Theory of a two-mode entanglement amplifier

Physics Letters A 366 (2007) 596–599 www.elsevier.com/locate/pla

Theory of a two-mode entanglement amplifier Yunxia Ping ∗ , Bo Zhang, Ze Cheng Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Received 5 November 2006; received in revised form 31 January 2007; accepted 5 February 2007 Available online 14 March 2007 Communicated by P.R. Holland

Abstract We develop a theory of two-mode entanglement amplifier in a cascade three-level atomic system. Two photons of a strong external pump field induce atomic coherence between the top and bottom levels. We study the dynamics of the system in the presence of losses, concluding that the creation of strongly entangled states with very large photon numbers seems achievable. © 2007 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.67.Mn; 03.65.Ud

1. Introduction In 1960, the invention of the laser allowed classical light to be amplified inside a cavity. In the past two decades, much attention has been given to the research on quantum amplifier. Scully and Zubairy [1] presented a theory of two-photon phase-sensitive amplification by a three-level atomic system in a cascade configuration. Many other studies about the phasesensitive amplification have also been investigated [2–5]. On the other hand, quantum entanglement plays an essential role in quantum information processing such as quantum teleportation [6–8], quantum computation [9,10], and quantum cryptography [11]. In 2001, Lamas-Linares et al. [12] first showed that photons in a special quantum entanglement state can be amplified. Simon and Bouwmeester [13] proposed a scheme to generate entanglement laser based on the parametric downconversion process. Xiong, Scully and Zubairy [14] proposed a scheme for an entanglement amplifier based on a two-mode correlated spontaneous emission laser. In a very recent paper, Tesfa [15] investigated the entanglement amplification in a nondegenerate three-level cascade laser, where the atomic coherence is induced by the initial atomic coherence superposition of the upper and the bottom levels. The entanglement ampli* Corresponding author.

E-mail address: [email protected] (Y. Ping). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.02.096

fication in a laser system maybe has significant application in quantum information processing, especially in quantum communications using optical techniques. In this Letter, we further extend the theory of two-mode entanglement amplifier in a two-mode three-level cascade atomic system when the top and bottom levels of the system are coupled by a strong external pump field. The transition from the top to the bottom level via the intermediate level comes out in the form of two photons of equal frequency. We derive the equations of motion of expectation values for different second-order moments, from the density-matrix equation of motion for the field and study the dynamics of this system. We show that entanglement of the two modes can be built in the cavity and the total average photon numbers in the two modes still increase dramatically even in the presence of cavity losses. This leads to an entanglement amplifier for the driven system. The organization of this Letter is as follows. Section 2 gives the master equation for the two-mode field. Section 3 is devoted to the entanglement analysis of the two modes in the cavity. In Section 4, we present a summary of our main results. 2. Model and the master equation We consider a two-photon three-level cascade configuration as shown in Fig. 1. The upper level a and the bottom level c have the same parity, but the intermediate level b has an opposite one. The dipole-allowed transitions a ↔ b and b ↔ c with

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where r denotes the relaxation processes. By considering the slowly varying field modes and taking traces over the atomic states, the density matrix equation of motion for the field modes as obtained in Ref. [4] is

Fig. 1. Systematic diagram for a three-level atomic system in cascade configuration.

frequencies ν1 and ν3 , respectively, are considered weak and treated quantum mechanically up to second order in coupling constant. The transition a ↔ c requires two pump photons of frequency ν2 . Strong pump field is treated classically up to all orders. We assume that the one-photon pump detuning ωbc − ν2 is sufficiently large that the dipole transition c ↔ b with pump frequency ν2 is negligible. The pump frequency ν2 is exactly one-half the atomic transition frequency ωac ≡ ωa − ωc . The side-mode frequencies ν1 and ν3 are assumed to satisfy the conservation condition ν1 + ν3 = 2ν2 , which gives the relation between the side-mode detuning  and the beat frequency  ≡ ν2 − ν1 as  = (ωbc − ν2 ) − . The Hamiltonian for the atom-field system is [4] H = H0 + V ,

(1)

where the unperturbed part of the Hamiltonian is 

H0 =

h¯ ωi |ii| +

3 

† hν ¯ j a j aj ,

(2)

V=

+ H.c.,

(3)

j =1

where a1 and a3 are the annihilation operators for the field modes 1 and 3, a2 is the effective two-photon annihilation operator for the pump mode, Uj = Uj (r) is the spatial mode factor for the j th field mode, and gj is the corresponding atom-field coupling constant. The matrices σj† are 

0 = 0 0  0 σ3† = 0 0

σ1†

1 0 0 0 0 0

 0 0 , 0  0 1 . 0



σ2†

 0 0 1 = 0 0 0 , 0 0 0 (4)

The time dependence of the atom-field density operator ρa−f can be obtained from the basic density operator equation of motion, as i d ρa−f = − [H, ρa−f ] + r, dt h¯

1 + I22 1 + I22 D1 D3∗ /4T1 T2

,

N g12 D1

fb , 2 2 1 + I2 1 + I2 D1 D3∗ /4T1 T2 N g32 D3 fb , A3 = 2 2 ∗ 1 + I2 1 + I2 D1 D3 /4T1 T2 N g32 D3 fc − I22 D1∗ D2 /4T1 T2 , B3 = 1 + I22 1 + I22 D1∗ D3 /4T1 T2 iNg32 D3 −fa D1∗ + D2 I2 e−iφ , C3 = 1 + I22 2(T1 T2 )1/2 1 + I22 D1∗ D3 /4T1 T2 fc D3∗ + D2 iNg12 D1 I2 e−iφ . D1 = 2 1/2 1 + I2 2(T1 T2 ) 1 + I22 D1 D3∗ /4T1 T2 B1 =

D1,3 =

D2 = † hg ¯ j a j Uj σ j

N g12 D1 fa + I22 D3∗ D2 /4T1 T2

(7a) (7b) (7c) (7d) (7e) (7f)

1 , γ1,3 + i1,3

(8)

where 1 = ωa − ωb − ν1 = − and 3 = ωb − ωc − ν3 =  .

and the perturbed part is 3 

A1 =

The complex Lorentzian for the field modes 1 and 3 is

j =1

i=a,b,c

    d ρ = −A1 ρa1 a1† − a1† ρa1 − (B1 + κ1 ) a1† a1 ρ − a1 ρa1† dt     − A3 ρa3 a3† − a3† ρa3 − (B3 + κ3 ) a3† a3 ρ − a3 ρa3†     + C3 a3† a1† ρ − a1† ρa3† + D1 ρa3† a1† − a1† ρa3† + H.c., (6) with κj (j = 1, 3) is the damping constant of each mode. Different coefficients are given by

(5)

1 , γ2

(9)

where γ2 ≡ 1/T2 is the two-photon coherent decay rate between the levels a and c. The dimensionless pump intensity I2 is defined by I2 = 2|V2 |(T1 T2 )1/2 ,

(10)

where V2 = g2 U2 (n2 )1/2 is the effective two-photon interaction energy. The population difference decay time T1 is   1 Γ1 T1 = (11) 1+ , Γa 2Γ3 where Γa (= Γ1 + Γ2 ) is the upper level decay rate to the lower levels b and c. Γ1 and Γ3 are the decay constants for the a → b and b → c transitions and Γ2 allows for nonradiative decay of level a to c. The probability factors fk are Γ3 I 2, Γ1 + 2Γ3 2 Γ1 fb = I 2, Γ1 + 2Γ3 2 fc = 1 + f a .

fa =

(12a) (12b) (12c)

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Also φ is the phase of the classical pump field which can be obtained from the relation V2 = |V2 |e−iφ ,

(13)

and N is the total number of interacting atoms. The physical interpretation of these coefficients is as follows. The terms Aj and Bj with their complex conjugates are the gain and absorption coefficients for the j th mode, respectively, and C3 and D1 represent the coupling between the two modes. 3. Entanglement analysis of the system In this section we seek to study the two-mode entanglement in the cavity. A quantum system is said to be entangled if it is not separable. That is, if the density operator for the combined state cannot be expressed as a combination of the product density operators of the constituents,  pj ρja ⊗ ρjb ρ = (14) j

with pj  0 and j pj = 1. A question from a practical point of view is whether one can formulate criteria for measuring entanglement in a given system. Recently, different criteria have been proposed [16–20]. Here, we choose the summation of the quantum fluctuations proposed in Ref. [17]. According to this criterion, a state is entangled if the sum of the quantum fluctuations of the two Einstein–Podolsky–Rosen (EPR)-like operators uˆ and vˆ satisfies the following inequality ˆ < 2. (u) ˆ + (v) 2

2

(15)

Here uˆ = xˆ1 + xˆ2 ,

(16a)

vˆ = pˆ 1 − pˆ 2 ,

(16b)

with  √  xˆ1 = a1 + a1† / 2,   √ xˆ2 = a3 + a3† / 2,  √  pˆ 1 = a1 − a1† /i 2,  √  pˆ 2 = a3 − a3† /i 2,

(17a) (17b) (17c) (17d)

being the quadrature operators for the two modes 1 and 3. They satisfy the relation [x, ˆ p] ˆ = iδjj  (j, j  = 1, 2). We also notice that if we substitute the definition of uˆ and vˆ into Eq. (15), we obtain 







ˆ 2 = a1† , a1 + a1 , a1† + a3† , a3 + a3 , a3† (u) ˆ 2 + (v) 

 + 2 a1 , a3  + a1† , a3† , (18) where we use the notation a, b = ab − ab. If we assume the field modes are initially in vacuum states, the expectation values for all first-order moments of the field will never show up. We can obtain

Fig. 2. (a) The time evolution of the variance (u) ˆ 2 + (v) ˆ 2 and (b) the total ˆ for initial vacuum states for the two modes with I2 = 20 photon numbers N (solid), I2 = 30 (dashed), I2 = 40 (dotted). Curves in (b) are truncated when ˆ 2 = 2 and the state is not necessarily entangled. The parameters (u) ˆ 2 + (v) are  = 0, φ = π/2, Q = 750. Parameters are chosen such that they correspond to the four-wave mixing experiment [22].







 ˆ 2 = 2 a1† a1 + a3† a3 + a1† a3† (u) ˆ 2 + (v)  + a1 a3  + 1 .

(19)

The equations of motion for the second-order moments of the fields can be obtained from the master equation (6)



 d †  (20a) a1 a1 = −α1 a1† a1 − D1 a3† a1† + A1 + H.c., dt



 d †  (20b) a3 a3 = −α3 a3† a3 + C3 a3† a1† + A3 + H.c., dt



 d a1 a3  = −(α1 + α3 )a1 a3  + C3 a1† a1 − D1 a3† a3 + C3 , dt (20c) where αj = Bj − Aj + κj (j = 1, 3). These four timedependent equations can be solved by using the standard techniques such as those based on Laplace transform method. We ˆ 2 and the total avcan then evaluate the variance (u) ˆ 2 + (v) ˆ ˆ ˆ erage photon numbers N  = N1  + N2 . These solutions of Eqs. (20a)–(20c) are long and tedious and we do not reproduce them here. We now discuss how the above system leads to entanglement amplifier. For simplicity, we assume g1 = g3 = g, κ1 = κ3 = κ, Γa = 1, Γ1 = Γ3 = 1, γ1 = γ3 = γ2 = γ = 1 and introduce a cooperativity parameter [21] Q = C/2κ, where C = Ng 2 /γ . In Figs. 2(a) and 2(b), we plot the time evolution of the variˆ 2 and the total average photon numbers Nˆ  ance (u) ˆ 2 + (v) as a function of Ct for the vacuum input, where we take  = 0, Q = 750 and φ = π/2. The choice of the phase for the pump field is such that the condition a1 a3  = −|a1 a3 | is satisfied.

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from Fig. 3(a) that perfect and durable entanglement is obtained when  = 0. With the increase of  , both the period and degree of entanglement are decreased. In addition, it is clear that, in Fig. 3(b), the maximally entangled photon numbers are obtained when  = 0. 4. Conclusion In summary, we have developed a theory of two-mode entanglement amplifier in a three-level cascade atomic system. Two photons of a strong external pump field are responsible for transition between the top and bottom levels. The generation and evolution of the two-mode entanglement in the cavity is investigated, even in the presence of the cavity losses. It shows that the field evolves into an entangled state. The entanglement increases initially, then decreases and vanishes eventually as the two mode fields evolve from initial vacuum states. The entanglement degree and period have a close relation with the pump intensity and the side-mode detuning. During the entanglement period, the intensity of the entangled light is amplified, thus leading to an entanglement amplifier. ˆ for the two modes Fig. 3. (a) The time evolution of (u) ˆ 2 + (v) ˆ 2 and (b) N with I2 = 30,  = 0 (solid),  = 4 (dashed),  = 8 (dotted). The other parameters are the same as in Fig. 2.

Solid, dashed, and dotted curves represent the pump field intensity for I2 = 20, 30, and 40, respectively. The parameter values are such that they correspond to the four-wave mixing experˆ 2 is iment [22]. It is clear from Fig. 2(a) that (u) ˆ 2 + (v) less than 2 and the two-mode field in the cavity evolves into an entangled state. The entanglement increases initially, then decreases and finally vanishes after a period that depends on I2 . Both entanglement and entanglement period are extended as the pump field intensity increases. As a result, in order to have entanglement with high intensity and longer entanglement time, high pump field intensity is needed. On the other hand, it is worthwhile to point out from ˆ increase Fig. 2(b) that the total average photon numbers N with the increasing of the pump field intensity I2 . Moreover, during the entanglement period, the total average photon numbers grow persistently and the intensity of the entangled light is amplified dramatically for the driven system. Thus we obtain both entanglement and amplification at the same time even in the presence of cavity losses. In order to see the dependence of the entanglement on the side-mode detuning  , in Figs. 3(a) and 3(b), we plot the time ˆ 2 and the total average evolution of the variance (u) ˆ 2 + (v) photon numbers Nˆ  as a function of Ct for different values of  . Solid, dashed, and dotted curves represent the side-mode detuning for  = 0, 4, and 8, respectively. One can directly see

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