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Squeezing and entanglement of a two-mode field in a four-level tripod atomic system
Optics Communications 284 (2011) 2949–2954
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Squeezing and entanglement of a two-mode field in a four-level tripod atomic system J.L. Ding a, B.P. Hou a,b,⁎ a b
School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Chengdu 610068, China
a r t i c l e
i n f o
Article history: Received 26 December 2010 Received in revised form 17 February 2011 Accepted 19 February 2011 Available online 11 March 2011 Keywords: Squeezing Entanglement Electromagnetically induced transparency
a b s t r a c t The interaction of a collection of N four-level tripod configuration atoms with two orthogonally polarized probe fields is investigated. Under the condition of electromagnetically induced transparency (EIT), we calculate the squeezing and entanglement spectra of the output probe fields. By analyzing the output spectrum, we find that the squeezing and entanglement of the probe fields can be well-preserved after passing through the optically thick medium. Additionally, the effects of the ground state dephasing rates of the atoms on the entanglement and squeezing of the output two-mode squeezed fields are investigated. It is shown that the dephasing rates will degrade the entanglement and squeezing, and these quantum properties can be lost when the dephasing rates increase up to a certain value. This will be useful in the quantum computation and quantum communication. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The phenomenon of electromagnetically induced transparency (EIT) in atomic media is a quantum interference effect that allows for the transmission of light with little absorption through an otherwise opaque atomic medium [1]. Medium exhibiting EIT has some remarkable properties, such as ultra slow propagation of a light through atomic cloud [2] and nonlinear optical processes [3], lasing without inversion [4]. Recently, the potential application of EIT has been reported in quantum information science. For example, it can be used to store and release the quantum information in an atomic ensemble [5], to generate the correlated photon pairs [6] and even the entanglement of remote atomic pairs generation [7]. These form the building blocks of the quantum communication and the quantum computation. Squeezing and entanglement are two representative quantities characterizing the amount of quantum information of a light field. Conventionally, the continuous variable entanglement can be easily studied using two-mode squeezing. Also, the two-mode squeezed states can be used to entangle distant atoms as well expected to lead to efficient distribution of entanglement and implementation of quantum channels by improving the yet low squeeze parameters [8]. Two-mode (polarization) squeezing has already been realized by
⁎ Corresponding author at: School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China. Tel.: +86 02884480791; fax: +86 02884480787. E-mail address: [email protected] (B.P. Hou). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.02.056
exploiting the Kerr non-linearity of a glass fiber [9] and with cold atoms in a high finesse optical cavity [10]. Besides, a method of generating unitarily two-mode field squeezing in an optical cavity with an atomic cloud has been proposed in Ref. [11]. Recently, the experimental investigation has shown that based on the EIT technique the quadrature squeezing was maintained within the transparency window when the squeezed light passed through an optically dense rubidium gas cell [12]. Additionally, the works in Ref. [13] showed that the squeezing of a single mode squeezed light can be well-preserved throughout the EIT atomic system in a three-level Λ configuration. The entanglement for the field throughout the EIT medium was also been studied by considering another field traveling in vacuum. Another work examined the preservation of the squeezing and the continuous-variable entanglement of a two-mode squeezed light when one of the modes transmitted through an EIT medium while the other mode propagated through an optical fiber [14]. It was found that loss of squeezing occurs when the mismatch in the transmission of the two modes is greater than 40%, while near-ideal squeezing is preserved when the transmission is equal. Most recently, we have investigated the squeezing survival and transfer of one probe field in the single and double EIT [15]. It has been pointed out in Ref. [14] that the squeezing or entanglement of the two mode fields will be lost due to their mismatching transmission. Taking into account the delay in the EIT medium for one field corresponding to the other field traveling in vacuum or in an optical fiber, as were proposed in Refs. [13] and [14], it cannot be convenient to realize the matching transmission which is needed to quantify the entanglement between these fields. In this paper, we consider two squeezed fields as probe fields to inject into a
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medium of four-level tripod configuration atoms driven by one control field. The two probe fields will propagate at the same slow velocities by using the atomic inherent symmetry and the symmetric action of the two probe fields. This can overcome the mismatching transmission in the investigated schemes. We examine the preservation of the continuous variable squeezing and entanglement of the two modes of the squeezed light after their simultaneous propagation through the EIT medium. By calculating the output squeezing spectrum and entanglement spectrum of the two-mode squeezed light, it is shown that the preservation of two-mode squeezing and entanglement can be also well-preserved for small dephasing rates.
Due to its perfectly preservation and its convenience, this will offer some promising perspectives for the realization of quantum communication and the quantum computation. The paper is arranged as follows. In Section 2, the model of the interaction of the four-level tripod-scheme atomic system with the two-mode squeezed field and a control field is briefly described, and its equation of motion for the two components of the weak probe field is shown. In Section 3, the squeezing and entanglement of the two-mode output squeezed light are discussed. In Section 4 the discussion and the summary of the main results of this work are made.
2. Model and equation of motion Consider a quasi-one-dimensional mode consisting of four-level atoms in the tripod configuration with a single upper state |0〉 and three lower states |1〉, |2〉 and |3〉 interacting with three optical fields, as shown in Fig. 1. We assume that all the atoms are equally located in the states |1〉 and |2〉 by optical pumping. A linearly polarized weak quantum field ε with the carrier frequency ωs has two circularly left and right-polarized components ε1, 2 which couple the atomic transitions |1〉 ↔ |0〉 and |2〉 ↔ |0〉, respectively. The part of the positive frequency of the electric field related to the weak quantum field εˆ 1;2 ðz; t Þ is given by ð+Þ Eˆ i ðz; t Þ =
sffiffiffiffiffiffiffiffiffiffiffi h ω i ℏωs εˆi ðz; t Þ exp i s ðz−ct Þ ði = 1; 2Þ 2ε0 V c
ð1Þ
where quantization volume of the electromagnetic field V is assumed to be the interaction volume. The detunings from atomic resonance for the two mode fields are defined by Δ1 = ωs − ω01 and Δ2 = ωs − ω02. The transition |3〉 ↔ |0〉 is driven by a classical coherent control field Ωc with the frequency ωc, and the corresponding detuning is denoted by Δ3 = ωc − ω03, and ωμν = (Eμ − Eν)/ℏ the atomic resonance frequency of the transition |μ〉 ↔ |ν〉. The effect of Doppler shift is minimized by using co-propagating configuration for the coherent fields in the present model. If the slowly-varying amplitude εˆi ðz; t Þ does not change much in a length internal Δz containing Nz(≫ 1) atoms, we can introduce the locallyaveraged slowly-varying atomic operators
σˆ μ ν ðz; t Þ =
1 NZ j i ω = c ðz−ct Þ ∑ σˆ ðt Þe ð μ ν Þ NZ j = 1 μ ν
ð2Þ
j
where σˆ μ ν = j μ j ðt Þ〉〈ν j ðt Þj is the flip operator for the jth atom. The interaction Hamiltonian of the system can be written as n o ˆ = ∫ Nℏ Δ σˆ + Δ σˆ + Δ σˆ −gσˆ ðz; t Þεˆ ðz; t Þ +g σˆ ðz; t Þεˆ ðz; t Þ +Ω ðz; t Þ σˆ ðz; t Þ +H:c dz Η I 1 11 2 22 3 33 01 1 02 2 c 03 L
ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g = g1 = g 2 with gi = }0i ℏωs = 2ε 0 V (i = 1,2) being the atom-field coupling constant, and }01 ð}02 Þ is the atomic dipole moment for the |1〉 ↔ |0〉 (|2〉 ↔ |0〉) transition. The two coupling constants are the same due to the atomic inherent symmetry and the symmetric actions of two component fields, N is the number of atoms and L is the interaction length in the propagation direction of the quantum field.
Fig. 1. Schematic diagram of a four-level system in tripod configuration.
J.L. Ding, B.P. Hou / Optics Communications 284 (2011) 2949–2954
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The Maxwell's propagation equations for the quantum fields in the slowly-varying envelope approximation are given by
∂ ∂ +c εˆ ðz; t Þ = igN σˆ 10 ðz; t Þ; ∂t ∂z 1
ð4aÞ
∂ ∂ +c εˆ ðz; t Þ = igNσˆ 20 ðz; t Þ: ∂t ∂z 2
ð4bÞ
The atomic Heisenberg–Langevin equations can be obtained by following our previous work [15]. Now, we assume that the Rabi frequencies of the quantum fields are much smaller than the classical control field, and the number density of photons in the quantum fields is much less than ˆ 11 i ≈〈 σ ˆ 22 i ≈1 =2 are that of atoms. In such a case, the atomic equations can be treated perturbatively in εˆ 1;2 ðz; t Þ. In the zeroth order only 〈σ different from zero. Therefore, one finds the atomic motion equations in the first order of εˆ 1;2 ðz; t Þ i ˆ 10 + g εˆ 1 + iΩc σ ˆ 13 + Fˆ 10 ; ˆ˙ 10 = −ðγ10 −iΔ1 Þ σ σ 2
ð5aÞ
i ˆ 20 + g εˆ 2 + iΩc σ ˆ 23 + Fˆ 20 ; ˆ˙ 20 = −ðγ20 −iΔ2 Þ σ σ 2
ð5bÞ
ˆ 13 + iΩ⁎ ˆ 10 + Fˆ 13 ; ˆ˙ 13 = −ðγ13 −iðΔ1 −Δ3 ÞÞ σ σ c σ
ð5cÞ
ˆ 23 + iΩ⁎ ˆ 20 + Fˆ 23 : ˆ˙ 23 = −ðγ23 −iðΔ2 −Δ3 ÞÞ σ σ c σ
ð5dÞ
where γμ is the decay rate of the atomic dipole operators, and Fˆ μ ν is the associated Langevin noise operators which describe the effect of spontaneous decay caused by the coupling of atoms to all the vacuum field modes. The decay rates γ12, γ13 and γ23 of the ground state coherence are modeling collisions or accounting for the transit of the atoms outside the interaction area with the light beams. In order to solve these equations, we transform them into Fourier space in the frequency domain according to 1 ∞ ˆ iωt ðz; t Þe dt G˜ ðz; ωÞ = pffiffiffiffiffiffi ∫−∞ G 2π
ð6Þ
A 80 000
60 000
40 000
20 000
0 15
10
5
0
5
10
15
15
10
5
0
5
10
15
B 40 000
20 000
0
20 000
40 000
Fig. 2. The absorption curve Im(K) (A) and dispersion curve Re(K) (B) of the probe field ε1 (dashed line) and ε2 (solid line) as functions of ω in the case Δ3 = 0 ; Δ2 = − Δ1 = 6π MHz, where Im[K(ω)] = Re[Λ(ω)] and Re[K(ω)] = − Im[Λ(ω)]. The values of the other parameters are given by γ10 = γ20 = 6π MHz, γ13 = γ23 = 5 KHz, and Ωc = 30π MHz.
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ˆ 10 ( σ ˆ 20 ) in terms of εˆ 1 ( εˆ 2 ) by solving where ω = 0corresponds to the carrier frequency ω01(ω02) in the interaction picture. The expressions of σ Eqs. (5) are substituted into Eq. (4), then the final integration over frequency domain gives the field at the exit of the cell after the interaction with the EIT medium igN L −Λi ðωÞðL−sÞ ½γi3 −iðΔi + ω−Δ3 Þ F˜ i0 ðs; ωÞ + iΩc F˜ i3 ðs; ωÞ ∫e ds × c 0 ½γi0 −iðΔi + ωÞ½γi3 −iðΔi + ω−Δ3 Þ + jΩc j2
−Λ ðωÞL ε˜ i ðL; ωÞ = e i ε˜i ð0; ωÞ +
2
ð7Þ
γ −iðΔ + ω−Δ Þ
g N
i3 i 3 − iω where Λi ðωÞ = 2c c (i = 1, 2). ½γi0 −iðΔi + ωÞ½γi3 −iðΔi + ω−Δ3 Þ + jΩc j2 Here the displaced detuning Δ0 + ω is considered as the detuning of probe field in the frequency-domain. The absorption and dispersion spectra of the two components of the probe field ε1, 2, which are characterized by Im(K1, 2) and Re(K1, 2) for the case of Δ3 = 0, are exhibited in Fig. 2(A) and (B) with Im[K(ω)]= Re[Λ(ω)], Re[K(ω)]= − Im[Λ(ω)], respectively [16]. Additionally, the group velocities for the corresponding fields are also obtained by expanding Λ1, 2(ω)L about the carrier frequency ω = − Δi, which are given by
Λi ðωÞL = Ki L−
iωL 2 ðω + Δi Þ + ο jωj ði = 1; 2Þ υgi
ð8Þ
and υg1 =
c
h
i ; 2 2 2 2 2 2 2 2 Δ21 jΩc j −γ 13 + jΩc j −γ10 jΩc j −2γ10 γ13 γ10 γ 13 + jΩc j Ng γ 10 γ13 + jΩc j h i 1+ 2 2 2 γ γ + jΩ j 2 γ γ + jΩ j2 + 2γ2 Δ2 2
10
υg2 =
c
13
10
c
13
10
1
c
h
i : 2 2 2 2 2 2 2 2 Δ22 jΩc j −γ 23 + jΩc j −γ20 jΩc j −2γ20 γ23 γ20 γ 23 + jΩc j Ng 2 γ20 γ 23 + jΩc j h i 1+ 2 2 2 γ γ + jΩ j 2 γ γ + jΩ j2 + 2γ2 Δ2 20
c
23
20
c
23
20
2
Here the cube and the higher order terms of the detunings (Δ1, 2) are ignored. For example, the group velocities of υg1 = υg2 = 5990 ms− 1 can be got for the values of the parameters used in Fig. 2. When the probe fields are resonantly applied on the atomic transitions (Δ1 = Δ2 = 0), the first terms on the right-hand side in Eq. (8) reduces to K1 =
Ng 2 γ13 ; 2c γ10 γ 13 + jΩc j2
K2 =
Ng2 γ 23 2c γ20 γ 23 + jΩc j2
which represent the absorption of the probe fields at resonant points. Correspondingly, the group velocities are given by c
υg1 = 1+
Ng2 ðjΩc j2 −γ213 Þ
2ðγ10 γ 13 + jΩc j
Þ
2 2
;
c
υg2 = 1+
:
Ng2 ðjΩc j2 −γ223 Þ
2ðγ20 γ23 + jΩc j
Þ
2 2
The absorption and dispersion curves of the two components εˆ 1;2 of the probe field are plotted in Fig. 3, and they are completely overlapped due to the symmetry of the system. Using the value of parameters in (Fig. 3), we can get υg1 = υg2 = 6200 ms− 1. They are two times faster than the single mode probe light in Λ configuration atomic system [13]. This is due to the fact that the group velocity is determined by the variation of the dispersion with the frequency, which is related to the Rabi frequencies of the coherent fields and the atomic ground population. The atomic ground population for the transition driven by the probe field in the tripod system is equal to one half of that in the Λ configuration atomic system.
80 000 60 000 40 000 20 000 0 20 000 40 000 15
10
5
0
5
10
15
Fig. 3. The dispersion curve Re(K) (dashed line) and absorption curve Im(K)(solid line) of the probe field as functions of ω in the case Δ1 = Δ2 = Δ3 = 0. Other parameters are the same as Fig. 2.
J.L. Ding, B.P. Hou / Optics Communications 284 (2011) 2949–2954
3. Squeezing and entanglement of output two-mode field From above discussions, the components of the two-mode quantum field can be transparent in the tripod-configuration atoms. It is natural to know whether the quantum properties of the field, such as squeezing and entanglement, can be well preserved after propagation through the EIT medium. First we consider the squeezing of the outgoing two-mode field. The Fourier transform for two-mode field quadrature operator is given by h i ˜ θ;α ðz; ωÞ = p1ffiffiffi ε̃ ðz; ωÞe−iθ + ε̃ ðz; ωÞe−iα + ε̃þ ðz; −ωÞeiθ + ε̃þ ðz; −ωÞeiα : U 1 2 out 1 2 2
ð9Þ
2953
where n and A denote the atomic density and the cross section area of the beam, respectively. According
to the1 weak probe approximation, it can be assumed ˆ 22 ≈ =2 and the expectation values of the other atomic ˆ 11 ≈ σ that σ operators are zero. In the following discussions, we adopt the assumption of γ13 = γ23 = γb, γ10 = γ20 = γa which is reasonable based on the atomic symmetry. This assures that the two probe fields propagate with an equal group velocity so that they simultaneously arrive at one place for the entanglement measurement. Then one can get Λ1 ðωÞ = Λ2 ðωÞ = ΛðωÞ =
2 ðγb −iωÞ g N iω − : 2c ðγa −iωÞðγb −iωÞ + jΩc j2 c
π π
0 ˜ 2; 2 Particularly, Ũ 0, out and U out are the standard field amplitude and phase quadrature operators, and they obey the communicate relation π π
0;0 ; 22 U˜ out ; U˜ out = 2i. Then the difference operator between the amplitude
quadratures (θ = 0, α = π) and the sum of the phase quadratures (θ = π2; α = π2) (standard phase quadrature operators in the field of squeezing) of the two modes are defined as [17] i 1 h out 0;π þ þ I˜ þ ðz; ωÞ = U˜ out = pffiffiffi ε̃1 ðz; ωÞ + ε̃1 ðz; −ωÞ−ε̃2 ðz; ωÞ−ε̃2 ðz; −ωÞ ; 2 ð10aÞ i ππ i h out ;2 þ þ = − pffiffiffi ε̃1 ðz; ωÞ−ε̃1 ðz; −ωÞ + ε̃2 ðz; ωÞ−ε̃2 ðz; −ωÞ : I˜− ðz; ωÞ = U˜ 2out 2 ð10bÞ out which obey the communicate relation [Ĩout + (z, ω), Ĩ− (z, ω)] = 0. These expressions are the Einsten–Padolsky–Rosen (EPR)-like entanglement between the two-mode output fields [18]. The entanglement out out out spectra Aout þ ðωÞ for Ĩ+ and A− ðωÞ (squeezing spectrum) for Ĩ− (phase quadrature) are defined as [17]
EE out c D ˜out out out out 2A ðωÞδ ω + ω′ = I ðωÞI˜ ω′ + I˜ ω′ I˜ ðωÞ : L
ð11Þ
j αβ
out
EE ˆ iμ ν D σ ˆ jαβ ˆ iμ ν σ ˆ jαβ −σ −D σ δðt1 −t2 Þδij ;
ð12Þ i
ˆ μ ν denotes the deterministic part of the Heisenberg where D σ
out
Aþ ðωÞ + A− ðωÞ b 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi out Aout þ ðωÞA− ðωÞ b 1:
out
h i in −2ReΛðωÞL ðωÞ = 1− 1−A ðωÞ e :
1.1 1.0 0.9 0.8
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ 13 σ ˆ 11 ; γ1 σ F˜13 ðz1 ; ω1 ÞF˜13 ðz2 ; ω2 Þ = nA
0.7
ð13aÞ
0.6
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ23 σ ˆ 22 ; γ2 σ F˜23 ðz1 ; ω1 Þ F˜23 ðz2 ; ω2 Þ = nA
0.5
ð13bÞ
0.4 2
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ 10 σ ˆ 11 ; γ1 σ F˜10 ðz1 ; ω1 ÞF˜10 ðz2 ; ω2 Þ = nA
ð13dÞ
ð16Þ
The output spectrum Aout ðωÞ of the two-mode squeezed field is plotted in Fig. 4 for an initial phase squeezing of 0.4 (1.0 is the standard quantum limit) for the different values of γb. It is clear from Fig. 3 that the output squeezing spectrum is within the transparency region. By comparing with the case of the single mode squeezed field in a Λ configuration atom [13], the squeezing spectrum of the two mode fields holds wider region. This is due to the fact that the
i
ð13cÞ
ð15Þ
out out Based on the relation Aout ðωÞ, both criteria þ ðωÞ = A− ðωÞ≡A forms imply that the two output modes of the probe light are EPR-like entangled when Aout ðωÞb1, which is consistent with the squeezing criterion [17]. Substitution of Eq. (10) into Eq. (11) leads to the output entanglement or squeezing spectrum in virtue of Eq. (13)
ˆ μ ν without considering the Langevin force term. The equation for σ Dirac delta function in Eq. (12) represents the short memory of the vacuum reservoir modes. By using Eq. (3), the nonzero continuous Langevin correlations in the frequency domain are
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ20 σ ˆ 22 ; γ2 σ F˜20 ðz1 ; ω1 Þ F˜20 ðz2 ; ω2 Þ = nA
ð14Þ
The criterion can also be given by the product form as [17]
A
To solve Eq. (11), it is necessary to calculate the correlation functions of the involved Langevin operators. By using the generalized Einstein relation, the generalized Einstein relation for the atomic operators can be written as [19] E D D i j ˆ ˆ iμ ν σ Fˆ μ ν ðt1 Þ Fˆ αβ ðt2 Þ = D σ
Now, we focus on the output entanglement and squeezing of the two mode probe fields. The two modes of the outgoing field are out entangled when both Ĩout + and Ĩ− are squeezed [20,21]. From the and the input squeezed vacuum probe fields, we can definitions Ĩout ± out out ðωÞ. The entanglement or squeezing get Aout þ ðωÞ = A− ðωÞ≡A spectrum has been connected with entanglement criterion [17]. The “sum” criterion defined by Duan et al. for the two mode fields can be given in virtue of the entanglement spectrum Aout ðωÞ [20]
1
0
1
2
Fig. 4. Output squeezing spectrum of two-mode probe field in tripod configuration with ππ
;
2 2 Sin = 0:4 and Δ1 = Δ2 = Δ3 = 0, and with different dephasing rates of the ground states:
γ13 = γ23 = 50KHz (dotted curve); γ13 = γ23 = 5 KHz (dashed curve); γ13 = γ23 = 50Hz (solid curve). Other parameters are the same as Fig. 2.
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J.L. Ding, B.P. Hou / Optics Communications 284 (2011) 2949–2954
EIT medium with equal velocities. Then our protocol will be more useful in quantum information processing.
1.0 0.8
4. Discussion and conclusion
0.6
In conclusion, the squeezing and the entanglement of a two-mode quantum field which propagates through an optically thick medium of coherently driven four-level atoms in a tripod configuration have been investigated. When the values of the phases in the two-mode π field quadrature operator are set by θ = π 2 and α = 2, the entanglement spectrum is the same as the squeezing spectrum for the input squeezed vacuum probe fields. Although the squeezing and entanglement for the output fields are degraded by the ground state dephasing rates, these quantum properties can be well preserved after propagating through the atomic system for small real dephasing rates. In contrast to the schemes in which one-mode field propagates through a three-level atomic system in a Λ configuration, and the other mode field travels in vacuum or optical fiber, the present scheme is more operational in experiment. This allows some applications in the quantum information, such as quantum computation, dense coding, and teleportation.
0.4 0.2 0.0
0
50 000
100 000
150 000
200 000
250 000
300 000
Fig. 5. Output entanglement spectrum of two-mode probe field in tripod configuration as functions of γb(= γ13 = γ23) with different initial conditions: Ain = 0:4 (dashed curve); Ain = 0:6 (dotted curve); Ain = 0:8 (solid curve). Other parameters are the same as Fig. 3.
squeezing spectrum width is related to the intensities of the coherent fields driving the atoms. The coherent fields including two probe and one control fields in the tripod scheme are stronger than those in the Λ configuration atom. At the same time, we investigate the effect of the ground dephasing rates on the entanglement and squeezing of the output two-mode squeezed field in Fig. 4. It is shown that the entanglement or squeezing of output fields can be almost well preserved after passing through an optically thick medium for small ground state dephasing rates γ13(γ23), although the ground state dephasing degrades these quantum properties. The preservation of the entanglement and squeezing is more perfect at the low dephasing rates than that in Λ configuration [13,14]. To investigate the effects of the ground state dephasing rates and the initial values Ain on these quantum properties in detail, we plot the output spectrum Aout ð0Þ at ω = 0 as a function of the ground state dephasing rates γb (= γ13 = γ23) with different initial values in Fig. 5. It is clear that the output spectrum gradually tends to 1.0 (the standard quantum limit) as the ground state dephasing rates increase. That means the output two-mode squeezed field will lose the quantum properties such as entanglement and squeezing when the ground state dephasing rates γ13(γ23) increase up to about 3 KHz in the present system. Fig. 6 depicts how the output spectrum Aout ð0Þ depends on the initial values Ain with different ground state dephasing rates γb. It is shown that the output spectrum varies linearly with the initial values and its slop is determined by the ground state dephasing rates. The investigation of the entanglement for the two mode fields in our case is more convenient due to their propagations through the same
1.0 0.8 0.6 0.4 0.2 0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 6. Output entanglement spectrum of two-mode probe field in tripod configuration as functions of Ain with different dephasing rates of the ground states: γ13 = γ23 = 50 KHz (dotted curve); γ13 = γ23 = 5 KHz (dashed curve); γ13 = γ23 = 50 Hz (solid curve). Other parameters are the same as Fig. 3.
Acknowledgements This work was supported in part by the National Natural Science Foundations of China under Grant No. 10647007, the Key Project of Chinese Ministry of Education under Grant No. 210192, and by the Young Foundation of Sichuan Province, China under Grant No. 09ZQ026-008, and by the Education Foundation of Sichuan Province, China under Grant No. 10ZA001. References [1] S.E. Harris, Phys. Today 50 (1997) 36. [2] L.V. Hau, S.E. Harris, Z. Dutton, et al., Nature 397 (1999) 594; C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Nature 409 (2001) 490. [3] H. Schmidt, A. Imamoglu, Opt. Lett. 21 (1996) 1936; L. Deng, M. Kozuma, E.W. Hagley, M.G. Payne, Phys. Rev. Lett. 88 (2002) 143902; Y. Wu, J. Saldana, Y.F. Zhu, Phys. Rev. A 67 (2003) 013811; Y. Wu, L. Wen, Y. Zhu, Opt. Lett. 28 (2003) 631. [4] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. [5] M. Fleischhauer, M.D. Lukin, Phys. Rev. Lett. 84 (2000) 5094; M. Fleischhauer, M.D. Lukin, Phys. Rev. A 65 (2002) 022314. [6] A. Boozer, A. Boca, C. Chou, et al., Nature 423 (2003) 731; S.Z. Wei, Y.X. Dong, H.B. Wang, X.D. Zhang, Phys. Rev. A 81 (2010) 053830. [7] D. Felinto, S. Polyakov, S. van Enk, et al., Nature 438 (2005) 828. [8] S.G. Clark, A.S. Parkins, Phys. Rev. Lett. 90 (2003) 047905; B. Kraus, J.I. Cirac, Phys. Rev. Lett. 92 (2004) 013602. [9] O. Glockl, et al., J. Opt. B 5 (2003) S492. [10] V. Josse, et al., Phys. Rev. Lett. 92 (2004) 123601. [11] R. Guzman, J.C. Retamal, E. Solano, N. Zagury, Phys. Rev. Lett. 96 (2006) 010502. [12] D. Akamatsu, K. Akiba, M. Kozuma, Phys. Rev. Lett. 92 (2004) 203602. [13] A. Peng, M. Johnsson, W.P. Bowen, P.K. Lam, H.-A. Bachor, J.J. Hope, Phys. Rev. A 71 (2005) 033809. [14] S. Thanvanthri, J. Wen, M.H,. Rubin, Phys. Rev. A 72 (2005) 023822. [15] J.L. Ding, B.P. Hou, S.J. Wang, J. Phys. B 43 (2010) 225502. [16] C. Hang, G. Huang, Phys. Lett. A 372 (2008) 3129. [17] D. Vitali, G. Morigi, J. Eschner, Phys. Rev. A 74 (2006) 053814; Ling Zhou, Mu. Qing-Xia, Zhong-Ju Liu, Phys. Lett. A 373 (2009) 2017. [18] A. Einstein, B. Podolsky, R. Rosen, Phys. Rev. 47 (1935) 777. [19] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, London, 1997. [20] L.M. Duan, G. Giedke, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 84 (2000) 2722. [21] W.P. Bowen, R. Schnabel, P.K. Lam, T.C. Ralph, Phys. Rev. Lett. 90 (2003) 043601.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Squeezing and entanglement of a two-mode field in a four-level tripod atomic system J.L. Ding a, B.P. Hou a,b,⁎ a b
School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Chengdu 610068, China
a r t i c l e
i n f o
Article history: Received 26 December 2010 Received in revised form 17 February 2011 Accepted 19 February 2011 Available online 11 March 2011 Keywords: Squeezing Entanglement Electromagnetically induced transparency
a b s t r a c t The interaction of a collection of N four-level tripod configuration atoms with two orthogonally polarized probe fields is investigated. Under the condition of electromagnetically induced transparency (EIT), we calculate the squeezing and entanglement spectra of the output probe fields. By analyzing the output spectrum, we find that the squeezing and entanglement of the probe fields can be well-preserved after passing through the optically thick medium. Additionally, the effects of the ground state dephasing rates of the atoms on the entanglement and squeezing of the output two-mode squeezed fields are investigated. It is shown that the dephasing rates will degrade the entanglement and squeezing, and these quantum properties can be lost when the dephasing rates increase up to a certain value. This will be useful in the quantum computation and quantum communication. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The phenomenon of electromagnetically induced transparency (EIT) in atomic media is a quantum interference effect that allows for the transmission of light with little absorption through an otherwise opaque atomic medium [1]. Medium exhibiting EIT has some remarkable properties, such as ultra slow propagation of a light through atomic cloud [2] and nonlinear optical processes [3], lasing without inversion [4]. Recently, the potential application of EIT has been reported in quantum information science. For example, it can be used to store and release the quantum information in an atomic ensemble [5], to generate the correlated photon pairs [6] and even the entanglement of remote atomic pairs generation [7]. These form the building blocks of the quantum communication and the quantum computation. Squeezing and entanglement are two representative quantities characterizing the amount of quantum information of a light field. Conventionally, the continuous variable entanglement can be easily studied using two-mode squeezing. Also, the two-mode squeezed states can be used to entangle distant atoms as well expected to lead to efficient distribution of entanglement and implementation of quantum channels by improving the yet low squeeze parameters [8]. Two-mode (polarization) squeezing has already been realized by
⁎ Corresponding author at: School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China. Tel.: +86 02884480791; fax: +86 02884480787. E-mail address: [email protected] (B.P. Hou). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.02.056
exploiting the Kerr non-linearity of a glass fiber [9] and with cold atoms in a high finesse optical cavity [10]. Besides, a method of generating unitarily two-mode field squeezing in an optical cavity with an atomic cloud has been proposed in Ref. [11]. Recently, the experimental investigation has shown that based on the EIT technique the quadrature squeezing was maintained within the transparency window when the squeezed light passed through an optically dense rubidium gas cell [12]. Additionally, the works in Ref. [13] showed that the squeezing of a single mode squeezed light can be well-preserved throughout the EIT atomic system in a three-level Λ configuration. The entanglement for the field throughout the EIT medium was also been studied by considering another field traveling in vacuum. Another work examined the preservation of the squeezing and the continuous-variable entanglement of a two-mode squeezed light when one of the modes transmitted through an EIT medium while the other mode propagated through an optical fiber [14]. It was found that loss of squeezing occurs when the mismatch in the transmission of the two modes is greater than 40%, while near-ideal squeezing is preserved when the transmission is equal. Most recently, we have investigated the squeezing survival and transfer of one probe field in the single and double EIT [15]. It has been pointed out in Ref. [14] that the squeezing or entanglement of the two mode fields will be lost due to their mismatching transmission. Taking into account the delay in the EIT medium for one field corresponding to the other field traveling in vacuum or in an optical fiber, as were proposed in Refs. [13] and [14], it cannot be convenient to realize the matching transmission which is needed to quantify the entanglement between these fields. In this paper, we consider two squeezed fields as probe fields to inject into a
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medium of four-level tripod configuration atoms driven by one control field. The two probe fields will propagate at the same slow velocities by using the atomic inherent symmetry and the symmetric action of the two probe fields. This can overcome the mismatching transmission in the investigated schemes. We examine the preservation of the continuous variable squeezing and entanglement of the two modes of the squeezed light after their simultaneous propagation through the EIT medium. By calculating the output squeezing spectrum and entanglement spectrum of the two-mode squeezed light, it is shown that the preservation of two-mode squeezing and entanglement can be also well-preserved for small dephasing rates.
Due to its perfectly preservation and its convenience, this will offer some promising perspectives for the realization of quantum communication and the quantum computation. The paper is arranged as follows. In Section 2, the model of the interaction of the four-level tripod-scheme atomic system with the two-mode squeezed field and a control field is briefly described, and its equation of motion for the two components of the weak probe field is shown. In Section 3, the squeezing and entanglement of the two-mode output squeezed light are discussed. In Section 4 the discussion and the summary of the main results of this work are made.
2. Model and equation of motion Consider a quasi-one-dimensional mode consisting of four-level atoms in the tripod configuration with a single upper state |0〉 and three lower states |1〉, |2〉 and |3〉 interacting with three optical fields, as shown in Fig. 1. We assume that all the atoms are equally located in the states |1〉 and |2〉 by optical pumping. A linearly polarized weak quantum field ε with the carrier frequency ωs has two circularly left and right-polarized components ε1, 2 which couple the atomic transitions |1〉 ↔ |0〉 and |2〉 ↔ |0〉, respectively. The part of the positive frequency of the electric field related to the weak quantum field εˆ 1;2 ðz; t Þ is given by ð+Þ Eˆ i ðz; t Þ =
sffiffiffiffiffiffiffiffiffiffiffi h ω i ℏωs εˆi ðz; t Þ exp i s ðz−ct Þ ði = 1; 2Þ 2ε0 V c
ð1Þ
where quantization volume of the electromagnetic field V is assumed to be the interaction volume. The detunings from atomic resonance for the two mode fields are defined by Δ1 = ωs − ω01 and Δ2 = ωs − ω02. The transition |3〉 ↔ |0〉 is driven by a classical coherent control field Ωc with the frequency ωc, and the corresponding detuning is denoted by Δ3 = ωc − ω03, and ωμν = (Eμ − Eν)/ℏ the atomic resonance frequency of the transition |μ〉 ↔ |ν〉. The effect of Doppler shift is minimized by using co-propagating configuration for the coherent fields in the present model. If the slowly-varying amplitude εˆi ðz; t Þ does not change much in a length internal Δz containing Nz(≫ 1) atoms, we can introduce the locallyaveraged slowly-varying atomic operators
σˆ μ ν ðz; t Þ =
1 NZ j i ω = c ðz−ct Þ ∑ σˆ ðt Þe ð μ ν Þ NZ j = 1 μ ν
ð2Þ
j
where σˆ μ ν = j μ j ðt Þ〉〈ν j ðt Þj is the flip operator for the jth atom. The interaction Hamiltonian of the system can be written as n o ˆ = ∫ Nℏ Δ σˆ + Δ σˆ + Δ σˆ −gσˆ ðz; t Þεˆ ðz; t Þ +g σˆ ðz; t Þεˆ ðz; t Þ +Ω ðz; t Þ σˆ ðz; t Þ +H:c dz Η I 1 11 2 22 3 33 01 1 02 2 c 03 L
ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g = g1 = g 2 with gi = }0i ℏωs = 2ε 0 V (i = 1,2) being the atom-field coupling constant, and }01 ð}02 Þ is the atomic dipole moment for the |1〉 ↔ |0〉 (|2〉 ↔ |0〉) transition. The two coupling constants are the same due to the atomic inherent symmetry and the symmetric actions of two component fields, N is the number of atoms and L is the interaction length in the propagation direction of the quantum field.
Fig. 1. Schematic diagram of a four-level system in tripod configuration.
J.L. Ding, B.P. Hou / Optics Communications 284 (2011) 2949–2954
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The Maxwell's propagation equations for the quantum fields in the slowly-varying envelope approximation are given by
∂ ∂ +c εˆ ðz; t Þ = igN σˆ 10 ðz; t Þ; ∂t ∂z 1
ð4aÞ
∂ ∂ +c εˆ ðz; t Þ = igNσˆ 20 ðz; t Þ: ∂t ∂z 2
ð4bÞ
The atomic Heisenberg–Langevin equations can be obtained by following our previous work [15]. Now, we assume that the Rabi frequencies of the quantum fields are much smaller than the classical control field, and the number density of photons in the quantum fields is much less than ˆ 11 i ≈〈 σ ˆ 22 i ≈1 =2 are that of atoms. In such a case, the atomic equations can be treated perturbatively in εˆ 1;2 ðz; t Þ. In the zeroth order only 〈σ different from zero. Therefore, one finds the atomic motion equations in the first order of εˆ 1;2 ðz; t Þ i ˆ 10 + g εˆ 1 + iΩc σ ˆ 13 + Fˆ 10 ; ˆ˙ 10 = −ðγ10 −iΔ1 Þ σ σ 2
ð5aÞ
i ˆ 20 + g εˆ 2 + iΩc σ ˆ 23 + Fˆ 20 ; ˆ˙ 20 = −ðγ20 −iΔ2 Þ σ σ 2
ð5bÞ
ˆ 13 + iΩ⁎ ˆ 10 + Fˆ 13 ; ˆ˙ 13 = −ðγ13 −iðΔ1 −Δ3 ÞÞ σ σ c σ
ð5cÞ
ˆ 23 + iΩ⁎ ˆ 20 + Fˆ 23 : ˆ˙ 23 = −ðγ23 −iðΔ2 −Δ3 ÞÞ σ σ c σ
ð5dÞ
where γμ is the decay rate of the atomic dipole operators, and Fˆ μ ν is the associated Langevin noise operators which describe the effect of spontaneous decay caused by the coupling of atoms to all the vacuum field modes. The decay rates γ12, γ13 and γ23 of the ground state coherence are modeling collisions or accounting for the transit of the atoms outside the interaction area with the light beams. In order to solve these equations, we transform them into Fourier space in the frequency domain according to 1 ∞ ˆ iωt ðz; t Þe dt G˜ ðz; ωÞ = pffiffiffiffiffiffi ∫−∞ G 2π
ð6Þ
A 80 000
60 000
40 000
20 000
0 15
10
5
0
5
10
15
15
10
5
0
5
10
15
B 40 000
20 000
0
20 000
40 000
Fig. 2. The absorption curve Im(K) (A) and dispersion curve Re(K) (B) of the probe field ε1 (dashed line) and ε2 (solid line) as functions of ω in the case Δ3 = 0 ; Δ2 = − Δ1 = 6π MHz, where Im[K(ω)] = Re[Λ(ω)] and Re[K(ω)] = − Im[Λ(ω)]. The values of the other parameters are given by γ10 = γ20 = 6π MHz, γ13 = γ23 = 5 KHz, and Ωc = 30π MHz.
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ˆ 10 ( σ ˆ 20 ) in terms of εˆ 1 ( εˆ 2 ) by solving where ω = 0corresponds to the carrier frequency ω01(ω02) in the interaction picture. The expressions of σ Eqs. (5) are substituted into Eq. (4), then the final integration over frequency domain gives the field at the exit of the cell after the interaction with the EIT medium igN L −Λi ðωÞðL−sÞ ½γi3 −iðΔi + ω−Δ3 Þ F˜ i0 ðs; ωÞ + iΩc F˜ i3 ðs; ωÞ ∫e ds × c 0 ½γi0 −iðΔi + ωÞ½γi3 −iðΔi + ω−Δ3 Þ + jΩc j2
−Λ ðωÞL ε˜ i ðL; ωÞ = e i ε˜i ð0; ωÞ +
2
ð7Þ
γ −iðΔ + ω−Δ Þ
g N
i3 i 3 − iω where Λi ðωÞ = 2c c (i = 1, 2). ½γi0 −iðΔi + ωÞ½γi3 −iðΔi + ω−Δ3 Þ + jΩc j2 Here the displaced detuning Δ0 + ω is considered as the detuning of probe field in the frequency-domain. The absorption and dispersion spectra of the two components of the probe field ε1, 2, which are characterized by Im(K1, 2) and Re(K1, 2) for the case of Δ3 = 0, are exhibited in Fig. 2(A) and (B) with Im[K(ω)]= Re[Λ(ω)], Re[K(ω)]= − Im[Λ(ω)], respectively [16]. Additionally, the group velocities for the corresponding fields are also obtained by expanding Λ1, 2(ω)L about the carrier frequency ω = − Δi, which are given by
Λi ðωÞL = Ki L−
iωL 2 ðω + Δi Þ + ο jωj ði = 1; 2Þ υgi
ð8Þ
and υg1 =
c
h
i ; 2 2 2 2 2 2 2 2 Δ21 jΩc j −γ 13 + jΩc j −γ10 jΩc j −2γ10 γ13 γ10 γ 13 + jΩc j Ng γ 10 γ13 + jΩc j h i 1+ 2 2 2 γ γ + jΩ j 2 γ γ + jΩ j2 + 2γ2 Δ2 2
10
υg2 =
c
13
10
c
13
10
1
c
h
i : 2 2 2 2 2 2 2 2 Δ22 jΩc j −γ 23 + jΩc j −γ20 jΩc j −2γ20 γ23 γ20 γ 23 + jΩc j Ng 2 γ20 γ 23 + jΩc j h i 1+ 2 2 2 γ γ + jΩ j 2 γ γ + jΩ j2 + 2γ2 Δ2 20
c
23
20
c
23
20
2
Here the cube and the higher order terms of the detunings (Δ1, 2) are ignored. For example, the group velocities of υg1 = υg2 = 5990 ms− 1 can be got for the values of the parameters used in Fig. 2. When the probe fields are resonantly applied on the atomic transitions (Δ1 = Δ2 = 0), the first terms on the right-hand side in Eq. (8) reduces to K1 =
Ng 2 γ13 ; 2c γ10 γ 13 + jΩc j2
K2 =
Ng2 γ 23 2c γ20 γ 23 + jΩc j2
which represent the absorption of the probe fields at resonant points. Correspondingly, the group velocities are given by c
υg1 = 1+
Ng2 ðjΩc j2 −γ213 Þ
2ðγ10 γ 13 + jΩc j
Þ
2 2
;
c
υg2 = 1+
:
Ng2 ðjΩc j2 −γ223 Þ
2ðγ20 γ23 + jΩc j
Þ
2 2
The absorption and dispersion curves of the two components εˆ 1;2 of the probe field are plotted in Fig. 3, and they are completely overlapped due to the symmetry of the system. Using the value of parameters in (Fig. 3), we can get υg1 = υg2 = 6200 ms− 1. They are two times faster than the single mode probe light in Λ configuration atomic system [13]. This is due to the fact that the group velocity is determined by the variation of the dispersion with the frequency, which is related to the Rabi frequencies of the coherent fields and the atomic ground population. The atomic ground population for the transition driven by the probe field in the tripod system is equal to one half of that in the Λ configuration atomic system.
80 000 60 000 40 000 20 000 0 20 000 40 000 15
10
5
0
5
10
15
Fig. 3. The dispersion curve Re(K) (dashed line) and absorption curve Im(K)(solid line) of the probe field as functions of ω in the case Δ1 = Δ2 = Δ3 = 0. Other parameters are the same as Fig. 2.
J.L. Ding, B.P. Hou / Optics Communications 284 (2011) 2949–2954
3. Squeezing and entanglement of output two-mode field From above discussions, the components of the two-mode quantum field can be transparent in the tripod-configuration atoms. It is natural to know whether the quantum properties of the field, such as squeezing and entanglement, can be well preserved after propagation through the EIT medium. First we consider the squeezing of the outgoing two-mode field. The Fourier transform for two-mode field quadrature operator is given by h i ˜ θ;α ðz; ωÞ = p1ffiffiffi ε̃ ðz; ωÞe−iθ + ε̃ ðz; ωÞe−iα + ε̃þ ðz; −ωÞeiθ + ε̃þ ðz; −ωÞeiα : U 1 2 out 1 2 2
ð9Þ
2953
where n and A denote the atomic density and the cross section area of the beam, respectively. According
to the1 weak probe approximation, it can be assumed ˆ 22 ≈ =2 and the expectation values of the other atomic ˆ 11 ≈ σ that σ operators are zero. In the following discussions, we adopt the assumption of γ13 = γ23 = γb, γ10 = γ20 = γa which is reasonable based on the atomic symmetry. This assures that the two probe fields propagate with an equal group velocity so that they simultaneously arrive at one place for the entanglement measurement. Then one can get Λ1 ðωÞ = Λ2 ðωÞ = ΛðωÞ =
2 ðγb −iωÞ g N iω − : 2c ðγa −iωÞðγb −iωÞ + jΩc j2 c
π π
0 ˜ 2; 2 Particularly, Ũ 0, out and U out are the standard field amplitude and phase quadrature operators, and they obey the communicate relation π π
0;0 ; 22 U˜ out ; U˜ out = 2i. Then the difference operator between the amplitude
quadratures (θ = 0, α = π) and the sum of the phase quadratures (θ = π2; α = π2) (standard phase quadrature operators in the field of squeezing) of the two modes are defined as [17] i 1 h out 0;π þ þ I˜ þ ðz; ωÞ = U˜ out = pffiffiffi ε̃1 ðz; ωÞ + ε̃1 ðz; −ωÞ−ε̃2 ðz; ωÞ−ε̃2 ðz; −ωÞ ; 2 ð10aÞ i ππ i h out ;2 þ þ = − pffiffiffi ε̃1 ðz; ωÞ−ε̃1 ðz; −ωÞ + ε̃2 ðz; ωÞ−ε̃2 ðz; −ωÞ : I˜− ðz; ωÞ = U˜ 2out 2 ð10bÞ out which obey the communicate relation [Ĩout + (z, ω), Ĩ− (z, ω)] = 0. These expressions are the Einsten–Padolsky–Rosen (EPR)-like entanglement between the two-mode output fields [18]. The entanglement out out out spectra Aout þ ðωÞ for Ĩ+ and A− ðωÞ (squeezing spectrum) for Ĩ− (phase quadrature) are defined as [17]
EE out c D ˜out out out out 2A ðωÞδ ω + ω′ = I ðωÞI˜ ω′ + I˜ ω′ I˜ ðωÞ : L
ð11Þ
j αβ
out
EE ˆ iμ ν D σ ˆ jαβ ˆ iμ ν σ ˆ jαβ −σ −D σ δðt1 −t2 Þδij ;
ð12Þ i
ˆ μ ν denotes the deterministic part of the Heisenberg where D σ
out
Aþ ðωÞ + A− ðωÞ b 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi out Aout þ ðωÞA− ðωÞ b 1:
out
h i in −2ReΛðωÞL ðωÞ = 1− 1−A ðωÞ e :
1.1 1.0 0.9 0.8
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ 13 σ ˆ 11 ; γ1 σ F˜13 ðz1 ; ω1 ÞF˜13 ðz2 ; ω2 Þ = nA
0.7
ð13aÞ
0.6
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ23 σ ˆ 22 ; γ2 σ F˜23 ðz1 ; ω1 Þ F˜23 ðz2 ; ω2 Þ = nA
0.5
ð13bÞ
0.4 2
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ 10 σ ˆ 11 ; γ1 σ F˜10 ðz1 ; ω1 ÞF˜10 ðz2 ; ω2 Þ = nA
ð13dÞ
ð16Þ
The output spectrum Aout ðωÞ of the two-mode squeezed field is plotted in Fig. 4 for an initial phase squeezing of 0.4 (1.0 is the standard quantum limit) for the different values of γb. It is clear from Fig. 3 that the output squeezing spectrum is within the transparency region. By comparing with the case of the single mode squeezed field in a Λ configuration atom [13], the squeezing spectrum of the two mode fields holds wider region. This is due to the fact that the
i
ð13cÞ
ð15Þ
out out Based on the relation Aout ðωÞ, both criteria þ ðωÞ = A− ðωÞ≡A forms imply that the two output modes of the probe light are EPR-like entangled when Aout ðωÞb1, which is consistent with the squeezing criterion [17]. Substitution of Eq. (10) into Eq. (11) leads to the output entanglement or squeezing spectrum in virtue of Eq. (13)
ˆ μ ν without considering the Langevin force term. The equation for σ Dirac delta function in Eq. (12) represents the short memory of the vacuum reservoir modes. By using Eq. (3), the nonzero continuous Langevin correlations in the frequency domain are
D EE
δðz1 −z2 Þδðω1 + ω2 Þ þ ˆ 00 + 2γ20 σ ˆ 22 ; γ2 σ F˜20 ðz1 ; ω1 Þ F˜20 ðz2 ; ω2 Þ = nA
ð14Þ
The criterion can also be given by the product form as [17]
A
To solve Eq. (11), it is necessary to calculate the correlation functions of the involved Langevin operators. By using the generalized Einstein relation, the generalized Einstein relation for the atomic operators can be written as [19] E D D i j ˆ ˆ iμ ν σ Fˆ μ ν ðt1 Þ Fˆ αβ ðt2 Þ = D σ
Now, we focus on the output entanglement and squeezing of the two mode probe fields. The two modes of the outgoing field are out entangled when both Ĩout + and Ĩ− are squeezed [20,21]. From the and the input squeezed vacuum probe fields, we can definitions Ĩout ± out out ðωÞ. The entanglement or squeezing get Aout þ ðωÞ = A− ðωÞ≡A spectrum has been connected with entanglement criterion [17]. The “sum” criterion defined by Duan et al. for the two mode fields can be given in virtue of the entanglement spectrum Aout ðωÞ [20]
1
0
1
2
Fig. 4. Output squeezing spectrum of two-mode probe field in tripod configuration with ππ
;
2 2 Sin = 0:4 and Δ1 = Δ2 = Δ3 = 0, and with different dephasing rates of the ground states:
γ13 = γ23 = 50KHz (dotted curve); γ13 = γ23 = 5 KHz (dashed curve); γ13 = γ23 = 50Hz (solid curve). Other parameters are the same as Fig. 2.
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J.L. Ding, B.P. Hou / Optics Communications 284 (2011) 2949–2954
EIT medium with equal velocities. Then our protocol will be more useful in quantum information processing.
1.0 0.8
4. Discussion and conclusion
0.6
In conclusion, the squeezing and the entanglement of a two-mode quantum field which propagates through an optically thick medium of coherently driven four-level atoms in a tripod configuration have been investigated. When the values of the phases in the two-mode π field quadrature operator are set by θ = π 2 and α = 2, the entanglement spectrum is the same as the squeezing spectrum for the input squeezed vacuum probe fields. Although the squeezing and entanglement for the output fields are degraded by the ground state dephasing rates, these quantum properties can be well preserved after propagating through the atomic system for small real dephasing rates. In contrast to the schemes in which one-mode field propagates through a three-level atomic system in a Λ configuration, and the other mode field travels in vacuum or optical fiber, the present scheme is more operational in experiment. This allows some applications in the quantum information, such as quantum computation, dense coding, and teleportation.
0.4 0.2 0.0
0
50 000
100 000
150 000
200 000
250 000
300 000
Fig. 5. Output entanglement spectrum of two-mode probe field in tripod configuration as functions of γb(= γ13 = γ23) with different initial conditions: Ain = 0:4 (dashed curve); Ain = 0:6 (dotted curve); Ain = 0:8 (solid curve). Other parameters are the same as Fig. 3.
squeezing spectrum width is related to the intensities of the coherent fields driving the atoms. The coherent fields including two probe and one control fields in the tripod scheme are stronger than those in the Λ configuration atom. At the same time, we investigate the effect of the ground dephasing rates on the entanglement and squeezing of the output two-mode squeezed field in Fig. 4. It is shown that the entanglement or squeezing of output fields can be almost well preserved after passing through an optically thick medium for small ground state dephasing rates γ13(γ23), although the ground state dephasing degrades these quantum properties. The preservation of the entanglement and squeezing is more perfect at the low dephasing rates than that in Λ configuration [13,14]. To investigate the effects of the ground state dephasing rates and the initial values Ain on these quantum properties in detail, we plot the output spectrum Aout ð0Þ at ω = 0 as a function of the ground state dephasing rates γb (= γ13 = γ23) with different initial values in Fig. 5. It is clear that the output spectrum gradually tends to 1.0 (the standard quantum limit) as the ground state dephasing rates increase. That means the output two-mode squeezed field will lose the quantum properties such as entanglement and squeezing when the ground state dephasing rates γ13(γ23) increase up to about 3 KHz in the present system. Fig. 6 depicts how the output spectrum Aout ð0Þ depends on the initial values Ain with different ground state dephasing rates γb. It is shown that the output spectrum varies linearly with the initial values and its slop is determined by the ground state dephasing rates. The investigation of the entanglement for the two mode fields in our case is more convenient due to their propagations through the same
1.0 0.8 0.6 0.4 0.2 0.0
0.0
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Fig. 6. Output entanglement spectrum of two-mode probe field in tripod configuration as functions of Ain with different dephasing rates of the ground states: γ13 = γ23 = 50 KHz (dotted curve); γ13 = γ23 = 5 KHz (dashed curve); γ13 = γ23 = 50 Hz (solid curve). Other parameters are the same as Fig. 3.
Acknowledgements This work was supported in part by the National Natural Science Foundations of China under Grant No. 10647007, the Key Project of Chinese Ministry of Education under Grant No. 210192, and by the Young Foundation of Sichuan Province, China under Grant No. 09ZQ026-008, and by the Education Foundation of Sichuan Province, China under Grant No. 10ZA001. References [1] S.E. Harris, Phys. Today 50 (1997) 36. [2] L.V. Hau, S.E. Harris, Z. Dutton, et al., Nature 397 (1999) 594; C. Liu, Z. Dutton, C.H. Behroozi, L.V. Hau, Nature 409 (2001) 490. [3] H. Schmidt, A. Imamoglu, Opt. Lett. 21 (1996) 1936; L. Deng, M. Kozuma, E.W. Hagley, M.G. Payne, Phys. Rev. Lett. 88 (2002) 143902; Y. Wu, J. Saldana, Y.F. Zhu, Phys. Rev. A 67 (2003) 013811; Y. Wu, L. Wen, Y. Zhu, Opt. Lett. 28 (2003) 631. [4] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. [5] M. Fleischhauer, M.D. Lukin, Phys. Rev. Lett. 84 (2000) 5094; M. Fleischhauer, M.D. Lukin, Phys. Rev. A 65 (2002) 022314. [6] A. Boozer, A. Boca, C. Chou, et al., Nature 423 (2003) 731; S.Z. Wei, Y.X. Dong, H.B. Wang, X.D. Zhang, Phys. Rev. A 81 (2010) 053830. [7] D. Felinto, S. Polyakov, S. van Enk, et al., Nature 438 (2005) 828. [8] S.G. Clark, A.S. Parkins, Phys. Rev. Lett. 90 (2003) 047905; B. Kraus, J.I. Cirac, Phys. Rev. Lett. 92 (2004) 013602. [9] O. Glockl, et al., J. Opt. B 5 (2003) S492. [10] V. Josse, et al., Phys. Rev. Lett. 92 (2004) 123601. [11] R. Guzman, J.C. Retamal, E. Solano, N. Zagury, Phys. Rev. Lett. 96 (2006) 010502. [12] D. Akamatsu, K. Akiba, M. Kozuma, Phys. Rev. Lett. 92 (2004) 203602. [13] A. Peng, M. Johnsson, W.P. Bowen, P.K. Lam, H.-A. Bachor, J.J. Hope, Phys. Rev. A 71 (2005) 033809. [14] S. Thanvanthri, J. Wen, M.H,. Rubin, Phys. Rev. A 72 (2005) 023822. [15] J.L. Ding, B.P. Hou, S.J. Wang, J. Phys. B 43 (2010) 225502. [16] C. Hang, G. Huang, Phys. Lett. A 372 (2008) 3129. [17] D. Vitali, G. Morigi, J. Eschner, Phys. Rev. A 74 (2006) 053814; Ling Zhou, Mu. Qing-Xia, Zhong-Ju Liu, Phys. Lett. A 373 (2009) 2017. [18] A. Einstein, B. Podolsky, R. Rosen, Phys. Rev. 47 (1935) 777. [19] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, London, 1997. [20] L.M. Duan, G. Giedke, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 84 (2000) 2722. [21] W.P. Bowen, R. Schnabel, P.K. Lam, T.C. Ralph, Phys. Rev. Lett. 90 (2003) 043601.