In this paper, the authors review the use of parity as a detection observable in quantum metrology and introduce some original findings with regard to measurement resolution in Ramsey spectroscopy and quantum nondemolition measures of atomic parity. Parity was first introduced in the context of Ramsey spectroscopy as an alternative to atomic state detection. It was later adapted for use in quantum optical interferometry where it has been shown to be the optimal detection observable saturating the quantum Cramér–Rao bound for path symmetric states. The authors include a brief review on the basics of phase estimation and the connection between parity-based detection and the quantum Fisher information as it applies to quantum optical interferometry. The authors also discuss the efforts made in experimental methods of measuring photon-number parity and close the paper with a discussion on the use of parity, leading to enhanced measurement resolution in multi-atom spectroscopy. The authors show how this may be of use in the construction of high-precision multi-atom atomic clocks.

Observables in quantum mechanics, which are generally based on their classical counterparts, are represented as Hermitian operators in Hilbert space due to the fact that such operators possess real eigenvalues. According to orthodox quantum mechanics, the measurement of an observable returns a corresponding eigenvalue of the operator with probabilities of obtaining specific eigenvalues determined by the prepared state vector. Usual quantum observables most commonly discussed are energy, position, momentum, angular momentum, etc., all of which have classical analogs. The spin degree of freedom of an electron is often taken as a quantum observable with no classical analog, which, of course, is true in the sense that it is not possible to make sense classically out of the notion of a point particle having spin angular momentum. Yet, spin angular momentum itself surely exists in the classical world of macroscopic objects.

The same cannot be said, however, for the concept of photon-number parity. Let |n, where n+, be a Fock state for a single-mode quantized electromagnetic field. The photon-number parity of the state is defined as the evenness or oddness of the number as quantified by (1)n. We define the usual boson operators â and â, the annihilation and creation operators of the field, respectively, which satisfy the usual commutation relation [â,â]=1 and the number operator n̂=ââ such that n̂|n=n|n. In terms of these operators, one can introduce the photon number parity operator Π̂=(1)ââ=eiπââ such that Π̂|n=(1)n|n. The eigenvalues of this operator are dichotomic and thus highly degenerate. While it is clear that Π̂ is Hermitian and thus constitutes an observable, it can also be shown that there exists no classical analog to photon number parity. This can be demonstrated by considering the energies of the quantized field: while these energies are discrete (i.e., En=ω(n+12)), the energies of a classical field are continuous.

The parity operator makes frequent appearances in quantum optics. For example, the use of photonic parity measurements has led to the distillation of single photons with a fidelity of 66(1)% (Ref. 1) using laser pulses reflected from a single atom strongly coupled to an optical resonator. The parity operator has also appeared in various proposals for testing highly excited entangled two-mode field states for violations of Bell's inequality.2 It has also been pointed out that the quasi-probability distribution known as the Wigner function3,4 can be expressed in terms of the expectation value of the displaced parity operator5 given for a single-mode field as

W(α)=2πD̂(α)Π̂D̂(α),
(1)

where D̂(α)=eαâα*â is the usual displacement operator familiar from quantum optics and α is the displacement amplitude generally taken to be a complex number. This relationship is the basis for reconstructing a field state through quantum state tomography.6–9 Methods for a direct measurement of the Wigner function for trapped ions and fields states in cavity quantum electrodynamics (QED) have been discussed in the literature.10 Work has been done on generalized parity operators and their applications to phase-space representations,11 the development of s-parameterized phase-spaces for SU(2),)12,13 and methods for computing the SU(2) Wigner function for large (up to 500 qubits in symmetric states) qubit systems.14 SU(2) Wigner functions have also been used to characterize optimal metrological states.15 

Another field in which parity sees use is quantum metrology, where it serves as a suitable detection observable for reasons we will endeavor to address. Quantum metrology is the science of using quantum mechanical states of light or matter in order to perform highly resolved and sensitive measurements of weak signals like those expected by gravitational wave detectors (see, for example, Barsotti et al.16 and references therein) and for the precise measurements of transition frequencies in atomic or ion spectroscopy.17 This is often done by exploiting an inherently quantum property of the state such as entanglement and/or squeezing. The goal of quantum metrology is to obtain greater sensitivities in the measurement of phase-shifts beyond what is possible with classical resources alone, which at best can yield sensitivities at the shot noise, or standard quantum limit (SQL). For an interferometer operating with classical (laser) light, the sensitivity of a phase-shift measurement, Δφ, scales as 1/n¯, where n¯ is the average photon number of the laser field. This defines the SQL as the greatest sensitivity obtainable using classical light: ΔφSQL=1/n¯. In cases where phase-shifts are due to linear interactions, the optimal sensitivity allowed by quantum mechanics is known as the Heisenberg limit (HL), defined as ΔφHL=1/n¯. Only certain states of light having no classical analog, including entangled states of light, are capable of breaching the SQL level of sensitivity. In many cases, perhaps even most, reaching the HL requires not only a highly nonclassical state of light but also a special observable to be measured. That observable turns out to be photon-number parity measured at one of the output ports of the interferometer. This is because the usual technique of subtracting irradiances of the two output beams of the interferometers fails for many important nonclassical states. Furthermore, as to be discussed in Sec. III, consideration of the quantum Fisher information (QFI) indicates that photon number parity serves as the optimal detection observable for path symmetric states in quantum optical interferometry.

This paper is organized as follows: Sec. II begins with a discussion on the origin of parity-based measurement in the context of atomic spectroscopy. In Sec. III, we present a brief review of the basics of phase estimation including a concise derivation of the quantum Cramér–Rao bound (qCRB) and the related quantum Fisher information and discuss how these relate to the parity-based measurement, particularly in the field of quantum optical interferometry. In Sec. IV, we highlight the use of several relevant states of light in quantum optical interferometry such as N00N states and coherent light. We show in the latter case that the use of parity does not yield sensitivity (i.e., reduced phase uncertainty) beyond the classical limit but does enhance measurement resolution. In Sec. V, we discuss the experimental efforts made in performing photonic parity measurements. Finally, in Sec. VI, we briefly return to the atomic population parity measurements, this time in the context of atomic coherent states (ACSs), and show that such measurements could lead to high-resolution multi-atom atomic clocks, i.e., atomic clocks of greater precision than is currently available. We conclude the paper with some brief remarks.

We first introduce the Dicke atomic pseudo-angular momentum operators for a collection of N two-level atoms. These are given by18 

Ĵx,y,z=12i=1Nσx,y,z(i),
(2)

satisfying the SU(2) commutation relation of Eq. (A13) (see Appendix A 2), where σx,y,z(i) are the Pauli operators for the ith atom as given by σx(i)=|eig|+|gie|,σy(i)=i(|eig||gie|),σz(i)=|eie||gig| and where the ground and excited states for the ith atom are denoted as |gi,|ei, respectively. It is obvious that the operators given in Eq. (2) are additives over all atoms. We now introduce the corresponding collective atomic states, the Dicke states, expressed in terms of the SU(2) angular momentum states |j,m, |j,m where j=N/2 and m{j,j+1,,j}, which can be given as superpositions of the product states of all atoms. For N=2j atoms, with j=1/2,1,..., the Dicke states |j,m are defined in terms of the individual atomic states as |j,j=|e1,e2,...,eN=|eN and |j,j=|g1,g2,...,gN=|gN with intermediate steps consisting of superpositions of all permutations with consecutively more atoms being found in the ground state all the way down to |j,j, where all atoms are in the ground state. The ladder operators given as

Ĵ+=Ĵx+iĴy=i=1N|eig|,Ĵ=ĴxiĴy=i=1N|gie|
(3)

can be used repeatedly to generate expressions for all the nonextremal states in terms of the individual atoms. We relegate further discussion of the mapping between the two sets of states to  Appendix A.

In what follows, we denote the exact transition frequency between the excited and ground states as ω0. The goal of Ramsey spectroscopy is to determine the frequency with as high a sensitivity (or as low an uncertainty) as possible. We first go through the Ramsey procedure with a single atom assumed initially in the ground state such that |in=|g. The atom is subjected to a π/2 pulse described by the operator ei(π/2)Ĵy and implemented with radiation of frequency ω. For one atom, the rotation operator about the y-axis for an arbitrary angle is given by1 

eiβĴy=ei(β/2)σy=I2cosβ/2iσysinβ/2,
(4)

where I2=|ee|+|gg| is the identity operator in this two-dimensional subspace. This results in the transformations

eiβĴy[|e|g]=[cos(β/2)|e+sin(β/2)|gcos(β/2)|gsin(β/2)|e],
(5)

which for β=π/2 we have the balanced superpositions

eiπ2Ĵy|e=12(|g+|e),eiπ2Ĵy|g=12(|g|e).
(6)

Assuming that the atom is initially in the ground state, the state of the atom after the first π/2 pulse becomes |ψtπ/2=ei(π/2)Ĵy|in=12|g|e. This is followed by a period T of free evolution (precession) governed by the operator ei(ω0ω)TĴz, ei(ω0ω)TĴz where once again ω is the frequency of the radiation field implementing π/2 pulses. The state after free evolution is

|ψtπ/2+T=12ei(ϕ/2)|gei(ϕ/2)|e,
(7)

where we have set ϕ=(ω0ω)T. After the second π/2 pulse following free evolution, we use Eq. (6) to find the final state

|ψ(tf)=isin(ϕ/2)|gcos(ϕ/2)|e,
(8)

where tf=2tπ/2+T and where it is assumed T2tπ/2 such that tfT. A diagram of the transformations in Bloch-sphere representation is provided in Fig. 1 and an idealized sketch of the Ramsey technique in Fig. 2. The expectation value of Ĵz for this state is

Ĵz(ϕ)tf(1)=12cosϕ=12cos[(ω0ω)T].
(9)

By tuning the frequency of the driving field so as to maximize Ĵz(ϕ)tf(1), one can estimate the transition frequency ω0.

If we consider N atoms one at a time or collectively through the Ramsey procedure,19 then the initial state with all atoms in their ground state is the Dicke state |j,j, and after the first π/2 pulse, the state generated |ψ=ei(π/2)Ĵy|j,j is an example of an atomic coherent state (ACS).20 For a brief review of the Dicke states and a derivation of the ACS, see Appendix A 1. In such a state, there is no entanglement among the atoms: each atom undergoes the same evolution through the Ramsey process, and because the Ĵz operator is the sum Ĵz=12iNσ(i)z, one easily finds that

Ĵz(ϕ)tf(ACS)=NĴz(ϕ)tf(1)=N2cos[(ω0ω)T].
(10)

The propagation of error in the estimation of the phase ϕ is given by (dropping scripts for notational convenience)

ΔϕACS=ΔĴz(ϕ)|ϕĴz(ϕ)|=1N,
(11)

showing the best sensitivity one can obtain with unentangled atoms in the SQL. In terms of the transition frequency, the error is given by Δω0=1/(TN). Before closing this section, it is worth summarizing the operator sequence required to implement Ramsey spectroscopy, which we do here in terms of an arbitrary initial state |in. In the Schröinger picture, this amounts to writing the final state in terms of the initial as

|ψfϕ=ei(π/2)ĴyeiϕĴzei(π/2)Ĵy|in,
(12)

where once again ϕ=(ω0ω)T. It is worth noting that Ramsey spectroscopy is mathematically equivalent to optical interferometry in that both can be described through the Lie algebra of SU(2) (see  Appendix A). However, the transformations of Eq. (12) is slightly different from how we will describe an interferometer in Sec. IV.

Now, we suppose that we have N atoms prepared in a maximally entangled state (MES) of the form

|ΨM=12(|eN+|gN)=12(|j,j+|j,j).
(13)

To implement the use of this state for Ramsey spectroscopy as described by the sequence of operators in Eq. (12), we should take the actual initial state to be |in=e+i(π/2)Ĵy|ΨM. After a free evolution time T, our state is now

|ΨMT=12(eiNϕ/2|j,j+eiNϕ/2|j,j).
(14)

After the second π/2 pulse, we arrive at the final state

|Ψtf=12eiNϕ/2ei(π/2)Ĵy|j,j+eiNϕ/2ei(π/2)Ĵy|j,j=12j2m=jj2jj+m1/2eiNϕ/2+1j+meiNϕ/2|j,m,
(15)

where we have used the fact that ei(π/2)Ĵy|j,±j are atomic coherent states generated from different extremal, or fiducial, states |j,jand|j,j (see Appendixes A 1 a and  B for more details). The expression of Eq. (15) is not particularly informative when it comes to the evaluation of Ĵz(ϕ)M as it can be shown that

Ĵzϕtf=TΨM|ei(π/2)ĴyĴzei(π/2)Ĵy|ΨMT=TΨM|Ĵx|ΨMT=0.
(16)

Thus, this expectation value furnishes no information on the phase ϕ.

To address this issue, Bollinger et al.17 proposed measuring the quantity (1)Ng, where Ng is the number of atoms in the ground state. As an operator for a given j value, this reads Π̂j=(1)jĴz=eiπ(jĴz). This requires us to calculate

Π̂jtf=TΨM|ei(π/2)ĴyeiπjĴzei(π/2)Ĵy|ΨMT=1jTΨM|eiπĴx|ΨMT=1jTΨM|ei(π/2)ĴzeiπĴyei(π/2)Ĵz|ΨMT,
(17)

where we have used the relations ei(π/2)ĴyĴzei(π/2)Ĵy=Ĵx and ei(π/2)ĴzĴyei(π/2)Ĵz=Ĵx. Setting Û=eiπ2ĴzeiπĴyeiπ2Ĵz, Û=ei(π/2)ĴzeiπĴyei(π/2)Ĵz, we have

Π̂jtf=(1)j12[j,j|Û|j,j+j,j|Û|j,j+eiNϕj,j|Û|j,j+eiNϕj,j|Û|j,j].
(18)

Then, with

j,j|Û|j,j=dj,jj(π)=0,j,j|Û|j,j=dj,jj(π)=0,j,j|Û|j,j=(1)jdj,jj(π)=(1)j,j,j|Û|j,j=(1)jdj,jj(π)=(1)j,
(19)

where dm,mj(β) are the Wigner-d matrix elements discussed in  Appendix B, we finally have

Π̂jtf=(1)Ncos[N(ω0ω)T].
(20)

Note the appearance of the factor N in the argument of the cosine. This is the consequence of maximal entanglement between the N atoms, leading to an increase in the sensitivity of a frequency measurement by a factor of 1/N over the SQL. Again, setting ϕ=(ω0ω)T, the error propagation calculus in this case results in

ΔϕM=ΔΠ̂j|ϕΠ̂jtf|=1N,
(21)

or that Δω0=1/NT. This is considered a Heisenberg-limited uncertainty as it scales with the inverse of N. A scheme for the generation of the maximally entangled state of Eq. (13) was discussed by Bollinger et al.,17 and a different scheme was discussed by Steinbach and Gerry.21 An experimental realization of parity-measurements has been performed by Leibfried et al.22 with three trapped ions prepared in a maximally entangled state, and a cavity-QED scheme for achieving Heisenberg-limited sensitivity with determined atom number parity was put forward by Vitali et al.23 

Next, we will detail how the fundamental limits on phase uncertainty change in the presence of decoherence. This was first discussed by Huelga et al.24 in the context of Ramsey spectroscopy with N trapped ions. Let us start by assuming that the ion trap is loaded with N ions, each with two internal degrees of freedom |0 and |1 and a transition frequency ω0, all initially prepared in the |0 state. Much like the previous discussion, a Ramsey pulse of frequency ω is applied to the state, preparing each ion in an equally weighted superposition state. The internal state of each ion is measured after a period of free evolution T followed by a second Ramsey pulse of frequency ω (see Fig. 2). The probability of finding an ion in the |1 state and the corresponding uncertainty are found to be P=12(1+cosϕT) and ΔP=P(1P)/dN, respectively, where ϕ=ωω0 and dN=Nttot/T. Here, ttot denotes the total duration over which the scheme is repeated and dN is the number of experimental data points. The uncertainty in atomic transition frequency can be obtained readily by error propagation Δω0=ΔP/|ωP|=1/NttotT. This is simply the SQL or classical bound on atomic frequency transition sensitivity. Huelga et al. went on to detail the same calculation for the case of two ions prepared in the maximally entangled (Bell) state |ψMES=12(|00+|11), which can be generated by applying a Ramsey pulse on the first ion followed by a C-NOT gate.25 The ions are disentangled through the same mechanism after the second Ramsey pulse. For this case, the probability of detecting one excitation becomes PN=12(1+cosNϕT), where the oscillation frequency scales as now scales as Nϕ. The frequency uncertainty is now Δω0=1/NttotT; an improvement over the SQL by a factor of 1/N, i.e., the HL.

The main source of decoherence in an experimental setup such as this is due to dephasing, which can be caused by ion collisions, extraneous fields, and laser instabilities.24 Working in the rotating frame (rotating at frequency ω), the master equation describing the time evolution of the reduced density operator for a single ion in the presence of decoherence is given by26 

ρ̇(1)(T)=iϕ(ρ(0)|11|+|11|ρ(0))+γ2(σzρ(0)σzρ(0)),
(22)

where once again σz=|00||11| is the Pauli spin operator in the z-direction and γ=1/τdec, where τdec is the decoherence time. For the case of initially unentangled ions, this simply leads to a broadening of the signal P̃=12(1+eγTcosϕT), leading to the frequency uncertainty

Δω0=1e2γTcos2ϕTNttotTe2γTsin2ϕT.
(23)

This can be optimized for a given free evolution time T to find the minimum uncertainty. The minimum value is obtained for ϕT=kπ/2,k{2n+1:n} and T=τdec/2 for which it was found Δω0,min.=2γe/Nttotmail=2e/NτdecT. Likewise, for the maximally entangled two-ion case, the probability of detecting excitation is modulated by the decoherence term, yielding P̃N=12(1+eNγTcosNϕT) with a phase uncertainty optimized by the conditions ϕT=kπ/2N,k{2n+1:n} and T=τdec/2N. Combining these results yields the same exact minimum frequency uncertainty as the unentangled ion case: Δω0,min.=2e/NτdecT. This exercise indicates that in the case of unentangled ions and a maximally entangled pair of ions, both preparations reach the same degree of precision when decoherence is taken into account. This makes intuitive sense as maximally entangled states are considerably more fragile and consequently more susceptible to decoherence. Furthermore, it has been shown that this limit cannot be overcome by employing a different form of measurement. The limit can, however, be overcome by considering ions initially prepared in highly symmetric but only partially entangled states.24 

In Sec. III, we will briefly review the basics of phase estimation and make the connection between parity-based measurement and the minimum phase uncertainty: the quantum Cramér–Rao bound.

Phases cannot be measured, only approximated. This is due to the fact that there exists no Hermitian phase operator.27 Consequently, within the realm of quantum mechanics, the phase is treated as a classical parameter rather than a quantum observable. The general approach to interferometry, be it optical or atomic, is to encode a suitable “probe state” with a classically treated phase and determine the optimal detection observable for estimating its value. The art lies in determining the best combination of the probe state and detection observable that yields the highest resolution and smallest phase uncertainty. In what follows, we will discuss some of the basics of phase estimation and arrive at a relation between parity-based detection and the upper-bound on phase estimation, which determines the greatest sensitivity afforded to a given quantum state.

In the broadest sense, an interferometer is an apparatus that can transform an input “probe state” ρ̂ in a manner such that the transformation can be parameterized by a real, unknown, parameter φ. A measurement is then performed on the output state ρ̂(φ) from which an estimation of the parameter φ takes place. The most general formulation of a measurement in quantum theory is a positive-operator valued measure (POVM). A POVM consists of a set of non-negative Hermitian operators satisfying the unity condition εÊ(ε)=1. Following the work of Pezzé and Smerzi28 the conditional probability to observe the result ε for a given value φ, known as the likelihood, is

P(ε|φ)=Tr[Ê(ε)ρ̂(φ)].
(24)

If the input state is made up of m independent uncorrelated subsystems such that ρ̂=ρ̂(1)ρ̂(2)ρ̂(3)ρ̂(4).ρ̂(m) and we restrict ourselves to local operations such that the phase φ is encoded into each subsystem and assuming that independent measurements are performed on each, then the likelihood function becomes the product of single-measurement probabilities

P(ε|φ)=i=1mPi(εi|φ),
(25)

where Pi(εi|φ)=Tr[Ê(i)(εi)ρ̂(i)(φ)]. For the case of independent measurements, as described in Eq. (25), often one considers the log-Likelihood function

L(ε|φ)lnP(ε|φ)=i=1mlnPi(εi|φ).
(26)

We define the estimator Φ(ε) as any mapping of a given set of outcomes, ε, onto parameter space in which an estimation of the phase is made. A prevalent example is the maximum-likelihood estimator (MLE),28 defined as the phase value that maximizes the likelihood function

ΦMLE(ε)=arg[maxφP(ε|φ)].
(27)

An estimator can be characterized by its phase dependent mean value

Φφ=εP(ε|φ)Φ(ε)
(28)

and its variance

(ΔΦ)φ2=εP(ε|φ)[Φ(ε)Φφ]2.
(29)

We will now discuss what it means for an estimator to be “good,” which in this case refers to an estimator that provides the smallest uncertainty. These are known as unbiased estimators and are defined as estimators whose average value coincides with the true value of the parameter in question, that is, Φ(ε)φ=φ is true for all values of the parameter φ, while estimators that are unbiased in the limit of m, such as the MLE, are considered asymptotically consistent. Estimators not satisfying this condition are considered biased, while estimators that are unbiased for a certain range of the parameter φ are considered locally unbiased.

We now move on to perhaps one of the most important tools in the theory of phase estimation: the Cramér–Rao bound (CRB). The CRB serves to set a lower bound on the variance of any arbitrary, locally unbiased, estimator and is given formally as

(ΔΦCRB)φ2=(Φφφ)2F(φ)(ΔΦ)φ2,
(30)

where the quantity F(φ) is the classical Fisher information (FI), which is given by

F(φ)(L(ε|φ)φ)2=ε1P(ε|φ)(P(ε|φ)φ)2,
(31)

where the sum extends over all possible values of the measurement values, ε. The derivation of the CRB is straightforward: First, we have Φφφ=εφP(ε|φ)Φ(ε)=ΦLφ, where L(ε|φ) is given by Eq. (26). Noting that εφP(ε|φ)=Lφ=0, we have

(Φφφ)2=(ΦΦφ)Lφφ2(ΦΦφ)2φ(Lφ)2φ=(ΔΦ)φ2F(φ),
(32)

where we have invoked the Cauchy–Schwarz inequality ABφ2A2φB2φ. Dividing by F(φ) yields the CRB in Eq. (30). While Eq. (30) is the most general form the CRB, it is most useful for the cases of unbiased estimators where the numerator on the right-hand side φΦφ=1. For this case, the CRB is simply given as the inverse of the Fisher information F(φ). An estimator that saturates the CRB is said to be efficient. The existence, however, of an efficient estimator depends on the properties of the probability distribution. It is also worth noting that the derivation of the Fisher information of Eq. (31) assumed an initial state ρ composed of m-independent subsystems. It is straight forward to show the additivity of the Fisher information F(φ)=i=1mF(i)(φ) using Eq. (31) and plugging in Eqs. (25) and (26), where F(i)(φ) is the Fisher information of the ith subsystem. For m identical subsystems and measurements, this yields F(φ)=mF(1)(φ), where F(1)(φ) is the Fisher information for a single-measurement and m is the total number of measurements. This is the form of the Fisher information most often used in the literature.

We now turn our attention toward discussing an upper bound29 on phase estimation, known as the quantum Cramér–Rao bound (qCRB), which in turn will be dependent on the quantum Fisher information FQ (QFI). We obtain this upper bound by maximizing the FI over all possible POVMs,30,31

FQ[ρ̂(φ)]max{Ê(ε)}F[ρ̂(φ),{Ê(ε)}],
(33)

where this quantity is known as the quantum Fisher information. It is important to note that the quantum Fisher information is independent of the POVM used. This quantity can be expressed as32–34 

FQ[ρ̂(φ)]=Tr[ρ̂(φ)L̂φ2],
(34)

where L̂φ is known as the symmetric logarithmic derivative (SLD)35 defined as the solution to the equation

ρ̂(φ)φ=ρ̂(φ)L̂φ+L̂φρ̂(φ)2.
(35)

The chain of inequalities is now

(ΔΦ)φ2(ΔΦ)CRB2(ΔΦ)qCRB2,
(36)

where it follows that the quantum Cramér–Rao bound (also known as the Helstrom bound36) is given by

(ΔΦ)qCRB2(Φφ)2FQ[ρ̂(φ)].
(37)

Since the qCRB is inversely proportional to the QFI, which itself is a maximization over all possible POVMs, it is clear to see how the qCRB serves as an upper bound on phase estimation.

1. Calculating the QFI for pure and mixed states

Here, we work through a suitable expression for the QFI, using our definition of the SLD given in Eq. (35), in terms of the complete basis {|n}, where our density operator is now given generally as ρ̂(φ)=npn|nn|. Following the work of Pezzé and Smerzi28 the quantum Fisher information can be written in this basis as

FQ[ρ̂(φ)]=Tr[ρ̂(φ)L̂φ2]=k,kpk|k|L̂φ|k|2=k,kpk+pk2×|k|L̂φ|k|2.
(38)

Thus, it is sufficient to know the matrix elements of the SLD, k|L̂φ|k, in order to calculate the QFI. Using Eq. (35) and our general density operator, it is easy to show

k|L̂φ|k=(2pk+pk)×k|φρ̂(φ)|k,
(39)

which makes Eq. (38)37 

FQ[ρ̂(φ)]=k,k2pk+pk×|k|φρ̂(φ)|k|2.
(40)

We proceed further through the use of the definition

φρ̂(φ)=k(φpk)|kk|+kpk|φkk|+kpk|kφk|,
(41)

which is a simple application of the chain rule for derivatives. Using the identity φk|k=φk|k+k|φk0, the matrix elements in Eq. (40) become

k|φρ̂(φ)|k=(φρ̂(φ))δk,k+(pkpk)φk|k.
(42)

The SLD and QFI, then, become

L̂φ=kφpkpk×|kk|+2k,kpkpkpk+pk×φk|k|kk|,FQ[ρ̂(φ)]=k(φpk)2pk+2k,k(pkpk)2pk+pk×|φk|k|2,
(43)

respectively. These results, we show next, simplify in the case of pure states where we can write ρ̂(φ)=|ψ(φ)ψ(φ)| and consequently φρ̂(φ)=φρ̂2(φ)=ρ̂(φ)[φρ̂(φ)]+[φρ̂(φ)]ρ̂(φ). Using this, and a cursory glance at Eq. (35), it is clear that the SLD becomes

L̂φ=2[φρ̂(φ)]=2[φ|ψ(φ)ψ(φ)|]=2|φψψ|+2|ψφψ|,
(44)

where in the last step, the φ-dependency of |ψ is implicit for notational convenience. Plugging this directly into the first line of Eq. (38) yields

FQ[ρ̂(φ)]=Tr[ρ̂(φ)L̂φ2]=ψ|L̂φ2|ψ=4{φψ|φψ|φψ|ψ|2},
(45)

which is the form of the QFI most often used in the quantum metrology literature. Next, we move on to discussing a specific detection observable: photon number parity.

2. Connection to parity-based detection

The central theme discussed throughout this paper is the use of the quantum mechanical parity operator as a detection observable in quantum optical interferometry. The use of parity as a detection observable first came about in conjunction with high precision spectroscopy by Bollinger et al.17 It was first adapted and formally introduced for use in quantum optical interferometry by Gerry et al.38,39 A detection observable is said to be optimal if for a given state, the CRB achieves the qCRB, that is,

(ΔΦ)φ2(ΔΦ)CRB2=(ΔΦ)qCRB2.
(46)

Furthermore, parity detection achieves maximal phase sensitivity at the qCRB for all pure states that are path symmetric.40,41 For the purposes of this paper, it is sufficient to derive the classical Fisher information. We start from the expression for the classical Fisher information, assuming that a single measurement is performed, given by Eq. (31). For parity, the measurement outcome ε can either be positive + or negative − and satisfies P(+|φ)+P(|φ)1.42 The expectation value of the parity operator can, then, be expressed as a sum over the possible eigenvalues weighted with the probability of that particular outcome, leading to

Π̂=iP(i|φ)λi=P(+|φ)P(|φ)=2P(+|φ)1=12P(|φ).
(47)

Likewise, we can calculate the variance

(ΔΠ̂)2=Π̂2Π̂2=1Π̂2=1(P(+|φ)P(|φ))2=1(P(+|φ)+P(|φ))2+4P(+|φ)P(|φ)=4P(+|φ)P(|φ).
(48)

Finally, from Eq. (47), it follows that

P(+|φ)φ=12Π̂φ=P(|φ)φ.
(49)

Combining Eqs. (48) and (49), we find

F(1)(φ)=ε11P(ε1|φ)(P(ε1|φ)φ)2=1(ΔΠ̂)2|Π̂φ|2,
(50)

making the CRB/qCRB

ΔΦCRB/qCRB=1F(1)(φ)=1Π̂2|φΠ̂|.
(51)

Equation (51) is simply the phase uncertainty obtained through the error propagation calculus. This is particularly advantageous over other means of detection because the use of parity does not require any pre- or post-data processing.42 By comparison, photon number counting typically works by construction of a phase probability distribution conditioned on the outcome of a sequence of m measurements.43 After a sequence of detection events, the error in the phase estimate is determined from this distribution. While this provides phase estimation at the qCRB, it lacks the advantage of being directly determined from the signal. There are, however, disadvantages to using parity. The performance. of photon number parity is highly susceptible to losses (a point that will be discussed in Sec. IV G). Parity also achieves maximal phase sensitivities at particular values of the phase, restricting its use to estimating local phases.44 Restricting its use to local parameter estimation, however, is not terribly problematic in interferometry, as one is interested in measuring small changes to parameters that are more-or-less known.

It is worth pointing out that the optimal POVM depends, in general, on φ. This is somewhat problematic as it requires one to already know the value of the parameter φ in order to choose an optimal estimator. Some work has been done to overcome this obstacle,45 which concludes that the QFI can be asymptotically obtained in a number of measurements without any knowledge of the parameter. For all cases considered throughout this paper, we will use parity as our detection observable (except in cases where we wish to draw comparisons between observables), which we know saturates the qCRB. We will now move on to discuss how one calculates the QFI in quantum optical interferometry.

We use the Schwinger realization of the SU(2) algebra with two sets of boson operators, discussed in detail in Appendix A 2, to describe a standard Mach–Zehnder interferometer (MZI).18 In this realization, the quantum mechanical beam splitter can be viewed as a rotation about a given (fictitious) axis determined by the choice of angular momentum operator, i.e., the generator Ĵx corresponds to a rotation about the x-axis, while Ĵy corresponds to a rotation about the y-axis. An induced phase shift, assumed to be in the b-mode, is described by a rotation about the z-axis characterized by the use of the Ĵz-operator. The state just before the second beam splitter is given as

|ψφ=eiφĴzei(π/2)Ĵx|in,
(52)

where we are assuming the beam splitters to be 50:50. This in turn makes the derivative

|ψφ=ieiφĴzĴzei(π/2)Ĵx|in,
(53)

leading to

ψ(φ)|ψ(φ)=iin|Ĵy|in
(54)

and

ψ(φ)|ψ(φ)=in|Ĵy2|in,
(55)

where we have made use of the Baker–Hausdorf identity in simplifying

ei(π/2)ĴxĴzei(π/2)Ĵx=Ĵy.
(56)

Combining Eqs. (54) and (55) into Eq. (45) yields for the QFI

FQ[ρ̂(φ)]=4{φψ|φψ|φψ|ψ|2}=4(ΔĴy)2in,
(57)

which is simply the variance of the Ĵy-operator with respect to the initial input state |in. This is the form of the QFI used in all of the following interferometric calculations. One important thing to notice is that the quantum Fisher information depends solely on the initial state and not on the value of the phase φ to be measured.

Note that Eq. (57) is a general result. Let Ô be a generator of a flow parameterized by ζ such that the wave function evolves according to the Schrödinger equation iζ|ψ=Ô|ψ with solution |ψ(ζ)=eiÔζ|ψ(0). Then, |ζψ=iÔ|ψ so that ζψ|ζψ=Ô2 and |ψ|ζψ|2=Ô2. Hence, FQ(ζ)=4[ζψ|ζψ|ψ|ζψ|2]=4(ΔÔ)2ζ. Thus, both Eqs. (30) and (37) yield a Fourier-like uncertainty relation for unbiased estimators of the form (ΔΦ)ζ(ΔÔ)ζ1/2 (if we do not use the maximum FI) reminiscent of ΔxΔk1/2 so that a precise measurement ΔΦζ1/2 of ζ requires a large uncertainty (variance) (ΔÔ)ζ1/2 in its generation. Noting that Π̂b=(1)b̂b̂=eiπn̂b is both unitary and Hermitian, define Π̂b(ϕ)eiϕn̂b with Ôn̂b and ζϕ (which we take to be ϕ=π for the parity operator). Thus, a measurement Π̂b(ϕ)|ψ(0)=eiϕn̂b|ψ(0) can be thought of as a Schrödinger-like evolution in ϕ with generator n̂b. Then, from the above, FΠb(ϕ)=4(Δn̂b)2ϕ. This result is independent of the parameter ϕ. This leads to the insightful interpretation of Eq. (51) as ΔΦCRB/qCRBΔn̂b=12, i.e., a more formal statement of the classical number-phase uncertainty relationship ΔϕΔN12. In fact, we now see that the parity operator Π̂b leads to the minimal phase uncertainty relationship possible since it saturates the inequality. This relation between phase estimation and the generator of phase shifts has an extensive history in the literature.30,31,46,47 A similar result was obtained by Zwierz et al.46 who pointed out that the Heisenberg limit cannot be likened to an uncertainty relation since it relates the uncertainty of the parameter to the first moment of the conjugate observable. Rather, they go on to point out that the HL is more closely related to the Margolus–Levitin bound48 on the time it takes a quantum state to evolve to an orthogonal state.

Next, we will discuss the phase uncertainty and measurement resolutions obtained using parity-based detection in interferometry for a number of cases in which input classical and/or quantum mechanical states of light are considered.

In this section, we highlight several different interferometric schemes involving both classical and nonclassical states of light comparing the use of several different detection observables. Here, we show that in cases where the bound on phase sensitivity is not saturated, parity-based detection yields sub-SQL limited phase sensitivity and can, in certain cases, approach or out-perform the HL of phase sensitivity. We also draw attention to the correlation between parity-based detection and the saturation of the qCRB. Before we get into certain cases, we will provide the reader with a concise derivation of the output state of an interferometer for arbitrary initial states and the average value of an arbitrary detection observable and the subsequent phase uncertainty.

Consider an interferometer transforming an initial state |in according to

|out=ei(π/2)ĴxeiϕĴzei(π/2)Ĵx|in=eiϕĴy|in,
(58)

where we employ the Schwinger realization of the SU(2) Lie algebra (see Appendix A 2), and a measurement is performed on the output state. A generalized schematic of the set-up can be seen in Fig. 3. It is worth comparing the expression for the output state of Eq. (58), with that of the final state obtained when one performs Ramsey spectroscopy, as shown in Eq. (12). The π/2 pulses performed on atomic states is analogous to a beam splitter transformation affecting two boson modes in that both can be described in terms of the SU(2) Lie algebra. The same can be said of how the phase ϕ is encoded in both procedures, though they have very different physical interpretations. For atomic systems, the phase shift arises during the period of free evolution, while in interferometry it stems from a relative path length difference between the two arms of the interferometer.

For the most general of separable initial states, |in=|ψ0(1)a|ψ0(2)b, where in the photon number basis |ψ0(i)=n=0Cn(i)|n,n+, the input state can be expressed as

|in=n=0q=0Cn(1)Cq(2)|na|qb=j=0,1/2,  m=jjCj+m(1)Cjm(2)|j,m.
(59)

Working in the Schrödinger picture, the transformation of Eq. (58) acting on this initial state yields the output state

|out=j=0,1/2,  m=jjj,m|eiϕĴy|in|j,m=j=0,1/2,  m=jjΓj,m(ϕ)|j+ma|jmb,
(60)

where we have inserted a complete set of states in the angular momentum basis and where the phase-dependent state coefficients are given by

Γj,m(ϕ)=m=jjCj+m(1)Cjm(2)dm,mj(ϕ).
(61)

The phase-dependent term in Eq. (61) are the well-known Wigner-d matrix elements discussed in some detail in Appendix B 1. Note that if our initial state is an entangled two-mode state, the corresponding coefficients would be of the form An,p where An,pCn(1)Cp(2). For an arbitrary detection observable Ô, the expectation value can be calculated directly as

Ôout=j,mJ,MΓJ,M*(ϕ)Γj,m(ϕ)J,M|Ô|j,m.
(62)

From this, the phase uncertainty can be found through the use of the usual error propagation calculus to be

ΔϕÔ=ΔÔout|ϕÔout|.
(63)

From the perspective of the Heisenberg picture, the transformed observable Ô is given by Ô=eiϕĴyÔeiϕĴy and the derivative of its expectation value

ϕÔout=ϕÔin=i[Ĵy,Ô]in.
(64)

We remind the reader that the greatest phase sensitivity afforded by classical states is the standard quantum limit (SQL), ΔϕSQL=1/n¯, while for quantum states, the phase sensitivity is bounded by the Heisenberg limit (HL) ΔϕHL=1/n¯. We note that while the HL serves as a bound on phase sensitivity, it has been demonstrated to be beaten by some quantum states for low (but still >1) average photon numbers.49 The goal of this section is to highlight the effectiveness of parity-based measurement performed on one of the output ports. Unless otherwise stated, we assume that the measurement is performed on the output b-mode without loss of generality.50 We are now ready to discuss several different cases.

The optical N00N state has been extensively studied for use in high-precision quantum metrology.39,51–53 It is defined as the superposition state in which N photons are in one mode (labeled the a-mode), while none are in the either (b-mode) |N,0a,b and where no photons are in the a-mode and N photons are in the b-mode |0,Na,b and can be written generally as

|ψN00N=12(|N,0+eiΦN|0,N),
(65)

where ΦN is a relative phase factor that may depend on N and whose value will generally depend on the method of state generation. The origin of the moniker “N00N state” is obvious, though such states are also known as maximally path-entangled number states as the path of the definite number of photons N in the superposition state of Eq. (65) can be interpreted as being objectively indefinite.

Let us consider an interferometric scheme in which the initial state is given by |in=|N,0a,b and the first beam splitter is replaced by an optical device that transforms the initial state into the N00N state of Eq. (65) (a “magic” beam splitter, so to speak). The state after the phase shift (taken to be in the b-mode) is

|ψN(ϕ)=12(|N,0a,b+ei(Nϕ+ΦN)|0,Na,b),
(66)

amounting to an additional relative phase shift of Nϕ. Finally, the state after the second beam splitter is39 

|out=12N+1N![(â+ib̂)N+ei(Nϕ+ΦN)(b̂+iâ)N]|0,0a,b,
(67)

which for the case of N =1 yields the same phase sensitivity as the |in=|1,0a,b state when implementing intensity-difference measurements in a regular MZI scheme. For the case of N =2, the output state can be written as

|out,2=122[(1ei(2ϕ+Φ2))|2,0a,b+2i(1+ei(2ϕ+Φ2))|1,1a,b+(ei(2ϕ+Φ2)1)|0,2a,b].
(68)

Interestingly, for all values N >1, an intensity-difference measurement fails to capture any phase-shift dependence; this results in a measurement of zero irrespective of the value of ϕ, making it an unsuitable detection observable for this choice of input state.

A Hermitian operator was introduced by Dowling et al.51,52 of the form Σ̂N=|N,0a,b0,N|+|0,Na,bN,0| whose expectation value with respect to the state Eq. (65), Σ̂N=cosNϕ, depends on the phase Nϕ with interference fringes whose oscillation period is N times shorter than that of the single-photon case. The phase uncertainty obtained through the error propagation calculus can be found easily to be

(Δϕ)2=Σ̂N2ΣN̂2(ϕΣN̂)2=1cos2NϕN2sin2Nϕ=ϕ01N2,
(69)

which is an improvement over the classical (SQL) limit by a factor of 1/N. The result of Eq. (69) follows from the heuristic number-phase relation ΔϕΔN1. For the N00N state of Eq. (65), the uncertainty in the photon number is N, the total average photon number, immediately giving the equality Δϕ=1/N. Extrapolating for arbitrary states, we define the Heisenberg limit (HL) ΔϕHL=1/n¯tot, where n¯tot is the total average photon number inside the interferometer.

It was found by Gerry and Mimih39 that the results of the projection operator employed by Dowling et al.51,52 can be realized through the implementation of photon-number parity measurements performed on one of the output modes. As a demonstration, consider the N =2 case described by the output state given by Eq. (68). The action of the parity operator on the b-mode, Π̂b=(1)b̂b̂, results in a sign flip on the center term

Π̂b|out,2=122[(1ei(2ϕ+Φ2))|2,0a,b2i(1+ei(2ϕ+Φ2))|1,1a,b+(ei(2ϕ+Φ2)1)|0,2a,b],
(70)

leading to the expectation value Π̂b=cos(2ϕ+Φ2), similar to the result obtained through the use of the projection operator employed by Dowling et al. Unlike the method of measuring the intensity-difference between modes, this carries relevant phase information from which an estimation can be made. For the arbitrary N case, it was found that

Π̂b=iN2eiNϕ+ΦN+1NeiNϕ+ΦN=1N/2cosNϕ+ΦNNeven1N+1/2sinNϕ+ΦNNodd,
(71)

and consequently, Δϕ=1/N. It is important to note that while photon-number parity functions similarly to the projection operator put forth by Dowling et al., they are not equivalent. This is clear as Σ̂N is not directly connected to an observable. Furthermore, there is presently no physical realization of the projector Σ̂N capable of being utilized experimentally.

Clearly, states similar in form to Eq. (65) after the first beam splitter are favorable in interferometry. Let us begin by considering a case in which the state after beam splitting can be written in terms of a superposition of N00N states.

One such case known to yield Heisenberg-limited phase sensitivity is the case in which the action of the first beam splitter results in an entangled coherent state (ECS).54 It has been shown by Israel et al.55 that such a state can be generated through the mixing of coherent light with a squeezed vacuum at the first beam splitter (a case to be discussed in greater detail in Sec. IV D). Another method involves the mixing of coherent and catstates. The coherent state is given by |α=e|α|2/2n=0αnn!|n and constitutes the most classical of quantized fields states, characterized as light from a well phase-stabilized laser. It is important to point out that while coherent light maintains classical properties, it is still a quantum state of light as it is defined in terms of a quantized electromagnetic field. The generalized cat state56 is expressed as a superposition of equal-amplitude coherent states differing by a π-phase shift. Such states have been studied extensively in the context of phase-shift measurements.57 The initial state can be written as

|in=|βaN(|γb+eiθ|γb),
(72)

where N is the cat state normalization factor given by N=1/2(1+e2|γ|2cosθ). The QFI can be calculated immediately for this input state using Eq. (57). For large γ, J+̂=J¯̂=Jŷ0 and

J+2=β*2γ2,J2=β2γ*2,
(73)
J+Ĵ=|β|2(1+|γ|2),JĴ+=|γ|2(1+|β|2),
(74)

resulting in

FQ=4(ΔĴy)2in=4Ĵy2=|β|2+|γ|2+2|β|2|γ|2(1cos[2(θβθγ)]),
(75)

where (θβθγ) is the phase difference between coherent states |β and |γ. Setting β=α/2 and γ=iβ=iα/2, the total average photon number in the interferometer in the limit of large coherent state amplitude is |β|2+|γ|2=|α|2. Plugging these into Eq. (75) and noting now that θβθγ=π/2, the minimum phase uncertainty is given by

Δϕminlargeα1|α|2=ΔϕHL.
(76)

Let us calculate the state after the first beam splitter for this choice of input state. Through the usual beam splitter transformations for coherent states, we have

|βa|γbBS1|12(β+iγ)a|12(γ+iβ)b,|βa|γbBS1|12(βiγ)a|12(γ+iβ)b,
(77)

making the state after the first beam splitter

|out,BS1=N(|12(β+iγ)a|12(γ+iβ)b+eiθ|12(βiγ)a|12(γ+iβ)b].
(78)

Once again setting β=α/2 and γ=iβ=iα/2, this becomes

|out,BS1=N(|αa|0b+eiθ|0a|iαb),
(79)

the entangled coherent state. This state was studied by Gerry et al.58 where the first beam splitter was replaced by an asymmetric nonlinear interferometer (ANLI) yielding the state |out,ANLI(|0,iαa,b+i|α,0a,b), similar to Eq. (79). This state can be written as a superposition of N00N states as per

|out,ANLI=e12|α|2N=0αNN!(1)N×12(|N,0a,b+eiΦN|0,Na,b),
(80)

where ΦN=π2(N+1). Parity-based detection was considered on the output b-mode after acquiring a phase shift ϕ and passing through the second beam splitter to find

Π̂b=en¯(1+en¯cosϕsin(n¯sinϕ)),
(81)
ϕΠ̂b=n¯en¯(1cosϕ)cos(ϕ+n¯sinϕ),
(82)

and from the error propagation calculus, Eq. (63), the phase uncertainty for ϕ0 is

ΔϕΠ̂b=1e2n¯n¯,
(83)

which subsequently yields the HL in the limit of large |α|. Plots of the expectation value of the parity operator, Eq. (81), and corresponding phase uncertainty, Eq. (83), found by Gerry et al.58 are given in Figs. 4 and 5, respectively. Clearly, distributions after the first beam splitter, which are reminiscent in form to the N00N state, tend toward providing greater phase sensitivity.

Another such case to consider is the case of the input twin-Fock states. As per the very well-known Hong–Ou–Mandel (HOM) effect, when the initial state |1,1a,b is incident upon a 50:50 beam splitter, the resulting state is the two-photon N00N state |2,0a,b+|0,2a,b. Let us now consider the case in which the arbitrary state |N,Na,b is taken as the initial state.

It was first pointed out by Holland and Burnett59 that interferometric phase measurements when considering input twin-Fock states |N,Na,b asymptotically approach the HL. They found this by studying the phase-difference distribution for the states inside the MZI just prior to the second beam splitter. On the other hand, Bollinger et al.17,50 showed that if the state just after the first beam splitter is somehow a maximally entangled state (MES) of the form |2N,0a,b+eiΦN|0,2Na,b,N+, then the phase uncertainty is exactly the HL: ΔϕHL=1/2N. The problem is that such a state is incredibly difficult to produce; in fact, it cannot generally be done with an ordinary beam splitter. Schemes for generating such states using both nonlinear devices and linear devices used in conjunction with conditional measurements have been proposed.38,58,61–65 For an initial state described by state coefficients Cn(1)Cq(2)=δN,nδN,q as per Eq. (59), the state after the first beam splitter is the well-known arcsine (AS) state,66 given by

|out,BS1=k=0N1Nk[2kk2N2kNk122N]1/2×|2ka|2N2kb,
(84)

where in this case a Ĵy-type beam splitter was considered rather than a Ĵx-type of Sec. III. This simply amounts to a relative phase difference between terms in the sum in Eq. (84). Clearly, for the N =1 case, we recover the well-known two-photon N00N state that has long been available in the laboratory.67 For N >1, the state of Eq. (84) does not clearly result in the N-photon N00N state but instead a superposition of the N-photon N00N state and other (but not all) permutations of the state |p,q where p+q=2N. Due to the strong correlations between photon number states of the two modes, the only nonzero elements of the joint-photon number probability distribution are the joint probabilities for finding 2k photons in the a-mode and 2N2k photons in the b-mode, given by

PAS(2k,2N2k)=|a,b2k,2N2k|out,BS1|2=(2kk)(2N2kNk)(12)2N,k[0,N],
(85)

forming a distribution known as the fixed-multiplicative discrete arcsine law of order N,68 deriving the name of the state. Such a distribution is characterized by the “bathtub” shape of an arcsine distribution, with peaks occurring for the |N,0a,b and |0,Na,b states, as shown in Fig. 6. The phase properties of this state were studied by Campos et al.66 

Next, we will compare the qCRB for this choice of initial state against the phase uncertainty obtained via parity-based measurements. Once again, the qCRB can be calculated directly from the initial state through Eq. (57). It is plainly evident that J+=J=Jy=0,Ĵ+Ĵ=ĴĴ+=N(N+1) and consequently

Δ2Ĵy=N2(1+N)Δϕmin=12N(1+N).
(86)

Right away, it is clear that for the case of N =1, the minimum phase uncertainty provides the HL: ΔϕHL=1/2N=1/2. Next, we consider the use of parity performed on the output b-mode as a detection observable. The expectation value of the parity operator can be calculated directly with respect to the state of Eq. (84), accounting for the phase shift and assuming that the second beam splitter is of the Ĵx-type, to be

Π̂bAS=in|Π̂b|in=out,BS1|eiϕĴzeiπ2ĴxΠ̂beiπ2ĴxeiϕĴz|out,BS1=k=0Nei2ϕ(N2k)(2kk)(2N2kNk)(12)2N.
(87)

The imaginary part of Eq. (87) sums identically to zero as it is the product of an even times and odd function of k. The real part is identically a Legendre polynomial, making

Π̂bAS=PN[cos(2ϕ)].
(88)

With this, the phase uncertainty can be computed directly from the error propagation calculus given by Eq. (63). For N =1, the expectation value of the parity operator is Π̂bAS=cos(2ϕ), leading to Δϕ=1/2, the HL. For N =2, it can be shown that Π̂bAS=1/4+3/4cos(4ϕ), which in the limit of ϕ0 yields Δϕ=1/12=0.2886, close to the HL of 1/4=0.25. These results are in agreement with the minimum phase uncertainty obtained through the calculation of the qCRB for this state, Eq. (86). A plot of the parity-based phase uncertainty and corresponding qCRB can be found in Fig. 7. Another interesting feature of Eq. (88) occurs when considering measurements around ϕ=π/2. Through the use of standard identities involving Legendre polynomials, it can be shown that Eq. (88) becomes Π̂bAS(ϕ=π/2)=(1)N, corresponding to a peak in the curve, yielding the same degree of resolution as ϕ=0, but takes the maximal (minimal) value of ±1 depending on the value of N. This may prove to be of use in verifying that one has lossless conditions within the interferometer as typically the experimenter would have foreknowledge of the value N and therefore know what the value in a π/2-shifted interferometer should be. The measurement to the contrary can point toward the presence of losses in the system.

In Sec. IV C 1, we will briefly consider states composed of superpositions of twin-Fock states.

1. Superpositions of twin-Fock states

Highly photon-number correlated continuous variable two-mode states have been investigated for use in quantum optical interferometry. It is immediately clear that such states are entangled as they are already in Schmidt form |Ψ=n=0Bn|n,na,b with total average photon number n¯tot=2n=0|Bn|2n. In terms of the initial state of Eq. (59), the coefficients for such a state are given by Cn(1)Cq(2)=Bnδq,n. The expectation value of the parity operator is readily calculable for arbitrary correlated two-mode states of this form using the results for the case of a twin-Fock state input of Eq. (88),

Π̂bcorr=n=0|Bn|2Pn[cos(2ϕ)],
(89)

where once again Pn(x) are the Legendre polynomial. The most well-known and studied correlated two-mode state is the two-mode squeezed vacuum state (TMSVS). The TMSVS is a laboratory standard, routinely produced through parametric down conversion:69 a second order nonlinear effect in which a pump photon of frequency 2ω is annihilated and two photons, each of frequency ω, are produced. The correlation between modes is due to the pair-creation of photons resulting from the down-conversion process. Note that under the parametric approximation, the pump is treated as a classical and nondepleting field. Consequently, the pump is not treated as a quantized mode. The TMSVS state coefficients are given by Bn(TMSVS)=(1|z|2)1/2zn. The parameter z is a complex number constrained to be |z|<1 and can be expressed in terms of the pump field parameters, γ and 2ϕ being the pump amplitude and phase, respectively, as z=ei2ϕtanhr, where r=|γ|t is the squeeze parameter. Interestingly, coupling the TMSVS coefficients with Eq. (89) yields for the phase value ϕ=π/2

Π̂btmsvs(ϕ=π/2)=1cosh(2r)={1,r02e2r,r1.
(90)

In other words, the measurement of photon-number parity on one of the output modes provides a direct measure of the degree of squeezing, i.e., determination of the squeeze parameter r. For the TMSVS, parity-based detection yields a phase uncertainty that falls below the HL for ϕ=0, an effect that has been pointed out in the past by Anisimov et al.,49 explained in terms of the Fisher information. Once again using Eq. (57) and noting for such a correlated state that FQ=2Ĵ+Ĵin, it can be shown that the TMSVS minimum phase uncertainty is

Δϕmin=1n¯tot(2+n¯tot),
(91)

where for the TMSVS, n¯tot=2sinh2r. This means that the TMSVS has the potential for super sensitive phase estimation; clearly, the phase estimate of Eq. (91) is sub-HL. This is seemingly a violation of the bound set for quantum states. It has been argued that such a limitation, based on the heuristic photon number-phase relation ΔϕΔN1, is reasonable in the case of definite photon number (finite energy) but proves to be an incomplete analysis when considering the effect of photon-number fluctuations. Hoffman40 suggested a more direct definition of the limit on phase estimation for quantum states in terms of the second moment of n̂,Δφ=1/n̂2. This provides better sensitivity in phase measurements than the HL as n̂2 contains direct information about fluctuations that n̂2 does not. In fact, this is why parity-based measurement yields greater sensitivity: it contains all moments of n̂. For the case of a TMSVS, the Hoffman limit is given by Δϕ=1/2n¯(n¯+1). With parity-based detection, the sensitivity of the phase estimate is better than that allowed by the HL but is never better than the Hoffman limit [the sub-HL sensitivity is prominent for low (but still >1) average photon numbers but asymptotically converges to the HL for ϕ=0]. In general, it has been shown that interferometric schemes attaining sub-HL sensitivities are infeasible in practice,70 requiring prior knowledge of the parameter to be measured.47 Furthermore, a more rigorous and loop-hole free form of the Heisenberg limit, considering the average error over all phase shifts, that is both constraint-free and non-asymptotic has been developed by Hall et al.71 

It was pointed out by Anisimov et al.49 and Gerry and Mimih72 that the TMSVS, using parity measurements, has superior phase sensitivity near ϕ=0 but degrades rapidly as the phase difference deviates from zero as shown in Fig. 8, making the state suboptimal for interferometry. On the other hand, the case in which one has parametric down-conversion with coherent states seeding the signal and idler modes, or two-mode squeezed coherent states (TMSCS), was considered by Birrittella et al.73 who found a measurement resolution and phase sensitivity dependent on a so-called cumulative phase (a combination of the initial field phases) that yields, for low squeezing, sub-SQL phase sensitivity that does not degrade like the TMSVS does as the phase deviates from zero for the optimal choice of cumulative phase. The cumulative-phase-dependent state statistics and entanglement properties of the state resulting from coherently stimulated down-conversion with a quantized pump field have been studied by Birrittella et al.;74 however, the state has not yet been considered in the context of interferometry.

Another correlated two-mode state that has been extensively studied in the context of interferometry and metrology is the pair coherent states (PCS),75 or circle states, which have the form

|ζ=N02π|ζeiθ|ζeiθdθ,
(92)

where |ζe±iθ are the Glauber coherent states. In terms of the number state basis, the PCS can be written as

|ζ=1I0(2|ζ|)n=0ζnn!|n,na,b,
(93)

where I0(2|ζ|) is the modified Bessel function of order zero and ζ is a complex number defined such that |ζ is a right-eigenstate of the joint photon-annihilation operators âb̂|ζ=ζ|ζ and (ââb̂b̂)|ζ=0.76 Such a state has been shown to exhibit sub-Poissonian statistics, which results in sub-SQL phase uncertainty, enhanced measurement resolution, and a high signal-to-noise ratio, results very close to those obtained from input twin-Fock states.72 Furthermore, the PCS does not display the degradation of phase sensitivity as ϕ deviates from zero as the TMSVS does, making it more stable for interferometric measurements. The problem, however, lies in generating such a state. Several schemes have been proposed, most notably a scheme involving the use of third order cross-Kerr coupling between coherent states and the implementation of a state-reductive measurement,77 resulting in the projection of the PCS in bursts. Currently, the PCS has yet to be experimentally realized.

The original interferometric scheme for reducing measurement error was proposed by Caves78 in the context of gravitational wave detection. Here, we consider the case in which the input state is a product of coherent light in one port and a single-mode squeezed vacuum state (SVS) in the other, given by |SVS=N=0SN|N with average photon number n¯SVS=sinh2r and with state coefficients,

SN=(1)N/2[N!2N(N2!)2(tanhNrcoshr)]1/2cos2(Nπ/2),
(94)

where once again r is the squeeze parameter. Such a state can be generated by beam splitting a TMSVS; the resulting two-mode state is a product of two single-mode SVS offset from each other by a π-phase shift.79 The initial state is, then, |in=|αa|SVSb with state coefficients as per Eq. (59) given by Cn(1)Cp(2)=e|α|2/2αnN!×Sp. This choice of input state has been shown to produce the ECS after the first beam splitter55 and has been studied extensively by Pezzé and Smerzi80 who considered a Bayesian phase inference protocol and showed that the phase sensitivity saturates the CRB and go on to demonstrate that the phase sensitivity can reach the HL Δϕ1/n¯tot independently of the true value of the phase shift. As the input state is path-symmetric,41 it is sufficient to determine the CRB through the calculation of the classical Fisher information, which they found for a single measurement to be

F(1)(ϕ)=Nc,Nd1P(Nc,Nd|ϕ)(P(Nc,Nd|ϕ)ϕ)2=|α|2e2r+sinh2r,
(95)

where P(Nc,Nd|ϕ) is the conditional probability of measuring Nc photons in one output mode and Nd in the other and is given in terms of the Wigner-d rotation elements (see  Appendix B) by

P(Nc,Nd|ϕ)=e|α|2|n=02jα2jn(2jn)!Sndm,jnj(ϕ)|2,
(96)

with j=(Nc+Nd)/2 and m=(NcNd)/2. For the regime in which both input ports are of roughly equal intensity |α|2sinh2r=n¯tot/2 and assuming large r such that n¯tote2r/2, it can be shown that

Δϕmin=1n¯tot(n¯tot+12)largen¯tot1n¯tot=ΔϕHL.
(97)

The same scheme was considered by Birrittella et al.81 using photon-number parity performed in the output b-mode as the detection observable. In particular, they studied a regime in which the two-mode joint-photon number distribution was parameterized such that it was symmetrically populated along the borders with no population in the interior, mimicking the case of the N00N state generated after the first beam splitter. The parameters α and r were chosen to be relevant to an experiment performed by Afek et al.82 based on the N00N state within the superposition of N00N states (found in the output state of the first beam splitter) in which they obtained high phase sensitivity and super-resolution. They achieved this measurement scheme by counting only the coincident counts where the total photon numbers counted added up to the selected value of N. In other words, they measured ââb̂b̂out but retained only the counts where if one detector detects m photons, the other detects Nm, and where all other counts N are discarded. They reported sub-SQL phase sensitivity with this scheme and super-resolved measurement resolution. It is worth pointing out, however, that this tends to work better for low average photon numbers as the large-photon-number case cannot be reasonably expressed as a superposition of N00N states.

With parity measurements performed on one of the output beams, it is not necessary nor possible to restrict oneself to a definite N-photon N00N state, which can be advantageous. It has been shown by Seshadreesan et al.83 that photon-number parity-based interferometry reaches the HL in the case of equal-intensity light incident upon a 50:50 beam splitter. The use of photon-subtracted squeezed light in one of the input ports has also been studied by Birrittella et al.84 who showed that the state after the first beam splitter resembled an ECS of higher average photon number than that generated through the use of a squeezed vacuum state.

Next, we will consider the case in which purely classical light is initially in one mode and the most quantum of quantized field states, a Fock (or number state), is initially in the other.

Next, we consider the choice of input state |in=|αa|Nb, that is, coherent light in the input a-mode and a Fock state of N photons in the input b-mode. This was studied by Birrittella et al.81 in the context of parity-based phase estimation. In terms of Eq. (59), the two-mode state coefficients are Cn(1)Cq(2)=e|α|2/2αn/n!×δq,N, which in terms of the angular momentum basis states can be written as

|in=e12|α|2j=N/2,α2jN(2jN)!|j,jN,
(98)

where the sum over j includes all half-odd integers. It is worth pointing out a characteristic of this particular state upon beam splitting. Consider the state after the first beam splitter, given by (see Appendix A 2) |out,BS1=ei(π/2)Ĵx|in. For the special case of N =1, this state can be written as

|out,BS1N=1=e(1/2)|α|2n=0q=0(α2)n(iα2)qn!p!×1δn,0δq,0γn,q|n,qa,b,
(99)

where the factor γn,q is given by

γn,q=4i(1)1q2n+q+1n!α(n+q+1)!2F̃1(2,n+1;2q;1)
(100)

and where 2F̃1(a,b;c;z) is a regularized hypergeometric function (see Appendix B 1). For n = q, this function is identically zero q0. Note that the term n=q=0 is a term that does not appear in the sum in Eq. (99) (due to the presence of the N =1 initial state). This coincides with a line of destructive interference along the diagonal line n = q, resulting in a bimodal distribution. This effect persists for odd values of N, resulting in a symmetric (assuming 50:50 beam splitter) (N+1)-modal distribution with the peaks of the distribution migrating toward their respective axes. The same structure of distribution occurs for even N as well; however, lines of contiguous zeroes do not occur. This is worth noting since a distribution like this is reminiscent of the well-known twin-Fock state input case, discussed earlier, in which the state after beam splitting are the so-called arcsine, or “bat,” states.66 It has long been known that the input twin-Fock state case, and states with similar distributions post-beam splitter, leads to sub-SQL sensitivity.39,51,52,66

1. Measurement resolution using parity-based detection

Once again we consider the use of photon-number parity performed on the output b-mode as our detection observable. Using in the input state of Eq. (98), the expectation value of the parity operator can be computed from Eqs. (61) and (62) to find

Π̂b(ϕ)N=(1)Ne|α|2j=N/2,|α|2(2jN)(2jN)!djN,jNj(2ϕ).
(101)

In the special case of N =0, we obtain the result found by Chiruvelli and Lee85 and discussed by Gao et al.,86 which will be expanded upon in greater detail in Sec. IV E 2. For small angles, ϕ0,Π̂b(0)01 but becomes narrower about ϕ=0 for increasing values of |α|, as seen in Fig. 9. The signal is not super-resolved in the usual sense of having oscillation frequencies scaling as Mϕ for integer M >1. However, compared with the corresponding result for output subtraction, we can see the signal for the parity measurement is much narrower, seen in Fig. 11. It is in this sense that Gao et al.86 interpreted the parity-based measurement to be super-resolved. Furthermore, for arbitrary N, the parity of the state is reflected by the expectation value of the parity operator evaluated at ϕ=0: Π̂b(0)N=(1)N. The peak (or trough) centered at ϕ=0 also narrows for increasing values of N. This can also be seen in Fig. 9.

2. Phase uncertainty: Approaching the Heisenberg limit

We begin with an analysis of the phase uncertainty obtained by computation of Eq. (63) with Ĵz as the detection observable. Plugging in the coefficients obtained using Eq. (61) for this choice of initial state and using Eq. (62) yield a phase uncertainty

ΔϕĴz=|α|2+N(1+2|α|2sin2ϕ)|(|α|2N)sinϕ|.
(102)

For N =0, we recover the well-known SQL. What is important to note about Eq. (102) is that for fixed |α|, the optimal noise reduction achievable is the SQL and occurs only for the case of N =0. For other values of N, the noise level rises to above the SQL. In particular, if the initial average photon numbers of the two modes are around the same value, i.e., |α|2N, the noise level becomes very high.

Parity-based detection fares quite a bit better. The phase uncertainty in this case is computed numerically using Eq. (63) with Π̂b as the detection observable. The phase uncertainty, along with the corresponding SQL and HL, ΔϕSQL=1/|α|2+N and ΔϕHL=1/(|α|2+N), respectively, is plotted for several different values of N in Fig. 10. The largest gain in sensitivity is achieved in the case of going from N =0, wherein the phase uncertainty falls along the SQL, to N =1, where the phase uncertainty is sub-SQL. The noise reduction approaches the HL for the increasing values of N.

One can also calculate the QFI using Eq. (57) and subsequent qCRB, which we remind the reader is independent of detection observable and depends solely on the choice of initial state, to find the minimum phase uncertainty attainable as follows:81 

Δϕmin=1|α|2+N(1+2|α|2).
(103)

This result follows the blue curve corresponding to parity-based phase sensitivity in Fig. 10 precisely for small values of the phase. It is worth pointing out that for small α, the phase uncertainty scales as 1/N, which follows the SQL.

So far, we have investigated the use of several different quantum states of light as our initial state to the MZI and have shown sub-SQL phase sensitivity in all cases when considering parity-based detection. Now, we will turn our attention to the case where strictly classical light is used. It is well known that this case will yield phase sensitivities at the SQL, which is the greatest sensitivity attainable using classical light, for the optimal choice of phase when considering intensity-difference measurements. So, the question is: what is gained by considering the use of photon-number parity? Here we will endeavor to shed some light on this question.

We start by considering coherent light in one of the input ports of the interferometer, making our initial state |in=|αa|0b, where once again the coherent state is given by |α=e|α|2/2n=0αnn!|n. Considering the transformation of Eq. (58) in which the beam splitters are of Ĵx-type, the state after the first beam splitter is given by (see Appendix A 3)

|αa|0bBS1|α2a|iα2b,
(104)

and the state after the accumulated phase shift in the b-mode (the Ĵz operation introduces an anti-symmetric phase shift of ±ϕ/2 in each mode, which can be treated as a phase shift of ϕ in one arm of the MZI without loss of generality) is given by

|α2a|iα2bϕ¯shift|α2a|iαeiϕ2b.
(105)

Finally, the state after the second beam splitter is

|α2a|iαeiϕ2bBS2|α2(1+eiϕ)a|iα2(1eiϕ)b.
(106)

Next, we compare the phase uncertainty obtained using two different detection observables: taking the intensity-difference between modes and performing photon number parity on one of the output modes.

1. Difference in output mode intensities

The intensity of a quantized field is proportional to the average photon number of the quantum state of the field,87In̂. Consequently, the difference between mode intensities at the output of the second beam splitter can be written as δIââb̂b̂. In terms of the SU(2) Lie algebra, this amounts to the expectation value of the operator 2Ĵz. Using Eq. (106), the mode intensities are given by n̂a(b)=n¯02(1±cosϕ), where n¯0=|α|2 (note that we are assuming a lossless interferometer such that n¯0=n̂a+n̂b is conserved), leading to the average value 2Ĵz=n¯0cosϕ. It is also straight-forward to show that (2Ĵz)2=n¯0(1+2Ĵzcosϕ). Combining these expressions and Eq. (63) yields a phase uncertainty

Δϕ2Ĵz=Ĵz2Ĵz2|ϕĴz|=1n¯0|sinϕ|,
(107)

which yields the SQL of phase sensitivity for the value of the phase ϕ=π/2, which means that the detection of small phase shifts, such as what would be expected in gravitational wave detectors, would have a high degree of uncertainty. Of course, one could compensate for this by inserting a π/2-phase-shifting element, which would have the effect of replacing sinϕ with cosϕ in Eq. (107). It is worth pointing out that ΔĴz does not vanish, which is an indication that the quantum fluctuations of the vacuum (the coherent state has the same quantum fluctuations as the vacuum) have the effect of limiting the precision of the phase-shift measurement. Next, we consider the use of parity, performed on the output b-mode, as our detection observable.

2. Parity-based detection

We define parity with respect to the b-mode as Π̂b=(1)b̂b̂ and likewise for the a-mode Π̂a=(1)ââ. From this, the corresponding expectation values and their first derivatives are found to be85 

Π̂a(b)=en¯0(1±cosϕ),
(108)
ϕΠ̂ab=±n¯0sinϕΠ̂ab.
(109)

Noting that Π̂a(b)21, the phase uncertainty can immediately be found from the error propagation calculus as follows:

ΔϕΠ̂a=1Π̂a2|ϕΠ̂a|=e4n¯0cos2ϕ/21n¯0|sinϕ|,
(110)
ΔϕΠ̂b=1Π̂b2|ϕΠ̂b|=e4n¯0sin2ϕ/21n¯0|sinϕ|.
(111)

The curves for both Π̂a(b) are displaced from one another by a π-phase shift, as evident by Eq. (108). This implies that while the peak for Π̂b occurs at ϕ=0, the peak for Π̂a occurs at ϕ=π. We can expand Eqs. (110) and (111) about their respective optimal phase values to find

ΔϕΠ̂a=1n¯0|ϕπ,ΔϕΠ̂b=1n¯0|ϕ0,
(112)

which is in agreement with the minimum phase uncertainty attainable for this choice of initial state, the qCRB. This can be quickly verified by calculating the QFI using Eq. (57) to immediately give the SQL Δϕmin=1/n¯0=ΔϕSQL.

For input coherent light, both the use of photon-number parity and intensity-difference measurements yield the same phase sensitivity: the SQL. What is gained by using photon-number parity is measurement resolution. In general, increased resolution tends toward yielding greater phase sensitivity. However, the distinction between the two is of particular importance when considering instances in which the bound limiting sensitivity is saturated, such as the case of input coherent light. In the broadest sense, measurement resolution refers to the ability to distinguish between two nearby peaks of the output signal, while sensitivity relates to how well the center of a lone peak can be found.88 Consequently, higher resolution results in narrower signal peaks, a property that we will be using heavily in Sec. VI. A demonstration using N00N states can be found in works by Dowling and collaborators89 in which the projection operator Σ̂N, realizable as photon-number parity and discussed in Sec. IV A of this paper, was considered. The distance between peaks in the output signal goes from λλ/N (i.e., the quantum lithography effect90), resulting in a sub-Rayleigh-diffraction limit resolution commonly referred to as “super-resolution.” The output signal using coherent light, given in Eq. (108) and plotted in Fig. 11, is not super-resolved in the usual sense of having Nϕ (N >1) oscillation frequency scaling.91 However, the narrowing of the peak when considering parity-based measurement as opposed to the usual method of taking the intensity-difference has been defined as a form of super-resolution by Gao et al.86 It is worth pointing out that for input states displaying quantum properties, such as all of the cases considered throughout the preceding subsections of Sec. IV, super-resolution tends towards providing a greater degree of phase sensitivity. On the other hand, super-resolution has also been demonstrated in the absence of entangled states using light exhibiting strictly classical interference.91 

One of the more substantial challenges facing experimental realizations of the interferometric schemes detailed throughout this section is the effect of losses and decoherence on phase sensitivity. Indeed, this section considered strictly ideal scenarios in which we assumed lossless interferometers were used. In an experimental setting, this is often an unrealistic assumption, and the effect of losses often limits the degree to which one may obtain high measurement precision. Schemes utilizing the quadratic scaling improvement of the HL over the SQL (i.e., an improvement by a factor of 1/n¯) have proven to be quite fragile in practice due to the effects of decoherence. It has been rigorously shown in the literature that even a small degree of noise in the system can degrade the HL into the SQL such that the quantum gain amounts to a constant factor rather than the quadratic factor suggested in the theory.92–95 One advantage of using strictly coherent light in interferometry is that coherent light is robust to losses (being a right eigenstate of the annihilation operator). A scheme in which strictly coherent light was used for sub-Rayleigh ranging remote-sensing and quantum illumination where homodyne detection was used.96 The authors that show such a scheme can yield longitudinal and angular super-resolution below the Rayleigh diffraction limit with SQL phase sensitivity. Parity-based measurements using coherent light have also been realized experimentally by Cohen et al.;97 this will discussed in greater detail in Sec. V B.

The phase sensitivity and resolution obtained in the presence of losses for several configurations discussed in this section using quantum states of light have been covered in the literature. The performance of the optical N00N state in a realistic setting in which losses were taken into account was considered by Gilbert et al.98 who found, using the detection observable Σ̂N (see again Sec. IV A), that an attenuation of the signal results in a dramatic degradation in estimation precision. They go on to show that for low medium transmittance, the sensitivity by the attenuated N00N state is worse than that of an equally attenuated separable N photon state. They conclude that for realistic experimental conditions, the N00N states not only fail to obtain Heisenberg-limited sensitivity but also fair worse than the SQL.

Work has been done on specifically addressing the effects of losses in parity-based schemes. Huver et al.99 and Jiang et al.100 considered the use of input m,m-entangled Fock states, which are a generalization of the N00N state with the form |m::m=12(|m,m+|m,m) and average photon number N¯=m+m. Defining the loss factor for each mode La(b)=1Ta(b), where Ta(b) is the transmission rate of the beam splitter modeling loss, the phase sensitivity can be expanded out for La=Lb=L as

Δϕ=1Δm+m+mΔmcscΔmϕL+OL2,Δmeven1Δm+m+mΔmsecΔmϕL+OL2,Δmodd,
(113)

where Δm=mm,m>m. In the limit of L0, Eq. 113 minimizes to Δϕ=1/Δm for the optimal value of phase,

ϕ2n1π2Δm,ΔmevennπΔm,Δmodd.
(114)

Taking losses into account, the SQL and HL are given by 1/N¯ and 1/N¯, respectively, where N¯=N¯(112La12Lb). Consequently, the SQL is beaten provided that the criterion Δm>m+m is satisfied. Jiang et al.100 provided an analysis of the sensitivity obtained under losses for a comparison between two states that yield sub-SQL sensitivity with no losses present: the |6::0N00N state and the |8::2 entangled Fock state, assuming equal losses in both modes. For losses up to 10%, both states yield sub-SQL sensitivity with the |6::0 state, providing greater sensitivity (both in general and as an improvement over the SQL). For high loss (>25%), the |6::2 entangled Fock state outperforms the N00N state, though both perform worse than their respective SQLs. It was also shown by Huver et al.99 that with a certain choice of detection observable, the mm-entangled Fock states outperform the N00N states (and the SQL) no matter how high of loss in the system. Furthermore, they show the trade-off between phase sensitivity and visibility (i.e., detection visibility for observable Ô,Vdet=Ôϕ=0) in the face of losses when considering different permutations of the superposition state for a given N=mm. They conclude that while |m::m states are more robust, they have loss-induced limitations: for 70% loss in one arm of the interferometer and perfect transmission in the other, they find the visibility drops to 10% for many |m::m states.

By comparison, work done by Joo et al.54 showed that entangled coherent states yield better phase sensitivity and resolution in lossy conditions compared to N00N and twin-Fock (bat) states, though the advantage is more substantial for modest average photon numbers; more specifically, they show this advantage applies to parity measurements for a narrow transmission window 0.995T1 (where T is the transmission rate of the beam splitter used to model losses). They also show that states of the form |ψ=N+(|α,01,2+|0,α1,2), where |α is a coherent state, outperform N00N states of the same average photon number for nonlinear phase shifts described by Û(k,ϕ)=exp[i(b̂b̂)k] with k =2 when small losses are considered.101 The work by Joo et al.54 was expanded on by Knott et al.102 who devised a read-out scheme utilizing the robust nature of the ECS using currently existing technologies capable of attaining sensitivities close to theoretical predictions, even with loss taken into account.

As emphasized in the Introduction, the parity operator, whether in the context of atomic (or spin) systems or in the context of photon number, is a Hermitian operator and is, therefore, an observable but one that does not have a classical analog. This being the case, the question becomes how can parity be measured or least determined through some measurement process? The obvious way to do that is through counting the number of atoms103 in the excited (or ground), something that can be done through a process known as electron shelving, or counting the number of photons in an optical field and raising –1 to that power. Of course, this means that the counting itself must be possible with a resolution at the level of a single atom or a photon: a challenging prospect for the cases where the number or atoms or photons is large. Ideally, one would like to be able to determine parity directly, by which we mean through a technique with a readout of ±1 without directly measuring the number of atoms or photons. Some work has been done on measuring parity in multi-qubit systems in circuit QED104 and microwave cavity-QED105,106 and using two-zone linear Paul traps.107 The parity of atomic ensembles can also be determined using quantum nondemolition (QND) measurements. It is worth noting that QND measurements can be used to measure the number of atoms or photons as well, but what we have in mind is the measurement of parity wherein the detector does no counting at all. As we will show, these two mentioned methods of measuring parity, while ultimately yielding the same parity values, amount to different kinds of measurements if used to perform, for example, state-projective measurements.

Here, we will discuss several methods of performing a QND measure of atomic parity through the use of coupling between the atomic system with an ancillary subsystem. We begin by defining the even/odd atomic parity projection operators. That is, we define the operators that project the atomic state into even/odd numbers of atoms in the ground state. These projectors are given by

Π̂even=m=jjcos2[(jm)π2]|j,mj,m|,
(115)
Π̂odd=m=jjsin2[(jm)π2]|j,mj,m|,
(116)

satisfying the POVM condition iΠ̂i=Î. Note that j + m represents the total number of atoms found in the excited state, while jm represents the total number of atoms found in the ground state. Consequently, j±m takes on only integer values such that for a given j±m, only one of the projectors will be nonzero. These projection operators can be used to express the atomic parity operator as

Π̂=iλiΠ̂i=Π̂evenΠ̂odd=m=jjcos[(jm)π]|j,mj,m|,
(117)

where λi are the eigenvalues of the parity operator, respectively. From Eqs. (115) and (116), it follows that

Π̂even=12(1+Π̂),
(118)
Π̂odd=12(1Π̂),
(119)

where Π̂odd+Π̂even=1. We note here that “parity” is defined with respect to the number of atoms found in the ground state as described in Sec. II. That is, “even” parity denotes an even number of atoms found in the ground state and “odd” parity denotes an odd number of atoms found in the ground state. Let us move on to consider a couple of different cases.

1. Coupling to an ancillary atomic system

Given an atomic system in which we wish to measure, denoted by the a-mode, we introduce an ancillary atomic system (occupying the b-mode) prepared in an atomic coherent state |τ,jbb and the coupling Hamiltonian and corresponding evolution operator,

Ĥ=χ(Ĵ0a+Ĵza)(Ĵ0b+Ĵzb)Û=eitĤ/,
(120)

where Ĵ0|j,m=j|j,m (see Appendix A 2) and where χ is the coupling strength. This interaction Hamiltonian can be thought of as the atomic analog to the field coupling cross-Kerr interaction. We define the initial state as

|initial=|ψa|τ,jbb=m=jajaCm(ja)|ja,ma(1+|τ|2)jbm=jbjb(2jbjb+m)τjb+m|jb,mb.
(121)

Note that we wish to make a projective measurement on the ancillary atomic system (b-system) in order to determine the atomic parity of the target system (a-system). The final state is, then,

|final=eitĤ/|initial=m=jajaCm(ja)|ja,ma|τeitχ(ja+m),jbb.
(122)

For the choice of χtπ, this becomes

|final=|ψevena|τ,jbb+|ψodda|τ,jbb=m=jaja[cos2((ja+m)π2)|τ,jbb+sin2((ja+m)π2)|τ,jbb]Cm(ja)|ja,m=m=jajaCm(ja)|ja,ma|Φmb.
(123)

The system is entangled such that a projection onto the ancillary atomic system yielding the ACS characterized by parameter τ will project out odd atomic states in the target system and projection onto the ACS characterized by parameter +τ will project out even atomic states in the target system. What we require is a means of determining which state the ancillary atomic system is in. Let us assume that the ancillary atomic coherent state is prepared such that τ=1; this corresponds to a separable state in which all atoms of the ancillary atomic system are in the same superposition state,

eiπ2Jŷ|j,j=|1,j=|ψN=12N/2(|g|e)N.
(124)

Similarly, we can also define the phase-rotated atomic coherent state

eiπ2Jŷ|j,j=|1,j=|ψ+N=12N/2(|g+|e)N.
(125)

With this, we can rewrite the state |Φmb, assuming τ=1, as

|Φmb=cos2((ja+m)π2)|1,jbb+sin2((ja+m)π2)|1,jbb=cos2((ja+m)π2)eiπ2Jyb̂|jb,jbb+sin2((ja+m)π2)eiπ2Jyb̂|jb,jbb.
(126)

It is important to note that only a single term in the superposition state Eq. (126) can be present at a time. Performing a single π/2-pulse yields the ancillary atomic system in the state

eiπ2Ĵyb|Φmb={cos2((ja+m)π2)|jb,jbb,ja+mevensin2((ja+m)π2)|jb,jbb,ja+modd,
(127)

where ja+m corresponds to the number of excited atoms. A state-reductive measurement performed on the ancillary atomic system would inform the experimenter of the parity of the target system without explicitly providing the value of ja+m. Note that this works for an arbitrary value of jb; the ancillary atomic system can be as small as a single atom. Next, we will discuss a method involving a known coupling Hamiltonian readily capable of being experimentally implemented.

2. Coupling to a field state

Some work has been done in developing a QND measure of photon number parity and a means of projecting out parity eigenstates in optical fields.108 Similarly, QND parity measurements have beenperformed in the context of error correction for a hardware-efficient protected quantum memory using Schrödinger cat states109 and a fault-tolerant detection of quantum error.110 Parity measurements have also been utilized in the entanglement of bosonic modes through the realization of the eSWAP operation111 using bosonic qubits stored in two superconducting microwave cavities.112 Here, we will consider a similar scheme in which one couples the target atomic system to an ancillary field state. The interaction Hamiltonian coupling the two subsystems is given by

Ĥ=χb̂b̂ĴzÛ=eitĤ,
(128)

where {b̂,b̂} are the usual boson annihilation and creation operators, respectively. Consider an ancillary field state, given by the usual coherent state

|αb=e12|α|2n=0αnn!|nb.
(129)

Following the same procedure as Sec. V A 1 and setting χt=π, this coupling Hamiltonian yields the final state

|final=Û|initial=m=jjCm(j)|j,ma|(1)mαb.
(130)

The final entangled state will depend greatly on the value of j. More specifically, noting N=2j, this simplifies to

|final={|ψodda|αb+|ψevena|αb,j=0,2,4,,N=0,4,8,|ψevena|αb+|ψodda|αb,j=1,3,5,,N=2,6,10,|ψevena|iαb+|ψodda|iαb,j=12,52,92,,N=1,5,9,|ψodda|iαb+|ψevena|iαb,j=32,72,112,,N=3,7,11,.
(131)

In order for this procedure to be applicable, the total number of atoms involved must be known with certainty as the resulting superposition state, and the subsequent detection scheme will greatly depend on having this information. As a proof of concept, let us consider the case where the number of atoms is N=0,4,8,. As per Eq. (131), the final state is given by |final=|ψodda|αb+|ψevena|αb. Mixing the field at a 50:50 beam splitter with an equal-amplitude phase-adjusted coherent state |αc. Assuming a 50:50Ĵy-type beam splitter of angle π/2 (see Appendix A 3), the beam splitter (labeled BS 2 for reasons that will become clear) results in the transformation

|αb|αcBS2|2αb|0c,|αb|αcBS2|0b|2αc.
(132)

The total state is now given by

|finalBS2=|ψodda|0b|2αc+|ψevena|2αb|0c,
(133)

in which all photons are in either one optical mode or the other. A simple detection scheme informs the experimenter of the parity of the target atomic system. Upon a state reductive measurement performed on the optical modes, the atomic system becomes

|Ψa=|ψevenabmodeclick|ψoddacmodeclick.
(134)

The optical fields |αb and |αc used in this procedure can be derived from the same beam if one starts with the state |2αc and uses a 50:50Ĵy-type beam splitter of angle +π/2 such that

|2αb|0cBS1|αb|αc.
(135)

After beam splitting, the b-mode coherent state can be coupled with the target atomic system before being photomixed with the coherent state |αc at the second beam splitter. This method would allow one to determine the parity of the atomic system without explicit knowledge of the number of excited atoms j + m.

As mentioned above, one way to obtain photon number parity is to perform photon number counts ncount and raise –1 to that power: Πmeasured=(1)ncount. That raises the issue of the general lack of photon number counting techniques having resolution at the level of one photon. Yet, there are ways around this problem. In fact, an experiment to detect a phase shift through optical interferometry using coherent light and parity measurements was performed a few years ago by Eisenberg's group at Hebrew University.97 Coherent light is particularly advantageous as it, by definition, does not suffer decoherence due to losses as it is a right eigenstate of the annihilation operator (i.e., â|α=α|α). Recall from above, and from Gao et al.,86 that parity measurements performed in this context are predicted to result in phase-shift detections that are super-resolved even though the phase sensitivity is at the SQL. The experiment of Cohen et al.97 confirms this.

In the experiment reported, the phase shift to be detected was set to be π. If we look at Eqs. (110) and (111), and based on the discussion to follow, we see that with our labeling scheme, we should be performing parity measurements on the output a-mode. That is, for ϕ near π, we have Π̂a=en¯01+cosϕe(n¯0/2)ϕπ2, where n¯0 is the total number of photons within the interferometer, which peaks at unity for ϕπ. In this limit Δϕ1/n¯0, the SQL.

These authors also measured a different kind of parity in which the outcomes are either no photons detected, as described by the projector |00|, or any number of photons detected but without resolution, n=0|nn|=I|00|. Defining Ẑ=|0a0|, from the output state given in Eq. (106,) we find the probability of there being no photon detections to be

P0ϕ=Ẑout=e(n¯0/2)1+cosϕϕπe(n¯0/4)ϕπ2.
(136)

From the error propagation calculus, one easily finds that for ϕπ, the phase uncertainty is given by Δϕa1/n¯0, which is also shot-noise limited. Evidently, the two observables give similar results for the case of input coherent light.

The experimental setup used by Cohen et al.97 is detailed in Fig. 12. The coherent inputs are produced by a Ti:sapphire laser with a calibrated variable neutral density filter (NDF) employed to control the average photon numbers. The MZI consists of two polarizers at 45°, and the phase-shift is produced by tilting the calcite crystal. The output mode to be measured is band pass filtered (BPF) and spatially filtered by a single mode fiber (SMF). This mode was detected by a silicon photomultiplier consisting of an array of beam splitters and single-photon detectors. Such an arrangement for photon counting with a resolution at the single-photon level is described in Kok and Lovett.113 See also the review of photon detection by Silberhorn.114 For the fine details of the experiment, we refer the reader to the paper by Cohen et al.97 

The expectation values of the parity operator Π̂a (in our notation) and Ẑ are plotted in Fig. 13 for several values of the average photon number. The predicted narrowing around the peak at ϕ=π is evident, though the degradation of the visibility for a high average photon number is due to the imperfect visibility of the interferometer itself and dark counts. Plots of the corresponding phase uncertainties for n¯0=200 are found in Fig. 14. Note that the peak at ϕ=π is wider for P0 than for parity. Finally, in Fig. 15, the results are summarized for resolution and sensitivity showing that, in part (a), parity yields greater resolution than P0. That is, for the parity-based measurement, resolution reaches λ288, where λ=780nm is the wavelength of the laser light, which is a factor of 1/144 improvement (smaller) over the Rayleigh limit λ/2. The resolutions in the two cases differ by the expected amount 2, On the other hand, as can be seen in part (b) of Fig. 15, parity has a larger deviation from the SQL as compared to P0. In fact, up to 200 photons, the sensitivity of P0 is maintained at the SQL.

We mention that in recent years, considerable effort has been directed toward the development of photon-number-resolving detectors. These include superconducting transition edge detectors,115 loop detectors,116 detection by multiplexing,117 an array of avalanche photodiodes (APDs),118 and an array of single-photon detectors.119 This list by no means exhausts the literature on this topic. In principle, one could avoid photon counting entirely and instead perform a quantum nondemolition measurement of photonic parity in the same vain as discussed in Sec. V A pertaining to atomic parity. This possibility has been discussed by Gerry et al.108 as an extension of a technique proposed for the QND measurement of photon number.120 The problem with these techniques is that they depend on a cross-Kerr interaction with a large third order nonlinearity, which does not exist in optical materials.

However, there is yet another possibility first mentioned by Campos et al.66 in their analysis of interferometry with twin-Fock states. Recall that the Wigner function is given by W(α)=2πD̂(α)Π̂D̂(α); it follows for α = 0 that Π̂π2W(0), thus showing that the expectation value of the parity operator can be found directly by value of the Wigner function at the origin of phase space. Now, the Wigner function can be constructed by the techniques of quantum state tomography, which uses a balanced homodyning and the inverse Radon transformation to perform filtered back projections.6 However, one does not need the entire Wigner function: only its value at the origin of phase space is required. Plick et al.121 have examined this prospect in detail for Gaussian states of light.

We close this section with a brief discussion of the prospect of using photon-number resolved detection as a basis for a quantum random number generator (QRNG) (for a discussion on QRNGs, see Herrero-Collantes and Garcia-Escartin122 and references therein) with laser light. As is well known, laser light shone on a 50:50 beam splitter cannot be the basis of a QRNG as a matter of principle no matter the intensity of the light. The ideal source of light is a single photon generated on demand, where the photon falls on the beam splitter and then appears randomly, i.e., which is detected randomly, in one or the other output beams. Since on-demand single photon generation is problematic in practice, QRNG devices use weak (coherent) laser light, with the idea being that on average only one photon is present at any given time. This is a fiction, of course, as the probability of having two photons at a time is not zero due to the photon bunching effect.87 Now, consider light prepared in a coherent state |α by a well phase-stabilized laser. The expectation value of the parity operator is

α|Π̂|α=e|α|2n=0|α|2nn!(1)n=en¯n=0(n¯)nn!=e2n¯,
(137)

where n¯=|α|2. It is easy to show numerically that Π̂0 as n¯ becomes large. Another way to put it is that the probabilities of getting even or odd photon-number parity must be equal to 1/2 for sufficiently large n¯. For arbitrary α, these probabilities are

Pe=α|P̂e|α=e|α|2m=0|α|4m(2m)!=en¯m=0n¯2m(2m)!=en¯coshn¯,
(138)
Po=α|P̂o|α=e|α|2m=0|α|4m+2(2m+1)!=en¯m=0n¯2m+1(2m+1)!=en¯sinhn¯,
(139)

where the even and odd photon number projectors are given, respectively by P̂e=m=0|2m2m| and P̂o=m=0|2m+12m+1| and where

Π̂=P̂eP̂o=PePo=e2n¯.
(140)

The average photon number need not be very high for the probabilities in Eqs. (138) and (139) to equalize close to 1/2. In fact, for n¯=9, n¯=9 one has Pe=Po=0.5 after retaining seven decimal places. For n¯=16, this is true to thirteen decimal places. Thus, a viable QRNG could be implemented with efficient parity detection in a regime where the average photon number need not be very high.

Finally, we mention that laser light prepared in a pure coherent state is not necessary: a statistical mixture of coherent states of the same amplitude will suffice. Consider the phase-averaged coherent states given by the density operator

ρ=12π02πdϕ|reiϕreiϕ|,
(141)

where r=|α|=n¯. It is easily shown that

Π̂=Tr[ρΠ̂]=12π02πdϕn=0(1)n|n|reiϕ|2=en¯n=0(n¯)nn!=e2n¯,
(142)

which is identical to Eq. (137).

In Sec. VI we will discuss parity-based measurements in the context of multi-atom spectroscopy with a particular interest in the construction of high-precision atomic clocks. We will consider both the classical case in which the atoms are initially unentangled and prepared in the same initial state and the case in which the atoms are initially entangled and spin-squeezed. A comparison between parity and population difference is also made in the context of enhanced measurement resolution.

Previous sections, specifically Secs. II, III, and IV, have discussed the benefit of parity-based detection in the context of measurement resolution in two-mode optical systems as it pertains to interferometry. Enhanced measurement resolution may also be beneficial in the construction of highly precise multi-atom atomic clocks. Collections of two level atoms have seen use in quantum information123 (in the context of cold atoms) and quantum metrology.124 In this section, we will endeavor to show how parity-based measurements yield finer resolution for increasingly large numbers of atoms, whereas simply considering the population difference between ground and excited states does not. While it is true that for entangled states, this enhanced resolution tends toward providing greater phase sensitivity, this will not be directly discussed in this section: the emphasis of this section is on enhancing measurement resolution through the use of parity as a detection observable. We will initially restrict the discussion to unentangled ensembles of atoms, where each atom is initially prepared in the same atomic state. We will, then, consider the case of an initially entangled (spin-squeezed) ensemble. The latter cases provide even greater resolution when parity measurements are performed.

Consider a general atomic state expressed in terms of a superposition of all Dicke states |j,m, where the Dicke states are defined relative to the individual atomic states as per Sec. A 1, given by

|ψπ/2=m=jjCm(j)|j,m.
(143)

Let Eq. (143) describe the state after the first π/2-pulse in a Ramsey interferometer, where Cm(j) are the probability amplitudes for obtaining each state |j,m. The state after the time evolution and the final π/2-pulse is given by

|ΨF=eiπ2ĴyeiϕĴz|ψπ/2=m=jj[m=jjCm(j)eiϕmdm,mj(π2)]|j,m=m=jjC̃m(j)|j,m,
(144)

where C̃m(j) are defined within the square brackets of Eq. (144) and dm,mj(β) are the usual Wigner-d matrix elements given by j,m|eiβĴy|j,m. We consider the measurement resolution obtained for this general case when using Ĵz as our detection observable. Physically, this corresponds to taking the difference between the number of atoms in the excited and ground states, respectively. The optical analog would be taking the difference in intensity between the two output modes of an MZI. The expectation value of Ĵz is given by

Ĵz=m=jjp=jjCp(j)*Cm(j)eiϕ(mp)×[m=jjmdm,pj(π2)dm,mj(π2)]=m=jjp=jjCp(j)*Cm(j)eiϕ(mp)×Γm,p(π2),
(145)

where the function Γm,p(λ) is defined by Eq. (B14). Assuming that the coefficients Cm(j) are real,125 we obtain the result

Ĵz=γ(j)cosϕ+α(j),
(146)

where

γ(j)=2m=jj1m=jjCm+1(j)Cm(j)mdm,m+1j(π2)dm,mj(π2),
(147)
α(j)=m=jjm=jj|Cm(j)dm,mj(π2)|2m.
(148)

A cursory glance informs α(j)0 for all states by symmetry. In addition to this, γ(j) can be simplified by resolving the sum over m in Eq. (147) to find γ(j)=Ĵ+in as one would expect. For the case of an initial atomic coherent state, Eq. (147) yields the expected result of Eq. (10),

ĴzACS=jcosϕ,
(149)

which we plot in Fig. 16 for several values of j. Similarly, an expression can be found for Ĵz2 by using Eq. (B15), which yields

Ĵz2ACS=Ω(j)cos2ϕ+κ(j)cosϕ+β(j),
(150)

where

Ωj=2m=jj2m=jjCm+2jCmjm2dm,m+2jπ2dm,mjπ2,κj=2m=jj1m=jjCm+1jCmjm2dm,m+1jπ2dm,mjπ2,βj=m=jjm=jj|Cmjdm,mjπ2m|2.
(151)

Note that in neither case do we see a narrowing of the peak regardless of the state used. The phase dependency goes as a cosine function regardless of the state coefficients and the number of atoms used. This particular detection observable cannot yield enhanced measurement resolution for any initial state taken under consideration.

Let us assume that our state is initially prepared such that the state after the first π/2 pulse is once again given by Eq. (143). The action of the Ramsey interferometer transforms the state to

|ΨF=eiπ2ĴyeiϕĴz|ψπ/2=m=jjm=jjCm(j)dm,mj(π2)eiϕm|j,m,
(152)

where the Wigner-d matrix elements dm,mj(β) are discussed in  Appendix B.

Here, we reintroduce the atomic parity operator discussed in Sec. II as Π̂g=(1)Ng=exp[iπ(Ĵ0Ĵ3)], defined with respect to the number of atoms found in the ground state Ng, where Ĵ0|j,m=j|j,m (see Appendix A 2). We can likewise define atomic parity in terms of the number of atoms found in the excited state as Π̂e=(1)Ne=exp[iπ(Ĵ0+Ĵ3)], where Ne is the number of atoms in the excited state. The relationship between the two expressions, Π̂g=(1)2jΠ̂e, is clear to see when considering the total number of atoms can be expressed as N=Ne+Ng=2j.

The expectation value of the parity operator using the final state described by Eq. (152) can be expressed as

Π̂g=m=jjp=jjCmjCpj*eiϕmp×[m=jj1jmdm,mjπ2dm,pjπ2],=m=jjp=jjCmjCpj*eiϕmpFm,pπ2,
(153)

where the term in the square brackets is given by Eq. (B13). Noting that Fm,p(π2)=(1)2jδm,p, we find for an arbitrary state

Π̂g=(1)2jm=jjCm(j)*Cm(j)e2iϕm.
(154)

To aid in motivating the use of atomic parity as a detection observable, let us first revisit the result for the most ideal of cases, that is, we analyze the measurement resolution obtained in a Ramsey interferometer when the state after the first π/2-pulse is the atomic N00N state, given by

|ψπ/2=|ψN00N=12(|j,j+eiθ|j,j),
(155)

including an arbitrary relative phase θ. The representation of a collection of two-level atoms in terms of the SU(2) Lie algebra, known as the Dicke model, is described in detail in  Appendix A. This can be expressed as a finite sum over all Dicke states as

|ψN00N=m=jjΛm(j)|j,m,
(156)

where the coefficients Λm(j) are given by

Λm(j)=12(δm,j+eiθδm,j).
(157)

This state is of interest as it is the atomic analog of the optical N00N state |ψN00N=12(|N,0+eiθ|0,N) (discussed in Sec. IV A), which yields the lower bound on phase uncertainty in quantum optical interferometry: the Heisenberg limit. This follows from the heuristic uncertainty relation between the photon number and phase ΔϕΔN1, where the N00N state maximizes the uncertainty in photon number, that is, ΔN=NΔϕ1/ΔN=1/N. The action of the interferometer is once again given by Eq. (152) where we assume that some means of experimentally generating the atomic N00N state is employed prior to free evolution. It is, then, a straightforward exercise to calculate the expectation value of the parity operator using Eq. (154) to find

Π̂gN00N=(1)Ncos(Nϕ+θ),
(158)

where N=2j, the total number of atoms. Here, the relative phase θ acts as a phase shift of the interference pattern, which now oscillates at the enhanced rate Nϕ. Finer measurement resolution, characterized by a narrowing of the peak of Π̂ at ϕ=0 for increasing values of N, tends toward enhancing phase sensitivity in interferometric measurements. This state would serve as the ideal since it is maximally entangled and yields the greatest measurement resolution for a given detection observable. The problem, however, is that atomic N00N states are incredibly difficult to make. We endeavor to find a means of enhancing measurement resolution by considering parity-based measurements performed on both unentangled ensembles and initially entangled atomic systems.

1. Unentangled ensemble of atoms

Let us assume that our state is initially prepared with all atoms in the ground state, that is |in=|j,j. The action of the Ramsey interferometer transforms an initial state |ψ by

|ΨF=eiπ2ĴyeiϕĴzeiπ2Ĵy|in=eiπ2ĴyeiϕĴz|τ=1,j,
(159)

where eiπ2Ĵy|j,j=|τ=1,j is the atomic coherent state in which all of the atoms in the ensemble are in the superposition state 12(|g|e), that is,

|τ=1,j=12N/2(|e|g)N=2j=2jm=jj(2jj+m)1/2(1)j+m|j,m.
(160)

Let us recover the known result for an initial atomic coherent state. Taking the state coefficients of Eq. (160) and plugging them into Eq. (154), we find

Π̂gACS=4jm=jj(2jj+m)ei2ϕm(1)4j=dj,jj(2ϕ)=cos2jϕ=cosNϕ.
(161)

This is plotted in Fig. 17. Unlike the case in which we take the population difference, larger numbers of atoms yield greater measurement resolution, characterized by a narrowing of the peak about ϕ=0. This effect can be demonstrated by making the approximation cosNϕeNϕ2/2 for small ϕ, which is a Gaussian that narrows for increasing N. We remind the reader that while the parity-based measurement yields greater resolution, this does not translate to better phase sensitivity beyond the SQL as the initial state is classical and therefore bounded by the SQL.

2. Entangled atomic ensemble

Now, let us next consider the case where we act with the Ĵz operator after the first π/2-pulse. The resulting state is both entangled at the atomic-state level and is known to exhibit spin-squeezing, which has been shown to reduce projection noise in high-precision population spectroscopy. In this case, the initial state is transformed by

|ΨF=eiπ2ĴyeiϕĴzĴzeiπ2Ĵy|in=eiπ2ĴyeiϕĴz|τ=1,j;1,
(162)

where the state

|τ=1,j;1Ĵz|τ=1,j=(22j1j)1/2m=jj(2jj+m)1/2(1)j+mm|j,m.
(163)

Once again, we use Eq. (154) to find the expectation value of the parity operator

Π̂gĴzopACS=22j1j1m=jj2jj+mm2ei2ϕm=2πΓj+12Γj+2ei2ϕ[3F22,2,1j;1,j+2;ei2ϕ+ei4ϕ3F22,2,1j;1,j+2;ei2ϕ,
(164)

where 3F2 is the hypergeometric function given generally by Eq. (B6). Simplifying for several different values of j=N/2 yields

j=1/2Π̂gĴzopACS=cosϕ,j=1Π̂gĴzopACS=cos2ϕ(2cos2ϕ)j=3/2Π̂gĴzopACS=cos3ϕ(32cos2ϕ),j=2Π̂gĴzopACS=cos4ϕ(43cos2ϕ),j=5/2Π̂gĴzopACS=cos5ϕ(54cos2ϕ),j=3Π̂gĴzopACS=cos6ϕ(65cos2ϕ),
(165)

from which we can extrapolate for N1,

Π̂gĴzopACS(N)=cosNϕ[N(N1)cos2ϕ].
(166)

We see that for the case of an initially entangled atomic ensemble, there is greater measurement resolution with the increasing number of atoms over the initially unentangled ACS case, except for the case of j =1/2 (N =1) in which case both initial states yield the same resolution. This is demonstrated explicitly in Fig. 18 for the value of N=2j=6, where we can see a narrowing of the peak about the phase origin for the same number of atoms when considering the initially entangled case. This trend of enhanced resolution persists for q >1 applications of Ĵz prior to free evolution, which further entangles the initial atomic ensemble. For the general case of q applications of Ĵz prior to free evolution, or |inĴzq|ζ=1,j, we can write the expectation value of the parity operator (unnormalized) as Π̂gĴzopACS(q)m=jj(2jj+m)m2qei2ϕm. Computations show greater resolutions obtained for the same (large) value of j for increasing values of q, shown in Fig. 19.

In this Review, we discussed the prospect of using parity-based measurements in phase estimation. In the context of Ramsey spectroscopy, atomic parity is defined in terms of number of atoms found in the excited state, while in the context of optical interferometry, it is defined with regard to the average photon number in one output mode of the interferometer. We provide a brief discussion on the basics of phase estimation and demonstrate how parity-based detection saturates the quantum Cramér–Rao bound for path symmetric states in quantum optical interferometry, making parity the optimal observable for interferometric measurements. We highlight the use of parity-measurements for several different initial states comprising both quantum and classical light and show agreement between the phase uncertainty when considering parity-based measurements and the quantum Cramér–Rao bound. For all quantum states considered, sub-SQL phase sensitivity is achieved when the parity-based measurement is considered. We also note that the method of photon-number parity detection has recently been discussed in the context of SU(1, 1) interferometry.126 

We also discuss the experimental efforts made toward measuring photon number parity in quantum optical interferometry using classical (coherent) light where the comparison is made to another dichotomic operator corresponding to a “click”/“no click” detection scheme. QND measures of atomic parity are also considered through coupling to both an ancillary atomic system that can be as arbitrarily small as a single atom and coupling to a light field prepared in a coherent state. We highlight that the latter case does not require a large third order nonlinearity and in fact utilizes a readily available means of coupling.

We return to atomic spectroscopy in the closing section where the emphasis is placed on the enhanced measurement resolution obtained through the parity-based measurement compared to the usual method of taking the population difference between the two internal degrees of freedom available to each atom. We show in the latter case that for arbitrary N atom initial states, the Ramsey technique cannot yield super-resolved measurements, while the former shows greater resolution for increasingly large numbers of atoms N. Of the cases discussed, we show that an initially entangled atomic ensemble yields greater resolution for the same number of atoms than the case in which all atoms are initially unentangled and in the same atomic state.

The authors dedicate this paper to the memory of Jonathan P. Dowling, whose body of work had a tremendous impact on the fields of quantum optical interferometry and quantum metrology.

R.J.B. acknowledges support from the National Research Council Research Associate Program (NRC RAP). C.C.G. acknowledges support under the AFRL Summer Faculty Fellowship Program (SFFP). P.M.A. and C.C.G. acknowledge support from the Air Force Office of Scientific Research (AFOSR). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Air Force Research Laboratory (AFRL).

The data that support the findings in this paper are available from the corresponding author upon request.

1. The Dicke states

In what follows, we prepare a brief review of the so-called Dicke states,127 which provides a representation of a collection of N two-level atoms in terms of the SU(2) “angular momentum” states |j,m, where j=N/2,m[j,j] and where the internal basis states of the ith atom are {|ei,|gi}. The quantity j in this context is sometimes called the cooperation parameter. Only states symmetric with respect to the permutations of the N atoms come into play in the cases considered here. That is, the direct product of the spaces of the two-level atoms, as given by 2122232N, has one symmetric and generally several anti-symmetric decompositions, but the anti-symmetric states are dynamically decoupled from the symmetric states via the selection rules. To demonstrate the construction of the symmetric states and connect them to the angular momentum states, we proceed by example where we use the ladder operators given in terms of the individual atoms as

Ĵ+=i=1N|eig|,Ĵ=i=1N|gie|
(A1)

and use the relations

Ĵ±|j,m=j(j+1)m(m±1)|j,m±1
(A2)

acting on the collective angular momentum states. In actuality, multiple applications of the raising operator are used starting from the state in which all atoms are in their ground state.

For two atoms, we have j=N/2=1. The state in which both atoms are in their ground state is given by m=j=1 such that |j=1,m=1=|g1|g2 or more simply

|1,1=|g1|g2.
(A3)

Applying Ĵ+ to both sides and using the above relations, we find

|1,0=12(|e1|g2+|g1|e2),
(A4)

and applying it again yields

|1,1=|e1|e2.
(A5)

We could have obtained the same set of states had we started with the state |1,1=|e1|e2 and laddered down to |1,1=|g1|g2 through consecutive applications of Ĵ. These are the symmetric states, or Dicke states, for the case of two atoms. Likewise, one can see that for three atoms, j=N/2=3/2, j=N/2=3/2 and the Dicke states are

|32,32=|g1|g2|g3,|32,12=13(|e1|g2|g3+|g1|e2|g3+|g1|g2|e3),|32,12=13(|e1|e2|g3+|g1|e2|e3+|e1|g2|e3),|32,32=|e1|e2|e3.
(A6)

The process can be continued for any number of atoms. Note that the extremal states |j,±j are (separable) product states, whereas all other states m±j are entangled at the atomic level.

a. The atomic coherent state

Two definitions of the atomic coherent state have been given in the literature: one by Radcliffe128 and another by Arrechi et al.20 Both definitions involve the action of the SU(2) “displacement” operator R(θ,φ)=eζĴ+ζ*Ĵ on the extremal states j,±ĵ, where ζ=θ2eiφ. The angles θ,φ are the usual angles that parameterize the Bloch sphere with 0θπ and 0φ2π where we follow Arrechi et al. in taking θ = 0 to be the south pole of the sphere. This operator acts on the |j,j state in the case of the Radcliffe definition and on the |j,j state in the case of Arrechi et al. For these two separate definitions, we require two different orderings of the “disentanglement” expressions of the operator R(θ,φ). These are

eζĴ+ζ*Ĵ=eτĴ+eln1+|τ|2Ĵzeτ*Ĵeτ*Ĵeln1+|τ|2ĴzeτĴ+,
(A7)

where τ=eiφtanθ2. In starting with the state |j,j, we use the upper expression on the right-hand side of Eq. (A7) to arrive at

|τ,j=(1+|τ|2)jeτĴ+|j,j=(1+|τ|2)jm=jj(2jj+m)1/2τj+m|j,m,
(A8)

and using the bottom line of the right-hand side of Eq. (A7), we obtain

|τ*,j=(1+|τ|2)jeτ*Ĵ+|j,j=(1+|τ|2)jm=jj(2jj+m)1/2(τ*)j+m|j,m.
(A9)

Note that under the replacement ττ*, these states are the same.

For the special case used throughout Secs. II and VI where the rotation operator eiπ2Ĵy acts on the states |j,±j states, we can set

R(θ,φ)=eζĴ+ζ*Ĵeiπ2Ĵy,
(A10)

informing the value ζ=π/4 and consequently the angles θ=π/2 and φ=π. Thus, we have τ=1 and τ*=1, which, for example, is demonstrated in Eq. (15). More simply, one can insert a full set of states to find

ei(π/2)Ĵy|j,±j=m=jj|j,mj,m|ei(π/2)Ĵy|j,±j=m=jjdm,±jjπ2|j,m=1+|τ|2jm=jj2jj+m±1j+m|j,m,
(A11)

where the matrix elements dm,±jj(π2) are the Wigner-d matrix elements found in Rose129 and given for this special case in Eq. (B11).

2. The Schwinger realization of SU(2)

Consider a two mode field with creation and annihilation operators satisfying the usual boson commutation relations [âi,âj]=[âi,âj]=0 and [âi,âj]=δi,j. One can introduce the Hermitian operators

Ĵx=12(â1â2+â2â1),Ĵy=i2(â1â2â2â1),Ĵz=12(â1â1â2â2),
(A12)

and N̂=12(â1â1+â2â2), satisfying the commutation relations of the Lie algebra of SU(2),

[Ĵi,Ĵj]=iĴkεi,j,k.
(A13)

The Casimir invariant of the group can be found to be of the form Ĵ2=Ĵx2+Ĵy2+Ĵz2=N̂2(N̂2+1). Note that N̂ commutes will all operators in Eq. (A12). One can also define the operator Ĵ0=12N̂ such that Ĵ0|j,m=j|j,m where j{0,12,1,...,} and m{j,j+1,...,j}. It is also useful to recall the action of the angular momentum operators Ĵi on the states |j,m, |j,m,

Ĵ2|j,m=jj+1|j,m,Ĵz|j,m=m|j,m,Ĵ±|j,m=jj+1mm±1|j,m±1,
(A14)

where the ladder operators can be written as Ĵ±=Ĵx±iĴy. We can also express a two-mode state in the Fock basis in terms of the angular momentum states using the above equations to find

Ĵz|j,m=m|j,mĴz|n,n1,2=nn2|n,n1,2,Ĵ2|j,m=j(j+1)|j,mĴ2|n,n1,2=n+n2(n+n2+1),
(A15)

which informs us that |n,n1,2|j,m where the values of j and m are given by j=n+n2 and m=nn2. Inversely, |j,m|n,n1,2, |j,m|n,n1,2 where n=j+m and n=jm. Finally, the SU(2) ladder operators act in Fock basis as

Ĵ±|j,m=jj+1mm±1|j,m±1Ĵ±|n,n1,2=12n+nn+n+2nnnn±2×n±1,n11,2.
(A16)

A beam splitter transforms the boson operators associated with each input mode according to the scattering matrix for the device, that is,

âout=Ûâin(â1â2)out=(U11U12U21U22)(â1â2)in.
(A17)

Note that since the boson creation and annihilation must satisfy the commutation relations both before and after beam splitting, the matrix Û must be unitary. Let us see how this transforms the operators of SU(2), J=(Ĵx,Ĵy,Ĵz). Consider the Ĵx-type beam splitter scattering matrix

Û=(cosθ2isinθ2isinθ2cosθ2)=(tirirt),
(A18)

which corresponds to a beam splitter with transmittance and reflectivity T=cos2θ2 and R=sin2θ2, respectively. For this scattering matrix, J transforms to

ĴxĴyĴzout=1000cosθsinθ0sinθcosθĴxĴyĴzin=eiθĴxĴxĴyĴzineiθĴx,
(A19)

which amounts to a rotation about the fictitious x-axis. Note that the last line of Eq. (A19) can be verified via the use of the Baker–Hausdorff identity,

eτÂB̂eτÂ=B̂+τ[Â,B̂]+12τ2[Â,[Â,B̂]]+.
(A20)

Working in the Schrödinger picture, the action of the beam splitter corresponds to a transformation of the initial state given by

|out,BS=eiθĴx|in.
(A21)

With this, the connection between two modes of a boson field and the algebra of SU(2) is complete. In Appendix A 3, we will delve a bit more deeply into beam splitter transformations.

3. Beam splitter transformations

Throughout this paper, several different beam splitter transformations are utilized in conjunction with optical interferometry. Here, we endeavor to show how these particular transformations affect the corresponding boson mode operators. First, let us reconsider the scattering matrix of Eq. (A18) corresponding to a Ĵx-type beam splitter such that |out,BS=eiθĴx|in as per Eq. (A21). For the choices of angle θ=±π/2, both corresponding to a 50:50 beam splitter but parameterized by different rotations about the fictitious x-axis, the mode operators transform according to

(â0â1)=12(1ii1)(â0â1).
(A22)

Let us consider two modes of an optical field labeled a- and b-modes for which operators {â0,â0} act on the a-mode and {â1,â1} on the b-mode. For θ=±π/2 and using Eq. (A22), we can write

â0=12(â0iâ1),â0=12(â0±iâ1),â1=12(â1iâ0),â1=12(â1±iâ0).
(A23)

Direct substitution of the mode operators in Eq. (A23) yields for the displacement operators

D̂aα=eαâ0α*â0e(α/2)(â0iâ1)(α*/2)(â0±iâ1)e(α/2)â0(α*/2)â0e(iα/2)â1(iα*/2)â1D̂a(α2)D̂b(iα2),
(A24)
D̂bβ=eβâ1β*â1e(β/2)(â1iâ0)(β*/2)(â1±iâ0)eiβ/2â0(iβ*/2)â0e(β/2)â1(β*/2)â1D̂a(iβ2)D̂b(β2),
(A25)

or noting the mode operators transform as ei(π/2)Ĵxâ0e±i(π/2)Ĵx=12(â0±iâ1) and ei(π/2)Ĵxâ1e±i(π/2)Ĵx=12(â1±iâ0), we can write more succinctly

eiπ2ĴxD̂a(α)e±iπ2Ĵx=D̂a(α2)D̂b(iα2),eiπ2ĴxD̂b(β)e±iπ2Ĵx=D̂a(iβ2)D̂b(β2).
(A26)

For the case in which coherent light is incident on a beam splitter from each input port, |in=|αa|βb, the resulting output state is straight forward to work out from the discussion above,130 

|out,BS=eiθĴx|in=θ±π2ei(π/2)Ĵx|αa|βb=|12(αiβ)a|12(βiα)b.
(A27)

The resulting state is an equal-intensity distribution of light in each output mode but where the reflected beam picks up a phase shift of ±π/2. For the case in which β0, the initial state is |in=|αa|0b, |in=|αa|0b and the output state is simply |out=eiπ2Ĵx|in=|α2a|iα2.

For a Ĵy-type beam splitter such that |out=eiθĴy|in, the action of the scattering matrix transforms the mode operators according to

(â0â1)=(cosθ2sinθ2sinθ2cosθ2)(â0â1)=θ±π212(11±11)(â0â1),
(A28)

leading to

â0=12(â0â1),â0=12(â0±â1),â1=12(â1±â0),â1=12(â1â0).
(A29)

Direct substitution into the single-mode displacement operators yields

D̂aα=eαâ0α*â0e(α/2)(â0±â1)(α*/2)(â0±â1)e(α/2)â0(α*/2)â0e±(α/2)â1(α*/2)â1D̂a(α2)D̂b(±α2),
(A30)
D̂bβ=eβâ1β*â1e(β/2)(â1â0)(β*/2)(â1â0)e(β/2)â1(β*/2)â1e(β/2)â0±(β*/2)â0D̂aβ2D̂bβ2.
(A31)

Once again assuming coherent light initially in each port incident on a 50:50 beam splitter, the output state is given by

|out,BS=eiθĴy|in=θ±π2eiπ2Ĵy|αa|βb=|12(αβ)a|12(β±α)b,
(A32)

where once again the result is an equal-intensity distribution in both output modes. For β0, the output state is given by |out=|α2a|±α2, where the choice of θ=π/2 results in a phase shift of π in the reflected beam.

1. Defined

Here, we provide a brief discussion on the matrix elements of an arbitrary rotation specified by an axis of rotation n̂ and angle of rotation ϕ. The matrix elements, with 1 for convenience, are

Dm,mj(R)=j,m|eiϕJ·n̂|j,m.
(B1)

Note that the rotation operator commutes with the Casimir invariant Ĵ2. The (2j+1)×(2j+1) matrix formed by Dm,mj(R) is referred to as the (2j+1)-dimensional irreducible representation of the rotation operator D(R). We now consider the matrix realization of the Euler rotation,

Dm,mj(α,β,γ)=j,m|Rzf(α)Ryf(β)Rzf(γ)|j,m=j,m|eiαĴzeiβĴyeiγĴz|j,m.
(B2)

These matrix elements are referred to as the Wigner-D matrix elements. Notice that the first rotation and last rotation only add a phase factor to the expression, thus making only the rotation about the fixed y-axis, the only nontrivial part of the matrix. For this reason, the Wigner-D matrix elements are written in terms of a new matrix,

Dm,mj(α,β,γ)=j,m|eiαĴzeiβĴyeiγĴz|j,m=ei(mα+mγ)j,m|eiβĴy|j,m=ei(mα+mγ)dm,mj(β),
(B3)

where the matrix elements dm,mj(β)=j,m|eiβĴy|j,m are known as the Wigner-d matrix elements and are given by

dm,mj(β)=((jm)!(j+m)!(j+m)!(jm)!)1/2×(1)mmcos2j+mm(β2)sinmm(β2)(mm)!×2F1(mj,mj;mm+1;tan2(β2)),
(B4)

with the property

dmmj(β)={dm,mj(β),mmdm,mj(β),m<m,
(B5)

and where 2F1(a,b;c;z) is the hypergeometric function defined formally by

pFq(a1,,ap;b1,,bq;z)=n=0(a1)n(ap)n(b1)n(bq)nznn!.
(B6)

The above Pochhammer symbol is used to express (x)n=x(x+1)(x+2)(x+n1)=Γ(x+n)/Γ(x) for n1 with the Gamma (generalized factorial) function Γ(x)=(x1)!. It is worth noting that in our interferometric calculations, we naturally end up with an expression that depends on the Wigner-d matrix elements. However, when simply dealing with a single Ĵx beam splitter of angle θ, one encounters the matrix elements j,m|eiθĴx|j,m. This can be simplified using the relations to

j,m|eiθĴx|j,m=j,m|eiπ2ĴzeiθĴyeiπ2Ĵz|j,m=Dm,mj(π2,θ,π2)=immdm,mj(θ).
(B7)

In our closing section, we state a handful of useful identities used throughout this paper.

2. Useful identities

Within the body of this review article, several simplified forms of the Wigner-d matrix are used. The cases relevant to the material herein are listed as follows

dj,jj(β)=dj,jj(β)=cos2j(β2),
(B8)
dj,jj(β)=(1)2jdj,jj(β)=(1)2jsin2j(β2),
(B9)

where we have used the identity

dm,mj(β)=(1)mmdm,mj(β).
(B10)

We have also used the simplified form

dm,jj(π2)=2j(2jj+m)1/2(1)j+m=(1)j+mdm,jj(π2),
(B11)

in our derivations pertaining to Ramsey spectroscopy. Through applications of the standard relations involving the Wigner-d rotation elements,131 it can be shown that

m=jj(1)jmdjn,mj(β)dm,jnj(β)=(1)ndjn,jnj(2β),
(B12)

so that in particular for n =0, we find dj,jj(2ϕ)=cos2jϕ. This identity is used in Eq. (101) in the derivation of the expectation value of the parity operator Π̂b for the input state |αa|Nb.

The following functions are defined in conjunction with the Ramsey spectroscopy derivations of Eqs. (161), (149), and (150), respectively:

Fm,pλ=m=jj1jmdm,mjλdm,pjλ=12jmdm,mjπλdm,pjλ=12jmdp,mjλdm,mjπλ=12jdp,mjπ2λ,
(B13)
Γm,p(λ)=m=jjmdm,pj(λ)dm,mj(λ)=m=jjmdm,pj(λ)dm,mj(λ)(δp,m+1+δp+1,m+δp,m),
(B14)
Λm,p(λ)=m=jjm2dm,pj(λ)dm,mj(λ)=m=jjm2dm,pj(λ)dm,mj(λ)×(δp,m+2+δp+2,m+δp,m+1+δp+1,m+δp,m).
(B15)
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