3.4 Conclusion and outlook
5.0.3 Amplification of path entangled state
This section presents the generalisation of the amplified path-entangled state pre-sented in eq 3.21. It now takes into accounts the coupling and transmission of the signal photon ηph and of the auxiliary photon ηaux, the detection efficiency at the BSMηBSM and the detection efficiencies of detectors in pathb (Alice) ande (Bob), respectivelyηA and ηB.
ρf = 1
Nf[a|00i h00|+b|10i h10|+c|01i h01|+d(|10i h01|+|01i h10|)], (5.2) where,
a=ηLηp(1−τ) +ηaux[1−ηp(ηAτ+ηBSMηL(1−τ)(1 +χ))
+t(1−ηph(ηauxτ +ηL(1−τ)(ηBSM+ηBSMχ−ηB)))] (5.3) b =ηAηauxηph(1−t)τ (5.4) c=ηBηauxηphηL(1−τ)t (5.5) d =q
ηAη2auxηph2 (1−t)τ ηBηL(1−τ)t (5.6) Nf =ηLηph(1−τ) + (1−t)ηaux(1−ηBSMηLηp(1−τ)(1 +χ)) (5.7) The parameterχ accounts here for photon number resolving at the BSM:χ= 0 for non-PNRD and χ = 1 for PNRD. It seems counter intuitive to lose a term when using non-PNRD but this is due to some simplifications which appear only when the extra vacuum term from the use of non-PNRD is present.
5.0.4 Calculation of the no-click joint probability after the swapping
In this section, I will derive the joint probability of no-click events at Alice and Bob and show how it gives rise to the requirement of locking the central interferometer.
The no-click joint probability p00 is given by:
p00 =| h00|ghD(αA)D(αB) 1
√2[ei(φ0a+φ1g+φ2c)|10igh+ei(φ0b+φ1h+φ2d)|01igh]|2, (5.8) where D(αA) corresponds to a displacement αA = |α|ei(φ0a−φ2a+φ1g) and D(αB) to αB = |α|ei(φ0b−φ2b+φ1h). Applying the displacements on the states |0ig and |0ih, we obtain:
p00 = 1
2|ei(φ0a+φ1g+φ2c)h00|ghD(αA)|1ig|αBih+ei(φ0b+φ1h+φ2d)h00|ghD(αB)|αAig|1ih|2. (5.9)
After expanding the coherent states in basis of Fock states and simplifying, this becomes:
p00= 1
4|ei(φ0a+φ1g+φ2c)e−|α|
2
2 h0|gD(αA)|1ig+ei(φ0b+φ1h+φ2d)e−|α|
2
2 h0|hD(αB)|1ih|2. (5.10) Applying again the displacements on the remaining h0| states followed by the ex-pansion of the resulting coherent state in Fock states and simplification, we obtain:
p00= 1
4e−|α|2|ei(φ0a+φ1g+φ2c)e−|α|
2
2 α∗A+ei(φ0b+φ1h+φ2d)e−|α|
2
2 α∗B|2. (5.11) Replacing for α∗A and α∗B:
p00= 1
4e−2|α|2|ei(φ0a+φ1g+φ2c)|α|e−i(φ0a−φ2a+φ1g)+ei(φ0b+φ1h+φ2d)|α|e−i(φ0b−φ2b+φ1h)|2. (5.12) p00= 1
4|α|2e−2|α|2|ei(φ0a+φ1g+φ2c)e−i(φ0a−φ2a+φ1g)+ei(φ0b+φ1h+φ2d)e−i(φ0b−φ2b+φ1h)|2. (5.13) After simplification of the phases, this is:
p00= 1
4|α|2e−2|α|2|ei(φ2c+φ2a)+ei(φ2d+φ2b)|2. (5.14) We can notice that, as expected, the phasesφ0aandφ0b given by the pump cancel out in both paths. We see here thatp00 depends only on the average photon number in the coherent states and on the phasesφ2c, φ2a,φ2d andφ2b. The central interferometer can be locked by imposing the following constrain:
φ2c+φ2a−(φ2d+φ2b) =constant=k. (5.15) Equation5.14 can be written:
p00= 1
4|α|2e−2|α|2|ei(k+φ2d+φ2b)+ei(φ2d+φ2b)|2. (5.16) p00 = 1
4|α|2e−2|α|2|ei(φ2d+φ2b)(eik+ 1)|2. (5.17)
• For k = 0:
p00 =|α|2e−2|α|2. (5.18)
• For k 6= 0:
p00= 1
2|α|2e−2|α|2(cos(k) + 1). (5.19)
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