35.1 Introduction

The concept of quantum interference is at the very heart of quantum physics and appears in many areas. For example, it demonstrates the wave nature of massive particles more than anything else, and it is the basic ingredient in emerging quantum technologies. Young's double slit experiment with electrons and electron diffraction on a crystal surface by Davisson and Germer (1) have been the first experimental proofs. Recently, similar diffraction was observed even for much larger particles (2). The impact of quantum physics on light experiments is associated with the quantization of the field where the energy of a light mode is quantized to be multiples of a basic unit, the photon energy. For light, quantum effects become apparent, for example, when performing experiments with single photons. The Hong–Ou–Mandel interference (3) is a striking example where two photons simultaneously impinging on a beam splitter, one in each of the two input ports, will never result in one photon at each of the two output ports. In general, one can say that it is the field statistics, that is, the higher moments of observables, that are modified or even dominated by quantum effects. The field statistics are crucial for understanding the basic sensitivity limitation of interferometers (4) even when operating at high‐intensity levels.

The number of applications of interferometric light interference, and in particular quantum interference, is overwhelming. It spans from spectroscopic measurements of fragile biological samples to the fascinating detection of gravitational waves using large‐scale detectors. Most of these applications are concerned with the estimation of a linear phase shift between the two arms of a Mach–Zehnder interferometer, and to quantify the performance of the interferometer, we need to address the precision by which this phase difference can be estimated. Using a classical theory, there is no fundamental limit to the precision; it can be arbitrarily high. Therefore, to find the actual limitations in phase estimation we must resort to quantum mechanics, which imposes some fundamental limits to its precision due to the intrinsic quantum noise of the probing laser. These quantum limits, however, depend on the statistics of the quantum noise as well as on the actual detection system. For example, using a standard Mach–Zehnder interferometer with standard laser beams (i.e., coherent states of light) and simple intensity measurements at the output, the precision scales as , where is the average number of photons. This is referred to as the shot noise limit. In 1981, Caves showed that by using squeezed state of light at the input of the empty port of the interferometer, the precision in phase estimation can go beyond the shot noise limit (5). The ultimate limit in the two‐arm interferometer is and is known as the Heisenberg limit. This has been shown to be reached by a number of different approaches, often based on complicated input states or complicated detection systems (6–11).

Many good reviews on quantum interferometry have been published, see, for example (12–14). In contrast to many of these reviews, in the present chapter, we will mainly focus on one simple setup with different combinations of Gaussian states at the input and using simple calculus to determine the interferometric precision. Specifically, we consider the Mach–Zehnder interferometer and averaged photon number difference measurements at the output while varying the input Gaussian states. In all previous reviews, these different cases with different Gaussian inputs to a simple interferometer have not been treated in a single account. In this lecture we will discuss all these different cases using basic quantum calculus combined with some pictorial explanations. We also address the fundamental limits for some selected cases using the quantum Fisher information (QFI). Finally, we end the chapter with a short discussion of photon loss and quantum interferometry using non‐Gaussian states.

35.2 The Interferometer

Probably the most famous two‐beam interferometer is the Mach–Zehnder interferometer, depicted in Figure 35.1. A beam is divided into two different spatial paths using a beam splitter. The beams run through different arms of the interferometer and finally recombine at a second beam splitter. The interferometer is thus a four‐port device since fields can enter through two different ports of the first beam splitter and leave through two different ports of the second beam splitter. Relative phase changes between the two optical paths in the interferometer can be extracted by measuring the intensity in one or both of the output ports. To realize that this is indeed the case, suppose that a coherent light beam enters the interferometer. If the two paths have exactly equal lengths, the interference at the second beam splitter creates a dark and a bright output, that is all photons leave one output only. A relative phase change, however, results in a division of the photons among the two outputs, the exact fraction being related to the relative phase shift in the interferometer. Therefore, by measuring the intensity in the two output ports, knowledge about the relative phase can be gained. However, this phase estimation process will inevitably be influenced by some noise, which in turn give rise to a statistical error. The precision by which the interferometer phase can be estimated depends on the states injected into the first beam splitter ( and in Figure 35.1) and the measurement strategy used to detect the phase changes.

The two‐beam splitters are the working horses of the interferometer. The beam splitter has two input and two output ports, each port being associated with a mode of the quantized electromagnetic field. If we denote these field operators as , , , and , the input–output relations for a beam splitter are

35.1

35.2

where the transmission and reflection coefficients and are real values, and the complex “ ”s are included to satisfy the commutation relation for the various fields involved (15,16). A particular important beam splitter for interferometry is the one that splits a beam in equal portions, namely the 50/50 beam splitter with . Let us now address the sensitivity of interferometers.

35.2.1 Sensitivity

Heisenbergs uncertainty relation for the phase and photon number is

35.3

where and are the standard deviations of the noise for the phase and the photon number, respectively. For shot noise limited light, where , the optimum phase resolution is

35.4

From this expression it is clear that with an unlimited amount of energy, we can obtain phase measurements with an arbitrary accuracy, since by increasing the power the phase resolution becomes smaller. In practice, however, the amount of energy is finite, and a certain resolution limit will be attained. Furthermore, at very high powers, radiation pressure on the interferometer mirrors and heating‐induced effects add additional noise, which eventually will limit the overall performance of the interferometer. Therefore, the following analysis of the resolution of interferometers will be made under the power constraint assumption, that is, only a limited amount of photons is available. But can we still improve the sensitivity with this power constraint? The answer is yes, the above limit can indeed be surpassed. Quantum mechanics does not put any restriction on further improvements, and it has been found that the ultimate precision in phase measurements is the so‐called Heisenberg limit (17), given by

35.5

This is a great improvement, since the number of photons needed to achieve the same sensitivity as the shot noise limited interferometers is greatly reduced. As we will show in the following sections, this limit can in principle be reached using manifestly nonclassical states of the light field.

To compute the sensitivity of an interferometer in a given setting, a careful quantum mechanical analysis of the interferometer must be carried out. The sensitivity of the Mach–Zehnder interferometer depends basically on two things: the prepared input states and the measurement strategy. In this chapter we consider the measurement strategy outlined earlier (where the intensity difference of the outputs is measured) while considering various input states. We begin the analysis by deriving a simple input–output relation for the Mach–Zehnder interferometer in the Heisenberg picture. Two arbitrary modes, and , enter via the two input ports of the first beam splitter; the modes interfere at the 50 : 50 beam splitter, and the corresponding output modes are given by

35.6

35.7

We allow for an arbitrary phase shift between the two input modes. After the introduction of a relative phase shift of , both modes interfere at a second beam splitter. To simplify the resulting expression, we consider a phase shift in both arms equal to but with opposite signs yielding the overall phase shift :

35.8

35.9

Rewriting these expressions in terms of the input states by inserting 35.6 and 35.7 in 35.8 and 35.9, we find, up to some global phase factor, the general input–output relation of a Mach–Zehnder interferometer

35.10

35.11

It is interesting to note that these two input–output relations are similar to the beam splitter equations in which the beam splitting ratio is controlled by the relative phase change between the two optical paths in the interferometer. These simplified equations make the analysis simple, and the effect on changing the input states can easily be computed.

Information about the phase change is now extracted by detecting the intensities of the output beams, and , and subsequently generating the difference of the photocurrents:

35.12

The noise variance that is associated with measurements of the difference signal is calculated as follows:

35.13

where indicates the quantum–mechanical expectation value taken over the two input states . Operators with index “1” act only on mode , and those with index “2” on .

The accuracy of the phase measurements can then be estimated using the calculus of error propagation:

35.14

Inserting 35.12 in 35.14 we find the statistical error in estimating a phase change when using the abovementioned measurement strategy and employing two arbitrary input states, and .

35.3 Interferometer with Coherent States of Light

The first scenario that we will consider is when a coherent state enters through one input port and a vacuum state enters through the other input port. In this simple case the expectation values have to be taken over these two states. For the standard deviation, we find

35.15

35.16

and the partial derivative of the mean photon number is

35.17

The phase resolution is thus

35.18

as expected for a coherent input state.

35.3.1 Geometrical Visualization

We now introduce a pictorial description of the propagation of noise in an interferometer. Such a visualization tool is helpful in understanding the various noise transforming mechanisms inside an interferometer (18), and it has also been shown to facilitate the understanding of the generation of intense quantum entangled light beams (19). For a general introduction, see, for example, Leuchs (20). We closely follow the description presented in (21).

In quantum optics, the field operator of a mode can be written as a superposition of a classical mean field and an operator describing the field uncertainty:

35.19

with . The state is best visualized in a phase space diagram (see Figure 35.2). The classical amplitude of the field is represented by the “stick” , the optical phase is given by its orientation in phase space. Hence, in this diagram the imaginary part of the field is plotted versus the real part. The fluctuations lead to a region of uncertainty, which can be considered as the contour of the Wigner function (22). For a field in a coherent state, the uncertainties in amplitude and phase direction are the same, and the contour is circular as shown in Figure 35.2. The noise that contributes to signals in direct detection corresponds to the projection of the noise arrows onto the direction along the classical excitation. In the figure this corresponds to the arrow , which represents the amplitude fluctuations, while the perpendicular arrow represents the phase noise.

These two arrows, which describe stochastic variables, span the circular region of uncertainty of the field. For the coherent state, for example, there will be no correlation between the two stochastic variables. The action of a beam splitter will be to transfer each arrow from an input port to both output ports with reduced amplitudes. In the model we have to properly take into account the beam splitter relations. If the same stochastically varying input arrow contributes to two output ports, one may expect correlations between these two partial output fields.

Let us now use this pictorial approach to understand the function of the interferometer. A coherent state enters through one input port, and the other input mode is not excited and, therefore, in a vacuum state (Figure 35.3). Coherent and vacuum states have a circular region of uncertainty in phase space and as a result all four stochastic arrows describing the two coherent states have the same variance.

The beam splitter relations are obeyed by associating a phase shift to each reflection, that is, the factor “ ,” and phase shift to each transmission. The amplitude reduction is not shown, for simplicity. The four input arrows , , , and determine the field uncertainties in the two output ports I and II right after the first beam splitter. The amplitude uncertainty in output I is determined by the projections of all arrows onto the amplitude direction: . Each arrow would then still have to be multiplied with its individual stochastic coefficient. Likewise, the amplitude uncertainty at output II is determined by .

Before going to the second beam splitter, we now introduce a phase shift (i.e., clockwise). This is done, for example, by introducing a path length difference between the two arms. It ensures that both output ports are at half fringe height, i.e., they are equally intense. At the outputs 3 and 4, again following the rules introduced earlier, we now have altogether eight arrows. Two arrows marked with the same letter derive from one and the same stochastic input variable, so they can be added vectorially. As can be seen in Figure 35.3, arrows contribute to correlated amplitude uncertainties in the two outputs 3 and 4, arrows to correlated phase, arrows to anticorrelated amplitude, and arrows to anticorrelated phase uncertainties. Another way to say this is that the amplitudes in output ports 3 and 4 are given by and , respectively. Although the uncertainties in both output ports are governed by the same four arrows, they are not correlated. The reason for this lack of correlation can be traced back to the sum of two statistically independent stochastic variables and their difference being again statistically independent. Due to this uncorrelation, the interferometer performs measurements at the shot noise limit, and the resolution is limited by as expected. In the following section, we will show how this limit can be crossed.

35.4 Interferometer with Squeezed States of Light

Carefully designed interferometers can beat the shot noise limit, for example, by injecting squeezed states into the interferometer. We will consider three different scenarios: (i) Input ports 1 and 2 are illuminated with a coherent state and vacuum squeezed state, respectively ( ). (ii) Both ports are illuminated with bright squeezed states ( ). (iii) A bright squeezed state and a squeezed vacuum state are injected into the interferometer ( ). In the following section, we will see that all these realizations beat the shot noise limit. However, only one of them reaches the Heisenberg limit.

35.4.1 Interferometer Operating with a Coherent State and a Squeezed Vacuum State

In our analysis, we assume that mode 1 is in a coherent state , while mode 2 is a vacuum state that is squeezed , where is the complex squeeze parameter . The strength of the squeezing is given by the parameter , and the orientation of the squeezing ellipse is given by . Basic expectation values required for the calculation are (23–25)

35.20

Again, the operators labeled with index “1” act only on mode one, while the index “2” acts on the second mode and denotes the photon number operator. The amplitude of the coherent state is assumed to be real (see Figure 35.2), as only the relative phase between the input modes matters. Using these relations, the noise of the photon number difference of the output modes can be calculated

The orientation of the squeezing ellipse is , corresponding to an amplitude squeezed vacuum mode. The resolution of the mean photon number is

35.21

By choosing the phase , we maximize the resolution while minimizing the noise. The error is then found to be

35.22

where denotes the number of classical photons in the interferometer. Let us discuss this result in detail. With no squeezing ( ), the expression 35.22 reduces to , in agreement with the result of the previous section 35.18. For quite moderate squeezing, where the number of “squeezed” photons, , is negligible compared to the photons of the bright input mode, , Eq. 35.22 is reduced to

35.23

This expression also follows from the linearized approach (26). However, for very strong squeezing, the number of photons due to squeezing becomes comparable to while we still may assume . With this approximation 35.22 can be rewritten as

35.24

Using the approximate solution, one can easily find the squeezing level at which the phase resolution is optimized. For a squeezing value of , Eq. 35.24 has a minimum, therefore the statistical error is

35.25

This result should be compared with , the case where no squeezing was present in the scheme. Squeezing the vacuum into the setup may significantly enhance the phase resolution properties; however, quite high squeezing is required to reach the optimum. These results are summarized in Figure 35.4, where the exact solution is plotted together with the approximate calculation and the results from the linearization.

The sensitivity improvement using a squeezed vacuum was first proposed by Caves (5), and later the effect of imperfections of the interferometer such as losses and nonunity fringe visibility were discussed by Gea‐Banacloche and Leuchs (27). The idea has also been experimentally demonstrated several times. Xiao et al. (28) and Grangier et al. (29) demonstrated a sensitivity improvement of a standard Mach–Zehnder interferometer, while McKenzie et al. (30) and Vahlbruch et al. (31) have demonstrated an improvement in a power‐recycled and a signal‐ and power‐recycled interferometer, respectively. These last experiments were predicted in Ref. (32). All these demonstrations were performed on single optical tables, and it was therefore an important milestone when the quantum‐enhanced techniques were applied to the large‐scale quantum interferometers for the detection of gravitational waves. In 2011 (33), the LIGO team used squeezed vacuum for improving the precision of GEO600 (a large Michelson interferometer with 600‐m arm length), while in 2013 (34), it was applied to the larger LIGO interferometer (with arm lengths of 4 km). When gravitational waves were detected in 2016 (35), squeezed light was not used. However, the LIGO team is planning to use it in a future version of the interferometer. We also note that in future versions of the gravitational wave interferometers, radiation pressure noise might play a role in the precision of phase estimation: The radiation pressure noise (stemming from the interaction between light and the interferometer mirrors) will increase the noise in the measurement and thereby degrade the sensitivity. Using standard approaches, this limits the sensitivity to what is known as the standard quantum limit. This limit can, however, be surpassed using different approaches ⁰¹.

The improvement of the interferometer sensitivity by the use of squeezed vacuum states can also be easily understood from the geometrical representation introduced in the previous section. Let us return to Figure 35.3, but now we consider the case where the input vector is suppressed due to the squeezing of the input field in input port 2. Recalling that the amplitude uncertainties of output ports 3 and 4 are governed by and , respectively, we clearly see that the two outputs are proportional to (and thus correlated) while reducing . When measuring the difference of the intensities at the two output ports, one finds a quantum noise suppressed signal and hence an improvement in the sensitivity for measuring arm length differences.

In the above discussion we assumed a phase shift in one of the interferometer arms in order to maximize the signal for the given measurement strategy. If instead we set the phase shift to be zero, one output will be dark and the other one will be bright. In this case the interferometer can attain the same sensitivity as before; however; another measurement strategy must be employed: homodyne detection in the dark port or a modulation technique (43). Again the pictorial argument can be made for the noise, and again one finds a sensitivity improvement by quantum noise reduction. To achieve this, has to be suppressed at input port 2 at the expense of increasing , which in turn does not affect the close‐to‐zero amplitude at port 4. The bright output beam at port 3 recovers the noise properties of input 1 and it can be recycled, that is, reinjected into the interferometer to enhance the total power inside the interferometer.

We have seen that the sensitivity of an interferometer can be increased by suppressing the noise in the dark, vacuum, input port. In the next two sections we will address the question whether the sensitivity can be increased further by squeezing the other bright input state as well.

35.4.2 Interferometer Operating with Two Bright Squeezed States

The expectation values of the photon number and the photon number squared (as needed to calculate the phase resolution) for two bright input squeezed states can be determined by making use of the squeezing operators, (to squeeze the vacuum) and the displacement operator, (to displace the squeezed vacuum), both acting on the vacuum state . Using this relation and the fact that , the basic expectation values required for the analysis of this type of interferometer can be calculated. For example, we have

and

35.26

We allowed for an arbitrary phase of mode and used , where is the phase‐shifting operator. With these relations at hand and assuming (i) that the two input states are equally amplitude squeezed (described by the parameter ), (ii) the excitation of the two inputs are identical, denoted , (iii) the relative phase shift between them is , and (iv) there is a zero relative phase shift between the two arms in the interferometer ( ), we can calculate the following photon number uncertainty:

The partial derivative of the average photon number with respect to the phase is , and the statistical error in phase estimation is

35.27

From this expression we see that there are two competitive terms in the denominator. The first term reduced the error in phase estimation, while the second term increases this error. The latter term is a function of the number of “squeezed” photons and thus relatively small for low degrees of squeezing. However, for high squeezing degrees this term might dominate, hereby deteriorating the performance of the interferometer. Again, we find the minimum for the phase resolution via an approximate solution with but :

35.28

The optimum squeezing where the phase resolution is optimized is then given by . Inserting this into equation 35.28, we find that the optimal sensitivity for this scenario is given by approximately . It is therefore better to use a squeezed vacuum state and a coherent state at the input, since in this case the statistical error was , which is indeed smaller than the abovementioned result.

35.4.3 Interferometer Operating with a Bright Squeezed State and a Squeezed Vacuum State

Now we consider the last scenario where one input state is bright squeezed, whereas the other input state is vacuum squeezed. To simplify the derivation on the phase resolution, we assume the two input states to be equally squeezed in the same quadrature, their relative phase shift to be and the biased phase shift in the interferometer to be . With these choices and using equations 35.27, we find the uncertainty

35.29

and . The phase resolution is then

35.30

which is plotted in Figure 35.5. Is this resolution approaching the Heisenberg limit? We answer this question by comparing expression 35.30 with the Heisenberg limit given by the total number of photons in the interferometer for a certain squeezing parameter

35.31

We maximize with respect to the squeezing parameter under the assumption that and using . Hence, for , the best resolution is achieved, and we find from 35.30 the limit

35.32

We compare this expression with the Heisenberg limit for , which is found to be

35.33

that is, we do not reach the Heisenberg limit exactly, but we find . This situation is displayed in Figure 35.5: The phase resolution for is plotted together with the Heisenberg limit and . Best resolution is achieved for a certain squeezing value .

We have now discussed various schemes with which the shot noise limit for interferometers can be surpassed. However, in the above descriptions two realizations with squeezed light were missing, namely the cases where a bright squeezed input beam is mixed with vacuum and the case where a coherent beam is mixed with a bright squeezed beam. The former realization reaches a sensitivity identical to the shot noise limit, that is, with the total photon number . In the latter case, we get the general solution: , and an approximative solution for the minimum is . This solution is rather complex; however, by comparing it to the previous strategies, we conclude that this strategy is in general worse.

35.5 Fundamental Limits

Up to this point, we have been only considering one specific measurement approach at the output of the interferometer. We have deduced the phase sensitivity for different combinations of input states, but we have fixed the detection strategy to a photon number difference detector. However, an intriguing question is whether there exist other measurement strategies that yield better phase sensitivities for fixed input states. This question can be addressed using a quantity called the quantum Fisher information (QFI).

Finding the optimal measurement strategy for a given input state is often a nontrivial task. However, it is possible to find a lower bound for the sensitivity, known as the quantum Cramer–Rao (QCR) bound (44),

35.34

where is the QFI. The QFI is a measure that quantifies our ability to discriminate different probability distributions: Different relative phases in the interferometer will produce different probability distributions, and the question is how well these distributions can be discriminated. The better the discrimination, the larger is the QFI, and for a specific input state the QFI attains a certain value that then yields the fundamental limit in phase sensing for this particular input state. The challenge is, however, to find the particular measurement strategy that saturates the QCR bound. In the following discussion, we will consider two specific cases: coherent state input and squeezed state input.

Let us first define the QFI in mathematical terms. It is given by

35.35

where is the density matrix describing the input state and the so‐called symmetric logarithmic derivative. For a thorough mathematical discussion of the quantity, see (12). Importantly, for pure states, , the QFI reduces to the simple relation

35.36

where is the generator of the phase change. Since the generator of a linear phase change in an interferometer is well known to be the photon number operator (as the phase‐shifting operator is ), we simply get that the QFI is given by

35.37

where is the photon number variance of the part of the beam that traverses the phase shift.

We now turn to the first example, namely a coherent state at the input of the interferometer. In this case, we simply find that a coherent state with average photon number and variance has the QFI , and thus a fundamental phase sensitivity limit of

35.38

where is the mean number of photons at the entrance to the Mach–Zehnder interferometer. This bound was in fact saturated with the particular measurement strategy discussed in Section 35.3 (leading to Eq. 35.18), and so we now know that this detection scheme is optimal for phase sensing using coherent states.

Let us now consider the more interesting case with squeezed vacuum injected into one input port and a coherent state at the other port. For this pair of input states, we deduce a photon number variance of , and by fixing the average number of photons, the maximal QFI is for , and thus the optimal phase sensitivity bound is

35.39

We see that this bound is identical to the Heisenberg limit and that it is lower than what we found in Section 35.4.1 where a specific photon number difference measurement was treated. This detection strategy is therefore not optimal as it does not saturate the QCR bound. It has, however, been found that using a more sophisticated measurement system that relies on photon number resolving detectors, it is possible to reach the QCR bound (45), although on the experimental side neither the QCR bound nor the bound has been reached. In all implementations with squeezed vacuum, the coherent states have been very bright, and therefore the requirements of (for the QCR bound) or (for Eq. 35.25) have not been fulfilled. Since , the phase sensitivity is given by as found in Eq. 35.23.

	First input port	Coherent state	Bright squeezed
Second input port
Vacuum state
Squeezed vacuum state
Bright squeezed state		02

.

35.6 Summary and Discussion

In the previous sections we have shown that the application of squeezed states in interferometry may enhance the phase sensitivity. The results of our analysis are displayed in Figure 35.6 in which we compare the phase sensitivity for the different scenarios discussed earlier. We clearly see that all the schemes employing squeezed states at the input surpass the shot noise limit for the interferometer. It is also evident that for high squeezing values the strategy where a bright squeezed state is mixed with a vacuum squeezed state is superior and eventually approaches the Heisenberg limit. The optimal phase resolutions for the various schemes are summarized in Table 35.1.

In this chapter we have considered only the Gaussian states of light as inputs to the interferometer. However, there exists also a large class of non‐Gaussian states (46) by which sub‐shot‐noise interferometry can be attained. For example, using the maximally entangled state and photon number difference measurements, Heisenberg scaling can be attained (7). Moreover, Holland and Burnett have shown that by using a slightly different detection strategy, the Heisenberg scaling can also be obtained using Fock states, , at the input to the interferometer but considering a more complicated measurement strategy (10). However, one of the most famous states for quantum interferometry is the so‐called NOON state (47) since the resulting sensitivity becomes identical to the Heisenberg limit. NOON state interferometry has been experimentally realized in numerous experiments ((48–50)) but most often in the coincidence basis (i.e., based on postselection), except in one case where a two‐photon NOON state was heralded (51). These entangled non‐Gaussian states are, however, notoriously difficult to generate, and thus, Gaussian states have so far produced the best sensitivities in the laboratories and are likely to do so also in real‐life experiments (52).

One point that has not been addressed so far, but is very important in real‐life experiments, is photon loss. For instance, in the gravitational wave observatory GEO600, 38% of all photons are lost due to absorption and scattering inside the interferometer (33). While nonclassical quantum states in general severely suffer from photon loss, NOON states in particular are extremely vulnerable and show an exponential decrease in their QFI with increasing photon loss (53). It has, however, been shown that for the GEO600 configuration with and the aforementioned photon loss, the combination of coherent and squeezed vacuum input is close to optimal (52). In general, for the Gaussian schemes mentioned earlier, sub‐shot‐noise sensitivity is obtained for any nonvanishing squeezing even in the presence of photon loss. Except for the case with a coherent and a vacuum state in the input, the analysis of the phase sensitivity bounds is more sophisticated due to the mixed nature of the thermal squeezed vacuum (or bright) state. The detection strategy chosen above deviates more and more from the optimal detection strategy with increasing photon loss since it does not optimally extract the phase information from the thermal photons in the state.

As a final note, we point out that in this chapter we have been concerned with phase sensing in contrast to ab initio phase estimation. The task of phase sensing is to measure a tiny phase shift relative to a known starting phase. This is highly relevant in a situation where the phase shift is very small like in gravitational wave interferometers. Ab initio phase estimation (or global phase estimation) is of relevance when large phase shifts, in principle from zero to , is to be measured as might be the case in quantum communication or in the realization of Shor quantum algorithm where phase estimation is a key element. In this case, a different strategy must be applied such as real‐time feedback or Bayesian estimation (54,55). Real‐time feedback for global phase estimation has been experimentally realized with Gaussian states, first for coherent (56) and later for squeezed states (57,58). Global phase estimation based on real‐time feedback has also been realized with non‐Gaussian states; both for NOON states (59) as well as single photons states combined with multipass interferometry (60). It is also possible to avoid real‐time feedback using highly complex input states such as the so‐called non‐Gaussian sine state (12).

Problems

1. 35.1 Derive Eq. 35.23 using the linearization approach. Hint: The annihilation operator can be decomposed into two terms: , where is the classical steady‐state field and is an operator associated with the quantum noise. Use Eq. 35.12 and 35.14.
2. 35.2 Derive Eq. 35.14 assuming that the minimum resolvable signal‐to‐noise ratio is unity.
3. 35.3
   1. Use the expansion of coherent states in terms of Fock states
      35.40
      to derive the mean photon number (see Eq. 35.20) and its fluctuations . Hint: , , .
   2. Use the relations and , where is the squeeze operator to derive and . Show that for a squeezed state yields

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