Precise qubit control beyond the rotating wave approximation

The ability to manipulate spins very fast allows one to increase the number of qubit operations before detrimental effects of decoherence take place, and it may further increase the bandwidth of spin based magnetometers. Optimal control theory (see for instance [1, 2]), provides powerful tools to work in this regime by finding the optimal way to transform the system from the initial to the desired state and to synthesize the target unitary gate with high fidelity. Here, we demonstrate a precisely-controlled ultra-fast (compared to the energy level spitting beyond rotating wave approximation (RWA)) single electron spin rotation using specially designed microwave (MW) fields without resorting to the standard RWA condition. To achieve this we employ an optimal control method, namely chopped random basis (CRAB) quantum optimization algorithm [3, 4]. It is used to numerically design and optimize our microwave control (see appendix A for further detailed information).

We implement our method using an electron spin associated with a single nitrogen-vacancy (NV) centre in diamond as a test bed. The centre shows remarkable physical properties: optical spin initialization and readout at room temperature [5], coherent spin control via MW fields [6], and long spin coherence time up to several milliseconds [7]. This NV system is a promising candidate for a nanoscale ultra-sensitive magnetometer [8–11], and for a solid state quantum processor [12, 13]. Strongly driven dynamics (beyond RWA) of two level systems in different cases has been theoretically analysed previously [14–19] and has been experimentally realized with NV using conventional pulses [20], where fast flipping of the NVʼs electron spin has been demonstrated.

The NV consists of a substitutional nitrogen atom and an adjacent vacancy with a triplet ground state (S = 1) and a strong optical transition, enabling the detection of single centres. Its fluorescence depends on the electron spin state, allowing one to perform coherent single spin control [5, 6]. The Hamiltonian of the NVʼs ground state in the presence of MW control ${{\Gamma }_{x}}(t)$ can be written as

$$\begin{eqnarray}&&\hat{\mathcal{H}}/\left( 2\pi \hbar \right)=D\widehat{S_{z}^{2}}+{{\omega }_{z}}{{\hat{S}}_{z}}+\sqrt{2}{{\Gamma }_{x}}(t){{\hat{S}}_{x}},\end{eqnarray} \tag{ 1 }$$

where $D\approx 2.87$ GHz is the NVʼs electron zero-field splitting, ${{\hat{S}}_{x}}$ and ${{\hat{S}}_{z}}$ are the x and z components of the electron spin operator and ${{\omega }_{z}}=g{{\mu }_{B}}{{B}_{0}}/\hbar$ is the Zeeman splitting due to a constant magnetic field B₀ along the NV axis (z-axis) with g the electron gyromagnetic ratio and ${{\mu }_{B}}$ the Bohr magneton [21]. For the description of our experiments the Hamiltonian has to be written in the lab frame since the control amplitude is comparable to the Larmor frequency of the spin ${\rm max} \{|{{\Gamma }_{x}}(t)|\}\sim {{\omega }_{L}}$ (where ${{\omega }_{L}}=D-{{\omega }_{z}}$), i.e. the counter-propagating term of the control can not be neglected [20, 22]. To fulfill this condition beyond the RWA and to approximate a two-level spin system of $|{{m}_{s}}=0\rangle$ and $|{{m}_{s}}=-1\rangle$, we apply a magnetic field B₀ = 1017.3 G (101.73 mT) and set the working transition frequency ${{\omega }_{L}}=30$ MHz (figure 1(a)). B₀ is aligned along the NV crystal axis in order to suppress mixing between the $|0\rangle$ and $|1\rangle$ states.

Zoom In Zoom Out Reset image size

We begin by measuring Rabi oscillations at different MW amplitudes and observe the spin dynamics (figure 1(c)). When the driving field $\Omega \lt {{\omega }_{L}}/2$ the system is in the RWA regime and a nice harmonic signal is obtained (figure 1(c), lower right inset). However, when $\Omega \gt {{\omega }_{L}}/2$ the signal becomes anharmonic as shown in figure 1(c) (upper left inset) and the precise control over the spin rotations is not trivial. Moreover, often the spin is not flipped within certain time as it can be seen from the upper left inset of figure 1(c), where the normalized fluorescence does not drop to zero and the $|{{m}_{s}}=-1\rangle$ state is not reached.

To perform a desired transformation of the spin state $|\psi (t)\rangle$, which follows the Schrödinger equation, ${\rm d}|\psi (t)\rangle /{\rm d}t=-{\rm i}\hat{\mathcal{H}}|\psi (t)\rangle$ (assuming ℏ = 1), we optimally engineer the time dependence of the control ${{\Gamma }_{x}}(t)$, such that at the final time T the state fidelity between the final spin state $|\psi (T)\rangle$ and the target $|{{\psi }_{T}}\rangle$ is maximized (see also appendix A). The fidelity F is defined as [23]

$$\begin{eqnarray}&&F=\sqrt{\left\langle {{\psi }_{{\rm target}}}\left| {{\rho }_{{\rm CRAB}}} \right|{{\psi }_{{\rm target}}} \right\rangle }\end{eqnarray} \tag{ 2 }$$

with ${{\psi }_{{\rm target}}}$ and ${{\rho }_{{\rm CRAB}}}$ being respectively the target state and the state expected after the CRAB pulse. Here, we used CRAB controls to implement the two most important single qubit rotations: flipping the qubit (π pulse) and creating a superposition between the qubit states ($\pi /2$ pulse). The experimental realization of these rotations is shown schematically in figure 1(b) while figure 2 shows the calculated trajectories of the spin during the CRAB-π and −π/2 pulses.

Zoom In Zoom Out Reset image size

We set the pulse lengths ${{T}_{\pi }}=15.4$ ns and ${{T}_{\pi /2}}=7.7\;$ ns, shorter than $T_{\pi }^{{\rm Rabi}}={{(2\Omega )}^{-1}}=16.67$ ns and $T_{\pi /2}^{{\rm Rabi}}={{(4\Omega )}^{-1}}=8.33$, where $\Omega =30$ MHz is the extrapolated Rabi frequency. It is important to note that the latter cannot be used for qubit rotations due to the breakdown of the RWA (see figure 1). The shortest possible pulse length for the π pulse is given by the bang–bang condition $T_{\pi }^{{\rm Bang}}=\pi /\sqrt{{{(\pi {{\omega }_{L}})}^{2}}+{{(2\pi \Omega )}^{2}}}=14.9$ ns [24]. We were not able to approach closer to this limit due to the limited bandwidth and gain curve of the amplifier (refer to appendix B).

We performed a density matrix tomography in order to determine the quality of the optimized pulses. Both off-diagonal elements have been measured by applying a low power ($\Omega =8$ MHz) $\pi /2$ pulse along the x- and y-axis of the rotating frame, followed by a laser pulse for read out (see figure 1(b)). To measure the diagonal elements the MW pulses have been omitted. After the CRAB-π pulse, the theoretically expected and the experimentally measured density matrices are

$$\begin{eqnarray*}\rho _{{\rm theory}}^{\pi }=\left( \begin{array}{ccccccccccccccc} 0 & 0 \\ 0 & 1 \\ \end{array} \right),\;\rho _{{\rm exp} }^{\pi }=\left( \begin{array}{ccccccccccccccc} 0.01 & 0.16-0.15{\rm i} \\ 0.16+0.15{\rm i} & 0.99 \\ \end{array} \right).\end{eqnarray*}$$

After the application of optimized CRAB-$\pi /2$ pulse, we expect

$$\begin{eqnarray*}\rho _{{\rm theory}}^{\pi /2}=\left( \begin{array}{ccccccccccccccc} 0.5 & 0.5 \\ 0.5 & 0.5 \\ \end{array} \right).\end{eqnarray*}$$

This is the state of the system expected directly after the MW pulse. However, due to limited time resolution of our apparatus, the measurement can be performed only after some dead time. We set this time to ${{t}_{{\rm evol}}}=100$ ns during which the spin rotates in the $xy$ plane of the lab frame and acquires a phase $\phi ={{{\rm e}}^{-{\rm i}{{\omega }_{L}}{{t}_{{\rm evol}}}}}$ (see figure 2, right). The density matrix after ${{t}_{{\rm evol}}}$ is then

$$\begin{eqnarray*}\rho _{{\rm theory}}^{\pi /2}=\left( \begin{array}{ccccccccccccccc} 0.5 & 0.48-0.14{\rm i} \\ 0.48+0.14{\rm i} & 0.5 \\ \end{array} \right).\end{eqnarray*}$$

From the tomography, we obtain

$$\begin{eqnarray*}\rho _{{\rm exp} }^{\pi /2}=\left( \begin{array}{ccccccccccccccc} 0.57 & 0.42-0.02{\rm i} \\ 0.42+0.02{\rm i} & 0.43 \\ \end{array} \right).\end{eqnarray*}$$

The expected fidelities of the CRAB pulses are $F_{{\rm theory}}^{\pi }=0.9986$ and $F_{{\rm theory}}^{\pi /2}=0.9545$, whilst from the experiment we obtain $F_{{\rm exp} }^{\pi }=(0.993\pm 0.016)$ and $F_{{\rm exp} }^{\pi /2}=(0.947\pm 0.007)$. We find an excellent agreement between the theoretical prediction and the experimental result. The theoretical values are lower than 1 due to the constrains set on the pulse duration. If the pulse length is increased, even higher numerical fidelities can be achieved [25]. The deviation from the prediction can be explained by distortion of the pulse shape, mainly due to the limited bandwidth of the MW amplifier and measurement error (see appendix C for details).

The pulses we have developed in this study are important not only for quantum information processing, but also for most of the pulsed nuclear magnetic resonance (NMR) and electron spin resonance (ESR) experiments. They performed the desired spin rotations (π or $\pi /2$ as we demonstrate below) up to a global phase factor no matter what the initial states is, although they were optimized for a $|{{m}_{s}}=0\rangle$ starting state. One of the most important NMR (and ESR) pulse sequences consists of a single $\pi /2$ pulse, where the spin magnetization is rotated from the z-axis to the $xy$ plane in the rotating frame. The spin then precesses and can be detected by the NMR detector thus giving a free induction decay (FID) signal. The Fourier transform of the latter is the spectrum of the sample [26, 27]. This experiment has been already implemented using NV and can be applied for detecting dc magnetic fields [28–30]. Here, we show that we can perform it both in the lab and in the rotating frame by using both CRAB and conventional rectangular pulses. The main difference here compared to previously reported studies, e.g. [30–32], is that we can perform these experiments beyond the RWA.

All sequences begin with a laser pulse for polarising the NV (figure 3(a)). Then we apply a CRAB-$\pi /2$ pulse, which aligns the spin magnetization along the x-axis of the lab frame. After a free evolution time τ we apply another CRAB-$\pi /2$ pulse to rotate the spin back to the z-axis and we then read out the spin state. The signal oscillates with the Larmor frequency ${{\omega }_{L}}$ (figure 3(a), see also figure 2, right). Here, we measure the signal in the laboratory frame. Now, if we replace the second CRAB-$\pi /2$ pulse with a low power rectangular $\pi /2$ pulse (here ${{\omega }_{L}}=4193$ MHz and $\Omega =48$ MHz) and keep its phase fixed for all τ (equivalent to having the B₁ parallel to y), again the FID signal oscillates with ${{\omega }_{L}}$ (figure 3(b), blue curve). However, if we set the phase of the second pulse to $\phi ={{{\rm e}}^{-{\rm i}{{\omega }_{L}}t}}$, we can 'follow' the magnetization in the equatorial plane of the Bloch sphere (figure 2, right) and the measurement is performed in the rotating frame and the oscillations at ${{\omega }_{L}}$ disappear (figure 3(b), red markers).

Zoom In Zoom Out Reset image size

The same experiment can be performed using two rectangular pulses with low MW amplitude (within the RWA) where again the phase of the second pulse is kept constant. In this case the experiment is run in the laboratory frame and the FID again oscillates with the Larmor frequency (figure 3(c), blue markers). If the phase of the MW is not the same for different τ, then we obtain the typical FID in the rotating frame as shown in figure 3(c) (red markers).

Another important sequence is the Hahn echo [33], which has found wide application in NMR and ESR. It is the basis of all dynamical decoupling techniques since all static inhomogeneous shifts (and fluctuations on the time scale of the coherence time T₂) are effectively canceled out [26]. It has been implemented with NV [6] and also for NV based ac magnetometry [7, 9]. The Hahn echo pulse sequence is depicted in figure 4(b). After a CRAB-$\pi /2$ pulse and a free evolution time τ₀ a CRAB-π pulse is applied. After a time τ the spin state is rotated to the z-axis by a CRAB-$\pi /2$ pulse and then it is read out by a laser pulse. The spin signal oscillates with the Larmor frequency of the NV transition (in this case ${{\omega }_{L}}=120$ MHz) as shown in figure 4(a) since the experiment is performed in the lab frame. If we use rectangular pulses in the rotating frame we obtain the envelope of the echo without the oscillations at ${{\omega }_{L}}$ (figure 4(c), red markers). However, we can again 'switch' to the lab frame by adjusting the phase ϕ of the last MW pulse (figure 4(c), blue curve). ESR experiments in the lab frame with low power MW pulses have been already reported [34, 35], but the 'switching' between the two frames in the strong driving regime has not been yet demonstrated. These results demonstrate that we can perform 'textbook' magnetic resonance experiments using the optimally designed pulses.

Zoom In Zoom Out Reset image size

In summary, we have developed a novel technique based on optimal control for precise spin qubit rotations in the ultra fast driving regime where standard pulses are not applicable. We design our qubit transformations by using the quantum optimization algorithm CRAB and find an excellent agreement with the experimental implementation. Moreover, we show even when the RWA breaks down the pulses developed here can be used for magnetic resonance experiments. Additionally we demonstrate on demand 'switching' between the rotating and the lab frame, using both CRAB and conventional (low power) pulses. This provides a precise control over the spin evolution and can be easily transferred to any other two level system, e.g. trapped atoms, trapped ions, quantum dots or superconducting qubits. Our results can find wide application in quantum computation and broadband magnetometry.

We thank Florian Dolde for the experimental support. This work is supported by EU STREP Project DIAMANT, SIQS, DIADEMS, DFG (SPP 1601/1, SFB TR21) and BMBF (QUOREP). Numerical optimization were performed on the bwGRiD (http://www.bw-grid.de).

In this section we shortly review the theoretical background of the CRAB method in the context of our current work, and furthermore describe the optimization procedures and the numerical results.

We recall the ground state Hamiltonian of the single electron NV spin in the presence of a single MW control ${{\Gamma }_{x}}(t)$, as discussed in the manuscript

$$\begin{eqnarray}&&\hat{\mathcal{H}}/\left( 2\pi \hbar \right)=D\widehat{S_{z}^{2}}+{{\omega }_{z}}{{\hat{S}}_{z}}+\sqrt{2}{{\Gamma }_{x}}(t){{\hat{S}}_{x}}.\end{eqnarray} \tag{ A.1 }$$

The CRAB method designs the control ${{\Gamma }_{x}}(t)$ by correcting an initial guess ${{\Gamma }_{0}}(t)$ with an optimized continuous function g(t), following ${{\Gamma }_{x}}(t)={{\Gamma }_{0}}(t)\times g(t)$ [3, 4]. We use here a simple constant initial guess ${{\Gamma }_{0}}(t)={{g}_{0}}$. Following [4], the correcting function g(t) is expanded into a Fourier-like basis function

$$\begin{eqnarray}&&g(t)=\frac{1}{2N\lambda (t)}\mathop{\sum }\limits_{n=1}^{N}\{{{a}_{n}}{\rm sin} \left( {{\omega }_{n}}t \right)+{{b}_{n}}{\rm cos} \left( {{\omega }_{n}}t \right)\},\end{eqnarray} \tag{ A.2 }$$

where N denotes a number of basis expansion having N-randomized discrete frequencies. It is noteworthy to state that the range of frequencies $({{\omega }_{1}},{{\omega }_{N}})$ directly corresponds to the real bandwidth of the apparatus. Therefore, we can pre-set in advance the frequency range for numerical optimization to meet the experimental limitations, e.g. the amplifierʼs or arbitrary waveform generatorʼs working bandwidth. The additional function $\lambda (t)$, is used to impose the control boundary such that ${{\Gamma }_{x}}(t)=0$ at the initial time t = 0, and the final time T. Here, we choose the bounding function $\lambda (t)={{h}^{p}}/({{h}^{p}}-{{(t-h)}^{p}})$, where $h=T/2$. Using this function we can also qualitatively vary the rising and falling times of the MW control by adjusting the even-numbered parameter p.

Our optimization objective is to find the set of CRAB parameters {${{\vec{a}}_{n}},{{\vec{b}}_{n}},{{\vec{\omega }}_{n}}$}, which minimizes the figure of merit, $\mathcal{F}=(1-f)+{{c}_{f}}{\rm max} \{|{{\Gamma }_{x}}(t)|\}$, where the quantity $f=|\langle \psi (T)|{{\psi }_{T}}\rangle {{|}^{2}}$ is the fidelity of the final state $|\psi (T)\rangle$, against the desired state $|{{\psi }_{T}}\rangle$. A dimensionless parameter c_f is incorporated in the figure of merit to limit the control amplitude during optimization. We employ the direct search simplex (Nelder–Mead) algorithm to find the optimal CRAB parameters.

The numerical optimization is initiated by setting some parameters obtained from the experimental preparations and apparatus calibrations: the measured Larmor transition ${{\omega }_{L}}$, the maximum control amplitude ${\rm max} \{|{{\Gamma }_{x}}(t)|\}=\Omega$, and the CRAB frequency range. The control time is fixed (the same as the desired rotation time) to be faster than the extrapolated rotation time if the RWA would be valid, e.g. for the spin $\pi$-rotation we have $T\lt$ 1/2Ω⁻¹, where Ω is the extrapolated Rabi frequency (see figure 1(c)). However, the $\pi$-rotation time in our case is limited by the minimum time of the theoretically proposed optimal bang–bang control, $T_{\pi }^{{\rm Bang}}=\pi /\sqrt{{{(\pi {{\omega }_{L}})}^{2}}+{{(2\pi \Omega )}^{2}}}$, where the control ${{\Gamma }_{x}}(t)$ takes only a constant value of $\pm \Omega$ for a certain time interval [24]. To obtain the optimized pulse for one target rotation, we do the following steps:

• (1)  
  Perform the parallel simplex search algorithm with an S number of random initial values of the CRAB parameters for j small positive real numbers of $\{c_{f}^{j}\}$, and k positive small integers $\{{{N}^{k}}\}$, typically $\{c_{f}^{j}\}\in (0.01,0.5)$ and $\{{{N}^{k}}\}\in (3,7)$.
• (2)  
  Obtain from step 1, $(S\times j\times k)$ sets of CRAB parameters, than construct $(S\times j\times k)$ numbers of control pulses $\{{{\Gamma }_{x}}(t)\}$.
• (3)  
  Investigate the numerical values of $\mathcal{F}$ and ${\rm max} \{|{{\Gamma }_{x}}(t)|\}$ for each pulse, and pick the best one out of $(S\times j\times k)$ pulses which satisfies $\mathcal{F}\leqslant {{\kappa }_{f}}$ and ${\rm max} \{|{{\Gamma }_{x}}(t)|\}\leqslant {{\kappa }_{\Gamma }}$. The preset quantities κ_f and ${{\kappa }_{\Gamma }}$ are the numerical infidelity and the maximum control amplitude, respectively. If the best pulse can not be obtained, return to step 1 with different values of $\{c_{f}^{j}\}$ and increase $\{{{N}^{k}}\}$.

The actual numerical calculations were carried out on the bwGRiD cluster where we utilized its multi-nodes multi-cores (eight cores per node) computational features to run the parallel Nelder–Mead searches that corresponds to various experimental parameters and random initial values. To find a single parallel sample in one-core, i. e. one set of CRAB optimized parameters, the typical computational time required to meet the experimentally acceptable fidelity is approximately less than 30 min. This allows one to perform a single optimization run in just a decent commercial personal computer. Hence, it is feasible in the future to apply our numerical CRAB optimization in standard close-loop control system involving directly the control apparatuses. For both cases of $\pi$-rotation and $\pi /2$-rotation we set the parameters as the following: N = 5, S = 30, ${{\omega }_{L}}={{g}_{0}}={\rm max} \{|{{\Gamma }_{x}}(t)|\}=30$ MHz, and ${{\omega }_{n}}\in (10\;100)$ MHz. We present the best obtained CRAB parameters for each rotation in table A.1 , while the corresponding MW pulses are shown in figure B.1.

Zoom In Zoom Out Reset image size

Table A.1.  The optimal CRAB parameters obtained via the Nelder–Mead simplex algorithm for π- and $\pi /2$-rotations. The corresponding MW-CRAB pulses are shown in figure B.1.

π-rotation			$\pi /2$-rotation
T = 15.4071 ns, p = 60, cf = 0.35			T = 7.7036 ns, p = 38, cf = 0.23
an	bn	ωn (GHz)	an	bn	ωn (GHz)
${{a}_{1}}=-5.4865$	${{b}_{1}}=0.2812$	${{\omega }_{1}}=0.0201$	${{a}_{1}}=2.1123$	${{b}_{1}}=9.6205$	${{\omega }_{1}}=0.0149$
${{a}_{2}}=2.4803$	${{b}_{2}}=1.8823$	${{\omega }_{2}}=0.0415$	${{a}_{2}}=-5.5973$	${{b}_{2}}=-28.7365$	${{\omega }_{2}}=0.0401$
${{a}_{3}}=-0.5404$	${{b}_{3}}=5.8533$	${{\omega }_{3}}=0.0513$	${{a}_{3}}=-9.7577$	${{b}_{3}}=-3.9425$	${{\omega }_{3}}=0.0464$
${{a}_{4}}=1.5659$	${{b}_{4}}=-2.2123$	${{\omega }_{4}}=0.0687$	${{a}_{4}}=26.3464$	${{b}_{4}}=5.4267$	${{\omega }_{4}}=0.0664$
${{a}_{5}}=1.4673$	${{b}_{5}}=3.6469$	${{\omega }_{5}}=0.0892$	${{a}_{5}}=-10.4212$	${{b}_{5}}=7.2445$	${{\omega }_{5}}=0.0909$

The pulses optimized by the CRAB algorithm is a superposition of ten periodic functions. In the experiment, these pulses are synthesized directly by an arbitrary waveform generator (AWG, Tektronix AWG7122C) with a sampling rate of 24 GS s⁻¹ and then sent to an amplifier (Mini-Circuits, ZHL-42W-SMA). The pulse shapes measured via an oscilloscope (Tektronix, TDS6804B) are displayed in figure B.1.

Optical measurements were obtained via a self-made confocal microscope, the AWG triggered both the acousto-optic modulator for laser pulse control and the photon-count card (FastComtec P7887).

In order to determine the fidelity of the final state in respect to the target state we performed a state tomography. For this purpose we evaluated all three components of the Bloch vector. The z-component is measured directly from the fluorescence level. The x- and y-components have to be projected into the measurable z-component. This is done by applying a MW with different phases and measure Rabi oscillations or in other words: rotate the Bloch vector around the x-axis to determine the y-component and vice versa, as shown in figure C.1.

Zoom In Zoom Out Reset image size

The density matrix of a single qubit can be written as

$$\begin{eqnarray}\rho =\frac{1}{2}\left( \begin{array}{ccccccccccccccc} 1+z & x+{\rm i}y \\ x-{\rm i}y & 1-z \\ \end{array} \right).\end{eqnarray} \tag{ C.1 }$$

In this definition, $x,y$ and z have values between −1 and 1. The first point of the measurements ${\rm Rab}{{{\rm i}}_{x}}(1)$ and ${\rm Rab}{{{\rm i}}_{y}}(1)$, where no MW was applied, can be understood as the z-component. The three components of the Bloch vector are calculated following

$$\begin{eqnarray}&&z=2\cdot \left\langle {\rm Rab}{{{\rm i}}_{x}}(1)+{\rm Rab}{{{\rm i}}_{y}}(1) \right\rangle -1={\rm Rab}{{{\rm i}}_{x}}(1)+{\rm Rab}{{{\rm i}}_{y}}(1)-1,\end{eqnarray} \tag{ C.2 }$$

$$\begin{eqnarray}&&y=\sqrt{{{\left( 2{\rm Am}{{{\rm p}}_{y}} \right)}^{2}}-{{z}^{2}}},\end{eqnarray} \tag{ C.3 }$$

$$\begin{eqnarray}&&x=\sqrt{{{\left( 2{\rm Am}{{{\rm p}}_{x}} \right)}^{2}}-{{z}^{2}}},\end{eqnarray} \tag{ C.4 }$$

where $x,y$ and z are the three components of the Bloch vector and ${\rm Am}{{{\rm p}}_{x,y}}$ are the amplitudes of the Rabi oscillations rotating around the y, x-axis respectively. In formula C.3 and C.4, values for x and y were set to 0 if the uncertainty was greater than the actual value. Another possibility to obtain the x- and y-component is to calculate ${\rm Rab}{{{\rm i}}_{x}}(\pi /2)$ and ${\rm Rab}{{{\rm i}}_{y}}(\pi /2)$ respectively.

To normalize the data we performed an additional, bare Rabi measurement. The normalization is done by ${\rm Rab}{{{\rm i}}_{x}}(1)=({\rm rawdata}(1)-({{y}_{0}}-A))/2A$, where A is the amplitude and y₀ is the offset of the normalization measurement.

In order to calculate the fidelity between the experimental state ρ and the target state $|\Psi \rangle$ the definition for pure states is used: $F=\sqrt{\langle \Psi |\rho |\Psi \rangle }$ which in this case is equivalent to the general definition $F=tr\sqrt{\sqrt{\sigma }\rho \sqrt{\sigma }}$ [23].

Due to experimental limitations we had to wait for 100 ns between the CRAB-pulse and the Rabi measurement, hence the target state after the time evolution on the $x,y$ plane becomes

$$\begin{eqnarray}&&|\Psi (t)\rangle ={{{\rm e}}^{-\frac{{\rm i}}{2}{{\sigma }_{z}}{{\omega }_{L}}t}}|\Psi (0)\rangle ,\end{eqnarray} \tag{ C.5 }$$

with the Pauli matrix σ_z and the Larmor frequency ${{\omega }_{L}}$. For the error calculation of the fidelity the noise of the Poisson distributed photon collection and fitting errors were taken into account. The error was determined by using the general law of error propagation [36]

$$\begin{eqnarray}&&\Delta F=\mathop{\sum }\limits_{k=1}^{N}{{\left( \frac{\partial F}{\partial {{f}_{k}}} \right)}^{2}}\operatorname{var}({{f}_{k}})+2\mathop{\sum }\limits_{l=1}^{N-1}\mathop{\sum }\limits_{m=l+1}^{N}\left( \frac{\partial F}{\partial {{f}_{l}}} \right)\quad \left( \frac{\partial F}{\partial {{f}_{m}}} \right)\;\;\operatorname{cov}\;\;({{f}_{l}},{{f}_{m}}).\end{eqnarray} \tag{ C.6 }$$