Generation of a two photon entangled states from a quantum dot embedded in a microcavity

# Generation of a two photon entangled states from a quantum dot embedded in a microcavity

## Abstract

We propose methods for generating entangled two photon state. Where, a single quantum dot is embedded in a photonic microcavity having two orthogonal modes. The quantum dot is off-resonantly coupled to the cavity through phonon mediated coupling. The emitted photons are entangled either in polarization degrees of freedom or in numbers. We also discuss effect of exciton-phonon coupling on entanglement.

###### pacs:
03.65.Ud, 03.67.Mn, 42.50.Dv

## I Introduction

A source of NOON states has various applications in quantum metrology(1); (2), quantum lithography(3); (4), quantum imaging(5) and quantum information processing(6). A photonic NOON state is an entangled Fock state of two orthogonal modes defined by,

 |ψ⟩NOON=1√2(|Nx,0y⟩+|0x,Ny⟩) (1)

where the subscripts and denote the two orthogonal mode. So far, most of the study for generating the NOON states deal with the parametric down converters(7); (8), photoselection(9); (10) and linear optical process(11). The rate of generation of entangled photons from such sources is very low and probabilistic. However, the quantum information processing(12) requires the deterministic on demand scalable sources of entangled photons. In last decade, quantum dots(QDs) have emerged as potential candidate for realization of scalable solid state source of quantum light. Further, emission rate from QDs can be increased by embedding them inside the photonic microcavities due to the purcell effect (13). In QDs, biexciton state decays to the ground state by generating a photon pair the emitted photon pair has either x-polarized or y-polarized depending on the biexciton decays via two bright exciton states having angular momenta . The emitted photon pair is entangled in frequency and polarization degrees of freedom. However, fine structure splitting(FSS) between two excitons in semiconductor QDs destroy the maximum achievable entanglement. This difficulty can be overcome by sevral ways, such as, by applying the external electric and magnetic field individually or simultaneously to the QDs (14); (15); (16); (17), by applying the proper combination of unaxial stress or strain (18); (19) and reduction of FSS by thermal annealing (20).

Phonon induced dephasings in QDs decay the coherence of excitons which is another main obstacle to achieve the maximum entanglement(21). Acoustic electron-phonon interaction in cavity QD systems have introduced the phonon mediated coupling between the exciton and cavity mode(23); (24), which contribute to the spectral broadening(25), off-resonant cavity mode feeding? and pump dependent broadening in Mollow triplets(27). The influence of electron - phonon interaction on QD-cavity coupling depends on the detuning between QD and cavity and phonon resevoir properties.

The effect of exciton phonon interaction in the systems like single quantum dot coupled to a high quality cavity(28); (29); (30) have been studied experimentally and theoretically in details. In this paper, we propose a scheme for generating two photon NOON state using single QD embedded inside a photonic microcavity and discuss the effect of electron phonon coupling on the generated state. We also investigate the role of exciton-phonon coupling on polarization entangled state.

## Ii Theory

We consider a single quantum dot embedded inside a photonic microcavity. The quantum dot consists a ground state , two exciton energy levels and with frequencies and respectively, and a biexciton energy level with frequency .

The fine structure splitting between the exciton energy levels and is . The cavity has two orthogonal modes of frequencies and having polarization along and directions respectively. We consider both cavity modes have same energy (), the energy of these modes can be tuned experimentally by electron-beam lithography techniques(31). The biexciton energy level is red detuned from both the cavity modes and , the exciton energy level and the exciton energy level are blue detuned from the x-polarized and y-polarized cavity modes respectively. One mode of cavity having polarization along x direction is off resonantly coupled to the to transition with coupling constant and the to transition with coupling constant respectively. Another mode of cavity having polarization along y direction is off resonantly coupled to the to transition with coupling constant and the to transition with coupling constant respectively. As QD have large biexciton binding energy so resonant coupling of biexciton and exciton are not possible simultaneously with same cavity mode but it provides opportunity of off-resonant coupling of these energy levels from same cavity mode simultaneously. Therefore, we consider the frequencies of cavity modes half of the biexciton frequency () to satisfy the two photon resonance condition and while , , and (22). The system is prepared in such way that initially the QD is in the excited state and cavity is in vaccum. The hamiltonian of the system in rotating frame is given by,

 H =ℏ2(Δx1+Δx2+Δy1+Δy2)|u⟩⟨u|+ℏ4(3Δx1−Δx2+Δy1+Δy2)|x⟩⟨x|+ℏ4(3Δy1−Δy2+Δx1+Δx2)|y⟩⟨y|−ℏ4(Δx1+Δx2 (2) −Δy1−Δy2)ax†caxc+ℏ4(Δx1+Δx2−Δy1−Δy2)ay†cayc+ℏgx1(|x⟩⟨g|axc+ax†c|g⟩⟨x|)+ℏgx2(|u⟩⟨x|axc+ax†c|x⟩⟨u|) +ℏgy1(|y⟩⟨g|ayc+ay†c|g⟩⟨y|)+ℏgy2(|u⟩⟨y|ayc+ay†c|y⟩⟨u|)+∑kℏωkb†kbk+∑kℏλxk(b†k+bk) +∑kℏλyk(b†k+bk)+∑kℏλuk(b†k+bk)

Where is the detuning between exciton energy level and cavity mode , is the detuning between the biexciton energy level and cavity mode , is the detuning between exciton energy level and cavity mode , and is the detuning between the biexciton energy level and cavity mode . ( ) and ( ) are annihilation(creation) operators for x and y polarized cavity field, and ( ) are phonon frequency and phonon annihilation(creation) operators respectively. , and are the coupling constants of biexciton, x-exciton and y-exciton to phonon modes. In our calculations, for an ideal QD, we used (32); (33). Next, we perform the polaron transformation(34); (34)to separate the electron-phonon coupling and QD-cavity terms using with . Here, we arrange the terms following the reference(27). The transformed Hamiltonian is given by,

 ˜Hs=ℏ2(Δx1+Δx2+Δy1+Δy2−2Δp)|u⟩⟨u|+ℏ4(3Δx1− (4) Δx2+Δy1+Δy2−Δp)|x⟩⟨x|+ℏ4(3Δy1−Δy2+Δx1+ Δx2−Δp)|y⟩⟨y|−ℏ4(Δx1+Δx2−Δy1−Δy2)ax†caxc+ ℏ4(Δx1+Δx2−Δy1−Δy2)ay†cayc+⟨B⟩xg ˜Hph=∑kℏωkb†kbk,˜Hs−ph=xgηg+xuηg (5)

with,

 xg=ℏgx1⟨B⟩(|x⟩⟨g|axc+ax†c|g⟩⟨x|)+ℏgx2⟨B⟩(|u⟩⟨x|axc (6) +ax†c|x⟩⟨u|)+ℏgy1⟨B⟩(|y⟩⟨g|ayc+ay†c|g⟩⟨y|)+ ℏgy2⟨B⟩(|u⟩⟨y|ayc+ay†c|y⟩⟨u|) xu=iℏgx1⟨B⟩(|x⟩⟨g|axc−ax†c|g⟩⟨x|)+iℏgx2⟨B⟩(|u⟩⟨x|axc (8) −ax†c|x⟩⟨u|)+iℏgy1⟨B⟩(|y⟩⟨g|ayc−ay†c|g⟩⟨y|)+ iℏgy2⟨B⟩(|u⟩⟨y|ayc−ay†c|y⟩⟨u|) Missing dimension or its units for \hskip (10) B±=e±∑kλkωk(b†k−bk),Δp=∑kλ2kωk=∫∞0dωdph(ω)ω (12) ⟨B⟩=⟨B+⟩=⟨B−⟩=e−12∫∞0dωdph(ω)ω2coth(ℏω/2KBT) (13)

where, is the polaron shift, which induces due to the polaron transformation and we assumed that it is absorbed in the detunings of biexciton and excitons to the cavity modes. and are the phonon fluctuation operators(25); (33). is the thermal average of coherent phonon displacement operator(36). Further, the simulations has been performed using time local master equation following the reference(27); (33), where terms in are considered up to second order (Born approximation). The time convolutionless master equation(27) is defined by,

 ∂ρ∂t =−iℏ[˜Hs,ρ]−κx2L[axc]−κy2L[ayc]−γ12L[|g⟩⟨x|]−γ12L[|g⟩⟨y|]−γ22L[|x⟩⟨u|]−γ22L[|y⟩⟨u|]−γdL[|u⟩⟨u|] (14) −γd2L[|x⟩⟨x|]−γd2L[|y⟩⟨y|]−1ℏ2∫∞0dτ∑m=g,u(Gm(τ)[xm,e−i˜Hsτxmei˜Hsτρ(t)]+H.c.)

where are the Lindblad operators. and are the spontaneous decay rate of excitons and biexciton respectively, and are the decay rate from the x and y polarized cavity modes respectively, is the pure dephasing rate. , where, and are the polaron Greens functions(27) with (27). , is the spectral function (measures the exciton-longitudinal acoustic phonon coupling)(27), where and are coupling constant and phonon cutoff frequency(37) respectively. This electron-phonon coupling is responsible for destroying the coherence of excitons in self assembled QDs(38). The master equation is simulated using the quantum optics toolbox in matlab(39).

## Iii Population dynamics and cavity emission spectrum

Further, we simulate the master and display the population of the states of coupled system in Fig. 2. When the x- polarized and y- polarized cavity modes satisfy the two photon resonance condition at and respectively under strong coupling regime i.e. then the QD decays via three paths. In first decay path, the QD decays from to via ( ) and generate two photons spontaneously. The spontaneous emission effect the cavity assisted generation so we consider very small spontaneous emission rate in our calculations. In second decay path, the single photon transition follow the path from to , to , to and to , to , to through the x and y polarized cavity modes respectively. In third decay path, the two photon transition follow the path to via and to via through x and y polarized cavity modes respectively.

In Fig. 2. represent the population in biexciton state, is the population in after leakage of one photon from , is the population in after leakage of one photon from . We also show the probabilities of single photon emissions as from the state , from the state , from the state , from the state and probabilities of resonant two photon emissions as from the state , from the state . In Fig. 2(a). the results are shown with spontaneous emission, cavity decay rate and pure dephasing rate at while in Fig. 2(b,c,d). the results are shown with electron-phonon coupling at , and , respectively. In Fig. 2(a), we found that the maximum population in the state and is marginally less than 0.1 and becomes zero in long time. The single photon emission probabilities , , and from , , and are around 0.073, 0.072, 0.461 , and 0.473 respectively in long time.

The two photon emission probabilities and are 0.395 and 0.407 respectively. We found that and , which is due to the leakage of photon from into and into as well as single photon transition from to and into respectively. In Fig. 2(b), when exciton-phonon coupling is present at , the maximum population in and are still less than 0.1 and becomes zero in long time. The single photon emission probabilities , , and become around 0.088, 0.088, 0.476 and 0.488 respectively in long time, which are marginally more than in the absence of exciton-phonon coupling. The two photon emission probabilities become around 0.379 and 0.391, which are marginally less than in the absence of electron-phonon coupling. In Fig. 2(c,d), the single photon emission probabilities , , and increases further and resonant two photon emission probabilities , decreases.

Further we calculate the spectrum of emitted photons from the x and y polarized cavity modes which is given by,

 S(ω)=∫∞0dt∫∞0dτ⟨a†x,y(t)ax,y(t+τ)⟩eiωτ (15)

The two time correlation is calculated by using the quantum regression theorem. In Fig. 3(a,b) the central big peaks (blue dashed lines and red solid lines) are corresponding to the two photon resonant emission at cavity frequencies and respectively. The small peaks which are situated at right to the central peaks are corresponding to the single photon emission at and small peaks which are situated at left to the central peaks are corresponding to the single photon emission at . In Fig. 3(a). electron-phonon coupling are not considered while in Fig. 3(b,c,d). electron-phonon coupling are considered. In Fig. 3(b,c,d) it is clear that on increasing the temperature, broadening of peaks corresponding to single photon emission is increasing.

## Iv Influence of Electron-Phonon Coupling on the Generated Two Photon Noon State

In our case, as we discussed in section II., initially the QD is prepared in the biexciton state. The QD decays radiatively via two paths ( and ) and relaxes in ground state after emitting the maximally entangled two photon NOON state,

 |ψ⟩NOON=1√2(|2x,0y⟩+|0x,2y⟩) (16)

The degree of entanglement is measured by the concurrence(40) which is given by,

 C=|∫∞0dτG2x,y(τ)||∫∞0dτG2x(τ)|+|∫∞0dτG2y(τ)| (17)

Where, , and are the second order two time correlation functions. , and are calculated with the help of quantum regression theorem.

The full entanglement can be achieved for in case of no dephasing and zero fine structure splitting and no entanglement for , but it is not possible to achieve the maximum entanglement because of the fine structure splitting between two exciton levels and electron phonon interaction in QDs. However, the fine structure splitting can be reduced up to large extent by applying various technique as already discussed in section I. Therefore to see the effect of electron phonon interaction on the entanglement, in Fig. 4, we show the concurrence on dephasing rate at fixed fine structure splitting (). We get the maximum concurrence when and decreases on increasing the . It has been also found that concurrence is greater in the absence of phonon assisted process (red solid line) than in the presence of phonon assisted process (blue and green solid line). The maximum concurrence goes around at (red line), around 0.83 (blue line) at and 0.79 (green line) at T = 10K respectively.

## V Influence of electron-phonon coupling on with in generation of entangled photons

In this case, the frequencies of photons emitted from x-polarized and y-polarized cavity modes should match in same generations(41). Here, we consider the detunings between biexciton and cavity modes equal to the binding energy and FSS is considered equal to the QD and cavity coupling. The to transition is detuned by, and to transition is detuned by, . In Fig.5, we show the concurrence on FSS(). When spontaneous emission, pure dephasing and phonon mediated coupling is not considered then the maximum achievable concurrence is at and it decays on increasing (shown by red solid line). But, in the presence of cavity decay, spontaneous emission and pure dephasing the maximum achievable concurrence is above than (blue solid line). When, the phonon mediated coupling is also considered with cavity decay, spontaneous emission and pure dephasing, the maximum concurrence at still remains above than but on increasing it decays rapidly (green solid line at and pink dashed line at ).

## Vi Conclusions

We have presented a method in which a single QD is coupled to the photonic microcavity through off-resonant phonon mediated process. Where, the entanglement for the generated two photon NOON state and with in generation of two photons could be very high.

## Vii Acknowledgements

This work was supported by DST grant.

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