Dissipation of a two-mode squeezed vacuum state in the single-mode amplitude damping channel

Transcription

1 Dissipation of a two-mode squeezed vacuum state in the single-mode amplitude damping channel Zhou Nan-Run( ) a), Hu Li-Yun( ) b), and Fan Hong-Yi( ) c) a) Department of Electronic Information Engineering, Nanchang University, Nanchang , China b) School of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 3300, China c) Department of Physics, Shanghai Jiao Tong University, Shanghai 00030, China (Received 5 April 011; revised manuscript received 17 June 011) We explore how a two-mode squeezed vacuum state sech e a b tanh 00 evolves when it undergoes a singlemode amplitude dissipative channel with rate of decay κ. We find that in this process not only the squeezing parameter decreases, tanh e κt tanh, but also the second-mode vacuum state evolves into a chaotic state exp{b b ln`1 e κt tanh }. The outcome state is no more a pure state, but an entangled mixed state. Keywords: amplitude dissipative channel, two-mode squeezed vacuum state, two-mode entangled state, quantum communication PACS: w, p DOI: / /0/1/ Introduction In recent years, quantum operation formalism has been a hot topic in quantum information theory, 1 to be specific, quantum operations describe quantum noise and the behaviour of open quantum systems (both the environment and system under our review). This topic is important and realistic since no quantum systems are ever perfectly closed. The role of quantum operations is transforming an input pure state to output mixed state, denoted by ρ 0 i M iρ 0 M i ρ (t), such a transformation also defines a quantum channel, M i is a kind of Kraus operator. From the viewpoint of mathematical formalism, the density operators ρ (t) at time t are solutions of master equations. For example, for an amplitude-damping channel the master equation is 4 dρ dt κ ( aρa a aρ ρa a ), (1) κ is the decaying rate, a, a 1. As we have experienced that the evolution of many quantum signals is related to quantum entanglement, for example, a two-mode squeezed state is composed of a signal mode and an idler mode, which is simultaneously an entangled state. In this work we explore how a twomode squeezed vacuum state, sech e a b tanh 00, evolves when it undergoes a single-mode (say, a-mode) amplitude dissipative channel, so that we may see how the b-mode affects a-mode s damping, or vice versa. In the following, first, we shall solve the master equations in the entangled state representation which is denoted by η, 3,4 (see Eq. () below), whose one mode is a fictitious one. The fictitious mode is a counterpart mode of the system-mode under review, then we can arrange density operator master equations as state-vector evolution equations. In this way the master equation for different physical systems can be concisely solved, 7 and the corresponding Kraus operator can be obtained. Second, we examine when the initial ρ 0 is a two-mode squeezed vacuum state, then how it evolves. As one can see later, we find that in this process not only the squeezing parameter decreases with a factor e κt, but also the second-mode vacuum state evolves into a chaotic state exp{b b ln ( 1 e κt) tanh }. The outcome state is no more a pure state, but an entangled mixed state. Project supported by the National Natural Science Foundation of China (Grant Nos and ), the Natural Science Foundation of Jiangxi Province of China (Grant Nos. 009GQS0080 and 010GQW007), and the Research Foundation of the Education Department of Jiangxi Province of China (Grant Nos. GJJ11339 and GJJ10097). Corresponding author. znr1@163.com 011 Chinese Physical Society and IOP Publishing Ltd

2 . ρ (t) obtained by the entangled state representation To solve the above master equation, similar to Ref. 5 we introduce the two-mode entangled state 6 ( η exp 1 ) η + ηa η ã + a ã 0 0, () ã is a fictitious mode independent of the real mode a, ã 0 0. Historically, Takahashi and Umezawa 8 first introduced the fictitious Fock space to treat the ensemble average of a mixed state as a pure state average. Using the technique of integration within ordered product (IWOP) of operators, we can prove the completeness η η 1. The state η 0 possesses the following properties: a η 0 ã η 0, a η 0 ã η 0, (3) (a a) n η 0 (ã ã) n η 0. (4) Acting both sides of Eq. (1) on the state η 0 I, and denoting ρ ρ I, we have d dt ρ κ ( aρa a aρ ρa a ) I κ ( aã a a ã ã ) ρ, (5) so its formal solution is given by ρ exp κt ( aã a a ã ã ) ρ 0, (6) ρ 0 ρ 0 I, ρ 0 is the initial density operator. Noticing that the operators in Eq. (5) obey the following commutative relation aã, a a aã, ã ã ãa, a a + ã ã, aã ãa, (7) and using the operator identity e λ(a+σb) e λa exp σ ( 1 e λτ ) B/τ (8) (which is valid for A, B τb), we have ( a a + ã ) ã exp κt aã exp κt ( a a + ã ã ) exp T aã, (9) Then using the operator identity T 1 e κt. (10) exp ( λa a ) : exp ( e λ 1 ) a a : (11) and the IWOP technique, we can express exp κt ( a a + ã ã ) exp T aã ρ 0 : exp ( e κt 1 ) ( a a + ã ã ) + ( 1 e κt) aã: ρ 0 : exp η + η ( a e κt ã ) + η ( e κt a ã ) + a ã + aã a a ã ã: ρ 0 e 1 T η η η e κt ρ 0. (1) Comparing Eq. (1) with ρ (t) η η ρ (t), (13) we can see η ρ (t) e 1 T η η e κt ρ 0, (14) which manifestly shows that the wave function of the mixed state ρ (t) in η representation is proportional to that of the initial state ρ 0 in the decayed entangled state η e κt, accompanied by a Gaussian damping factor e T η. Thus we see clearly how the dissipative channel plays its role in time evolution. This is the advantage of employing η representation. Further, substituting Eq. (9) into Eq. (6) yields ρ exp κt ( a a + ã ã ) e κta a an ã n ρ 0 an ρ 0 a n e κtã ã I e κta a a n ρ 0 a n e κta a I, (15) which leads to the infinitive operator-sum representation of ρ (t), ρ (t) M n ρ 0 M n, (16) T n M n e κta a a n. (17) Using Eq. (11) we can prove M nm n n n a n e κta a a n

3 n e nκt : a n a n : e κta a : e T e κt a a : e κta a 1. (18) Thus M n is a kind of Kraus operator, and ρ (t) in Eq. (16) is qualified to be a density operator, i.e., Tr ρ (t) Tr M n ρ 0 M n Tr ρ 0. (19) Therefore, for any given initial state ρ 0, the density operator ρ (t) can be directly calculated from Eq. (16). The entangled state representation provides us with an elegant way of deriving the infinitive sum representation of the density operator as a solution of the master equation. 3. Evolution of a two-mode squeezed vacuum state in the single-mode amplitude damping channel The realistic two-mode squeezed vacuum state is expressed by S () 00 sech exp a b tanh 00, (0) the vacuum state 00 is annihilated by either a or b, a, a 1 b, b, and S () is the squeezing operator with the squeezing parameter, S () exp ( a b ab ). When the initial state is the two-mode squeezed vacuum then ρ (0) S () S (), (1) ρ (t) e κta a a n S () S () a n e κta a. () Substituting Eq. (0) into Eq. (16), and using we obtain ρ (t) a n S () 00 ( b tanh ) a n 1 S () 00 ( b tanh ) n S () 00, (3) tanh n e κta a b n S () S () b n e κta a sech tanh n e κta a b n e a b tanh e ab tanh b n e κta a. (4) Then using the relation we obtain ρ (t) sech e κta a a e κta a e κt a, (5) tanh n b n e e κt a b tanh e e κt ab tanh b n. (6) By introducing another damping squeezing parameter, tanh e κt tanh, (7) we can introduce a new time-dependent squeezed vacuum state sech e e κt a b tanh 0 0 S ( ) 00, (8) S ( ) exp ( a b ab ). The state S ( ) 00 includes less real photons, since 00 S ( ) a as ( ) 00 sinh With Eq. (8), ρ (t) becomes ρ (t) sech sech tanh e κt tanh < sinh. (9) tanh n b n S ( ) S ( ) b n, (30) which indicates that in the dissipation process, the a-mode photon decreases, while the b-mode photon increases, since b n S ( ) 00 is an excitation of S ( ) 00. Further, using the normal product form : e a a b b :, 0 aa 0 : e a a :, 0 bb 0 : e b b :, (31) we can rewrite ρ (t) in Eq. (6) as

4 ρ (t) sech : Chin. Phys. B Vol. 0, No. 1 (011) tanh n b n b n e e κt a b tanh e e κt ab tanh a a b b : sech : e b b(1 e κt ) tanh + e κt tanh (a b +ab) a a b b : sech e e κt a b tanh 0 aa 0 e b b ln(1 e κt ) tanh e e κt ab tanh, (3) in the last step we have used the operator identity : e b b( e λ 1) : e λb b. To confirm the validity of the above conclusion, using the completeness relation of the two-mode coherent state 9,10 d z 1 d z z 1, z z 1, z 1, (33) z 1, z exp 1 ( z1 + z ) + z 1 a + z b 00, (34) we check the trace-preserving by evaluating d Tr ρ (t) sech z 1 d z z 1, z 0 aa 0 : e b b(1 e κt ) tanh 1 : z 1, z e e κt (z 1z +z 1 z ) tanh d sech z exp ( tanh 1 ) z 1, (35) we have used the following integration formula d z eζ z +ξz+ηz 1 ( ζ exp ξη ), ζ Re (ζ) < 0. (36) 4. Evolution of the Wigner function We now see how the Wigner function evolves in this process. Using the Wigner operator coherent state representation 11 (α, β) e α + β d z 1 d z 4 z 1, z z 1, z e (αz 1 α z 1) e (βz β z ), (37) and noticing that z e b b ln(1 e κt ) tanh z exp( ( 1 e κt) tanh 1)z z, as well as using Eq. (3), we can calculate the Wigner function at time t as: W (α, β) Tr ρ (t) (α, β) 1 sech + 4 e κt tanh 1 + R (α β + αβ) { 1 + R exp (1 R) β + α (1 + tanh ) 1 + R }, (38) R ( 1 e κt) tanh. In particular, when κt 0, i.e., the case of twomode squeezed state, R tanh, equation (41) reduces to the initial state Wigner function 1,13 W (α, β) 1 exp (α β + αβ) sinh ( α + β ) cosh. (39) On the other hand, when κt, then R tanh, equation (38) becomes W (α, β) 1 e α sech exp β sech. (40) It is interesting to note that the Wigner function of the thermal vacuum state is W th 1 e ω/kt ( 1 + e ) ω/kt ( ) 1 e ω/kt exp ( ) 1 + e ω/kt β, (41) thus equation (40) is just the product of the Wigner function W 0 0 of the vacuum state and the W th of the thermal vacuum state, i.e., W (α, β) W 0 0 (α) W th (β), (4) W 0 0 (α) 1 e α,

5 sech W th (β) e β sech, (43) with e ω/kt tanh, a a th sinh. From Eq. (4) we can see that the output state is a product of a vacuum state and a thermal state after a long interaction time. In fact, this point can also be understood from Eq. (9). 5. Conclusion and discussion As shown in Eq. (9), e b b ln(1 e κt ) tanh represents a time-dependent chaotic field of b-mode which is in sharp contrast to the initial pure state 0 bb 0 in ρ (0). The formula ρ (0) ρ (t) exhibits how a pure two-mode squeezed vacuum state evolves into the mixed state, i.e., not only its squeezing parameter tanh e κt tanh, but also evolves into 0 aa 0 e b b ln(1 e κt ) tanh. The outcome state is no more a pure state, but an entangled mixed state. This is because a two-mode squeezed state is simultaneously an entangled state, while the a-mode undergoes a damping, the entangled b-mode also varies correspondingly. For this channel, the associated loss mechanism for the two-mode squeezed state is in an entangled way. References 1 Bouwmeester D, Ekert A and Zeilinger 000 The Physics of Quantum Information (Berlin: Springer) Gardiner C W and Zoller P 000 Quantum Noise nd edn. (New York: Springer-Verlag) 3 Fan H Y and Hu L Y 009 Chin. Phys. B Fan H Y and Hu L Y 009 Commun. Theor. Phys Fan H Y and Klauder J R 1994 Phys. Rev. A Fan H Y and Fan Y 1998 Phys. Lett. A Fan H Y and Hu L Y 008 Mod. Phys. Lett. B Takahashi Y and Umezawa H 1975 Collecive Phenomena 55 9 Loudon R and Knight P L 1987 J. Mod. Opt Mandel L and Wolf E 1995 Optical Coherence and Quantum Optics (New York: Cambridge University Press) 11 Hu L Y, Xu X X, Wang Z S and Xu X F 010 Phys. Rev. A Hu L Y, Xu X X, Guo Q and Fan H Y 010 Opt. Commun Hu L Y and Fan H Y 009 Chin. Phys. B