PHYSICAL REVIEW A 76, 013809 共2007兲

Continuous source of phase-controlled entangled two-photon laser C. H. Raymond Ooi* Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 305-701 Korea; Frick Laboratory, Department of Chemistry, Princeton University, New Jersey 08544, USA; and Institute for Quantum Studies and Department of Physics, Texas A&M University, Texas 77843-4242, USA 共Received 14 February 2007; published 11 July 2007兲 We show that an absolute coherent phase of a laser can be used to manipulate the entanglement of photon pairs in two-photon laser. We present simple physics behind a general master equation for two-photon laser. Our focus is on the generation of a continuous source of macroscopically entangled photon pairs in the double ⌳ 共or Raman兲 scheme. We show how the steady-state photon numbers and entanglement depend on the laser parameters, especially the phase. We obtain a relationship between entanglement and two-photon correlation. We derive conditions that give steady-state entanglement for the spontaneous Raman-electromagnetic-induced transparency scheme and use it to identify the region with macroscopic entanglement. No entanglement is found for the double resonant Raman scheme. DOI: 10.1103/PhysRevA.76.013809

PACS number共s兲: 42.50.Dv, 03.67.Mn, 03.65.Ud, 42.50.Gy

II. MASTER EQUATION AND PHYSICS OF TWO-PHOTON LASER

I. INTRODUCTION

Entangled photon pairs is an integral asset to quantum communication technology with continuous variables 关1兴. A bright source of entangled photon pairs could be useful also for quantum lithography 关2兴. Transient entanglement of a large number of photon pairs has been shown to exist for a cascade scheme 关3,4兴, and double Raman scheme 关5兴. The transient regime does not provide a continuous source of entangled photon pairs that could be as useful and practical as typical lasers in continuous wave 共cw兲 operation. One might wonder whether the entanglement still survives in the long time limit. In this paper, we control the coherent phase of the lasers to generate a continuous source of a large number 共macroscopic兲 of entangled photon pairs. This provides the possibility of coherently controlling the degree of entanglement in the steady state. We focus on the double Raman configuration 关Fig. 1共a兲兴. The Raman-EIT 共electromagnetic induced transparency兲 scheme has been shown to produce nonclassically correlated photon pairs in the single atom 关6兴 and many atoms 关7兴 cases. First, we discuss the physics of a two-photon emission laser using the master equation in Sec. II. The physical significance of each term in the master equation is elaborated and related to the quantities of interests 共in Sec. III兲 such as two-photon correlation and Duan’s 关8兴 entanglement measure. In Sec. IV, we show the importance of laser phase for acquiring entanglement. In Sec. V, the steady-state solutions for the photon numbers and correlation between photon pairs are given. We show that the laser phase provides a useful knob for controlling entanglement. We then use the results to derive a condition for entanglement in the double Raman scheme. By using proper values of cavity damping and laser parameters based on analysis of the entanglement condition, we obtain a macroscopic number of entangled photon pairs in the steady state. We also analyze the double resonant Raman scheme but find no entanglement.

*[email protected] 1050-2947/2007/76共1兲/013809共7兲

We consider a single atom with double Raman scheme localized in a double cavity that are resonantly tuned to the Stokes 共aˆ1兲 and anti-Stokes 共aˆ2兲 modes, as in Fig. 1共a兲. The atom is driven by the pump “p” and control “c” lasers with Rabi frequencies ⍀ p,c. The Stokes and anti-Stokes photons spontaneously emitted into the cavity modes are amplified by stimulated emissions into strong lasing modes, thus the Hamiltonian in the interaction picture is Vˆ = −ប共⍀ p␴ˆ dce−i⌬pt + g1␴ˆ dbaˆ1e−i⌬1t + ⍀c␴ˆ abe−i⌬ct + g2␴ˆ acaˆ2e−i⌬2t兲 + adj where ␴ˆ ␣␤ = 兩␣典具␤兩, ⍀q = 兩⍀q 兩 ei␸q共q = p , c兲, g j = 兩g j 兩 ei␸ j共j = 1-Stokes, 2-anti-Stokes兲. The derivation of the laser master equation follows the usual approach 关9兴, starting from dtd ␳ˆ tot = ih1 关Vˆ , ␳ˆ tot兴 |dñ Dp Wp

|añ Dc

Wc Stokes

| n1 - 1,n 2 +1ñ | n1 ,n 2 +1ñ | n1 + 1,n 2 +1ñ antiStokes

|bñ |cñ double Raman scheme

C4

C1

| n1 + 1,n 2 ñ | n1 - 1,n 2 ñ

| n1 ,n 2 ñ

C2

trapping pump and drive lasers

C3

| n1 - 1,n 2 -1ñ | n1 ,n 2 -1ñ | n1 + 1,n 2 -1ñ

a)

b)

FIG. 1. 共Color online兲 共a兲 Double Raman atom in a doubly resonant optical cavity. The atom is trapped by an optical dipole force and driven by a pump laser and a control laser. The RamanEIT scheme 关⍀c , ⌬ p共=⌬兲 Ⰷ ⍀ p , ␥ and ⌬c = 0兴 and double resonant Raman scheme 共⍀c = ⍀ p , ⌬c = ⌬ p = 0兲 would be the focus for analysis. 共b兲 Photon number states for the Stokes and anti-Stokes are shown in two dimensions. The four arrows correspond to the offdiagonal density matrix elements for two-photon emission with their respective coefficients Ck in Eq. 共1兲.

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©2007 The American Physical Society

PHYSICAL REVIEW A 76, 013809 共2007兲

C. H. RAYMOND OOI

for the atom-field density operator ␳ˆ tot, the master equation for the radiation state ␳ˆ = 兺x=a,b,c具x 兩 ␳ˆ tot 兩 x典 is obtained by tracing over the internal states, which gives dtd ␳ˆ = i共g1aˆ1␳ˆ bd + g2aˆ2␳ˆ ca + g*1aˆ†1␳ˆ db + g*2aˆ†2␳ˆ ac兲 + adj where ␳ˆ ␤␣ = 兺x=a,b,c具x 兩 ␴ˆ ␣␤␳ˆ tot 兩 x典 are the atom-field coherence operators projected out by the operators ␴ˆ ␣␤. Since the atomic dynamics vary much faster than the fields, it is possible to express ␳ˆ ␤␣ in terms of ␳ˆ , aˆ j and aˆ†j from the quasi-steady-state solutions of the coupled equations for dtd ␳ˆ ␤共␤ = ac , ad , bc , bd兲 that contain the four channels of spontaneous emissions. By st st st ␳ˆ , ␳ˆ ab ⯝ pab ␳ˆ , ␳ˆ dc ⯝ pdc ␳ˆ , where “st” implies using ␳ˆ ␣␣ ⯝ p␣␣ the steady-state solutions of the density matrix equations 共in the interaction picture兲 without the quantum fields aˆ j 共the zeroth-order approximation兲, we obtain the master equation

III. RELATION BETWEEN ENTANGLEMENT AND TWO-PHOTON CORRELATION

Two-photon correlation for the Raman-EIT 共large detuning and weak pump兲 case for single atom 关6兴 and extended medium 关7兴 show nonclassical properties such as antibunching and violation of Cauchy-Schwarz inequality. It is useful to show how nonclassical correlation relates to entanglement. The normalized two-photon correlation at zero time delay is g共2兲共t兲 ⬟

兩具aˆ2aˆ1典兩 = 冑¯n1¯n2关g共2兲共t兲 − 1兴.

d ␳ˆ = 关Closs1共aˆ1␳ˆ aˆ†1 − ␳ˆ aˆ†1aˆ1兲 + Cgain1共aˆ†1␳ˆ aˆ1 − aˆ1aˆ†1␳ˆ 兲 dt + e−i␸t共C1aˆ2␳ˆ aˆ1 − C2␳ˆ aˆ1aˆ2 + C3aˆ1␳ˆ aˆ2 − C4aˆ1aˆ2␳ˆ 兲兴

具aˆ2aˆ1典 = 兩具aˆ2aˆ1典兩ei␾21 .

共1兲

with the effective phase ␸t = ␸ p + ␸c − 共␸1 + ␸2兲. The phases ␸␣共z兲 = k␣z + ␾␣ of the lasers depend on both the position z of the atom and the controllable absolute phases ␾␣ of the lasers. So, ␾1 = ␾2 = 0. Since k p + kc − ks − ka = 0, the effective phase becomes ␸t = ␾ p + ␾c = ␾. The explicit expressions for Clossj, Cgainj, and Ck 共where j = 1 , 2 and k = 1 , 2 , 3 , 4兲 are given in Appendix A. Equation 共1兲 already includes the cavity damping Liouvillean L␳ˆ = −兺 j=1,2␬ j共aˆ†j aˆ j␳ˆ + ␳ˆ aˆ†j aˆ j − 2aˆ j␳ˆ aˆ†j 兲 since Clossj depend on ␬ j, the cavity damping rates for the Stokes 共j = 1兲 and anti-Stokes 共j = 2兲 modes. The Cgainj are due to the emissions processes of the atom in the excited levels and Raman process via the laser fields which provide gain. On the other hand, the Clossj are due to cavity dissipation ␬ j and absorption processes of the atom in the ground levels which create loss. The terms with Ck coefficients correspond to the coherence between n j and n j ± 1 such that the difference between the total photon number in the bra and in the ket is always 2. These terms give rise to squeezing and will be elaborated on in future presentations. Figure 1共b兲 illustrates the essence of each diagonal term in Eq. 共1兲 in two-dimensional photon number space. We find that the relation holds, C1 + C3 = C2 + C4 .

共3兲

共4兲

Thus, the g共2兲共t兲 does not provide phase ␾21 information of the correlation 具aˆ2aˆ1典. We bring out the phase information by writing

+ Closs2共aˆ2␳ˆ aˆ†2 − aˆ†2aˆ2␳ˆ 兲 + Cgain2共aˆ†2␳ˆ aˆ2 − ␳ˆ aˆ2aˆ†2兲 + adj,

兩具aˆ2aˆ1典兩2 + 1, 具aˆ†2aˆ2典具aˆ†1aˆ1典

共2兲

The consequence of this relation for a large number of photons n j Ⰷ 1 is that the coherences due to the terms aˆ2␳ˆ aˆ1 , ␳ˆ aˆ1aˆ2 , aˆ1␳ˆ aˆ2 , aˆ1aˆ2␳ˆ and their adjoint are approximately equal. Hence, the contribution of the off-diagonal terms vanish and the master equation reduces to the classical rate equation. Since the off-diagonal terms give rise to entanglement 共as we show below兲, we can understand that there will be no entanglement for very large n j. Note that Eq. 共1兲 generalizes the master equation for the cascade scheme 关10兴 in which C3 = Cgain2 = 0, and Clossj = ␬ j.

共5兲

Among the various measures of entanglement for continuous variables 关11兴, we choose to employ the Duan’s criteria due to its convenience for the present problem and applicability to Gaussian states 关12兴 such as in the present case where the bosonic operators in the master equation come in pairs. Besides, it has been used in Refs. 关3–5兴. The sufficient condition 共and necessary condition for Gaussian states兲 for entanglement between the two modes aˆ1 and aˆ2 is D共t兲 = 具共⌬uˆ兲2典 + 具共⌬vˆ 兲2典 ⬍ 2, where uˆ = xˆ1 + xˆ2 and vˆ = pˆ1 − pˆ2 are the EPR-type operators, with the real operators xˆ j = 冑12 共aˆ j + aˆ+j 兲, and pˆ j = i冑12 共aˆ j − aˆ+j 兲, g共2兲共t兲 ⬟ 兩具aˆ2aˆ1典兩2 / 具aˆ†2aˆ2典具aˆ†1aˆ1典 + 1. Hence, the D共t兲 function may be written as D共t兲 = 2兵1 + ¯n1 + ¯n2 + 2冑¯n1¯n2关g共2兲共t兲 − 1兴cos ␾21 − 兩具aˆ2典兩2 − 兩具aˆ1典兩2 − 具aˆ2典具aˆ1典 − 具aˆ†2典具aˆ†1典其.

共6兲

Clearly, the presence of inseparability or entanglement is entirely determined by the phase ␾21 in Eq. 共6兲. We now find a knob for controlling entanglement, i.e., cos ␾21 must be negative or ␲ / 2 ⬍ ␾21 ⬍ 3␲ / 2. If the two modes are in coherent states, the second line in Eq. 共6兲 becomes −兩␣1兩2 − 兩␣2兩2 − 共␣1␣2 + ␣*1␣*2兲. Here, there is no correlation, i.e., g共2兲共t兲 = 1, and we have D共t兲 = 2关1 − 共␣1␣2 + ␣*1␣*2兲兴,

共7兲

which shows that if the phases of the coherent modes ␣ j = r j exp共i␪ j兲 are locked such that cos共␪1 + ␪2兲 ⬎ 0, the modes can be entangled. In the following, we consider initial vacuum state and the modes that have not evolved into the coherent state, in which the second line of Eq. 共6兲 vanishes. Then, the condition for inseparability or entanglement 0 ⬍ D共t兲 ⬍ 2 can be rewritten in terms of the phase and the two-photon correlation,

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PHYSICAL REVIEW A 76, 013809 共2007兲

具aˆ2aˆ1典共t兲 = i

冕

t

e−K共t−t⬘兲兵␣12具aˆ†2aˆ2典共t⬘兲

0

+ ␣21关具aˆ†1aˆ1典共t⬘兲 + 1兴其dt⬘ ⯝ iX,

共10兲

where K = ␤22 − ␤11 + ␬2 + ␬1 and X is real, whose expression is not important for the present discussion. For initial conditions 具aˆ j共0兲典 = 0, the Duan’s parameter becomes 共2兲

FIG. 2. The 具共⌬uˆ兲 典 + 具共⌬vˆ 兲 典 and g 共t兲 vary in a similar manner as a function of time for decoherence ␥bc = 0.6␥ac 共thin line兲 and without decoherence ␥bc = 0 共thick line兲. Parameters used are ␬1 = ␬2 = 0.001g1, ⍀ p = 2g1, ⌬ p = 40g1, ⍀c = 25g1, ⌬c = 0. We have assumed g2 = g1, with ␥ac = ␥dc = ␥ab = ␥db = ␥. 2

−

¯n1 + ¯n2 + 1

2冑¯n1¯n2共g共2兲 − 1兲

2

⬍ cos ␾21 ⬍ −

¯n1 + ¯n2

2冑¯n1¯n2共g共2兲 − 1兲

D = 2共1 + 具aˆ†1aˆ1典 + 具aˆ†2aˆ2典 + 具aˆ2aˆ1典 + 具aˆ†1aˆ†2典兲

which clearly shows there is no entanglement 共D ⬎ 2兲. For finite phase ␾ associated to the pump laser, the correlation 具aˆ2aˆ1典 becomes 具aˆ2aˆ1典ei␾ but the photon numbers are not affected. The parameter becomes D = 2共1 + 具aˆ†1aˆ1典 + 具aˆ†2aˆ2典 + 2X sin ␾兲

, 共8兲

= 1. When ¯n1 = ¯n2, we have Note that for a small correlation g ⲏ 1 the entanglement window for ␾21 can be quite large when ¯n is small. For large two-photon correlation g共2兲 Ⰷ 1 and large photon numbers ¯n1 ⯝ ¯n2 Ⰷ 1, the range for entanglement becomes quite restrictive, cos ␾21 ⯝ − 冑 共2兲1 becomes very small in magnitude g −1 共but negative兲 and from Eq. 共6兲 we have D共t兲 ⱗ 2, i.e., the entanglement is small. This explains the results in Fig. 2 where large transient correlation is accompanied by small entanglement. In the long time limit, Fig. 2 shows that the correlation vanishes 共corresponding to photon antibunching兲 and the entanglement increases, D Ⰶ 2. Although both the correlation and entanglement are quantum mechanical properties they do not vary in the same way. The entanglement increases with time while the correlation decreases with time. This clearly shows that correlation and entanglement are distinct terminologies and should be carefully discerned from each other. Figure 2 also shows that the decoherence ␥bc tends to reduce the degree of entanglement and the magnitude of correlation, as expected. IV. LASER PHASE FOR ENTANGLEMENT

Here, we show by using a simple example from the resonant cascade work of Ref. 关3兴 that the nonzero phase of the paired correlation 具aˆ2aˆ1典 is necessary for entanglement. Let us analyze the transient equation 共written in their notations with zero laser phase兲 d具aˆ2aˆ1典 * * = − 具aˆ2aˆ1典共␤22 − ␤11 + ␬2 + ␬1兲 − ␤21 共具aˆ†1aˆ1典 + 1兲 dt + ␤12具aˆ†2aˆ2典.

共12兲

which gives maximum entanglement when ␾ = −␲ / 2 or 3␲ / 2, and no entanglement when ␾ = 0.

where the lower limit corresponds to maximum entangle¯n +n ¯ +1/2 ment. The midpoint value cos ␾21 = − 冑¯1 ¯ 2 共2兲 gives D共t兲 2 n1n2共g −1兲 ¯ − 冑1+1/2n ⬍ cos ␾21 ⬍ − 冑g共2兲1 −1 . g共2兲−1 共2兲

共11兲

V. STEADY-STATE ENTANGLEMENT

The master equation 共1兲 is linear and does not include saturation. One might wonder whether steady-state solutions exist. We find that there are steady-state solutions when the photon numbers ¯n j do not increase indefinitely but saturate at large times. Parameters that give non-steady-state solutions manifest as a negative value of D and should be disregarded. The study of entanglement via nonlinear theory will be presented elsewhere. In the case of initial vacuum, the coupled equations for † † d¯n1 d¯n2 d具aˆ2aˆ1典 d具aˆ1aˆ2典 dt , dt , dt , dt

are sufficient to compute the Duan’s entanglement parameter, where ¯n j = 具aˆ†j aˆ j典, j = 1 , 2. The full expressions for the coupled equations and the corresponding steady-state solutions for ¯n1, ¯n2, and 具aˆ2aˆ1典 are given in Appendix B. From Eq. 共11兲, together with the steady-state solutions 具aˆ2aˆ1典 = Eei␾, ¯n1 and ¯n2 given by Eqs. 共B6兲–共B9兲, the neces¯ 1 + ¯n2兲. sary condition for entanglement is Eei␾ + E*e−i␾ ⬍ −共n If E is real positive there would be no entanglement in the region cos ␾ ⬎ 0. Entanglement is still possible even if ␾ ¯ 1 + ¯n2兲. Thus, the phase ␾ is not = 0 provided 2 Re兵E其 ⬍ −共n necessary for entanglement, but it provides an extra knob for controlling entanglement. Let us search for entanglement conditions in the limiting cases of Raman-EIT scheme which produces nonclassically correlated photon pairs, and the double resonant Raman 共DRR兲 scheme. A. Raman-EIT case

For this scheme, ⍀c, ⌬ p = ⌬1共=⌬兲 Ⰷ ⍀ p, ␥␣␤ 共␣ , ␤ = a , b , c , d兲 and ⌬c = ⌬2 = 0. Thus, we have pba

共9兲

The coefficients for the resonant case are such that ␤11 , ␤22 are real while ␤12 = i␣12 and ␤21 = i␣21 are purely imaginary. Clearly we have an imaginary value for

−i⍀*

= ␥abc 共pbb − paa兲 ⯝ 0 since the population is primarily in level −⍀ c 共pcc ⯝ 1 , pbb ⯝ paa ⯝ 0兲 and pcd = ⌬ p = pdc. The coefficients for the Raman-EIT case are given in Appendix C. For mod兩g 兩2 ⍀2 erate cavity damping, typically ⍀j2 ⌬p Ⰶ ␬ j. Thus, the only

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c

PHYSICAL REVIEW A 76, 013809 共2007兲

C. H. RAYMOND OOI

á(Du ) 2 ñ + á(Dv) 2 ñ

á(Du ) 2 ñ + á(Dv) 2 ñ

a)

n1 , n2

b)

c)

FIG. 3. 共Color online兲 Entanglement parameter D versus ⍀c and ␾ for Raman-EIT scheme in 共a兲 wide view, 共b兲 magnified view of the highly entangled region in 共a兲, and 共c兲 mean photon numbers ¯n1 共solid line兲 and ¯n2 共dots兲, for ␾ = ␲ / 2. The parameters are ⍀ p = ␥ac, ⌬ p = 40␥ac, ⌬c = 0. Cavity damping values ␬2 = ␬1 = 1.01兩 C2兩 ensures a minimum value of the denominator of the condition in Eq. 共18兲. We have assumed g2 = g1 = ␥, ␥bc = 0 and ␥ac = ␥dc = ␥ab = ␥db = ␥.

significant terms are Closs1 ⯝ ␬1 and Closs2 ⯝ ␬2. Since ⌬ Ⰷ ␥bc we have C1 Ⰶ C2 ⯝ −C4 ⯝ i⌶, where ⌶=

− g2g1⍀c⍀ p关⌬2 − 共⍀2c − ⍀2p兲兴/⌬ 共⍀2c

− ⌬ 兲␥ac␥bc − ⌬ 2

which becomes g2g1 ⌬共␥ K 1 = − 2 ␬ 1,

⍀ p⍀ c 2 ac␥bc+⍀c 兲

2

⍀2c

+

共⍀2c

−

, ⍀2p兲2

共13兲

when ⍀c Ⰷ ⍀ p. It follows that

K 2 = − 2 ␬ 2,

K12 = − 共␬1 + ␬2兲.

共14兲

Hence, we have the steady-state solutions n1 = ⌶2

␬2 , 共␬1 + ␬2兲共␬2␬1 − ⌶2兲

共15兲

n2 = ⌶2

␬1 , 共␬1 + ␬2兲共␬2␬1 − ⌶2兲

共16兲

具aˆ1aˆ2典 = ei␾i⌶

␬ 2␬ 1 . 共␬1 + ␬2兲共␬2␬1 − ⌶2兲

共17兲

The entanglement criteria can be rewritten as ⌶− ⌶

␬ 2␬ 1 2 sin ␾ 共 ␬ 1 + ␬ 2兲 ⬍ 0. ␬ 2␬ 1 − ⌶ 2

␬␬

c

p

minimum. This seems to prevent the generation of steadystate macroscopic entanglement. The region of maximum entanglement occurs around ␾ = 90°. We verify that if we change to a negative detuning ⌬ = −40␥ac there is no entanglement. Although entanglement can occur over a wide range of large ⍀c, the photon numbers ¯n j decrease as ⍀c increases. Figure 4 shows that it is possible to obtain a continuous bright source of entangled photons. We realize that the number of nonclassical photon pairs in the Raman-EIT case is limited by the weak pump field. Thus, by increasing the pump field we can generate more Stokes photons 关Fig. 4共a兲兴. At the same time, the detuning is increased as well to ensure that the scheme remains in the Raman-EIT regime 共⌬ p Ⰷ ⍀ p兲. By further applying the condition Eq. 共18兲 we obtained a larger 共macroscopic兲 number of entangled photon pairs 关Fig. 4共b兲兴. B. Double resonant Raman

共18兲

Note that the sign of the detuning ⌬ in Eq. 共13兲 is important for entanglement generation. There are many ways for arranging the laser parameters and the cavity rates ␬ j in Eq. 共18兲 to obtain entanglement. For negative detuning ⌶ ⬍ 0, there are two possibilities: 共i兲 if ␬2␬1 ⬍ ⌶2 entanglement occurs in the region ␬ 2␬ 1 共␬1+␬2兲 2 sin ␾ ⬎ ⌶, 共ii兲 if ␬2␬1 ⬎ ⌶2 we have entanglement in ⬍ ⌶. Similarly, for positive detuning ⌶ ⬎ 0: ␬␬ 共i兲 if ␬2␬1 ⬍ ⌶2 we need ⌶ ⬎ 共␬12+␬12兲 2 sin ␾,

sin ␾ ⬃ 1. Figure 3 is plotted using ␬2 = ␬1 = 1.01兩 C2兩 and ⌬ = 40␥ac for ␾ ⬃ 90°. The region ⍀c ⬃ ⌬ gives a large en⍀ c⍀ p tanglement where ⌶ → 共2⍀2−⍀ 2 兲⌬ , but the photon numbers are

␬ 2␬ 1 共 ␬ 1+ ␬ 2兲 2

sin ␾

Numerical results seem to show that steady-state entanglement in the double resonant Raman case 共⍀c = ⍀ p , ⌬c = ⌬ p = 0兲 is hardly possible. In the following, we verify this analytically. Here, the coefficients 共given in Appendix D兲 Cac,ac, Cac,bd, Cbd,ac, Cbd,bd are real and positive while Cac,ad, Cac,bc, Cbd,ad, Cbd,bc are purely imaginary 共positive or negative兲. Since pcd = −i 兩 pcd兩, pba = −i 兩 pba兩, all C j, Clossj, and Cgainj are real but could be negative. Thus, we have K j = 2Cgainj − 2Clossj and K12 = 21 共K1 + K2兲. For symmetric system ⍀ p ⯝ ⍀c, we find pab = −pba = pdc = −pcd, pcc ⯝ pbb, and paa ⯝ pdd 关13兴. Then, Cac,ac = Cbd,bd and Cac,bd = Cbd,ac. If we take Tac = Tdc = Tab = Tdb = ␥ 共spontaneous decay rate兲 with Tbc = ␥bc and Tad = 2␥ we further have Cbd,ad = −Cac,ad, Cbd,bc = −Cac,bc. The resulting steady-state solutions for the DRR scheme can be written as

共ii兲 if ␬2␬1 ⬎ ⌶2 then we need ⌶ ⬍ 共␬12+␬12兲 2 sin ␾. To obtain large entanglement, we tune the cavity damping such that the denominator ␬2␬1 − ⌶2 in Eq. 共18兲 is small and 013809-4

¯n1 = ¯n2 =

Cgain共Cgain − Closs兲 + 21 C2C12 2 C12 − 共Cgain − Closs兲2

,

共19兲

CONTINUOUS SOURCE OF PHASE-CONTROLLED …

á ( Du ) 2 ñ + á ( Dv ) 2 ñ

PHYSICAL REVIEW A 76, 013809 共2007兲

á(Du ) 2 ñ + á(Dv) 2 ñ

a)

n1 , n2

Macroscopic entanglement

W c / g ac

b)

c)

FIG. 4. 共Color online兲 Macroscopic entanglement with larger pump field and detuning ⍀ p = 10␥ac, ⌬ p = 400␥ac, and cavity damping ␬2 = ␬1 = 1.001兩 C2兩. Other parameters are the same as Fig. 3. This gives larger mean photon numbers, i.e., macroscopically entangled photon pairs. 共a兲 A wide view of D versus ⍀c and ␾ with an entangled region 共red circle兲. 共b兲 Magnified view of the entangled region in 共a兲. 共c兲 Photon numbers ¯n1 ⯝ ¯n2 up to 200 can be generated for ␾ = ␲ / 2.

具aˆ1aˆ2典 = − ei␾

冉

C1Cgain − 21 C2共Cgain + Closs兲 2 C12 − 共Cgain − Closs兲2

冊

,

共20兲

and the corresponding entanglement condition becomes ¯n1 + ¯n2 ⬍ 2␰ cos ␾ ,

共21兲

where ␰ is the term in the bracket 共¯兲 of Eq. 共20兲. In order to determine whether Eq. 共21兲 can be met, we consider a simpler case where ␥bc = 0. From the coefficients in Appendix D, we have Closs − Cgain = ␬ and C12 = ␩共pcc − paa兲 with ␩ = g2 / ␥, C1 = ␩ paa, C2 = ␩共2paa − pcc兲, Closs = 21 ␩ pbb + ␬, and Cgain = 21 ␩ pcc. These results are used to rewrite Eqs. 共19兲 and 共20兲 as 1 ␩共2paa − pcc兲共pcc − paa兲 − pcc␬ ¯n j = ␩ , 2 关␩共pcc − paa兲兴2 − ␬2 具aˆ1aˆ2典 =

ei␾ 共2paa − pcc兲␬ − ␩共pcc − paa兲pcc ␩ . 关␩共pcc − paa兲兴2 − ␬2 2

共22兲

VI. CONCLUSION

We have shown that two-photon laser can produce a continuous source of entangled photon pairs based on the steady-state solutions and an entanglement criteria. We obtained a relationship between entanglement and two-photon correlation, and found that both do not vary with time in the same manner. We have derived a condition for steady-state entanglement in the Raman-EIT scheme and found macroscopic steady-state entanglement. Thus, we have bypassed the constraint that a large steady-state entanglement is at the expense of a small number of photons. We reinforce the significance of the Raman-EIT scheme, by showing that the double resonant Raman scheme does not generate steadystate entangled photon pairs for any laser parameters. Finally, we foresee that a continuous source of entangled two-photon laser could be a practical tool in optics that would spawn new applications.

共23兲

For strong fields, the populations in the upper and lower levels are equally distributed, i.e., pcc ⯝ paa = 0.25. The steady solutions become ¯n1 = ¯n2 = g2 / 8␥␬, 具aˆ1aˆ2典 = −共g2 / 8␥␬兲ei␾, and D = 2关1 + 共g2 / 2␥␬兲sin2 21 ␾兴, i.e., no entanglement. For weak fields, one-half of the population is in level b and one-half in level c, so pcc ⯝ pbb ⯝ 0.5, paa ⯝ pdd ⯝ 0. The 2 2 1 e i␾ steady solutions are ¯n1 = ¯n2 = 4g␥ ␬−g2/2␥ , 具aˆ1aˆ2典 = 4g␥ ␬−g2/2␥ with

ACKNOWLEDGMENT

The author wishes to thank Professor M. Suhail Zubairy for stimulating discussions.

2g ␬ ⬎ g2 / 2␥ and hence D = 2共1 + 2␥␬ sin2 21 ␾兲, again no en−g2 tanglement. Here, the cavity damping must be sufficiently large 共␬ ⬎ g2 / 2␥兲 to ensure ¯n1 and ¯n2 are positive, i.e., the existence of steady-state solutions. If the cavity damping is small ␬ ⬍ g2 / 2␥, negative values of D and ¯n j would appear, corresponding to the non-steady-state regime. Thus, we have shown that there is no steady-state entanglement for the DRR scheme in both weak field and strong field regimes, in contrast to the Raman-EIT photon pairs, which are entangled in the steady state. This is compatible with the corresponding G共2兲, which shows classical two-photon correlation 关14兴. 2

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APPENDIX A: COEFFICIENTS FOR DOUBLE RAMAN SCHEME

The coefficients in Eq. 共1兲 are Closs1 = 兩g1兩2共Cbd,ad pab + Cbd,bd pbb兲 + ␬1 ,

共A1兲

Cgain1 = 兩g1兩2共Cbd,bd pdd + Cbd,bc pdc兲,

共A2兲

Closs2 = 兩g2兩2共Cac,ac pcc + Cac,ad pcd兲 + ␬2 ,

共A3兲

Cgain2 = 兩g2兩2共Cac,ac paa + Cac,bc pba兲,

共A4兲

C1 * * = Cbd,ac pcc + Cac,bd pdd + 共Cbd,ad + Cac,bc 兲pcd , g 2g 1 共A5兲

PHYSICAL REVIEW A 76, 013809 共2007兲

C. H. RAYMOND OOI

C2 * * = Cbd,ac paa + Cac,bd pdd + Cbd,bc pba + Cac,bc pcd , g 2g 1

¯1 dn = ¯n1K1 + e−i␾共C1 − C2兲具aˆ2aˆ1典 + ei␾共C*1 − C*2兲具aˆ†1aˆ†2典 dt 共A6兲

C3 * * = Cbd,ac paa + Cac,bd pbb + 共Cbd,bc + Cac,ad 兲pba , g 2g 1

¯2 dn = ¯n2K2 + e−i␾共C3 − C2兲具aˆ2aˆ1典 + ei␾共C*3 − C*2兲具aˆ†1aˆ†2典 dt 共A7兲

冉

共A8兲 C Jk = g2gk1 ,

g2, g1 are atom-field coupling strengths, where C␣␤,␥␦ 共␣ , ␤ , ␥ , ␦ = a , b , c , d兲 are complex coefficients that depend on decoherence rates ␥␣␤, laser detunings ⌬ p, ⌬c and Rabi frequencies ⍀ p, ⍀c. The p␣␣, pab, pcd 共␣ = a , b , c , d兲 are steady-state populations and coherences. The C␣␤ coefficients are * * * * TbcTdb + I pTad + IcTbc Tad , Z

* Tbc Tdb + I p − Ic , Cac,ad = − i⍀ p Z

Cac,bc = − i⍀c

−

* Tad Tdb

* Tbc

+ Z

* Tad

共A13兲

* * − Tac Tbc + I p − Ic , Z

共A14兲

* * Tac Tad + I p − Ic , Z

共A15兲

* * T* T* T* + I pTbc + IcTad Cbd,bd = ac ad bc , Z

where 2 Re Cgainj = Cgainj + C*gainj and the gain and/or loss coefficients are

I p = ⍀2p,

共B4兲

* * − 共Closs2 + Closs1 兲. K12 = Cgain2 + Cgain1

共B5兲

The steady-state solution for the correlation is * * * * 具aˆ1aˆ2典 = 关− C32 共K2K12 + C12 C32 − C12C32 兲2 Re Cgain1 * * * * − C12 共K1K12 + C12C32 − C12 C32兲2 Re Cgain2 * * * + C*2共K1C12C32 + K2C32C12 兲 − C*2K1K2K12 * * − C2C32 C12共K1 + K2兲兴

e i␾ , M

共B6兲

* * * * M = 共K1K12 + K2K12兲共C12 C32兲 + c.c. − 共C12C32 − C12 C32兲2

共B7兲

* − K1K2K12K12 .

The steady-state solutions for the photon numbers are ¯n1 = 2 Re Cgain1

共A16兲

* * * * * * * * * Z = Tac TadTbcTdb + I pTac Tad + I pTbc Tdb + IcTac Tbc + IcTad Tdb

+ 共I p − Ic兲2 ,

K j = 2 Re共Cgainj − Clossj兲,

where

* * Tbc + Tad , Z

Cbd,bc = − i⍀ p

共B3兲

共A11兲

共A12兲

Cbd,ac = ⍀ p⍀c

冊

d ¯ 1共C*3 − C*2兲 + ¯n2共C*1 − C*2兲 − C*2兴, − K12 具aˆ2aˆ1典 = ei␾关n dt

共A10兲

,

Cac,bd = ⍀c⍀ p

Cbd,ad = − i⍀c

+ I p − Ic , Z

共A9兲

共B2兲

+ 2 Re Cgain2 ,

C4 * * = Cbd,ac pcc + Cac,bd pbb + Cbd,ad pcd + Cac,ad pba , g 2g 1

Cac,ac =

共B1兲

+ 2 Re Cgain1 ,

* C12 + 2 Re Cgain2C12

where the complex decay rates Tac = i⌬2 + ␥ac, Tad = i共⌬c − ⌬1兲 + ␥ad, Tbc = i共⌬ p − ⌬1兲 + ␥bc, Tdb = i⌬1 + ␥db, and the detunings ⌬ p = ␯ p − ␻dc, ⌬c = ␯c − ␻ab, ⌬1 = ␯s − ␻db, ⌬2 = ␯a − ␻ac. APPENDIX B: COUPLED EQUATIONS AND SOLUTIONS

From the master equation 共1兲, we obtain 013809-6

* K12 + K12 M

+ C*2C12

* * K2K12 + 共C12 C32 − c.c.兲 M

* + C2C12

* K2K12 + 共C12C32 − c.c.兲 , M

¯n2 = 2 Re Cgain2

共A17兲 Ic = ⍀2c ,

* * * K2K12K12 − 共C12 C32K12 + c.c.兲 M

共B8兲

* * * − 共C32 C12K12 + c.c.兲 K1K12K12 M

* C32 + 2 Re Cgain1C32

* K12 + K12 M

+ C*2C32

* * K1K12 + 共C12C32 − c.c.兲 M

* + C2C32

* K1K12 + 共C12 C32 − c.c.兲 . M

共B9兲

CONTINUOUS SOURCE OF PHASE-CONTROLLED …

PHYSICAL REVIEW A 76, 013809 共2007兲

Z ⯝ 共Ic − ⌬2兲␥ac␥bc − ⌬2Ic + 共Ic − I p兲2 + iI p⌬共␥ac + ␥bc兲.

APPENDIX C: COEFFICIENTS FOR RAMAN-EIT SCHEME

共C7兲

By noting that most parameters would be zero from pcc ⯝ 1 , pbb ⯝ paa ⯝ pbd ⯝ 0, and pcd = −⍀ p / ⌬ = pdc for the REIT scheme, the coefficients reduce to Closs1 ⯝ ␬1 ,

Closs2 ⯝ 兩g2兩2

冉

J2 ⯝

共C2兲

冊

Ip − ␥bc 关⌬2 − 共Ic − I p兲兴 ⌬ + ␬2 , Z

冉

冊

共C3兲

⍀ p⍀ c i␥ac ␥bc 1 − , Z ⌬

共C4兲

− i⍀c⍀ p ⌬2 − 共Ic − I p兲 , Z* ⌬

共C5兲

J1 ⯝

冋 冉

冊

From Appendix A, we obtain the coefficients

共C1兲

I p ␥aci⌬ + I p − Ic , Cgain1 ⯝ 兩g1兩2i ⌬ Z i

APPENDIX D: COEFFICIENTS FOR DRR SCHEME

共D1兲

I C2 = g2g1 2关Tbc paa + Tad共2paa − pcc兲兴, Z

共D2兲

C12 = C1 − C2 = C3 − C2 = g2g1⍀2

Tbc + Tad 2共pcc − paa兲, Z 共D3兲

where Z = ␥关TadTbc␥ + 2I共Tad + Tbc兲兴 and I = ⍀2. Taking ␬1 = ␬2 = ␬ we also have Closs1 = Closs2, Cgain1 = Cgain2 and K2 = K1 = K12 = 2共Cgain − Closs兲, and hence Closs = 兩g1兩2

册

i关⌬2 − 共Ic − I p兲兴 ⍀ p⍀ c i␥ac J4 ⯝ ␥bc 1 − + , 共C6兲 Z ⌬ ⌬

冉

冊

TTbc + I ITbc pbb + ␬ , paa + Tad Z Z

共D4兲

冉

共D5兲

Cgain = 兩g1兩2 Tbc

where

关1兴 S. L. Braunstein and P. van Look, Rev. Mod. Phys. 77, 513 共2005兲. 关2兴 A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 共2000兲. 关3兴 H. Xiong, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 94, 023601 共2005兲. 关4兴 H.-T. Tan, S.-Y. Zhu, and M. S. Zubairy, Phys. Rev. A 72, 022305 共2005兲; L. Zhou, H. Xiong, and M. S. Zubairy, ibid. 74, 022321 共2006兲. 关5兴 M. Kiffner, M. S. Zubairy, J. Evers, and C. H. Keitel, Phys. Rev. A 75, 033816 共2007兲. 关6兴 M. O. Scully and C. H. Raymond Ooi, J. Opt. B: Quantum Semiclassical Opt. 6, S816 共2004兲. 关7兴 C. H. Raymond Ooi, Q. Sun, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 75, 013820 共2007兲. 关8兴 L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 共2000兲. 关9兴 M. O. Scully and M. S. Zubairy, Quantum Optics 共Cambridge University Press, Cambridge, 1997兲. 关10兴 N. A. Ansari, J. Gea-Banacloche, and M. S. Zubairy, Phys. Rev. A 41, 5179 共1990兲; C. A. Blockley and D. F. Walls, ibid. 43, 5049 共1991兲.

I C1 = g2g1 2共Tbc pcc + Tad paa兲, Z

冊

TTad + I ITad pdd + pcc . Z Z

关11兴 Entanglement criteria for bipartite Gaussian states by R. Simon, Phys. Rev. Lett. 84, 2726 共2000兲 involves the PeresHorodecki criterion that is less practical for the present formulation. Entanglement measures for bipartite non-Gaussian states have been obtained by G. S. Agarwal and A. Biswas, New J. Phys. 7, 211 共2005兲; M. Hillery and M. S. Zubairy, Phys. Rev. Lett. 96, 050503 共2006兲; H. Nha and J. Kim, Phys. Rev. A 74, 012317 共2006兲. 关12兴 M. Stobinska and K. Wódkiewicz, Phys. Rev. A 71, 032304 共2005兲. 关13兴 When the fields are weak, the populations are distributed only in the lower levels, so pcc, pbb = 0.5 and paa, pdd = 0. When the fields are very large, all levels are equally populated; paa = pbb = pcc = pdd = 0.25. So, the ranges 0.25艋 pcc, pbb 艋 0.5 and 0 艋 paa, pdd 艋 0.25 apply for the double resonant Raman scheme. 关14兴 C. H. Raymond Ooi, “Nonclassical correlation of macroscopic Raman photon pairs,” Proceeding of quantum communications and measurement conference (QCMC), 2006. Further results for DRR scheme will be presented elsewhere as a sequel to Ref. 关7兴.

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