G x, x E x E x E x E x. a a a a. is some matrix element. For a general single photon state. ), applying the operators.

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1 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Proble Set # Quetio With G x, x E x E x E x E x G pqr p q r where G pqr i oe trix eleet For geerl igle photo tte k k k f (with fk ), pplyig the opertor r will ecerily ihilte the tte, ie, Thu, we coclude tht r Now coider ited Altertive view G p r q p q r qr p * qr f j fk jpk k, j * qr j k k jp k, j f f * qr p f f For y tte creted by booic cretio opertor, it i Proof: eigette of the totl uber opertor N k k k N N k N k N Therefore tte with differet uber of re orthogol oider y opertor Ô which ct o oly the two-prticle ubpce jk jk j k k j We c lwy write I j, k j, k ˆ ˆ O IOI O pqr pqr r q p Pge of 8

2 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Note tht O Tr O ˆ pqr r p q Tr O ˆ p q r where Tr i the trce opertor For deity opertor Ô, we the ee tht it i lwy relted to the orl-ordered expecttio vlue Tr p q r p q r Quetio () For, G d G Thu g (b) For, G d G Thu g / (c) For, G (ote tht ) d G Thu g (d) (i) For A e E where E h d / kt B, G ˆ A A A A A h / kbt A e (ii) oider, Pge of 8

3 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 AA A A G A A A A A A A A A A A A A where we iplicitly dded the ter k G A k kq i ech erie, d ued q q q We therefore hve g Quetio H ~ A H H A () Be plitter effect o the tte: i i H H H ; H H ( H H ) where _ d _ e repectively triio d reflectio r of the be plitter Probbilitie for differet outcoe: Detector (triio r): photo (oly) probbility i ~ A/+A /+A / photo probbility ~ A /*, where the fctor of pper becue of the two reolved photo S=A/+A /+A / Detector h the e probbilitie The coicidece cout:,=a /+ A / ( photo o ech detector) (), A / g, if A S S ( A/ A / A / ) / ( A) (b) Exctly the e ) (c) Up to the order A, HV ~ A H H AV V A A VV VV A HV HV, where oe freedo i tke i the tte ottio HV, for exple, e Pge of 8

4 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 H V d e H V i After be plitter, HV HV H V ( HV H V ) Hece, logouly to ), g () S S, A / A ( A/ ( A / A / / ) A A / / A / ) Quetio () oider it it E ( e e ) i ( )co t ( )i t i( X co t X i t) i Thee two opertor re ocited with phe fctor which ocillte out of phe with ech other by /, d hece clled qudrture opertor (b) co, d iilrly * oider X X i Next, X * X So X X X / Siilrly, d hece X / X X Pge of 8

5 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Altertive ethod With the diplceet opertor D, or equivletly, D The D, we kow tht D D & D D &, o * * D XD X X X D X D X X X i * X X X X Siilrly we foud X / D X X D X Quetio 5 () i i A ( LO LO LO LO LO LO LO LO A ( LO LO) ) (b) A, e i LO LO ), e i A ( e i e i ), uig tht α i rel, o ll the phe i tke ito ccout i e iφ We lo ued the reltio i i i e d e e e i e i e i NB: e i or e i i thi ce e the e (jut ottio) They both repreet br-vector for the ket-tte e i (c) A e i e i i A ( X ix ) e ( X ix ) e i ) A X co X i Pge 5 of 8

6 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Quetio 6 () Derivig the followig i otrivil detil re give i the ppedix:! S th r coh r! Note tht the odd uber tte re ot preet oe c be coviced by explicitly expdig the queezig opertor S exp r Perhp, it i obviou tht X X, ice the o-zero tte re ll eve uber tte Oe c lo ee thi directly (uig the reult fro prt (c)), ice X oly deped o ter like d, which equl zero (b) Fortutely, to clculte the vrice we do t eed to del with the ey expreio bove We hll eed to obti the propertie of the queezig opertor logou to tht of the diplceet opertor, ie, A A ivoke the forul e Be B A B A A B The, X S X X S S X S S S,,,!, o S S? r,!!! r r r r coh r ih r S S coh r ih r S X SS X S coh ih coh ih r r r r r r e e e e r, r To proceed, we which iplie tht X e r r / Siilrly, X e / Note tht XX / Pge 6 of 8

7 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Appedix We hll derive for the followig expreio: r /! e th r coh r! Method (ref Gerry & Kight, Itro Qutu Optic) where cohr d ihr SS S S, d S Fro it we hve!! th r!! where the orliztio coditio require The reiig prt re purely rithetic ivolvig oe pecil erie reltio, d hll be kipped (plee refer to the referece for further detil) Method (ref Brett & Rdore, Method i Theoreticl Qutu Optic ) There i geerl theore: for opertor d tifyig K K K, K K & K, K K, K K K K K K exp exp exp l exp where coh ih, ih coh ih d / K / Here oe c how tht, K / d K / tify the couttio reltio tted bove We put r d The Pge 7 of 8

8 Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 r, coh r d th r We therefore hve S r r r which give exp th exp l coh exp th,! S th r coh r! Pge 8 of 8