Filtering of the absolute value of photon-number difference for two-mode macroscopic quantum superpositions

0.26 MB·English

by  Magdalena Stobińska

Filtering of the absolute value of photon-number difference
for two-mode macroscopic quantum states
M. Stobin´ska,1,2 F. T¨oppel,1,3 P. Sekatski,4 A. Buraczewski,5
M. Z˙ukowski,6,7 M. V. Chekhova,1 G. Leuchs,1,3 and N. Gisin4
1Max Planck Institute for the Science of Light, Erlangen, Germany
2Institute for Theoretical Physics II, Erlangen-Nu¨rnberg University, Erlangen, Germany
3Institute for Optics, Information and Photonics,
Erlangen-Nu¨rnberg University, Erlangen, Germany
4Group of Applied Physics, University of Geneva, Geneva, Switzerland
5Faculty of Electronics and Information Technology,
1 Warsaw University of Technology, Warsaw, Poland
1 6Institute for Theoretical Physics and Astrophysics, University of Gdan´sk, Gdan´sk, Poland
0 7Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
2 University of Science and Technology of China, Hefei, Anhui, China

g
We propose a filter that selects two-mode high numberFock states whose photon-numberdiffer-
u
enceexceedsacertain value. Suchafilterisimportant fortheengineeringofmacroscopic quantum
A
states of light and for the control of bright light beams. It improves distinguishability of some
statesbutpreservesmacroscopicsuperpositions. Wesuggestanoperationalimplementationofsuch
4
a filter. To this end we employ an interference effect in which two input ports of a beamsplitter
2
areenteredbyhighlypopulatedmulti-photonstates. Ifthephotonnumbersat thetwoinputports

are almost equal, the photons are highly asymmetrically distributed between the two outputs. If
]
h photon numbersdiffera lot, theexit ports get nearly equal photon occupation numbers.
p
-
t
n Thesetofquantumstatesthatareefficientlyproduced ter (MDF). It filters out two-mode states of light whose
a is limited. Nevertheless,with quantumstate engineering modepopulationsdifferbymorethanacertainthreshold.
u
one can modify or enhance certain properties of exper- Its optical implementation will be based on the interfer-
q
imentally accessible states. In particular, measurement enceofmulti-photonstatesenteringabeamsplitter,each
[
induced state operations allow one to filter out states modeviaadifferentinputport. Theoretically,theaimed

1
of required properties. In the case of intensity mea- filter should perform the projection operation
v
surement, of considerable help are threshold detectors,
6 ∞
0 selecting Fock states or superpositions with sufficiently = k,l k,l , (1)
9 highpopulationnumbers. Examplesoflow-thresholdde- Pδth k,l=0;X|k−l|≥δth| ih |
4 tectors are known: single photon on-off detectors and
8. humaneyes[1,2]. Suchdetectorscanbe utilizedinfeed- where |k,li is a two-mode Fock state. Further on, with-
0 forwardtechniquesbasedontapmeasurements,inwhich out the loss of generality, we will assume that the two
1 effectively POVM measurements [3] are performed lead- modesoftheinitialstatearedistinguishedbyorthogonal
1 ing to quantum operations. As a result, it is possible to polarizations,whileallothercharacteristicsareidentical.
v: block light of unwanted properties (too low or too high For δth >0 the filter acts as “quantum scissors” [15]. It
i intensity). More complicated filters for Fock states are cuts out those terms of an expression of the state in the
X
based on the interference effects [4, 5]. Fock basis for which the occupation difference is below
r
the threshold and preservesthose where the difference is
a
A more challenging task is to construct a filter that
above it.
would select states with certain properties without de-
We would like to comment on the two key features
stroying quantum superpositions. This is especially im-
of such a filter. It estimates the absolute value of the
portant for macroscopic superpositions of Schr¨odinger-
differenceinsteadofthe differenceitself. This hasanad-
cat type, which suffer from low distinguishability and
vantage. Since the spectrum ofthe operator is fully
are easily destroyed by inefficient detection [6–8]. Such Pδth
degenerate,thefilterdoesnotprovideanyinformationon
macroscopicsuperpositions combine quantumproperties
which mode was more populated. Further on we discuss
with macroscopic population and could enable efficient
the action of a MDF on a macroscopic qubit encoded in
light-matter coupling, and thereforeare of much interest
polarization(seeformula(3)). Weshowthatitisnotable
forvariousprotocolsinquantuminformationtechnology,
todistinguishthestatesandfiltersthemfairly. Secondly,
such as quantum memory [9, 10], quantum key distribu-
the estimationis performedina “yes”-“no”manner: the
tion [11], macroscopic Bell inequality tests [12, 13], and exact value of the modulus is never measured and thus,
quantum metrology [14]. this informationdoes not exist. This property is ofgreat
Here, we propose a blueprint of such a device which importance for all quantum protocols that require state
is further called a modulus of intensity difference fil- engineeringpreservingthesuperposition. Thisiswhywe
2
l (S1) l −1 −1 −1(S1) (a)−4×10 3.5×10−2
(a) YES −−11 −−11 −1 (b) .142 0
NO (S2) −1 0 (S2) 10−2
80 10−6
+1 +1 10−4
δth YES δth +1 +1 +1 160 10−8
0 δth k 0 δth +1 +1 +1 +1 k l240 S S1 10−10 1100−−86
2
320 10−6 10−10
10−8 10−10
FIG.1: (Coloronline)Comparisonoftwofilteringtechniques: 400 10−12
0 80 160 240 320 400
absolute difference (MDF) (a) and orthogonality filter (b). k
−5 1.69×10−2
(b) 10
×
call this device a filter. .795 0
Thesetwofeaturesarethemaindifferencebetweenthe 10−2
MDFandtheorthogonalityfilter(OF)basedonintensity 40 10−6 10−4
difference measurements[16]. OFis the basic element in 80 10−8
the setups performing measurement induced operations l S110−10 10−6
on the macroscopic polarization states [12]. It splits the 120 S 10−8
2
stateintwoorthogonalpolarizationsonaPBSandmea- 160 10−6 10−10
10−8 10−10
sures their intensities I and I⊥ by photon multipliers 200 10−12
0 40 80 120 160 200
(PM) separately. This is a von Neumann measurement k
πkl =|k,lihk,l|anditprojectsthe stateonasingleFock (c)−410 5.77×10−2
state. Next,theelectronicsignalsfromPMsarepassedto ×
the OFandelectronic“dichotomization”is applied. The .635 0
intensitydifferenceiscalculatedbytakingthesignalsdif- 10 10−6 10−2
10−8
ference and it is compared with a threshold value. The 20 10−10 10−4
+1 (−1) value is signaled if I−I⊥ >δth (I⊥−I >δth), l30 S 10−6
and 0 otherwise. Thus, this is a comparative threshold 40 1
measurement. ContrarytoMDF,suchaschemedestroys 50 101−06−8 S2 10−8
superpositions. Measurement induced operations on the 60 10−10 10−10
70
multiphoton states based on the OF device allow only 10−12
0 10 20 30 40 50 60 70
for efficient state discrimination, not filtering. Thus, OF k
is not suitable for preselection strategies in Bell tests, of
the kind suggested in [12]. There, polarization analy-
FIG. 2: Distribution pΦ computed for g = 1.87 and filtering
sis on the reflected beam leads to state identification in
threshold δth =200 with 0% (a), 50% (b),90% (c) losses.
thetransmittedbeamandbreakstheentanglement. The
action of MDF and OF is compared in Fig. 1. The for-
mer projects onto S1 and S2 areas simultaneously and coshg−2((tanhg)/2)i+j (1+2i)!(2j)!/i!/j!, g is the
all superpositionterms belonging to them are preserved. parametric gain. Pleasepalso note that, due to the dif-
Thelattercombinedwithphotonmultipliersprojectsthe ferent parity of occupation in the Fock basis, Φ and
| i
state on a Fock state either in S1 or S2 (red or blue dot Φ⊥ are orthogonal and thus form a basis for a macro-
| i
in the figure). scopicqubit. Inarecentexperiment[16],eachpulsecon-
Let us analyze the action of the MDF on specific tained up to 4sinh(g) 104 photons on the average.
≃
“macroscopicqubits,”producedbytheoptimalphaseco- However,in such a photon number regime, detectors are
variantquantumclonersintheprocessofphasesensitive not single photon resolving,but distinguish counts vary-
coherent parametric amplification [2, 16, 17], ing by 150 photons in the best case [17]. This makes
±
the macroqubits hardly distinguishable through direct
∞
Φ = γij (2i+1)ϕ,(2j)ϕ⊥ , (2) detection [16]. Distinguishability may be quantified in
| i iX,j=0 (cid:12)(cid:12) i taenrdmsp of(kth,le)pghivoitnogntdhiestprirboubtaiobnilitpiΦes(ko,fl)fin=din|hgk,sli|mΦui|l2-
∞ Φ⊥
Φ⊥ = γij (2j)ϕ,(2i+1)ϕ⊥ , taneously k photons in ϕ and l in ϕ⊥
| i i
iX,j=0 (cid:12) ∞
(cid:12)
p (k,l)= γ2 δ δ , p (k,l)=p (l,k), (3)
where ϕ and ϕ⊥ are the two orthogonal polar- Φ ij k,2i+1 l,2j Φ⊥ Φ
iX,j=0
izations from the equatorial plane of the Poincar´e
sphere, the real-valued probability amplitude γ = where the Kronecker delta δ equals 1 if a = b or 0
ij a,b
3
1
0.8
0.6
v
0.4 δth=0
δth=40
0.2 δδtthh==240000
0
0 0.2 0.4 0.6 0.8 1
R
FIG. 3: Distinguishability v evaluated for the gain g = 1.87
and different threshold values δth as a function of the losses FIG.4: Animplementationofmodulusofintensitydifference
R. filter. The description is in themain text.
otherwise. Distinguishability is given by aswellandapproachesunitywithreasonableprobability,
e.g. v = 0.96 with the probability of success p = 10−4.
s
v =P(S1) P(S1) =P(S1) P(S2), (4) Obviously, for R=1 we get v =0.
Φ − Φ⊥ Φ − Φ
A physicalimplementationof anMDF canbe realized
where PΦ(Si) = k,l∈SipΦ(k,l) is the probability of find- by a tap measurement, interference of Fock states on a
ing Φ inS ofPFig.1andP(S1)+P(S2) =1. Itincreases beamsplitter and the feed forward technique, see Fig. 4.
if Φ| i(Φ i) starts to occupΦy mostΦly one of S regions, Let us consider a superposition of two mode polariza-
⊥ i
e.g|. Si |(S i), with increasing δ . Fully distinguishable tion Fock states. The tap measurement is realized by a
1 2 th
(indistinguishable) states have v =1 (v =0). highly biased BS with low reflectivity r, e.g. r = 10%,
Our analysis is performed taking into account possi- which reflects only a small fraction of the state leav-
ble losses, which are modeled by a BS with reflectivity ing the transmitted part almost unchanged. The anal-
R giving the loss. The p distributions evaluated for ysis of probability distribution for a BS shows that, for
Φ
g = 1.87 and δ = 200 in the presence of 0%, 50% and highly populated input with super-Poissonian statistics,
th
90%oflossesaredepictedinFig. 2. The filtering results such BS should not reflect more than 20% of the input
incutting outastripe √2δ wide locatedsymmetrically intensity to preserve most of the character of the ini-
th
along the k =l line. The state Φ occupies two disjoint tial state in the transmitted beam. Under such condi-
regionsofspace: the top(S )an|dibottom(S )triangles, tions, an estimation of the population difference for the
1 2
but increasing the threshold reduces the contribution of reflected beam will give us an estimation for the trans-
p in S : the peak value goes downoriginally from10−2 mitted beam. One should employ here the “worst” case
Φ 2
to2 10−4 forδ =200andR=0. Atthesametimethe assumption. The reflected part will be filtered via an
th
distr·ibution peak in S increases from 10−2 to 4 10−2. interferenceand measurementprocess,andthe measure-
1
Similarbehaviorisobservedforhighergains. Beha·viorof ment result will determine a feed forward action on the
p is identicalbutmirrorreflected. Thus,distinguisha- transmitted beam, which will be opening or closing of a
Φ⊥
bility increases. The particle loss results in shifting the shutter.
distribution towards the origin of the coordinates, i.e. Let us describe in detail the filtering setup. The anni-
vacuum state. The distribution peaks become smooth hilationoperatorsofthetwopolarizationmodesofthere-
and symmetric. The edges along the threshold lines are flectedbeamwillbedenotedasar,ar⊥. Thecorrespond-
blurredandthebiggerlossesare,thesmallerwidthofthe ing intensities are given by Ir = a†rar, Ir⊥ = a†r⊥ar⊥.
gap is. It disappears completely for 90% of losses. With They enter the input ports of the polarizing beamsplit-
increasing losses the height of the upper and left peak ter (PBS). We shall assume that the PBS is oriented in
first, drops due to the fact that the distribution “melts” linear polarization HV basis with the output operators
andnext,increasesbecausethetotalprobabilityoverthe being aH =1/√2(ar +ar⊥), aV =1/√2(ar⊥ ar). The
−
shrinking area has to be 1. twoexitbeams,polarizedHandV,propagatetoapairof
For states (2) we computed their distinguishability v detectors,whichmeasuretheintensities. Ifatagivenmo-
for severalfiltering thresholds δ as a function of losses, menttherespectivedetectors’readingsgivephotonoccu-
th
seeFig.3. Ifnofilteringisapplied,thedistinguishability pationnumbers K and L, this is equivalent to a collapse
v =0.64 is independent of the gain value and losses, but ofthestateinto Ψ = K,L =1/√K!L!a† Ka† L 0 .
| iout | i H V | i
dropsquicklyto0ifR>0.9. Ifδ increases,v increases This state, according to the aforementionedmode trans-
th
4
ber s = 200. We also calculated the probabilities that
(a) 0.05 p(|δ|≥δth)=0.028
p(|δ|<δth)=0.972 δ was below or above a given δth (vertical dashed lines
0.04 | |
δ = 30). The filter works probabilistically. For any
th
p(δ)0.03 outcome K and L all values of δ are possible, but not
0.02 equally probable. For the majority of the cases, for a
reasonable threshold the probability of high population
0.01
difference is higher than the probability of the low one.
0.00
−200 −100 −300 30 100 200 If the probability is satisfactory e.g. above 90%, as is for
δ
K =L=100, the feed forward signal opens a shutter in
(b) 0.025 p(|δ|≥δth)=0.896 thetransmittedbeamoftheinitialBS,Fig.4. Otherwise,
p(|δ|<δth)=0.104
0.020 we reject it and the shutter remains closed.
p(δ)0.015 Theproposedschemeofmodulusofintensitydifference
filter is feasible. This is a threshold measurement which
0.010
projects the input state on a certain multi-dimensional
0.005
subspaceoftheHilbertspace,thusenablingquantumsu-
0.000
−200 −100 −300 30 100 200 perpositions and further state processing. The filter em-
δ
ploys the interference of multi-photon states on a beam-
(c) 0.05 p(|δ|≥δth)=0.905 splitteranditworksforanytwo-modepolarizationstate
p(|δ|<δth)=0.095
0.04 withsuper-Poissonianstatistics. This propertyis impor-
tantduetothepropertiesofthetappingBS.Realization
p(δ)0.03
ofsuchmeasurementswithcommonlyusedphotondetec-
0.02
tors is quite challenging,howeverproperties of the MDF
0.01 are worth the possible effort. The filter could be applied
0.00 in the engineering of macroscopic states of light. In the
−200 −100 −300 30 100 200
δ case of macro-qubits it circumvents the problem of in-
efficient detection, and improves their distinguishability,
thusmakingthemgoodquantuminformationcarriersfor
FIG. 5: Distribution of the population difference p(δ) in the
quantum information protocolsand quantum metrology.
input state with s = 200 photons conditioned on the mea-
surement of K = 0, L = 200 (a), K = 60, L = 140 (b), MS acknowledges MNiSW grant No.
K =L=100 (c) at thePBS output. 2619/B/H03/2010/38. Calculations are performed
at CI TASK in Gdan´sk. MZ acknowledges EU Q-
ESSENCE project, MNiSW (NCN) grant N202 208538.
formation at our PBS can be rewritten in the terms of
the input modes as the following superposition [18]
K L
1 K L
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