Input-output relations for multiport ring cavities

0.14 MB·English

by  Matteo G. A. Paris

Input-output relations for multiport ring cavities
Matteo G. A. Paris
DipartimentodiFisicadell’Universita`diMilano,Italia
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0 Abstract. Quantuminput-outputrelationsforagenericn-portringcavityare
2 obtainedbymodelingtheringasacascadeofninterlinkedbeamsplitters. Cavity
response to abeam impingingon one port is studied as a function of the beam-
n splitterreflectivitiesandtheinternalphase-shifts. Interferometricsensitivityand
a stabilityareanalyzedasafunctionofthenumberofports.
J

2
1 Multiport ring cavities represent the natural generalization of two-port Fabry-Perot
interferometers [1] to several modes of the radiation field. They find application in

1 advanced interferometry, division multiplexing and optical cross connect. Recently,
v
a three-port fiber ring laser was suggested and demonstrated to improve sensing
8
resolution[2],whereasmultiportopticalcirculatorshavebeenusedforinterconnecting
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0 single-fiber bidirectionalring networks[3]. Inaddition, a three-portreflectiongrating
1 wasdemonstrated[4]andthe correspondinginput-output relationshavebeen derived
0 [5]. Froma more fundamentalperspective, multiport couplers,either multiport beam
7
splitters or ring cavities, are crucial devices to generate and engineer multiphoton
0
entangled states [6]. In fact, the cavity response is linear in the input modes for both
/
h kindofdevices,withringcavitiesofferingtheadditionalfeatureofahighnonlinearity
p
with respect to the internal phase-shifts of the cavity. The use of ring cavities,
-
t supplemented by nonlinear media, has been also suggested to realize nondemolitive
n
measurement and photon filtering [7].
a
u Inthisletter,fullyquantuminput-outputrelationsforagenericn-portringcavity
q are obtained by modeling the ring as a cascade of n interlinked, suitably matched,
:
v beam splitters. In this way, the cavity response to an impinging beam, as well as the
i useofthecavityininterferometry,canbe evaluatedasafunctionofthebeam-splitter
X
reflectivities, the internal phase-shifts, and the number of ports.
r
Let us first illustrate the results in details for the case ofa three-portring cavity,
a
which has been schematically depicted in Fig. 1. We assume that the three beam
splitters used to build the cavity have the same transmissivity τ. We also assume
thatlossesatthe beamsplitters arenegligible. Thereflectivityofeachcoupleris thus
given by ρ = 1 τ. The input-output relation for the three beam splitters are given
−
by
BS : bk =τ21dk+ρ12ak (1)
k (cid:26) c1⊕k =−ρ12dk+τ21ak
with k = 1,2,3 and denoting sum modulo 3. Any additional phase-shift at the
⊕
beam splitters may be absorbed into the internal phase-shifts φ . In order to build
k
the cavity the matching relations dk = eiφk ck should be also satisfied, together with
Input-output relations for multiport ring cavities 2
Eqs. (1). After lengthy but straightforward calculations one arrives at the input-
output relations for the cavity
1
b = √ρ 1+√ρeiφ a τ√ρeiφ13a +τeiφ1a
1 1 2 3
A −
3
1 (cid:8) (cid:2) (cid:3) (cid:9)
b = τeiφ2a +√ρ 1+√ρeiφ a τ√ρeiφ12a
2 1 2 3
A −
3
1 (cid:8) (cid:2) (cid:3) (cid:9)
b = τ√ρeiφ23a +τeiφ3a +√ρ 1+√ρeiφ a , (2)
3 1 2 3
A −
3
(cid:8) (cid:2) (cid:3) (cid:9)
where φjk = φj +φk, φ = φ1+φ2+φ3 and A3 = 1+ρ32eiφ. Unitarity of the mode
transformations(2)canbe explicitlycheckedthroughthenormalizationoftheoutput
modes [b ,b†]=δ .
j k jk
Figure1. Three-portringcavityasacascadeofthreeinterlinkedbeamsplitters.
The cavity is built by three beam splitters with equal transmissivity τ. The
matchingrelationsdk =eiφkckshouldbesatisfiedtogetherwiththeinput-output
relations(1)foreachbeamsplitter.
The above model can be generalized to a cavity with an arbitrary number of
ports, see Fig. 2. We have
n
b(n) = M(n)a (3)
k kj j
j=1
X
where
1
M1(1n) =A √ρ 1+ (−)1+nρn2−1eiφ
n
(cid:2) (cid:3)
n
M(n) = 1 ( )nτ ρn2 ( )jρ−2jeiθ1(nj) (4)
1j A  − − 
n
 Xj=2 
and cyclic transformations for M , k = 2,...,n, with φ = n φ , A = 1 +
kj k=1 k n
ρn2(−1)1+neiφ and θt(jn) =φt+ nk=j+1φk. P
P
Input-output relations for multiport ring cavities 3
Figure 2. Multiportringcavityasacascadeofinterlinkedbeamsplitters.
Explicitly, for a four-port cavity we have
1
b(4) = √ρ 1 ρeiφ a +τρeiφ134a
1 A − 1 2−
4
nτ√ρe(cid:2)φ14a +τ(cid:3)eiφ1a (5)
3 4
o
and cyclic transformations, where φ θ(4) =φ +φ +φ .
134 ≡ 12 1 3 4
Let us now consider the situation in which one of the port (say, port 1) is fed
by a coherent beam α , whereas the other ports are left unexcited. Using mode
| i
transformations (3) one may analyze the cavity response to a given excitation, i.e.
how the input mean energy a†a αa†a α = α2 is distributed among the n
h 1 1i ≡ h | 1 1| i | |
output photocurrents I(n) = b(n)†b(n), k = 1,...,n obtained by detecting light at the
k k k
n output ports of the cavity. Being the mode transformations linear also the output
(n)
beams are coherent state with amplitudes β . Upon defining the cavity response
| k i
as
b†(n)b(n)
f(n)(ρ,φ)= h k k i ,
k a†(n)a(n)
h 1 1 i
one has β(n) =α f(n)exp iθ(n) with θ(n) =argM(n) and
k k { k } k k1
q
ρ
f1(n) = A 2 1+ρn−2+2(−)1+ncosφρn2−1 (6)
n
| |
(1 ρ(cid:2))2 (cid:3)
f(n) = − ρn−k 2 k n (7)
k A 2 ≤ ≤
n
| |
where An 2 =1+ρn+2( )1+nρn2 cosφ. The cavity response explicitly depends on
| | −
the mirror reflectivity ρ, while, remarkably,it depends on the internalphase-shifts φ
k
only through the total phase-shift φ. The following sum-rule holds
n
f(n)+ f(n) =1 n, φ, ρ,
1 k ∀ ∀ ∀
k=2
X
which, in turn, assures energy conservation. In Fig. 3 we show the cavity responses
(n)
f for n = 4 as a function of the mirror reflectivity for different values of the total
k
internalphase-shift. Notice that0 f(4) 1for k =1,4and0 f(4) 1 otherwise.
≤ k ≤ ≤ k ≤ 4
Input-output relations for multiport ring cavities 4
Figure 3. Cavity responses f(4), k = 1,..,4 of a four-port ring-cavity as a
k
functionofthemirrors’reflectivityfordifferentvaluesofthetotalinternalphase-
shift. In plot of f(4) (f(4) for k6=1), from bottom to top (from top to bottom)
1 k
thecurvescorrespondingtoφ=0, π, π,π,π,π respectively.
20 10 5 2
In general, the minimum of the cavity response at the first port is achieved for
φ = 0 for n even and for φ = π for n odd. For these values (cavity at resonance) we
have
f(n) =ρ 1−ρn2−1 2 (8)
1 1 ρn2
(cid:18) − (cid:19)
2
1 ρ
f(n) =ρn−k − 2 k n, (9)
k 1 ρn2 ≤ ≤
(cid:18) − (cid:19)
either for n even or odd. In the high-reflectivity limit ρ 1 we have f(n) =(1 2)2
→ 1 − n
and f(n) = 4 k = 1. In other words, in a two-port cavity at resonance the energy
k n2 ∀ 6
is completely transferred to the second mode, while increasing the number of ports
the energy is unavoidably “more distributed”. For large n the input beam is mostly
(n)
reflectedonthebeamb andthecavitybecomesopaque. Anequaldistributionatthe
1
outputisobtainedforn=4(f(4) = 1). Thecavityresponsef(n) atthefirstport(last
k 4 1
port f(n) respectively) monotonically increases (decreases) as the mirror reflectivity
n
approaches unit value. On the other hand, f(n), k = 1,n show a maximum value,
k ∀ 6
whose location depends on the internal phase-shift, as well as the number of ports of
the cavity.
Thesensitivityofthecavityindetectingperturbationstotheinternalphase-shift
decreases as the number of ports increases. This is true either monitoring the cavity
output at resonance or doing the same at a fixed working point in an interferometric
setup. InFig. 4weshowthecavityresponsesf(n)andf(n) asafunctionoftheinternal
1 n
phase-shift for different number of ports. As it is apparent from the plot the curves
flatten as the number of ports increases. The full-width half-minimum (maximum) of
f(n) (f(n)), for a generic value of n, is given by
1 n
δφ(n) 1−ρn/2 ρ→1 n(1 ρ), (10)
HW ≃ 2ρn/4 ≃ 4 −
showing a linear increases of the half-width.
More generally, if one aims to detect the fluctuations of the internal phase-shift
around a fixed working point φ = φ∗ by monitoring the output photocurrents I(n)
k
Input-output relations for multiport ring cavities 5
Figure4. Cavityresponsesf1(n)(left)andfn(n)(right)foramultiportringcavity
as a function of the internal phase-shift φ. The responses for n = 2,3,4,5 and
ρ=0.99arereported. Thesmallerisn,thepeakedarethecurves.
then the minimum detectable fluctuation corresponds to the quantity [8]
−1
δ I(n)
δφ(n) = h k i ∆I(n)2 , (11)
k (cid:12)(cid:12) δφ !φ=φ∗(cid:12)(cid:12) qh k i
(cid:12) (cid:12)
(cid:12) (cid:12)
where ∆I(n)2 = (b(n)†b(cid:12)(n))2 b(n)†b(n(cid:12)) 2 denotethermsfluctuationsoftheoutput
h k i h k (cid:12)k i−h k k (cid:12)i
photocurrents. When a single input port is excited in a coherent state α also the
i
output signals are coherent and Eq. (11) rewrites as
−1
f(n) ∂f(n)
δφ(n) = k k , (12)
k qα (cid:12) ∂φ (cid:12)
| | (cid:12) !φ=φ∗(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where α corresponds to the squ(cid:12)are root of the(cid:12)incoming averagenumber of photons.
The op|tim| al working point φ∗, c(cid:12)orresponding to(cid:12) maximum sensitivity, is the internal
phase-shiftthatminimizesthe valueofδφ(n) Wefoundthatφ∗ isclose,butnotequal,
k
to φ = 0 for n even and to φ = π for n odd. Only slight differences are observed for
different values of k, which vanishes for in the high-reflectivity regime. As a matter
of fact, by increasing n the optimal workingpoint φ∗ movesawayfrom φ=0 (φ=π)
and the minimum value of δφ increases. Since in the high-reflectivity regime ρ 1
→
the quantities δφ(n) do not depends on k at fixed n, i.e δφ(n) = δφ(n), k = 1,...,n
k k 1 ∀
the overall sensitivity of the cavity may be evaluated as δφ(n) = δφ(n)/√n. In Fig.
1
5 we report the rescaled sensitivity y(n) = αδφ(n) as function of φ for ρ = 0.99 and
| |
for differentvalues of n. As it is apparentfromthe plotthe overallsensitivity slightly
degrades with increasing n, despite the factor 1/√n decreases. The curves versus φ
flatten for increasing n and this implies that the need of tuning of the cavity at the
optimal working point also becomes less stringent, i.e stability slightly increases.
Figure5. Rescaledsensitivityy(n)=|α|δφ(n) asfunctionofworkingpointφfor
ρ=0.99andfordifferentvaluesofn. Frombottom totopn=2,3,4,5.
Input-output relations for multiport ring cavities 6
Inconclusion,bymodelingan-portring-cavityasacascadeofninterlinkedbeam
splitters we obtained its input-output relations in terms of the involved modes of the
quantized radiation field. Using this approach, the cavity response to an impinging
beam, as well as sensitivity to perturbations, can be straightforwardly evaluated as
a function of the beam splitters reflectivity and the internal phase-shifts. We found
that increasing the number of ports the input energy is unavoidably distributed over
theoutputports. Thesensitivityofthecavityindetectingfluctuationsoftheinternal
phase-shift, either at resonance or at a fixed optimal working point, slightly degrades
as the number of ports increases while, on the contrary, stability slightly increases.
This work has been supported by MIUR through the project PRIN-2005024254-002.
The author thanks Maria Bondani for several discussions.
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