Generation of three-dimensional entangled states between a single [PDF]

arXiv:1201.0248v2 [quant-ph] 19 May 2015

Generation of three-dimensional entangled state between a single atom and a Bose-Einstein condensate via adiabatic passage Li-Bo Chen,1,2 Peng Shi,1 Chun-Hong Zheng,1,2 and Yong-Jian Gu 1,∗ 1 Department 2 School

of Physics, Ocean University of China, Qingdao 266100, China of Science, Qingdao Technological University, Qingdao 266033, China ∗ [email protected]

Abstract: Inspired by a recently experiment by M. Lettner et al. [Phys. Rev. Lett. 106, 210503 (2011)], we propose a robust scheme to prepare three-dimensional entanglement state between a single atom and a Bose-Einstein condensate (BEC) via stimulated Raman adiabatic passage (STIRAP) technique. The atomic spontaneous radiation, the cavity decay, and the fiber loss are efficiently suppressed by the engineering adiabatic passage. Our strictly numerical simulation shows our proposal is good enough to demonstrate the generation of three-dimensional entanglement with high fidelity and within the current experimental technology. © 2018 Optical Society of America OCIS codes: (270.0270) Quantum optics; (270.5585) Quantum information and processing.

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1.

Introduction

Quantum entanglement plays a vital role in many practical quantum information system, such as quantum teleportation [1], quantum dense coding [2], and quantum cryptography [3]. Entangled states of higher-dimensional systems are of great interest owing to the extended possibilities they provide, which including higher information density coding [4], stronger violations of local realism [5,6], and more resilience to error [7] than two dimensional system. Over the past few years, fairish attention has been paid to implement higher-dimensional entanglement with trapped ions [8, 9], photons [10–12], and cavity QED [13–16]. Moreover, it has been shown that entanglement between two spatially separated subsystems is very useful for distributed quantum computation [17, 18]. Recently, a large number of schemes have been proposed for generating entangled state of atoms, which are individually trapped in distant optical cavities connected by fibers [19–25]. The main problems in entangling atoms in these schemes are the decoherence due to leakage of photons from the cavity and fiber modes, and spontaneous radiation of the atoms [26]. By using the stimulated Raman adiabatic passage (STIRAP) [27–34], our scheme can overcome these problems. The idea of STIRAP is that the system is initially prepared in a decoherence-free state (dark state), and evolve adiabatically along the dark state to the required state by two delayed but partially overlapping pulses. Many schemes have been proposed to prepare entanglement state via STIRAP [35–43]. The Bose-Einstein condensate (BEC) has many advantages over other systems such as long storage times, the high write-read efficiencies, and excellent internal-state preparation [44, 45]. Recently, remote entanglement between a single atom and BEC was experimentally realized [46]. But the efficiency is very low due to the photon loss. In this paper, we takes both the advantages of cavity-fiber system and STIRAP in order to create three-dimensional entanglement state between a single 87 Rb atom and a 87 Rb BEC at a distance. The atom and BEC are placed inside two high-finesse optical cavities respectively, which connected by an optical fibre. The atom–light interaction is identical for all atoms of the BEC and enhanced greatly because

the atoms collectively couple to the same light mode [47, 48]. The entanglement state can be generated with highly fidelity even in the range that the cavity decay and spontaneous radiation of the atoms are comparable with the atom-cavity coupling strength. Our scheme is also robust to the variation of atom number in the BEC. As a result, the highly fidelity three-dimensional entanglement state of the BEC and atom can be realized base on our proposed scheme. This paper is organized as follows. In Sec. 2, we introduce the basic model of our system. In Sec. 3, the generation of the three-dimensional entanglement state is provided. In Sec. 4, we demonstrate the influences of atomic spontaneous radiation, photon leakage out of the cavities and fiber on the implementation. Finally, in Sec. 5, we discuss experimental feasibility of our scheme and conclude our results. 2.

The fundamental model

We consider the situation describe in Fig. 1, where a single 87 Rb atom and a 87 Rb BEC are trapped in two distant double-mode optical cavities, which are connected by an optical fiber (see Fig. 1). The 87 Rb atomic levels and transitions are also depicted in this Fig. [46, 49, 50]. The states |gL i, |g0 i, |gR i and |ga i correspond to |F = 1, mF = −1i, |F = 1, mF = 0i, |F = 1, mF = 1i of 5S1/2 and |F = 2, mF = 0i of 5S1/2 , while |eL i, |e0 i and |eR i correspond to |F = 1, mF = −1i, |F = 1, mF = 0i and |F = 1, mF = 1i of 5P3/2. The atomic transition |ga i ↔ |e0 i of atom in cavity A is driven resonantly by a π -polarized classical field with Rabi frequency ΩA ; |e0 iA ↔ |gL iA (|e0 iA ↔ |gR iA ) is resonantly coupled to the cavity mode aA,L (aA,R) with coupling constant gA . The atomic transition |gL iB ↔ |eL iB (|gR iB ↔ |eR iB ) of BEC in cavity B is driven resonantly by a π -polarized classical field with Rabi frequency ΩB ; |eR iB ↔ |g0 iB (|eL iB ↔ |g0 iB ) is resonantly coupled to the cavity mode aB,L (aB,R ) with coupling constant gB . Here we consider BEC for a single excitation, the and single ex ground √
  citation states are described by the state vectors G f = 1/ N ∑Nj=1 g f j ⊗Nk=1,k6= j |g0 i j and   √
E f = 1/ N ∑N e f ⊗N |g i ( f = 0, L, R), where |...i j describe the state of the jth j=1 j k=1,k6= j 0 j atom in the BEC [46]. Initially, if the atom and BEC are prepared in the states |ga iA and |G0 iB respectively, and the cavities and fiber modes are in the vacuum states. In the rotating wave approximation, the interaction Hamiltonian of the atom (BEC)-cavity system can be written as (setting h¯ = 1) [47] √ Hac = ∑ (ΩA (t) |e0 iA hga | + gA (t)aA,k |e0 iA hgk | + NΩB (t) |Ek iB hGk |) k=L,R

√ √ + NgB (t)aB,L |ER iB hG0 | + NgB (t)aB,R |EL iB hG0 | + H.c.,

(1)

In the short fibre limit, the coupling between the cavity fields and the fiber modes can be written as the interaction Hamiltonian [20, 23, 24] h   i + Hc f = ∑ νk bk a+ (2) A,k + aB,k + H.c. . k=L,R

In the interaction picture the total Hamiltonian now becomes HI = Hac + Hc f . 3.

(3)

Generation of the three-dimensional entanglement state

In this section, we begin to investigate the generation of the three-dimensional entangled state in detail. The time evolution of the whole system state is governed by the Schr¨odinger equation i

∂ |ψ (t)i = HI |ψ (t)i . ∂t

(4)

A

B

atom BEC

F =1

eR

e0

eL

laser A

5S1/2 F = 1

aL

ga

F = 2 5S1/2

ga gL

g0

gR

gL

F = 1 5P3/2

π

π

π 5S1/2 F = 2

eR

e0

aR

laser B

aR

aL

eL

laser B

5P3/2

g0

gR

F = 1 5S1/2

Fig. 1. A single 87 Rb atom and a 87 Rb BEC are trapped in two distant double-mode optical cavities, which are connected by an optical fiber. The states |gL i, |g0 i, |gR i and |ga i correspond to |F = 1 , mF = −1i, |F = 1 , mF = 0i, |F = 1 , mF = 1i of 5S1/2 and |F = 2 , mF = 0i of 5S1/2 , while |eL i, |e0 i and |eR i correspond to |F = 1 , mF = −1i, |F = 1 , mF = 0i and |F = 1 , mF = 1i of 5P3/2 . The atomic transition |ga i ↔ |e0 i of atom in cavity A is driven resonantly by a π -polarized classical field with Rabi frequency
ΩA ; |e0 iA ↔ |gL iA (|e0 iA ↔ |gR iA ) is resonantly coupled to the cavity mode aA,L aA,R with coupling constant gA . The atomic transition |gL iB ↔ |eL iB (|gR iB ↔ |eR iB ) of BEC in cavity B is driven resonantly by a π -polarized classical field with Rabi frequency
ΩB ; |eR iB ↔ |g0 iB (|eL iB ↔ |g0 iB ) is resonantly coupled to the cavity mode aB,L aB,R with coupling constant gB .

The single excitation subspace can be spanned by the following state vectors [51] |φ1 i |φ2 i |φ3 i

= |ga iA |G0 iB |0000ic |00i f ,

= |e0 iA |G0 iB |0000ic |00i f , = |gL iA |G0 iB |1000ic |00i f ,

|φ4 i

= |gR iA |G0 iB |0100ic |00i f ,

|φ7 i |φ8 i

= |gL iA |G0 iB |0010ic |00i f , = |gR iA |G0 iB |0001ic |00i f ,

|φ5 i |φ6 i

|φ9 i |φ10 i

= |gL iA |G0 iB |0000ic |10i f , = |gR iA |G0 iB |0000ic |01i f ,

= |gL iA |ER iB |0000ic |00i f , = |gR iA |EL iB |0000ic |00i f ,

|φ11 i = |gL iA |GR iB |0000ic |00i f , |φ12 i = |gR iA |GL iB |0000ic |00i f ,

(5)

where |nAL nAR nBL nBR ic denotes the field state with nAi (i = L, R) photons in the i polarized mode of cavity A, nBi in the i polarized mode of cavity B, and |nL nR i f represents ni photons in i polarized mode of the fiber. The Hamiltonian HI has the following dark state: |D(t)i =

K{2gA(t)ΩB (t) |φ1 i − ΩA(t)ΩB (t) [|φ3 i + |φ4 i − |φ7 i − |φ8 i] −gB (t)ΩA (t) [|φ11 i + |φ12 i]},

(6)

which is the eigenstate of the Hamiltonian corresponding to zero eigenvalue. Here and in the following gi , Ωi are real, and K −2 = g2A Ω2B + 4Ω2AΩ2B + 2g2BΩ2A . Under the condition

we have

gA (t), gB (t) ≫ ΩA (t), ΩB (t),

(7)

|D(t)i ∼ 2gA (t)ΩB (t) |φ1 i − gB(t)ΩA (t) [|φ11 i + |φ12 i] .

(8)

Suppose the initial state of the system is |φ1 i, if we design pulse shapes such that gB (t)ΩA (t) gA (t)ΩB (t) gA (t)ΩB (t) lim t→+∞ gB (t)ΩA (t) lim

t→−∞

= 0, = 12 ,

(9)

we can adiabatically transfer the initial state |φ1 i to a equal superposition of |φ1 i, |φ11 i and √ |φ12 i, i.e., 1/ 3(|ga iA |G0 iB − |gL iA |GR iB − |gR iA |GL iB ) |0000ic |00i f , which is a product state of the three-dimensional atom-BEC entangled state, the cavity mode vacuum state, and the fiber mode vacuum state. The pulse shapes and sequence can be designed by an appropriate choice of the parameters. The coupling rates are chosen such that gA (t) = ghB (t) = g, νL = νR =i

ν = 100g, N = 104 , laser Rabi frequencies are chosen as ΩA (t) = Ω0 exp − (t − t0 )2 /200τ 2 h i   and ΩB (t) = Ω0 exp −t 2 /200τ 2 + Ω20 exp − (t − t0 )2 /200τ 2 , with t0 = 20τ being the delay between pulses [52]. Fig. 2 shows the simulation results of the entanglement generation process, where we choose g = 5Ω0 , τ = Ω−1 0 . With this choice, conditions (7) and (8) can be well satisfied. The Rabi frequencies of ΩA (t), ΩB (t) are shown in Fig. 2(a). Fig. 2(b) and 2(c) shows the time evolution of populations. In Fig. 2(b) P1 , P11 , and P12 denote the populations of the states |φ1 i, |φ11 i, and |φ12 i. Fig. 2 (c) show the time evolution of populations of other states {|φ2 i, |φ3 i, |φ4 i, |φ5 i, |φ6 i, |φ7 i, |φ8 i, |φ9 i, |φ10 i}, which are almost zero during the whole dynamics. Finally P1 , P11 , and P12 arrive at 1/3, which means the successful generation of the 3-dimensional entangled state. Figure 2(d) shows the error probability defined by [53]: Pe (t) = 1 − |hD (t)| ϕs (t)i|2 ,

(10)

here |ϕs (t)i is the state obtained by numerical simulation of Hamiltonian (3) and |D(t)i is the dark state defined by Eq. (6). From the Fig. 2(a)-(d) we conclude that we can prepare the threedimensional entanglement state between single atom and a BEC with high success probability.

4.

Effects of spontaneous emission and photon leakage

To evaluate the performance of our scheme, we now consider the dissipative processes due to spontaneous decay of the atoms from the excited states and the decay of cavity. We assess the effects through the numerical integration of the master equation for the system in the Lindblad form. The master equation for the density matrix of whole system can be expressed as [25] dρ = dt


κfk + + bk bk ρ − 2bk ρ b+ k + ρ bk bk k=L,R 2
+ + a+ ik aik ρ − 2aik ρ aik + ρ aik aik ]

−i [HI , ρ ] − −

κik i=A,B 2

∑

∑

[

Ω0

1.0

ΩB

0.5

(a) Ω

Populations

Populations

A

0.0 1.0 P1 P11(P12) 0.0 5.0

−3

x10

(c)

2.5 0.0 1.0

Pe

(b)

0.5

−3

x10

(d)

0.5 0.0 −30

−20

−10

0

10 Ω0t

20

30

40

50

Fig. 2. The numerical simulation of Hamiltonian (3) in the entanglement generation process, where we choose g = 5Ω0 , τ = Ω−1 0 . (a): the Rabi frequency of ΩA (t), ΩB (t). (b): the time evolution of populations of the states |φ1 i, |φ11 i, and |φ12 i is denoted by P1 , P11 , and P12 respectively. (c): time evolution of populations of other states {|φ2 i, |φ3 i, |φ4 i, |φ5 i, |φ6 i, |φ7 i, |φ8 i, |φ9 i, |φ10 i} , which are almost zero during the whole dynamics. (d): error probability Pe (t) defined by Eq. (6).

−

∑

j=a,L,R

γ0Aj  2

σeA0 e0 ρ − 2σgAj e0 ρσeA0 g j + ρσeA0 e0



 γkBhj  Bh ∑ ∑ 2 σek ek ρ − 2σgBhj ek ρσeBhk g j + ρσeBhk ek , h=1 k=L,R j=k,0 N

−∑

(11)

 where γ0Aj and γkBhj denote the spontaneous radiation rates from state |e0 iA to g j A and |ek iB to  g j of the hth atom in the BEC, respectively; κik and κ f k denote the photon leakage rates from B i = |mi hn| (m, n = e , e , g ) are the usual the cavity fields and fiber modes, respectively; σmn j 0 k i Pauli matrices. Starting with the initial density matrix |φ1 i hφ1 |, by solving numerically Eq. (11) in the subspace spanned by the vectors (5) and |φ13 i = |gL iA |G0 iB |0000ic |00i f , |φ14 i = |gR iA |G0 iB |0000ic |00i f . Fig. 3 shows the fidelity of the entanglement state as a function of
the photon leakage rate κ κ = κAk = κBk = κf k and for the atom spontaneous radiation rate

γ γ = ∑ j=a,L,R γ0Aj = ∑ j=L,0 γkBhj = ∑ j=R,0 γkBhj = 0, 0.2g, 0.4g, 0.6g, 0.8g, 1.0g (from the top

A = γ A = γ /3, γ Bh = γ Bh = to the bottom). In the calculation, for simplicity we choose γ0a 0k k0 kk γ /2 (k = l, r), the other parameters same as in Fig. 2. From the Fig. 3 we can see that the entanglement state can be generated with highly fidelity even in the range of γ , κ ∼ g.

5.

Discussion and conclusion

It is necessary to briefly discuss the experimental feasibility of our scheme. Firstly, trapping 87 Rb BEC in cavity QED has also been realized in recently experiment [47]. In this experiment, the atom number can be selected between 2, 500 and 200, 000 and the relevant cavity QED parameter (g, κ , γ ) = 2π × (10.6, 1.3, 3.0) MHz is realizable. So the condition γ , κ < 0.4g can be satisfied with these system parameters for entangling the BEC and atom with fidelity larger than 98%. Secondly, the classical fields Rabi frequency can be selected by changing the laser density in principle. The strong coupling between two cavities by a waveguide has been experimental realized [54]. The coupling strength can be reached as high as 25 GHz, which is much larger than atom-cavity coupling strength and the strength of the classical fields.

1.000

γ=0

0.995 γ = 0.2g 0.990 fidelity

γ = 0.4g 0.985 γ = 0.6g 0.980 γ = 0.8g 0.975 γ=g 0.970 0

0.2

0.4

κ/g

0.6

0.8

1

Fig. 3. Fidelity of the entanglement state (obtained by numerical simulation of master equation (8)) as a function of the photon leakage rate κ and for the atom spontaneous radiation rate γ = 0, 0.2g, 0.4g, 0.6g, 0.8g, 1.0g (from the top to the bottom).

Finally, atoms in BEC do not fulfill the requirement of identical coupling, but it shows a similar energy spectrum, which can be modeled by the Tavis-Cummings Hamiltonian with an effective √
collective coupling geBf f = gµ (N), here µ (N) = 0.5 1 − 0.0017N 0.34 is the overlap between BEC spatial atomic mode and cavity mode [47,55]. So the coupling strength gB (t) will decrease with increasing atom number N. We can increase the ΩB (t) accordingly to compensate this. One challenge here is photoassociation driven by the classical laser because it gradually reduces the BEC atom number N [46]. The fidelity as a function of the atom number N of the BEC is plotted in Fig. 4 with the parameters γ = κ = 0.4g, and the other parameters same as in Fig. 2. From the Fig. 2, we can see that our scheme is robust to the variation of atom number in the BEC. Of course if the lost atoms carry away the single excitation, the scheme will be fail. 1

0.995

0.99

Fidelity

0.985

0.98

0.975

0.97

0.965

0.96

1

10

100

1000

10000

100000

Atom Number (N)

Fig. 4. Fidelity vs the atom number N of the BEC with the parameters γ = κ = 0.4g, and the other parameters same as in Fig. 2.

In summary, based on the STIRAP technique, we propose a scheme to prepare threedimensional entanglement state between a BEC and a atom. In this scheme, the atomic spontaneous radiation and photon leakage can be efficiently suppressed, since the populations of the excited states of atoms and cavity (fiber) modes are almost zero in the whole process. We also

show that this scheme is highly stable to the variation of atom number in the BEC. Recently, strong atom–field coupling for Bose–Einstein condensates in an optical cavity on a chip [56] and strong coupling between distant photonic nanocavities [54] have been experimentally realized. So our scheme is considered as a promising scheme for realizing entanglement between BEC and atom on a photonic chip. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 60677044, 11005099), the Fundamental Research Funds for the central universities (Grant No. 201013037). L. Chen was also supported in part by the Government of China through CSC (Grant No. 2009633075).