Quantum sensing

Many properties of more complex quantum systems can be modeled through a qubit sensor.

“The focus on two-level systems (1) is not a severe restriction because many properties of more complex quantum systems can be modeled through a qubit sensor (Goldstein et al. , 2010).”

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Alkali atoms are suitable sensing qubits fulfilling the definitions of quantum sensing.

“Alkali atoms are suitable sensing qubits fulfilling the above definitions (Kitching et al. , 2011).”

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Trapped atoms have been employed to detect and image magnetic fields at the microscale.

“Trapped atoms have been employed to detect and image magnetic fields at the microscale, by replicating Larmor precession spectroscopy on a trapped Bose-Einstein condensate (Vengalattore et al. , 2007) and by direct driving of spin-flip transitions by microwave currents (Ockeloen et al. , 2013) or thermal radiofrequency noise in samples (Fortagh et al. , 2002; Jones et al. , 2003).”

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• cites Ockeloen et al., 2013Not checked yet

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The most advanced applications of trapped ions employ quantized motional levels as sensing qubits for electric fields and forces.

“The most advanced applications employ the quantized motional levels as sensing qubits for electric fields and forces.”

Rydberg states have become accessible in atomic vapour cells.

“Recently, Rydberg states have become accessible in atomic vapour cells (K¨ ubler et al. , 2010).”

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Electrical field noise is a key limiting factor in ultrasensitive force microscopy.

“Electrical field noise is a severe source of decoherence for ion traps and superconducting quantum processors (Labaziewicz et al. , 2008) and a key limiting factor in ultrasensitive force microscopy (Kuehn et al. , 2006; Tao and Degen, 2015).”

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• cites Tao and Degen, 2015Not checked yet

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The sensitivity of a quantum sensor scales with the transduction parameter γ and the decoherence or relaxation time Tχ.

“In Section V, we will see that the sensitivity scales as”

• Also stated in: § B. Quantum sensors · p. 4 · plain

  “where γ is the above transduction parameter and T χ is a decoherence or relaxation time that reflects the immunity of the quantum sensor against noise.”

Relaxation and coherence times of spins in atomic vapors can be pushed to the second to minute range.

“This is owing to the surprising fact that relaxation and coherence times of spins in atomic vapors can be pushed to the second to minute range (Balabas et al. , 2010).”

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The use of trapped ions in practical applications has proven difficult.

“Their use in practical applications, however, has proven difficult.”

In the simplest implementation, a thermal vapor of atoms serves as a quantum sensor for magnetic fields.

“In the simplest implementation, a thermal vapor of atoms serves as a quantum sensor for magnetic fields (Budker and Romalis, 2007; Kominis et al. , 2003).”

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The ground state spin of alkali atoms can be both prepared and read out optically by the strong spin-selective optical dipole transition linking their s-wave electronic ground state to the first (p-wave) excited state.

“Their ground state spin - a coupled angular momentum of electron and nuclear spin - can be both prepared and read out optically by the strong spin-selective optical dipole transition linking their s-wave electronic ground state to the first (p-wave) excited state.”

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Spin squeezing has improved the sensitivity of magnetometry devices beyond the shot noise limit.

“It has improved the sensitivity of magnetometry devices beyond the shot noise limit (Ockeloen et al. , 2013; Sewell et al. , 2012) and has increased their bandwidth (Shah et al. , 2010).”

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Electrical field noise is a severe source of decoherence for ion traps and superconducting quantum processors.

“Electrical field noise is a severe source of decoherence for ion traps and superconducting quantum processors (Labaziewicz et al. , 2008) and a key limiting factor in ultrasensitive force microscopy (Kuehn et al. , 2006; Tao and Degen, 2015).”

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Vapor cells promise complementary access to high-energy physics, detecting anomalous dipole moments from coupling to exotic elementary particles and background fields beyond the standard model.

“Vapor cells also promise complementary access to high-energy physics, detecting anomalous dipole moments from coupling to exotic elementary particles and background fields beyond the standard model (Pustelny et al. , 2013; Smiciklas et al. , 2011; Swallows et al. , 2013).”

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• cites Swallows et al. , 2013Not checked yet

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Trapped ions have demonstrated magnetic sensitivities down to 4.6 pT/√Hz.

“Being an extremely clean system, trapped ions have demonstrated sensitivities down to 4 . 6 pT / √ Hz (Baumgart et al. , 2016) and served as a testbed for advanced sensing protocols such as dynamical decoupling (Biercuk et al. , 2009; Kotler et al. , 2011) and entanglement-enhanced sensing (Leibfried et al. , 2004).”

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Spin squeezing has increased the bandwidth of magnetometry devices.

“It has improved the sensitivity of magnetometry devices beyond the shot noise limit (Ockeloen et al. , 2013; Sewell et al. , 2012) and has increased their bandwidth (Shah et al. , 2010).”

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Entanglement in the form of spin squeezing has been produced by optical non-destructive measurements of atomic population and atomic interactions.

“Entanglement - in the form of spin squeezing (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. , 2016; Hosten et al. , 2016a; Leroux et al. , 2010a; LouchetChauvet et al. , 2010; Schleier-Smith et al. , 2010b) and atomic interactions (Esteve et al. , 2008; Riedel et al. , 2010).”

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• cites Esteve et al., 2008Not checked yet

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Trapped ions have served as a testbed for advanced sensing protocols such as dynamical decoupling and entanglement-enhanced sensing.

“Being an extremely clean system, trapped ions have demonstrated sensitivities down to 4 . 6 pT / √ Hz (Baumgart et al. , 2016) and served as a testbed for advanced sensing protocols such as dynamical decoupling (Biercuk et al. , 2009; Kotler et al. , 2011) and entanglement-enhanced sensing (Leibfried et al. , 2004).”

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• cites Kotler et al., 2011Not checked yet

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• cites Leibfried et al., 2004Not checked yet

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Trapped ions have been extensively used to study electric field noise above surfaces.

“Trapped ions have been extensively used to study electric field noise above surfaces (Brownnutt et al. , 2015), which could arise from charge fluctuations induced by adsorbents.”

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Rydberg atom single-photon detector sensitivity has been improved by employing Schrödinger cat states to reach 300 nV/m/√Hz.

“Their sensitivity has recently been improved by employing Schr¨ odinger cat states to reach a level of 300nV/m/ √ Hz (Facon et al. , 2016).”

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Rydberg states in vapour cells have been applied to sense weak electric fields, mostly in the GHz frequency range.

“They have been applied to sense weak electric fields, mostly in the GHz frequency range (Fan et al. , 2015; Sedlacek et al. , 2012), and have been suggested as a candidate for a primary traceable standard of microwave power.”

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In atomic gravimeters and gyrometers, an atomic cloud measures acceleration by sensing the spatial phase shift of a laser beam along its freely falling trajectory.

“In these devices an atomic cloud measures acceleration by sensing the spatial phase shift of a laser beam along its freely falling trajectory.”

Vapor cells have been used to demonstrate entanglement-enhanced sensing.

“Vapor cells have been miniaturized to few mm 3 small volumes (Shah et al. , 2007) and have been used to demonstrate entanglement-enhanced sensing (Fernholz et al. , 2008; Wasilewski et al. , 2010).”

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Trapped ions' small size could compensate for their lower absolute sensitivity in applications like microscopy where high spatial resolution is required.

“Their small size could compensate for this downside in applications like microscopy, where high spatial resolution is required.”

A high atomic density suppresses decoherence from atomic interactions because frequent collisions average out their effect, similar to motional narrowing.

“Somewhat counterintuitively, a high density suppresses decoherence from atomic interactions, since collisions occur so frequently that their effect averages out, similar to motional narrowing of dipolar interactions in nuclear magnetic resonance (Happer and Tang, 1973).”

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Vapor cells have found use in magnetoencephalography.

“The most advanced application of vapor cells is arguably the detection of neural activity (Jensen et al. , 2016; Livanov et al. , 1978), which has found use in magnetoencephalography (Xia et al. , 2006).”

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Ground state spin sublevels of ions are magnetic-field-sensitive qubits analogous to neutral atoms.

“Independently, the ground state spin sublevels of ions are magnetic-field-sensitive qubits analogous to neutral atoms discussed above (Baumgart et al. , 2016; Kotler et al. , 2011; Maiwald et al. , 2009).”

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Rydberg atoms in vacuum have been employed as single-photon detectors for microwave photons in a cryogenic cavity.

“As their most spectacular sensing application, Rydberg atoms in vacuum have been employed as single-photon detectors for microwave photons in a cryogenic cavity in a series of experiments that has been highlighted by the Nobel prize in Physics in 2012 (Gleyzes et al. , 2007; Haroche, 2013; Nogues et al. , 1999).”

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The quadratic relation for variance detection holds for small γV_rms t ≪ 1.

“This relation holds for small γV rms t ≪ 1.”

If the magnitude of δV fluctuates between measurements so that 〈δV〉=0, readout at p0=0.5 will yield no information about δV, since 〈p〉≈p0=0.5.

“If the magnitude of δV fluctuates between measurements so that 〈 δV 〉 = 0, readout at p 0 = 0 . 5 will yield no information about δV , since 〈 p 〉 ≈ p 0 = 0 . 5.”

Only time-dependent signals with frequency ω ≈ ω0 affect the transition probability p in a Rabi measurement, a condition known as resonance.

“Hence, only time-dependent signals with frequency ω ≈ ω 0 affect the transition probability p , a condition known as resonance.”

Spin ensembles have served as gyroscopes exploiting Larmor precession frequency shift in a rotating laboratory frame.

“Spin ensembles have equally served as gyroscopes (Fang and Qin, 2012; Woodman et al. , 1987), exploiting the fact that Larmor precession occurs in an independent frame of reference and therefore appears frequencyshifted in a rotating laboratory frame.”

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Today's most advanced atomic clocks employ optical transitions in single ions or atomic clouds trapped in an optical lattice.

“Today's most advanced atomic clocks employ optical transitions in single ions (Huntemann et al. , 2016) or atomic clouds trapped in an optical lattice (Bloom et al. , 2014; Hinkley et al. , 2013; Takamoto et al. , 2005).”

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NV ensemble advantages are partially offset by a reduced coherence time (T2 < 1 ms at room temperature, compared to T2 > 1 s for vapor cells).

“Although these advantages are partially offset by a reduced coherence time ( T 2 &lt; 1 ms at room temperature, as compared to T 2 &gt; 1 s for vapor cells), the predicted sensitivity of diamond magnetometers (250 aT / √ Hz / cm -3 / 2 ) (Taylor et al. , 2008) or gyroscopes (10 -5 rad / s / √ Hz / mm 3 / 2 ) (Ajoy and Cappellaro, 2012; Ledbetter et al. , 2012) would be competitive with their”

Some advanced clocks employ multi-qubit quantum logic gates for readout of highly stable but optically inactive clock ions.

“Interestingly, even entanglement-enhanced sensing has found use in actual devices, since some advanced clocks employ multi-qubit quantum logic gates for readout of highly stable but optically inactive clock ions (Rosenband et al. , 2008; Schmidt et al. , 2005).”

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Quantum sensors can be regarded as clocks operating on a bad, environment-sensitive clock transition to measure external fields.

“Vice versa, quantum sensors discussed above can be regarded as clocks that operate on purpose on a bad, environment-sensitive clock transition in order to measure external fields.”

Translation of NV ensemble potential into actual devices remains challenging.

“Translation of this potential into actual devices remains challenging, with two technical hurdles standing out.”

By recording the transition probability p as a function of time t in a Ramsey measurement, an oscillatory output ('Ramsey fringes') is observed with a frequency given by ω0.

“By recording p as a function of time t , an oscillatory output ('Ramsey fringes') is observed with a frequency given by ω 0 .”

Readout of single spins in the solid state has been achieved both by electrical and optical schemes.

“Readout of single spins in the solid state -a major milestone on the road towards quantum computers -has been achieved both by electrical and optical schemes.”

The Ramsey interferometer is most sensitive to small perturbations δV around V0=ω0/γ when operated at the point of maximum slope where p0=0.5.

“The Ramsey interferometer is most sensitive to small perturbations δV around V 0 = ω 0 /γ when operated at the point of maximum slope where p 0 = 0 . 5, indicated by the filled red dot in Fig. 3(a).”

Quantum projection noise is the most fundamental source of uncertainty in quantum sensing.

“Quantum projection noise is the most fundamental source of uncertainty in quantum sensing.”

Atomic clock operation employs identical protocols as quantum sensors to compare the qubit's transition to an unstable local oscillator and lock the latter to the former.

“Their operation as clocks, however, employs identical protocols as the operation of quantum sensors, in order to repeatedly compare the qubit's transition to the frequency of an unstable local oscillator and subsequently lock the latter to the former.”

Slope detection has a limited linear range because phase wrapping occurs for |γδV t| > π/2.

“Note that slope detection has a limited linear range because phase wrapping occurs for | γδV t | &gt; π/ 2.”

A Rabi measurement can provide information not only on the magnitude V⊥, but also on the frequency ω of a signal.

“From this condition it is clear that a Rabi measurement can provide information not only on the magnitude V ⊥ , but also on the frequency ω of a signal (Aiello et al. , 2013; Fedder et al. , 2011).”

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The Ramsey measurement can directly provide a measurement of the energy splitting ω0.

“Thus, the Ramsey measurement can directly provide a measurement of the energy splitting ω 0 .”

If the fluctuation is Gaussian, the variance detection result can be extended to any large value of γV_rms t.

“If the fluctuation is Gaussian, the result can be extended to any large value of γV rms t ,”

Densely-doped diamond crystals promise to deliver 'frozen vapor cells' of spin ensembles that combine strong electronic magnetic moment and efficient optical readout of atomic vapor cells with high spin densities achievable in solid state.

“Densely-doped diamond crystals promise to deliver 'frozen vapor cells' of spin ensembles that combine the strong (electronic) magnetic moment and efficient optical readout of atomic vapor cells with the high spin densities achievable in the solid state.”

Spin coherence times are reduced 100-1000 times in high-density NV ensembles owing to interaction of NV spins with parasitic substitutional nitrogen spins incorporated during high-density doping.

“Second, spin coherence times are reduced 100 -1000 times in highdensity ensembles owing to interaction of NV spins with parasitic substitutional nitrogen spins incorporated during high-density doping (Acosta et al. , 2009).”

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Magnetic field sensors have been built that infer field strength from Larmor precession of nuclear spins.

“Magnetic field sensors have been built that infer field strength from their Larmor precession, analogous to neutral atom magnetometers described above (Kitching et al. , 2011; Packard and Varian, 1954; Waters and Francis, 1958).”

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If the interferometer is tuned to p0=0, a signal with variance 〈δV^2〉=V_rms^2 gives rise to a mean transition probability that is quadratic in V_rms and t.

“If the interferometer is tuned to p 0 = 0, a signal with variance 〈 δV 2 〉 = V 2 rms gives rise to a mean transition probability that is quadratic in V rms and t (Meriles et al. , 2010),”

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NMR ensemble magnetometers have found broad use in geology, archaeology and space missions thanks to their simplicity and robustness.

“Although the sensitivity of these devices (10 pT / √ Hz) (Lenz, 1990) is inferior to their atomic counterparts, they have found broad use in geology, archaeology and space missions thanks to their simplicity and robustness.”

Phase wrapping restricts the dynamic range of the quantum sensor.

“The phase wrapping restricts the dynamic range of the quantum sensor.”

An atomic clock can be regarded as a quantum sensor measuring and stabilizing the phase drift of a local oscillator.

“Therefore, an atomic clock can be equally regarded as a quantum sensor measuring and stabilizing the phase drift of a local oscillator.”

The detection of AC fields can be extended to the more general task of sensing and reconstructing arbitrary time dependent fields.

“The detection of AC fields can be extended to the more general task of sensing and reconstructing arbitrary time dependent fields.”

For weak signals with long averaging times T ≫ Tχ, variance detection is typically much less sensitive than slope detection.

“Hence, for weak signals with long averaging times T ≫ T χ , variance detection is typically much less sensitive than slope detection.”

The weighting functions of CP and PDD sequences resemble narrow-band filters around center frequencies f_ac = f_k = k/(2τ), where k = 1,3,5,... is the harmonic order.

“Because of the first (sinc) term, these weighting functions resemble narrow-band filters around the center frequencies f ac = f k = k/ (2 τ ), where k = 1 , 3 , 5 , ... is the harmonic order.”

For slope detection, where p0=0.5, the projective readout adds noise of order 1/(2√N) to the probability value p.

“Thus, the projective readout adds noise of order 1 / (2 √ N ) to the probability value p .”

• AssumptionThe authors state this as an assumption: The readout is projective.

These basis-function methods can be combined with compressive sensing techniques to reduce the number of acquisitions needed to reconstruct the time-dependent signal.

“In addition, they can be combined with compressive sensing techniques (Cand´ es et al. , 2006; Magesan et al. , 2013a; Puentes et al. , 2014) to reduce the number of acquisition needed to reconstruct the time-dependent signal.”

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• cites Magesan et al. , 2013aNot checked yet

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• cites Puentes et al. , 2014Not checked yet

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If C(t_m) improves as C ∝ √t_m, the optimum choice is t_m ≈ t.

“In particular, if C ( t m ) does improve as C ∝ √ t m - which is a typical situation when operating in the averaged readout regime - the optimum choice is t m ≈ t .”

Since the qubit is parked in |0> and |1> during the free evolution period, relaxation is no longer governed by T2, but by the typically much longer T1 relaxation time.

“Since the qubit is parked in | 0 〉 and | 1 〉 during the free evolution period, relaxation is no longer governed by T 2 , but by the typically much longer T 1 relaxation time (Laraoui and Meriles, 2013).”

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In continuous sampling, the output signal can be Fourier transformed to extract the undersampled frequency of the original signal.

“The output signal can then be Fourier transformed to extract the undersampled frequency of the original signal.”

Multipulse sensing sequences allow detailed shaping of the frequency response of the quantum sensor.

“These sequences are commonly referred to as multipulse sensing sequences or multipulse control sequences, and allow for a detailed shaping of the frequency response of the quantum sensor.”

For a Ramsey measurement with an exponential dephasing e^-χ(t)=e^-t/T2*, the optimum evolution time is t=T2*/2.

“Specifically, for a Ramsey measurement with an exponential dephasing e -χ ( t ) = e -t/T ∗ 2 , the optimum evolution time is t = T ∗ 2 / 2 and”

• AssumptionThe authors state this as an assumption: t_m ≪ t for the exact optimum solution in slope detection.

In this way, a Fourier-limited spectral resolution of Δf ∼ 1/T1 is possible.

“In this way, a Fourierlimited spectral resolution of ∆ f ∼ 1 /T 1 is possible.”

Sensitivity to alternating signals can be restored by using time-reversal ('spin echo') techniques.

“Sensitivity to alternating signals can be restored by using time-reversal ('spin echo') techniques (Hahn, 1950).”

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Decoherence and relaxation cause random transitions between states or random phase pick-up during coherent evolution of the qubit, leading to a reduction of the observed probability δp with increasing sensing time t.

“Decoherence and relaxation cause random transitions between states or random phase pick-up during coherent evolution of the qubit (for more detail, see Section VII). The two processes lead to a reduction of the observed probability δp with increasing sensing time t ,”

The Allan variance can also be used to reveal the performance of a sensor beyond the standard quantum limit.

“The Allan variance can also be used to reveal the performance of a sensor beyond the standard quantum limit (Leroux et al. , 2010b), and its extension to and limits in quantum metrology have been recently explored (Chabuda et al. , 2016).”

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The Ramsey sequence will only be sensitive to slowly varying signals up to some cut-off frequency ≈ t^-1.

“To answer our question, the Ramsey sequence will only be sensitive to slowly varying signals up to some cut-off frequency ≈ t -1 .”

Since the multipulse sequence is phase sensitive, constructive or destructive phase build-up occurs between the two sequences depending on whether the free evolution period t1 is a half multiple or full multiple of the AC signal period T_ac = 1/f_ac.

“Since the multipulse sequence is phase sensitive, constructive or destructive phase build-up occurs between the two sequences depending on whether the free evolution period t 1 is a half multiple or full multiple of the AC signal period T ac = 1 /f ac .”

Continuous sampling has recently led to the demonstration of μHz spectral resolution.

“Continuous sampling has recently led to the demonstration of µ Hz spectral resolution (Boss et al. , 2017; Jelezko et al. , 2017).”

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The time response of the transition probability for a signal with fixed amplitude and random phase is given by Eq. (75), where J0 is the Bessel function of the first kind.

“The time response of the transition probability is (Kotler et al. , 2013)”

• Also stated in: § 4. AC signals with random phase · p. 22 · plain

  “where J 0 is the Bessel function of the first kind and where the second equation reflects the resonant case.”

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For composite signals consisting of several harmonics with different frequencies and amplitudes, the accumulated phase is the sum of individual tone amplitudes multiplied by the respective weighting functions.

“For composite signals consisting of several harmonics with different frequencies and amplitudes, Eq. (61), the accumulated phase simply represents the sum of individual tone amplitudes multiplied by the respective weighting functions.”

This simple waveform reconstruction approach is limited to the bandwidth of the sequence and by readout deadtimes.

“This approach is, however, limited to the bandwidth of the sequence and by readout deadtimes.”

By measuring N Walsh coefficients (by applying N different sequences), one can reconstruct an N-point functional approximation to the field V(t') from the N-th partial sum of the Walsh-Fourier series.

“By measuring N Walsh coefficients (by applying N different sequences) one can reconstruct an N -point functional approximation to the field V ( t ′ ) from the N -th partial sum of the Walsh-Fourier series (Cooper et al. , 2014; Magesan et al. , 2013b)”

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It is possible to correlate the phase acquired during two subsequent multipulse sequences to perform high-resolution spectroscopy of AC signals.

“Furthermore, it is possible to correlate the phase acquired during two subsequent multipulse sequences to perform high-resolution spectroscopy of AC signals (see Section VI.E).”

A simple approach to waveform reconstruction is to record the time response p(t) under a specific sensing sequence, such as a Ramsey sequence, and to reconstruct the phase φ(t) and signal V(t) from the time trace.

“A simple approach is to record the time response p ( t ) under a specific sensing sequence, such as a Ramsey sequence, and to reconstruct the phase φ ( t ) and signal V ( t ) from the time trace (Balasubramanian et al. , 2009).”

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The Fisher information for the Ramsey protocol oscillates between its minimum, where γV t = (k+1/2)π and F=0, and its optimum, where γV t = kπ and F=t^2 e^-2χ.

“The Fisher information thus oscillates between its minimum, where γV t = ( k + 1 / 2) π and F = 0, and its optimum, where γV t = kπ and F = t 2 e -2 χ .”

• AssumptionThe authors state this as an assumption: A projective measurement in the σ_x basis is performed for the Fisher information.

The spectral resolution of dynamical decoupling spectroscopy is determined by the bandwidth of the weighting function W(f_ac, τ), which is given by Δf ≈ 1/t.

“The spectral resolution of dynamical decoupling spectroscopy is determined by the bandwidth of the weighting function W ( f ac , τ ), which is given by ∆ f ≈ 1 /t (see Eq. (69)).”

To reach the quantum Cramér-Rao bound, local measurements and at most adaptive estimators are sufficient, without the need for entanglement.

“Thus, to reach the QCRB, local measurements and at most adaptive estimators are sufficient, without the need for entanglement.”

The uncertainty in p is given by the variance of the binomial distribution.

“The uncertainty in p is given by the variance of the binomial distribution (Itano et al. , 1993),”

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• AssumptionThe authors state this as an assumption: The readout is projective.

Because t can only be made as long as the decoherence time T2, the spectral resolution is limited to Δf ≈ 1/T2.

“Because t can only be made as long as the decoherence time T 2 , the spectral resolution is limited to ∆ f ≈ 1 /T 2 .”

The precision of frequency estimation, which also depends on the signal-to-noise ratio, is directly proportional to Δf.

“The precision of the frequency estimation, which also depends on the signal-to-noise ratio, is directly proportional to ∆ f .”

Because the decay function χ(t) exponentially penalizes the sensitivity for t > Tχ, the optimum sensing time is reached when t ≈ Tχ.

“However, because the decay function χ ( t ) exponentially penalizes the sensitivity for t &gt; T χ , the optimum sensing time is reached when t ≈ T χ .”

The phase φ can be recovered by Bayesian estimation.

“The phase φ can be recovered by Bayesian estimation.”

The Heisenberg limit can only be reached if the number of resources (qubits or repetitions) spent on each bit are carefully optimized.

“The Heisenberg limit can only be reached if the number of resources (qubits or repetitions) spent on each bit are carefully optimized (Berry et al. , 2009; Cappellaro, 2012; Said et al. , 2011).”

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The transition rate equals the spectral density of the noise evaluated at the transition frequency, multiplied by a matrix element of order unity.

“The last equation (97) is an extremely simple, yet powerful and quantitative relationship: the transition rate equals the spectral density of the noise evaluated at the transition frequency, multiplied by a matrix element of order unity (Abragam, 1961; Schoelkopf et al. , 2003).”

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To provide an estimate of φ with 2^{-M} accuracy, 2^M applications of the phase shift U are required.

“To provide an estimate of φ with 2 -M accuracy, 2 M applications of the phase shift ˆ U are required.”

Given a redundant set of phases, a post-processing algorithm can mimic the adaptive algorithm by postselecting those results that have been measured using the phase most closely resembling the correct adaptive choice.

“given a redundant set of phases, a post-processing algorithm can mimic the adaptive algorithm by postselecting those results that have been measured using the phase most closely resembling the correct adaptive choice.”

T1 relaxometry is only sensitive to the transverse component of V.

“Thus, T 1 relaxometry is only sensitive to the transverse component of ⃗ V .”

The smallest detectable signal is approximately V_min ≈ 2/(γC√(T2* T)), where T is total measurement time and C is readout efficiency.

“The smallest detectable signal is approximately V min ≈ 2 / ( γC √ T ∗ 2 T ), where T is the total measurement time and C the readout efficiency parameter (see Eqs. 43 and 47).”

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A detuning increases the accessible spectral range towards higher frequencies.

“A detuning therefore increases the accessible spectral range towards higher frequencies.”

The limitation of QFT-based phase estimation can be circumvented by an adaptive measurement scheme that reads the qubits sequentially, feeding back the classical measurement result into the quantum circuit.

“This limitation can be circumvented by an adaptive measurement scheme that reads the qubits sequentially, feeding back the classical measurement result into the quantum circuit (Griffiths and Niu, 1996).”

• cites Griffiths and Niu, 1996Not checked yet

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The Ramsey sequence acts as a simple sinc filter for the noise spectrum S_V(ω), centered at zero frequency and with a lowpass cut-off frequency of approximately π/t.

“The Ramsey sequence hence acts as a simple sinc filter for the noise spectrum S V ( ω ), centered at zero frequency and with a lowpass cut-off frequency of approximately π/t .”

This set of measurements contains all the information required to extract φ.

“This set of measurements still contains all the information required to extract φ , which can be motivated by the following arguments:”

Given a measured transition probability p, there is an infinite number of possible signal amplitudes V that can correspond to this value of p.

“Given a measured transition probability p , there is an infinite number of possible signal amplitudes V that can correspond to this value of p”

The basic measurement protocol can be applied only to small changes of a quantity around a fixed known value, frequently zero.

“Hence, the basic measurement protocol can be applied only to small changes of a quantity around a fixed known value, frequently zero.”

The basic quantum sensing protocol cannot simultaneously achieve high sensitivity and measure signals over a large amplitude range.

“The basic quantum sensing protocol cannot simultaneously achieve high sensitivity and measure signals over a large amplitude range .”

Because ω1 ≪ ω0, dressed states provide useful means for covering the medium frequency range of the spectrum.

“Because ω 1 ≪ ω 0 , dressed states provide useful means for covering the medium frequency range of the spectrum (see Fig. 10).”

The dynamic range of the basic protocol only improves with the square root of the total measurement time T.

“Moreover, the dynamic range only improves with the square root of the total measurement time T .”

In a Ramsey measurement, phase wrapping occurs once γVt exceeds ±π/2, where t is the sensing time.

“For the example of a Ramsey measurement, p oscillates with the signal amplitude V and phase wrapping occurs once γV t exceeds ± π/ 2, where t is the sensing time.”

A family of advanced sensing techniques can achieve a dynamic range that scales close to DR ∝ T.

“Interestingly, a family of advanced sensing techniques can efficiently address this problem and achieve a dynamic range that scales close to DR ∝ T .”

Relaxometry is based on first-order perturbation theory and Fermi's golden rule.

“Relaxometry is based on first-order perturbation theory and Fermi's golden rule.”

• cites Abragam, 1961Not checked yet

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This result provides a simple strategy for reconstructing the noise spectrum: by sweeping the time τ between pulses, the spectrum can be probed at various frequencies.

“This result provides a simple strategy for reconstructing the noise spectrum. By sweeping the time τ between pulses the spectrum can be probed at various frequencies.”

A unique assignment of V from p requires a priori knowledge that V lies within ±π/(2γt) of an expected signal amplitude.

“A unique assignment hence requires a priori knowledge - that V lies within ± π/ (2 γt ), or within half a Ramsey fringe, of an expected signal amplitude.”

At least one qubit of every digit will perform its measurement on the slope of a Ramsey fringe, allowing for a precise measurement of 2^m φ regardless of its value.

“Hence, at least one qubit of every digit will perform its measurement on the slope of a Ramsey fringe, allowing for a precise measurement of 2 m φ regardless of its value.”

The origin of the limited dynamic range problem lies in the limited range of values for the probability p, which must fall between 0 and 1.

“The origin of this problem lies in the limited range of values for the probability p , which must fall between 0 and 1.”

Most resources should be allocated to the most significant bit, because errors at this stage are most detrimental to sensitivity.

“Clearly, most resources should be allocated to the most significant bit, because errors at this stage are most detrimental to sensitivity.”

Detuned drive fields have been used to map the 1/f noise spectrum of transmon qubits up to the GHz range.

“Detuned drive fields have been used to map the 1 /f noise spectrum of transmon qubits up to the GHz range (Slichter et al. , 2012).”

• cites Slichter et al., 2012Not checked yet

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Every measurement x_j = ±1 provides information about φ, described by the a posteriori probability.

“Every measurement x j = ± 1 provides information about φ , which is described by the a posteriori probability”

The central idea of advanced sensing is to combine measurements with different sensing times such that the least sensitive measurement with the highest V_max yields a coarse estimate, subsequently refined by more sensitive measurements.

“The central idea is to combine measurements with different sensing times t such that the least sensitive measurement with the highest V max yields a coarse estimate of the quantity of interest, which is subsequently refined by more sensitive measurements (Fig. 11).”

The inverse quantum Fourier transform (QFT) can be implemented with polynomial effort, using O(M^2) control gates.

“This algorithm can be implemented with polynomial effort ( i.e. , using O ( M 2 ) control gates).”

• cites Nielsen and Chuang, 2000Not checked yet

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From a spectroscopic point of view, measurements with different phases correspond to Ramsey fringes with different quadratures.

“From a spectroscopic point of view, measurements with different phases correspond to Ramsey fringes with different quadratures.”

Efficient quantum phase estimation can also be implemented without adaptive feedback, with the advantage of technical simplicity.

“Efficient quantum phase estimation can also be implemented without adaptive feedback, with the advantage of technical simplicity (Higgins et al. , 2009).”

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The basic protocol does not apply to determining the value of a large and a priori unknown quantity.

“The protocol does not apply to the problem of determining the value of a large and a priori unknown quantity.”

In practice, it is sometimes difficult to achieve a √M scaling because instrumental stability is more critical for ensemble sensors.

“In practice, it is sometimes difficult to achieve a √ M scaling because instrumental stability is more critical for ensemble sensors (Wolf et al. , 2015).”

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Already for small M, it is possible to show sensitivity beyond the standard quantum limit.

“Already for small M , it is possible to show sensitivity beyond the standard quantum limit (Kuzmich and Mandel, 1998).”

• cites Kuzmich and Mandel, 1998Not checked yet

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Heisenberg scaling in the temporal dimension shifted into focus with an experiment published in 2007, where adaptive phase estimation was employed in a single-photon interferometer.

“Heisenberg scaling in the temporal dimension has shifted into focus with an experiment published in 2007, where adaptive phase estimation was employed in a single-photon interferometer (Higgins et al. , 2007).”

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For ensemble sensors such as atomic vapor magnetometers or spin arrays, the quantity of interest is more likely the number density of qubits, rather than the absolute number of qubits M.

“For ensemble sensors such as atomic vapor magnetometers or spin arrays, the quantity of interest is more likely the number density of qubits, rather than the absolute number of qubits M .”

High-dynamic-range protocols have been demonstrated on NV centers in diamond using both nonadaptive and adaptive implementations.

“Meanwhile, high-dynamic-range protocols have been demonstrated on NV centers in diamond using both nonadaptive implementations (Nusran et al. , 2012; Waldherr et al. , 2012) and an adaptive protocol based on quantum feedback (Bonato, C. et al. , 2016).”

• cites Nusran et al., 2012Not checked yet

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• cites Waldherr et al., 2012Not checked yet

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• cites Bonato, C. et al., 2016Not checked yet

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The use of M qubits accelerates the measurement by a factor of M, because the basic quantum sensing cycle can now be operated in parallel rather than sequentially.

“The use of M qubits accelerates the measurement by a factor of M , because the basic quantum sensing cycle (Steps 1-5 of the protocol, Fig. 2) can now be operated in parallel rather than sequentially.”

The maximum density of qubits is limited by internal interactions between the qubits.

“The maximum density of qubits is limited by internal interactions between the qubits.”

Bayesian estimation is slower than adaptive estimation due to additional redundant measurements.

“Bayesian estimation in turn is slower than adaptive estimation due to additional redundant measurements.”

The scaling can be seen as arising from the projection noise associated with measuring the quantum system, where it is often called the Standard Quantum Limit (SQL) or shot noise limit.

“The scaling can be seen as arising from the projection noise associated with measuring the quantum system, where it is often called the Standard Quantum Limit (SQL) (Braginskii and Vorontsov, 1975; Giovannetti et al. , 2004) or shot noise limit.”

• cites Braginskii and Vorontsov, 1975Not checked yet

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• cites Giovannetti et al., 2004Not checked yet

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Adaptive estimation is slower than quantum phase estimation by the QFT since it trades spatial resources (entanglement) into temporal resources (measurement time).

“Adaptive estimation is slower than quantum phase estimation by the QFT since it trades spatial resources (entanglement) into temporal resources (measurement time).”

These entangled states promise Heisenberg scaling in the spatial dimension (number of qubits) rather than time.

“These states promise Heisenberg scaling in the spatial dimension (number of qubits) rather than time (Giovannetti et al. , 2004, 2006; Lee et al. , 2002)”

• cites Giovannetti et al., 2004, 2006Not checked yet

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• cites Lee et al., 2002Not checked yet

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This scaling is equivalent to the situation where M classical sensors are operated in parallel.

“This scaling is equivalent to the situation where M classical sensors are operated in parallel.”

The use of multiple qubits opens up many additional possibilities that cannot be implemented on a single qubit sensor.

“The use of multiple qubits opens up many additional possibilities that cannot be implemented on a single qubit sensor.”

This in turn requires preparing the quantum sensors in an entangled state.

“This in turn requires preparing the quantum sensors in an entangled state.”

Equivalently, M parallel qubits can improve the sensitivity by √M per unit time.

“Equivalently, M parallel qubits can improve the sensitivity by √ M per unit time.”

Heisenberg-limited sensitivity with a GHZ state was demonstrated using three entangled Be ions.

“Heisenberg-limited sensitivity with a GHZ state was demonstrated using three entangled Be ions (Leibfried et al. , 2004).”

• cites Leibfried et al., 2004Not checked yet

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Assuming each probe is subjected to uncorrelated dephasing noise, the rate of decoherence of the GHZ state is M times faster than for a product state.

“Assuming, for example, that each probe is subjected to uncorrelated dephasing noise, the rate of decoherence of the GHZ state is M time faster than for a product state.”

If the phase is applied only to mode a of the interferometer, the phase accumulated is M times larger than for a one-photon state.

“If the phase is applied only to mode a of the interferometer, the phase accumulated is then”

• Also stated in: § 1. GHZ and N00N states · p. 33 · plain

  “that is, M times larger than for a one-photon state.”

More recently, atom-light interactions in high-quality cavities have enabled squeezing of large ensembles atoms that can perform as atomic clocks beyond the standard quantum limit.

“More recently, atom-light interactions in high-quality cavities have enabled squeezing of large ensembles atoms (Bohnet et al. , 2014; Cox et al. , 2016; Gross et al. , 2010; Hosten et al. , 2016a; Leroux et al. , 2010a; LouchetChauvet et al. , 2010; Schleier-Smith et al. , 2010a) that can perform as atomic clocks beyond the standard quantum limit.”

• cites Bohnet et al., 2014Not checked yet

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• cites Cox et al., 2016Not checked yet

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• cites Gross et al., 2010Not checked yet

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The quantum state |ψ〉 can be stored and retrieved in an auxiliary qubit.

“The quantum state | ψ 〉 can be stored and retrieved in the auxiliary qubit.”

• cites Rosskopf et al. , 2016Not checked yet

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More generally, one can measure a many-body Hamiltonian term with a strength proportional to the signal to be estimated.

“More generally, one can measure a many-body Hamiltonian term with a strength proportional to the signal to be estimated (Herrera-Mart´ ı et al. , 2015).”

• cites Herrera-Mart´ ı et al. , 2015Not checked yet

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In dynamical decoupling schemes, a separation in the frequency of noise and signal is required.

“For example, in dynamical decoupling schemes, a separation in the frequency of noise and signal is required.”

This imposes an additional condition on a QEC code: the quantum Fisher information in the code subspace must be non-zero.

“This imposes an additional condition on a QEC code: the quantum Fisher information (Giovannetti et al. , 2011; Lu et al. , 2015) in the code subspace must be non-zero.”

• cites Giovannetti et al. , 2011Not checked yet

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• cites Lu et al. , 2015Not checked yet

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Similar to GHZ and N00N states, a key ingredient to this sensitivity enhancement is entanglement.

“Similar to GHZ and N00N states, a key ingredient to this sensitivity enhancement is entanglement (Sørensen and Mølmer, 2001; Wang and Sanders, 2003).”

• cites Sørensen and Mølmer, 2001Not checked yet

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• cites Wang and Sanders, 2003Not checked yet

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A challenge in achieving the full potential of multiqubit enhanced metrology is the widespread inefficiency of quantum state readout.

“A challenge in achieving the full potential of multiqubit enhanced metrology is the widespread inefficiency of quantum state readout.”

Repetitive (quantum non-demolition) readout of the final qubit state can be used to reduce classical readout noise.

“A second example includes a repetitive (quantum non-demolition) readout of the final qubit state, which can be used to reduce the classical readout noise (Jiang et al. , 2009).”

• cites Jiang et al. , 2009Not checked yet

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These states can help improving the sensitivity of quantum interferometry, as demonstrated by Wineland et al. (1992, 1994).

“Thus, these states can help improving the sensitivity of quantum interferometry, as demonstrated by Wineland et al. , 1992,Wineland et al. , 1994.”

• cites Wineland et al., 1992Not checked yet

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• cites Wineland et al., 1994Not checked yet

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Storage can be achieved by a SWAP gate (or more simply two consecutive c-NOT gates) on the sensing and auxiliary qubits.

“Storage can be achieved by a SWAP gate (or more simply two consecutive c-NOT gates) on the sensing and auxiliary qubits, respectively (Rosskopf et al. , 2016).”

• cites Rosskopf et al. , 2016Not checked yet

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The simplest scheme for QEC is to use a single good qubit (unaffected by noise) to protect the sensor qubit.

“The simplest scheme for QEC is to use a single good qubit (unaffected by noise) to protect the sensor qubit (Arrad et al. , 2014; Hirose and Cappellaro, 2016; Kessler et al. , 2014; Ticozzi and Viola, 2006).”

• cites Arrad et al. , 2014Not checked yet

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• cites Hirose and Cappellaro, 2016Not checked yet

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• cites Kessler et al. , 2014Not checked yet

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If the signal to be detected couples to the sensor in a similar way as the noise, the QEC code also eliminates the effect of the signal.

“In particular, if the signal to be detected couples to the sensor in a similar way as the noise, the QEC code also eliminates the effect of the signal.”

Beyond single qubits, QEC has also been applied to N00N states.

“Beyond single qubits, QEC has also been applied to N00N states (Bergmann and van Loock, 2016).”

• cites Bergmann and van Loock, 2016Not checked yet

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Equal superpositions of these two logical basis states are optimal to achieve Heisenberg-limited sensitivity in estimating a global phase.

“Note that (D¨ ur et al. , 2014; Ozeri, 2013) equal superpositions of these two logical basis states are also optimal to achieve Heisenberglimited sensitivity in estimating a global phase (Bollinger et al. , 1996; Leibfried et al. , 2004).”

• cites D¨ ur et al. , 2014Not checked yet

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• cites Ozeri, 2013Not checked yet

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• cites Bollinger et al. , 1996Not checked yet

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Qubit protection can typically only be applied against errors that are orthogonal to the signal, because otherwise the signal itself would be 'corrected'.

“On the other hand, qubit protection can typically only be applied against errors that are orthogonal to the signal, because otherwise the signal itself would be 'corrected'.”

Since this can typically only be achieved in a perturbative way, this scheme still leads to a compromise between noise suppression and effective signal strength.

“Since this can typically only be achieved in a perturbative way, this scheme still leads to a compromise between noise suppression and effective signal strength.”

Since the noise strength is typically much weaker than the noise fluctuation rate, the correction steps can be performed at a much slower rate compared to dynamical decoupling.

“Since the noise strength is typically much weaker than the noise fluctuation rate, the correction steps can be performed at a much slower rate compared to dynamical decoupling.”

• AssumptionRests on an assumption: The noise strength is typically much weaker than the noise fluctuation rate.

For vector fields, quantum error correction can be used to protect against noise in one spatial direction while leaving the sensor responsive to signals in the orthogonal spatial direction.

“In particular, for vector fields, quantum error correction can be used to protect against noise in one spatial direction while leaving the sensor responsive to signals in the orthogonal spatial direction.”

This compromise between error suppression and preservation of signal sensitivity is common to other error correction methods.

“This compromise between error suppression and preservation of signal sensitivity is common to other error correction methods.”

QEC often cannot discriminate between signal and noise.

“While this seems to indicate that QEC codes could be extremely useful for metrology, the method is hampered by the fact that QEC often cannot discriminate between signal and noise.”

Squeezing has also been extended to fermionic degrees of freedom (spin squeezing) to reduce the uncertainty in spectroscopy measurements of ensemble of qubit probes.

“Squeezing has also been extended to fermionic degrees of freedom (spin squeezing, Kitagawa and Ueda, 1993) to reduce the uncertainty in spectroscopy measurements of ensemble of qubit probes.”

• cites Kitagawa and Ueda, 1993Not checked yet

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Because the 'good' ancillary qubit can only protect against one error event (or, equivalently, suppress the error probability for continuous error), the signal acquisition must be periodically interrupted to perform a corrective step.

“Because the 'good' ancillary qubit can only protect against one error event (or, equivalently, suppress the error probability for continuous error), the signal acquisition must be periodically interrupted to perform a corrective step.”