On-demand source of maximally entangled photon-pairs using the biexciton-exciton radiative cascade

TL;DR

This paper demonstrates a method to generate on-demand, maximally entangled photon pairs from a semiconductor quantum dot using a biexciton-exciton radiative cascade, with entanglement quality depending on temporal resolution.

Contribution

It provides a deterministic initiation of biexciton states and full tomographic measurement showing maximal entanglement of emitted photon pairs.

Findings

Photon pairs are maximally entangled during the cascade.

Entanglement degree depends on temporal resolution.

Method enables on-demand entangled photon source.

Abstract

We perform full time resolved tomographic measurements of the polarization state of pairs of photons emitted during the radiative cascade of the confined biexciton in a semiconductor quantum dot. The biexciton was deterministically initiated using a $\pi$-area pulse into the biexciton two-photon absorption resonance. Our measurements demonstrate that the polarization states of the emitted photon pair are maximally entangled. We show that the measured degree of entanglement depends solely on the temporal resolution by which the time difference between the emissions of the photon pair is determined. A route for fabricating an on demand source of maximally polarization entangled photon pairs is thereby provided.

Tables

Table 1. Table 1: Various two photon polarization sensitive emission probability rates.

Criteria	# of cases	$p(t,P_1,P_2)/p_{X^0}\left(t\right)$
${\theta }_1={\theta }_2=0\left(\pi \right)$	2	$\frac{1}{2}$
${\theta }_1=\pi \left(0\right)$, ${\theta }_2=0\left(\pi \right)$	2	0
${\theta }_1=0or\pi$,${\theta }_2=\pi /2$	8	$\frac{1}{4}$
${\theta }_1=\pi /2$,${\theta }_2=0or\pi$	8	$\frac{1}{4}$
${\theta }_1={\theta }_2=\pi /2$	16	$\frac{1}{4}\left[1+\cos \left({\varphi }_1+{\varphi }_2+2\pi \frac{t}{T_P}\right)\right]$

Equations

$$|{\psi }_{P_1{X}^0}⟩=\frac{1}{\sqrt{2}}\left(|H_1{X}_H^0⟩+|V_1{X}_V^0⟩\right)$$

$$|{\psi }_{P_1{P}_2}(t)⟩=\frac{1}{\sqrt{2}}\left(|H_1{H}_2⟩+e^{-i2\pi \frac{t}{T_P}}|V_1{V}_2⟩\right)$$

$${\rho }_{P_1{P}_2}(t)=\frac{1}{2}\left(\begin{array}{cccc}1& 0& 0& e^{-i2\pi \frac{t}{T_P}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ e^{i2\pi \frac{t}{T_P}}& 0& 0& 1\end{array}\right)$$

$$\mathcal{N}[{\rho }_{P_1{P}_2}(t)]=\frac{1}{2}|e^{i\frac{\Delta }{\hslash }t}|=\frac{1}{2}|e^{i2\pi \frac{t}{T_P}}|=\frac{1}{2}$$

$$\mathcal{N}[{\rho }_{P_1{P}_2}(t,\Delta T)]=\frac{1}{2\Delta T}\left|{\int }_t^{t+\Delta T}{e}^{i2\pi \frac{t^{\prime }}{T_P}}d{t}^{\prime }\right|$$

$$\mathcal{N}[{\rho }_{P_1{P}_2}(\Delta T)]=\frac{1}{2}\left|\text{sinc}\left(\pi \frac{\Delta T}{T_P}\right)\right|$$

$$|P(\theta ,\varphi )⟩=\cos (\theta /2)|H⟩+e^{i\varphi }\sin (\theta /2)|V⟩$$

$$|H⟩=|P(\theta =0,\varphi )⟩$$

$$|V⟩=|P(\theta =\pi ,\varphi ⟩$$

$$|D⟩=(|H⟩+|V⟩)/\sqrt{2}=|P(\theta =\pi /2,\varphi =0)⟩$$

$$|\bar{D}⟩=(|H⟩-|V⟩)/\sqrt{2}=|P(\theta =\pi /2,\varphi =\pi )⟩$$

$$|L⟩=(|H⟩+i|V⟩)/\sqrt{2}=|P(\theta =\pi /2,\varphi =\pi /2)⟩$$

$$|R⟩=(|H⟩-i|V⟩)/\sqrt{2}=|P(\theta =\pi /2,\varphi =3\pi /2)⟩$$

$$p[t,P_1({\theta }_1,{\varphi }_1),P_2({\theta }_2,{\varphi }_2)]=p_{X^0}(t)\cdot |⟨P_1{P}_2⟩{\psi }_{P_1{P}_2}(t){|}^2$$

$$\displaystyle=\frac{e^{-\frac{t}{\tau_{R}}}}{2\tau_{R}}\Bigl{|}\cos\frac{\theta_{1}-\theta_{2}}{2}\cos\left(\frac{\phi_{1}+\phi_{2}}{2}+\frac{\pi t}{T_{P}}\right)+$$

$$\displaystyle+i\cos\frac{\theta_{1}+\theta_{2}}{2}\sin\left(\frac{\phi_{1}+\phi_{2}}{2}+\frac{\pi t}{T_{P}}\right)\Bigl{|}^{2}$$

$$P_{HH(VV)}={\int }_0^{\infty }p[t^{\prime },HH(VV)]d{t}^{\prime }=\frac{1}{2}$$

Full text

We present here a novel study of a single semiconductor quantum dot, optically depleted Schmidgall2015 and then resonantly excited on-demand by a $\pi$-area pulse to the biexciton two photon absorption resonance  muller2014 . The resulting pairs of biexciton and exciton photons are detected by two superconducting detectors synchronized to the exciting laser pulse. By performing synchronized time resolved polarization tomography of the two emitted photons, we unambigously show that the photons remain maximally polarization entangled during the whole radiative decay, and that the measured degree of entanglement does not depend on the QD source, but rather depends on the temporal resolution by which the time difference between the two photon emissions can be determined. Since during the radiative decay the exciton does not lose coherence, there is no need to eliminate the excitonic fine structure splitting. A relatively simple arrangement  jones2006photon ; wang2010demand should therefore provide a reliable source of on demand pairs of maximally entangled photons from a single quantum dot regardless of its non-vanishing excitonic fine structure splitting.

Theoretical background

The biexciton-exciton radiative cascade is schematically described in Fig. 1(a). In the figure, resonant excitation of the biexciton using a $\pi$-area pulse to the virtual two photon excitation (TPE) resonance  brunner1994sharp ; stufler2006two ; jayakumar2013deterministic ; muller2014 , is described by an upward green arrow. This 12 ps laser pulse follows a few nanosecond long optical depletion pulse Schmidgall2015 which empties the QD from charges and long lived dark excitons. The $\pi$-area pulse therefore deterministically photogenerates a confined ground state biexciton ($\ket{XX^{0}}$) in the QD brunner1994sharp ; muller2014 . The biexciton spontaneously radiatively decays, leaving in the QD an exciton in a coherent superposition of its two eigenstates: the $\ket{X^{0}_{H}}$ and the $\ket{X^{0}_{V}}$ states gammon1996 ; kulakovskii1999fine . The optical selection rules for the biexciton radiative recombination and the lack of information by “which path” the recombination proceeds result in entanglement between the exciton state and the polarization state of the emitted photon benny2011prl ; poem2011 ; ward2014coherent . Their mutual wavefunction is given by:

[TABLE]

where $\ket{H_{1}}$ and $\ket{V_{1}}$ are the two rectilinear polarization states of the first (biexciton) photon. Since the two exciton eigenstates are not degenerate, the relative phase between these eigenstates precesses in time with a period of $T_{P}=h/\Delta$ where $h$ is the Planck constant and $\Delta$ is the exciton fine structure splitting (see Fig. 1(a)). This precession is schematically described on the exciton Bloch sphere in Fig. 1(b). The precession “stops” when the exciton recombines and the radiative cascade is completed with the emission of a second photon. The two photons are thus entangled. Their mutual wavefunction depends on the recombination time and is given by:

[TABLE]

where $\ket{H_{2}}$ and $\ket{V_{2}}$ are the second (exciton) photon polarization states and $t=t_{X^{0}}-t_{XX^{0}}$ is the time between the emission of the biexciton photon $t_{XX^{0}}$ to that of the exciton $t_{X^{0}}$. It follows that the normalized two photon polarization density matrix is expressed in the basis $\ket{H_{1}H_{2}}$, $\ket{H_{1}V_{2}}$, $\ket{V_{1}H_{2}}$ and $\ket{V_{1}V_{2}}$ as:

[TABLE]

To evaluate the degree of entanglement between the two photon polarization states we used a standard measure: the negativity $\mathcal{N}$ of their polarization density matrix peres1996separability , where $\mathcal{N}$ is defined as the magnitude of the negative eigenvalue of the partially transposed density matrix. For $\mathcal{N}>0$, the two photons are entangled with fidelity of about $2\mathcal{N}$ to a two photon Bell-state. $\mathcal{N}=\frac{1}{2}$ corresponds to maximal entanglement with unit fidelity to two photon Bell state. It is easy to see from Eq. (3) that the negativity of the two photon density matrix is given by

[TABLE]

and thus the two photon polarization states are always maximally entangled. The information contained in the different colors of the photon pairs, which reveals the decay path of the radiative cascade, remains in the temporal dependence of the phase of the non-diagonal elements of the polarization density matrix.

In an actual experiment, the temporal resolution is limited. Thus, if one cannot define the time difference between the two photons to a better resolution than $\Delta T$ the negativity in the measured polarization density matrix is given by:

[TABLE]

where for simplicity we assumed that the temporal resolution is represented by a square temporal window of width $\Delta T$. It follows that the measured negativity is independent of the time $t$:

[TABLE]

As can be seen from Eq. 6 the negativity and hence the measured degree of entanglement vanish for $\Delta T=T_{P}$, but it revives before vanishing for a wider temporal window ($\Delta T\gg T_{P}$ ).

We define a general polarization state of light as a point on the Poincaré sphere in which the horizontal, $\ket{H}$ (vertical, $\ket{V}$) rectilinear polarization forms the north (south) pole of the unit sphere as shown in Fig. 1(c) scully1997quantum . Thus any polarization can be described in terms of the rectilinear basis using two angles $\theta$ and $\phi$

[TABLE]

In this representation, the rectilinear polarization basis is given by:

[TABLE]

and two additional orthogonal bases, the diagonal linear and the circular polarization bases are given by:

[TABLE]

and:

[TABLE]

These bases are schematically described in Fig. 1(c).

Once the first photon is detected, then the probability rate to detect the first photon with polarization $P_{1}$ and the second cascading photon in polarization $P_{2}$ at time $t$ later is given by:

[TABLE]

where $p_{X^{0}}(t)=e^{-\frac{t}{\tau_{R}}}/\tau_{R}$ is the recombination rate of the exciton and $\tau_{R}$ is the exciton radiative lifetime. Note that the probability $P_{HH}$ ($P_{VV}$) that the QD emits both photons co-linearly polarized H (V) is given by

[TABLE]

as expected for an on-demand generated biexciton radiative cascade. The probability to detect the two photons in a given polarization state $P^{D}_{P_{1}P_{2}}$ is given by $P^{D}_{P_{1}P_{2}}=\eta^{2}P_{P_{1}P_{2}}$ where $\eta$ is the light harvesting efficiency of our experimental system schmidgall2014deterministic .

For polarization tomography measurements one detects two photons, which temporally belong to the same cascade, and projects the polarization state of each photon on the three orthogonal polarization bases. In at least one of the bases, two measurements on both polarization states are required. Thus $4\times 4=16$ independent time resolved polarization sensitive two photon correlation measurements are needed in order to fully characterize the temporal evolution of the two photons polarization density matrix james2001measurement . Considering only the polarization bases of Eq. 8-10 one gets 36 possible different two photon polarization sensitive probability rates, which divide essentially to 4 different cases (See Table 1). In the 16 cases where $\theta_{1}=\theta_{2}=\pi/2$ oscillatory temporal dependence of the probability rate is expected with a relative phase which is given by $\phi_{1}+\phi_{2}$.

The sample was grown by molecular beam epitaxy on $[001]$-oriented GaAs substrate. A layer of strain-induced InAs QDs was deposited in the center of an intrinsic GaAs layer. The GaAs layer was placed between two AlAs/GaAs distributed Bragg mirrors of quarter wavelength facilitating a microcavity. The microcavity design provides efficient collection of the light emitted due to recombination of QD confined electrons and holes from their respective lower energy levels. The QD is resonantly excited to the two photon biexciton absorption resonance using a synchronously pumped dye laser pulse with a repetition rate of $76\,\mathrm{MHz}$. The temporal width of the laser pulse is $\sim 12\,\mathrm{ps}$ and the spectral width is $\sim 100\,\mathrm{\mu eV}$. The resulting two photon emission is measured using a polarization selective Hanbury Brown-Twiss (HBT) arrangement as schematically described in Fig. 2(a). The emitted PL is divided by a non-polarizing beam splitter into two separated beams. A pair of liquid crystal variable retarders (LCVR) and a polarizing beam splitter on each beam is used to project the light on the desired polarization direction. The energy of the collected light on each beam is then filtered by a 1 meter long monochromator (MC) followed by a superconducting nanowire single photon detector (SNSPD) and a time-correlated single photon counter (TCSPC).

Results

In Fig. 2(b) we present polarization sensitive PL spectra of the photoluminescence from the QD under resonant excitation into the biexciton two-photon absorption resonance muller2014 ; brunner1994sharp . The energy of the exciting laser is tuned exactly between the $X\!X^{0}$ and $X^{0}$ spectral line energies. Each of these spectral lines is composed of two cross linearly polarized components split by $\Delta=34\,\mathrm{\mu eV}$ as can be clearly seen in Figs. 2(c) and (d), which present the $X^{0}$ and $XX^{0}$ spectral lines, respectively, on expanded energy scales. This energy splitting translates into an exciton precession time of $T_{P}=h/\Delta=122\,\mathrm{ps}$.

Our setup was carefully designed to facilitate polarization sensitive time resolved correlation measurements between photons emitted from the $XX^{0}$ and the $X^{0}$ spectral lines. The correlation measurements were performed after deterministic generation of the $XX^{0}$ biexciton using a $\pi$-area pulse resonantly tuned into the biexciton two-photon absorption resonance brunner1994sharp ; muller2014 . Prior to the generating pulse, a 4 ns long depletion pulse was used Schmidgall2015 , verifying that the QD is empty from charges and dark excitons, ready for biexciton generation. The depletion pulse was produced using a continuous wave Ti:Sapphire laser light, temporally shaped by an electro optical modulator (EOM). The modulator was synchronized to the pulsed, synchronously pumped ps dye laser, which generated the biexciton (see Fig. 2(a)).

In Fig. 3 the correlation measurements are presented as two dimensional images showing by the false color scale the number of events in which two photons are detected as a function of their detection times after the excitation pulse. Fig. 3 (a) shows the events when the two photons are H linearly co-polarized. Fig. 3 (b) and (c), show the events when the two photons are cross and co-circularly polarized, respectively. Fifteen more similar maps (not shown) were measured for different combinations of the two photon polarizations, providing the data required for constructing the temporal evolution of the two photon polarization density matrix. The temporal evolution of the signal integrated over the exciton detection time $t_{X^{0}}$ showed polarization independent simple exponential decay with a characteristic biexciton decay time of $260\pm 10$ ps (not shown). Therefore, in order to increase the measurement statistics, we summed over the detection times of the biexciton $t_{XX^{0}}$ in Fig. 3 and obtained the number of coincidences as a function of the time difference between the photon detections.

Typical measurements are presented by the blue symbols and error bars in Fig. 4. In Fig. 4 (a) the integrated data from Fig. 3 (a) where the two photons are H co-linearly polarized are presented and in Fig. 4 (c) the data are obtained from Fig. 3 (c) where the two photons are co-circularly polarized. In both figures, the green solid line presents the system temporal response as obtained by detecting the picosecond laser pulse, on both detectors. The system response is best-fitted by a Gaussian function with full width at half maximum of $42\,\mathrm{ps}$.

The solid black lines in Fig. 4 (a) and (c) represent the best fitted calculations using Eq. 11. We note here that in fitting all the 16 different polarization sensitive correlation measurements, only one fitting parameter was used - $\tau_{R}$ the radiative lifetime of the exciton. We used $\tau_{R}=410\pm 10$ ps. The exciton precession period of $T_{P}=122\pm 12$ ps was directly obtained from the spectral measurement of the exciton fine structure. Since all the correlation measurements were performed together, as the system automatically changed the polarization projections every minute, all the curves are automatically normalized and no attempt was made to fit their relative intensities. The overall coincidence accumulation rate of about $10^{-6}$ coincidences per pulse was also in agreement with the known light harvesting efficiency of our experimental setup schmidgall2014deterministic . The solid black lines represent the calculations using Eq. 11. The red lines overlaying the data are obtained by convolving the calculations by the system response function. Fig. 4 (b) and (d) show the difference between the data and the convolved calculations. These differences are within the experimental uncertainty and imply excellent agreement between the measured data and the convolved calculations.

In Fig. 5 (a) we present by solid black lines sixteen different polarization sensitive time resolved probability rates calculated using Eq.11 (or Table 1).These sixteen different polarization projections are required for tomographic reconstruction of the two photon polarization density matrix. For these calculations we used $T_{P}=h/\Delta=122$ ps as deduced from the PL spectra, and $\tau_{R}=410$ ps as the only fitted parameter. In Fig. 5 (b) solid blue lines represent the corresponding sixteen time resolved two photon coincidence rate measurements. Solid red lines in (b) overlaid on the measured data represent the calculations of Fig. 5 (a) convolved with the system response function from Fig. 4 (a) and (c). As can be seen in Fig. 5 (b), the agreement between the calculations and measurements is excellent.

The two-photon density matrix as function of time was reconstructed james2001measurement both from the measured data and from the model calculations only. The absolute values of four such matrices for various time differences after the detection of the biexciton photon are presented in Fig. 6(a). In both cases, the temporal window width, $\Delta T$, over which the data was integrated was set to $24$ ps. In Fig. 6(a), blue bars represent the measured (deduced from the data in Fig. 5 (b)) absolute values of the density matrix elements. Empty bars represent the matrix elements obtained from the fitted model calculations (deduced from Fig. 5 (b)), which consider the measured temporal response of our detectors.

The negativity of the density matrix as a function of time between the photons is presented in Fig. 6(b), in blue for the as measured matrices and in black for the calculated ones, considering the detectors temporal response. Here as well, the temporal window width $\Delta T$, was set to $24$ ps.

Fig. 6(c) presents the negativity of the two photon polarization density matrix as a function of the temporal window width, $\Delta T$. The solid blue line in the figure presents the negativity as deduced directly from the measured raw data. The solid black line presents the negativity deduced from the fitted calculations, considering the detectors temporal response. Shaded areas with corresponding colors in both Fig. 6(b) and (c) represent uncertainties of one standard deviation. Note the excellent agreement between the solid black line and the theoretical dependence predicted by Eq. 6, represented in Fig. 6(c) by the green solid line.

Summary

In summary, we have demonstrated experimentally that the biexciton-exciton radiative cascade in semiconductor quantum dots is an excellent deterministic source of maximally polarization entangled photon pairs. We showed that: a) The polarization states of the emitted two photons are maximally entangled during the whole radiative decay. b) The measured degree of entanglement between the polarization states of the two photons depends only on the temporal resolution by which the time difference between the two photon emissions is determined. A relatively simple arrangement, which provides periodic retardation change compatible with the exciton precession period, can therefore transform any quantum dot, into an on demand source of maximally entangled photons specht2009phase ; jones2006photon ; wang2010demand .

Acknowledgements.

We thank N. H. Lindner, O. Kenneth and J. A. Avron for helpful discussions and P.M. Petroff for the sample. The support of the Israeli Science Foundation (ISF), the Technion’s RBNI and the Israeli Focal Technology Area on “Nanophotonics for Detection” is gratefully acknowledged. This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 695188).