Quantum metrology and its application in biology

Quantum metrology and its application in biology

Abstract

Quantum metrology provides a route to overcome practical limits in sensing devices. It holds particular relevance to biology, where sensitivity and resolution constraints restrict applications both in fundamental biophysics and in medicine. Here, we review quantum metrology from this biological context, focussing on optical techniques due to their particular relevance for biological imaging, sensing, and stimulation. Our understanding of quantum mechanics has already enabled important applications in biology, including positron emission tomography (PET) with entangled photons, magnetic resonance imaging (MRI) using nuclear magnetic resonance, and bio-magnetic imaging with superconducting quantum interference devices (SQUIDs). In quantum metrology an even greater range of applications arise from the ability to not just understand, but to engineer, coherence and correlations at the quantum level. In the past few years, quite dramatic progress has been seen in applying these ideas into biological systems. Capabilities that have been demonstrated include enhanced sensitivity and resolution, immunity to imaging artifacts and technical noise, and characterisation of the biological response to light at the single-photon level. New quantum measurement techniques offer even greater promise, raising the prospect for improved multi-photon microscopy and magnetic imaging, among many other possible applications. Realization of this potential will require cross-disciplinary input from researchers in both biology and quantum physics. In this review we seek to communicate the developments of quantum metrology in a way that is accessible to biologists and biophysicists, while providing sufficient detail to allow the interested reader to obtain a solid understanding of the field. We further seek to introduce quantum physicists to some of the central challenges of optical measurements in biological science. We hope that this will aid in bridging the communication gap that exists between the fields, and thereby guide the future development of this multidisciplinary research area.

keywords:
Quantum metrology, biology, cell, coherence, quantum correlations, squeezed state, NOON state, shot noise, quantum Fisher information
1

1 Introduction

Fundamentally, all measurement processes are governed by the laws of quantum mechanics. The most direct influence of quantum mechanics is to impose constraints on the precision with which measurements may be performed. However, it also allows for new measurement approaches with improved performance based on phenomena that are forbidden in a purely classical world. The field of quantum metrology investigates the influence of quantum mechanics on measurement systems and develops new measurement technologies that can harness non-classical effects to their advantage.

Quantum metrology broadly began with the discovery that quantum correlated light could be used to suppress quantum shot noise in interferometric measurements, and thereby enhance precision (1). To this day, the development of techniques to enhance precision in optical measurements remains a primary focus of the field. Such enhancement is particularly relevant in situations where precision cannot be improved simply by increasing optical power, due, for example, to power constraints introduced by optical damage or quantum measurement back-action (2). One such situation is gravitational wave measurement, where kilometre-scale interferometric observatories operate with power near the damage threshold of their mirrors, yet still have not achieved the extreme precision required to directly observe a gravitational wave (3); (4); (5); (6). Biological measurements are another prominent application area which has been discussed from the earliest days of quantum metrology (7); (8); (9); (10), since biological samples are often highly photosensitive and optical damage is a limiting factor in many biophysical experiments (11); (12); (13).

Although these applications were recognised in the 1980s (7), at that time the technology used for quantum metrology was in its infancy and unsuited to practical measurements. Since then, both the technology and theory of quantum metrology has advanced dramatically. In recent years, quantum measurement techniques based on quantum correlated photons have made in-roads in significant application areas within the biological sciences (see, for example, Refs. (14); (15); (16)), while new biologically relevant measurement technologies are under rapid development. With researchers continuing to explore the possibilities of biological quantum metrology, it may soon be possible to achieve advantages over classical measurement strategies in practical settings. The promise of quantum techniques is recognised within the bioscience community, featuring prominently in recent reviews of advances in optical instrumentation (for instance, see Refs. (17); (18); (19)). However, no dedicated review yet exists that bridges the gap between the fields. While a number of review articles are available which focus on quantum metrology (see, for example, Refs. (20); (21); (22)), they are generally targeted towards researchers within that community, and tend to be inaccessible to readers who are uninitiated in advanced quantum mechanics. At its broadest level, therefore, the aim of this review is to act as a bridge. We aim to explain the concepts of quantum metrology and their implications in as accessible a manner as possible, while also introducing a range of state-of-the-art approaches to biological measurement and imaging, along with their associated challenges. Such a review cannot hope to be exhaustive. However, we do seek to introduce the techniques of most relevance to near-future developments in this multi-disciplinary area, and to identify the key technological and practical challenges which must be overcome to see those those developments realised. In this way, we hope that the review will contribute in a positive way to the field.

Although quantum metrology is generally considered to have begun in the 1980s, positron emission tomography (PET) has been utilizing entangled photon pairs in imaging since the 1960s (23); (24). In PET, a radioactive marker undergoes decay to produce a positron. The positron annihilates with a nearby electron to produce a high energy entangled photon pair. Since the photons propagate in near-opposite directions, the position of the annihilation event can be estimated to occur along a chord connecting coincident photon detections. With sufficient coincident detection events, a full three dimensional profile of the radioactive marker density can be reconstructed. This is now used routinely in clinical applications to image cancerous tumours and to observe brain function (25).

The development of modern quantum technologies allows quantum correlated states of light to be engineered, in contrast to the uncontrolled generation of entangled photons which is used in PET. This enables a far broader range of applications. Entangled photon pairs have now been applied in tissue imaging (15), absorption imaging (14), and refractive index sensing of a protein solution (26); squeezed states of light have been used both to measure dynamic changes (16) and image spatial properties (27) of sub-cellular structure; and single photons have been used to stimulate retinal rod cells, thus allowing the cellular response to single photons to be deterministically characterized (28). These experiments have demonstrated the prospects of quantum correlations for new capabilities and unrivalled precision in practical biological measurements. Furthermore, a broad range of technologies have already been demonstrated in non-biological measurements which could soon have important applications in biology. These near-future applications include cellular imaging with both multi-photon microscopy (29); (30) and super-resolution of fluorescent markers (31); (32), enhanced phase contrast (33); (34) microscopy, and measurement of biomagnetic fields (35).

The review begins with a semi-classical explanation of quantum noise in optical measurements, which allows a qualitative, but not rigorous, derivation of the limits imposed on optical measurements by the quantisation of light, including the standard quantum limit, the Heisenberg limit, and the limit imposed by optical inefficiencies (Section 2). It then proceeds to a quantum mechanical description of photodetection, quantum coherence and quantum correlations (Section 3). Section 4 introduces the theoretical tools of quantum metrology for the example case of optical phase measurement, and introduces the commonly used squeezed and NOON states. Section 5 then describes the unique challenges associated with practical biological experiments, including resolution requirements, optical damage, and the non-static nature of living cells. The experiments which have applied quantum metrology in biology are described in Section 6. Section 7 overviews a range of promising technologies which in future may have important biological applications. Section Section 8, departs briefly from optical approaches to quantum metrology, providing a brief overview of spin-based quantum metrology experiments that hold promise for future biological applications. Finally, Section 9 concludes the review with a broad summary of the potential of quantum metrology for biological measurements in the near future.

2 Semi-classical treatment of optical phase measurement

This review introduces the theory of quantum measurement in the context of interferometric phase measurements. In doing so, we seek to provide the simplest possible example quantum limits to measurements, and how they may be overcome using quantum correlations. This example is particularly relevant, both since it has been comprehensively studied in the quantum metrology literature, and due to its many applications ranging from optical range-finding to phase-contrast imaging. We would emphasise that the concepts introduced are quite general and can be naturally applied in other contexts – as is particularly well seen for optical nanoparticle tracking in Section 6.3.

The measurement process quite generally involves the preparation of a probe, its interaction with a system of interest, and finally measurement of the probe to extract relevant information about the system. In optical phase measurements, this typically involves using a laser to produce a coherent optical field, propagating the field though an interferometer, and measuring the power in the two output ports to estimate the phase shift applied within one arm of the interferometer (see Fig. 1). More generally, the probe need not be laser light; optical measurements can be carried out with states of light ranging from thermal light (36) to non-classical states of light (2), while non-optical measurements can be carried out with probes such as coherent matter waves (37), spin states of atoms (38), or mechanical states of a cantilever (39). The field of quantum metrology explores the influence of the input state on the achievable precision, as well as the advantage which can be gained from use of non-classical states.

Before introducing a full quantum treatment of the problem of phase measurement, we consider a semi-classical scenario where the optical electric field is treated classically and can, in principle, be deterministic and carry no noise, with photon quantization (or ‘shot noise’) introduced phenomenologically in the detection process. This approach to quantization results in a stochastic output photocurrent with mean proportional to the intensity of the measured field. This semi-classical approach has the advantage of illustrating the deleterious effect of shot noise on optical measurements, and allows straightforward (though not rigorous) derivations of quantum limits to the precision of phase measurements which should be readily understood by scientists from outside the quantum metrology community.

2.1 Standard quantum limit of optical phase measurement

Figure 1: A classical analysis of a phase sensing experiment based on a Mach-Zehnder interferometer. An incident optical field is split into a reference and probe field at the input port of the interferometer by a beam splitter with transmission . A differential phase shift is then applied between the fields in the two arms. The fields are then recombined on a second beam splitter to produce two output fields (termed and ). Detection of these fields allows the phase shift to be estimated.

Optical phase is generally measured via interference. Here we consider the case where an optical field propagates through the arms of a Mach-Zehnder interferometer, as shown in Fig. 1. We term the arm in which the system of interest is placed the “signal arm”, and the other arm of the interferometer the “reference arm”, since its role in the measurement is solely to provide a phase reference. The field in the signal arm experiences a phase shift from its interaction with the system, which we often refer to alternatively as the “sample” or “specimen”. This phase shift is estimated from intensity measurements at the two interferometer outputs, labelled “” and “” here. In a classical treatment of this problem, taking the case where the two beam splitters in the interferometer each have 50% reflectivity and no absorption, the electric fields of the light that reaches the detectors are given by

(1a)
(1b)

where is the incident optical field. Each field is then detected with a photodiode. Quantized photocurrents and are produced as valence band electrons in each photodiode are independently excited into the conduction band, with probability proportional to the optical intensity. As such, each photocurrent fluctuates stochastically about a mean that is proportional to the intensity of light incident on the photodiode ( and ). In a quantum treatment of photodetection, each photoelectron is excited by a single photon (see Section 3). Consequently, the photocurrents can equivalently be thought of as the photon flux of the detected fields. Evaluating the mean detected intensities, one finds that

(2a)
(2b)

where is the photon flux of the input field to the interferometer. Information about the phase can be extracted from the difference photocurrent, which has a mean value of

(3)

The phase sensitivity is optimized when where , since this maximises the derivative of the difference photocurrent with respect to a small change in . Interferometers are generally actively stabilised to ensure operation near one of these optimal points. For small displacements about such a point, the phase shift is given to first order as

(4)

Consequently, the relative phase may be estimated as

(5)

The statistical variance of the phase estimate is then given by

(6)

where is the variance of the variable , and the covariance quantifies the correlations between the variables and . In our semi-classical treatment, photoelectrons are generated through stochastic random processes at each photodetector. In the limit that the optical fields are stationary in time, the photon detection events are uncorrelated both on one photodetector, and between the photodetectors. The latter property means that the covariance is zero, while the former results in Poissonian photon counting statistics on each detector. As discussed in Section 4.2, this prediction of Poissonian statistics is consistent with a fully quantum treatment of a coherent state, which is the state generated by a perfectly noise-free laser. Due to the Poissonian counting statistics, the variance of each photocurrent is equal to its mean, i.e., with . Assuming that the interferometer is operating very close to its optimal point, with the phase deviation away from this point being much smaller than one, the input photon flux is split approximately equally between the two interferometer outputs, so that . Substituting for the photocurrent variances and covariance in Eq. (6), the achievable phase precision is then given by

(7)

We see that the phase precision improves as the square-root of the photon flux input into the interferometer (see Fig. 2). Even though the approach used here is semi-classical and treats the optical electric field as a perfectly deterministic quantity, Eq. (7) reproduces the standard quantum limit for phase measurements, which quantifies the best precision that can be reached without the use of quantum correlations for any optical phase measurement using a mean photon flux of . As discussed in Section 4.2, the standard quantum limit can be achieved using coherent states. In fact, coherent states achieve the best precision possible without quantum correlations for many forms of measurement, not just phase estimation (2). Consequently, the sensitivity achievable using coherent light of a given power generally provides an important benchmark for quantum metrology experiments.

Examination of Eq. (6) suggests that precision could be improved if the detection events are correlated, such that . As we will see later (first in Section 2.3), this is the case for quantum correlated light. However, a detailed calculation shows that when classical correlations are introduced – such as those introduced by a temporal modulation of the input optical intensity – they also increase the photon number variances and , and ultimately the precision is not enhanced.

It is worthwhile to note that the above derivation only assumes quantization in the photocurrent which provides the electronic record of the light intensity. As such, the limit of Eq. (7) can be arrived at either either by considering a perfectly noiseless optical field which probabilistically excites photoelectrons, or a quantized field with each photon exciting a single electron. Violation of this limit, however, requires electron correlations in the detected photocurrents which cannot follow from probabilistic detection, and therefore necessitates a quantum treatment of the optical fields (40).

2.2 Variations on the standard quantum limit

The quantum noise limit

So far in the review we have considered only the restricted class of measurements that can be performed with perfect efficiency. That is, we have assumed that no photons are lost in transmission of the optical fields, both through free-space and through the sample, or their detection. In practice, this is never the case, with inefficiencies being a particular concern for biological applications of quantum measurement.

For the moment, constraining our analysis to the case of uncorrelated photons, the analysis in the previous section can be quite straightforwardly extended to include optical inefficiencies. This results in the so-termed quantum noise limit of interferometric phase measurement. Here, we consider the particular case where the two arms of the interferometer exhibit balanced losses, each having transmissivity . Within our semi-classical treatment, such balanced losses have the same effect as loss of the same magnitude prior to the interferometer,2 and are therefore equivalent to a reduced input photon flux from to . They therefore modify the standard quantum limit of Eq. (7) to the quantum noise limit:

(8)

In the general case where unbalanced loss is present, the quantum noise limit is modified, but retains the general properties of constraining the measurement to precision inferior to the standard quantum limit, and scaling as .

The quantum noise limit should be interpreted as quantifying the precision that is achievable in a given (imperfect) apparatus without access to quantum correlations between photons. As such, it is particularly relevant to biological applications where experimental non-idealities are commonly unavoidable and prevent the standard quantum limit from being reached with coherent light. Quantum metrology experiments often compare their achieved precision to the quantum noise limit, since violation of this limit proves that quantum resources have enabled an improvement in precision. The standard quantum limit provides a more stringent bound, defining the precision that is achievable without quantum correlations in an ideal apparatus that has no loss and perfect detectors. Violation of this limit therefore proves that the experiment operates in a regime that is classically inaccessible not only for a given apparatus, but for any apparatus in general. The standard quantum limit and quantum noise limit are generally used in different contexts, with continuous measurements on bright fields often compared to the quantum noise limit (41); (16); (35); (42); (5), and photon counting measurements typically compared to the standard quantum limit (26); (43); (44).

Although the quantum noise limit is a widely used benchmark, it is worth noting that there is no clear consensus as to its name. It is most often referred to as either the quantum noise limit (41); (16); (42) or the shot noise limit (6); (45); (35), often interchangeably (5), though other names are also used (46); (47); (48). It is also important to note that the phrase “standard quantum limit” carries two distinct meanings in different communities. While much of the quantum metrology community uses the definition introduced here, the optomechanics community defines the standard quantum limit as the best sensitivity possible with arbitrary optical power, which occurs when quantum back-action from the measurement is equal to the measurement imprecision (2).

Power constraints

As can be seen from Eqs. (7) and (8), the precision of an optical phase measurement can, at least in principle, be enhanced arbitrarily by increasing the optical power input to the interferometer. Consequently, quantum limits on precision are only relevant in circumstances where the optical power is constrained.

In circumstances where an experimental constraint existed on the total photon flux , due, for example, to limitations in available laser output power or detector damage thresholds, Eq. (7) defines the standard quantum limit to precision. In many other experiments – and particularly in biological measurements where the sample is often susceptible to photo-induced damage and photochemical intrusion (see Section 5.4) – however, the constraint is instead placed on the power incident on the sample. In this case, the precision can be improved by unbalancing the interferometer such that the reference arm carries more power than the signal arm. This suppresses the noise contribution from photon fluctuations in the reference arm, and thereby improves the precision achievable for a fixed power at the sample. A similar treatment to that given in Section 2.1 shows that, when the power incident on the sample is constrained, the standard quantum limit becomes

(9)

where is the photon flux in the signal arm. As discussed briefly above and more thoroughly in Section 5.4, biological specimens can be significantly influenced by the optical fields used to probe them. In this review we will therefore generally use this sample-power-constrained standard quantum limit, rather than the perhaps more conventional standard quantum limit of Eq. (7). It is shown, as a function of in Fig. 2. Since, for the balanced interferometer considered in Section 2.1, , we see by comparison of Eqs. (7) and (9) that, with a constraint on power within the sample, the standard quantum limit is improved by a factor of .

Figure 2: Sensitivity limits of interferometric phase measurement as a function of the number of photons used in the probe arm . Here, we consider the scenario where the power constraint is imposed by the possibility of damage to the sample, rather than a constraint on total power within the interferometer. The solid black lines show the standard quantum limit (Eq. (9)) and the Heisenberg limit (Eq. (10)), where in the later case . The dashed line shows the Cramer-Rao Bound for squeezed states (Eq. (57)), defining the best precision achievable with such states and ideal measurement. The grey shaded regions indicate levels of precision that are unattainable given a loss at a level of within the probe arm of the interferometer (see Section 2.4).

Thus far, our discussion has centred entirely on interferometric phase measurements. A similar semi-classical analysis can also be performed to find standard quantum limits for most other optical measurements. For instance, Poissonian detection statistics also introduce noise to amplitude or intensity measurements, thus setting a lower limit to precision (we will see in Section 6.3 for the case of optical particle tracking). Although the exact form of the quantum limits can depend on the type of measurement, the precision achievable with coherent light is almost always used as a benchmark in quantum metrology experiments.

2.3 The Heisenberg limit

The standard quantum limit derived above cannot be violated with purely probabilistic photon detection. However, as we briefly discussed at the end of Section 2.1, the presence of correlations () between the output photocurrents from the interferometer provides the prospect to suppress statistical noise and therefore improve the measurement precision (see Eq. (6)). In particular, the two detected fields could, in principle, be entangled such that quantum correlations exist between photodetection events,3 suppressing the statistical variance in the difference signal. Taken to its extreme, it can be conceived that perfect entanglement may allow the elimination of all statistical noise in Eq. (6). In this case, the phase precision would only be limited by the requirement that and be integers. A phase shift can only be resolved if it changes the difference signal by at least one photon. Under a constraint on total power, this can be seen to limit the possible precision to

(10)

Once again, this semi-classical derivation reproduces an important and fundamental result. Eq. (10) is generally referred to as the Heisenberg limit, and is the absolute limit to precision possible in a phase measurement using exactly photons, as discussed in more detail in Section 4.4. Notice that the Heisenberg limit scales faster in than the standard quantum limit, as shown graphically in Fig. 2. This scaling is generally referred to as “Heisenberg scaling”. In principle, it promises a dramatic improvement in precision. For instance, a typical 1 mW laser has a photon flux of around  s. If the Heisenberg limit could be reached with this photon flux, it would be possible, within a one second measurement time, to achieve a phase sensitivity times superior to the best sensitivity possible without quantum correlations. Unfortunately, as it turns out, entanglement tends to become increasingly fragile as the number of photons involved increases. This places a prohibitive limitation on the absolute enhancements that are possible, as discussed in Section 6.2.1.

The Heisenberg limit of Eq. (10) is often described as a fundamental lower limit to the precision of phase measurement achievable using any quantum state (2). In fact, it is possible to outperform this limit using states with indeterminate total photon number – we consider the specific case of squeezed states in Section 4.3. However, no linear phase estimation scheme has been found that provides scaling that is superior to . For completeness, we note that nonlinear parameters can in principle be estimated with scaling that exceeds the Heisenberg limit (49); (50). However, it is not currently clear what benefits such approaches might offer in biological applications, and they will not be discussed further in this review.

2.4 Fundamental limit introduced by inefficiencies

As we saw in Section 2.2.1, the precision of optical phase measurements is degraded in the presence of optical inefficiencies. For measurements that utilise quantum correlated photons, this degradation enters in two ways. First, inefficiencies reduce the magnitude of observed signals and, second, they degrade the correlations used to reduce the noise-floor of the measurement. We consider the effect of inefficiencies on two specific approaches to quantum measurement in Sections 6.2.1 and 6.3.4. Here, without restricting ourselves to a specific class of quantum measurement, we introduce – again via a semi-classical treatment – a fundamental limit introduced by the presence of inefficiencies that is applicable for all phase measurements. This limit is rigorous, although our derivation is not. For a rigorous derivation of this limit we refer the reader to Refs. (51); (20).

Let us return to the balanced optical interferometer from Section 2.1, for which we derived the standard quantum limit with a constraint on total power. We found, there, an expression for the statistical variance of phase estimation (Eq. (6)), which depends on the mean injected photon number as well as the variances and of the photon number arriving at the detectors placed at each interferometer output. As in the previous section, let us imagine that it was possible in some way to achieve photon number variances (and therefore, also, ). With the caveat that and must be integers which leads to the Heisenberg limit discussed in the previous section, this would allow an exact measurement of phase.

Now, consider the effect of inefficiencies on this (unrealistic) perfect phase measurement. By probabilistically removing photons from the fields incident on each of the two detectors, such inefficiencies will introduce statistical uncertainty in the detected photon number, and therefore degrade the precision of the phase measurement. Let us define the detection efficiency at each detector to be , so that each photon arriving at the detector had a probability of not being registered. Within our semi-classical treatment, if half of the photons input to the interferometer are incident on each of the two detectors, the probability distribution of observed photons can be calculated straightforwardly from basic probability theory. This results in the binomial distribution for detector :

(11)

where here, must be even to allow a deterministic and equal number of photons to arrive at each of the two detectors. is identical to , except for the replacement , throughout. Since this is a binomial distribution, its variance is well known, and given by

(12)

Substituting this expression into Eq. (6), and also making the substitution on the bottom line to account for the reduction in the signal due to the presence of inefficiency, we arrive at the bound on phase measurement precision

(13)

which applies under a constraint on total power injected into the interferometer. This exactly corresponds to the fundamental bound due to inefficiency more rigorously derived in Ref. (51). Comparing this expression to the relevant standard quantum limit in Eq. (7), we see that the scaling with input photon number is identical, but with an efficiency-dependent coefficient introduced which, for fundamentally constrains the measurement precision above the standard quantum limit.

This shows that the Heisenberg scaling discussed in the previous section cannot be achieved with arbitrarily high photon numbers, but rather the scaling returns to the usual dependence for sufficiently high photon numbers. Equating Eq. (13) with Eq. (10), we find that this transition back to dependence occurs at a mean input photon number of .

So far in this section we have considered the case where the power constraint on the measurement is placed on the total power in the interferometer. In scenarios where the constraint is instead on the power within the sample, the measurement precision is improved by a factor of two compared to Eq. (13(20), similarly to the case of the standard quantum limit discussed in Section 2.2.2.

3 Quantum coherence and quantum correlations

In the previous section, we introduced several important quantum limits on optical measurements that arise due to the quantised nature of the photon. In order to do this in the simplest possible way, we have treated optical fields from a semiclassical perspective, imagining that they consist of a train of discrete photons, or even as a noise-free electromagnetic field with quantisation only occurring via the production of electrons within the detection apparatus. While this semiclassical analysis is helpful to develop intuition, the quantum correlations at the heart of quantum-enhanced measurements require a full quantum mechanical treatment.

Figure 3: Layout of a Hong-Ou-Mandel interferometer (52). A nonlinear crystal is pumped to produce entangled photon pairs by parametric down-conversion. These photons are then recombined with a beam splitter and detected. A classical treatment would predict that the relative phase would dictate the detection events, and would determine whether the photons are more likely to arrive at detector A or detector B. In reality, non-classical interference ensures that both photons arrive at one detector, with equal probabilities that the photon pair arrives at detector A and B. This violates the classical wave treatment of light, and can only be predicted with quantum theory.

A quantum treatment of photodetection and optical coherence was first performed by Glauber and Sudarshan in Refs. (53); (54). Following such a treatment, one finds that there are some optical phenomena which exhibit classically forbidden behaviour, such as two-photon interference in a Hong-Ou-Mandel (HOM) interferometer (52) (see Fig. 3). These non-classical phenomena rely on quantum correlations, which can also be used to surpass quantum limits to measurement such as the quantum noise limit and standard quantum limit introduced above. This section seeks to provide a basic introduction to the quantum theory of light, and quantum correlations, in the context of quantum metrology. We will utilise standard techniques in introductory quantum mechanics. We refer the uninitiated reader to one of many undergraduate quantum mechanics textbooks for details of these techniques (for example, Ref. (55)). A reader more interested in the applications of quantum measurements than the underlying physics could reasonably skip both this and the following section of the review.

3.1 Quantum treatment of the electric field

In a full quantum treatment of light, the deterministic classical description of the optical electric field introduced earlier must be replaced with an operator description. The electric field is decomposed into a sum of contributions from a complete set of orthonormal spatial modes

(14)

where , , and are, respectively, the spatial mode function, frequency, and volume of mode ; and and are, respectively, the annihilation and creation operators. As their names suggest, when acted upon an optical field, the annihilation operator removes one photon from mode , while the creation operator adds one photon to mode . The spatial mode function is normalised such that . It is straightforward to show that , where is the photon number operator for mode . Therefore, similarly to our previous classical description, is proportional to the total mean photon number in the field.

3.2 Quantum treatment of photodetection

Following the approach of Glauber (53), the process of photodetection can be viewed as the annihilation of photons from the optical field, with corresponding generation of photoelectrons that can be amplified to generate a photocurrent. Annihilation of one photon collapses the state of the optical field from its initial state into a new state, defined by the initial state acted upon by the annihilation operator

(15)

where for simplicity, here, and for the majority of the review, we limit our analysis to only one spatial mode of the field and drop the subscript . We return to multimode fields when treating approaches to quantum imaging in Sections  6.3.3 and 5.1.2. Fermi’s Golden Rule tells us that the transition rate to an arbitrary final state due to photon annihilation is proportional to

(16)

Since the optical field is destroyed in the photodetection process, we are ultimately uninterested in the transition rate to a specific final state, but rather the overall decay of the field. This is given by the sum over all possible final states

(17)
(18)

where is the photon number operator. We see, as expected, a direct correspondence to the classical case where the photon number in the field is viewed as a classical stochastic process, with the rate at which photons are detected being proportional to the photon number in the field.

3.3 Higher order quantum coherence

Let us now consider a second order process where two photons are annihilated. In general, the annihilation events can occur at distinct locations in space and time. However, here we restrict our analysis to co-located events to display the essential physics in the simplest way. In this case

(19)

Repeating a similar calculation as that performed above, we find using Fermi’s Golden Rule that the rate of two-photon detection is proportional to

(20)
(21)

where we have used the commutation relation

(22)

We see that, due to the non-commutation of the annihilation operators, the rate of two-photon detection is fundamentally different than would be predicted for a classical stochastic variable , since it includes the additional term .

To quantify the second order coherence it is conventional to define the normalised second order coherence function

(23)

where is the variance of and the sub-script ‘11’ is used to indicate that the annihilation events are co-located in space and time. In the more general case of non-coincident annihilation, both in time and space, the second order coherence function can be easily shown to be

(24)

where the subscript is used to label an annihilation event at some spatial location and time . When the annihilation events are spatially co-located (), this expression reduces to the well known with the substitution and

(25)

Higher order coherence functions may be defined analogously to Eq. (24) for annihilation events involving more than two photons (see Ref. (56) for further details).

3.4 Classically forbidden statistics

It is illuminating to compare the second order correlation functions in Eq. (23) and (24) to those obtained by modelling the photon number as a classical stochastic process described by a well defined probability distribution . For a field which can fluctuate in time, a classical treatment finds that and . The classical second order coherence function is then

(26)

Similarly, the two-point second order correlation function would classically be given by

(27)

As pointed out by Glauber (53), there exist rigorous bounds on the values that these classical coherence functions can take. Firstly, since it is immediately clear from Eq. (26) that

(28)

A classical field can only saturate this limit if it is perfectly noise-free. By inspection of Eq. (23) it is clear that a quantum mechanical field can violate this limit. Similarly, the Cauchy-Schwarz inequality can be applied to Eq. (27) to show that

(29)

No process with a well defined classical probability distribution can exceed either of these bounds. However, both may be violated with a non-classical field due to the additional term in Eq. (23) which acts to reduce the second order coherence function in co-incident detection. Higher order coherence functions exhibit similar classical bounds, which also may be violated with non-classical fields. Violation both indicates that the field cannot be described fully by a classical probability distribution (and consequently is ill-behaved in phase space in the Glauber–Sudarshan –representation (57)), and is an unambiguous signature that quantum correlations are present. Importantly, the inability to describe such non-classical states via a classical probability distribution provides the prospect for measurements whose performance exceeds the usual bounds introduced by classical statistics.

3.5 Photon bunching and anti-bunching

The second order coherence function quantifies pair-wise correlations between photons in an optical field. If no pair-wise correlations are present, for all . If the average intensity fluctuates, due, for example, to temperature fluctuations in a thermal source, temporally co-incident photons are more likely at times of high intensity. This is known as photon bunching, and is quantified by a second order coherence function for which . Photon co-incidences can also be suppressed, which is known as photon anti-bunching and characterised by . Photon anti-bunching occurs naturally in atomic emission, since an atom is a single-photon emitter. Such a system enters the ground state upon emission of a photon, and must first be excited before it can emit another photon. A non-classical field with (see Eq. (26)) must also exhibit anti-bunching, since at sufficiently long delays any correlations between photons must decay away with approaching unity. Photon bunching and anti-bunching, and their corresponding second order coherence functions are illustrated in Fig. 4.

Figure 4: Photon statistics can be formally quantified with higher order correlation functions, such as (see Eq. (25)), as shown here. The example profiles here correspond to (upper red) thermal light with photon bunching (), (blue) coherent light with , and (bottom green) anti-bunched light ().

3.6 Phase space representations of optical states

Similarly to classical fields, it is often convenient to represent quantum fields as a vector in phase space. However, there exist fundamental differences in the phase space representations of classical and quantum fields. Classical fields may, in principle, be noiseless and represented as a deterministic vector in phase space. Furthermore, in the presence of noise they can be represented as a positive-definite well behaved probability distribution. As we will see in this section, quantum fields, by contrast, exhibit a fundamental minimum level of phase space uncertainty due to the well known Heisenberg uncertainty principle, and have quasi-probability distributions (the quantum analog of a classical probability distribution) that can be badly behaved and even negative over small regions of phase space. Several different but closely related quasi-probability distributions are commonly used to describe quantum fields, including the -representation (58)4, the -representation (59), and the Wigner representation (60). In this review we focus on the Wigner representation, since it is most analogous to a usual classical probability distribution. For simplicity, we avoid a mathematical definition of the Wigner representation, but rather discuss qualitative aspects of the distribution.

Optical amplitude and phase quadratures

Figure 5: Phase space representations of some important quantum states of light. Dashed-lines illustrate the characteristic phase uncertainty of state. a) Fock state (Note: Wigner-negativity associated with Fock states is not depicted), b) vacuum state, c) coherent state, d) squeezed vacuum state, e) amplitude squeezed state, f) phase squeezed state.

The amplitude and phase quadratures which form the axes of the Wigner representation are the quantum mechanical analog of the real and imaginary parts of a classical electric field. They may be defined in terms of the creation and annihilation operators as

(30)
(31)

Using the boson commutation relation given in Eq. (22), it is possible to show that the optical amplitude and phase quadratures do not commute, and satisfy

(32)

As a result, it is not possible to simultaneously measure both quadratures with arbitrary precision. For any optical field, the quadratures are subject to an uncertainty principle given by

(33)

Within the Wigner phase space representation, this enforces a minimum area for the fluctuations of any optical field (56). This places a fundamental constraint on measurements of both the amplitude and phase of the field.

Ball and stick diagrams

Wigner distributions of an optical state may be represented qualitatively on a ball and stick diagram, such as those shown in Fig. 5. Diagrams of this kind illustrate in a clear way the consequences of quantum noise on optical measurements. This is particularly true for Gaussian states that are displaced far from the origin (see Fig. 5c, e, and f), such as the coherent and squeezed states to be discussed in Sections 4.2 and 4.3, where the extent of the noise balls graphically illustrates the precision constraints quantum noise introduces to both amplitude and phase measurements.

4 Quantum treatment of optical phase measurement

In this section we introduce the quantum Fisher information, which allows a full quantum treatment of the precision achievable in general measurements. Its use is illustrated for the phase estimation experiment considered in Section 2. Sections 4.3 and 4.4 then extend the treatment to squeezed states and NOON states, which are quantum correlated states that are regularly applied in the context of quantum metrology.

4.1 Quantum Fisher Information

Figure 6: A quantum treatment of a general measurement process.

One of the key goals of quantum metrology is to study the fundamental limits to precision. The optimal precision achievable with any measurement is quantified by the Cramer-Rao Bound, which states that the Fisher information limits the precision with which a general parameter may be determined,

(34)

For a classical measurement system, the Fisher information is defined by the probability distribution of measurement outcomes. As discussed in the previous section, however, both quantum states and the quantum measurement process are fundamentally different than their classical counterparts. For quantum measurement processes, the quantum Fisher information is determined by the quantum state of the probe. For any quantum measurement procedure, an input probe state interacts with the system of interest. The quantum Fisher information quantifies the information contained within the final state about the parameter (see Fig. 6) and is defined for pure states as (61)

(35)

where primes denote derivatives with respect to . The achievable precision improves when a small change in induces a large change in the final state, as this maximizes the first term . The second term can be understood by considering a first order Taylor expansion of the final state,

(36)

with the small perturbation estimated via measurement of the occupation of the state . The precision of this estimation is limited by , which defines the overlap of the state with the unperturbed state and therefore establishes a noisy background to the measurement.

In the specific case of phase measurement, a phase shift within the signal arm of the interferometer is generated by the unitary operator

(37)

where is the photon number operator for the field in the signal arm. This transforms an arbitrary probe state to

(38)

To evaluate the quantum Fisher information, we first note that

(39)

Using this relation, it is evident that

(40)

and

(41)

The quantum Fisher information is therefore given by

(42)

Since the photon number is conserved by the phase shift operator , the Fisher information can be evaluated either for the input state or the final state . This very elegant result shows that, fundamentally, the sensitivity of phase measurement on a pure quantum state is dictated solely by the photon number variance of the state in the signal arm of the interferometer. Of course, technical limitations can also limit the sensitivity. Most significantly, the Cramer-Rao Bound may only be reached through an optimal measurement on the optical field, and this measurement may prove intractable, for example requiring perfect photon number resolving detectors.

4.2 Phase sensing with coherent states

We begin our quantum treatment of phase sensing by considering interferometry with coherent states.

Coherent states

Coherent states were introduced to the field of quantum optics in 1963 simultaneously by Sudarshan (54) and Glauber (53); (56); (62), who were motivated by the goal to provide an accurate quantum mechanical description of the field emitted by a laser. Coherent states are eigenstates of the annihilation operator

(43)

where is the coherent amplitude of the state, given by a complex number. As can be easily shown from Eq. (43), the coherent amplitude is related to the mean photon number in the field by

(44)

Of all quantum states, coherent states most closely resemble semi-classical fields such as those considered in Section 2. For instance, photon detection on coherent states exhibits the Poisson distribution expected of random arrivals of uncorrelated photons or random production of photoelectrons. This can be seen by expanding the state in the Fock basis (63)

(45)

where is a Fock state consisting of exactly photons. The photon number distribution is then given by

(46)
(47)
(48)

where in the last step, we have used Eq. (44) and the orthogonality of the Fock states . This is the usual form of a Poisson distribution, with variance

(49)

Eqs. (44) and (49) allow the second order coherence function of a coherent state to be determined from Eq. (23), with the result that . This indicates that coherent states exhibit no pair-wise correlations between photons, and indeed their pair-wise statistics may be fully understood within a classical framework.

The variances of the amplitude and phase quadratures of a coherent state are

(50)
(51)

where we have used the Boson commutation relation (Eq. (22)), and the definitions of the quadrature operators (Eqs. (30) and (31)). We see that the coherent state is a minimum uncertainty state, whose quadrature variances are equal and saturate the uncertainty principle (Eq. (33)). As discussed earlier in Section 3, coherent states provide the minimum uncertainty possible without quantum correlations, and therefore establish an important lower bound on measurement precision. Indeed, a coherent state can be thought of as a perfectly noiseless classical field with additional Gaussian noise introduced with the statistics of vacuum noise. For this reason they are often referred to as “classical light” (2); (41); (16), with the standard quantum limit described as the best precision which can be achieved classically (64); (65), even though truly classical light could, in principle, be entirely noise free.

Cramer-Rao Bound on phase sensing with coherent states

From Eqs. (34), (42), and (49) it is possible to immediately determine the Cramer-Rao Bound for phase sensing with coherent states. This, unsurprisingly, reproduces the semi-classical result given in Eq. (9), and can be reached using the same linear estimation strategy given in Eq. (5). Since this represents the best precision that is classically achievable, it follows that any state with higher quantum Fisher information than a coherent state must necessarily exhibit quantum correlations; though the value of the quantum Fisher information does not necessarily quantify the degree of entanglement (66).

It is important to note that the Cramer-Rao Bound, as derived here, establishes the fundamental limit to the precision achievable for a given photon number at the phase-shifting sample. However, achieving this bound generally requires that additional photons are introduced within the measurement device. If the total power is constrained, these additional photons must be included in the derivation of the quantum Fisher information.

4.3 Phase sensing with squeezed states

The field of quantum metrology broadly began when Caves showed theoretically that squeezed states of light could be used to suppress quantum noise in an inteferometric phase measurement (1). This principle is currently used in gravitational wave observatories, with squeezed vacuum used to enhance precision beyond the limits of classical technology (4); (5). In this sub-section we introduce the concept of phase sensing with squeezed states.

Squeezed states

As we have already shown, the uncertainty principle places a fundamental constraint on the product of amplitude and phase quadrature variances of an optical field (see Eq. (33)). Coherent states spread this uncertainty equally across both quadratures. By contrast, as their name suggests, squeezed states trade-off reduced uncertainty in one quadrature with increased uncertainty in the other. This is an important capability in quantum metrology, and can be applied to enhance the precision of a broad range of optical measurements. If the amplitude is squeezed as illustrated in Fig 5e, for instance, the photons tend to arrive more evenly spaced than in a coherent field, which is a classically forbidden phenomena known as photon anti-bunching (see Section 3.4). This can be used to reduce the variance in amplitude or intensity measurements, and thus enable precision better than that possible with coherent states. Alternatively, as illustrated in Fig. 5f, the precision of phase sensing can be improved by orientating the squeezed quadrature to be orthogonal to the coherent amplitude of the field.

Mean photon number and photon number variance

The mean photon number in a squeezed state is given by

(52)

where we have used the definitions of and in Eqs. (30) and (31), and the commutation relation between them (Eq. (32)). We see that, unlike the coherent state, squeezed states have non-zero photon number even when their coherent amplitude is zero (see Fig. 5d). While we do not derive it here, in general, the photon number variance of a squeezed state is given by (63)

(53)

where is the angle between the squeezed quadrature of the state and its coherent amplitude.

Cramer-Rao Bound for squeezed vacuum

As shown in Section 4.1, to maximise the precision of a phase measurement the photon number variance should be made as large as possible. Without loss of generality, we take the quadrature to be squeezed, with the quadrature then maximally anti-squeezed. For fixed mean photon number , it can be shown from Eqs. (52) and (53) that the maximum photon number variance is achieved when . Rearranging Eq. (52), we then find that

(54a)
(54b)

where we have used the relation which is valid for pure squeezed states. Substituting these expressions into Eq. (53) and simplifying results in the photon number variance:

(55)

Substitution into Eq. (42) for the quantum Fisher information then yields

(56)

so that the Cramer-Rao Bound (Eq. (34)) on phase sensing with squeezed light is given by

(57)

This squeezed state bound is compared with the Heisenberg and standard quantum limits in Fig. 2. It exceeds the Heisenberg limit at all photon numbers, though only by a relatively small margin. For large photon number , it has Heisenberg scaling () as has been shown previously in Ref. (67). Furthermore, Ref. (68) shows that of all possible choices of input state, squeezed vacuum achieves the optimal sensitivity. It should be emphasized, however, that proposals to reach the bound given in Eq. (57) require perfectly efficient photon-number resolving detectors and nonlinear estimation processes, and are not achievable with existing technology (67). The influence of inefficiencies on the precision of squeezed-state based phase measurement is considered for the specific case of linear detection and estimation in Section 6.3.4.

Cramer-Rao Bound in the large coherent amplitude limit

In general, increasing levels of quantum correlation are required to achieve a large photon number variance in a pure quantum state that has no coherent amplitude (). As a result such states quickly become fragile to the presence of loss, which removes photons and therefore degrades the correlations between them (see Sections 6.2.1 and 6.3.3). As a result, most state-of-the-art measurements rely on coherent states with large coherent amplitudes rather than non-classical states. However, with squeezed states it is possible to benefit from both large coherent amplitude and quantum correlations. Considering the limit , Eqs. (52) and (53) become:

(58)
(59)

These expressions allow us to calculate the second order coherence function for bright squeezed states, as described in Eq. (23), with the result that

(60)

Remembering that we have chosen to orientate our squeezed state such that , we can thus see that phase squeezed light with is bunched (); by contrast, amplitude squeezed light with is anti-bunched (), which is one clear indicator of quantum correlations (see Section 3.4).

Using Eqs. (58) and (59) the quantum Fisher information for a bright squeezed state can be calculated from Eq. (35). It is given by

(61)

The optimal phase precision is clearly achieved when the antisqueezed quadrature is aligned in phase space with the coherent amplitude of the state (i.e., ), such that the phase variance is minimized while the amplitude variance is maximized. In this limit, the achievable phase precision is given by

(62)

where the relevant standard quantum limit for phase measurement is given in Eq. (9). Since here , we see that squeezed light allows precision beyond the standard quantum limit.

Similarly to the case of phase sensing with coherent states, the optimum phase precision in Eq. (62) may be reached using a straight-forward linear estimation strategy. Let us describe the squeezed probe field in the Heisenberg picture via the annihilation operator , where as usual the coherent amplitude and the quantum noise operator has zero expectation value. The action of a phase shift is to transform the field to the output field given by

(63)
(64)
(65)

where in the first approximation, we have taken a first order Taylor expansion of the exponential, constraining our analysis to a small phase shift, ; and in the second approximation, since we are considering the case of a bright field with large, we neglect the product of two small terms . This expression shows that, to first order, the action of the phase shift is to introduce an imaginary component to the coherent amplitude of the field – i.e., it displaces the field in phase space along the axis of the phase quadrature . This is illustrated in Fig. 7, which compares coherent and phase squeezed fields.

Figure 7: Phase space representations of the first-order effect of a phase shift on (a) bright coherent and (b) phase squeezed fields. () and () label the states of the fields prior to, and after, the phase shift, respectively.

Using Eq. (31), the phase quadrature of the output field is given by

(66)

so that, rearranging,

(67)

We see that measurement of , which can be achieved quite straightforwardly, for instance, using homodyne detection, allows an estimate of the phase . The uncertainty of the estimate is determined, in combination, by the uncertainty of the phase quadrature of the probe field , and the magnitude of the coherent amplitude of the probe field, which, in the bright field limit taken here, is simply related to the probe photon number via Eq. (58). Taking the standard deviation of the measurement noise term in Eq. (67), we find that this measurement exactly saturates the phase precision inequality of Eq. (62), demonstrating that a standard noise-free homodyne measurement can – in principle – reach the absolute Cramer-Rao Bound to phase precision for a bright squeezed field.

Interferometry combining squeezed vacuum with a coherent field

Figure 8: A Mach–Zehnder phase sensing experiment enhanced via the injection of a squeezed vacuum. The squeezed vacuum entangles the fields within the interferometer, and can be used to suppress quantum noise.

The scenario described in the previous sub-section corresponds to a single bright squeezed state propagating through a phase shifting element. Most experimental demonstrations apply an alternative approach, in which a squeezed vacuum and coherent state are combined on a beam splitter and then propagate through an interferometer (see Fig. 8). The resulting two fields in the interferometer are an entangled pair of bright squeezed states, with anticorrelated quantum fluctuations. The phase of the input fields is chosen to align the coherent input to the antisqueezed quadrature, which maximizes the quantum Fisher information for a differential phase shift measurement. A full analysis of this approach shows that the phase precision can be enhanced by , similar to Eq. (62(1).

This is the basic approach considered by Caves in the first quantum metrology proposal (1), and which is now used in gravitational wave observatories (4); (5). It is particularly useful since the quantum state preparation can be separated from the generation of a large coherent amplitude, allowing both states to be independently optimized. Within this framework, the non-classical state provides a fixed enhancement in precision for the arbitrarily bright coherent state. It is of note that, for constrained photon occupation in the non-classical state, of all possible choices of input state squeezed vacuum provides the largest possible precision enhancement to the bright interferometric phase measurement (68).

Generation of squeezed states

Squeezed states of light are produced via nonlinear optical interactions. The essential feature of such nonlinear interactions is a reversible phase sensitive amplification of the optical field, such that the fluctuations of one quadrature are noiselessly amplified while the fluctuations of the orthogonal quadrature are noiselessly de-amplified. Phase sensitive amplification can be achieved using a wide range of nonlinear processes, such as optical parametric oscillation and amplification (69); (42), optical Kerr nonlinearities (70), second harmonic generation (71), and four-wave mixing in an atomic vapour (72); (73); and each of these approaches has been used to generate squeezed light. To take one example, amplitude squeezed light can be generated via second harmonic generation, where two photons combine to produce a single frequency-doubled photon with a probability proportional to the square of the intensity of the incident field. The quantum fluctuations in the intensity of the field then translate to fluctuations in the probability of second harmonic generation. Thus, when the field intensity fluctuates toward a larger value, the second harmonic generation becomes more efficient and more of the light is lost. A fluctuation of field intensity toward a smaller value will result in less efficient second harmonic generation, and consequently smaller loss. The net effect is to suppress the intensity fluctuations, which can result in fluctuations below the vacuum level, with a consequent increase in the phase fluctuations.

Much of the effort in the development of squeezed light sources over the past decade has focussed on achieving sources useful to enhance measurements in gravitational wave observatories (3). One important requirement for this application is a high degree of squeezing which can allow an appreciable enhancement in precision. In absolute terms, the strongest single-mode squeezing is currently achieved in optical parametric oscillators that are pumped with light at 532 nm or 430 nm and which produce vacuum squeezed fields at 1064 nm (74); (69); (42) and 860 nm (75), respectively. Such sources are capable of squeezed quadrature variances as small as  (74). A second key requirement is for the squeezed light source to allow quantum enhanced precision at low frequencies (76); (77); (78). Squeezed light has a finite frequency band of enhancement, and many sources only provide squeezing in the MHz regime (16); (79). This is inadequate for gravitational wave detection, where most of the signals are expected at hertz and kilohertz frequencies (3). This frequency band is also important to biological experiments, as it encompasses most biophysical dynamics studied to date (see Section 5.3). Consequently, biological quantum metrology could directly benefit from the advances toward gravitational wave detection. Squeezed light has now been demonstrated at frequencies as low as 1 Hz (80), though light which is squeezed to 10 Hz is still considered state-of-the-art (5); (42).

In a typical squeezed light source, the nonlinear medium is placed within an optical resonator to increase the strength of the nonlinear interaction and thus the magnitude of the squeezing. This generally allows only a single mode of the optical field to be squeezed, as is required to enhance single-parameter measurements such as phase or intensity measurements. However, efforts have also been made to generate strong multimode squeezing for applications such as quantum enhanced imaging. In particular, multimode degenerate optical resonators have been developed to allow resonant enhancement of as many as eight modes (81); while the strong optical nonlinearity near the optical resonances of an atomic vapour allows strong squeezing without use of an optical cavity, allowing generation of a multimode squeezed vacuum state over hundreds of orthogonal spatial modes (72); (73). Another source of highly multimode quantum correlated light is parametric down-conversion, which produces a photon pair from a single high-energy photon. To conserve momentum, the photons always propagate with opposing directions, such that they occupy coupled spatial modes.

4.4 Phase sensing with NOON states

As discussed in Section 4.1, in general, the achievable precision of optical phase measurements improves as the photon number variance of the field in the signal arm of the interferometer increases (see Eq. (4.1)). In the limit that exactly photons are used,5 the precision is maximized for a NOON state, in which the constituent photons are in a superposition of all occupying either the signal or reference arm of the interferometer, with the other unoccupied (64). In the number state basis, the NOON state can be represented as

(68)

where signifies a state containing photons, and the subscripts “sig” and “ref” label the signal and reference arms of the interferometer, respectively. When using a NOON state, the quantum Fisher information for phase estimation (Eq. (42)) is determined by

(69)
(70)
(71)
(72)
(73)
(74)

such that

(75)

and

(76)

This is exactly the Heisenberg limit on phase estimation which we derived via a semi-classical approach earlier (see Eq. (10)). It has been shown that the NOON state achieves the optimal phase precision for states which contain exactly photons; although states with indeterminate photon number such as the squeezed vacuum have been shown to also achieve Heisenberg scaling and allow superior precision (82); (83); (84); (85) (compare Eq. (76) with Eq. (57) in Section 4.3).

To understand why NOON states allow enhanced precision, it is instructive to consider the evolution of the state through the interferometer. When the phase shifting unitary of Eq. (37) is applied to a number state, it evolves as

(77)

Consequently, the phase shifting operator applies a phase shift of to a number state. NOON states acquire this amplified phase shift. By contrast, using the Fock state expansion of coherent states given in Eq. (45), we see that a coherent state evolves as

(78)

which corresponds to the same phase shift for any mean photon number. The enhanced phase precision achievable with a NOON state originates from this amplification of the relative phase shift (64). Based on this behaviour, one might expect that a NOON state would achieve sub-wavelength interference features. This intuition is correct, but not in the manner that may be expected. NOON states do not allow sub-wavelength interference of intensity, but rather allow sub-wavelength interference features in the higher order photon statistics, such as two-photon coincidence detection.

Generation of NOON states

A single photon NOON state can be generated simply by splitting a single photon on a 50/50 beam splitter, which places the photon into an equal superposition of reflecting and transmitting, i.e., . However, the Heisenberg limit coincides with the standard quantum limit for (compare Eq. (7) with Eq. (10)), so that such a state cannot be used to enhance precision. Most sensing applications, instead, apply two photon NOON states, which are generally produced from entangled photon pairs via two-photon interference in a Hong-Ou-Mandel (HOM) interferometer (52) (see Fig. 9, and also Fig. 3). In principle this allows the standard quantum limit to be surpassed by a factor of . However, this is only true for constrained total power. An important and often overlooked distinction arises if we only constrain power through the sample, and allow arbitrary power in the reference arm; in this case the relevant limit is Eq. (9) and a two-photon NOON state can only equal the standard quantum limit. Put another way, NOON states must contain at least three photons before they can outcompete coherent states of similar mean intensity at the sample.

The entangled photon pairs can be generated via spontaneous parametric down conversion in a nonlinear crystal that is pumped with light at twice the optical frequency. The spatial modes of the photons generated by this down conversion process are dictated by energy and momentum conservation. In general, many pairs of spatial modes are each populated with correlated photons. These photons are termed to be “hyper-entangled” (86), since they can exhibit not only temporal correlations, but also correlations in their momentum, position, and polarization. This enables a range of imaging applications, some of which are discussed in Sections 6.16.4, and 7.4. For simplicity, here, however, we restrict our analysis to a single pair of energy and momentum conserving spatial modes (labelled with the subscripts “1” and “2”), in which case the ideal state generated by the process is  (87)

(79)

where characterises the strength of the nonlinear interaction. It can be seen that this state includes components of vacuum (), the desired photon pairs (), as well as higher order terms. While, in some circumstances, the higher order terms can be useful, generally they are undesirable, introducing noise to the measurements. Consequently, most quantum metrology experiments operate with very weak nonlinear interaction strength, such that . Furthermore, since photon counters will clearly register no clicks if the state is vacuum, the vacuum term in the expansion can be neglected. In this case, the output state from a parametric down converter can be well approximated as the ideal photon pair

(80)

Interfering the photon pairs generated via parametric down conversion on a balanced beam splitter results in Hong-Ou-Mandel interference (52). For generality, we include a relative phase shift on the input arm via the phase shifting unitary operator (Eq. (37)), though, ultimately, we will see that this does not affect the outcome. The input quantum state therefore becomes