Phonon-assisted decoherence in the production of polarization-entangled photons in a single semiconductor quantum dot
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Phonon-assisted decoherence in the production of polarization-entangled photons in asingle semiconductor quantum dot
Ulrich Hohenester, ∗ Gernot Pfanner, and Marek Seliger
Institut f¨ur Physik, Karl–Franzens–Universit¨at Graz, Universit¨atsplatz 5, 8010 Graz, Austria (Dated: October 31, 2018)We theoretically investigate the production of polarization-entangled photons through the biex-citon cascade decay in a single semiconductor quantum dot. In the intermediate state the entangle-ment is encoded in the polarizations of the first emitted photon and the exciton, where the excitonstate can be effectively “measured” by the solid state environment through the formation of a lat-tice distortion. We show that the resulting loss of entanglement becomes drastically enhanced ifthe phonons contributing to the lattice distortion are subject to elastic scatterings at the deviceboundaries, which might constitute a serious limitation for quantum-dot based entangled-photondevices.
PACS numbers: 78.67.Hc,42.50.Dv,71.35.-y,63.22.+m
Single quantum dots provide a viable source for en-tangled photons [1, 2, 3], as needed for quantum com-putation [4] and quantum cryptography [5]: a biexciton,consisting of two electron-hole pairs with opposite spinorientations, decays radiatively through two intermedi-ate optically active exciton states. If the exciton statesare degenerate, the two decay paths of the cascade dif-fer in polarization but are indistinguishable otherwise.Therefore, the emitted photons are entangled in polar-ization [1]. Such ideal performance is usually spoiled bythe electron-hole exchange interaction [6, 7], that splitsthe intermediate exciton states by a small amount andattaches a which-path information to the photon fre-quencies. This process deteriorates the photon entan-glement. Spectral filtering [2], energetic alignment of theexciton states by means of external magnetic [3] or elec-tric [8] fields, or growth optimization [9] allow in princi-ple to erase this information and to recover a high de-gree of entanglement. While all these schemes mask thewhich-path information to an outside observer, who couldgain information about the photon polarization by simplymeasuring its frequency, it might also be the solid-stateenvironment within which the quantum dot is embedded(e.g., lattice degrees of freedom or charging centers) thatacts as an “observer” of the intermediate exciton stateand thereby diminishes the entanglement degree.It is the purpose of this Letter to theoretically inves-tigate the loss of photon entanglement due to couplingsto the solid state environment. We consider the situa-tion that the interaction of the exciton with the latticedegrees of freedom, i.e., phonons, depends slightly on theexciton polarization. This is expected for self-organizedquantum dots owing to the strong strain-induced piezo-electric fields [9, 10] that are also responsible for thefine-structure splitting. We show that the biexciton cas-cade decay is accompanied by the formation of a polaron[11, 12, 13, 14, 15], i.e., a lattice distortion in the vicinityof the quantum dot, leading to a partial loss of coherenceand photon entanglement. This effect becomes drasti- cally magnified for single-dot devices, such as mesas ormicrocavities. The phonons contributing to the polaroncan scatter elastically at the device boundaries [16]. Inconsequence, the phonons continuously “measure” theexciton state and einselect [17] the system from a max-imally entangled state to a mixed one. We show thatthis effect leads to a significant reduction of the degreeof photon entanglement, and might constitute a seriouslimitation for quantum-dot based entangled-photon de-vices.
FIG. 1: (color online) Schematic sketch of the creation of po-larization entangled photons in a single semiconductor quan-tum dot through the cascade decay of the biexciton XX , asdescribed in text. The two intermediate exciton states X H and X V have dipole moments oriented along x and y , respec-tively; 0 is the quantum dot groundstate. The different panelsreport the excitonic wavefunctions and the distortion of thelattice in the vicinity of the quantum dot, which is assumedto be slightly asymmetric for states X H and X V . H XX ( H X )and V XX ( V X ) denote the polarizations of the photons emittedin the biexciton (exciton) decay. We assume a well-definedpolarization of the emitted photons, which can be achievedby placing the quantum dot into a microcavity [1, 2, 3], andequal magnitude of the dipole moments. The frequencies ofthe photons created in the biexciton and exciton decay differby a few meV owing to the biexciton binding energy. In our theoretical approach we consider a single quan-tum dot which is initially in the biexciton state | i (two electron-hole pairs with opposite spin orientations[1, 2, 3], XX in Fig. 1). In a typical experiment, thesystem will eventually pass after photoexcitation or elec-trical injection of electrons and holes through this state.It thus suffices to consider this initial condition. Thebiexciton is optically coupled to the two exciton states | i ( X H ) and | i ( X V ) which are polarized along x and y , re-spectively [7, 9, 10]. We presume that the fine-structuresplitting due to the anisotropic electron-hole exchangeinteraction is corrected for by appropriate external mag-netic or electric fields [3, 8], such that the energies E and E are identical. In the ideal case, the cascade decay | i → | H i + | V i → | HH i + | V V i (1)then generates a maximally entangled two-photon state[1, 18, 19], where for simplicity we have suppressed allphase and normalization factors. Under realistic con-ditions, however, the state of the emitted radiation isaffected by a number of uncertainties, such as the pho-ton emission time (time jitter [20]) or environment cou-plings. In particular, if the two states 1 and 2 coupledifferently to the solid state environment, for reasons de-tailed below, the intermediate entangled photon-excitonstate will gradually decay, with a given rate γ , into amixed state. To quantify the resulting entanglement loss,we introduce the two-photon correlation function [21] G (2) ( t, τ ) = h : I XX ( t ) I X ( t + τ ) : i , describing the polar-ization correlations between the photons emitted in thebiexciton decay at time t and the exciton decay at time t + τ , respectively. It can be computed by means of thequantum regression theorem [18, 21]. In the two-photonsubspace spanned by the basis HH , HV , V H , and
V V ,we then obtain G (2) ( t, τ ) ∝ e − t − Γ τ e − γτ e − γτ , (2)with Γ the radiative decay rate. The first exponentialon the right-hand side accounts for the gradual buildupof the two-photon state due to the radiative biexcitonand exciton decay, while the matrix describes the (condi-tional) two-photon density matrix. For the specific formgiven in Eq. (2) one can easily check that the entangle-ment of formation or concurrence [22], which providesa quantitative measure of the photon entanglement, issimply given by e − γτ . Thus, the longer the interme-diate exciton interacts with the environment, the morethe entanglement becomes diminished. The overall de-gree of polarization entanglement 1 / (1+ γ/ Γ) is obtainedthrough an average over all times t and τ [18].Having established the consequences of polarization-dependent exciton couplings, we shall now propose a mi-croscopic model for such asymmetry. Our starting point P o l a r i z a t i on a m p li t ude
10 K20 K30 K P o l a r i z a t i on a m p li t ude FIG. 2: (color online) Decay of polarization amplitude |h | ρ ( t ) | i| in presence of phonon couplings and for three dif-ferent temperatures. At time zero the system is prepared inthe superposition state | i + | i . The inset shows for compar-ison the time evolution of the optically induced polarization |h | ρ ( t ) | i| (initial state | i + | i ), that is directly accessible infour-wave mixing experiments [12]. The gray lines correspondto simulations where elastic phonon scatterings are ignored,the dark lines to simulations with γ d = 50 µ eV [16]. In thecalculations we assume an asymmetry η = 0 . is provided by the usual independent Boson model [11, 13] H = H + X i =1 X λ (cid:16) g iλ ∗ a λ + g iλ a † λ (cid:17) | i ih i | , (3)which describes the coupling of the biexciton and theexcitons to the phonons. Here, H = P i =1 E i | i ih i | + P λ ω λ a † λ a λ , with λ labeling the different acoustic phononmodes of energy ω λ , a † λ are the bosonic creation opera-tors, and g iλ the (bi)exciton-phonon couplings. Since thebiexciton approximately consists of two excitons with op-posite spin orientations, we set g λ = g λ + g λ . In anideal quantum dot the coupling constants g λ = g λ aregiven by the form factors of the exciton wavefunctions[13, 23] and the phonon coupling constants of the bulksemiconductor. These are entirely real for deformationpotential coupling and entirely imaginary for piezoelec-tric couplings [11]. For self-organized InAs/GaAs or In-GaAs/GaAs quantum dots strain fields result in strongpiezoelectric fields [9, 10], which are in part responsi-ble for the large observed fine-structure splittings, andpartially mix the deformation potential and piezoelec-tric couplings. Although the strength (and even sign)of these internal fields is still a matter of intense debate[9, 10, 24] and the calculation of the coupling asymme-try beyond the scope of this paper, we shall make thereasonable assumption that the phonon-exciton matrixelements differ by a small phase factor η viz. g λ = e iη g λ .We shall show next that even small asymmetries η canresult in a significant entanglement loss. In doing so, weexploit the fact that the independent Boson model canbe solved analytically [11]. Since no transitions betweendifferent excitonic states are induced by the phonon cou-pling (3), the time evolution operator is diagonal [23, 25] U ( t ) = e − iH t X i Y λ e i | α iλ | sin ω λ t D λ ( α iλ [1 − e iω λ t ]) | i ih i | , (4)with α iλ = g iλ /ω λ and D λ the usual bosonic displace-ment operator. The time evolution described by eq. (4)accounts for the polaron buildup. Let us consider firsta four-wave mixing experiment [12], where the quantumdot is initially brought into a superposition state | i + | i and the polarization decay (cid:12)(cid:12) h | U ( t ) ρ e iH t | i (cid:12)(cid:12) throughphonon dephasing is subsequently monitored. The graylines in the inset of Fig. 2 show results of simulations withtypical parameters for small quantum dots. The polariza-tion amplitude drops on a picosecond timescale to a valuethat is several ten percent smaller than its initial value.At later times it remains constant, indicating the forma-tion of a stable polaron. It was shown in Refs. [23, 26]that this initial drop is due to the emission of a phononwavepacket away from the dot and the resulting imprintof the superposition state into the solid-state environ-ment. The gray lines in the main panel of Fig. 2 report re-sults of simulations where the system is initially broughtinto the superposition state | i + | i , corresponding tothe buildup of a photon-exciton entanglement throughthe biexciton decay, see Eq. (1). The polarization am-plitude (cid:12)(cid:12) h | U ( t ) ρ U † ( t ) | i (cid:12)(cid:12) slightly drops at early timesdue to the asymmetric phonon coupling through whichthe environment acquires information about the inter-mediate exciton state (see panels X H and X V in Fig. 1).However, because of the small η value used in our sim-ulations the resulting loss of entanglement is extremelysmall (less than a percent).Things change considerably if additional phonon cou-plings are considered. Indeed, it is well known thatthe polarization amplitude further drops at later times[12]. This has been attributed to higher-order phononprocesses [27], extrinsic charging effects [28], or elasticphonon scatterings at the device boundaries [16]. Thelatter are particularly important for single quantum-dotdevices, as the one considered in this work. Typicalphonon linewidths due to elastic boundary scatteringsare of the order of a few tens of µ eV [16], where in thiswork we shall discard all directional and wavevector de-pendencies and use for simplicity a constant value γ d .We model the elastic phonon scatterings within a masterequation approach [29]˙ ρ = − i [ H, ρ ] − γ d X λ [ N λ , [ N λ , ρ ]] , (5)with N λ = a † λ a λ . In accordance with elastic scatter-ings, this form conserves the number of phonon excita- η γ ( n s − ) (a) γ d = 10 µ eV γ d = 25 µ eV γ d = 50 µ eV0 0.05 0.1 0.1500.20.40.60.81 Asymmetry parameter η C on c u rr en c e (b) HHHVVHVVHHHVVHVV0 0.5
FIG. 3: (color online) (a) Entanglement decay rate [seeEq. (2)] as a function of asymmetry parameter η for differentelastic phonon scattering rates γ d . The lattice temperature is10 K (dark lines) and 20 K (gray lines). (b) Concurrence as afunction of η and for different values of γ d and lattice temper-ature, and for a radiative decay rate Γ = 1 ns − . The insetreports a typical two-photon density matrix of the biexcitoncascade, where the off-diagonal elements decay according to e − γτ . The overall concurrence 1 / (1 + γ/ Γ) results from thisdensity matrix through time averaging over the whole decayprocess. tions and gives in thermal equilibrium an exponentiallydamped correlation function h a † λ ( t ) a λ (0) i = e ( iω λ − γ d ) t ¯ n λ ,where ¯ n λ is the Bose-Einstein distribution. We solve themaster equation (5) numerically by discretizing the timedomain into subintervals of length δt and expanding ρ and U ( δt ) in the basis of phonon number states. Thedark lines in Fig. 2 report results of simulations includ-ing elastic phonon scatterings, showing a constant decayof the polarization amplitude at later time. This is at-tributed to the combined effect of polaron formation andelastic phonon scatterings. In the formation of a po-laron, the exciton becomes partially entangled with thephonon degrees of freedom. In turn, the elastic boundaryscatterings of the phonons contributing to the polaronlead to an effective “measurement” of the environmenton the superposition properties of the intermediate exci-ton state. Thereby the system becomes einselected froma maximally entangled to a mixed state. Thus, after agiven time τ , when the second photon is emitted in theexciton decay, the reduced degree of photon-exciton en-tanglement is swapped to a reduced photon-photon en-tanglement.The decrease of the polarization amplitude with timeturns out to be reasonably well described by an expo-nential decay with rate γ [see also Eq. (2)], which isreported in Fig. 3(a) for different asymmetry param-eters and phonon broadenings γ d . One observes thateven small asymmetries η in the phonon coupling resultin a significant entanglement loss. For instance, using γ ( η = 0 . ∼ .
25 ns − at 10 K for a moderate phononbroadening γ d = 25 µ eV, we obtain for a typical radiativedecay rate Γ ∼ − a decrease of the concurrence byabout twenty percent [see Fig. 3(b)]. Increased values of η , γ d , or temperature even result in higher losses.The mechanism described above might explain thesmall degree of photon entanglement observed in recentexperiments [3], and is at the same time compatible withthe long measured spin scattering [30] and dephasing [31]times observed in large quantum dot samples. Theoreti-cally, the magnitude of the phonon asymmetry might beestimated from calculations for realistic quantum dots[9, 10], including strain and piezoelectric fields. Experi-mentally, a time-resolved measurement of G (2) ( τ ) wouldallow a direct observation of concurrence decay. Quitegenerally, the degree of photon entanglement can be en-hanced by decreasing the radiative decay times throughstronger microcavity couplings [19, 32]. In addition theremight be other coupling channels to the environment,such as optical phonons or localized vibrations decayingthrough anharmonic processes, or impurity defects andcharging centers in the vicinity of the dot [28] that pro-duce fluctuating electric fields, which couple differentlyto the exciton states and thereby result in entanglementlosses.In summary, we have theoretically investigated the ef-fects of environment couplings in the cascade decay of abiexciton confined in a single quantum dot. We haveshown that a slightly polarization-dependent exciton-phonon coupling results, through the formation of a po-laron in the vicinity of the dot, in a small loss of photonentanglement. This effect becomes drastically enhancedwhen the phonons contributing to the polaron scatter atthe device boundaries. In consequence, the concurrencemay easily drop by several ten percent even for moderateelastic phonon scatterings and low temperature. Our re-sults suggest that in addition to the fine structure split-ting, which can be controlled by external magnetic orelectric fields, also extrinsic effects of the environmentmight provide a limiting factor in quantum-dot basedentangled-photon sources.We gratefully acknowledge helpful discussions with Fil-ippo Troiani. This work has been supported in part bythe Austrian Science Fund FWF under projet P18136–N13. ∗ Electronic address: [email protected] [1] O. Benson, C. Santori, M. Pelton, and Y. Yamamoto,Phys. Rev. Lett. , 2513 (2000).[2] N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky,J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff,Phys. Rev. Lett. , 130501 (2006).[3] R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper,D. A. Ritchie, and A. J. Shields, Nature (London) ,179 (2006).[4] E. Knill, R. Laflamme, and G. J. Milburn, Nature ,46 (2001).[5] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev.Mod. Phys. , 145 (2002).[6] D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer,and D. Park, Phys. Rev. Lett. , 3005 (1996).[7] M. Bayer et al., Phys. Rev. B , 195315 (2002).[8] B. D. Gerardot, S. Seidl, P. A. Dalgarno, R. J. Warbur-ton, D. Granados, J. M. Garcia, K. Karrai, A. Badolato,and P. M. Petroff, Appl. Phys. Lett. , 041101 (2007).[9] R. Seguin, A. Schliwa, S. Rodt, K. P¨otschke, U. W. Pohl,and D. Bimberg, Phys. Rev. Lett. , 257402 (2005).[10] G. Bester and A. Zunger, Phys. Rev. B , 045318(2005).[11] G. D. Mahan, Many-Particle Physics (Plenum, NewYork, 1981).[12] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L.Sellin, D. Ouyang, and D. Bimberg, Phys. Rev. Lett. ,157401 (2001).[13] B. Krummheuer, V. M. Axt, and T. Kuhn, Phys. Rev. B , 195313 (2002).[14] S. Hameau, Y. Guldner, O. Verzelen, R. Ferreira, G. Bas-tard, J. Zeman, A. Lemaitre, and J. M. G´erard, Phys.Rev. Lett. , 4152 (1999).[15] O. Verzelen, R. Ferreira, and G. Bastard, Phys. Rev.Lett. , 146803 (2002).[16] S. Rudin, T. L. Reinecke, and M. Bayer, Phys. Rev. B , 161305 (2006).[17] W. H. Zurek, Rev. Mod. Phys. , 715 (2003).[18] F. Troiani, J. I. Perea, and C. Tejedor, Phys. Rev. B ,035316 (2006).[19] F. Troiani, J. I. Perea, and C. Tejedor, Phys. Rev. B ,235310 (2006).[20] A. Kiraz, M. Atature, and A. Imamoglu, Phys. Rev. A , 032305 (2004).[21] L. Mandel and E. Wolf, Optical Coherence and QuantumOptics (Cambridge University Press, Cambridge, 1995).[22] W. K. Wooters, Phys. Rev. Lett. , 2245 (1998).[23] U. Hohenester, J. Phys. B , S315 (2007).[24] M. Ediger, G. Bester, B. D. Gerardot, A. Badolato, P. M.Petroff, K. Karrai, A. Zunger, and R. J. Warburton,Phys. Rev. Lett. , 036808 (2007).[25] L. Viola and S. Lloyd, Phys. Rev. A , 2733 (1998).[26] U. Hohenester, Phys. Rev. B , 161307 (2006).[27] E. A. Muljarov and R. Zimmermann, Phys. Rev. Lett. , 237401 (2004).[28] I. Favero, A. Berthelot, G. Cassabois, C. Voisin, C. De-lalande, P. Roussignol, R. Ferreira, and J. M. Gerard,Phys. Rev. B , 073308 (2007).[29] G. J. Milburn, Phys. Rev. A , 5401 (1991).[30] M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler,D. Schuh, G. Abstreiter, and J. J. Finley, Nature ,81 (2004).[31] A. Greilich et al., Science , 341 (2006).[32] T. M. Stace, G. J. Milburn, and C. H. W. Barnes, Phys.Rev. B67