A Proposal for Experimetal Detectio of Amplitude th-power Squeezig Rajaa Prakash ǂ ad Ajay K. Yadav ǂǂ Physics Departmet, Uiversity of Allahabad, Allahabad-00 (U.P.), Idia ǂ prakash_rajaa97@rediffmail.com, ǂǂ ajaypdau@gmail.com Abstract Recetly, i several theoretical ivestigatios, amplitude th-power squeezig has bee studied with =, 3,, 5, although a scheme for experimetal detectio is kow for = oly. I the preset paper, we give a proposal for experimetal detectio of amplitude th-power squeezig usig ordiary homodyig with coheret light for arbitrary power ad discuss i detail its theory. The proposed scheme requires oly repeated measuremets of the factorial momets of umber of photos i the light obtaied after homodyig, with various shifts of phase of coheret light, ad ivolves o approximatios, whatsoever.. Itroductio A sigle mode light has two compoets i quadrature ad the operators correspodig to these i quatum mechaics are o-commutig [-3] ad satisfy a ucertaity relatio. For classical optical fields, which have a o-egative weight fuctio i Sudarsha-Glauber diagoal represetatio [], the variaces of the quadrature operators have equal lower bouds. Optical fields may have a o-classical feature ad variace of oe quadrature amplitude may be less tha this lower boud at the expese of icreased variace for the other. This o-classical feature, called squeezig, was studied earlier i academic iterest [5] but its importace has ow bee uderstood because of its applicatio to optical commuicatio [6], optical waveguide tap [7], gravitatioal wave detectio [8], iterferometric techiques [9], ehacig the chael capacity [0], quatum teleportatio [], quatum dese codig [], quatum cryptography [3], Nao-displacemet measuremet [], optical storage [5] ad amplificatio of sigals [6]. Geeratio of squeezed states has bee reported i a variety of oliear optical processes, e.g., multiphoto absorptio [7], degeerate parametric amplifier [8], free electro laser [9], harmoic geeratio [0], degeerate parametric oscillatio [], degeerate four-wave mixig [], degeerate hyper-rama scatterig [3], resoace fluorescece [], the sigle atom-sigle field mode iteractio [5], superposed coheret states [6], ad oliear beam splitter [7]. Squeezig is uderstood by writig the aihilatio operator â i terms of the two hermitia quadrature amplitude operators ˆX ad ˆX i the form, â X ix or X ˆ ( a a), ˆ i X ( a a), which gives i [ X, X] or ( Xˆ ˆ ) ( X), where 6 coical brackets deote expectatio values ad X ˆ, X, X,. ˆX is said to be squeezed if ˆ ( X) ad ˆX is said to be squeezed if ( Xˆ ). Istead of cosiderig Xˆ ˆ ad X oe ca cosider the most geeral quadrature amplitude operator, ˆ i i X Xcos X si ( a e ae ), the commutatio relatio, i [ X, X ], which gives ucertaity relatio, ( Xˆ ) ( ˆ X ), leads to idea 6 of squeezig of the geeral compoet ˆX whe ( ˆ ) <. That, squeezig is a o- X
classical feature, is clear from the fact that, for Sudarsha-Glauber diagoal represetatio [] d P, we have ˆ i ( X ) d P[Re{( ) e }] 0, d P. () This cocept has bee geeralized i a umber of ways. I the first, Hog ad Madel [8] cosider powers of variace of quadrature amplitudes ad the field is called squeezed to order wheever ( Xˆ ) is less tha its value for coheret state. This type of squeezig has bee studied by several authors [9]. I the secod geeralizatio, itroduced by Hillery [30] for the lowest order ( =, referred to as amplitude squared squeezig) ad by Zhag [3] for higher-orders, Hermitia ad ati-hermitia parts of the square or th-power of the aihilatio operator have bee cosidered ad the two Hermitia operators thus obtaied have bee used to defie squeezig. Explicitly, i the geeral case, we defie ˆ i i X ( e e ), () which gives commutatio relatio ˆ ˆ [, ˆ ] i i X X [ a, a ] W, (3) where ˆ r r r r W r!( C ) a a. () The ucertaity relatio is the ˆ ( X ) ( Xˆ ˆ ) W, (5) 6 ˆ ad therefore the defiitio of squeezig of X is ˆ ˆ ( X ) W. (6) This type of squeezig is called amplitude th-power squeezig. Sice we ca write ˆ ˆ ˆ : ( X ) : ( X ) W, we have ˆ ˆ ˆ : ( X ) : ( X ) W, where double dots (: :) deote ormal orderig of operators. This tells that Eq. (6) which gives the defiitio of th-order squeezig ca also be writte i the simpler form ˆ : ( X ) : 0. (7) Amplitude-squared squeezig has bee studied for a aharmoic oscillator [3], the iteractio betwee atom ad radiatio field [33], mixig of waves [3], Kerr medium [35], superposed coheret states [36]. Also ehacemet of amplitude-squared squeezig is studied by mixig it with coheret light beam [37]. Amplitude th-power squeezig has bee studied by several authors [38-]. It may be oted that sice the Hog ad Madel s cocept of th order squeezig is ot based o ay ucertaity relatio, th order squeezig ca be obtaied for both ˆX ad ˆX simultaeously []. Simultaeous squeezig of ˆ X ad ˆ X is however ruled out because of the ucertaity relatios. Hillery also cosidered a secod type of amplitude squared squeezig by separatig ( ) ito its Hermitia ad ati-hermitia parts. A differet type of amplitude th-power squeezig has bee defied by Bužek ad Jex [3]. Other geeralizatios of squeezig ivolve multi-mode operators ad their commutatio relatios; the commo examples beig sum ad differece squeezig [], spi squeezig [5], atomic squeezig [6], ad polarizatio squeezig [7]. Proposal for experimetal detectio of ordiary squeezig was give by Madel [8] usig homodyig with itese light with adjustable phase from a local oscillator ad measurig the umber of photos ad its square i oe of the outputs. Such homodyig of
squeezed light coverts squeezig ito sub-poissoia photo statistics [9] ad the degree of squeezig is obtaied from measuremets of expectatio values of photo umber operator ad its square. Prakash ad Kumar [50] showed a similar coversio of fourth-order squeezig ito secod-order sub-poissoia photo statistics ad proposed a balaced homodye method for detectio of fourth-order squeezed light i the similar fashio. Prakash ad Mishra [5] exteded the proposal of ordiary homodyig for experimetal detectio of amplitude-squared squeezig by measurig higher order momets of umber operator of mixed light with shifted phases. They also studied higher-order sub-poissoia photo statistics coditios for o-classicality ad discussed its use for the detectio of Hog ad Madel s squeezig of arbitrary order. Prakash et al reported recetly [5] a ordiary homodye method for detectio of secod type of amplitude-squared squeezig of Hillery by measurig the higher-order momets of the umber operator i light obtaied by homodyig with itese coheret light. A very large umber of theoretical papers has bee published o the study of amplitude th-power squeezig with = 3 [38], [39], 5 [0], ad for = k [] but o geeral method for detectio has bee reported for >. We preset here a proposal for experimetal detectio of amplitude th-power squeezig for ay arbitrary power by usig the ordiary homodye detectio method. It may be oted that we work without takig ay approximatio whatsoever like replacig operator for coheret state by c-umber or takig trasmittace or reflectace of the beam splitter very small.. The detectio scheme Schematic diagram for proposed detectio scheme, show i Fig., cotais the experimetal outlie ad also the coceptual meaig of quatum efficiecy of the experimetal detector []. The beam splitter ad ideal detector placed iside dotted rectagle model the real photodetector []. The sigal represeted by operator â is mixed usig beam splitter with sigal ˆ i be, obtaied by shiftig by φ the phase of sigal from a local oscillator represeted by operator b ˆ, to give output sigal ĉ ad c ˆ. Oe of the output sigals, say c ˆ, is detected. If the beam splitter has trasmittace T ad we write t T ad r T as coefficiets of trasmissio ad reflectio for the amplitudes, respectively, we ca write [3] ˆ i c ta rbe, (8) with t ad r real. Number operator of the mixed light is the ˆ ˆ i ( ˆ )( i Nc c c ta rb e ta rbe ). (9) I this model [] of photodetector with quatum efficiecy η, the beam splitter mixes (i) iput sigal ĉ ad (ii) vacuum sigal a ˆv, ad oe of the outputs ˆd give by dˆ cˆ v (0) is detected by a ideal detector havig 00% efficiecy. It is iterestig to ote that this model [] explais that the detected couts, dd, are η times the icidet umber of cc ˆˆ, photos, i.e., ˆ ˆ d d cˆˆ c () which is also obtaied directly from Eq. (0). For factorial momets of order, we get similarly ˆ ˆ d d cˆ cˆ. () The relatioship for th-power of photo umber operator is ot a simple oe. This is oe reaso why we ivolve factorial momets of photo umber operator ad ot powers of the photo umber operator i our aalysis ad result, as was doe earlier [5].
LO Local Oscillator ˆb Phase Shifter (φ) PS be ˆ i Squeezed Light â ĉ Beam Splitter ĉ v ˆd ˆd Detector with 00% Efficiecy Couter Photodetector Fig.. Schematic diagram of detectio of squeezig via ordiary homodyig For the setup uder cosideratio, the observed factorial momets of couts with phase shift φ is the If observatios are doe for l m l m l m ˆ l ˆm i( ml) l m. lm, 0 M c c C C t r a a b b e (3) M for φ = kπ/ with k = 0,,, - we ca easily fid the values of observables P M e ( tr ) [ a e i a e ], () ik i k k 0 Q M t ( C ) ( t r ) a a, l l l k l k0 l0 (5) if the local oscillator gives i the coheret state with e is the complex amplitude. Eq. () gives a method for measurig ˆ, usig θ β = θ/, but a method for measurig Wˆ l l ad therefore for momets is still to be desired. To achieve this ed, we ca solve Eq. (5) for factorial momets of photo umber operator by iteratio. This is doe i Appedix ad gives where K s is defied by Straight forward calculatios lead to P P ( tr ) s ( s) s( s) ( ) s s0 P P K C r Q X s s t Ks( Cs) ( r ) Q s, s0 (6) s s s l ( l ) with 0. l 0 K K C K (7) ( tr ) [ : ( Xˆ ) : cos ( ) : ( Xˆ ) : si ( ) ˆ ˆ ˆ ˆ ( X X X X )si ( )] i
ˆ ( tr ) : ( X ) :, if ( ) 0; ( tr ) : ( Xˆ ) :, if ( ). This equatio shows that amplitude th-power squeezig for ay value of θ, defied by Eqs. (6) or (7), ca be detected by measuremet of observables P, Q, η, T ad β. 3. Discussio of results We explaied how P ad Q ca be foud from measuremets of (8) ˆ ( ) N (see Eqs. () ad (5)). Measuremet of quatum efficiecy η for a give detector is somewhat tricky ad requires use of spotaeous parameter dow coversio [53]. If a photo of a large eergy ħω breaks to create two photos of eergies ħω ad ħω (with ω + ω = ω), the latter two photos should give coicidece couts oly ideally. If N photos break ad the quatum efficiecies for modes ω ad ω are η ad η (both < ), the experimet registers couts N = η N, N = η N ad coicidece couts N c = η η N. The quatum efficiecies are the η = N c /N ad η = N c /N. Oce η is determied for ay detector, the detector ca be used to measure T ad β easily. It should be oted that as icreases the choice of θ β becomes more sesitive ad for detectio of : ( ˆ X ) :, θ β is to be set as θ/. For =, i.e., for ordiary squeezig detectio was first studied by Madel [9]. His formula is differet from that obtaied from Eq. (8) after substitutig =. This is uderstadable as Madel s result is approximate, ad is obtaied uder the approximatio, r, i our otatios. Also, for =, i.e., for amplitude-squared squeezig, the results of Prakash ad Mishra [5] are differet from the results obtaied from Eq. (8) as their results are also approximate, the same approximatio ( r, i our otatios) beig used. Ackowledgemets We would like to thak Prof. H. Prakash ad Prof. N. Chadra for their iterest ad critical commets. We would like to ackowledge Dr. R. Kumar, Dr. P. Kumar, Dr. D. K. Mishra, Namrata Shukla, Ajay K. Maurya, Vikram Verma, ad Maoj K. Mishra for their valuable ad stimulatig discussios. Oe of the authors (Ajay K. Yadav) is grateful to the Uiversity Grats Commissio, New Delhi, Idia for fiacial support uder a UGC-JRF fellowship. Appedix We ca separate the ˆ ˆ a a term o right had side of Eq. (5) ad write l l l l 0( 0) ( l) ( ) l t Q K C C t r (A) with K 0 =. I the first term i summatio o the right had side we substitute for the expressio obtaied from Eq. (A) usig K 0 ( l C 0 ) = ad this gives with t Q K ( C ) ( t r ) [ t Q m m m l l l l ( C ) ( ) ] m t r a a K0( C0) ( Cl) ( t r ) a a m l K K ( C ). This ca be simplified ad writte as 0 0
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