Frequency-modulated pulses for quantum bits coupled to time-dependent baths

We consider the coherent control of a quantum bit by the use of short pulses with finite duration ${\tau }_{\mathrm{p}}$. By shaping the pulse, we perturbatively decouple the dynamics of the bath from the dynamics of the quantum bit during the pulse. Such shaped pulses provide single quantum bit gates robust against decoherence which are useful for quantum-information processing. We extend previous results in two ways: (i) we treat frequency-modulated pulses and (ii) we pass from time-independent baths to analytically time-dependent baths. First- and second-order solutions for $\pi$ and $\pi /2$ pulses are presented. They are useful in experiments where amplitude modulation is difficult to realize.

Figure 1

First-order $\pi$ pulse FM-1-PI. We also plot $\sin \Phi \left(t\right)\propto v_y\left(t\right)$ and $\cos \Phi \left(t\right)\propto v_x\left(t\right)$ to illustrate the pulse shape in spin space. The left scale refers to $\Phi \left(t\right)$ and the right scale to $\sin \Phi \left(t\right)$ and $\cos \Phi \left(t\right)$, respectively. The Fourier coefficients for this pulse are given in Table .

Figure 2

First-order $\pi /2$ pulse FM-1-PI2. We also plot $\sin \Phi \left(t\right)\propto v_y\left(t\right)$ and $\cos \Phi \left(t\right)\propto v_x\left(t\right)$ to illustrate the pulse shape in spin space. The left scale refers to $\Phi \left(t\right)$ and the right scale to $\sin \Phi \left(t\right)$ and $\cos \Phi \left(t\right)$, respectively. The Fourier coefficients for this pulse are given in Table .

Figure 3

Second-order $\pi$ pulse FM-2-PI. We also plot $\sin \Phi \left(t\right)\propto v_y\left(t\right)$ and $\cos \Phi \left(t\right)\propto v_x\left(t\right)$ to illustrate the pulse shape in spin space. The left scale refers to $\Phi \left(t\right)$ and the right scale to $\sin \Phi \left(t\right)$ and $\cos \Phi \left(t\right)$, respectively. The Fourier coefficients for this pulse are given in Table .

Figure 4

Second-order $\pi /2$-pulse FM-2-PI2. We also plot $\sin \Phi \left(t\right)\propto v_y\left(t\right)$ and $\cos \Phi \left(t\right)\propto v_x\left(t\right)$ to illustrate the pulse shape in spin space. The left scale refers to $\Phi \left(t\right)$ and the right scale to $\sin \Phi \left(t\right)$ and $\cos \Phi \left(t\right)$, respectively. The Fourier coefficients for this pulse are given in Table .

Figure 5

Minimized second-order $\pi$ pulse FM-2-MIN-PI; $\sin \Phi \left(t\right)\propto v_y\left(t\right)$ and $\cos \Phi \left(t\right)\propto v_x\left(t\right)$ show the pulse shape in spin space. The left scale refers to $\Phi \left(t\right)$ and the right scale to $\sin \Phi \left(t\right)$ and $\cos \Phi \left(t\right)$, respectively. The Fourier coefficients for this pulse are given in Table .

Figure 6

Minimized second-order $\pi /2$ pulse FM-2-MIN-PI2. We also plot $\sin \Phi \left(t\right)\propto v_y\left(t\right)$ and $\cos \Phi \left(t\right)\propto v_x\left(t\right)$ to illustrate the pulse shape in spin space. The left scale refers to $\Phi \left(t\right)$ and the right scale to $\sin \Phi \left(t\right)$ and $\cos \Phi \left(t\right)$, respectively. The Fourier coefficients for this pulse are given in Table .