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Photons which are generated in a two-photon cascade process have an underlying time correlation since the spontaneous emission of the upper level populates the intermediate state. This correlation leads to a reduction of the purity of the photon emitted from the intermediate state. Here we characterize this time correlation for the biexciton-exciton cascade of an InAs/GaAs quantum dot. We show that the correlation can be reduced by tuning the biexciton transition in resonance to a planar distributed Bragg reflector cavity. The enhanced and inhibited emission into the cavity accelerates the biexciton emission and slows down the exciton emission thus reduces the correlation and increases the purity of the exciton photon. This is essential for schemes like creating time-bin entangled photon pairs from quantum dot systems.
© 2013 Optical Society of America
1. Introduction
Photon entanglement is an important tool for various quantum communication protocols [1–3]. A traditional way to produce entangled photon pairs is spontaneous parametric down-conversion [4]. However, down-conversion sources obey thermal photon statistics [5]. Quantum dots are semiconductor integrated sources with deterministic single photon emission [6–9]. The quantum dot biexciton-exciton cascade can be used as a source of single (polarization) entangled photon pairs [10–12]. Due to the fine-structure splitting of the exciton state and a low pair collection efficiency it is experimentally demanding to get polarization entangled photon pairs from a single quantum dot [13, 14]. In addition, polarization entangled photons have a generally limited transmission length when travelling through an optical fiber due to the polarization mode dispersion [15]. An alternative type of entanglement that overcomes this problem is time-bin entanglement [16]. Also, time-bin entanglement can be used to achieve higher dimensional entanglement in a straightforward way [17]. Recently, it has been proposed to extend this scheme to quantum dot systems [18, 19].
In this paper we investigate the modification of the temporal correlations of a biexciton-exciton cascade emission from an InAs/GaAs quantum dot. These correlations are essential for the state-purity of the photons produced in the time-bin entanglement scheme. The ability to change the temporal correlations is essential for time-bin entanglement experiments as detailed in the following section.
2. Influence of temporal correlations on time-bin entanglement
In [18, 19] a quantum dot is proposed as the system to create time-bin entanglement (see Fig. 1(a)). Initially, the quantum dot is prepared in the metastable state |m〉 from which it is excited to the level |b〉 with two successive coherent pump pulses. From here, the system decays to the ground state by emitting a pair of photons. If the probability for exciting the system with the first pulse is and the probability to excite the system with the second pulse is p2 = 1, the obtained entangled state is of the form:
where ϕ is the phase between the two pump pulses, |early〉b |early〉x is the photon cascade from the first excitation pulse and |late〉b |late〉x is the photon cascade from the second excitation pulse. Here, b and x stand for the biexciton and exciton photon, respectively [18]. If the system is not excited from the metastable state |m〉 but from the ground state, the resulting state is a probabilistic mixture of |early〉b |early〉x and |late〉b |late〉x states because the system could be excited in each pulse. The authors of the proposals [18, 19] suggested the metastable state |m〉 to be a dark exciton state.
Fig. 1 (a) Proposed level scheme for time-bin entanglement [18]. The system consists of four states: the initially prepared metastable state |m〉 from where the system is excited to the excited state |b〉 and decays in a cascade over the intermediate state |x〉 to the ground state |g〉, sending out two photons b and x. (b) Experimental Setup. Light derived from a pulsed wavelength tunable Ti:Sapphire laser is used to excite the quantum dot sample from the side using a microfocusser. The emitted photons are collected with an objective. A grating in connection with two collecting single mode fibers serves as a monochromator for the photoluminescence from the quantum dots. The recombination photons from the exciton and biexciton decay are detected with APD1 and APD2, respectively. Photodiode PD1 is used to synchronize incoming laser pulses with the emitted photons.
Unfortunately, in a real system the photon cascade |early〉b |early〉x is not a product state, but a state of the form:
where tb(x) is the emission time of the biexciton (exciton) photon and τb(x) is the lifetime of the biexciton (exciton) state [18]. The factor θ(tx − tb) describes the temporal ordering of the two emitted photons and leads to an entanglement of the single cascade, which necessarily reduces the purity of the exciton state and therefore impedes multi-pair experiments [18]. Also, the use of the biexciton-exciton cascade as a heralded single photon source demands the single cascade not be entangled. The purity [20] of the exciton state can be calculated by [18]: where ρx is the reduced density matrix of the exciton for a single cascade and τx(b) is the lifetime of the exciton (biexciton) state, respectively. The matrix ρx is derived by performing a partial trace of the biexciton emission time tb over the density matrix corresponding to the single cascade wave-function Φ(tb, tx). For multi-pair experiments to work perfectly, the purity of the exciton emission should be one [18].3. Experimental setup
In our experiment we measured the correlation of the arrival times of the biexciton and the exciton of an InAs/GaAs quantum dot embedded in a weakly coupled planar distributed Bragg reflector (DBR) cavity (dot1, dot2, dot3) consisting of alternating layers of AlAs and GaAs. Here, dot1 is a randomly picked quantum dot which emits far away from the cavity resonance. For dot2 the biexciton is red-shifted with respect to the cavity resonance at 6 K and can be red-tuned by increasing the temperature. Dot3 is blue-shifted with respect to the cavity at 6 K and is in resonance with the cavity at 30 K (see Table 1). From the measured correlation we extracted the true lifetimes, whereby we mean the lifetime of a decay process regardless of the level loading dynamics, be it an excitation laser or a higher level feeding the one under consideration. As a reference we performed the same measurement with an InAs/GaAs quantum dot without a cavity (dot0).
Table 1. Detunings Δb for the biexciton to the cavity for the different investigated quantum dots at different temperatures. The error for all values is 20 GHz since the cavity is very broad and its central wavelength is not accurately determinable. The exciton has a detuning to the cavity of Δx ≈ Δb − 640(30) GHz for all investigated quantum dots
Our set-up is depicted in Fig. 1(b). A pulsed, wavelength-tunable Ti:Sapphire laser with a pulse length of 2 ps creates successive pump pulses with a repetition rate of 76 MHz. This light excites the quantum dot sample non-resonantly from the side using a microfocusser. The laser is focused onto the cleaved edge of the sample, allowing the light to be guided between the DBRs. The emission of the quantum dots is collected with an objective and is directed onto a grating, which serves as a monochromator in connection with two collecting single mode fibers. A multichannel event timer synchronized to the laser via a photodiode (PD1) registers the detection events from two avalanche photodiodes (APDs) with a single photon time resolution of 35 ps.
The quantum dot emission can be tuned with respect to the cavity by changing the temperature (Fig. 2). With the increase of the temperature we observe increased emission intensity [21, 22] and a reduced emitter lifetime [23]. Since we would like the fraction in Eq. (3) to be close to one, either the biexciton lifetime should be shortened by the cavity with respect to the exciton lifetime or the exciton lifetime should be lengthened by suppressing the exciton emission.
Fig. 2 PL spectrum of quantum dots (a) dot2 and (b) dot3 and the cavity reflection white-light measurements for two different temperatures each. The PL measurements were performed with an excitation power density of 125 W/cm2. The quantum dot emission shifts red faster than the cavity emission. When the emission of the quantum dot is in resonance with the cavity emission its intensity gets enhanced and its lifetime gets shortened due to the Purcell effect [26].
4. Results
We measured the arrival times of the biexciton photon and the exciton photon with respect to the pump laser pulse and plotted the correlations of their arrival times in a 2D diagram. From this diagram we extracted the lifetimes of both emitters. Figure 3(a) shows such a diagram for quantum dot dot3 at 6 K where both the transitions - the exciton and the biexciton - are out of resonance, and Fig. 3(b) shows the decay extracted from (a). The measurements resulted in a biexciton lifetime of τb = 0.56(1) ns and an exciton lifetime of τx = 0.66(1) ns. The resulting value for the purity is .
Fig. 3 (a, b) dot3 at 6K, where the biexciton is not resonant to the cavity. (a) shows a 2D-histogram of the exciton and biexciton photon arrival times with respect to the exciting laser pulse. The plotted coincidence counts are placed on the left from the graph diagonal which indicates the time ordering of the detected events. (b) shows the decay of the exciton and biexciton state with an exponential fit. The data was extracted from (a) as followed. The 2D-histogram we denote as f (tb, tx). The biexciton decay is expressed as f(tb) = ∑txf(tb, tx) which is the projection onto the x-axis (see blue arrows). The exciton decay is calculated as f (tx) = ∑tbf (tb, tx + tb), which corresponds to a projection onto the y-axis with a shear transformation (see red arrows). The shear mapping sets the diagonal as the new x-axis, leading to an exciton decay that is not influenced by the decay of the biexciton. The shown parameters τx(b) are the lifetimes of the states and correspond to an exponential decay fit. (c, d) shows the same for dot3 at 30K, where the biexciton is in resonance with the cavity. The shape of the temporal correlation in (c) changed compared to (a).
Figure 3(c) shows the same measurement as Fig. 3(a) at a temperature of 30K. We measured a biexciton lifetime of τb = 0.37(2) ns and an exciton lifetime of τx = 0.98(1) ns. The resulting value for the purity is . We observe the biexciton lifetime to become shorter and the exciton lifetime to become longer. Although a shortening of the lifetime was attributed to the influence of temperature in Ref. [24], their system is quite different from ours. In Ref. [25], whose system is similar to ours, an unchanged lifetime up to 100K was reported. Also, the measurements presented in Ref. [24] showed a shortening of the lifetime for all states. Therefore we attribute the shortened lifetime to an enhancement by the cavity. The exciton lifetime becomes longer, despite being more than 1 nm detuned from the cavity resonance. This is unexpected since the change in detuning is small compared to its absolute value. Temperature effects can be excluded by the same reasons as above. A detailed explanation of the lengthening of the exciton lifetime will require further investigation.
In Fig. 4 values for the exciton purity are plotted for all the investigated quantum dots and for various temperatures. The green triangle is a reference measurement made on dot0. The red square is a reference measurement from a randomly picked quantum dot (dot1) on the sample with the DBR cavity. The blue filled circles represent a series of measurements on the quantum dot2. Here, at the temperature 6K, the biexciton emission is red-shifted but proximate to cavity resonance. With the temperature increase the biexciton emission was tuned away from the resonance. Therefore, the value of the trace presented in Fig. 4 decreases. The open black circles are measurements for dot3, where two single measurements for 6 K and 30 K were shown before. The emission from this dot is at 6 K blue shifted with respect to the cavity. With the temperature increase we can tune the biexciton emission in resonance to the cavity.
Fig. 4 Change of with temperature. The green triangle is the result of dot0. The red square is of dot1. The blue filled circles are the results from dot2 and the black open circles are measurements from dot3. The dot names are explained in the text. Lines are connecting the data points for better clarity.
We estimated the enhancement factor using the data of dot3, which could be tuned in and out of resonance with the cavity. For this we took the ratio between the unaltered lifetime of the biexciton and its enhanced lifetime. The resulting enhancement factor is . This shows that the purity of the exciton can go up to with an enhancement factor of only 1.51 ± 0.09 and a suppression of the exciton by about the same factor.
5. Comparison to a theoretical estimate
The spontaneous emission of a system, which is initially in an excited state, is caused by the interaction with the vacuum of the electromagnetic field. The mode structure of the vacuum can thus influence and change the dynamics of the emission. In particular, the spontaneous emission is controlled through the photon density of states in an optical cavity, which is called Purcell effect. If the transition dipole is in resonance with a single cavity mode the density of states is increased relative to free space and the spontaneous emission is enhanced, which leads to a faster radiative decay than in free space. On the other hand, if the transition is off resonance then the density of states is lower than in vacuum and the spontaneous emission is reduced. In the frame of the Fermi golden rule the enhancement and the inhibition of the spontaneous emission can be calculated perturbatively in the light-matter weak coupling regime [27].
Considering quantum dots as artificial atoms within an optical cavity we can adopt some of the results for atomic systems using appropriate coupling parameters. The enhanced spontaneous emission of quantum dots in an optical micro-cavity has been observed in several experiments [28]. In free space the damping rate of a two-level system is , where ωa is the transition frequency and μ the transition dipole. For a quantum dot between planar cavity mirrors, such that it is in resonance with the lowest cavity mode at zero in-plane wave vector, and is taken to be localized at the maximum of the mode function, the damping rate within a cavity is Γc = FPΓf. The Purcell factor is given by , where Veff is the effective cavity mode volume, λc the cavity mode wavelength at resonance, and n the refractive index of the medium. Here, Q is the cavity quality factor that is defined by Q = ωc/Δω. From the reflection spectrum of the cavity we extracted a Q factor of 900(30). Our sample contained self-assembled InAs quantum dots of low density (≈ 10 μm−2) grown by molecular beam epitaxy. The quantum dots were embedded in a distributed Bragg reflector (DBR) micro-cavity consisting of 15.5 lower and 10 upper λ/4 thick DBR layer pairs of AlAs and GaAs, with a cavity mode at λ = 920 nm. The physical thickness of the central cavity layer is L = 542.7 nm. For an estimated Purcell factor of FP = 1.5 the effective mode volume is a cylinder with the height equal to the cavity length L and a diameter of 1.4 μm.
6. Conclusion
In this work we showed that both biexciton and exciton lifetimes can be modified with a planar micro-cavity. We measured a Purcell factor for the biexciton emission of 1.51(9) for the planar micro-cavity. While the biexciton emission is accelerated the exciton emission is slowed down by approximately the same factor. The modification of both lifetimes leads to a significant improvement of the purity of the exciton photon from to 0.73(1). The purity is important for schemes generating time-bin entanglement using such a biexciton exciton cascade. In particular the fidelity of the entangled state will depend on the remaining correlation of the two photons. To improve on this result one could use quantum dots embedded in micro-pillar cavities which have a smaller mode volume. As inhibition of the exciton emission as well as the enhancement of the biexciton emission increases the purity of the emitted photon, a sample design where both processes are granted would be optimal. As such devices we could propose coated microresonators [29] or quantum dots embedded in a photonic crystal cavity [30]. Nevertheless, planar microcavities allow for laser-scattering free schemes of coherent excitation and control of the biexciton population [31]. Such a scheme is essential for probabilistic time-bin entanglement [32]. With a Purcell factor of five the calculated value for the purity is already above 85%. Such a Purcell factor is easily reachable in low finesse micropillar cavities which do not show effects of strong coupling. The practical purity requirements depend on the specific use of the entangled photon pairs [33, 34].
Acknowledgments
This work was funded by the European Research Council (project EnSeNa) and the Canadian Institute for Advanced Research through its Quantum Information Processing program. A.P. would like to thank the Austrian Science Fund (FWF) for the support provided through Lise Meitner Postdoctoral Fellowship M-1243. G.S.S. acknowledges partial support through the Physics Frontier Center at the Joint Quantum Institute (PFC@JQI).
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