MANUAL DE SEGURIDAD

The situation changes if the desired field state is a superposition of number states. Then, the state|ψ₀, Eq. (4.7), is in general not an eigenstate ofM(ρ0)

(unlike the situation for the number states, see above), which implies that the

∗_{Also at the even|1trapping state, i.e.,φ=}√_{2π, the maximum fidelityF}

maxdiffers from F. However, since alreadyF is extremely large in this case (f=−log(1−F)12), the deviation is not visible on the scale of Fig. 5.1.

5.1 Test of the Conjecture 41

0 2 4 6

vacuum Rabi angle φ

0 2 0 2 4 0 5 0 5 10 0 5 10 |χ₁> |χ₂> |χ₃> |χ₄> |χ₅>

−log(1−F)

Figure 5.2: Maximum fidelity Fmax (solid line), fidelityF achieved with |ψ0,

Eq. (4.7) (dotted), andF, Eq. (4.17), (dashed), for the preparation of the phase states|χn =

n i=0|i/

√

n+ 1 of the cavity field, truncated atn= 1, . . .5, with N = 10 atoms injected into the resonator. The initial field state is the vacuum

|0. The agreement of Fmax with F is fairly good, on the logarithmic scale.

The|m-trapping states withm < n are denoted by the vertical dashed lines.

fidelity F achieved with |ψ₀ as initial atomic state is strictly smaller than the maximum fidelity, and strictly larger than the lower bound F, i.e., F < F < Fmax. As an example, we consider the truncated phase states |χn =

_n

i=0|i/ √

n+ 1. In some sense, those states are the complement of the number states examined above: whereas the latter possess a well defined field intensity (given by the photon number), but with completely undetermined phase, the truncated phase state|χn has a uniform photon number distribution (in the

finite dimensional space of at mostnphotons), and is as close as possible to a state with a well defined phase [88].

The numerical results (Fmax, F and F as a function of φ) for the phase

states as target states are shown in Fig. 5.2. Comparing with Fig. 5.1, we see that the maximum fidelity (solid line) is higher for the phase states than for the corresponding number states, especially for the higher photon numbers. The reason is obvious: the phase states are easier to prepare since not the whole field population has to be transferred to the maximum photon number. Fur- thermore, the fidelity for the preparation of the phase states does not exhibit any zeros since the target state has a nonvanishing overlap with the vacuum (in

42 Chapter 5. Numerical Results 0

θ

a)

b)

c)

−log(1−F)

Figure 5.3: Maximum fidelity Fmax (solid line), fidelity F achieved with |ψ0

(dotted), Eq. (4.7), and F (dashed), Eq. (4.17), for the preparation of the state |χ = (|0+|1+eiθ|2)/√3, with N = 10, as a function of the phase θ. We chose three different vacuum Rabi angles: (b) the optimal valueφ(2)_opt= 1.3 (compare Fig. 5.2 and chapter 4.2.2), (a) φ= 1.0, and (c) φ= 1.6. The initial field state is the vacuum |0. Whereas in (b) and (c) the maximum fidelity for the preparation of (|0+|1−|2)/√3 (i.e., θ = π) is slightly higher than the one of (|0+|1+|2)/√3 (i.e.,θ= 0), the opposite is true for the lowest value φ= 1.0 of the vacuum Rabi angle in (a).

contrast to the number states, where zeros are observed at trapping states, see the vertical dashed lines in Fig. 5.1). Apart from that, however, the behavior of Fmax is in both cases quite similar, which indicates that the fidelity is predom-

inantly determined by the maximum photon number of the target field state. In particular, the optimal values of φ, estimated by the small arrows according to Eq. (4.23), are almost the same as in Fig. 5.1.

In contrast to Fig. 5.1, however, Fig. 5.2 shows small deviations (on a loga- rithmic scale) of the lower bound F from the maximum fidelity Fmax (dashed

vs. solid line), even in those regions where no deviations are present for the number states. Nevertheless, we find that - in the optimal regime ofφbelow the first trapping state - the initial atomic state|ψ₀ from Eq. (4.7) gives nearly the optimum result, i.e., F Fmax (dotted vs. solid line). In contrast, for higher

5.1 Test of the Conjecture 43

Varying the phases of the coherent superposition

Since, in general,Fmax > F, whereas Fmax =F in the case of number states

(at least ifφis not close to a trapping state), the fidelity for the preparation of a coherent superposition of number states is higher than the correspondingly weighted sum of fidelities for the preparation of the various number states, see Eq. (4.17). Hence, we may expect that the maximum fidelity not only depends on the photon number populations|n|χ|2 of the target field state, but also on the phases of the photon number amplitudes. An example is shown in Fig. (5.3). Here, we varied the phase of the 2-photon amplitude of the truncated phase state

|χ2, i.e., we considered target states of the form|χθ= (|0+|1+eiθ|2)/

√

3. [Note that any state eiθ0|_0+eiθ1|_1+eiθ2|_{2 can be written in this form by}

(i) multiplication with a global phase, and (ii) multiplication of the n-photon amplitude, n = 0,1,2, with a relative phase einθ˜. This leaves the Jaynes- Cummings interaction (2.1) invariant, if, simultaneously, the atomic states are transformed according to|d →e−iθ˜_{|d.] As can be seen in Fig. 5.3, the influence}

of the phaseθ depends on the vacuum Rabi angle: whereas for the two higher valuesφ= 1.3 (b) and 1.6 (c), the maximum fidelityFmaxof the preparation of

(|0+|1 − |2)/√3 is higher than the one of (|0+|1+|2)/√3, the opposite is true for φ = 1.0 (a). In the latter case, the fidelity F achieved by |ψ₀ is very close to Fmax, while it is closer toF for the highest value ofφ. We have

checked that the mean value ofF (averaged over a uniform distribution of the phase θ) agrees exactly with the estimation (4.21). This is not surprising if we look at the two assumptions made in the derivation of Eq. (4.21): (i) the interference between the different final atomic states in Eq. (4.20) is cancelled by the phase average, and (ii) the distribution of the coefficientsd(mn),m < nis

irrelevant if the photon number distribution|cm|2of the target state is constant.

As for point (i), the fact thatF is almost independent of θ in (c) shows that the different final atomic states are (almost) orthogonal to each other in this case.

Note, however, the small scale of the fidelity axis in Fig. 5.3: the variation ofFmax is not larger than about 10% on the logarithmic scale. This underlines

the approximate validity of the conjectureFmaxF for states with arbitrarily

chosen phases. Below, we will give evidence that this is not only true in the example of Fig. 5.3, but also for states with an arbitrarily chosen photon number distribution (in a finite dimensional photon subspace).