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p s scurrent topics in solid state physics

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caphys. stat. sol. (c) 6, No. 1, 276–279 (2009) / DOI 10.1002/pssc.200879802

Entangled-photon generation froma quantum dot in cavity QED

Hiroshi Ajiki*,1 and Hajime Ishihara2

1 Department of Materials Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan2 Department of Physics and Electronics, Osaka Prefecture University, 1-1 Gakuen-cho Naka-ku, Sakai, Osaka 599-8531, Japan

Received 12 June 2008, revised 2 July 2008, accepted 5 July 2008Published online 30 September 2008

PACS 03.65.Ud, 03.67.Mn, 78.67.Hc

∗ Corresponding author: e-mail ajiki@mp.es.osaka-u.ac.jp, Phone: +81-6-6850-6454, Fax: +81-6-6850-6454

We theoretically study polarization-entangled photongeneration from a single quantum dot in a microcavity.Entangled-photon pairs with singlet or triplet Bell statesare generated in the resonant-hyperparametric scatteringvia dressed states in the cavity QED.

Although co-polarized non-entangled photons are alsogenerated, the generation is dramatically suppressed inthe strong-coupling limit owing to the photon blockadeeffect. Finite binding energy of biexciton is also impor-tant for the generation of photon pairs with high degreeof entanglement.

1 Introduction Nonlocal entangled states are the im-portant resources for implementing quantum informationprocessing such as quantum key distribution, quantum tele-portation, and quantum logic gates [1]. Among the severaltypes of entangled states, polarization-entangled photonsare expected to be good carriers for quantum communica-tion because they do not interact with each other.

In order to generate polarization-entangled photons, itis necessary to have two-photon cascade-emission pro-cesses through degenerate intermediate states with differ-ent polarizations. An early technique for entangled-photongeneration has been developed by two-photon cascade de-cay of an atomic excitation [2–6]. An analogous cascade-decay process has been pointed out by Benson et al. [7] ina semiconductor, i.e., a cascade decay of a biexciton viadegenerate excitons. For the first time, entangled-photongeneration via biexcitons has been observed in the reso-nant hyper-parametric scattering (RHPS) [8,9]. The gener-ation method has been predicted by Savasta et al. [10]. Adrastic enhancement in the generation due to a microcav-ity has been predicted for a quantum well in terms of cavityquantum electrodynamics (cavity QED) [11,12]. Optimumenhancement occurs when the excited dressed states con-tain the biexciton and cavity photon components at compa-rable levels. Recently, entangled-photon generation fromthe biexciton in a single quantum dot (QD) has been re-

ported [13–15]. This system provides event-ready entan-gled photon generation.

In this paper, we provide theoretical study on cross-polarized entangled-photon generation by means of RHPSin a QD embedded in a microcavity. In the RHPS of thecavity-QD system, entangled-photon pairs are generatedfrom cascade decay processes via dressed states in the cav-ity QED. We have reported entangled-photon generationin this mechanism [16]. A rate of entangled-photon gen-eration has been evaluated by a visibility for coincidencecount as a function of relative angle between two polariza-tion filters. However, widely known measure of the degreeof entanglement is a concurrence. Here, we shall compre-hensively present the concurrence of the entangled-photonpairs. We shall also derive the entangled-photon states(singlet or triplet Bell states), which has not been reportedin our previous work.

2 Entangled photons from a QD in microcavityWe consider the situation that right- (ER) and left-polarizedlaser fields (EL), which have frequencies ΩR and ΩL, re-spectively, are applied to the cavity embedding a singleQD. The QD is modeled by a four-level system consist-ing of a ground state |G〉, degenerate right- (left-)polarizedexciton |XR〉 (|XL〉), and a biexciton |B〉 with bindingenergy hΔB (see Fig. 1). The frequency ω0 of the lowest

© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

cavity mode is tuned to that of the excitons. Then, the fre-quency of biexciton is given by ωB = 2ω0−ΔB . The cou-pling energy between the exciton (biexciton) and the cav-ity mode is denoted by hgX (hgB). In the rotating-waveapproximation, the situation is well described by the fol-lowing Hamiltonian:

HV =hω0

∑ξ=R,L

(|Xξ〉〈Xξ| + a†

ξaξ

)+ hωB |B〉〈B|

+[ihgX(a†

R|G〉〈XR| + a†L|G〉〈XL|) + H.c.

]

+[ihgB(a†

R|XL〉〈B| + a†L|XR〉〈B|) + H.c.

]

+(

ih√

Γ∑

ξ=R,L

Eξa†ξe

iΩξt + H.c.)

, (1)

where aR (aL) is an annihilation operator of the cavitymode with right (left) polarization and Γ = ω0/2Q with acavity quality factor Q is a coupling constant of the cavitymode and continuous modes outside the cavity.

Although some exciton states in a QD lie around thecavity-mode frequency, we take into account only two ex-citon states of |XR〉 and |XL〉 in this model. This approx-imation is valid when the level separation of excitons aregreater than the vacuum Rabi splitting. Similarly, level sep-aration of two-exciton states should be greater than theRabi splitting in order to be valid for the one-biexcitonmodel. These situations are realized, for example, in GaAsQDs. It has been observed that the vacuum Rabi splittingis 0.4 meV for a GaAs QD with a lateral dimension of22 nm [17], while the exciton-level separation is a fewmeV [18]. Usually, the cross-polarized excitons in a QDexhibit a fine structure splitting due to their anisotropicconfinement. In the case of the GaAs QD, the fine struc-ture splitting ranges from 20 to 50 μeV [18]. Althoughthe degeneracy of the cross-polarized excitations is nec-essary for the entangled-photon generation, the problem isovercome by applying a magnetic field [13,14] or by us-ing spectral filtering [15]. The Bohr radius of the excitonin GaAs is approximately 20 nm, which is comparable tothe lateral confinement size. In such a strong-confinementregime, the level separation of the two-exciton states is ex-

Entangled

Non-entangledphoton pairs

photon pair

Figure 1 Schematic illustration of a model for entangled-photongeneration.

pected to be greater than the vacuum Rabi splitting, and thepresent model would be valid for such a QD.

The eigenstates of Hamiltonian (1) without the inputfields Eξ represent the dressed states, which are classifiedinto n-excitation (ne) manifold, where n is the number ofenergy quanta inside the cavity. The 1e manifold consistsof two dressed states (1e states) with frequencies ω1e

± =ω0±gX ; each state has two-fold degeneracy with respect tothe polarization. Each 1e state with right polarization, forexample, is represented as |R±〉 = (1/

√2)(|1R〉∓i|XR〉),

where |1R〉 denotes the Fock state with one right-polarizedphoton in the cavity. The 2e manifold is further classifiedinto cross-polarized and co-polarized 2e manifolds. Thecross-polarized 2e manifold consists of four dressed states(cross-polarized 2e states), one of which is given by

|RL; s〉 = (1/√

2)(|XR; 1L〉 − |XL; 1R〉), (2)

with a frequency ω2es = 2ω0. Other three cross-polarized

2e states with frequencies ω2eti (i=1,2,3) are represented as

|RL; ti〉 = ci1|1R1L〉

+(|XR; 1L〉 + |XL; 1R〉

)+ ci

2 + ci3|B〉, (3)

where cij denote expansion coefficients for |RL; ti〉. The

co-polarized 2e manifold consists of two dressed states(co-polarized 2e states) with frequencies ω2e

± ; each statehas two-fold degeneracy. The co-polarized 2e states withright polarization, for example, are given by

|RR±〉 = (1/√

2)(|2R〉 ∓ i|XR; 1R〉). (4)

An interaction Hamiltonian Hvacint between the cavity

photons and the vacuum field outside the cavity is writtenas follows: Hvac

int = ih√

Γ∫ ∞−∞ dω

∑ξ={R,L} b†ξ(ω)aξ +

H.c., where bR(ω) [bL(ω)] is an annihilation operator ofthe right (left)-polarized photons outside the cavity. We ap-proximately recast Hvac

int in terms of the dressed states up tothe cross-polarized 2e states, where frequency in bR(L)(ω)and b†R(L)(ω) are chosen as the transition frequencies be-tween dressed states. This approximation is valid in thestrong-coupling regime, where the dressed states are wellseparated from each other. Entangled-photon pairs are ob-tained by operating (Hvac

int )2 to |Ω〉|RL; s〉 or |Ω〉|RL; ti〉,where |Ω〉 represents a vacuum state outside the cavity.Then, generated photon states |Ψ〉 are written as follows:

|Ψ〉 ∝∑i=±

si,tj(|Lω1ei 〉|Rωi:tj〉 + |Rω1e

i 〉|Lωi:tj〉),

(emission from |RL; tj〉, j = 1, 2, 3), (5)

|Ψ〉 ∝∑i=±

si,s(|Lω1ei 〉|Rωi:s〉 − |Rω1e

i 〉|Lωi:s〉),

(emission from |RL; s〉), (6)

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Figure 2 (Color online) Concurrence of photon pairs as a func-tion of gX = gB and ΔΩR = ΩR − ω0, where ΩL = ω1e

− isfixed. Dotted lines indicate the resonant conditions for excitingthe cross-polarized 2e states. The binding energy of the biexcitonis ΔB = 0.5 meV.

with ωi:tj = ω2etj − ω1e

i , ωi:s = ω2es − ω1e

i , |R(L)ω〉 =b†R(L)(ω)|Ω〉, and si,tj , si,s being constants. The index i inthe summation in Eqs. (5) and (6) indicates the intermedi-ate 1e states, and thus, the photon pairs are entangled as asuperposition of four two-photon states.

Since the exciton and biexciton states are strongly cou-pled with the single-cavity mode, most of the photons arescattered in the same direction similar to the input fields.Therefore, the generated entangled photons should be sep-arated from the scattered photons in the linear responseregime. The entangled photons can be extracted by usingspectral filtering. Let us assume that either of the input-field frequencies ΩR or ΩL is tuned to the lower 1e state.Then, the photon pairs generated via the upper 1e statewould have the frequencies that are different from ΩR andΩL (see the level diagram in Fig. 2). After the filtering, theentangled-photon pairs are represented by Eq. (5) or (6)without the summation of i = −. Similarly, spectral filter-ing of the entangled photons is also possible when the 2estates are excited via the upper 1e state. In both cases, thesign of superposition is positive (triplet Bell states) for theentangled photons via |RL; ti〉 [see Eq. (5)] and negative(a singlet Bell state) for those via |RLs〉 [see Eq. (6)].

3 Concurrence It should be noted that non-entangledco-polarized photons are also generated in this configura-tion because the classical laser fields contain more than onephoton. Then, a degree of entanglement of the photon pairsas a mixed state would be suppressed by the co-polarizednon-entangled photon pairs. In order to investigate the de-gree of entanglement, we evaluate a concurrence C. Theconcurrence monotonically increases from 0 to 1 withdegree of entanglement. A photon pair is maximally en-tangled for C = 1. The concurrence is calculated from thereduced density matrix ρ of two photons after the spectral

Figure 3 Concurrence of photon pairs as a function of gX =gB along the dotted lines in Fig. 2. The binding energy of thebiexciton is ΔB = 0.5 meV.

filtering, where we extract the photons from the cascadeemission via the upper 1e states. In this model, the densitymatrix has the following form:

ρ =

⎛⎜⎜⎜⎝

ρa ρb 0 0ρ∗b ρc 0 00 0 ρd 00 0 0 ρe

⎞⎟⎟⎟⎠ , (7)

where the bases are in the following order |Lω1e+ 〉|Rω2e

+ 〉,|Rω1e

+ 〉|Lω2e+ 〉, |Lω1e

+ 〉|Lω2e+ 〉, and |Rω1e

+ 〉|Rω2e+ 〉, with

ω2e+ = ΩR +ΩL −ω1e

+ . Then, the concurrence C(ρ) is ob-tained from C(ρ) = max

{(0,

√λ1−

√λ2−

√λ3−

√λ4

};

here, λis are the eigenvalues, in decreasing order, of theHermitian matrix ρ(σy ⊗ σy)ρ∗(σy ⊗ σy) with σy beingthe Pauli matrix.

Figure 2 shows the concurrence of the photon pairsgenerated from the cascade emission from the cross-polarized 2e states via the upper 1e states, where the 2estates are excited via the lower 1e state. The concurrence iscalculated as a function of gX = gB and ΔΩR = ΩR−ω0

under the condition ΩL = ω1e− . A phenomenological

damping energy 10 μeV for the excitons, biexciton, andcavity photons are taken into account by using a masterequation. The binding energy of the biexciton is chosenas ΔB = 0.5 meV. The master equation is solved in theweak-field limit of Eξ under the stationary-state condi-tion. The dotted lines in the figure indicate the resonantexcitation conditions for the cross-polarized 2e states.

Figure 3 shows the concurrence as a function of gX =gB under the resonant excitation conditions of the cross-polarized 2e states. The concurrences for 2e states |RL; t2〉

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278 H. Ajiki and H. Ishihara: Entangled-photon generation from a quantum dot in cavity QED

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Figure 4 Concurrence of photon pairs as a function of ΔB underthe resonant excitation conditions of the cross-polarized 2e stateswith gX = gB = 0.5 meV.

and |RL; s〉 approach 1 with an increase in gX = gB . Thisbehavior originates from the significant reduction in theexcitation of the co-polarized 2e states. The reduction isdue to the photon blockade effect, where the excitation ofa dressed state by a first photon blocks the transmissionof a second photon that has the same polarization and fre-quency [19,20]. This effect arises from the anharmonicityof the dressed states; the Rabi splitting of the co-polarized2e manifold is 2

√2gX , while that of the 1e manifold (vac-

uum Rabi splitting) is 2gX . As a result, the excitation ofthe co-polarized 2e states becomes non-resonant when thephoton frequency is tuned to a dressed state in the 1e man-ifold. The anharmonicity increases with gX , and thus, sig-nificant reduction in the co-polarized photon generationoccurs for large gX .

Figure 4 shows the concurrence as a function of thebinding energy of biexciton ΔB under the resonant exci-tation conditions with gX = gB = 0.5 meV. The con-currences increase with |ΔB |. For ΔB = 0, photon pairsgenerated from all cross-polarized 2e states are not en-tangled (C = 0). The 2e states |RL; s〉 and |RL; t2〉for ΔB = 0 are degenerate with eigenfrequency 2ω0.It is noted that degenerate |RL; s〉 and |RL; t2〉 provide|L(ω0+gX)〉|R(ω0−gX)〉−|R(ω0+gX)〉|Lω0−gX)〉 and|L(ω0 +gX)〉|R(ω0−gX)〉+ |R(ω0 +gX)〉|L(ω0−gX)〉,respectively. Therefore, the states of the photon pairs gen-erated under the condition ΩR+ΩL = 2ω0 are representedas |L(ω0 + gX)〉|R(ω0 − gX)〉, namely, the photon pairsare not entangled.

4 Summary In conclusion, we have theoreticallystudied cross-polarized entangled-photon pairs generated

from a QD in a microcavity in the RHPS via dressed states.Both singlet and triplet Bell states of entangled photons canbe generated. The entangled photons can be extracted bythe spectral filtering owing to the vacuum Rabi splittingin the 1e manifold. Although co-polarized non-entangledphotons are also generated, their generation is dramaticallysuppressed in the strong-coupling regime owing to the pho-ton blockade effect. Thus, the concurrence approaches 1with an increase in the coupling constant. However, entan-gled photons cannot be generated when the binding energyof biexciton is zero. The degree of entanglement increaseswith the binding energy. It should be noted that these char-acteristic features originate from the cavity QED.

This study was partially supported by the Japan Societyfor the Promotion of Science, a Grant-in-Aid for ScientificResearch (C), 19540339, 2007, a Grant-in-Aid for CreativeScience Research, 17GS1204, 2005.

References

[1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge Univ. Press, Cam-bridge, 2000).

[2] C. A. Kocher and E. D. Commins, Phys. Rev. Lett. 18, 575(1967).

[3] S. J. Freedman , Phys. Rev. Lett. 28, 938 (1972).[4] J. F. Clauser, Phys. Rev. Lett. 36, 1223 (1976).[5] E. S. Fry and R. C. Thompson, Phys. Rev. Lett. 37, 465

(1976).[6] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 47,

460 (1981).[7] O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, Phys.

Rev. Lett. 84, 2513 (2000).[8] K. Edamatsu, Nature 431, 167 (2004).[9] G. Oohata, R. Shimizu, and K. Edamatsu, Phys. Rev. Lett.

98, 140503 (2007).[10] S. Savasta, G. Marino, and R. Girlanda, Solid State Com-

mun. 111, 495 (1999).[11] H. Ajiki and H. Ishihara, J. Phys. Soc. Jpn. 76, 053401

(2007).[12] H. Oka and H. Ishihara, Phys. Rev. Lett. 100, 170505

(2008).[13] R. M. Stevenson, Nature 439, 179 (2006).[14] R. J. Young, New J. Phys. 8, 029 (2006).[15] N. Akopian, Phys. Rev. Lett. 96, 130501 (2006).[16] H. Ajiki and H. Ishihara, Physica E 40, 371 (2007).[17] E. Peter, P. Senellart, D. Martrou, A. Lemaıtre, J. Hours,

J. M. Gerard, and J. Bloch, Phys. Rev. Lett. 95, 067401(2005).

[18] D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer,and D. Park, Phys. Rev. Lett. 76, 3005 (1996).

[19] A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch,Phys. Rev. Lett. 79, 1467 (1997).

[20] K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E.Norhup, and H. J. Kimble, Nature 436, 87 (2005).

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