Table Of ContentA Quantum Pulse Gate based on Spectrally Engineered Sum Frequency Generation
Andreas Eckstein,1 Benjamin Brecht,2,1 and Christine Silberhorn2,1
1Max Planck Institute for the Science of Light, Günther-Scharowsky-Strasse 1, 91054 Erlangen, Germany∗
2Applied Physics, University of Paderborn, Warburgerstrasse 100, 33098 Paderborn, Germany
(Dated: January 27, 2011)
We introduce the concept of a quantum pulse gate (QPG), a method for accessing the intrinsic
broadbandspectralmodestructureofultrafastquantumstatesoflight. Thismodestructurecannow
beharnessedforapplicationsinquantuminformationprocessing. Weproposeanimplementationin
aPPLNwaveguide,basedonspectrallyengineeredsumfrequencygeneration(SFG).Itallowsusto
pick well-defined spectral broadband modes from an ultrafast multi-mode state for interconversion
toabroadbandmodeatanotherfrequency. Bypulse-shapingthebrightSFGpumpbeam,different
orthogonalbroadbandmodescanbeaddressedindividuallyandextractedwithnearunitefficiency.
1
1 Ultrashortpulsesoflightplayanever-increasingrolein
0
modern quantum information and communications. To-
2
day,theyalreadyenablehighdatatransmissionratesfor
n
secure quantum key distribution[1] and high precision
a
positioning and clock synchronization protocols[2]. In
J
recent years there has been increased interest in a finer
6
2 control over the rich temporal and spectral structure of
quantum light pulses, for applications such as ultrafast
] probingofthetemporalwavefunctionofphotonpairs[3],
h
orefficientlycouplingsinglephotonstotrappedatoms[4].
p
- Given direct access, this structure could also be utilized
t
n to encode more information into or extract more quan- FIG.1. QuantumPulseGateschema: Gatingwithapulsein
a tum information from one pulse of light. spectralbroadbandmodeu convertsonlythecorresponding
k
u It is possible to decompose any pulse form into any modefromtheinputpulsetoaGaussianwavepacketatsum
q
completesetoforthogonalbasisfunctions, orbroadband frequency.
[
modes[5]. Thus it can be considered to be made up of
2 an infinite number of temporally overlapping but inde-
v pendent pulses. While for classical light all basis sets characteristics are lost.
5
are equivalent, for quantum light there may be one spe- The idea of using broadband modes as quantum infor-
1
2 cial, intrinsic basis choice[6]. For photon pair states this mation carriers is especially compelling because of their
3 choice determined by a Schmidt decomposition of their natural occurrence in ultrafast pulses, and their stability
7. bi-photon spectral amplitude into two correlated basis in transmission: Centered around one frequency within
0 sets of broadband pulse forms, the Schmidt modes[7]. a relatively small bandwidth typically, they allow for op-
0 Heralding one of those photons by detecting the other tical components that are highly optimized for a small
1
with a single photon detector (SPD), this correlation re- spectralrange. Sinceallbroadbandmodesexperiencethe
:
v sults in the preparation of a photon in a mixed state of same chromatic dispersion in optical media, they exhibit
i all Schmidt modes present[8]. But with a SPD sensitive thesamephasemodulationandthusstayexactlyorthog-
X
to a certain Schmidt mode, it opens up the possibility onal to each other. So a light pulse’s broadband mode
r
a to prepare pure single photons in the correlated Schmidt structureisresilienttotheeffectsofchromaticdispersion,
mode. Typically though, SPDs and optical detectors in makingamulti-channelprotocolbasedonthemidealfor
general exhibit very broad spectral response and are not optical fiber transmission. Additionally they allow for
able to discern between different pulse forms. high transmission rates, as they inherit the ultrashort
To compensate for the detectors’ shortcomings, one properties of their ’carrier’ pulse, when compared to the
needstoincludeafilteroperationsensitivetobroadband ’long’ pulses used for classical, narrow-band frequency
modes. Ithasalreadybeenshownthatordinaryspectral multiplexing techniques. However, it is extremely chal-
filters cannot fulfill this role[9, 10]: They always trans- lenging to actually access them in a controlled manner:
mit part of all impinging broadband modes at once, and Ordinary spectral filters and standard optical detectors
thus cannot be matched to a single broadband mode. A destroy the mode structure of a beam. A homodyne de-
sufficiently narrow spectral filter can be used to select tector with an ultrafast pulsed local oscillator beam is
a monochromatic mode, however this way, high purity able to select a single broadband mode by spectral over-
heralded quantum states are impossible[10, 11]. Also, lap, but only at the cost of consuming the whole input
mostoftheoriginalbeam’sbrightnessaswellasitspulse beam.[12, 13]
2
For discrete spatial modes, complete control of a α(ω −ω )×Φ(ω ,ω ) maps the input frequencies ω to
o i o i i
beam’s multi-mode structure can be accomplished with thesumfrequenciesω ,whereαisthespectralamplitude
o
linear optics, as combining them to synthesize multi- ofthegatingpulseandΦthephasematchingdistribution
mode beams and separating constituents without losses of the SFG process.
is possible[14]. In order to exploit the pulse form degree In parametric down-conversion (PDC), the Schmidt
offreedom, wemustbeabletoexactsimilarcontrolover decomposition of the joint spectral amplitude of the
broadband modes. generated photon pairs reveals their broadband mode
An important step towards this goal is to selectively structure[7]. Applying the same approach to SFG[20]
targetasinglebroadbandmodeforinterconversionintoa to decompose the spectral transfer function we find
moreaccessiblechannel,forinstancetoshiftittoanother (cid:88)
f(ω ,ω )= κ ϕ (ω ) ψ (ω ). (2)
frequency with SFG. On the single photon level, in the i o k k i k o
SFG process two single photons “fuse” into one photon k
at their sum frequency inside a χ(2)-nonlinear material. The decomposition is well-defined and yields two corre-
Well known in classical nonlinear optics, in recent years lated sets of orthonormal spectral amplitude functions
it has seen increasing adoption in quantum optics for ef- {ϕk(ω)} and {ψk(ω)} and the real Schmidt coefficients
ficient NIR single photon detection[15], all-optical fast κk which satisfy the relation (cid:80)kκ2k = 1. If the gat-
switching[16], super high resolution timing of quantum ing pulse has the form of a weighted Hermite function
pulses[17], and quantum information erasure[18]. More- uk(ω) ∝ e(ω−2σω20)2Hk(cid:0)ω−σω0(cid:1) with Hk the Hermite poly-
over,combinedwithspectralengineering[8,19],itenables nomials, the basis functions of both sets are in good ap-
a new type of quantum interference between photons of proximation Hermite functions as well. In the Schmidt-
different color[20]. decomposedform,thetransferfunctiondescribesamap-
In this Letter we introduce the Quantum Pulse Gate ping between pairs of broadband modes ϕ (ω)→ψ (ω).
k k
(QPG): A device based on spectrally engineered SFG to By defining broadband mode operators A =
k
extract photons in a well-defined broadband mode from (cid:82)dωϕ (ω) a(ω) and C = (cid:82)dωψ (ω) c(ω) correspond-
k k k
alightbeam. Weoverlapanincomingweak, multi-mode ing to the Schmidt bases, the effective Hamiltonian from
input pulse with a bright, classical gating pulse inside a Eq. 1 can be rewritten as
nonlinear optical material (Fig. 1). Spectral engineering
(cid:88) (cid:16) (cid:17)
ensures that only the fraction of the input pulse which H=θ κ A C† +A†C , (3)
k k k k k
follows the gating pulse form is converted. The residual k
pulse, orthogonal to the gating pulse, is ignored. An in- AnopticalbeamsplitterhasaHamiltonianoftheform
putquantumlightpulse’squantumpropertiescanbepre- H =θac†+h. c.Sowithrespecttobroadbandmodes,
BS
served in conversion by mode-matching the gating pulse SFG can be formally interpreted as a set of beam split-
to its intrinsic mode structure. ters, independently operating on one pair of broadband
SFG conversion efficiency can be tuned with gating modeseach,suchthatA →cos(θ )A +ısin(θ )C . The
k k k √ k k
pulse power, and unit efficiency is in principle reachable. effectivecouplingconstantθ =θ·κ ∝ P takestherole
k k
ThustheQPGisabletounconditionallyfilterbroadband ofthebeamsplitterangle. Itstransmissionprobability–
modes from arbitrary input states, and to convert them theprobabilitytofindaphotonintheup-convertedmode
into a well-defined Gaussian wave packet at the sum fre- C if it initially has been in mode A – is η =sin2(θ ).
k k k k
quency. By pulse-shaping the gating pulse we are able In Fig. 2 A1-C1, we illustrate an example for a non-
toswitchbetweendifferenttargetbroadbandmodesdur- engineered SFG process, as commonly found in pulsed
ing the experiment. By superimposing gating pulses for SFG experiments: The transfer function f(ω ,ω )(Fig.
i o
two different broadband modes, we create interference 2 A1) exhibits spectral correlations, causing more than
betweenthosepreviouslyorthogonalpulses. Incombina- one non-zero Schmidt coefficient (Fig. 2 B1). This leads
tion with a standard single photon detector we are able to the simultaneous conversion of multiple modes A at
√ k
to herald pulsed, pure, single-mode single photons from once with non-zero coupling constants θ ∝ P for any
k
a multi-mode photon pair source. given gating pulse power P (Fig. 2 C1). Hence a SFG
Forabrightclassicalgatingpulse, theeffectiveHamil- process in general is not mode-selective.
tonian of SFG that up-converts a photon in mode ’a’ to However, spectral engineering can make SFG mode-
mode ’c’ is given by selective by eliminating its spectral correlations so that
(cid:90) the frequency of an up-converted photon gives no infor-
H=θ dωi dωo f(ωi,ωo)a(ωi)c†(ωo)+h. c. (1) mation about its original frequency. Now, Schmidt de-
composition yields one predominant parameter κ ≈ 1
√ k
Here we introduced the coupling constant θ ∝ χ(2) P withallothersclosetozeroandaseparabletransferfunc-
with χ(2) denoting the second order nonlinear polariza- tionf(ω ,ω )≈κ ϕ (ω )ψ (ω ). Also,nowthefullcou-
i o k k i k o
tion tensor element of the SFG process and P the gat- plingθ ≈θisexploited,allowingforrelativelyweakgat-
k
ing pulse power. The SFG transfer function f(ω ,ω ) = ing beams for unit conversion efficiency. We achieve this
i o
3
FIG.3. AQPGapplication: Generatingpureheraldedbroad-
bandsinglephotonsindifferentmodesfromaPDCsourceof
FIG. 2. (A1-A3) SFG transfer function f(ω ,ω ) with (A1)
i o multi-mode photon pairs
and without (A2,A3) frequency correlations. (B1-B3) Coeffi-
cientsκ forthefirstfourSchmidtmodepairsofthetransfer
k
functions. (C1-C3)SFGefficienciesA →C forthefirstfour
k k ally not in pure, but spectrally mixed states[8], and thus
SchmidtmodesagainstgatingpowerdependentSFGcoupling
of limited usefulness for most quantum optical applica-
constant θ
tions. We feed the signal photon (containing all broad-
band modes A˜ ) from the PDC source into the QPG
k
by choosing a SFG process with an already correlation- whichismode-matchedsuchthatA˜ =A . Wenotethat
0 0
free phasematching function Φ. If the phasematching
for heralding pure single photons or pure Fock states[9],
bandwidth is narrow compared to gating pulse width,
mode-matching is not necessary and an engineered SFG
spectral correlations are negligible (Fig. 2 A2-A3), and
process according to Eq. 4 is sufficient. In that case
wecanapproximateaseparabletransferfunction(Fig. 2
however, the resulting pulse shape is a coherent super-
B2-B3). The effective SFG Hamiltonian is now formally
position of all input modes. Here, only the 0th mode is
a beam splitter Hamiltonian
selected, and the higher modes do not interact with the
H =θ A C† +h. c. (4) QPG because the according beam splitter transforma-
QPG k k 0 tions yield the identity A → A for k > 0. We choose
k k
meaning that only mode A is accepted for conver- the gating pulse power such that θ = π for optimal
k 0 2
sion. Becauseofthehorizontalphasematching,thetarget conversion efficiency. Combining the PDC source with
mode is always the Gaussian pulse C . a subsequent QPG results transforms the PDC Hamil-
0
This process implements the QPG, with the bright in- tonian as HPDC → H(cid:48) = e−ıHQPGHPDCeıHQPG, and we
putpulseusedasgatepulsetoselectaspecificbroadband obtain
mode. Bytuningthecentralwavelengthandspectraldis-
∞
tribution of the gating pulse, we can control the selected H(cid:48) =ıχB†C† +χ(cid:88)c A˜†B† +h. c. (6)
0 0 k k k
broadband mode’s shape, width and central wavelength.
k=1
We compare the effect of different gating pulse forms:
Gating with mode u0 (i. e. a Gaussian spectrum, Fig. 2 Since mode C0 is centered at the sum frequency of input
A2-C2)selectsinputmodeA ,gatingwithmodeu (Fig. and gating pulse, it can be split off easily into a sepa-
0 1
2 A3-C3) selects A from the input pulse for frequency rate beam path with a dichroic mirror. Conditioning on
1
up-conversion. single photon events on the path of C0 provides us with
Pure heralded single photons are a crucial resource in pure heralded single photons in mode B0. Fig. 3 illus-
many quantum optical applications, but the widely used trates this scheme: A photon detection event heralds a
PDCphotonpairsourcesemitmixedheraldedphotonsin pure single photon pulse in broadband mode u1. This
general. We now consider the application of the QPG to process can be cascaded to successively pick off several
“purify” those photons. In type-II PDC, a pump photon modes Ak from the input beam. Note that if we in-
decays inside a χ(2) -nonlinear medium into one horizon- sert a mode matched QPG into the vertically polarized
tally polarized signal and one vertically polarized idler PDCbeamtoconvertB0intoD0,wecanunconditionally
photon. For a collinear type-II PDC source pumped single out an ultrafast two-mode squeezed vacuum state
by ultrafast pulses the general effective Hamiltonian in eıχc0(C†0D†0+C0D0)|0(cid:105) from a multi-mode squeezer[21].
terms of broadband modes reads Finally we give realistic parameters to show the fea-
H =χ(cid:88)c (cid:16)A˜†B† +A˜ B (cid:17). (5) sibility of an experimental implementation of the QPG,
PDC k k k k k and analyze the mode selection performance. For the
k SFGprocessweuseaperiodicallypoledLiNbO (PPLN)
3
Using such a photon pair source for the preparation of waveguidewithanareaof8×5µm2,alengthofL=50mm,
heralded single photons, one finds that those are usu- a Λ = 4.2µm periodic poling period and at 175◦C to
4
We proposed as an initial application the preparation of
pure heralded single photons from an arbitrary type II
PDC source. For pulsed QKDschemes[1], it canact as a
de-multiplexer of multiple quantum channels within one
physical pulse, and in metrology it may be used to fur-
therenhancemeasurementaccuracybeyondtheclassical
limit by replacing multiple squeezed pulses[2] with one
multi-mode squeezed pulse of light.
We would like to thank Andreas Christ2 for helpful
discussions and acknowledge support of this work under
FIG. 4. Overlap between input pulse mode u˜ and QPG
l
the EC grant agreement CORNER (FP7-ICT-213681).
Schmidt mode ϕ for mode-matched (left) and non-mode-
k
matched (right) case.
achieve phasematching for SFG of an input pulse at
1550nm to about 557nm. It is gated by coherent laser ∗ [email protected]
[1] A. R. Dixon et al., Applied Physics Letters 96, 161102
pulses at 870nm with 2ps pulse length or a spectrum
(2010).
with 0.635nm FWHM to ensure a transfer function sep-
[2] V. Giovannetti, S. Lloyd, and L. Maccone, Nature 412,
arability. The uncorrelated, separable transfer functions
417 (Jul. 2001), ISSN 0028-0836.
in Fig. 2 (A2-A3) are calculated from these parameters, [3] A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg,
usinggatingpulseswithu andu asspectralamplitude, Phys. Rev. Lett. 94, 073601 (Feb 2005).
0 1
respectively. [4] M. Stobińska, G. Alber, and G. Leuchs, Europhys. Lett.
In Fig. 4 we illustrate the switching capabilities of 86, 14007 (2009).
[5] U.M.TitulaerandR.J.Glauber,Phys.Rev.145,1041
our QPG, as well as the impact of mode matching. For
(May 1966).
the given material parameters, we employ gating pulses
[6] M. Martinelli et al., Phys. Rev. A 67, 023808 (2003).
withpulseformu tou ,determinetheSchmidtdecom-
0 10 [7] C.K.Law,I.A.Walmsley,andJ.H.Eberly,Phys.Rev.
position of the resulting transfer function f(ωi,ωo), and Lett. 84, 5304 (2000).
plottheoverlapofthepredominantSchmidtfunctionϕ [8] W. P. Grice, A. B. U’Ren, and I. A. Walmsley, Phys.
k
(κ ≈ 1) with an Hermitian input mode u˜ from an in- Rev. A 64, 063815 (Nov 2001).
k l
cident light pulse. On the left, gating and input pulse [9] P. P. Rohde, W. Mauerer, and C. Silberhorn, New Jour-
nal of Physics 9, 91 (2007).
haveequalfrequencyFWHM,whichisessentialforgood
[10] A.M.Branczyk,T.C.Ralph,W.Helwig,andC.Silber-
mode matching. Now, by switching the order k of the
horn, New Journal of Physics 12, 063001 (2010).
gating mode (and without changing the physical param-
[11] K. Laiho, K. N. Cassemiro, and C. Silberhorn, Opt. Ex-
eters of the QPG), we select with high fidelity only the press 17, 22823 (Dec 2009).
input mode k to be converted. For k ≤ 10, the overlap [12] H.P.YuenandV.W.S.Chan,Opt.Lett.8,177(1983).
(cid:12)(cid:12)(cid:82)dωu˜∗k(ω)ϕk(ω)(cid:12)(cid:12)2 exceeds 99%, and the overlap for all [[1143]] JB..LLe.aScchheutmaalk.,erP,hOysp.t.RLevet.tL.e9t,t.18898,(12958749)0.1 (Jun 2002);
other input modes combined therefore is less than 1%:
N.Trepsetal.,Science301,940(2003); M.Lassenetal.,
Only a negligible fraction of modes other than the se-
Phys. Rev. Lett. 98, 083602 (2007); T. Yarnall, A. F.
lected input mode are converted.
Abouraddy, B. E. A. Saleh, and M. C. Teich, ibid. 99,
In contrast, Fig. 4 (right) has no mode matching, the 250502 (Dec 2007).
gatingpulseFWHMistwicethatoftheinputpulse. Mul- [15] R.V.Roussev,C.Langrock,J.R.Kurz,andM.M.Fejer,
tiplestrongoverlapsbetweenSFGSchmidtmodesϕ and Opt. Lett. 29, 1518 (2004); M. A. Albota and F. C.
k
input modes u˜ appear: A wide range of modes is con- Wong, ibid. 29, 1449 (2004); A. P. Vandevender and
l
P. G. Kwiat, J. Mod. Opt. 51, 1433 (2004), ISSN 0950-
verted for any given input spectrum. The checkerboard
0340; S. Tanzilli et al., Nature 437, 116 (2005), ISSN
pattern reflects the fact that only Hermite modes of the
0028-0836.
same parity overlap, and the SFG Schmidt modes are in
[16] A. P. VanDevender and P. G. Kwiat, Opt. Express 15,
good approximation Hermite modes. 4677 (2007).
In conclusion, we have introduced the concept of the [17] O. Kuzucu, F. N. C. Wong, S. Kurimura, and
QPG, a flexible device to split well-defined broadband S. Tovstonog, Phys. Rev. Lett. 101, 153602 (2008).
modes from a light pulse based on spectrally engineered [18] H. Takesue, Phys. Rev. Lett. 101, 173901 (Oct 2008).
[19] A. B. U’Ren et al., Laser Physics 15 (2005), ISSN 1054-
SFG. The selected mode can be switched by shaping
660X; P. J. Mosley et al., Phys. Rev. Lett. 100, 133601
the gating pulse spectrum and converted with high ef-
(2008).
ficiency. Further, we have given a realistic set of ex-
[20] M.G.Raymer,S.J.vanEnk,C.J.McKinstrie,andH.J.
perimental parameters for a QPG realized in a PPLN McGuinness, Opt. Comm. 283, 747 (2010).
waveguide and demonstrated the high flexibility of the [21] A. Eckstein, A. Christ, P. J. Mosley, and C. Silberhorn,
QPG achieved through shaping the gating pulse form. Phys. Rev. Lett. 106, 013603 (Jan 2011).
