Using a biased qubit to probe complex systems

Complex mesoscopic systems play increasingly important roles in modern science, from understanding biological functions at the molecular level to designing solid-state information processing devices. The operation of these systems typically depends on their energetic structure, yet probing their energy landscape can be extremely challenging; they have many degrees of freedom, which may be hard to isolate and measure independently. Here, we show that a qubit (a two-level quantum system) with a biased energy splitting can directly probe the spectral properties of a complex system, without knowledge of how they couple. Our work is based on the completely positive and trace-preserving map formalism, which treats any unknown dynamics as a “black-box” process. This black box contains information about the system with which the probe interacts, which we access by measuring the survival probability of the initial state of the probe as function of the energy splitting and the process time. Fourier transforming the results yields the energy spectrum of the complex system. Without making assumptions about the strength or form of its coupling, our probe could determine aspects of a complex molecule's energy landscape as well as, in many cases, test for coherent superposition of its energy eigenstates.

Figure 1

The controlled two-level probe scenario: the properties of an uncharacterized complex system $\mathcal{S}$ can be inferred from the way in which it influences dynamics of a two-level probe $\mathcal{P}$, with which it interacts. By varying a controllable bias of magnitude $\lambda$, more information about the system becomes accessible, including aspects of its state and spectrum.

Figure 2

Probe evolution as a black box: (a) a two-level probe is prepared in state $\left|\alpha \right.〉$; (b) the probe is acted on by CPTP map ${\mathrm{\Lambda }}_{\tau :0}(\lambda ,\theta ,\varphi )$, which has a set of controllable parameters corresponding to the magnitude and basis of an imposed splitting of the probe's energy levels $H_{\mathcal{P}}=\frac{1}{2}\lambda {\sigma }_{(\theta ,\varphi )}$; (c) the probe is measured in the basis defined by state $\left|\beta \right.〉$.

Figure 3

Transition probability. Typical behavior of $p_{\beta :\alpha }$ as $\lambda$ is varied from small to large (for reference the matrix elements of $V_{\mathcal{PS}}$ are sampled such that $|\left.〈j_{0\left(1\right)}\right|B\left|k_{0\left(1\right)}\right.〉|\lesssim \sqrt{2})$. The results shown are for a system comprising two qubits; $V_{\mathcal{PS}}$ and ${\rho }_{\mathcal{S}}$ were sampled randomly, such that they were effectively unknown prior to the simulations. Different panels show effects at different orders for large $\lambda$, corresponding to different choices of preparation and measurement. Panels (b) and (c) both correspond to first-order effects, but with different choices of preparation. Solid red represents the values of $p_{\beta :\alpha }$, and bounding envelopes are shown with dashed blue lines.

Figure 4

Simulated spin spectra: (a) a superconducting flux qubit probe $\mathcal{P}$ (with magnetic moment $0.3\mathrm{meV}{\mathrm{T}}^{-1})$ is placed at four different randomly chosen positions (i)–(iv) within a radius of $100\mathrm{nm}$ around a spin system $\mathcal{S}$, which consists of four spins (with magnetic moments $\sim {\mu }_N)$ randomly arranged within a $0.01\text{−}\mathrm{nm}$ sphere; a $\left|\mathbf{B}\right|=1\mathrm{mT}$ magnetic field is applied perpendicular to the magnetic moment of $\mathcal{P}$. Probe positions are shown in the plane defined by $\mathbf{B}$ and the vector joining $\mathcal{P}$ and $\mathcal{S}$. (b) Reconstructed spectra for the different probe positions. Red peaks indicate the spectrum that would be obtained with perfect measurement of the transition probabilities $p_{1:0}$ for all values of $\lambda$ and $\tau$. The other curves are simulated reconstructions, where the experiment was performed only a finite number $\left(\sim {10}^6\right)$ of times for each choice of parameters. $\tau$ was sampled evenly in $100\mathrm{ns}$ steps for different total evolution times: $80\mu \mathrm{s}$ (dotted green); $160\mu \mathrm{s}$ (dashed purple); $2\mathrm{ms}$ (solid blue). 100 different values of $\lambda$ were tested in each case.