Letter pubs.acs.org/NanoLett
Highly Entangled Photons from Hybrid Piezoelectric-Semiconductor Quantum Dot Devices Rinaldo Trotta,*,† Johannes S. Wildmann,† Eugenio Zallo,‡ Oliver G. Schmidt,‡ and Armando Rastelli† †
Institute of Semiconductor and Solid State Physics, Johannes Kepler University, Altenbergerstr. 69, A-4040 Linz, Austria Institute for Integrative Nanosciences, IFW Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany
‡
S Supporting Information *
ABSTRACT: Entanglement resources are key ingredients of future quantum technologies. If they could be efficiently integrated into a semiconductor platform, a new generation of devices could be envisioned, whose quantum-mechanical functionalities are controlled via the mature semiconductor technology. Epitaxial quantum dots (QDs) embedded in diodes would embody such ideal quantum devices, but a fine-structure splitting (FSS) between the bright exciton states lowers dramatically the degree of entanglement of the sources and hampers severely their real exploitation in the foreseen applications. In this work, we overcome this hurdle using strain-tunable optoelectronic devices, where any QD can be tuned for the emission of photon pairs featuring the highest degree of entanglement ever reported for QDs, with concurrence as high as 0.75 ± 0.02. Furthermore, we study the evolution of Bell’s parameters as a function of FSS and demonstrate for the first time that filtering-free violation of Bell’s inequalities requires the FSS to be smaller than 1 μeV. This upper limit for the FSS also sets the tuning range of exciton energies (∼1 meV) over which our device operates as an energy-tunable source of highly entangled photons. A moderate temporal filtering further increases the concurrence and the tunability of exciton energies up to 0.82 and 2 meV, respectively, though at the expense of 60% reduction of count rate. KEYWORDS: Quantum dots, entanglement, Bell’s inequalities, piezoelectric substrates, fine structure splitting
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proposed scheme conceals the very stringent requirement set on the QD structural properties: the absence of an energetic splitting between the intermediate X states, commonly referred to as the fine structure splitting (FSS, s). The FSS, caused by mixing of the ±1 spin states due to the anisotropic electron− hole exchange interaction12 in QDs with reduced structural symmetry (lower than D2d), induces dephasing and dramatically affects the degree of entanglement. More precisely,13,14 the presence of a finite FSS turns the state into ψ′ = 1/√2(|HXXHX⟩ + ei(Sτ/ℏ)|VXXVX⟩), where τ is the time interval between the emission of the two photons and ℏ is the reduced Planck’s constant. During the time τ, the FSS causes the superposition of the photon-pair state to oscillate, thus reducing the time-averaged fidelity of ψ′ to the maximally entangled Bell state ψ, possibly revealing only classical correlations among the emitted photons.14 Despite impressive progress in the fabrication of highly symmetric QDs11 and in the postgrowth control of their optical properties via external perturbations,5,15−19 it remains extremely difficult to systematically drive the FSS to vanishing small values, mainly due to the coherent coupling of the two bright exciton states.20 As a result, the high degree of entanglement needed for the implementation of QDs in real applications and in advanced
he potential of semiconductor quantum dots (QDs) as entanglement resources was highlighted for the first time in 2000 by Benson and co-workers.1 The proposal relies on the generation of polarization entangled photon pairs during the radiative decay of the biexciton (XX) to the exciton (X) to the crystal ground state (0), and it is analogous to the “atomic concept” used by Aspect and co-workers2 to demonstrate the nonlocal nature of entanglement. In QDs featuring high structural symmetry (D2d or higher), the intermediate exciton level, built up from an electron and a heavy-hole, consists of two degenerate bright states with total spin projection ±1. The XX (total spin projection 0) cascade to the ground state can thus take place following two different decay paths: (i) with the emission of a right-circularly polarized biexciton photon (RXX) followed by a left-circularly polarized exciton photon (LX), or (ii) with the emission of a LXX photon followed by a RX photon. Since the two decay paths have equal probability, the photonpair results entangled in polarization and predicted to be in the maximally entangled Bell state ψ = 1/√2(|RXXLX⟩ + |LXXRX⟩), which can be equivalently rewritten as ψ = 1/√2(|HXXHX⟩ + |VXXVX⟩) or ψ = 1/√2(|DXXDX⟩ + |AXXAX⟩), where H (V) and D (A) indicate horizontally (vertically)-polarized and diagonally (antidiagonally)-polarized photons, respectively. This theoretical proposal has led to an explosion of research efforts,3−11 mainly motivated by the possibility of having at hand solid-state triggered entangled photon sources for quantum cryptography and quantum computation. However, the simplicity of the © XXXX American Chemical Society
Received: March 14, 2014 Revised: May 19, 2014
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Figure 1. (a) Sketch of InGaAs semiconductor quantum dots (QDs) embedded in a dual-knob device where in-plane anisotropic biaxial strain, induced by the electric fields across the piezoelectric actuator (Fp), and vertical electric fields across the diode (Fd) are used to correct for the QD structural asymmetries that prevent them from emitting polarization entangled photons (shown by the two photons connected via a chain). (b) Behavior of the fine structure splitting (FSS, s) as a function of Fd for a QD whose polarization direction of the exciton emission (θ) has been previously aligned via Fp along the effective direction of application of the electric field. The blue line corresponds to the prediction of the theoretical model, see ref 18. (c) Same as that in panel b, but for θ. The inset shows the dependence (in polar coordinates) of ΔE as a function of the angle the linear polarization analyzer forms with the [110] crystal axis (polarization angle), where ΔE is half of the difference between XX and X energies minus its minimum value (see ref 18). The length and orientation of the petals give the value of s and θ, respectively. Blue and red circles are obtained respectively at (Fd, Fp) = (−30 kV/cm, 0 kV/cm) and (Fd, Fp) = (−30 kV/cm, 17.3 kV/cm), The latter value of Fp (= F*p) corresponds to the critical strain that allows the bright exciton level degeneracy to be restored via Fd. For the blue (red) curve s = 8 μeV (s = 11 μeV). (d) Microphotoluminescence spectrum of a representative QD at s = 0. Radiative recombinations ascribed to the exciton (X), biexciton (XX), and positively (X+) and negatively (X−) charged excitons are also indicated. The inset shows the intensity (in arb. units) of the exciton emission as a function of the polarization angle; 0° correspond to the [110] direction of the GaAs crystal.
quantum optics experiments,21,22 such the violation of Bell’s inequalities,23 has been achieved only by few groups worldwide by cherry-picking the very rare QDs featuring s ≈ 024 or by using temporal9,25 and spectral4 filtering techniques, which inevitably reduce the brightness of the source. In this work, we demonstrate that virtually any QD embedded in hybrid semiconductor-piezoelectric devices can be used for the generation of highly polarization-entangled photons. This novel class of electrically controlled entangledphoton sources violates Bell’s inequalities without the need of temporal or spectral filtering. By means of quantum state tomography we prove that our source features the highest degree of entanglement ever reported for QD-based entanglement resources, with a concurrence as high as 0.75 ± 0.02. Furthermore, we clarify that although moderately small FSS (s < 4 μeV) are sufficient to overcome the classical limit, a much tighter control is required to violate Bell’s inequalities (s < 1 μeV), which also sets the range of X energies (∼1 meV) over which our device operates as an energy-tunable source of highly entangled photons. A moderate temporal filtering further increases the level of concurrence (the tunability of X energies) to 0.82 (2 meV), though at the expense of 60% reduction of count rate. The key feature of our device is the capability to provide two independent external perturbations for engineering the energy levels of the quantum emitters on demand. This is achieved by embedding InGaAs QDs in the intrinsic region of n−i−p diode nanomembranes and by their subsequent integration onto [Pb(Mg1/3Nb2/3)O3]0.72−[PbTiO3]0.28 (PMN−PT) piezoelectric actuators.26 Therefore, the device features two electrical tuning knobs (see Figure 1a): the voltage applied to the diode allows the electric field (Fd) across the QDs to be controlled. Simultaneously, the electric field (Fp) across the piezoelectric actuator induces in-plane anisotropic biaxial strain fields to the
QDs. Further details on the device fabrication can be found in the Supporting Information. In order to control and suppress the coherent coupling of the bright exciton states, it is fundamental to achieve full control over two QD parameters: the magnitude of the FSS and the polarization direction of the exciton emission (θ). Controlling only one of them generally results in an anticrossing of the two X states as the magnitude of the external field is varied.15−18,20 In our experiment, we use strain (Fp) to precisely align the polarization direction of the exciton emission (θ) along the “effective” axis of application of the electric field (Fd). Fd is then employed, at fixed Fp, to compensate completely for in-plane asymmetries in the QD confinement potential and thus to drive the FSS through zero. For the particular QD reported in Figure 1b,c, the strain needed to properly rotate θ is achieved at F*p= 17.3 kV/cm, and the minimum value of s = (0.2 ± 0.3) μeV is reached at the critical electric field of F*d = −93 kV/cm. The behavior of s and θ follows closely the theoretical predictions of a mesoscopic model developed for describing the field-induced cancellation of the coherent coupling of the two bright exciton states, see the blue solid lines in Figure 1b,c and ref 18. Several important points need to be discussed at this stage: (i) The behavior of s and θ reported in Figure 1b,c is universal and always reproduced by the theoretical model. (ii) The values of Fd and Fp at which the FSS reaches s = 0, i.e., F*p and F*d, depend on the particular QD under study (see Figure 3). This is connected to fluctuations in the composition, size, and shape among the different QDs that arise inevitably during growth.27 (iii) In order to restore the X level degeneracy, F*p and F*d have to fall within the tuning range, i.e., for a fixed sample, the probability P of finding QDs that can be driven to s = 0 is determined by the tuning range. We found that P can be as high as 100%, and it varies among different devices (the discussion of the differences among the devices is beyond the B
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scope of the present work and will be presented elsewhere). The last two points are particularly relevant in the perspective of extending our approach to QDs emitting at 1.3 or 1.5 μm,28,29 where the large tunability of the FSS offered by our dual-knob device may lead to a significant increase of the number of QDs suitable for entangled photon generation at telecom wavelengths.29 We now use polarization-resolved photon correlation spectroscopy to investigate in more detail the dynamics of the XX−X cascade in QDs whose FSS has been fine-tuned to s = 0. The microphotoluminescence spectrum of a typical QD acquired in these conditions is shown in Figure 1d. The presence of the negatively (positively) charged exciton X− (X+) appearing at the low (high) energy side of the X−XX doublet is very common in our QDs.30 Their relative intensity with respect to neutral states could be reduced using diode membranes with a thicker intrinsic region so as to suppress tunneling of carriers (mainly electrons) into the dot from the doped layers of the device. This would also increase the net number of correlated photon pairs generated in the XX−X cascade.9 The inset of Figure 1d displays the intensity of the exciton emission as a function of the angle between the linear polarization analyzer and the [110] direction (see Supporting Information): it proves that the source is unpolarized (within 4%) at s = 0. The first clear signature of entangled photon emission is shown in Figure 2a−f, where cross-correlation measurements between XX and X are reported for circular (Figure 2a,d), linear (Figure 2b,e), and diagonal (Figure 2c,f) polarization basis for one of the investigated QDs. We observe strong bunching when recording coincidence counts with the following polarizations: HXX HX (or VXX VX), DXX DX (or AXX AX) and RXX LX (or LXX RX). However, the bunching peaks disappear for HXX VX (or VXX HX), DXX AX (or AXX DX) and RXX RX (or LXX LX). This is exactly the predicted behavior of photon pairs emitted in the maximally entangled Bell state ψ, where strong correlations for colinear, codiagonal, and cross-circular polarization are expected.3 For an ideal source of entangled photons the degree of correlation (or correlation visibility) CAB, defined as the difference between copolarized and crosspolarized correlations divided by their sum, should be CHV = |CRL| = CDA = 1. In our experiment, we find (see Figure 2g−i): CHV = 0.72 ± 0.05, |CRL| = 0.82 ± 0.02, CDA = 0.72 ± 0.05. Although these values suggest that the photons emitted during the XX−X cascade are entangled, the deviations from the expected values indicate that the two-photon state is not exactly the Bell state ψ. We estimate the fidelity to ψ as13 f = (1 + CHV + CDA + |CRL|)/4 and obtain f = 0.82 ± 0.04 at zero time delay (see Figure 2j). This value is well above the classical limit of 0.5 and highlights that the application of large external fields does not prevent QDs from emitting entangled photons. In order to completely characterize the entangled state, we have reconstructed the two-photon density matrix ρ̂ measuring XX− X correlations in 36 polarization settings and with the help of a maximum likelihood method.31 The real and imaginary parts of ρ̂ for a selected QD are displayed in Figure 3a,b, respectively. The presence of the strong off-diagonal terms ρ̂HH,VV and ρ̂VV,HH illustrates immediately the superposition of the twophoton pair wave functions. The matrix satisfies the Peres inseparability criterion32 for entanglement, being −0.35 ( F*d. The fidelity first increases, it reaches the highly nonclassical value of 0.82 ± 0.02, and then it decreases again. As shown by Hudson and co-workers,13 this behavior can be approximated by a Lorentzian function, which, in our case, has a full width at half-maximum (fwhm) equal to 3 μeV. This value nicely matches the lifetime of the exciton transition (τ ≈ 1 ns, being roughly constant for the range of electric fields explored during the experiment). For the particular QD under study, violation of the classical limit (see dashed line in Figure 4a) can be observed even for s values as large as ∼4 μeV. These results confirm the general idea that entanglement persists despite a nonzero separation between the two bright excitonic states13 and explain the large spread of f values that can be found in the literature. However, overcoming the classical limit is not sufficient for applications relying on nonlocal correlations between the emitted photons, such as entanglement swapping21 (the key element of quantum relays and repeaters) as well as quantum cryptography protocols.23 One possible criterion to define the “useful” entanglement degree is the violation of Bell’s inequalities, initially proposed to demonstrate the entanglement nonlocality and then used as a base for quantum cryptography.23 The high level of concurrence measured at s = 0 suggests that our source violates Bell’s inequalities, but the relationship between Bell’s inequalities and FSS has never been investigated so far. Following refs 9 and 25, we use three different Bell’s inequalities for three different planes of the Poincaré sphere: SRC = √2(CHV − CRL), SRD = √2(CHV + CDA), and SCD = √2(CDA − CRL). A value larger than 2 of one of these parameters ensures violation of Bell inequalities, even if our state deviates from the maximally entangled Bell state (see above). Figure 4b shows the evolution of these parameters as a function of s. Paralleling the fidelity, a Lorentzian-like behavior is observed. The different fwhms reflect the different values of the correlations in the linear, diagonal, and circular bases; see Figure 2. Around the minimum splitting we measure SRD = 2.04 ± 0.05, SCD = 2.24 ± 0.07, and SRC = 2.33 ± 0.04. The latter parameter shows violation of Bell’s inequalities by more than 8 standard deviations and proves unambiguously that our electrically controlled source can produce nonlocal states of light. Most importantly, Figure 4b highlights that violation of Bell’s inequalities can be achieved only in a very narrow range of FSS, s < 1 μeV, thus pointing out that tight control of the electronic structure of QDs is fundamental for achieving the high level of entanglement required for the application of these quantum emitters. It is important to notice that we achieve these results using the raw data, without any temporal8,9,25 or
spectral4 filtering and without any background light subtraction. The values of the three Bell parameters further increase well above the limit of 2 using temporal postselection of the emitted photons. In particular, by reducing the temporal window used to integrate the coincidence counts to w = 1 ns (corresponding to 60% reduction of the count rate, see Supporting Information), we find SRD = 2.29 ± 0.07, SCD = 2.55 ± 0.09, and SRC = 2.39 ± 0.06 (see Figure4d). At the same time, the fidelity to the maximally entangled Bell state increases to a maximum value of f = 0.88 ± 0.02 (See Figure 4c). Having achieved such a high level of entanglement, it is worth discussing the possibility to use our device as an electrically controlled energy-tunable source of entangled photons, i.e., a device where the energy of the X (or XX) transition is modified electrically while maintaining a high level of entanglement. This is indeed a crucial prerequisite for entanglement swapping experiments using dissimilar QD-based qubits and cannot rely on as-grown QDs only, i.e., without the aid of external perturbations. Figure 4a,b shows that violation of the classical limit can be easily achieved when the X energy is tuned over a spectral range as large as ∼4 meV. However, highly entangled photons can be emitted only in a narrower spectral range (0.6 meV), which can be further increased using a moderate temporal filtering to 2 meV (see Figure 4d). Despite that these values could be sufficient for proof of principle experiments aiming at interfacing distant QD-based entanglement resources, a different device concept is required to further increase the tunabilty of the X energy, especially in the perspective of real applications, where one would prefer to exploit the electric field across the diode-membrane to inject carriers electrically9 rather than to control the QD optical properties. We leave this point to future studies. We have demonstrated that virtually any semiconductor QD embedded into strain-tunable optoelectronic devices can be used for the generation of entangled photons featuring the highest degree of entanglement reported to date for QD-based photon sources, with concurrence as high as 0.75 ± 0.02. We have achieved these results by merging the semiconductor and piezoelectric technologies, which allow a long-standing problem plaguing optically active quantum dots, the presence of a FSS between the two bright exciton states, to be systematically solved. Furthermore, we demonstrate that filtering-free violation of Bell’s inequalities can be achieved in a range of FSS (s < 1 μeV) significantly smaller than the one required to overcome the classical limit (s < 4 μeV). This upper limit for the FSS also sets the range of exciton or biexciton energies (∼1 meV) over which our device can be used as an energytunable source of entangled photons. This spectral range increases up to 2 meV using moderate temporal postselection of the emitted photons, which also brings the level of concurrence up to 0.82. After more than a decade from the first proposal,1 we have demonstrated that the implementation of QD-based optoelectronic devices into a solid-state quantum technology is feasible, and we envisage that our device concept is going to play a key role in future experiments and applications built up around entanglement nonlocality. The usefulness of the method is not limited to the InGaAs QDs investigated here. In particular, its implementation on QDs emitting at 1.3 or even 1.5 μm may lead to a significant increase of the number of QDs suitable for entangled photon generation at telecomm wavelengths.28,29 However, the use of strain-free GaAs QDs10,24,35 may yield E
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(15) Plumhof, J. D.; Trotta, R.; Rastelli, A.; Schmidt, O. G. S. Nano Res. Lett. 2012, 7, 336. (16) Bennett, A. J.; Pooley, M. A.; Stevenson, R. M.; Ward, M. B.; Patel, R. B.; Boyer de la Giroday, A.; Sköld, N.; Farrer, I.; Nicoll, C. A.; Ritchie, D. A.; Shields, A. J. Nat. Phys. 2010, 6, 947. (17) Gong, M.; Zhang, W.; Guo, G.-C.; He, L. Phys. Rev. Lett. 2011, 106, 227401. (18) Trotta, R.; Zallo, E.; Ortix, C.; Atkinson, P.; Plumhof, J. D.; Brink, J. V. D.; Rastelli, A.; Schmidt, O. G. Phys. Rev. Lett. 2012, 109, 147401. (19) Muller, A.; Fang, W.; Lawall, J.; Solomon, G. Phys. Rev. Lett. 2009, 103, 217402. (20) Singh, R.; Bester, G. Phys. Rev. Lett. 2010, 104, 196803. (21) Pan, J.-W.; Bouwmeester, D.; Weinfurter, H.; Zeilinger, A. Phys. Rev. Lett. 1998, 80, 3891. (22) Zhao, Z.; Zhang, A.-N.; Chen, Y.-A.; Zhang, H.; Du, J.-F.; Yang, T.; Pan, J.-W. Phys. Rev. Lett. 2005, 94, 030501. (23) Ekert, A. K. Phys. Rev. Lett. 1991, 67, 661. (24) Kuroda, T.; Mano, T.; Ha, N.; Nakajima, N.; Kumano, H.; Urbaszek, B.; Jo, M.; Abbarchi, M.; Sakuma, Y.; Sakoda, K.; Suemune, I.; Marie, X.; Amand, T. Phys. Rev. B 2013, 88, 041306. (25) Young, R. J.; Stevenson, R. M.; Hudson, A. J.; Nicoll, C. A.; Ritchie, D. A.; Shields, A. J. Phys. Rev. Lett. 2009, 102, 030406. (26) Trotta, R.; Atkinson, P.; Plumhof, J. D.; Zallo, E.; Rezaev, R. O.; Kumar, S.; Baunack, S.; Schroter, J. R.; Rastelli, A.; Schmidt, O. G. Adv. Mater. 2012, 24, 2668. (27) Rastelli, A.; Stoffel, M.; Malachias, A.; Merdzhanova, T.; Katsaros, G.; Kern, K.; Metzger, T. H.; Schmidt, O. G. Nano Lett. 2008, 8, 1404. (28) Sapienza, L.; Malein, R. N. E.; Kuklewicz, C. E.; Kremer, P. E.; Srinivasan, K.; Griffiths, A.; Clarke, E.; Gong, M.; Warburton, R. J.; Gerardot, B. D. Phys. Rev. B 2013, 88, 155330. (29) Ward, M. B.; Dean, M. C.; Stevenson, R. M.; Bennett, A. J.; Ellis, D. J. P.; Cooper, K.; Farrer, I.; Nicoll, C. A.; Ritchie, D. A.; Shields, A. J. Nat. Commun. 2014, 5, 3316. (30) Trotta, R.; Zallo, E.; Magerl, E.; Schmidt, O. G.; Rastelli, A. Phys. Rev. B 2013, 88, 155312. (31) James, D. F. V.; Kwait, P. G.; Munro, W. J.; White, A. G. Phys. Rev. A 2001, 64, 052312. (32) Peres, A. Phys. Rev. Lett. 1996, 77, 1413. (33) Jayakumar, H.; Predojević, A.; Huber, T.; Kauten, T.; Solomon, G. S.; Weihs, G. Phys. Rev. Lett. 2013, 110, 135505. (34) Müller, M.; Bounouar, S.; Jöns, K. D.; Glässl, M.; Michler, P. Nat. Photonics 2014, 8, 224. (35) Sallen, G.; Kunz, S.; Amand, T.; Bouet, L.; Kuroda, T.; Mano, T.; Paget, D.; Krebs, O.; Marie, X.; Sakoda, K.; Urbaszek, B. Nat. Commun. 2014, 5, 3268.
even larger entanglement levels due to the reduced nuclear magnetic fields and associated fluctuations.
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ASSOCIATED CONTENT
S Supporting Information *
Sample growth and device fabrication; experimental setup; data analysis for correlation measurements; sources of entanglement degradation; temporal postselection of the emitted photons. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*(R.T.) E-mail:
[email protected]. Author Contributions
R.T. and A.R. conceived and designed the experiment. R.T. and J.S.W. fabricated the device and performed measurements and data analysis with help from A.R. E.Z. and O.G.S carried out sample growth. R.T. wrote the manuscript with help from all the authors. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank G. Bauer and A. Predojević for fruitful discussions, P. Atkinson for help with the sample design and growth, and F. Binder, A. Halilovic, U. Kainz, E. Vorhauer, and S. Brauer for technical assistance. The work was supported financially by the European Union Seventh Framework Programme 209 (FP7/ 2007−2013) under Grant Agreement No. 601126 210 (HANAS).
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REFERENCES
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