User:OtherColeKP/Noh Foegères Mandel Experiment

The Noh-Foegères-Mandel (NFM) experiment was a quantum optics study carried out by J. W. Noh, A. Fougères, and L. Mandel in 1991 to investigate the phase of the electromagnetic field in quantum mechanics. Although experimental work on the subject had been done before, this high-precision experiment is known for its pragmatic philosophy of interpretation, which yielded a well-known and lasting legacy of Comments  , Replies   , and follow-ups.

The "phase" the authors were interested in refers not the argument of the (unobservable) global phase of a quantum state, or the (observable) relative phase of superposed states, but rather the optical phase of the electromagnetic field. A mode of a classical plane wave may be described by the vector function

$$\mathbf{u}(\mathbf{x},t)=\mathbf{u}_0e^{i(\mathbf{k}\cdot\mathbf{x}-\omega t+\phi)}$$,

where $$\mathbf{x}$$ and $$t$$ are the position and time parameters, $$\mathbf{u}_0$$ is the vector amplitude, the argument of the exponent $$\mathbf{k}\cdot\mathbf{x}-\omega t$$ encodes the time and space dependence, and $$\phi$$ is the phase of the wave. The corresponding quantum mechanical idea of this last quantity was the focus of this experiment.

Classical Harmonic Oscillator
Observables in classical mechanics are conveniently described as smooth functions on phase space -- that is, elements of $$C^\infty(T^*M)$$, where $$M$$ is the configuration space of interest and $$T^*M$$ its cotangent bundle. Phase is nothing special in this setting : for the simple harmonic oscillator: the Hamiltonian in (q,p) coordinates may be written


 * $$ H=\tfrac{p^2}{2m}+\tfrac{1}{2}m\omega^2q^2$$.

Performing a type 2 canonical transformation effects a change to action-angle coordinates $$(I=H\omega^{-1},\theta)$$, where the equations of motion are readily integrated. One may compactly write


 * $$ \theta(t)=-\omega t+\theta_0,\quad I(t)=I(0),$$
 * $$\alpha(t)=\sqrt{\tfrac{m\omega}{2}} (q(t)+ \tfrac{i}{m\omega}p(t)) =\sqrt{I(t)}e^{i\theta(t)}$$,

where $$\alpha(t)$$ is the complex amplitude. The system flows in phase space along level surfaces of conserved energy. Time dependence is encoded entirely within the linear evolution of the phase variable $$\theta(t)$$.

Quantum Harmonic Oscillator
The quantum harmonic oscillator is a standard problem  typically solved with the technology of ladder operators. One writes


 * $$ \widehat{a}=\sqrt{\tfrac{m\omega}{2\hbar}} (\widehat{q} + \tfrac{i}{m\omega}\widehat{p}),\quad \widehat{a}^\dagger=\sqrt{\tfrac{m\omega}{2\hbar}} (\widehat{q} - \tfrac{i}{m\omega}\widehat{p})$$
 * $$\widehat{H}=\tfrac{\widehat{p}^2}{2m}+\tfrac{1}{2}m\omega^2\widehat{q}^2=\hbar\omega(\widehat{a}^\dagger\widehat{a}+\tfrac{1}{2})$$,

and then defines the vacuum state $$\vert 0 \rangle$$ to be such that it's annihilated by the lowering operator: $$\widehat{a}\vert 0 \rangle=0$$. The tower of eigenstates $$\{\vert n\rangle\}$$ with eigenenergies $$E_n=\hbar\omega(n+\tfrac{1}{2})$$ is constructed by acting on this vacuum with the raising operator: $$\sqrt{n!}\vert n\rangle = \widehat{a}^\dagger\vert 0 \rangle$$.

The expressions for the lowering and raising operators greatly resemble those of the amplitudes in the classical case. This motivates definitions for quantum action and phase operators by analogy :


 * $$\sqrt{\hbar}\widehat{a}=\sqrt{\widehat{I}}\widehat{e^{i\theta}},\quad \sqrt{\hbar}\widehat{a}^\dagger=\left(\widehat{e^{i\theta}}\right)^\dagger\sqrt{\widehat{I}}$$.

If one demands that $$\left(\widehat{e^{i\theta}}\right)^\dagger \widehat{e^{i\theta}}=\widehat{1}$$ it makes sense to write $$\widehat{I}=\widehat{n}\hbar=\widehat{H}\omega^{-1}-\tfrac{1}{2}$$. This is a reasonable quantum generalization of the classical action with a correction, and since $$\widehat{n}$$ is a positive operator it follows that $$\widehat{I}$$ has a well-defined square root. However, from the commutation relation $$\left[\widehat{a},\widehat{a}^\dagger\right]=1$$ it can be found that $$\widehat{e^{i\theta}}\left(\widehat{e^{i\theta}}\right)^\dagger \ne\widehat{1}$$. $$\widehat{e^{i\theta}}$$ therefore cannot be a unitary operator. In this sense one cannot define a Hermitian operator $$\theta$$ exponentiating in such a way that permits the expression $$\widehat{e^{i\theta}}=e^{i\widehat{\theta}} $$.

Modern Quantum Observables
The definition of observables in older quantum mechanics literature requires them to be Hermitian operators. Such objects have real eigenvalues and can be spectrally decomposed into orthogonal projections. This leads to the notion of a projection-valued measure, which can be used to define a probability distribution for an observable with respect to a specified quantum state. Given a state $$\widehat{\rho}$$ and a Hermitian operator $$\widehat{O}$$ this probability distribution may be defined via currying:


 * $$P(\widehat{O}=\lambda|\widehat{\rho})=\text{Tr}(\widehat{\rho}\widehat{\pi}_\lambda)$$.

This expression holds literally in discrete case, and spiritually in the continuous part of the operator spectrum $$\sigma(\widehat{O})$$ as well. $$\widehat{\pi}_O$$ is the projection onto the eigensubspace corresponding to eigenvalue $$\lambda\in\sigma(\widehat{O})$$, and the set of all such projectors formally satisfies the completeness relation $$\sum\limits_{\lambda}\widehat{\pi}_\lambda=\widehat{1}$$.

For non-Hermitian operators this decomposition no longer holds. Consequently, this statistical construction can't be carried out. In this strict sense, the above mathematical exposition suggests that phase is an observable in classical mechanics that fails to be one in quantum mechanics. This fault singles phase out as a standard classical observable without a proper quantum analogue.

Modern quantum mechanics, however, has developed a more general measurement theory through the technology of positive operator-valued measures (POVMs). From this perspective one speaks of the probability of some effect $$E_i$$ on a quantum state $$\widehat{\rho}$$ being observed with probability


 * $$P(i|\widehat{\rho})=\text{Tr}(\widehat{\rho}\widehat{E}_i)$$.

Like $$\widehat{\pi}_\lambda$$ before, the $$\widehat{E}_i$$ satisfy the sum rule $$\sum\limits_{i}\widehat{E}_i=\widehat{1}$$, which reflects the unitary principle of probability. The $$\widehat{E}_i$$, however, are more general: they are only required to be positive semi-definite operators, whereas the $$\widehat{\pi}_\lambda$$ must be orthogonal projections.

POVMs describe so-called generalized measurements: which may be thought of as observations of open quantum systems. They are particularly natural when considering measurements of the electromagnetic field, whose photons are often absorbed by detectors. Measurements based on such events cannot be described in terms of a simple projection, since the experiment can't be repeated on the post-measurement state; absorbed photons are lost.

It is important to note that although this perspective may appear more general, it is in fact not at odds with the original principle that only Hermitian operators are observables. Any POVM may be theoretically realized (non-uniquely) by a standard projective measurement carried out on a Hilbert space larger than the system of interest after it's interacted with some ancilla. In a sense, this is what's already done in the von Neumann model of measurement. This process may coarsely be described in terms of the system becoming correlated via some interaction with a classical measuring apparatus. The post-measurement quantum state is inferred from the state of the classical system, whose projected post-measurement state in turn is taken as self-evident.

This principle of generalized measurements is important in the context of NFM's experiment because they are in effect performing quantum homodyne detection. This corresponds to measurement of $$\widehat{a}_\theta=\tfrac{1}{2}\left(\widehat{a}e^{-i\theta}+\widehat{a}^\dagger e^{i\theta}\right)$$. This is a Hermitian operator, but it has support on a continuous spectrum. What is measured is a difference of normalized intensities -- that is, physically, a (renormalized) difference of photon counts between two detectors. More interesting, however, is the ultimate transformation : taking the beam splitter's unitary transform into account, the infinitesimal effect corresponds to


 * $$d\widehat{E}_x=dx\delta(x-\widehat{a}_\theta)\equiv dx\int_{-\infty}^\infty\frac{d\lambda}{2\pi}e^{i\lambda\left(x-\tfrac{1}{2}\left(\widehat{a}e^{-i\theta}+\widehat{a}^\dagger e^{i\theta}\right)\right)}$$,

where $$ x$$ here labels the renormalized measurement that corresponds to a quadrature measurement. That is to say, the output of homodyne detection is a marginal of one input state's Wigner function, depending on the phase of the other. Mathematically, this is not considerably different from idealizations of position measurement in the von Neumann scheme. What is significant is how the actual measurement is a collective measurement on the entire state (of both beam outputs) that is subsequently marginalized (only differences in photon number are important).

The similar quantum heterodyne scheme is more radically different than the standard apparatus. Compared with homodyne, the frequency of the ancilla is detuned from the input state of interest, yielding the effects


 * $$ \widehat{E}_{\alpha}=\tfrac{1}{\pi}\vert \alpha\rangle\langle \alpha \vert$$,

where $$\vert \alpha\rangle$$ is a coherent state. The operator effectively measured in this case is the non-Hermitian $$\widehat{a}_S+\widehat{a}_I^\dagger$$ sum of signal and image modes. Consequently, the set of effects is nonorthogonal, and the measurement statistics draw from the Husimi Q distribution.

Defining Quantum Phase
In their paper, NFM compare their experimental results against two leading (and competing) theoretical frameworks: The standard framework of Susskind and Glogower, and the then-newer work by Pegg and Barnett. These historical perspectives are operator-based, in the sense that the authors sought to define operators whose decomposition yielded projectors that could be used to define phase probability distributions. This contrasts with the more modern understanding of measurements via POVMs, and with the phase-space perspective that considers Q distributions, P functions, or Wigner functions. NFM's experimental results differ considerably from the predictions of both the Susskind-Glogower and Pegg-Barnett theories in the quantum regime. They advocate the eponymous operational approach, where observables are defined by the experimental routine employed to measure them.

Susskind-Glogower formalism
Leonard Susskind and Jonathan Glogower, along with Jack Sarfatt, were inspired by a homework problem assigned by Peter Carruthers to find a logically consistent definition of a phase operator $$\widehat{\phi}$$. A key idea guiding their work was that while $$\widehat{e^{i\theta}}$$ wasn't a unitary operator, it could be suitably added or subtracted from its conjugate to obtain Hermitian operators that looked like sines and cosines of phase. In their paper, Susskind and Glogower define
 * $$\widehat{\text{cos}\phi}=\frac{1}{2}\left[\tfrac{1}{\sqrt{\widehat{n}+1}}\widehat{a}+\widehat{a}^\dagger\tfrac{1}{\sqrt{\widehat{n}+1}}\right],\quad \widehat{\text{sin}\phi}=\frac{1}{2i}\left[\tfrac{1}{\sqrt{\widehat{n}+1}}\widehat{a}-\widehat{a}^\dagger\tfrac{1}{\sqrt{\widehat{n}+1}}\right] $$.

These operators notably do not commute: it can be shown that $$\left[\widehat{\text{cos}\phi},\widehat{\text{sin}\phi}\right]=\tfrac{i}{2}\vert 0\rangle\langle 0 \vert$$. Conventional quantum mechanics suggests that the values of the observables corresponding to these two operators therefore cannot be known simultaneously. This is an obstacle to inferring the phase $$\phi$$, since classically knowledge of both the sine and cosine of an angle is sufficient to determine the angle. A further complication arises from the fact the equality $$\widehat{\text{cos}\phi}^2+\widehat{\text{sin}\phi}^2=1-\tfrac{1}{2}\vert 0\rangle\langle 0 \vert$$. This is contrary to conventional trigonometry.

Ultimately, however, it is phase differences that one hypothetically measures. In the same paper, Susskind and Glogower define phase difference operators to match the familiar trigonometric addition formulas:


 * $$\widehat{\text{cos}}(\phi_1-\phi_2)=\widehat{\text{cos}\phi_1}\widehat{\text{cos}\phi_2}-\widehat{\text{sin}\phi_1}\widehat{\text{sin}\phi_2},\quad \widehat{\text{sin}}(\phi_1-\phi_2)=\widehat{\text{cos}\phi_1}\widehat{\text{sin}\phi_2}+\widehat{\text{sin}\phi_1}\widehat{\text{cos}\phi_2}$$.

However, these operators still do not commutate or obey the standard square sum formula:
 * $$\left[\widehat{\text{cos}}(\phi_1-\phi_2),\widehat{\text{sin}}(\phi_1-\phi_2)\right]=\tfrac{i}{2}\left(\vert 0\rangle\langle 0 \vert_1-\vert 0\rangle\langle 0 \vert_2\right),\quad \widehat{\text{cos}}(\phi_1-\phi_2)^2+\widehat{\text{sin}}(\phi_1-\phi_2)^2=1-\tfrac{1}{2}\left(\vert 0\rangle\langle 0 \vert_1-\vert 0\rangle\langle 0 \vert_2\right)$$.

Remarkably, despite these apparent defects, the eigenkets $$\vert\widehat{e^{i\theta}}\rangle$$ of the Susskind Glogower $$\widehat{e^{i\theta}}$$ are in a sense canonical objects, in that they lead to maximum likelihood phase estimation. These states are commonly written in the unnormalizable form


 * $$\vert\widehat{e^{i\theta}}\rangle=\sum\limits_{n=0}^\infty e^{in\theta}\vert n\rangle $$,

which are nonorthogonal on account of the lower limit of the sum and the nonunitary nature of $$\widehat{e^{i\phi}}$$. They are nevertheless overcomplete, and a POVM may be constructed from them with elements $$\widehat{E}_{\theta}=\tfrac{1}{2\pi}\vert\widehat{e^{i\theta}}\rangle\langle\widehat{e^{i\theta}}\vert$$.

Use of these effects, in conjunction with optimal states, yields a theoretically optimal phase estimation scheme, known as canonical phase measurement. This insight came after the work of NFM, along with broader understanding of the POVM formalism: from this later perspective, the value of the Susskind-Glogower formalism is solely in providing a means of coming up with the kets $$\vert e^{i\theta}\rangle$$ "that task accomplished, its job is over." Coming up with realistic instrumentation realizing this, however, is more difficult. It has been shown that it may be asymptotically realized using several identically prepared input states via homodyne detection, while adaptive methods can theoretically realize single-shot canonical phase measurement.

Pegg-Barnett formalism
Pegg and Barnett begin by first considering the finite dimensional states
 * $$\vert \theta_m\rangle_s=\tfrac{1}{\sqrt{s+1}}\sum\limits_{n=0}^s e^{in\theta_m}\vert n\rangle $$,

where $$\theta_m=\theta_0+\tfrac{2m\pi}{s+1}$$ for $$m=0,1,...,s$$. $$\theta_0$$ is an arbitrary reference angle. These states may be understood as states of maximal uncertainty for the number operator. With these, a well-defined finite dimensional Hermitian phase operator may be constructed as $$\widehat{\phi}_s=\sum\limits_{m=0}^s\theta_m\vert \theta_m\rangle_s\langle\theta_m\vert_s$$. Physical matrix elements are then defined using this operator via a limit: for a function $$F(\phi)$$ of the phase one writes
 * $$F_{\psi\psi'}=\underset{s\to\infty}\text{lim}\langle \psi\vert F(\widehat{\phi}_s)\vert\psi'\rangle$$.

The states $$\vert \theta_m\rangle_s$$ themselves become poorly-defined in the norm limit (or otherwise improperly normalized): this is why the strong limit is taken on the matrix elements. However, the same is true of the standard momentum and position eigenkets $$\vert \mathbf{x}\rangle$$, $$\vert \mathbf{p}\rangle$$: this situation may be formally ameliorated through the technology of the rigged Hilbert space.

In contrast to the Susskind-Glogower operator, trigonometric functions of the Pegg-Barnett operator satisfy $$\text{cos}^2\widehat{\phi}_s+\text{sin}^2\widehat{\phi}_s=1$$ and $$\left[\text{cos}\widehat{\phi}_s,\text{sin}\widehat{\phi}_s \right]=0$$.

Pegg and Barnnet's approach, along with other physically equivalent methods, is often referred to as the canonical approach, with the phase distribution $$P(\theta|\widehat{\rho})=\text{Tr}\left( \hat{\rho}\vert\theta\rangle\langle\theta\vert|\right)$$ known as the canonical phase distribution. This is because the physical phase in this formalism is canonically conjugate to the photon number, much as momentum is canonically conjugate to position. This distribution is physically the same as the POVM defined via the $$\vert e^{i\theta}\rangle $$ kets in the Susskind-Glogower formalism.

The Experiment
The authors considered two separate schemes for inferring the phase of light. Since it is never total phase, but only relative phase that can be observed, their schemes were necessarily interference experiments. The first scheme was not experimentally implemented in the first paper, but acts as a function of rhetoric in framing their quantum-classical analysis.

The central idea was the following: to consider procedures that yield, in some limit, a classical measure of phase, and then tune to a quantum regime and compare with theory. What they found is that both the Susskind-Glogower and Pegg-Barnett formalisms -- two of the most predominant theoretical frameworks at the time -- failed to describe their experimental results in the quantum limit. Moreover, the two separate schemes the authors considered disagreed with each other in the quantum limit, in the sense that the authors constructed distinct theoretical frameworks for the two to model their results. It was these discrepancies that lead the authors to promote their titular philosophy of an "Operational approach to the phase of a quantum field," wherein "appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme" and accordingly that "different schemes correspond to different operators."

Scheme 1
In the first scheme, two quasimonochromatic beams of the same polarization are combined at a balanced beam splitter, then measured by two detectors. If the complex fields corresponding to the two inputs have arguments differing by $$\phi_2-\phi_1$$, then the quantity $$\text{cos}(\phi_2-\phi_1)$$ can be inferred from the integrated signals of the two detectors and their correlations. By inserting a phase-shifter it is possible to obtain $$\text{sin}(\phi_2-\phi_1)$$. The setup is shown in the figure above with a classical interpretation. The specific relationship between phase and measured intensity may be written


 * $$\text{cos}(\phi_2-\phi_1)= \frac{W_4-W_3}{(W_4-W_3)^2+(W_6-W_5)^2},\quad \text{sin}(\phi_2-\phi_1)= \frac{W_6-W_5}{(W_4-W_3)^2+(W_6-W_5)^2},$$
 * $$W_i=\alpha\int_0^T I_i(t')dt',\quad i=3,4,5,6,$$

where $$I_i(t')$$ is the measured intensity and $$\alpha$$ is the  quantum detector efficiency.

For the quantum theory, which becomes relevant for low intensity inputs where fluctuations become important, it's to be understood that what's actually registered by the detectors are individual photon counts. Consequently, the experimental record is related to the difference of these photon counts. The operators considered are written


 * $$\widehat{\text{sin}}(\phi_2-\phi_1)=\frac{\widehat{n}_4-\widehat{n}_3}{\sqrt{(\widehat{n}_4-\widehat{n}_3)^2+(\widehat{n}_6-\widehat{n}_5)^2}},\quad \widehat{\text{cos}}(\phi_2-\phi_1)=\frac{\widehat{n}_6-\widehat{n}_5}{\sqrt{(\widehat{n}_4-\widehat{n}_3)^2+(\widehat{n}_6-\widehat{n}_5)^2}}$$,

where $$\widehat{n}_i$$ is a number operator at a detector. In terms of the creation and annihilation operators at the input ports, these may expressed as


 * $$\widehat{n}_4-\widehat{n}_3=\tfrac{1}{2i}\left(\widehat{a}^\dagger_1\widehat{a}_2-\widehat{a}^\dagger_2\widehat{a}_1\right)$$,
 * $$\widehat{n}_6-\widehat{n}_5=\tfrac{1}{2}\left(\widehat{a}^\dagger_2\widehat{a}_1+\widehat{a}^\dagger_1\widehat{a}_2\right)$$.

These are Hermitian operators: what makes their measurement less standard is the fact that we don't care about the specific number of photons seen by each detector (i.e. their individual intensities), but only their difference. In conjunction with this, the beam splitters provide an additional unitary process that must be incorporated into the measurement process. When this is all put together, the ultimate infinitesimal POVM effect corresponds to a delta function of a field quadrature -- this is quite distinct from the familiar rank one projectors encountered in introductory quantum mechanics.

Scheme 2
For second scheme considered by the authors, the incident beams whose relative phase is of interest are initially sent through different separate balanced beam splitters -- each has the vacuum as the second input. The output beams are combined by two more beam splitters (combining beams from different sources, with a quarter-wave plate inserted along one beam), and then finally collected at detectors. Equations similar to the first setup -- both classical and quantum -- permit the deduction of $$\text{sin}(\phi_2-\phi_1)$$ and $$\text{cos}(\phi_2-\phi_1)$$ from photon counts. An important difference is that now the (3,4) detectors yield the cosine, and (5,6) the sine. In the classical theory,


 * $$\text{cos}(\phi_2-\phi_1)= \frac{W_6-W_5}{(W_4-W_3)^2+(W_6-W_5)^2},\quad \text{sin}(\phi_2-\phi_1)= \frac{W_4-W_3}{(W_4-W_3)^2+(W_6-W_5)^2},$$

while in the quantum theory,


 * $$\widehat{\text{sin}}(\phi_2-\phi_1)=\frac{\widehat{n}_6-\widehat{n}_5}{\sqrt{(\widehat{n}_4-\widehat{n}_3)^2+(\widehat{n}_6-\widehat{n}_5)^2}},\quad \widehat{\text{cos}}(\phi_2-\phi_1)=\frac{\widehat{n}_4-\widehat{n}_3}{\sqrt{(\widehat{n}_4-\widehat{n}_3)^2+(\widehat{n}_6-\widehat{n}_5)^2}}$$,

A further feature is that the number operators in the quantum theory are more complicated, courtesy of the additional beam splitters. The modes of the vacuum input must also be taken into account to preserve unitarity:


 * $$\widehat{n}_4-\widehat{n}_3=\tfrac{1}{2}\left(\widehat{a}^\dagger_2+i\widehat{a}^\dagger_{\text{vac} 2}\right)\left(\widehat{a}_1+i\widehat{a}_{\text{vac} 1}\right)+h.c.$$,
 * $$\widehat{n}_6-\widehat{n}_5=\tfrac{1}{2}\left(\widehat{a}^\dagger_2-i\widehat{a}^\dagger_{\text{vac} 2}\right)\left(i\widehat{a}_1+\widehat{a}_{\text{vac} 1}\right)+h.c.$$.

Another important difference is that $$ [\widehat{n}_4-\widehat{n}_3,\widehat{n}_6-\widehat{n}_5]=0$$. In this sense it may be said that this scheme allows the sine and cosine to be measured simultaneously, whereas in the first they are obtained separately.

Instrumentation
Only the second scheme was carried out experimentally in the original paper. Being a Physical Review Letters paper, it is relatively light on the precise details of the setup. As inputs, the authors used two coherent light beams derived from a Helium–neon_laser. A relative phase was induced between the beams by a piezeo-mounted mirror. The authors report their beam splitters as identical 50:50 within 1%. To approximately equalize quantum efficiency, they placed filters in front of the detectors, which were cooled to dark counts below 50/s. Finally, they effectively varied the input photon count from $$10^{-2}$$ to 30 by tuning the counting interval over which measurements were taken from 0.5μs to 1ms.

Follow-up work by the authors provided a great deal more detail, and also realized the first scheme. They specified that their laser had a frequency and intensity stability of 1 MHz/h and 0.1%/h respectively. They showed that two pinholes and an additional neutral density filter were placed in front of the original laser source, along with a pair of polarizers. Photoelectric pulses from detectors were fed to amplifiers, followed by an NIM, which allowed the signals to be sent to a counter. They also stated that their beamsplitter mirrors were coated with a multilayer dielectric on the input side and antireflective coating on the obverse.

This follow-up paper reported dark counts of about 200/s, so not all setup statistics are comparable to the original paper's. They reported "typical data-acquisition time for one data point is about 8s" and counting rates on the order of $$10^4$$ counts/s for each photodetector. They also claimed that the input photon number could be tuned from 0.01 to $$10^5$$ by adjusting the counting interval from 100ns to "more than 1s."

It was also stated that the individual schemes could be executed in two different ways. In the first case, the filter placed in front of the source laser is fixed, while the counting interval is changed, whereas the roles are swapped in the second case. For the first, which was employed in the original paper, the authors state that the renormalization of the distribution caused by throwing out the zero counts had a significant effect in the quantum limit. They reported that this effect was not so pronounced in the second case.

Published Responses
Although experiments on quantum phase had been performed prior to Noh, Foegères, and Mandel, their work stands out for three reasons: the abundance and accuracy of their data, the pragmatic philosophy accompanying their work, and its series of published Comments, Replies, and follow-ups.

Stephen M. Barnett and D. T. Pegg
Barnett and Pegg observe that NFM's experiment is "in excellent agreement with the predictions of their theoretical analysis," and furthermore that "the agreement is so good that there is little point in attempting to explain their experimental results in terms of any other theory." However, they contest the interpretation of the experiment as a proper measurement of quantum phase.

To argue their point, Barnett and Pegg construct alternative trigonometric operators whose measured values can be expressed in terms of the detector photon counts $$\widehat{n}_i$$. These operators respect the same classical limit as those employed by NFM. They go on to observe that in the limit where only a single photon enters the Scheme 2 interferometer, these various trigonometric operators lead to inconsistent values of the measured phase difference.

They argue that if NFM's experiment "had a consistent interpretation as quantum phase-different measurement," then it should be applicable to any input state. They also observe that the phase difference of number states should be completely random. They therefore conclude that NFM's interpretation "as a phase-difference measurement in the quantum regime is inappropriate."

NFM's response to this criticism is to reaffirm their operational philosophy: that different measurement schemes define different phase operators, which may accordingly yield different results in the quantum regime. They observe that Barnett and Pegg's constructions all define new schemes, so the apparent contradiction is not surprising, and even expected. They also state, similarly to the response to Hradil's first comment, that the matter of the random phase may be ameliorated by modifying the phase shifter.

Zdeněk Hradil and Jiří Bajer
Hradil made two Comments on NFM's experimental work, the second in conjunction with Bajer. In the first, Hradil claims that although "the agreement between experiment and developed theory is convincing," that there are flaws with identifying the measurement as the quantum phase difference.

He starts by observing that for a periodic quantity, a modified definition of dispersion $$D^2=1-|\langle e^{i\phi}\rangle|^2$$ is more appropriate than the one used by NFM. He goes on to observe that NFM's procedure of discarding "ambiguous phase data" (when detectors 3 and 4 and detectors 5 and 6 have matching counts) and renormalizing "effectively changes the measured quantum state." He suggests defining $$\cos(\phi_1-\phi_2)=\sin(\phi_1-\phi_2)=0 $$ in this special case to fix this.

Because of their renormalization procedure, Hradil insists that the quantum measurements of NFM "cannot be treated as phase measurement." By analyzing the dispersion, he then claims that the discrepancy between the Pegg-Barnett and Susskind-Glogower in one of NFM's figures disappears, and the combined curve will be below NFM's experimental curve "since the ideal phase measurement is the minimum dispersion estimator." Finally, Hradil raises the objection that a phase observable should take on a continuum of values, while NFM's procedure, counting only individual photons, is necessarily discrete.

NFM respond that for their Scheme 2, the dispersion is equal to the variance they consider, and then contest that calling $$D^2$$ dispersion is potentially confusing. They explain the positions of the Susskind-Glogower and Pegg-Barnett curves in their figures as being a consequence of them rescaling the incident beam intensity due to the presence of the beam splitters.

With regards to Hradil's suggestion concerning ambiguous phase data, NFM say that his prescription "makes little sense," since there's "no phase angle whose cosine and sine are both zero." Finally, NFM say that the matter of measuring a continuum of phase values may be accomplished by inserting a phase shifter in front of one of the input ports.

In his second Comment on a follow-up paper by NFM, Hradil, along with Bajer, reiterate several of the same contentions as the first Comment, and add to them. They first assert that since the sine and cosines in NFM's Scheme 1 are statistically independent (being carried out in separate experiments), they should be represented by commutating operators. They suggest that the input field should be represented as a four mode state, consisting of two sets of identical coherent states. With this change, and an accompanying alteration of Scheme 2, they assert that the two methods become equivalent. They finally insist that NFM's treatment of normalization is incorrect, leading to a value twice higher than the classical limit. They observe that this issue extends to measurements of higher moments of the cosine and sine, and should be fixed in those cases as well.

NFM respond to Hradil and Bajer's claim that the cosine and sine operators should commute by arguing as follows. They observe that it is also possible to measure the position $$\widehat{q}$$ and momentum $$\widehat{p}$$ independently with identically prepared initial states, yet still it is generally said that the two cannot be measured together and do not commutate. The cosine and sine operators for Scheme 1, they argue, should be interpreted the same way.

With this response, NFM then say that regarding Scheme 1 as "fully equivalent" to Scheme 2 is incorrect, and then provide an example of a valid measurement outcome in the second scheme that would produce incompatible sine and cosine values according to the first. They finally reiterate their contention that Hradil's prescription assigning $$\cos(\phi_1-\phi_2)=\sin(\phi_1-\phi_2)=0 $$ in the case of ambiguous phase measurement since this "breaks all connection with what one generally understands by phase, and leads to meaningless values at times."