REPORTS The content of a collection of notes is often more imp more import ortant ant tha than n the their ir ord order. er. Cho Chords rds can theref the refore ore be mod modele eled d as mul multis tisets ets of eit either her pitches or pitch classes. (BChord [ will henceforth refer to a multiset of pitch classes unless otherwise other wise noted.) The music musical al term BtranspoDmitri Tymoczko sition[ sition [ is synon synonymous ymous with the mathematical mathematical term Btranslation translation[ [ and is represented by addiA musical chord can be represented as a point in a geometrical space called an orbifold. Line tion tio n in pit pitch ch or pit pitchch-cla class ss spa space. ce. Tra Transp nsposi osi-segments represent mappings from the notes of one chord to those of another. Composers in a tion onal ally ly re rela late ted d ch chor ords ds ar aree th thee sa same me up to wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using ti transl nslati ation; on; thu thus, s, the C maj major or cho chord, rd, AC, E, short line segments between structurally similar chords. Such line segments exist only when chords tra GZ or A0, 4, 7Z 7Z, is transpositionally related to are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant the F major maj or chord, cho rd, AF, A, CZ CZ or A5, 9, 0Z 0Z, and dissonant chords possess different near-symmetries and suggest different musical uses. because A5, 9, 0Z 0Z K A0 þ 5, 4 þ 5, 7 þ 5Z ester est ern n mu music sic lie liess at th thee in inter tersec secti tion on of cy f wit with h a rea reall num number ber p acc accord ording ing to the modulo 12Z. The musical term Binversion inversion[ [ is synonymo syno nymous us wit with h the math mathemat ematical ical term two seem seemingl ingly y inde independ pendent ent dis discici- equation plines: plines: harmo harmony ny and count counterpo erpoint. int. Breflection reflection[ [ and corr correspo esponds nds to subt subtract raction ion p 0 69 þ 12 log2 ð f =440Þ ð1Þ fro from m a con consta stant nt val value. ue. Inv Invers ersion ionall ally y rel relate ated d Harmony Harm ony del delimit imitss the accep acceptabl tablee chor chords ds (si (simulmultaneously occurring notes) and chord sequences. chords are the same up to reflection; thus, the C Counterpoint (or voice leading) is the technique Th Thee re resu sult lt is a li line near ar sp spac acee (p (pit itch ch sp spac ace) e) in wh whic ich h maj major or cho chord rd is inv invers ersion ionall ally y rel relate ated d to the C octavess have size 12, semitones (the distance minor chord AC, E>, GZ, or A0, 3, 7Z of connecting the individual notes in a series of octave 7Z, because chords cho rds so as to fo form rm sim simult ultane aneous ous mel melodi odies. es. between adjacent keys on a piano) have size 1, A0, 3, 7Z 7Z K A7 – 7, 7 – 4, 7 – 0Z 0 Z modulo 12Z. Chords are usually connected so that these lines and middle middle C is ass assig igned ned the nu numb mber er 60 60.. Musically, transposition and inversion are im(or voices) move independently (not all in the Distance in this space reflects physical distance portant because they preserve the character of a keyboard ard instr instrument uments, s, ortho orthograph graphical ical dis- chord same direction by the same amount), efficiently on keybo chord:: Trans Transposit positional ionally ly relat related ed chord chordss soun sound d (by short distances), and without voice crossings tance in Western musical notation, and musical extremely similar, inversionally related chords distance ce as measu measured red in psych psycholog ological ical experi- fairly so (movie S1). (along (alon g nonintersecti nonintersecting ng paths) (Fig. 1, A to C). distan (14,, 15 15). ). A voice leading between two multisets A x1, These featu features res facil facilitate itate musica musicall perf performan ormance, ce, ments (14 engage explicit aesthetic norms (1 (1, 2), and enMusical Mus ically, ly, the chr chroma oma of a not notee is oft often en x2, I, xmZ and and A A y1, y2, I, ynZ is a multiset of able liste listeners ners to disti distinguis nguish h multi multiple ple simul simultata- more important than its octave. It is therefore ordered pairs ( xi, y j ), such that every member of useful to identify all pitches p and and p p þ 12. The each multiset is in some pair. A trivial voice neous melodies (3 (3). How is it tha thatt Wes Wester tern n mus music ic can satisfy satisfy resul resultt is a circu circular lar quotient space (pitc (pitch-cla h-class ss leading contains only pairs of the form ( x, x, x). mathematici maticians ans call R /12Z (fig. The notation ( x1, x2, I, xn ) Y ( y1, y2, I, yn) harmonic harmo nic and contr contrapunt apuntal al const constraint raintss at once? space) that mathe Whatt det Wha determ ermine iness whe whethe therr two chords chords can be S3). (For a glossary of terms and symbols, see identifies the voice leading that associates the connec con nected ted by eff effici icient ent voi voice ce lea leadin ding? g? Mus Musii- tab table le S1. S1.)) Poi Points nts in thi thiss spa space ce (pi (pitch tch clas classes ses)) cor corres respon pondin ding g ite items ms in eac each h lis list. t. Thu Thus, s, the provide numerical alternatives alternatives to the familiar vo cians have been investigating investigating these quest questions ions provide voic icee le lead adin ing g (C (C,, C, E, G) Y (B, D, F, G) for almost three centuries. The circle of fifths letter-names of Western music theory: C 0 0, associates C with B, C with D, E with F, and G (fig. (fi g. S1) S1),, fir first st pub publis lished hed in 172 1728 8 (4), depict depictss C /D> 0 1, D 0 2, D quarter-tone sharp 0 2.5, with G. Music theorists have proposed numerous effici eff icient ent voi voice ce lea leadin dings gs amo among ng the 12 maj major or etc. Western Western mus music ic typ typica ically lly uses onl only y a way wayss of meas measuri uring ng voi voicece-lead leading ing size size.. Rath Rather er than an ado adopt ptin ing g on one, e, I wi will ll req requi uire re on only ly th that at a scales. The Tonnetz (fig. S2), originating with discrete lattice of points in this space. Here I th Euler Eul er in 173 1739, 9, rep repres resent entss ef effic ficien ientt voi voice ce lea leadin dings gs consi consider der the more gene general ral,, cont continu inuous ous case case.. meas measure ure sati satisfy sfy a few cons constra traints ints refl reflecti ecting ng because use the symmetrica symmetricall chor chords ds that widely acknowledged features of Western music among the 24 major and minor triads (2 (2, 5). This is beca Recent work (5–13 (5–13)) investigates efficient voice influence voice-leading behavior need not lie (16 ). ). These constraints make it possible to idenleadin lea ding g in a var variet iety y of spe specia ciall cas cases. es. Des Despit pitee on the discrete lattice. tify, in polynomial time, a minimal voice leading tantalizing hints (6–10 (6–10), ), however, no theory has articulated artic ulated general princ principles iples expla explaining ining when and why eff effici icient ent voi voice ce lea leadin ding g is pos possib sible. le. A C D B This report provides such a theory, describing geometrical geome trical spaces in which points represent chords and line segments represent voice leadingss bet ing between ween the their ir end endpoi points nts.. The These se spa spaces ces show us precisely how harmony and counter point are related. Human pitch perception is both logarithmic and periodic: Frequencies f f and and 2 f f are are heard to be separated by a single distance (the octave) and to possess the same quality or chroma. To model the logarithmic aspect of pitch percep- Fig Fig.. 1. Effi Efficient cient voic voicee lead leading ing betw between een tran transpos spositio itionally nally and inve inversio rsionally nally rela related ted chor chords. ds. Thes Thesee tion, I associate a pitch_ pitch_s fundamental frequen- progress progressions ions exploit three near-symmetries: transposition (A and B), inversion (C), and permutation (D).
The Geometry of Musical Chords
W
Department of Music, Princeton University, Princeton, NJ 08544, USA, and Radcliffe Institute for Advanced Study, 34 Concord Avenue, Cambridge, MA 02138, USA. E-mail:
[email protected]
72
Sources: a common 18th-century upper-voice voice-leading pattern (A), a common jazz-piano voiceleading pattern, which omits chord roots and fifths and adds ninths and thirteenths (B), Wagner’s Parsifal (C), Debussy’s Pre´ lude a` l’ap l’apre re`s-mi s-midi di d’un d’ un faune faun e (C), and contemporary atonal voice leading in the style of Ligeti and Lutoslawski (D) (soundfile S1). Chord labels refer to unordered sets of pitch classes and do not indicate register.
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REPORTS (not nec (not necess essari arily ly bij bijecti ective) ve) bet betwee ween n arb arbitr itrary ary chords cho rds (16 ). ). Every musicmusic-theor theoretical etical measur measuree of voice-leading size satisfies these constraints. I no now w de desc scri ribe be th thee ge geom omet etry ry of mu musi sica call chords. An ordered sequence of n pitches can be re repr pres esen ente ted d as a po poin intt in Rn (fi (fig. g. S4) S4).. Directed line segments in this space represent voice leadings. A measure of voice-leading size assign ass ignss len length gthss to the these se lin linee seg segmen ments. ts. I wil willl use quotient spaces to model the way listeners abstract from octave and order information. To model an ordered sequence of n pitch classes, n form the quotient space (R/12Z) , also known n as the n-torus T . To model unordered n-note chords of pitch classes, identify all points ( x1, x2, I xn) and ( x (1), x (2), I x (n)), where s is any per permu mutat tatio ion. n. Th Thee re resul sultt is th thee glo global bal-n n quotient orbifold T /S n (17 , 18 18), ), the n-torus T modulo modul o the symme symmetric tric group S n. It con contai tains ns singularities at which the local topology is not that of Rn. 2 Figure 2 shows the orbifold T /S 2, the space of un unor orde dere red d pa pair irss of pi pitc tch h cla class sses es.. It is a MP bius strip, a square whose left edge is given a hal halff twi twist st and identifi identified ed wit with h its right. right. The orbifold is singular at its top and bottom edges, which act like mirrors (18 (18). ). Any bijective voice leadin lea ding g bet betwee ween n pai pairs rs of pit pitche chess or pai pairs rs of pitch classes can be associated with a path on Fig. 2 (movie S2). Measures of voice-leading size determine these paths paths__ lengt lengths. hs. They are the images of line segments in the parent spaces n n T and R , and are either line segments in the orbifo orb ifold ld or Breflected [ lin linee segm segment entss tha thatt bounc bouncee off its sin singul gular ar edg edges. es. For exa exampl mple, e, the voice leading (C, D>) Y (D>, C) reflects off the orbifold _s upper mirror boundary (Fig. 2). Generalizing Genera lizing to higher dimensions dimensions is straig straighthtforward. To construct the orbifold T n/S n, take an n-di -dimen mensio sional nal prism whose base is an (n – 1) simplex, twist the base so as to cyclically permute its vertices, and identify it with the opposite face (figs. S5 and S6) (16 (16 ). ). The boundaries of the orb orbifo ifold ld are sin singul gular, ar, act acting ing as mir mirror rorss and containing chords with duplicate pitch classes. Chords that divide the octave evenly lie at the center of the orbifold and are surrounded by the familiar famil iar sonor sonorities ities of Weste Western rn tonal tonality. ity. Voice leadings parallel to the height coordinate of the prism prism act as tra transp nsposi ositio tions. ns. A fre freee com comput puter er program written by the author allows readers to explore these spaces (19 (19). ). In man many y Wes Wester tern n sty styles les,, it is des desira irable ble to find efficient, efficient, indep independent endent voice leadi leadings ngs between transpositionally or inversionally related chords. The progressions in Fig. 1 are all of this typee (mo typ (movie vie S3). A cho chord rd can par partic ticipa ipate te in such progressions only if it is nearly symmetrical under tran transposit sposition, ion, permu permutation tation,, or inversio ver sion n (16 ). ). I con conclu clude de by des descri cribin bing g the these se symmetries, explaining how they are embodied in the orbifolds orbifolds_ geomet geometry, ry, and showi showing ng how they the y hav havee bee been n exp exploi loited ted by Wes Wester tern n com compos posers ers.. A chord is transpositionally symmetrical (Tsymmetrical) if it either divides the octave into s
s
s
equall pa equa part rtss or is th thee un unio ion n of eq equa uall lly y si size zed d subsets that do so (20 (20). ). Nearly T-symmetrical chords are close to these T-symmetrical chords. Both types of chord can be linked to at least some of their transpositions by efficient bijective voice leadings. As one moves toward the center of the orbifold, chords become increasingly T-symmetrical and can be linked to their transpositions by increasingly efficient bijective voice leading. The perfectly even chord at the center of the orbifold can be linked to all of its transpositions by the smallest possible bijective voice leading; a relat related ed resul resultt cover coverss discr discrete ete pitch-class pitchclass spaces (16 ). ). Efficient voice leadings betwee between n per perfec fectly tly T-s T-symm ymmetr etrica icall cho chords rds are typicall typi cally y not ind indepen ependent dent.. Thu Thus, s, comp composer oserss
have reason to prefer near T-symmetry to exact T-symmetry. It fol follow lowss tha thatt the aco acoust ustica ically lly con conson sonant ant chords of traditional Western music can be connected by efficient voice leading. Acoustic consonance sonan ce is incom incompletel pletely y under understood stood;; howev however, er, theorists have long agreed that chords approximating the first few consecutive elements of the harmonic series are particularly consonant, at least when played with harmonic tones (21 (21). ). Because elements n to 2n 2 n of the harmonic series evenly divide an octave in frequency space, they divide the octave nearly evenly in log-frequency space. These chords are therefore clustered near thee cen th cente terr of th thee or orbi bifo fold ldss (T (Tabl ablee 1) an and d can typical typ ically ly be lin linked ked by eff effici icient, ent, inde independ pendent ent
transposition 00
11 01
e1
12 02
t2
13
t4
93
48 58
68
57
78
67 77
Fig. 2. The orbifold
8t
79
88
9e
99
tt
[t2]
[e1] major second
t0
minor second
e0 ee
major third minor third
t1
te
tritone perfect fourth
92
90
9t
89
[39]
91
major third perfect fourth
38
81
8e
[48]
28
80
7e 7t
69
17
major second minor third
47
27
70
6e
[57]
37
unison minor second
56
36
16
[66]
46
26
06
6t
59
35
15
e5
55 45
25
05
5t
49
34
14
e4
44
24
04
e3 t3
33 23
03
e2
66
22
[00]
unison
T2 / S S2 .
C 0 0, C 0 1, etc., with B> 0 t, and B 0 e. The left edge is given a half twist and identified with the right. The voice leadings (C, D >) Y (D>, C) and (C, G) Y (C , F ) are shown; the first reflects off the singular boundary. Table Ta ble 1. Common sonorities in Western tonal music. The center column lists the best equal-
tempered approximation to the first n pitch classes of the harmonic series; the right column lists other good approximations. All divide the octave evenly or nearly evenly. Size
Best approximation
Other approximations
2 notes 3 notes
C, G C, E, G
4 notes
C, E, G, B>
5 notes
C, D, E, G, B>
6 notes
C, D, E, F , G, B>
7 notes
C, D, E, F , G, A, B>
C, F C, E>, G> C, E>, G C, E, G C, E>, G>, A C, E>, G>, B> C, E>, G, B> C, E, G, B C, D, E, G, A C, D, E, G, B C, D, E>, F, G, B > C, D, E, F , G , B > C, D, E, F, G, A, B > C, D, E>, F , G, A, B >
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REPORTS voice leadings. Traditional tonal music exploits throughout the orbifolds (16 (16 ). ). For example, the this possibility (Fig. 1, A to C, and movie S4). F halfhalf-dimin diminished ished seventh seventh chord A6, 9, 0, 4Z 4Z This central feature of Western counterpoint is and the F dominant seventh chord A5, 9, 0, 3Z 3Z made possible by compo composers sers__ int intere erest st in the are related by inversion and are very close to harmonic property of acoustic consonance. the I-s I-symm ymmetri etrical cal cho chord rd A5. 5.5, 5, 9, 0, 3. 3.5 5Z. A cho chord rd wi with th du dupl plic icat atee pi pitch tch cl clas asses ses is Conseq Consequentl uently, y, we can find an effi efficient cient voice permutationall permutationally y symmetri symmetrical cal (P-sy (P-symmetrica mmetrical) l) leading between them, (6, 9, 0, 4) Y (5, 9, 0, 3) because because there is some nontrivial nontrivial permu permutation tation (Fi (Fig. g. 1C) (16 ). ). Nearl Nearly y T-sym T-symmetri metrical cal chor chords, ds, of its notes that is a trivial voice leading. These su such ch as th thee C ma majo jorr tr tria iad, d, an and d ne near arly ly Pchords cho rds lie on the singular singular bou bounda ndarie riess of the symmetrical chords, such as AC, D>, E>Z, can orbifolds. Nearly P-symmetrical chords, such as also be nearly I-symmetrical. Consequently, IAE, F, G>Z, are near these chords and contain symmetry is exploited in both tonal and atonal several notes that are clustered close together. music. It plays a salient role in the 19th century, Efficient voice leadings permuting the clustered part partic icul ular arly ly in th thee mu musi sicc of Sc Schu hube bert rt (22), 22), notes bounce off the nearby boundaries (Fig. 2 Wagner (23 (23), ), and Debussy (Fig. 1C). and movies movies S2 and S4). Suc Such h voi voice ce lea leadin dings gs Thee pr Th prec eced edin ing g id idea eass ca can n be ex exte tend nded ed in can be independent and nontrivial. Trivial voice several directions. First, one might examine in leadings are musically inert; therefore, as with detail how composers have exploited the geomT-symmetry T-sym metry,, compo composers sers have reaso reason n to pref prefer er etry of musical chords. Second, one could gennear P-symmetry to exact P-symmetry. eralize the geometrical approach by considering Nearly P-symmetrical chords such as AB, C, quoti quotient ent spaces that ident identify ify trans transposit positional ionally ly D>Z are considered to be extremely dissonant. and inv invers ersion ionall ally y rel relate ated d cho chords rds (24 24). ). Thir Third, d, They are well-suited to static music in which because cyclical rhythmic patterns can also be voices move by small distances within an un- modeled as points on T n/S n, one could use these changing harmony (Fig. 1D). Such practices are spaces to study African and other non-Western characteristic of recent atonal composition, par- rhy rhythm thms. s. Fou Fourth rth,, one cou could ld inv invest estiga igate te how ticularly the music of Ligeti and Lutoslawski. dist distances ances in the orbifolds orbifolds rela relate te to perc perceptu eptual al From the present perspective, these avant-garde judg judgment mentss of chor chord d simi similari larity. ty. Fin Finally ally,, unde underrtechniques are closely related to those of tradi- sta standi nding ng the rel relati ation on bet betwee ween n har harmon mony y and tion ti onal al to tona nali lity ty:: Th They ey exp explo loit it on onee of th thre reee coun counterp terpoint oint may sugg suggest est new techn technique iquess to fundamenta fund amentall symme symmetries tries permi permitting tting effic efficient, ient, contemporar contemporary y composers. independen indep endentt voice leadin leading g betwe between en trans transposiposiReferences and Notes tionally or inversionally related chords. 1. C. Masson,, Nouveau Traite´ des Re`gles Masson `gles pour la Composition A chor chord d is inve inversio rsionall nally y symm symmetri etrical cal de la Musique (Da Capo, New York, 1967). (I-symmetrical) if it is invariant under reflecngen 2. O. Hostinsky Hostinsky´, Die Lehre von den musikalischen Kla¨ ¨ ngen tion in pitch-class space. Nearly I-symmetrical (H. Dominicus, Prague, 1879). 3. D. Huron Huron,, Mus. Percept. 19, 1 (2001). chords are near these chords and can be found
A High-Brightness Source of Narrowband, Identical-Photon Pairs James K. Thompson,1* Jonathan Simon,2 Huanqian Loh,1 Vladan Vuletic´1
We generated narrowband pairs of nearly identical photons at a rate of 5 Â 104 pairs per second from a laser-cooled atomic ensemble inside an optical cavity. A two-photon interference experiment demonstrated that the photons could be made 90% indistinguishable, a key requirement for quantum information-processing protocols. Used as a conditional single-photon source, the system operated near the fundamental limits on recovery efficiency (57%), Fourier transform–limited bandwidth, and pair-generation-rate–limited suppression of two-photon events (factor of 33 below the Poisson limit). Each photon had a spectral width of 1.1 megahertz, ideal for interacting with atomic ensembles that form the basis of proposed quantum memories and logic.
T
he ge gener nerat atio ion n of ph phot oton on pai pairs rs is us usef eful ul fo forr a conv converter erterss base based d on nonl nonlinear inearcrys crystals tals are excel excellent lent broad range of applications, from the fun- sou source rcess of pho photon tonpai pairs rs,, bu butt th they ey are com compar parati ativel vely y damental Eexclu exclusion sion of hidd hidden-v en-varia ariable ble dim because their photon bandwidths range up to formulations of quantum mechanics (1 (1)^ to the hundreds of GHz. However, new applications are more prac practical tical Equan quantum tum cry cryptog ptograph raphy y (2) an and d emerging that demand large pair-generation rates quantum computation (3 (3)^. A key parameter de- into the narrow bandwidths (5 MHz) suitable for termin ter mining ingthe the us usefu efuln lness ess of a par partic ticula ularr sou source rce is it itss strong interaction of the photons with atoms and brigh br ightne tness, ss, i.e i.e., ., how man many y pho photon ton pai pairs rs per sec second ond molecules (2 (2, 4–7 ). are gener generated ated into a part particul icular ar elect electrom romagnet agnetic ic We report the development of a source of mode and fre frequen quency cy band bandwidt width. h. Para Parametr metric ic down down-- photon pairs with spectral brightness near fun74
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4. J. D. Heinichen Heinichen,, Der General-Bass in der Composition (G. Olms, New York, 1969). 5. R. Cohn, Cohn, J. Mus. Theory 41, 1 (1997). 6. J. Roeder, thesis, Yale Yale University University (1984). 7. E. Agmon, Agmon, Musikometrica 3, 15 (1991). 8. R. Cohn, Cohn, Mus. Anal. 15, 9 (1996). 9. C. Callender Callender,, Mus. Theory Online 10 (2004) (http:// mto.societymusictheory.org/issues/mto.04.10.3/ mto.04.10.3.callender.pdf). 10. G. Mazzola, Mazzola, The Topos of Music (Birkha¨ user, Boston , 2002). 11. R. Morris, Morris, Mus. Theory Spectrum 20, 175 (1998). 12. J. Douthett, P. P. Steinbach, J. Mus. Theory 42, 241 (1998). 13. J. Straus, Straus, Mus. Theory Spectrum 25, 305 (2003). 14. F. Attneave, Attneave, R. Olson, Am. Psychol. 84, 147 (1971). 15. R. Shepard, Shepard, Psychol. Rev. 89, 305 (1982). 16. See supporting supporting material material on Science Online. 17. I. Satake, Satake, Proc. Natl. Acad. Sci. U.S.A. 42, 359 (1956). 18. W. Thurston, Thurston, The Geometry and Topology of Three Manifolds (www.msri.org/publications/books/gt3m). 19. ChordGeometries 1.1 1.1 (http://music.princeton.edu/ dmitri/ChordGeometries.html). 20. R. Cohn, Cohn, J. Mus. Theory 35, 1 (1991). 21. W. Sethares, Sethares, Tuning, Timbre, Spectrum, Scale (Springer, New York, 2005). 22. R. Cohn, Cohn, 19th Cent. Mus. 22, 213 (1999). 23. B. Boretz, Boretz, Perspect. New Mus. 11, 146 (1972). 24. C. Callender, Callender, I. Quinn Quinn,, D. Tymo Tymoczko, czko, paper presented presented at the John Clough Memorial Conference, Conference, University University of Chicago, 9 July 2005. 25. Thanks to D. Biss, C. Calle Callender nder,, E. Camp, K. Easwaran, Easwaran, N. Elkies, P. Godfrey Smith, R. Hall, A. Healy, I. Quinn, N. Weiner, and M. Weisberg. È
Supporting Online Material www.sciencemag.org/cgi/content/full/313/5783/72/DC1 Materials and Methods Figs. S1 to S12 Table S1 Movies S1 to S4 Soundfile S1 References 15 February 2006; accepted 26 May 2006 10.1126/science.1126287
damental physi damental physical cal limit limitation ationss and appro approximat ximately ely three orders of magnitude greater than the best current devices based on nonlinear crystals (8 (8). Unlike paramet parametric ric downco downconverter nverters, s, however however,, the atomic ensemble can additionally act as a quantum memory and store the second photon, allowing allow ing trigg triggered ered (i.e. (i.e.,, deter determinis ministic) tic) gener generaation tio n of the sec second ond pho photon ton.. Tri Trigge ggered red del delays ays of up to 20 ms have been demonstrated (9–15 (9–15), ), and it is exp expect ected ed tha thatt opt optica icall lat lattic tices es hol hold d the potential to extend the lifetime of these quantum memories to seconds (9 (9). Lastly, proposed ap plications in quantum information (2 (2, 3) rely on joint measurements of single photons for which indisting indis tinguisha uishabilit bility y is cruci crucial al for high fidel fidelity. ity. We obs observ ervee lar large ge degr degrees ees of ind indist isting inguis uishhabilit abi lity y in the tim time-r e-reso esolve lved d int interf erfere erence nce between the two generated photons (16–19 (16–19). ). 1
Department of Physics, MIT–Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, Massachusetts Institute Insti tute of Tech Technolog nology, y, 77 Mass Massachus achusetts etts Aven Avenue, ue, Cambridge,, MA 02139, USA. 2Depar bridge Department tment of Phys Physics, ics, MIT– Harvard Center for Ultracold Atoms, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA. *To whom corres corresponden pondence ce shoul should d be addre addressed. ssed. E-mail:
[email protected]
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