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Effective generation of polarization-entangled photon pairs in a cavity-QED system
Physics Letters A 372 (2008) 5959–5963
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Effective generation of polarization-entangled photon pairs in a cavity-QED system Pengbo Li a , Ying Gu a,∗ , Qihuang Gong a , Guangcan Guo a,b a b
State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
a r t i c l e
i n f o
Article history: Received 7 April 2008 Received in revised form 16 July 2008 Accepted 30 July 2008 Available online 5 August 2008 Communicated by P.R. Holland
a b s t r a c t We propose a cavity-QED scheme to effectively generate Einstein–Podolsky–Rosen polarization-entangled photon pairs. Assisted by a classical π -polarized pump field, a tripod four-level atom successively couples to two high-Q optical cavities possessing polarization degeneracy. Through stimulated Raman adiabatic passage process the polarization-entangled photon pairs can be produced. © 2008 Elsevier B.V. All rights reserved.
PACS: 03.67.Mn 42.50.Dv 42.50.Pq Keywords: EPR photon pairs Cavity QED
Quantum entanglement is one of the most valuable resources in quantum information science, which has many applications in the fields of quantum computation and quantum communication [1]. Recently great efforts have been made to controllably generate and detect entangled states, such as the Einstein–Podolsky–Rosen (EPR) state of two qubits [2,3], Greenberger–Horne–Zeilinger (GHZ) and W states of three qubits [4,5], and other multipartite entangled states [6–8]. Since photons are the ideal carriers of quantum information, a large number of theoretical and experimental schemes have been proposed for producing entangled photons. The traditional optical parametric down conversion method is used to produce entangled photon pairs [9], yet the process is stochastic in nature. In order to controllably generate the entangled photons, the cavity-QED scheme utilizing the coherent interaction and cavity field modes is proposed [10]. Cavity-QED offers an ideal system for the generation of entangled states and the implementation of quantum information processing. Experimental and theoretical progress in the entanglement with cavity-QED within the strong-coupling limit [11,12] has been made, such as entangled atoms [13–16], atom–photon entanglement [17] and entangled photons [18]. Here we propose a cavity-QED scheme composing of a four-level tripod atom and two cavities to produce the EPR entangled photon pairs.
*
Corresponding author. Tel.: +86 10 62752882; fax: +86 10 62756567. E-mail address: [email protected] (Y. Gu).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.07.065
To implement the cavity-QED schemes for generating entangled photons, one has to consider the effect of atomic spontaneous decay on the coherent evolution of the system. This decoherence process may induce harmful effects on entangled state. The stimulated Raman adiabatic passage (STIRAP) can be used to overcome this problem. STIRAP was first used to coherently control the dynamical processes in atoms and molecules [19]. It used the partially overlapping pulses to produce complete population transfer between two quantum ground states of an atom or molecule. In STIRAP, the population adiabatically follows the evolution of the dark state [20,21] and the excited state is never involved. Therefore, it is particularly robust against atomic spontaneous decay. The STIRAP technique is now widely used in the chemical reaction dynamics, laser-induced cooling, atom optics [19] and cavity-QED systems [22–25]. In the following, we make use of the STIRAP technique to prohibit the spontaneous decay in the process of producing entangled photon pairs. In this Letter, we present a cavity QED scheme which can effectively produce EPR polarization-entangled photon pairs. A fourlevel tripod atom successively couples to two single longitudinal mode high- Q optical cavities possessing polarization degeneracy, and this process is assisted by a classical π -polarized pump field. The spatial profiles of the two cavity modes and the pump field have to be overlapped, which can provide a counterintuitive pulse sequence and maintain the two-stage STIRAP process [19]. Stage 1 is to produce a σ + or σ − polarized photon in cavity 1 entangled with the atom by the first STIRAP. Stage 2 is to make the atom swap its entanglement with the photon in cavity 1 to the pho-
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the dipole and rotating wave approximations [26], the interaction Hamiltonian of the cavity-atom system at the present is (let h¯ = 1) †
†
H I1 = Ω(t )σeg + g 1 (t )a1+ σae + g 1 (t )a1− σbe + H.c.,
(1)
where σα β = |α β|, and ai ± is the σ ± circularly polarized photon creation operator in the corresponding mode. The system has the following dark state [20,21] †
1 | D 1 = sin θ √ |a|101 + |b|011 − cos θ| g |001 ⊗ |112 , 2
(a)
(2)
√
Ω(t )
†
where tan θ = g˜ (t ) , g˜ 1 (t ) = 2g 1 (t ), |101 = a1+ |001 , and |011 = 1 † a1− |001 . We then consider the details of STIRAP process. The pulse sequence is counterintuitive in the sense that the two initially empty levels are coupled first, and then the initially populated level is driven by the pump field. Moreover, the two field modes must overlap partially. If the couplings Ω(t ) and g 1 (t ) g 1 (t ) change slowly enough, and limt →∞ Ω( = 0, the system will t) start in the state | g ⊗ |001 ⊗ |112 and end up in the state √1 (|a|101 + |b |011 ) ⊗ |112 , following the adiabatic eigenstate 2
by Eq. (2). That is, when θ : 0 → π2 , (b) Fig. 1. (a) Tripod four-level atomic system under consideration. (b) Proposed setup for the production of polarization EPR entangled photon pairs.
ton in cavity 2 by a second STIRAP. At this stage a two-photon polarization-entangled state is generated and the atoms return to their ground states. The stimulated Raman adiabatic passage process has been utilized in the cavity–QED system, thus it is robust against atomic spontaneous decay. This proposal should have potential applications in quantum information processing. The system under investigation is shown in Fig. 1. It is composed of an atom and two identical high- Q cavities possessing polarization degeneracy. The atom has a tripod configuration and successively couples to the two high- Q optical resonators to produce the EPR photon pairs. The ground state of the atom is labeled as | g , the two metastable states as |a, |b, and the excited state as |e . The transitions |a → |e , and |b → |e are coupled by the cavity polarization degeneracy modes with the coupling coefficient g i (i = 1, 2), where i denotes the ith cavity. The transition | g → |e is driven by a classical π -polarized pump field with Rabi frequency Ω . The detunings for these transitions are Δ1 = ωe − ωa − ωc , Δ2 = ωe − ωb − ωc , and Δ3 = ωe − ω g − ω p , where ωc and ω p denote the cavity mode and the pump field frequency respectively, and ωα (α = a, b, e) denotes the atomic level energy. In order to implement the two-stage STIRAP processes, the pump field and the cavity modes have to be overlapped spatially. We assume that all of them have the Gaussian modes. We focus on the situation where the two-photon resonance is satisfied, i.e., Δ1 = Δ2 = Δ3 = Δ. The dark-state condition can make sure that the STIRAP in the cavity–QED system takes place. In the next paragraphs, we present the details of generating polarization-entangled photon pairs by STIRAP. Stage 1: Producing a photon in cavity 1 entangled with the four-level atom. Suppose that the atom is initially prepared in the ground state | g , cavity 1 in the vacuum state |001 , and cavity 2 in |112 = †
†
a2+ a2− |002 [22–25]. There are two pathways that the atom transfers from the ground state to the metastable states. After undergoing the STIRAP, the atom is prepared in state |a or |b with the same probability, and emits a σ + or σ − polarized photon. Under
1
| D 1 : | g ⊗ |001 ⊗ |112 → √
2
|a|101 + |b|011 ⊗ |112 .
(3)
At last, the atom emits a polarized photon and is entangled with that photon. Stage 2: The atom swapping its entanglement with the photon in cavity 1 to the photon in cavity 2. The atom enters cavity 2 which has been prepared in a twomode Fock state with just one photon in each mode. It then interacts with the photons in cavity 2. After undergoing the second STIRAP process, the atom absorbs one of the photons. Now the atom swaps its entanglement with the photon in cavity 1 to the photon left in cavity 2 and returns to the ground state. At this stage a two-photon EPR polarization-entangled state is prepared. The corresponding Hamiltonian described the coherent interaction is †
†
H I2 = Ω(t )σeg + g 2 (t )a2+ σae + g 2 (t )a2− σbe + H.c.
(4)
In this case, the dark state is
1
| D 2 = sin β √
2
|a|101 + |b|011 ⊗ |112
1 − cos β √ | g |101 |012 + |011 |102 , 2
where tan β =
Ω(t ) . g 2 (t )
(5)
If the couplings Ω(t ) and g 2 (t ) change
slowly, and let limt →∞
Ω(t ) g 2 (t )
= 0, the system will begin from
the state √1 (|a|101 + |b|011 ) ⊗ |112 and reach the state 2
√1
| g ⊗ (|101 |012 + |011 |102 ), following the adiabatic eigenstate by Eq. (5). That is, when β : π2 → 0, 2
1
|D2 : √
2
|a|101 + |b|011 ⊗ |112
1 → √ | g ⊗ |101 |012 + |011 |102 . 2
(6)
Finally, the atom returns to its ground state | g , and the two cavity photons of different polarization have been entangled with each other. This is the central result of this work.
P. Li et al. / Physics Letters A 372 (2008) 5959–5963
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In order to verify the above STIRAP processes, we solve the Schrödinger equation numerically. The coherent dynamics of the system is governed by i
d dt
Ψ (t ) = H I Ψ (t ) ,
(7)
where |Ψ is the state vector described the atom–cavity system, and H I is given by, H I = σee + Ω(t )σeg +
2
i =1
†
†
g i (t )ai + σae + g i (t )ai − σbe + H.c.
(8)
Fig. 2 displays the numerical results of the Schrödinger equation (7) under the basis {| A , | B , |C , | D , | E , | F , |G , | H }, where (a)
| A = | g ⊗ |001 ⊗ |112 , | B = |e ⊗ |001 ⊗ |112 , 1
|C = √ (|a|101 + |b|011 ) ⊗ |112 , 2 1
| D = √ (|101 |012 + |011 |102 ) ⊗ |e , 2
1
| E = √ (|101 |012 + |011 |102 ) ⊗ | g , 2 1
| F = √ (|a|101 − |b|011 ) ⊗ |112 , 2 1
|G = √ (|101 |012 − |011 |102 ) ⊗ |e , 2 1
| H = √ (|101 |012 − |011 |102 ) ⊗ | g .
(b)
2
The time evolution of the two cavity coupling g 1 (t ), g 2 (t ) and the Rabi frequency Ω(t ) of the pump field is shown in Fig. 2(a). Both the cavity modes and pump beam are assumed to have a Gaussian transverse shape, i.e., Ω(t ) = Ω0 e −2δt 2 −( t τc )
t 2 −( t −δ τp )
, g 1 (t ) = g 10 e
−( tτc )2
,
. In order to obtain the optimal results, and g 2 (t ) = g 20 e we have performed many numerical simulations with different parameters. We find that first the necessary condition for adiabatic following must be maintained, i.e., Ω0 τ p , 2g 0 τc 1 [24,25]; second, the transit time T for the atom passing through the cavities should be within their characteristic life time, i.e., T κ 1. Taking account of these factors, we find that a reasonable range of parameters giving the optimal results is 2 τc κ 1, and Ω0 T 1, g 0 T 1. Here we choose Ω0 = 50Γ , g 0 = 10Γ , τ p = 2.5Γ −1 , τc = 2.5Γ −1 , and δt = 4.5Γ −1 . Γ −1 is a characteristic time, with the value Γ ∼ κ 2π MHz for optical CQED and Γ ∼ 10κ 2π kHz for microwave CQED [17]. They are in line with the recent CQED experiments with high finesse optical resonators [11,12, 22,23] or microwave resonators [17]. In optical CQED experiments, the waists of the cavities may be about w c ∼ 20 μm and the velocity of the atom could be v ∼ 20 m/s, then the width of cavity modes would be about τc = w c / v ∼ 1 μs. In microwave CQED experiments, the cavity waist may be w c ∼ 6 mm, and the velocity of the atom could be v ∼ 0.5 km/s, then the cavity modes width would be about 12 μs [13]. The photon life time of microwave resonators could reach 1 ms [17]. In Fig. 2(b) the dynamics of the system is displayed with the time evolution of the cavity couplings g 1 (t ), g 2 (t ) and the pump Rabi frequency Ω(t ). The system starts from state | A , via the state |C , and eventually reaches the state | E . During the process, the states | B , | D , | F , |G , and | H are never involved. In order to see the dependence of the generation of polarization entangled photons on the initial Fock state in cavity 2, in Fig. 2(c) we also simulate the evolution of the system starting from an imperfectly
(c) Fig. 2. (a) Time evolution of the coupling g 1 (t ), g 2 (t ) and Rabi frequency Ω(t ). The parameters are chosen as, Ω0 = 50Γ , g 0 = 10Γ , τ p = 2.5Γ −1 , τc = 2.5Γ −1 , and δt = 4.5Γ −1 . (b) Coherent evolution of the cavity-atom system from initial state, | g |001 |112 . (c) The system evolution from an imperfectly prepared Fock state in cavity 2, ρ2 (0) = (1 − p )|002 00| + p |112 11| with different success probabilities p = 0.9 (Star) and p = 0.8 (UpTriangle).
achieved state ρ2 (0) = (1 − p )|002 00| + p |112 11|. Here p refers to the success probability of preparing the Fock state |11 in cavity 2. From Fig. 2(c) we see that if the initial Fock state in cavity 2 is not perfectly achieved in the first place, the final state could reach the polarization entangled-photon state with a probability up to p. It is necessary to check whether the results given above are sensitive to change of parameters. The mechanism of this proposal relies on STIRAP technique, which is insensitive to fluctuations of experimental parameters. For example, if we expect g 2 (t ) to be g 2 f at the end of this process, it is actually g 2 f + δ g 2 f due to some
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modes. As it has been discussed previously, the STIRAP process for generating entangled photon pairs is immune to atomic spontaneous decay. So now only the damping of cavity modes is considered. The evolution of density operator ρ (t ) in the presence of the cavity decay is described by the master equation [27]:
∂ρ = −i [ H I , ρ ] + L 1 ρ + L 2 ρ , ∂t
(9)
where the cavity dissipative terms are L1ρ = κ
†
†
†
†
†
†
2a1ξ ρ a1ξ − a1ξ a1ξ ρ − ρ a1ξ a1ξ ,
ξ =+,−
L2ρ = κ
2a2ξ ρ a2ξ − a2ξ a2ξ ρ − ρ a2ξ a2ξ ,
(10) (11)
ξ =+,− (a)
(b)
2κ is the one side decay of the two cavities, while the other side of the cavities are assumed to be perfectly reflecting. To solve the master equation numerically, we have used the Monte Carlo wave function (MCWF) formalism from the quantum trajectory methods [27,28]. The following results are averaged over enough realizations of quantum trajectories. The success probability of producing an entangled pair of photons after sending one atom through the cavities is calculated with respect to the full Hilbert space. However, fidelity and concurrence will be evaluated after postselection of events that yield one photon from each cavity. Fig. 3 depicts the numerical results of the master equation (9) in the presence of cavity dissipation. Here the parameters are chosen as in Fig. 2. We consider the evolution of the system toward the entangled states | E with the different cavity decay rate κ , i.e., κ 0.01g0 and 0.1g0 (g0 10Γ ). In Fig. 3(a), the cavity decay rate is κ 0.01g 0 , which represents the high- Q strong coupling situation. The system starts from the state | A , evolves into the entangled state | E with a probability P 0.80, i.e., the success probability of generating entangled photon pairs is 80%. The fidelity between the final state and the EPR state F = EPR|ρ f |EPR is higher than 90%, where the EPR state is defined as |EPR = √1 (|101 |012 + |011 |102 ). In Fig. 3(b), the cavity decay rate is 2
κ 0.1g0 , which generally corresponds to a strong coupling situ-
(c) Fig. 3. (a)–(b) Evolution of the system from exact calculations of the master equations. Parameters are chosen as in Fig. 2, but with different cavity decay rates, i.e., κ ∼ 0.01g 0 for (a), and 0.1g 0 for (b). (c) Probabilities for the different photon emission processes through the cavity mirrors and their ultimate photon-detection, parameters are as in (a).
fluctuations, such as the velocity fluctuation of the atom. However, the mechanism for generating the polarization entangled photons still works well because the condition | g 2 (t )|/|Ω(t )| 1 is still satisfied at this moment when the Rabi frequency Ω(t ) is switched off. Thus the results are insensitive to small change of parameters provided that the necessary conditions for STIRAP are satisfied. We now consider the dissipative effect on the coherent interaction of the four-level tripod atom and two cavity modes. This includes the spontaneous decay of atom and damping of cavity
ation. The success probability of producing the entangled photon pairs is about 50%. Thus we see that the cavity decay strongly affects the process of producing polarization entangled photons. Fig. 3(c) shows the probabilities of the different photon emission processes for the parameters from Fig. 3(a). In column (i) the success probability P of generating the state | E is displayed after sending one atom through the setup, while columns (ii)–(iv) show the probabilities for other physical processes. To further quantify the mixed state entanglement of the two polarized photons, we exploit the concurrence C (ρ ) [29], whose value goes from 0 to 1, meaning the mixed states go from factorizable to maximally entangled states. The quantity C (ρ ) is calculated after postselection of events that yield one photon from each cavity, which allows to confine to the subspace of two qubit defined by the polarizations of the photons. Let p 2 be the probability for coincidence photodetection. Then after postselection, the final density matrix of the photons will be
1 † † † † a a |00|a2+ a1− + a2− a1+ |00|a2− a1+ 2 2+ 1− + α |EPREPR|,
ρ f = (1 − α )
(12)
with α = P / p 2 and |0 = |001 |002 . In the two-qubit subspace † † † † † † † † {a1− a2− |0, a1− a2+ |0, a1+ a2− |0, a1+ a2+ |0}, we can calculate the √ √ √ √ concurrence C (ρ ) = max{ λ1 − λ2 − λ3 − λ4 , 0}, with λi the eigenvalues of the operator ρ f (σ y ⊗ σ y )ρ ∗f (σ y ⊗ σ y ). Trough direct calculations, the concurrence between the polarized photons are simply obtained as C (ρ f ) = α . In Fig. 4 we display
P. Li et al. / Physics Letters A 372 (2008) 5959–5963
5963
In summary, we have proposed a cavity quantum electrodynamics scheme that can effectively generate EPR polarizationentangled photon pairs, by means of a four-level tripod atom successively coupling to two high- Q optical cavities presenting polarization degeneracy. This proposal relies on the cavity–QED system and counterintuitive stimulated Raman adiabatic passage process. It is robust against atomic spontaneous decay and should have potential applications in quantum information processing. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants Nos. 10674009, 10521002, 10434020 and National Key Basic Research Program No. 2006CB921601. The discussions with Prof. Tiancai Zhang are appreciated. Pengbo Li acknowledges the useful discussions with Hongyan Li. References
(a)
(b) Fig. 4. (a) Concurrence C at the end of the process including post-selection vs. (b) Concurrence C over the process with cavity decay κ 0.1g.
κ / g0 .
the quantity C (ρ f ) as a function of cavity decay and time, respectively. Fig. 4(a) shows C (ρ f ) vs. cavity decay after postselection events. It shows that cavity decay can strongly influence the production of entangled photons even if postselection is utilized. When the cavity decay is large, the photons in cavity 2 are very likely to leak out before interacting with the atom. Then it will be much easier to detect two non-entangled photons. Fig. 4(b) illustrates C as a function of the time t over the process. We implement the STIRAP process within the relatively strong coupling regime. At the end of the experiment and after post-selecting the events that yield one photon from each cavity, we obtain the concurrence C = 0.81 for the parameters of Fig. 2. The parameters from the recent cavity QED experiments with high finesse optical and microwave resonators are ( g 0 , κ )/2π (16, 1.4) MHz [30], ( g 0 , κ )/2π (16, 3.8) MHz [31], and ( g 0 , κ )/2π ∼ (47, 1) kHz [17], in line with the regime of the present scheme.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Effective generation of polarization-entangled photon pairs in a cavity-QED system Pengbo Li a , Ying Gu a,∗ , Qihuang Gong a , Guangcan Guo a,b a b
State Key Laboratory for Mesoscopic Physics, Department of Physics, Peking University, Beijing 100871, China Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
a r t i c l e
i n f o
Article history: Received 7 April 2008 Received in revised form 16 July 2008 Accepted 30 July 2008 Available online 5 August 2008 Communicated by P.R. Holland
a b s t r a c t We propose a cavity-QED scheme to effectively generate Einstein–Podolsky–Rosen polarization-entangled photon pairs. Assisted by a classical π -polarized pump field, a tripod four-level atom successively couples to two high-Q optical cavities possessing polarization degeneracy. Through stimulated Raman adiabatic passage process the polarization-entangled photon pairs can be produced. © 2008 Elsevier B.V. All rights reserved.
PACS: 03.67.Mn 42.50.Dv 42.50.Pq Keywords: EPR photon pairs Cavity QED
Quantum entanglement is one of the most valuable resources in quantum information science, which has many applications in the fields of quantum computation and quantum communication [1]. Recently great efforts have been made to controllably generate and detect entangled states, such as the Einstein–Podolsky–Rosen (EPR) state of two qubits [2,3], Greenberger–Horne–Zeilinger (GHZ) and W states of three qubits [4,5], and other multipartite entangled states [6–8]. Since photons are the ideal carriers of quantum information, a large number of theoretical and experimental schemes have been proposed for producing entangled photons. The traditional optical parametric down conversion method is used to produce entangled photon pairs [9], yet the process is stochastic in nature. In order to controllably generate the entangled photons, the cavity-QED scheme utilizing the coherent interaction and cavity field modes is proposed [10]. Cavity-QED offers an ideal system for the generation of entangled states and the implementation of quantum information processing. Experimental and theoretical progress in the entanglement with cavity-QED within the strong-coupling limit [11,12] has been made, such as entangled atoms [13–16], atom–photon entanglement [17] and entangled photons [18]. Here we propose a cavity-QED scheme composing of a four-level tripod atom and two cavities to produce the EPR entangled photon pairs.
*
Corresponding author. Tel.: +86 10 62752882; fax: +86 10 62756567. E-mail address: [email protected] (Y. Gu).
0375-9601/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.07.065
To implement the cavity-QED schemes for generating entangled photons, one has to consider the effect of atomic spontaneous decay on the coherent evolution of the system. This decoherence process may induce harmful effects on entangled state. The stimulated Raman adiabatic passage (STIRAP) can be used to overcome this problem. STIRAP was first used to coherently control the dynamical processes in atoms and molecules [19]. It used the partially overlapping pulses to produce complete population transfer between two quantum ground states of an atom or molecule. In STIRAP, the population adiabatically follows the evolution of the dark state [20,21] and the excited state is never involved. Therefore, it is particularly robust against atomic spontaneous decay. The STIRAP technique is now widely used in the chemical reaction dynamics, laser-induced cooling, atom optics [19] and cavity-QED systems [22–25]. In the following, we make use of the STIRAP technique to prohibit the spontaneous decay in the process of producing entangled photon pairs. In this Letter, we present a cavity QED scheme which can effectively produce EPR polarization-entangled photon pairs. A fourlevel tripod atom successively couples to two single longitudinal mode high- Q optical cavities possessing polarization degeneracy, and this process is assisted by a classical π -polarized pump field. The spatial profiles of the two cavity modes and the pump field have to be overlapped, which can provide a counterintuitive pulse sequence and maintain the two-stage STIRAP process [19]. Stage 1 is to produce a σ + or σ − polarized photon in cavity 1 entangled with the atom by the first STIRAP. Stage 2 is to make the atom swap its entanglement with the photon in cavity 1 to the pho-
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the dipole and rotating wave approximations [26], the interaction Hamiltonian of the cavity-atom system at the present is (let h¯ = 1) †
†
H I1 = Ω(t )σeg + g 1 (t )a1+ σae + g 1 (t )a1− σbe + H.c.,
(1)
where σα β = |α β|, and ai ± is the σ ± circularly polarized photon creation operator in the corresponding mode. The system has the following dark state [20,21] †
1 | D 1 = sin θ √ |a|101 + |b|011 − cos θ| g |001 ⊗ |112 , 2
(a)
(2)
√
Ω(t )
†
where tan θ = g˜ (t ) , g˜ 1 (t ) = 2g 1 (t ), |101 = a1+ |001 , and |011 = 1 † a1− |001 . We then consider the details of STIRAP process. The pulse sequence is counterintuitive in the sense that the two initially empty levels are coupled first, and then the initially populated level is driven by the pump field. Moreover, the two field modes must overlap partially. If the couplings Ω(t ) and g 1 (t ) g 1 (t ) change slowly enough, and limt →∞ Ω( = 0, the system will t) start in the state | g ⊗ |001 ⊗ |112 and end up in the state √1 (|a|101 + |b |011 ) ⊗ |112 , following the adiabatic eigenstate 2
by Eq. (2). That is, when θ : 0 → π2 , (b) Fig. 1. (a) Tripod four-level atomic system under consideration. (b) Proposed setup for the production of polarization EPR entangled photon pairs.
ton in cavity 2 by a second STIRAP. At this stage a two-photon polarization-entangled state is generated and the atoms return to their ground states. The stimulated Raman adiabatic passage process has been utilized in the cavity–QED system, thus it is robust against atomic spontaneous decay. This proposal should have potential applications in quantum information processing. The system under investigation is shown in Fig. 1. It is composed of an atom and two identical high- Q cavities possessing polarization degeneracy. The atom has a tripod configuration and successively couples to the two high- Q optical resonators to produce the EPR photon pairs. The ground state of the atom is labeled as | g , the two metastable states as |a, |b, and the excited state as |e . The transitions |a → |e , and |b → |e are coupled by the cavity polarization degeneracy modes with the coupling coefficient g i (i = 1, 2), where i denotes the ith cavity. The transition | g → |e is driven by a classical π -polarized pump field with Rabi frequency Ω . The detunings for these transitions are Δ1 = ωe − ωa − ωc , Δ2 = ωe − ωb − ωc , and Δ3 = ωe − ω g − ω p , where ωc and ω p denote the cavity mode and the pump field frequency respectively, and ωα (α = a, b, e) denotes the atomic level energy. In order to implement the two-stage STIRAP processes, the pump field and the cavity modes have to be overlapped spatially. We assume that all of them have the Gaussian modes. We focus on the situation where the two-photon resonance is satisfied, i.e., Δ1 = Δ2 = Δ3 = Δ. The dark-state condition can make sure that the STIRAP in the cavity–QED system takes place. In the next paragraphs, we present the details of generating polarization-entangled photon pairs by STIRAP. Stage 1: Producing a photon in cavity 1 entangled with the four-level atom. Suppose that the atom is initially prepared in the ground state | g , cavity 1 in the vacuum state |001 , and cavity 2 in |112 = †
†
a2+ a2− |002 [22–25]. There are two pathways that the atom transfers from the ground state to the metastable states. After undergoing the STIRAP, the atom is prepared in state |a or |b with the same probability, and emits a σ + or σ − polarized photon. Under
1
| D 1 : | g ⊗ |001 ⊗ |112 → √
2
|a|101 + |b|011 ⊗ |112 .
(3)
At last, the atom emits a polarized photon and is entangled with that photon. Stage 2: The atom swapping its entanglement with the photon in cavity 1 to the photon in cavity 2. The atom enters cavity 2 which has been prepared in a twomode Fock state with just one photon in each mode. It then interacts with the photons in cavity 2. After undergoing the second STIRAP process, the atom absorbs one of the photons. Now the atom swaps its entanglement with the photon in cavity 1 to the photon left in cavity 2 and returns to the ground state. At this stage a two-photon EPR polarization-entangled state is prepared. The corresponding Hamiltonian described the coherent interaction is †
†
H I2 = Ω(t )σeg + g 2 (t )a2+ σae + g 2 (t )a2− σbe + H.c.
(4)
In this case, the dark state is
1
| D 2 = sin β √
2
|a|101 + |b|011 ⊗ |112
1 − cos β √ | g |101 |012 + |011 |102 , 2
where tan β =
Ω(t ) . g 2 (t )
(5)
If the couplings Ω(t ) and g 2 (t ) change
slowly, and let limt →∞
Ω(t ) g 2 (t )
= 0, the system will begin from
the state √1 (|a|101 + |b|011 ) ⊗ |112 and reach the state 2
√1
| g ⊗ (|101 |012 + |011 |102 ), following the adiabatic eigenstate by Eq. (5). That is, when β : π2 → 0, 2
1
|D2 : √
2
|a|101 + |b|011 ⊗ |112
1 → √ | g ⊗ |101 |012 + |011 |102 . 2
(6)
Finally, the atom returns to its ground state | g , and the two cavity photons of different polarization have been entangled with each other. This is the central result of this work.
P. Li et al. / Physics Letters A 372 (2008) 5959–5963
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In order to verify the above STIRAP processes, we solve the Schrödinger equation numerically. The coherent dynamics of the system is governed by i
d dt
Ψ (t ) = H I Ψ (t ) ,
(7)
where |Ψ is the state vector described the atom–cavity system, and H I is given by, H I = σee + Ω(t )σeg +
2
i =1
†
†
g i (t )ai + σae + g i (t )ai − σbe + H.c.
(8)
Fig. 2 displays the numerical results of the Schrödinger equation (7) under the basis {| A , | B , |C , | D , | E , | F , |G , | H }, where (a)
| A = | g ⊗ |001 ⊗ |112 , | B = |e ⊗ |001 ⊗ |112 , 1
|C = √ (|a|101 + |b|011 ) ⊗ |112 , 2 1
| D = √ (|101 |012 + |011 |102 ) ⊗ |e , 2
1
| E = √ (|101 |012 + |011 |102 ) ⊗ | g , 2 1
| F = √ (|a|101 − |b|011 ) ⊗ |112 , 2 1
|G = √ (|101 |012 − |011 |102 ) ⊗ |e , 2 1
| H = √ (|101 |012 − |011 |102 ) ⊗ | g .
(b)
2
The time evolution of the two cavity coupling g 1 (t ), g 2 (t ) and the Rabi frequency Ω(t ) of the pump field is shown in Fig. 2(a). Both the cavity modes and pump beam are assumed to have a Gaussian transverse shape, i.e., Ω(t ) = Ω0 e −2δt 2 −( t τc )
t 2 −( t −δ τp )
, g 1 (t ) = g 10 e
−( tτc )2
,
. In order to obtain the optimal results, and g 2 (t ) = g 20 e we have performed many numerical simulations with different parameters. We find that first the necessary condition for adiabatic following must be maintained, i.e., Ω0 τ p , 2g 0 τc 1 [24,25]; second, the transit time T for the atom passing through the cavities should be within their characteristic life time, i.e., T κ 1. Taking account of these factors, we find that a reasonable range of parameters giving the optimal results is 2 τc κ 1, and Ω0 T 1, g 0 T 1. Here we choose Ω0 = 50Γ , g 0 = 10Γ , τ p = 2.5Γ −1 , τc = 2.5Γ −1 , and δt = 4.5Γ −1 . Γ −1 is a characteristic time, with the value Γ ∼ κ 2π MHz for optical CQED and Γ ∼ 10κ 2π kHz for microwave CQED [17]. They are in line with the recent CQED experiments with high finesse optical resonators [11,12, 22,23] or microwave resonators [17]. In optical CQED experiments, the waists of the cavities may be about w c ∼ 20 μm and the velocity of the atom could be v ∼ 20 m/s, then the width of cavity modes would be about τc = w c / v ∼ 1 μs. In microwave CQED experiments, the cavity waist may be w c ∼ 6 mm, and the velocity of the atom could be v ∼ 0.5 km/s, then the cavity modes width would be about 12 μs [13]. The photon life time of microwave resonators could reach 1 ms [17]. In Fig. 2(b) the dynamics of the system is displayed with the time evolution of the cavity couplings g 1 (t ), g 2 (t ) and the pump Rabi frequency Ω(t ). The system starts from state | A , via the state |C , and eventually reaches the state | E . During the process, the states | B , | D , | F , |G , and | H are never involved. In order to see the dependence of the generation of polarization entangled photons on the initial Fock state in cavity 2, in Fig. 2(c) we also simulate the evolution of the system starting from an imperfectly
(c) Fig. 2. (a) Time evolution of the coupling g 1 (t ), g 2 (t ) and Rabi frequency Ω(t ). The parameters are chosen as, Ω0 = 50Γ , g 0 = 10Γ , τ p = 2.5Γ −1 , τc = 2.5Γ −1 , and δt = 4.5Γ −1 . (b) Coherent evolution of the cavity-atom system from initial state, | g |001 |112 . (c) The system evolution from an imperfectly prepared Fock state in cavity 2, ρ2 (0) = (1 − p )|002 00| + p |112 11| with different success probabilities p = 0.9 (Star) and p = 0.8 (UpTriangle).
achieved state ρ2 (0) = (1 − p )|002 00| + p |112 11|. Here p refers to the success probability of preparing the Fock state |11 in cavity 2. From Fig. 2(c) we see that if the initial Fock state in cavity 2 is not perfectly achieved in the first place, the final state could reach the polarization entangled-photon state with a probability up to p. It is necessary to check whether the results given above are sensitive to change of parameters. The mechanism of this proposal relies on STIRAP technique, which is insensitive to fluctuations of experimental parameters. For example, if we expect g 2 (t ) to be g 2 f at the end of this process, it is actually g 2 f + δ g 2 f due to some
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P. Li et al. / Physics Letters A 372 (2008) 5959–5963
modes. As it has been discussed previously, the STIRAP process for generating entangled photon pairs is immune to atomic spontaneous decay. So now only the damping of cavity modes is considered. The evolution of density operator ρ (t ) in the presence of the cavity decay is described by the master equation [27]:
∂ρ = −i [ H I , ρ ] + L 1 ρ + L 2 ρ , ∂t
(9)
where the cavity dissipative terms are L1ρ = κ
†
†
†
†
†
†
2a1ξ ρ a1ξ − a1ξ a1ξ ρ − ρ a1ξ a1ξ ,
ξ =+,−
L2ρ = κ
2a2ξ ρ a2ξ − a2ξ a2ξ ρ − ρ a2ξ a2ξ ,
(10) (11)
ξ =+,− (a)
(b)
2κ is the one side decay of the two cavities, while the other side of the cavities are assumed to be perfectly reflecting. To solve the master equation numerically, we have used the Monte Carlo wave function (MCWF) formalism from the quantum trajectory methods [27,28]. The following results are averaged over enough realizations of quantum trajectories. The success probability of producing an entangled pair of photons after sending one atom through the cavities is calculated with respect to the full Hilbert space. However, fidelity and concurrence will be evaluated after postselection of events that yield one photon from each cavity. Fig. 3 depicts the numerical results of the master equation (9) in the presence of cavity dissipation. Here the parameters are chosen as in Fig. 2. We consider the evolution of the system toward the entangled states | E with the different cavity decay rate κ , i.e., κ 0.01g0 and 0.1g0 (g0 10Γ ). In Fig. 3(a), the cavity decay rate is κ 0.01g 0 , which represents the high- Q strong coupling situation. The system starts from the state | A , evolves into the entangled state | E with a probability P 0.80, i.e., the success probability of generating entangled photon pairs is 80%. The fidelity between the final state and the EPR state F = EPR|ρ f |EPR is higher than 90%, where the EPR state is defined as |EPR = √1 (|101 |012 + |011 |102 ). In Fig. 3(b), the cavity decay rate is 2
κ 0.1g0 , which generally corresponds to a strong coupling situ-
(c) Fig. 3. (a)–(b) Evolution of the system from exact calculations of the master equations. Parameters are chosen as in Fig. 2, but with different cavity decay rates, i.e., κ ∼ 0.01g 0 for (a), and 0.1g 0 for (b). (c) Probabilities for the different photon emission processes through the cavity mirrors and their ultimate photon-detection, parameters are as in (a).
fluctuations, such as the velocity fluctuation of the atom. However, the mechanism for generating the polarization entangled photons still works well because the condition | g 2 (t )|/|Ω(t )| 1 is still satisfied at this moment when the Rabi frequency Ω(t ) is switched off. Thus the results are insensitive to small change of parameters provided that the necessary conditions for STIRAP are satisfied. We now consider the dissipative effect on the coherent interaction of the four-level tripod atom and two cavity modes. This includes the spontaneous decay of atom and damping of cavity
ation. The success probability of producing the entangled photon pairs is about 50%. Thus we see that the cavity decay strongly affects the process of producing polarization entangled photons. Fig. 3(c) shows the probabilities of the different photon emission processes for the parameters from Fig. 3(a). In column (i) the success probability P of generating the state | E is displayed after sending one atom through the setup, while columns (ii)–(iv) show the probabilities for other physical processes. To further quantify the mixed state entanglement of the two polarized photons, we exploit the concurrence C (ρ ) [29], whose value goes from 0 to 1, meaning the mixed states go from factorizable to maximally entangled states. The quantity C (ρ ) is calculated after postselection of events that yield one photon from each cavity, which allows to confine to the subspace of two qubit defined by the polarizations of the photons. Let p 2 be the probability for coincidence photodetection. Then after postselection, the final density matrix of the photons will be
1 † † † † a a |00|a2+ a1− + a2− a1+ |00|a2− a1+ 2 2+ 1− + α |EPREPR|,
ρ f = (1 − α )
(12)
with α = P / p 2 and |0 = |001 |002 . In the two-qubit subspace † † † † † † † † {a1− a2− |0, a1− a2+ |0, a1+ a2− |0, a1+ a2+ |0}, we can calculate the √ √ √ √ concurrence C (ρ ) = max{ λ1 − λ2 − λ3 − λ4 , 0}, with λi the eigenvalues of the operator ρ f (σ y ⊗ σ y )ρ ∗f (σ y ⊗ σ y ). Trough direct calculations, the concurrence between the polarized photons are simply obtained as C (ρ f ) = α . In Fig. 4 we display
P. Li et al. / Physics Letters A 372 (2008) 5959–5963
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In summary, we have proposed a cavity quantum electrodynamics scheme that can effectively generate EPR polarizationentangled photon pairs, by means of a four-level tripod atom successively coupling to two high- Q optical cavities presenting polarization degeneracy. This proposal relies on the cavity–QED system and counterintuitive stimulated Raman adiabatic passage process. It is robust against atomic spontaneous decay and should have potential applications in quantum information processing. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants Nos. 10674009, 10521002, 10434020 and National Key Basic Research Program No. 2006CB921601. The discussions with Prof. Tiancai Zhang are appreciated. Pengbo Li acknowledges the useful discussions with Hongyan Li. References
(a)
(b) Fig. 4. (a) Concurrence C at the end of the process including post-selection vs. (b) Concurrence C over the process with cavity decay κ 0.1g.
κ / g0 .
the quantity C (ρ f ) as a function of cavity decay and time, respectively. Fig. 4(a) shows C (ρ f ) vs. cavity decay after postselection events. It shows that cavity decay can strongly influence the production of entangled photons even if postselection is utilized. When the cavity decay is large, the photons in cavity 2 are very likely to leak out before interacting with the atom. Then it will be much easier to detect two non-entangled photons. Fig. 4(b) illustrates C as a function of the time t over the process. We implement the STIRAP process within the relatively strong coupling regime. At the end of the experiment and after post-selecting the events that yield one photon from each cavity, we obtain the concurrence C = 0.81 for the parameters of Fig. 2. The parameters from the recent cavity QED experiments with high finesse optical and microwave resonators are ( g 0 , κ )/2π (16, 1.4) MHz [30], ( g 0 , κ )/2π (16, 3.8) MHz [31], and ( g 0 , κ )/2π ∼ (47, 1) kHz [17], in line with the regime of the present scheme.
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