Nonclassicality and entanglement for wavepackets
NNonclassicality and entanglement for wavepackets
Mehmet Emre Tasgin, ∗ Mehmet Gunay, and M. Suhail Zubairy Institute of Nuclear Sciences, Hacettepe University, 06800, Ankara, Turkey Institute of Quantum Studies and Department of Physics,Texas A & M University, College Station, TX 77 843-4242, USA (Dated: May 1, 2019)Mode-entanglement based criteria and measures become insufficient for broadband emission, e.g.from spasers (plasmonic nano-lasers). We introduce criteria and measures for the (i) total entan-glement of two wavepackets, (ii) entanglement of a wavepacket with an ensemble and (iii) totalnonclassicality of a wavepacket (WP). We discuss these criteria in the context of (i) entanglementof two WPs emitted from two initially entangled cavities (or two initially entangled atoms) and(ii) entanglement of an emitted WP with the ensemble/atom for the spontaneous emission and thesingle-photon superradiance. We also show that, (iii) when the two constituent modes of a WP areentangled, this creates nonclassicality in the WP as a noise reduction below the standard quantumlimit. The criteria we introduce are, all, compatible with near-field detectors.
I. INTRODUCTION
Quantum entanglement, once appeared as a science-fiction phenomenon, became easily observable both inthe macroscopic [1] and microscopic scales [2]. Achieve-ments like quantum teleportation [3] with satellites [4]or detection of stealth jets [5] with entangled microwavephotons [6] (quantum radars) made also the non-scientificcommunity become aware of the importance of non-classical phenomena which certainly will revolutionizethe current technology. This makes the generation, de-tection and quantification of nonclassical states —suchas quadrature/number-squeezed[7], two-mode entangledand many-particle entangled states [8, 9]— much moreimportant than the past century.The last two decades witnessed stunning progress alsoin another research field: plasmonics and quantum plas-monics. Plasmonics affected all the fields of sciencefrom sub-wavelength imaging of surfaces (SNOM) [10, 11]to sub-nm imaging of a molecule [12] and Raman-selective detection of ingredients via surface enhancedRaman scattering (SERS) [13]. Observation of phe-nomena analogous to electromagnetically induced trans-parency (EIT) [14] via path interference effects, e.g. Fanoresonances [15–17] and nonlinearity enhancement [18–20], made plasmonic systems more attractive. Whileplasmons decay much faster ( τ p ∼ − - − sec [21])compared to quantum emitters (QE, τ QE = 10 − - − sec [22]), experiments show that they are capable of han-dling quantum entanglement and nonclassical states fortimes longer than τ ent = 10 − sec [23–25]. Entangledand nonclassical states, once observable in the far-field-coupled photons, are now producible in the near-fieldelectromagnetic radiation, e.g. in the form of plasmonoscillations [24]. Fano resonances can also enhance thedegree of entanglement [26].It is well-demonstrated that presence of a metal ∗ [email protected] nanoparticle (MNP) near a quantum emitter (QE) modi-fies (increases) the bandwidth of the QE about 3-4 ordersof magnitude [27], i.e. the Purcell effect. Even thoughradiation bandwidth of a bare QE, or a standard laser, isvery narrow compared to the optical radiation; a QE cou-pled with a MNP, spaser (surface plasmon amplificationby stimulated emission of radiation) nano-lasers [28, 29],radiate/lase in a very broad bandwidth [30]. This band-width modification enables the fast-turn on/off nano-dimensional lasers (spasers), on one hand, enables minia-turized ultrafast-response [31] technologies. On the otherhand, they introduce a problem in the definition andquantification of entanglement/nonclassicality in such ra-diators.Quantum entanglement witnesses and measures, weusually deal in quantum optics, rely on the inseparabilityof the two modes which are commonly represented by asingle wave-vector k , i.e. ˆ a k and ˆ a k . Here, ω , = ck , are the carrier frequencies of the two nonclassical beams.Such a treatment is acceptable for narrow-frequency-width pulses, especially when the detector is placed (mea-surement is performed) in the far-field, where choice ofsingle component k is justified also with the directional(small solid angle) arguments. Such a simplification, twosingle k modes, can be applied also to the modified (very-broadened) emission of a QE-MNP hybrid for the far-fielddetection. Because a specific k value is detected, again,due to the small solid angle argument. However, quantifi-cation of the entanglement/nonclassicality via detectingthe inseparability of only the two modes, e.g. carrier fre-quencies of the two beams, is highly insufficient in the de-tection and "use" of the whole entanglement potential ofthe two pulses. Maximum entanglement/nonclassicalityharvesting, e.g. in quantum teleportation [3] and quan-tum thermodynamics (heat engines) [32, 33], is importantin the efficiency of such devices. The situation (insuffi-ciency) becomes even more adverse, if the quantificationis tried to perform via two near-field detectors [34], wherepronunciation of two modes becomes impossible.Therefore, entanglement of two wavepackets, oncecould be questioned due to curiosity, now, became a ne- a r X i v : . [ qu a n t - ph ] A p r cessity [35] with the development of fast-response nano-control [29] and nano-imaging techniques [34]. In thispaper, we aim to extend the notion of, i.e., (i) two-mode entanglement (TME) to the entanglement of twowavepackets (WPs) each containing a broadband of fre-quency components. (ii) We also introduce a notion forthe nonclassicality (Nc) of a WP, which is referred assingle-mode nonclassicality (SMNc), e.g. squeezing, foran almost single-mode beam. Furthermore, (iii) we ex-tend the definition of entanglement between an ensembleof QEs and the emitted-mode [36] to the ensemble-WPentanglement [37].After a survey among the possible exten-sions/generalizations of the entanglement into WPs, wedemonstrate that the most meaningful definition couldbe performed via making a replacement, ˆ a → (cid:80) r ˆ a r ,from a single-mode to a WP. The summation (cid:80) r standsfor the volume/area of the detector for the measurementvia a near-field detector and (cid:80) r stands for the wholespace for the calculation of the total entanglementexisting between the two WPs. ˆ a r is the operatorannihilating a photon (could as well be a plasmon) atposition r . In particular, we study the entanglement ofWPs emitted either from two initially-entangled cavitiesor initially-entangled atoms.The paper is organized as follows. First, in Sec. II, weintroduce the entanglement of two WPs using the electricfields of the two WPs, i.e. ˆ a i → ˆ E (+) i = (cid:80) k ε k e i k · r ˆ a k .We show that generation (onset) of entanglement be-tween the two pulses, at positions r and r , propagateswith the speed of light, c . This definition is demonstratedto be not useful for two purposes. ( ) Entanglement doesnot quantify the inseparability of the two WPs, but it wit-nesses on the inseparability (correlations) of the electricfield measurements at the positions r and r . ( ) Us-ing such a definition, we face with a divergence problem,in (cid:80) k ε k , when we desire to use the analogues of thestronger criterion Simon-Peres-Horedecki (SPH) [38] orthe criterion by Duan-Giedke-Cirac-Zoller (DGCZ) [39]for the WPs. We face the same divergence problem whenwe introduce ˆ a i → (cid:80) k i ˆ a i, k i , although this definitionhas the potential to detect the inseparability of any twomodes selected from each WPs. Next, in Sec. III, werealize that, by introducing ˆ a i → (cid:80) r i ˆ a i, r i , we can bothcircumvent the divergence problem in item ( ) and calcu-late the total entanglement which two near-field detectorsmeasure. We can also calculate the whole entanglement(potential) between the two WPs. Here, i = 1 , refersto the two WPs.In Sec. IV, we define the total entanglement be-tween two WPs by introducing the annihilation oper-ator ˆ A i = (cid:80) r i ˆ a i, r i . We introduce the analogues ofSPH [38] and Hillery&Zubairy (HZ) [40], also derivedby Shchukin&Vogel priorly [41], criteria for WP-WP en-tanglement. We study the time development of the totalentanglement of two WPs, emitted from two initially en-tangled cavities/atoms; using both HZ and SPH criteria.In Sec. V, we introduce ensemble-WP entanglement cri- teria by replacing ˆ a i → ˆ A i . We study the spontaneousemission of a single atom and superradiant single-photonemission from a many-particle entangled ensemble. InSec. VI, we define the nonclassiality (Nc) of a WP bothvia noise matrix of ˆ X , ˆ P operators defined over ˆ A and viaa beam-splitter (BS): by measuring the WP-WP entan-glement this nonclassical WP generates at the BS output.We show that (a) when some of the constituent modesof the WP are squeezed or (b) when two modes of theWP are entangled, WP becomes nonclassical, i.e. withreduced noise in a ˆ X φ operator, with ˆ A φ = e iφ ˆ A . Sec-tion VII contains our summary. II. CORRELATIONS OF ELECTRIC-FIELDMEASUREMENTS
Arriving a convenient definition, or a notion, for theentanglement of two wavepackets (WPs) necessitates theexploration of the correlations between the electric (E)fields of he two WPs at different positions r and r . Itis straight forward to see that one can obtain the sameforms with the two criteria, DGCZ [38] and HZ [40, 41],for ˆ a → ˆ E (+)1 ( r ) and ˆ a → ˆ E (+)2 ( r ) where ˆ E (+) i ( r i ) = (cid:88) k i ε k i e i k i · r i ˆ a i, k i (2.1)are the positive part of the electric field operators asso-ciated with the two WPs, i = 1 , . Each WP has themomentum components ˆ a i, k i . ε k i = (cid:112) ¯ hck i /(cid:15) V i is theelectric field of a single photon, depending on the quanti-zation volume V i of the i th WP. Following the same stepsgiven in Ref. [40], the analogous form of the HZ criterioncan be written as λ HZ = (cid:104) ˆ E (+)2 ( r ) ˆ E ( − )2 ( r ) ˆ E (+)1 ( r ) ˆ E ( − )1 ( r ) (cid:105)−|(cid:104) ˆ E (+)2 ( r ) ˆ E ( − )1 ( r ) (cid:105)| , (2.2)where λ HZ < witnesses the inseparability of the twoWPs, or the presence of nonlocal correlations between E-field measurements of the two WPs at positions r and r . ˆ E ( − ) i ( r i ) is the hermitian conjugate of ˆ E (+) i ( r i ) . HZcriterion, also derived by Shchukin&Vogel priorly [41] pri-orly in another context, does not lead to any divergenceproblem since it does not necessitate the evaluation ofa term like (cid:104) ˆ E (+) i ( r i ) ˆ E ( − ) i ( r i ) (cid:105) , in difference to SPH orDGCZ criteria.One can also derive DGCZ criterion for the entangle-ment of two WPs with the replacement ˆ x → ˆ E ( r ) and ˆ x → ˆ E ( r ) using the same arguments in Ref. [39],i.e. Cauchy-Schwarz inequality for separable states. Here ˆ E i ( r i ) = ˆ E (+) i ( r i ) + ˆ E ( − ) i ( r i ) is the electric field opera-tor. This criterion, however, is not a useful one since itcontains terms like (cid:104) ˆ E (+) i ( r i ) ˆ E ( − ) i ( r i ) (cid:105) which do diverge.SPH criterion also includes similar divergent terms anddoes not have any practical use here.Our experience shows us that DGCZ criterion worksgood for quadrature-squeezed like states, while HZ crite-rion works good mainly for number-squeezed like statesand superpositions of Fock states [42]. Here, in thissection, we consider the entanglement of two WPs,emitted from two initially entangled cavities, | ψ (0) (cid:105) = a (0) | (cid:105) c | (cid:105) c + a (0) | (cid:105) c | (cid:105) c into two different reser-voirs, or from two initially entangled atoms | ψ (0) (cid:105) = a (0) | e (cid:105)| g (cid:105) + a (0) | g (cid:105) | e (cid:105) . (We study the extended ver-sion of the system in Ref. [43] where the reservoirs aretreated as two single modes.) Fortunately, we can studythe correlations in such a system. Because the systememits the superpositions of Fock states, where HZ crite-rion, do not diverge, can be used.In Fig. 1, the two cavities are ini-tially in an entangled state, | ψ (0) (cid:105) =( a (0) | (cid:105) c | (cid:105) c + a (0) | (cid:105) c | (cid:105) c ) | (cid:105) R | (cid:105) R , where | (cid:105) c , and | (cid:105) R , are the Fock states for the two entan-gled cavities and the two reservoirs the cavities decay,respectively. The solution of the interaction picturehamiltonian [43] ˆ V = (cid:88) i =1 (cid:88) k i ¯ hg k i ˆ a † i, k i ˆ c i e − i (Ω i − ω k i ) t + H.c. (2.3)in subspace of possible states | ψ ( t ) (cid:105) =( b ( t ) | (cid:105) c | (cid:105) c + b ( t ) | (cid:105) c | (cid:105) c ) | (cid:105) R | (cid:105) R + | (cid:105) c | (cid:105) c (cid:16) (cid:88) k d , k ( t ) | k (cid:105) R | (cid:105) R + | (cid:105) R (cid:88) k d , k ( t ) | k (cid:105) R (cid:17) (2.4)is determined by the coefficients b i ( t ) = e − γ i t/ a i (0) , (2.5) d i, k i ( t ) = g k i a i (0) 1 − e − i (Ω i − ω ki ) t − γ i t/ ( ω k i − Ω i ) + iγ i / , (2.6)where Ω i and γ i are the cavity resonance and dampingrate, respectively. g k i is the coupling strength betweenthe i th cavity and the i th reservoir. When we considersufficiently long two cavities, and thin mirrors which cou-ple the cavities to the reservoirs, HZ criterion for theentanglement of the two WPs can be calculated as λ HZ ( t ) (cid:39) − (2 π ) g (Ω ) D (Ω ) g (Ω ) D (Ω ) ε K ε K × e − γ | z − ct | / c e − γ | z − ct | / c Θ( t − z /c ) Θ( t − z /c ) , (2.7)where we assume that dispersion of the cavity emission isnegligible in the transverse directions, ˆ x i and ˆ y i . D (Ω i ) is the density of states at the cavity resonance Ω i and canbe related to the damping rate as γ i = πD i (Ω i ) g (Ω i ) . ε K = (cid:112) ¯ h Ω i /(cid:15) V i with K i = Ω i /c . The step functionsin Eq. (2.7), Θ( t − z i /c ) reveal the luminal "onset" ofcorrelations (entanglement) between the two WPs, at z and z . We note that this approximate result for en-tanglement is realistic in the following aspect. For two Cavity 1
Reservoir 1 initially
ENTANGLED
Cavity 2
WP-1 WP-2
Reservoir 2
FIG. 1. The two cavities are initially in an entangled stateand they decay into two different reservoirs. We examine thetime evolution of the onset of the entanglement of the tworeservoirs, or in other words, correlations in the electric fieldmeasurements of the emitted wavepackets (WPs) in the tworeservoirs. We also calculate the total entanglement of thetwo WPs in Sec. IV. collimated wavepackets of narrow frequency band, theentanglement does not decay (or decays negligibly) with z -propagation. We also evaluate the λ HZ ( t ) for an un-collimated emission, where we find that absolute value ofits negativity decreases with spatial spreading.Such a definition of entanglement (correlations) be-tween two WPs is instructive especially for exploring theonset of the entanglement in spatial dimensions. How-ever, such a definition fails to work for most useful non-classical states, the Gaussian states, which are the onesconvenient to generate and use in the experiments.Moreover, it has a potential only to quantify the WP-WP entanglement on a position-to-position basis. Thatis, it does not quantify the "total" entanglement betweenthe two WPs. A candidate for quantifying the total en-tanglement, i.e. between all of the modes, could be ˆ a i → (cid:88) k i ˆ a i, k i or ˆ a i → (cid:88) k i ε k i ˆ a i, k i , (2.8)which has the potential to address the entanglement ofany two modes, ˆ a , k and ˆ a , k , between the two WPs .Such a definition however is again not useful for Gaussianstates since it leads to divergence in SPH and DGCZcriteria. III. CONVENIENCE OF WORKING IN THESPATIAL DOMAIN —CONVERGENCE
Next, we realize that we cannot avoid the divergence of (cid:80) k summation, since we cannot adopt a bound for the k -space. In difference to momentum space, fortunately,a (cid:80) r summation is bound by the volume V which canbe handled theoretically or can be limited in the exper-iments. Thus, we choose to work in the spatial domainby introducing the mode expansion [44, 45] ˆ a ( r ) = (cid:88) k e i k · r ˆ a k , (3.1) We use the phrase "has the potential to detect entanglement"on purpose. Because noise reduction due to ˆ a , k ↔ ˆ a , k en-tanglement can be screened by a noise increase due to two othermodes ˆ a , k (cid:48) ↔ ˆ a , k (cid:48) . which can be Fourier transformed as (cid:88) r ˆ a ( r ) e − i k · r = (cid:88) k (cid:48) (cid:32)(cid:88) r e i ( k − k (cid:48) ) · r (cid:33) ˆ a k (cid:48) = ˆ a k (3.2)by defining the normalized summation (cid:80) r → (cid:82) d r /V and using (cid:80) k → V (2 π ) (cid:82) d k as usual [14]. Hermitianconjugates of Eqs. (3.1) and (3.2) can be used, appliedon vacuum, to relate the spatial and momentum Fockspaces, e.g., as | r (cid:105) = (cid:88) k e − i k · r | (cid:105) k and | k (cid:105) = (cid:88) k e i k · r | r (cid:105) . (3.3)The advantage of working in the spatial domain, bydefining the annihilation operator ˆ a i → ˆ A i = (cid:88) r i ˆ a i ( r i ) (3.4)is, now, the quantity (cid:104) ˆ A i ˆ A † i (cid:105) does not diverge! Here, i = 1 , enumerates the two WPs. Moreover, Eq. (3.4),when used in an entanglement criterion, has the po-tential to detect correlations between any two spatialmodes, ˆ a , r ↔ ˆ a , r , of the two WPs. One can obtainthe commutation [ ˆ A, ˆ A † ] = 1 (3.5)from the relation [ˆ a ( r ) , ˆ a ( r (cid:48) )] = V δ ( r − r (cid:48) ) which deducesfrom Eq. (3.1) and [ˆ a k , ˆ a k (cid:48) ] = δ k , k (cid:48) . Commutation (3.5)remains convergent and dimensionless via normalizeddefinition of the spatial integration (cid:80) r → V (cid:82) d r .In the next section, we use the annihilation opera-tor ˆ A , defined in Eq. (3.4), to obtain WP analoguesof DGCZ [39], HZ [40] and SPH [38] criteria. We alsouse the same form, ˆ A , for introducing the ensemble-WPentanglement (Sec. V) and nonclassicality of a WP, inSec. VI). IV. WAVEPACKET–WAVEPACKETENTANGLEMENT
In order to obtain a "convergent" entanglement cri-terion which has the potential to address a kind of"total" entanglement, e.g. taking all spatial or k -modecorrelations into account, we introduce ˆ A i = (cid:80) r i ˆ a i (r i ) ,for instance, for the DGCZ criterion [39] λ DGCZ = (cid:104) (∆ˆ u ) (cid:105) + (cid:104) (∆ˆ v ) (cid:105) − ( α + β ) , (4.1)where λ DGCZ < witnesses the inseparability of the twoWPs. Here, the operators are ˆ u = α ˆ X + β ˆ X , (4.2) ˆ v = α ˆ P − β ˆ P , (4.3) where ˆ X i = ( ˆ A † i + ˆ A i ) / √ (cid:88) r i ˆ x i ( r i ) , (4.4) ˆ P i = i ( ˆ A † i − ˆ A i ) / √ (cid:88) r i ˆ p i ( r i ) . (4.5) ˆ X i and ˆ P i satisfy the usual commutation relation [ ˆ X i , ˆ P i ] = i. (4.6)Eq. (4.6) is a central result of the paper. Because itindicates that any two-mode entanglement (TME) crite-rion derived for ˆ a ↔ ˆ a , see also Ref. [46], are valid alsofor the inseparability of the two WPs, when ˆ X i and ˆ P i are defined as in Eqs. (4.4) and (4.5).More explicitly, if one defines the operators ˆ ξ = [ ˆ X ˆ P ˆ X ˆ P ] (4.7)and calculates the noise matrix V ij = 12 (cid:104) ˆ ξ i ˆ ξ j + ˆ ξ j ˆ ξ i (cid:105) = (cid:104) ˆ ξ i (cid:105)(cid:104) ˆ ξ j (cid:105) , (4.8)the SPH criterion [38] λ SPH =det A det B + (cid:18) − | det C | (cid:19) − tr( AJCJBJC T J ) −
14 (det A + det B ) (4.9)is also valid for the entanglement of two WPs. Here, A , B and C are 2 × × V = [ A , C ; C T , B ] . SPH criterion [38] is a partic-ularly important one, since it accounts any intra-moderotations, i.e. ˆ A φ = e iφ ˆ A , in the X i - P i plane [46].In Sec. III.3 of Ref. [46], we show that such a strongcriterion is possible to be derived also for number-phasesqueezed like states [7]. Similar to SPH criterion [38], itaccounts intra-mode rotations in the n - Φ , number-phase,plane. This new criterion is also valid for detecting theentanglement of two WPs.Similarly, Hillery&Zubairy (HZ) criterion [40], also for-merly introduced by Shchukin and Vogel [41], λ HZ = (cid:104) ˆ A † ˆ A ˆ A † ˆ A (cid:105) − |(cid:104) ˆ A † ˆ A (cid:105)| (4.10)can be derived, using the same arguments in Ref. [40],for two WPs. IV.1. Two entangled cavities
In the following, we calculate the total entanglementbetween two wavepackets (WPs) emitted from two ini-tially entangled cavities into two different reservoirs.This is depicted in Fig. 1. First, we calculate the λ HZ ( t ) given in Eq. (4.10), since the emitted pulses are super-positions of Fock states. Second, we preform the samecalculation for λ SPH given in Eq. (4.9). Similar resultscan be obtained also for the emission of two initially en-tangled atoms.
IV.1.1. HZ criterion:
The solution of the emission from two entangled cavi-ties, Eq. (2.4), can be transformed to the spatial domainof the two reservoirs as | ψ ( t ) (cid:105) =( b ( t ) | (cid:105) c | (cid:105) c + b ( t ) | (cid:105) c | (cid:105) c ) | (cid:105) R | (cid:105) R + | (cid:105) c | (cid:105) c (cid:34) | (cid:105) R (cid:16) (cid:88) r I ( r , t ) | r (cid:105) R (cid:17) (4.11) + (cid:16) (cid:88) r I ( r , t ) | r (cid:105) R (cid:17) | (cid:105) R (cid:35) , (4.12)where I i ( r i , t ) = (cid:80) k i d i, k i ( t ) e i k i · r i with d i, k i ( t ) is givenin Eq. (2.6). Using the contour-integration method, mo-mentum integral can be calculated as I i ( r i , t ) = V b i (0)2 πcr i K i g i (Ω i ) e − ( i Ω i + γ i / r i /c Θ( ct − r i ) , (4.13)where K i = Ω i /c and g i (Ω i ) is the cavity-reservoir cou-pling evaluated at the cavity resonance ω = Ω i . Weremark that, in the evaluation of I i we did not makea collimated-beam approximation, i.e. k (cid:39) k z , whichwe performed in Eq. (2.7). In Eq. (2.7), we performcollimated-beam approximation for providing an easierunderstanding on the experiments. The notion of entan-glement would not change if we were/were not performedsuch an approximation.When ˆ A operator is acted on the | ψ ( t ) (cid:105) , we obtain ˆ A | ψ ( t ) (cid:105) = (cid:16) (cid:88) r (cid:88) r (cid:48) I ( r , t )ˆ a ( r (cid:48) ) | r (cid:105) R (cid:17) | (cid:105) R | (cid:105) c | (cid:105) c = (cid:16) (cid:88) r I ( r , t ) (cid:17) | (cid:105) R | (cid:105) R | (cid:105) c | (cid:105) c . (4.14)The same form appears for ( ˆ A | ψ ( t ) (cid:105) ) † = (cid:104) ψ ( t ) | ˆ A † . Ifwe define the spatial integral in Eq. (4.14) as J i ( t ) = (cid:80) r i I i ( r i , t ) , the second term of the λ HZ , in Eq. (4.10) canbe identified as |(cid:104) ˆ A † ˆ A (cid:105)| . It is evident from Eq. (4.14) isthat ˆ A ˆ A | ψ ( t ) (cid:105) = 0 . Hence, the first term in Eq. (4.10)is zero. Then, HZ criterion for two WPs reduces to λ HZ ( t ) = −| J ( t ) | | J ( t ) | , (4.15)where spatial integrals can be evaluated as J i ( t ) = 2 b i (0) c K i g i (Ω i ) 1 − e α i ct + e α i ct α i ctα i , (4.16)with α i ct = − ( i Ω i + γ i / t , which do not depend on thereservoir volume. In Fig. 2, we plot λ HZ ( t ) . The total en-tanglement increases till the two WPs leave the two cavi-ties (or the two atoms) completely. Then, it drops but ap-proaches a constant value as γt (cid:29) . We scale the y -axisof Fig. 2 with a (0) b (0) K K g (Ω ) g (Ω ) /c α α . Weconsider emission from a plasmonic cavity, thus choose γ = 10 − Ω with Ω is in the optical regime. FIG. 2. Hillery&Zubairy and Simon-Peres-Horodecki criteria, λ HZ ( t ) = λ SPH ( t ) , for the two wavepackets emitted from twoinitially entangled cavities of Fig. 1. In difference to point-wise, E ( r , t ) ↔ E ( r , t ) , E-field correlations studied inSec. II, λ HZ , SPH ( t ) < witnesses a kind of total entanglementbetween the two WPs emitted into two different reservoirs. IV.1.2. SPH criterion:
We can also calculate the total entanglement be-tween the two WPs, using the SPH criterion defined inEq. (4.9). The terms like (cid:104) ˆ A i (cid:105) and (cid:104) ˆ A ˆ A (cid:105) do vanish.So, the 2 × A = (cid:20) (cid:96) (cid:96) (cid:21) , B = (cid:20) (cid:96) (cid:96) (cid:21) , and C = (cid:20) a b − b a (cid:21) , (4.17)where (cid:96) , = + | J , | , a = ( J ∗ J + J ∗ J ) / and b = i ( J J ∗ − J J ∗ ) / . The SPH criterion is evaluated as λ SPH = (cid:96) (cid:96) + (cid:16) − ( a + b ) (cid:17) − (cid:96) (cid:96) ( a + b ) −
14 ( (cid:96) + (cid:96) ) , (4.18)which reduces to λ SPH ( t ) = −| J ( t ) | | J ( t ) | = λ HZ (4.19)for the particular system we consider here. V. ENSEMBLE-WAVEPACKETENTANGLEMENT
Similarly, we can introduce an entanglement criterionbetween an ensemble and a (e.g. emitted) WP. When wechange ˆ a → ˆ A in the Eq. (4) of Ref. [9], it is straightfor-ward to obtain the criterion µ HZ = (cid:104) ˆ S + ˆ S − ˆ A † ˆ A (cid:105) − |(cid:104) ˆ S + ˆ A (cid:105)| , (5.1)which works better for the entanglement of number(Fock) like states with an ensemble. This is the casefor the spontaneous emission of a single atom [14] orsuperradiant single-photon emission from an ensembleof many-particle entangled atoms [9, 47, 48]. Here, ˆ S + = (cid:80) Nj =1 σ (+) j is the collective raising operator for theensemble containing N two-level atoms with σ (+) j is thePauli matrix of the j th atom, and ˆ S − = ˆ S † + .One can also obtain the analogue of DGCZ criterionfor ensemble-WP entanglement, ˆ a → ˆ A in Ref. [36], byexamining the uncertainty bound for (cid:104) (∆ˆ u ) (cid:105) + (cid:104) (∆ˆ v ) (cid:105) using ˆ u = ˆ S x + ˆ X and ˆ v = ˆ S y − ˆ P , (5.2)where ˆ X = ( ˆ A † + ˆ A ) / √ , ˆ P = i ( ˆ A † − ˆ A ) / √ , ˆ S x =( ˆ S + + ˆ S − ) / and ˆ S y = i ( ˆ S − − ˆ S + ) / . Such a criterionhas already been studied for the entanglement betweenan ensemble and a single mode of light [36], in the con-text of squeezing transfer from a nonclassical light to anensemble resulting in spin squeezing. Here, we only makethe replacement ˆ a → ˆ A and introduce ensemble-WP en-tanglement. DGCZ criterion works fine for Gaussian orquadrature-squeezed like states.Below, first, we calculate the µ HZ ( t ) for the sponta-neous emission of a single atom. Next, we evaluate µ HZ ( t ) for single-photon superradiant emission [47, 48] from aninitially entangled ensemble of atoms [9]. V.1. Spontaneous emission of a single atom
The wave function of a two-level atom, initially in theexcited state, is give by [14] | ψ ( t ) (cid:105) = β ( t ) | e (cid:105)| (cid:105) + | g (cid:105) (cid:88) k γ k ( t ) | k (cid:105) , (5.3)where spontaneous emission is possible into many k modes with probability amplitudes γ k ( t ) = e − i k · r g k − e i ( ω k − ω eg ) t − Γ t/ ( ω k − ω eg ) + i Γ / , (5.4)where r is the position of the atom and β ( t ) = e − Γ t/ . ω eg and Γ are the level-spacing and damping rate of theatom, respectively. g k is the coupling strength of the k vacuum mode with the atomic dipole. When ˆ A acts onthis state, it results ˆ A | ψ ( t ) (cid:105) = (cid:34)(cid:88) r (cid:32)(cid:88) k e i k · r γ k ( t ) (cid:33)(cid:35) | g (cid:105)| (cid:105) , (5.5)where (cid:80) k integration in the inner parenthesis, I A , yields I A ( r , t ) = V π cr s g ( ω eg ) K eg e − ( iω eg +Γ / r s /c Θ( ct − r s ) , (5.6)with r s = | r − r | , K eg = ω eg /c and Θ( x ) is the step-function. Then, the (cid:80) r spatial integration results J A ( t ) = 2 g ( ω eg ) K eg c − e αct + e αct αctα , (5.7)similar to Eq. (4.16) of the previous section. Here, αct = − ( iω eg + Γ / t . It is easy to see from Eq. (5.5) t | H Z ( t )) | en s e m b l e - W P en t. FIG. 3. Spontaneous emission from a single atom. Evolutionof the entanglement µ HZ ( t ) <0 between the atom and the emit-ted wavepacket. Superradiant single-photon emission from anensemble shows a similar behaviour except emission time de-termined by collective decay γ N in place of single atom decay γ . that ˆ S − ˆ A | ψ ( t ) (cid:105) = 0 which turns the first term in µ HZ ,Eq. (5.1), equal to zero. The ( (cid:104) ψ ( t ) | ˆ S + ) † = ˆ S − | ψ ( t ) (cid:105) is ˆ S − | ψ ( t ) (cid:105) = β ( t ) | g (cid:105)| (cid:105) . (5.8)So, HZ criterion becomes µ HZ ( t ) = −| β ( t ) | | J A ( t ) | = − e − Γ t | J A ( t ) | . (5.9)In Fig. 3a, we plot µ HZ ( t ) . V.2. Superradiant emission from an ensemble
We also study the entanglement of the superradiantlyemitted single photon from an initially entangled ensem-ble of atoms | φ (0) (cid:105) ens = (cid:80) Nj =1 e i k · r j | e j (cid:105) , where | e j (cid:105) in-dicates that the j th atom is in the excited state and allother ( N − ones are in the ground state. The methodfor the generation of such a state is described in Ref. [48]. r j are the positions of the atoms in the ensemble whichcan be much larger than the emission wavelength λ =2 π/k . In Fig. (4) of Ref. [9], we demonstrated the entan-glement between the central mode (carrier frequency) ofthe emitted light and the ensemble. Here, in difference,we examine the entanglement of the ensemble with thewhole emitted light, the wavepacket (WP).Time evolution, superradiant emission, of this initialstate is given [47] by | ψ ( t ) (cid:105) = N (cid:88) j =1 β j ( t ) | e j (cid:105)| (cid:105) + (cid:16) (cid:88) k γ k ( t ) | k (cid:105) (cid:17) | g (cid:105) , (5.10)where β j ( t ) = 1 √ N e − γ N t e i k · r j , (5.11) γ k ( t ) = g k √ N − e − γ N t + i ( ω k − ω eg ) t ( ω k − ω eg + iγ N ) N (cid:88) j =1 e i ( k − k ) · r j . (5.12)This emission, from an extended ( L > λ ) entangled en-semble, is referred as timed superradiance and the initialstate is called as timed-Dicke states. Here, γ N is thecollective (superradiant) decay rate, which can be muchlarger than the decay rate of a single atom [47]. ˆ A | ψ ( t ) (cid:105) can be calculated similar to the spontaneousemission case, where now I A in Eq. (5.6) becomes I (SR) A ( r , t ) = N (cid:88) j =1 e i k · r j √ N V π cr j g ( ω eg ) K eg × e − ( iωeg + γ N / r j /c Θ( ct − r j ) . (5.13) J (SR) A ( t ) = (cid:80) r I (SR) A can also be calculated similarlywhich results J (SR) A ( t ) = J A ( t, γ N ) N (cid:88) j =1 e i k · r j √ N , (5.14)where J A ( t, γ N ) is the integral calculated for a singleatom emission in Eq. (5.7), with Γ / → γ N . We de-fine the last term of Eq. (5.14), a phase coherence term,as ζ = (cid:80) Nj =1 e i k · r j / √ N .Similar to the spontaneous emission of a single atom ˆ S − ˆ A | ψ ( t ) (cid:105) = 0 and ( (cid:104) ψ ( t ) | ˆ S + ) † = ˆ S − | ψ ( t ) (cid:105) yields ˆ S − | ψ ( t ) (cid:105) = (cid:16) N (cid:88) j =1 β j ( t ) (cid:17) | g (cid:105)| (cid:105) = e − γ N t ζ | g (cid:105)| (cid:105) . (5.15)Therefore, the ensemble-WP entanglement criterion µ HZ becomes µ (SR) HZ ( t ) = e − γ N t | ζ | | J A ( t, γ N ) | , (5.16)where J A ( t, γ N ) is given in Eq. (5.7) with Γ / → γ N .We note that, one cannot tell if a larger µ HZ implies astronger entanglement or not; neither in the WP-WPentanglement nor in ensemble-WP entanglement. Thisis because, unlike logarithmic negativity [49] such entan-glement criteria are not demonstrated to be used as anentanglement measure. VI. NONCLASSICALITY OF A WAVEPACKET
In this section, we introduce the nonclassicality (Nc)of a wavepacket (WP). We show that a WP possessesnonclassicality both (a) when some of the constituent( k ) modes are squeezed or (b) when, e.g., two constituentmodes k ↔ k are entangled. Below, we first express thetwo methods used for the quantification/witness of thesingle-mode nonclassicality (SMNc) of a detected mode.Then, we apply these two methods for introducing thenonclassicality of a WP.We remind that, single-mode nonclassicality of a lightmode can be defined in two different ways. (i) Onemay, e.g. for Gaussian states, examine the noise ma-trix, i.e. V ij = (cid:104) ˆ ξ i ˆ ξ j + ˆ ξ j ˆ ξ i (cid:105) / − (cid:104) ˆ ξ i (cid:105)(cid:104) ˆ ξ j (cid:105) for the real vari-ables ξ (r) = [ x , p ] or using the complex representation ξ (c) = [ α , α ∗ ] [50, 51]. One can show that quadrature-squeezing, a SMNc, exists if |(cid:104) ˆ a (cid:105)| > (cid:104) ˆ a † ˆ a (cid:105) [46], whichderives from the eigenvalues of the noise matrix.(ii) Alternatively, one can also witness/quantify thenonclassicality of a single-mode ˆ a via checking if it cre-ates two-mode entanglement (TME) at a BS output [52–54]. For instance, SPH criterion [38] —not only a neces-sary&sufficient condition for Gaussian states, but alsoa criterion working well for superpositions of numberstates— can be used to determine the TME at the BSoutput. This approach may work better in witnessingthe SMNc for a wider range of nonclassical states, seeFig. 2(c) in Ref. [55].Both approaches can be used in defining the nonclas-sicality of a WP. We first use the method (i) to examinethe states (a) and (b), expressed in the first paragraphof the present section. At the end of the section, we alsomention briefly about the use of the second method (ii). VI.1. (i) Examining the noise matrix
Analogous to a single-mode (SM) state, we can definethe noise-matrix of a WP as (cid:20) + (cid:104) ˆ A † ˆ A (cid:105) (cid:104) ˆ A (cid:105)(cid:104) ˆ A (cid:105) ∗ + (cid:104) ˆ A † ˆ A (cid:105) (cid:21) (6.1)in the complex representation, and as (cid:20) (cid:104) ˆ X (cid:105) − (cid:104) ˆ X (cid:105) (cid:104) ˆ X ˆ P + ˆ P ˆ X (cid:105) / − (cid:104) ˆ X (cid:105)(cid:104) ˆ P (cid:105)(cid:104) ˆ X ˆ P + ˆ P ˆ X (cid:105) / − (cid:104) ˆ X (cid:105)(cid:104) ˆ P (cid:105) (cid:104) ˆ P (cid:105) − (cid:104) ˆ P (cid:105) (cid:21) (6.2)in the real variables. Similar to SM case [46], λ sm =1 / (cid:104) ˆ A † ˆ A (cid:105)−|(cid:104) ˆ A (cid:105)| determines the minimum noise (max-imum squeezing) in the quadratures ˆ X φ = ( ˆ A † φ + ˆ A φ ) / √ with ˆ A φ = e iφ ˆ A . Here, φ is chosen along the min noisedirection. VI.1.1. (i.a) Constituent modes of a WP are squeezed
As an example, we first examine the nonclassicality ofa WP, whose some of the modes are squeezed, but themodes are all separable.
Only two modes are squeezed — For simplicity, asa warm up, first we assume that only two modes ofthe WP are in squeezed vacuum state, i.e. | ψ (cid:105) = | ξ (cid:105) k | ξ (cid:105) k | (cid:105) k | (cid:105) k . . . , and other modes are in vacuumstate . Here, ξ i are squeezed vacuum states. In such a Actually, this is equivalent to assuming that all other modes arein coherent state. Because only the noise operators δ ˆ a i determinethe the Nc features. ˆ D ( α i ) displacement of each state does notalter the Nc features [50] for Gaussian states. case, only four terms non-vanish in (cid:104) ˆ A (cid:105)(cid:104) ψ | ˆ A | ψ (cid:105) = (cid:104) ξ |(cid:104) ξ | (cid:88) r (cid:88) r (cid:48) (cid:104) e i k · ( r + r (cid:48) ) ˆ a k + e i k · ( r + r (cid:48) ) ˆ a k +2 e i ( k · r + k · r (cid:48) ) ˆ a k ˆ a k (cid:105) | ξ (cid:105)| ξ (cid:105) . (6.3)We remark that, here, k and k are not variables, butthey refer to two modes which are entangled with eachother. All other modes are separable. The expectationvalues can be calculated by transforming the annihilationoperators as ˆ a i ( ξ i ) = C i ˆ a i − S i ˆ a † i where C i ≡ cosh r i and S i ≡ sinh r i , with r i are squeezing parameters [14] ξ i = r i e iθ i . We set the squeezing angles θ i = 0 for simplicity.In Eq. (6.3), only ˆ a k i terms survive and we obtain (cid:104) ψ | ˆ A | ψ (cid:105) = − (cid:88) r (cid:88) r (cid:48) (cid:16) e i k · ( r + r (cid:48) ) S C + e i k · ( r + r (cid:48) ) S C (cid:17) . (6.4)Similarly, (cid:104) ψ | ˆ A † ˆ A | ψ (cid:105) yields (cid:104) ψ | ˆ A † ˆ A | ψ (cid:105) = (cid:88) r (cid:88) r (cid:48) (cid:16) e i k · ( r + r (cid:48) ) S + e i k · ( r + r (cid:48) ) S (cid:17) . (6.5)One can note that (cid:88) r (cid:48) e i k i · ( r + r (cid:48) ) = (cid:88) r (cid:48) e i k i · ( r − r (cid:48) ) = (cid:12)(cid:12)(cid:12) (cid:88) r e i k i · r (cid:12)(cid:12)(cid:12) = (cid:16) (cid:88) r sin( k i · r ) (cid:17) + (cid:16) (cid:88) r cos( k i · r ) (cid:17) . (6.6)We remark that in the evaluation of (cid:104) ˆ A (cid:105) , in Eq. (6.3),we consider only the two modes k , k among the summa-tion, or ω -integral, over an infinite number of modes. Ascould be anticipated, the contribution of the two modesremains only infinitesimal. Hence, a | (cid:80) r e i k i · r | sum-mation, when converted to integration | (cid:82) d r e i k i · r /V | ,vanishes. Still, we can account the infinitesimal contri-butions (squeezing) of the two modes to the nonclassi-cality of the WP as follows. sin( k i · r ) summation inEq. (6.6) gives exactly zero, since it is zero at r = 0 and symmetric/periodic terms cancel each other. In the cos( k i · r ) summation, however, the central term at r = 0 , cos(0) = 1 , does not vanish. Hence, following our (cid:80) r definition in Sec. III, Eq. (6.6) becomes (cid:12)(cid:12)(cid:12) (cid:88) r e k i · r (cid:12)(cid:12)(cid:12) = (∆ r ) V , (6.7)which is dimensionless and becomes zero in a standardcontinuous integration, i.e. (∆ r ) /V → .When we include this infinitesimal constribution to thenoise of our WP, we obtain λ sm = 12 + (cid:104) ˆ A † ˆ A (cid:105) − |(cid:104) ˆ A (cid:105)| = 12 + (∆ r ) V (cid:2) ( S − S C ) + ( S − S C ) (cid:3) , (6.8) which is always less than / since S i − S i C i < andbecomes more negative as r i increases. Many modes are squeezed — We are aware that, in-troducing the contribution from a single nonzero point, (∆ r ) around r = 0 , leaves an ambiguity. However, weconduct this treatment because we do need it unavoid-ably in the case (i,b), below. In order to leave the am-biguity, now, we also present the same treatment for acontinuous distribution of the squeezing to many modes.We use the experience we obtained in our treatment withtwo modes.When | ξ k (cid:105) is a continuous function of k modes, weobtain (cid:104) ψ | ˆ A | ψ (cid:105) = (cid:104) | (cid:88) r , r (cid:48) (cid:88) k , k (cid:48) e i k · ( r + r (cid:48) ) δ k , k (cid:48) ˆ a k ( ξ k ) | (cid:105) . (6.9)We know from Eq. (6.3) that ˆ a k ˆ a k (cid:48) does not contribute.So, (cid:104) ψ | ˆ A | ψ (cid:105) becomes (cid:104) ψ | ˆ A | ψ (cid:105) = (cid:88) r , r (cid:48) (cid:88) k e i k · ( r ± r (cid:48) ) ( − S k C k ) , (6.10)where S k ≡ sinh r k and C k ≡ cosh r k , and r k , squeezingparameter for the k -mode, is a continuous function of k .If we consider a simple function, e.g. with S k C k doesnot have any poles anywhere in the complex k -plane,then the k -integration in Eq. (6.10) vanishes unless r = r which leads to a single r summation (cid:104) ψ | ˆ A | ψ (cid:105) = (cid:88) r (cid:88) k ( − S k C k ) = − V (2 π ) (cid:90) d k S k C k , (6.11)where (cid:80) r = 1 , see Sec. III, and (cid:80) k → V (2 π ) (cid:82) d k asusual [14]. (cid:104) ˆ A † ˆ A (cid:105) can be calculated similarly as (cid:104) ψ | ˆ A | ψ (cid:105) = V (2 π ) (cid:90) d k S k , (6.12)which gives a finite squeezing (reduction in noise) λ sm = 12 + (cid:104) ψ | ˆ A † ˆ A | ψ (cid:105) − |(cid:104) ψ | ˆ A | ψ (cid:105)| = 12 + V (2 π ) (cid:90) d k ( S k − S k C k ) (6.13)for the WP. We note that ( S k − S k C k ) < and we remindthat S k ≡ sinh r k and C k ≡ cosh r k . VI.1.2. (i.b) Entanglement of two constituent modes
We raise the following question. Does the entangle-ment between two constituent modes, let them again be k and k , contribute to the nonclassicality of the WP?We consider a state, where there is no squeezing in themodes, but only the two modes k and k are entangledvia two-mode squeezing operator ˆ E = e β ˆ a † ˆ a † − β ∗ ˆ a ˆ a , | ψ ent (cid:105) = | β (cid:105) k , k | (cid:105) k | (cid:105) k . . . (6.14)The reason we consider the entanglement due to ˆ E opera-tor is it creates "pure entanglement" between the k and k modes. That is, it does create single-mode nonclas-sicality in the modes, see Sec. II.5.(iii) in Ref. [46] andalso Ref. [54].We can transform the ˆ a i ( β ) operators as ˆ a ( β ) = C ˆ a + S ˆ a † (6.15) ˆ a ( β ) = C ˆ a + S ˆ a † (6.16)in stead of working with the entangled state | β (cid:105) k , k .Here, C ≡ cosh r and S ≡ sinh r where r determinesthe degree of the entanglement.In this case, only the ˆ a k ( β )ˆ a k ( β ) and ˆ a k ( β )ˆ a k ( β ) terms contribute with CS in the calculation of (cid:104) ˆ A (cid:105) andonly ˆ a † k , ( β )ˆ a k , ( β ) terms contribute with S in the cal-culation of (cid:104) ˆ A † ˆ A (cid:105) . Thus, we find (cid:104) ˆ A (cid:105) β = 2 (∆ r ) V CS, (6.17) (cid:104) ˆ A † ˆ A (cid:105) β = 2 (∆ r ) V S , (6.18)which creates an infinitesimal squeezing in the WP as (cid:104) (∆ ˆ X φ ) (cid:105) = λ sm = 12 + (cid:104) ˆ A † ˆ A (cid:105) − |(cid:104) ˆ A (cid:105)| = 12 + 2 (∆ r ) V ( S − SC ) , (6.19)which is always less than the SQL / . So, it creates asqueezed uncertainty WP. VI.2. (ii) WP nonclassicality via entanglement at abeam-splitter output
It is a known fact that single-mode nonclasical-ity (SMNc) criterion (cid:104) ˆ a † ˆ a (cid:105) < |(cid:104) ˆ a (cid:105)| , so (cid:104) ˆ A † ˆ A (cid:105) < |(cid:104) ˆ A (cid:105)| ,works good for quadrature squeezed (and Gaussian) likestates. For more general states, such a nonclassicalitycriterion fails. In these cases, a beam-splitter (BS) canhelp us very much. When a nonclassical state is inputto a BS, mixed with vacuum or a coherent state, it gen-erates two-mode entanglement (TME) at the BS output.Hence, we can also decide that a WP is nonclassical, if it produces WP-WP entanglement at the BS output. BStransformation for a WP is given in Refs. [56].It is well-experienced that the SPH, two-mode entan-glement, criterion [38] is able to reveal the TME in somestates other than the Gaussian ones, e.g. some superpo-sitions of two-mode Fock states. Hence, determining theWP-nonclassicality via BS provides us the advantage ofbeing able to detect some of the non-Gaussian states, e.g.superposed number states, using the strength (enhancedgenerality) of the SPH criterion .For instance, use of a BS can resolve the SMNc ofa superradiant-phase single-mode state, see Fig.2(c) inRef. [55], whose nature is extremely different than theGaussian-like states. It is a straightforward process todevelop the same method, see Sec. II.b in Ref. [55], with ˆ a → ˆ A , also for WP-nonclassicality.Even though SPH criterion [38] is a strong one whichis able to determine also some of the other states; inthe Sec. III.3 of Ref. [46], we developed an SPH-like(strong, invariant) criterion for number-phase squeezedlike states. This new criterion is invariant under the ro-tations in the number-phase ( n - Φ ) plane. Although SPHis a strong criterion, it is defined with quadrature vari-ables, while the new criterion is defined with ˆ n and ˆΦ operators. VII. SUMMARY
Developments in the current technology necessitate en-tanglement/nonclassicality criteria for broadband emit-ting sources, e.g. like spasers [28, 29, 35]. Currentmode-based criteria can still be used for the broadbandstates. However, they detect/measure the entanglementof only between the two carrier frequencies. We intro-duce criteria and measures for the "total" entanglementof two wavepackets (WPs). That is, the newly intro-duced criteria can measure the entanglement among allof the modes of the two WPs. Wee also develop a "total"nonclassicality for a WP, which accounts the nonclassi-cality of a WP both due to squeezing of the constituentmodes and entanglement present among the constituentmodes. In analogy with WP-WP entanglement and WP-nonclassicality, we also introduce criteria for ensemble-WP entanglement. All the criteria/measure we introducecan also be used for measurements with near-field detec-tors [34]. 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