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Wei Quantum Optoelectronics Laboratory and School of Physics and Technology, Southwest Jiaotong University, Chengdu 610031 China 1. Introduction An ion can be trapped in a small region in three dimensional space by the electromagnetic trap, and can be effectively manipulated by the applied laser pulses (1; 2; 3). The system of trapped ions driven by laser pulses provides a powerful platform to engineer quantum states (4; 5; 6; 7) and implement quantum information processings (QIPs) (8; 9; 10). Typically, manipulations and detections quantum information can be utilized to test certain fundamental principles (such as EPR and Shr ¨ odinger cat ”paradox”) in quantum mechanics and implement quantum computation, quantum telegraph, and quantum cryptography, etc.(see, e.g., (11; 12; 13; 14)). Up to now, several kinds of physical systems, e.g., cavity-QEDs (15), superconducting Josephson junctions (16), nuclear magnetic resonances (NMRs) (17), and coupled quantum dots (18), etc., have been proposed to physically implement the QIPs. The system of trapped ions is currently one of the most advanced models for implementing the QIPs, due to its relatively-long coherence time. Indeed, the coherent manipulations up to eight trapped ions had already been experimentally demonstrated (14). However, most of the quantum manipulations with the trapped cold ions are within the Lamb-Dicke (LD) approximation, e.g., the famous Cirac-Zoller model (8). Such an approximation requires that the coupling between the quantum bit (encoded by two atomic levels of a trapped ion) and the data bus (the collective vibration mode of the ions) should be sufficiently weak. Sometimes, the LD limits could be not rigorously satisfied for typical single trapped-ion system, and thus higher-order powers of the LD parameter must be taken into account (19). Alternatively, the laser-ion interaction beyond the LD approximation might be helpful to reduce the noise in the ion-trap and improve the cooling rate(see, e.g., (20; 21; 22)), and thus could be utilized to high-efficiently realize QIPs. In this chapter, we summarize our works (23; 24; 25; 26; 27; 28; 29; 30; 31) on how to design proper laser pulses for the desirable quantum-state engineerings with single trapped-ions, including the preparations of various typical quantum states of the data bus and the implementations of quantum logic gates beyond the LD approximation. The chapter is organized as: In Sec. 2 we derivate the dynamical evolutions of a single trapped ion (driven by a classical laser beam) with and beyond the LD approximation. Based on the quantum dynamics beyond the LD approximation, in Sec. 3, we discuss how to use a series of laser pulses to generate various vibrational quantum states of the trapped ion, e.g., coherence states, squeezed coherent states, squeezed odd/even coherent states and squeezed vacuum states, etc. We also present the approach (by properly setting the laser pulses) to realize control-NOT 4 2 Laser Pulses Fig. 1. (Color online) A sketch of an two-level ion trapped in one-dimensional electromagnetic trap (Only the vibrational motion along the x-direction is considered) and driven by a classical laser field propagating along the x direction. (CNOT) gates (between the internal and external freedoms of a single trapped ion) beyond the LD approximation. In Sec. 4 we give a brief conclusion. 2. Dynamics of single trapped ions driven by laser beams An electromagnetic-trap can provides a three-dimensional potential to trap an ion (with mass m and charge e). The potential function takes the form (3) U(x, y, z) ≈ 1 2 (αx 2 + βy 2 + γz 2 ), (1) near the trap-center (x, y, z = 0), with α, β, and γ bing the related experimental parameters. Therefore, a trapped ion (see, Fig.1) has two degrees of freedom: the vibrational motion around the trap-center and the internal atomic levels of the ion. For simplicity, we consider that the trap provides a pseudopotential whose frequencies satisfy the condition v x = v  v y , v z . This implies that only the quantized vibrational motion along the x-direction (i.e., the principal axes of the trap) is considered. For the internal degree of freedom of the trapped ion, we consider only two selected levels, e.g., the ground state |g and excited state |e. The Hamiltonian describing the above two uncoupled degrees of freedom can be written as ˆ H 0 = ¯hν( ˆ a † ˆ a + 1 2 )+ ¯h 2 ω a ˆ σ z , (2) where, ¯h is the Planck’s constant divided by 2π, ˆ a † and ˆ a are the bosonic creation and annihilation operators of the external vibrational quanta of the cooled ion with frequency ν. ˆ σ z = |ee|−|gg| is the Pauli operator. The transition frequency ω a is defined by ω a =(E e − E g )/¯h with E g and E e being the corresponding energies of the two selected levels, respectively. In order to couple the above two uncoupled degrees of freedom of the trapped ion, we now apply a classical laser beam E (x,t)=A cos(k l x − ω l t − φ l ) (propagating along the x direction) to the trapped ion (see Fig. 1), where A, k l , ω l , and φ l are its amplitude, wave-vector, frequency, and initial phase, respectively. This yields the laser-ion interaction V = erE (x, t) . (3) Certainly, x = √ ¯h/2mν( ˆ a + ˆ a † ) and r = g|r|e(|ge| + |eg|), and thus the above interaction can be further written as ˆ H int = ¯hΩ 2 ˆ σ x  e iη( ˆ a + ˆ a † )−iω l t−iϑ l + e −iη( ˆ a + ˆ a † )+iω l t+iϑ l  , (4) 76 Coherence and Ultrashort Pulse Laser Emission Quantum Manipulations of Single Trapped-Ions Beyond the Lamb-Dicke Limit 3 where, ˆ σ x = ˆ σ − + ˆ σ + with ˆ σ − = |ge| and ˆ σ − = |eg| being the usual raising and lowering operators, respectively. The Rabi frequency Ω = eAg|r|e/¯h describes the strength of coupling between the applied laser field and the trapped ion. Finally, η = k l √ ¯h/2mν is the so-called LD parameter, which describes the strength of coupling between external and internal degrees of freedom of the trapped ion. Obviously, the total Hamiltonian, describing the trapped ion driven by a classical laser field, reads ˆ H = ˆ H 0 + ˆ H int = ¯hν( ˆ a † ˆ a + 1 2 )+ ¯h 2 ω a ˆ σ z + ¯hΩ 2 ˆ σ x  e iη( ˆ a + ˆ a † )−iω l t−iϑ l + e −iη( ˆ a + ˆ a † )+iω l t+iϑ l  . (5) This Hamiltonian yields a fundamental dynamics in this chapter for engineering the quantum states of trapped cold ion. 2.1 Within Lamb-Dicke approximation Up to now, most of the experiments (see, e.g., (10)) for engineering the quantum states of trapped ions are operated under the LD approximation, i.e., supposing the LD parameters are sufficiently small (η  1) such that one can make the following approximation: e ±iη( ˆ a + ˆ a † ) ≈ 1 ±iη( ˆ a + ˆ a † ). (6) Under such an approximation, the Hamiltonian (5) can be reduced to ˆ H LDA = ˆ H 0 + ¯hΩ 2 ˆ σ x  e −iω l t−iϑ l + e iω l t+iϑ l + iηe −iω l t−iϑ l ( ˆ a + ˆ a † ) − iηe iω l t+iϑ l ( ˆ a + ˆ a † )  , (7) and can be further written as ˆ H  LDA = ¯hΩ 2  e −iω a t ˆ σ − + ˆ σ + e iω a t  ×  e −iω l t−iϑ l + e iω l t+iϑ l + iηe −iω l t−iϑ l (e −iνt ˆ a + e iνt ˆ a † ) − iηe iω l t+iϑ l (e −iνt ˆ a + e iνt ˆ a † )  (8) in the interaction picture defined by the unity operator ˆ U = exp(−it ˆ H 0 /¯h). Note that, to obtain the interacting Hamiltonian (8), we have used the following relations ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ e −iα ˆ σ z ˆ σ z e iα ˆ σ z = ˆ σ z e −iα ˆ σ z ˆ σ + e iα ˆ σ z = e i2α ˆ σ + e −iα ˆ σ z ˆ σ − e iα ˆ σ z = e −i2α ˆ σ − e −iα ˆ a † ˆ a f ( ˆ a † , ˆ a)e iα ˆ a † ˆ a = f ( e iα ˆ a † ,e −iα ˆ a ). (9) Above, α is an arbitrary parameter and f ( ˆ a † , ˆ a) is an arbitrary function of operators ˆ a † and ˆ a. Now, we assume that the frequencies of the applied laser fields are set as ω l = ω 0 + Kν, with K = 0, ±1 corresponding to the usual resonance (K = 0), the first blue- (K = 1), and red- (K = 1) sideband excitations (4), respectively. As a consequence, the Hamiltonian (8) can be specifically simplified to (30) ˆ H 0 LDA = ¯hΩ 2  e iϑ l ˆ σ − + e −iϑ l ˆ σ +  for K = 0, (10) ˆ H r LDA = ¯hΩ 2 iη  e −iϑ l ˆ σ + ˆ a −e iϑ l ˆ σ − ˆ a †  for K = −1, (11) 77 Quantum Manipulations of Single Trapped-Ions Beyond the Lamb-Dicke Limit 4 Laser Pulses ˆ H b LDA = ¯hΩ 2 iη  e −iϑ l ˆ σ + ˆ a † −e iϑ l ˆ σ − ˆ a  for K = −1, (12) under the rotating-wave approximation (i.e., neglected the rapidly-oscillating terms in Eq. (8)). Obviously, the Hamiltonian ˆ H 0 LDA describes a Rabi oscillation and corresponds to a one-qubit operation, and Hamiltonians ˆ H r LDA and ˆ H b LDA correspond to the Jaynes-Cummings (JC) model (JCM) and anti-JCM (4), respectively. It is well-known that the JCM describes the basic interaction of a two-level atom and a quantized electromagnetic field (15). Here, the quantized electromagnetic field is replaced by the quantized vibration of the trapped ion, and thus the entanglement between the external motional states and the internal atomic states of the ions can be induced. Certainly, these JC interactions take the central role in the current trap-ion experiments for implementing the QIPs (10). 2.2 Beyond the LD approximation In principle, quantum motion of the trapped ions beyond the usual LD limit ( η  1) is also possible. Utilizing the laser-ion interaction outside the LD regime might be helpful to reduce the noise in the ion-trap and improve the cooling rate, and thus could be utilized to efficiently realize QIPs. Indeed, several approaches have been proposed to coherently operate trapped ions beyond the LD limit (see, e.g., (19)). Because [ ˆ a, ˆ a † ]=1, the term exp[±iη( ˆ a + ˆ a † )] in Eq. (5) can be expanded as e ±iη( ˆ a + ˆ a † ) = e −η 2 /2 e ±iη ˆ a † e ±iη ˆ a = e −η 2 /2 ∞ ∑ n,m (±iη ˆ a † ) n (±iη ˆ a) m n!m! . (13) Comparing to the expansion Eq. (6), the above expansion is exactly, and the LD parameter can be taken as an arbitrary value. With the help of relation (9) and (13), the Hamiltonian (5) can be written as: ˆ H  NLDA = ¯hΩ 2 ˆ σ +  e i(ω a +ω l )t e iϑ l e −η 2 /2 ∑ ∞ n,m (−iη ˆ a † e iνt ) n (−iη ˆ ae −i νt ) m n!m! +e i(ω a −ω l )t e −iϑ l e −η 2 /2 ∑ ∞ n,m (iη ˆ a † e iνt ) n (iη ˆ ae −i νt ) m n!m!  + H.c (14) in the interaction picture defined by unity operator ˆ U = exp(−it ˆ H 0 /¯h). For the so-called sideband excitations ω l = ω 0 + Kν with K being arbitrary integer, the above Hamiltonian can be reduced to (23; 26) ˆ H r NLDA = ¯hΩ 2 e −η 2 /2 ⎡ ⎣ e −iϑ l (iη) k ˆ σ + ∞ ∑ j=0 (iη) 2j ( ˆ a † ) j ˆ a j+k j!(j + k)! + H.c ⎤ ⎦ for K ≤ 0 (15) ˆ H b NLDA = ¯hΩ 2 e −η 2 /2 ⎡ ⎣ e −iϑ l (iη) k ˆ σ + ∞ ∑ j=0 (iη) 2j ( ˆ a † ) j+k ˆ a j j!(j + k)! + H.c ⎤ ⎦ for K ≥ 0 (16) under the rotating wave approximation. Above, K = 0, K < 0, and K > 0 correspond to the resonance, the kth blue-, and red-sideband excitations (with k = |K|), respectively. Comparing with Hamiltonian (10)-(12) with LD approximation, the above Hamiltonian (without performing the LD approximation) can describes various multi-phonon transitions (i.e., k > 1) (19). This is due to the contribution from the high order effects of LD parameter. 78 Coherence and Ultrashort Pulse Laser Emission Quantum Manipulations of Single Trapped-Ions Beyond the Lamb-Dicke Limit 5 2.3 Time evolutions of quantum states The time evolution of the system governed by the above Hamiltonian can be solved by the time-evolution operator ˆ T = exp(−it ˆ H/¯h), with ˆ H taking ˆ H 0 LDA , ˆ H r LDA , ˆ H b LDA , ˆ H r NLDA , and ˆ H b NLDA , respectively. For an arbitrary initial state |ϕ(0) , the evolving state at time t reads |ϕ( t) = ˆ T |ϕ(0) = ∞ ∑ n=0 1 n!  −it ¯h  n ˆ H n |ϕ(0). (17) Typically, if the external vibrational state of the ion is initially in a Fock state |m and the internal atomic state is initially in the atomic ground sate |g or excited one |e, then the above dynamical evolutions can be summarized as follows (23; 26): i) For the resonance or red-sideband excitations K ≤ 0: ⎧ ⎨ ⎩ |m|g−→|m|g, m < k, |m|g−→cos(Ω m−k,k t)|m|g + i k−1 e −iϑ l sin(Ω m−k,k t)|m − k|e; m ≥ k, |m|e−→cos(Ω m,k t)|m|e−(−i) k−1 e iϑ l sin(Ω m,k t)|m + k|g (18) ii) For the resonance or blue-sideband excitations K ≥ 0: ⎧ ⎨ ⎩ |m|g−→cos(Ω m,k t)|m|g + i k−1 e −iϑ l sin(Ω m,k t)|m + k|e, |m|e−→|m|e, m < k , |m|e−→cos(Ω m−k,k t)|m|e−(−i) k−1 e iϑ l sin(Ω m−k,k t)|m − k|g, m ≥k. (19) The so-called effective Rabi frequency introduced above reads Ω m,k = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Ω L m,k = Ω 2 η k  (m+k)! m! , k = 0, 1, Ω N m,k = Ω 2 η k e −η 2 /2  (m+k)! m! ∑ m j =0 (iη) 2j m! (j+k)!j!(m−j)! , k = 0, 1, 2, 3, . (20) The above derivations show that the dynamics either within or beyond the LD approximation has the same form (see, Eqs. (18) and (19)), only the differences between them is represented by the specifical Rabi frequencies Ω L m,k and Ω N m,k . Certainly, the dynamical evolutions without LD approximation are more closed to the practical situations of the physical processes. Furthermore, comparing to dynamics with LD approximation (where k = 0, 1), the dynamics without LD approximation (where k = 0, 1, 2, 3, ) can describes various multi-phonon transitions. Certainly, when the LD parameters are sufficiently small, i.e., η  1, the rate γ = Ω L m,k Ω N m,k = e η 2 /2 ∑ m j =0 (iη) 2j m! (j+k)!j!(m−j)! ∼ 1, (21) and thus the dynamics within LD approximation works well. Whereas, if the LD parameter are sufficiently large, the quantum dynamics beyond the LD limit must be considered. 79 Quantum Manipulations of Single Trapped-Ions Beyond the Lamb-Dicke Limit 6 Laser Pulses 3. Quantum-state engineerings beyond the LD limit 3.1 Preparations of various motional states of the single ions The engineering of quantum states has attracted considerable attention in last two decades. This is in order to test fundamental quantum concepts such as non-locality, and for implementing various potential applications, including sensitive detections and quantum information processings. Laser-cooled single trapped ions are the good candidates for various quantum-state engineering processes due to their relatively-long decoherence times. Indeed, various engineered quantum states of trapped cold ions have been studied. The thermal, Fock, coherent, squeezed, and arbitrary quantum superposition states of motion of a harmonically bound ion have been investigated (4; 5; 6; 7). However, most of the related experiments are operated under the LD approximation. In this section we study the engineerings of various typical vibrational states of single trapped ions beyond the LD limit. There are serval typical quantum states in quantum optics: coherence states, odd/even coherent states and squeezed states, etc They might show highly nonclassical properties, such as squeezing, anti-bunching and sub-Poissonian photon statistics. In the Fock space, coherence state |α can be regarded as a displaced vacuum state, i.e., |α = D(α)|0 = e −|α| 2 /2 ∞ ∑ n=0 α n √ n! |n, (22) where D (α)=exp(α ˆ a † − α ∗ ˆ a ) is the so-called displace operator, with α being a complex parameter and describing the strength of displace. Similarly, the squeezed states |ψ s  are generated by applying the squeeze operator to a quantum state |ψ s  = ˆ S (ξ)|ψ, (23) where ˆ S = exp(ξ ∗ ˆ a 2 /2 −ξ ˆ a †2 /2), and ξ is a complex parameter describing the strength of the squeezes. On the other hand, the odd/even coherent states are the superposed states of two coherent states with different phases |α o  = C o (|α−|−α), (24) |α e  = C e (|α + |−α). (25) Above, C o =[2 − 2 exp(−2|α| 2 )] −1/2 and C e =[2 + 2 exp(−2|α| 2 )] −1/2 are the normalized coefficients. According to the above definitions: Eqs. (22)-(25), one can obtain the squeezed coherent state: |α s  = ˆ S (ξ)|α = ∞ ∑ n=0 G n (α, ξ)|n, (26) squeezed vacuum state: |0 s  = ˆ S (ξ)|α = 0 = ∞ ∑ n=0 G n (0, ξ)|n, (27) squeezed odd state: |α o,s  = ˆ S (ξ)|α o  = ∞ ∑ n=0 O n (α, ξ)|n, (28) 80 Coherence and Ultrashort Pulse Laser Emission [...]... L 3) A resonance pulse with the initial phase 3 = −π/2 and duration t3 = π/(4Ω0,0 ) is again applied to implement the following operation (2) | ψa (2) |ψb (2) |ψc (2) |ψd (3) −→ |ψa (3) −→ |ψb (3) −→ |ψc (3) −→ |ψd (3) = α11 |0 (3) = α21 |0 (3) = 31 |0 (3) = α41 |0 (1) (3) |g |g |g |g + α12 |0 (3) + α22 |0 (3) + 32 |0 (3) + α42 |0 (1) |e |e |e |e (3) + α 13 |1 (3) + α 23 |1 (3) + 33 |1 (3) + α 43. .. 457.4064 32 8.61 93 351.1877 490.0675 30 4.5596 34 7.0706 4 43. 4884 38 5.56 13 316.7005 37 7.2728 276.8880 436 . 539 2 471.0198 31 8.7519 t2 Ω 9 93. 3470 1410.200 14 63. 000 1192.800 32 6.7189 438 .1591 284 .36 25 170.16 23 2 83. 1649 190.9979 492.7649 481.9516 39 5.8756 292.1111 281.21 43 486.4 535 460.5527 39 4.2894 θ1 = ± π 2 + + + + + + + + + + F 0.9907 0.9958 0.9984 0.9978 0.9967 0.99 93 0.9994 0.9975 0.9980 0.9970 0.99 93 0.9967... 94 [27] [28] [29] [30 ] [31 ] [32 ] [33 ] [34 ] [35 ] [36 ] [37 ] [38 ] [39 ] Laser Pulses Coherence and Ultrashort Pulse Laser Emission L F Wei, M Zhang, H.Y Jia, and Y Zhao, Phys Rev A 78, 01 430 6 (2008) M Zhang, X H Ji, H Y Jia, and L F Wei, J Phys B42, 035 501 (2009) M Zhang, H Y Jia, and L F Wei, Opt Communications 282, 1948 (2009) H J Lan, M Zhang, and L F Wei, Chin Phys Lett 27, 01 030 4 (2010) M Zhang, “quantum-state... N Ωt1 (ϑ1 ) Ωt2 (ϑ2 ) Ωt3 ( 3 ) Ωt4 (ϑ4 ) Ωt5 (ϑ5 ) odd 1 0.0 5 3. 24 13( 0) 9.69 43( 3π/2) 0(0) 436 .5962(π/2) 9 83 4065.1(0) Ωt6 (ϑ6 ) F 0.9927 odd 2 0.5 6 3. 24 13( 0) 7. 732 8 (3 /2) 0(0) 438 .6774(π/2) 0(0) 36 35.9(0) odd 2 0.8 6 3. 24 13( 0) 9.9520 (3 /2) 0(0) 85 .34 71(π/2) 0(0) 36 35.9(0) 0.9974 even 1 0.0 5 1 .31 06(0) 0(0) 59.9889(0) 0(0) 4065.1(0) 0.9995 even 2 0.5 5 2.2687(0) 0(0) 61 .33 54(0) 0(0) 4065.1(0) 0.9870... cos(Ω0,0 t3 ) (3) α 23 = −eiϑ2 cos(Ω0,0 t1 ) sin(Ω0,1 t2 ) cos(Ω1,0 t3 ) (3) α24 = −ei(ϑ2 − 3 ) cos(Ω (68) 0,0 t1 ) sin( Ω0,1 t2 ) sin( Ω1,0 t3 ) (3) 31 = −iei( 3 −ϑ2 ) cos(Ω1,0 t1 ) sin(Ω0,1 t2 ) sin(Ω0,0 t3 ) (3) 32 = e−iϑ2 cos(Ω1,0 t1 ) sin(Ω0,1 t2 ) cos(Ω0,0 t3 ) (3) 33 = cos(Ω1,0 t1 ) cos(Ω0,1 t2 ) cos(Ω1,0 t3 ) − ei( 3 −ϑ1 ) sin(Ω1,0 t1 ) cos(Ω1,1 t2 ) sin(Ω1,0 t3 ) (3) 34 = −ie−i 3 cos(Ω1,0... exist and the CNOT gate could be approximately generated The fidelity F for such an approximated realization is defined as the minimum among the probability amplitudes: (3) (3) (3) (3) α11 , α22 , 34 , and α 43 , i.e (27), 12 86 Laser Pulses Coherence and Ultrashort Pulse Laser Emission Ωt1 = Ωt3 192.01 292. 03 1.6690 1.7287 9.0200 31 .000 177.00 2.0260 97 .30 4 61.710 66.900 175.99 73. 110 12.400 25 .30 0 η... 0.9950 0.99 53 0.9980 0.9999 0.9971 0.9969 Table 4 CNOT gates implemented with sufficiently high fidelities for arbitrarily selected parameters (28) 14 88 Laser Pulses Coherence and Ultrashort Pulse Laser Emission η 0.18 0.20 0.22 0.24 0.26 0.28 0 .30 0 .32 0 .34 0 .36 0 .38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 t1 Ω 267.75 2 43. 65 179.76 168. 13 129.49 130 .92 104.95 92 .35 79.49 80.64 67 .33 68.44... 1.0859(0) 0(0) 43. 9507(π) 0(0) 4065.1(π) 0.9816 1 0.0 5 1.8966(0) 7.1614 (3 /2) 46.0741(0) 34 3.5682(π/2) 4065.1(0) 0.9981 2 0.5 5 2.5666(0) 5 .32 67 (3 /2) 43. 7265(0) 37 1.8479(π/2) 4065.1(π) 0.9882 2 0.5 6 2.5666(0) 5 .32 67 (3 /2) 43. 7265(0) 37 1.8479(π/2) 1826 .3( π) 36 35.9(0) 0.9910 2 0.8 6 2.2975(0) 6.8280 (3 /2) 43. 9627(0) 59.89 73( π/2) 1877.7(π) 36 35.9(0) 0.9821 F Table 1 Parameters for generating squeezed coherent... sin(Ω1,0 t3 ) − ie−iϑ1 sin(Ω1,0 t1 ) cos(Ω1,1 t2 ) cos(Ω1,0 t3 ) (3) 35 = iei(ϑ2 −ϑ1 ) sin(Ω1,0 t1 ) sin(Ω1,1 t2 ) cos(Ω2,0 t3 ) (3) 36 = ei(ϑ2 −ϑ1 − 3 ) sin(Ω1,0 t1 ) sin(Ω1,1 t2 ) sin(Ω2,0 t3 ) (69) (3) α41 = −ei(ϑ1 −ϑ2 + 3 ) sin(Ω1,0 t1 ) sin(Ω0,1 t2 ) sin(Ω0,0 t3 ) (3) α42 = −iei(ϑ1 −ϑ2 ) sin(Ω1,0 t1 ) sin(Ω0,1 t2 ) cos(Ω0,0 t3 ) (3) α 43 = −iei 3 cos(Ω1,0 t1 ) cos(Ω1,1 t2 ) sin(Ω1,0 t3 ) − ieiϑ1... sin(Ω0,0 t3 ) (3) α12 = −ie−i 3 cos(Ω0,0 t1 ) sin(Ω0,0 t3 ) − ie−iϑ1 sin(Ω0,0 t1 ) cos(Ω0,1 t2 ) cos(Ω0,0 t3 ) (3) α 13 = iei(ϑ2 −ϑ1 ) sin(Ω0,0 t1 ) sin(Ω0,1 t2 ) cos(Ω0,1 t3 ) (3) α14 = iei(ϑ2 −ϑ1 − 3 ) sin(Ω (67) 0,0 t1 ) sin( Ω0,1 t2 ) sin( Ω0,1 t3 ) (3) α21 = −ieiϑ1 sin(Ω0,0 t1 ) cos(Ω0,0 t3 ) − iei 3 cos(Ω0,0 t1 ) cos(Ω0,1 t2 ) sin(Ω0,0 t3 ) (3) α22 = −ei(ϑ1 − 3 ) sin(Ω0,0 t1 ) sin(Ω0,0 t3 ) + cos(Ω0,0 . α (3) 14 |1|e | ψ (2) b −→|ψ (3) b  = α (3) 21 |0|g+ α (3) 22 |0|e + α (3) 23 |1|g+ α (3) 24 |1|e | ψ (2) c −→|ψ (3) c  = α (3) 31 |0|g+ α (3) 32 |0|e + α (3) 33 |1|g+ α (3) 34 |1|e + α (3) 35 |2|g+. Ωt 2 (ϑ 2 ) Ωt 3 (ϑ 3 ) Ωt 4 (ϑ 4 ) Ωt 5 (ϑ 5 ) Ωt 6 (ϑ 6 ) F odd 1 0.0 5 3. 24 13( 0) 9.69 43( 3π/2) 0(0) 436 .5962(π/2) 4065.1(0) 0.9927 odd 2 0.5 6 3. 24 13( 0) 7. 732 8 (3 /2) 0(0) 438 .6774(π/2) 0(0) 36 35.9(0). 7.1614 (3 /2) 46.0741(0) 34 3.5682(π/2) 4065.1(0) 0.9981 2 0.5 5 2.5666(0) 5 .32 67 (3 /2) 43. 7265(0) 37 1.8479(π/2) 4065.1(π) 0.9882 2 0.5 6 2.5666(0) 5 .32 67 (3 /2) 43. 7265(0) 37 1.8479(π/2) 1826 .3( π) 36 35.9(0)