Birefringent spin-photon interface generates polarization entanglement
Nikita Leppenen Dmitry S. Smirnov













nikita.leppenen@weizmann.ac.il
Department of Chemical & Biological Physics, Weizmann Institute of Science, Rehovot 7610001, Israel


Ioffe Institute, 194021 St. Petersburg, Russia



  A spin-photon interface based on the luminescence of a singly charged quantum dot in a micropillar cavity allows for the creation of photonic entangled states. Current devices suffer from cavity birefringence, which limits the generation of spin-photon entanglement.
  
  In this paper, we theoretically study the light absorption and emission by the interface with an anisotropic cavity and derive the maximal excitation and spin-photon entanglement conditions. We show that the concurrence of the spin-photon state equal to one and complete quantum dot population inversion can be reached for a micropillar cavity with any degree of birefringence by tuning the quantum dot resonance strictly between the cavity modes. This sweet spot is also valid for generating a multiphoton cluster state, as we demonstrate by calculating the three-tangle and fidelity with the maximally entangled state.


Birefringent spin-photon interface generates polarization entanglement
    Dmitry S. Smirnov
    April 29, 2024
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§ INTRODUCTION


Spin entanglement is an essential resource for quantum information processing due to the excellent isolation of spin degree of freedom from the environment <cit.>. A basic example of the entanglement production is given by absorption of a linearly polarized photon in GaAs-based nanostructures when a Bell state of electron and hole spins is optically generated <cit.>. However, flying qubits naturally represented by photons are more useful for quantum information transfer. Their polarization entanglement is routinely produced by biexciton recombination <cit.>. 

Photons can be coupled to the localized qubits — electron spins in quantum dots (QDs), by zero-dimensional spin-photon interfaces <cit.>, which can be efficiently realized using micropillar cavities <cit.> (see Refs. senellart2017high,10.1063/PT.3.4962 for recent reviews). These interfaces allow for the photon polarization manipulation by electron spin <cit.> and fast control of electron spin by photons <cit.>. But most importantly, electron-photon entanglement can be created in these structures <cit.>.

An auspicious task is the generation of entangled photon cluster states <cit.>. Among different approaches <cit.> including parametric down conversion <cit.>, the most scalable one is the sequential entanglement of spin of a photon and localized electron <cit.>. After the first realization of the theoretical proposal <cit.>, the intensive investigations in this direction continue until now <cit.>.



An important factor limiting the generation of many entangled photons is the intrinsic electron spin relaxation <cit.>. An approach for its investigation by measurement of the equilibrium electron spin noise through the photon correlation functions was proposed theoretically <cit.> and recently realized experimentally <cit.>. However, a significant obstacle for the spin dynamics investigation and spin-photon entanglement generation remains due to the splitting or birefringence of the micropillar cavity modes <cit.> despite multiple attempts to reduce it <cit.>.



In this work, we theoretically address this issue. After formulating the model in Sec. <ref>, we show proof of principle numerical calculation in Sec. <ref> and then describe optical excitation of the quantum dot in Sec. <ref>, spin-photon entanglement generation in Sec. <ref>, and multiphoton entanglement in Sec. <ref>. In conclusion (Sec. <ref>), we stress that there is a “sweet spot” at which trion can be efficiently excited and complete spin-photon entanglement can be realized for arbitrary strength of the micropillar cavity birefringence.



§ MODEL



We describe a micropillar cavity with an electron-charged quantum dot [Fig. <ref>(a)] by the following general Hamiltonian <cit.>:

    H = ∑_H,V H^(cav)_H,V+∑_±[ H^(trion)_±+ H^(int)_±+ H^(exc)_±].

Let us describe each of these terms. The first one reads H^(cav)_H,V = ω_H,V c^†_H,Vc_H,V. It describes the energy of photons in two linearly polarized cavity modes, H and V, with the corresponding annihilation operators c_H,V and frequencies ω_H,V (hereafter, we set ħ=1). It is convenient to represent these frequencies as ω_H,V=ω_c±Δ, where ω_c is the central frequency and 2Δ is the splitting between cavity modes, as shown in Fig. <ref>(b).

Absorption of a photon by a quantum dot with an electron leads to the formation of a negatively charged trion. Its energy is given by H^(trion)_± = ω_0 a^†_± 3/2 a_± 3/2, where ω_0 is the trion resonance frequency and a_± 3/2 are the annihilation operators of trions with the heavy hole spin ±3/2, which correspond to the trion pseudospin J_z=±1/2.

The third term in Eq. (<ref>) 

    H^(int)_± = g[a^†_± 3/2 c_± a_± 1/2+a^†_± 1/2 c_±^† a_± 3/2],

describes light-matter coupling with the strength g in a circular basis with

    c_±=c_H±ı c_V/√(2).

The operators a_±1/2 destroy an electron in the ground state with the spin S_z=±1/2. The selection rules described by this Hamiltonian can also be rewritten in the basis of H and V modes for the electron and trion states S_x,J_x=±1/2. The corresponding transitions are shown in Fig. <ref>(c).

The last term of the general Hamiltonian H^(exc)_± =  E_±^*(t)c_±+ E_±(t)c^†_± stands for the resonant cavity excitation by external light pulse described by the amplitudes E_±(t) of two circular components. A train of pulses is needed to generate many entangled photons. But if the intervals between the pulses are long enough (longer than the trion recombination time γ^-1), one can separately consider each pulse. We assume the length of the pulses to be so short that their spectrum covers both cavity modes, and take E_±(t)= E_±δ(t) in what follows. Note that in the opposite limit of the pump pulses longer than the inverse cavity decay rate, the birefringence does not play a significant role; it only leads to the polarization-dependent photon injection rate.

We consider only one, the most important incoherent process, the escape of photons from the cavity with the polarization-independent rate 2ϰ. It is assumed to be larger than the coupling strength, ϰ≫ g. This is the so-called fast cavity regime, which is realized in most structures. Typically, in the experiments, the decay rates of the two cavity modes are similar, so we leave the study of the effect of their difference for future works. We neglect the incoherent trion decay, assuming it to be slower than γ. For this reason, the trion state is represented by a narrow Lorentzian in Fig. <ref>(b). The transmission spectrum of the system calculated within the input-output approach is shown in Appendix <ref>. In experiments, the reflection geometry is commonly employed. Moving forward, our focus is on studying spin entanglement with the photons emitted from the microcavity, with the emission direction being inconsequential for the theoretical analysis.

The dynamics of the system can be described on equal footing by the Lindblad equation or by the method of quantum trajectories <cit.>. In the latter approach, the non-Hermitian Hamiltonian should be considered

    H_ nh =  H-iϰ∑_± c_±^† c_±

and the jump operators are C_± = √(2ϰ) c_±. Averaging over individual quantum trajectories gives the expectation values of physical observables in the system.





§ NUMERICAL CALCULATION



To illustrate the following analytical theory, we present a numerical calculation for a particular set of parameters. In Fig. <ref>, we consider a σ^+ circularly polarized excitation and initial electron spin polarization along the x axis, S_x=1/2. The other parameters are given in the figure caption. For the simulations, we use the quantum Monte-Carlo trajectory method <cit.> realized in QuTiP python framework <cit.> and perform averaging over 300 individual quantum trajectories.

Panel (a) shows the numbers of σ^+ and σ^- polarized photons in the cavity mode as functions of time. The beatings with the period π/Δ are related to the splitting between linearly polarized cavity modes. After the initial injection of photons, their number exponentially decays due to the escape of photons from the cavity.



During this time, a trion gets excited in the quantum dot. Its population N_ tr=a_+3/2^† a_+3/2+a_-3/2^† a_-3/2 is shown in Fig. <ref>(b) by the red circles. It starts from zero and then quickly (t ∼ 1/ϰ) increases almost to one, corresponding to the π pulse. Then, the trion starts to decay slowly, giving rise to a small population of the cavity mode at the long time scales. The black circles show the concurrence between a photon emitted from the cavity at the given time and the spin of the electron left in the quantum dot. It is close to one, which demonstrates complete spin-photon entanglement.

The analytical expressions for these quantities are given below. However, the main results can already be seen from the figure: (i) Despite splitting of the cavity modes and their detuning from the trion resonance, a complete population inversion (π pulse) is possible. (ii) Even strong birefringence of the microcavity does not spoil spin photon entanglement so that concurrence reaches one. Below, we describe both results analytically and in more detail.





§ TRION EXCITATION



Light quantization does not play a role in the weak coupling regime (g≪ϰ). Then, the average amplitudes of the linearly polarized light components in the cavity obey the classical equations

    c_H,V(t) = c_H,V(0)e^-iω_H,Vt-ϰ t,

with the initial conditions determined by the pump pulse: c_H(0) = i/√(2)( E_++ E_-), c_V(0) = 1/√(2)( E_+- E_-). These expressions are used to calculate the blue and green solid lines in Fig. <ref>(a). The total number of photons decays at the rate of 2ϰ.

The trion decay can be neglected during the photon lifetime in the cavity so that the trion excitation can be described by the Schrodinger equation <cit.>. The quantum dot state is characterized by the four quantum amplitudes ψ_±1/2(t) and ψ_±3/2(t) of the corresponding electron and trion spin states. From the Hamiltonian (<ref>), one can see that the Schrodinger equation for them reads

  
  
    ıψ̇_± 3/2=ω_0ψ_±3/2+gc_±(t)ψ_±1/2,

  
    ıψ̇_± 1/2=gc_±^†(t)ψ_±3/2,


where the dot denotes the time derivative.  One can see that the excitations of the trions with J_z=±1/2 are independent of each other. The solution of these equations for t→∞ gives the trion state as a function of the initial electron state, amplitude, and polarization of the exciting light.

In the general case, an analytical solution of these equations is complicated. We solve them numerically for the different splittings of the cavity modes, Δ, and different trion resonance frequencies, ω_0. For each Δ and ω_0, we find the maximum (as a function of the pump amplitudes E_+ and E_-) occupancy of the trion state N_ tr^ max for the initial electron in-plane state. This gives the average probability of excitation of the two trion states from the corresponding electron states.










The result is shown in Fig. <ref>(a). Surprisingly, the trion can be excited with the probability close to one for the most range of the parameters Δ and ω_0. The cyan curves show the levels of N_ tr^ max=0.9. Several limiting cases can explain this result.

First, if the splitting between the modes is zero, then the quantum dot represents a pair of two-level systems. If the quantum dot frequency coincides with the cavity frequency, ω_0=ω_c, the populations of the trion states experience Rabi oscillations as functions of the pump amplitudes <cit.>:

    a_±3/2^† a_± 3/2=sin^2(g| E_±|/ϰ).

So the π pulse is realized, for example, by a linearly polarized excitation with E_±=πϰ/(2g). If the trion resonance is detuned from the cavity frequency, one can show that the maximum trion population is given by

    N_ tr^ max=1/2[1+(π(ω_0-ω_c)/2ϰ)].

This expression describes the trion population along the line Δ=0 in Fig. <ref>(a). For small trion detunings, it reaches one, and for large detunings, it tends to one-half. The trion population for the strong pump pulses experiences the Rabi oscillations from zero to one. Still, when the light amplitude inside the cavity decays according to Eq. (<ref>), the amplitude of the oscillations decreases, so a finite trion population N_ tr<1 is left in the QD after the escape of all photons from the cavity.

Another simple limit is realized for ω-ω_c=±Δ [diagonals of Fig. <ref>(a)]. The trion resonance coincides, in this case, with one of the two cavity modes. So, the pump polarization along this mode can realize the complete population inversion. Now, it becomes natural that in the whole region of |ω-ω_c|<|Δ|, the π pulses can be realized for some amplitudes and polarizations of the pump pulse.

The most important case is the trion resonance exactly between the cavity modes, ω_0=ω_c. Due to the detuning from both modes, one could expect only a limited maximum possible trion population. However, we find that if the pump amplitudes E_+ and E_- are shifted in phase by π/2 (E_+=| E_+|, E_-=-ı| E_-|), the solution of Eq. (<ref>) reads

  
   
    ψ_± 3/2(t) = ψ_± 1/2(0)sin(g Ẽ_±(t)Δ^2+ϰ^2),

   where

    Ẽ_±(t) =  E_±ϰ∓ E_∓Δ+[(± E_∓Δ- E_±ϰ)cos(Δ t).  
    .  +( E_±Δ+ E_∓ϰ)sin(Δ t)]e^-ϰ t.


From this, one can see the conditions for complete trion excitation (π pulses):

    sin^2[g E_±ϰ∓ E_∓Δ/Δ^2+ϰ^2]=  1.

This proves that N_ tr^ max=1 along the vertical line in the middle of Fig. <ref>(a), and this will be important for the spin-photon entanglement in the next section.

For example, at Δ=ϰ, Eq. (<ref>) is fulfilled for circularly polarized light with E_-=0, E_+=πϰ/g. This particular case was used for the numerical calculations in Fig. <ref>.

Finally, we note that the π pulses completely transfer populations of electron spin-up and spin-down states to the trion spin-up and -down states, respectively. This means that after excitation, the trion spin J_z equals the initial electron spin S_z. Also, for the electron in-plane spin orientation, the trion spin will also be completely polarized in the (xy) plane.




§ SPIN-PHOTON ENTANGLEMENT



Having described the trion excitation at the short time scales, we now analyze the slow trion radiative recombination and the spin-photon entanglement generation. Making use again of the weak coupling regime, we use the Schrödinger equation with the non-Hermitian Hamiltonian (<ref>) to describe the trion recombination <cit.>. We consider a trion state described by the amplitudes ψ_+3/2,0 and ψ_-3/2,0 (the subscript 0 indicates the absence of photons in the cavity mode). Then the Schrödinger equation gives us

  
    ıψ̇_±1/2,±=(ω_c-ıϰ)ψ_±1/2,±+Δψ_±1/2,∓+gψ_±3/2,0,

  
    ıψ̇_±1/2,∓=(ω_c-ıϰ)ψ_±1/2,∓+Δψ_±1/2,±,


where the first subscript ±1/2 in the wave functions refers to the electron spin, and the second subscript ± denotes a single σ^± photon in the cavity.

The solution of these equations reads

  
  
    ψ_±1/2,±=g(ω_0-ω_c+ıϰ)ψ_±3/2,0/(ω_0-ω_c+ıϰ)^2-Δ^2,

  
    ψ_±1/2,∓=gΔψ_±3/2,0/(ω_0-ω_c+ıϰ)^2-Δ^2,


where we took into account that the trion wave functions ψ_±3/2,0 have a phase factor of ^-ıω_0 t. In Appendix <ref>, we show that the same result can be obtained from the solution of the Lindblad equation for the density matrix of the system.

The numbers of photons in the cavity modes are given by 

    c_±^† c_±=|ψ_+1/2,±^2|+|ψ_-1/2,±^2|,

and this expression is shown by the black dashed line in Fig. <ref>(a). It allows us to calculate the trion (amplitude) decay rate as:

    γ=ϰ|ψ_±1/2,+^2|+|ψ_±1/2,-^2|/|ψ_± 3/2,0^2|,

because it equals the photon escape rate. Thus we obtain

    γ=1/2∑_H,Vg^2ϰ/(ω_0-ω_H,V)^2+ϰ^2,

which is generally of the order of g^2/ϰ or smaller. The slow trion decay N_ tr=exp(-2γ t) is shown by the red dashed line in Fig. <ref>(b).

Because of the assumption of equal decay rates of the two cavity photon modes, the trion decay rate does not depend on the trion spin. So, the wave function of an electron in the quantum dot and an emitted photon is also given by Eqs. (<ref>). It can be written (up to a phase factor) as

    Ψ=ψ_+3/2,0|↑,+̃⟩+ψ_-3/2,0|↓,-̃⟩,

where ↑ and ↓ denote the electron spin state, and the photon states are

    |±̃⟩=cosα|±⟩-ısinα^ıβ|∓⟩

with |±⟩ denoting σ^± photons and

  
    tanα=Δ/√((ω_0-ω_c)^2+ϰ^2),

  
    tanβ=ω_0-ω_c/ϰ.


Both angles α and β are in the range (-π/2,π/2) and depend on the trion resonance frequency and cavity mode splitting.









We stress that the electron-photon state is pure. Moreover, it has the same form as for the optically isotropic cavity <cit.> (with Δ=0) but with the rotated photon states (<ref>). So, we can easily calculate spin-photon concurrence
  
    𝒞=2√(J_x^2+J_y^2)ℱ_c

with
  
    ℱ_c=√(1-sin^2(2α)sin^2β).

  One can see that the spin-photon entanglement requires in-plane trion spin polarization as usual <cit.>. The cavity birefringence is accounted for by the factor ℱ_c≤1, shown in Fig. <ref>(b) as a function of the system parameters. It is related to the scalar product of the two modified photon states as ℱ_c^2=1-|<+̃|-̃>^2|, so the largest concurrence requires the minor overlap of the emitted photon states. The overlap vanishes in the absence of cavity mode splitting, Δ=0, and one arrives at 𝒞=2√(J_x^2+J_y^2) <cit.>.

  But strikingly, at ω_0=ω_c we obtain β=0, <+̃|-̃>=0, and ℱ_c=1 also. So, despite the cavity birefringence, the perfect spin-photon entanglement is achieved when the trion resonance is located exactly between the two cavity modes. Qualitatively, perfect spin-photon entanglement requires zero overlap between two possible emitted photon states |±̃⟩, which is possible for the equal probabilities of photon emission to H and V modes only. From Eq. (<ref>), one can see that this corresponds exactly to ω_0=ω_c, which was used already in Ref. coste_high-rate_2023.

We note that complete trion excitation and in-plane spin polarization are also realized in this case, as shown in the previous section. For this reason, we call the condition ω_0=ω_c a “sweet spot” for the entanglement generation in the birefringent microcavities. This is the main result of our work.
  


§ MULTIPHOTON ENTANGLEMENT



Spin-photon entanglement is useful by itself and indispensable for generating many-body entangled photon cluster states. The form of the electron-photon wave function (<ref>) shows that the protocol of Ref. lindner_proposal_2009 can be successfully applied to the birefringent micropillar cavities. But in the result, the circularly polarized σ^± photon states should be replaced with |±̃⟩.

For example, an electron spin rotation by π/2 between the two excitation pulses leads to the entangled state of an electron and two photons

    Ψ_3=|↑+̃+̃⟩+|↓-̃+̃⟩+|↓-̃-̃⟩-|↑+̃-̃⟩/2.

The straightforward calculation shows that the three-tangle <cit.> of this state is given by

    τ=ℱ_c^4.

It is shown in Fig. <ref>(c). It has more narrow regions with high entanglement than the concurrence in panel (b) but generally has the same form. In particular, it reaches one at the sweet spot ω_0=ω_c.

The three tangle does not reach zero for any system parameters. This suggests that the cluster-like states generated by the birefringent cavity are always genuine entangled. In particular, localizable entanglement <cit.> between any two emitted photons in the cluster state equals ℱ_c. In this sense, the entanglement length of the multiphoton state is infinite for any system parameters.

To quantify the degree of entanglement, we introduce the orthogonal photon states

    |±⟩=cos(θ/2)|±⟩-ısin(θ/2)|∓⟩,

where the angle θ is defined by

    tanθ=2ϰΔ/(ω_0-ω_c)^2+ϰ^2-Δ^2.

These states are the closest orthogonal states to |±̃⟩ (see Appendix <ref>), so they determine the optimal basis for the measurement of multiphoton entanglement.

We denote the cluster-like states with n photons produced by the birefringent cavity as Ψ_n, and the ideal cluster states in the basis of |±⟩ states as Ψ_n^(0). Then, the fidelity between these states is

    |<Ψ_n|Ψ_n^(0)>|^2=(1+ℱ_c/2)^n.

In particular, for n=1, the fidelity equals (1+ℱ_c)/2 in agreement with the general relations between concurrence and fidelity <cit.>. Eq. (<ref>) shows that the fidelity exponentially decreases with the number of photons, except for the sweet spot, where the maximally entangled multiphoton states are generated.





§ CONCLUSION



We have developed a microscopic theory of the optical resonant excitation of a charged quantum dot in a birefringent micropillar cavity and its radiative recombination. The results show that a sweet spot exists for the system when the frequency of the trion resonance is placed exactly between the two cavity modes. At this point, even for the large cavity modes splitting, a complete trion excitation (π pulse) and in-plane spin polarization are possible, as well as the complete spin-photon entanglement after trion recombination. It shows the way to increase the fidelity of the present spin-photon interfaces.


We thank A. A. Toropov for fruitful discussions and the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS.” The derivation of the analytical expressions for the spin-photon entanglement by D.S.S was supported by the Russian Science Foundation Grant No. 21-72-10035.






§ TRANSMISSION SPECTRA


Here, we study the transmission of the continuous wave coherent light with frequency ω through the birefringent spin-photon interface. We modify the input-output relations <cit.> for the case of the asymmetric cavity using the Hamiltonian (<ref>)

    ĉ̇_+ = -iω_c ĉ_+-iΔĉ_–ϰĉ_+-ig â_+1/2^†â_3/2+√(ϰ)ĉ^ in_+
    ĉ̇_- = -iω_c ĉ_–iΔĉ_+-ϰĉ_-+√(ϰ)ĉ^ in_-
    â̇_+3/2 = -iω_0 â_+3/2-ig ĉ_+ â_+1/2
    â̇_+1/2 = -ig ĉ^†_+â_+3/2
    ĉ_+^ out = √(ϰ)ĉ_+
    ĉ_-^ out = √(ϰ)ĉ_-
        .

Here ĉ^ in,out_± are input and output σ^± light field operators, which are proportional to ^-ıω t, and we assume the electron and trion spin-down states to be not populated. These equations can be solved in the first order in the amplitude of the incident light by replacing a_+1/2^† a_+1/2 and a_+3/2^† a_+1/2 with one and zero, respectively <cit.>.

Without the cavity mode splitting, Δ = 0, we arrive at the previously established results <cit.>: Transmission of σ^- polarized light is given by

    t_0 = ĉ_-^ out/ĉ_-^ in = i ϰ/ω-ω_c+iϰ,

which corresponds to the transmission when the optical selection rules forbid photon interaction with the QD, while for the σ^+ light polarization, we have 

    t_1 = ĉ_+^ out/ĉ_+^ in  = iϰ/ω-ω_c+iϰ-g^2/ω-ω_0,

relevant for the case when the QD can absorb the circularly polarized photons.

In the birefringent cavite case, Δ≠ 0, from the solution of Eqs. (<ref>) we obtain 

    t_++ = t_1/1+Δ^2t_1t_0/ϰ^2,   t_– = t_0/1+Δ^2t_1t_0/ϰ^2, 
     t_-+ = -ıΔ/ϰt_++t_0,     t_+- = -iΔ/ϰt_–t_1,

where t_αβ = ĉ_β^ out/ĉ_α^ in for α ,β = ±. As one can see, the mode splitting results in the light polarization conversion. We note that these expressions are valid in both weak and strong coupling regimes.



The intensity transmission coefficient for unpolarized light is
  
    T=1/2(|t_++^2|+|t_+-^2|+|t_-+^2|+|t_–^2|).

  It does not depend on the electron spin. The same expression describes the transmission of polarized light for unpolarized electron spin. Its frequency dependence is shown in Fig. <ref>. One can see that in the fast cavity regime ϰ≫ g, the transmission has two Lorentzians peaks at the cavity eigenfrequencies ω_c ±Δ and an asymmetric Fano-like peak at the trion resonance frequency ω_0. 






§ TRION RECOMBINATION AFTER MASTER EQUATION




Let us describe trion recombination studied in Sec. <ref> within a different approach similar to Ref. Leppenen2022. The Lindblad equation for the density matrix of the system ρ accounting for the photon escape from the cavity can be written in the form

    ρt = -i( H_ nhρ-ρ H_ nh^†) -∑_±C_±ρ C_±^†,

 with the non-Hermitian Hamiltonian and jump operators introduced in Sec. <ref>.
 
In the fast cavity regime, the density matrix elements between trion and electron with photon states are determined by the trion density matrix only:


    ρ̇_⇑, ↑+ = -[i(ω_0-ω_c)+ϰ] ρ_⇑, ↑ ++
    +iΔρ_⇑,↑-+igρ_⇑,⇑,


    ρ̇_⇑, ↑- = -[i(ω_0-ω_c)+ϰ]ρ_⇑, ↑ -+iΔρ_⇑,↑+.


Equations for ρ_⇑, ↓- and ρ_⇑, ↓+ can be obtained by replacing in these equations in the second subscript ↑, ⇑, and ± with ↓, ⇓, and ∓, respectively. Similarly, the equations for the density matrix elements ρ_⇓, ↑+, ρ_⇓, ↑-, ρ_⇓, ↓-, and ρ_⇓, ↓+ can be obtained from the equations for ρ_⇑, ↑+, ρ_⇑, ↑-, ρ_⇑, ↓-, and ρ_⇑, ↓+ simply by replacing ⇑ with ⇓ in the first subscript. The rest of the off-diagonal density matrix elements with the trion states follow from the Hermitian conjugation.

We solve these equations in the steady state and substitute the solution to the equations for the 4× 4 density submatrix ρ_ e-ph in the basis of states |↑+⟩, |↑-⟩, |↓+⟩, |↓-⟩. This gives
 
    ρ̇_ e-ph = -2ϰρ_ e-ph-i[ H_Δ,ρ_ e-ph]+ρ_ rec,

  where H_Δ = Δ (ĉ_-^†ĉ_++ĉ_+^†ĉ_-) is a part of the Hamiltonian (<ref>) proportional to the cavity modes splitting and the last term ρ_ rec depends on the trion density matrix

    ρ_ rec = 2ϰ g^2/Δ^2+(ϰ+ı(ω_0-ω_c))^2×
    ×[ Aρ_⇑,⇑   ıΔ/2ϰρ_⇑,⇑   ıΔ/2ϰρ_⇑,⇓    Aρ_⇑,⇓
     -ıΔ/2ϰ Bρ_⇑,⇑    0     0     -ıΔ/2ϰ Bρ_⇑,⇓
      -ıΔ/2ϰ Bρ_⇓,⇑    0     0     -ıΔ/2ϰ Bρ_⇓,⇓
     Aρ_⇓,⇑   ıΔ/2ϰρ_⇓,⇑   ıΔ/2ϰρ_⇓,⇓    Aρ_⇓,⇓],

with 

    A = Δ^2+(ϰ+ı(ω_0-ω_c))^2/Δ^2+ϰ^2+(ω_0-ω_c)^2 B = 
     = Δ^2+ϰ^2+(ω_0-ω_c)^2/Δ^2+(ϰ-ı(ω_0-ω_c))^2.

Finally, we solve Eq. (<ref>) in the steady state and normalize the result, obtaining

    ρ_ e-ph = (ω _0-ω _c)^2+ϰ ^2/(ω _0-ω _c)^2+Δ ^2+ϰ ^2×
    ×[ρ_⇑,⇑    Gρ_⇑,⇑    Gρ_⇑,⇓   ρ_⇑,⇓
     G^* ρ_⇑,⇑    G^2 ρ_⇑,⇑    G^2ρ_⇑,⇓    G^* ρ_⇑,⇓
     G^* ρ_⇓,⇑    G^2 ρ_⇓,⇑    G^2 ρ_⇓,⇓    G^* ρ_⇓,⇓
    ρ_⇓,⇑    Gρ_⇓,⇑    Gρ_⇓,⇓   ρ_⇓,⇓],

where G = Δ/(ω_0-ω_c-ıϰ). One can readily check that this is the density matrix of the pure state given in Eq. (<ref>).



§ CLOSEST ORTHOGONAL PHOTON STATES



The electron-photon state (<ref>) has the same form as for the optically isotropic micropillar cavity except for the replacement of circular photon states |±⟩ with |±̃⟩, see Eq. (<ref>). These states are generally not orthogonal. The Stockes parameters <cit.> for them are

    ξ_1,± = ∓cosβsin 2α,  ξ_2,±  = ±cos 2α, 
    ξ_3,± =  sinβsin 2α.

The latter parameter is the same for the two states, which evidences their overlap. Generally, by the appropriate rotation of the polarization basis of the Poincaré sphere, one can obtain the Stokes parameters of the form

    ξ_1',± = 0,    ξ_2',± = ±ξ,     ξ_3,± = ξ_3

Where the parameter ξ = √(ξ_1,±^2+ξ_2,±^2) and the polarization basis is rotated about the axis 3 by the angle ϑ which is defined by

    cosϑ = ξ_2,+/√(ξ_2,+^2+ξ_1,+^2) = 1/√(1+cos^2 βtan^2 (2α)).

The closest orthogonal states |±⟩ are defined by the conditions ⟨±||∓⟩ = 0 and the overlap ⟨±||±̃⟩ is maximal. Therefore they should satisfy ξ_1',±=0, ξ_2',±=±1, and ξ_3',±=0. Such states are given by Eq. (<ref>).




