Quantum sensing

In the paper

“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).”

What the sources say

• Not checked yetcites Goldstein et al. , 2010

In the paper

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

What the sources say

• Not checked yetcites Kitching et al. , 2011

In the paper

“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).”

What the sources say

• Not checked yetcites Vengalattore et al., 2007
• Not checked yetcites Ockeloen et al., 2013
• Not checked yetcites Fortagh et al., 2002

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Kübler et al., 2010

In the paper

“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).”

What the sources say

• Not checked yetcites Kuehn et al., 2006
• Not checked yetcites Tao and Degen, 2015

In the paper

“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.”

In the paper

“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).”

What the sources say

• Not checked yetcites Balabas et al. , 2010

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Budker and Romalis, 2007
• Not checked yetcites Kominis et al. , 2003

In the paper

“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.”

What the sources say

• Not checked yetcites Kitching et al. , 2011

In the paper

“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).”

What the sources say

• Not checked yetcites Ockeloen et al., 2013
• Not checked yetcites Sewell et al., 2012

In the paper

“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).”

What the sources say

• Not checked yetcites Labaziewicz et al., 2008

In the paper

“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).”

What the sources say

• Not checked yetcites Pustelny et al. , 2013
• Not checked yetcites Smiciklas et al. , 2011
• Not checked yetcites Swallows et al. , 2013

In the paper

“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).”

What the sources say

• Not checked yetcites Baumgart et al., 2016

In the paper

“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).”

What the sources say

• Not checked yetcites Shah et al., 2010

In the paper

“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).”

What the sources say

• Not checked yetcites Wineland et al., 1992
• Not checked yetcites Appel et al., 2009
• Not checked yetcites Esteve et al., 2008

In the paper

“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).”

What the sources say

• Not checked yetcites Biercuk et al., 2009
• Not checked yetcites Kotler et al., 2011
• Not checked yetcites Leibfried et al., 2004

In the paper

“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.”

What the sources say

• Not checked yetcites Brownnutt et al., 2015

In the paper

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

What the sources say

• Not checked yetcites Facon et al., 2016

In the paper

“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.”

What the sources say

• Not checked yetcites Fan et al., 2015
• Not checked yetcites Sedlacek et al., 2012

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Fernholz et al. , 2008
• Not checked yetcites Wasilewski et al. , 2010

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Happer and Tang, 1973

In the paper

“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).”

What the sources say

• Not checked yetcites Xia et al. , 2006

In the paper

“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).”

What the sources say

• Not checked yetcites Baumgart et al., 2016
• Not checked yetcites Kotler et al., 2011
• Not checked yetcites Maiwald et al., 2009

In the paper

“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).”

What the sources say

• Not checked yetcites Gleyzes et al., 2007
• Not checked yetcites Haroche, 2013
• Not checked yetcites Nogues et al., 1999

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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.”

What the sources say

• Not checked yetcites Fang and Qin, 2012
• Not checked yetcites Woodman et al., 1987

In the paper

“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).”

What the sources say

• Not checked yetcites Huntemann et al., 2016
• Not checked yetcites Bloom et al., 2014
• Not checked yetcites Takamoto et al., 2005

In the paper

“Although these advantages are partially offset by a reduced coherence time ( T 2 < 1 ms at room temperature, as compared to T 2 > 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”

In the paper

“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).”

What the sources say

• Not checked yetcites Rosenband et al., 2008
• Not checked yetcites Schmidt et al., 2005

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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).”

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Aiello et al., 2013
• Not checked yetcites Fedder et al., 2011

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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).”

What the sources say

• Not checked yetcites Acosta et al., 2009

In the paper

“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).”

What the sources say

• Not checked yetcites Kitching et al., 2011
• Not checked yetcites Packard and Varian, 1954
• Not checked yetcites Waters and Francis, 1958

In the paper

“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),”

What the sources say

• Not checked yetcites Meriles et al., 2010

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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.”

What the sources say

• Not checked yetcites Cand´ es et al. , 2006
• Not checked yetcites Magesan et al. , 2013a
• Not checked yetcites Puentes et al. , 2014

In the paper

“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 .”

In the paper

“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).”

What the sources say

• Not checked yetcites Laraoui and Meriles, 2013

In the paper

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

In the paper

“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.”

In the paper

“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”

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Hahn, 1950

In the paper

“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 ,”

In the paper

“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).”

What the sources say

• Not checked yetcites Leroux et al., 2010b
• Not checked yetcites Chabuda et al., 2016

In the paper

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

In the paper

“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 .”

In the paper

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

What the sources say

• Not checked yetcites Boss et al. , 2017
• Not checked yetcites Jelezko et al. , 2017

In the paper

“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.”

What the sources say

• Not checked yetcites Kotler et al. , 2013

In the paper

“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.”

In the paper

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

In the paper

“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)”

What the sources say

• Not checked yetcites Cooper et al. , 2014 source
• Not checked yetcites Magesan et al. , 2013b

In the paper

“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).”

In the paper

“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).”

What the sources say

• Not checked yetcites Balasubramanian et al. , 2009

In the paper

“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 .”

In the paper

“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)).”

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Itano et al., 1993

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Berry et al., 2009
• Not checked yetcites Cappellaro, 2012
• Not checked yetcites Said et al., 2011

In the paper

“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).”

What the sources say

• Not checked yetcites Abragam, 1961
• Not checked yetcites Schoelkopf et al. , 2003

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Eqs. 43 and 47

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Griffiths and Niu, 1996

In the paper

“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 .”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Abragam, 1961

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Slichter et al., 2012

In the paper

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

In the paper

“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).”

In the paper

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

What the sources say

• Not checked yetcites Nielsen and Chuang, 2000

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Higgins et al., 2009

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Wolf et al., 2015

In the paper

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

What the sources say

• Not checked yetcites Kuzmich and Mandel, 1998

In the paper

“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).”

What the sources say

• Not checked yetcites Higgins et al., 2007

In the paper

“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 .”

In the paper

“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).”

What the sources say

• Not checked yetcites Nusran et al., 2012
• Not checked yetcites Waldherr et al., 2012
• Not checked yetcites Bonato, C. et al., 2016

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

“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.”

What the sources say

• Not checked yetcites Braginskii and Vorontsov, 1975
• Not checked yetcites Giovannetti et al., 2004

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Giovannetti et al., 2004, 2006
• Not checked yetcites Lee et al., 2002

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites Leibfried et al., 2004

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

What the sources say

• Not checked yetcites Bohnet et al., 2014
• Not checked yetcites Cox et al., 2016
• Not checked yetcites Gross et al., 2010

In the paper

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

What the sources say

• Not checked yetcites Rosskopf et al. , 2016

In the paper

“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).”

What the sources say

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

In the paper

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

In the paper

“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.”

What the sources say

• Not checked yetcites Giovannetti et al. , 2011
• Not checked yetcites Lu et al. , 2015

In the paper

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

What the sources say

• Not checked yetcites Sørensen and Mølmer, 2001
• Not checked yetcites Wang and Sanders, 2003

In the paper

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

In the paper

“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).”

What the sources say

• Not checked yetcites Jiang et al. , 2009

In the paper

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

What the sources say

• Not checked yetcites Wineland et al., 1992
• Not checked yetcites Wineland et al., 1994

In the paper

“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).”

What the sources say

• Not checked yetcites Rosskopf et al. , 2016

In the paper

“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).”

What the sources say

• Not checked yetcites Arrad et al. , 2014
• Not checked yetcites Hirose and Cappellaro, 2016
• Not checked yetcites Kessler et al. , 2014

In the paper

“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.”

In the paper

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

What the sources say

• Not checked yetcites Bergmann and van Loock, 2016

In the paper

“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).”

What the sources say

• Not checked yetcites D¨ ur et al. , 2014
• Not checked yetcites Ozeri, 2013
• Not checked yetcites Bollinger et al. , 1996

In the paper

“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'.”

In the paper

“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.”

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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.”

What the sources say

• Not checked yetcites Kitagawa and Ueda, 1993

In the paper

“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.”