3.2 Heralded Amplification of Single Photon Path Entanglement
3.2.2 Effects of experimental imperfections on the amplification . 45
It is important to look at how experimental imperfections, such as transmission loss, non-unitary detection efficiency at the BSM and non photon number resolving detectors (non-PNRD) affect the amplification. We define ηaux as the probability of having an auxiliary photon in path d, accounting for coupling and transmission losses up to the BS, and ηBSM as the detection efficiency of the BSM. The output state heralded by a detection at the BSM can be re-written as:
ρout = λ0|0i h0|+λ1|1i h1|
λ0+λ1 , (3.21)
where,
λ0 =ηaux(1−ηph)(1−t)ηBSM+ (1−ηaux)ηphηBSM +ηauxηph(1−t)(1−(1−ηBSM)2)
λ1 =ηauxηphηBSMt.
(3.22)
The vacuum component has now two extra terms. The first one is heralded when the signal photon is detected at the BSM but the auxiliary photon is lost. The second one is heralded when both signal photon and auxiliary photon enter the BSM and exit through the same port of the 50:50 BS. If the detector is a non-PNRD, we must also include events given by two photons: the probability that at least one of the two photons is detected is given by 1−(1−ηBSM)2.
To evaluate the effects of the experimental imperfections on the amplification, we can calculate the gain of the amplifier, which is defined as the ratio between the input and output probability of the component |1i h1|:
G= λ1
λ1+λ0 × 1
ηph = ηauxt
ηaux(1−t)(1−ηphηBSM) +ηph (3.23)
t
0.5 0.6 0.7 0.8 0.9 1
Gain
0 1 2 3 4 5
ηaux=1,η BSM=1 ηaux=0.6,ηBSM=1 ηaux=1,η
BSM=0.6
Figure 3.6: Gain of the amplifier as a function of the beam splitter transmissiontfor different values of the input probabilityηaux of the auxiliary photon and detection efficiencyηBSM at the BSM. The input probability ηph of the photon is set here to 0.2, which is equivalent to the transmission of 35 km of optical fibres.
Figure3.6 shows the gain of the amplifier as a function of the transmission t of the BS for different values of ηaux and ηBSM. First of all, we can observe that the minimumtneeded to amplify (G > 1) is now larger than 0.5 due to the heralding of
3.2. Heralded Amplification of Single Photon Path Entanglement 47 vacuum from the losses on the auxiliary photon and the use of non-PNRD. Secondly, we can notice that the gain is much more affected by the coupling and losses on the auxiliary photon than by the detection inefficiency at the BSM. The maximum gain, which in perfect conditions is 1/ηph, is achieved for a transmission t → 1.
However, a transmission t close to unity will considerably reduce the probability of a detection at the BSM. One must carefully choose the parameter t to find a good compromise between amplification and heralding rates. It is important to notice that if t = 1, there is no longer any amplification as no auxiliary photon reaches the BSM. The gain, as defined above, does not give an indication about the preservation of the coherence. One method to verify that the photon amplifiers preserves the coherence, is to recombined some of the input state with the output amplified state and do an interference measure. The result in [102, 118] showed that the amplifier preserves the coherence and could therefore be applied on one mode of the path-entangled state to compensate for the losses in that mode while still preserving entanglement.
3.2.3 Amplification of Path Entanglement
We consider here a simplified model of path-entanglement amplification in which we assume that ηph = ηaux = ηBSM = 1, we use non-PNRD and neglect photon double pair contributions. A more complete model including the experimental im-perfections can be found in Appendix 5.0.3.
50:50
BSM
Loss
a b
c
d
t
e f
n m
Alice
Bob
Figure 3.7: Possible set-up to amplify path entangled state. One of the mode of the path entangled state is sent to a lossy channel with transmissionηL. An amplifier is then used to increase the component of the path entangled state in Bob’s direction.
Figure 3.7 shows one implementation of path-entanglement amplification. The single photon is sent to a beam-splitter with transmission (1−τ) and the transmitted part passes through a lossy channel of transmissionηL before entering the amplifier.
The path-entangled state to be amplified becomes:
ρi = (1−τ)(1−ηL)|00i h00|+ (τ+ (1−τ)ηL)|φii hφi|, (3.24) with,
|φii=
√τ|10ibc+p
(1−τ)ηL|01ibc
pτ + (1−τ)ηL
. (3.25)
A detection at the BSM will herald an amplified state ρf: ρf = λf|00i h00|+ (Nf −λf)|φfi hφf|
Nf
, (3.26)
where,
|φfi=
pτ(1−t)|10ibe+p
(1−τ)ηLt|01ibe
pτ(1−t) + (1−τ)ηLt , (3.27)
λf = (1−τ)(1−t), (3.28)
Nf = (1−t) + (1−τ)ηLt. (3.29) Here, the amplifier has two objectives, to compensate for the losses of the channel and to herald a path entangled state as close as possible to a maximally entangled state |ψi = √1
2(|10i+|01i). By inspecting eq. 3.27, we realise that blindly setting t→1 to achieve the maximum gain is not a direct solution as it will, in most cases, unbalance the state. Indeed, the probability of having a photon heralded in pathb depends also on the value (1−t). A solution to this is to optimise the initial ratio τ : (1−τ) for given values oft andηL. Figure3.8 (left) shows the fidelity computed before the amplification Fi =hψ|ρi|ψi and after the amplification Ff =hψ|ρf|ψi as a function of the parameter t, for different values of τ. We can first notice that for a range of parameter t, the fidelity to the maximally entangled state is higher after amplification. Secondly, we observe that an initial unbalance of the path-entangled state (e.g by settingτ = 0.8 instead of 0.5) can increase the fidelity of the final state. In fact, the optimal value τ for given parameters t and ηL is given by τopt = 1−t(1−ηηLt
L). Initial and final fidelities computed with τopt are shown in Figure 3.8 (right), and approach unity as t →1. However, setting both τ and t close to 1, will greatly diminish the detections rate at the BSM, hence the heralding of the amplified state. The probability of a detection at the BSM is given by the transmissions of the signal and auxiliary photons:
PBSM= 1
2[(1−t) +ηL(1−τ)]. (3.30) This probability will further decrease when including non-unitary detection effi-ciency. Thus, the choice of the different parameters must take into account time of acquisition and stability constraints of the experiment.
This model has shown that for a certain choice of parameters τ, ηL and t, the amplified path entangled state can be brought close to a maximally entangled
3.2. Heralded Amplification of Single Photon Path Entanglement 49
t
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Ff,τ
opt
Fi,τ
opt
t
0 0.2 0.4 0.6 0.8 1
Fidelity
0.2 0.4 0.6 0.8 1
Ff,τ= 0.8 Ff,τ= 0.5 Fi,τ= 0.5 Fi,τ= 0.8
Figure 3.8: Fidelity to a maximally entangled state, before (Fi) and after (Ff)