Entanglement on demand through time reordering

0.3 MB·English

by  J. E. Avron

Entanglement on demand through time reordering
J. E. Avron, G. Bisker, D. Gershoni, N. H. Lindner, E. A. Meirom
Department of Physics, Technion—Israel Institute of Technology, 32000 Haifa, Israel
R. J. Warburton
School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
Entangled photons can be generated “on demand” in a novel scheme involving unitary time
reordering of the photons emitted in a radiative decay cascade. The scheme yields polarization
entangledphotonpairs,eventhoughpriortoreorderingtheemittedphotonscarrysignificant“which
pathinformation”andtheirpolarizationsareunentangled. Thisshowsthatquantumchronologycan
be manipulated in a way that is lossless and deterministic (unitary). The theory can, in principle,
8 betested and applied to thebiexciton cascade in semiconductor quantum dots.
0
0
2 Entangled quantum states are an important resource
n in quantum information and communication [1]. Entan-
a gled photons are particularly attractive for applications
J duetotheirnoninteractingnatureandtheeasebywhich

7 they can be manipulated. There is, therefore, consider-

able interest in the development of sources for reliable
]
h (non-random)polarizationentangledphoton pairs. Cur-
p rently,the mostimportantandpracticalsourceofpolar-
-
ization entangled pairs is down conversion [2, 3] which
t
n has a large random component. Furthermore, since the FIG. 1: Two alternative level schemes that can be used to
a entanglementiscreatedbyacoincidencedetectionofthe generateentangled pairs. Thesolid (dashed-dotted)line rep-
u
resentsthedecaychannelthatyieldstwox(y)polarizedpho-
q pair,the entangledstate becomes unavailablefor further
tons. The energies correspond to the colors of the emitted
[ manipulations.
photons. (a) Represents the situation where photon colors

2 Quantum dots are a source of single photons “on de- match in the first generation, and has thegeometry of a kite
v mand” [4, 5, 6]. Recently, it has been shown [7, 8] that (deltoid); (b) Represents the case where colors in different
8 theycanbeusedassourcesofpolarizationentangledpho- generation match, and is geometrically a parallelogram.
0
ton pairs. The entangled photon pair is obtained from
3
the decay cascade of a biexciton in a quantum dot. The
2
. biexciton has two decay channels, each emitting a pho- elusive goal so far for both practical and fundamental
0
tonpairwithapolarizationcharacteristictothechannel. reasons. The fundamental reason is that quantum me-
1
7 Perfect “which path ambiguity” requires that the inter- chanics has the principle of level repulsion: In quantum
0 mediate exciton state is doubly degenerate as illustrated dotsthe scaleofenergyresponsible forthe detuning (ex-
: inFig.1(a). Inthisidealizedsetting,thefirstgeneration change)dominatesthescaleoftheradiativewidthwhich
v
i photons have identical colors (energies) and the second is the smallestenergy scale in the problem[10, 12]. This
X
generationphotonsalsohaveidentical,thoughingeneral putsan“inprinciple”obstacletosubstantial“whichpath
r different, colors. With perfect “which path ambiguity” ambiguity” in quantum dots.
a
the stateofpolarizationofthe twophotonsismaximally
Entangled photons from quantum dots have been ob-
entangled, and each pair can, in principle, be produced
tained by selectively filtering the photons that conform
“on demand” [9].
to the which pathambiguity [7]. The entanglementthen
Quantum dots do not have perfect cylindrical symme- comesatthe pricethata substantialfractionofthe pho-
try and this lifts the degeneracy of the intermediate ex- ton pairs are lost and the quantum dot does not furnish
citon states [10]. We shall refer to this splitting as “de- entangled pairs “on demand”.
tuning”. Since the detuning is large (compared with the
Analternatestrategyproposedin[13,14]istotunethe
radiative width), the two decay paths are effectively dis- dotspectrumtohavecoincidenceofcolorsacrossgenera-
tinguished by the distinct colors of the emitted photons. tions, rather than within generations. This is illustrated
This adversely affects the “raw” entanglement which is
in Fig. 1 (b) where the colors of the photons in the first
then negligible [7]. generation match, in pairs, the colors of the photons in
Inprinciple,thedetuningcanbemanipulatedbyStark the second generation. Since a coincidence of colors in
and Zeeman effects, by stress etc. and much experimen- different generations does not require degeneracy, there
tal effort has been devoted into reducing it to small val- isnofundamentalobstacletotuningtheleveldiagramto
ues (below the radiative width) [11]. This has been an a precise coincidence of colors [12, 13, 14]. When this is
2
the case the two decay channels are identical up to time (negativity) [17, 18]
ordering. The different time ordering of the two decay
pathsofthe‘raw’statebetraythepathwhich,asitturns |hαx|W |αyi| ∗
γ(∆,β;W)= , W =U U . (4)
out,completelykillstheentanglement. Inordertoregain hαx|αxi+hαy|αyi x y
entanglementoneneedstomanipulatethetimeordering.
The theoryof re-orderingthe quantumchronologyis de- Themaximalvalueofγ is 1 correspondingtomaximally
2
velopedinthisLetter. Itallowsustoderivethemeasure entangled (Bell) states. Note that the denominator is
of entanglement of the reordered state and its depen- just the normalization of the state |ψi.
denceonthespectralpropertiesoftheradiativecascade. The |αji are fully determined by the (complex) ener-
Perhaps the most important result is that it shows that gies of the level diagram, Za = Ea −iΓa. In the limit
the reorderingcanbe made in a way that conforms with that the dipole approximation holds, the (normalized)
the requirement of “on demand”. The theory then also wave packets are given by [15, 19]:
allows an optimization of the entanglement and it leads
hk ,k |α i=A(|k |+|k |,Z ) A(|k |,Z )+A(|k |,Z )
to a proposal of a practical experimental realization. 1 2 j 1 2 u 1 j 2 j
We denote by ∆ the detuning, the (dimensionless) (cid:0) (5)(cid:1)
where:
measure of the color matching of the photons in a given
generation. Perfect matching within generationis repre-
Γ 1
sented by ∆=0. Similarly, we denote by β the (dimen- A(k,Z)= . (6)
rπ k−Z
sionless) spectral control which measures the matching
of the colors across generations (the “biexciton binding (We use units where ~=c=1.) The photon of the first
energy”ina quantumdot). Perfectcrosscolormatching generation has energy near E −E while the photon of
u j
in this case is represented by β =0. More precisely:
the second generation has energy near E and relative
j
time delay of order 1/Γ . Note that positive (negative)
E −E E −E −E −E j
y x u 0 x y
∆= , β = . (1) delayisassociatedwithZ inthelower(upper)halfcom-
2Γ 2Γ j
plex energy-plane.
Ex, Ey, Eu areasinFig.1andΓisthehalfwidthofthe Inpractice,the smallestenergyscaleinthe problemis
intermediate levels. As we have explained, in dots there the radiative width, Γ. We treat it as a small parameter
are fundamental reasonsthat force|∆|≥1, while β can, inthetheoryandthuscansafelydroptheabsolutevalue
in principle, be tuned to zero. The issue is: Can one inEq.(5). Thisallowsforananalyticcalculationofsome
generate entangled photon pairs by tuning β = 0 even of the integrals that arise. In particular, when the two
when |∆|≫1? As we show the answer is yes. emitted photons have different colors, i.e. when ||k |−
1
Suppose that the two photons are emitted along the |k ||≫Γ, one has
2
z-axisand havetwo decay modes with equalamplitudes.
Then,the(possiblyun-normalized)photonwavefunction hα |α i=2 (7)
j j
has the form [15]:
to leading order in Γ.
|ψi= |α i⊗|jji (2) ThenumeratorinEq.(4),hα |W |α icannowbewrit-
j x y
jX=x,y ten as a sum of the two integrals:
where |αji describes the photons’ wave packet and |jji 2ΓΓu W(k1,k2)dk1dk2
y = (8)
their state of polarization. 1 π2 Z |k +k −Z |2(k −Z∗)(k −Z )
1 2 u 1 x 2 y
Forreasonsthatwillbecomeclearbelowweneedtoal-
lowforaunitarypostprocessingofthe‘raw’stateemerg- and
ingfromthecascade. Themanipulation,U ,isdescribed
j
2ΓΓ W(k ,k )dk dk
by a unitary operation that depends on the polarization y = u 1 2 1 2 (9)
state (i.e. the decay channel)and is then formally of the 2 π2 Z |k1+k2−Zu|2(k1−Zx∗)(k1−Zy)
form
where Γ =Γ =Γ. The first term, y , may be thought
x y 1
of as the contribution to the entanglement from the co-
|α i→U |α i. (3)
j j j
incidence of colors across generations while the second
Inthelanguageofquantuminformationthiscorresponds term, y , as the contribution to entanglement from the
2
to applying single qubit unitary gates on each of the matching of colors within a generation.
two polarization states (this operation can be made by Consider first the ‘raw’ outgoing state where W =
whomeverpreparesthe state,but canalsobe made later 1. One expects that entanglement can arise only from
and so falls under the class of local operations [16]). matchingcolorswithinageneration. Byfirstshiftingthe
As a measure of the entanglement we take the abso- integration variables, k −E → k and k −E → k
1 x 1 2 y 2
lute value of the negative eigenvalue in the Peres test in Eq. (8), and using the residue theorem to compute
3
the integrals,one indeed finds that the cross-generations a)
contribution vanishes: 3π/2
)mal
y1(β;W =1)=0. (10) wopti π
g(
For the second integral, shifting k → k + Ex+Ey first, ar
and using similar elementary manijpulatijons, on2e finds π/2−3 Γ −2Γ −Γ k0−E Γ 2Γ 3Γ
2 y
b) c)
−2i
y (∆;W =1)= . (11) 0.5 0.4
2
∆−i
0.48
0.35
0.46
In particular, combining with Eqs. (4,7), we find
0.44 0.3
1 0.42
γ2(∆,β;1)= . (12) 0.25
4(∆2+1) γ 0.4 γ
0.2
0.38
The ‘raw’ entanglement is independent of β (to leading
0.36 0.15
orderinΓ)anddoesnotbenefitfrommatchingthecolors
0.34
of photons in different generations (tuning to β = 0). 0.1
0.32
Since typically |∆|≫1, the ‘raw’entanglement is small.
0.05
Fortunately,asuitablechoiceofW canyieldanentan- 0 1 Γ2 / Γ 3 4 −20 −10 β0 10 20
u
gled state of polarization even when |∆| ≫ 1. Since W
is unitary, W(k ,k ) is a phase. The optimal choice of
phaseisonethat1wo2uldmakey aslargeaspossible. This FIG. 2: a) The argument of Wopt as a function of k2−Ey
canbeachievedwhentheinteg1randinEq.(8)hasafixed for k1 = Ex. The dashed line represents a feasible linear
approximation, generated by an optical delay. b) The off
phase(sothattheoscillationsleadingtothecancellation
diagonalmatrixelementγ forWopt,asafunctionoftheratio
in Eq. (10) are eliminated), e.g.: Γu/Γ. For typical biexciton decays, Γu/Γ ≈ 2. c) γ as a
function of the color matching dimensionless parameter, β,
∗
W =−(k1−Zx)(k2−Zy). (13) for Γu/Γ = 2. The maximal value occurs when β = 0. The
opt |k −Z∗||k −Z | width of thepeak is of order Γ.
1 x 2 y
W (E ,k −E ) is plotted in Fig. 2 (a).
opt x 2 y
By inserting Eq. (13) into Eq. (8) and shifting the in-
Physically, choosing W may be thought of as letting
tegration variables, this choice for W gives for y
opt 1 the two polarizations go through different gates that in-
troducedifferent,butfixed,timedelaysonthetwocolors.
2 g dk dk
1 2
y1 = π2 Z |k +k −β−ig|2|k −i||k +i|,(14) To see this, we note first that each of the two factors of
1 2 1 2 Eq. (13) can indeed be interpreted as a time shift. This
where g = Γ /Γ. Let us study the case of β = 0, where follows from the fact that
u
this function (which is evenin β) achievesits maximum. k−Z
≈ie−ik/ΓeiE/Γ (16)
Combining this with Eq. (4,7) one finds for the optimal |k−Z|
W
for k ≈ E, see Fig. 2 (a). This represents a shift of the
γ(g;|∆|≫1,β =0;W )= 1 +f(g), (15) wavefunction incoordinate spaceby 1/Γ whichcanalso
opt 2
be interpreted as a (non-random) shift in time by 1/Γ.
which is plotted, as a function of g for perfect color Therefore, the two factors in W may be implemented,
opt
matching,β =0, in Fig.2 (b). The function f(g)is neg- approximately, by manipulating the optical paths.
ative, monotonically decreasing and vanishes for g = 0, It is important to understand that the manipulation
where maximal entanglement, γ = 1, is achieved. For of the quantum chronology proposed here is a fixed uni-
2
systemssuchasthebiexcitonradiativecascades,onecan tary manipulation of the wave function, which is a non-
notgetmaximalentanglementevenwhenthe colorsper- randomobject. Itisnotamanipulationoftheindividual
fectly match, since g ≈ 1.5−2, however, one does get detection events which are random and uncontrollable.
substantial entanglement, γ ≈0.4. However, the probability distribution for these random
The entanglement γ is, of course, sensitive to the events is determined by the wave function according to
matching of colorsacrossgenerations,so that when β ≫ the rules of quantum mechanics.
1 the entanglement becomes small. This is illustrated in FromEq.(13),weseethatW hastwofactors,affect-
opt
Fig.2(c) whichshowsγ asa functionofβ for g =2,the ing the two photons. Fig. 3 explains why both photons
value relevant to biexciton decay. mustbemanipulated: toyieldentangledphotons,allthe
4
properties distinguishing the x and y polarized photons
must be erased. This requires that: the arrival times
at the detector D of photons with energies E (the red
x
photons in Fig. 3) must be independent of polarization,
and likewise for the photons with energy E (the blue
y
photons in the figure).
Suppose that we let the y-polarized photon emitted
first (red arrow on left side of the the figure), travel a
distance longer by ℓ+1/Γ from the photon emitted sec-
ond(blueontheleft)therebyreversingtheirorder(ℓ≥0
is arbitrary). Now the time order (chronology) of both FIG. 4: An optical setup which introduces the appropriate
polarizationsagrees,the firstphotonis blue andthe sec- delays to each of the photons. BS (PBS) stands for a (po-
ondred. However,the whichpathinformationis not yet larizing) beam splitter, and MC for a monochromator. The
MC(approximatelylinear)dispersioncanbesettoobtainan
erased. For the red photon to arrive at the same time,
the x-polarized photon emitted second must be delayed approximationtoWoptofEq.(13). Thephotonspassthefirst
beam splitter and MC, are reflected back by the mirrors and
by a distance ℓ. For the blue photon to arrive at the
afterpassingthroughtheMCagain, theyaremeasured. The
same time, the x-polarized photon emitted first must be optical path are chosen as 2Dy = D+1/Γ+ℓ, 2Dy = D,
1 2
delayed by a distance 1/Γ. The average delay between 2D1x =D+1/Γ, and 2D2x =D+ℓ where D is arbitrary.
the time of arrivalof the blue and redphoton is then 2ℓ.
The extra optical paths can be represented as unitary
gatesactingonthe photonstatesby eikL,whereL isthe
pathlength. Theimplementationofthedelaysdescribed In summary: Entanglement can be created by a non-
above is given by invasive(unitary)manipulation ofthe quantumchronol-
ogy. This provides a possible and practical avenue for
Ux =eik2/Γeik1ℓ Uy =eik1(ℓ+1/Γ). (17) creating entangled photon pairs on demand.
It follows that W = Ux∗Uy = eik1/Γe−ik2/Γwhich, by Acknowledgment: TheresearchissupportedbyISF,
Eq. (16), approximates Wopt. by RBNI, by the fund for promotion of research at the
TechnionandbyEPSRC(UK).WethankOdedKenneth
and Terry Rudolph for helpful discussions. D. Gershoni
thanks Jonathan Finely for pointing out his success in
tuning colors across generations in biexciton decays.
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