be obtained from Eqs. (S11) and (S12) using the generalized fluctuation-dissipation theorem [3], assuming that the QDs are coupled with the vacuum reservoir. It can be shown that the diffusion coefficients D αβ,α β have the following forms D 21,12 = Γ 23 Ŝ 33 , (S14a) Note that, for the discussions of the propagation and the storage and retrieval of single photon wavepackets presented in Sec. 3.1 and Sec. 3.2 in the manuscript, a reduced version of the above equations, i.e., the one for the three-level Λ-type system obtained by taking ∆ 4 → ∞, is used. In this case, one hasÊ p2 = 0, Ω c2 = 0,Ŝ 4α =Ŝ α4 = 0,F 4α = 0 (α = 1, 2, 3, 4); however, for the discussion on the routing of single-photon wavepackets presented in Sec. 3.3, the above equations for the four-level double Λ-type system must be employed.

A. Solution of Eq. (4) in the manuscript
Equation (4) in the manuscript can be exactly solved by using the Fourier transformation where X representsÊ p ,Ŝ α1 , andF α1 (α = 1, 2, 3). The solution readŝ Substituting Eq. (S16) into Eq. (3b) in the manuscript under the Fourier transformation and solve the closed equation forẼ p , one can obtain the solution ofÊ p , given by Eq. (5) in the main context.

B. Expression of S 33
To give an explicit expression of the photon number of the probe field described by Eq. (7) in the main text, we have to know S 33 . For this aim, we need to solve the reduced equations of Eqs. (3a)-(3c) in the manuscript obtained under the condition ∆ 4 → ∞, and carry out the calculation to the second-order in g pÊp , which results in the following equations Here an ensemble average on the equations have been made and hence the Langevin noise terms do not appear because F αβ = 0. By solving Eq. (S17) with the help of the solution (5) in the main text ofÊ p , and the correlation function for Langevin operators (S13) and (S14), the solution of Ŝ 33 can be expressed by with Here A * B ≡ +∞ −∞ dω Ã * (ω − ω)B(ω ) and Substituting Eq. (5) in the main text into Eq. (S18) and using Eqs. (S13) and (S14) again, we obtain the following equation under the condition of one-photon input. Here The solution of Eq. (S22) is given by The normalized second-order coherence function of the probe field is defined by

EQUATIONS OF MOTION OF THE SINGLE-PARTICLE WAVEFUNCTIONS DERIVED FROM THE HLM EQUATIONS
For the probe fields working at single-photon levels,Ê p1 andÊ p2 can be taken as small quantities and hence the HLM equations can be linearized around the initial state solution (i.e.,Ŝ (0) αβ = δ α1 δ β1 ). Then the HLM equations (3a)-(3c) in the main text in the absence of the microwave field can be simplified as i ∂ ∂x i ∂ ∂x NŜ 31 ,ψ 4 = √ NŜ 41 , andψ 5 = √ NŜ 21 , Equations (S28a)-(S28e) can be written into the form withψ α satisfying the bosonic commutation relations As shown in Sec. 3.1 in the main text, due to the EIT effect the Langevin noise plays a negligible role, and hence the quantum statistical property of the incident single photons can be well preserved during propagation. The physical reasons for this are the following: (i) Since there is no optical pumping, the noiseF will not be amplified during the propagation. (ii) The population of the excited state |3 , given by Ŝ 33 , is negligible under the condition of the EIT [see the quantitative analysis given in Sec. III(A)]. Thus the impact of spontaneous emission from |3 on the photon number of the probe field is negligible. (iii) The energy of the optical transition |1 ↔ |3 , given byhω 31 , is much larger than that of thermal noises coming from the coupled reservoirs, given by k B T (k B is the Boltzmann constant and T is the experimental temperature). Typically, one hashω 31 /(k B T) ≈ 1600 1 at T ≈ 10 mK, thereby the incoming thermal noises can be safely considered to be at the vacuum states. As a result, the normally-ordered correlation functions F †F is zero, leading to the absent of the incoherent pump of atoms from their ground state |1 to the excited state |3 [4,5].
Comparing with the operator equation of motion (S29), Eq. (S34) is more convenient for the calculation of photon dynamics because it is a c-number one. Here we use it to calculate the time evolution of a single-photon wavepacket in two cases: • the storage and retrieval of the single-photon wavepacket, where one takes ∆ 4 → ∞ and henceÊ p2 = 0,ψ 2 =ψ 4 = 0, equivalent to a Λ-type three-level system with Φ = (Φ 1 , Φ 3 , Φ 5 ) T .
Note that Eq. (S34) can be generalized to the case when the microwave field coupling with the two lower states |1 and |2 is present.

THE PHYSICAL MECHANISM OF SUPPRESSING THE DEPHASING
The microwave field is used to realize a coherent population transfer between the two lower states |1 and |2 . In this way, a quantum coherence between these two lower states can be acquired.
with microwave field without microwave field Coherence Fig. S1. Wavefunction Φ 5 of the coherenceŜ 21 between the two lower quantum states |1 and |2 as a function of t/τ 0 , for cases with (solid red line) and without (dashed blue line) the use of microwave field at x = 400 µm. The system parameters are the same as those used in Fig. 3 of the main text.
To illustrate the role of the microwave field for creating the quantum coherence, for simplicity we consider the three-level configuration of QDs with the quantum states given by |1 , |2 , and |3 ; see Fig. S1(a). The timing-sequences of the control and microwave fields are shown in the lower part of the figure. By solving Eqs. (S11) and (S12), we can obtain the wavefunction Φ 5 for the coherenceŜ 21 between the two lower quantum states |1 and |2 at time t = T on , which reads Φ 5 (x, T on ) = Φ 1 (x, T off )e id 21 (T on −T off ) + Ω C Ω M (x) where Φ 1 (x, T off ) is the wavefunction of the probe field at time t = T off , T off (T on ) is the switchingoff (switching-on) time of the control field, T M off (T M on ) is the switching-off (switching-on) time of the microwave field. The first term of the above expression is the contribution from the input probe photon; the second term is the contribution by the microwave field. We see that the shape, intensity, and time duration of the microwave field plays an important role for the coherence wavefunction Φ 5 .
Shown in Fig. S1 is Φ 5 as a function of t/τ 0 , for cases with (solid red line) and without (dashed blue line) the use of the microwave field at x = 400 µm. The system parameters are the same as those used in Fig. 3 of the main text. We see that when the microwave field is present the coherence is significantly enhanced; while when the microwave field is absent the coherence is very small.
Consequently, the application of the microwave field can increase the quantum coherence between the two lower states, by which the system can acquire high retrieval and routing efficiencies of the probe photon after its storage. In fact, such a technique was already used for realizing an efficient classical light memory in a three-level atomic gas [7].