Having at disposal the microscopic master equations (3.30), (3.34) and (3.38), describing the dynamics of the three-qubit system, we have found the density matrix ⇢(ttot) of the system at the
time instant ttot = t1+ tint+ t3, supposing that at t = 0 the initial condition was| (0)i = |000i.
Moreover, we have assumed that all the three baths are characterized by the same spectral density given in particular by the ohmic one
(!) = (
0 ! = 0
↵! !6= 0 (3.39)
where 0 is introduced in order to take into account a non zero decay rate for ! = 0. In my
master thesis work I studied the same problem with the hypothesis of a flat spectrum. Now we are improving the calculation, since transitions with higher Bohr frequencies are here more damped. The ohmic spectrum implies that the ratio between the decay rates is equal to the ratio between the correspondent Bohr frequencies.
To quantify the e↵ects of the bosonic baths we can consider the fidelity F
F = T r{⇢exp⇢(ttot)} (3.40)
that gives an idea of the di↵erence existing between the density matrix ⇢exp, obtained when the
0 10 20 30 40 w
g
0.88 0.90
Figure 3.7: Fidelity F as a function of !/g when we assume that all the three baths are characterized by the same ohmic spectral density given in (3.39) with 0/g = ↵ = 10 3 and the parameters assume the values
˜
g/g = 0.1. The inset shows the Fidelity for values 0 < !/g < 0.08.
obtained are given in figure 3.7 where we plot F as a function of the ratio !/g assigning to the parameters 0and ↵ physically reasonable values. In particular we have chosen 0/g = ↵ = 10 3
[7, 23, 66, 95] .
The fidelity shown in figure 3.7 slowly decreases for increasing values of !/g, as it is expected from the form of the dissipation constants given in (3.39). As we can see, at least for !. 20g the presence of bosonic baths at zero temperature does not a↵ect in a significative way the dynamics of the system during the di↵erent steps of the procedure, the fidelity not being less than 0.9. One should expect that the fidelity F is a monotonically decreasing function of !/g. The model we have used for the decay rate (see eq.(3.39), however, is discontinuous for zero frequency because we want to consider also possible dephasing channels. Due to this discontinuity one is not allowed to perform the limit !/g tending to zero in the fidelity. Anyway this is not a problem as for ! = 0 our scheme is meaningless since in this limit no rotations are performed. Moreover, as the inset in figure 1 shows, the increase of F is rapid with respect to !/g.
Let us now observe that increasing the bath decay rates by an order of magnitude, the fidelity F remains experimentally significative as shown in figure 3.8.
Both figures make evident that the presence of the three independent bosonic baths do not a↵ect in a dramatic way the results reached under the hypothesis of perfect isolation.
We are interested in the generation of GHZ states as given in equation (3.9), which, as we have previously seen, can be obtained only if the condition (3.23) is satisfied. Thus it is of
3.2. GHZ state generation of three Josephson qubits in presence of bosonic baths 51 10 20 30 40 w g 0.4 0.5 0.6 0.7 0.8 0.9 F
Figure 3.8: Fidelity F as a function of !/g when we assume that all the three baths are characterized by the same ohmic spectral density given in (3.39) with 0/g = ↵ = 10 2 and the parameters assume the values
˜
g/g = 0.1.
interest for us to analyze the fidelity FGHZ defined as
FGHZ = T r{|GHZihGHZ|⇢(ttot)} (3.41)
and reported in figures 3.9 and 3.10 as a function of the ratio !/g.
Figure 3.9 is obtained for realistic bath decay rates generally reported in literature [7, 23, 66, 95]. The results shown in figure 3.10 are obtained supposing worse conditions. As expected, the fidelity FGHZ shows maxima at values of !/g which satisfy condition (3.23). The value of
such maxima moreover decreases increasing the ratio !/g. This circumstance is in turn related to the fact that the decay rates appearing in the master equations (3.30), (3.34) and (3.38), are increasing functions of !. However, also considering the worst case we may conclude that it is possible to choose an interval of values of the ratio !/g for which FGHZ is greater than 0.7. On
the other hand for experimentally reasonable values of the decay rates 0 and ↵ we can obtain
values of FGHZ greater than 0.9 also fixing !/g in di↵erent intervals, see figure 3.9.
We thus may conclude that the scheme to generate GHZ states discussed here, (3.9) is robust enough with respect to the presence of noise sources describable as independent ohmic bosonic baths [92].
10 20 30 40 w
g
0.2
Figure 3.9: Fidelity FGHZas a function of !/g when we assume that all the three baths are characterized by the
same ohmic spectral density given in (3.39) with 0/g = ↵ = 10 3 and the parameters assume the
values ˜g/g = 0.1. To evaluate FGHZ we put ' = 3⇡2 (p3 +(3+ p
3) 4
!
g g˜) for the GHZ state in equation
(3.41). 10 20 30 40 w g 0.4 0.6 0.8 F GHZ
Figure 3.10: Fidelity FGHZas a function of !/g when we assume that all the three baths are characterized by the
same ohmic spectral density given in (3.39) with 0/g = ↵ = 10 2 and the parameters assume the
values ˜g/g = 0.1. To evaluate FGHZ we put ' =3⇡2 (p3 +(3+ p3) 4
!
g ˜g) for the GHZ state in equation
3.2. GHZ state generation of three Josephson qubits in presence of bosonic baths 53