Entanglement measurements at zero fine structure splitting

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot

0 200 400 600

0 200 400 600

0 200 400 600

Coincidences(counts)

H HXX X

V HXX XX

0 10 10

-10

Delay (ns) D DXX X

D AXX XX

R RXX X

R LXX XX

0 10 10

-10

Delay (ns)

0 10 10

-10

Delay (ns)

Figure 3.19: Measurement of the entanglement fidelity with a reduced mea-surement set on QD1. Cross-correlation between biexciton (XX) and ex-citon (X) photons. VXX,X (HXX,X),DXX,X (AXX,X) andRXX,X (LXX,X) indicate vertical (horizontal), diagonal (antidiagonal) and circularly right (circularly left) polarized photons. The graphs for copolarized (red) and

cross-polarized (blue) photons are temporally shifted by 3 ns for clarity.

study on a second, randomly selected QD (QD2) (see Fig. 3.18 (c) and (d)) and obtain f⁺= 0.953(2),e= 0.960(2) andC= 0.919(4).

3.4.5 Background light correction

Now we are interested in answering the following questions: What is preventing the degree of entanglement from being ideal? And, most importantly, can QDs be really considered as a decoherence-free entangled photon source? While previous works have theoretically suggested that the answer to the latter question is positive [30, 92], an experimental demonstration of near-maximally entangled photons from QDs is still lacking. In order to answer these questions, we first have a closer look at the experimental setup. We identify three sources of errors: (i) The detector dark counts, (ii) the retardance of the waveplates used for the reconstruction of the density matrix, and (iii) background photons.

Subtracting the dark counts leads to a 0.3% improvement for the fidelity and 0.8% for the concurrence for both measured QDs (the dark count rate of our detector is < 20 Hz). (ii) The retardance of the waveplates is instead a more delicate issue. According to the formalism presented in Ref. [72], a tomographically complete measurement set is required for the calculation of the density matrix. Due to imperfections of the waveplates used to project the two-photon state into the different bases, the real measurement basis will deviate from the one assumed in the calculation. Therefore, we incorporate the retardance of our achromatic waveplates at the emission wavelength of the QD into the computation with 0.516 waves and 0.258 waves (according to the data sheet provided by the constructor Thorlabs, Inc.) for the HWP and QWP, respectively. The position accuracy of the fast axis, which in principle can also influence the projective set, is 0.02^◦ and thus can safely be neglected. In Fig. 3.20 the change in fidelity and concurrence, respectively, versus the waveplate retardance used for the computation is illustrated. We want to point out that it is not possible to find the correct projection

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot

set by maximizing the fidelity (concurrence) via varying the retardance, because the mapping of the correlationsnν to the elements in the density matrix is not an unique transformation. Hence, to optimize the parameters for the reconstruction knowing the exact value of the HWP and QWP retardance is obligatory. Taking into account the

0.48 0.52 0.56 0.60 0.20

0.24 0.28 0.32

HWP retardance (waves)

0.844 0.877 0.910 0.943 0.977 fidelity

QWPretardance(waves)

0.48 0.52 0.56 0.60 0.20

0.24 0.28 0.32

HWP retardance (waves) 0.844 0.860 0.893 0.927 0.977

QWPretardance(waves) concurrence

a b

Figure 3.20: Effect of the waveplate retardance onto the measured degree of entanglement. (a) Entanglement fidelity and (b) concurrence versus the half-waveplate (HWP) and quarter-waveplate (QWP) retardance.

dark counts and the effect of the waveplates, the imaginary elementshHH|ρ|V Vi and hV V|ρ|HHi of the DM shown in Fig. 3.18 disappear and the resulting values for fidelity and concurrence aref⁺= 0.968(2) and C= 0.936(5) and f⁺ = 0.958(2) and C= 0.925(5) for QD1 and QD2, respectively.

To investigate the effect of (iii) we first derive a statistical model to estimate the influence of the background onto a polarization resolved cross-correlation measurement between XX and X. Therefore, we assume a photon pair extraction efficiency<<1 (we estimate it to be around 1%). Further, to increase the readability we do not handle the detection and transmission efficiencies of the setup during the calculation as their contributions will anyway cancel out. The probability to create a X or a background photon and transmit it through a polarizer in the stateiwithin the spectral range of the X notch filter is given by:

px,i=axtx,i (3.59)

and

p_bx,i =a_bxt_bx,i, (3.60)

respectively. Herea_x(bx) is the generation probability of a X (one or more background) photon(s) within a time bin considered in the g⁽²⁾(t) and t_x(xb),i its probability to be transmitted through the analyzer statei. In a Hanbury-Brown-Twiss like experiment in a start-stop mode the g⁽²⁾(0) - autocorrelation function att= 0 is given by

g⁽²⁾(0) =N(cx,i+px,ip_bx,i+p_bx,ip_bx,i), (3.61) where N is a constant andcx,i is the probability to get a coincidence by two emitted X photons. Hereby, we assume that c_x,i= 0 as the X transition emits only single photons

[101]. Inserting Eq. 3.59 and Eq. 3.60 in Eq. 3.61 yields:

g⁽²⁾(0) =N(a_xt_x,ia_bxt_bx,i+a²_bxt²_bx,i) (3.62) In the same way the first side peak of the g²-function can be calculated to

g⁽²⁾(τ) =N(p_x,i(0)p_x,i(τ) +p_x,i(0)p_bx,i(τ) +p_x,i(τ)p_bx,i(0) +p_bx,i(0)p_bx,i(τ))

=N(a²_xt²_x,i+a²_bxt²_bx,i+ 2axtx,iabxtbx,i)

(3.63)

assuming thatp_x,i is constant over time. Due to the low background photon generation rateabx, the ratio between g⁽²⁾(0) andg⁽²⁾(τ) can be expanded in a Taylor series around a_bx= 0, which leads to

g²(0)

g²(τ) = a_bxt_bx,i

a_bxt_bx,i+a_xt_x,i ≈ a_bxt_bx,i

a_xt_x,i −t²_bx,ia²_bx

a²_xt²_x,i +O[a_bx]³. (3.64) By taking into account that ax ≈1 in π−pulse and abx << ax we can safely neglect the quadratic and other higher order terms. Further,t_x,i = ¹₂, if we assume two modes i={H, V} for an unpolarized QD source and thus Eq. 3.64 simplifies to

g⁽²⁾_X,i= g²(0)

g²(τ) = 2·t_bx,ia_bx. (3.65) For the XX a similar expression can be derived:

g_XX,i⁽²⁾ = 2·tbxx,iabxx. (3.66) This allows us to estimate the amount of time correlated background emission by mea-suring theg⁽²⁾(t)- autocorrelation function for X and XX separately in all 6 polarization bases (H,V,D,A,R,L). To correct the raw data with respect to the background emission, we have to calculate the probability b_ik to get a coincidence related to a background photon in the polarization passing an analyzer in the state i,k. It follows that

b_ik = axxtxx,ig_X,k⁽²⁾

2 +axtx,kg_XX,i⁽²⁾

2 +g_XX,i⁽²⁾ g⁽²⁾_X,k

4 (3.67)

with a_xxt_xx,i =a_xt_x,k≈ ¹₂ and g⁽²⁾_XX,ig_X,k⁽²⁾ ≈0 this equation simplifies into b_ik= 1

4(g_X,k⁽²⁾ +g_XX,i⁽²⁾ ). (3.68) The number of coincidences in the central peak of a cross-correlation measurement in the basisi, k is then given by

n_c,ik=N(4^e hi, k|ρ|i, kig⁽²⁾_X,XX+ b_i,kg_b⁽²⁾+Rg_R⁽²⁾)

(3.69)

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot

where Ne is a constant, ρ is the two-photon density matrix of the entangled state,g_X,XX⁽²⁾ , g⁽²⁾_b andg_R⁽²⁾ are the values of the cross-correlation function between unpolarized X and XX photons, X and XX background photons and non entangled X and XX photons at τ=0, respectively, andR is the probability to generate a non entangled photon pair not related to background. Since the background is dominated by coherent laser light, it follows that g_b⁽²⁾ = 1. Furthermore, we assume that g⁽²⁾_R ≈ g⁽²⁾_X,XX, which reflects the emitter properties of the QD. The number of coincidences in a side peak instead is given by

n_s,ik =Ne(1/2 +b_i,k+R). (3.70) The relation

nc,ik

n_s,ik = pe+b_ik+R

1

2 +b_ik+R, (3.71)

where p_e is the probability to have a coincidence from an entangled photon pair, allows us to get an estimate forR and to calculate ⁿ_n^c,ik

s,ik without background to n⁰_c,ik

n⁰_s,ik = pe+R

1

2 +R. (3.72)

In order to correct the density matrix, we perform additional autocorrelation measure-ments for XX and X in all the polarization bases (see Tab. 3.4.5) needed to reconstruct the density matrix and find that the background is primarily linear vertical polarized.

This confirms that the background photons originate from the excitation laser, which is also polarized vertically. With the help of our statistical model we calculated the corrected density matrix and found a fidelityf⁺ = 0.978(5), which is an increase of 2.6%.

The largest eigenvalue improves to e = 0.981(5) and the concurrence to C = 0.97(1).

From a fundamental point of view, it is interesting to check whether there is a remaining decoherence mechanism occurring during the cascade decay. To do so, we investigate the degree of entanglement as a function of the FSS by using a spin scattering model, as discussed in the next section.

3.4.6 Spin scattering model

We start by considering a possible residual FSS S0=250 neV (corresponding to the resolution of the used setup), and a background according to theg⁽²⁾ measurements as discussed above. Additional FSS fluctuations are expected because of the fluctuating Overhauser fieldBOH(t) [108]. Since our FSS measurements are performed on timescales of seconds, we cannot quantify such fluctuations experimentally. To estimate their amplitude we assume a maximum field ofB_max= 4 T [89] with a standard deviation of σ=Bmax/√

N ≈6 mT, whereN is the number of spin-3/2 nuclei in the QD material (N ≈4·10⁵ for our QDs). For QDs, the effect of the Overhauser field on the FSS is dominated by its vertical component inz-direction (Faraday-configuration) [96], because

measurement g⁽²⁾(0) X_H 0.009(3) X_V 0.025(5) X_D 0.017(4) XA 0.010(3) XR 0.015(3) XL 0.010(4) XXH 0.003(2) XXV 0.036(7) XX_D 0.012(3) XX_A 0.014(4) XX_R 0.015(4) XX_L 0.021(5)

Table 3.3: Polarization-dependent autocorrelation measurements. The table shows the values of the second order autocorrelation function at zero time delay (g²(0)) for biexciton (XX) and exciton (X) for linear horizontal (H), vertical (V), diagonal (D) and antidiagonal (A) as well as circular right (R) and left (L) polarized photons.

0 0.1 0.2 0.3 0.4 0.5 Re(ρ)

0 0.1 0.2 0.3 0.4 0.5 Im(ρ)

VV

VH HV HH VV

VH HVHH

VV

VH HV HH VV

VH HVHH

0 0.1 0.2 0.3 0.4 0.5 Re(ρ)

0 0.1 0.2 0.3 0.4 0.5 Im(ρ)

VV

VH HV HH VV

VH HVHH

VV

VH HV HH VV

VH HVHH

a b

Figure 3.21: Background photon corrected density matrix of QD2. Real (a) and imaginary part (b) of the two-photon density matrix measured on QD2.

The gray bars show the results calculated from the raw data, while the red bars are related to the matrix after the correction for background photons, waveplate retardance and dark counts.

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot

of the small in-plane g−factors. The effective FSS is then according the Hamiltonian in Eq. 3.42 given by ˜S=S₀+µ_B(ge,z +g_h,z)B_OH,z(t), withe⁻ (HH) g-factor ge,z =−0.15 (g_h,z = 1.1) according to Ref. [109]. We simulate the fidelity in a time integrated experiment by calculating the density matrix ˆρB using the entangled state in Eq. 3.31 and the expression for S from above. The Overhauser field is typically varying on a timescale ofµs so that we can assume a constant value of B_OH,z, while integrating over the time window [0,∞] in Eq. 3.32. However, the integration time to obtain an useful number of coincidences nν is in the order of 15 minutes. In fact, theses coincidences represent an average over different magnitudes ofB_OH,z. We take this into account by calculating an average fidelity by assuming a Gaussian distribution of the Overhauser field:

f⁺= Z b

a

f⁺( ˆρ(BOH,z)) 1 σ√

2πe^{−12(BOH,zσ )}dBOH,z. (3.73) By numerical integration on the interval [−3 T,+3 T] we obtain a fidelityf⁺ = 0.99 and find that the measurement data for QD1 (QD2) still deviates by 4.5 (2.4) standard deviations from our theory.

This result shows that an additional dephasing mechanism could play a role, where we suggest spin scattering as a possible source. For verification we make use of the spin-scattering model as presented in Ref. [75]. Therefore, we introduce a factorγs= _1+τ¹

1/τss, where τ_ss is the characteristic scattering time, describing the fraction of photons un-affected by exciton spin scattering. As discussed in Sec. 3.1 such a spin scattering randomizes the polarization correlation between XX and X. Starting from the density matrix given Eq. 3.49 we expand the background model by including the amount of spin scattered photons via γ_s into the density matrix:

ρˆ⁰_kss= 1 4











1 +kγs 0 0 2kγsz^∗

0 1−kγ_s 0 0

0 0 1−kγ_s 0

2kγsz^∗ 0 0 1 +kγs











. (3.74)

We then find an explicit formula for the fidelity by using Eq. 3.19:

f = 1

4(1 +kγs+ 2kγs

1 + (^γsSτ_¯h ¹)²). (3.75) The model can be used to obtain an estimate for τss by measuring the trend of the fidelity versus the FSS. Therefore, we use one leg of the piezoelectric device to increase the FSS. However, this is a challenging task, as tuning with a single leg typically affects not only the FSS, but also the emission energy. As a consequence, one also has to adapt the Bragg gratings BG1 and BG2 by rotating them. Thereby, the polarization response of the setup is altered and the fidelity measurements become inconsistent. In principle, there are two solutions to this problem: (i) one can use all three legs of the device to keep the energy constant or (ii) for every value of the FSS the polarization compensations

has to be readjusted. Both strategies are highly time consuming making these kind of measurements impossible in a reasonable amount of time. Nevertheless, with QD1 we found that one leg of the piezoelectric actuator is mainly changing the FSS, while leaving the photon emission energy nearly unaffected. The reason for this unexpected behavior is that the stress configuration exerted by the legs is strongly position depended due to a non-uniform bonding of the membrane [58]. Finally, this allows us to measure the curve presented in Fig. 3.22. To reduce the number of free parameters in Eq. 3.75, we measure

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1.0

theory incl. bg. t_ss®¥ fit

fit with rotation Fidelit+y f

FSS (µeV)

0 1

0.9 1.0

FSS (µeV) Fidelit+y f

Figure 3.22: Spin scattering model. Entanglement fidelity f⁺ versus fine structure splitting (S) for QD1. The black data points represent the measurement data. The red curve is a fit according to Eq. 3.75, while the green curve is a fit taking into account a rotation of the state, as explained in the text.

The blue curve takes background laser-light (bg) into account, but no spin dephasing (τss→ ∞). The inset shows the evolution of the fidelity versus the FSS for background-free entangled photons and no spin dephasing (orange line). The data points (black) are the measured data from the main figure, but corrected for the estimated background.

the X decay time and found τ₁= 241(10) ps. Further, we measure the unpolarized g⁽²⁾ for XX and X, respectively, to g⁽²⁾_XX(0) = 0.014(3) and g⁽²⁾_X (0) = 0.008(2) and estimate k= 0.978(4) according to Eq. 3.49. The fit of the measurement data (red curve in Fig.

3.22) yields a value for the spin dephasing ofτ_ss= 11(8) ns. In addition to the fit also the theoretical curve without the presence of spin dephasing (τss→ ∞) but in presence of the measured laser-photons background is plotted (see blue curve). The latter one (theoretical curve for decoherence-free entanglement) shows a larger deviation at small FSS, while the former (fit) reveals a deviation at FSS>2µeV. The deviation between fit and measurement can be explained by the fact that the fidelity is only estimated using the correlation visibilitiesC_µ in the linear, diagonal, and circular base (see Eq. 3.23). In

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot

case of zero FSS the DM as well as Eq. 3.23 yield the same fidelity. However, this does not hold if the FSS6= 0 and the entangled state contains an additional phase factorω introduced by the measurement setup. The entangled state than can be written as:

|χ⁰_roti=U(ω)·1/√

2(|H_XXi |H_Xi+e^iSth¯ |V_XXi |V_Xi), (3.76) where U is a rotation matrix as a function of the angle ω.

Now we want to derive a model considering ω in the fidelity as well as taking the background and X - spin scattering into account. Starting from the definition of the density matrix in Eq. 3.8, we can rewrite any density matrix in the form:

ρˆ=^X

i

p_iρˆ_i, (3.77)

where ˆρ_i is a density matrix of a pure stateψ_i. As the matrix multiplication is distributive we can write the fidelity f⁺ of the matrix ˆρ according to its definition in Eq. 3.19 as:

f⁺=T r( ˆρ·ρˆ0) =^X

i

piT r( ˆρi·ρˆ0) =^X

i

pif_i⁺, (3.78) which shows that the fidelity is additive. The fidelity measured via the reduced set is then³:

f⁺ =^X

i

p_i(hHH|ρˆ_i|HHi+hDD|ρˆ_i|DDi − hRR|ρˆ_i|RRi), (3.79) which we can rewrite to:

f⁺=^X

i

pi(| hHH|ψ_ii |²+| hDD|ψ_ii |²− | hRR|ψ_ii |²). (3.80) Finally, we have to think about which states contribute to the density matrix and their probability p_i. In total, we consider three contributions: (i) photons stem-ming from spin-scattered XX-X transitions with the quantum states ˜φij = U(ω)|iji) (ij =HH,HV,VH,VV) and p= ^1−γ₄ ^s. (ii) background light, where we assume the same states as for (i) andp= ^1−k₄ . This is true for unpolarized background light. However, for the fit the polarization of the background does not play a role. Although, the matrix elementshii|ρ|iii (ii=HH, DD, RR) are sensitive to the background polarization the fidelity f⁺ depends only on the its total amount. Therefore we can estimate k by an unpolarizedg⁽²⁾ measurement. (iii) the entangled light with the state|χ⁰_rotiand p=kγ_s. Finally, we obtain a function for the fidelity of the mixed state:

f_total⁺ =kγ_sf⁺(|χ⁰_roti) + 1

4(1−kγ_s)^X

ij

f⁺( ˜φ_ij) (3.81) In Fig. 3.23 the fidelity versus FSS with ω varying between −4^◦ and +4^◦ is plotted by using Eq. 3.81. Already a small value of ω can dramatically influence the values of the fidelity at S > 0. If we use Eq. 3.81 to fit the data points in Fig. 3.22 (green

3We use the definition off⁺according to Eq. 3.22.

+4°

-4°

0°

0 2 4 6

0.5 0.6 0.7 0.8 0.9 1.0

Fidelity

FSS (µeV)

Figure 3.23: Fidelity of a rotated state. The plot shows the deviation between the unrotated (ω = 0^◦) entangled state (green curve) and a rotation up to ω=±4^◦ (red and blue curve, respectively). The fidelity calculated from the density matrix is robust against such a rotation.

curve) we obtain ω=−9(4)^◦ andτ_ss= 14(10) ns. The origin ofω can be explained as follows: The fine tuning of the liquid crystal retarders via the correlation measurements allows to make the X and XX light path equal in their polarization response with a high accuracy. However, an overall rotation can only be corrected by a precise measurement of the polarization response, which is given by the accuracy to determine a Stokes vector and additionally the approximation method to calculate the correction parameters. The large error of ±10 ns in τss does not allow us to draw a definite conclusion about the origin of the spin scattering. A plausible explanation is the interaction between the confined X and charges in the vicinity of the QD [34, 30]. If we use Eq. 3.49 to estimate a background correction for the data points in Fig. 3.22 the fidelity at FSS= 0 shows a significant deviation from the ideal case (see inset Fig. 3.22). By considering the measured X lifetime of QD 2, which isτ1= 290(5) ps, and the fitted spin dephasing time, we can estimate the highest achievable fidelity using Eq. 3.75, with S= 0 and k= 1 to f=0.98(1), which is within the error of the corrected fidelity presented above.

3.4.7 Discussion

The results presented in this section show for the first time that semiconductor QDs can generate nearly-maximally entangled photons pairs. Although it already has been speculated that this is the case [30], there was no experimental verification before. By looking at the concurrence (fidelity), the level of entanglement reported here represents a 10% (4%) increase as compared to the best QD source of entangled photons to date [22], which we presented in Sec. 3.3. Further, even with temporal post selection such a high degree of fidelity has not been observed [27]. However, the data indicate the presence of an almost-negligible, albeit non-zero, decoherence mechanisms, likely related to spin-scattering. Here we assume that the interaction with charges in the vicinity of the QD play a major role [34]. The origin of these free charges is not clear yet, but the

3.4 Observation of nearly maximal entangled photons in a GaAs quantum dot

following scenarios are possible: (i) The charges stem from defect or/and impurity states, which are populated by the resonant laser or the cw support laser. (ii) The cw laser itself is exciting charges in the barrier, which scatter with the confined X (excess charge scattering). (iii) Instead of a radiative emission during the decay of an excited state an Auger recombination takes place, which releases charges from the confinement potential to the barrier. We want to point out that we observe charged states under resonant excitation (see Fig. 3.7), which are known to initiate such a process [110]. In principle, one could exclude (ii) by repeating the measurements without using an above band gap support laser. However, this leads to a loss in count rate, which in the end prevents us to perform the measurements. To verify the other hypotheses additional experiments are required, which are beyond the scope of this work. Nevertheless, we suggest the use of a photonic structure to overcome this problem. In particular, by increasing the Purcell factor from ≈ 1 in the used device to 3 - a value which may be achieved in photonic structures compatible with nondegenerate entangled photon generation [47] -the expected entanglement fidelity would surpass 0.99 and lift QD entanglement to -the same level as PDC [111, 112, 113, 114]. It is also worth to mention that, compared to previous works [41, 40], the device reported here uses membranes with a thickness of 30 microns instead of few hundred nanometers. Such an approach is compatible with the processing steps required to fabricate state of the art photonic structures [115] and would allow for boosting the flux of photons so as to realize the ideal source of entangled photons.

4 Two photon interference experiments with GaAs quantum dots

4.1 Fundamentals of two-photon interference

Two-photon interference —also referred as the Hong-Ou-Mandel effect [116]— is a quantum mechanical effect, which occurs when two identical single photon wave packets enter a 50:50 beam splitter, one in each input port. Due to an interference or, more precisely, a coalescence of the wave functions, one observes that the two photons always leave the beam splitter pairwise on one of the outputs. We will discuss the fundamentals of this effect and its application for characterization of the indistinguishability of photons emitted by QDs [117] according to the experimental scheme in Fig. 4.1. At first, we want

1

BS1 BS2

QD

D1

D2 3

4 2 δt

δt Laser

Figure 4.1:Schematics of a two-photon interference experiment. The QD is ex-cited via two laser pulses with a temporal separation ofδt, which consequently gives a sequence of two single photons with the same time delay. Subsequently, an unbalanced Mach-Zehnder interferometer (red dotted frame) is used to overlap the two photons in space for the two-photon interference at the beam splitter BS2. The resulting photon statistics is recorded at the outputs of BS2 with the detectors D1 and D2.

to focus only on the two-photon interference (TPI) itself and ignore for the moment the

properties of the single photon source. Therefore, we restrict our considerations to the green dotted area in Fig. 4.1. Two single photon wave packets are impinging on the 50:50 beam splitter (BS2) at input port 1 and 2, respectively. We assume that the two photons are identical, so that we can describe the quantum state with a simple Fock-state picture as:

|1₁,1₂i=a^†₁a^†₂|0i, (4.1) with the photon-creation operator a^†_i and the vacuum state |0i. The index i = 1,2 describes the mode and the beam splitter input, respectively. In quantum mechanics the beam splitter can be described by an unitary transformation of the input states:

a^†₁= 1

√2(a^†₃+a^†₄) (4.2)

a^†₂ = 1

√

2(a^†₃−a^†₄), (4.3)

where 3 and 4 denotes the output ports of the beam splitter. We want to point out, that this unitary transformation depends on the construction of the beam splitter (half transparent mirror, evanescence wave beam splitter, ... ). Nevertheless, an observation of TPI is possible with any kind of beam splitters. By expressing a^†₁ anda^†₂ in Eq. 4.1 by Eq. 4.2 and Eq. 4.3 we obtain:

|1₁1₂i= 1

2([a^†₃]²−[a^†₄]²)|0i= 1

√2(|2₃0₄i − |0₃2₄i). (4.4) The resulting entangled state reveals that both input photons end up in the same output mode. Hence, either the detector D1 or detector D2 detects two photons simultaneously, while the situation of detecting a photon on each detector is forbidden.

As already implied, the interference takes only place if the two photons are indis-tinguishable in all degrees of freedom (polarization, spatio-temporal mode structure, and energy). In the following we want to expand the discussion to a more general scenario where we can model the properties of the interfering wave packets. Therefore, we follow the calculations from Ref. [118] to find an expression for the joint detection probability P34(τ), which describes the probability of detecting a photon on output 3 and 4 with a time delay τ. First of all, we have to choose a spectral distribution function describing the wave package of the photons. For a QD system we choose a Lorentzian distribution [34]:

χ(ω) = 1 π

∆

(ω−ω₀)²+ (∆/2)², (4.5)

where ω0 is the central frequency and δ the linewidth. For further calculation we need to derive the spatio-temporal mode-function of the photon i, which we obtain by a Fourier transformation of Eq. 4.5:

ζi(t) = 1 2π

Z

χi(ω)e^−iωtdω. (4.6)

4.1 Fundamentals of two-photon interference

The mode function allows us to write the electric field operators as:

E_i⁺(t) =ζ_ia_i (4.7)

and

E_i⁻(t) =ζia^†_i. (4.8)

Similar to Eq. 4.2 and 4.3 we can describe the transformation of the field operators at the the beam splitter inputs (E₁⁺ and E₂⁺) and outputs (E₃⁺ and E₄⁺) according to:

E₃⁺(t) = 1

√

2[E₁⁺(t) +E₂⁺(t)] (4.9) and

E⁺₄(t) = 1

√2[E₁⁺(t)−E₂⁺(t)]. (4.10) If we consider that the two photons at input 1 and 2 are equally polarized and described by the mode ζ₁(t) and ζ₂(t) the input state reads |ψ_ini =a^†₁a^†₂|0i. Consequently, the joint detection probability is given by:

P₃₄=h0|a₁a₂E₃⁻(0)E₄⁻(τ)E₄⁺(τ)E₃⁺(0)a^†₁a^†₂|0i. (4.11) By using Eq. 4.7, 4.8, 4.9 and 4.10 we can write it in the form:

P34= 1

4|ζ₁(τ)ζ2(0)−ζ2(τ)ζ1(0)|²

=|ζ₁(τ)ζ2(0)|²+|ζ₂(τ)ζ1(0)|²−2|ζ₁(τ)ζ2(0)||ζ₂(τ)ζ1(0)|.

(4.12) From this equation we immediately see that for τ = 0 we always obtain P₃₄ = 0 independent from the from of ζ₁ andζ₂. However, if τ >> τ_c, where τ_c is the mutual coherence time of the incoming photons, the last term in Eq. 4.12 averaged over an ensemble of different photons becomes zero [118]. In this case the joint detection probability reduces to:

P34= 1

4(P1(0)P2(τ) +P2(0)P1(τ)) (4.13) with P_i(t) =ζ_i^∗(t)ζ_i(t). We can interpretP₃₄ as the joint detection probability of fully distinguishable photons.

In the following, we use Eq. 4.12 to simulate the TPI of photons from a non-ideal source. For a QD we are particularly interested in the effect of an energy and linewidth mismatch, respectively, among the interfering photons as well as spectral diffusion. For the simulation we use a linewidth of 10 µeV and a central emission energy of 1.58 eV, which are typical values for the investigated QDs [36]. In Fig. 4.2 (a)P34 for different energy mismatchesδE between the two interfering photon wave packets are shown. For a perfect energy matchP₃₄ equals zero at any value ofτ, while for a finite δE we observe

0.0 0.5 1.0

P34(a.u.)

t (ps) dE=0 µeV dE=5 µeV dE=10 µeV dE=20 µeV

-80 -40 0 40

t (ps) dD=5 µeV dD=10 µeV dD=20 µeV

V=1 V=0.8 V=0.5 V=0.2

V=0.96 V=0.89 V=0.75

80

a b

-80 -40 0 40 80

Figure 4.2: Simulation of two-photon interference with non-equal wave packets.

(a) The solid lines show P₃₄ for interfering wave packets with a difference in the central energyδE. The black dotted line gives the trend ofP34, for the parameters as given in the text. (b) Same as in (a), but with a difference linewidth (δ∆) of the photon wave packets atδE = 0. The calculated values for the TPI visibilityV are given in the plots.

non-zero values for τ 6= 0. However, at τ = 0 P₃₄ always equals zero independent from the energy mismatch (see Eq. 4.12). The same behavior is observed for different values of linewidth among the photons as shown in Fig. 4.2 (b). In a QD a typical dephasing mechanism is spectral diffusion via fluctuating electric fields. Such fluctuations, which origin from charges or defects in the vicinity of the QD [119], cause a Gaussian broadening of the spectral line. We simulate this dephasing mechanism by using Monte Carlo method, where we vary the energy E0= ¯hω0 (withδ∆ = 0) of interfering photons within a Gaussian distribution with a variance σ. The simulation results of this inho-mogeneous broadening forσ = 20µeV and 30µeV is shown in Fig. 4.3. We used 10000 samples for the calculation. The trend of P34 is comparable to the one shown in Fig.

4.2 (a), which is expected as both originate from an energy mismatch between the photons.

According to the simulations, we expect in a TPI experiment non-ideal values of P34 due to nonidealities of the QD system and consequent non-perfect photon indistinguishability.

Hence, it is necessary to define a measure for the degree of indistinguishability of photons.

Therefore we introduce the interference visibility V, which is given by:

V = A−A

A , (4.14)

where A andA are the integrated areas ofP₃₄ andP₃₄, respectively. As already men-tioned the latter one describes the joint detection probability of a fully distinguishable photon pair as one would expect for two cross-polarized photons. In Fig. 4.2 and Fig.

4.3 the trend of P₃₄ is plotted (black dotted line) for δE = 0, δ∆ = 0 and σ = 20

4.1 Fundamentals of two-photon interference

-80 -40 0 40 80

t (ps) s=20 µeV s=30 µeV

V=0.35 V=0.25

0.0 0.5 1.0

P34(a.u.)

Figure 4.3:Simulation of two-photon interference with spectral wandering.

The solid line shows a simulation of P₃₄ assuming a Gaussian distribution of the central wavelength E0 of the interfering photons for different values of standard deviation. The black dotted line is gives the trend of P₃₄. The simulation parameters are given in the text.

µeV, respectively. The calculated values of V are given in the plots. As expected, V is dropping with an increasing mode mismatch between the interfering photons. The visibility V can be interpreted in the following way: If we perform a TPI experiment with a stream of single photon pairs, V is the probability of finding an indistinguishable photon pair within this stream. The quantityV is then a figure of merit to characterize the ability of a photon source to generate indistinguishable photons.

In the following, we want to discuss the scheme of a TPI experiment with photons from a QD. The full setup is shown in Fig. 4.1. As already mentioned the X (XX) transition gives at most a single photon per excitation pulse. However, two photons impinging simultaneously on the beam splitter are required for a TPI. By interfering the a XX-X pair no conclusion about indistinguishability can be drawn as they differ by the binding energy. Therefore, the QD is excited with two laser pulses separated by a delay time δt. This leads to the generation of two single XX and X photons also delayed byδt. A monochromator (not shown in the sketch) allows us to select either the XX or X photons. We use an unbalanced Mach-Zehnder interferometer (see red dotted frame in Fig. 4.1) with a path length differenceδl=c·δt, to ensure a temporal overlap between the two wave packets at the beam splitter BS2. In Fig. 4.4 (a) a coincidence measurement on the outputs of a beam splitter with co- (red) and cross-polarized (gray) XX photons from a QD under pulsed resonant two-photon excitation with δt= 2 ns is shown. The region around zero delay consist of 5 peaks (labeled as 1-5), where the peaks 1, 2, 4 and 5 are due to the coincidences of photon pairs not arriving at the same time at the interference beam splitter BS2. This happens because there is a 50 % chance of

-15 -10 -5 0 5 10 15 Delay (ns)

Coincidences (a. u.)

Co-polarized Cross-polarized

1 2 3 4

5

-5 0 5

Delay (ns)

1 3

4

5 2

a b

Coincidences (a. u.)

Figure 4.4: Two-photon interference under pulsed two-photon excitation. (a) Two-photon interference with co-polarized (red) and cross-polarized (grey) XX photons. (b) Fit of the five central peaks (labeled as 1-5 in (a)). The coloured lines are the fits of peak 1 (orange), 2 (green), 3 (red), 4 (green) and 5 (orange), respectively. The black line is the sum of all single peak fits.

The areas under the peaks 2, 3 and 4 are used for the calculations of the two-photon interference visibility.

the early (late) generated photon to take the short (long) path in the Mach-Zehnder interferometer [117].

The central peak (3) represents the coincidences of photons arriving at the same time and allows us to calculateV according to Eq. 4.14, withA (A) its area in co-polarization (cross-polarization). We want to point out that peak 3 is not zero at τ = 0, which is in contradiction to the simulations shown above. The reason for this is the limited time resolution of the used detectors, which is in the order of 500 ps. A closer inspection of the peak (3) reveals that there is some overlap with the neighboring peaks (2) and (4). As a consequence one would underestimate V by calculating the area by integration.

To avoid this problem we fit the experimental data as shown in Fig. 4.4 (b) using the following equation:

f(t) =y0+

5

X

i=1

2Aiw

π·(w²+ 4(t−(t₀+i·d))²) (4.15) where y0 is the offset,Ai the area of peaki,t0 the position of the first peak,w the width of the peaks and d the temporal distance between the peaks. From the experimental conditions we expect that the distancedbetween the peaks is the same and all peaks should have the same width. The width is mainly determined by the time jitter of the detector, therefore one would expect a Gaussian peak function in Eq. 4.15. However, after testing several fit functions it turns out that the one in Eq. 4.15 matches best to the experimental data. In general, one would also expect that peak 1 and 5 as well as 2 and 4 are equal in intensity. However, we have to consider the slightly different intensity between the two excitation pulses as well as the different detection efficiency for both

4.2 Two-photon interference experiments with GaAs quantum dots

fiber outputs. This is taken into account by leaving the Ai as free parameters. Then, we can calculate the TPI visibility by using (see Ref. [117, 21]):

V_TPE = 1− 2·A₃ A2+A4

. (4.16)

4.2 Two-photon interference experiments with GaAs quantum dots

4.2.1 Study of the two-photon interference visibility at short time scales

-5 0 5 -5 0 5

Delay (ns) Delay (ns)

X XX

XX X

Coincidences (a.u.) Coincidences (a.u.)

a

b

Figure 4.5:Two-photon interference experiments with a quantum dot. Two-photon interference with co-polarized Two-photons from two different QDs around zero time delay. The data for both the X and XX co-polarized photons are reported. The dashed lines are Lorentzian fits of the peaks. The solid lines show the total sum of the single fits according to Eq. 4.15.

The measurements are performed with an experimental setup as sketched in Fig. 4.1.

The QDs are resonantly excited every 12.5 ns by two π-pulses separated by δt= 2 ns and emit ideally two XX–X photon pairs per cycle via the XX-X cascade. As discussed in the previous chapter, we use a CW laser at low power to improve the dressing of the XX state. This method not only allows us to increase the detector count rate, it also improves the TPI visibility for XX and X photons as shown by Reindl et al. in Ref. [56].

The effect is supposed to be more dominant for the X as here an improvement of about 20% has been observed, while XX only 8% has been reported.

Im Dokument GaAs quantum dots: A Dephasing-Free Source of Entangled Photon Pairs / submitted by DI Daniel Huber, BSc (Seite 65-125)