Implementation of Controlled Geometric Phase Gate

Commun. Theor. Phys. (Beijing, China) 54 (2010) pp. 835–839 c Chinese Physical Society and IOP Publishing Ltd

Vol. 54, No. 5, November 15, 2010

Implementation of Controlled Geometric Phase Gate for Two Trapped Neutral Atoms∗

Ç~“),

WU Huai-Zhi (

†


I), and ZHENG Shi-Biao (x¸I)

‡

YANG Zhen-Biao (

Department of Physics and State Key Laboratory Breeding Base of Photocatalysis, Fuzhou University, Fuzhou 350002, China

(Received March 18, 2010)

Abstract We propose a scheme for realizing a controlled geometric phase gate for two neutral atoms. We apply the stimulated Raman adiabatic passage to transfer atoms from their ground states into Rydberg excited states, and use the Rydberg interaction induced energy shifts to generate geometric phase and construct quantum gates. PACS numbers: 03.67.Lx, 32.80.Ee, 32.80.Rm

Key words: neutral atom, geometric phase gate, Rydberg interaction The observation of Rydberg excitation blockade in many-body atomic system[1] and single-atom setup[2] has recently been achieved. Remarkably, the entanglement[3] and controlled-NOT gate[4] between two Rydberg atoms have also been demonstrated. The experimental progress prompts the scientists to further explore the physical realization of quantum information processing in neutral atomic system. The implementation of quantum phase gate using Rydberg interaction was firstly proposed by Jaksch et al.[5] The mechanism is then generalized for preparing collective spin states and constructing scalable quantum logic gates in mesoscopic atomic ensemble.[6] The important application of Rydberg blockade excitation in quantum network has been realized by Saffman and Walk, who used it to create the single-photon sources and entangle the single-atom qubits with collective atomic ensemble.[7] The energy shifts induced by dipole-dipole interaction can not only be used for implementing twoqubit dynamic phase gates and geometric phase gates,[8−9] but also for construction of multi-qubit quantum gates[10] and preparation of multi-qubit entangled states.[11] More recently, two interesting theoretical schemes for implementing mesoscopic Rydberg gates based on electromagnetically induced transparency (EIT)[12] and generating Greenberger–Horne–Zeilinger (GHZ) states using asymmetric Rydberg interaction[13] are presented, which promoted the quantum computation with neutral atom towards the more promising and scalable way. In this paper, a theoretical scheme for realizing a twoqubit controlled geometric phase gate via Rydberg interaction is proposed. The neutral atoms are excited to Rydberg states via the stimulated Raman adiabatic passage (STIRAP). The energy level shifts due to Rydberg interaction, which is used for constructing geometric phase gate. The detailed process is discussed in what follows. Consider two atoms 1 and 2 stored in two separated dipole traps collectively coupling to the driving fields. The relevant single-atom energy level structure is shown in ∗ Supported

Fig. 1. The atoms have two ground states |0i and |1i, and have two Rydberg excited states |si and |pi. The ground state |1i and Rydberg state |pi are resonantly coupled to the high-lying Rydberg state |si by the smooth laser pulses with Rabi frequencies Ωs and Ωp , respectively. The strong Rydberg interaction between states |si leads to an energy gap Uss = ~∆ss . The asymmetric Rydberg interaction X Hsp = ~∆sp |sii hs| ⊗ |pij hp | (i 6= j) i,j=1,2

results in energy shifts when the states |si and |pi are simultaneously populated. The atoms in state |pi interact weakly with each other giving rise to energy shift ∆pp . The interaction strengths fulfill the relation 0 < ∆pp ≪ ∆ss , ∆sp . On the other hand, in what follows we denote the atomic ground states |0i and |1i as logic zero and one.

Fig. 1 Level scheme of the atoms resonantly interacting with laser pulses of Rabi frequencies Ωs,p on the transitions |1i ↔ |si and |si ↔ |pi, respectively. ∆ss , ∆sp , and ∆pp are energy shifts induced by Rydberg interaction.

We firstly assume that the two-atom system is initially in the state |1i1 |1i2 . Then the temporal evolution of the system remains in the subspace spanned by S9 = {|1i1 |1i2 , |1i1 |si2 , |si1 |1i2 , |si1 |si2 , |1i1 |pi2 , |pi1 |1i2 , |si1 |pi2 , |pi1 |si2 , |pi1 |pi2 }. The associated Hamiltonian, in the interaction picture, can be depicted by

by the National Natural Science Foundation of China under Grant No. 10974028, the Doctoral Foundation of the Ministry of Education of China under Grant No. 20093514110009, the Natural Science Foundation of Fujian Province under Grant No. 2009J06002, and Funds from the State Key Laboratory Breeding Base of Photocatalysis, Fuzhou University † E-mail: [email protected] ‡ Corresponding author, E-mail: [email protected]

836

WU Huai-Zhi, YANG Zhen-Biao, and ZHENG Shi-Biao

H=

hΩ

Vol. 54

i Ωp (I1 ⊗ |pi2 hs| + |pi1 hs| ⊗ I2 ) + H.c. + ∆pp |pi1 hp| ⊗ |pi2 hp| 2 2 + ∆ss |si1 hs| ⊗ |si2 hs| + ∆sp (|si1 hs| ⊗ |pi2 hp| + |pi1 hp| ⊗ |si2 hs|) , s

(I1 ⊗ |1i2 hs| + |1i1 hs| ⊗ I2 ) +

(1)

where Ik = |1ik h1| + |sik hs| + |pik hp|, k = 1, 2. In order to see the physical insight of the geometric phase, we switch to the interaction picture with respect to Hpp = ∆pp |pi1 hp| ⊗ |pi2 hp| and rewrite H as hΩ Ωp i∆pp t s (I1 ⊗ |1i2 hs| + |1i1 hs| ⊗ I2 ) + (e |pi1 hp| ⊗ |pi2 hs| + |1i1 h1| ⊗ |pi2 hs| HI = 2 2 i + |si1 hs| ⊗ |pi2 hs| + |pi1 hs| ⊗ |1i2 h1| + |pi1 hs| ⊗ |si2 hs| + ei∆pp t |pi1 hs| ⊗ |pi2 hp| + H.c. + ∆sp (|si1 hs| ⊗ |pi2 hp| + |pi1 hp| ⊗ |si2 hs|) + ∆ss |si1 hs| ⊗ |si2 hs| .

(2)

The above Hamiltonian can be expressed as a matrix in the subspace spanned by S9 ,   0 Ωs /2 Ωs /2 0 0 0 0 0 0  Ω /2 0 0 Ωs /2 Ωp /2 0 0 0 0   s     Ωs /2 0 0 Ωs /2 0 Ωp /2 0 0 0       0  Ω /2 Ω /2 ∆ 0 0 Ω /2 Ω /2 0 s s ss p p     HI =  0 Ωp /2 0 0 0 0 Ωs /2 0 0 ,    0 0 Ωp /2 0 0 0 0 Ωs /2 0      ′∗   0 0 0 Ω /2 Ω /2 0 ∆ 0 Ω /2 p s sp p       0 0 0 Ωp /2 0 Ωs /2 0 ∆sp Ω′∗ /2 p 0

0

0

0

where Ω′p = Ωp ei∆pp t and we have assumed that Ωs and Ωp are real. The Hamiltonian HI has a unique dark state, which is the normalized eigenvector corresponding to the E = 0 null eigenvalue, |D11 (t)i = cos2 θ|1i1 |1i2 − cos θ sin θ(|1i1 |pi2 + |pi1 |1i2 ) + ei∆pp t sin2 θ|pi1 |pi2 ,

where

Ωs sin θ = q , Ω2s + Ω2p

φ=i

Z

0

t

hD11 (t)|D˙ 11 (t)idt = −∆pp

Z

t

sin4 θdt .

(5)

(6)

0

Then, we rewrite the system state in the Schr¨odinger picture, |Ψs (t)i = e−iHpp t |ΨI (t)i

= eiφ (cos2 θ|1i1 |1i2 − cos θ sin θ(|1i1 |pi2 + |pi1 |1i2 ) + sin2 θ|pi1 |pi2 ) .

Ω′p /2 Ω′p /2

0

During the time period that all lasers are turned off, the Hamiltonian of the entire system is reduced to H = ∆pp |pi1 hp| ⊗ |pi2 hp| .

(9)

|Ψs (t1 + δt)i −→ e−i∆pp δt eiφ(t1 ) |pi1 |pi2 ,

(10)

ϕ = φ(t1 ) + φ(t2 ) − ∆pp δt

(12)

Thus the system state introduces another phase factor e−i∆pp δt , that is,

Applying the counterintuitive pulse sequences of Fig. 2(a) to the two atoms, under a certain condition, the system state will adiabatically follow the evolution of the trapped state |D11 i. Due to the Rydberg interaction between states |pi, we consequently obtain a geometric phase factor eiφ , i.e., with

0

ends at time t = t1 , then the system is transferred to the state |Ψs (t1 )i = eiφ(t1 ) |pi1 |pi2 . (8)

(4)

Ωp cos θ = q . Ω2s + Ω2p

|ΨI (0)i = |1i1 |1i2 → |ΨI (t)i = eiφ |D11 i ,

0

(3)

(7)

The temporal evolution of the system state has been showed in Fig. 2(b). Suppose the first STIRAP sequence

where δt is the time interval between the two STIRAP sequences. After the second STIRAP sequence with time span t2 , the system is eventually transferred back to the state |Ψs (t1 + δt + t2 )i = eiϕ |1i1 |1i2 , (11) with being the global phase accumulated in the whole process. We note that the geometric phase ϕ comes about as a result of the Rydberg interaction induced energy shift ∆pp . Now we turn to the adiabatic condition, which requires the nonadiabatic coupling between |D11 i and other eigenstates of HI to be small compared to the laser-induced splitting of the eigen energies of these states. Considering the energy shift ∆pp , the adiabatic condition can be depicted by[14] |hE|D˙ 11 i| ≪ δ(E, ∆pp ) , (13)

No. 5

Implementation of Controlled Geometric Phase Gate for Two Trapped Neutral Atoms

where δ(E, ∆pp ) = ||E|2 /E − ∆pp |. E (E 6= 0) is an arbitrary eigenvalue of HI , and |Ei is the corresponding eigenstate. Since our goal is to realize the transformation |1i1 |1i2 → eiϕ |1i1 |1i2 and ϕ is proportional to ∆pp , the gate operation time can be shorten if we increase the magnitude of ∆pp . However, the criteria of adiabaticity requires δ(E, ∆pp ) ≫ 0. On the other hand, insufficient coupling by the laser pulses in reversed order (and thereby the lesser Rabi frequencies) may prevent the

837

system state from adiabatically following the evolution of the dark state. Thus, loss of population due to nonadiabatic transfer to the other eigenstates may occur. We have searched the practicable parameters for the perfect transfer of the system state by performing extensive numerical studies of the fidelity of the final state. It is found that the criterion of adiabaticity can be satisfied for ∆pp < 1 MHz with (Ωs0 , Ωp0 )/2π > 30 MHz, τ = 1 µs and td = 0.78 µs.

Fig. 2 Numerical simulation of the two-qubit geometric π-phase gate. (a) Profiles of the Rabi frequencies Ωp and Ωs for the transitions |pi ↔ |si and |1i ↔ |si, respectively. The laser pulses are modeled by gauss pulses, Ωp (t) = Ωp0 exp [−[(t + T0 /2 + td )/(τ /2)]2 ], Ωs (t) = Ωs0 exp [−[(t + T0 /2)/(τ /2)]2 ] for −5 < t < 0, and Ωp (t) = Ωp0 exp [−[(t − T0 /2)/(τ /2)]2 ], Ωs (t) = Ωs0 exp [−[(t − T0 /2 − td )/(τ /2)]2 ] for 0 ≤ t < 5. The optimum pulse delay and the optimum time delay of the STIRAP sequences are considered. The amplitudes of the Rabi frequencies are (Ωp0 , Ωs0 )/2π = 50 MHz. The FWHM of each pulse is τ = 1 µs, and the time delay between pulses within one STIRAP process is td = 0.78 µs. The two Ωs pulses in the two different STIRAP sequences are seperated by T0 = 3.325 µs; (b) Temporal evolution of the probability amplitudes of √ the states |pi1 |pi2 , |0i1√ |0i2 , |1i1 |pi2 , |1i1 |1i2 and |0i1 |1i2 . The system is assumed to be initially in the state (1/ 2) (|0i1 + |1i1 ) ⊗ (1/ 2)(|0i2 + |1i2 ), and the atomic distance d is 2.2 µm.

Second, the two-atom system initially in the state |0i1 |1i2 (|1i1 |0i2 ) can only evolve in the subspace spanned by S3 = {|0i1 |1i2 , |0i1 |si2 , |0i1 |pi2 } (R3 = {|1i1 |0i2 , |si1 |0i2 , |pi1 |0i2 }). This is exactly the case that a threelevel atom is coherently coupled to the counterintuitive laser pulses.[12] Thus, the Rydberg excitation blockade does not exist and the geometric phase is equal to zero. The correlative Hamiltonian can be easily given by Ωs Ωp H3 = |1ik hs| + |pik hs| + H.c. , (14) 2 2 where k = 1 or 2. It is straightforward to read the dark

states |D10 i = cos θ|1i1 |0i2 − sin θ|pi1 |0i2 ,

(15)

|D01 i = cos θ|0i1 |1i2 − sin θ|0i1 |pi2 .

(16)

To ensure the time evolution of the system adiabatically follows the Eqs. (15) and (16), we have to choose 1q 2 Ωs + Ω2p ∆τ ≫ 1 , 2 where ∆τ is the time period during which the pulses overlap. In this case, there are no Rydberg interactions between the two atoms. Thus, no geometric phases can be

838

WU Huai-Zhi, YANG Zhen-Biao, and ZHENG Shi-Biao

obtained, i.e., Z t Z t hD01 (t)|D˙ 01 (t)idt = hD10 (t)|D˙ 10 (t)idt = 0 . (17) 0

0

In Fig. 2(b), we have numerically testified that the populations of the states |0i1 |1i2 and |1i1 |0i2 finally revert to initial distribution after the four smooth laser pulses. Finally, the interaction of the atoms with laser fields can not happen if the two atoms are initially in the state |0i. The system keeps staying in |D00 i = |0i1 |0i2 . In general, only the state |1i1 |1i2 can obtain the geometric phase ϕ. Therefore, by appropriately choosing the parameters of the STIRAP sequences, we can successfully implement the two-qubit geometric phase gate with the transformation |0i1 |0i2 → |0i1 |0i2 , |0i1 |1i2 → |0i1 |1i2 ,

|1i1 |0i2 → |1i1 |0i2 , |1i1 |1i2 → eiϕ |1i1 |1i2 .

(18)

Now we firstly address the experimental feasibility of the geometric phase gate. Then, we numerically demonstrate the realization of two-qubit π-phase gate. The scheme is based on strong asymmetric Rydberg interaction, i.e., ∆ss , ∆sp ≫ ∆pp , which can be found by using Rydberg states |si = |41s1/2 , m = 1/2i and |pi = |40 p3/2 , m = 1/2i in Rb with the interatomic separation d = 2.2 µm.[15] The spontaneous emission rates for states |si and |pi are approximately equal to (γs , γp ) = 1/57 MHz. Applying a weak external magnetic field to all atoms, the Rydberg interaction will couple the atomic Zeeman states with different magnetic quantum numbers. Thus, the interaction strength ∆sp never equals to zero regardless of the angle between dipoles, and scales with 1/R3 . The energy shift ∆pp is induced by F¨orster process,[16] and has a scaling of 1/R6 in the van der Waals limit. Therefore, we have ∆sp = ∆sp (d)(d/R)3 and ∆pp = ∆pp (d)(d/R)6 , where d and R denote the characteristic and practical interatomic distance, respectively. The angle-dependent ratio ∆sp /∆pp is larger than 150 for arbitrary angles. We will characterize the interactions, after averaging all spatial angles between dipoles, ¯ sp (d), ∆ ¯ pp (d))/2π = (36.5, 0.122) MHz for simplicby (∆ ity. It is also found that the interaction strength between Rydberg states |si1 and |si2 is ∆ss (d)/2π = 23.8 MHz, which is isotropic but decreases with 1/R6 .[17] Taking into account the dissipation due to spontaneous emission of the atoms, the master equation of motion for density matrix ̺ of the entire system is then given by 2 γs X − + + − + − (2σjk ̺σjk − σjk σjk ̺ − ̺σjk σjk ) ̺˙ = −i[H, ̺] + 2

Vol. 54

final density matrix of the system. The fidelity of the geometric π-phase gate as a function of T0 is plotted in Fig. 3. It can be found that the gate error is less than 10−4 for T0 = 3.325 µs without considering the spontaneous decay (rs = rp = 0). However, the gate operation time is limited by the strength of Rydberg interaction and the adiabatic condition. The effect of spontaneous emission decay on the gate fidelity can not be neglected. Thus, the fidelity overall decreases and the maximum is limited to 0.932. The gate fidelity is associated with the time delay of the two STIRAP sequences, however, it is still larger than 0.91 with the deviation of time delay being ∆T0 = 0.1T0 . On the other hand, fluctuation of the interatomic separation may dramatically change the gate fidelity, as shown in Fig. 4. Thus, the atoms should be trapped in deep Lamb–Dicke regime.

Fig. 3 The gate fidelity F as a function of T0 . The upper line (dashed): rs = rp = 0. The lower line (solid): (γs , γp ) = (1/57) MHz. The STIRAP sequences are given in Fig. 2(a).

Fig. 4 The gate fidelity F versus the interatomic separation R with (γs , γp ) = (1/57) MHz. The STIRAP sequences are given in Fig. 2(a).

j,k=1

γp X − + − + + (2σ + ̺σ − − σj2 σj2 ̺ − ̺σj2 σj2 ) , 2 j=1,2 j2 j2

(19)

− − where σj1 = |1ij hs|, σj2 = |pij hs|. Suppose the system is √ √ initially in the state (1/ 2) (|0i1 + |1i1 ) ⊗ (1/ 2)(|0i2 + |1i2 ). We then calculate the fidelity of the geometric phase gate with ϕ = π by Tr(̺ideal ̺), where ̺ideal is the ideal

It is noted that the asymmetric interaction between |si and |pi scales as ∆sp ∼ n4 (n is the principle quantum number) and the second order F¨orster process induced ∆pp scales as ∆pp ∼ n11 . Moreover, the blackbody limited spontaneous emission lifetime scales as τs,p ∼ n2 . Now if we can use n = 80 states for |si and n = 79 states for |pi, the lifetime of the Rydberg levels will be

No. 5

Implementation of Controlled Geometric Phase Gate for Two Trapped Neutral Atoms

839

larger than 200 µs. Setting R = 8 µm, then the interaction strengths are on the order of (∆sp , ∆ss , ∆pp )/2π = (11, 16, 0.1) MHz. Combining these parameters, we can also implement the geometric phase gate by choosing the more smooth laser pulses with (Ωs0 , Ωp0 )/2π > 8 MHz. In summary, we have presented a scheme for implementing a controlled geometric phase gate for two trapped neutral atoms. When the atoms are initially in the state

|1i, the energy shifts due to Rydberg interaction ∆sp and ∆ss remove the degeneracy of dark states, which make it possible to simply control the unique dark state |D11 i. The magnitude of the geometric phase is determined by the product of ∆pp and the time scale of the STIRAP sequences. We have theoretically analyzed the gate errors induced by nonadiabatic coupling and spontaneous emission decay.

References

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