Inference of the photon probabilities for obtaining y c and ∆

BSField 2

8.9 Inference of the photon probabilities for obtaining y c and ∆

frequency ofE_auxis the same as the single photon’s. To stabilizeβ_1,3andβ_2,4, we set(λ/2)_vat0^◦and create interference fringes at the outputs of BS₁and BS₂. Phase-modulation spectroscopy allows us to lock the relative phases so that a high contrast interference (V >0.99) is achieved for quantum fields (Fig. 8.5). The control of the relative path lengths for the modes, as well as their modulation, is afforded by piezoelectric fiber stretcher modules (Fig. 8.6a, see also Appendix A for the locking servo) located between PBS1and BS1, and PBS2and BS2, in Fig. 8.1c. These devices provide up to50×2πof dynamic range enabling the interferometers to remain continuously locked for several days (Fig. 8.6b).

Importantly, to avoid noise associated with the auxiliary laser in the single-photon detectors{D1,· · · , D4}, E_auxmust be filtered out. In our work, phase stabilization is performed asynchronously with entanglement generation and its verification in the fiber-based network of interferometers (Appendix A). This eliminates the need to wavelength filterE_aux as was necessary in previous experiments²⁷. During the21ms of our40 Hz experimental cycle that the MOT is activated,E_auxis switched on, the output modes of the sum uncer- tainty setup are directed toward an auxiliary set of detectors with MEMS fiber multiplexers (Sercalo), and our servo electronics stabilizeβ1,3 andβ2,4. To prepare for measurements of∆, we switch offE_aux and reroute the output modes to the single-photon detectors with the MEMS multiplexers, and we use dynamic polarization rotators (nematic liquid-crystal waveplates from Meadowlark^b) to set(λ/2)vat22.5^◦. Further, we apply calibrated feedforward signals to the servo electronics that can precisely scan the relative phases of modes{1,3}and{2,4} to explore the dependencies of our∆measurements. Fig. 8.5 shows the number of photons (n_c) detected at D1and D2as a function ofβ_1,3. Here, all other relevant optical phases in our setup were optimized to achieve minimum∆. Therefore, at the minima and maxima ofn_ccorresponding to β_1,3= 0,180^◦, 360^◦, we find that∆is0.06±0.01.

p₁=P₁₀₀₀+P₀₁₀₀+P₀₀₁₀+P₀₀₀₁. Likewise,11elements comprise the subspace with two or more photons, ˆ

ρ_≥2, subject to the restriction of one photon per mode, withp_≥2=P₁₁₀₀+P₁₀₁₀+· · ·+P₁₁₁₀+· · ·+P₁₁₁₁. In the case ofycmeasurements, the typical detection efficiency including the photodetector quantum efficiency is≈20%. To infer the photon probabilities at the outputs of the verification interferometers (Fig. 8.1c) for our measurements of∆, we follow a similar procedure, but we confine our analysis to the subspacesρˆ0and ˆ

ρ1. In this case, the typical photon detection efficiency is≈30%.

Similarly, due to the uses of photon non-resolving photodetectors and lossy paths for our projectors (Eq. 8.2), the measured sum uncertainty∆m includes spurious contributions from multiple photons p≥2. To account for this, we follow the procedure described in section 7.6.2 (ref.³⁸, chapter 7), which leads to a conservative estimation of the photon sum uncertainty∆arising only fromρˆ₁. In the case of balanced losses, the correction factorcis expressed in terms of two-photon suppressiony_cand transmission efficiencyηwith

c≈(1 + 3

8(2−η)p1yc), (8.3)

where we apply c∆m > ∆ to obtain a conservative estimate of the 1-photon∆ (section 7.6.2; see also chapter 9 for a more efficient method for obtaining the upper bound of∆). For our experimental parameters, the correction factor(c−1) ≈ 6%is obtained for yc = 1, as depicted in black line of Fig. 8.7. This is significantly smaller than the fractional uncertainties ^δ(∆_∆^m)

m ≈ 25%of our data. Furthermore, since the correction factor scales asyc, the correction factor gives(c−1)<1%for the relevant data sets ofyc <0.2 for four-mode entanglement (Fig. 8.4). Following the standard procedures for loss propagations¹⁹³, we also account for the effect of differential losses and imbalanced beamsplitter ratios (red line in Fig. 8.7).

8.9.1 Imbalances and threshold detectors

In chapter 7, we developed a method to account for losses and imbalances. Here, we obtain an explicit formulas ofq₁in the case of differential losses and imbalanced beamsplitter ratios. In order to propagateρˆ^(r)_W through the imbalanced verification interferometers, we rewriteρˆ^(r)_W,

ˆ

ρ^(r)_W =p0ρˆ0+ ˆρ⁰₁+ ˆρ⁰₂, (8.4) in terms of mode operatorsˆaiwhere

ˆ ρ⁰₁=X

i,j

P_i+P_j

2 V(i, j)ˆa^†_i|0ih0|ˆa_j (8.5)

ˆ ρ⁰₂=

4

X

j>i 4

X

l>k

P_ij+P_kl

2 V(ij, kl)ˆa^†_iˆa^†_j|0ih0|ˆa_kˆa_l. (8.6)

Figure 8.7:Correction factorcas a function of two-photon contaminationyc.The black line corresponds to the calculation ofc for balanced lossesη = 36%and fixedp1 = 9%. The red line is a calculation ofc including differential losses{η}and imbalanced beamsplitter ratios{α}(Table 8.1). The uncertainty of c due to the systematic uncertainties in{η, α} is shown as purple bands. The filled circles showcusing the data points in the experiment (i.e., using the parametersPijklof each points). The uncertainty in the vertical direction includes the systematic uncertainties in{η, α}as well as the statistical uncertainties inPijkl. Here,V(i, j) = V(j, i)andV(i, i) = 1. Through the lossy and imbalanced setup in Fig. 8.1, the mode operatorsaˆiare transformed into following forms,

ˆ

ai7→X

i⁰

e^iφ(i)i0 q

α_i⁽ⁱ⁾0 ( q

η_i⁽ⁱ⁾0 ˆai⁰+ q

1−η_i⁽ⁱ⁾0 vˆi⁰). (8.7)

Here,ˆvi⁰is the vacuum mode operator. The precise correspondences between the imbalances{α⁽ⁱ⁾_i0 , η⁽ⁱ⁾_i0 , φ⁽ⁱ⁾_i0 } and experimental parameters (Table 8.1 and Fig. 8.1) are not shown for clarity.

The stateρˆ^(r)_W, then, is transformed to a state (see section 7.6.2 for the balanced case), ˆ

ρ^(r)_η =p⁰₀ρˆ0+p⁰₁(q₁⁰ρˆ⁽¹⁾₁ + (1−q₁⁰) ˆρ⁽²⁾₁ ) +p⁰_≥2ρˆ⁽²⁾_≥2, (8.8) where the relevant parameters{p⁰₁q₁⁰, p⁰₁(1−q⁰₁), p⁰₂}are given as^c

p⁰₁q⁰₁=X

i,j

Pi+Pj

2 V(i, j)X

κ

e^{−i(φ(i)κ −φ(j)κ )} q

η⁽ⁱ⁾κ ηκ^(j)

q

α⁽ⁱ⁾κ α^(j)κ (8.9)

cHere, we have assumed that the two-photon subspace is fully coherent,V(ij, kl) = 1, thereby leading to∆( ˆρ⁽²⁾₁ ) = ∆( ˆρ⁽²⁾₂ ) = 0 and a conservative estimate of∆.

p⁰₁(1−q⁰₁) =X

i>j

X

k>l

Pij+Pkl

2 1 2

X

κ₁,κ₂

q

α⁽ⁱ⁾κ1α^(j)κ2α^(k)κ1α^(l)κ2e^{−i(φ(k)κ1+φ(l)κ2−φ(i)κ1−φ(j)κ2)}×

( q

ηκ⁽ⁱ⁾1(1−ηκ^(j)2) + q

η^(j)κ2(1−η⁽ⁱ⁾κ1))(

q

ηκ^(k)1 (1−η^(l)κ2) + q

η^(l)κ2(1−ηκ^(k)1))

(8.10)

p⁰_≥2=X

i>j

X

k>l

P_ij+P_kl 2

X

κ1,κ2

q

ακ⁽ⁱ⁾₁α^(j)κ₂α^(k)κ₁α^(l)κ₂e^{−i(φ(k)κ1+φ(l)κ2−φ(i)κ1−φ(j)κ2)} q

ηκ⁽ⁱ⁾₁η^(j)κ₂ηκ^(k)₁η^(l)κ₂. (8.11) Therefore, correction factorc= 1/q1is given as (chapter 7)

q1= p⁰₁q⁰₁

p⁰₁+p⁰_≥2. (8.12)

In the case of balanced lossesη, it can be confirmed that Eqs. 8.3 and 8.12 are equivalent.

Here, we give the definitions of{α⁽ⁱ⁾_i0 , η_i⁽ⁱ⁾0 , φ⁽ⁱ⁾_i0 }following the notations in Fig. 8.9.

α⁽ⁱ⁾_i0 =











α⁰₁₄α12 α⁰₁₄(1−α12) (1−α⁰₁₄)α34 (1−α⁰₁₄)(1−α34) α⁰₂₃(1−α12) α⁰₂₃α12 (1−α⁰₂₃)(1−α34) (1−α⁰₂₃)α34

(1−α⁰₂₃)(1−α12) (1−α⁰₂₃)α12 α⁰₂₃(1−α34) α⁰₂₃α34

(1−α⁰₁₄)α12 (1−α⁰₁₄)(1−α12) α⁰₁₄α34 α⁰₁₄(1−α34)











(8.13)

e^iφ(i)i0 =











e^iφi1 e^iφi2 −e^iφi3 −e^iφi4

e^iφi1 −e^iφi2 −e^iφi3 e^iφi4 e^iφi1 −e^iφi2 e^iφi3 −e^iφi4

e^iφi1 e^iφi2 e^iφi3 e^iφi4











(8.14)

η⁽ⁱ⁾_i0 =











η1η₁⁰ η2η₁⁰ η3η⁰₄ η4η⁰₄ η1η₂⁰ η2η₂⁰ η3η⁰₃ η4η⁰₃ η1η₂⁰ η2η₂⁰ η3η⁰₃ η4η⁰₃ η1η₁⁰ η2η₁⁰ η3η⁰₄ η4η⁰₄











(8.15)