Quantum sensing

“Quantum sensing” describes the use of a quantum system, quantum properties, or quantum phenomena to perform a measurement of a physical quantity. Historical examples of quantum sensors include magnetometers based on superconducting quantum interference devices and atomic vapors or atomic clocks. More recently, quantum sensing has become a distinct and rapidly growing branch of research within the area of quantum science and technology, with the most common platforms being spin qubits, trapped ions, and flux qubits. The field is expected to provide new opportunities—especially with regard to high sensitivity and precision—in applied physics and other areas of science. This review provides an introduction to the basic principles, methods, and concepts of quantum sensing from the viewpoint of the interested experimentalist.

Figure 1

Basic features of a two-state quantum system. $|0⟩$ is the lower energy state and $|1⟩$ is the higher energy state. Quantum sensing exploits changes in the transition frequency ${\omega }_0$ or the transition rate $\mathrm{\Gamma }$ in response to an external signal $V$.

Figure 2

Basic steps of the quantum sensing process.

Figure 3

Transition probability $p$ for a Ramsey experiment as a function of the signal $V$ picked up by the sensor. (a) Slope detection: The quantum sensor is operated at the $p_0=0.5$ bias point (filled red circle). A small change in the signal $\delta V$ leads to a linear change in the transition probability $\delta p=\delta \varphi /2=\gamma \delta Vt/2$ (empty red circle). The uncertainty ${\sigma }_p$ in the measured transition probability leads to an uncertainty in the estimated signal $V_{\min }$ (gray shade). (b) Variance detection: The quantum sensor is operated at the $p_0=0$ bias point (filled blue square). A small change in the signal $\delta V$ leads to a quadratic change $\delta p=\delta {\varphi }^2/4={\gamma }^2\delta V^2{t}^2/4$ (empty blue square). The information on the sign of $\delta V$ is lost. The experimental readout error ${\sigma }_p$ translates into an uncertainty in the estimated signal $V_{\min }$, according to the slope or curvature of the Ramsey fringe (gray shade).

Figure 4

Illustration of the sensor readout. $N$ measurements are performed producing $\{x_j{\}}_{j=1,\dots ,N}$ readings on the physical measurement apparatus. The readings $\{x_j\}$ are then binned into a histogram. (a) Ideal readout. Only two values are observed in the histogram, $x_{|0⟩}$ and $x_{|1⟩}$, which correspond to the qubit states $|0⟩$ and $|1⟩$. All $\{x_j\}$ can be assigned to 0 or 1 with 100% fidelity. (b) Single shot readout. Most $\{x_j\}$ can be assigned, but there is an overlap between histogram peaks leading to a small error. (c) Averaged readout. $\{x_j\}$ cannot be assigned. The ratio between 0 and 1 is given by the relative position of the mean value $\overline{x}$ and the error is determined by the histogram standard deviation ${\sigma }_x$. $R$ is the ratio of readout and projection noise, and $C$ is an overall readout efficiency parameter that is explained in the text.

Figure 5

Pulse diagrams for dc and ac sensing sequences. Narrow blocks represent $\pi /2$ pulses and wide blocks represent $\pi$ pulses, respectively. $t$ is the total sensing time and $\tau$ is the interpulse delay. (a) Ramsey sequence. (b) Spin-echo sequence. (c) Carr-Purcell (CP) multipulse sequence. (d) PDD multipulse sequence.

Figure 6

Modulation and weight functions of a $CP$ multipulse sequence. (a) $CP$ multipulse sequence. (b) Signal $V(t^{\prime })$, modulation function $y(t^{\prime })$, and “rectified” signal $V(t^{\prime })\times y(t^{\prime })$. The accumulated phase is represented by the area under the curve. (c) Weight function ${\overline{W}}^2(f_{\mathrm{ac}})$ associated with the modulation function. $k$ is the harmonic order of the filter resonance.

Figure 7

Transition probability $p(t)$ as a function of phase accumulation time $t$. (a) Ramsey oscillation [Eq. (14)]. (b) ac signal with fixed amplitude and optimum phase [Eq. (71)]. (c) ac signal with fixed amplitude and random phase [Eq. (75)]. (d) ac signal with random amplitude and random phase [Eq. (76)]. (e) Broadband noise with $\chi =\mathrm{\Gamma }t$ [Eq. (83)].

Figure 8

Correlation spectroscopy. (a) ac signal $V(t^{\prime })$. (b) Correlation sequence. Two multipulse sequences are interrupted by an incremented delay time $t_1$. Because the multipulse sequences are phase sensitive, the total phase accumulated after the second multipulse sequence oscillates with $f_{\mathrm{ac}}{t}_1$. The maximum $t_1$ is limited by the relaxation time $T_1$, rather than the typically short decoherence time $T_2$. (c) Continouous sampling. The signal is periodically probed in intervals of the sampling time $t_s$. The frequency can be estimated from a sample record of arbitrary duration, permitting an arbitrarily fine frequency resolution.

Figure 9

Common relaxometry protocols. (a) $T_1$ relaxometry. (b) $T_2^{*}$ relaxometry. (c) Dressed state relaxometry. Narrow black boxes represent $\pi /2$ pulses and the gray box in (c) represents a resonant or off-resonant drive field.

Figure 10

Typical spectral range of noise spectroscopy protocols. Scales refer to the quantum sensors discussed in Sec. 3.

Figure 11

High dynamic range sensing. A series of measurements with different interrogation times $t$ is combined to estimate a signal of interest. The shortest measurement (lowest line) has the largest allowed signal $V_{\max }$ and provides a coarse estimate of the quantity, which is subsequently refined by longer and more sensitive measurements. Although the $p(V)$ measured in a sensitive measurement (top line) can correspond to many possible signal values, the coarse estimates allow one to extract a unique signal value $V$.

Figure 12

Phase estimation algorithms. (a) Quantum phase estimation by the inverse quantum Fourier transform, as it is employed in prime factorization algorithms ([309]; [187]). (b) Adaptive phase estimation. Here the quantum Fourier transform is replaced by measurement and classical feedback. Bits are measured in ascending order, subtracting the lower digits from measurements of higher digits by phase gates that are controlled by previous measurement results. (c) Nonadaptive phase estimation. Measurements of all digits are fed into a Bayesian estimation algorithm to estimate the most likely value of the phase. $H$ represents a Hadamard gate, $R(\mathrm{\Phi })$ a $Z$ rotation by the angle $\mathrm{\Phi }$, and $U$ the propagator for one time element $t_0$. The boxes labeled by “x” represent readouts.

Figure 13

Left: Ramsey scheme. Right: entangled Ramsey scheme for Heisenberg-limited sensitivity