Entangling power of holonomic gates in atom-cavity systems
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Entangling power of holonomic gates in atom-cavity systems
Vahid Azimi Mousolou
1, 2 and Erik Sj¨oqvist Department of Mathematics, Faculty of Science,University of Isfahan, Box 81745-163 Isfahan, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden
Our goal is to provide a new approach to the construction of geometry-induced entanglementbetween a pair of Λ type atoms in a system consists of N identical atoms by means of nonadiabaticquantum holonomies. By employing the quantum Zeno effect, we introduce a tripod type interac-tion Hamiltonian between two selected atoms trapped in an optical cavity, which allows arbitrarygeometric entangling power. This would be a substantial step toward resolving the feasibility ofrealizing universal nonadiabatic holonomic entangling two-qubit gates. I. INTRODUCTION
Environment induced noise and decoherence is a greatbane in realizing quantum computers. There are diversetheoretical proposals developed for physical implemen-tations, which try to avoid noise completely or at leastprotect against the effect of noise. Decoherence free sub-spaces [1–3], dynamical decoupling [4, 5], noiseless sub-systems [6, 7], topological [8, 9] and geometric [10–13] ap-proaches, and quantum error-correction methods [14, 15]are among these proposals.As one of the key approaches in achieving fault-tolerantquantum computation, geometric/holonomic quantumcomputation has caught a great deal of interest in recentyears. Holonomic quantum computation was initially in-troduced in the adiabatic regime [10, 16–19] and subse-quently developed for nonadiabatic processes [11–13, 20–22], the latter being compatible with the short coherencetime of quantum bits (qubits). Nonadiabatic holonomicgates have been experimentally implemented in variousphysical settings, such as NMR [23, 24], superconductingtransmon [25], NV centers in diamond [26–29]. Furtherfeasible schemes have been established for nonadiabaticgeometric processing with spin qubit systems [30], andpseudo-spin charge qubits [31]. Moreover, nonadiabaticholonomic quantum computation has been incorporatedwith decoherence free subspaces [32–39], noiseless subsys-tems [40], and dynamical decoupling [41]. Nonetheless,the construction of an externally controlled multipartitesystem with full entangling power for nonadiabatic holo-nomic processing is one of the main and challenging ob-stacles from a practical perspective.In this paper, we discuss an atom-cavity system, whichnot only allows for combining two fault-tolerant meth-ods in quantum computing, namly, decoherence-free sub-spaces and holonomic quantum processing, but also al-lows for full geometry-induced entangling power. Thesystem we present here consists of separated three-levelatoms placed at fixed positions inside an optical cavity,which can be implemented by using current technology.Taking the advantage of the quantum Zeno effect, weestablish a tripod interaction in a decoherence-free sub- space corresponding to two selected atoms in this cav-ity, each of which represents a qubit system. We thendemonstrate that this tripod arrangement permits im-plementation of nonadiabatic holonomic two-qubit gateswith arbitrary entangling power. The generic nature ofthe proposed scheme would help to overcome the practi-cal challenges in realizing universal holonomic quantuminformation processing.
II. ATOM-CAVITY SYSTEM
The system we have in mind consists of N identicalatoms, arranged in a line and trapped along the symme-try axis of an optical cavity so that each atom can beaddressed individually (see Fig. 1). To create geometry-induced entanglement between atom pairs, we further as-sume that the selected atoms have fixed positions insidethe cavity. Without loss of generality, we select the atomsfixed at the first and second position in the chain.Each atom exhibits a three-level Λ-type structure, withthe atomic ground states | i and | i coupled to an ex-cited state | e i . The ground state levels | i and | i spana qubit state space. The atomic transitions | i ↔ | e i and | i ↔ | e i are assumed to be in resonance with thefield mode in the cavity. For simplicity, the atom-cavitycoupling constant is taken to be g for all atoms. Forno photon in the cavity mode, one finds that the com-putational states | i , | i , | i , | i and the maximallyentangled trapped state | α i = ( | e i − | e i ) / √ (cid:12)(cid:12) ( i ) (cid:11) ↔ (cid:12)(cid:12) e ( i ) (cid:11) and (cid:12)(cid:12) ( i ) (cid:11) ↔ (cid:12)(cid:12) e ( i ) (cid:11) transitions in atom i = 1 ,
2, are taken to be Ω ( i )0 and Ω ( i )1 ,respectively. These frequencies are generally complex-valued. Thus, the laser part of the conditional Hamilto-nian that describes the dynamics of the system is givenby H laser = ~ X i =1 1 X j =0 Ω ( i ) j (cid:12)(cid:12)(cid:12) j ( i ) E D e ( i ) (cid:12)(cid:12)(cid:12) + H . c . (1) (1) (1) e (1) Ω Ω g (2) (2) e (2) Ω Ω g FIG. 1. (Color online) The upper panel shows a system ofidentical atoms arranged in a line and trapped along the sym-metry axis of an optical cavity. The lower panel illustratestwo selected Λ type atoms trapped in the cavity tuned withstrength g on resonance with the (cid:12)(cid:12)(cid:12) ( i ) E ↔ (cid:12)(cid:12)(cid:12) e ( i ) E , i = 1 , ( i )0 and Ω ( i )1 , respectively inducing (cid:12)(cid:12)(cid:12) ( i ) E ↔ (cid:12)(cid:12)(cid:12) e ( i ) E and (cid:12)(cid:12)(cid:12) ( i ) E ↔ (cid:12)(cid:12)(cid:12) e ( i ) E transitions in the corre-sponding atom i = 1 , If the amplitude of the Rabi frequencies are muchsmaller than κ and g /κ , where κ is the decay rate ofa single photon inside the resonator, then with the helpof the environment-induced quantum Zeno effect, the sys-tem can be kept inside the DFS during the evolution ofthe system [42, 43]. In this regime, the effective Hamil-tonian H eff = P H laser P , (2)where P is the projection operator on the DFS, leads tothe following tripod configuration (see Fig. 2) H eff = ~ √ (cid:16) − Ω (1)0 | i h α | + Ω (2)0 | i h α | +(Ω (2)1 − Ω (1)1 ) | i h α | + H . c . (cid:17) , (3)where we have used the short-hand notation (cid:12)(cid:12) j (1) k (2) (cid:11) ≡| jk i , j, k = 0 , H eff , we obtain or-thonormal eigenstates | D i = e − iφ sin θ | i + e − iφ cos θ | i , | D i = e − iφ cos ϕ | x i − sin ϕ | i , | B ± i = | y i ± | α i√ , (4) α − Ω Ω Ω − Ω FIG. 2. (Color online) The conditional tripod interactionpicture, where the atoms, the cavity mode as well as theenvironment-induced quantum Zeno effect are taken into ac-count. and corresponding eigenenergies E D = E D = 0, and E B ± = ± ~ ω √ . Here, ω = q | Ω (1)0 | + | Ω (2)0 | + | Ω (2)1 − Ω (1)1 | , Ω (1)0 = ωe iφ sin ϕ cos θ, Ω (2)0 = ωe iφ sin ϕ sin θ, Ω (2)1 − Ω (1)1 = ωe iφ cos ϕ, | x i = e iφ sin θ | i − e iφ cos θ | i , | y i = sin ϕ | x i + e iφ cos ϕ | i . (5)These parameters are kept constant for the duration [0 , τ ]of the laser pulses, resulting in the time evolution opera-tor of the DFS U ( τ,
0) = e − i ~ R τ H eff dt = | D i h D | + | D i h D | + cos a τ (cid:0) | y i h y | + | α i h α | (cid:1) − i sin a τ (cid:0) | y i h α | + | α i h y | (cid:1) , (6)where a τ = ωτ √ . III. QUANTUM HOLONOMIES ANDGEOMETRY-INDUCED ENTANGLEMENT
By choosing the run time τ = √ π/ω such that a τ = π ,the three dimensional subspace Span {| i , | i , | i} un-dergoes cyclic evolution in the four dimensional partSpan {| i , | i , | i , | α i} of the DFS, while the remain-ing two-qubit state | i is fully decoupled. Moreover,along this evolution we would have U ( t, P c U † ( t, H eff U ( t, P c U † ( t,
0) = 0 (7)for the projection P c = | i h | + | i h | + | i h | . In other words, the three dimensional subspaceSpan {| i , | i , | i} evolves along a loop C in the Grass-mannian G (4 , U ( C ) = P c U ( τ, P c (8)is the nonadiabatic quantum holonomy of the loop C inthe Grassmannian G (4 ,
3) [44].It is instructive to compare the present scheme with thetripod-based single-qubit architecture for adiabatic geo-metric manipulation proposed in Ref. [17]. In additionto being based on adiabatic evolution, the loop corre-sponding to each geometric single-qubit gate in Ref. [17]resides in the Grassmannian G (3 ,
2) since the qubit levelsare encoded in the two dark states evolving in the three dimensional space Span {| i , | i , | i} .Since the computational basis state | i does not con-tribute to the dynamics described by the effective Hamil-tonian in Eq. (3), it remains unchanged during the evo-lution. Therefore, the evolution results in the followingtwo-qubit nonadiabatic holonomic gate U = | i h | + U ( C ) , (9)which takes the following form in the computational or-dered basis {| i , | i , | i , | i} U = − ϕ cos θ e − iφ sin 2 θ sin ϕ e − iφ sin 2 ϕ cos θ e iφ sin 2 θ sin ϕ − ϕ sin θ − e − iφ sin 2 ϕ sin θ e iφ sin 2 ϕ cos θ − e iφ sin 2 ϕ sin θ − cos 2 ϕ , (10)where φ lk = φ l − φ k , l, k = 1 , , U is that it provides geometric gates with arbitrarily largeentangling power. To see this, let us look at some entan-gling characteristics of U . Evaluating the local invari-ances [45], one obtains G = − sin ϕ sin θ,G = cos 2 ϕ + 2 sin ϕ (cid:0) cos ϕ + cos 4 θ sin ϕ (cid:1) , (11)which consequently results in the entangling power [46,47] e p ( U ) = 29 (1 − | G | )= 29 (cid:0) − sin ϕ sin θ (cid:1) . (12) FIG. 3. (Color online) Entangling power, e p ( U ), of the two-qubit nonadiabatic holonomic gate U as a function of thecontrol parameters θ and ϕ in a single period. As shown in Fig. 3, a careful tuning of the laser pulsescan provide any entangling power.From Eqs. (11) and (12), one may note that the en-tangling nature of the gate U does not in general dependon the complex nature of the Rabi frequencies Ω ( i )0 andΩ ( i )1 , i = 1 ,
2. Extracting the corresponding symmetry re-duced geometric coordinate ( c , c , c ) of U on the Weylchamber [48], which classifies non-local two-qubit gates,we have ( c , c , c ) = (cid:16) π , c, c (cid:17) , (13)where c = arcsin (cid:16) (cid:12)(cid:12)(cid:12) Ω (1)0 Ω (2)0 (cid:12)(cid:12)(cid:12) ω − (cid:17) . (14)This indicates that the geometric gate U covers the wholeequivalence classes of two-qubit gates along the line seg-ment connecting the equivalence class of special per-fect entanglers [CNOT], represented by the coordinate (cid:0) π , , (cid:1) , to the class of local gates represented by thecoordinate (cid:0) π , π , π (cid:1) on the Weyl chamber. Moreover, ifthe lasers are tuned so that (cid:12)(cid:12)(cid:12) Ω (1)0 Ω (2)0 (cid:12)(cid:12)(cid:12) = ω √ , (15)then the geometric gate U belongs to the equivalenceclass of perfect entanglers corresponding to the point (cid:0) π , π , π (cid:1) on the Weyl chamber. The entangling na-ture in fact depends on the absolute frequency ratios (cid:12)(cid:12)(cid:12) Ω (2)0 / Ω (1)0 (cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:0) Ω (2)1 − Ω (1)1 (cid:1) /ω (cid:12)(cid:12)(cid:12) . The gate U tends tothe equivalence class of special perfect entanglers, de-noted as [CNOT], with maximum entangling power of ,when (cid:12)(cid:12)(cid:12) Ω (2)0 / Ω (1)0 (cid:12)(cid:12)(cid:12) → , ∞ or (cid:12)(cid:12)(cid:12)(cid:0) Ω (2)1 − Ω (1)1 (cid:1) /ω (cid:12)(cid:12)(cid:12) →
1. Ta-ble I specifies some frequencies to achieve different classof entangling gates.
TABLE I. Entanglement characteristics of the geometric two-qubit entangling gate U for some specific frequencies (1)0 , Ω (2)0 , Ω (2)1 − Ω (1)1 ) G G e p ( U ) Weyl chambercoordinate1 (0 , , = 0) 0 1 2/9 ( π/ , ,
0) = [CNOT]2 (0 , = 0 ,
0) 0 1 2/9 ( π/ , ,
0) = [CNOT]3 ( = 0 , ,
0) 0 1 2/9 ( π/ , ,
0) = [CNOT]4 (0 , = 0 , = 0) 0 1 2/9 ( π/ , ,
0) = [CNOT]5 ( = 0 , , = 0) 0 1 2/9 ( π/ , ,
0) = [CNOT]6 ( = 0 , = 0 , − sin θ θ − / − sin θ ) ( π/ , θ, θ ), 0 < θ ≤ π = 0 , = 0 , = 0) − sin ϕ − ϕ (1 − sin ϕ ) ( π/ , c, c )0 < c = arcsin(sin ϕ ) ≤ π/ Note that the tripod configuration in Fig. 2 reduces toa two level interaction system for the three first uppercases in the table I. Therefore, in these cases, the loop C would effectively reside in the Grassmannian G(2, 1) andits corresponding nonadiabatic quantum holonomy givenin Eq. (8) would describe only an Abelian nonadiabaticgeometric phase [49]. However, in the other cases listedin the table, the tripod structure reduces to a three-levelΛ structure, which would instead correspond to the ef-fective loop C reside in the Grassmannian G(3, 2) an theaccompanying non-Abelian quantum holonomy. In otherwords, Tab. I shows that perfect geometry-induced en-tanglement can be achieved through both Abelian andnon-Abelian quantum holonomies in the above proposedinteraction picture.One may notice that the approach in Ref. [42] is a spe-cial example of the case listed in row four of the table I.The present work, in other words, is an expansion of theproposal in Ref. [42] introducing a wider class of entan-gling gates with more freedom in the choice of frequen-cies. Our analysis shows that nonadiabatic holonomieshave full entangling power. IV. CONCLUSIONS
We have followed the nonadiabatic geometric approachto study the entangling power of quantum holonomy in an atom-cavity system. We have established a nonadi-abatic holonomic manipulation of two decoherence-freequbits, described in terms of quantum Zeno effect in thestudy of a chain of identical atoms trapped in an opti-cal cavity. We achieved arbitrary geometry-induced en-tangling power through the proposed nonadiabatic holo-nomic approach. Moreover, the proposed system bene-fits from both decoherence-free subspace and holonomicmanipulation methods to gain robustness to decoherenceeffects and parameter noises, respectively, and to intro-duce an efficient way of entangling qubit systems. Ourscheme is generic, scalable, and can be implemented in awide range of atomic and ionic systems trapped in cavi-ties [50, 51].
V. ACKNOWLEDGMENT
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