Experimental Hamiltonian identification for controlled two-level systems

We present a strategy to empirically determine the internal and control Hamiltonians for an unknown two-level system (black box) subject to various (piecewise constant) control fields when direct readout by measurement is limited to a single, fixed observable.

Figure 1

Rotation about axis $d$ and observable $z\left({\alpha }_t\right)$. The expectation value of the measurement operator will oscillate sinusoidally, amplitude and frequency are directly related to the declination and length of the vector representing the applied Hamiltonian.

Figure 2

Bloch sphere and arrangement of rotation axes. We can define the relative angular displacement of the vectors $d_k$ representing applied Hamiltonians with respect to a reference vector $d_r(\ne z\mathrm{axis})$.

Figure 3

Identification of $\parallel d\parallel$ and $\theta$ for $d=d_0+f_2^{\left(10\right)}{d}_2$. Shown is simulated data for $d_{0+2}^{\left(10\right)}$ for $f_2^{\left(10\right)}=0.5$. The top graph is used to estimate the rotation frequency using the discrete Fourier transform, the bottom left plot shows the Fourier transform, and the bottom right plot the magnification of the region of interest as well as the parabola that provides the best least-squares fit to the data.

Figure 4

Identification of $\phi$ for $d=d_0+f_2^{\left(10\right)}{d}_2$. The points ∗ are a coarse sampling of the curve from which the first axis crossing can be estimated (red circle). This then gives estimates of the turning points ${\alpha }_{min}^{est}$ and ${\alpha }_{max}^{est}$ which can be refined by resampling (dots) in their vicinity and fitting a parabola (red curves). From this we determine the vertices $({\alpha }_{min},z_{min})$ and $({\alpha }_{max},z_{max})$.

Figure 5

Identification of the components of $d_0$ and $d_m$. Typical simulated experimental run for the system defined by Eq. (11) and the estimated components of $d$ for different field settings. Shown are the best straight line fit (dashed lines) and the actual values (solid line). Note the excellent agreement between the actual values and the best fit obtained. A typical run of the program yielded $d_0={(0.1987,0.0064,0.0992)}^T$, $d_1={(0.9859,0.9122,0.1149)}^T$, $d_2={(0.2081,0.0170,0.8957)}^T$; i.e., the Hilbert-Schmidt norm errors $\parallel d_m^{est}-d_m^{act}\parallel$ were less than 3% (0.0066, 0.0238, 0.0193 for $m=0,1,2$, respectively).