aa r X i v : . [ qu a n t - ph ] D ec Amplification of NOON States
G. S. Agarwal, S. Chaturvedi, and Amit Rai We examine the behavior of a Non Gaussian state like NOON state under phase insensitiveamplification. We derive analytical result for the density matrix of the NOON state for arbitrarygain of the amplifier. We consider cases of both symmetric and antisymmetric amplification of thetwo modes of the NOON state. We quantitatively evaluate the loss of entanglement by the amplifierin terms of the logarithmic negativity parameter. We find that NOON states are more robust thantheir Gaussian counterparts.
PACS numbers: 42.50.Ex, 42.50.Nn, 03.65.Ud
I. INTRODUCTION
Among the continuous variable (CV) entangled states[1], non Gaussian states are generally believed to possessmuch more robust entanglement vis a vis the Gaussianstates– states characterized by a Gaussian quasi probabil-ity distributions and hence by their first and second mo-ments. Though mathematically not as well understoodas the Gaussian states [2] in so far as their entanglementproperties are concerned, non Gaussian states, by virtueof the robustness of their entanglement have in recentyears emerged as strong contenders for potential applica-tions in quantum information technology. The fact thatthe non Gaussian states are defined by what they are notmakes a general discussion of their entanglement proper-ties impossible and one is forced to restrict oneself to subfamilies of non Gaussian states such as states obtainedby adding or subtracting a fixed number of photons toGaussian states [3] and forming suitable superpositionsthereof. One such widely discussed family of non Gaus-sian states parameterized by an integer N is the familyof NOON states [4] | N OON > = 1 √ | N, > + | , N > ] . (1)These states, similar in structure to the EPR states,have attracted much attention in recent years and can beviewed as a two mode state consisting of a superpositionof states containing N photons in one mode and none inthe other and vice versa. Schemes for reliable productionof such states have been proposed [5, 6] and their useful-ness as a practical tool in making super-precision mea-surements in optical interferometry, atomic spectroscopyand in sensing extremely small magnetic fields than hith-erto possible have been highlighted [7]. This circum-stance makes it imperative to investigate their behav-ior under attenuation and amplification. While studieson the decoherence effects on NOON states under spe-cific models for system-bath interactions already existin the literature [8], in the present work we focus onthe question of amplification of NOON states and theconsequent degradation of entanglement therein and to compare and contrast it with the behavior of entangle-ment in Gaussian states under amplification investigatedin [9, 10]. It is important to understand the amplifica-tion of NOON states as quantum communication proto-cols would use both amplifiers and attenuators [11]. Inthe present work we consider only phase insensitive am-plification and model this in the standard way as a bathconsisting of N two level atoms of which N are in theexcited state and N in the ground state with N > N .Under the assumptions that atomic transitions have alarge width and that the bath is maintained in a steadystate, the time evolution of the density operator ρ fora single mode of radiation field on resonance with theatomic transition is described, in the interaction picture,by the master equation ∂ρ∂t = − κN ( aa † ρ − a † ρa + ρaa † ) − κN ( a † aρ − aρa † + ρa † a ) , (2)where a and a † are the annihilation and creation opera-tors of the field mode.A brief outline of this work is as follows. In Section II,with the NOON state as the input to the amplifier, weobtain expressions for the output density operator forthe case when both the modes are symmetrically ampli-fied and for the case when only one mode is subjected toamplification and the other is not amplified at all. We in-vestigate how the entanglement in the output state varieswith the amplifier gain using logarithmic negativity asthe quantifier for entanglement [12]. Section III containsour concluding remarks and further outlook. Our studycomplements the work done by Vitelli et al [6] on ampli-fication of NOON state by a phase sensitive amplifier. II. EVOLUTION OF THE NOON STATEUNDER PHASE INSENSITIVE AMPLIFICATION
In our earlier work [9], we found that for a two modesqueezed vacuum as the input, there are limits on thegain beyond which the output of the amplifier has noentanglement between the two modes and the limitingvalues of the gain in the two cases considered, symmetricand asymmetric amplification, were found to be G = (cid:18) η η + e − r (cid:19) Symmetric Amplification (3) G = 1 + 1 η Asymmetric Amplification . (4)where the gain G = exp(( N − N ) κt ) and η = N / ( N − N ). In particular, when η →
0, the limiting gain in thesymmetric case remains finite, that for the asymmetriccase the limiting value moves off to infinity [13].In this section we discuss how an input NOON stateevolves under the action of a phase insensitive amplifieras modeled by the master equation (2) confining our-selves for simplicity to the η → Q ( α ) ≡ π < α | ρ | α > (5)The master equation (2) for η = 0 then leads to ∂Q∂t = − G ∂ ( αQ ) ∂α − G ∂ ( α ∗ Q ) ∂α ∗ (6)Note that this differential equation for Q function in-volves only the first order derivatives with respect tophase space variables and hence its solution is simple [17] Q in ( α ) ≡ π < α | ρ in | α > → Q out ( α ) = 1 G Q in ( α/G )(7)Let us see what the result (7) means. Let us consider theinput state to be vacuum then the output would be Q in ( α ) = 1 π < α | >< | α > = 1 π e −| α | → Q out ( α ) ≡ πG e −| α | /G (8)Such an output Q function is equivalent to a thermaldensity matrix with mean number of photons equal to( G − a and b in the input NOON state are sym-metrically amplified and the asymmetric case in whichonly one mode, say a , is amplified. Before discussing the amplification of the NOON state we examine quantita-tively the entanglement in the state (1). We compute thelog negativity parameter which is defined as E N = log (2 N + 1) , where N is the absolute value of the sum of all the neg-ative eigenvalues of the partial transpose of the densitymatrix ρ . It is clear that the partial transpose of thedensity matrix associated with the state (1) is ρ pt = 12 [ | N, >< N, | + | , N >< , N | + | N, N >< , | + | , >< N, N | ] , (9)which can be written in the diagonal form as ρ pt = 12 ( | N, >< N, | + | , N >< , N | )+ 14 ( | N, N > + | , > )( < N, N | + < , | ) −
14 ( | N, N > −| , > )( < N, N |− < , | ) (10)The partial transpose has a negative eigenvalue − / E N = 1. Symmetric case : The Q - function corresponding to thedensity operator for the input NOON state ρ in = 12 [ | N, >< N, | + | N, >< , N | + | , N >< N, | + | , N >< , N | ]= 12 N ! [ a † N ρ a N + a † N ρ b N + b † N ρ a N + b † N ρ b N ] ,ρ = | , >< , | , (11)is found to be Q in ( α, β ) ≡ π < α, β | ρ in | α, β > = 12 N ! π | α N + β N | exp[ − ( | α | + | β | )](12)Following the prescription in (7), under symmetric phaseinsensitive amplification, the Q-function evolves as fol-lows: Q in ( α, β ) → Q out ( α, β ) = 1 G Q in ( α/G, β/G )= 12 N ! π G N +4 | α N + β N | × exp[ − ( | α | + | β | ) /G ](13)The NOON state is highly nonclassical. A quantitativemeasure for nonclassicality is obtained by examining ze-roes of the Q function [18]. We note that the zeroes ofthe function Q out are identical to the zeroes of the func-tion Q in and hence we have the remarkable result thatthe nonclassical character of the input NOON state ispreserved.We can now find the density matrix after amplificationby using the results (8) and (13) : ρ in → ρ out = 12 N ! G N [ a † N ρ G a N + a † N ρ G b N + b † N ρ G a N + b † N ρ G b N ] ,ρ G = 1 G e − β ( a † a + b † b ) ,β = ln (cid:18) G G − (cid:19) . (14)We note that the structure of (14) is such that it can notbe written in a separable form. This is seen more clearlyif we write (14) as ρ out = 12 N ! G N { a † N + b † N } ρ G { a N + b N } (15)We further note that the output state has the structure ofa two mode photon added thermal state in which eithermode has added photons. The single mode version ofphoton added thermal state was introduced by Agarwaland Tara [19]. These states have been experimentallystudied recently [20].Writing ρ G in the number state basis as ρ G = 1 G ∞ X n,m =0 (cid:18) G − G (cid:19) n + m | n, m >< n, m | (16)we have ρ out = 12 N ! G N +4 ∞ X n,m =0 (cid:18) G − G (cid:19) n + m × [ a † N | n, m >< n, m | a N + a † N | n, m >< n, m | b N + b † N | n, m >< n, m | a N + b † N | n, m >< n, m | b N ]= 12 N ! G N +4 ∞ X n,m =0 (cid:18) G − G (cid:19) n + m × [ ( n + N )! n ! | n + N, m >< n + N, m | + ( m + N )! m ! | n, m + N >< n, m + N | + r ( n + N )! n ! ( m + N )! m ! ( | n + N, m >< n, m + N | + | n, m + N >< n + N, m | )] (17) which immediately gives us the expression for the oper-ator ρ P Tout obtained by partially transposing ρ out ( withrespect to the b mode): ρ P Tout = 12 N ! G N +4 ∞ X n,m =0 (cid:18) G − G (cid:19) n + m × [ ( n + N )! n ! | n + N, m >< n + N, m | + ( m + N )! m ! | n, m + N >< n, m + N | + r ( n + N )! n ! ( m + N )! m ! ( | n + N, m + N >< n, m | + | n, m >< n + N, m + N | )]= 12 N ! G N [ a † N ρ G a N + b † N ρ G b N + a † N b † N ρ G + ρ G a N b N ] (18) N=2N=4N=6
FIG. 1: Behavior of the logarithmic negatively as a functionof G for the symmetric case. The object of interest now is to calculate the logarith-mic negativity E N , the sum of the logarithmic negativityeigen-values of ρ P Tout and to see how it varies as a functionof G . We carry out this task numerically and the resultsare displayed in Fig. 1 where we plot E N as a functionof G for N = 2 , Asymmetric case : Proceeding as above, one finds that Q in ( α, β ) → Q out ( α, β ) = 1 G Q in ( α/G, β )= 12 N ! π G N +2 | ( α/G ) N + β N | × exp[ − ( | α | /G + | β | )](19)and hence ρ in → ρ out = 12 N ! G N [ a † N ˜ ρa N + G N a † N ˜ ρb N + G N b † N ˜ ρa N + G N b † N ˜ ρb N ]˜ ρ = 1 G e − β ( a † a ) | >< | ; β = ln (cid:18) G G − (cid:19) (20) N=2N=4N=6
FIG. 2: Behavior of the logarithmic negatively as a functionof G for the asymmetric case. Writing ˜ ρ as˜ ρ = 1 G ∞ X n =0 (cid:18) G − G (cid:19) n | n, >< n, | (21)we can write ρ out in terms of number states as ρ out = 12 N ! G N +2 ∞ X n =0 (cid:18) G − G (cid:19) n × [ a † N | n, >< n, | a N + G N a † N | n, >< n, | b N + G N b † N | n, >< n, | a N + G N b † N | n, >< n, | b N ]= 12 N ! G N +2 ∞ X n =0 (cid:18) G − G (cid:19) n × [ ( n + N )! n ! | n + N, >< n + N, | + G N N ! | n, N >< n, N | + G N r ( n + N )! n ! N !( | n + N, >< n, N | + | n, N >< n + N, | )] (22)and hence ρ P Tout = 12 N ! G N +2 ∞ X n =0 (cid:18) G − G (cid:19) n × [ ( n + N )! n ! | n + N, >< n + N, | + G N N ! | n, N >< n, N | + G N r ( n + N )! n ! N !( | n + N, N >< n, | + | n, >< n + N, N | )]= 12 N ! G N [ a † N ˜ ρa N + G N b † N ˜ ρb N + G N a † N b † N ˜ ρ + G N ˜ ρa N b N ] (23) The logarithmic negativity of ρ P Tout is computed numeri-cally and the results are shown in Fig. 2 for N = 2 , , | Φ i ∝ a † b † exp { ζa † b † − ζ ∗ ab }| i (24)This is a non Gaussian state. The input and output Q functions are found to be Q in ∝ | α | | β | Q sq (25) Q out ∝ | α | | β | G Q sq,G (26)where Q sq is the Q function for the two mode squeezedvacuum and Q sq,G is the Q function obtained by ampli-fication of the squeezed vacuum. Thus the density oper-ator after amplification of the non Gaussian state can bewritten as ρ out ∝ a † b † ( ρ sqout ) ab, (27)where ρ sqout is the density operator for the squeezed vac-uum after amplification. Now ρ sqout becomes separablefor G greater than that given by (3) and hence ρ out be-comes separable if G > (2 + 2 η ) / (1 + 2 η + e − r ). Thusthe non Gaussian states obtained from Gaussian statesby the addition of photons would behave under symmet-ric amplification in a manner similar to Gaussian states.We have therefore found that the NOON states behavequite differently under amplification. III. CONCLUSIONS
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