Intermodal entanglement in Raman processes Biswajit Sen1 , Sandip Kumar Giri2, Swapan Mandal3, C. H. Raymond Ooi4, Anirban Pathak5,6 1Department of Physics, Vidyasagar Teachers’ Training College, Midnapore-721101, India 2Department of Physics, Panskura Banamali College, Panskura-721152, India 3Department of Physics, Visva-Bharati, Santiniketan, India 4Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia 5Jaypee Institute of Information Technology, A-10, Sector-62, Noida, UP-201307, India 6RCPTM, Joint Laboratory of Optics of Palacky University and Institute of Physics of Academy of Science of the Czech Republic, Faculty of Science, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic The operator solution of acompletely quantummechanical Hamiltonian of theRaman processes is used here to investigate the possibility of obtaining intermodal entanglement between different 3 modes involved in the Raman processes (e.g. pump mode, Stokes mode, vibration (phonon) mode 1 andanti-Stokesmode). Intermodalentanglementisreportedbetweena)pumpmodeandanti-Stokes 0 mode, b) pump mode and vibration (phonon) mode c) Stokes mode and vibration phonon mode, 2 d) Stokes mode and anti-stokes mode in the stimulated Raman processes for the variation of the n phaseangleofcomplex eigenvalueα1 ofpumpmodea. Someincidentsofintermodal entanglement a in the spontaneous and the partially spontaneous Raman processes are also reported. Further it J is shown that the specific choice of coupling constants may produce genuine entanglement among 2 Stokes mode, anti-Stokes mode and vibration-phonon mode. It is also shown that the two mode entanglement not identified by Duan’s criterion may be identified by Hillery-Zubairy criteria. It ] is furthershown that intermodal entanglement,intermodal antibunchingand intermodal squeezing h p are independentphenomena. - t n I. INTRODUCTION separability criteria in detail but to study the possibil- a u ity of generation of multi-partite entangled state in two- q photon stimulated Raman processes, as depicted in Fig. Entanglement is one of the most important resources [ 1. The scheme is essentially a sequential double Raman for quantum communication and quantum information process that can produce Stokes and antiStokes photons 1 processing. For example, it is well known that entangle- v thatshowhighlynonclassicalcorrelation[11]andmacro- ment is essential for teleportation, dense coding, quan- 6 scopic entanglement in two-photon laser [12]. To study tum information splitting etc. Thus we need entan- 8 the two-mode entanglement in the two-photon Raman gled states to perform various important tasks related 2 processesitwouldbereasonabletousethreecriteriafrom 0 to quantum information theory. To do so, first we need set B. To be precise, we have chosen the two criteria of . a protocol to check, whether a state generally mixed is 1 Hillery and Zubairy [7], [8] and the criterion of Duan entangledornot? Thisisaveryimportantissueinquan- 0 et al. [3]. Since all these three criteria are only suffi- 3 tum information science and several inseparability cri- cient, a particular criterion can detect only a subset of 1 teria have been proposed for this purpose ([1] and ref- all sets of entangled states. Consequently, application : erences therein). In 1996, Peres [2] proposed the first v of a single criterion may yield incomplete result. This inseparability criterion based on negative eigenvalues of i is why we have used three experimentally testable in- X partialtransposeofthecompositedensityoperator. This separability criteria for our investigation of intermodal r criterion is sufficient and necessary for the detection of a entanglement in stimulated, spontaneous and partially entanglement in (2x2) and (2x3) dimensional states, but spontaneous Raman processes. is not necessary for higher dimensional states (see [3] and references therein). Since the pioneering work of Nonclassicalpropertiesofthese Ramanprocesseshave Peres, severalother inseparability inequalities have been been extensively studied. Initial studies were restricted reported for two mode and multi-mode states [3]-[13]. to the short-time approximation [14]-[16]. But recently Most of these criteria only provide sufficient condition some of the present authors have reported different non- of inseparability. Further, these criteria may be classi- classical effects (such as squeezing, antibunching, inter- fied into two sets [10]: A) set ofcriteria which cannotbe modal antibunching and sub-shot noise photon number directly tested through experiments [4]-[5] and B) set of correlation) in stimulated and spontaneous Raman pro- criteriawhichcanbetestedexperimentally[2],[3],[6]-[10]. cesses [17]-[20] without using traditional short-time ap- Experimentally testable inequalities involve variance or proximation technique. Our solution of Raman pro- higherordermomentsofsomeobservables. Since the ex- cesses, which does not involve short-time approxima- pectationvaluesofphysicalobservablescanbemeasured tion, is found to reveal many facets of nonclassical ef- experimentally,these setof inseparabilitycriteriacanbe fects which were undetected by short-time approxima- tested experimentally. tion technique. However, the possibility of observing in- The aim of the present work is not to study the in- termodal entanglement is not rigorously studied so far. 2 This fact has motivated us to study the intermodal en- icaltreatmentwhereallfourmodesareconsideredquan- tanglement in the double Raman processes. The present tummechanical. If welookcloselyinto the methodology investigation is relevant for quantum communication for adoptedin the earlierstudies we would quickly find that two reasons: Firstly because entanglement is an essen- theapproachadoptedinthepresentpaperissimplerand tial resource for quantum communication and secondly easily extensible to the other physical systems which are because spontaneous Raman process is reported to be described by bosonic Hamiltonians. useful in the realization of quantum repeaters [21]-[22] The paper is organized as follows. In the Section II which has its application in long distance high-fidelity we have described the Hamiltonian of spontaneous and quantum communication. stimulated Raman processes and its operator solution. In Section III we have used the solution to show that it is possible to observe intermodal entanglement in Ra- man processes. The inseparability criteria used for this purpose are also described in this section. Finally, Sec- tionIVis dedicatedto conclusionswherewe havebriefly summarizedtheresultofthepresentstudyandhavealso discussed the mutual relations among different nonclas- sical phenomena observed in the Raman processes. II. MODEL HAMILTONIAN Figure 1: (Color online) Two-photon stimulated Raman The Hamiltonian [14]-[20], [26] of our interest is scheme. The pump photon is converted into a Stokes photon and a phonon. The pump photon can also mix H = ω a a+ω b b+ω c c+ω d d with a phonon to produce an anti-Stokes photon. a † b † c † d † (1) + g ab c +h.c. +χ acd +h.c. , † † † (cid:0) (cid:1) (cid:0) (cid:1) Here it is worthy to note that V Pe˘rinová et al. [23] whereh.c. standsfortheHermitianconjugate. Through- have recently studied the possibility of observing entan- out the present paper, we use ¯h = 1. The annihila- glementinRamanprocessusingthe method ofinvariant tion(creation)operatorsa(a ), b(b ), c c , d(d )corre- † † † † (cid:0) (cid:1) subspace. They have followed an independent approach spondtothe laser(pump) mode, Stokesmode,vibration and have numerically computed the time dependence of (phonon)modeandanti-Stokesmode,respectively. They a measure of entanglement. Earlier S V Kuznetsov et obey the well-known boson commutation relations. The al. [24] studied the entanglement in the stimulated Ra- quantities ω , ω , ω and ω correspond to the frequen- a b c d man process considering only two modes (Stokes mode cies ofpumpmode a,Stokesmode b,vibration(phonon) and phonon mode) and taking the pump mode as the mode c and anti-Stokes mode d, respectively. The pa- classical light source. Naturally Kuznetsov et al.’s work rametersg andχaretheStokesandanti-Stokescoupling illustrated an incomplete scenario and failed to observe constants, respectively. Coupling constant g (χ) denotes intermodalentanglementinvolvinganti-Stokesmodeand thestrengthofcouplingbetweentheStokes(anti-Stokes) pump mode. To circumvent this limitation we have mode and the vibrational (phonon) mode and depends used here a completely quantum mechanical Hamilto- on the actual interaction mechanism. The dimension of nian. Further, Pathak, K˘repelka and Pe˘rina [25] have g and χ are that of frequency and consequently gt and recently investigated the possibilities of observing inter- χt are dimensionless. Further, gt and χt are very small modal entanglement in the Raman processes using the comparedtounity. Inordertostudythepossibilityofin- same Hamiltonian but with a short-time approximated termodal entanglement, we need simultaneous solutions solution. Their work is restricted by the intrinsic lim- ofthe followingHeisenbergoperatorequationsofmotion itations of the short-time approximation. Such limita- for various field operators tions may be circumvented by the analytical methods developed by us in recent past to study the stimulated a˙ = −i(ωaa+gbc+χcd) Raman scheme [17]-[20]. Those methods are systemati- b˙ = −i ω b+gac (cid:0) b †(cid:1) (2) cally used here and a relatively complete scenario of in- c˙ = −i ω c+gab +χa d c † † (cid:0) (cid:1) termodal entanglement in Raman processes is presented d˙ = −(ω d+χac). d here. Interestingly, we have observed intermodal entan- glement between i) pump mode and anti-Stokes mode, The above set of coupled nonlinear differential operator and ii) Stokes mode and anti-stokes mode. These two equations (2) are not exactly solvable in the closed ana- intermodal entanglement was not observedin the earlier lytical form under weak pump condition. But when the analyticstudies[24,25]. Thebeautyofthepresentstudy pump is very strong the operator a can be replaced by lies in the fact that analytic expressions for separability a c-number and the above set of equations (2) can be criterionareobtainedbyacompletelyquantummechan- solved exactly [16]. In order to solve these equations 3 under weak pump approximation we use the perturba- culationsaregiveninourpreviouspapers[17]-[20]. Here tive approach. Our solutions are more general than the we just note that under weak pump approximation, the well-knownshort-timeapproximation. Detailsofthecal- solutions of Eq. (2) assume the following form: a(t) = f a(0)+f b(0)c(0)+f c (0)d(0)+f a (0)b(0)d(0)+f a(0)b(0)b (0) 1 2 3 † 4 † 5 † + f a(0)c (0)c(0)+f a(0)c (0)c(0)+f a(0)d (0)d(0) 6 † 7 † 8 † b(t) = g b(0)+g a(0)c (0)+g a2(0)d (0)+g c 2(0)d(0)+g b(0)c(0)c (0) 1 2 † 3 † 4 † 5 † + g b(0)a(0)a (0) 6 † . (3) c(t) = h c(0)+h a(0)b (0)+h a (0)d(0)+h b (0)c (0)d(0)+h c(0)a(0)a (0) 1 2 † 3 † 4 † † 5 † + h c(0)b(0)b (0)+h c(0)d (0)d(0)+h c(0)a (0)a(0) 6 † 7 † 8 † d(t) = l d(0)+l a(0)c(0)+l a2(0)b (0)+l b(0)c2(0)+l c (0)c(0)d(0) 1 2 3 † 4 5 † + l a(0)a (0)d(0) 6 † The functions f , g , h and l are evaluated from the ular nature is a direct outcome of the perturbation the- i i i i dynamics under the initial conditions. In order to apply ory. Inthepresentinvestigationtheseculartermisnota the boundary condition, we put t = 0, in the first term problem since we consider small interaction time. Small of the Eq. (3). It is clear that f (0) = g (0) = h (0) = interactiontimealsoensuresthatthedampingtermcon- 1 1 1 l (0) = 1 and f (0) = g (0) = h (0) = l (0) = 0 (for tributes insignificantly. Here ∆ω = ω +ω −ω and 1 i i i i 1 b c a i=2, 3, 4, 5, 6, 7 and 8). Under these initial conditions ∆ω = ω +ω −ω . Normally, the detunings ∆ω and 2 a c d 1 thecorrespondingsolutionsforf (t), g (t), h (t)andl (t) ∆ω are extremely small. In the present investigation, i i i i 2 are obtained as given in the Appendix. we however assume that the small (non-zero) detuning The solutions Eqs. (3), (A1)-(A4) are valid up to the is present and hence ∆ω 6= 0 and ∆ω 6= 0. Here we 1 2 second orders in g and χ. Interestingly, there is no re- have used |∆ω | = 0.1 MHz and |∆ω | = 0.19 MHz. Of 1 2 strictionontimet. Forexample,f risesindefinitelywith course, in Eqs. (A1)-(A4) we have neglected the terms 2 theincreaseoftimet.Clearly,thedivergentnatureofthe beyond the second order in g and χ. Now we may use parameters f , g , h and l become more explicit as the Eq. (3) to obtain the temporal evolution of the number i i i i order of the perturbation theory is increased. The sec- operators of various modes as N (t) = |f |2a (0)a(0)+|f |2b (0)c (0)b(0)c(0)+|f |2c(0)d(0) c (0)d(0) a 1 † 2 † † 3 † † + f f a (0)b(0)c(0)+f f a (0)c (0)d(0)+f f a (0)a (0)b(0)d(0) (cid:2) 1∗ 2 † 1∗ 3 † † 1∗ 4 † † + f1∗f5(a†(0)a(0)+a†(0)a(0)b†(0)b(0))+f1∗f6a†(0)a(0)c†(0)c(0) (4) + f f a (0)a(0)c (0)c(0)+f f a (0)a(0)d (0)d(0) 1∗ 7 † † 1∗ 8 † † + f f b (0)c 2(0)d(0)+h.c. , 2∗ 3 † † i N (t) = |g |2b (0)b(0)+|g |2a (0)c(0)a(0)c (0)+ g g b (0)a(0)c (0) b 1 † 2 † † (cid:2) 1∗ 2 † † + g g b (0)a2(0)d (0)+g g b (0)c 2(0)d(0)+g g (b (0)b(0) 1∗ 3 † † 1∗ 4 † † 1∗ 5 † (5) + b (0)b(0)c (0)c(0))+g g (b (0)b(0)+b (0)b(0)a (0)a(0)) † † 1∗ 6 † † † + h.c.], N (t) = |h |2c (0)c(0)+|h |2a (0)b(0)a(0)b (0)+|h |2a(0)d (0)a (0)d(0) c 1 † 2 † † 3 † † + h h c (0)a(0)b (0)+h h c (0)a (0)d(0) (cid:2) ∗1 2 † † ∗1 3 † † + h∗1h4c†(0)b†(0)c†(0)d(0)+h∗1h5(c†(0)c(0)+c†(0)c(0)a†(0)a(0)) (6) + h h (c (0)c(0)+c (0)c(0)b (0)b(0))+h h c (0)c(0)d (0)d(0) ∗1 6 † † † ∗1 7 † † + h h c (0)c(0)a (0)a(0)+h h a 2(0)b(0)d(0)+h.c. , ∗1 8 † † ∗2 3 † i and 4 N (t) = |l |2d (0)d(0)+|l |2a (0)a(0)c (0)c(0)+ l l a(0)c(0)d (0) d 1 † 2 † † (cid:2) 1∗ 2 † + l l a2(0)b (0)d (0)+l l b(0)c2(0)d (0)+l l c (0)c(0)d (0)d(0). (7) 1∗ 3 † † 1∗ 4 † 1∗ 5 † † + l l d (0)d(0)(1+a (0)a(0))+h.c. . 1∗ 6 † † (cid:3) In the following section these number operators will be On the other hand, the second criterion is given by used to study the intermodal entanglement in Raman processes. hNaihNbi−|habi|2 <0. (11) From here onward we will refer to these criteria as HZ- 1 and HZ-2 criterion respectively. In addition to these III. INTERMODAL ENTANGLEMENT two criteria, we will also use the Duan’s inseparability criterion due to Duan et al. [3] : Inordertoinvestigatetheintermodalentanglementfor various coupled modes, we assume that all photon and phononmodesareinitially coherent. In otherwords,the h∆a†∆aih∆b†∆bi−|h∆a∆bi|2 <0. (12) composite boson field consisting of photons and phonon areintheinitialcoherentstate. Therefore,thecomposite In the criteria Eqs. (10)-(12), a and b are annihilation coherent state arises from the product of the coherent operators for two arbitrary modes. They are not limited states |α i, |α i, |α i, and |α i which are the eigenkets to the pump mode and the Stokes mode only. 1 2 3 4 of a, b, c and d respectively. Thus the initial composite We note that all the above criteria are only sufficient state is (not necessary) for detection of entanglement. Keeping this fact in mind, we have applied all these criteria to |ψ(0)i=|α i⊗|α i⊗|α i⊗|α i. (8) 1 2 3 4 study intermodal entanglement between different modes Itisclearthattheinitialstateisseparable. Nowthefield ofRamanHamiltonianandhaveobservedintermodalen- operator a(t) operating on such a multi-mode coherent tanglement in various situations. state gives rise to the complex eigenvalue α (t). Hence Let us first investigate the possibility of two mode en- 1 we have, tanglementinRamanprocessusingHZ-1criterion. From Eqs. (3), (4), (5) and (8) we obtain a(0)|ψ(0)i=α |ψ(0)i, (9) 1 where |α1|2 is the number of input photons in the pump hN N i− ab 2 = |f |2|α |2|α |2 modea.Inasimilarfashionwehavethreemorecomplex a b (cid:12)(cid:10) †(cid:11)(cid:12) 3 2 4 amplitudes α2(t), α3(t) and α4(t) corresponding to the (cid:12) (cid:12) + |g2|2(cid:16)|α1|4−|α1|2|α2|2(cid:17). Stokes, vibrational (phonon) and anti-Stokes field mode (13) operators b, c and d respectively. Clearly, for a sponta- Consequently, for spontaneous Raman process Eq. (13) neous process, the complex amplitudes are α2 = α3 = reduces to α = 0 and α 6= 0. For partial spontaneous process, 4 1 the complex amplitude α and any one of the remaining hN N i− ab 2 = |g |2|α |4 (14) 1 a b (cid:12)(cid:10) †(cid:11)(cid:12) 2 1 three eigenvalues are not equal to zero while the other (cid:12) (cid:12) It is evidently clear that the right hand side of the two complex amplitudes are zero. On the other hand, Eq. (14) is always positive. Hence HZ-1 criterion does forastimulatedprocess,the complexamplitudesarenot not show any signature of intermodal entanglement be- necessarilyzero. Inourpresentinvestigationweconsider α = |α |e iφ and the other eigenvalues for the Stokes, tween pump and Stokes modes in the spontaneous Ra- 1 1 − man process. For partially spontaneous Raman process vibrational(phonon)andanti-Stokesfieldmodesarereal. (|α | 6= 0, |α | 6= 0, |α | = |α | = 0), the Eq. (13) re- The aim of the present work is to investigate the pos- 1 2 3 4 duces to sibility of intermodal entanglement in the spontaneous, partially spontaneous and stimulated Raman processes. hN N i− ab 2 =|g |2 |α |2−|α |2 |α |2. (15) To do so let us begin with the investigationof two mode a b (cid:12)(cid:10) †(cid:11)(cid:12) 2 (cid:16) 1 2 (cid:17) 1 (cid:12) (cid:12) entanglement using Hillery and Zubairy’s criteria. It is clear that the entanglement is possible in the par- tially spontaneous Raman process only when |α |2 > 2 |α |2, i.e. the number of Stokes photon is more than the A. Two mode entanglement 1 number of pump photons, which is not the usual case. According to the HZ-1 criterion of Eq. (10), it is clear There are two criteria due to Hillery and Zubairy [7]- that the negative values on the right hand side of Eq. [8]. The first one is (13) would indicate the presence of intermodalentangle- hN N i−|hab i|2 <0. (10) ment between the pump mode and the Stokes mode in a b † 5 stimulated Raman process. To investigate the possibil- ria in stimulated Raman process,it is straightforwardto ity ofintermodalentanglementinthe stimulatedRaman study the special cases of: i) spontaneous Raman pro- processwehaveusedχ=g =105Hz,|α |=10,|α |=8, cess, where α =α =α =0 but α 6=0 and ii) partial 1 2 2 3 4 1 |α | = 0.01 and |α |= 1 [27]. We have plotted the right spontaneous Raman process, where α 6= 0 and any one 3 4 1 handsideof(13)inFig. 2awhichdoesnotshowanysig- of the other three α (i=2, 3, 4) is non-zero. i nature of intermodal entanglement between pump mode The same technique used in the above case is now and the Stokes mode in the stimulated Raman process. adopted to obtain the following equations for the study Here we would like to note that once we have an an- ofintermodalentanglementinstimulatedRamanprocess alytic expression for the HZ-1 or HZ-2 or Duan crite- using HZ-1 criterion: hN N i− bc 2 = |g |2 3|α |2|α |2+3|α |2|α |2+|α |2−|α |2|α |2 +|h |2|α |2|α |2 b c (cid:12)(cid:10) †(cid:11)(cid:12) 2 (cid:16) 1 3 1 2 1 2 3 (cid:17) 3 2 4 (16) (cid:12) (cid:12) + h h α α α +2g g α α2α +h h α2α α +c.c. (cid:2)(cid:8) ∗1 2 1 ∗2 ∗3 4∗ 1 2 3 ∗4 2 ∗3 1 ∗2 ∗4(cid:9) (cid:3) hN N i− ad 2 = |f |2 |α |2+|α |4+|α |2|α |2−|α |2|α |2 −|l |2 |α |2+|α |2|α |2 (17) a d (cid:12)(cid:10) †(cid:11)(cid:12) 3 (cid:16) 3 4 1 3 1 4 (cid:17) 2 (cid:16) 3 1 3 (cid:17) (cid:12) (cid:12) hN N i− bd 2 = |g |2|α |2|α |2+ l l α2α α +c.c. (18) b d (cid:12)(cid:10) †(cid:11)(cid:12) 2 1 4 (cid:2)(cid:8) 1∗ 3 1 ∗2 ∗4 (cid:9)(cid:3) (cid:12) (cid:12) hN N i− cd 2 = |l |2 2|α |2+2|α |2|α |2−2|α |2−|α |4−|α |2|α |2 +|h |2|α |2|α |2 (19) c d (cid:12)(cid:10) †(cid:11)(cid:12) 2 (cid:16) 1 1 4 4 4 3 4 (cid:17) 2 1 4 (cid:12) (cid:12) hN N i− ac 2 = |f |2 2|α |2+|α |4+|α |2|α |2−4|α |2−2|α |2|α |2−2|α |2|α |2 a c (cid:12)(cid:10) †(cid:11)(cid:12) 2 (cid:16) 1 1 1 3 2 1 2 2 3 (cid:17) (cid:12) (cid:12) + |f |2 |α |2+3|α |2|α |2+3|α |2|α |2−|α |2|α |2 +[{f f α α α (20) 3 (cid:16) 4 3 4 1 4 1 3 (cid:17) 1∗ 3 ∗1 ∗3 4 + h h α 2α α +f f α α 2α +c.c. . ∗2 3 ∗1 2 4 2∗ 3 ∗2 ∗3 4(cid:9) (cid:3) The right hand sides (RHS) of Eqs. (16)-(20) are plot- three modes of the system. Interestingly, with the suit- ted in Fig. 2b - Fig. 2f. It is interesting to note that able choice of the complex eigenvalues α it is possible i the presence of intermodal entanglement in stimulated to observe the signature of the intermodal entanglement Raman process is observed between i) the Stokes mode using HZ-1 criteria in partially spontaneous Raman pro- andthevibration(phonon)mode(Fig. 2b),ii)thepump cessinseveralwaysbutnosuchsignatureisobservedfor mode and the anti-Stokesmode (Fig. 2c), iii) the Stokes the completely spontaneous Raman process. mode and the anti-Stokes mode (Fig. 2d) and iv) the Since the HZ-1 criterion is only sufficient, we might pump mode and vibration mode (Fig. 2f). However, no have failed to detect some intermodal entanglement. In signature of intermodal entanglement is observed in the anattempttodetectsuchintermodalentanglementusing other two cases (Figs. 2a and e). Further, it does not HZ-2 criterion (11), we have used Eqs. (3), (4)-(7) and show the presence of genuine entanglement between any (8) to yield: hN ihN i−|habi|2 = |g |2|α |4+|f |2|α |2|α |2−[(g g a b 2 1 3 2 4 1∗ 6 , (21) + f f g g )|α |2|α |2+c.c. 1∗ 2 1∗ 2 1 2 i and hN ihN i−|hbci|2 = |g |2|α |2|α |2−|h |2 1+|α |2 |α |2+|h |2|α |2|α |2 b c 2 1 3 2 (cid:16) 2 (cid:17) 1 3 2 4 − h h α α α +(h h +g g h h )α α2α +h h α 2α α (22) (cid:2)(cid:0) ∗1 2 1 ∗2 ∗3 1 ∗4 1 2∗ 1 ∗3 2 3 ∗4 ∗2 3 ∗1 2 4 + h h |α |2|α |2+g g h h |α |2|α |2 +c.c. , ∗1 6 2 3 1 2∗ ∗1 2 1 3 (cid:17) i 6 hN ihN i−|hadi|2 = |f |2|α |4− l l |α |2|α |2+c.c , (23) a d 3 4 h1∗ 6 1 4 i hN ihN i−|hbdi|2 = |g |2|α |2|α |2− l l α2α α +c.c , (24) b d 2 1 4 (cid:2) 1∗ 3 1 ∗2 ∗4 (cid:3) hNcihNdi−|hcdi|2 = |h2|2|α1|2|α4|2+|h3|2 |α4|4−hl1∗l5|α3|2|α4|2+c.ci, (25) hN ihN i−|haci|2 = |h |2|α |4−|h |2 |α |2+|α |2|α |2 +|f |2|α |2|α |2−[(h h α α α a c 2 1 3 (cid:16) 4 1 4 (cid:17) 3 3 4 ∗1 3 ∗1 ∗3 4 + h h |α |2|α |2+h h α 2α α −h h |α |2|α |2 (26) ∗1 8 1 3 ∗2 3 ∗1 2 4 ∗1 5 1 3 + f f h h α α2α +f f h h |α |2|α |2 +c.c. . 1∗ 2 ∗3 1 2 3 ∗4 1∗ 3 ∗3 1 3 4 (cid:17) i From the closed form analytic expressions Eqs. (21)- Using Eqs. (3), (8) and (27)-(29) we can obtain analytic (26), it is possible to obtain the signature of intermodal expressionforDthelefthandsideofDuanetal. criterion entanglement in various cases. Interestingly, we obtain Eq. (27), intermodal entanglement between Stokes mode and vi- For mode a and b, bration mode for spontaneous Raman process. The in- termodalentanglementEqs. (21)-(26)forstimulatedRa- D = 2 |f ||α |2+|g |2|α |2 ab h 3 4 2 1 man processes are illustrated in the Fig. 3a-Fig. 3f. In + 1{(f g +f g )α α (30) accordancetoHZ-2criterion,negativevaluesoftheordi- 2 1 6∗ 5 1∗ 1 ∗2 + (2f g +f g +f g )α α +c.c.}], nates indicate the signature of entanglement. Therefore, 1 3∗ 4 1∗ 3 2∗ ∗1 4 the intermodal entanglement is observed in: i) Stokes for mode a and c mode and vibration mode, ii) Stokes mode and anti- Stokes mode and iii) pump mode and vibration mode. However, there is no signature of intermodal entangle- D = 2 |f ||α |2+|h |2|α |2+|h |2|α |2 ac h 3 4 2 1 3 4 ment in the remaining cases for Stimulated Raman pro- + 1{(f h +f h +f h (31) cesses. It is possible to obtain the intermodal entangle- 2 1 ∗5 6 ∗1 3 ∗3 + f h +f h )α α }+c.c.], mentforvariouspartiallyspontaneousRamanprocesses. 7 ∗1 1 ∗8 1 ∗3 However, these results are not exhibited in the present for mode b and c text. ItisinterestingtonotethatHZ-2criterionfailedto detectintermodalentanglementbetween,pumpandanti- D = 2 |g |2|α |2+|h |2|α |2+|h |2|α |2 bc h 2 1 2 1 3 4 (32) Stokes mode. Thus the Raman process provides a very + 1{(g h +g h +g h )α α }+c.c. , nice example of physical system where it can be shown 2 1 ∗6 5 ∗1 2 ∗2 ∗3 2 (cid:3) with physical example that these inseparability criteria for mode a and d are only sufficient. Still there are two situations where we have not found the signature of intermodal entangle- D = 2 |f ||α |2+ 1{(f l +f l )α ad h 3 4 2 1 2∗ 3 1∗ ∗3 ment. Let us see what happens when we apply another + (2f l +f l +f l )α α +f l α α +c.c.}], sufficient but not necessary criterion of inseparability. 1 3∗ 4 1∗ 2 2∗ ∗1 2 8 1∗ 1 ∗4 (33) Now Duan criterion Eq. (12) for the intermodal en- for mode c and d tanglement can also be written as [28] D = 2 |h |2|α |2+|h |2|α |2+ D =D(∆u)2E+D(∆v)2E −2< 0 (27) cd + 1h{(22l h +1 l h +3l h4)α α (34) 2 4 ∗1 2 ∗2 1 ∗4 2 3 + (l h +l h )α α ++c.c.}], where 5 ∗1 1 ∗7 ∗3 4 uˆ = √12(cid:2)(cid:0)a+a†(cid:1)+(cid:0)b+b†(cid:1)(cid:3) (28) and for mode b and d and D = 2 |g |2|α |2+ 1 (l g +l g +l g )α2+c.c. . bd h 2 1 2(cid:8) 4 1∗ 2 2∗ 1 4∗ 3 (cid:9)i vˆ = i√12(cid:2)(cid:0)a−a†(cid:1)+(cid:0)b−b†(cid:1)(cid:3) . (29) (35) 7 inter- HZ-1 HZ-2 Duan antibunching Squeezing[17] mode [18] φ=0 π/2 π φ=0 π/2 π φ=0 π/2 π ab nc nc nc nc nc nc nc nc nc possible possible ac entangled entangled entangled nc entangled nc nc nc nc bunching possible (time- dependent) ad entangled entangled entangled nc nc nc entangled nc nc anti- possible bunching bc nc entangled nc entangled nc entangled nc nc nc bunching possible (time- dependent) bd entangled nc entangled nc entangled nc nc nc nc antibunching notpossible cd nc nc nc nc nc nc nc nc nc bunching possible Table I: Relation between different nonclassicalphenomena observed in stimulated Raman process. Here nc stands for non-conclusive. . Right hand sides of equations (30)-(35) are plotted andanti-Stokes mode. The criterionis non-conclusivein in Fig. 4a- Fig 4f. It is clear that the intermodal- all other cases. This is so because Duan criterion is only entanglement is observed only between the pump mode sufficient. IV. CONCLUSIONS the Raman processes. Further, if we look at the possi- bilities of different kinds of nonclassicalities summarized in Table I (see the rows corresponding to ac, bc, and WehaveclearlyestablishedthatthestimulatedRaman bd modes), then we can quickly recognize that the ex- process can produce intermodal entanglement. The ob- istence of any one of the nonclassical phenomenon does servations are summarized in the Table I. Here it would not depend on the presence of the other. To be precise, beapttonotethatrecentlyPathak,K˘repelkaandPe˘rina entanglement,antibunching andsqueezingarenonclassi- [25] have investigated the possibilities of observing in- cal phenomena but they are independent of each other. termodal entanglement in the Raman processes using a However, Duan criterion in the present form implies in- approximated short-time solution. They have identified termodalsqueezinginoneofthequadraturevariablebut intermodalentanglementinpump-phonon acandStokes- the converse is not true. This fact can be clearly seen in phonon bc modes only. However, in addition to those the Table I, where we note that exceptin bd mode inter- two modes we have observed intermodal entanglement modal squeezing is possible in all other coupled modes. in pump-antiStokes ad and Stokes-antiStokes bd modes However,the Duancriterionof intermodalentanglement too. In addition to these, we explore the various pos- is satisfied only for the ac mode. sibilities of getting intermodal entanglement in partial spontaneous Raman processes too. In this way, our so- In quantum optics, physical systems (matter-field in- lution is found more powerful compared to those of the teractions) are usually described by multi-mode bosonic solutions of Raman processes under short-time approxi- Hamiltonians. The procedure followed in the present mation. Further,theuseofshort-timesolutionledtothe work may be applied directly to those systems to study monotonic nature of entanglement parameter as seen in theintermodalentanglement. Itisexpectedthatmostof Eqs. (11) and (12) of ref. [25]. As our solution is valid these systems will show intermodal entanglement. This for all times and hence the entanglement parametersare is so because most of the quantum states are entangled. free from this particular problem which is generally a Separability is a very special case. A natural question characteristicofshort-time solutions. Another earlieref- should arise at this point: If entanglementis so common fortto study the intermodalentanglementin the Raman why are we looking for it? The answer lies in the fact processes by S. V. Kuznetsov [24] was restricted to the that entanglement is one of the most important resources study of intermodal entanglement between Stokes mode for quantum information processing and quantum com- and the vibrationmode as they had considereda simpli- munication but still it is not very easy to produce useful fiedtwo-modeHamiltonian. Thustheuseofacompletely multi-partiteentanglement. Asmanyofthequantumop- quantum mechanical description of the Raman process, tical systems described by bosonic Hamiltonian (includ- our solution, and the strategy to use more than one in- ing the system studied here) are experimentally achiev- separabilitycriterionhavehelpedustoobtainarelatively able, useful intermodal entanglement may be produced morecompletepictureoftheintermodalentanglementin by them. Entanglement obtained in such a system is 8 expected to find application in controlled quantum tele- portation, quantum information splitting, dense coding, directsecuredquantumcommunicationetc. Weconclude this paper withanoptimistic view thatthe presentwork will motivate others to look for theoretical and experi- mental generation of useful multi-mode entanglement in other quantum optical systems. h = exp(−iω t) 1 c ge iωct V. ACKNOWLEDGMENTS h2 = − − ei∆ω1t−1 ∆ω (cid:2) (cid:3) 1 χe iωct AP thanks Department of Science and Technology h = − − ei∆ω2t−1 3 ∆ω (cid:2) (cid:3) (DST), India for support provided through the DST 2 project No. SR/S2/LOP-0012/2010 and he also thanks χge−iωct ei(∆ω1+∆ω2)t 1 − ei∆ω1t theOperationalProgramEducationforCompetitiveness h4 = −χ∆gωe2−iωcht e∆i(∆ω1ω+1+∆∆ωω22−)t 1 −∆eωi∆1ω2it (A3) - European Social Fund project CZ.1.07/2.3.00/20.0017 ∆ω1 h ∆ω1+∆ω2− ∆ω2 i of the Ministry of Education, Youth and Sports of the g2e iωct ig2te iωct Czech Republic. RO thanks the Ministry of Higher Ed- h5 = − ∆−ω2 (cid:2)ei∆ω1t−1(cid:3)+ ∆ω− ucation (MOHE)/University of Malaya HIR Grant No. 1 1 h = −h A-000004-50001for support. 6 5 χ2e iωct iχ2te iωct h = − − ei∆ω2t−1 + − 7 ∆ω2 (cid:2) (cid:3) ∆ω 2 2 Appendix A: Parameters for the solutions in Eq. (3) χ2e iωct iχ2te iωct h = − ei∆ω2t−1 − − 8 ∆ω2 (cid:2) (cid:3) ∆ω 2 2 f = exp(−iω t), 1 a ge iωat f2 = − e−i∆ω1t−1 , ∆ω (cid:2) (cid:3) 1 χe iωat f = − − ei∆ω2t−1 , 3 ∆ω (cid:2) (cid:3) 2 −χge−iωat e−i(∆ω1−∆ω2)t 1 + ei∆ω2t f4 = −χg∆e−ωi1ωat he−i(∆∆ωω11−−∆∆ωω22)t−1 − e−∆iω∆2ω1it , (A1) ∆ω2 h ∆ω1−∆ω2 − ∆ω1 i g2e iωat ig2te iωat f5 = ∆−ω2 (cid:2)e−i∆ω1t−1(cid:3)+ ∆ω− , 1 1 f = f , 6 5 χ2e iωat iχ2te iωat l = exp(−iω t) f = − ei∆ω2t−1 − − , 1 d f78 = −f∆7.ω22 (cid:2) (cid:3) ∆ω2 l2 = χe∆−ωiω2dt (cid:2)e−i∆ω2t−1(cid:3) g = exp(−iω t), l3 = +∆ωχ∆1gχω(eg∆2−e∆ω−iω1ωidω−1td∆t(cid:2)ωe2−)i∆(cid:2)eωi2(∆t−ω1−1(cid:3)∆ω2)t−1(cid:3) (A4) g12 = −ge∆−ωiω1btb(cid:2)ei∆ω1t−1(cid:3), l4 = −∆ωχ∆1gχω(eg∆2−e∆ω−iω1ωidω+1td∆t(cid:2)ωe2−)i∆(cid:2)eω−2it(∆−ω11(cid:3)+∆ω2)t−1(cid:3) g3 = −∆ωχ2gχ(eg∆−eω−iω1ibω−tb∆tωei2∆)ω(cid:2)e1ti(−∆ω11−,∆ω2)t−1(cid:3) (A2) l5 = iχ2∆teω−2iωdt + χ2∆e−ωi22ωdt (cid:2)e−i∆ω2t−1(cid:3) ∆ω2∆ω1 (cid:2) (cid:3) l = l χge−iωbt ei(∆ω1+∆ω2)t−1 6 5 g4 = −∆ωχ2g(e∆−ωiω1b+t∆ωe2i∆)ω(cid:2)1t−1 , (cid:3) ∆ω2∆ω1 (cid:2) (cid:3) g2e iωbt ig2te iωbt g = − ei∆ω1t−1 − − , 5 ∆ω2 (cid:2) (cid:3) ∆ω 1 1 g = −g . 6 5 9 [1] O.Gühne, G. Tóth, Phys.Rep. 474 (2009) 1. [16] J Pe˘rina, Quantum Statistics of Linear and Nonlinear [2] A Peres, Phys. Rev.Lett. 77 (1996) 1413. Optical Phenomena (Kluwer,Dordrecht, 1991). [3] LMDuan,GGiedke,JICiracandPZollar,Phys.Rev. [17] B Sen and S Mandal, J. Mod. Opt.52 (2005) 1789. Lett.84 (2000) 2722. [18] BSen,SMandalandJPe˘rina,J.Phys.B:At.Mol. Opt. [4] H Hungand G S Agarwal, Phys. Rev.A,49 (1994) 52. Phys. 40 (2007) 1417. [5] J Lee, M S Kim and H Jeong, Phys. Rev. A, 62 (2000) [19] B Sen and S Mandal, J. Mod. Opt.55 (2008) 1697. 032305. [20] BSen,VPe˘rinov´a,JPe˘rina,ALuk˘s,andJK˘repelka,J. [6] R Simon, Phys.Rev.Lett. 84 (2000) 2726. Phys. B: At.Mol. Opt. Phys.44 (2011)105503. [7] M Hillery and M S Zubairy, Phys. Rev. Lett. 96 (2006) [21] P Grangier, Nature 438 (2005) 749. 050503. [22] LMDuan,MLukin,JICiracandPZoller,Nature414 [8] M Hillery and M S Zubairy, Phys. Rev. A 74 (2006) (2005) 413. 032333. [23] VPe˘rinov´a,Luk˘s,and JK˘repelka,J. Phys.A44(2011) [9] M Hillery, H T Dung and H Zheng, Phys. Rev. A 81 035303. (2010) 062322. [24] SVKuznetsov,OVMan’koandNVTcherniega,J.Opt. [10] GS Agarwal and A Biswas, New J. Phys. 7(2005) 211. B: QuantumSemiclass. Opt.5 (2003) S503–S512. [11] C. H. Raymond Ooi, Q. Sun, M. Suhail Zubairy and M. [25] A Pathak, J K˘repelka and J Pe˘rina , quant- O.Scully,Phys. Rev.A 75, 013820 (2007). ph/1210.3779v1. [12] C. H. Raymond Ooi, Phys. Rev. A 76, 013809 (2007); [26] D FWalls, Z. Phys.237 (1970) 224. Eyob A. Sete, and C. H. Raymond Ooi, Phys. Rev. A [27] Samevaluesof χ,g and|αi|areusedin theentirepaper 85, 063819 (2012). (unless otherwise specified). For spontaneous and par- [13] A Miranowicz et al., Phys.Rev.A 82, 013824 (2010). tially spontaneous Raman proceses these values of |αi| [14] PSzlachetka,SKielich,JPe˘rinaandVPe˘rinov´a,J.Phys are used used for non-zero|αi|’s. A:Math. Gen 12 (1979) 1921. [28] S Sivakumar,J. Phys.B: At.Mol. Opt.Phys. 42 (2009) [15] P Szlachetka, S Kielich, J Pe˘rina and V Pe˘rinov´a, Opt. 095502. Acta27 (1980) 1609. 10 Figure 2: (Color online) Intermodal entanglement in stimulated Raman process by using HZ-1 criterion: The dotted line, dash-dotted line and the smooth line are used for the phase angle of the input complex amplitude α for φ=0, 1 πand π/2 respectively. a) intermodal entanglement is observedbetween Stokes mode and vibration-phononmode for φ=π/2, b) intermodal entanglement is observed between pump mode and anti-Stokes mode, c) intermodal entanglement is observed between Stokes mode and anti-Stokes mode for φ=0 and π/2, d) signature of intermodal entanglement is not observed between vibration-phonon mode and anti-Stokes mode, e) intermodal entanglement is observed between pump mode and vibration-phonon mode. Figure 3: (Color online) Intermodal entanglement in stimulated Raman process by using HZ-2 criterion: The dotted line, dash-dotted line and the smooth line are used for the phase angle of the input complex amplitude α for φ=0, 1 π/2 and π respectively. a) intermodal entanglement is not observed between Stokes mode and pump mode b) intermodal entanglement is observed between Stokes mode and vibration-phonon mode, c) signature of intermodal entanglement is not observed between pump mode and anti-Stokes mode, d) intermodal entanglement is observed between Stokes mode and anti-Stokes mode only for φ=π/2, e) intermodal entanglement is not observed between vibration-phonon mode and anti-Stokes mode, f) intermodal entanglement (time dependent) is observed between pump mode and vibration-phonon mode for φ=π/2.