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Phase-dependent squeezing and entanglement in a cavity QED J.J. Mu a , B.P. Hou a,b,∗ a b

College 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 2 September 2012 Accepted 22 January 2013

PACS: 42.50.Gy 42.50.Hz 42.50.−p Keywords: Squeezing Entanglement Cavity QED Relative phase

a b s t r a c t The output squeezed spectrum of the output two modes from a cavity QED system which is embedded by a four-level atom driven by two classical ﬁelds are calculated. The effects of the cavity decay rates, relative phase between two classical ﬁelds, and effective coupling constants between the atom and the cavity on the entanglement and squeezing of the two cavity modes are investigated. It is shown that the depth of the squeezing or entanglement spectrum is insensitive to the cavity decay rates while its width is dependent on the cavity decay rates and the coupling constants. In particular, the squeezing or entanglement spectrum including their maximal values and widths is remarkably affected by the relative phase between the classical ﬁelds. It is found that the output spectrum varies with the relative phase in a cosine-like behavior with a period of 2. The results can indicate how to suitably adjust the relative phase for acquiring maximal squeezing or perfect entanglement for the two cavity ﬁelds. This will be useful in quantum communication and computation. © 2013 Elsevier GmbH. All rights reserved.

1. Introduction Entanglement and squeezing are not only the most interesting properties in quantum mechanical but also the important resources in quantum technology. Since the EPR paradox was proposed by Einstein, Podolsky and Rosen proposed [1] and the concept of squeezed light state was introduced by Walls [2], much work about entanglement and squeezing has been investigated deeply and widely, including preparation and application. As we known that the entanglement of continuous variable is widely used in quantum information and quantum computation [3–5], such as quantum dense coding, quantum teleportation and quantum swapping. Those applications have been successful realized in experiments. Highly efﬁcient quantum dense coding for continuous variables was experimentally realized by exploiting a bright Einstein–Podolsky–Rosen (EPR) beam [6]. The experiment of quantum teleportation of a polarization state with a complete bell state measurement is reported [7]. Subsequently, Jia et al. [8] demonstrated the unconditional entanglement swapping by utilizing squeezed entanglement state generated by two nondegenerate optical parametric ampliﬁers in experiment.

∗ Corresponding author. College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China. E-mail address: [email protected] (B.P. Hou). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.01.107

The preparation and manipulation of entanglement and squeezing state have also attracted a great interest due to the wide applications in quantum information and quantum computation. In recent years, many theoretical models based on the interaction between atomic and cavity modes are used to prepare the entanglement and squeezed state [9–11]. That is because cavity quantum electrodynamics (QED) system is one of the effective carriers processing the quantum information. Solano et al. proposed a method to generate a great number of multipartite entangled states by exploiting the interaction of a system of N two-level atoms in a cavity of high quality factor with an additional strong classical driving ﬁeld [12]. Subsequently, Zhou et al. [13] proposed a protocol to generate a macroscopic entangled state by utilizing a single cascade three-level atom in a cavity QED system which can be applied in quantum secure communication. Conventionally, utilizing cavity QED system to prepare entanglement squeezed states needs highQ factor and strong coupling constant. Recently, Mu et al. [14] have investigated the output squeezing and entanglement generation from a low-Q cavity with a single atom. It is well known that the physical properties in an atomic system with a closed-loop conﬁguration are strongly dependent on the relative phase between the ﬁelds driving the atom. This can provide a new method to reveal more plentiful physical phenomenon. For example, a wide variety of spectral behavior of spontaneous emission can be realized by controlling the phase and the amplitude of the driving ﬁelds [15,16]. The phase-dependent electromagnetically induced transparency in closed-loop schemes of interaction

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H1 = +

g12 a+ a 1 1 1

˝22

+

2

|g g| +

g22 a+ a 2 2 2

+

˝12

|e e|

1

1 eiϕ1 a+ e−iı1 t + 2 e−iϕ2 a2 eiı2 t |g e| + H.c. 1

(2)

where i = (gi ˝i /i ) (i = 1, 2). We also assume that the classical ﬁelds is strong, i.e. ˝1 , ˝2 g1 , g2 , the Hamiltonian can be further simpliﬁed by discarding the smaller terms, which is given by

H2 = Fig. 1. Level scheme of the atomic system which is embedded in the quantum cavity. Two orthogonal cavity modes (annihilation operators a1 and a2 ) couple to atomic transitions (|e ↔ |c, |g ↔ |d) with the coupling constants g1 and g2 , respectively. The transitions |c ↔ |e and |d ↔ |g are driven by two classical ﬁelds with Rabi frequencies ˝1 and ˝2 .

of atoms with the radiation was experimentally demonstrated [17]. Subsequently, experimental observations of phase-controlled light switching at low light levels in a four-level system were reported [18]. Recently the phase-sensitive manipulations of a squeezed vacuum ﬁeld in an optical parametric ampliﬁer inside an optical cavity have been experimentally demonstrated [19]. Additionally, the inﬂuence of the relative phase on the single- and two-photon absorption was discussed [20]. This stimulate us to know how the relative phase between the ﬁelds driving the atomic system with a closed-loop conﬁguration inﬂuence the squeezing and entanglement for the outgoing ﬁelds from the cavity embedded by this atom. Our paper will focus on the phase dependent squeezing and entanglement generation for the outputting ﬁelds in a cavity QED system embedded by a four-level atom, which are driven by two orthogonal polarization cavity modes and two classical ﬁelds. We shall investigate the effects of the relative phase as well as other physical parameters on the output squeezing spectrum and the entanglement. The paper is arranged as follows. In Section 2, the cavity QED is brieﬂy described and its equation of motion is shown. In Section 3, output squeezing and entanglement for the outgoing cavity ﬁelds are discussed. In Section 4, the main results of this work are summarized. 2. The model and equation of motion

˝22 2

1 eiϕ1 a+ e−iı1 t 1

(3)

H3 = (1 eiϕ1 a+ + 2 e−iϕ2 a2 ) |g e| + H.c. 1

(4)

After using another unitary transformation U = e the corresponding Hamiltonian is given by

iϕ1 (a+ a1 +a+ a2 ) 1 2

H4 = (1 a+ + 2 a2 ei ) − + H.c., 1

,

(5)

where = ϕ1 − ϕ2 , − = |g e| and + = ( − Using the input-output theory [21], the Langevin equations of motion for the system are )+ .

√

1 a1 − 1 ain 1, 2

a˙ 1 = −i1 − −

a˙ 2 = −i2 e−i + −

(6a)

√

2 a2 − 2 ain 2, 2

(6b)

e−i ) z , ˙ − = (i1 a1 + i2 a+ 2

(6c)

where z = |e e| − |g g| and 1 and 2 are decaying rates for the (j = 1, 2) associated two cavities, respectively. The operator ain j

with the input ﬁelds satisﬁes the correlation − t)

ain (t), ain+ (t) j j

=

j

ıjj ı(t for j, = 1, 2. Here, the atomic decays are neglected because the upper levels have been adiabatically eliminated and decaying rate for the transition between |e ↔ |g is very small. In order to prepare the atom initially in the ground state |g, we assume the cavity decay rate 1 and 2 are large enough and choose Rabi frequency ˝1 ˝2 . With the approximation z (t) ≈ −1 for all time [22], we can substitute the average value of z (t) into Eq. (6). By using the Fourier transformation

HI = g1 a1 ei1 t |c g| + g2 a2 ei2 t d e| + ˝1 eiϕ1 ei(1 +ı1 )t |c e|

+∞

eiωt G(t) dt,

(7)

−∞

Eq. (6) can be transformed into the Langevin equations in the frequency domain for a1 (ω) and a2 (ω)

2

ω2 − 1 + 2

ω2 + 2 +

(1)

where the cavity coupling constants (g1 and g2 ) and the laser Rabi frequencies (˝1 and ˝2 ) are real, and the detuning of the two cavity modes are denoted by 1 = ωc − ω1 = ωc − ωe − ωL1 − ı1 and 2 = ωd − ωe − ω2 = ωd − ωL2 − ı2 . If detuning i (i = 1, 2) is sufﬁciently large, the upper levels |c and |d can be adiabatically eliminated. With the assumptions of 1 , 2 g1 , g2 , ˝1 , ˝2 , ı1 , ı2 , the effective Hamiltonian are obtained [14]

By using a unitary transformation U = e−iAt with A = (˝22 /2 ) |g g| + (˝12 /1 ) |e e|, and choosing ı1 = −ı2 = (˝12 /1 ) − (˝22 /2 ), the effective Hamiltonian can be given by

1 G(ω) = √ 2

+ ˝2 eiϕ2 ei(2 +ı2 )t d g| + H.c.

1

|e e| +

+2 e−iϕ2 a2 eiı2 t ) |g e| + H.c.

We consider a four-level atomic system interacting with two orthogonal polarization cavity modes and two classical ﬁelds, which is embedded in the quantum cavity. The energy level scheme of the atomic system is shown in Fig. 1. The levels |e and |g are ground states, the levels |c and |d are upper levels, which have free energies ωe , ωc and ωd , respectively. The transitions |c ↔ |e and |d ↔ |g are driven by classical ﬁelds of frequency ωLi with Rabi frequency ˝i and phase ϕi (i = 1, 2). The transitions |c ↔ |g and |d ↔ |e are coupled by two orthogonal polarization cavity modes (annihilation operators a1 and a2 ) of frequency ωi (i = 1, 2) with coupling constant gi (i = 1, 2). In the interaction picture the Hamiltonian of the system can be written as

˝12

|g g| +

√ iω 1 a1 (ω) = 1 2 e−i a+ (−ω) − iω 1 ain 1 (ω), (8a) 2 2 iω 2 2

√ a2 (ω) = 1 2 e−i a+ (−ω) − iω 2 ain 2 (ω), (8b) 1

Under the cavity input-output relation aout (ω) = ain + i i √

i ai (i = 1, 2), we can get the output ﬁelds [23], aout 1 (ω) =

H(ω) in A a (ω) + ain+ (−ω), (ω) 1 (ω) 2

aout 2 (ω) = −

H ∗ (ω) in A a (ω) − ain+ (−ω), (ω) 2 (ω) 1

(9a)

(9b)

J.J. Mu, B.P. Hou / Optik 124 (2013) 4659–4663

√ where A = 1 2 1 2 e−i , and H(ω) = −iω iω +

(ω) = −iω iω −

1

2

iω −

− 1

2

iω −

2

+ 2

2

iω +

1

1.0 ,

0.8

1

2

2

1 2 2 iω − − 1 iω − + 2 iω − . 2 2 2 2

0.6

2

2

2

2

S

4661

0.4

3. Output squeezing and entanglement

0.2

In the following, we shall concentrate on the non-classical properties, especially entanglement and squeezing of the outgoing cavity ﬁelds. In order to analyze the squeezing of outgoing cavity ﬁelds we calculate the output squeezing spectrum which is deﬁned by [24]

0.0

out out out out out (ω)ı(ω + ω ) = I± (ω)I± (ω ) + I± (ω )I± (ω) , 2S±

(10)

2

1

0 ω/λ1

1

2

Fig. 2. The output squeezing spectrum S as a function of frequency ω (in units of 1 ) with different decay rates: 1 = 2 = 2 (solid curve); 1 = 2 = 4 (dashed line);

1 = 2 = 6 (dotted line); 1 = 2 = 8 (dashed-dotted line). The others parameters are set by 2 = 0.4, = .

out (ω) and I out (ω) are deﬁned by [25] where I+ −

1 out+ out+ out I+ (ω) = √ aout (−ω) − aout (−ω) , 1 (ω) + a1 2 (ω) − a2 2

(11a)

−i out+ out+ out (ω) = √ aout (−ω) + aout (−ω) , I− 1 (ω) − a1 2 (ω) − a2 2

(11b)

1.0 0.8

S

0.6 0.4

out (ω) and I out (ω) are squeezed, the two output cavity modes If I+ − will be entangled [26,27]. The entanglement between two output cavity modes is related to their squeezing through the entanglement criterion [28]. The sum criterion can be deﬁned by [26]

S+ (ω) + S− (ω) < 2

S+ (ω)S− (ω) < 1

(13)

From Eqs. (9) and (11), the relation can be easily obtained as out (ω) = S out (ω) = S out (ω). Based on the above discussions, it is S+ − shown that the output two modes are of EPR-like entanglement if Sout (ω) < 1. This is coincided with the squeezing criterion [28]. In other words, the criterion Sout (ω) < 1 implies that the two cavity modes are not only entangled but also squeezed. The substitution of Eq. (9) into Eq. (11) leads to the expressions out (ω) and I out (ω), which are given by of the operators I+ − 1 I+out (ω) = √ 2

+

−

A∗ H(ω) + (ω) (ω)

A∗ H ∗ (ω) + (ω) (ω)

i I−out (ω) = − √ 2

0.0 2

1

0 ω/λ1

(12)

Mean while the criterion can be also given by the product form [28]

0.2

ain (ω) + 1

ain (ω) + 2

A∗ H(ω) + (ω) (ω)

A∗ H ∗ (ω) + (ω) (ω)

A H(ω) + (ω) (ω)

A∗ H ∗ (ω) + (ω) (ω)

ain (ω) − 1

ain (ω) + 2

ain+ (−ω) 1

A H(ω) + (ω) (ω)

A∗ H ∗ (ω) + (ω) (ω)

ain+ (−ω) 2

,

(14a)

ain+ (−ω) 1

1

2

Fig. 3. The output squeezing S as a function of frequency ω (in unit of 1 ) with different 2 : 2 = 0.2 (solid curve); 2 = 0.4 (dashed curve); 2 = 0.6(dotted curve); 2 = 0.8 (dashed-dotted curve). Other parameter values are set by 1 = 2 = 3 and = .

the generality, we assume that the dissipation rates of the two cavity modes are equal. To verify the results obtained by calculating output spectrum valid, we recover the dependences of the entanglement on the decay rate 1 = 2 = 1 and the coupling constants investigated by calculating entanglement criterion in paper [14], which are shown by Figs. 2 and 3, respectively. It is shown in Fig. 2 that the decay rates of the two mode cavity only affect the bandwidth of the squeezing and entanglement spectra, but not affect the maximum of squeezing and entanglement. While the depth of squeezing spectrum increases and its bandwidth becomes narrow when the coupling described by 2 becomes stronger, which is shown by Fig. 3. We plot the output squeezing spectrum as a function of 2 in Fig. 4. We can see that the maximum of squeezing spectrum decreases with 2 for 2 < 1 , after which its maximum increases. The coupling value of 2 = 1 is its turning point.

ain+ (−ω) 2

.

(14b)

1.0 0.8

After substituting Eq. (14) into Eq. (10), the squeezing spectrum are obtained as S

(ω) =

out S+ (ω)

=

out S− (ω)

= n − c,

(15)

where |H(ω)|2 + |A|2 , n= |(ω)|2

0.6

S

out

0.4 0.2

2Re[A]Re[H(ω)] c=− . |(ω)|2

From Eq. (15) we know that the squeezing spectrum and entanglement of the two cavity modes are dependent on the relative phase between two classical ﬁelds, the cavity decay rates 1 and

2 , and the effective coupling constants 1 and 2 . Without loss of

0.0 0

2

4

6

8 λ2/λ1

10

12

14

Fig. 4. The output squeezing spectrum as a function of 2 (in unit of 1 ) with = ,

1 = 2 = 3 and ω = 0.

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J.J. Mu, B.P. Hou / Optik 124 (2013) 4659–4663

a function of the relative phase , which is shown in Fig. 7. The ﬁgure displays that the output spectrum varies with in a cosinelike behavior with a period of 2. Meanwhile, we can see that the two cavity modes are squeezed and entangled in the interval of (2n + 0.74223) < < (2n + 1.24223) (n = 0, 1, 2, . . .), beyond which the ﬁelds display noise. Additionally, it is exhibited that when the relative phase is set as = (2n + 1) (n = 0, 1, 2, . . .) the output ﬁelds display perfect squeezing and entanglement.

5 4

S

3 2 1

4. Conclusion

0 2

1

0

1

2

Fig. 5. The output squeezing spectrum S as a function of frequency ω (in unit of 1 ) with different relative phases: = 0 (solid curve); = 0.4 (dashed curve); = 0.6 (dotted curve) and = 0.7 (dashed-dotted curve), respectively. The other parameters are given by 1 = 2 = 3, 2 = 0.4.

1.0 0.8

S

0.6 0.4 0.2 0.0 2

1

0

1

2

In conclusion, the output entanglement and squeezing in system with two cavity modes interaction with four levels atom system are discussed. We have calculated the expression of the outgoing ﬁelds spectrum. It is found that the depth of squeezing spectrum and entanglement are both insensitive to the cavity decay rate, but can be controlled by adjusting the relative phase and the effective coupling constant. In particular, the squeezing spectrum and entanglement is strongly dependent on the relative phase between the classical ﬁelds driving the atom. By adjusting the relative phase, the two cavity ﬁelds can be changed into squeezing or entanglement from noise. It is found that the output spectrum varies with the relative phase in a cosine-like behavior with a period of 2. Its variation with the relative phase can instruct to suitably adjust the relative phase to get the maximal squeezing and entanglement in their applications. This will be useful in quantum communication and computation.

ω/λ1 Fig. 6. The output squeezing spectrum S as a function of frequency ω (in unit of 1 ) with different relative phase: = 0.75 (solid curve); = 0.8 (dashed curve); = 0.9 (dotted curve); = (dashed-dotted curve). The other parameters are set by the same values as those in Fig. 5.

In the following, we shall discuss the dependence of the outgoing-ﬁelds squeezing spectrum on the relative phase between two classical driving ﬁelds. We plot the squeezing spectrum with different relative phases in Figs. 5 and 6. It is shown that the output spectrum displays noise (Sout (ω) > 1) for the given values of : 0, 0.4, 0.6 and 0.7, and their maximal magnitudes at central frequency of the output spectrum decrease with . However when the values of the relative phase are set as 0.75, 0.8, 0.9, and in Fig. 6, the output ﬁelds are squeezed (Sout (ω) < 1), and their maximal squeezing magnitudes increase with . The output squeezing spectrum satisfying Sout (ω) < 1 indicates the outgoing ﬁelds are both entangled and are squeezed, i.e. the two cavity modes are of EPRlike entanglement. In order to further investigate the inﬂuence of the relative phase on the output spectrum, we plot the output spectrum as

5

S

4 3 2 1 0 0

π 2

π

3π 2

2π

5π 2

3π

φ Fig. 7. The output squeezing spectrum S as a function of the relative phase (in unit of 1 ).

Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 10647007 and 10974137, the Key Project of Chinese Ministry of Education under Grant No. 210192, and by the Young Foundation of Sichuan Province, China under Grant No. 092Q026-008, and by the Education Foundation of Sichuan Province, China under Grant No. 10ZA001. References [1] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 (1935) 777–780. [2] D.F. Walls, Squeezed states of light, Nature 306 (1983) 141–146. [3] L. Samuel, Braunstein, Teleportation of continuous quantum variables, Phys. Rev. Lett. 80 (1998) 869–872. [4] S.L. Braunstein, P. van Look, Quantum information with continuous variables, Rev. Mod. Phys. 77 (2005) 513. [5] W.-L. Li, C.-F. Li, G.-G. Guo, Probabilistic teleportation and entanglement matching, Phys. Rev. A 61 (2000) 034301. [6] X. Li, J. Jing, J. Zhang, C. Xie, K. Peng, Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam, Phys. Rev. Lett. 88 (2002) 047904. [7] Y.-H. Kim, S.P. Kulik, Y. Shih, Experiment demonstration of unconditional entanglement swapping for continuous variables, Phys. Rev. Lett. 86 (2001) 1370. [8] X. Jia, X. Su, Q. Pan, J. Gao, C. Xie, K. Peng, Experiment demonstration of unconditional entanglement swapping for continuous variables, Phys. Rev. Lett. 93 (2004) 250503. [9] Z.Q. Yin, Y.-J. Han, Generating EPR beams in a cavity optomechanical system, Phys. Rev. A 79 (2009) 024301. [10] J.-H. An, M. Feng, C.H. Oh, Quantum information processing with a single photon by an input–output process, Phys. Rev. A. 79 (2009) 032303. [11] M.J. Collett, C.W. Gardiner, Squeezing of intracavity and traveling-wave light ﬁelds produced in parametric ampliﬁcation, Phys. Rev. A 30 (1984) 1386. [12] E. Solano, G.S. Agarwal, H. Walther, Strong-driving-assisted multipartite entanglement in cavity QED, Phys. Rev. Lett. 90 (2003) 027903. [13] L. Zhou, H. Xiong, M.S. Zubairy, Single atom as a macroscopic entanglement source, Phys. Rev. A 74 (2006) 022321. [14] Q.-X. Mu, Y.-H. Ma, L. Zhou, Output squeezing and entanglement generation from a single atom with respect to a low-Q cavity, Phys. Rev. A 81 (2010) 024301. [15] E. Paspalakis, P.L. Knight, Phase control of spontaneous emission, Phys. Rev. Lett. 81 (1998) 293. [16] F. Ghafoor, S.-Y. Zhu, M. Suhail Zubairy, Amplitude and phase control of spontaneous emission, Phys. Rev. A 62 (2000) 013811.

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