Effect of loss on photon-pair generation in nonlinear waveguides arrays

1.6 MB·

by  Diana A. Antonosyan

Effect of loss on photon-pair generation in nonlinear
waveguides arrays
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Diana A. Antonosyan, Alexander S. Solntsev and
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n Andrey A. Sukhorukov
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Nonlinear Physics Centre, Research School of Physics and Engineering, Australian
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8 National University, Canberra 0200 Australia
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E-mail: [email protected]

]
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c Abstract. We describe theoretically the process of spontaneous parametric down-
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t conversion in quadratic nonlinear waveguide arrays in the presence of linear loss. We
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derive a set of discrete Schrodinger-type equations for the biphoton wave function,
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. and the wave function of one photon when the other photon in a pair is lost.
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c We demonstrate effects arising from loss-affected interference between the generated
i photon pairs and show that nonlinear waveguide arrays can serve as a robust loss-
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y tolerant integrated platform for the generation of entangled photon states with non-
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classical spatial correlations.
p
[

1 Keywords: spontaneous parametric down-conversion, waveguide array, photon pair, loss,
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quantum walk
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0 PACS numbers: 42.65.-k, 42.65.Wi, 42.65.Lm
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Effect of loss on photon-pair generation in nonlinear waveguides arrays 2
1. Introduction
Optical quantum communications and computation schemes rely on controlled
preparation of well-defined photonic states [1, 2]. Spontaneous parametric down-
conversion (SPDC) in nonlinear crystals [3, 4, 5] has become a source of choice for
experimental generation of correlated and entangled photon pairs with demonstrations
of such effects as quantum teleportation [6, 7, 8, 9], quantum cryptography [10], Bell-
inequality violations [11] and quantum imaging [12].
The mode confinement in a waveguide enables a significant increase of the SPDC
source brightness in comparison to bulk crystal setups [13]. Even more importantly,
waveguide integration provides interferometric stability, which is essential for quantum
simulations and cryptography. SPDC in nonlinear waveguides can be implemented to
produce photon pairs in distinct spatial modes [14, 15, 16, 17, 18]. Overall, nonlinear
waveguides can serve as photon-pair sources ideally suited for applications in quantum
communications [19].
Recently, there has been growing interest in the study of the propagation of
nonclassical light in coupled waveguides: quantum gates were implemented using pairs
ofwaveguides actingasintegratedbeamsplitters[20], andlatticesofcoupledwaveguides
were used for the study of Bloch oscillations [21] and propagation of squeezed light [22].
Overall integratedoptical quantum circuits utilising coupledwaveguides areincreasingly
gainingattentionasapossiblesolutionforscalablequantumtechnologieswithimportant
applications to quantum simulations. A key mechanism for quantum simulations can be
provided by the process of quantum walks in an optical waveguide array (WGA) [23],
with applications to boson sampling [24, 25, 26, 27]. Furthermore, it was recently
suggested [28, 29, 30, 31] that a nonlinear waveguide array can be used for both photon-
pair generation through spontaneous parametric down-conversion and quantum walks
of the generated biphotons with strong spatial entanglement between the waveguides.
Importantly, such integrated scheme avoids input losses, since in anintegrated nonlinear
waveguide array photon pairs can be generated inside the quantum walk circuit. The
internal losses in the waveguides however may still be present. In this work, we address
an important question of the tolerance of the biphoton generation to possible losses
in the waveguides. We focus our attention on Markovian losses, as this is the most
common type of losses in waveguides, which can be associated in particular with leaky
modes [32, 33, 34].
This paper is organised as follows. Section 2 contains detailed investigation
of spontaneous parametric down-conversion in a single lossy quadratic nonlinear
waveguide. We explore the dependence of photon-pair intensity on losses and phase
mismatch and demonstrate a number of counter-intuitive effects. For example we
show that the increase in idler losses can lead to the increase of signal intensity, and
that the signal intensity becomes independent on nonlinear waveguide length after a
particular propagation distance. We also demonstrate that signal and idler losses lead
to the transformation of common sinc-shaped photon-pair correlation spectrum into
Effect of loss on photon-pair generation in nonlinear waveguides arrays 3
Figure 1. Scheme of photon-pair propagation involving SPDC and losses in a single
waveguide. The first step shows the probability of photon-pair generation though
SPDC,the secondstep correspondsto the probabilityto lose a signalphoton,and the
third step corresponds to the probability to lose an idler photon.
a Lorenzian shape, and that this transformation can be fully reversed by the specific
increase in pump losses. The results related to WGAs are presented in Sec. 3. We
derive a model of the SPDC and photon-pair propagation in finite quadratic nonlinear
WGAs with losses and present the detailed analysis of the generated photon-pair spatial
correlations, entanglement andspatialintensitydistributions. Weshowthatphoton-pair
spatial entanglement generated in nonlinear WGAs remains strong even in the presence
of high losses.
2. Spontaneous parametric down-conversion in a single χ(2) waveguide with
losses
The process of SPDC can occur in a χ(2) nonlinear waveguide pumped by a pump laser,
where a pump photon at frequency ω can be spontaneously split into signal and idler
p
photons with corresponding frequencies ω and ω , such that ω = ω +ω . The effect of
s i p s i
linear losses on SPDC was previously considered in various contexts [4, 35, 36]. Here,
we perform a detailed analysis of the emerging photon intensities and correlations, in
the regime of photon-pair generation.
To describe waveguide losses, it is possible to introduce them through series of
virtual asymmetric beam-splitters in an otherwise conservative medium [35, 37], see
Fig. 1. At each step during propagation from z to z + ∆z the photon pairs can be
generated through SPDC. On the other hand there is a probability for signal and
idler photons to be reflected by beam-splitters, corresponding to the loss of photons
from the waveguide. Then, according to the general principles [38], the photon
dynamics is governed by a sum of Hamiltonians which individually describe SPDC in
lossless nonlinear medium (Hˆ ) and linear losses due to virtual beam-splitters (Hˆ ),
nl bs
Hˆ = Hˆ +Hˆ .
nl bs
The SPDC process in the absence of losses, in the undepleted classical pump
approximation, is governed by a Hamiltonian [5]:
Effect of loss on photon-pair generation in nonlinear waveguides arrays 4
Hˆ (z) = dω β(0)(ω )a†(ω )a (ω )+ dω β(0)(ω )a†(ω )a (ω ) (1)
nl s s s s s s s i i i i i i i
Z Z
+ dω dω E (z,ω +ω )a†(ω )a†(ω )+E∗(z,ω +ω )a (ω )a (ω ) ,
s i p s i s s i i p s i s s i i
Z Z h i
where a† and a are the creation and annihilation operators for the signal and idler
s,i s,i
photons with the commutators [a (ω ),a∗(ω )] = δ(ω − ω ) and [a (ω ),a∗(ω )] =
s 1 s 2 1 2 i 1 i 2
δ(ω − ω ), δ(z) is a Kronecker delta-function, E (z,ω ) is proportional to the pump
1 2 p p
(0)
amplitude at frequency ω and quadratic nonlinearity, and β are the signal and idler
p s,i
propagation constants relative to the pump.
We assume Markovian losses and negligible thermal fluctuations. Then, the
Hamiltonian corresponding to a series of beam-splitters [35, 37, 39] can we written
as:
Hˆ (z) = dω 2γ (ω ) a (ω )b†(z,ω )+a†(ω )b (z,ω ) (2)
bs s s s s s s s s s s s
Z
p (cid:2) (cid:3)
+ dω 2γ (ω ) a (ω )b†(z,ω )+a†(ω )b (z,ω ) ,
i i i i i i i i i i i
Z p h i
where the operators b† (z,ω) describe creation of photons which are lost from
s,i
a waveguide after a beam-splitter at coordinate z, with the commutators
[b (z ,ω ),b∗(z ,ω )] = δ(z −z )δ(ω −ω ) and [b (z ,ω ),b∗(z ,ω )] = δ(z −z )δ(ω −
s 1 1 s 2 2 1 2 1 2 i 1 1 i 2 2 1 2 1
ω ), and γ are the linear loss coefficients.
2 s,i
We focus on the generation of a single photon pair and consider multi-photon-
pair processes to be negligible for appropriately attenuated pump power. Then, the
generation of photon pairs with different frequencies occurs independently, due to the
absence of cascading processes. We will therefore omit ω in the following analysis to
s,i,p
simplify the notations. Then, we seek a solution for a two-photon state at distance z
as:
z
|Ψ(z)i = Φ(z)a†a†|0i+ dz Φ˜(s)(z,z )a†b†(z )|0i
s i l l s i l
Z
0
z
+ dz Φ˜(i)(z,z )b†(z )a†|0i (3)
l l s l i
Z
0
z z
+ dz dz Φ˜(si)(z ,z )b†(z )b†(z )|0i,
Z lsZ li ls li s ls i li
0 0
where |0i denotes a vacuum state with zero number of signal and idler photons. The
ˆ
equation for the evolution of the state vector is dΨ(z)/dz = −iH(z)[|0i + |Ψ(z)i],
assuming undepleted vacuum state. Then, we obtain the following equations for the
two-photon wave functions:
∂Φ(z)
= −(i∆β(0) +γ +γ )Φ(z)+Ae−γpz, Φ(z = 0) = 0, (4)
s i
∂z
∂Φ˜(s)(z,z )
l = −(iβ(0) +γ )Φ˜(s)(z,z ) = 0, z ≥ z , (5)
∂z s s l l
Effect of loss on photon-pair generation in nonlinear waveguides arrays 5
∂Φ˜(i)(z,z )
l = −(iβ(0) +γ )Φ˜(i)(z,z ) = 0, z ≥ z , (6)
∂z i i l l
Φ˜(s)(z ,z ) = −i 2γ Φ(z ), Φ˜(i)(z ,z ) = −i 2γ Φ(z ), (7)
l l i l l l s l
p p
where ∆β(0) = β(0)+β(0), and we take into account possible pump absorption with the
s i
loss coefficient γ by putting E (z) = Aexp(−γ z). We disregard the evolution of Φ˜(si)
p p p
wavefunction, since it corresponds to the case when both photons are lost.
Equation (4) can be solved analytically:
z
Φ(z) = zAsinc ∆β(0) −i(γ +γ −γ )
s i p
(cid:26)z (cid:27)
(cid:2) (cid:3)
iz
×exp − ∆β(0) −i(γ +γ +γ ) . (8)
s i p
(cid:26) 2 (cid:27)
(cid:2) (cid:3)
We now calculate the normalized intensity of photons generated through SPDC,
which is proportional to an average number of photons per unit time. The expressions
for the signal and idler photons are analogous, and to be specific we consider the signal
mode. The total signal intensity I (z) is found as:
s
z 2
I (z) = I(0)(z)+I˜(z), I(0)(z) = |Φ(z)|2, I˜(z) = dz Φ˜(s)(z,z ) , (9)
s s s s s Z l(cid:12) l (cid:12)
0 (cid:12) (cid:12)
(cid:12) (cid:12)
where I(0)(z) is the contribution when both photons are not absorbe(cid:12)d and I˜(z(cid:12)) is a
s s
contribution from the states with lost idler photons. Note that there is no interference
between the photons with lost pairs [36, 40]. The intensity contributions can be
calculated analytically:
2A2e−(γs+γi+γp)z cosh[(γ +γ −γ )z]−cos ∆β(0)z
I(0) = s i p , (10)
s (∆(cid:8)β(0))2 +(γ +γ −γ )2 (cid:0) (cid:1)(cid:9)
s i p
4A2γ e−2γsz
I˜ = i G[z,i(γ +γ −γ )]−G(z,∆β(0)) ,(11)
s s i p
(∆β(0))2 +(γ +γ −γ )2
s i p n o
where
L
G(z,p) = cos(ξp)e−ξ(γi+γp−γs)dξ
Z
0
γ +γ −γ +e−z(γi+γp−γs) psin(zp)−cos(zp)(γ +γ −γ )
i p s i p s
(cid:20) (cid:21)
= . (12)
p2 +(γ +γ −γ )2
i p s
The total intensity can be measured by a sensitive camera, which will provide an overall
number of detected photons per unit time. The intensity contributions can be separated
(0)
using a scheme with single-photon detectors: I will be proportional to the number of
s
coincidence counts of signal and idler photons, and I˜ will be proportional to the signal
s
counts without the corresponding idler photon.
It is instructive to consider a number of limiting cases. In particular, zero pump
loss (γ = 0) can be achieved in various conventional waveguides, where losses at pump
p
frequency can be significantly smaller than losses at signal and idler frequencies due
to the difference in the fundamental mode cross-section sizes for different wavelengths.
Effect of loss on photon-pair generation in nonlinear waveguides arrays 6
Figure 2. Normalized number of photon pairs, Is(0), generated through SPDC in a
single waveguide vs. the phase mismatch ∆β(0) for z = 5, A = 1 and different losses:
(a) γ =γ =γ =0, (b) γ =0, γ =γ =0.5, (c) γ =γ =0.5, γ =γ +γ =1.
p s i p s i s i p s i
In this case both components of signal intensity I(0)(z) and I˜(z) approach stationary
s s
values for large distances:
A2
lim[I(0)(z)] = lim[I˜(z)]γ γ−1 = , (13)
z→∞ s z→∞ s s i (∆β(0))2 +(γ +γ )2
s i
We see that if there is no idler loss (γ = 0), then I˜(z) → 0, which means that all signal
i s
photons are paired with an idler photon, as expected. If the signal and idler exhibit the
same loss (γ = γ ), then half of signal photons remains paired.
s i
For degenerate SPDC regime with indistinguishable signal and idler photons
(γ = γ = γ) and no pump losses (γ = 0), we have:
s i p
2A2e−2zγ cosh(2zγ)−cos(z∆β(0))
I(0)(z) = , (14)
s (cid:2)(∆β(0))2 +4γ2 (cid:3)
2A2e−2zγ
I˜(z) = sinh(2zγ)−2zγsinc(z∆β(0)) . (15)
s
(∆β(0))2 +4γ2
(cid:2) (cid:3)
In the case of strongly non-degenerate SPDC, when signal and idler photons are
generated with significantly different frequencies, pump and signal losses may become
negligible γ = γ = 0, while idler absorption may be substantial [4]. In this case the
p s
biphoton-related component of the signal intensity for long propagation distances is:
2A γ
lim[I(0)(z)] = s,i i . (16)
z→∞ s (∆β(0))2 +γ2
i
We check that Eq. (16) agrees with the result derived in Ref. [4] through the application
of fluctuation-dissipation theorem.
It is interesting to analyze the dependence of the biphoton-related component
of the signal intensity I(0) on the phase mismatch ∆β(0). When losses are absent
s
(γ = γ = γ = 0), it has a well-known [4] shape of sinc-function [Fig. 2(a)]:
p s i
∆β(0)z
I(0)(z) = A2L2sinc2 . (17)
s (cid:18) 2 (cid:19)
Effect of loss on photon-pair generation in nonlinear waveguides arrays 7
Figure 3. (a,c,e) totalsignalintensityI (z)and(b,d,f)ratioofintensitycontribution
s
when both photons are not absorbed and the full intensity I(0)(z)/I (z) vs. the
s s
signal and idler losses in a single waveguide for different values of phase mismatch
(a,b) ∆β(0) = 0, (c,d) ∆β(0) = 3, (e,f) ∆β(0) = 6. Parameters are γ = 0, z = 5,
p
A=1.
For negligible pump losses (γ = 0) and large signal or idler losses {exp[−(γ +γ )z] ≪
p s i
1} the dependence is transformed into a Lorenz shape [Fig. 2(b)] according to Eq. (13).
Interestingly, when pump losses are increased to match the combined idler and signal
losses (γ = γ +γ ) the spectrum returns to a sinc shape [Fig. 2(c)]:
p s i
∆β(0)z
I(0)(z) = A2z2e−2(γs+γi)zsinc2 . (18)
s (cid:18) 2 (cid:19)
Next we present a detailed investigation of the signal mode intensity depending
on the loss (Figs. 3 and 4) and propagation distance (Fig. 5) in the absence of pump
loss γ = 0. Figures 3 (a,c,e) show that the signal intensity I is decreasing with the
p s
increase of signal loss, however the dependence on the idler loss in relation to the phase
mismatch ∆β(0) is nontrivial due to additional signal intensity component I˜ related to
s
the disruption of interference when the idler photon is lost. The ratio between the pure
(0)
biphoton and the full signal intensity, I /I , depends weakly on the phase mismatch,
s s
see Figs. 3 (b,d,f). Indeed, Fig. 4 demonstrates that regardless of the phase mismatch
∆β(0) the proportion of signal photons paired with idler to all signal photons, I(0)/I ,
s s
Effect of loss on photon-pair generation in nonlinear waveguides arrays 8
1
∆ β(0)=0
∆ β(0)=3
0.8 ∆ β(0)=6
s
(0)I/Is
0.6
0.4
0 1 2 3 4 5
γ
Figure 4. Ratio of intensity contribution when both photons are not absorbed and
the full intensity I(0)(z)/I (z) vs. the signal and idler loss γ (γ = γ =γ, γ =0) in
s s s i p
a single waveguide. Parameters are A = 1, z = 5, and ∆β(0) = {0,3,6} as indicated
by labels.
1
γ =γ=0
s i
0.8
γ =γ=0.3
s i
0.6 γ =γ=0.6
s i
s
I
0.4
0.2
0
0 1 2 3 4 5 6 7 8
z
Figure 5. Total signal mode intensity I vs. the propagation distance in a single
s
waveguide for different signal and idler losses γ = γ = γ = {0,0.3,0.6}. Parameters
s i
are A=1, ∆β(0) =0, γ =0.
p
becomes independent on the loss above certain loss threshold.
Figure5showsthebehaviorofthetotalsignalintensityvs. thepropagationdistance
for different losses in the regime of phase-matching. Total signal intensity exhibits fast
growth in the absence of losses. However when moderate of high losses are present, the
total signal intensity I approaches a fixed value at large distances, see Eq. (13).
s
3. SPDC in Waveguide Array with Losses
It was shown that nonlinear WGAs can serve as a reconfigurable on-chip source of
spatially entangled photon pairs [28, 29, 30, 31]. Since internal generation of photon
pairs in nonlinear waveguide arrays solves the problem of input losses, it is important
to understand the effect of internal losses on photon-pair propagation and resulting
entanglement and correlations.
For the theoretical analysis, we combine the one-waveguide Hamiltonians
introduced in the previous section, and the linear coupling between the waveguides
Effect of loss on photon-pair generation in nonlinear waveguides arrays 9
through the Hamiltonian Hˆ . If the waveguide parameters are identical across the whole
c
array, then the Hamiltonian is:
Hˆ(z) = Hˆ (z)+Hˆ (z)+Hˆ (z), (19)
nl bs c
Hˆ (z) = β(0)a†(n )a (n )+ β(0)a†(n )a (n ) (20)
nl s s s s s i i i i i
Xns Xni
+ E (z,n )a†(n )a†(n )+E∗(z,n )a (n )a (n )
p p s p i p p p s p i p
Xnp h i
Hˆ (z) = 2γ a (n )b†(z,n )+a†(n )b (z,n ) (21)
bs s s s s s s s s s
Xns p (cid:2) (cid:3)
+ 2γ a (n )b†(z,n )+a†(n )b (z,n )
i i i i i i i i i
Xni p h i
Hˆ (z) = C a (n )a†(n +1)+a†(n )a (n +1) (22)
c s s s s s s s s s
Xns (cid:2) (cid:3)
+ C a (n )a†(n +1)+a†(n )a (n +1) . (23)
i i s i i i i i i
Xni h i
Here n and n are the waveguide numbers for the signal and idler photons, a† (n)
s i s,i
and a (n) are the creation and annihilation operators for the signal and idler photons
s,i
in a waveguide number n, b† (z,n) describe creation of photons which are lost from
s,i
a waveguide number n at a coordinate z, C are the coupling constants between the
s,i
neighboring waveguides, E (z,n ) is proportional to pump amplitude in waveguide n .
p p p
Then, we seek a solution for a biphoton state as:
|Ψ(z)i = Φ (z)a†(n )a†(n )|0i
ns,ni s s i i
Xns Xni
z
+ dz Φ˜(s) (z,z )a†(n )b†(z ,n )|0i
Z l ns,ni l s s i l i
Xns Xni 0
z
+ dz Φ˜(i) (z,z )b†(z ,n )a†(n )|0i (24)
Z l ns,ni l s l s i i
Xns Xni 0
z z
+ dz dz Φ˜(si) (z ,z )b†(z ,n )b†(z ,n )|0i.
Z lsZ li ns,ni ls li s ls s i li i
Xns Xni 0 0
The resulting set of equations for the evolution of the biphoton wave functions is:
∂Φ (z)
ns,ni = −i∆β(0)Φ −(γ +γ )Φ +A δ e−γpz
∂z ns,ni s i ns,ni ns ns,ni
− iC (Φ +Φ )−iC (Φ +Φ ), (25)
s ns−1,ni ns+1,ni i ns,ni−1 ns,ni+1
∂Φ˜(s) (z,z )
ns,ni l = −(iβ(0) +γ )Φ˜(s) −iC (Φ˜(s) +Φ˜(s) ),z ≥ z ,(26)
∂z s s ns,ni s ns−1,ni ns+1,ni l
∂Φ˜(i) (z,z )
ns,ni l = −(iβ(0) +γ )Φ˜(i) −iC (Φ˜(i) +Φ˜(i) ),z ≥ z ,(27)
∂z i i ns,ni i ns,ni−1 ns,ni+1 l
Φ˜(s) (z ,z ) = −i 2γ Φ (z ), Φ˜(i) (z ,z ) = −i 2γ Φ (z ),(28)
ns,ni l l i ns,ni l ns,ni l l s ns,ni l
p p
where we do not consider the evolution of Φ˜(si) wavefunction corresponding to both
Effect of loss on photon-pair generation in nonlinear waveguides arrays 10
lost photons. The real-space representation can be Fourier-transformed into spatial
k-space [30]:
Φ = Φ einskseiniki. (29)
ks,ki ns,ni
nXs,ni
Then the biphoton propagation equations in k-space can be written as follows:
∂Φ
ks,ki = −(i∆β +γ +γ )Φ +A e−γpz, (30)
∂z s i ks,ki ks,ki
∂Φ˜(s) (z,z ) ∂Φ˜(i) (z,z )
ns,ni l = −(iβ +γ )Φ˜(s) , ns,ni l = −(iβ +γ )Φ˜(s) , (31)
∂z s s ns,ni ∂z i i ns,ni
Φ˜(s) (z ,z ) = −i 2γ Φ (z ), Φ˜(i) (z ,z ) = −i 2γ Φ (z ), (32)
ks,ki l l i ks,ki l ks,ki l l s ks,ki l
p p
(0) (0)
where β = β +2C cos(k ), β = β +2C cos(k ), and ∆β = β +β . These equations
s s s s i i i i s i
have the same form as Eqs. (4)-(7) for a single waveguide. Accordingly, a solution for
the wave function Φ can be formulated analogous to Eq. (8).
ks,ki
Finally, the real-space wave functions can be calculated by applying the inverse
Fourier transform:
1 π π
Φ = dk dk Φ e−iksnse−kini, (33)
ns,ni (2π)2 Z Z s i ks,ki
−π −π
We can also determine the reduced density matrixes, for instance, of the subsystem
corresponding to the signal photons when idler photon is not lost, ρ(0)(k ,k ,z), and
s1 s2
when the idler photon is lost, ρ (k ,k ,z), as follows:
s s1 s2
ρ(0)(k ,k ,z) =e dk Φ∗ (z)Φ (z), (34)
s1 s2 Z i ks1,ki ks2,ki
z ∗
ρ (k ,k ,z) = dz dk Φ˜(s) (z,z ) Φ˜(s) (z,z ). (35)
s s1 s2 Z lZ i ks1,ki l ks2,ki l
0 h i
e
Taking into account Eq. (30) and Eq. (31) we can write the master equations for
ρ (k ,k ,z) as:
s s1 s2
∂ρ (k ,k ,z)
e s s1 s2 = 2γ ρ(0)(k ,k ,z)−2γ ρ (k ,k ,z). (36)
i s1 s2 s s s1 s2
∂z
e
This equation represents the propagationof the signal pheotonwith the lost idler photon,
where the first term corresponds to the probability of the idler photon loss, while the
second term accounts for the possibility of the signal photon to be lost as well.
The dependence of the intensity for the signal mode on the propagation distance
can be written in the following form for k-space:
I (k ,z) = I(0)(k ,z)+I˜(k ,z), (37)
s s s s s s
π
I(0)(k ,z) = dk |Φ (z)|2, (38)
s s Z i ks,ki
−π
z π 2
˜ ˜(s)
I (k ,z) = dz dk Φ (z,z ) , (39)
s s Z lZ i(cid:12) ks,ki l (cid:12)
0 −π (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)