Dipole-dipole interaction driven antiblockade of two Rydberg atoms
RRydberg antiblockade with resonant dipole-dipole interactions
Shi-Lei Su
School of Physics, Zhengzhou University, Zhengzhou 450001, China
We perform a comprehensive investigation of the resonant Rydberg dipole-dipole interaction basedantiblockade regimes for different Rydberg-Rydberg interaction types that have been observed inexperiment. By using the dressed state method, the laser coupled terms were rewritten with respectto the dressed state formed by the strong and resonant dipole-dipole interaction, based on which wecan calculate the effective dynamics and further get the Rydberg antiblockade (RAB) condition. Wethen study the possible applications of the proposed RAB regimes, including the geometric quantumcomputation, dissipative dynamics based entanglement preparation, and possible applications insome physical parameters estimation. Our study enriches the RAB regime since it goes beyond theusual vdW-type based RAB, and may be get more attention for the experimental and theoreticalstudy in neutral atoms in the near future.
I. INTRODUCTION
Long-range interaction would be exhibited betweenhighly excited Rydberg atoms with principal quantumnumber n (cid:29) V d is much smaller than driv-ing Rabi frequency Ω ( V d (cid:29) Ω), Ref. [19] showed thatonce the dark state which contains three excited Rydbergatoms was populated, it would keep invariant. The two-qubit case of the atoms are interacting with a zero-areaphase-jump pulse was also studied in Ref. [20]. Besides,under the parameter range V d ∼ Ω, the dynamics thattwo-excitation Rydberg states involve in the evolution arealso discussed to limit the blockade error [21]. Along withdevelopment of the technology in optical control [22], mi-crowave control [23–28] and electric field control [3, 29–43] of the resonant dipole-dipole RRI, the implementa-tion of these schemes may also be guaranteed.In fact, with the condition V v (cid:28) Ω, where V v de-notes van der Waals (vdW)-type RRI strength, the two-atom-excitation was considered earlier for the construc-tion of quantum logic gate in Ref. [2]. Also, the RABwas studied under the condition V v (cid:29) Ω [44, 45], andhas been studied for motional effects [46], dissipative dy-namics [47–49], periodically driving [50] as well as con-struction of quantum gates [51]. Very recently, RAB was also studied in strongly interacting Rydberg atom experi-ment [52], trapped Rydberg ion chain [53], and cold atomensemble [54].In this manuscript, we would perform comprehensiveinvestigations for the resonant dipole-dipole-RRI-basedantiblockade under the regime V d (cid:29) Ω and briefly discussits applications. The resonant two-body dipole-dipoleRRIs modulated by the external fields can be roughlyclassified as the following types. The first type is theF¨orster resonance, such as | d (cid:105)| d (cid:105) ↔ | p (cid:105)| f (cid:105) + | f (cid:105)| p (cid:105) [24, 33–35], and | p (cid:105)| p (cid:105) ↔ | s (cid:105)| s (cid:48) (cid:105) + | s (cid:48) (cid:105)| s (cid:105) [36, 38]. The secondtype is “spin-exchange”-type RRI interaction | s (cid:105)| p (cid:105) ↔| p (cid:105)| s (cid:105) [37], | p (cid:105)| d (cid:105) ↔ | d (cid:105)| p (cid:105) [55] or | s (cid:105)| p (cid:48) (cid:105) ↔ | p (cid:105)| s (cid:48) (cid:105) [28].And the third type is “collective exchange”-type RRI, | s (cid:105)| s (cid:48) (cid:105) ↔ | p (cid:105)| p (cid:48) (cid:105) [36, 39–42] or | d (cid:105)| d (cid:48) (cid:105) ↔ | f (cid:105)| f (cid:48) (cid:105) [43].In addition, there may be another form of RRI type, | p (cid:105)| p (cid:48) (cid:105) ↔ | s (cid:105)| d (cid:105) [31] or | d (cid:105)| d (cid:48) (cid:105) ↔ | p (cid:105)| f (cid:105) [43]. The strongdipole-dipole interactions facilitate the dressed states,based on which we show how to achieve the RAB forall of the above mentioned resonant types.The rest content of the manuscript is organized as fol-lows: In Sec. II, we illustrate the basic theory of themanuscript, including the models extracted from experi-ment and the dressed state method. In Sec. III, we studyhow to achieve the RAB regime and plot the populationevolution dynamics with the models shown in Sec. II. InSec. IV, we show the potential applications of the pro-posed RAB with one model as an example. The conclu-sion is given in Sec. V. II. BASIC THEORYA. Models
As shown in Fig. 1, we consider how to achieve theeffective RAB dynamical process (a) for three types ofresonant dipole-dipole interactions (b) (c) and (d), re-spectively, with dressed state method. For panel (b),we consider the experimental configuration [34] | p (cid:105) ≡| P / , m J = 1 / (cid:105) , | d (cid:105) ≡ | D / , m J = 3 / (cid:105) and | f (cid:105) ≡ | F / , m J = 5 / (cid:105) of two Rb atoms. By ap- a r X i v : . [ qu a n t - ph ] J un Effective 𝑉 𝑑 𝑉 𝑑 𝑉 𝑑 ۧ|𝑝 Dressed states ۧ|1ۧ|0 ۧ|1ۧ|0 ۧ|1ۧ|0 ۧ|1 ۧ|0 ۧ|1ۧ|0 ۧ|1ۧ|0ۧ|11 ۧ|10 ۧ|00ۧ|01 (a) (b) (c) (d) ۧ|𝑑ۧ|𝑓 ۧ|𝑝 ۧ|𝑑 ۧ|𝑓 ۧ|𝑑ۧ|𝑝 ۧ|𝑑 ۧ|𝑝 ۧ|𝑝 ۧ|𝑠 ۧ|𝑠′ ۧ|𝑝′
FIG. 1. The diagrammatic sketch of the RAB with resonantdipole-dipole interaction. (a) The effective dynamics undertwo-atom basis with dressed states. Panels (b) (c) and (d)show the laser drivings to achieve the RAB for the f¨orsterresonance, “spin exchange” and “collective exchange” RRIs,respectively. V d means the RRI strength. | s (cid:105) , | s (cid:48) (cid:105) , | p (cid:105) , | d (cid:105) and | f (cid:105) denote Rydberg states. | (cid:105) and | (cid:105) are two ground states.For simplicity, we label the left atom as atom 1 and the rightatom as atom 2 throughout this manuscript for panels (b),(c) and (d). plying an electric fields (cid:15) (cid:39)
32 mV cm − , these Ry-dberg states can be brought to exact resonance with C = 2 .
54 GHz µm . One of the states in computationalspace is chosen as | (cid:105) ≡ | S / , F = 2 , m F = 2 (cid:105) [34] andthe other states | (cid:105) in computational subspace is decou-pled with the excitation process and may be chosen as | (cid:105) ≡ | S / , F = 1 , m F = 0 (cid:105) . The excitation is accom-plished by two-photon process with two lasers of wave-lengths 795 nm( π polarization) and 474 nm( σ + polariza-tion) [34].For panel (c), we consider the experimental config-uration as [55] | d (cid:105) ≡ | D / , m J = 3 / (cid:105) , | p (cid:105) ≡| P / , m J = 1 / (cid:105) . These two Rydberg states are res-onant with each other and C = 7 .
965 GHz µm . Oneof the ground states are chosen as | (cid:105) ≡ | S / , F =2 , m F = 2 (cid:105) [55] and the rest computational state canbe chosen as | (cid:105) ≡ | S / , F = 1 , m F = 0 (cid:105) . The exci-tation process from | (cid:105) to state | d (cid:105) is accomplished bytwo-photon transition with wavelengths 795 nm ( π po-larization) and 474 nm ( σ + polarization), respectively.In this manuscript, we also consider the single-photonexcitation process from | (cid:105) to | p (cid:105) [55].For panel (d), we consider the experimental config-uration as [39], | s (cid:105) ≡ | S / , m J = 1 / (cid:105) , | p (cid:105) ≡| P / , m J = 1 / (cid:105) , | s (cid:48) (cid:105) ≡ | S / , m J = 1 / (cid:105) , | p (cid:48) (cid:105) ≡| P / , m J = 1 / (cid:105) . These states are resonant with eachother when the electric fields (cid:15) = 710 mV cm − andthe value of C is about 0.6 GHz µm . Two groundstates in computational subspace can be chosen as | (cid:105) ≡| S / , F = 2 , m F = 0 (cid:105) and | (cid:105) ≡ | S / , F = 1 , m F =0 (cid:105) [39]. The excitation process from | (cid:105) to | s (cid:105) or | s (cid:48) (cid:105) canbe implemented by two-photon process [39]. B. Dressed states
To show the basis of dressed states, we here temporar-ily consider Hamiltonianˆ H = ˆ H Ω + ˆ H d , (1)where ˆ H Ω = (cid:80) k =1 Ω / | (cid:105) k (cid:104) R | +H.c. denote the exci-tation process from | (cid:105) to Rydberg state and ˆ H d = V d | Rr (cid:105)(cid:104) rR | denotes the resonant dipole-dipole interac-tions. If V d (cid:29) Ω, the eigenstates of ˆ H v , i.e., | Π ± (cid:105) witheigenvalues E ± , can be used to rewrite Eq. (1) asˆ H = (cid:88) j =+ , − ˆ H Ω | Π j (cid:105)(cid:104) Π j | + E j | Π j (cid:105)(cid:104) Π j | . (2)For concrete Rydberg atom system, one can calculate thefirst term of Eq. (2) and rotate with respect to the sec-ond term to see the systematic dynamics more clearlyand further choose parameters to achieve the desired dy-namics. In this process, | Π ± (cid:105) can be called as dressedstates. III. RYDBERG ANTIBLOCKADE WITHRESONANT DIPOLE-DIPOLE INTERACTIONSA. RAB with f¨orster resonance
As shown in Fig. 1(b), consider two Rydberg atomsand each atom has two ground states | (cid:105) and | (cid:105) , andthree Rydberg states | s (cid:105) , | p (cid:105) and | d (cid:105) . Bichromaticclassical fields are imposed on these two atoms to off-resonantly drive the transition | (cid:105) ↔ | d (cid:105) with an identicalRabi frequency Ω but opposite detuning ∆ through two-photon process. After the rotating-wave approximation,the Hamiltonian for this concrete system can be writtenas (let (cid:126) = 1)ˆ H Ω = Ω2 (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105) (cid:104) d | + | (cid:105) (cid:104) d | ) + H . c . ˆ H d = √ V d | dd (cid:105)(cid:104) r pf | + H . c . (3)where | mn (cid:105) denotes the abbreviation of | m (cid:105) ⊗ | n (cid:105) andwould be used throughout this manuscript. | r pf (cid:105) is de-fined as | r pf (cid:105) ≡ ( | pf (cid:105) + | f p (cid:105) ) / √
2. Following the processin Sec. II B, one can diagonalize ˆ H d as √ V d ( | + (cid:105)(cid:104) + | −|−(cid:105)(cid:104)−| ) with |±(cid:105) ≡ ( | dd (cid:105) ± | r pf (cid:105) ) / √
2. Then the Hamil-tonian can be written asˆ H Ω = Ω √ (cid:0) e i ∆ t + e − i ∆ t (cid:1) (cid:2) | (cid:105)(cid:104) Ψ | + 1 √ | Ψ (cid:105) ( (cid:104) + | + (cid:104)−| ) (cid:3) + Ω2 (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105)(cid:104) d | + | (cid:105)(cid:104) d | ) + H . c . ˆ H d = √ V d ( | + (cid:105)(cid:104) + | − |−(cid:105)(cid:104)−| ) , (4)in which | Ψ (cid:105) ≡ ( | d (cid:105) + | d (cid:105) ) / √
2. When ∆ = 0, Hamil-tonian ˆ H Ω itself describes resonant interactions. How-ever, when V d (cid:29) Ω is satisfied, after rotating the to-tal Hamiltonian ˆ H with respect to ˆ H d , one can see thatthe two-excitation Rydberg states would be coupled off-resonantly with large detuning. Thus the Rydberg block-ade is produced. In the following we would show how to achieve the RAB even when V d (cid:29) Ω.More clearly, we here introduce an energy operatorˆ h ≡ δ ( | + (cid:105)(cid:104) + | − |−(cid:105)(cid:104)−| ) to perform the unitary operationˆ U = exp( i ˆ ht ) so that the total Hamiltonian ˆ H Ω + ˆ H d becomes [50, 56]ˆ H = (cid:8) Ω2 (cid:2) √ (cid:0) e i ∆ t + e − i ∆ t (cid:1) | (cid:105)(cid:104) Ψ | + (cid:16) e i (∆ − δ ) t + e − i (∆+ δ ) t (cid:17) | Ψ (cid:105)(cid:104) + | + (cid:16) e i (∆+ δ ) t + e − i (∆ − δ ) t (cid:17) | Ψ (cid:105)(cid:104)−| + (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105)(cid:104) d | + | (cid:105)(cid:104) d | ) (cid:3) + H . c . (cid:9) + ˆ H d − ˆ h (5)If { ∆ , ∆ ± δ } (cid:29) Ω, δ = 2∆ and the RAB condition V d = √ − Ω / (3 √ H e = Ω | (cid:105)(cid:104) r pf | + H . c .. (6)From Eq. (6), one can see that the collective groundstate | (cid:105) is resonantly coupled with two-excitation Ryd-berg state | r pf (cid:105) with the effective Rabi frequency Ω eff ≡ Ω / ∆, which indicates that the RAB is achieved underthe case of f¨orster resonance.Here we should mention that in Ref. [34], the groundstate | gg (cid:105) (Corresponding to | (cid:105) in our manuscript) isexcited to Rydberg state | dd (cid:105) firstly via π pulse throughthe detuned laser. Then the electric field is tuned tomake state | dd (cid:105) resonant with ( | pf (cid:105) + | f p (cid:105) ) / √
2. In thissubsection of our manuscript, we consider the strongf¨orser resonant interactions from beginning to end, anddesigned schemes to achieve the Rabi oscillation fromcollective ground state to two-excitation Rydberg state( | pf (cid:105) + | f p (cid:105) ) / √
2. Meanwhile, the states | (cid:105) , | (cid:105) and | (cid:105) are decoupled with the two-excitation Rydbergstates, which is very convenient to apply this model tothe quantum information processing field. B. RAB with “spin-exchange” interaction
As shown in Fig. 1(c), consider Rydberg atoms withtwo ground states | (cid:105) and | (cid:105) , and two Rydberg states | p (cid:105) and | d (cid:105) . For left(right) Rydberg atom, bichromatic clas-sical fields are imposed to off-resonantly drive the tran-sition | (cid:105) ↔ | p ( d ) (cid:105) through single(two)-photon processwith an identical Rabi frequency Ω but opposite detuning∆. After the rotating-wave approximation, the Hamilto-nian for this concrete system can be written as (let (cid:126) = 1)ˆ H Ω = Ω2 (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105) (cid:104) p | + | (cid:105) (cid:104) d | ) + H . c . ˆ H d = V d | pd (cid:105)(cid:104) dp | + H . c . (7) Through using the dressed states | (cid:101) ±(cid:105) ≡ ( | pd (cid:105) ± | dp (cid:105) ) / √ H Ω = Ω √ (cid:0) e i ∆ t + e − i ∆ t (cid:1) [ | (cid:105)(cid:104) Φ | + 1 √ | Φ (cid:105) ( (cid:104) (cid:101) + | + (cid:104) (cid:101) −| )]+ Ω2 (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105)(cid:104) d | + | (cid:105)(cid:104) d | ) + H . c . ˆ H d = V d ( | (cid:101) + (cid:105)(cid:104) (cid:101) + | − | (cid:101) −(cid:105)(cid:104) (cid:101) −| ) (8)with | Φ (cid:105) ≡ ( | d (cid:105) + | p (cid:105) ) / √
2. Follow the similar processin Sec. III A, if the relation ∆ (cid:29)
Ω and RAB condition V d = 2∆ − Ω / (3∆) are satisfied, one can get the effectiveHamiltonian as ˆ H e = Ω | (cid:105)(cid:104) dp | + H . c ., (9)which means the Rabi oscillation between collectiveground state | (cid:105) and two-excitation Rydberg state | pf (cid:105) emerges and the RAB would be implemented if Ω t/ ∆ = π is fulfilled.Now we discuss the differences of excitation process be-tween our scheme and that in Ref. [55]. In Ref. [55], theatoms are excited step by step. Firstly, one of the Ry-dberg atoms is excited to | d (cid:105) state through two-photonprocess. Then the state of the excited Rydberg atom istransferred from | d (cid:105) to | p (cid:105) through microwave field. Im-mediately, the rest Rydberg atom is excited to state | d (cid:105) with Ω (cid:39) . V d and along with this process the spin-exchange process also happens. The blockade effect inRef. [55] does not work because V d is less than Ω. For ourscheme in Sec. III B, by using the dressed state methodand appropriately choosing parameters, RAB can be ac-complished in one step with the condition V d (cid:29) Ω. C. RAB with “collective-exchange” interaction
As shown in Fig. 1(d), consider two Rydberg atoms,each has two ground states | (cid:105) and | (cid:105) . The left (right)atom has two Rydberg states | s ( s (cid:48) ) (cid:105) and | p ( p (cid:48) ) (cid:105) . Thebichromatic classical fields are imposed to off-resonantlydrive the transition | (cid:105) ↔ | s ( s (cid:48) ) (cid:105) through two-photonprocess with an identical Rabi frequency Ω but oppo-site detuning ∆. With the consideration of rotating-wave approximation and set the electric field strength (cid:15) = 710 mV cm − , the Hamiltonian for this concretesystem can be written as (let (cid:126) = 1)ˆ H Ω = Ω2 (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105) (cid:104) s | + | (cid:105) (cid:104) s (cid:48) | ) + H . c . ˆ H d = V d | ss (cid:48) (cid:105)(cid:104) pp (cid:48) | + H . c . (10)Through using the dressed states |± (cid:48) (cid:105) ≡ ( | ss (cid:48) (cid:105) ±| pp (cid:48) (cid:105) ) / √
2, one can rewrite Eq. (10) asˆ H Ω = Ω √ (cid:0) e i ∆ t + e − i ∆ t (cid:1) [ | (cid:105)(cid:104) Ξ | + 1 √ | Ξ (cid:105) ( (cid:104) + (cid:48) | + (cid:104)− (cid:48) | )]+ Ω2 (cid:0) e i ∆ t + e − i ∆ t (cid:1) ( | (cid:105)(cid:104) s (cid:48) | + | (cid:105)(cid:104) s | ) + H . c . ˆ H d = V d ( | + (cid:48) (cid:105)(cid:104) + (cid:48) | − |− (cid:48) (cid:105)(cid:104)− (cid:48) | ) (11)with | Ξ (cid:105) ≡ ( | s (cid:48) (cid:105) + | s (cid:105) ) / √
2. Follow the similar process inSec. III A, if the parameters satisfy the relations ∆ (cid:29)
Ωand RAB condition V d = 2∆ − Ω / (3∆), one can get theeffective Hamiltonian asˆ H e = Ω | (cid:105)(cid:104) pp (cid:48) | + H . c ., (12)which means the Rabi oscillation between collectiveground state | (cid:105) and two-excitation Rydberg state | pp (cid:48) (cid:105) emerges and the RAB would be implemented if Ω t/ ∆ = π is fulfilled. In addition to the cases discussed above, theRAB with the resonant dipole-dipole interaction type inRef. [31, 43] can also be constructed in the similar way.In Ref. [39], optically trapped cloud of 2 × Rbgate and source atoms are used for studying the enhance-ment of single-photon nonlinearity. At zero electric field,the interaction between the | ss (cid:48) (cid:105) pair which is of vdWtype and much less than the dipole-dipole interaction | ss (cid:48) (cid:105)(cid:104) pp (cid:48) | . Thus the collective ground state can be excitedto | ss (cid:48) (cid:105) and the single-photon nonlinearity was observedto be enhanced by electrically tuning | ss (cid:48) (cid:105) and | pp (cid:48) (cid:105) pairstates into resonant interactions [39]. In this subsection,the resonant dipole-dipole interaction is an initial consid-eration and on that basis we design the pulse to achievethe RAB in one step with the condition V d (cid:28) Ω. D. Dynamics with partial experimental parameters
In this subsection, we discuss the dynamics of theproposed RAB schemes through numerically solving themaster equation˙ˆ ρ = i [ˆ ρ, ˆ H ] + 12 (cid:88) k [2 ˆ L k ˆ ρ ˆ L † k − ˆ L † k ˆ L k ˆ ρ − ˆ ρ ˆ L † k ˆ L k ] (13)where ρ denote the density matrix of system state, ˆ L k is the k -th Lindblad operator described the dissipationprocess, and ˆ H is the initial Hamiltonian of the whole system. For the model in Sec. III A, the full Hamiltonianis ˆ H = ˆ H Ω + ˆ V d that have been shown in Eq. (3). Thelifetimes for | p (cid:105) , | d (cid:105) and | f (cid:105) are about 0.53 ms, 0.22 msand 0.13 ms respectively [57, 58]. The Lindblad opera-tors are ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | , ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | , ˆ L = (cid:112) γ f / | (cid:105) (cid:104) f | ,ˆ L = (cid:112) γ f / | (cid:105) (cid:104) f | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | , ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | ,ˆ L = (cid:112) γ f / | (cid:105) (cid:104) f | , ˆ L = (cid:112) γ f / | (cid:105) (cid:104) f | , where γ j de-notes the atomic spontaneous emission rate. For themodel in Sec. III B, the full Hamiltonian is shown inEq. (7). The lifetimes for | p (cid:105) and | d (cid:105) are about 0.59ms and 0.25 ms respectively [57, 58]. The Lindbladoperators are ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | ,ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | , ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | ,ˆ L = (cid:112) γ d / | (cid:105) (cid:104) d | . For the model in Sec. III C, the fullHamiltonian is shown in Eq. (10). The lifetimes for | s (cid:105) , | s (cid:48) (cid:105) , | p (cid:105) and | p (cid:48) (cid:105) are about 0.12 ms, 0.13 ms, 0.25 msand 0.27ms, respectively [57, 58]. The Lindblad opera-tors are ˆ L = (cid:112) γ s / | (cid:105) (cid:104) s | , ˆ L = (cid:112) γ s / | (cid:105) (cid:104) s | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ p / | (cid:105) (cid:104) p | , ˆ L = (cid:112) γ s (cid:48) / | (cid:105) (cid:104) s (cid:48) | ,ˆ L = (cid:112) γ s (cid:48) / | (cid:105) (cid:104) s (cid:48) | , ˆ L = (cid:112) γ p (cid:48) / | (cid:105) (cid:104) p (cid:48) | , ˆ L = (cid:112) γ p (cid:48) / | (cid:105) (cid:104) p (cid:48) | . t/T P o p u l a t i o n s (a) t/T (b) |11 |01(10) |00 t/T (c) 〉 〉 〉 FIG. 2. (a)[(b), (c)] Population of the states for RAB schemein Sec. III A, (III B, III C) under one evolution period T withthe consideration of atomic spontaneous emission. Parame-ters are chosen as Ω = 2 π × µ m(3 µ m, 2 µ m). IV. POTENTIAL APPLICATIONS OF THEPROPOSED RAB
In this section, for simplicity we only consider the RABin Sec. III A, based on which one can generalize the ap-plications to other RRI cases. F i d e li t y θ = π θ = 3 π /4 θ = π /2 θ = π /4 θ = π /6 (cid:127)|11⟩ (a) (b) FIG. 3. (a) Evolution of the fidelity of the geometriccontrolled-arbitrary-phase gate with the consideration of dis-sipation and the parameters are the same as that in Fig. 2(a).(b) Bloch sphere representation for the conceptual explana-tion of geometric quantum operation.
A. Geometric gate with unitary dynamics
We now consider how to construct the controlled-arbitrary-phase geometric gate with the form asˆ U CP = e iθ . (14)in the computational space {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} . Bymodulating the Rabi frequencies of the initial Hamilto-nian at the half evolution time T /
T / , T ] as ˆ H e = e iθ Ω | (cid:105)(cid:104) r pf | + H . c ., (15)Based on which the desired gate can be achieved.The fidelity of the gate with specific θ is shown inFig. 3(a) by numerically solving the master equationwith initial Hamiltonian, in which the initial state is setas | ψ (0) (cid:105) = ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) / | ψ ( t ) (cid:105) = ˆ U | ψ (0) (cid:105) . With theconsideration of dissipation, the final fidelity are 0.9969,0.9962, 0.9949, 0.9938 and 0.9936 when θ equals π , 3 π/ π/ π/ π/
6, respectively. The geometric featureof the phase can be easily verified since | (cid:105) → | r pf (cid:105) → e iθ | (cid:105) is achieved and (cid:104) Ψ j | ˆ H e | Ψ k (cid:105) = 0 [59–63] is satis-fied, where | Ψ j (cid:105) ( | Ψ k (cid:105) ) is any one of the four states in {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)} . Thus, θ is the non-adiabatic geo-metric phase, which is half of the solid angle enclosed bythe evolution path [64], as shown in Fig. 3(b). B. Steady entanglement with dissipative dynamics
In Rydberg atom system, the creation of steady-state entanglement via dissipation has been studied inRefs. [47, 65], where a weak microwave field is needed -4 -2 I n f i d e li t y -4 × |+〉 ( a ) ( b ) γ ωω ω/ FIG. 4. (a) Dynamical processes to generate the steady en-tangled state through combining the unitary and dissipa-tive dynamics. (b) Infidelity of the steady entangled state( | (cid:105) − | (cid:105) ) / √ ω/ Ω (cid:48) eff . The inter-atomic distance isset as 3 µ m. And the Rabi frequency is set as Ω = 2 π × V d = √ to drive resonantly the transition between two groundstates | (cid:105) and | (cid:105) . Following the basic ideas of theschemes [47, 65], we here also consider that the two atomsare interacting with the microwave field asˆ H mw = √ ω | (cid:105) + | (cid:105) ) (cid:104) T | + H . c ., (16)where | T (cid:105) ≡ ( | (cid:105) + | (cid:105) ) / √ | S (cid:105) ≡ ( | (cid:105)−| (cid:105) ) / √ | (cid:105) , | T (cid:105) , and | (cid:105) , but keeps | S (cid:105) invariant. Sincethe stark shifts do not influence the dissipative dynam-ics, we thus consider to turn the red-detuned laser off andmodify the RAB condition as V d = √ H (cid:48) e = (Ω (cid:48) eff / | (cid:105)(cid:104) + | + H . c . ) + ˆ S , whereΩ (cid:48) eff = √ / (2∆) and ˆ S denotes the stark shifts.Combining the effective Hamiltonian ˆ H (cid:48) e with the mi-crowave Hamiltonian ˆ H mw in Eq. (16), and the dissipa-tive dynamics as depicted in Fig. 4(a), the desired state | S (cid:105) would be prepared as the steady state of the system.In other words, once | S (cid:105) is occupied through the dissipa-tive dynamics, the entangled state is created successfully.Otherwise, if the other three states are occupied, the uni-tary dynamics will excite the two atoms into | r pf (cid:105) andthen it decays to the ground subspace again. In Fig. 4(b),we plot the infidelity of the steady state via numericallysolving the master equation (13) with initial Hamilto-nian, and the practical parameters of RRI and atomicspontaneous emission rate. One can see that the fidelityis higher than 0.999 when the x-axis value range is in[0.26, 0.54]. C. Measurement of parameters If C and the distance as well as laser parameters Ω and∆ are known, one can change the electric field strengthto observe where the collective Rabi oscillation with theeffective frequency Ω / ∆ emerges. And inversely deter-mine that whether the tuned electric field strength makesdipole-dipole interaction resonant or not.Besides, if the electric field strength that make the res-onant dipole-dipole interaction are known and set well inadvance, one can scan the values of Ω and ∆ to observewhether the RAB-based Rabi oscillation are achieved.And thus can inversely calculate the RRI strength andcan further calculate the C parameter when the inter-atomic distance is known. Also, one can roughly estimatethe inter-atomic distance if C is known. V. CONCLUSION
In conclusion, we have proposed how to constructthe RAB dynamics with several types of resonant Ryd-berg dipole-dipole interaction by using the dressed statemethod. In contrast to the usual vdW-based RAB whichis valid when the inter-atomic distance is larger than thecharacteristic distance R c [4], our study is valid when theinter-atomic distance is less than the characteristic dis-tance R c , which makes the layout of RAB more complete.We also show the potential applications of the proposedRAB in geometric quantum computation, dissipative-dynamics based entanglement preparation, and the pa-rameter estimation. ACKNOWLEDGEMENTS
The author would like to thank Dr. Jin-Lei Wu andProf. Xiao-Qiang Shao for discussions, and Dr. Bao-Jie Liu for useful suggestions. This work was sup-ported by National Natural Science Foundation of China(NSFC) under No.11804308 and China Postdoctoral Sci-ence Foundation (CPSF) under No. 2018T110735. [1] T. F. Gallagher,
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