Modelos de fiabilidad en lenguajes de descripción arquitectural

In chapter 5.4.1, we have examined the convergence of the fidelity towards 1 as a function of the N. Now, we want to see how the corresponding optimal initial atomic states change withN, and what happens when we approach the limit N → ∞. Thereby, we will obtain a more detailed picture of the limit of asymptotic completeness than by only looking at the maximum fidelity.

As we have seen in chapter 5.2, in many cases the state |ψ₀, Eq. (4.7), gives a good approximation for the optimal initial atomic state. Furthermore, we may use the explicit expression, Eq. (4.7), in order to study the behavior of |ψ₀ as a function of N. In order to distinguish the states|ψ₀ for different N’s, we write|ψ₀(N). Then, we can try to establish a relation between|ψ₀(N) and the state|ψ₀(N−1)for N−1 atoms.

Starting from the definition of |ψ₀(N), Eq. (4.7), we have:

F(N) |_ψ(N)

0 = 0|UN†|χ, d . . . d

= 0|U₁†U_N†₋₁|χ, d . . . d, (5.5) whereU₁†operates on the first atom [but is appliedafterU_N†₋₁, since the dagger reverses the order of the N atoms in Eq. (2.4)], and U_N†₋₁ on the remaining N −1 atoms. The latter operation results in a final state of the field and the lastN −1 atoms which we write as follows:

U_N†₋₁|χ, d . . . d N−1 = ∞ n=0 Fn(N−1) |n, ψn(N−1). (5.6)

This equation defines the atomic state|ψn(N−1) obtained when projecting the

final stateU_N†₋₁|χ, d . . . d onto the field state|n. Forn= 0, this is the state

|ψ₀(N−1)given by Eq. (4.7). The coefficientsFn(N−1) are required to normalize

the states |ψn(N−1), and give the fidelity of the state (5.6) with respect to

the photon number state |n. Hence, they fulfill the normalization condition

nF

(N−1)

n = 1, and, forn = 0, the fidelity F₀(N−1) =F(N−1) is identical to

the lower bound F as defined by Eq. (4.9).

Next, following Eq. (5.5), we calculate the operation of U₁† on the first atom, which is in state |d, and on the field, which is entangled with the last N −1 atoms, as a consequence of Eq. (5.6). [Since the dagger transforms−i in Eq. (2.1) into +i, the operation of U₁† is similar to Eq. (2.3), but with +i instead of−i.] After projecting onto the field vacuum, we obtain:

F(N) _|ψ(N) 0 = F(N−1) _|ψ(N−1) 0 , d + isin(φ) F₁(N−1) |ψ₁(N−1), u. (5.7) Since F(N−1) is very close to 1 for largeN, and consequently F₁(N−1) almost zero, the main contribution to the state |ψ₀(N) consists of the state |ψ₀(N−1) for the lastN−1 atoms, which is supplemented by the first atom in the ground state. As a consequence, for largeN, the first few atoms enter the cavity almost

5.4 Reaching the Limit of Asymptotic Completeness 67 0.8 1 0.8 1 0.8 1 p_i 0.8 1 0.8 1 0 1 0 1 0 1 p_i(d) 0 1 0 1 first last atom first last atom N=6 N=7 N=8 N=9 N=10

Figure 5.15: Optimal initial atomic state for the preparation of the (randomly chosen) state|χ= (0.34−0.36i)|0+ (−0.14−0.31i)|1+ (−0.02 + 0.28i)|2+ (−0.16 + 0.004i)|3+ 0.29|4, withN = 6, . . .10 atoms (top to bottom), starting from the vacuum as initial field state. The symbols (connected by the solid lines) represent the optimal initial atomic state which is almost identical to the state|ψ₀ (dotted lines), Eq. (4.7). As in Figs. 5.7-5.10, the ground state population of the i-th atom (right hand side) and its entanglement with the other ones (left hand side) are shown (remember: pi = 1/2 indicates maximal

and pi = 1 no entanglement). Vacuum Rabi angle: φ(4)opt = 1.05, according to

the estimation (4.23), withn = 4. When increasing N, the first atoms enter the cavity almost exactly in the ground state, whereas the last atoms remain essentially unchanged. Note that the logarithmic fidelity increases fromf = 2.5 atN = 6 to f = 4.6 at N = 10, whereas it would remain constant if the first atoms entered the cavitypreciselyin the ground state.

68 Chapter 5. Numerical Results 5 10 i 0.7 0.8 0.9 1 p_i(d) 0.995 1 p_i 1 N=8 N=13 N=8 N=13 a) b)

Figure 5.16: Optimal initial atomic state for the preparation of the coherent state|αwith mean photon number|α|2 = 1, withN = 8,9, . . . ,13 atoms, start- ing from the vacuum as initial field state. As in Figs. 5.7-5.15, the ground state population of thei-th atom (bottom) and its entanglement with the other ones (top) are shown for the optimal vacuum Rabi angleφ= 0.95. The convergence behavior deviates from the one of |ψ₀ depicted in Fig. 5.15: when increasing N, the ground state population of the last atom considerably changes, whereas the first atom approaches the ground state only very slowly.

exactly in the ground state, as we already argued in chapter 4.3. This is also confirmed by Fig. 5.7 for smaller photon numbers (where N = 10 is a ‘large’ number of atoms). This behavior holds for any value ofφ(except for trapping state conditions). If φ is not close to its optimal value, however, we need a higher number N of atoms to reach a value of F(N−1) close to 1.

Note that if the first atom would enter the cavityexactlyin the ground state, the maximum fidelity for N atoms would obviously be the same as for N −1 atoms (since the first atom in the ground state does not have any effect on the field vacuum). Hence, it is the tiny part of the optimal atomic state (5.7) with the first atom in the upper state, which is responsible for the increase of the maximum fidelity.

Since the state|ψ₀is equal to the optimal initial atomic state|ψ(opt)for the preparation of number states, and at least gives a good approximation in most other cases, we expect that the above conclusions are also valid for the optimal atomic state: at large N, the main contribution to|ψ(opt) should be the first atom in the ground state and the optimal state forN−1 atoms. Fig. 5.15 shows an example, which, indeed, confirms this prediction. In order to emphasize

5.4 Reaching the Limit of Asymptotic Completeness 69

that this rule not only holds for special field states such as number states or the truncated phase states, we have randomly chosen a target field state |χ including up to 4 photons according to Eq. (5.1), see caption of Fig. 5.15.‡ Since, indeed, the optimal atomic state is almost identical to the state |ψ₀ (solid and dotted lines in Fig. 5.15), its convergence for N → ∞ follows the above predicted behavior.

Is this also the case if the optimal atomic state deviates more strongly from

|ψ₀, as, e.g., for the coherent states? Fig. 5.16 shows the answer. As already observed in Fig. 5.9 for N = 10, the ground state population of the first few atoms is not as high as for the state|ψ₀. Although the ground state population p(₁d) of the first atom slightly increases with increasingN, see Fig. 5.16(b), it is unclear whether it will converge to 1 in the limitN → ∞. Instead, it is rather the ground state population of thelastatoms, which is influenced most strongly by the numberN of atoms - in contrast to the behavior of the state|ψ₀, where the state of the last atoms is almost unchanged when increasing N [see the above discussion of Eq. (5.7) and Fig. 5.15]. In all cases, the state remains quite close to a product state (note the scale of Fig. 5.16a). Furthermore, the smallest one of the eigenvaluespi of the reduced density matrix [which gives an

upper bound for the overlap with a product state, see Eq. (2.10)] is always the last one, i=N. Based on the range ofN = 8, . . . ,13 covered in Fig. 5.16, we cannot draw precise conclusions about the limitN → ∞of the overlap with a product state: although, from N = 8 to N = 11, the smallest eigenvalue pN

increases, indicating an increasing overlap with a product state, this trend is not continued for larger values ofN.