CRITERIOS DE DEFINICION DE UN ADL

Nevertheless, one might argue that, from a physical point of view, the limit N → ∞ in Eq. (3.2) is not so important, as long as we are able to prepare the desired field state with high fidelity. As discussed in chapter 3.2, however, asymptotic completeness is more than the ability to prepare a given field state. It also implies independence of the final field state from the initial field state.

5.4 Reaching the Limit of Asymptotic Completeness 75

We have already examined this aspect in chapter 5.3. There, we always used the optimal initial atomic state and studied how the maximum fidelity of the state preparation starting from mixed initial field states converges to the ideal value 1. According to Eq. (3.2), the independence of the final from the initial field state should also hold for all other initial atomic states.

In order to verify this aspect of the asymptotic completeness numerically, we quantify the difference ∆ betweenM|00|andM(ρ0) _{by the Hilbert-Schmidt}

norm [85]: ∆2 = Tr M(ρ0)−_M|00| 2 /2N, (5.8)

i.e., ∆2 is the average over the square of all eigenvalues of M(ρ0)−_M|00|_{. In}

order to compensate for the increasing number of eigenvalues, the normaliza- tion factor 2N is required. Asymptotic completeness is fulfilled if and only if ∆→0 withN → ∞, for all final and initial field states. Indeed, Fig. 5.21 con- firms this prediction in all the four cases (for the two target states|2 and|α, α = 1, starting from thermal and truncated maximally mixed initial states). Furthermore, the convergence is again exponentially fast. Note, however, that the range of the log(∆)-axis corresponds to only one order of magnitude. Hence, the convergence is much slower than for the maximum fidelity (Figs. 5.13 and 5.14). Furthermore, the rate of convergence does not depend mainly on the target and initial field states, but rather on the vacuum Rabi angle, which is almost the same in the three cases (b), (c), and (d). For the preparation of number states (a and c), in some cases a zig-zag structure is observed, which indicates a dependence of ∆ on whetherN is even or odd. This feature is not yet understood.

In summary, although universal preparability is equivalent to asymptotic completeness in the limit N → ∞, the first property is reached much faster than the second.

Chapter 6

The influence of noise

Whereas we have so far assumed idealized experimental conditions, in a real laboratory we have to deal with various noise sources: the initial atomic state cannot be prepared with perfect fidelity, the vacuum Rabi angle is not precisely the same for all atoms (e.g., due to a finite velocity spread of the atomic beam), and the photon field decays during the interaction with the cavity walls. In this chapter, we will examine the influence of those noise sources upon the fidelity of the state preparation.

6.1

Cavity dissipation

Since, under realistic experimental conditions, the cavity field is not perfectly isolated from its environment, the field decays due to the interaction with the cavity walls. This decay can be treated using standard techniques (see, e.g., chapter 15.1 in [59]): the environment is treated as a heat bath at temperature T, which has no memory (Markov approximation). Furthermore, the coupling between cavity field and heat bath is assumed to be weak, and mediated by the photon annihilation and creation operatorsa and a†. Under these general conditions, one arrives at the following master equation of the damped harmonic oscillator: ˙ ρ = γ 2 (nb+ 1) (2aρa †_−a†_aρ−ρa†_{a) +} γ 2 nb (2a †_ρa−aa†_ρ−ρaa†_{). (6.1)} Here, γ is the decay rate of the cavity, and nb the mean photon number at

thermal equilibrium. The latter is connected to the temperatureT of the heat bath via the familiar Boltzmann factor, i.e.,

nb = e~ω/kT −1 ₋1 . (6.2)

In the laboratory, temperatures of about T 0.3 K can be realized, corre- sponding to nb 0.03 in the microwave regime (ω 20 GHz). Furthermore,

with the high quality microwave cavities presently at use in the laboratory [93], average photon lifetimes as high asγ−1 = 0.2 s can be reached. On the other hand, the interaction timestint of a single atom with the field are of the order of

microseconds, and assuming a coupling constant of Ω40 kHz [93], a vacuum 77

78 Chapter 6. The Influence of Noise

Rabi angle of φ= Ωtint 1 is realized with tint 25 µs, which is about 4 or-

ders of magnitudes smaller than the cavity decay rateγ−1. Hence, it is a good approximation to neglect the decay during the atom-field interaction, which is therefore still described by Eq. (2.1). Only during the intervals between two successive atoms will we account for the decay via Eq. (6.1). For simplicity, we assume that those intervals are of constant length tp. We do not expect that

fluctuations oftp significantly change the results presented below.∗

In general, any interaction of the field with the environment will reduce the purity of the field state, and therefore also reduce the fidelity of the state preparation. The question is: can we do something against it by choosing a different initial atomic state? For example, this could be an atomic state with higher excitation number, compare Eq. (2.6), in order to compensate for the expected photon losses. Our numerical calculations (see below) give a negative answer: the optimal initial atomic state is nearly the same with or without dissipation. In order to explain this result, we will first examine how the decay alone affects the cavity field, without any atoms passing through the cavity.