Quantum sensing

In the paper

“Freely falling atoms have enabled the development of atomic gravimeters (Kasevich and Chu, 1992; Peters et al. , 1999) and gyrometers (Gustavson et al. , 1997, 2000).”

What the sources say

• Not checked yetcites Kasevich and Chu, 1992
• Not checked yetcites Peters et al. , 1999
• Not checked yetcites Gustavson et al. , 1997, 2000

In the paper

“These long relaxation and coherence times are achieved by coating cell walls to preserve the atomic spin upon collisions, and by operating in the spin exchange relaxation-free ('SERF') regime of high atomic density and zero magnetic field.”

In the paper

“The sensor has a predicted sensitivity of 500 nV / m / √ Hz or 1 yN / √ Hz for the force acting on the ion (Biercuk et al. , 2010; Maiwald et al. , 2009).”

What the sources say

• Not checked yetcites Biercuk et al., 2010
• Not checked yetcites Maiwald et al., 2009

In the paper

“The reduced velocity spread of cold atoms enabled sensing with longer interrogation times using spatially confined atoms, freely falling along specific trajectories in vacuum or trapped.”

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 Jensen et al. , 2016
• Not checked yetcites Livanov et al. , 1978

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 Shah et al. , 2007

In the paper

“Sensing with cold atoms has found application in solid state physics by elucidating current transport in microscopic conductors (Aigner et al. , 2008).”

What the sources say

• Not checked yetcites Aigner et al., 2008

In the paper

“However, operation of ion traps in close proximity to surfaces remains a major challenge.”

In the paper

“Rydberg atoms - atoms in highly excited electronic states - are remarkable quantum sensors for electric fields for a similar reason as trapped ions: In a classical picture, the loosely confined electron in a highly excited orbit is easily displaced by electric fields. In a quantum picture, its motional states are coupled by strong electric dipole transitions and experience strong Stark shifts (Her”

What the sources say

• Not checked yetcites Herrmann et al., 1986
• Not checked yetcites Osterwalder and Merkt, 1999

In the paper

“In order to optimize the sensitivity, γ should be large (for example, by choice of an appropriate physical realization of the sensor) and the decoherence time T χ must be made as long as possible.”

In the paper

“Practically all sensing demonstrations have focused on single ions, which, in terms of absolute sensitivity, cannot compete with ensemble sensors such as atomic vapors.”

In the paper

“Some of the earliest quantum sensors have been based on ensembles of nuclear spins.”

In the paper

“Among all solid state spins, NV centers in diamond have received by far the most attention for sensing purposes.”

In the paper

“Although the above protocol is generic and simple, it is sufficient to describe most sensing experiments.”

In the paper

“Variance detection is especially important for detecting ac signals when their synchronization with the sensing protocol is not possible (Section VI.C.4), or when the signal represents a noise source (Section VII).”

In the paper

“As a technically less demanding application, NV centers in a magnetic field gradient have been employed as spectrum analyzer for high frequency microwave signals (Chipaux et al. , 2015).”

What the sources say

• Not checked yetcites Chipaux et al., 2015

In the paper

“In the most advanced implementation, nuclear spin precession is read out by an atomic magnetometer, which is equally used for compensation of the Zeeman shift (Kornack et al. , 2005).”

What the sources say

• Not checked yetcites Kornack et al., 2005

In the paper

“In addition, mechanical detection of single paramagnetic defects in silica (Rugar et al. , 2004) and real-time monitoring of few-spin fluctuations (Budakian et al. , 2005) have been demonstrated.”

What the sources say

• Not checked yetcites Rugar et al., 2004
• Not checked yetcites Budakian et al., 2005

In the paper

“As a consequence, even the most advanced devices are currently limited to ∼ 1 pT / √ Hz (Wolf et al. , 2015) and operate several orders of magnitude above the theory limit.”

What the sources say

• Not checked yetcites Wolf et al., 2015

In the paper

“Optical readout was shown with single organic molecules (Wrachtrup et al. , 1993a,b), optically active quantum dots (Atature et al. , 2007; Kroutvar et al. , 2004; Vamivakas et al. , 2010), and defect centers in crystalline materials including diamond (Gruber et al. , 1997) and silicon carbide (Christle et al. , 2015; Widmann et al. , 2015).”

What the sources say

• Not checked yetcites Wrachtrup et al., 1993a,b
• Not checked yetcites Atature et al., 2007
• Not checked yetcites Gruber et al., 1997

In the paper

“More recently, NMR sensor probes have been developed for in-situ and dynamical field mapping in clinical MRI systems (Zanche et al. , 2008).”

What the sources say

• Not checked yetcites Zanche et al., 2008

In the paper

“Electrical readout has been demonstrated with phosphorus dopants in silicon (Morello et al. , 2010) and electrostatically-defined semiconductor quantum dots (Elzerman et al. , 2004).”

What the sources say

• Not checked yetcites Morello et al., 2010
• Not checked yetcites Elzerman et al., 2004

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”

What the sources say

• Not checked yetcites Taylor et al., 2008
• Not checked yetcites Ajoy and Cappellaro, 2012
• Not checked yetcites Ledbetter et al., 2012

In the paper

“Most prominently, this line of research has enabled imaging of magnetic organelles in magnetotactic bacteria (Le Sage et al. , 2013) and microscopic magnetic inclusions in meteorites (Fu et al. , 2014), as well as contrast-agent-based magnetic resonance microscopy (Steinert et al. , 2013).”

What the sources say

• Not checked yetcites Le Sage et al., 2013
• Not checked yetcites Fu et al., 2014
• Not checked yetcites Steinert et al., 2013

In the paper

“For example, classical continuous-wave absorption and transmission spectroscopy can be considered as an averaged variety of this protocol.”

In the paper

“Also, the time evolution can be replaced by a spatial evolution to describe a classical interferometer.”

In the paper

“Much excitement has recently been sparked by ensembles of nitrogen-vacancy centers (NV centers) - electronic spin defects in diamond that can be optically initialized and read out.”

In the paper

“These experiments reached a sensitivity of 5 · 10 -7 rad / s / √ Hz, which is comparable to compact implementations of atomic interferometers and optical Sagnac interferometers.”

What the sources say

• Not checked yetcites Kornack et al., 2005

In the paper

“First, efficient fluorescence detection of large NV ensembles is difficult, while absorptive and dispersive schemes are not easily implemented (Clevenson et al. , 2015; Jensen et al. , 2014; Le Sage et al. , 2012).”

What the sources say

• Not checked yetcites Clevenson et al., 2015
• Not checked yetcites Jensen et al., 2014
• Not checked yetcites Le Sage et al., 2012

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.”

What the sources say

• Not checked yetcites Lenz, 1990

In the paper

“Noise spectroscopy relies on the systematic analysis of decoherence and relaxation under different control sequences.”

In the paper

“We notice that variance detection improves only ∝ T 1 / 4 with the integration time, while slope detection improves ∝ T 1 / 2 .”

In the paper

“The decay time T χ , also known as the decoherence time or relaxation time depending on the noise process, is an important figure-of-merit of the qubit, as it sets the maximum possible evolution time t available for sensing.”

In the paper

“Often, the multipulse sequence cannot be synchronized with the signal or the phase α cannot be controlled. Then, incoherent averaging leads to phase cancellation, 〈 φ 〉 = 0.”

In the paper

“Although the underlying noise processes may be very complex, χ ( t ) can often be approximated by a simple power law,”

In the paper

“For variance detection, where ideally p 0 = 0, the projection noise would in principle be arbitrarily low. In any realistic experiment, however, decoherence will shift the fringe minimum to a finite value of p (see below), where Eq. (25) holds up to a constant factor.”

In the paper

“Because continuous sampling no longer relies on quantum state lifetimes, the Fourier-limited resolution can be extended to arbitrary values and is only limited by total experiment duration T , and ultimately the control hardware.”

In the paper

“Due to the widespread inefficiency of quantum state readout, classical readout noise is often the dominating source of error.”

In the paper

“Thus, variance detection profits more from a long coherence time T χ than slope detection (but is, in turn, more vulnerable to decoherence).”

In the paper

“For large pulse numbers n , the sinc term becomes very peaked and the filter bandwidth ∆ f ≈ 1 / ( nτ ) = 1 /t (full width at half maximum) becomes very narrow.”

In the paper

“If the amplitude V pk is not fixed, but slowly fluctuating between different measurements, the variance 〈 φ 2 〉 must be integrated over the probability distribution of V pk .”

In the paper

“A particularly important situation is a Gaussian amplitude fluctuation with an rms amplitude V rms . In this case, the resonant time response of the transition probability is”

• Also stated in: § 5. AC signals with random phase and random amplitude · p. 22 · plain

  “where I 0 is the modified Bessel function of the first kind.”

In the paper

“The multipulse sequence is subdivided into two equal sensing periods of duration t a = t/ 2 that are separated by an incremented free evolution period t 1 .”

In the paper

“In particular, good knowledge of the noise spectrum can help the adoption of suitable sensing protocols (like dynamical decoupling or quantum error correction schemes) to maximize the sensitivity of the quantum sensor.”

In the paper

“In particular, the pass-band frequency can be selected via the interpulse delay τ , while the filter width can be adjusted via the number of pulses n = t/τ (up to a maximum value of n ≈ T 2 /τ ).”

In the paper

“For white noise, the dephasing is Markovian and χ ( t ) = Γ t , where Γ is the decay rate. For Lorentzian noise centered at zero frequency the decoherence function is χ ( t ) = (Γ t ) 3 . For a generic 1 /f -like decay, where the noise falls of ∝ ω a (with a around unity), the decoherence function is χ ( t ) = (Γ t ) a +1 (Cywinski et al. , 2008; Medford et al. , 2012) with a logarithmic correct”

What the sources say

• Not checked yetcites Cywinski et al. , 2008
• Not checked yetcites Medford et al. , 2012
• Not checked yetcites Dial et al. , 2013

In the paper

“An imperfect initialization leads to a similar reduction in the observed probability δp obs as with decoherence,”

• Also stated in: § 3. Errors due to initialization and qubit manipulations · p. 15 · plain

  “Contrary to the case of decoherence, this reduction does not depend on the sensing time t .”

In the paper

“We note that the scaling of the QCRB with the number of copies, N -1 / 2 , is a consequence of the additivity of the quantum Fisher information for tensor states ρ ⊗ N . This is the well-known standard quantum limit (SQL).”

In the paper

“In addition, the number of measurements N is equal to T/ ( t + t m ), where T is the total available measurement time and t m is the extra time needed to initialize, manipulate and readout the sensor.”

In the paper

“Under the assumption of a Gaussian, stationary noise, the loss of coherence can be captured by an exponential decay of the transition probability with time t ,”

• Also stated in: § 1. Decoherence function χ ( t ) · p. 25 · plain

  “where χ ( t ) is the associated decay function or decoherence function that was already discussed in the context of sensitivity (Section V).”

In the paper

“where C = 1 / √ 1 + R 2 ≈ 1 / √ 1 + 4 κ is an overall readout efficiency parameter (Taylor et al. , 2008).”

What the sources say

• Not checked yetcites Taylor et al., 2008

In the paper

“The resonance order k can also be used to select the pass-band frequency, however, because k = 1 provides the strongest peak transmission, most reported experiments used this resonance.”

In the paper

“A suitable basis are Walsh dynamical decoupling sequences (Hayes et al. , 2011), which apply a π pulse every time the corresponding Walsh function (Walsh, 1923) flips its sign.”

What the sources say

• Not checked yetcites Hayes et al. , 2011
• Not checked yetcites Walsh, 1923

In the paper

“Noise spectroscopy is an important tool in quantum sensing, as it can provide much insight into both external signals and the intrinsic noise of the quantum sensor.”

In the paper

“This defines for any multipulse sequence a weighting function W ( f ac , α ).”

In the paper

“To go beyond the SQL, one then needs to use entangled states (see Section IX) in particular, simply using correlated POVMs is not sufficient.”

In the paper

“Kotler et al. , 2011, have shown that both quadratures of a signal can be recovered, allowing one to perform lock-in detection of the signal.”

What the sources say

• Not checked yetcites Kotler et al. , 2011

In the paper

“In quantum sensing, most frequency estimation schemes are based on dynamical decoupling sequences.”

In the paper

“C ≤ 1 describes the reduction of the signal-to-noise ratio compared to an ideal readout ( C = 1), see Fig. 4.”

In the paper

“The resolution can be further enhanced by long-lived auxiliary memory qubits (see Section X) and resolution improvements by two-to-three orders of magnitude over dynamical decoupling spectroscopy have been demonstrated (Pfender et al. , 2016; Rosskopf et al. , 2016; Zaiser et al. , 2016).”

What the sources say

• Not checked yetcites Pfender et al. , 2016
• Not checked yetcites Rosskopf et al. , 2016
• Not checked yetcites Zaiser et al. , 2016

In the paper

“and the peak transmission at f ac = k/ 2 τ is W 2 = 2 / ( kπ ) 2 .”

In the paper

“Conversely, if C is independent of t m - for example, because the sensor is operated in the single-shot regime or because readout resets the sensor t m should be made as short as possible.”

In the paper

“Quite generally, χ ( t ) can be identified with an rms phase accumulated during time t ,”

• Also stated in: § 1. Decoherence function χ ( t ) · p. 25 · plain

  “according to the expression for variance detection, Eq. (20).”

In the paper

“To further specify the SNR, the change in probability δp can be related to the change in signal δV as δp = δV q | ∂ q V p ( t ) | ∝ ( γtδV ) q , with q = 1 for slope detection and q = 2 for variance detection (see Fig. 3).”

In the paper

“where V rms = V pk / √ 2 is the rms amplitude of the signal and W 2 ( f ac ) is the average over α = 0 . . . 2 π of the weighting functions,”

In the paper

“Relaxation can also be analyzed in the presence of a resonant drive field. This method is known as 'spin locking' in magnetic resonance (Slichter, 1996).”

What the sources say

• Not checked yetcites Slichter, 1996

In the paper

“All three protocols can be considered variations of the phase estimation scheme depicted in Fig. 12(a).”

In the paper

“Assuming the control sequence has a modulation function y ( t ) (see Section VI.C), the decay function is determined by the correlation integral (Biercuk et al. , 2011; de Sousa, 2009)”

• Also stated in: § 2. Filter function Y ( ω ) · p. 25 · plain

  “where G V ( t ) is the autocorrelation function of V ( t ) (Eq. 81).”

What the sources say

• Not checked yetcites Biercuk et al. , 2011
• Not checked yetcites de Sousa, 2009

In the paper

“The dynamic range is then given by the maximum allowed signal divided by the minimum detectable signal,”

In the paper

“The scheme was originally put forward by Shor, 1994, in the seminal proposal of a quantum algorithm for prime factorization and has been interpreted by Kitaev, 1995, as a phase estimation algorithm.”

What the sources say

• Not checked yetcites Shor, 1994 source
• Not checked yetcites Kitaev, 1995

In the paper

“This is possible because the controlledU gate does not change the eigenvector ψ , so that it can be re-used as often as required.”

In the paper

“In the frequency domain the decay function can be expressed as”

• Also stated in: § 2. Filter function Y ( ω ) · p. 25 · plain

  “where | Y ( ω ) | 2 is the so-called fi lter function of y ( t ), defined by the finite-time Fourier transform”

In the paper

“Since S V ( ω ) is often dominated by low-frequency noise, S V || (0) is typically much larger than S V ⊥ ( ω 0 ) and the high-frequency contribution can often be neglected.”

In the paper

“The joint probability distribution of all measurements is obtained from the product”

• Also stated in: § 3. Non-adaptive phase estimation · p. 31 · plain

  “from which the most likely value of φ is picked as the final result (Nusran et al. , 2012; Waldherr et al. , 2012).”

What the sources say

• Not checked yetcites Nusran et al., 2012
• Not checked yetcites Waldherr et al., 2012

In the paper

“This scaling is sometimes referred to as another instance of the Heisenberg limit, because it can be regarded as a 1 /T scaling of sensitivity at a fixed V max .”

In the paper

“While quantum phase estimation based on the QFT can be performed with polynomial time effort, the algorithm requires two-qubit gates, which are difficult to implement experimentally, and the creation of fragile entangled states.”

In the paper

“The aim of relaxometry is to connect the spectral density S V ( ω ) of a noise signal V ( t ) to the relaxation rate Γ in first-order kinetics, χ ( t ) = Γ t .”

In the paper

“T ∗ 2 relaxometry probes the transition rate between the superposition states |±〉 = ( | 0 〉 ± e -iω 0 t | 1 〉 ) / √ 2. This corresponds to the free induction decay observed in a Ramsey experiment”

In the paper

“Due to the presence of the resonant field the degeneracy between |±〉 is lifted and the states are separated by the energy ℏ ω 1 , where ω 1 ≪ ω 0 is the Rabi frequency of the drive field.”

In the paper

“Since the filter function is dominated by the first harmonic ( k = 1) the frequency corresponding to a certain τ is 1 / (2 τ ).”

In the paper

“The QFT is thus replaced by measurement and classical feedback, which can be performed using a single qubit sensor.”

In the paper

“This problem can be generalized to estimating the phase shift φ imparted by passage through an interferometer or exposure to an external field.”

In the paper

“Hence, the algorithm scales linearly with the number of applications of ˆ U which in turn is proportional to the total measurement time T .”

In the paper

“Because T 1 can be very long, very high sensitivities are in principle possible, assuming that the resonance is not skewed by low-frequency noise.”

In the paper

“The key idea of adaptive phase estimation is to first measure the least significant bit of φ , represented by the lowest qubit in Fig. 12(b). In the measurement of the next significant bit, this value is subtracted from the applied phase.”

• Also stated in: § 2. Adaptive phase estimation · p. 30 · plain

  “This procedure is then repeated in ascending order of the bits.”

In the paper

“The sensitivity of the measurement, on the other hand, is proportional to the slope of the Ramsey fringe and reaches its optimum when t ≈ T ∗ 2 .”

In the paper

“In addition, since dressed state relaxometry does not require sweeping a static control field for adjusting the probe frequency, it is more versatile than standard T 1 relaxometry.”

In the paper

“The implementation by Bonato, C. et al. , 2016, for example, scaled the number of resources N m linearly according to”

• Also stated in: § 2. Adaptive phase estimation · p. 31 · plain

  “with typical values of G = 5 and F = 2.”

What the sources say

• Not checked yetcites Bonato, C. et al., 2016

In the paper

“The maximum allowed signal is then set by the shortest sensing time, V max = π/ ( γt 0 ), while the smallest detectable signal is determined by the longest sensing time, V min ≈ 2 / ( γC √ t M T ).”

In the paper

“Because a phase flip does not require energy, the spectral density is probed at zero frequency.”

In the paper

“For a very large detuning the effective frequency becomes similar to the detuning ω eff ≈ ∆ ω , and the drive field only enters as a scaling factor for the spectral density.”

In the paper

“Because T ∝ t M due to the exponentially growing interrogation times, the dynamic range of the improved protocol scales as”

In the paper

“The QFT recovers the phase φ from the Fourier series (109) and places the register in state”

• Also stated in: § 1. Quantum phase estimation · p. 30 · plain

  “A measurement of the register directly yields a digital representation of the phase φ .”

In the paper

“In quantum sensing, phase tagging by back-action is replaced by the exposure of each qubit to an external field for a time 2 m t 0 (or passage through an interferometer of length 2 m l 0 ).”

In the paper

“Quantum phase estimation is the core component of Shor's algorithm, where it is used to compute discrete logarithms with polynomial time effort (Shor, 1994).”

What the sources say

• Not checked yetcites Shor, 1994 source

In the paper

“For large cycle numbers, Y n c ,τ c ( ω ) presents sharp peaks at multiples of the inverse cycle time τ -1 c , and it can be approximated by a series of δ functions.”

In the paper

“Although other scheduling is possible, this choice allows for an intuitive interpretation: subsequent measurements estimate subsequent digits of a binary expansion of the signal.”

In the paper

“Thus, the filter function plays the role of a transfer function, and the decay of coherence is captured by the overlap with the noise spectrum S V ( ω ).”

In the paper

“This scaling is obvious from an order-of-magnitude estimate: adding an additional measurement step increases both precision and measurement duration t by a factor of two, such that precision scales linearly with total acquisition time T .”

In the paper

“In practical implementations (Higgins et al. , 2007), the measurement of each digit is repeated multiple times or performed on multiple parallel qubits.”

What the sources say

• Not checked yetcites Higgins et al., 2007

In the paper

“Adaptive estimation is technically more demanding since real-time feedback is involved, which often requires dedicated hardware (such as field-programmable gate arrays or a central processing unit) for the fast decision making.”

In the paper

“They have been used not only for sensing but also for enhanced lithography (Boto et al. , 2000).”

What the sources say

• Not checked yetcites Boto et al., 2000

In the paper

“The proposals of Shor (Shor, 1994), Kitaev (Kitaev, 1995) and Griffiths (Griffiths and Niu, 1996) were followed by a decade where research towards Heisenberglimited measurements has focused mostly on the use of entangled states, such as the N00N state (see Section IX).”

What the sources say

• Not checked yetcites Shor, 1994 source
• Not checked yetcites Kitaev, 1995
• Not checked yetcites Griffiths and Niu, 1996

In the paper

“Then, the interrogation time also needs to be reduced by a factor M and no net advantage in sensitivity can be obtained.”

In the paper

“One of the most impressive application of squeezed states of light (Ligo Collaboration', 2011; Schnabel et al. , 2010; Walls, 1983) has been the sensitivity enhancement of the LIGO gravitational wave detector (Collaboration, 2013), obtained by injecting vacuum squeezed states in the interferometer.”

What the sources say

• Not checked yetcites Ligo Collaboration', 2011
• Not checked yetcites Schnabel et al., 2010
• Not checked yetcites Walls, 1983

In the paper

“All of the above variants of phase estimation achieve a DR ∝ T scaling of the dynamic range.”

In the paper

“Shortly after, both variants have been translated into protocols for spin-based quantum sensing (Said et al. , 2011).”

What the sources say

• Not checked yetcites Said et al., 2011

In the paper

“In particular, squeezed states (Caves, 1981; Kitagawa and Ueda, 1993; Wineland et al. , 1992) have been proposed early on to achieve the Heisenberg limit and thanks to experimental advances have recently shown exceptional sensitivity (Hosten et al. , 2016a).”

What the sources say

• Not checked yetcites Caves, 1981
• Not checked yetcites Kitagawa and Ueda, 1993
• Not checked yetcites Wineland et al., 1992

In the paper

“Similar to GHZ states, N00N states have been conceived to improve interferometry (Lee et al. , 2002).”

What the sources say

• Not checked yetcites Lee et al., 2002

In the paper

“Optimal densities have been calculated both for atomic vapor magnetometers (Budker and Romalis, 2007) and ensembles of NV centers (Taylor et al. , 2008; Wolf et al. , 2015).”

What the sources say

• Not checked yetcites Budker and Romalis, 2007
• Not checked yetcites Taylor et al., 2008
• Not checked yetcites Wolf et al., 2015

In the paper

“This allows using an M -times shorter interrogation time or achieving an improvement of the sensitivity (calculated from the QCRB) by a factor √ M .”

In the paper

“Here, too, acquisition time scales with the significance of the bit measured to achieve the Heisenberg limit.”

In the paper

“The standard quantum limit can be overcome by using quantum-enhanced sensing strategies to reach a more fundamental limit where the uncertainty σ p (Eq. ( 25)) scales as 1 /M .”

In the paper

“The sensitivity is then expressed per unit volume ( ∝ meters -3 / 2 ).”

In the paper

“Still, 'N00N' states are very fragile (Bohmann et al. , 2015) and they are afflicted by a small dynamic range.”

What the sources say

• Not checked yetcites Bohmann et al., 2015

In the paper

“Experimentally, Bayesian estimation is usually simple to implement because no real-time feedback is needed and the phase estimation can be performed a posteriori .”

In the paper

“Squeezed states are promising for quantum-limited sensing as they can reach sensitivity beyond the standard quantum limit.”

In the paper

“Unfortunately, the √ M advantage in sensitivity is usually compensated by the GHZ state's increased decoherence rate (Huelga et al. , 1997), which is an issue common to most entangled states.”

What the sources say

• Not checked yetcites Huelga et al., 1997

In the paper

“The experiment has subsequently been extended to a nonadaptive version (Higgins et al. , 2009).”

What the sources say

• Not checked yetcites Higgins et al., 2009

In the paper

“Weak measurements have been more generally proposed to enhance sensing protocols (), but their ultimate usefulness is still under debate ().”

In the paper

“Using an M -particle interferometer, one can prepare an entangled Fock state,”

In the paper

“Entanglement-enhanced sensing has been pioneered with atomic systems, especially atomic clocks (Giovannetti et al. , 2004; Leibfried et al. , 2004).”

What the sources say

• Not checked yetcites Giovannetti et al., 2004
• Not checked yetcites Leibfried et al., 2004

In the paper

“As a final remark, we note that a similar performance - 1 /T scaling and an increased dynamic range - may be achieved by weak measurement protocols, which continuously track the evolution of the phase over the sensing sequence (Kohlhaas et al. , 2015; Shiga and Takeuchi, 2012).”

What the sources say

• Not checked yetcites Kohlhaas et al., 2015
• Not checked yetcites Shiga and Takeuchi, 2012

In the paper

“Achieving the Heisenberg limit requires reducing the variance of a chosen quantum observable at the expenses of the uncertainty of a conjugated observable.”

In the paper

“This limit is also known as the Heisenberg limit .”

In the paper

“To understand the benefits that an entangled state can bring to quantum sensing, the simplest example is given by Greenberger-Horne-Zeilinger (GHZ) states.”

In the paper

“Squeezed states of light have been introduced by Caves, 1981, as a mean to reduce noise in interferometry experiments.”

What the sources say

• Not checked yetcites Caves, 1981

In the paper

“They differ, however, by a constant offset.”

In the paper

“The sensing scheme is similar to a Ramsey protocol, however, if M qubit probes are available, the preparation and readout pulses are replaced by entangling operations (Fig. 13).”

In the paper

“This has led to the quest for different entangled states that could be more resilient to decoherence.”

In the paper

“They were first introduced by Yurke, 1986, in the context of neutron Mach-Zender interferometry as the fermionic 'response' to the squeezed states proposed by Caves, 1981, for Heisenberg metrology.”

What the sources say

• Not checked yetcites Yurke, 1986
• Not checked yetcites Caves, 1981

In the paper

“In parallel, state squeezing is routinely applied in optical systems, such as optical interferometers (Ligo Collaboration', 2011).”

What the sources say

• Not checked yetcites Ligo Collaboration', 2011

In the paper

“Quantum phase estimation by the QFT, finally, requires many entangled qubits.”

In the paper

“Apart from an obvious gain in readout sensitivity, multiple qubits allow for the implementation of second-generation quantum techniques, including entanglement and state squeezing, which provide a true 'quantum' advantage that cannot be realized with classical sensors.”

In the paper

“Comparing this result with what obtained in Sec. IV.C, 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 one qubit.”

In the paper

“While this is difficult to obtain for large qubit numbers, recent advances show great promise (Hume et al. , 2013; Zhang et al. , 2012) (see also next section on alternative detection methods).”

What the sources say

• Not checked yetcites Hume et al., 2013
• Not checked yetcites Zhang et al., 2012

In the paper

“However, imperfections in this measurement limits the sensitivity, since achieving the Heisenberg limit requires single-particle state detection.”

In the paper

“Spin squeezing can be also implemented in qubit systems (Auccaise et al. , 2015; Bennett et al. , 2013; Cappellaro and Lukin, 2009; Sinha et al. , 2003) following the original proposal by Kitagawa and Ueda, 1993.”

What the sources say

• Not checked yetcites Auccaise et al., 2015
• Not checked yetcites Bennett et al., 2013
• Not checked yetcites Cappellaro and Lukin, 2009

In the paper

“Compared to dynamical decoupling schemes, qubit protection covers the entire noise spectrum and is not limited to low-frequency noise.”

In the paper

“Since QEC works independently of noise frequency, distinct operators for the signal and noise interactions are required.”

In the paper

“In this scheme, which has recently been implemented with NV centers (Unden et al. , 2016), the qubit sensor detects a signal along one axis ( e.g. , a phase) while being protected against noise along a different axis ( e.g. , against bit flip).”

What the sources say

• Not checked yetcites Unden et al. , 2016

In the paper

“The metrology squeezing parameter, proposed by Wineland et al. , 1992, is then”

What the sources say

• Not checked yetcites Wineland et al., 1992

In the paper

“For example, the code | 0 〉 L = | 000 〉 and | 1 〉 L = | 111 〉 can correct against a single qubit flip error.”

In the paper

“For example, from the commutation relationship for the collective angular momentum, ∆ J α ∆ J β ≥ |〈 J γ 〉| , one can naturally define a squeezing parameter”

In the paper

“Retrieval uses the same two c-NOT gates in reverse order.”

In the paper

“Then, during the free evolution (step 3) an enhanced phase can be acquired, similar to what happens for entangled states.”

In the paper

“The most common sensing protocols with squeezed states forgo step 4, and directly measure the population difference between the state | 0 〉 and | 1 〉 .”

In the paper

“To quantify the advantage of squeezed states in sensing, it is advantageous to directly relate squeezing to the improved sensitivity.”

In the paper

“Early demonstrations of spin squeezing were obtained by entangling trapped ions via their shared motional modes (Meyer et al. , 2001), using repulsive interactions in a Bose-Einstein condensate (Esteve et al. , 2008), or partial projection by measurement (Appel et al. , 2009).”

What the sources say

• Not checked yetcites Meyer et al., 2001
• Not checked yetcites Esteve et al., 2008
• Not checked yetcites Appel et al., 2009

In the paper

“The degree of spin squeezing can be measured by several parameters.”

In the paper

“A first example includes correlation spectroscopy, where two sensing periods are interrupted by a waiting time t 1 (Laraoui and Meriles, 2013; Rosskopf et al. , 2016; Zaiser et al. , 2016).”

What the sources say

• Not checked yetcites Laraoui and Meriles, 2013
• Not checked yetcites Rosskopf et al. , 2016
• Not checked yetcites Zaiser et al. , 2016

In the paper

“A different strategy is to follow more closely the sensing protocol for entangled states, and refocus the squeezing (reintroducing step 4) before readout (Davis et al. , 2016).”

What the sources say

• Not checked yetcites Davis et al., 2016

In the paper

“These recent results hint at the potential of QEC for sensing which has just about begun to being explored.”

In the paper

“Squeezed states are then characterized by a reduced uncertainty in one observable at the expense of another observable.”

In the paper

“For the example of an electron-nuclear qubit pair, c-NOT gates have been implemented both by selective pulses (Pfender et al. , 2016; Rosskopf et al. , 2016) and using coherent rotations (Zaiser et al. , 2016).”

What the sources say

• Not checked yetcites Pfender et al. , 2016
• Not checked yetcites Rosskopf et al. , 2016
• Not checked yetcites Zaiser et al. , 2016

In the paper

“Thus, quantum error correction suppresses noise according to spatial symmetry, and not according to frequency.”

In the paper

“The simplest code is the 3-qubit repetition code, which corrects against one-axis noise with depth one (that is, it can correct up to one error acting on one qubit).”

In the paper

“However, the description of squeezed states is simplified by the use of a single collective angular-momentum variable.”

In the paper

“In both cases, coupling of the quantum system to the detection apparatus is inefficient, often because strong coupling would destroy the very quantum state used in the metrology task.”

In the paper

“In this context, a simple quantum sensing scheme, following the procedure in Sec. IV.C, could be constructed by replacing step 2 with the preparation of a squeezed state, so that | ψ 0 〉 is a squeezed state.”

In the paper

“Metrology schemes often require single qubit readout or the measurement of complex, many-body observables.”

In the paper

“While for typical states the uncertainty in the two observables is on the same order, it is possible to redistribute the fluctuations in the two conjugate observables.”

In the paper

“Overall, quantum sensing has the potential to fundamentally transform our measurement capabilities, enabling higher sensitivity and precision, new measurement types, and covering atomic up to macroscopic length scales.”

In the paper

“In particular, these allow operation of the sensor over a wide signal frequency range, can be adjusted to maximize sensitivity and dynamic range, and allow discrimination of different types of signals by symmetry or vector orientation.”

In the paper

“A better understanding of decoherence in a particular implementation of a quantum system can help the adoption of strategies to protect the qubit, and guide the engineering and materials development.”

In the paper

“Vice versa, quantum sensing has become an important resource for quantum technologies as it provides much insight into the 'environment' of qubits, especially through decoherence spectroscopy.”

In the paper

“While many experiments so far made use of single qubit sensors, strategies to implement entangled multi-qubit sensors with enhanced capabilities and higher sensitivity are just beginning to be explored.”

In the paper

“The border region between quantum sensing and quantum simulation, in addition, is becoming a fertile playground for emulating and detecting many-body physics phenomena.”

In the paper

“Over the last decade, and especially in the last few years, a comprehensive toolset has been established that can be applied to any type of quantum sensor.”