Quantum Sensing

C. L. Degen∗ Department of Physics, ETH Zurich, Otto Stern Weg 1, 8093 Zurich, Switzerland. F. Reinhard† Walter Schottky Institut and Physik-Department, Technische Universit¨atM¨unchen, Am Coulombwall 4, 85748 Garching, Germany. P. Cappellaro‡ Research Laboratory of Electronics and Department of Nuclear Science & Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge MA 02139, USA.

(Dated: June 7, 2017)

“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. In this review, we provide an introduction to the basic principles, methods and concepts of quantum sensing from the viewpoint of the interested experimentalist.

CONTENTS 2. Neutrons9 I. Other sensors9 I. Introduction2 1. Single electron transistors9 Content2 2. Optomechanics 10 3. Photons 10 II. Deﬁnitions3 A. Quantum sensing3 IV. The quantum sensing protocol 11 B. Quantum sensors3 A. Quantum sensor Hamiltonian 11 1. Internal Hamiltonian 11 III. Examples of quantum sensors4 2. Signal Hamiltonian 11 A. Neutral atoms as magnetic ﬁeld sensors4 3. Control Hamiltonian 12 1. Atomic vapors4 B. The sensing protocol 12 2. Cold atomic clouds4 C. First example: Ramsey measurement 13 B. Trapped ions6 D. Second example: Rabi measurement 13 C. Rydberg atoms6 E. Slope and variance detection 14 arXiv:1611.02427v2 [quant-ph] 6 Jun 2017 D. Atomic clocks6 1. Slope detection (linear detection) 14 E. Solid state spins – Ensemble sensors7 2. Variance detection (quadratic detection) 14 1. NMR ensemble sensors7 2. NV center ensembles7 V. Sensitivity 14 F. Solid state spins - Single spin sensors7 A. Noise 15 G. Superconducting circuits8 1. Quantum projection noise 15 1. SQUIDs8 2. Decoherence 15 2. Superconducting qubits8 3. Errors due to initialization and qubit H. Elementary particle qubits9 manipulations 15 1. Muons9 4. Classical readout noise 16 B. Sensitivity 17 1. Signal-to-noise ratio 17 2. Minimum detectable signal and sensitivity 17

∗ 3. Signal integration 18 [email protected] C. Allan variance 18 † [email protected] D. Quantum Cram´erRao Bound for parameter ‡ [email protected] estimation 18 2

VI. Sensing of AC signals 19 Interestingly, however, this belief might turn out to A. Time-dependent signals 19 be incomplete. In recent years a different class of ap- B. Ramsey and Echo sequences 20 plications has emerged that employs quantum mechan- C. Multipulse sensing sequences 20 1. CP and PDD sequences 20 ical systems as sensors for various physical quantities 2. Lock-in detection 21 ranging from magnetic and electric fields, to time and 3. Other types of multipulse sensing sequences 21 frequency, to rotations, to temperature and pressure. 4. AC signals with random phase 22 5. AC signals with random phase and random “Quantum sensors” capitalize on the central weakness amplitude 22 of quantum systems – their strong sensitivity to external D. Waveform reconstruction 22 disturbances. This trend in quantum technology is curi- E. Frequency estimation 23 ously reminiscent of the history of semiconductors: here, 1. Dynamical decoupling spectroscopy 23 2. Correlation sequences 23 too, sensors – for instance light meters based on selenium 3. Continuous sampling 24 photocells (Weston, 1931) – have found commercial applications decades before computers. VII. Noise spectroscopy 24 A. Noise processes 24 Although quantum sensing as a distinct field of re- B. Decoherence, dynamical decoupling and filter search in quantum science and engineering is quite recent, functions 24 many concepts are well-known in the physics community 1. Decoherence function χ(t) 25 and have resulted from decades of developments in high- 2. Filter function Y (ω) 25 3. Dynamical decoupling 25 resolution spectroscopy, especially in atomic physics and C. Relaxometry 26 magnetic resonance. Notable examples include atomic 1. Basic theory of relaxometry 26 clocks, atomic vapor magnetometers, and superconduct- 2. T1 relaxometry 27 ∗ ing quantum interference devices. What can be consid- 3. T2 and T2 relaxometry 28 4. Dressed state methods 28 ered as “new” is that quantum systems are increasingly investigated at the single-atom level, that entanglement VIII. Dynamic range and adaptive sensing 28 is used as a resource for increasing the sensitivity, and A. Phase estimation protocols 29 1. Quantum phase estimation 29 that quantum systems and quantum manipulations are 2. Adaptive phase estimation 30 specifically designed and engineered for sensing purposes. 3. Non-adaptive phase estimation 31 The focus of this review is on the key concepts and 4. Comparison of phase estimation protocols 31 methods of quantum sensing, with particular atten- B. Experimental realizations 31 tion to practical aspects that emerge from non-ideal IX. Ensemble sensing 32 experiments. As “quantum sensors” we will consider A. Ensemble sensing 32 mostly qubits – two-level quantum systems. Although an B. Heisenberg limit 32 C. Entangled states 32 overview over actual implementations of qubits is given, 1. GHZ and N00N states 32 the review will not cover any of those implementation in 2. Squeezing 33 specific detail. It will also not cover related fields includ- 3. Parity measurements 34 ing atomic clocks or photon-based sensors. In addition, 4. Other types of entanglement 34 theory will only be considered up to the point necessary X. Sensing assisted by auxiliary qubits 35 to introduce the key concepts of quantum sensing. The A. Quantum logic clock 35 motivation behind this review is to offer an introduction B. Storage and retrieval 35 C. Quantum error correction 35 to students and researchers new to the field, and to provide a basic reference for researchers already active in the XI. Outlook 36 field. Acknowledgments 37

Appendix A: Table of symbols 37 Content References 37 The review starts by suggesting some basic definitions for “quantum sensing” and by noting the elementary cri- I. INTRODUCTION teria for a quantum system to be useful as a quantum sensor (SectionII). The next section provides an overview Can we find a promising real-world application of of the most important physical implementations (Section quantum mechanics that exploit its most counterintuitive III). The discussion then moves on to the core concepts properties? This question has intrigued physicists ever of quantum sensing, which include the basic measure- since quantum theory development in the early twentieth ment protocol (SectionIV) and the sensitivity of a quan- century. Today, quantum computers (Deutsch, 1985; Di- tum sensor (SectionV). SectionsVI andVII cover the Vincenzo, 2000) and quantum cryptography (Gisin et al., important area of time-dependent signals and quantum 2002) are widely believed to be the most promising ones. spectroscopy. The remaining sections introduce some ad- 3 vanced quantum sensing techniques. These include adaptive methods developed to greatly enhance the dynamic range of the sensor (SectionVIII), and techniques that involve multiple qubits (SectionsIX andX). In particular, entanglement-enhanced sensing, quantum storage and quantum error correction schemes are discussed. The review then concludes with a brief outlook on possible future developments (SectionXI). FIG. 1 Basic features of a two-state quantum system. |0i There have already been several reviews that covered is the lower energy state and |1i is the higher energy state. different aspects of quantum sensing. Excellent intro- Quantum sensing exploits changes in the transition frequency ω0 or the transition rate Γ in response to an external signal ductions into the field are the review (Budker and Ro- V . malis, 2007) and book (Budker and Kimball, 2013) by Budker, Romalis and Kimball on atomic vapor magnetometry, and the paper by Taylor et al., 2008, on mag- as for type-III sensors, they already can provide several netometry with nitrogen-vacancy centers in diamond. advantages, most notably operation at nano-scales that Entanglement-assisted sensing, sometimes referred to as are not accessible to classical sensors. “quantum metrology”, “quantum-enhanced sensing” or Because type-III quantum sensor rely on entanglement, “second generation quantum sensors” are covered by more than one sensing qubit is required. A well-known Bollinger et al., 1996, Giovannetti et al., 2004, Giovan- example is the use of maximally entangled states to reach netti et al., 2006, and Giovannetti et al., 2011. In ad- a Heisenberg-limited measurement. Type III quantum dition, many excellent reviews covering different imple- sensors are discussed in SectionX. mentations of quantum sensors are available; these will be noted in SectionIII. B. Quantum sensors

II. DEFINITIONS In analogy to the DiVincenzo criteria for quantum computation (DiVincenzo, 2000), a set of four necessary A. Quantum sensing attributes can be listed for a quantum system to function as a quantum sensor. These attributes include three “Quantum sensing” is typically used to describe one of original DiVincenzo criteria: the following: (1) The quantum system has discrete, resolvable en- I. Use of a quantum object to measure a physical ergy levels. Specifically, we will assume it to be a quantity (classical or quantum). The quantum two-level system (or an ensemble of two-level sys- object is characterized by quantized energy lev- tems) with a lower energy state 0 and an upper els. Specific examples include electronic, magnetic energy state 1 that are separated| i by a transition or vibrational states of superconducting or spin | i 1 energy E = ~ω0 (see Fig.1) . qubits, neutral atoms, or trapped ions. (2) It must be possible to initialize the quantum system II. Use of quantum coherence (i.e., wave-like spatial or into a well-known state and to read out its state. temporal superposition states) to measure a physical quantity. (3) The quantum system can be coherently manip- ulated, typically by time-dependent fields. This III. Use of quantum entanglement to improve the sensi- condition is not strictly required for all proto- tivity or precision of a measurement, beyond what cols; examples that fall outside of this criterion is possible classically. are continuous-wave spectroscopy or relaxation rate measurements. Of these three definitions, the first two are rather broad and cover many physical systems. This even includes The focus on two-level systems (1) is not a severe restric- some systems that are not strictly “quantum”. An exam- tion because many properties of more complex quantum ple is classical wave interference as it appears in optical or systems can be modeled through a qubit sensor (Gold- mechanical systems (Faust et al., 2013; Novotny, 2010). stein et al., 2010). The fourth attribute is specific to The third definition is more stringent and a truly “quan- quantum sensing: tum” definition. However, since quantum sensors according to definitions I and II are often close to applications, we will mostly focus on these definitions and discuss them 1 extensively in this review. While these types of sensors Note that this review uses ~ = 1 and expresses all energies in might not exploit the full power of quantum mechanics, units of angular frequency. 4

(4) The quantum system interacts with a relevant 1. Atomic vapors physical quantity V (t), like an electric or magnetic field. The interaction is quantified by a coupling or In the simplest implementation, a thermal vapor of transduction parameter of the form γ = ∂qE/∂V q atoms serves as a quantum sensor for magnetic fields which relates changes in the transition energy E (Budker and Romalis, 2007; Kominis et al., 2003). Held to changes in the external parameter V . In most in a cell at or above room temperature, atoms are spin- situations the coupling is either linear (q = 1) or polarized by an optical pump beam. Magnetic field sens- quadratic (q = 2). The interaction with V leads to ing is based on the Zeeman effect due to a small external a shift of the quantum system’s energy levels or to field orthogonal to the initial atomic polarization. In a transitions between energy levels (see Fig.1). classical picture, this field induces coherent precession of the spin. Equivalently, in a quantum picture, it drives Experimental realizations of quantum sensors can be spin transitions from the initial quantum state to a dis- compared by some key physical characteristics. One tinct state, which can be monitored by a probe beam, e.g. characteristic is to what kind of external parameter(s) via the optical Faraday effect. Despite their superficial the quantum sensor responds to. Charged systems, simplicity, these sensors achieve sensitivities in the range like trapped ions, will be sensitive to electrical fields, of 100 aT/√Hz (Dang et al., 2010) and approach a theory while spin-based systems will mainly respond to mag- limit of < 10 aT/√Hz, placing them on par with Super- netic fields. Some quantum sensors may respond to sev- conducting Quantum Interference Device (SQUIDs, see eral physical parameters. below) as the most sensitive magnetometers to date. This A second important characteristic is a quantum sen- is owing to the surprising fact that relaxation and coher- sor’s “intrinsic sensitivity”. On the one hand, a quan- ence times of spins in atomic vapors can be pushed to tum sensor is expected to provide a strong response to the second to minute range (Balabas et al., 2010). These wanted signals, while on the other hand, it should be long relaxation and coherence times are achieved by coat- minimally affected by unwanted noise. Clearly, these are ing cell walls to preserve the atomic spin upon collisions, conflicting requirements. In SectionV, we will see that and by operating in the spin exchange relaxation-free the sensitivity scales as (“SERF”) regime of high atomic density and zero magnetic field. Somewhat counterintuitively, a high density 1 suppresses decoherence from atomic interactions, since sensitivity p , (1) ∝ γ Tχ collisions occur so frequently that their effect averages out, similar to motional narrowing of dipolar interactions where γ is the above transduction parameter and Tχ is a in nuclear magnetic resonance (Happer and Tang, 1973). decoherence or relaxation time that reflects the immunity Vapor cells have been miniaturized to few mm3 small vol- of the quantum sensor against noise. In order to optimize umes (Shah et al., 2007) and have been used to demon- the sensitivity, γ should be large (for example, by choice strate entanglement-enhanced sensing (Fernholz et al., of an appropriate physical realization of the sensor) and 2008; Wasilewski et al., 2010). The most advanced ap- the decoherence time Tχ must be made as long as pos- plication of vapor cells is arguably the detection of neural sible. Strategies to achieve the latter are discussed at activity (Jensen et al., 2016; Livanov et al., 1978), which length in the later sections of this review. has found use in magnetoencephalography (Xia et al., 2006). Vapor cells also promise complementary access to high-energy physics, detecting anomalous dipole mo- III. EXAMPLES OF QUANTUM SENSORS ments from coupling to exotic elementary particles and background fields beyond the standard model (Pustelny We now give an overview of the most important exper- et al., 2013; Smiciklas et al., 2011; Swallows et al., 2013). imental implementations of quantum sensors, following the summary in TableI. 2. Cold atomic clouds

A. Neutral atoms as magnetic field sensors The advent of laser cooling in the 1980s spawned a rev- olution in atomic sensing. The reduced velocity spread Alkali atoms are suitable sensing qubits fulfilling the of cold atoms enabled sensing with longer interrogation above definitions (Kitching et al., 2011). Their ground times using spatially confined atoms, freely falling along state spin - a coupled angular momentum of electron and specific trajectories in vacuum or trapped. nuclear spin - can be both prepared and read out opti- Freely falling atoms have enabled the development cally by the strong spin-selective optical dipole transition of atomic gravimeters (Kasevich and Chu, 1992; Peters linking their s-wave electronic ground state to the first et al., 1999) and gyrometers (Gustavson et al., 1997, (p-wave) excited state. 2000). In these devices an atomic cloud measures ac- 5

Implementation Qubit(s) Measured Typical Initalization Readout Typea quantity(ies) frequency Neutral atoms Atomic vapor Atomic spin Magnetic field, DC–10 GHz Optical Optical II–III Rotation, Time/Frequency Cold clouds Atomic spin Magnetic field, DC–10 GHz Optical Optical II–III Acceleration, Time/Frequency Trapped ion(s) Long-lived Time/Frequency THz Optical Optical II-III electronic state Rotation Optical Optical II Vibrational mode Electric field, Force MHz Optical Optical II Rydberg atoms Rydberg states Electric field DC, GHz Optical Optical II-III Solid state spins (ensembles) NMR sensors Nuclear spins Magnetic field DC Thermal Pick-up coil II NVb center ensembles Electron spins Magnetic field, DC–GHz Optical Optical II Electric field, Temperature, Pressure, Rotation Solid state spins (single spins) P donor in Si Electron spin Magnetic field DC–GHz Thermal Electrical II Semiconductor Electron spin Magnetic field, DC–GHz Electrical, Optical Electrical, Optical I–II quantum dots Electric field Single NVb center Electron spin Magnetic field, DC–GHz Optical Optical II Electric field, Temperature, Pressure, Rotation Superconducting circuits SQUIDc Supercurrent Magnetic field DC–10 GHz Thermal Electrical I–II Flux qubit Circulating Magnetic field DC–10 GHz Thermal Electrical II currents Charge qubit Charge Electric field DC–10 GHz Thermal Electrical II eigenstates Elementary particles Muon Muonic spin Magnetic field DC Radioactive decay Radioactive decay II Neutron Nuclear spin Magnetic field, DC Bragg scattering Bragg scattering II Phonon density, Gravity Other sensors SETd Charge Electric field DC–100 MHz Thermal Electrical I eigenstates Optomechanics Phonons Force, Acceleration, kHz–GHz Thermal Optical I Mass, Magnetic field, Voltage Interferometer Photons, (Atoms, Displacement, – II-III Molecules) Refractive Index

TABLE I Experimental implementations of quantum sensors. aSensor type refers to the three definitions of quantum sensing on page3. b NV: nitrogen-vacancy; c SQUID: superconducting quantum interference device; dSET: single electron transistor. celeration by sensing the spatial phase shift of a laser mor precession spectroscopy on a trapped Bose-Einstein beam along its freely falling trajectory. condensate (Vengalattore et al., 2007) and by direct driv- Trapped atoms have been employed to detect and im- ing of spin-flip transitions by microwave currents (Ock- age magnetic fields at the microscale, by replicating Lar- eloen et al., 2013) or thermal radiofrequency noise in 6 samples (Fortagh et al., 2002; Jones et al., 2003). Sens- atomic vapors. Their small size could compensate for ing with cold atoms has found application in solid state this downside in applications like microscopy, where high physics by elucidating current transport in microscopic spatial resolution is required. However, operation of ion conductors (Aigner et al., 2008). traps in close proximity to surfaces remains a major chal- Arguably the most advanced demonstrations of lenge. Recent work on large ion crystals (Arnold et al., entanglement-enhanced quantum sensing (“Definition 2015; Bohnet et al., 2016; Drewsen, 2015) opens however III”) have been implemented in trapped cold atoms and the potential for novel applications to precise clocks and vapor cells. Entanglement – in the form of spin squeez- spectroscopy. ing (Wineland et al., 1992) – has been produced by optical non-destructive measurements of atomic population (Appel et al., 2009; Bohnet et al., 2014; Cox et al., C. Rydberg atoms 2016; Hosten et al., 2016a; Leroux et al., 2010a; Louchet- Chauvet et al., 2010; Schleier-Smith et al., 2010b) and Rydberg atoms – atoms in highly excited electronic atomic interactions (Esteve et al., 2008; Riedel et al., states – are remarkable quantum sensors for electric fields 2010). It has improved the sensitivity of magnetome- for a similar reason as trapped ions: In a classical picture, try devices beyond the shot noise limit (Ockeloen et al., the loosely confined electron in a highly excited orbit is 2013; Sewell et al., 2012) and has increased their band- easily displaced by electric fields. In a quantum picture, width (Shah et al., 2010). its motional states are coupled by strong electric dipole transitions and experience strong Stark shifts (Herrmann et al., 1986; Osterwalder and Merkt, 1999). Preparation B. Trapped ions and readout of states is possible by laser excitation and spectroscopy. Ions, trapped in vacuum by electric or magnetic fields, As their most spectacular sensing application, Rydberg have equally been explored as quantum sensors. The atoms in vacuum have been employed as single-photon most advanced applications employ the quantized mo- detectors for microwave photons in a cryogenic cavity tional levels as sensing qubits for electric fields and forces. in a series of experiments that has been highlighted by These levels are strongly coupled to the electric field by the Nobel prize in Physics in 2012 (Gleyzes et al., 2007; dipole-allowed transitions and sufficiently (MHz) spaced Haroche, 2013; Nogues et al., 1999). Their sensitivity to be prepared by Raman cooling and read out by laser has recently been improved by employing Schrödinger spectroscopy. The sensor has a predicted sensitivity of cat states to reach a level of 300nV/m/√Hz (Facon et al., 500 nV/m/√Hz or 1 yN/√Hz for the force acting on the 2016). ion (Biercuk et al., 2010; Maiwald et al., 2009). Trapped Recently, Rydberg states have become accessible in ions have been extensively used to study electric field atomic vapour cells (Kübler et al., 2010). They have been noise above surfaces (Brownnutt et al., 2015), which applied to sense weak electric fields, mostly in the GHz could arise from charge fluctuations induced by adsor- frequency range (Fan et al., 2015; Sedlacek et al., 2012), bents. Electrical field noise is a severe source of decoher- and have been suggested as a candidate for a primary ence for ion traps and superconducting quantum proces- traceable standard of microwave power. sors (Labaziewicz et al., 2008) and a key limiting factor in ultrasensitive force microscopy (Kuehn et al., 2006; Tao and Degen, 2015). D. Atomic clocks Independently, the ground state spin sublevels of ions are magnetic-field-sensitive qubits analogous to neutral At first sight, atomic clocks – qubits with transitions atoms discussed above (Baumgart et al., 2016; Kotler so insensitive that their level splitting can be regarded as et al., 2011; Maiwald et al., 2009). Being an extremely absolute and serve as a frequency reference – do not seem clean system, trapped ions have demonstrated sensi- to qualify as quantum sensors since this very definition tivities down to 4.6 pT/√Hz (Baumgart et al., 2016) violates criterion (4). Their operation as clocks, however, and served as a testbed for advanced sensing proto- employs identical protocols as the operation of quantum cols such as dynamical decoupling (Biercuk et al., 2009; sensors, in order to repeatedly compare the qubit’s tran- Kotler et al., 2011) and entanglement-enhanced sensing sition to the frequency of an unstable local oscillator and (Leibfried et al., 2004). Recently, trapped ions have subsequently lock the latter to the former. Therefore, an also been proposed as rotation sensors, via matter-wave atomic clock can be equally regarded as a quantum sen- Sagnac interferometry (Campbell and Hamilton, 2017). sor measuring and stabilizing the phase drift of a local Their use in practical applications, however, has proven oscillator. Vice versa, quantum sensors discussed above difficult. Practically all sensing demonstrations have fo- can be regarded as clocks that operate on purpose on cused on single ions, which, in terms of absolute sen- a bad, environment-sensitive clock transition in order to sitivity, cannot compete with ensemble sensors such as measure external fields. 7

Today’s most advanced atomic clocks employ optical of diamond magnetometers (250 aT/√Hz/cm−3/2)(Tay- transitions in single ions (Huntemann et al., 2016) or lor et al., 2008) or gyroscopes (10−5 rad/s/√Hz/mm3/2) atomic clouds trapped in an optical lattice (Bloom et al., (Ajoy and Cappellaro, 2012; Ledbetter et al., 2012) would 2014; Hinkley et al., 2013; Takamoto et al., 2005). Inter- be competitive with their atomic counterparts. estingly, even entanglement-enhanced sensing has found Translation of this potential into actual devices re- use in actual devices, since some advanced clocks employ mains challenging, with two technical hurdles standing multi-qubit quantum logic gates for readout of highly sta- out. First, efficient fluorescence detection of large NV ble but optically inactive clock ions (Rosenband et al., ensembles is difficult, while absorptive and dispersive 2008; Schmidt et al., 2005). schemes are not easily implemented (Clevenson et al., 2015; Jensen et al., 2014; Le Sage et al., 2012). Second, spin coherence times are reduced 100 1000 times in high- E. Solid state spins – Ensemble sensors density ensembles owing to interaction− of NV spins with parasitic substitutional nitrogen spins incorporated dur- 1. NMR ensemble sensors ing high-density doping (Acosta et al., 2009). As a consequence, even the most advanced devices are currently Some of the earliest quantum sensors have been based limited to 1 pT √Hz (Wolf et al., 2015) and operate on ensembles of nuclear spins. Magnetic field sensors / several orders∼ of magnitude above the theory limit. As a have been built that infer field strength from their Lar- technically less demanding application, NV centers in a mor precession, analogous to neutral atom magnetome- magnetic field gradient have been employed as spectrum ters described above (Kitching et al., 2011; Packard and analyzer for high frequency microwave signals (Chipaux Varian, 1954; Waters and Francis, 1958). Initialization of et al., 2015). spins is achieved by thermalization in an externally applied field, readout by induction detection. Although the While large-scale sensing of homogeneous fields remains a challenge, micrometer-sized ensembles of NV sensitivity of these devices (10 pT/√Hz) (Lenz, 1990) is inferior to their atomic counterparts, they have found centers have found application in imaging applications, broad use in geology, archaeology and space missions serving as detector pixels for microscopic mapping of thanks to their simplicity and robustness. More recently, magnetic fields. Most prominently, this line of research NMR sensor probes have been developed for in-situ and has enabled imaging of magnetic organelles in magne- dynamical field mapping in clinical MRI systems (Zanche totactic bacteria (Le Sage et al., 2013) and microscopic et al., 2008). magnetic inclusions in meteorites (Fu et al., 2014), as well Spin ensembles have equally served as gyroscopes as contrast-agent-based magnetic resonance microscopy (Fang and Qin, 2012; Woodman et al., 1987), exploiting (Steinert et al., 2013). the fact that Larmor precession occurs in an independent frame of reference and therefore appears frequency- shifted in a rotating laboratory frame. In the most ad- F. Solid state spins - Single spin sensors vanced implementation, nuclear spin precession is read out by an atomic magnetometer, which is equally used Readout of single spins in the solid state – a ma- for compensation of the Zeeman shift (Kornack et al., jor milestone on the road towards quantum comput- 2005). These experiments reached a sensitivity of 5 ers – has been achieved both by electrical and optical schemes. Electrical readout has been demonstrated 10−7 rad/s/√Hz, which is comparable to compact imple-· mentations of atomic interferometers and optical Sagnac with phosphorus dopants in silicon (Morello et al., interferometers. 2010) and electrostatically-defined semiconductor quantum dots (Elzerman et al., 2004). Optical readout was shown with single organic molecules (Wrachtrup et al., 2. NV center ensembles 1993a,b), optically active quantum dots (Atature et al., 2007; Kroutvar et al., 2004; Vamivakas et al., 2010), and Much excitement has recently been sparked by ensem- defect centers in crystalline materials including diamond bles of nitrogen-vacancy centers (NV centers) – electronic (Gruber et al., 1997) and silicon carbide (Christle et al., spin defects in diamond that can be optically initialized 2015; Widmann et al., 2015). In addition, mechanical and read out. Densely-doped diamond crystals promise detection of single paramagnetic defects in silica (Rugar to deliver “frozen vapor cells” of spin ensembles that com- et al., 2004) and real-time monitoring of few-spin fluctu- bine the strong (electronic) magnetic moment and effi- ations (Budakian et al., 2005) have been demonstrated. cient optical readout of atomic vapor cells with the high Among all solid state spins, NV centers in diamond spin densities achievable in the solid state. Although have received by far the most attention for sensing pur- these advantages are partially offset by a reduced coher- poses. This is in part due to the convenient room- ence time (T2 < 1 ms at room temperature, as compared temperature optical detection, and in part due to their to T2 > 1 s for vapor cells), the predicted sensitivity stability in very small crystals and nanostructures. The 8 latter permits use of NV centers as sensors in high- and Braginski, 2004; Fagaly, 2006; Jaklevic et al., 1965). resolution scanning probe microscopy (Balasubramanian These devices – interferometers of superconducting con- et al., 2008; Chernobrod and Berman, 2005; Degen, ductors – measure magnetic fields with a sensitivity down 2008), as biomarkers within living organisms (Fu et al., to 10 aT/√Hz (Simmonds et al., 1979). Their sensing 2007), or as stationary probes close to the surface of dia- mechanism is based on the Aharonov-Bohm phase im- mond sensor chips. Quantum sensing with NV centers printed on the superconducting wave function by an en- has been considered in several recent focused reviews circled magnetic field, which is read out by a suitable (Rondin et al., 2014; Schirhagl et al., 2014). circuit of phase-sensitive Josephson junctions. Single NV centers have been employed and/or pro- From a commercial perspective, SQUIDs can be con- posed as sensitive magnetometers (Balasubramanian sidered the most advanced type of quantum sensor, with et al., 2008; Maze et al., 2008; Taylor et al., 2008), elec- applications ranging from materials characterization in trometers (Dolde et al., 2011), pressure sensors (Doherty solid state physics to clinical magnetoencephalography et al., 2014) and thermometers (Hodges et al., 2013; Kuc- systems for measuring tiny ( 100 fT) stray fields of elec- sko et al., 2013; Neumann et al., 2013; Toyli et al., 2013), tric currents in the brain. In∼ parallel to the development using the Zeeman, Stark and temperature shifts of their of macroscopic (mm-cm) SQUID devices, miniaturization spin sublevels. The most advanced nano-sensing experi- has given birth to sub-micron sized “nanoSQUIDs” with ments in terms of sensitivity have employed near-surface possible applications in nanoscale magnetic, current, and NV centers in bulk diamond crystals. This approach has thermal imaging (Halbertal et al., 2016; Vasyukov et al., enabled sensing of nanometer-sized voxels of nuclear or 2013). Note that because SQUIDs rely on spatial rather electronic spins deposited on the diamond surface (De- than temporal coherence, they are more closely related to Vience et al., 2015; Loretz et al., 2014; Lovchinsky et al., optical interferometers than to the spin sensors discussed 2016; Mamin et al., 2013; Shi et al., 2015; Staudacher above. et al., 2013; Sushkov et al., 2014b), of distant nuclear spin SQUIDs have been employed to process signals from clusters (Shi et al., 2014), and of 2D materials (Lovchin- the DC up to the GHz range (Hatridge et al., 2011; Mck sky et al., 2017). Other applications included the study of et al., 2003), the upper limit being set by the Joseph- ballistic transport in the Johnson noise of nanoscale con- son frequency. Conceptually similar circuits, dedicated ductors (Kolkowitz et al., 2015), phases and phase tran- to amplification of GHz frequency signals, have been ex- sitions of skyrmion materials (Dovzhenko et al., 2016; plored in great detail in the past decade (Bergeal et al., Dussaux et al., 2016), as well as of spin waves (van der 2010; Castellanos-Beltran et al., 2008; Ho Eom et al., Sar et al., 2015; Wolfe et al., 2014), and relaxation in 2012; Macklin et al., 2015). Arguably the most widely nanomagnets (Schafer-Nolte et al., 2014; Schmid-Lorch studied design is the Josephson parametric amplifier, et al., 2015). which has been pushed to a nearly quantum-limited input Integration of NV centers into scanning probes has en- noise level of only few photons and is now routinely used abled imaging of magnetic fields with sub-100 nm resolu- for spectroscopic single shot readout of superconducting tion, with applications to nanoscale magnetic structures qubits (Vijay et al., 2011). and domains (Balasubramanian et al., 2008; Maletinsky et al., 2012; Rondin et al., 2012), vortices and domain walls (Rondin et al., 2013; Tetienne et al., 2014, 2015), 2. Superconducting qubits superconducting vortices (Pelliccione et al., 2016; Thiel et al., 2016), and mapping of currents (Chang et al., Temporal quantum superpositions of supercurrents or 2017). charge eigenstates have become accessible in supercon- NV centers in 10-nm-sized nanodiamonds have also ducting qubits (Clarke and Wilhelm, 2008; Martinis been inserted into∼ living cells. They have been employed et al., 2002; Nakamura et al., 1999; Vion et al., 2002; for particle tracking (McGuinness et al., 2011) and in vivo Wallraff et al., 2004). Being associated with large mag- temperature measurements (Kucsko et al., 2013; Neu- netic and electric dipole moments, they are attractive mann et al., 2013; Toyli et al., 2013) and could enable candidates for quantum sensing. Many of the estab- real-time monitoring of metabolic processes. lished quantum sensing protocols to be discussed in Sec- tionsIV–VII have been implemented with superconducting qubits. Specifically, noise in these devices has been G. Superconducting circuits thoroughly studied from the sub-Hz to the GHz range, using Ramsey interferometry (Yan et al., 2012; Yoshihara 1. SQUIDs et al., 2006), dynamical decoupling (Bylander et al., 2011; Ithier et al., 2005; Nakamura et al., 2002; Yan et al., 2013; The Superconducting Quantum Interference Device Yoshihara et al., 2006), and T1 relaxometry (Astafiev (SQUIDs) is simultaneously one of the oldest and one et al., 2004; Yoshihara et al., 2006). These studies have of the most sensitive type of magnetic sensor (Clarke been extended to discern charge from flux noise by choos- 9 ing qubits with a predominant electric (charge qubit) are easily implemented by application of localized mag- or magnetic (flux qubit) dipole moment, or by tuning netic fields along parts of the neutron’s trajectory. As bias parameters in situ (Bialczak et al., 2007; Yan et al., a consequence, many early demonstrations of quantum 2012). Operating qubits as magnetic field sensors, very effects, such as the direct measurement of Berry’s phase promising sensitivities (3.3 pT/√Hz for operation at 10 (Bitter and Dubbers, 1987), have employed neutrons. MHz) were demonstrated (Bal et al., 2012). Extending Sensing with neutrons has been demonstrated in mul- these experiments to the study of extrinsic samples ap- tiple ways. Larmor precession in the magnetic field of pears simultaneously attractive and technically challeng- samples has been employed for three-dimensional tomog- ing, since superconducting qubits have to be cooled to raphy (Kardjilov et al., 2008). Neutron interferometry temperatures of only few tens of millikelvin. has put limits on the strongly-coupled chameleon field (Li et al., 2016). Ultracold neutrons have been employed as a probe for gravity on small length scales in a series of H. Elementary particle qubits experiments termed ”qBounce”. These experiments exploit the fact that suitable materials perfectly reflect the Interestingly, elementary particles have been employed matter wave of sufficiently slow neutrons so that they as quantum sensors long before the development of can be trapped above a bulk surface by the gravity of atomic and solid state qubits. This somewhat paradox- earth as a “quantum bouncing ball” (Nesvizhevsky et al., ical fact is owing to their straightforward initialization 2002). The eigenenergies of this anharmonic trap depend and readout, as well as their targeted placement in rele- on gravity and have been probed by quantum sensing vant samples by irradiation with a particle beam. techniques (Jenke et al., 2014, 2011). The most established technique, neutron spin echo, can reveal materials properties by measuring small (down 1. Muons to neV) energy losses of neutrons in inelastic scattering events (Mezei, 1972). Here, the phase of the neutron Muons are frequently described as close cousins of elec- spin, coherently precessing in an external magnetic field, trons. Both particles are leptons, carry an elementary serves as a clock to measure a neutron’s time of flight. charge and have a spin that can be employed for quantum Inelastic scattering in a sample changes a neutron’s ve- sensing. Sensing with muons has been termed “muon locity, resulting in a different time of flight to and from a spin rotation” (µSR). It employs antimuons (µ+) that sample of interest. This difference is imprinted in the spin are deterministically produced by proton-proton colli- phase by a suitable quantum sensing protocol, specifically sions, from decay of an intermediate positive pion by the a Hahn echo sequence whose π pulse is synchronized with + + reaction π µ + νµ. Here, parity violation of the passage through the sample. weak interaction→ automatically initializes the muon spin to be collinear with the particle’s momentum. Readout of the spin is straightforward by measuring the emis- I. Other sensors sion direction of positrons from the subsequent decay + + µ e + νe + νµ, which are preferably emitted along In addition to the many implementations of quantum → the muon spin (Blundell, 1999; Brewer and Crowe, 1978). sensors already discussed, three further systems deserved Crucially, muons can be implanted into solid state sam- special attention for their future potential or for their fun- ples and serve as local probes of their nanoscale envi- damental role in developing quantum sensing methodol- ronment for their few microseconds long lifetime. Lar- ogy. mor precession measurements have been used to infer the intrinsic magnetic field of materials. Despite its exotic nature, the technique of muon spin rotation (µSR) 1. Single electron transistors has become and remained a workhorse tool of solid state physics. In particular, it is a leading technique to mea- Single electron transistors (SET’s) sense electric fields sure the London penetration depth of superconductors by measuring the tunneling current across a submicron (Sonier et al., 2000). conducting island sandwiched between tunneling source and drain contacts. In the “Coulomb blockade regime” of sufficiently small (typically 100 nm) islands, tunneling 2. Neutrons across the device is only allowed≈ if charge eigenstates of the island lie in the narrow energy window between the Slow beams of thermal neutrons can be spin-polarized Fermi level of source and drain contact. The energy of by Bragg reflection on a suitable magnetic crystal. Spin these eigenstates is highly sensitive to even weak exter- readout is feasible by a spin-sensitive Bragg analyzer and nal electric fields, resulting in a strongly field-dependent subsequent detection. Spin rotations (single qubit gates) tunneling current (Kastner, 1992; Schoelkopf, 1998; Yoo 10 et al., 1997). SETs have been employed as scanning limited sensitivity by 3.5 dB (Ligo Collaboration”, 2011); probe sensors to image electric fields on the nanoscale, in a proof-of-principle experiment in the LIGO gravita- shedding light on a variety of solid-state-phenomena such tional wave detector, the injection of 10dB of squeezing as the fractional quantum Hall effect or electron-hole lowered the shot-noise in the interferometer output by ap- puddles in graphene (Ilani et al., 2004; Martin et al., proximately 2.15dB (28%)(Collaboration, 2013), equiva- 2008). In a complementary approach, charge sensing by lent to an increase by more than 60% in the power stored stationary SETs has enabled readout of optically inac- in the interferometer arm cavities. Further upgrades as- cessible spin qubits such as phosphorus donors in silicon sociated with Advanced LIGO could bring down the shot (Morello et al., 2010) based on counting of electrons (By- noise by 6dB, via frequency dependent squeezing (Oelker lander et al., 2005). et al., 2016).

2. Optomechanics In addition, quantum correlations between photons have proven to be a powerful resource for imaging. This Phonons – discrete quantized energy levels of vibration has been noted very early on in the famous Hanbury- – have recently become accessible at the “single-particle” Brown-Twiss experiment, where bunching of photons is level in the field of optomechanics (Aspelmeyer et al., employed to filter atmospheric aberrations and to per- 2014; O’Connell et al., 2010), which studies high-quality form “super-resolution” measurements of stellar diame- mechanical oscillators that are strongly coupled to light. ters smaller than the diffraction limit of the telescope While preparation of phonon number states and their employed (Hanbury Brown and Twiss, 1956). While coherent superpositions remains difficult, the devices this effect can still be accounted for classically, a recent built to achieve these goals have shown great promise for class of experiments has exploited non-classical correla- sensing applications. This is mainly due to the fact that tions to push the spatial resolution of microscopes be- mechanical degrees of freedom strongly couple to nearly low the diffraction limit (Schwartz et al., 2013). Vice all external fields, and that strong optical coupling en- versa, multi-photon correlations have been proposed and ables efficient actuation and readout of mechanical mo- employed to create light patterns below the diffraction tion. Specifically, optomechanical sensors have been em- limit for superresolution lithography (Boto et al., 2000; ployed to detect minute forces (12 zN/√Hz, Moser et al., D’Angelo et al., 2001). They can equally improve im- 2013), acceleration (100 ng/√Hz, Cervantes et al., 2014 age contrast rather than resolution by a scheme known and Krause et al., 2012), masses (2 yg/√Hz, Chaste as “quantum illumination” (Lloyd, 2008; Lopaeva et al., et al., 2012), magnetic fields (200 pT/√Hz, Forstner 2013; Tan et al., 2008). Here, a beam of photons is em- et al., 2014), spins (Degen et al., 2009; Rugar et al., 2004), ployed to illuminate an object, reflected light being de- and voltage (5 pV/√Hz, Bagci et al., 2014). While these tected as the imaging signal. Entangled twins of the illu- demonstrations have remained at the level of classical mination photons are conserved at the source and com- sensing in the sense of this review, their future exten- pared to reflected photons by a suitable joint measure- sion to quantum-enhanced measurements appears most ment. In this way, photons can be certified to be re- promising. flected light rather than noise, enhancing imaging contrast. In simpler schemes, intensity correlations between entangled photons have been employed to boost contrast 3. Photons in transmission microscopy of weakly absorbing objects (Brida et al., 2010) and the reduced quantum fluctua- While this review will not discuss quantum sensing tions of squeezed light have been used to improve optical with photons, due to the breadth of the subject, several particle tracking (Taylor et al., 2013). fundamental paradigms have been pioneered with optical sensors including light squeezing and photonic quantum correlations. These constitute examples of quantum- enhanced sensing according to our “Definition III”. The most advanced demonstrations of entanglement- Squeezing of light – the creation of partially-entangled enhanced sensing have been performed with single pho- states with phase or amplitude fluctuations below those tons or carefully assembled few-photon Fock states. Most of a classical coherent state of the light field – has been prominently, these include Heisenberg-limited interfer- proposed (Caves, 1981) and achieved (Slusher et al., ometers (Higgins et al., 2007; Holland and Burnett, 1993; 1985) long before squeezing of spin ensembles (Hald et al., Mitchell et al., 2004; Nagata et al., 2007; Walther et al., 1999; Wineland et al., 1992). Vacuum squeezed states 2004). In these devices, entanglement between photons have meanwhile been employed to improve the sensitivity or adaptive measurements are employed to push sensitiv- of gravitational wave detectors. In the GEO gravitational ity beyond the 1/√N scaling of a classical interferometer wave detector, squeezing has enhanced the shot-noise where N is the number of photons (see SectionIX). 11

IV. THE QUANTUM SENSING PROTOCOL qubits. In addition, the internal Hamiltonian contains time-dependent stochastic terms due to a classical envi- In this Section, we describe the basic methodology for ronment or interactions with a quantum bath that are performing measurements with quantum sensors. Our responsible for decoherence and relaxation. discussion will focus on a generic scheme where a measurement consists of three elementary steps: the initialization of the quantum sensor, the interaction with 2. Signal Hamiltonian the signal of interest, and the readout of the final state. ˆ Phase estimation (Kitaev, 1995; Shor, 1994) and param- The signal Hamiltonian HV (t) represents the coupling eter estimation (Braunstein and Caves, 1994; Braunstein between the sensor qubit and a signal V (t) to be mea- et al., 1996; Goldstein et al., 2010) techniques are then sured. When the signal is weak (which is assumed here) ˆ ˆ used to reconstruct the physical quantity from a series HV (t) adds a small perturbation to H0. The signal of measurements. Experimentally, the protocol is typi- Hamiltonian can then be separated into two qualitatively cally implemented as an interference measurement using different contributions, pump-probe spectroscopy, although other schemes are Hˆ (t) = Hˆ (t) + Hˆ (t) , (4) possible. The key quantity is then the quantum phase V V|| V⊥ picked up by the quantum sensor due to the interac- where Hˆ is the parallel (commuting, secular) and tion with the signal. The protocol can be optimized for V|| ˆ the transverse (non-commuting) component, respec- detecting weak signals or small signal changes with the HV⊥ highest possible sensitivity and precision. tively. The two components can quite generally be captured by ˆ 1 HV|| (t) = 2 γV||(t) 1 1 0 0 , A. Quantum sensor Hamiltonian n {| ih | − | ih |} o ˆ 1 † HV⊥ (t) = 2 γ V⊥(t) 1 0 + V⊥(t) 0 1 , (5) For the following discussion, we will assume that the | ih | | ih | quantum sensor can be described by the generic Hamil- where V||(t) and V⊥(t) are functions with the same units tonian of V (t). γ is the coupling or transduction parameter of the qubit to the signal V (t). Examples of coupling Hˆ (t) = Hˆ0 + HˆV (t) + Hˆcontrol(t) , (2) parameters include the Zeeman shift parameter (gyro- magnetic ratio) of spins in a magnetic field, with units ˆ ˆ where H0 is the internal Hamiltonian, HV (t) is of Hz/T, or the linear Stark shift parameter of electric the Hamiltonian associated with a signal V (t), and dipoles in an electric field, with units of Hz/(Vm−1). Al- ˆ Hcontrol(t) is the control Hamiltonian. We will assume though the coupling is often linear, this is not required. ˆ ˆ that H0 is known and that Hcontrol(t) can be deliberately In particular, the coupling is quadratic for second-order chosen so as to manipulate or tune the sensor in a con- interactions (such as the quadratic Stark effect) or when trolled way. The goal of a quantum sensing experiment operating the quantum sensor in variance detection mode is then to infer V (t) from the effect it has on the qubit (see Section IV.E.2). ˆ via its Hamiltonian HV (t), usually by a clever choice of The parallel and transverse components of a signal ˆ Hcontrol(t). have distinctly different effects on the quantum sensor. ˆ A commuting perturbation HV|| leads to shifts of the energy levels and an associated change of the transition 1. Internal Hamiltonian ˆ frequency ω0. A non-commuting perturbation HV⊥ , by contrast, can induce transitions between levels, manifest- ˆ H0 describes the internal Hamiltonian of the quantum ing through an increased transition rate Γ. Most often, sensor in the absence of any signal. Typically, the inter- this requires the signal to be time-dependent (resonant nal Hamiltonian is static and defines the energy eigen- with the transition) in order to have an appreciable effect states 0 and 1 , | i | i on the quantum sensor. An important class of signals are vector signal V~ (t), Hˆ0 = E0 0 0 + E1 1 1 , (3) | ih | | ih | in particular those provided by electric or magnetic fields. The interaction between a vector signal V~ (t) = where E and E are the eigenenergies and ω = E E 0 1 0 1 0 V ,V ,V (t) and a qubit can be described by the sig- is the transition energy between the states ( = 1).− Note x y z ~ nal{ Hamiltonian} that the presence of an energy splitting ω0 = 0 is not necessary, but it represents the typical situation6 for most ˆ HˆV (t) = γV~ (t) ~σ, (6) implementations of quantum sensors. The qubit internal · Hamiltonian may contain additional interactions that are where ~σ = σx, σy, σz is a vector of Pauli matrices. For { } specific to a quantum sensor, such as couplings to other a vector signal, the two signal functions V||(t) and V⊥(t) 12 are 1. Initialize

V||(t) = Vz(t) , V (t) = V (t) + iV (t), (7) ⊥ x y 2. Transform where the z direction is deﬁned by the qubit’s quantiza- tion axis. The corresponding signal Hamiltonian is

ˆ 3. Evolve for time HV (t) = γRe[V⊥(t)]ˆσx + γIm[V⊥(t)]ˆσy + γV||(t)ˆσz . (8)

3. Control Hamiltonian 4. Transform

For most quantum sensing protocols it is required to manipulate the qubit either before, during, or after the sensing process. This is achieved via a control Hamilto- 5. Project, Readout ˆ nian Hcontrol(t) that allows implementing a standard set “0” with probability of quantum gates (Nielsen and Chuang, 2000). The most “1” with probability common gates in quantum sensing include the Hadamard gate and the Pauli-X and Y gates, or equivalently, a set of 6. Repeat and average π/2 and π rotations (pulses) around different axes. Ad- vanced sensing schemes employing more than one sensor qubit may further require conditional gates, especially 7. Estimate signal controlled-NOT gates to generate entanglement, Swap gates to exploit memory qubits, and controlled phase shifts in quantum phase estimation. Finally, the control Hamiltonian can include control fields for systematically FIG. 2 Basic steps of the quantum sensing process. tuning the transition frequency ω0. This capability is frequently exploited in noise spectroscopy experiments. 4. The quantum sensor is transformed into a superposition of observable readout states α = Uˆb ψ(t) = B. The sensing protocol 0 0 0 0 | i | i c0 0 + c1 1 . For simplicity we assume that the initialization| i | i basis 0 1 and the readout basis Quantum sensing experiments typically follow a , 00 10 are the same{| i and| i} that ˆ = ˆ †, but this generic sequence of sensor initialization, interaction with , Ub Ua is{| noti | required.i} Under these assumptions, the co- the signal, sensor readout and signal estimation. This efficients 0 and 0 represent the overlap sequence can be summarized in the following basic pro- c0 c0 c1 c1 between the≡ initial and final≡ sensing states. tocol, which is also sketched in Fig.2: 5. The final state of the quantum sensor is read 1. The quantum sensor is initialized into a known ba- out. We assume that the readout is projective, sis state, for example 0 . | i although more general positive-operator-valued- 2. The quantum sensor is transformed into the desired measure (POVM) measurements may be possible (Nielsen and Chuang, 2000). The projective initial sensing state ψ0 = Uâ 0 . The transforma- tion can be carried out| usingi a| seti of control pulses readout is a Bernoulli process that yields an answer “0” with probability 1 p0 and an answer “1” with represented by the propagator Uâ. In many cases, probability p0, where p−0 = c0 2 p is proportional ψ0 is a superposition state. 1 | i to the measurable transition| | probability,∝

3. The quantum sensor evolves under the Hamiltonian 2 2 ˆ p = 1 c0 = c1 (10) H [Eq. (2)] for a time t. At the end of the sensing − | | | | period, the sensor is in the final sensing state that the qubit changed its state during t. The binary answer is detected by the measurement appa- ψ(t) = UˆH (0, t) ψ0 = c0 ψ0 + c1 ψ1 , (9) | i | i | i | i ratus as a physical quantity x, for example, as a voltage, current, photon count or polarization. where UˆH (0, t) is the propagator of Hˆ , ψ1 is the | i state orthogonal to ψ0 and c0, c1 are complex co- Steps 1-5 represent a single measurement cycle. Because efficients. | i step 5 gives a binary answer, the measurement cycle 13 needs to be repeated many times in order to gain a pre- up the relative phase φ = ω0t, and the state after cise estimate for p: the evolution is 1 6. Steps 1-5 are repeated and averaged over a large ψ(t) = ( 0 + e−iω0t 1 ) , (12) number of cycles N to estimate p. The repetition | i √2 | i | i can be done by running the protocol sequentially up to an overall phase factor. on the same quantum system, or in parallel by averaging over an ensemble of N identical (and non- 4. Using a second π/2 pulse, the state ψ(t) is con- interacting) quantum systems. verted back to the measurable state | i Step 6 only provides one value for the transition prob- 1 1 α = (1 + e−iω0t) 0 + (1 e−iω0t) 1 . (13) ability p. While a single value of p may sometimes be | i 2 | i 2 − | i sufficient to estimate a signal V , it is in many situations convenient or required to record a set of values pk : 5,6. The final state is read out. The transition proba- { } bility is 7. The transition probability p is measured as a function of time t or of a parameter of the control p = 1 0 α 2 − |h | i| Hamiltonian, and the desired signal V is inferred 2 1 = sin (ω0t/2) = [1 cos(ω0t)]. (14) from the data record pk using a suitable proce- 2 − dure. { } By recording p as a function of time t, an oscillatory More generally, a set of measurements can be optimized output (“Ramsey fringes”) is observed with a frequency to efficiently extract a desired parameter from the signal given by ω0. Thus, the Ramsey measurement can directly Hamiltonian (see SectionVIII). Most protocols presented provide a measurement of the energy splitting ω0. in the following implicitly use such a strategy for gaining information about the signal. Although the above protocol is generic and simple, it is D. Second example: Rabi measurement sufficient to describe most sensing experiments. For example, classical continuous-wave absorption and trans- A second elementary example is the measurement of mission spectroscopy can be considered as an averaged the transition matrix element V⊥ : variety of this protocol. Also, the time evolution can | | 1. The quantum sensor is initialized into ψ0 = 0 . be replaced by a spatial evolution to describe a classical | i | i interferometer. 3. In the absence of the internal Hamiltonian, Hˆ0 = 0, To illustrate the protocol, we consider two elementary ˆ 1 the evolution is given by HV⊥ = γV⊥σx = ω1σx, examples, one for detecting a parallel signal V and one 2 || where ω1 is the Rabi frequency. The state after for detecting a transverse signal V⊥. These examples will evolution is: serve as the basis for the more refined sequences discussed 1 1 in later Sections. ψ(t) = α = (1 + e−iω1t) 0 + (1 e−iω1t) 1 . (15) | i | i 2 | i 2 − | i

C. First example: Ramsey measurement 5,6. The ﬁnal state is read out. The transition probability is:

A ﬁrst example is the measurement of the static en- 2 2 p = 1 0 α = sin (ω1t/2). (16) ergy splitting ω0 (or equivalently, a static perturbation − |h | i| V||). The canonical approach for this measurement is a In a general situation where Hˆ0 = 0, the transi- Ramsey interferometry measurement (Lee et al., 2002; 6 Taylor et al., 2008): tion probability represents the solution to Rabi’s original problem (Sakurai and Napolitano, 2011), 1. The quantum sensor is initialized into 0 . | i ω2 q = 1 sin2 2 + 2 (17) 2. Using a π/2 pulse, the quantum sensor is trans- p 2 2 ω1 ω0 t . ω1 + ω0 formed into the superposition state Hence, only time-dependent signals with frequency ω 1 ≈ ω0 aﬀect the transition probability p, a condition known ψ0 = + ( 0 + 1 ) . (11) | i | i ≡ √2 | i | i as resonance. From this condition it is clear that a Rabi measurement can provide information not only on the 3. The superposition state evolves under the Hamilto- magnitude V⊥, but also on the frequency ω of a signal nian Hˆ0 for a time t. The superposition state picks (Aiello et al., 2013; Fedder et al., 2011). 14

1 t,

(a) 1 1 δp = [1 cos(ω0t + γδV t)] 2 2  p − − p .5 p(V) 1 γδV t, (18) (b) V V ≈ ±2

p p where the sign is determined by k. V V 0 min min Note that slope detection has a limited linear range Signal to be measured, V because phase wrapping occurs for γδV t > π/2. The phase wrapping restricts the dynamic| range| of the quan- FIG. 3 Transition probability p for a Ramsey experiment as tum sensor. SectionVIII discusses adaptive sensing tech- a function of the signal V picked up by the sensor. (a) Slope niques designed to extend the dynamic range. detection: The quantum sensor is operated at the p0 = 0.5 bias point (filled red circle). A small change in the signal δV leads to a linear change in the transition probability, δp = δφ/2 = γδV t/2 (empty red circle). The uncertainty 2. Variance detection (quadratic detection) σp in the measured transition probability, leads to an uncertainty in the estimated signal, Vmin (grey shade). (b) Variance If the magnitude of δV fluctuates between measure- detection: The quantum sensor is operated at the p0 = 0 bias ments so that δV = 0, readout at p0 = 0.5 will yield h i point (filled blue square). A small change in the signal δV no information about δV , since p p0 = 0.5. In this 2 2 2 2 leads to a quadratic change, δp = δφ /4 = γ δV t /4 (empty situation, it is advantageous toh detecti ≈ the signal vari- blue square). The information on the sign of δV is lost. The ance by biasing the measurement to a point of minimum experimental readout error σp translates into an uncertainty slope, ω0t = kπ, corresponding to the bias points p0 = 0 in the estimated signal, Vmin, according to the slope or curvature of the Ramsey fringe (grey shade). and p0 = 1 (filled blue square in Fig.3(b)). If the interferometer is tuned to p0 = 0, a signal with variance δV 2 = V 2 gives rise to a mean transition probability h i rms that is quadratic in Vrms and t (Meriles et al., 2010), E. Slope and variance detection 1 δp = p = [1 cos(ω0t + γδV t)] 2 − A central objective of quantum sensing is the detection 1 2 2 2 of small signals. For this purpose, it is advantageous to γ Vrmst . (19) measure the deviation of the transition probability from ≈ 4 a well-chosen reference point p0, which we will refer to This relation holds for small γVrmst 1. If the fluctua- as the bias point of the measurement, corresponding to tion is Gaussian, the result can be extended to any large a known value of the external signal V0 or reached by value of γVrmst, setting some additional parameters in the Hamiltonian under the experimenter’s control. The quantity of in- 1 2 2 2 p = 1 exp( γ Vrmst /2) . (20) terest is then the difference δp = p p0 between the 2 − − probability measured in the presence and− absence of the signal, respectively. Experimentally, the bias point can Variance detection is especially important for detecting be adjusted by several means, for example by adding a ac signals when their synchronization with the sensing small detuning to ω0 or by measuring the final state ψ(t) protocol is not possible (Section VI.C.4), or when the along different directions. | i signal represents a noise source (SectionVII).

V. SENSITIVITY

The unprecedented level of sensitivity offered by many 1. Slope detection (linear detection) quantum sensors has been a key driving force of the field. In this section, we quantitatively define the sensitivity. The Ramsey interferometer is most sensitive to small We start by discussing the main sources of noise that perturbations δV around V0 = ω0/γ when operated at enter a quantum sensing experiment, and derive expres- the point of maximum slope where p0 = 0.5, indicated sions for the signal-to-noise ratio (SNR) and the mini- by the filled red dot in Fig.3(a). This bias point is mum detectable signal, i.e., the signal magnitude that reached when ω0t = kπ/2, with k = 1, 3, 5, .... Around yields unit SNR. This will lead us to a key quantity of p0 = 0.5 , the transition probability is linear in δV and this paper: the sensitivity, vmin, defined as the minimum 15 detectable signal per unit time. In particular, we will Thus, the projective readout adds noise of order find in Sec. V.B.2 that vmin is approximately 1/(2√N) to the probability value p. For variance detection, where ideally p0 = 0, the projection noise would in √2e principle be arbitrarily low. In any realistic experiment, vmin p (21) ≈ γC Tχ however, decoherence will shift the fringe minimum to a finite value of p (see below), where Eq. (25) holds up to for slope detection and a constant factor.

√2e vmin q (22) ≈ 4 3 2. Decoherence γ√C Tχ A second source of error includes decoherence and re- for variance detection, where T is the sensor’s coherence χ laxation during the sensing time . Decoherence and re- time, C 1 is a dimensionless constant quantifying read- t laxation cause random transitions between states or ran- out efficiency,≤ and e is Euler’s number (see Eqs. 43 and 45). In the remainder of the section signal averaging and dom phase pick-up during coherent evolution of the qubit the Allen variance are briefly discussed, and a formal def- (for more detail, see SectionVII). The two processes lead inition of sensitivity by the quantum Cramér-Raobound to a reduction of the observed probability δp with increas- (QCRB) is given. ing sensing time t, −χ(t) δpobs(t) = δp(t)e , (26) A. Noise where δp(t) is the probability that would be measured in the absence of decoherence (see Eqs. (18,19)). χ(t) is a Experimental detection of the probability p will have a phenomenological decoherence function that depends on non-zero error σ . This error translates into an error for p the noise processes responsible for decoherence (see Sec- the signal estimate, which is determined by the slope or tion VII.B.1). Although the underlying noise processes curvature of the Ramsey fringe (see Fig.3). In order to may be very complex, ( ) can often be approximated by calculate SNR and sensitivity, it is therefore important χ t a simple power law, to analyze the main sources of noise that enter σp. χ(t) = (Γt)a , (27) 1. Quantum projection noise where Γ is a decay rate and typically a = 1...3. The decay rate can be associated with a decay time T = Γ−1 Quantum projection noise is the most fundamental χ that equals the evolution time t where δp /δp = 1/e source of uncertainty in quantum sensing. The projective obs 37%. The decay time T , also known as the decoherence≈ readout during “Step 5” of the quantum sensing proto- χ time or relaxation time depending on the noise process, col (Section IV.B) does not directly yield the fractional is an important figure-of-merit of the qubit, as it sets the probability p [0...1], but one of the two values “0” or maximum possible evolution time t available for sensing. “1” with probabilities∈ 1 p and p, respectively. In order to precisely estimate p−, the experiment is repeated N times and the occurrences of “0” and “1” are binned into 3. Errors due to initialization and qubit manipulations a histogram (see Fig.4(a)). The estimate for p is then Errors can also enter through the imperfect initializa- N1 p = , (23) tion or manipulations of the quantum sensor. An im- N perfect initialization leads to a similar reduction in the where N1 is the number of measurements that gave a re- observed probability δpobs as with decoherence, sult of “1”. The uncertainty in p is given by the variance of the binomial distribution (Itano et al., 1993), δpobs = β δp , (28)

2 1 where β < 1 is a constant factor that describes the re- σp,quantum = p(1 p) . (24) N − duction of the observed δpobs as compared to the ideal δp. Contrary to the case of decoherence, this reduction The uncertainty in p therefore depends on the bias point does not depend on the sensing time t. Errors in qubit of the measurement. For slope detection, where = p0 p0 manipulations can cause many eﬀects, but will typically 0.5, the uncertainty is also lead to a reduction of δp. A more general approach, considering, e.g., faulty initialization through a density 2 1 σ = for p0 = 0.5 (25) p,quantum 4N matrix approach, will be brieﬂy discussed in the context 16

(a) (b) (c) a histogram, two peaks are observed centered at x|0i and x|1i, respectively (see Fig.4(b)). However, compared to N1 the ideal situation (Fig.4(a)), the histogram peaks are N1 broadened and there is a ﬁnite overlap of between the 2x N N0 tails of the peaks. To obtain an estimate for p, all xj are 0 assigned to either “0” or “1” based on a threshold value xT chosen roughly midway between x|0i and x|1i,

N0 = number of measurements xj < xT (29) x x x xT x x x x N1 = number of measurements xj > xT , (30) “ideal readout” “single shot readout” “averaged readout” R=0 R<1 R>1 where p = N1/N. Note that the choice of the threshold C=1 C>0.707 C<0.707 is not trivial; in particular, for an unbiased measurement, xT depends itself on the probability p. FIG. 4 Illustration of the sensor readout. N measurements Because of the overlap between histogram peaks, some are performed producing {xj }j=1...N readings on the physical values xj will be assigned to the wrong state. The error measurement apparatus. The readings {x } are then binned j introduced due to wrong assignments is into a histogram. (a) Ideal readout. Only two values are observed in the histogram, x|0i and x|1i, which correspond to 2 1 the qubit states |0i and |1i. All {xj } can be assigned to “0” σp,readout = [κ0(1 κ0)p + κ1(1 κ1)(1 p)] , (31) or “1” with 100% ﬁdelity. (b) Single shot readout. Most {xj } N − − − can be assigned, but there is an overlap between histogram where κ0 and κ1 are the fraction of measurements that peaks leading to a small error. (c) Averaged readout. {xj } cannot be assigned. The ratio between “0” and “1” is given are erroneously assigned. The actual values for κ0,1 de- by the relative position of the mean valuex ¯ and the error pend on the exact type of measurement noise and are de- is determined by the histogram standard deviation σx. R is termined by the cumulative distribution function of the the ratio of readout and projection noise, and C is an overall two histogram peaks. Frequently, the peaks have an ap- readout eﬃciency parameter that is explained in the text. proximately Gaussian distribution, such that 1 x|0i xT κ0 1 + erf | − | , (32) of quantum limits to sensitivity (see Sec. V.D). In addi- ≈ 2 σx tion, the observed probability is sometimes reduced by the control sequence of the sensing protocol, for example and likewise for κ1, where erf(x) is the Gauss error func- if there is no one-to-one mapping between the initialization. Moreover, if κ κ0 κ1 1 are small and of tion, sensing and readout basis (“Step 2” and “Step 4” similar magnitude, ≡ ≈ in the protocol). Since β is a constant of time, we will κ assume β = 1 in the following for reasons of simplicity. σ2 . (33) p,readout ≈ N

4. Classical readout noise b. Averaged readout When the classical noise added during the quantum state readout is large, only one peak A ﬁnal source of error is the classical noise added dur- appears in the histogram and the xj can no longer be as- ing the readout of the sensor. Two situations can be signed to x|0i or x|1i. The estimate for p is then simply distinguished, depending on whether the readout noise is given by the mean value of x, small or large compared to the projection noise. We will denote them as the “single shot” and “averaged” readout N x¯ x|0i 1 X xj x|0i p = − = − , (34) regimes, respectively. Due to the widespread ineﬃciency x x N x x of quantum state readout, classical readout noise is often |1i − |0i j=1 |1i − |0i the dominating source of error. 1 P wherex ¯ = N xj is the mean of xj . The standard error of p is { }

2 2 a. Single shot readout In the “single shot” regime, classi- 2 σx R σp,readout = 2 = , (35) cal noise added during the readout process is small. The (x|1i x|0i) 4N physical reading x produced by the measurement appa- − ratus will be very close to one of the two values x and where x|1i x|0i is the measurement contrast and |0i | − | x|1i, which would have been obtained in the ideal case for the qubit states 0 and 1 , respectively. By binning σp,readout 2√Nσx | i | i R = (36) the physical readings xj of j = 1...N measurements into ≡ σp,quantum x x | |1i − |0i| 17 is the ratio between classical readout noise and quantum 2. Minimum detectable signal and sensitivity projection noise. As an example, we consider the optical readout of an The sensitivity is defined as the minimum detectable atomic vapor magnetometer (Budker and Romalis, 2007) signal vmin that yields unit SNR for an integration time or of NV centers in diamond (Taylor et al., 2008). For of one second (T = 1 s), this example, x and x denote the average numbers of |0i |1i χ(t)√ + χ(t)√ + photons collected per readout for each state. The stan- q e t tm e t tm vmin = q q q . (41) 2C(tm) ∂ p(t) ∝ 2C(tm)γ t dard error is (under suitable experimental conditions) | V | dominated by shot noise, σx √x¯. The readout noise Eq. (41) provides clear guidelines for maximizing the sen- ≈ parameter becomes sitivity. First, the sensing time t should be made as long p 2√x¯ 2 1 /2 2 as possible. However, because the decay function χ(t) R = − , (37) exponentially penalizes the sensitivity for t > Tχ, the ≈ x|1i x|0i √x|1i ≈ √x|1i | − | optimum sensing time is reached when t Tχ. Second, ≈ where = 1 x /x is a relative optical contrast the sensitivity can be optimized with respect to tm. In | − |0i |1i| between the states, 0 < < 1, and the last equation particular, if C(tm) does improve as C √tm – which is ∝ represents the approximation for 1. a typical situation when operating in the averaged readout regime – the optimum choice is tm t. Conversely, if ≈ C is independent of tm – for example, because the sensor c. Total readout uncertainty The classical readout noise is operated in the single-shot regime or because readout σp,readout is often combined with the quantum projection resets the sensor – tm should be made as short as possible. noise σp,quantum to obtain a total readout uncertainty, Finally, C can often be increased by optimizing the experimental implementation or using advanced quantum σ2 = σ2 + σ2 p p,quantum p,readout schemes, such as quantum logic readout. 2 σp,quantum 1 We now evaluate Eq. (41) for the most common ex- (1 + R2)σ2 = , (38) ≈ p,quantum ≈ C2 4C2N perimental situations: where C = 1/√1 + R2 1/√1 + 4κ is an overall readout efficiency parameter≈ (Taylor et al., 2008). C 1 a. Slope detection For slope detection, p = 0.5 and describes the reduction of the signal-to-noise ratio com-≤ 0 δp(t) 1 γV t (Eq. 18). The sensitivity is pared to an ideal readout (C = 1), see Fig.4. We will in ≈ 2 χ(t) the following use Eq. (38) to derive the SNR and mini- e √t + tm mum detectable signal. vmin = . (42) γC(tm)t Note that the units of sensitivity are then typically given B. Sensitivity by the units of the signal V to be measured times Hz−1/2. Assuming tm t, we can find an exact optimum solution 1. Signal-to-noise ratio with respect tot. Specifically, for a Ramsey measurement −χ(t) −t/T ∗ with an exponential dephasing e = e 2 , the op- The signal-to-noise ratio (SNR) for a quantum sensing ∗ timum evolution time is t = T2 /2 and experiment can be defined as √2e 1 ∗ δpobs v = for t = T . (43) SNR = = ( ) −χ(t)2 √ (39) min p ∗ 2 δp t e C N, γC T2 2 σp This corresponds to Eq. (21) highlighted in the introduc- where δpobs is given by Eq. (26) and σp is given by Eq. (38). To further specify the SNR, the change in prob- tion to this section. ability δp can be related to the change in signal δV as δp = δV q ∂q p(t) (γtδV )q, with q = 1 for slope de- V b. Variance detection For variance detection, δp tection and| q =| 2 ∝ for variance detection (see Fig.3). 1 γ2V 2 t2 (Eq. 19). The sensitivity is ≈ In addition, the number of measurements N is equal to 4 rms T/(t + t ), where T is the total available measurement 1/2 m 2eχ(t)√t + t time and t is the extra time needed to initialize, manip- v = m , (44) m min ( ) 2 2 ulate and readout the sensor. The updated SNR becomes C tm γ t In the limit of tm 0 and t Tχ, this expression simpli- q q −χ(t) √T ≈ ≈ SNR = δV ∂V p(t) e 2C(tm) , (40) fies to | | √t + tm √2e where C(tm) is a function of tm because the readout ef- vmin = q , (45) 4 3 ficiency often improves for longer readout times. γ√C Tχ 18 which corresponds to Eq. (22) highlighted in the intro- where N is the number of samples xj. One is typically 2 duction of the section. Thus, variance detection profits interested in knowing how σX varies with time, given 2 more from a long coherence time Tχ than slope detection the recorded sensor outputs. To calculate σX (t) one can (but is, in turn, more vulnerable to decoherence). Al- group the data in variable-sized bins and calculate the ternatively, for the detection of a noise spectral density Allan variance for each grouping. The Allan variance for 1 2 SV (ω), the transition probability is δp 2 γ SV (ω)Tχ each grouping time t = mts can then be calculated as (see Eqs. (84) and (98)) and ≈ N−2m 1 X e σ2 (mt ) = (x x )2. (50) S (ω) . (46) X s 2 2 j+m j vmin 2 p 2(N 2m)m ts − ≈ γ C Tχ − j=1 The Allan variance can also be used to reveal the perfor- 3. Signal integration mance of a sensor beyond the standard quantum limit (Leroux et al., 2010b), and its extension to and lim- The above expressions for sensitivity all refer to the its in quantum metrology have been recently explored minimum detectable signal per unit time. By integrating (Chabuda et al., 2016). the signal over longer measurement times T , the sensor performance can be improved. According to Eq. (40), the minimum detectable signal for an arbitrary time T is D. Quantum CramérRao Bound for parameter estimation V q (T ) = vq /√T . Therefore, the minimum detectable min min The sensitivity of a quantum sensing experiment can signal for slope and variance detection, Eqs. (42) and be more rigorously considered in the context of the (45), respectively, scale as Cramér-Rao bound applied to parameter estimation. −1/2 Vmin(T ) = vminT for slope detection, (47) Quantum parameter estimation aims at measuring the −1/4 value of a continuous parameter V that is encoded in the Vmin(T ) = vminT for variance detection. (48) state of a quantum system ρV , via, e.g., its interaction The corresponding scaling for the spectral density is with the external signal we want to characterize. The −1/2 estimation process consists of two steps: in a first step, SVmin = Svmin T . We notice that variance detection improves only T 1/4 with the integration time, while the state ρV is measured; in a second step, the estimate slope detection∝ improves T 1/2. Hence, for weak signals of V is determined by data-processing the measurement ∝ outcomes. with long averaging times T Tχ, variance detection is typically much less sensitive than slope detection. As we In the most general case, the measurement can be de- will discuss in SectionVIII, adaptive sensing methods scribed by a positive-operator-valued measure (POVM) N can improve on these limits. = Ex over the N copies of the quantum system. TheM measurement{ } yields the outcome x with conditional (N) ⊗N probability pN (x V ) = Tr[Ex ρV ]. C. Allan variance With some further| data processing, we arrive at the estimate v of the parameter V . The estimation uncer- Sensors are typically also characterized by their sta- tainty can be described by the probability PN (v V ) := bility over time. Indeed, while the sensitivity calcula- P (N) (N) | x pest (v x)pN (x V ), where pest (v x) is the probability tion suggests that one will always improve the minimum of estimating| v from| the measurement| outcome x. We detectable signal by simply extending the measurement can then define the estimation uncertainty as ∆VN := time, slow variations affecting the sensor might make this pP 2 v[v V ] PN (v V ). Assuming that the estimation not possible. These effects can be quantified by the Allan − | procedure is asymptotically locally unbiased, ∆VN obeys variance (Allan, 1966) or its square root, the Allan devi- the so-called Cramér-Raobound ation. While the concept is based on a classical analysis p of the sensor output, it is still important for characteriz- ∆VN 1/γ FN (V ) , (51) ing the performance of quantum sensors. In particular, ≥ the Allan variance is typically reported to evaluate the where performance of quantum clocks (Hollberg et al., 2001; 2 X 1 ∂pN (x V ) Leroux et al., 2010b). FN (V ) := | p (x V ) ∂V Assuming that the sensor is sampled over time at con- x N | 2 (52) (N) ⊗n ! stant intervals ts yielding the signal xj = x(tj) = x(jts), X 1 ∂ Tr[Ex ρ ] = V the Allan variance is defined as (N) ⊗n ∂V x Tr[Ex ρV ] N−1 2 1 X 2 σ (τ) = (xj+1 xj) , (49) is the Fisher information associated with the given X 2(N 1)t2 − − s j=1 POVM measurement (Braunstein and Caves, 1994). 19

By optimizing Eq. (51) with respect to all possible giving the outcome probabilities p(x± V ) = ρ(V ) , POVM’s, one obtains the quantum Cramér-Raobound the Fisher information is | h±| |±i (QCRB) (Braunstein, 1996; Braunstein and Caves, 1994; 2 2 −2χ X 1 2 t cos (γV t)e Goldstein et al., 2010; Helstrom, 1967; Holevo, 1982; F = [∂V p(x V )] = . p(x V ) | 1 e−2χ sin2(γV t) Paris, 2009) x | − (58) 1 1 The Fisher information thus oscillates between its min- ∆VN > p > p , (53) γ maxM(N) [FN (V )] γ N (ρV ) imum, where γV t = (k + 1/2)π and F = 0, and its F optimum, where γV t = kπ and F = t2e−2χ. The uncer- where the upper bound of max (N) [F (V )] is expressed M N tainty in the estimate δV = 1/γ√NF therefore depends in terms of the quantum Fisher information ( ), de- ρV on the sensing protocol bias point. In the optimum case fined as F F corresponds to the quantum Fisher information and −1 −1 we find the QCRB (ρV ) := Tr[ (∂V ρV )ρV (∂V ρV )], (54) F RρV RρV 1 eχ with ∆VN = = . (59) γ√N γt√N F −1 X 2Ajk j k ρ (A) := | ih | (55) Depending on the functional form of χ(t), we can further R λj + λk j,k:λj +λk6=0 find the optimal sensing time for a given total measurement time. Note that if we remember that N experiments being the symmetric logarithmic derivative written in the will take a time T = N(t + t ), and we add inefficiency basis that diagonalizes the state, = P . m ρV j λj j j due to the sensor readout, we can recover the sensitivity A simple case results when ρ is a pure state,| ih obtained| V v of Eq. (42). from the evolution of the reference initial state 0 un- min | i Similarly, we can analyze more general protocols, such −iHˆV t der the signal Hamiltonian, ψV = e 0 . Then, the | i | i as variance detection of random fields, simultaneous es- QCRB is a simple uncertainty relation (Braunstein, 1996; timation of multiple parameters (Baumgratz and Datta, Braunstein and Caves, 1994; Helstrom, 1967; Holevo, 2016) or optimized protocols for signals growing over time 1982), (Pang and Jordan, 2016). 1 ∆V , (56) N > √ 2γ N ∆H VI. SENSING OF AC SIGNALS p where ∆H := H2 H 2. We note that the scaling of the QCRB withh i the − h numberi of copies, N −1/2, is a So far we have implicitly assumed that signals are consequence of the additivity of the quantum Fisher in- static and deterministic. For many applications it is im- formation for tensor states ρ⊗N . This is the well-known portant to extend sensing to time-dependent signals. For standard quantum limit (SQL). To go beyond the SQL, example, it may be required to detect the amplitude, fre- one then needs to use entangled states (see SectionIX)– quency or phase of an oscillating signal. More broadly, in particular, simply using correlated POVMs is not suf- one may be interested in knowing the waveform of a time- ficient. Thus, to reach the QCRB, local measurements varying parameter or reconstructing a frequency spec- and at most adaptive estimators are sufficient, without trum. A diverse set of quantum sensing methods has the need for entanglement. been developed for this purpose that are summarized in While the quantum Fisher information (and the the following two sections. QCRB) provide the ultimate lower bound to the achiev- Before discussing the various sensing protocols in more able uncertainty for optimized quantum measurments, detail, it is important to consider the type of information the simpler Fisher information can be used to evaluate a that one intends to extract from a time-dependent signal given measurement protocol, as achievable, e.g., within V (t). In this SectionVI, we will assume that the signal experimental constraints. is composed of one or a few harmonic tones and our goal Consider for example the sensing protocols described will be to determine the signal’s amplitude, frequency, in SectionIV. For the Ramsey protocol, the quantum phase or overall waveform. In the following SectionVII, sensor state after the interaction with the signal V is we will discuss the measurement of stochastic signals with given by the intent of reconstructing the noise spectrum or measuring the noise power in a certain bandwidth. 1 i −iγV t −χ(t) ρ11(V, t) = ρ12(V, t) = e e . (57) 2 −2 A. Time-dependent signals Here, e−χ(t) describes decoherence and relaxation as discussed with Eq. (26). If we assume to perform a projec- As measuring arbitrary time-dependent signals is a 1 tive measurement in the σx basis, = √ ( 0 1 ), complex task, we first focus on developing a basic set of {|±i} { 2 | i±| i 20

Init   Readout t Sensitivity to alternating signals can be restored by us- (a) ing time-reversal (“spin echo”) techniques (Hahn, 1950).  To illustrate this, we assume that the AC signal goes (b) through exactly one period of oscillation during the sensing time t. The Ramsey phase φ due to this signal is zero (c)  because the positive phase build-up during the ﬁrst half of t is exactly canceled by the negative phase build-up (d)  during the second half of t. However, if the qubit is in- verted at time t/2 using a π pulse (see Fig.5(b)), the time t’ time evolution of the second period is reversed, and the FIG. 5 Pulse diagrams for DC and AC sensing sequences. accumulated phase is non-zero, Narrow blocks represent π/2 pulses and wide blocks represent Z t/2 Z t π pulses, respectively. t is the total sensing time and τ is 0 0 0 0 2 φ = γV (t ) dt γV (t ) dt = γVpkt cos α . the interpulse delay. (a) Ramsey sequence. (b) Spin-echo − π sequence. (c) CP multipulse sequence. (d) PDD multipulse 0 t/2 sequence. (63)

AC sensing protocols, assuming a single-tone AC signal C. Multipulse sensing sequences given by The spin echo technique can be extended to sequences 0 0 with many pulses. These sequences are commonly re- V (t ) = Vpk cos(2πfact + α) . (60) π ferred to as multipulse sensing sequences or multipulse This signal has three basic parameters, including the sig- control sequences, and allow for a detailed shaping of nal amplitude Vpk, the frequency fac and the relative the frequency response of the quantum sensor. To un- phase α. Our aim will be to measure one or several of derstand the AC characteristics of a multipulse sensing these parameters using suitable sensing protocols. sequence, we consider the phase accumulated for a gen- Signal detection can be extended to multi-tone signals eral sequence of n π pulses applied at times 0 < tj < t, by summing over individual single-tone signals, with j = 1...n. The accumulated phase is given by

t 0 X 0 Z V (t ) = Vpk,m cos(2πfac,mt + αm) , (61) φ = γV (t0)y(t0) dt0 , (64) m 0

0 where Vpk,m, fac,m and αm are the individual amplitudes, where y(t ) = 1 is the modulation function of the se- frequencies and phases of the tones, respectively. quence that changes± sign whenever a π pulse is applied. 0 0 For a harmonic signal V (t ) = Vpk cos(2πfact + α) the phase is B. Ramsey and Echo sequences γVpk n φ = [sin(α) ( 1) sin(2πfact + α) To illustrate the difference between DC and AC sens- 2πfac − − ing, we re-examine the Ramsey measurement from Sec- n X j tion IV.C. The corresponding pulse diagram is given in + 2 ( 1) sin(2πfactj + α)] − Fig.5(a). This protocol is ideally suited to measure j=1 static shifts of the transition energy. But is it also capa- = γVpkt W (fac, α) . (65) ble of detecting dynamical variations? In order to answer × this question, one can inspect the phase φ accumulated This defines for any multipulse sequence a weighting func- during the sensing time t due to either a static or a time- tion W (fac, α). For composite signals consisting of sev- dependent signal V (t), eral harmonics with different frequencies and amplitudes, Eq. (61), the accumulated phase simply represents the Z t sum of individual tone amplitudes multiplied by the re- φ = γV (t0) dt0 . (62) 0 spective weighting functions. For a static perturbation, the accumulated phase is simply φ = γV t. For a rapidly oscillating perturbation, by 1. CP and PDD sequences contrast, phase accumulation is averaged over the sensing time, and φ = γ V (t0) t 0. To answer our question, the The simplest pulse sequences used for sensing have Ramsey sequenceh willi only≈ be sensitive to slowly varying been initially devised in nuclear magnetic resonance signals up to some cut-off frequency t−1. (NMR) (Slichter, 1996) and have been further developed ≈ 21 in the context of dynamical decoupling (DD) (Viola and (a) t = n Lloyd, 1998). They are composed of n equally spaced    π-pulses with an interpulse duration τ. The most common types are Carr-Purcell (CP) pulse trains (Carr and Purcell, 1954) (Fig.5(c)) and periodic dynamical de- (b) Signal V(t’) coupling (PDD) sequences (Khodjasteh and Lidar, 2005) (Fig.5(d)). 0 For a basic CP sequence, = 2j−1 , and the weighting tj 2 τ Modulation function y(t’) function is (Hirose et al., 2012; Taylor et al., 2008) +1

sin(πfacnτ) -1 WCP(fac, α) = [1 sec(πfacτ)] cos(α+πfacnτ) . πfacnτ − Rectified signal (66) Similarly, for a PDD sequence, tj = jτ and 0 time t’ sin(πfacnτ) WPDD(fac, α) = tan(πfacτ) sin(α+πfacnτ) . 2 πfacnτ (c) W (fac) (67) 0.5 N=10 Because of the first (sinc) term, these weighting functions k=1 resemble narrow-band filters around the center frequen- 0.4 cies fac = fk = k/(2τ), where k = 1, 3, 5, ... is the har- 0.3 monic order. In fact, they can be rigorously treated as filter functions that filter the frequency spectrum of the 0.2 ~1/t signal V (t) (see SectionVII). For large pulse numbers n, the sinc term becomes very peaked and the filter band- 0.1 k=3 k=5 width ∆ 1 ( ) = 1 (full width at half maximum) f / nτ /t 0.0 becomes very≈ narrow. The narrow-band filter characteristics can be summed up as follows (see Fig.5(c)), 0123 456 frequency fac [1/(2)] fk = k/(2τ) center frequencies (68) ∆f 1/t = 1/(nτ) bandwidth (69) FIG. 6 Modulation and weight functions of a CP multipulse ≈ sequence. (a) CP multipulse sequence. (b) Signal V (t0), mod- 2 k−1 ulation function y(t0) and “rectified” signal V (t0) × y(t0). The WCP(α) = ( 1) 2 cos(α) πk − peak transmission (70) accumulated phase is represented by the area under the curve. 2 2 WPDD(α) = sin(α) (c) Weight function W (fac) associated with the modulation −πk function. k is the harmonic order of the filter resonance. The advantage of the CP an PDD sequences is that their filter parameters can be easily tuned. In particular, the pass-band frequency can be selected via the interpulse AC signal and the modulation function y(t). For a signal delay τ, while the filter width can be adjusted via the that is in-phase with y(t), the maximum phase accumu- number of pulses n = t/τ (up to a maximum value of lation occurs, while for an out-of-phase signal, φ = 0. This behavior can be exploited to add further capa- n T2/τ). The resonance order k can also be used to select≈ the pass-band frequency, however, because k = 1 bilities to AC signal detection. Kotler et al., 2011, have provides the strongest peak transmission, most reported shown that both quadratures of a signal can be recov- experiments used this resonance. The time response of ered, allowing one to perform lock-in detection of the the transition probability is signal. Furthermore, it is possible to correlate the phase acquired during two subsequent multipulse sequences to 1 p = [1 cos (W (fac, α)γVpkt)] perform high-resolution spectroscopy of AC signals (see 2 − Section VI.E). 1 2γV t cos α = 1 cos pk , (71) 2 − kπ where the last expression represents the resonant case 3. Other types of multipulse sensing sequences (fac = k/2τ) for CP sequences. Many varieties of multipulse sequences have been con- ceived with the aim of optimizing the basic CP design, 2. Lock-in detection including improved robustness against pulse errors, better decoupling performance, narrower spectral response The phase φ acquired during a CP or PDD sequence and sideband suppression, and avoidance of signal har- depends on the relative phase difference α between the monics. 22

A systematic analysis of many common sequences has 1.0 been given by Cywinski et al., 2008. One favorite has (a) been the XY4, XY8 and XY16 series of sequences (Gul- p(t) (b) lion et al., 1990) owing to their high degree of pulse error compensation. A downside of XY type sequences (c) are signal harmonics (Loretz et al., 2015) and the side- 0.5 (d) bands common to CP sequences with equidistant pulses. Other recent efforts include sequences with non-equal pulse spacing (Ajoy et al., 2017; Casanova et al., 2015; (e) Zhao et al., 2014) or sequences composed of alternat- 0.0 ing subsequences (Albrecht and Plenio, 2015). A Flo- 0  2 3 4 quet spectroscopy approach to multipulse sensing has accumulated phase  = Vt also been proposed (Lang et al., 2015). FIG. 7 Transition probability p(t) as a function of phase accumulation time t. (a) Ramsey oscillation (Eq. 14). (b) AC 4. AC signals with random phase signal with fixed amplitude and optimum phase (Eq. 71). (c) AC signal with fixed amplitude and random phase (Eq. 75). Often, the multipulse sequence cannot be synchronized (d) AC signal with random amplitude and random phase (Eq. with the signal or the phase α cannot be controlled. 76). (e) Broadband noise with χ = Γt (Eq. 83). Then, incoherent averaging leads to phase cancellation, φ = 0. In this case, it is advantageous to measure the varianceh i of the phase φ2 rather than its average φ . 5. AC signals with random phase and random amplitude (Although such a signalh technicallyi represents a stochas-h i tic signal, which will be considered in more detail in the If the amplitude Vpk is not fixed, but slowly fluctuating between different measurements, the variance φ2 must next section, it is more conveniently described here.) h i For a signal with fixed amplitude but random phase, be integrated over the probability distribution of Vpk.A the variance is particularly important situation is a Gaussian amplitude fluctuation with an rms amplitude Vrms. In this case, the 2 2 2 2 2 resonant time response of the transition probability is φ = γ V t W (fac), (72) h i rms " 2 2 2 2 ! 2 2 2 2 !# where Vrms = Vpk/√2 is the rms amplitude of the sig- 1 W γ Vrmst W γ Vrmst 2 p(t) = 1 exp 2 I0 2 nal and W (fac) is the average over α = 0 ... 2π of the 2 − − 2k 2k weighting functions, (76) where I is the modified Bessel function of the first kind. Z 2π 0 2 1 2 W (fac) = W (fac, α) dα (73) 2π 0 For the CP and PDD sequences, the averaged functions D. Waveform reconstruction are given by

2 The detection of AC fields can be extended to the more 2 sin (πfacnτ) 2 W CP(fac) = 2 [1 sec (πfacτ)] , general task of sensing and reconstructing arbitrary time 2(πfacnτ) − 2 dependent fields. A simple approach is to record the time 2 sin (πfacnτ) 2 response p(t) under a specific sensing sequence, such as W PDD(fac) = 2 tan (πfacτ) , (74) 2(πfacnτ) a Ramsey sequence, and to reconstruct the phase φ(t) and signal ( ) from the time trace (Balasubramanian 2 V t and the peak transmission at fac = k/2τ is W = et al., 2009). This approach is, however, limited to the 2/(kπ)2. The time response of the transition probability bandwidth of the sequence and by readout deadtimes. is (Kotler et al., 2013) To more systematically reconstruct the time dependence of an arbitrary signal, one may use a family of 1 h i p(t) = 1 J0 √2W (fac)γVrmst pulse sequences that forms a basis for the signal. A 2 − " !# suitable basis are Walsh dynamical decoupling sequences 1 2√2γVrmst (Hayes et al., 2011), which apply a π pulse every time = 1 J0 (75) 2 − kπ the corresponding Walsh function (Walsh, 1923) flips its sign. Under a control sequence with n π-pulses ap- where J0 is the Bessel function of the first kind and where plied at the zero-crossings of the n-th Walsh function 0 0 the second equation reflects the resonant case. y(t ) = wn(t /t), the phase acquired after an acquisition 23 period t is V(t’) (a) Z t time t’ 0 0 0 φ(t) = γ V (t )y(t )dt = γVnt , (77) 0 multipulse Init sequence t Readout which is proportional to the n-th Walsh coefficient Vn of (b) 1 V (t0). By measuring N Walsh coefficients (by applying N different sequences) one can reconstruct an N-point ta ta functional approximation to the field V (t0) from the N- ts th partial sum of the Walsh-Fourier series (Cooper et al., Init Readout 2014; Magesan et al., 2013b), (c)

ta N−1 0 X 0 VN (t ) = Vnwn(t /t), (78) FIG. 8 Correlation spectroscopy. (a) AC signal V (t0). (b) n=0 Correlation sequence. Two multipulse sequences are interrupted by an incremented delay time t1. Because the mul- 0 0 which can be shown to satisfy limN→∞ VN (t ) = V (t ). A tipulse sequences are phase sensitive, the total phase accu- similar result can be obtained using different basis func- mulated after the second multipulse sequence oscillates with tions, such as Haar wavelets, as long as they can be easily fact1. The maximum t1 is limited by the relaxation time implemented experimentally (Xu et al., 2016). T1, rather than the typically short decoherence time T2. (c) An advantage of these methods is that they provide Continouous sampling. The signal is periodically probed in intervals of the sampling time ts. The frequency can be esti- protection of the sensor against dephasing, while extract- mated from a sample record of arbitrary duration, permitting ing the desired information. In addition, they can be an arbitrarily fine frequency resolution. combined with compressive sensing techniques (Candés et al., 2006; Magesan et al., 2013a; Puentes et al., 2014) to reduce the number of acquisition needed to reconstruct 2. Correlation sequences the time-dependent signal. The ultimate metrology limits in waveform reconstruction have also been studied Several schemes have been proposed and demonstrated (Tsang et al., 2011). to further narrow the bandwidth and to perform high resolution spectroscopy. All of them rely on correlation- type measurements where the outcomes of subsequent E. Frequency estimation sensing periods are correlated. A first method is illustrated in Fig.8(a) in combina- An important capability in AC signal detection is the tion with multipulse detection. The multipulse sequence precise estimation of frequencies. In quantum sensing, is subdivided into two equal sensing periods of duration most frequency estimation schemes are based on dynam- ta = t/2 that are separated by an incremented free evo- ical decoupling sequences. These are discussed in the lution period t1. Since the multipulse sequence is phase following. Fundamental limits of frequency estimation sensitive, constructive or destructive phase build-up oc- based on the quantum Fisher information have been con- curs between the two sequences depending on whether sidered by Pang and Jordan, 2016. the free evolution period t1 is a half multiple or full multiple of the AC signal period Tac = 1/fac. The final transition probability therefore oscillates with t1 as 1. Dynamical decoupling spectroscopy 1 p(t1) = 1 sin[Φ cos(α)] sin[Φ cos(α + 2πfact1)] 2 { − } A simple approach for determining a signal’s frequency 1 2 is to measure the response to pulse sequences with dif- 1 Φ cos(α) cos(α + 2πfact1) (79) ≈ 2 − ferent pulse spacings τ. This is equivalent to stepping the frequency of a lock-in amplifier across a signal. The where Φ = γVpkt/(kπ) is the maximum phase that can spectral resolution of dynamical decoupling spectroscopy be accumulated during either of the two multipulse se- is determined by the bandwidth of the weighting function quences. The second equation is for small signals where W (fac, τ), which is given by ∆f 1/t (see Eq. (69)). sin Φ Φ. For signals with random phase, Eq. (79) ≈ Because t can only be made as long as the decoherence can be≈ integrated over α and the transition probability time T2, the spectral resolution is limited to ∆f 1/T2. simplifies to The precision of the frequency estimation, which≈ also depends on the signal-to-noise ratio, is directly proportional 1 Φ2 p(t1) 1 cos(2πfact1) (80) to ∆f. ≈ 2 − 2 24

Since the qubit is parked in 0 and 1 during the free A. Noise processes evolution period, relaxation| isi no longer| i governed by T2, but by the typically much longer T1 relaxation time For the following analysis we will assume that the (Laraoui and Meriles, 2013). In this way, a Fourier- stochastic signal V (t) is Gaussian. Such noise can be limited spectral resolution of ∆f 1/T1 is possible. described by simple noise models, like a semi-classical The resolution can be further enhanced∼ by long-lived Gaussian noise or the Gaussian spin-boson bath. In ad- auxiliary memory qubits (see SectionX) and resolution dition, we will assume that the autocorrelation function improvements by two-to-three orders of magnitude over of V (t), dynamical decoupling spectroscopy have been demon- 0 0 strated (Pfender et al., 2016; Rosskopf et al., 2016; Zaiser GV (t) = V (t + t)V (t ) , (81) h i et al., 2016). The correlation protocol was further shown to eliminate several other shortcomings of multipulse decays on a time scale tc that is shorter than the sensing sequences, including signal ambiguities resulting from time t, such that successive averaging measurements are the multiple frequency windows and spectral selectivity not correlated. The noise can then be represented by a (Boss et al., 2016). power spectral density (Biercuk et al., 2011), Z ∞ −iωt SV (ω) = e GV (t) dt , (82) 3. Continuous sampling −∞ that has no sharp features within the bandwidth ∆f A second approach is the continuous sampling of a 1/t of the sensor. The aim of a noise spectroscopy exper-≈ signal, illustrated in Fig.8(b). The output signal can iment is to reconstruct SV (ω) as a function of ω over a then be Fourier transformed to extract the undersam- frequency range of interest. pled frequency of the original signal. Because continuous Although this Section focuses on Gaussian noise with sampling no longer relies on quantum state lifetimes, the tc . t, the analysis can be extended to other noise mod- Fourier-limited resolution can be extended to arbitrary els and correlated noise. When tc t, the frequency values and is only limited by total experiment duration T , and amplitude of V (t) are essentially fixed during one and ultimately the control hardware. The original signal sensing period and the formalism of AC sensing can be frequency can then be reconstructed from the undersam- applied (see Section VI.C). A rigorous derivation for all pled data record using compressive sampling techniques ranges of tc, but especially tc t is given by Cum- (Nader et al., 2011). Continuous sampling has recently mings, 1962. More complex noise≈ models, such as 1/f led to the demonstration of µHz spectral resolution (Boss noise with no well-defined tc, or models that require a et al., 2017; Jelezko et al., 2017). cumulant expansion beyond a first order approximation on the noise strength can also be considered (Ban et al., 2009; Bergli and Faoro, 2007). Finally, open-loop con- VII. NOISE SPECTROSCOPY trol protocols have been introduced (Barnes et al., 2016; Cywiński, 2014; Norris et al., 2016; Paz-Silva and Viola, In this Section, we discuss methods for reconstruct- 2014) to characterize stationary, non-Gaussian dephasing the frequency spectrum of stochastic signals, a task ing. commonly referred to as noise spectroscopy. Noise spectroscopy is an important tool in quantum sensing, as it can provide much insight into both external signals and B. Decoherence, dynamical decoupling and filter functions the intrinsic noise of the quantum sensor. In particular, good knowledge of the noise spectrum can help the adop- There have been many proposals for examining deco- tion of suitable sensing protocols (like dynamical decou- herence under different control sequences to investigate pling or quantum error correction schemes) to maximize noise spectra (Almog et al., 2011; Faoro and Viola, 2004; the sensitivity of the quantum sensor. Young and Whaley, 2012; Yuge et al., 2011). In particu- Noise spectroscopy relies on the systematic analysis lar, dynamical decoupling sequences based on multipulse of decoherence and relaxation under different control se- protocols (Section VI.C) provide a systematic means for quences. This review will focus on two complementary filtering environmental noise (Alvarez´ and Suter, 2011; frameworks for extracting noise spectra. A first concept Biercuk et al., 2011; Kotler et al., 2011). These have is that of “filter functions”, where the phase pick-up due been implemented in many physical systems (Bar-Gill to stochastic signals is analyzed under different dynam- et al., 2012; Bylander et al., 2011; Dial et al., 2013; Kotler ical decoupling sequences. The second concept, known et al., 2013; Muhonen et al., 2014; Romach et al., 2015; as “relaxometry”, has its origins in the field of magnetic Yan et al., 2012, 2013; Yoshihara et al., 2014). A brief in- resonance spectroscopy and is closely related to Fermi’s troduction to the method of filter functions is presented golden rule. in the following. 25

1. Decoherence function χ(t) (Note that this definition differs by a factor of ω2 from the one by Biercuk et al., 2011). Thus, the filter func- Under the assumption of a Gaussian, stationary noise, tion plays the role of a transfer function, and the decay the loss of coherence can be captured by an exponential of coherence is captured by the overlap with the noise decay of the transition probability with time t, spectrum SV (ω). To illustrate the concept of filter functions we recon- 1 p(t) = 1 e−χ(t) . (83) sider the canonical example of a Ramsey sensing se- 2 − quence. Here, the filter function is where χ(t) is the associated decay function or decoher- sin2(ωt/2) ence function that was already discussed in the context 2 Y (ω) = 2 . (88) of sensitivity (SectionV). Quite generally, χ(t) can be | | ω identified with an rms phase accumulated during time t, The decoherence function χ(t) then describes the “free- induction decay”, 1 ( ) = 2 (84) χ t φrms . Z ∞ 2 2 2 2 sin (ωt/2) 1 2 χ(t) = γ SV (ω) 2 dω γ SV (0)t , according to the expression for variance detection, Eq. π 0 ω ≈ 2 (20). (89) Depending on the type of noise present, the decoher- where the last equation is valid for a spectrum that is flat around . The Ramsey sequence hence acts as a ence function shows a different dependence on t. For ω . π/t simple sinc filter for the noise spectrum ( ), centered white noise, the dephasing is Markovian and χ(t) = Γt, SV ω where Γ is the decay rate. For Lorentzian noise cen- at zero frequency and with a lowpass cut-off frequency of approximately . tered at zero frequency the decoherence function is χ(t) = π/t (Γt)3. For a generic 1/f-like decay, where the noise falls a of ω (with a around unity), the decoherence function 3. Dynamical decoupling is χ∝(t) = (Γt)a+1 (Cywinski et al., 2008; Medford et al., 2012) with a logarithmic correction depending on the ra- To perform a systematic spectral analysis of SV (ω), tio of total measurement time and evolution time (Dial one can examine decoherence under various dynamical et al., 2013). Sometimes, decoherence may even need to decoupling sequences. Specifically, we inspect the fil- be described by several decay constants associated with ter functions of periodic modulation functions yn ,τ (t), several competing processes. A thorough discussion of c c where a basic building block y1(t) of duration τc is re- decoherence is presented in the recent review by Suter peated n times. The filter function of y (t) is given ´ c nc,τc and Alvarez, 2016. by

nc−1 X iτck 2. Filter function Y (ω) Ync,τc (ω) = Y1,τc (ω) e k=0

The decoherence function χ(t) can be analyzed un- sin(ncωτc/2) = ( ) −i(nc−1)ωτc/2 (90) der the eﬀect of diﬀerent control sequences. Assuming Y1,τc ω e , sin(ωτc/2) the control sequence has a modulation function y(t) (see

Section VI.C), the decay function is determined by the where Y1,τc (ω) is the filter function of the basic building correlation integral (Biercuk et al., 2011; de Sousa, 2009) block. For large cycle numbers, Ync,τc (ω) presents sharp −1 peaks at multiples of the inverse cycle time τc , and it Z t Z t 1 0 00 0 00 2 0 00 can be approximated by a series of δ functions. χ(t) = dt dt y(t )y(t )γ GV (t t ) , (85) 2 0 0 − Two specific examples of periodic modulation functions include the CP and PDD sequences considered in where ( ) is the autocorrelation function of ( ) GV t V t Section VI.C, where τc = 2τ and nc = n/2. The filter (Eq. 81). In the frequency domain the decay function function for large pulse numbers n is can be expressed as 2 X 2π 2 Z ∞ Yn,τ sinc[(ω ωk)t/2] 2 2 2 | | ≈ (kπ)2 − χ(t) = γ SV (ω) Y (ω) dω , (86) k π | | 0 X 2π tδ(ω ωk) (91) 2 ≈ (kπ)2 − where Y (ω) is the so-called filter function of y(t), de- k fined by| the| finite-time Fourier transform where ωk = 2π k/(2τ) are resonances with k = 1, 3, 5, ... t × Z 0 (note that these expression are equivalent to the filters Y (ω) = y(t0)eiωt dt0 . (87) 0 Eq. (74) found for random phase signals). 26

The decay function can then be expressed by a simple where V (ω) = V †( ω). Next, we calculate the proba- − sum of different spectral density components, bility amplitude c1 that a certain frequency component V (ω) causes a transition between two orthogonal sensing Z ∞ 2 2 X 2π states ψ0 and ψ1 during the sensing time t. Since the χ(t) = γ SV (ω) tδ(ω ωk) dω π (kπ)2 − perturbation| i is weak,| i perturbation theory can be applied. 0 k 2 The probability amplitude c1 in first order perturbation 4t X γ SV (ωk) = (92) theory is π2 k2 k t Z 0 0 i(ω01−ω)t c1(t) = i dt ψ1 HˆV (ω) ψ0 e This result provides a simple strategy for reconstruct- − 0 h | | i ing the noise spectrum. By sweeping the time τ between i(ω01−ω)t ˆ e 1 pulses the spectrum can be probed at various frequen- = i ψ1 HV (ω) ψ0 − (94) − h | | i i(ω01 ω) cies. Since the filter function is dominated by the first − harmonic (k = 1) the frequency corresponding to a cer- where HˆV (ω) is the Hamiltonian associated with V (ω) tain τ is 1/(2τ). For a more detailed analysis the contri- and ω01 is the transition energy between states ψ0 and | i butions from higher harmonics as well as the exact shape ψ1 . The transition probability is of the filter functions has to be taken into account. The | i 2 spectrum can then be recovered by spectral decomposi- 2 ˆ 2 sin[(ω01 ω)t/2] ´ c1(t) = ψ1 HV (ω) ψ0 − tion (Alvarez and Suter, 2011; Bar-Gill et al., 2012). | | |h | | i| (ω01 ω)/2 − The filter analysis can be extended to more general 2 2π ψ1 HˆV (ω) ψ0 tδ(ω01 ω) (95) dynamical decoupling sequences. In particular, Zhao ≈ |h | | i| − et al., 2014, consider periodic sequences with more com- where the second equation reflects that for large t, the plex building blocks, and Cywinski et al., 2008, consider sinc function approaches a δ function peaked at ω01. The aperiodic sequences like the UDD sequence. associated transition rate is 2 ∂ c1(t) 2 | | 2π ψ1 HˆV (ω) ψ0 δ(ω01 ω) . (96) ∂t ≈ |h | | i| − C. Relaxometry This is Fermi’s golden rule expressed for a two-level sys- An alternative framework for analyzing relaxation and tem that is coupled to a radiation field with a continuous doherence has been developed in the field of magnetic frequency spectrum (Sakurai and Napolitano, 2011). resonance spectroscopy, and is commonly referred to as The above transition rate is due to a single frequency ˆ “relaxometry” (Abragam, 1961). The concept has later component of HV (ω). To obtain the overall transition been extended to the context of qubits (Schoelkopf et al., rate Γ, Eq. (96) must be integrated over all frequencies, 2003). The aim of relaxometry is to connect the spectral Z ∞ 1 ˆ 2 density S (ω) of a noise signal V (t) to the relaxation Γ = dω 2π ψ1 HV (ω) ψ0 δ(ω01 ω) V π |h | | i| − rate Γ in first-order kinetics, χ(t) = Γt. Relaxometry 0 ˆ 2 is based on first-order perturbation theory and Fermi’s = 2 ψ1 HV (ω01) ψ0 |h | | i| golden rule. The basic assumptions are that the noise 2 2 = 2γ SV01 (ω01) ψ1 σV /2 ψ0 (97) process is approximately Markovian and that the noise · |h | | i| strength is weak, such that first-order perturbation the- where in the last equation, SV01 is the spectral density ory is valid. Relaxometry has found many applications in of the component(s) of V (t) than can drive transitions magnetic resonance and other fields, especially for map- between ψ0 and ψ1 , multiplied by a transition matrix | i | i 2 ping high-frequency noise based on T1 relaxation time element ψ1 σV /2 ψ0 of order unity that represents |h | | î| measurements (Kimmich and Anoardo, 2004). the operator part of HV = V (t)σV /2 (see Eq.5). The last equation (97) is an extremely simple, yet powerful and quantitative relationship: the transition rate 1. Basic theory of relaxometry equals the spectral density of the noise evaluated at the transition frequency, multiplied by a matrix element of To derive a quantitative relationship between the de- order unity (Abragam, 1961; Schoelkopf et al., 2003). cay rate Γ and a noise signal V (t), we briefly revisit the The expression can also be interpreted in terms of the rms 2 elementary formalism of relaxometry (Abragam, 1961). phase φrms. According to Eq. (84), φrms = 2χ(t) = 2Γt, 2 1 which in turn yields (setting ψ1 σV /2 ψ0 = ) In a first step, V (t) can be expanded into Fourier com- |h | | i| 4 ponents, 2 2 φrms = γ SV01 (ω01)t . (98) 1 Z ∞ V (t) = dω V (ω)e−iωt + V †(ω)eiωt (93) The rms phase thus corresponds to the noise integrated 2π −∞ over an equivalent noise bandwidth of 1/(2πt). 27

Method Sensing states Sensitive to V|| at Sensitive to V⊥ at Frequency tunable via {|ψ0i, |ψ1i} frequency frequency Ramsey {|+i, |−i} 0 —a — Spin echo {|+i, |−i} 1/t —a — Dynamical decoupling {|+i, |−i} πk/τ, with k = 1, 3, .. —a Pulse spacing τ, resonance order k T1 relaxometry {|0i, |1i} — ω0 Static control field a Dressed states (resonant) {|+i, |−i} ω1 — Drive field amplitude ω1 a Dressed states (off-resonant) {|+i, |−i} ωeff ≈ ∆ω — Detuning ∆ω √ a TABLE II Summary of noise spectroscopy methods. |±i = (|0i ± |1i)/ 2. also affected by T1 relaxation.

The relation between the transition rate Γ and the Init Readout t spectral density can be further specified for vector signals (a) V~ . In this case the transition rate represents the sum of the three vector components of Vj, where j = x, y, z, Init   Readout (b) t X 2 Γ = 2 ψ1 HˆV (ωj) ψ0 |h | j | i| j=x,y,z Init  t  Readout X 2 2 (c) = 2 γ SV (ωj) ψ1 σˆj ψ0 (99) j |h | | i| j=x,y,z FIG. 9 Common relaxometry protocols. (a) T1 relaxometry. (b) T ∗ relaxometry. (c) Dressed state relaxometry. Narrow where S (ω ) is the spectral density of V , ω is a transi- 2 Vj j j j black boxes represent π/2 pulses and the grey box in (c) rep- tion frequency that reflects the energy exchange required resents a resonant or off-resonant drive field. for changing the state, andσ ˆj are Pauli matrices. Note that if ψ0 , ψ1 are coherent superposition states, Vx {| i | i} and Vy represent the components of V⊥ that are in-phase and out-of-phase with the coherence, rather than the 9(a)). The transition rate is static components of the vector signal V~ . Relaxation rates can be measured between any set −1 1 2 of sensing states , including superposition (T1) = γ SV⊥ (ω0) , (100) ψ0 , ψ1 2 states. This gives{| risei to| ai} great variety of possible relaxometry measurements. For example, the method can be used to probe different vector components Vj(t) (or where T1 is the associated relaxation time and SV⊥ = commuting and non-commuting signals V||(t) and V⊥(t), SVx + SVy . Thus, T1 relaxometry is only sensitive to the respectively) based on the selection of sensing states. transverse component of V~ . Because T1 can be very long, Moreover, different sensing states typically have vastly very high sensitivities are in principle possible, assum- different transition energies, providing a means to cover ing that the resonance is not skewed by low-frequency a wide frequency spectrum. If multiple sensing qubits are noise. By tuning the energy splitting ω0 between 0 available, the relaxation of higher-order quantum transi- and 1 , for example through the application of a static| i | i tions can be measured, which gives additional freedom to control field, a frequency spectrum of SV⊥ (ω) can be probe different symmetries of the Hamiltonian. recorded (Kimmich and Anoardo, 2004). For this reason An overview of the most important relaxometry pro- and because it is experimentally simple, T1 relaxometry tocols is given in TableII and Fig.9. They are briefly has found many applications. For example, single-spin discussed in the following. probes have been used to detect the presence of magnetic ions (Steinert et al., 2013), spin waves in magnetic films (van der Sar et al., 2015), high-frequency magnetic fluctuations near surfaces (Myers et al., 2014; Romach 2. T1 relaxometry et al., 2015; Rosskopf et al., 2014), and single molecules (Sushkov et al., 2014a). T1 relaxometry has also been ap- T1 relaxometry probes the transition rate between plied to perform spectroscopy of electronic and nuclear states 0 and 1 . This is the canonical example of en- spins (Hall et al., 2016). In addition, considerable effort ergy relaxation.| i | i Experimentally, the transition rate is has been invested in mapping the noise spectrum near measured by initializing the sensor into 0 at time t0 = 0 superconducting flux qubits by combining several relax- and inspecting p = 1 α 2 at time t0 |=it without any ometry methods (Bialczak et al., 2007; Bylander et al., further manipulation|h of| i| the quantum system (see Fig. 2011; Lanting et al., 2009; Yan et al., 2013). 28

phase ﬂip therefore is no longer energy conserving. The T1 relaxometry associated relaxation time T1ρ is given by Dressed states (off-resonant)

Dressed states −1 1 2 1 2 (T1ρ) γ SV (ω0) + γ SV (ω1) (102) ≈ 4 ⊥ 2 || Multipulse By systematically varying the Rabi frequency , the Hahn echo ω1 spectrum SV|| (ω1) can be recorded (Loretz et al., 2013; Ramsey Yan et al., 2013). Because ω1 ω0, dressed states provide useful means for covering the medium frequency range of the spectrum (see Fig. 10). In addition, since Electronic spins, superconducting qubits dressed state relaxometry does not require sweeping a DC 1 kHz 1 MHz 1 GHz static control field for adjusting the probe frequency, it is more versatile than standard T1 relaxometry. Nuclear spins, trapped ions (vibrational) Dressed state methods can be extended to include off- DC 1 Hz 1 kHz 1 MHz resonant drive fields. Specifically, if the drive field is detuned by ∆ω from ω0, relaxation is governed by a modi- FIG. 10 Typical spectral range of noise spectroscopy proto- fied relaxation time cols. Scales refer to the quantum sensors discussed in Section 2 2 III. −1 2 1 ∆ω 1 ω1 2 (T1ρ) γ 1 + 2 SV⊥ (ω0) + 2 γ SV|| (ωeff ) , ≈ 4 ωeff 2 ωeff

∗ (103) 3. T2 and T2 relaxometry p 2 2 ∗ where ωeff = ω1 + ∆ω is the effective Rabi frequency. T2 relaxometry probes the transition rate between the A detuning therefore increases the accessible spectral superposition states = ( 0 e−iω0t 1 )/√2. This |±i | i ± | i range towards higher frequencies. For a very large detun- corresponds to the free induction decay observed in a ing the effective frequency becomes similar to the detun- Ramsey experiment (Fig.9(b)). The associated dephas- ∗ ing ωeff ∆ω, and the drive field only enters as a scaling ing time T2 is given by factor for≈ the spectral density. Detuned drive fields have been used to map the 1/f noise spectrum of transmon ∗ −1 1 2 1 2 (T ) = γ SV (ω0) + γ SV (0) , (101) qubits up to the GHz range (Slichter et al., 2012). 2 4 ⊥ 2 ||

where SV|| = SVz (see also Eq. (89)). The transverse SV⊥ term in Eq. (101) involves a “bit ﬂip” and the parallel VIII. DYNAMIC RANGE AND ADAPTIVE SENSING

SV|| term involves a “phase flip”. Because a phase flip does not require energy, the spectral density is probed “Adaptive sensing” refers to a class of techniques ad- dressing the intrinsic problem of limited dynamic range at zero frequency. Since SV (ω) is often dominated by in quantum sensing: The basic quantum sensing protocol low-frequency noise, SV (0) is typically much larger than || cannot simultaneously achieve high sensitivity and mea- SV⊥ (ω0) and the high-frequency contribution can often be neglected. Note that Eq. (101) is exact only when sure signals over a large amplitude range. The origin of this problem lies in the limited range of the spectrum SV|| (ω) is flat up to ω π/t. ∗ ∼ values for the probability p, which must fall between 0 T2 relaxometry can be extended to include dephasing under dynamical decoupling sequences. The relevant re- and 1. For the example of a Ramsey measurement, p oscillates with the signal amplitude V and phase wrapping laxation time is then usually denoted by T2 rather than ∗ occurs once γV t exceeds π/2, where t is the sensing T2 . Dephasing under dynamical decoupling is more rig- ± orously described by using filter functions (see Section time. Given a measured transition probability p, there is VII.B.2). an infinite number of possible signal amplitudes V that can correspond to this value of p (see top row of Fig. 11). A unique assignment hence requires a priori knowledge 4. Dressed state methods – that V lies within π/(2γt), or within half a Ramsey fringe, of an expected± signal amplitude. This defines a Relaxation can also be analyzed in the presence of a maximum allowed signal range, resonant drive field. This method is known as “spin lock- π Vmax = . (104) ing” in magnetic resonance (Slichter, 1996). Due to the γt presence of the resonant field the degeneracy between |±i is lifted and the states are separated by the energy ~ω1, The sensitivity of the measurement, on the other hand, where ω1 ω0 is the Rabi frequency of the drive field. A is proportional to the slope of the Ramsey fringe and 29

∗ p(V) reaches its optimum when t T2 . The smallest de- /t ≈ p ∗ tectable signal is approximately Vmin 2/(γC T T ),   2 t where T is the total measurement time≈ and C the read- M out efficiency parameter (see Eqs. 43 and 47). The dy- least . significant namic range is then given by the maximum allowed signal . compatible V digit (LSD) divided by the minimum detectable signal, . init readout . Vmax πC√T t1 . DR = = √T. (105) p ∗ . Vmin 2 T2 ∝ t Hence, the basic measurement protocol can be applied 0 most significant only to small changes of a quantity around a fixed known V digit (MSD) value, frequently zero. The protocol does not apply to the problem of determining the value of a large and a priori maximum signal Vmax unknown quantity. Moreover, the dynamic range only minimum signal Vmin improves with the square root of the total measurement time T . FIG. 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 Vmax and provides a A. Phase estimation protocols 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 cor- Interestingly, a family of advanced sensing techniques respond to many possible signal values, the coarse estimates can efficiently address this problem and achieve a dy- allow one to extract a unique signal value V . namic range that scales close to DR T . This scaling is sometimes referred to as another instance∝ of the Heisen- berg limit, because it can be regarded as a 1/T scaling of 1. Quantum phase estimation sensitivity at a fixed Vmax. The central idea is to combine measurements with different sensing times t such that the All three protocols can be considered variations of the least sensitive measurement with the highest Vmax yields phase estimation scheme depicted in Fig. 12(a). The a coarse estimate of the quantity of interest, which is sub- scheme was originally put forward by Shor, 1994, in the sequently refined by more sensitive measurements (Fig. seminal proposal of a quantum algorithm for prime fac- 11). torization and has been interpreted by Kitaev, 1995, as In the following we discuss protocols that use ex- a phase estimation algorithm. m ponentially growing sensing times tm = 2 t0, where The original formulation applies to the problem of find- 2πiφ m = 0, 1,...,M and t0 is the smallest time element (see ing the phase φ of the eigenvalue e of a unitary opera- Fig. 11). Although other scheduling is possible, this tor Uˆ, given a corresponding eigenvector ψ . This prob- | i choice allows for an intuitive interpretation: subsequent lem can be generalized to estimating the phase shift φ im- measurements estimate subsequent digits of a binary ex- parted by passage through an interferometer or exposure pansion of the signal. The maximum allowed signal is to an external field. The algorithm employs a register of then set by the shortest sensing time, Vmax = π/(γt0), N auxiliary qubits (N = 3 in Fig. 12) and prepares them while the smallest detectable signal is determined by the into a digital representation φ = φ1 φ2 ... φM of a PM| i | −mi | i | i longest sensing time, Vmin 2/(γC√tM T ). Because binary expansion of φ = φm2 by a sequence of ≈ m=1 T tM due to the exponentially growing interrogation three processing steps: times,∝ the dynamic range of the improved protocol scales as 1. State preparation: All qubits are prepared in a coherent superposition state + = ( 0 + 1 )/√2 by an initial Hadamard gate.| Thei resulting| i | i state of √tM T DR T. (106) the full register can then be written as ∝ t0 ∝ 2M −1 1 X This scaling is obvious from an order-of-magnitude esti- j (107) mate: adding an additional measurement step increases √ M | iM 2 j=0 both precision and measurement duration t by a factor of two, such that precision scales linearly with total ac- where j denotes the register state in binary ex- | iM quisition time T . We will now discuss three specific im- pansion 0 M = 00 ... 0 , 1 M = 00 ... 1 , 2 M = plementations of this idea. 00 ... 10| ,i etc. | i | i | i | i | i 30

(a) Quantum phase estimation (c) Non-adaptive (Bayesian) H 0 MSD Φ -1 H 0 R( 1) H 0 QFT X MSD

H 0 LSD H 0 R(Φ1') X 4 2 Φ ψ U U U H 0 R( 2) Bayesian X estimator

H 0 R(Φ2') (b) Adaptive X

H 0 R(Φ3) π π H 0 R( /4) R( /2) MSD X LSD X H 0 R(Φ3') H 0 R(π/2) X X LSD 4 2 H 0 ψ U U U X ψ U4 U2 U

FIG. 12 Phase estimation algorithms. (a) Quantum phase estimation by the inverse Quantum Fourier transform, as it is employed in prime factorization algorithms (Kitaev, 1995; Shor, 1994). (b) Adaptive phase estimation. Here, the quantum Fourier transform is replaced by measurement and classical feedback. Bits are measured in ascending order, substracting the lower digits from measurements of higher digits by phase gates that are controlled by previous measurement results. (C) Non-adaptive 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(Φ) a Z-rotation by the angle Φ, and U the propagator for one time element t0. The box labeled by “x” represents a readout.

2. Phase encoding: The phase of each qubit is A measurement of the register directly yields a dig- tagged with a multiple of the unknown phase ital representation of the phase φ. To provide an φ. Specifically, qubit m is placed in state estimate of φ with 2−M accuracy, 2M applications m ( 0 + e2πi2 φ 1 )/√2. Technically, this can be of the phase shift Uˆ are required. Hence, the al- implemented| i by| i exploiting the back-action of a gorithm scales linearly with the number of applica- m controlled-Uˆ 2 -gate that is acting on the eigenvec- tions of Uˆ which in turn is proportional to the total tor ψ conditional on the state of qubit m. Since ψ measurement time T . is an eigenvector of Uˆ j for arbitrary j, this action transforms the joint qubit-eigenvector state as Quantum phase estimation is the core component of Shor’s algorithm, where it is used to compute discrete 1 √ ( 0 + 1 ) ψ logarithms with polynomial time effort (Shor, 1994). 2 | i | i ⊗ | i m √1 ( 0 + e2πi2 φ 1 ) ψ (108) → 2 | i | i ⊗ | i Here, the back-action on the control qubit m cre- 2. Adaptive phase estimation ates the required phase tag. The state of the full register evolves to While quantum phase estimation based on the QFT can be performed with polynomial time effort, the al- 2M −1 1 X M gorithm requires two-qubit gates, which are difficult to e2πiφj/2 j (109) √2M | i implement experimentally, and the creation of fragile en- j=0 tangled states. This limitation can be circumvented by In quantum sensing, phase tagging by back-action an adaptive measurement scheme that reads the qubits is replaced by the exposure of each qubit to an ex- sequentially, feeding back the classical measurement re- m ternal field for a time 2 t0 (or passage through an sult into the quantum circuit (Griffiths and Niu, 1996). m interferometer of length 2 l0). The scheme is illustrated in Fig. 12(b). The key idea of adaptive phase estimation is to first 3. Quantum Fourier Transform: In a last step, an in- measure the least significant bit of φ, represented by the verse quantum Fourier transform (QFT) (Nielsen lowest qubit in Fig. 12(b). In the measurement of the and Chuang, 2000) is applied to the qubits. This next significant bit, this value is subtracted from the ap- algorithm can be implemented with polynomial ef- plied phase. The subtraction can be implemented by fort (i.e., using (M 2) control gates). The QFT classical unitary rotations conditioned on the measure- recovers the phaseO φ from the Fourier series (109) ment result, for example by controlled R(π/j) gates as and places the register in state shown in Fig. 12(b). This procedure is then repeated in φ = φ1 φ2 ... φM . (110) ascending order of the bits. The QFT is thus replaced by | i | i | i | i 31 measurement and classical feedback, which can be per- Here, too, acquisition time scales with the significance of formed using a single qubit sensor. the bit measured to achieve the Heisenberg limit. In practical implementations (Higgins et al., 2007), the measurement of each digit is repeated multiple times or performed on multiple parallel qubits. This is possible 4. Comparison of phase estimation protocols because the controlled-U gate does not change the eigenvector ψ, so that it can be re-used as often as required. All of the above variants of phase estimation achieve The Heisenberg limit can only be reached if the number a DR T scaling of the dynamic range. They differ, of resources (qubits or repetitions) spent on each bit are however,∝ by a constant offset. Adaptive estimation is carefully optimized (Berry et al., 2009; Cappellaro, 2012; slower than quantum phase estimation by the QFT since Said et al., 2011). Clearly, most resources should be al- it trades spatial resources (entanglement) into temporal located to the most significant bit, because errors at this resources (measurement time). Bayesian estimation in stage are most detrimental to sensitivity. The implemen- turn is slower than adaptive estimation due to additional tation by Bonato, C. et al., 2016, for example, scaled the redundant measurements. number of resources Nm linearly according to Experimentally, Bayesian estimation is usually simple to implement because no real-time feedback is needed Nm = G + F (M 1 m). (111) − − and the phase estimation can be performed a posteriori. with typical values of G = 5 and F = 2. Adaptive estimation is technically more demanding since real-time feedback is involved, which often requires dedicated hardware (such as field-programmable gate arrays 3. Non-adaptive phase estimation or a central processing unit) for the fast decision making. Quantum phase estimation by the QFT, finally, requires Efficient quantum phase estimation can also be imple- many entangled qubits. mented without adaptive feedback, with the advantage of technical simplicity (Higgins et al., 2009). A set of measurements xj j=1...N (where N > M) is used to B. Experimental realizations separately determine{ } each unitary phase 2mφ with a set of fixed, classical phase shifts before each readout. This The proposals of Shor (Shor, 1994), Kitaev (Kitaev, set of measurements still contains all the information re- 1995) and Griffiths (Griffiths and Niu, 1996) were fol- quired to extract φ, which can be motivated by the fol- lowed by a decade where research towards Heisenberg- lowing arguments: given a redundant set of phases, a limited measurements has focused mostly on the use of post-processing algorithm can mimic the adaptive algo- entangled states, such as the N00N state (see Section rithm by postselecting those results that have been mea- IX). These states promise Heisenberg scaling in the spa- sured using the phase most closely resembling the cor- tial dimension (number of qubits) rather than time (Gio- rect adaptive choice. From a spectroscopic point of view, vannetti et al., 2004, 2006; Lee et al., 2002) and have measurements with different phases correspond to Ram- been studied extensively for both spin qubits (Bollinger sey fringes with different quadratures. Hence, at least et al., 1996; Jones et al., 2009; Leibfried et al., 2004, 2005) one qubit of every digit will perform its measurement and photons (Edamatsu et al., 2002; Fonseca et al., 1999; on the slope of a Ramsey fringe, allowing for a precise Mitchell et al., 2004; Nagata et al., 2007; Walther et al., m measurement of 2 φ regardless of its value. 2004; Xiang, G. Y. et al., 2011). The phase φ can be recovered by Bayesian estimation. Heisenberg scaling in the temporal dimension has Every measurement xj = 1 provides information about ± shifted into focus with an experiment published in 2007, φ, which is described by the a posteriori probability where adaptive phase estimation was employed in a single-photon interferometer (Higgins et al., 2007). The p(φ xj), (112) | experiment has subsequently been extended to a non- the probability that the observed outcome xj stems from adaptive version (Higgins et al., 2009). Shortly after, a phase φ. This probability is related to the inverse con- both variants have been translated into protocols for ditional probability p(xl φ) – the excitation probability spin-based quantum sensing (Said et al., 2011). Mean- | describing Ramsey fringes – by Bayes’ theorem. The joint while, high-dynamic-range protocols have been demon- probability distribution of all measurements is obtained strated on NV centers in diamond using both non- from the product adaptive implementations (Nusran et al., 2012; Waldherr Y et al., 2012) and an adaptive protocol based on quantum p(φ) p(φ xj) , (113) ∝ | feedback (Bonato, C. et al., 2016). As a final remark, j we note that a similar performance – 1/T scaling and an from which the most likely value of φ is picked as the increased dynamic range – may be achieved by weak mea- final result (Nusran et al., 2012; Waldherr et al., 2012). surement protocols, which continuously track the evolu- 32 tion of the phase over the sensing sequence (Kohlhaas 0 H H 0 et al., 2015; Shiga and Takeuchi, 2012). Weak measure- | | 0 H H 0 ments have been more generally proposed to enhance | | sensing protocols (), but their ultimate usefulness is still ...... entangling entangling 0 H free evolution H 0 under debate (). | | free evolution

FIG. 13 Left: Ramsey scheme. Right: entangled Ramsey IX. ENSEMBLE SENSING scheme for Heisenberg-limited sensitivity

Up to this point, we have mainly focused on single the sensitivity of each qubit. The sensitivity is then ex- qubit sensors. In the following two sections, quantum pressed per unit volume ( meters−3/2). The maximum sensors consisting of more than one qubit will be dis- density of qubits is limited∝ by internal interactions be- cussed. The use of multiple qubits opens up many ad- tween the qubits. Optimal densities have been calculated ditional possibilities that cannot be implemented on a both for atomic vapor magnetometers (Budker and Ro- single qubit sensor. malis, 2007) and ensembles of NV centers (Taylor et al., This section considers ensemble sensors where many 2008; Wolf et al., 2015). (usually identical) qubits are operated in parallel. Apart from an obvious gain in readout sensitivity, multiple qubits allow for the implementation of second-generation B. Heisenberg limit quantum techniques, including entanglement and state squeezing, which provide a true “quantum” advan- The standard quantum limit can be overcome by us- tage that cannot be realized with classical sensors. ing quantum-enhanced sensing strategies to reach a more Entanglement-enhanced sensing has been pioneered with fundamental limit where the uncertainty σp (Eq. ( 25)) atomic systems, especially atomic clocks (Giovannetti scales as 1/M. This limit is also known as the Heisen- et al., 2004; Leibfried et al., 2004). In parallel, state berg limit. Achieving the Heisenberg limit requires reduc- squeezing is routinely applied in optical systems, such as ing the variance of a chosen quantum observable at the optical interferometers (Ligo Collaboration”, 2011). expenses of the uncertainty of a conjugated observable. This in turn requires preparing the quantum sensors in an entangled state. In particular, squeezed states (Caves, A. Ensemble sensing 1981; Kitagawa and Ueda, 1993; Wineland et al., 1992) have been proposed early on to achieve the Heisenberg Before discussing entanglement-enhanced sensing tech- limit and thanks to experimental advances have recently niques, we briefly consider the simple parallel operation shown exceptional sensitivity (Hosten et al., 2016a). of M identical single-qubit quantum sensors. This imple- The fundamental limits of sensitivity (quantum mentation is used, for example, in atomic vapor magne- metrology) and strategies to achieve them have been dis- tometers (Budker and Romalis, 2007) or solid-state spin cussed in many reviews (Giovannetti et al., 2004, 2006, ensembles (Wolf et al., 2015). The use of M qubits ac- 2011; Paris, 2009; Wiseman and Milburn, 2009) and they celerates the measurement by a factor of M, because the will not be the subject of our review. In the following, basic quantum sensing cycle (Steps 1-5 of the protocol, we will focus on the most important states and methods Fig.2) can now be operated in parallel rather than se- that have been used for entanglement-enhanced sensing. quentially. Equivalently, M parallel qubits can improve the sensitivity by √M per unit time. This scaling is equivalent to the situation where M C. Entangled states classical sensors are operated in parallel. The scaling 1. GHZ and N00N states can be seen as arising from the projection noise associated with measuring the quantum system, where it is To understand the benefits that an entangled state can often called the Standard Quantum Limit (SQL) (Bra- bring to quantum sensing, the simplest example is given ginskii and Vorontsov, 1975; Giovannetti et al., 2004) or by Greenberger-Horne-Zeilinger (GHZ) states. The sens- shot noise limit. In practice, it is sometimes difficult to ing scheme is similar to a Ramsey protocol, however, if M achieve a √M scaling because instrumental stability is qubit probes are available, the preparation and readout more critical for ensemble sensors (Wolf et al., 2015). pulses are replaced by entangling operations (Fig. 13). For ensemble sensors such as atomic vapor magnetome- We can thus replace the procedure in Sec. IV.C with ters or spin arrays, the quantity of interest is more likely the following: the number density of qubits, rather than the absolute number of qubits M. That is, how many qubits can 1. The quantum sensors are initialized into 0 0 be packed into a certain volume without deteriorating ... 0 0 00 ... 0 . | i ⊗ | i ⊗ ⊗ | i ⊗ | i ≡ | i 33

2. Using entangling gates, the quantum sensors are were first introduced by Yurke, 1986, in the context brought into the GHZ state ψ0 = ( 00 ... 0 + of neutron Mach-Zender interferometry as the fermionic 11 ... 1 )/√2. | i | i “response” to the squeezed states proposed by Caves, | i 1981, for Heisenberg metrology. Using an M-particle in- 3. The superposition state evolves under the Hamilto- terferometer, one can prepare an entangled Fock state, nian Hˆ0 for a time t. The superposition state picks √ up an enhanced phase φ = Mω0t, and the state ψN00N = ( M a 0 b + 0 a M b)/ 2 , (118) after the evolution is | i | i | i | i | i where N indicates the N-particle Fock state in spatial 1 | ia ψ(t) = ( 00 ... 0 + eiMω0t 11 ... 1 ) , (114) mode a. Already for small M, it is possible to show | i √2 | i | i sensitivity beyond the standard quantum limit (Kuzmich and Mandel, 1998). If the phase is applied only to mode 4. Using the inverse entangling gates, the state ψ(t) a of the interferometer, the phase accumulated is then is converted back to an observable state, e.g. |α =i iMφa 1 iMω0t 1 iMω0t | i √ [ (e + 1) 01 + (e 1) 11 ] 0 ... 0 . ψN00N = (e M a 0 b+ 0 a M b)/ 2 , (119) 2 | i 2 − | i | i2,M | i | i | i | i | i 5,6. The final state is read out (only the first quantum that is, M times larger than for a one-photon state. Ex- probe needs to be measured in the case above). The perimental progress has allowed to reach “high N00N” transition probability is (with M > 2) states (Mitchell et al., 2004; Monz et al., 2011; Walther et al., 2004) by using strong nonlinearities p = 1 0 α 2 or measurement and feed-forward. They have been used − |h | i| 1 2 not only for sensing but also for enhanced lithography = [1 cos(Mω0t)] = sin (Mω0t/2). (115) 2 − (Boto et al., 2000). Still, “N00N” states are very fragile (Bohmann et al., 2015) and they are afflicted by a small Comparing this result with what obtained in Sec. IV.C, dynamic range. we see that the oscillation frequency of the signal is enhanced by a factor M by preparing a GHZ state, while the shot noise is unchanged, since we still measure only 2. Squeezing one qubit. This allows using an M-times shorter interrogation time or achieving an improvement of the sen- Squeezed states are promising for quantum-limited sitivity (calculated from the QCRB) by a factor √M. sensing as they can reach sensitivity beyond the stan- While for M uncorrelated quantum probes the QCRB of dard quantum limit. Squeezed states of light have been Eq. (59) becomes introduced by Caves, 1981, as a mean to reduce noise in interferometry experiments. One of the most impressive 1 eχ application of squeezed states of light (Ligo Collabora- ∆VN,M = = , (116) γ√N γt√MN tion”, 2011; Schnabel et al., 2010; Walls, 1983) has been F the sensitivity enhancement of the LIGO gravitational for the GHZ state, the Fisher information reflects the wave detector (Collaboration, 2013), obtained by inject- state entanglement to give ing vacuum squeezed states in the interferometer. 1 eχ Squeezing has also been extended to fermionic degrees ∆VN,GHZ = = (117) of freedom (spin squeezing, Kitagawa and Ueda, 1993) γ√N GHZ γMt√N F to reduce the uncertainty in spectroscopy measurements Heisenberg-limited sensitivity with a GHZ state was of ensemble of qubit probes. The Heisenberg uncertainty demonstrated using three entangled Be ions (Leibfried principle bounds the minimum error achievable in the et al., 2004). measurement of two conjugate variables. While for typ- Unfortunately, the √M advantage in sensitivity is usu- ical states the uncertainty in the two observables is on ally compensated by the GHZ state’s increased decoher- the same order, it is possible to redistribute the fluc- ence rate (Huelga et al., 1997), which is an issue common tuations in the two conjugate observables. Squeezed to most entangled states. Assuming, for example, that states are then characterized by a reduced uncertainty each probe is subjected to uncorrelated dephasing noise, in one observable at the expense of another observable. the rate of decoherence of the GHZ state is M time faster Thus, these states can help improving the sensitivity of than for a product state. Then, the interrogation time quantum interferometry, as demonstrated by Wineland also needs to be reduced by a factor M and no net advan- et al., 1992,Wineland et al., 1994. Similar to GHZ and tage in sensitivity can be obtained. This has led to the N00N states, a key ingredient to this sensitivity en- quest for different entangled states that could be more hancement is entanglement (Sørensen and Mølmer, 2001; resilient to decoherence. Wang and Sanders, 2003). However, the description of Similar to GHZ states, N00N states have been con- squeezed states is simplified by the use of a single collec- ceived to improve interferometry (Lee et al., 2002). They tive angular-momentum variable. 34

The degree of spin squeezing can be measured by 3. Parity measurements several parameters. For example, from the commuta- tion relationship for the collective angular momentum, A challenge in achieving the full potential of multi- ∆Jα∆Jβ Jγ , one can naturally define a squeezing qubit enhanced metrology is the widespread inefficiency parameter≥ |h i| of quantum state readout. Metrology schemes often require single qubit readout or the measurement of com- q ξ = ∆Jα/ Jγ /2 , (120) plex, many-body observables. In both cases, coupling of |h i the quantum system to the detection apparatus is inef- with ξ < 1 for squeezed states. To quantify the advan- ficient, often because strong coupling would destroy the tage of squeezed states in sensing, it is advantageous to very quantum state used in the metrology task. directly relate squeezing to the improved sensitivity. This To reveal the properties of entangled states and to take may be done by considering the ratio of the uncertainties advantage of their enhanced sensitivities, an efficient ob- on the acquired phase for the squeezed state ∆φsq and servable is the parity of the state. The parity observable for the uncorrelated state ∆φ0 in, e.g., a Ramsey mea- was first introduced in the context of ion qubits (Bollinger surement. The metrology squeezing parameter, proposed et al., 1996; Leibfried et al., 2004) and it referred to the by Wineland et al., 1992, is then excited or ground state populations of the ions. The parity has become widely adopted for the readout of

∆φ sq √N∆Jy(0) N00N states, where the parity measures the even/odd ξR = | | = . (121) number of photons in a state (Gerry and Mimih, 2010). ∆φ 0 Jz(0) | | |h i| Photon parity measurements are as well used in quan- Early demonstrations of spin squeezing were obtained tum metrology with squeezed states. While the simplest by entangling trapped ions via their shared motional method for parity detection would be via single pho- modes (Meyer et al., 2001), using repulsive interactions ton counting, and recent advances in superconducting in a Bose-Einstein condensate (Esteve et al., 2008), or single photon detectors approach the required efficiency partial projection by measurement (Appel et al., 2009). (Natarajan et al., 2012), photon numbers could also be More recently, atom-light interactions in high-quality measured with single-photon resolution using quantum cavities have enabled squeezing of large ensembles atoms non-demolition (QND) techniques (Imoto et al., 1985) (Bohnet et al., 2014; Cox et al., 2016; Gross et al., that exploit nonlinear optical interactions. Until recently, 2010; Hosten et al., 2016a; Leroux et al., 2010a; Louchet- parity detection for atomic ensembles containing more Chauvet et al., 2010; Schleier-Smith et al., 2010a) that than a few particles was out of reach. However, recent can perform as atomic clocks beyond the standard quan- breakthroughs in spatially resolved (Bakr et al., 2009) tum limit. Spin squeezing can be also implemented in and cavity-based atom detection (Hosten et al., 2016b; qubit systems (Auccaise et al., 2015; Bennett et al., 2013; Schleier-Smith et al., 2010b) enabled atom counting in Cappellaro and Lukin, 2009; Sinha et al., 2003) following mesoscopic ensembles containing M & 100 atoms. the original proposal by Kitagawa and Ueda, 1993. In this context, a simple quantum sensing scheme, following the procedure in Sec. IV.C, could be constructed 4. Other types of entanglement by replacing step 2 with the preparation of a squeezed state, so that ψ0 is a squeezed state. The state is pre- The key difficulty with using entangled states for sens- pared by evolving| i a reference (ground) state 0 under a ing is that they are less robust against noise. Thus, the | i 2 squeezing Hamiltonanian, such as the one-axis H1 = χJz advantage in sensitivity is compensated by a concurrent 2 2 or two-axis H1 = χ(Jx Jy ) squeezing Hamiltonians. reduction in coherence time. In particular, it has been Then, during the free evolution− (step 3) an enhanced demonstrated that for frequency estimation, any non- phase can be acquired, similar to what happens for en- zero value of uncorrelated dephasing noise cancels the tangled states. The most common sensing protocols advantage of the maximally entangled state over a classi- with squeezed states forgo step 4, and directly measure cally correlated state (Huelga et al., 1997). An analogous the population difference between the state 0 and 1 . result can be proven for magnetometry (Auzinsh et al., However, imperfections in this measurement| limitsi | thei 2004). sensitivity, since achieving the Heisenberg limit requires Despite this limitation, non-maximally entangled single-particle state detection. While this is difficult to states can provide an advantage over product states obtain for large qubit numbers, recent advances show (Shaji and Caves, 2007; Ulam-Orgikh and Kitagawa, great promise (Hume et al., 2013; Zhang et al., 2012) 2001). Optimal states for quantum interferometry in the (see also next section on alternative detection methods). presence of photon loss can, for example, be found by A different strategy is to follow more closely the sensing numerical searches (Huver et al., 2008; Lee et al., 2009). protocol for entangled states, and refocus the squeezing Single-mode states have also been considered as a more (reintroducing step 4) before readout (Davis et al., 2016). robust alternative to two-mode states. Examples include 35 pure Gaussian states in the presence of phase diffusion ble clock transition. To overcome this dilemma, quantum (Genoni et al., 2011), mixed Gaussian states in the pres- logic spectroscopy has been introduced (Schmidt et al., ence of loss (Aspachs et al., 2009) and single-mode vari- 2005). The key idea is to employ two ion species: a clock ants of two-mode states (Maccone and Cillis, 2009). ion that presents a stable clock transition (and represents Other strategies include the creation of states that the quantum sensor), and a logic ion (acting as auxil- are more robust to the particular noise the system is iary qubit) that is used to prepare, via a cooling tran- subjected to (Cappellaro et al., 2012; Goldstein et al., sition, and readout the clock ion. The resulting “quan- 2011) or the use of entangled ancillary qubits that are tum logic” ion clock can thus take advantage of the most not quantum sensors themselves (Demkowicz-Dobrzanski stable ion clock transitions, even when the ion cannot and Maccone, 2014; Dür et al., 2014; Huang et al., 2016; be efficiently read out, thus achieving impressive clock Kessler et al., 2014). These are considered in the next performance (Hume et al., 2007; Rosenband et al., 2008, section (SectionX). 2007). Advanced quantum logic clocks may incorporate multi-ion logic (Tan et al., 2015) and use quantum algorithms for more efficient readout (Schulte et al., 2016). X. SENSING ASSISTED BY AUXILIARY QUBITS

In the previous section we considered potential im- B. Storage and retrieval provements in sensitivity derived from the availability of multiple quantum systems operated in parallel. A dif- The quantum state ψ can be stored and retrieved in ferent scenario arises when only a small number of addi- the auxiliary qubit. Storage| i can be achieved by a SWAP tional quantum systems is available, or when the addi- gate (or more simply two consecutive c-NOT gates) on tional quantum systems do not directly interact with the the sensing and auxiliary qubits, respectively (Rosskopf signal to be measured. Even in this situation, however, et al., 2016). Retrieval uses the same two c-NOT gates “auxiliary qubits” can aid in the sensing task. Although in reverse order. For the example of an electron-nuclear auxiliary qubits – or more generally, additional quan- qubit pair, c-NOT gates have been implemented both tum degrees of freedom – cannot improve the sensitivity by selective pulses (Pfender et al., 2016; Rosskopf et al., beyond the quantum metrology limits, they can aid in 2016) and using coherent rotations (Zaiser et al., 2016). reaching these limits, for example when it is experimen- Several useful applications of storage and retrieval have tally difficult to optimally initalize or readout the quan- been demonstrated. A first example includes correlation tum state. Auxiliary qubits may be used to increase the spectroscopy, where two sensing periods are interrupted effective coherence or memory time of a quantum sensor, by a waiting time t1 (Laraoui and Meriles, 2013; Rosskopf either by operation as a quantum memory or by enabling et al., 2016; Zaiser et al., 2016). A second example in- quantum error correction. cludes a repetitive (quantum non-demolition) readout of In the following we discuss some of the schemes the final qubit state, which can be used to reduce the that have been proposed or implemented with auxiliary classical readout noise (Jiang et al., 2009). qubits.

C. Quantum error correction A. Quantum logic clock Quantum error correction (Nielsen and Chuang, 2000; Clocks based on optical transitions of an ion kept in a Shor, 1995) aims at counteracting errors during quantum high-frequency trap exhibit significantly improved accu- computation by encoding the qubit information into re- racy over more common atomic clocks. Single-ion atomic dundant degrees of freedom. The logical qubit is thus en- clocks currently detain the record for the most accurate coded in a subspace of the total Hilbert space (the code) optical clocks, with uncertainties of 2.1 1018 for a 87Sr such that each error considered maps the code to an or- ensemble clock (Nicholson et al., 2015) and× 3.2 1018 for thogonal subspace, allowing detection and correction of a single a single trapped 171Yb (Huntemann et al.× , 2016). the error. Compared to dynamical decoupling schemes, The remaining limitations on optical clocks are re- qubit protection covers the entire noise spectrum and is lated to their long-term stability and isolation from ex- not limited to low-frequency noise. On the other hand, ternal perturbations such as electromagnetic interference. qubit protection can typically only be applied against er- These limitations are even more critical when such clocks rors that are orthogonal to the signal, because otherwise are based on a string of ions in a trap, because of the as- the signal itself would be “corrected”. In particular, for sociated unavoidable electric field gradients. Only some vector fields, quantum error correction can be used to ion species, with no quadrupolar moment, can then be protect against noise in one spatial direction while leav- used, but not all of them present a suitable transition for ing the sensor responsive to signals in the orthogonal spa- laser cooling and state detection beside the desired, sta- tial direction. Thus, quantum error correction suppresses 36 noise according to spatial symmetry, and not according explored. to frequency. The simplest code is the 3-qubit repetition code, which corrects against one-axis noise with depth one (that is, it XI. OUTLOOK can correct up to one error acting on one qubit). For example, the code 0 L = 000 and 1 L = 111 can correct Despite its rich history in atomic spectroscopy and against a single| qubiti flip| error.i | Notei that| i (Dür et al., classical interferometry, quantum sensing is an excit- 2014; Ozeri, 2013) equal superpositions of these two log- ingly new and refreshing development advancing rapidly ical basis states are also optimal to achieve Heisenberg- along the sidelines of mainstream quantum engineer- limited sensitivity in estimating a global phase (Bollinger ing research. Like no other field, quantum sensing has et al., 1996; Leibfried et al., 2004). While this seems been uniting diverse efforts in science and technology to indicate that QEC codes could be extremely useful to create fundamental new opportunities and applica- for metrology, the method is hampered by the fact that tions in metrology. Inputs have been coming from tra- QEC often cannot discriminate between signal and noise. ditional high-resolution optical and magnetic resonance In particular, if the signal to be detected couples to the spectroscopy, to the mathematical concepts of parameter sensor in a similar way as the noise, the QEC code also estimation, to quantum manipulation and entanglement eliminates the effect of the signal. This compromise be- techniques borrowed from quantum information science. tween error suppression and preservation of signal sensi- Over the last decade, and especially in the last few years, tivity is common to other error correction methods. For a comprehensive toolset has been established that can example, in dynamical decoupling schemes, a separation be applied to any type of quantum sensor. In particu- in the frequency of noise and signal is required. Since lar, these allow operation of the sensor over a wide signal QEC works independently of noise frequency, distinct op- frequency range, can be adjusted to maximize sensitivity erators for the signal and noise interactions are required. and dynamic range, and allow discrimination of differ- This imposes an additional condition on a QEC code: the ent types of signals by symmetry or vector orientation. quantum Fisher information (Giovannetti et al., 2011; Lu While many experiments so far made use of single qubit et al., 2015) in the code subspace must be non-zero. sensors, strategies to implement entangled multi-qubit Several situations for QEC-enhanced sensing have been sensors with enhanced capabilities and higher sensitivity considered. One possible scenario is to protect the quan- are just beginning to be explored. tum sensor against a certain type of noise (e.g., single One of the biggest attractions of quantum sensors is qubit, bit-flip or transverse noise), while measuring the their immediate potential for practical applications. This interaction between qubits (Dür et al., 2014; Herrera- potential is partially due to the immense range of con- Mart´ı et al., 2015). More generally, one can measure ceived sensor implementations, starting with atomic and a many-body Hamiltonian term with a strength propor- solid-state spin systems and continuing to electronic and tional to the signal to be estimated (Herrera-Mart´ı et al., vibrational degrees of freedom from the atomic to the 2015). Since this can typically only be achieved in a per- macroscale. In fact, quantum sensors based on SQUID turbative way, this scheme still leads to a compromise magnetometers and atomic vapors are already in every- between noise suppression and effective signal strength. day use, and have installed themselves as the most sen- The simplest scheme for QEC is to use a single sitive magnetic field detectors currently available. Like- good qubit (unaffected by noise) to protect the sensor wise, atomic clocks have become the ultimate standard qubit (Arrad et al., 2014; Hirose and Cappellaro, 2016; in time keeping and frequency generation. Many other Kessler et al., 2014; Ticozzi and Viola, 2006). In this and more recent implementations of quantum sensors are scheme, which has recently been implemented with NV just starting to make their appearance in many different centers (Unden et al., 2016), the qubit sensor detects niches. Notably, NV centers in diamond have started a signal along one axis (e.g., a phase) while being pro- conquering many applications in nanoscale imaging due tected against noise along a different axis (e.g., against to their small size. bit flip). Because the “good” ancillary qubit can only What lies ahead in quantum sensing? On the one hand, protect against one error event (or, equivalently, suppress the range of applications will continue to expand as new the error probability for continuous error), the signal ac- types and more mature sensor implementations become quisition must be periodically interrupted to perform a available. Taking the impact quantum magnetometers corrective step. Since the noise strength is typically much and atomic clocks had in their particular discipline, it can weaker than the noise fluctuation rate, the correction be expected that quantum sensors will penetrate much of steps can be performed at a much slower rate compared the 21st century technology and find their way into both to dynamical decoupling. Beyond single qubits, QEC has high-end and consumer devices. Advances with quan- also been applied to N00N states (Bergmann and van tum sensors will be strongly driven by the availability Loock, 2016). These recent results hint at the potential of “better” materials and more precise control, allowing of QEC for sensing which has just about begun to being their operation with longer coherence times, more effi- 37 cient readout, and thus higher sensitivity. REFERENCES In parallel, quantum sensing will profit from efforts in quantum technology, especially quantum computing, Abragam, A. (1961), The Principles of Nuclear Magnetism, where many of the fundamental concepts have been de- International series of monographs on physics (Clarendon veloped, such as dynamical decoupling protocols, quan- Press). Acosta, V. M., E. Bauch, M. P. Ledbetter, C. Santori, K.- tum storage and quantum error correction, as well as M. C. Fu, P. E. Barclay, R. G. Beausoleil, H. Linget, quantum phase estimation. Vice versa, quantum sensing J. F. Roch, F. Treussart, S. Chemerisov, W. Gawlik, and has become an important resource for quantum technolo- D. Budker (2009), Phys. Rev. B 80 (11), 115202. gies as it provides much insight into the “environment” Aiello, C. D., M. Hirose, and P. Cappellaro (2013), Nat. of qubits, especially through decoherence spectroscopy. Commun. 4, 1419. A better understanding of decoherence in a particular Aigner, S., L. D. Pietra, Y. Japha, O. Entin-Wohlman, implementation of a quantum system can help the adop- T. David, R. Salem, R. Folman, and J. Schmiedmayer (2008), Science 319 (5867), 1226. tion of strategies to protect the qubit, and guide the en- Ajoy, A., and P. Cappellaro (2012), Phys. Rev. A 85, 042305. gineering and materials development. The border region Ajoy, A., Y.-X. Liu, K. Saha, L. Marseglia, J.-C. Jaskula, between quantum sensing and quantum simulation, in U. Bissbort, and P. Cappellaro (2017), Proc. Nat. Acad. addition, is becoming a fertile playground for emulat- Sc. 10.1073/pnas.1610835114. ing and detecting many-body physics phenomena. Over- Albrecht, A., and M. B. Plenio (2015), Phys. Rev. A 92, all, quantum sensing has the potential to fundamentally 022340. transform our measurement capabilities, enabling higher Allan, D. (1966), Proceedings of the IEEE 54 (2), 221. Almog, I., Y. Sagi, G. Gordon, G. Bensky, G. Kurizki, and sensitivity and precision, new measurement types, and N. Davidson (2011), Journal of Physics B: Atomic, Molec- covering atomic up to macroscopic length scales. ular and Optical Physics 44 (15), 154006. Alvarez,´ G. A., and D. Suter (2011), Phys. Rev. Lett. 107, 230501. ACKNOWLEDGMENTS Appel, J., P. J. Windpassinger, D. Oblak, U. B. Hoff, N. Kjar- gaard, and E. S. Polzik (2009), Proceedings of the National The authors thank Jens Boss, Dmitry Budker, Kevin Academy of Sciences 106, 10960. Chang, Kristian Cujia,Lukasz Cywiński,Simon Gus- Arnold, K., E. Hajiyev, E. Paez, C. H. Lee, M. D. Bar- tavsson, Sebastian Hofferberth, Dominik Irber, Fedor rett, and J. Bollinger (2015), Physical Review A 92 (3), Jelezko, Renbao Liu, Luca Lorenzelli, Tobias Rosskopf, 10.1103/physreva.92.032108. Arrad, G., Y. Vinkler, D. Aharonov, and A. Retzker (2014), Daniel Slichter, JörgWrachtrup and Jonathan Zopes Phys. Rev. Lett. 112, 150801. for helpful comments and discussions. CLD acknowl- Aspachs, M., J. Calsamiglia, R. MuñozTapia, and E. Bagan edges funding from the DIADEMS program 611143 of (2009), Phys. Rev. A 79, 033834. the European Commission, the Swiss NSF Project Grant Aspelmeyer, M., T. J. Kippenberg, and F. Marquardt (2014), 200021 137520, the Swiss NSF NCCR QSIT, and ETH Reviews of Modern Physics 86 (4), 1391. Research Grant ETH-03 16-1. FR acknowledges funding Astafiev, O., Y. A. Pashkin, Y. Nakamura, T. Yamamoto, from the Deutsche Forschungsgemeinschaft via Emmy and J. S. Tsai (2004), Phys. Rev. Lett. 93, 267007. Atature, M., J. Dreiser, A. Badolato, and A. Imamoglu Noether grant RE 3606/1-1. PC acknowledges fund- (2007), Nat Phys 3 (2), 101. ing from the U.S. Army Research Office through MURI Auccaise, R., A. G. Araujo-Ferreira, R. S. Sarthour, I. S. grants No. W911NF-11-1-0400 and W911NF-15-1-0548 Oliveira, T. J. Bonagamba, and I. Roditi (2015), Phys. and by the NSF PHY0551153 and PHY1415345. Rev. Lett. 114, 043604. Auzinsh, M., D. Budker, D. F. Kimball, S. M. Rochester, J. E. Stalnaker, A. O. Sushkov, and V. V. Yashchuk (2004), APPENDIX A: TABLE OF SYMBOLS Phys. Rev. Lett. 93 (17), 173002. Bagci, T., A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor, A. Sorensen, K. Us- ami, A. Schliesser, and E. S. Polzik (2014), Nature 507, 81. Bakr, W. S., J. I. Gillen, A. Peng, S. Folling, and M. Greiner (2009), Nature 462 (7269), 74. Bal, M., C. Deng, J.-L. Orgiazzi, F. Ong, and A. Lupascu (2012), Nat. Commun. 3, 1324. Balabas, M. V., T. Karaulanov, M. P. Ledbetter, and D. Bud- ker (2010), Physical Review Letters 105 (7), 070801. Balasubramanian, G., I. Y. Chan, R. Kolesov, M. Al-Hmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hemmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Bratschitsch, F. Jelezko, and J. Wrachtrup (2008), Nature 455 (7213), 648. 38

Quantity Symbol Unit Readout eﬃciency C 0 ≤ C ≤ 1 Dynamic range DR — AC signal: Frequency fac Hz Multipulse sensing: Bandwidth ∆f Hz Hamiltonian Hˆ (t) Hz

- internal Hamiltonian Hˆ0 - signal Hamiltonian HˆV (t) ˆ ... commuting part HV|| (t) ˆ ... non-commuting part HV⊥ (t) - control Hamiltonian Hˆcontrol(t) Number of qubits in ensemble; other uses M — Multipulse sensing: Filter order k — Multipulse sensing: No. of pulses n — Number of measurements N — Transition probability p p ∈ [0...1] - Bias point p0 - Change in transition probability δp = p − p0 Signal spectral density SV (ω) Signal squared per Hz Sensing time t s Signal autocorrelation time tc s Total measurement time T s Relaxation or decoherence time Tχ s - T1 relaxation time T1 ∗ - Dephasing time T2 - Decoherence time T2 - Rotating frame relaxation time T1ρ Signal V (t) varies - parallel signal V||(t) = Vz(t) 2 2 1/2 - transverse signal V⊥(t) = [Vx (t) + Vy (t)] - vector signal V~ (t) = {Vx,Vy,Vz}(t) - rms signal amplitude Vrms - AC signal amplitude Vpk - minimum detectable signal amplitude Vmin ... per unit time vmin Unit signal per second Multipulse sensing: Weighting function W (fac, α),W¯ (fac), etc. — Physical output of quantum sensor x, xj varies Multipulse sensing: Modulation function y(t)— Multipulse sensing: Filter function Y (ω) Hz−1 AC signal: Phase α — Coupling parameter γ Hz per unit signal Decoherence or transition rate Γ s−1 Quantum phase accumulated by sensor φ — - rms phase φrms — Pauli matrices ~σˆ = {σˆx, σˆy, σˆz} Uncertainty of transition probability σp — - due to quantum projection noise σp,quantum - due to readout noise σp,readout Multipulse sequence pulse delay τ s Transition frequency ω0 Hz Rabi frequency ω1 Hz - eﬀective Rabi frequency ωeﬀ Hz Decoherence function χ(t)— Basis states (energy eigenstates) {|0i, |1i} — Superposition states {|+i, |−i} — Sensing states {|ψ0i, |ψ1i} —

TABLE III Frequently used symbols. 39

Balasubramanian, G., P. Neumann, D. Twitchen, Braginskii, V. B., and Y. I. Vorontsov (1975), Soviet Physics M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, Uspekhi 17 (5), 644. J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, Braunstein, S. L. (1996), Physics Letters A 219 (3-4), 169. F. Jelezko, and J. Wrachtrup (2009), Nature Materials 8, Braunstein, S. L., and C. M. Caves (1994), Phys. Rev. Lett. 383. 72 (22), 3439. Ban, M., S. Kitajima, and F. Shibata (2009), Physics Letters Braunstein, S. L., C. M. Caves, and G. J. Milburn (1996), A 373 (40), 3614. Annals of Physics 247 (1), 135. Bar-Gill, N., L. Pham, C. Belthangady, D. Le Sage, P. Cap- Brewer, J. H., and K. M. Crowe (1978), Annual Review of pellaro, J. Maze, M. Lukin, A. Yacoby, and R. Walsworth Nuclear and Particle Science 28 (1), 239. (2012), Nat. Commun. 3, 858. Brida, G., M. Genovese, and I. R. Berchera (2010), Nature Barnes, E., M. S. Rudner, F. Martins, F. K. Malinowski, Photonics 4 (4), 227. C. M. Marcus, and F. Kuemmeth (2016), Physical Review Brownnutt, M., M. Kumph, P. Rabl, and R. Blatt (2015), B 93 (12), 10.1103/physrevb.93.121407. Reviews of Modern Physics 87 (4), 1419. Baumgart, I., J.-M. Cai, A. Retzker, M. B. Plenio, and Budakian, R., H. J. Mamin, B. W. Chui, and D. Rugar C. Wunderlich (2016), Phys. Rev. Lett. 116 (24), 240801. (2005), Science 307, 408. Baumgratz, T., and A. Datta (2016), Phys. Rev. Lett. Budker, D., and D. F. J. Kimball (2013), Optical Magne- 116 (3), 030801. tometry, by Dmitry Budker , Derek F. Jackson Kimball, Bennett, S. D., N. Y. Yao, J. Otterbach, P. Zoller, P. Rabl, Cambridge, UK: Cambridge University Press, 2013 . and M. D. Lukin (2013), Phys. Rev. Lett. 110, 156402. Budker, D., and M. Romalis (2007), Nat. Phys. 3, 227, pro- Bergeal, N., F. Schackert, M. Metcalfe, R. Vijay, V. E. vided by the Smithsonian/NASA Astrophysics Data Sys- Manucharyan, L. Frunzio, D. E. Prober, R. J. Schoelkopf, tem. S. M. Girvin, and M. H. Devoret (2010), Nature Bylander, J., T. Duty, and P. Delsing (2005), Nature 434, 465 (7294), 64. 361. Bergli, J., and L. Faoro (2007), Phys. Rev. B 75 (5), Bylander, J., S. Gustavsson, F. Yan, F. Yoshihara, 10.1103/physrevb.75.054515. K. Harrabi, G. Fitch, D. G. Cory, and W. D. Oliver (2011), Bergmann, M., and P. van Loock (2016), Phys. Rev. A 94, Nat. Phys. 7, 565. 012311. Campbell, W. C., and P. Hamilton (2017), Journal of Physics Berry, D. W., B. Higgins, S. Bartlett, M. Mitchell, G. Pryde, B: Atomic, Molecular and Optical Physics 50 (6), 064002. and H. Wiseman (2009), Phys. Rev. A 80, 052114. Cand´es,E. J., J. K. Romberg, and T. Tao (2006), Comm. Bialczak, R. C., R. McDermott, M. Ansmann, M. Hofheinz, Pure App. Math. 59 (8), 1207. N. Katz, E. Lucero, M. Neeley, A. D. O’Connell, H. Wang, Cappellaro, P. (2012), Phys. Rev. A 85, 030301(R). A. N. Cleland, and J. M. Martinis (2007), Phys. Rev. Lett. Cappellaro, P., G. Goldstein, J. S. Hodges, L. Jiang, J. R. 99 (18), 187006. Maze, A. S. Sørensen, and M. D. Lukin (2012), Phys. Rev. Biercuk, M. J., A. C. Doherty, and H. Uys (2011), J. of Phys. A 85, 032336. B 44 (15), 154002. Cappellaro, P., and M. D. Lukin (2009), Phys. Rev. A 80 (3), Biercuk, M. J., H. Uys, J. W. Britton, A. P. VanDevender, 032311. and J. J. Bollinger (2010), Nat Nano 5, 646. Carr, H. Y., and E. M. Purcell (1954), Phys. Rev. 94 (3), Biercuk, M. J., H. Uys, A. P. VanDevender, N. Shiga, W. M. 630. Itano, and J. J. Bollinger (2009), Nature 458 (7241), 996. Casanova, J., Z. Wang, J. F. Haase, and M. B. Plenio (2015), Bitter, T., and D. Dubbers (1987), Physical Review Letters Phys. Rev. A 92, 042304. 59 (3), 251. Castellanos-Beltran, M. A., K. D. Irwin, G. C. Hilton, L. R. Bloom, B. J., T. L. Nicholson, J. R. Williams, S. L. Campbell, Vale, and K. W. Lehnert (2008), Nat Phys 4 (12), 929. M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye Caves, C. M. (1981), Phys. Rev. D 23 (8), 1693. (2014), Nature 506 (7486), 71. Cervantes, F. G., L. Kumanchik, J. Pratt, and J. M. Taylor Blundell, S. J. (1999), Contemporary Physics 40 (3), 175. (2014), Applied Physics Letters 104 (22), 221111. Bohmann, M., J. Sperling, and W. Vogel (2015), Phys. Rev. Chabuda, K., I. D. Leroux, and R. Demkowicz-Dobrza´nski A 91, 042332. (2016), New Journal of Physics 18 (8), 083035. Bohnet, J. G., K. Cox, M. Norcia, J. Weiner, Z. Chen, and Chang, K., A. Eichler, J. Rhensius, L. Lorenzelli, and J. K. Thompson (2014), Nat. Phot. 8 (9), 731. C. L. Degen (2017), Nano Letters, Article ASAP Bohnet, J. G., B. C. Sawyer, J. W. Britton, M. L. Wall, 10.1021/acs.nanolett.6b05304. A. M. Rey, M. Foss-Feig, and J. J. Bollinger (2016), Sci- Chaste, J., A. Eichler, J. Moser, G. Ceballos, R. Rurali, and ence 352 (6291), 1297. A. Bachtold (2012), Nat. Nanotechnol. 7, 300. Bollinger, J. J., W. M. Itano, D. J. Wineland, and D. J. Chernobrod, B. M., and G. P. Berman (2005), Journal of Heinzen (1996), Phys. Rev. A 54 (6), R4649. Applied Physics 97 (1), 014903. Bonato, C.,, Blok, M. S., Dinani, H. T., Berry, D. W., Chipaux, M., L. Toraille, C. Larat, L. Morvan, S. Pezzagna, Markham, M. L., Twitchen, D. J., and Hanson, R. (2016), J. Meijer, and T. Debuisschert (2015), Applied Physics Nat Nano 11 (3), 247. Letters 107 (23), http://dx.doi.org/10.1063/1.4936758. Boss, J. M., K. Chang, J. Armijo, K. Cujia, T. Rosskopf, Christle, D. J., A. L. Falk, P. Andrich, P. V. Klimov, J. U. J. R. Maze, and C. L. Degen (2016), Phys. Rev. Lett. Hassan, N. Son, E. Janzon, T. Ohshima, and D. D. 116, 197601. Awschalom (2015), Nat Mater 14 (2), 160. Boss, J. M., K. Cujia, J. Zopes, and C. L. Degen (2017), . Clarke, J., and A. I. Braginski (2004), The SQUID handbook Boto, A. N., P. Kok, D. S. Abrams, S. L. Braunstein, C. P. (Wiley-VCH). Williams, and J. P. Dowling (2000), Phys. Rev. Lett. 85, Clarke, J., and F. K. Wilhelm (2008), Nature 453, 1031. 2733. 40

Clevenson, H., M. E. Trusheim, C. Teale, T. Schrder, Fang, J., and J. Qin (2012), Sensors 12 (5), 6331. D. Braje, and D. Englund (2015), Nature Physics 11 (5), Faoro, L., and L. Viola (2004), Phys. Rev. Lett. 922, 117905. 393. Faust, T., J. Rieger, M. J. Seitner, J. P. Kotthaus, and E. M. Collaboration, T. L. S. (2013), Nat Photon 7 (8), 613. Weig (2013), Nature Physics 9, 485. Cooper, A., E. Magesan, H. Yum, and P. Cappellaro (2014), Fedder, H., F. Dolde, F. Rempp, T. Wolf, P. Hemmer, Nat. Commun. 5, 3141. F. Jelezko, and J. Wrachtrup (2011), Applied Physics B: Cox, K. C., G. P. Greve, J. M. Weiner, and J. K. Thompson Lasers and Optics 102 (3), 497. (2016), Phys. Rev. Lett. 116, 093602. Fernholz, T., H. Krauter, K. Jensen, J. F. Sherson, A. S. Cummings, F. W. (1962), Am. J. Phys. 30, 898. Sørensen, and E. S. Polzik (2008), Phys. Rev. Lett. 101, Cywinski, L., R. M. Lutchyn, C. P. Nave, and S. DasSarma 073601. (2008), Phys. Rev. B 77 (17), 174509. Fonseca, E. J. S., C. H. Monken, and S. Pdua (1999), Phys. Dang, H. B., A. C. Maloof, and M. V. Romalis (2010), Ap- Rev. Lett. 82 (14), 2868. plied Physics Letters 97 (15), 151110. Forstner, S., E. Sheridan, J. Knittel, C. L. Humphreys, G. A. D’Angelo, M., M. V. Chekhova, and Y. Shih (2001), Phys. Brawley, H. Rubinsztein-Dunlop, and W. P. Bowen (2014), Rev. Lett. 87 (1), 013602. Advanced Materials 26 (36), 6348. Davis, E., G. Bentsen, and M. Schleier-Smith (2016), Phys. Fortagh, J., H. Ott, S. Kraft, A. Gunther, and C. Zimmer- Rev. Lett. 116 (5), 10.1103/physrevlett.116.053601. mann (2002), Phys. Rev. A 66, 041604. Degen, C. L. (2008), App. Phys. Lett 92 (24), 243111. Fu, C.-C., H.-Y. Lee, K. Chen, T.-S. Lim, H.-Y. Wu, P.-K. Degen, C. L., M. Poggio, H. J. Mamin, C. T. Rettner, and Lin, P.-K. Wei, P.-H. Tsao, H.-C. Chang, and W. Fann D. Rugar (2009), Proc. Nat Acad. Sc. 106 (5), 1313. (2007), Proc. Nat Acad. Sc. 104 (3), 727. Demkowicz-Dobrzanski, R., and L. Maccone (2014), Phys. Fu, R. R., B. P. Weiss, E. A. Lima, R. J. Harrison, X.-N. Bai, Rev. Lett. 113, 250801. S. J. Desch, D. S. Ebel, C. Suavet, H. Wang, D. Glenn, Deutsch, D. (1985), Proc. R. Soc. A 400 (1818), 97. D. L. Sage, T. Kasama, R. L. Walsworth, and A. T. Kuan DeVience, S. J., L. M. Pham, I. Lovchinsky, A. O. Sushkov, (2014), Science 346 (6213), 1089. N. Bar-Gill, C. Belthangady, F. Casola, M. Corbett, Genoni, M. G., S. Olivares, and M. G. A. Paris (2011), Phys. H. Zhang, M. Lukin, H. Park, A. Yacoby, and R. L. Rev. Lett. 106, 153603. Walsworth (2015), Nat Nano 10 (2), 129. Gerry, C. C., and J. Mimih (2010), Contemporary Physics Dial, O., M. Shulman, S. Harvey, H. Bluhm, V. Uman- 51 (6), 497. sky, and A. Yacoby (2013), Phys. Rev. Lett. 110 (14), Giovannetti, V., S. Lloyd, and L. Maccone (2004), Science 10.1103/PhysRevLett.110.146804, cited By 81. 306 (5700), 1330. DiVincenzo, D. P. (2000), Fortschr. Phys. 48, 771. Giovannetti, V., S. Lloyd, and L. Maccone (2006), Phys. Rev. Doherty, M. W., V. V. Struzhkin, D. A. Simpson, L. P. Lett. 96 (1), 010401. McGuinness, Y. Meng, A. Stacey, T. J. Karle, R. J. Hemley, Giovannetti, V., S. Lloyd, and L. Maccone (2011), Nat. Pho- N. B. Manson, L. C. L. Hollenberg, and S. Prawer (2014), ton. 5 (4), 222. Phys. Rev. Lett. 112 (4), 10.1103/physrevlett.112.047601. Gisin, N., G. Ribordy, W. Tittel, and H. Zbinden (2002), Dolde, F., H. Fedder, M. W. Doherty, T. Nobauer, F. Rempp, Rev. Mod. Phys. 74 (1), 145. G. Balasubramanian, T. Wolf, F. Reinhard, L. C. L. Hol- Gleyzes, S., S. Kuhr, C. Guerlin, J. Bernu, S. DelEglise,` lenberg, F. Jelezko, and J. Wrachtrup (2011), Nat. Phys. U. Busk Hoff, M. Brune, J.-M. Raimond, and S. Haroche 7 (6), 459. (2007), Nature 446 (7133), 297. Dovzhenko, Y., F. Casola, S. Schlotter, T. X. Zhou, F. Bttner, Goldstein, G., P. Cappellaro, J. R. Maze, J. S. Hodges, R. L. Walsworth, G. S. D. Beach, and A. Yacoby (2016), L. Jiang, A. S. Sorensen, and M. D. Lukin (2011), Phys. 1611.00673. Rev. Lett. 106 (14), 140502. Drewsen, M. (2015), Physica B: Condensed Matter 460, 105. Goldstein, G., M. D. Lukin, and P. Cappellaro (2010), Dür,W., M. Skotiniotis, F. Fröwis, and B. Kraus (2014), ArXiv:1001.4804. Phys. Rev. Lett. 112, 080801. Griffiths, R., and C.-S. Niu (1996), Phys. Rev. Lett. 76 (17), Dussaux, A., P. Schoenherr, K. Koumpouras, J. Chico, 3228. K. Chang, L. Lorenzelli, N. Kanazawa, Y. Tokura, Gross, D., Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert M. Garst, A. Bergman, C. L. Degen, and D. Meier (2016), (2010), Phys. Rev. Lett. 105, 150401. Nature Communications 7, 12430. Gruber, A., A. Drabenstedt, C. Tietz, L. Fleury, Edamatsu, K., R. Shimizu, and T. Itoh (2002), Phys. Rev. J. Wrachtrup, and C. v. Borczyskowski (1997), Science Lett. 89 (21), 213601. 276 (5321), 2012. Elzerman, J. M., R. Hanson, L. H. Willems van Beveren, Gullion, T., D. B. Baker, and M. S. Conradi (1990), J. Mag. B. Witkamp, L. M. K. Vandersypen, and L. P. Kouwen- Res. 89 (3), 479. hoven (2004), Nature 430 (6998), 431. Gustavson, T. L., P. Bouyer, and M. A. Kasevich (1997), Esteve, J., C. Gross, A. Weller, S. Giovanazzi, and M. K. Phys. Rev. Lett. 78, 2046. Oberthaler (2008), Nature 455 (7217), 1216. Gustavson, T. L., A. Landragin, and M. A. Kasevich (2000), Facon, A., E.-K. Dietsche, D. Grosso, S. Haroche, J.-M. Classical and Quantum Gravity 17 (12), 2385. Raimond, M. Brune, and S. Gleyzes (2016), Nature Hahn, E. L. (1950), Phys. Rev. 80 (4), 580. 535 (7611), 262. Halbertal, D., J. Cuppens, M. B. Shalom, L. Embon, Fagaly, R. L. (2006), Review of Scientific Instruments 77 (10), N. Shadmi, Y. Anahory, H. R. Naren, J. Sarkar, A. Uri, 101101. Y. Ronen, Y. Myasoedov, L. S. Levitov, E. Joselevich, Fan, H., S. Kumar, J. Sedlacek, H. Kbler, S. Karimkashi, A. K. Geim, and E. Zeldov (2016), Nature advance on- and J. P. Shaffer (2015), Journal of Physics B: Atomic, line publication, 10.1038/nature19843. Molecular and Optical Physics 48 (20), 202001. 41

Hald, J., J. L. Srensen, C. Schori, and E. S. Polzik (1999), Imoto, N., H. A. Haus, and Y. Yamamoto (1985), Phys. Rev. Physical Review Letters 83 (7), 1319. A 32, 2287. Hall, L. T., P. Kehayias, D. A. Simpson, A. Jarmola, Itano, W. M., J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, A. Stacey, D. Budker, and L. C. L. Hollenberg (2016), D. J. Heinzen, F. L. Moore, M. G. Raizen, and D. J. Nat. Commun. 7, Article. Wineland (1993), Phys. Rev. A 47 (5), 3554. Hanbury Brown, R., and R. Q. Twiss (1956), Nature Ithier, G., E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Es- 178 (4541), 1046. teve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, Happer, W., and H. Tang (1973), Physical Review Letters and G. Schon (2005), Phys. Rev. B 72, 134519. 31 (5), 273. Jaklevic, R. C., J. Lambe, J. E. Mercereau, and A. H. Silver Haroche, S. (2013), Reviews of Modern Physics 85 (3), 1083. (1965), Physical Review 140 (5A), A1628. Hatridge, M., R. Vijay, D. H. Slichter, J. Clarke, and I. Sid- Jelezko, F., A. Retzker, and et al. (2017), . diqi (2011), Phys. Rev. B 83 (13), 134501. Jenke, T., G. Cronenberg, J. Burgdörfer,L. A. Chizhova, Hayes, D., K. Khodjasteh, L. Viola, and M. J. Biercuk (2011), P. Geltenbort, A. N. Ivanov, T. Lauer, T. Lins, S. Rotter, Phys. Rev. A 84, 062323. H. Saul, U. Schmidt, and H. Abele (2014), Phys. Rev. Lett. Helstrom, C. W. (1967), Physics Letters A A 25, 101. 112, 151105. Herrera-Mart´ı,D. A., T. Gefen, D. Aharonov, N. Katz, and Jenke, T., P. Geltenbort, H. Lemmel, and H. Abele (2011), A. Retzker (2015), Phys. Rev. Lett. 115, 200501. Nat Phys 7 (6), 468. Herrmann, P. P., J. Hoffnagle, N. Schlumpf, V. L. Telegdi, Jensen, K., R. Budvytyte, R. A. Thomas, T. Wang, A. M. and A. Weis (1986), Journal of Physics B: Atomic and Fuchs, M. V. Balabas, G. Vasilakis, L. D. Mosgaard, H. C. Molecular Physics 19 (9), 1271. Stærkind, J. H. Mueller, T. Heimburg, S.-P. Olesen, and Higgins, B. L., D. W. Berry, S. D. Bartlett, M. W. Mitchell, E. S. Polzik (2016), Scientific Reports 6, 29638. H. M. Wiseman, and G. J. Pryde (2009), New J. Phys. 11, Jensen, K., N. Leefer, A. Jarmola, Y. Dumeige, V. Acosta, 073023. P. Kehayias, B. Patton, and D. Budker (2014), Physical Higgins, B. L., D. W. Berry, S. D. Bartlett, H. M. Wiseman, Review Letters 112 (16), 160802. and G. J. Pryde (2007), Nature 450 (7168), 393. Jiang, L., J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, Hinkley, N., J. A. Sherman, N. B. Phillips, M. Schioppo, N. D. D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, Lemke, K. Beloy, M. Pizzocaro, C. W. Oates, and A. D. A. S. Zibrov, and M. D. Lukin (2009), Science 326 (5950), Ludlow (2013), Science 341 (6151), 1215. 267. Hirose, M., C. D. Aiello, and P. Cappellaro (2012), Phys. Jones, J. A., S. D. Karlen, J. Fitzsimons, A. Ardavan, S. C. Rev. A 86, 062320. Benjamin, G. A. D. Briggs, and J. J. L. Morton (2009), Hirose, M., and P. Cappellaro (2016), Nature 532 (7597), 77. Science 324 (5931), 1166. Ho Eom, B., P. K. Day, H. G. LeDuc, and J. Zmuidzinas Jones, M. P. A., C. J. Vale, D. Sahagun, B. V. Hall, and (2012), Nat Phys 8 (8), 623. E. A. Hinds (2003), Phys. Rev. Lett. 91 (8), 080401. Hodges, J. S., N. Y. Yao, D. Maclaurin, C. Rastogi, M. D. Kardjilov, N., I. Manke, M. Strobl, A. Hilger, W. Treimer, Lukin, and D. Englund (2013), Phys. Rev. A 87, 032118. M. Meissner, T. Krist, and J. Banhart (2008), Nature Holevo, A. (1982), Probabilistic and Statistical Aspects of Physics 4 (5), 399. Quantum Theory (North-Holland, Amsterdam). Kasevich, M., and S. Chu (1992), Applied Physics B 54 (5), Holland, M. J., and K. Burnett (1993), Phys. Rev. Lett. 71, 321. 1355. Kastner, M. A. (1992), Rev. Mod. Phys. 64 (3), 849. Hollberg, L., C. Oates, E. Curtis, E. Ivanov, S. Diddams, Kessler, E. M., I. Lovchinsky, A. O. Sushkov, and M. D. T. Udem, H. Robinson, J. Bergquist, R. Rafac, W. Itano, Lukin (2014), Phys. Rev. Lett. 112, 150802. R. Drullinger, and D. Wineland (2001), IEEE Journal of Khodjasteh, K., and D. A. Lidar (2005), Phys. Rev. Lett. Quantum Electronics 37 (12), 1502. 95 (18), 180501. Hosten, O., N. J. Engelsen, R. Krishnakumar, and M. A. Kimmich, R., and E. Anoardo (2004), Prog. Nucl. Magn. Kasevich (2016a), Nature 529 (7587), 505. Reson. Spectrosc. 44, 257. Hosten, O., R. Krishnakumar, N. J. Engelsen, and M. A. Kitaev, A. Y. (1995), arXiv:quant-ph/9511026. Kasevich (2016b), Science 352 (6293), 1552. Kitagawa, M., and M. Ueda (1993), Phys. Rev. A 47 (6), Huang, Z., C. Macchiavello, and L. Maccone (2016), Phys. 5138. Rev. A 94, 012101. Kitching, J., S. Knappe, and E. Donley (2011), Sensors Jour- Huelga, S. F., C. Macchiavello, T. Pellizzari, A. K. Ekert, nal, IEEE 11 (9), 1749. M. B. Plenio, and J. I. Cirac (1997), Phys. Rev. Lett. Kohlhaas, R., A. Bertoldi, E. Cantin, A. Aspect, A. Landra- 79 (20), 3865. gin, and P. Bouyer (2015), Phys. Rev. X 5 (2), 021011. Hume, D. B., T. Rosenband, and D. J. Wineland (2007), Kolkowitz, S., A. Safira, A. A. High, R. C. Devlin, S. Choi, Phys. Rev. Lett. 99 (12), 120502. Q. P. Unterreithmeier, D. Patterson, A. S. Zibrov, V. E. Hume, D. B., I. Stroescu, M. Joos, W. Muessel, H. Strobel, Manucharyan, H. Park, and M. D. Lukin (2015), Science and M. K. Oberthaler (2013), Phys. Rev. Lett. 111, 253001. 347 (6226), 1129. Huntemann, N., C. Sanner, B. Lipphardt, C. Tamm, and Kominis, K., T. W. Kornack, J. C. Allred, and M. V. Romalis E. Peik (2016), Phys. Rev. Lett. 116, 063001. (2003), Nature 422, 596. Huver, S. D., C. F. Wildfeuer, and J. P. Dowling (2008), Kornack, T. W., R. K. Ghosh, and M. V. Romalis (2005), Phys. Rev. A 78 (6), 10.1103/physreva.78.063828. Phys. Rev. Lett. 95, 230801. Ilani, S., J. Martin, E. Teitelbaum, J. H. Smet, D. Mahalu, Kotler, S., N. Akerman, Y. Glickman, A. Keselman, and V. Umansky, and A. Yacoby (2004), Nature 427 (6972), R. Ozeri (2011), Nature 473 (7345), 61. 328. Kotler, S., N. Akerman, Y. Glickman, and R. Ozeri (2013), Phys. Rev. Lett. 110, 110503. 42

Krause, A. G., M. Winger, T. D. Blasius, Q. Lin, and Loretz, M., J. M. Boss, T. Rosskopf, H. J. Mamin, D. Rugar, O. Painter (2012), Nature Photonics 6, 768. and C. L. Degen (2015), Phys. Rev. X 5, 021009. Kroutvar, M., Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, Loretz, M., S. Pezzagna, J. Meijer, and C. L. Degen (2014), G. Abstreiter, and J. J. Finley (2004), Nature 432, 81. Appl. Phys. Lett. 104, 33102. Kübler,H., J. P. Shaffer, T. Baluktsian, R. Löw,and T. Pfau Loretz, M., T. Rosskopf, and C. L. Degen (2013), Phys. Rev. (2010), Nature Photonics 4 (2), 112. Lett. 110, 017602. Kucsko, G., P. C. Maurer, N. Y. Yao, M. Kubo, H. J. Noh, Louchet-Chauvet, A., J. Appel, J. J. Renema, D. Oblak, P. K. Lo, H. Park, and M. D. Lukin (2013), Nature N. Kjaergaard, and E. S. Polzik (2010), New J. Phys. 500 (7460), 54. 12 (6), 065032. Kuehn, S., R. F. Loring, and J. A. Marohn (2006), Phys. Lovchinsky, I., J. D. Sanchez-Yamagishi, E. K. Urbach, Rev. Lett. 96, 156103. S. Choi, S. Fang, T. I. Andersen, K. Watanabe, Kuzmich, A., and L. Mandel (1998), Quantum and Semiclas- T. Taniguchi, A. Bylinskii, E. Kaxiras, P. Kim, H. Park, sical Optics: Journal of the European Optical Society Part and M. D. Lukin (2017), Science 355 (6324), 503. B 10 (3), 493. Lovchinsky, I., A. O. Sushkov, E. Urbach, N. P. de Leon, Labaziewicz, J., Y. Ge, D. R. Leibrandt, S. X. Wang, R. Shew- S. Choi, K. De Greve, R. Evans, R. Gertner, E. Bersin, mon, and I. L. Chuang (2008), Phys. Rev. Lett. 101, C. Müller,L. McGuinness, F. Jelezko, R. L. Walsworth, 180602. H. Park, and M. D. Lukin (2016), Science 351 (6275), Lang, J. E., R. B. Liu, and T. S. Monteiro (2015), Phys. Rev. 836. X 5, 041016. Lu, X., S. Yu, and C. H. Oh (2015), Nature Communications Lanting, T., A. J. Berkley, B. Bumble, P. Bunyk, A. Fung, 6, 7282. J. Johansson, A. Kaul, A. Kleinsasser, E. Ladizinsky, Maccone, L., and G. D. Cillis (2009), Phys. Rev. A 79 (2), F. Maibaum, R. Harris, M. W. Johnson, E. Tolkacheva, 10.1103/physreva.79.023812. and M. H. S. Amin (2009), Phys. Rev. B 79, 060509. Macklin, C., K. OBrien, D. Hover, M. E. Schwartz, Laraoui, A., and C. A. Meriles (2013), ACS Nano 7 (4), 3403, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi pMID: 23565720, http://dx.doi.org/10.1021/nn400239n. (2015), Science 350 (6258), 307. Le Sage, D., K. Arai, D. R. Glenn, S. J. DeVience, L. Pham, Magesan, E., A. Cooper, and P. Cappellaro (2013a), Phys. L. M .and Rahn-Lee, M. D. Lukin, A. Yacoby, A. Komeili, Rev. A 88, 062109. and R. L. Walsworth (2013), Nature 496 (7446), 486. Magesan, E., A. Cooper, H. Yum, and P. Cappellaro (2013b), Le Sage, D., L. M. Pham, N. Bar-Gill, C. Belthangady, M. D. Phys. Rev. A 88, 032107. Lukin, A. Yacoby, and R. L. Walsworth (2012), Phys. Rev. Maiwald, R., D. Leibfried, J. Britton, J. C. Bergquist, B 85, 121202. G. Leuchs, and D. J. Wineland (2009), Nature Physics Ledbetter, M. P., K. Jensen, R. Fischer, A. Jarmola, and 5, 551. D. Budker (2012), Phys. Rev. A 86 (5), 052116. Maletinsky, P., S. Hong, M. S. Grinolds, B. Hausmann, M. D. Lee, H., P. Kok, and J. P. Dowling (2002), Journal of Modern Lukin, R. L. Walsworth, M. Loncar, and A. Yacoby (2012), Optics 49, 2325. Nat. Nanotech. 7 (5), 320. Lee, T.-W., S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, Mamin, H. J., M. Kim, M. H. Sherwood, C. T. Rettner, C. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and K. Ohno, D. D. Awschalom, and D. Rugar (2013), Sci- J. P. Dowling (2009), Phys. Rev. A 80 (6), 10.1103/phys- ence 339 (6119), 557. reva.80.063803. Martin, J., N. Akerman, G. Ulbricht, T. Lohmann, J. H. Leibfried, D., M. D. Barrett, T. Schaetz, J. Britton, J. Chi- Smet, K. von Klitzing, and A. Yacoby (2008), Nature averini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Physics 4 (2), 144. Wineland (2004), Science 304 (5676), 1476. Martinis, J., S. Nam, J. Aumentado, and C. Urbina (2002), Leibfried, D., E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, Phys. Rev. Lett. 89, 117901. J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, Maze, J. R., P. L. Stanwix, J. S. Hodges, S. Hong, J. M. C. Langer, R. Ozeri, R. Reichle, and D. J. Wineland Taylor, P. Cappellaro, L. Jiang, A. Zibrov, A. Yacoby, (2005), Nature 438 (7068), 639. R. Walsworth, and M. D. Lukin (2008), Nature 455, 644. Lenz, J. (1990), Proceedings of the IEEE 78 (6), 973. McGuinness, L. P., Y. Yan, A. Stacey, D. A. Simpson, L. T. Leroux, I. D., M. H. Schleier-Smith, and V. Vuletić(2010a), Hall, D. Maclaurin, S. Prawer, P. Mulvaney, J. Wrachtrup, Phys. Rev. Lett. 104, 073602. F. Caruso, R. E. Scholten, and L. C. L. Hollenberg (2011), Leroux, I. D., M. H. Schleier-Smith, and V. Vuletić(2010b), Nat. Nanotech. 6 (6), 358. Phys. Rev. Lett. 104, 250801. Mck, M., C. Welzel, and J. Clarke (2003), Applied Physics Li, K., M. Arif, D. Cory, R. Haun, B. Heacock, M. Huber, Letters 82 (19), 3266. J. Nsofini, D. Pushin, P. Saggu, D. Sarenac, C. Shahi, Medford, J., L. Cywiński,C. Barthel, C. M. Marcus, M. P. V. Skavysh, W. Snow, and A. Y. and (2016), Physical Hanson, and A. C. Gossard (2012), Phys. Rev. Lett. 108, Review D 93 (6), 10.1103/physrevd.93.062001. 086802. Ligo Collaboration”, (2011), Nat. Phys. 7 (12), 962. Meriles, C. A., L. Jiang, G. Goldstein, J. S. Hodges, J. Maze, Livanov, M., A. Kozlov, A. Korinevski, V. Markin, and M. D. Lukin, and P. Cappellaro (2010), J. Chem. Phys. S. Sinel’nikova (1978), Doklady Akademii nauk SSSR 133 (12), 124105. 238 (1), 253. Meyer, V., M. A. Rowe, D. Kielpinski, C. A. Sackett, W. M. Lloyd, S. (2008), Science 321 (5895), 1463. Itano, C. Monroe, and D. J. Wineland (2001), Phys. Rev. Lopaeva, E. D., I. Ruo Berchera, I. P. Degiovanni, S. Olivares, Lett. 86, 5870. G. Brida, and M. Genovese (2013), Phys. Rev. Lett. 110, Mezei, F. (1972), Zeitschrift fr Physik 255 (2), 146. 153603. Mitchell, M. W., J. S. Lundeen, and A. M. Steinberg (2004), Nature 429 (6988), 161. 43

Monz, T., P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, Pang, S., and A. N. Jordan (2016), arXiv:1606.02166. W. A. Coish, M. Harlander, W. Hänsel,M. Hennrich, and Paris, M. G. A. (2009), Int. J. Quant. Inf. 7, 125. R. Blatt (2011), Phys. Rev. Lett. 106, 130506. Paz-Silva, G. A., and L. Viola (2014), Phys. Rev. Lett. 113, Morello, A., J. J. Pla, F. A. Zwanenburg, K. W. Chan, K. Y. 250501. Tan, H. Huebl, M. Mottonen, C. D. Nugroho, C. Yang, Pelliccione, M., A. Jenkins, P. Ovartchaiyapong, C. Reetz, J. A. van Donkelaar, A. D. C. Alves, D. N. Jamieson, C. C. E. Emmanouilidou, N. Ni, and A. C. B. Jayich (2016), Escott, L. C. L. Hollenberg, R. G. Clark, and A. S. Dzurak Nature Nanotechnology 11, 700. (2010), Nature 467 (7316), 687. Peters, A., K. Y. Chung, and S. Chu (1999), Nature Moser, J., J. Guttinger, A. Eichler, M. J. Esplandiu, D. E. 400 (6747), 849. Liu, M. I. Dykman, and A. Bachtold (2013), Nature Nan- Pfender, M., N. Aslam, H. Sumiya, S. Onoda, P. Neu- otechnology 8, 493. mann, J. Isoya, C. Meriles, and J. Wrachtrup (2016), Muhonen, J. T., J. P. Dehollain, A. Laucht, F. E. Hudson, arXiv:1610.05675. R. Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. Puentes, G., G. Waldherr, P. Neumann, G. Balasubramanian, McCallum, A. S. Dzurak, and A. Morello (2014), Nat Nano and J. Wrachtrup (2014), Sci. Rep. 4, 10.1038/srep04677. 9 (12), 986. Pustelny, S., D. F. J. Kimball, C. Pankow, M. P. Ledbet- Myers, B. A., A. Das, M. C. Dartiailh, K. Ohno, D. D. ter, P. Wlodarczyk, P. Wcislo, M. Pospelov, J. R. Smith, Awschalom, and A. C. B. Jayich (2014), Phys. Rev. Lett. J. Read, W. Gawlik, and D. Budker (2013), Annalen der 113, 027602. Physik 525 (8-9), 659. Nader, C., N. Björsell, and P. Händel(2011), Signal Process- Riedel, M. F., P. Böhi,Y. Li, T. W. Hänsch, A. Sinatra, and ing 91 (5), 1347. P. Treutlein (2010), Nature 464, 1170. Nagata, T., R. Okamoto, J. L. O’Brien, K. Sasaki, and Romach, Y., C. Müller,T. Unden, L. J. Rogers, T. Isoda, S. Takeuchi (2007), Science 316, 726. K. M. Itoh, M. Markham, A. Stacey, J. Meijer, S. Pezza- Nakamura, Y., Y. Pashkin, and J. Tsai (1999), Nature 398, gna, B. Naydenov, L. P. McGuinness, N. Bar-Gill, and 786. F. Jelezko (2015), Phys. Rev. Lett. 114, 017601. Nakamura, Y., Y. Pashkin, and J. Tsai (2002), Phys. Rev. Rondin, L., J. P. Tetienne, T. Hingant, J. F. Roch, Lett. 88, 047901. P. Maletinsky, and V. Jacques (2014), Rep. Prog. Phys. Natarajan, C. M., M. G. Tanner, and R. H. Hadfield (2012), 77, 056503. Superconductor Science and Technology 25 (6), 063001. Rondin, L., J. P. Tetienne, S. Rohart, A. Thiaville, T. Hin- Nesvizhevsky, V. V., H. G. Borner, A. K. Petukhov, H. Abele, gant, P. Spinicelli, J. F. Roch, and V. Jacques (2013), Nat. S. Baeszler, F. J. Ruesz, T. Stoferle, A. Westphal, A. M. Commun. 4,. Gagarski, G. A. Petrov, and A. V. Strelkov (2002), Nature Rondin, L., J. P. Tetienne, P. Spinicelli, C. dal Savio, K. Kar- 415 (6869), 297. rai, G. Dantelle, A. Thiaville, S. Rohart, J. F. Roch, and Neumann, P., I. Jakobi, F. Dolde, C. Burk, R. Reuter, V. Jacques (2012), Appl. Phys. Lett. 100, 153118. G. Waldherr, J. Honert, T. Wolf, A. Brunner, J. H. Shim, Rosenband, T., D. B. Hume, P. O. Schmidt, C. W. Chou, D. Suter, H. Sumiya, J. Isoya, and J. Wrachtrup (2013), A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Nano Letters 13 (6), 2738. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, Cywiński,L. (2014), Phys. Rev. A 90, 042307. N. R. Newbury, W. M. Itano, D. J. Wineland, and J. C. Nicholson, T. L., S. L. Campbell, R. B. Hutson, G. E. Marti, Bergquist (2008), Science 319 (5871), 1808. B. J. Bloom, R. L. Mcnally, W. Zhang, M. D. Barrett, M. S. Rosenband, T., P. O. Schmidt, D. B. Hume, W. M. Itano, Safronova, G. F. Strouse, W. L. Tew, and J. Ye (2015), T. M. Fortier, J. E. Stalnaker, K. Kim, S. A. Diddams, Nature Communications 6, 6896. J. C. J. Koelemeij, J. C. Bergquist, and D. J. Wineland Nielsen, M. A., and I. L. Chuang (2000), Quantum com- (2007), Phys. Rev. Lett. 98 (22), 220801. putation and quantum information (Cambridge University Rosskopf, T., A. Dussaux, K. Ohashi, M. Loretz, R. Schirhagl, Press, Cambridge; New York). H. Watanabe, S. Shikata, K. M. Itoh, and C. L. Degen Nogues, G., A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. (2014), Phys. Rev. Lett. 112, 147602. Raimond, and S. Haroche (1999), Nature 400 (6741), 239. Rosskopf, T., J. Zopes, J. M. Boss, and C. L. Degen (2016), Norris, L. M., G. A. Paz-Silva, and L. Viola (2016), Phys. arXiv:1610.03253. Rev. Lett. 116, 150503. Rugar, D., R. Budakian, H. J. Mamin, and B. W. Chui Novotny, L. (2010), American Journal of Physics 78, 1199. (2004), Nature 430 (6997), 329. Nusran, M., M. M. Ummal, and M. V. G. Dutt (2012), Nat. Said, R. S., D. W. Berry, and J. Twamley (2011), Phys. Rev. Nanotech. 7 (2), 109. B 83, 125410. Ockeloen, C. F., R. Schmied, M. F. Riedel, and P. Treutlein Sakurai, J. J., and J. Napolitano (2011), Modern quantum (2013), Physical Review Letters 111 (14), 143001. mechanics (Addison-Wesley). O’Connell, A. D., M. Hofheinz, M. Ansmann, R. C. Bialczak, van der Sar, T., F. Casola, R. Walsworth, and A. Yacoby M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, (2015), Nat Commun 6,. M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland Schafer-Nolte, E., L. Schlipf, M. Ternes, F. Reinhard, (2010), Nature 464, 697. K. Kern, and J. Wrachtrup (2014), Phys. Rev. Lett. 113, Oelker, E., T. Isogai, J. Miller, M. Tse, L. Barsotti, N. Maval- 217204. vala, and M. Evans (2016), Phys. Rev. Lett. 116, 041102. Schirhagl, R., K. Chang, M. Loretz, and C. L. Degen (2014), Osterwalder, A., and F. Merkt (1999), Physical Review Let- Annu. Rev. Phys. Chem. 65, 83. ters 82 (9), 1831. Schleier-Smith, M. H., I. D. Leroux, and V. Vuletić(2010a), Ozeri, R. (2013), ArXiv:1310.3432. Phys. Rev. A 81, 021804. Packard, M., and R. Varian (1954), Physical Review 93 (4), Schleier-Smith, M. H., I. D. Leroux, and V. Vuletić(2010b), 941. Phys. Rev. Lett. 104, 073604. 44

Schmid-Lorch, D., T. Haberle, F. Reinhard, A. Zappe, Staudacher, T., F. Shi, S. Pezzagna, J. Meijer, J. Du, C. A. M. Slota, L. Bogani, A. Finkler, and J. Wrachtrup (2015), Meriles, F. Reinhard, and J. Wrachtrup (2013), Science Nano Lett. 15, 4942. 339 (6119), 561. Schmidt, P., T. Rosenband, C. Langer, W. Itano, Steinert, S., F. Ziem, L. T. Hall, A. Zappe, M. Schweikert, J. Bergquist, and D. Wineland (2005), Science 309 (5735), N. Gutz, A. Aird, G. Balasubramanian, L. Hollenberg, and 749. J. Wrachtrup (2013), Nat. Commun. 4, 1607. Schnabel, R., N. Mavalvala, D. E. McClelland, and P. K. Sushkov, A. O., N. Chisholm, I. Lovchinsky, M. Kubo, P. K. Lam (2010), Nat. Commun. 1 (8), 121. Lo, S. D. Bennett, D. Hunger, A. Akimov, R. L. Walsworth, Schoelkopf, R. J. (1998), Science 280 (5367), 1238. H. Park, and M. D. Lukin (2014a), Nano Letters 14 (11), Schoelkopf, R. J., A. A. Clerk, S. M. Girvin, K. W. Lehnert, 6443. and M. H. Devoret (2003), “Qubits as spectrometers of Sushkov, A., O., I. Lovchinsky, N. Chisholm, L. Walsworth, quantum noise,” in Quantum Noise in Mesoscopic Physics, R., H. Park, and D. Lukin, M. (2014b), Phys. Rev. Lett. edited by Y. V. Nazarov (Springer Netherlands, Dordrecht) 113, 197601. pp. 175–203. Suter, D., and G. A. Alvarez´ (2016), Rev. Mod. Phys. 88, Schulte, M., N. Lorch, I. D. Leroux, P. O. Schmidt, and 041001. K. Hammerer (2016), Phys. Rev. Lett. 116, 013002. Swallows, M. D., T. H. Loftus, W. C. Griffith, B. R. Heckel, Schwartz, O., J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, E. N. Fortson, and M. V. Romalis (2013), Physical Review and D. Oron (2013), Nano Letters 13 (12), 5832. A 87 (1), 10.1103/physreva.87.012102. Sedlacek, J. A., A. Schwettmann, H. Kbler, R. Lw, T. Pfau, Takamoto, M., F.-L. Hong, R. Higashi, and H. Katori (2005), and J. P. Shaffer (2012), Nature Physics 8 (11), 819. Nature 435, 321. Sewell, R. J., M. Koschorreck, M. Napolitano, B. Dubost, Tan, S.-H., B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, N. Behbood, and M. W. Mitchell (2012), Phys. Rev. Lett. L. Maccone, S. Pirandola, and J. H. Shapiro (2008), Phys. 109, 253605. Rev. Lett. 101 (25), 253601. Shah, V., S. Knappe, P. D. D. Schwindt, and J. Kitching Tan, T. R., J. P. Gaebler, Y. Lin, Y. Wan, R. Bowler, (2007), Nature Photonics 1 (11), 649. D. Leibfried, and D. J. Wineland (2015), Nature Shah, V., G. Vasilakis, and M. V. Romalis (2010), Phys. Rev. 528 (7582), 380. Lett. 104 (1), 013601. Tao, Y., and C. L. Degen (2015), Nano Letters Shaji, A., and C. M. Caves (2007), Phys. Rev. A 76, 032111. 10.1021/acs.nanolett.5b02885. Shi, F., X. Kong, P. Wang, F. Kong, N. Zhao, R. Liu, and Taylor, J. M., P. Cappellaro, L. Childress, L. Jiang, D. Bud- J. Du (2014), Nature Physics 10, 21. ker, P. R. Hemmer, A. Yacoby, R. Walsworth, and M. D. Shi, F., Q. Zhang, P. Wang, H. Sun, J. Wang, X. Rong, Lukin (2008), Nat. Phys. 4 (10), 810. M. Chen, C. Ju, F. Reinhard, H. Chen, J. Wrachtrup, Taylor, M. A., J. Janousek, V. Daria, J. Knittel, B. Hage, J. Wang, and J. Du (2015), Science 347 (6226), 1135, BachorHans-A., and W. P. Bowen (2013), Nat Photon http://science.sciencemag.org/content/347/6226/1135.full.pdf. 7 (3), 229. Shiga, N., and M. Takeuchi (2012), New Journal of Physics Tetienne, J. P., T. Hingant, J. Kim, L. H. Diez, J. P. Adam, 14 (2), 023034. K. Garcia, J. F. Roch, S. Rohart, A. Thiaville, D. Rav- Shor, P. W. (1994), in Foundations of Computer Science, 1994 elosona, and V. Jacques (2014), Science 344, 1366. Proceedings., 35th Annual Symposium on, pp. 124–134. Tetienne, J. P., T. Hingant, L. J. Martinez, S. Rohart, A. Thi- Shor, P. W. (1995), Phys. Rev. A 52, R2493. aville, L. H. Diez, K. Garcia, J. P. Adam, J. V. Kim, Simmonds, M., W. Fertig, and R. Giffard (1979), IEEE J. F. Roch, I. M. Miron, G. Gaudin, L. Vila, B. Ocker, Transactions on Magnetics 15 (1), 478. D. Ravelosona, and V. Jacques (2015), Nat. Commun. 6, Sinha, S., J. Emerson, N. Boulant, E. M. Fortunato, T. F. 10.1038/ncomms7733. Havel, and D. G. Cory (2003), Quantum Information Pro- Thiel, L., D. Rohner, M. Ganzhorn, P. Appel, E. Neu, cessing 2 (6), 433. B. Muller, R. Kleiner, D. Koelle, and P. Maletinsky (2016), Slichter, C. P. (1996), Principles of Magnetic Resonance, 3rd Nature Nanotechnology 11, 677. ed. (Springer-Verlag). Ticozzi, F., and L. Viola (2006), Phys. Rev. A 74 (5), 052328. Slichter, D. H., R. Vijay, S. J. Weber, S. Boutin, M. Boisson- Toyli, D. M., C. F. de las Casas, D. J. Christle, V. V. Do- neault, J. M. Gambetta, A. Blais, and I. Siddiqi (2012), brovitski, and D. D. Awschalom (2013), Proc. Nat Acad. Phys. Rev. Lett. 109, 153601. Sc. 110 (21), 8417. Slusher, R. E., L. W. Hollberg, B. Yurke, J. C. Mertz, and Tsang, M., H. M. Wiseman, and C. M. Caves (2011), Phys. J. F. Valley (1985), Physical Review Letters 55 (22), 2409. Rev. Lett. 106, 090401. Smiciklas, M., J. M. Brown, L. W. Cheuk, S. J. Smullin, and Ulam-Orgikh, D., and M. Kitagawa (2001), Phys. Rev. A M. V. Romalis (2011), Physical Review Letters 107 (17), 64 (5), 052106. 10.1103/physrevlett.107.171604. Unden, T., P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Sonier, J. E., J. H. Brewer, and R. F. Kiefl (2000), Reviews Plenio, M. Markham, D. Twitchen, A. Stacey, I. Lovchin- of Modern Physics 72 (3), 769. sky, A. O. Sushkov, M. D. Lukin, A. Retzker, B. Naydenov, Sørensen, A. S., and K. Mølmer (2001), Phys. Rev. Lett. L. P. McGuinness, and F. Jelezko (2016), Phys. Rev. Lett. 86 (20), 4431. 116, 230502. de Sousa, R. (2009), “Electron spin as a spectrometer of Vamivakas, A. N., C.-Y. Lu, C. Matthiesen, Y. Zhao, S. Flt, nuclear-spin noise and other fluctuations,” in Electron Spin A. Badolato, and M. Atatre (2010), Nature 467 (7313), Resonance and Related Phenomena in Low-Dimensional 297. Structures, edited by M. Fanciulli (Springer Berlin Heidel- Vasyukov, D., Y. Anahory, L. Embon, D. Halbertal, J. Cup- berg, Berlin, Heidelberg) pp. 183–220. pens, L. Neeman, A. Finkler, Y. Segev, Y. Myasoedov, M. L. Rappaport, M. E. Huber, and E. Zeldov (2013), 45

Nat. Nano. 8, 639. 180406. Vengalattore, M., J. M. Higbie, S. R. Leslie, J. Guzman, L. E. Woodman, K., P. Franks, and M. Richards (1987), Journal Sadler, and D. M. Stamper-Kurn (2007), Phys. Rev. Lett. of Navigation 40 (03), 366. 98 (20), 200801. Wrachtrup, J., C. von Borczyskowski, J. Bernard, M. Orrit, Vijay, R., D. H. Slichter, and I. Siddiqi (2011), Phys. Rev. and R. Brown (1993a), Nature 363, 244. Lett. 106, 110502. Wrachtrup, J., C. von Borczyskowski, J. Bernard, M. Orrit, Viola, L., and S. Lloyd (1998), Phys. Rev. A 58, 2733. and R. Brown (1993b), Phys. Rev. Lett. 71, 3565. Vion, D., A. Aassime, A. Cottet, P. Joyez, H. Pothier, Xia, H., A. B.-A. Baranga, D. Hoffman, and M. V. Romalis C. Urbina, D. Esteve, and M. Devorett (2002), Science (2006), Appl. Phys. Lett. 89 (211104). 296, 886. Xiang, G. Y.,, Higgins, B. L., Berry, D. W., Wiseman, H. M., Waldherr, G., J. Beck, P. Neumann, R. Said, M. Nitsche, and Pryde, G. J. (2011), Nat Photon 5 (1), 43. M. Markham, D. J. Twitchen, J. Twamley, F. Jelezko, and Xu, N., F. Jiang, Y. Tian, J. Ye, F. Shi, H. Lv, Y. Wang, J. Wrachtrup (2012), Nat. Nanotech. 7 (2), 105. J. Wrachtrup, and J. Du (2016), Phys. Rev. B 93, 161117. Wallraff, A., D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, Yan, F., J. Bylander, S. Gustavsson, F. Yoshihara, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf K. Harrabi, D. G. Cory, T. P. Orlando, Y. Nakamura, J.-S. (2004), Nature 431, 162. Tsai, and W. D. Oliver (2012), Phys. Rev. B 85, 174521. Walls, D. F. (1983), Nature 306 (5939), 141. Yan, F., S. Gustavsson, J. Bylander, X. Jin, F. Yoshihara, Walsh, J. L. (1923), Amer. J. Math. 45 (1), 5. D. G. Cory, Y. Nakamura, T. P. Orlando, and W. D. Oliver Walther, P., J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gaspa- (2013), Nat. Comms. 4, 2337. roni, and A. Zeilinger (2004), Nature 429 (6988), 158. Yoo, M. J., T. A. Fulton, H. F. Hess, R. L. Willett, L. N. Wang, X., and B. C. Sanders (2003), Phys. Rev. A 68 (1), Dunkleberger, R. J. Chichester, L. N. Pfeiffer, and K. W. 012101. West (1997), Science 276 (5312), 579. Wasilewski, W., K. Jensen, H. Krauter, J. J. Renema, M. V. Yoshihara, F., K. Harrabi, A. O. Niskanen, Y. Nakamura, Balabas, and E. S. Polzik (2010), Phys. Rev. Lett. 104, and J. S. Tsai (2006), Phys. Rev. Lett. 97 (16), 167001. 133601. Yoshihara, F., Y. Nakamura, F. Yan, S. Gustavsson, J. By- Waters, G. S., and P. D. Francis (1958), Journal of Scientific lander, W. Oliver, and J.-S. Tsai (2014), Physical Re- Instruments 35 (3), 88. view B - Condensed Matter and Materials Physics 89 (2), Weston, E. (1931), “Exposure Meter,”. 10.1103/PhysRevB.89.020503, cited By 11. Widmann, M., S.-Y. Lee, T. Rendler, N. T. Son, H. Fed- Young, K. C., and K. B. Whaley (2012), Phys. Rev. A 86, der, S. Paik, L.-P. Yang, N. Zhao, S. Yang, I. Booker, 012314. A. Denisenko, M. Jamali, S. A. Momenzadeh, I. Gerhardt, Yuge, T., S. Sasaki, and Y. Hirayama (2011), Phys. Rev. T. Ohshima, A. Gali, E. Janzén, and J. Wrachtrup (2015), Lett. 107, 170504. Nature Mat. 14, 164. Yurke, B. (1986), Phys. Rev. Lett. 56, 1515. Wineland, D. J., J. J. Bollinger, W. M. Itano, and D. J. Zaiser, S., T. Rendler, I. Jakobi, T. Wolf, S. Lee, S. Wagner, Heinzen (1994), Phys. Rev. A 50 (1), 67. V. Bergholm, T. Schulte-herbruggen, P. Neumann, and Wineland, D. J., J. J. Bollinger, W. M. Itano, F. L. Moore, J. Wrachtrup (2016), Nature Communications 7, 12279. and D. J. Heinzen (1992), Phys. Rev. A 46 (11), R6797. Zanche, N. D., C. Barmet, J. A. Nordmeyer-Massner, and Wiseman, H., and G. Milburn (2009), Quantum measurement K. P. Pruessmann (2008), Magn. Reson. Med. 60, 176. and control (Cambridge University Press). Zhang, H., R. McConnell, S. Cuk,´ Q. Lin, M. H. Schleier- Wolf, T., P. Neumann, K. Nakamura, H. Sumiya, T. Ohshima, Smith, I. D. Leroux, and V. Vuletić(2012), Physical Re- J. Isoya, and J. Wrachtrup (2015), Physical Review X view Letters 109 (13), 10.1103/physrevlett.109.133603. 5 (4), 041001. Zhao, N., J. Wrachtrup, and R. B. Liu (2014), Phys. Rev. A Wolfe, C. S., V. P. Bhallamudi, H. L. Wang, C. H. Du, 90, 032319. S. Manuilov, R. M. Teeling-Smith, A. J. Berger, R. Adur, F. Y. Yang, and P. C. Hammel (2014), Phys. Rev. B 89,