Odd- and even-order dispersion cancellation in quantum interferometry
Olga Minaeva, Cristian Bonato, Bahaa E. A. Saleh, David S. Simon, Alexander V. Sergienko
aa r X i v : . [ qu a n t - ph ] F e b Odd- and even-order dispersion cancellation in quantum interferometry
Olga Minaeva,
1, 2
Cristian Bonato,
1, 3
Bahaa E.A. Saleh, David S. Simon, and Alexander V. Sergienko
1, 4 Dept. of Electrical & Computer Engineering, Boston University, Boston, Massachusetts 02215 Department of Physics, Moscow State Pedagogical University, 119992 Moscow (Russia) CNR-INFM LUXOR, Department of Information Engineering, University of Padova, Padova (Italy) Dept. of Physics, Boston University, Boston, Massachusetts 02215
We describe a novel effect involving odd-order dispersion cancellation. We demonstrate that odd-and even-order dispersion cancellation may be obtained in different regions of a single quantuminterferogram using frequency-anticorrelated entangled photons and a new type of quantum inter-ferometer. This offers new opportunities for quantum communication and metrology in dispersivemedia.
PACS numbers: 03.67.Bg, 42.50.St, 42.50.Dv, 42.30.Kq
INTRODUCTION
The even-order dispersion cancellation effect based onnonclassical frequency-anticorrelated entangled photonshas been known in quantum optics for some time [1, 2].The nonlinear optical process of spontaneous parametricdown conversion (SPDC) traditionally provides a reliablesource of frequency-entangled photon pairs with anticor-related spectral components, as a consequence of energyconservation. If the frequency of the signal photon is ω s , then the frequency of its twin idler photon must be ω i = Ω p − ω s , where Ω p is the frequency of the pumpbeam. A quantum interferometer records the modula-tion in the rate of coincidence between pulses from twophoton-counting detectors at the output ports of a beam-splitter in response to a temporal delay between two spec-trally correlated photons entering its input ports sym-metrically. This type of quantum optics intensity corre-lation measurement, exhibited in the Hong-Ou-Mandel(HOM) interferometer [3], is manifested by an observeddip in the rate of coincidences. In previous demonstra-tions of dispersion cancellation, one photon of the down-converted pair travels through a dispersive material inone arm of the HOM interferometer while its twin trav-els only through air. The final coincidence interferencedip is not broadened in this case, demonstrating insensi-tivity to even-order dispersion coefficients [2, 4].Even-order dispersion cancellation has been used inquantum information processing, quantum communica-tion, and in quantum optical metrology. For example,it enhances the precision of measuring photon tunnelingtime through a potential barrier [5] and improves theaccuracy of remote clock synchronization [6]. The sameeffect provides superior resolution in quantum optical co-herence tomography [7] by eliminating the broadening ofinterference envelope resulting from group velocity dis-persion. The potential of quantum even-order disper-sion cancellation has recently stimulated efforts to mimicthis effect by use of classical nonlinear optical analogues[8, 9, 10].In this Letter we introduce a novel type of quantum in- terferometer that enables demonstration of the odd-orderdispersion cancellation as a part of new dispersion man-agement technique. In our design, both even-order andodd-order dispersion cancellation effects can be recordedas parts of a single quantum interference pattern. FIG. 1: Schematic diagram of the optical setup. The SPDCsource produces pairs of frequency anticorrelated photonscombining on a beamsplitter in a HOM interferometer con-figuration. Photons exiting one HOM port are fed into a MZinterferometer. Coincidence events are registered between twosingle-photon detectors at the output ports of the MZ inter-ferometer. A dispersive sample in one arm of the MZ inter-ferometer generates a phase delay ( φ ). HOM interferometers are commonly used to produceeither | Ψ i ∼ | , i − | , i state, when the delay τ is setto balance the two paths, ensuring destructive interfer-ence in the middle of the interference dip, or a super-position of | , i , | , i and | , i states, when the de-lay τ significantly unbalances two paths and shifts co-incidences to the shoulder of HOM interference pattern.Mach-Zehnder (MZ) interferometers fed by a particularquantum state have also been studied in detail [11].In the new design two interferometers work together:one output port of a HOM interferometer provides inputto a MZ interferometer. The state of light introduced intothe MZ interferometer is continuously modified when thedelay τ in the HOM interferometer is scanned. A signalfrom one of the HOM output ports is fed into a MZ in-terferometer with a dispersive sample providing a phaseshift φ in one arm, as shown in Fig.1. The delay τ insidethe MZ interferometer is kept at a fixed value. A peculiarquantum interference pattern is observed in the rate ofcoincidences between two photon-counting detectors D D τ . The interference profile has two dis-tinct patterns. The central interference pattern dependsonly on even-order dispersion coefficients, while the pe-ripheral pattern depends only on odd-order terms. Thisability to manipulate and evaluate odd-order and even-order dispersion terms independently in a single quantuminterferometer opens new perspectives in quantum com-munication and in precise optical measurement. THEORETICAL MODEL
For detectors D and D much slower than the tempo-ral coherence of the downconverted photons, the coinci-dence rate in such intensity correlation measurements is[12]: R c ( τ , τ ) = Z dt Z dt G (2) ( t , t ) , (1)with G (2) ( t , t ) second order correlation function G (2) ( t , t ): G (2) ( t , t ) = | h | ˆ E (+)1 ( t ) ˆ E (+)2 ( t ) | Ψ i | . (2) E (+)1 ( t ) and E (+)2 ( t ) are the electrical field operators atthe surfaces of detectors D and D , respectively.ˆ E (+) j ( t j ) = 1 √ π Z dω j e − iω j t j ˆ b j ( ω j ) , (3)where ˆ b j ( ω j ) is the mode operator at detector j , ex-pressed in terms of the input field operators ˆ a j ( ω j ) [12].The quantum state of light emitted in a frequency-degenerate non-collinear type-I phase-matching SPDCprocess with a monochromatic pump Ω p is: | Ψ i ∝ Z dωf ( ω )ˆ a † (Ω + ω )ˆ a † (Ω − ω ) | i , (4)where f ( ω ) is a photon wavepacket spectral function de-fined by the phase matching condition in the nonlin-ear material, Ω = Ω p / ω s = Ω + ω is the signal photon frequency,and ω i = Ω − ω is the idler frequency .The phase shift φ ( ω ) acquired by the broadband opti-cal wavepacket as it travels through a dispersive materialcould be expanded in a Taylor’s series [13]: φ ( ω s,i ) = c + c ( ω s,i − Ω )+ c ( ω s,i − Ω ) + c ( ω s,i − Ω ) + · · · (5)where the linear term c represents the group delay andthe second-order term c is responsible for group delay dispersion. In a conventional white-light interferome-ter, c is responsible for a temporal shift of the inter-ference pattern envelope, c causes its temporal broad-ening, while c provides a non-symmetric deformation ofthe wavepacket envelope. Higher-order terms might beincluded when a strongly dispersive material is used orin the case of extremely broadband optical wavepackets.In the optical setup of Fig.1, the dispersive materialproviding phase shift φ ( ω ) could be situated in threepossible locations. When the sample is placed an armof the HOM interferometer it leads to the well-knowneven-order dispersion cancellation effect [4]. It may beshown that the presence of a dispersive material betweenthe two interferometers does not affect the coincidenceinterferogram. We thus concentrate on the most inter-esting case: we place the dispersive sample of phase shift φ ( ω ) inside the MZ interferometer, with delay τ set toa fixed value, and τ as the variable parameter.Following the usual formalism [12], one can show thatthe coincidence rate between the detectors is: R c ( τ , τ ) = Z dω (Φ − Φ α ( ω, τ ) − Φ β ( ω, τ )) · ( f ( ω ) f ∗ ( ω ) + f ( ω ) f ∗ ( − ω ) e − iωτ ) , (6)where Φ is a constant,Φ α ( ω, τ ) = e − iωτ e iφ (Ω − ω ) e − iφ (Ω + ω ) + c.c., (7)andΦ β ( ω, τ ) = e − i Ω τ e − iφ (Ω − ω ) e − iφ (Ω + ω ) + c.c. (8)Although not obvious from the form of equation (6), R c ( τ , τ ) is a real function for any spectrum f ( ω ), as canbe seen by rewriting Eq. (6) in manifestly real form: R c ( τ , τ ) = Z dω (cid:8) | f ( ω ) | + | f ( − ω ) | + (cid:2) e − iωτ f ( ω ) f ∗ ( − ω ) + c.c. (cid:3)(cid:9) × [Φ − Φ α ( ω ) − Φ β ( ω )] (9)This fact ensures that the technique demonstratedhere applies to all types of broadband frequency-anticorrelated states of light, including those withnonsymmetric spectral profiles produced in chirpedperiodically-polled nonlinear crystals.The final coincidence counting rate R c ( τ , τ ) of Eq.(6) may also be written as a linear superposition: R c ( τ , τ ) = B + R ( τ ) − R even ( τ , τ ) − R odd ( τ , τ ) . (10)The first coefficient B incorporates all terms that are notdependent on the variable delay τ , providing a constantafter integration. It establishes a baseline level for thequantum interfererogram. The following terms: R ( τ ) = 4 Z dωf ( ω ) f ∗ ( − ω ) e − iωτ , (11) R even ( τ ,τ ) = Z dωf ( ω ) f ∗ ( − ω ) · e − iωτ [ e − i Ω τ e − iφ (Ω − ω ) e − iφ (Ω + ω ) + e i Ω τ e iφ (Ω − ω ) e iφ (Ω + ω ) ] , (12) R odd ( τ ,τ ) = Z dωf ( ω ) f ∗ ( − ω ) · [ e − iω ( τ + τ ) e iφ (Ω − ω ) e − iφ (Ω + ω ) + e − iω ( τ − τ ) e − iφ (Ω − ω ) e iφ (Ω + ω ) ] (13)are responsible for the shape of the interference pattern.The term R ( τ ) represents a peak centered at τ = 0that is simply a Fourier transform of the down convertedradiation spectrum and is insensitive to the dispersionassociated with φ ( ω ). Since R even ( τ , τ ) is dependenton the sum φ (Ω − ω ) + φ (Ω + ω ), it is sensitive onlyto even-order terms in the expansion Eq. (5). This man-ifests odd-order dispersion cancellation and generates adispersion-broadened function centered around τ = 0.The last term R odd ( τ , τ ), in contrast, is sensitive onlyto odd-order dispersion terms in φ ( ω ). This term demon-strates the well known even-order cancellation. The co-efficients e − iω ( τ + τ ) and e − iω ( τ − τ ) shift the two dipsaway from the center of the interference pattern in op-posite directions. Such decomposition of quantum inter-ference terms makes it possible to observe odd-order andeven-order dispersion cancellation effects in two distinctregions of the coincidence interferogram. EXAMPLE
Our results are illustrated by a numerical example ofquantum interference for a 3-mm thick slab of a strongly-dispersive optical material ZnSe, inserted in one arm ofthe MZ interferometer to provide the phase shift φ ( ω ).In this experiment we assume the use of frequency-entangled down-converted photons with 100-nm widespectrum. As illustrated in Fig. 2, one can identify thenarrow peak R ( τ ) in the center, which is insensitiveto dispersion, along with the component R even ( τ , τ ),which is broadened by even-order dispersion contribu-tions only. This central component of the interferogramillustrates the odd-order dispersion cancellation effect.Two symmetric side dips R odd ( τ , τ ) appear shiftedfar away from the central peak by the group velocity de-lay c acquired by entangled photons inside the disper-sive material. However, this shift can be controlled by FIG. 2: The normalized coincidence rate as a function of τ when a 3-mm thick ZnSe sample is placed in the MZ inter-ferometer. The fixed delay τ = 26 ps is used. The insertillustrates the odd-order dispersion contribution. properly adjusting the value of the fixed delay τ . Sucha simple adjustment moves both dips back closer to thecenter and makes it convenient for observing both disper-sion cancellation features in a single scan of the variabledelay line ( τ ) inside HOM interferometer (see Fig. 2).The appearance of asymmetric fringes on the side of twodips is a clear sign of the third-order dispersion. [13]. DISCUSSION
This result can also be understood physically by an-alyzing all possible probability amplitudes that lead tomeasured coincidence events between D D
2. TheMZ interferometer input is a pair of spectrally-entangledphotons separated by time delay τ ; if the leading pho-ton has a high frequency, the lagging photon will have alow frequency, and vice-versa. We consider first the casewhen no dispersive element is present, so that the MZinterferometer introduces only a time delay τ betweenits two arms. We assume that τ is much greater thanthe photon wave packet width, τ c . To explain the depen-dance of the photon coincidence rate on τ , as shown inFig.2, we consider three processes occurring at the inputports of the last beam splitter in the MZ interferometer:1) If | τ | > τ c and | τ − τ | > τ c , then the two photonsarriving at the final beam splitter will be distinguishable,so that no quantum interference is exhibited.2) If | τ | ≈ | τ | , so that | τ − τ | < τ c , then quantuminterference can occur when the leading photon takes thelong path of the MZ interferometer and the lagging pho-ton takes the short path. The two arrive almost simul-taneously (within a time τ c ) at the two ports of the finalbeam splitter. Then the Hong-Ou-Mandel (HOM) effectis exhibited at the beam splitter, albeit with only 25%visibility because of the presence of the other possibil-ity that both photons arrive at a single port, leading toa background coincidence rate independent of τ . Froma different perspective, one may regard this scenario assimilar to that obtained in a Franson interferometer [14],for which photon pairs follow long-long or short-shortpaths. This scenario explains the components of the co-incidence interferogram near τ = ± τ , and in this casethe two spectrally-entangled photons entering separateports of the final beam splitter lead to quantum interfer-ence accompanied by even-order dispersion cancellation.3) Finally, when | τ | < τ c , then one possibility is thatthe photons arrive at separate input ports of the finalbeam splitter. Since these photons are separated by atime τ >> τ c , they are distinguishable and do not con-tribute to quantum interference. The other possibilityis that the pair arrive at the same beam splitter inputport. In this case, upon transmission or reflection at thebeam splitter there are two alternatives for producingcoincidence: transmission of the high-frequency photonand reflection of the low-frequency photon, or vice-versa.This explains the component of the coincidence interfer-ogram near τ ≈
0. In this scenario, which involves twospectrally-entangled photons entering a single port of abeam splitter, quantum interference is accompanied byodd-order dispersion cancellation. We thus see that thequantum interference effects exhibited in scenarios 2) and3) are accompanied by dispersion cancellation – althoughin opposite manners in the two cases.In conclusion, we have demonstrated a new effect inwhich even- and odd-order dispersion cancellations ap-pear in different regions of a single interferogram. Thisis achieved via frequency-anticorrelated photons in anew quantum interferometer formed by a variable delayHOM interferometer followed by a single-input, fixed-delay Mach-Zehnder interferometer. The possibility ofindependently evaluating even- and odd-order dispersioncoefficients of a medium has potential for applicationsin quantum communication and in quantum metrologyof complex dispersive photonics structures. In partic-ular, the ability to accurately characterize higher-orderdispersion coefficients is of great interest in the study offlattened-dispersion optical fibers [15, 16] and in disper-sion engineering with metamaterials [17]. The demon-strated potential of even-order dispersion cancellationhas stimulated the search for classical analogues [8, 9].We expect that the scheme presented here would alsotrigger the similar development of nonlinear optical tech-niques mimicking this quantum effect. Finally, note that our apparatus may be extended by adding a secondMach-Zehnder to the unused HOM output port, allowingthe investigation of new four-photon interference effects.
ACKNOWLEDGMENTS