In this section, we show that ITT can be used to realize shortcuts to adiabaticity. As an example, we consider the case that (omega _1

(13)

where (E(omega _1)) is the eigenenergy, and again (m,lin {1,2}) and (lne m). When (omega _1

(14)

In this section, we show that ITT can be used to realize shortcuts to adiabaticity. As an example, we consider the case that (omega _1

(13)

where (E(omega _1)) is the eigenenergy, and again (m,lin {1,2}) and (lne m). When (omega _1

(14)

We assume that (phi _m^{text{FF}}) also satisfies Eq. (7). Using Eqs. (7), (13) and (14), we obtain

$$begin{aligned} frac{dphi _m(omega _1

(15)

and

$$begin{aligned} omega _m^{text{FF}}

(16)

where again, (m,lin {1,2}) and (lne m)we assume that (film) is real. Equation (15) is used to calculate the additional phase, (f_m

(17)

where (Delta omega _0) is constant, and (T_{text{F}}) is the final time of the control.

Figure 3 shows the intensity of (beta ^{text{FF}}_{text{STA}}) defined as

$$begin{aligned} beta ^{text{FF}}_{text{STA}}(t,f_2) = frac{dphi _1(omega _1

(18)

which is the difference of the left hand side and the right hand side of Eq. (15) for (f_1