The parity operator: Applications in quantum metrology

Birrittella, Richard J.; Alsing, Paul M.; Gerry, Christopher C.

doi:10.1116/5.0026148

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.

I. INTRODUCTION

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 $n \in ℤ^{+}$ ⁠, 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 $\hat{a}$ and ${\hat{a}}^{†}$ ⁠, the annihilation and creation operators of the field, respectively, which satisfy the usual commutation relation $[\hat{a}, {\hat{a}}^{†}] = 1$ and the number operator $\hat{n} = {\hat{a}}^{†} \hat{a}$ such that $\hat{n} | n ⟩ = n | n ⟩$ ⁠. In terms of these operators, one can introduce the photon number parity operator $\hat{Π} = {(- 1)}^{{\hat{a}}^{†} \hat{a}} = e^{i π {\hat{a}}^{†} \hat{a}}$ such that $\hat{Π} | n ⟩ = {(- 1)}^{n} | n ⟩$ ⁠. The eigenvalues of this operator are dichotomic and thus highly degenerate. While it is clear that $\hat{Π}$ 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., $E_{n} = ℏ ω (n + \frac{1}{2})$ ⁠), 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 function^3,4 can be expressed in terms of the expectation value of the displaced parity operator⁵ given for a single-mode field as

W (α) = \frac{2}{π} ⟨ \hat{D} (α) \hat{Π} {\hat{D}}^{†} (α) ⟩,

(1)

where $\hat{D} (α) = e^{α {\hat{a}}^{†} - α^{*} \hat{a}}$ 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, $Δ φ$ ⁠, scales as $1 / \sqrt{\bar{n}}$ ⁠, where $\bar{n}$ is the average photon number of the laser field. This defines the SQL as the greatest sensitivity obtainable using classical light: $Δ φ_{SQL} = 1 / \sqrt{\bar{n}}$ ⁠. 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 $Δ φ_{HL} = 1 / \bar{n}$ ⁠. 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.

II. RAMSEY SPECTROSCOPY WITH ENTANGLED AND UNENTANGLED ATOMS

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

{\hat{J}}_{x, y, z} = \frac{1}{2} \sum_{i = 1}^{N} σ_{x, y, z}^{(i)},

(2)

satisfying the SU(2) commutation relation of Eq. (A13) (see Appendix A 2), where $σ_{x, y, z}^{(i)}$ are the Pauli operators for the $i th$ atom as given by $σ_{x}^{(i)} = | e ⟩_{i} {⟨ g | + | g ⟩}_{i} {⟨ e |, σ_{y}^{(i)} = - i (| e ⟩_{i} {⟨ g | - | g ⟩}_{i} ⟨ e |), σ_{z}^{(i)} = | e ⟩}_{i} {⟨ e | - | g ⟩}_{i} ⟨ g |$ and where the ground and excited states for the $i th$ atom are denoted as $| g ⟩_{i}, | e ⟩_{i}$ ⁠, 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 $m \in {- j, - j + 1, \dots, j}$ ⁠, which can be given as superpositions of the product states of all atoms. For $N = 2 j$ atoms, with $j = 1 / 2, 1, . . .$ ⁠, the Dicke states $| j, m ⟩$ are defined in terms of the individual atomic states as $| j, j ⟩ = | e_{1}, e_{2}, . . ., e_{N} ⟩ = | e ⟩^{\otimes N}$ and $| j, - j ⟩ = | g_{1}, g_{2}, . . ., g_{N} ⟩ = | g ⟩^{\otimes N}$ 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

\begin{matrix} {\hat{J}}_{+} = {\hat{J}}_{x} + i {\hat{J}}_{y} = \sum_{i = 1}^{N} | e ⟩_{i} ⟨ g |, \\ {\hat{J}}_{-} = {\hat{J}}_{x} - i {\hat{J}}_{y} = \sum_{i = 1}^{N} | g ⟩_{i} ⟨ e | \end{matrix}

(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 $e^{- i (π / 2) {\hat{J}}_{y}}$ and implemented with radiation of frequency ω. For one atom, the rotation operator about the y-axis for an arbitrary angle is given by¹

e^{- i β {\hat{J}}_{y}} = e^{- i (β / 2) σ_{y}} = I_{2} cos (β / 2) - i σ_{y} sin (β / 2),

(4)

where $I_{2} = | e ⟩ ⟨ e | + | g ⟩ ⟨ g |$ is the identity operator in this two-dimensional subspace. This results in the transformations

e^{- i β {\hat{J}}_{y}} [\begin{matrix} | e ⟩ \\ | g ⟩ \end{matrix}] = [\begin{matrix} cos (β / 2) | e ⟩ + sin (β / 2) | g ⟩ \\ cos (β / 2) | g ⟩ - sin (β / 2) | e ⟩ \end{matrix}],

(5)

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

\begin{matrix} e^{- i \frac{π}{2} {\hat{J}}_{y}} | e ⟩ = \frac{1}{\sqrt{2}} (| g ⟩ + | e ⟩), \\ e^{- i \frac{π}{2} {\hat{J}}_{y}} | g ⟩ = \frac{1}{\sqrt{2}} (| g ⟩ - | e ⟩) . \end{matrix}

(6)

Assuming that the atom is initially in the ground state, the state of the atom after the first $π / 2$ pulse becomes $| ψ (t_{π / 2}) ⟩ = e^{- i (π / 2) {\hat{J}}_{y}} | in ⟩ = \frac{1}{\sqrt{2}} (| g ⟩ - | e ⟩)$ ⁠. This is followed by a period T of free evolution (precession) governed by the operator $e^{- i (ω_{0} - ω) T {\hat{J}}_{z}}$ ⁠, $e^{- i (ω_{0} - ω) T {\hat{J}}_{z}}$ where once again ω is the frequency of the radiation field implementing $π / 2$ pulses. The state after free evolution is

| ψ (t_{π / 2} + T) ⟩ = \frac{1}{\sqrt{2}} (e^{i (ϕ / 2)} | g ⟩ - e^{- i (ϕ / 2)} | e ⟩),

(7)

where we have set $ϕ = (ω_{0} - ω) T$ ⁠. After the second $π / 2$ pulse following free evolution, we use Eq. (6) to find the final state

| ψ (t_{f}) ⟩ = i sin (ϕ / 2) | g ⟩ - cos (ϕ / 2) | e ⟩,

(8)

where $t_{f} = 2 t_{π / 2} + T$ and where it is assumed $T ≫ 2 t_{π / 2}$ such that $t_{f} ≃ T$ ⁠. 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 ${\hat{J}}_{z}$ for this state is

{⟨ {\hat{J}}_{z} (ϕ) ⟩}_{t_{f}}^{(1)} = \frac{1}{2} cos ϕ = \frac{1}{2} cos [(ω_{0} - ω) T] .

(9)

Fig. 1.

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Bloch sphere representation for single-atom spectroscopy. The atom is initially in its ground state. The first $π / 2$ -pulse (at frequency ω) places the atom in a superposition state. The free evolution of time T corresponds to a rotation in the x–y plane. The final $π / 2$ -pulse (also at frequency ω) places the state of the atom near the excited state. For a frequency-spread in radiation $Δ ω$ ⁠, the state will be nearly at the north pole of the Bloch sphere. Note that $R_{i} [θ]$ represents rotation by θ about the i-axis.

FIG. 2.

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A sketch of the Ramsey technique employed for the case of a single atom initially prepared in the ground state. The first $π / 2$ pulse (frequency ω) places the atom in a superposition state. The free evolution over cavity length L results in a phase shift $ϕ = (ω_{0} - ω) T$ (⁠ $T = L / v$ ⁠, where v is the atom velocity). The final $π / 2$ pulse (frequency ω) places the state of the atom near the excited state.

By tuning the frequency of the driving field so as to maximize ${⟨ {\hat{J}}_{z} (ϕ) ⟩}_{t_{f}}^{(1)}$ ⁠, 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 $| ψ ⟩ = e^{- i (π / 2) {\hat{J}}_{y}} | j, - j ⟩$ 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 ${\hat{J}}_{z}$ operator is the sum ${\hat{J}}_{z} = \frac{1}{2} \sum_{i}^{N} σ^{{(i)}_{z}}$ ⁠, one easily finds that

{⟨ {\hat{J}}_{z} (ϕ) ⟩}_{t_{f}}^{(ACS)} = N {⟨ {\hat{J}}_{z} (ϕ) ⟩}_{t_{f}}^{(1)} = \frac{N}{2} cos [(ω_{0} - ω) T] .

(10)

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

Δ ϕ_{ACS} = \frac{Δ {\hat{J}}_{z} (ϕ)}{| \partial_{ϕ} ⟨ {\hat{J}}_{z} (ϕ) ⟩ |} = \frac{1}{\sqrt{N}},

(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 $Δ ω_{0} = 1 / (T \sqrt{N})$ ⁠. 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

| ψ_{f} (ϕ) ⟩ = e^{- i (π / 2) {\hat{J}}_{y}} e^{- i ϕ {\hat{J}}_{z}} e^{- i (π / 2) {\hat{J}}_{y}} | in ⟩,

(12)

where once again $ϕ = (ω_{0} - ω) T$ ⁠. 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

| Ψ_{M} ⟩ = \frac{1}{\sqrt{2}} (| e ⟩^{\otimes N} + | g ⟩^{\otimes N}) = \frac{1}{\sqrt{2}} (| 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 $| in ⟩ = e^{+ i (π / 2) {\hat{J}}_{y}} | Ψ_{M} ⟩$ ⁠. After a free evolution time T, our state is now

| Ψ_{M} ⟩_{T} = \frac{1}{\sqrt{2}} (e^{- i N ϕ / 2} | j, j ⟩ + e^{i N ϕ / 2} | j, - j ⟩) .

(14)

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

\begin{matrix} | Ψ ⟩_{t_{f}} = \frac{1}{\sqrt{2}} (e^{- i N ϕ / 2} e^{- i (π / 2) {\hat{J}}_{y}} | j, j ⟩ + e^{i N ϕ / 2} e^{- i (π / 2) {\hat{J}}_{y}} | j, - j ⟩) \\ = \frac{1}{2^{j} \sqrt{2}} \sum_{m = - j}^{j} {(\begin{matrix} 2 j \\ j + m \end{matrix})}^{1 / 2} (e^{- i N ϕ / 2} + {(- 1)}^{j + m} e^{i N ϕ / 2}) | j, m ⟩, \end{matrix}

(15)

where we have used the fact that $e^{- i (π / 2) {\hat{J}}_{y}} | j, \pm j ⟩$ are atomic coherent states generated from different extremal, or fiducial, states $| j, j ⟩ and | j, - j ⟩$ (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 ${⟨ {\hat{J}}_{z} (ϕ) ⟩}_{M}$ as it can be shown that

\begin{matrix} ⟨ {\hat{J}}_{z} (ϕ) ⟩_{t_{f}} =_{T} ⟨ Ψ_{M} | e^{i (π / 2) {\hat{J}}_{y}} {\hat{J}}_{z} e^{- i (π / 2) {\hat{J}}_{y}} | Ψ_{M} ⟩_{T} \\ =_{T} ⟨ Ψ_{M} | (- {\hat{J}}_{x}) | Ψ_{M} ⟩_{T} = 0. \end{matrix}

(16)

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

To address this issue, Bollinger et al.¹⁷ proposed measuring the quantity ${(- 1)}^{N_{g}}$ ⁠, where N_g is the number of atoms in the ground state. As an operator for a given j value, this reads ${\hat{Π}}_{j} = {(- 1)}^{j - {\hat{J}}_{z}} = e^{i π (j - {\hat{J}}_{z})}$ ⁠. This requires us to calculate

\begin{matrix} ⟨ {\hat{Π}}_{j} ⟩_{t_{f}} =_{T} ⟨ Ψ_{M} | e^{i (π / 2) {\hat{J}}_{y}} e^{i π (j - {\hat{J}}_{z})} e^{- i (π / 2) {\hat{J}}_{y}} | Ψ_{M} ⟩_{T} \\ = {(- 1)}^{j}_{T} ⟨ Ψ_{M} | e^{i π {\hat{J}}_{x}} | Ψ_{M} ⟩_{T} \\ = {(- 1)}^{j}_{T} ⟨ Ψ_{M} | e^{i (π / 2) {\hat{J}}_{z}} e^{i π {\hat{J}}_{y}} e^{- i (π / 2) {\hat{J}}_{z}} | Ψ_{M} ⟩_{T}, \end{matrix}

(17)

where we have used the relations $e^{i (π / 2) {\hat{J}}_{y}} {\hat{J}}_{z} e^{- i (π / 2) {\hat{J}}_{y}} = - {\hat{J}}_{x}$ and $e^{i (π / 2) {\hat{J}}_{z}} {\hat{J}}_{y} e^{- i (π / 2) {\hat{J}}_{z}} = {\hat{J}}_{x}$ ⁠. Setting $\hat{U} = e^{i \frac{π}{2} {\hat{J}}_{z}} e^{i π {\hat{J}}_{y}} e^{- i \frac{π}{2} {\hat{J}}_{z}}$ ⁠, $\hat{U} = e^{i (π / 2) {\hat{J}}_{z}} e^{i π {\hat{J}}_{y}} e^{- i (π / 2) {\hat{J}}_{z}}$ ⁠, we have

\begin{matrix} {⟨ {\hat{Π}}_{j} ⟩}_{t_{f}} = {(- 1)}^{j} \frac{1}{2} [⟨ j, j | \hat{U} | j, j ⟩ + ⟨ j, - j | \hat{U} | j, - j ⟩ \\ + e^{i N ϕ} ⟨ j, j | \hat{U} | j, - j ⟩ + e^{- i N ϕ} ⟨ j, - j | \hat{U} | j, j ⟩] . \end{matrix}

(18)

Then, with

\begin{matrix} ⟨ j, j | \hat{U} | j, j ⟩ = d_{j, j}^{j} (- π) = 0, \\ ⟨ j, - j | \hat{U} | j, - j ⟩ = d_{- j, - j}^{j} (- π) = 0, \\ ⟨ j, j | \hat{U} | j, - j ⟩ = {(- 1)}^{j} d_{j, - j}^{j} (- π) = {(- 1)}^{j}, \\ ⟨ j, - j | \hat{U} | j, j ⟩ = {(- 1)}^{- j} d_{- j, j}^{j} (- π) = {(- 1)}^{j}, \end{matrix}

(19)

where $d_{m', m}^{j} (β)$ are the Wigner-d matrix elements discussed in Appendix B, we finally have

{⟨ {\hat{Π}}_{j} ⟩}_{t_{f}} = {(- 1)}^{N} cos [N (ω_{0} - ω) 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 $1 / \sqrt{N}$ over the SQL. Again, setting $ϕ = (ω_{0} - ω) T$ ⁠, the error propagation calculus in this case results in

Δ ϕ_{M} = \frac{Δ {\hat{Π}}_{j}}{| \partial_{ϕ} {⟨ {\hat{Π}}_{j} ⟩}_{t_{f}} |} = \frac{1}{N},

(21)

or that $Δ ω_{0} = 1 / N T$ ⁠. 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 $P = \frac{1}{2} (1 + cos ϕ T)$ and $Δ P = \sqrt{P (1 - P) / d_{N}}$ ⁠, respectively, where $ϕ = ω - ω_{0}$ and $d_{N} = N t_{tot} / T$ ⁠. Here, $t_{tot}$ denotes the total duration over which the scheme is repeated and d_N is the number of experimental data points. The uncertainty in atomic transition frequency can be obtained readily by error propagation $Δ ω_{0} = Δ P / | \partial_{ω} P | = 1 / \sqrt{N t_{tot} T}$ ⁠. 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 $| ψ ⟩_{MES} = \frac{1}{\sqrt{2}} (| 00 ⟩ + | 11 ⟩)$ ⁠, 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 $P_{N} = \frac{1}{2} (1 + cos N ϕ T)$ ⁠, where the oscillation frequency scales as now scales as $N ϕ$ ⁠. The frequency uncertainty is now $Δ ω_{0} = 1 / N \sqrt{t_{tot} T}$ ⁠; an improvement over the SQL by a factor of $1 / \sqrt{N}$ ⁠, 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

\begin{matrix} {\dot{ρ}}^{(1)} (T) = i ϕ (ρ (0) | 1 ⟩ ⟨ 1 | + | 1 ⟩ ⟨ 1 | ρ (0)) \\ + \frac{γ}{2} (σ_{z} ρ (0) σ_{z} - ρ (0)), \end{matrix}

(22)

where once again $σ_{z} = | 0 ⟩ ⟨ 0 | - | 1 ⟩ ⟨ 1 |$ is the Pauli spin operator in the z-direction and $γ = 1 / τ_{dec}$ ⁠, where $τ_{dec}$ is the decoherence time. For the case of initially unentangled ions, this simply leads to a broadening of the signal $\tilde{P} = \frac{1}{2} (1 + e^{- γ T} cos ϕ T)$ ⁠, leading to the frequency uncertainty

Δ ω_{0} = \sqrt{\frac{1 - e^{- 2 γ T} {cos}^{2} ϕ T}{N t_{tot} T e^{- 2 γ T} {sin}^{2} ϕ T}} .

(23)

This can be optimized for a given free evolution time T to find the minimum uncertainty. The minimum value is obtained for $ϕ T = k π / 2, k \in {2 n + 1 : n \in ℤ}$ and $T = τ_{dec} / 2$ for which it was found $Δ ω_{0, min .} = \sqrt{2 γ e / N t_{totmail}} = \sqrt{2 e / N τ_{dec} T}$ ⁠. Likewise, for the maximally entangled two-ion case, the probability of detecting excitation is modulated by the decoherence term, yielding ${\tilde{P}}_{N} = \frac{1}{2} (1 + e^{- N γ T} cos N ϕ T)$ with a phase uncertainty optimized by the conditions $ϕ T = k π / 2 N, k \in {2 n + 1 : n \in ℤ}$ and $T = τ_{dec} / 2 N$ ⁠. Combining these results yields the same exact minimum frequency uncertainty as the unentangled ion case: $Δ ω_{0, min .} = \sqrt{2 e / N τ_{dec} T}$ ⁠. 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.

III. PHASE ESTIMATION AND 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.

A. The Cramér–Rao lower bound

In the broadest sense, an interferometer is an apparatus that can transform an input “probe state” $\hat{ρ}$ 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 $\hat{ρ} (φ)$ 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 $\sum_{ε} \hat{E} (ε) = 1$ ⁠. 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 [\hat{E} (ε) \hat{ρ} (φ)] .

(24)

If the input state is made up of m independent uncorrelated subsystems such that $\hat{ρ} = {\hat{ρ}}^{(1)} \otimes {\hat{ρ}}^{(2)} \otimes {\hat{ρ}}^{(3)} \otimes {\hat{ρ}}^{(4)} \otimes \dots . \otimes {\hat{ρ}}^{(m)}$ 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) = \prod_{i = 1}^{m} P_{i} (ε_{i} | φ),

(25)

where $P_{i} (ε_{i} | φ) = Tr [{\hat{E}}^{(i)} (ε_{i}) {\hat{ρ}}^{(i)} (φ)]$ ⁠. For the case of independent measurements, as described in Eq. (25), often one considers the log-Likelihood function

L (ε | φ) \equiv ln P (ε | φ) = \sum_{i = 1}^{m} ln P_{i} (ε_{i} | 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

Φ_{MLE} (ε) = arg [{max}_{φ} P (ε | φ)] .

(27)

An estimator can be characterized by its phase dependent mean value

{⟨ Φ ⟩}_{φ} = \sum_{ε} P (ε | φ) Φ (ε)

(28)

and its variance

{(Δ Φ)}_{φ}^{2} = \sum_{ε} P (ε | φ) {[Φ (ε) - {⟨ Φ ⟩}_{φ}]}^{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 $m \to \infty$ ⁠, 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

{(Δ Φ_{CRB})}_{φ}^{2} = \frac{{(\frac{\partial {⟨ Φ ⟩}_{φ}}{\partial φ})}^{2}}{F (φ)} \leq {(Δ Φ)}_{φ}^{2},

(30)

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

F (φ) \equiv ⟨ {(\frac{\partial L (ε | φ)}{\partial φ})}^{2} ⟩ = \sum_{ε} \frac{1}{P (ε | φ)} {(\frac{\partial P (ε | φ)}{\partial 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 $\frac{\partial {⟨ Φ ⟩}_{φ}}{\partial φ} = \sum_{ε} \partial_{φ} P (ε | φ) Φ (ε) = ⟨ Φ \frac{\partial L}{\partial φ} ⟩$ ⁠, where $L (ε | φ)$ is given by Eq. (26). Noting that $\sum_{ε} \partial_{φ} P (ε | φ) = ⟨ \frac{\partial L}{\partial φ} ⟩ = 0$ ⁠, we have

\begin{matrix} {(\frac{\partial {⟨ Φ ⟩}_{φ}}{\partial φ})}^{2} = {⟨ (Φ - {⟨ Φ ⟩}_{φ}) \frac{\partial L}{\partial φ} ⟩}_{φ}^{2} \\ \leq {⟨ {(Φ - {⟨ Φ ⟩}_{φ})}^{2} ⟩}_{φ} {⟨ {(\frac{\partial L}{\partial φ})}^{2} ⟩}_{φ} = {(Δ Φ)}_{φ}^{2} F (φ) \end{matrix},

(32)

where we have invoked the Cauchy–Schwarz inequality ${⟨ A B ⟩}_{φ}^{2} \leq {⟨ A^{2} ⟩}_{φ} {⟨ B^{2} ⟩}_{φ}$ ⁠. 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 $\partial_{φ} ⟨ Φ ⟩_{φ} = 1$ ⁠. 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 $F (φ) = \sum_{i = 1}^{m} F^{(i)} (φ)$ 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 $F (φ) = m F^{(1)} (φ)$ ⁠, 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.

B. Quantum Fisher information and the upper bound

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 F_Q (QFI). We obtain this upper bound by maximizing the FI over all possible POVMs,^30,31

F_{Q} [\hat{ρ} (φ)] \equiv max_{{\hat{E} (ε)}} F [\hat{ρ} (φ), {\hat{E} (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 as^32–34

F_{Q} [\hat{ρ} (φ)] = Tr [\hat{ρ} (φ) {\hat{L}}_{φ}^{2}],

(34)

where ${\hat{L}}_{φ}$ is known as the symmetric logarithmic derivative (SLD)³⁵ defined as the solution to the equation

\frac{\partial \hat{ρ} (φ)}{\partial φ} = \frac{\hat{ρ} (φ) {\hat{L}}_{φ} + {\hat{L}}_{φ} \hat{ρ} (φ)}{2} .

(35)

The chain of inequalities is now

{(Δ Φ)}_{φ}^{2} \geq {(Δ Φ)}_{CRB}^{2} \geq {(Δ Φ)}_{qCRB}^{2},

(36)

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

{(Δ Φ)}_{qCRB}^{2} \equiv \frac{{(\frac{\partial ⟨ Φ ⟩}{\partial φ})}^{2}}{F_{Q} [\hat{ρ} (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 $\hat{ρ} (φ) = \sum_{n} p_{n} | n ⟩ ⟨ n |$ ⁠. Following the work of Pezzé and Smerzi²⁸ the quantum Fisher information can be written in this basis as

\begin{matrix} F_{Q} [\hat{ρ} (φ)] = Tr [\hat{ρ} (φ) {\hat{L}}_{φ}^{2}] = \sum_{k, k'} p_{k} | ⟨ k | {\hat{L}}_{φ} | k' ⟩ |^{2} \\ = \sum_{k, k'} \frac{p_{k} + p_{k'}}{2} \times | ⟨ k | {\hat{L}}_{φ} | k' ⟩ |^{2} . \end{matrix}

(38)

Thus, it is sufficient to know the matrix elements of the SLD, $⟨ k | {\hat{L}}_{φ} | k' ⟩$ ⁠, in order to calculate the QFI. Using Eq. (35) and our general density operator, it is easy to show

⟨ k | {\hat{L}}_{φ} | k' ⟩ = (\frac{2}{p_{k} + p_{k'}}) \times ⟨ k | \partial_{φ} \hat{ρ} (φ) | k' ⟩,

(39)

which makes Eq. (38)³⁷

F_{Q} [\hat{ρ} (φ)] = \sum_{k, k'} \frac{2}{p_{k} + p_{k'}} \times | ⟨ k | \partial_{φ} \hat{ρ} (φ) | k' ⟩ |^{2} .

(40)

We proceed further through the use of the definition

\partial_{φ} \hat{ρ} (φ) = \sum_{k} (\partial_{φ} p_{k}) | k ⟩ ⟨ k | + \sum_{k} p_{k} | \partial_{φ} k ⟩ ⟨ k | + \sum_{k} p_{k} | k ⟩ ⟨ \partial_{φ} k |,

(41)

which is a simple application of the chain rule for derivatives. Using the identity $\partial_{φ} ⟨ k | k' ⟩ = ⟨ \partial_{φ} k | k' ⟩ + ⟨ k | \partial_{φ} k' ⟩ \equiv 0$ ⁠, the matrix elements in Eq. (40) become

⟨ k | \partial_{φ} \hat{ρ} (φ) | k' ⟩ = (\partial_{φ} \hat{ρ} (φ)) δ_{k, k'} + (p_{k} - p_{k'}) ⟨ \partial_{φ} k | k' ⟩ .

(42)

The SLD and QFI, then, become

\begin{matrix} {\hat{L}}_{φ} = \sum_{k} \frac{\partial_{φ} p_{k}}{p_{k}} \times | k ⟩ ⟨ k | + 2 \sum_{k, k'} \frac{p_{k} - p_{k'}}{p_{k} + p_{k'}} \times ⟨ \partial_{φ} k | k' ⟩ | k' ⟩ ⟨ k |, \\ F_{Q} [\hat{ρ} (φ)] = \sum_{k} \frac{{(\partial_{φ} p_{k})}^{2}}{p_{k}} + 2 \sum_{k, k'} \frac{{(p_{k} - p_{k^{'}})}^{2}}{p_{k} + p_{k'}} \times | ⟨ \partial_{φ} k | k' ⟩ |^{2}, \end{matrix}

(43)

respectively. These results, we show next, simplify in the case of pure states where we can write $\hat{ρ} (φ) = | ψ (φ) ⟩ ⟨ ψ (φ) |$ and consequently $\partial_{φ} \hat{ρ} (φ) = \partial_{φ} {\hat{ρ}}^{2} (φ) = \hat{ρ} (φ) [\partial_{φ} \hat{ρ} (φ)] + [\partial_{φ} \hat{ρ} (φ)] \hat{ρ} (φ)$ ⁠. Using this, and a cursory glance at Eq. (35), it is clear that the SLD becomes

\begin{matrix} {\hat{L}}_{φ} = 2 [\partial_{φ} \hat{ρ} (φ)] = 2 [\partial_{φ} | ψ (φ) ⟩ ⟨ ψ (φ) |] \\ = 2 | \partial_{φ} ψ ⟩ ⟨ ψ | + 2 | ψ ⟩ ⟨ \partial_{φ} ψ |, \end{matrix}

(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

\begin{matrix} F_{Q} [\hat{ρ} (φ)] = Tr [\hat{ρ} (φ) {\hat{L}}_{φ}^{2}] = ⟨ ψ | {\hat{L}}_{φ}^{2} | ψ ⟩ \\ = 4 {⟨ \partial_{φ} ψ | \partial_{φ} ψ ⟩ - | ⟨ \partial_{φ} ψ | ψ ⟩ |^{2}}, \end{matrix}

(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,

{(Δ Φ)}_{φ}^{2} \geq {(Δ Φ)}_{CRB}^{2} = {(Δ Φ)}_{qCRB}^{2} .

(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 $P (+ | φ) + P (- | φ) \equiv 1$ ⁠.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

\begin{matrix} ⟨ \hat{Π} ⟩ = \sum_{i} P (i | φ) λ_{i} = P (+ | φ) - P (- | φ) \\ = 2 P (+ | φ) - 1 = 1 - 2 P (- | φ) . \end{matrix}

(47)

Likewise, we can calculate the variance

\begin{matrix} {(Δ \hat{Π})}^{2} = ⟨ {\hat{Π}}^{2} ⟩ - {⟨ \hat{Π} ⟩}^{2} = 1 - {⟨ \hat{Π} ⟩}^{2} \\ = 1 - {(P (+ | φ) - P (- | φ))}^{2} \\ = 1 - {(P (+ | φ) + P (- | φ))}^{2} + 4 P (+ | φ) P (- | φ) \\ = 4 P (+ | φ) P (- | φ) . \end{matrix}

(48)

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

\frac{\partial P (+ | φ)}{\partial φ} = \frac{1}{2} \frac{\partial ⟨ \hat{Π} ⟩}{\partial φ} = - \frac{\partial P (- | φ)}{\partial φ} .

(49)

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

\begin{matrix} F^{(1)} (φ) = \sum_{ε_{1}} \frac{1}{P (ε_{1} | φ)} {(\frac{\partial P (ε_{1} | φ)}{\partial φ})}^{2} \\ = \frac{1}{{(Δ \hat{Π})}^{2}} | \frac{\partial ⟨ \hat{Π} ⟩}{\partial φ} |^{2}, \end{matrix}

(50)

making the CRB/qCRB

Δ Φ_{CRB / qCRB} = \frac{1}{\sqrt{F^{(1)} (φ)}} = \frac{\sqrt{1 - {⟨ \hat{Π} ⟩}^{2}}}{| \partial_{φ} ⟨ \hat{Π} ⟩ |} .

(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.

C. Calculating 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 ${\hat{J}}_{x}$ corresponds to a rotation about the x-axis, while ${\hat{J}}_{y}$ 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 ${\hat{J}}_{z}$ -operator. The state just before the second beam splitter is given as

| ψ (φ) ⟩ = e^{- i φ {\hat{J}}_{z}} e^{- i (π / 2) {\hat{J}}_{x}} | in ⟩,

(52)

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

| ψ' (φ) ⟩ = - i e^{- i φ {\hat{J}}_{z}} {\hat{J}}_{z} e^{- i (π / 2) {\hat{J}}_{x}} | in ⟩,

(53)

leading to

⟨ ψ' (φ) | ψ (φ) ⟩ = i ⟨ in | {\hat{J}}_{y} | in ⟩

(54)

and

⟨ ψ' (φ) | ψ' (φ) ⟩ = ⟨ in | {\hat{J}}_{y}^{2} | in ⟩,

(55)

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

e^{i (π / 2) {\hat{J}}_{x}} {\hat{J}}_{z} e^{- i (π / 2) {\hat{J}}_{x}} = {\hat{J}}_{y} .

(56)

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

\begin{matrix} F_{Q} [\hat{ρ} (φ)] = 4 {⟨ \partial_{φ} ψ | \partial_{φ} ψ ⟩ - | ⟨ \partial_{φ} ψ | ψ ⟩ |^{2}} \\ = 4 {⟨ {(Δ {\hat{J}}_{y})}^{2} ⟩}_{in}, \end{matrix}

(57)

which is simply the variance of the ${\hat{J}}_{y}$ -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 $\hat{O}$ be a generator of a flow parameterized by ζ such that the wave function evolves according to the Schrödinger equation $i \partial_{ζ} | ψ ⟩ = \hat{O} | ψ ⟩$ with solution $| ψ (ζ) ⟩ = e^{- i \hat{O} ζ} | ψ (0) ⟩$ ⁠. Then, $| \partial_{ζ} ψ ⟩ = - i \hat{O} | ψ ⟩$ so that $⟨ \partial_{ζ} ψ | \partial_{ζ} ψ ⟩ = ⟨ {\hat{O}}^{2} ⟩$ and $| ⟨ ψ | \partial_{ζ} ψ ⟩ |^{2} = {⟨ \hat{O} ⟩}^{2}$ ⁠. Hence, $F_{Q} (ζ) = 4 [⟨ \partial_{ζ} ψ | \partial_{ζ} ψ ⟩ - | ⟨ ψ | \partial_{ζ} ψ ⟩ |^{2}] = 4 {⟨ {(Δ \hat{O})}^{2} ⟩}_{ζ}$ ⁠. Thus, both Eqs. (30) and (37) yield a Fourier-like uncertainty relation for unbiased estimators of the form ${(Δ Φ)}_{ζ} {⟨ (Δ \hat{O}) ⟩}_{ζ} \geq 1 / 2$ (if we do not use the maximum FI) reminiscent of $Δ x Δ k \geq 1 / 2$ so that a precise measurement ${(Δ Φ)}_{ζ} ≪ 1 / 2$ of ζ requires a large uncertainty (variance) ${⟨ (Δ \hat{O}) ⟩}_{ζ} ≫ 1 / 2$ in its generation. Noting that ${\hat{Π}}_{b} = {(- 1)}^{{\hat{b}}^{†} \hat{b}} = e^{i π {\hat{n}}_{b}}$ is both unitary and Hermitian, define ${\hat{Π}}_{b}^{(ϕ)} \equiv e^{- i ϕ {\hat{n}}_{b}}$ with $\hat{O} \to {\hat{n}}_{b}$ and $ζ \to ϕ$ (which we take to be $ϕ = - π$ for the parity operator). Thus, a measurement ${\hat{Π}}_{b}^{(ϕ)} | ψ (0) ⟩ = e^{- i ϕ {\hat{n}}_{b}} | ψ (0) ⟩$ can be thought of as a Schrödinger-like evolution in $ϕ$ with generator ${\hat{n}}_{b}$ ⁠. Then, from the above, $F_{Π_{b}} (ϕ) = 4 {⟨ {(Δ {\hat{n}}_{b})}^{2} ⟩}_{ϕ}$ ⁠. This result is independent of the parameter $ϕ$ ⁠. This leads to the insightful interpretation of Eq. (51) as $Δ Φ_{CRB / qCRB} Δ {\hat{n}}_{b} = \frac{1}{2}$ ⁠, i.e., a more formal statement of the classical number-phase uncertainty relationship $Δ ϕ Δ N \geq \frac{1}{2}$ ⁠. In fact, we now see that the parity operator ${\hat{Π}}_{b}$ 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.

IV. QUANTUM OPTICAL INTERFEROMETRY

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 ⟩ = e^{i (π / 2) {\hat{J}}_{x}} e^{- i ϕ {\hat{J}}_{z}} e^{- i (π / 2) {\hat{J}}_{x}} | in ⟩ = e^{- i ϕ {\hat{J}}_{y}} | 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, $| in ⟩ = | ψ_{0}^{(1)} ⟩_{a} \otimes | ψ_{0}^{(2)} ⟩_{b}$ ⁠, where in the photon number basis $| ψ_{0}^{(i)} ⟩ = \sum_{n = 0}^{\infty} C_{n}^{(i)} | n ⟩, n \in ℤ^{+}$ ⁠, the input state can be expressed as

\begin{matrix} | in ⟩ = \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} C_{n}^{(1)} C_{q}^{(2)} | n ⟩_{a} \otimes | q ⟩_{b} \\ = \sum_{j = 0, 1 / 2, \dots}^{\infty} \sum_{m = - j}^{j} C_{j + m}^{(1)} C_{j - m}^{(2)} | j, m ⟩ . \end{matrix}

(59)

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

\begin{matrix} | out ⟩ = \sum_{j' = 0, 1 / 2, \dots}^{\infty} \sum_{m' = - j'}^{j'} ⟨ j', m' | e^{- i ϕ {\hat{J}}_{y}} | in ⟩ | j', m' ⟩ \\ = \sum_{j = 0, 1 / 2, \dots}^{\infty} \sum_{m' = - j}^{j} Γ_{j, m'} (ϕ) | j + m' ⟩_{a} \otimes | j - m' ⟩_{b}, \end{matrix}

(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

Γ_{j, m'} (ϕ) = \sum_{m = - j}^{j} C_{j + m}^{(1)} C_{j - m}^{(2)} d_{m', m}^{j} (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 $A_{n, p}$ where $A_{n, p} \neq C_{n}^{(1)} C_{p}^{(2)}$ ⁠. For an arbitrary detection observable $\hat{O}$ ⁠, the expectation value can be calculated directly as

{⟨ \hat{O} ⟩}_{out} = \sum_{j, m'} \sum_{J, M} Γ_{J, M}^{*} (ϕ) Γ_{j, m'} (ϕ) ⟨ J, M | \hat{O} | j, m' ⟩ .

(62)

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

Δ ϕ_{\hat{O}} = \frac{Δ {\hat{O}}_{out}}{| \partial_{ϕ} {⟨ \hat{O} ⟩}_{out} |} .

(63)

From the perspective of the Heisenberg picture, the transformed observable $\hat{O}'$ is given by $\hat{O}' = e^{i ϕ {\hat{J}}_{y}} \hat{O} e^{- i ϕ {\hat{J}}_{y}}$ and the derivative of its expectation value

\partial_{ϕ} {⟨ \hat{O} ⟩}_{out} = \partial_{ϕ} {⟨ \hat{O}' ⟩}_{in} = i {⟨ [{\hat{J}}_{y}, \hat{O}'] ⟩}_{in} .

(64)

We remind the reader that the greatest phase sensitivity afforded by classical states is the standard quantum limit (SQL), $Δ ϕ_{SQL} = 1 / \sqrt{\bar{n}}$ ⁠, while for quantum states, the phase sensitivity is bounded by the Heisenberg limit (HL) $Δ ϕ_{HL} = 1 / \bar{n}$ ⁠. 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.

A. The N00N states

The optical $N 00 N$ 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) $| N, 0 ⟩_{a, b}$ and where no photons are in the a-mode and N photons are in the b-mode $| 0, N ⟩_{a, b}$ and can be written generally as

| ψ ⟩_{N 00 N} = \frac{1}{\sqrt{2}} (| N, 0 ⟩ + e^{i Φ_{N}} | 0, N ⟩),

(65)

where $Φ_{N}$ 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 “ $N 00 N$ 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 $| in ⟩ = | N, 0 ⟩_{a, b}$ and the first beam splitter is replaced by an optical device that transforms the initial state into the $N 00 N$ 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

| ψ_{N} (ϕ) ⟩ = \frac{1}{\sqrt{2}} (| N, 0 ⟩_{a, b} + e^{i (N ϕ + Φ_{N})} | 0, N ⟩_{a, b}),

(66)

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

| out ⟩ = \frac{1}{\sqrt{2^{N + 1} N!}} [{({\hat{a}}^{†} + i {\hat{b}}^{†})}^{N} + e^{i (N ϕ + Φ_{N})} {({\hat{b}}^{†} + i {\hat{a}}^{†})}^{N}] | 0, 0 ⟩_{a, b},

(67)

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

\begin{matrix} | out, 2 ⟩ = \frac{1}{2 \sqrt{2}} [(1 - e^{i (2 ϕ + Φ_{2})}) | 2, 0 ⟩_{a, b} \\ + \sqrt{2} i (1 + e^{i (2 ϕ + Φ_{2})}) | 1, 1 ⟩_{a, b} + (e^{i (2 ϕ + Φ_{2})} - 1) | 0, 2 ⟩_{a, b}] . \end{matrix}

(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 ${\hat{Σ}}_{N} = | N, 0 ⟩_{a, b} {⟨ 0, N | + | 0, N ⟩}_{a, b} ⟨ N, 0 |$ whose expectation value with respect to the state Eq. (65), $⟨ {\hat{Σ}}_{N} ⟩ = cos N ϕ$ ⁠, depends on the phase $N ϕ$ 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

{(Δ ϕ)}^{2} = \frac{⟨ {\hat{Σ}}_{N}^{2} ⟩ - {⟨ \hat{Σ_{N}} ⟩}^{2}}{{(\partial_{ϕ} ⟨ \hat{Σ_{N}} ⟩)}^{2}} = \frac{1 - {cos}^{2} N ϕ}{N^{2} {sin}^{2} N ϕ} \overset{ϕ \to 0}{=} \frac{1}{N^{2}},

(69)

which is an improvement over the classical (SQL) limit by a factor of $1 / \sqrt{N}$ ⁠. The result of Eq. (69) follows from the heuristic number-phase relation $Δ ϕ Δ N \geq 1$ ⁠. For the $N 00 N$ state of Eq. (65), the uncertainty in the photon number is N, the total average photon number, immediately giving the equality $Δ ϕ = 1 / N$ ⁠. Extrapolating for arbitrary states, we define the Heisenberg limit (HL) $Δ ϕ_{HL} = 1 / {\bar{n}}_{tot}$ ⁠, where ${\bar{n}}_{tot}$ 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, ${\hat{Π}}_{b} = {(- 1)}^{{\hat{b}}^{†} \hat{b}}$ ⁠, results in a sign flip on the center term

\begin{matrix} {\hat{Π}}_{b} | out, 2 ⟩ = \frac{1}{2 \sqrt{2}} [(1 - e^{i (2 ϕ + Φ_{2})}) | 2, 0 ⟩_{a, b} \\ - \sqrt{2} i (1 + e^{i (2 ϕ + Φ_{2})}) | 1, 1 ⟩_{a, b} + (e^{i (2 ϕ + Φ_{2})} - 1) | 0, 2 ⟩_{a, b}], \end{matrix}

(70)

leading to the expectation value $⟨ {\hat{Π}}_{b} ⟩ = cos (2 ϕ + Φ_{2})$ ⁠, 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

\begin{matrix} ⟨ {\hat{Π}}_{b} ⟩ = \frac{i^{N}}{2} [e^{i (N ϕ + Φ_{N})} + {(- 1)}^{N} e^{- i (N ϕ + Φ_{N})}] \\ = \{\begin{matrix} {(- 1)}^{N / 2} cos (N ϕ + Φ_{N}) & N even \\ {(- 1)}^{(N + 1) / 2} sin (N ϕ + Φ_{N}) & N odd, \end{matrix}) \end{matrix}

(71)

and consequently, $Δ ϕ = 1 / N$ ⁠. 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 ${\hat{Σ}}_{N}$ is not directly connected to an observable. Furthermore, there is presently no physical realization of the projector ${\hat{Σ}}_{N}$ 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 $N 00 N$ states.

B. Entangled coherent 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 $| α ⟩ = e^{- | α |^{2} / 2} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} | n ⟩$ 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 ⟩ = | β ⟩_{a} \otimes N (| γ ⟩_{b} + e^{i θ} | - γ ⟩_{b}),

(72)

where $N$ is the cat state normalization factor given by $N = 1 / \sqrt{2 (1 + e^{- 2 | γ |^{2}} cos θ)}$ ⁠. The QFI can be calculated immediately for this input state using Eq. (57). For large γ, $⟨ \hat{J_{+}} ⟩ = ⟨ \hat{J_{¯}} ⟩ = ⟨ \hat{J_{y}} ⟩ ≃ 0$ and

⟨ J_{+}^{2} ⟩ = β^{* 2} γ^{2}, ⟨ J_{-}^{2} ⟩ = β^{2} γ^{* 2},

(73)

⟨ J_{+} {\hat{J}}_{-} ⟩ = | β |^{2} (1 + | γ |^{2}), ⟨ J_{-} {\hat{J}}_{+} ⟩ = | γ |^{2} (1 + | β |^{2}),

(74)

resulting in

\begin{matrix} F_{Q} = 4 {⟨ {(Δ {\hat{J}}_{y})}^{2} ⟩}_{in} = 4 ⟨ {\hat{J}}_{y}^{2} ⟩ \\ = | β |^{2} + | γ |^{2} + 2 | β |^{2} | γ |^{2} (1 - cos [2 (θ_{β} - θ_{γ})]), \end{matrix}

(75)

where $(θ_{β} - θ_{γ})$ is the phase difference between coherent states $| β ⟩$ and $| γ ⟩$ ⁠. Setting $β = α / \sqrt{2}$ and $γ = - i β = - i α / \sqrt{2}$ ⁠, 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 $θ_{β} - θ_{γ} = π / 2$ ⁠, the minimum phase uncertainty is given by

Δ ϕ_{min} \overset{large α}{\to} \frac{1}{| α |^{2}} = Δ ϕ_{HL} .

(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

\begin{matrix} | β ⟩_{a} | γ ⟩_{b} \overset{BS 1}{\to} | \frac{1}{\sqrt{2}} (β + i γ) ⟩_{a} | \frac{1}{\sqrt{2}} (γ + i β) ⟩_{b}, \\ | β ⟩_{a} | - γ ⟩_{b} \overset{BS 1}{\to} | \frac{1}{\sqrt{2}} (β - i γ) ⟩_{a} | \frac{1}{\sqrt{2}} (- γ + i β) ⟩_{b}, \end{matrix}

(77)

making the state after the first beam splitter

\begin{matrix} | out, BS 1 ⟩ = N (| \frac{1}{\sqrt{2}} (β + i γ) ⟩_{a} | \frac{1}{\sqrt{2}} (γ + i β) ⟩_{b} \\ + e^{i θ} | \frac{1}{\sqrt{2}} (β - i γ) ⟩_{a} | \frac{1}{\sqrt{2}} (- γ + i β) ⟩_{b}] . \end{matrix}

(78)

FIG. 3.

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A sketch of a standard Mach–Zehnder interferometer (MZI). Light propagates through one (or both) input ports of the MZI. A phase shift occurs after the first beam splitter due to a relative path length difference between the two arms of the MZI. The measurement occurs on one (or both) mode of the output.

Once again setting $β = α / \sqrt{2}$ and $γ = - i β = - i α / \sqrt{2}$ ⁠, this becomes

| out, BS 1 ⟩ = N (| α ⟩_{a} | 0 ⟩_{b} + e^{i θ} | 0 ⟩_{a} | i α ⟩_{b}),

(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 $| out, ANLI ⟩ \propto {(| 0, i α ⟩}_{a, b} + i | - α, 0 ⟩_{a, b})$ ⁠, similar to Eq. (79). This state can be written as a superposition of $N 00 N$ states as per

\begin{matrix} | out, ANLI ⟩ = e^{- \frac{1}{2} | α |^{2}} \sum_{N = 0}^{\infty} \frac{α^{N}}{\sqrt{N!}} {(- 1)}^{N} \\ \times \frac{1}{\sqrt{2}} (| N, 0 ⟩_{a, b} + e^{- i Φ_{N}} | 0, N ⟩_{a, b}), \end{matrix}

(80)

where $Φ_{N} = \frac{π}{2} (N + 1)$ ⁠. 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

{⟨ \hat{Π} ⟩}_{b} = e^{- \bar{n}} (1 + e^{\bar{n} cos ϕ} sin (\bar{n} sin ϕ)),

(81)

\partial_{ϕ} {⟨ \hat{Π} ⟩}_{b} = \bar{n} e^{- \bar{n} (1 - cos ϕ)} cos (ϕ + \bar{n} sin ϕ),

(82)

and from the error propagation calculus, Eq. (63), the phase uncertainty for $ϕ \to 0$ is

Δ ϕ_{{\hat{Π}}_{b}} = \frac{\sqrt{1 - e^{- 2 \bar{n}}}}{\bar{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 $N 00 N$ state, tend toward providing greater phase sensitivity.

Fig. 4.

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Expectation value of the parity operator $⟨ {\hat{Π}}_{b} ⟩$ for ${\bar{n}}_{tot} = 5$ (blue, solid) and ${\bar{n}}_{tot} = 25$ (red, dashed). The peaks of the curves narrow and migrate toward the phase origin, which serves as an inflection point. Note that for these curves, the relative phase shift in Eq. (72) is chosen as $θ = π / 2$ given our interferometric scheme of Eq. (58) to align with the results found by Gerry et al.⁵⁸

Fig. 5.

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Phase uncertainty obtained when the state after the first beam splitter is an entangled coherent state of the form Eq. (79) for a relative phase $θ = π / 2$ and a phase value $ϕ = π / 45$ ⁠. Note that aside from the spike at ${\bar{n}}_{tot} \sim 21$ ⁠, the phase uncertainty follows closely along the HL.

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 $| 1, 1 ⟩_{a, b}$ is incident upon a 50:50 beam splitter, the resulting state is the two-photon $N 00 N$ state $| 2, 0 ⟩_{a, b} + | 0, 2 ⟩_{a, b}$ ⁠. Let us now consider the case in which the arbitrary state $| N, N ⟩_{a, b}$ is taken as the initial state.

C. Twin-Fock state input

It was first pointed out by Holland and Burnett⁵⁹ that interferometric phase measurements when considering input twin-Fock states $| N, N ⟩_{a, b}$ 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 $\propto | 2 N, 0 ⟩_{a, b} + e^{- i Φ_{N}} | 0, 2 N ⟩_{a, b}, N \in ℤ^{+}$ ⁠, then the phase uncertainty is exactly the HL: $Δ ϕ_{HL} = 1 / 2 N$ ⁠. 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 $C_{n}^{(1)} C_{q}^{(2)} = δ_{N, n} δ_{N, q}$ as per Eq. (59), the state after the first beam splitter is the well-known arcsine (AS) state,⁶⁶ given by

\begin{matrix} | out, BS 1 ⟩ = \sum_{k = 0}^{N} {(- 1)}^{N - k} {[(\begin{matrix} 2 k \\ k \end{matrix}) (\begin{matrix} 2 N - 2 k \\ N - k \end{matrix}) {(\frac{1}{2})}^{2 N}]}^{1 / 2} \\ \times | 2 k ⟩_{a} \otimes | 2 N - 2 k ⟩_{b}, \end{matrix}

(84)

where in this case a ${\hat{J}}_{y}$ -type beam splitter was considered rather than a ${\hat{J}}_{x}$ -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 $N 00 N$ 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 $N 00 N$ state but instead a superposition of the N-photon $N 00 N$ state and other (but not all) permutations of the state $| p, q ⟩$ where $p + q = 2 N$ ⁠. 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 $2 N - 2 k$ photons in the b-mode, given by

\begin{matrix} P_{AS} (2 k, 2 N - 2 k) = |_{a, b} ⟨ 2 k, 2 N - 2 k | out, BS 1 ⟩ |^{2} \\ = (\begin{matrix} 2 k \\ k \end{matrix}) (\begin{matrix} 2 N - 2 k \\ N - k \end{matrix}) {(\frac{1}{2})}^{2 N}, k \in [0, N] \end{matrix},

(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 $| N, 0 ⟩_{a, b}$ and $| 0, N ⟩_{a, b}$ states, as shown in Fig. 6. The phase properties of this state were studied by Campos et al.⁶⁶

Fig. 6.

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Two-mode joint photon number distribution resulting from beam splitting the state $| in ⟩ = | N, N ⟩_{a, b}$ ⁠. The “bathtub” shape of the arcsine distribution is peaked at the $| N, 0 ⟩_{a, b}$ and $| 0, N ⟩_{a, b}$ states. This is plotted for N = 8.

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 ${⟨ J ⟩}_{+} = {⟨ J ⟩}_{-} = {⟨ J ⟩}_{y} = 0, ⟨ {\hat{J}}_{+} {\hat{J}}_{-} ⟩ = ⟨ {\hat{J}}_{-} {\hat{J}}_{+} ⟩ = N (N + 1)$ and consequently

Δ^{2} {\hat{J}}_{y} = \frac{N}{2} (1 + N) \to Δ ϕ_{min} = \frac{1}{\sqrt{2 N (1 + N)}} .

(86)

Right away, it is clear that for the case of N = 1, the minimum phase uncertainty provides the HL: $Δ ϕ_{HL} = 1 / 2 N = 1 / 2$ ⁠. 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 ${\hat{J}}_{x}$ -type, to be

\begin{matrix} {⟨ {\hat{Π}}_{b} ⟩}_{AS} = ⟨ in | {\hat{Π}}_{b} | in ⟩ \\ = ⟨ out, BS 1 | e^{- i ϕ {\hat{J}}_{z}} e^{- i \frac{π}{2} {\hat{J}}_{x}} {\hat{Π}}_{b} e^{i \frac{π}{2} {\hat{J}}_{x}} e^{i ϕ {\hat{J}}_{z}} | out, BS 1 ⟩ \\ = \sum_{k = 0}^{N} e^{i 2 ϕ (N - 2 k)} (\begin{matrix} 2 k \\ k \end{matrix}) (\begin{matrix} 2 N - 2 k \\ N - k \end{matrix}) {(\frac{1}{2})}^{2 N} . \end{matrix}

(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

{⟨ {\hat{Π}}_{b} ⟩}_{AS} = P_{N} [cos (2 ϕ)] .

(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 ${⟨ {\hat{Π}}_{b} ⟩}_{AS} = cos (2 ϕ)$ ⁠, leading to $Δ ϕ = 1 / 2$ ⁠, the HL. For N = 2, it can be shown that ${⟨ {\hat{Π}}_{b} ⟩}_{AS} = 1 / 4 + 3 / 4 cos (4 ϕ)$ ⁠, which in the limit of $ϕ \to 0$ yields $Δ ϕ = 1 / \sqrt{12} = 0.2886$ ⁠, 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 ${⟨ {\hat{Π}}_{b} ⟩}_{AS}^{(ϕ = π / 2)} = {(- 1)}^{N}$ ⁠, 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.

Fig. 7.

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Parity-based phase uncertainty with phase $ϕ = 10^{- 4}$ for the case of an input twin-Fock state. Note that the phase uncertainty obtained corresponds to the qCRB.

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 $| Ψ ⟩ = \sum_{n = 0}^{\infty} B_{n} | n, n ⟩_{a, b}$ with total average photon number ${\bar{n}}_{tot} = 2 \sum_{n = 0}^{\infty} | B_{n} |^{2} n$ ⁠. In terms of the initial state of Eq. (59), the coefficients for such a state are given by $C_{n}^{(1)} C_{q}^{(2)} = B_{n} δ_{q, n}$ ⁠. 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),

{⟨ {\hat{Π}}_{b} ⟩}_{corr} = \sum_{n = 0}^{\infty} | B_{n} |^{2} P_{n} [cos (2 ϕ)],

(89)

where once again $P_{n} (x)$ 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 $2 ω$ 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 $B_{n}^{(TMSVS)} = {(1 - | z |^{2})}^{1 / 2} z^{n}$ ⁠. 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 $2 ϕ$ being the pump amplitude and phase, respectively, as $z = e^{i 2 ϕ} \tanh r$ ⁠, where $r = | γ | t$ is the squeeze parameter. Interestingly, coupling the TMSVS coefficients with Eq. (89) yields for the phase value $ϕ = π / 2$

{⟨ {\hat{Π}}_{b} ⟩}_{tmsvs}^{(ϕ = π / 2)} = \frac{1}{\cosh (2 r)} = {\begin{matrix} 1, & r \to 0 \\ 2 e^{- 2 r}, & r ≫ 1 . \end{matrix}

(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 $F_{Q} = 2 {⟨ {\hat{J}}_{+} {\hat{J}}_{-} ⟩}_{in}$ ⁠, it can be shown that the TMSVS minimum phase uncertainty is

Δ ϕ_{min} = \frac{1}{\sqrt{{\bar{n}}_{tot} (2 + {\bar{n}}_{tot})}},

(91)

where for the TMSVS, ${\bar{n}}_{tot} = 2 \sinh^{2} r$ ⁠. 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 $Δ ϕ Δ N \geq 1$ ⁠, 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 $\hat{n}, Δ φ = 1 / ⟨ {\hat{n}}^{2} ⟩$ ⁠. This provides better sensitivity in phase measurements than the HL as $⟨ {\hat{n}}^{2} ⟩$ contains direct information about fluctuations that ${⟨ \hat{n} ⟩}^{2}$ does not. In fact, this is why parity-based measurement yields greater sensitivity: it contains all moments of $\hat{n}$ ⁠. For the case of a TMSVS, the Hoffman limit is given by $Δ ϕ = 1 / \sqrt{2 \bar{n} (\bar{n} + 1)}$ ⁠. 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.

Fig. 8.

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Phase uncertainty using parity-based detection for the input two-mode squeezed vacuum state. For phases close to zero, the phase uncertainty yields sub-HL sensitivity; however, it rapidly degrades as the phase deviates from zero.

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 ⟩ = N \int_{0}^{2 π} | \sqrt{ζ} e^{i θ} ⟩ | \sqrt{ζ} e^{- i θ} ⟩ d θ,

(92)

where $| \sqrt{ζ} e^{\pm i θ} ⟩$ are the Glauber coherent states. In terms of the number state basis, the PCS can be written as

| ζ ⟩ = \frac{1}{\sqrt{I_{0} (2 | ζ |)}} \sum_{n = 0}^{\infty} \frac{ζ^{n}}{n!} | n, n ⟩_{a, b},

(93)

where $I_{0} (2 | ζ |)$ 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 $\hat{a} \hat{b} | ζ ⟩ = ζ | ζ ⟩$ and $({\hat{a}}^{†} \hat{a} - {\hat{b}}^{†} \hat{b}) | ζ ⟩ = 0$ ⁠.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.

D. Coherent light mixed with a squeezed vacuum state

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 $| SVS ⟩ = \sum_{N = 0}^{\infty} S_{N} | N ⟩$ with average photon number ${\bar{n}}_{SVS} = \sinh^{2} r$ and with state coefficients,

S_{N} = {(- 1)}^{N / 2} {[\frac{N!}{2^{N} {(\frac{N}{2}!)}^{2}} (\frac{\tanh^{N} r}{\cosh r})]}^{1 / 2} {cos}^{2} (N π / 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, $| in ⟩ = | α ⟩_{a} \otimes | SVS ⟩_{b}$ with state coefficients as per Eq. (59) given by $C_{n}^{(1)} C_{p}^{(2)} = e^{- | α |^{2} / 2} \frac{α^{n}}{\sqrt{N!}} \times S_{p}$ ⁠. 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 $Δ ϕ \sim 1 / {\bar{n}}_{tot}$ 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

\begin{matrix} F^{(1)} (ϕ) = \sum_{N_{c}, N_{d}}^{\infty} \frac{1}{P (N_{c}, N_{d} | ϕ)} {(\frac{\partial P (N_{c}, N_{d} | ϕ)}{\partial ϕ})}^{2} \\ = | α |^{2} e^{2 r} + \sinh^{2} r, \end{matrix}

(95)

where $P (N_{c}, N_{d} | ϕ)$ is the conditional probability of measuring N_c photons in one output mode and N_d in the other and is given in terms of the Wigner-d rotation elements (see Appendix B) by

P (N_{c}, N_{d} | ϕ) = e^{- | α |^{2}} {| \sum_{n = 0}^{2 j} \frac{α^{2 j - n}}{(2 j - n)!} S_{n} d_{m, j - n}^{j} (ϕ) |}^{2},

(96)

with $j = (N_{c} + N_{d}) / 2$ and $m = (N_{c} - N_{d}) / 2$ ⁠. For the regime in which both input ports are of roughly equal intensity $| α |^{2} ≃ \sinh^{2} r = {\bar{n}}_{tot} / 2$ and assuming large r such that ${\bar{n}}_{tot} ≃ e^{2 r} / 2$ ⁠, it can be shown that

Δ ϕ_{min} = \frac{1}{\sqrt{{\bar{n}}_{tot} ({\bar{n}}_{tot} + \frac{1}{2})}} \overset{large {\bar{n}}_{tot}}{\to} \frac{1}{{\bar{n}}_{tot}} = Δ ϕ_{HL} .

(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 $N 00 N$ 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 $N 00 N$ state within the superposition of $N 00 N$ 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 ${⟨ {\hat{a}}^{†} \hat{a} {\hat{b}}^{†} \hat{b} ⟩}_{out}$ but retained only the counts where if one detector detects m photons, the other detects N − m, and where all other counts $\neq N$ 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 $N 00 N$ 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 $N 00 N$ 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.

E. Coherent light mixed with N photons

Next, we consider the choice of input state $| in ⟩ = | α ⟩_{a} \otimes | N ⟩_{b}$ ⁠, 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 $C_{n}^{(1)} C_{q}^{(2)} = e^{- | α |^{2} / 2} α^{n} / \sqrt{n!} \times δ_{q, N}$ ⁠, which in terms of the angular momentum basis states can be written as

| in ⟩ = e^{- \frac{1}{2} | α |^{2}} \sum_{j = N / 2, \dots}^{\infty} \frac{α^{2 j - N}}{\sqrt{(2 j - N)!}} | j, j - N ⟩,

(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) $| out, BS 1 ⟩ = e^{- i (π / 2) {\hat{J}}_{x}} | in ⟩$ ⁠. For the special case of N = 1, this state can be written as

\begin{matrix} | out, BS 1 ⟩_{N = 1} = e^{(- 1 / 2) | α |^{2}} \sum_{n = 0}^{\infty} \sum_{q = 0}^{\infty} \frac{{(\frac{α}{\sqrt{2}})}^{n} {(\frac{i α}{\sqrt{2}})}^{q}}{\sqrt{n! p!}} \\ \times (1 - δ_{n, 0} δ_{q, 0}) γ_{n, q} | n, q ⟩_{a, b}, \end{matrix}

(99)

where the factor $γ_{n, q}$ is given by

γ_{n, q} = \frac{4 i {(- 1)}^{1 - q} 2^{n + q + 1} n!}{α (n + q + 1)!}_{2} {\tilde{F}}_{1} (2, n + 1; 2 - q; - 1)

(100)

and where $_{2} {\tilde{F}}_{1} (a, b; c; z)$ is a regularized hypergeometric function (see Appendix B 1). For n = q, this function is identically zero $\forall q \neq 0$ ⁠. 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

{⟨ {\hat{Π}}_{b} (ϕ) ⟩}_{N} = {(- 1)}^{N} e^{- | α |^{2}} \sum_{j = N / 2, \dots}^{\infty} \frac{| α |^{2 (2 j - N)}}{(2 j - N)!} d_{j - N, j - N}^{j} (2 ϕ) .

(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, $ϕ \to 0, {⟨ {\hat{Π}}_{b} (0) ⟩}_{0} \to 1$ 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 $M ϕ$ 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 $ϕ = 0$ ⁠: ${⟨ {\hat{Π}}_{b} (0) ⟩}_{N} = {(- 1)}^{N}$ ⁠. The peak (or trough) centered at $ϕ = 0$ also narrows for increasing values of N. This can also be seen in Fig. 9.

Fig. 9.

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Expectation value of the parity operator taken with respect to the output b-mode for an input state $| in ⟩ = | α ⟩_{a} | N ⟩_{b}$ for (a) N = 1 and (b) N = 8. Note the narrowness of the peak for increasing $| α |$ and N.

2. Phase uncertainty: Approaching the Heisenberg limit

We begin with an analysis of the phase uncertainty obtained by computation of Eq. (63) with ${\hat{J}}_{z}$ 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

Δ ϕ_{{\hat{J}}_{z}} = \frac{\sqrt{| α |^{2} + N (1 + 2 | α |^{2} {sin}^{2} ϕ)}}{| (| α |^{2} - N) sin ϕ |} .

(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., $| α |^{2} ≃ N$ ⁠, 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 ${\hat{Π}}_{b}$ as the detection observable. The phase uncertainty, along with the corresponding SQL and HL, $Δ ϕ_{SQL} = 1 / \sqrt{| α |^{2} + N}$ and $Δ ϕ_{HL} = 1 / (| α |^{2} + N)$ ⁠, 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.

Fig. 10.

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Phase uncertainty computed with phase $ϕ = 10^{- 4}$ for the input state $| in ⟩ = | α ⟩_{a} | N ⟩_{b}$ ⁠, with accompanying SQL and HL, for (a) N = 1 and (b) N = 8. In the limit of $N \to \infty$ ⁠, the phase uncertainty approaches the HL. The blue line in each figure also corresponds to the qCRB for this choice of initial state.

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

Δ ϕ_{min} = \frac{1}{\sqrt{| 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 $1 / \sqrt{N}$ ⁠, which follows the SQL.

F. Interferometry with strictly classical (coherent) light

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 $| in ⟩ = | α ⟩_{a} \otimes | 0 ⟩_{b}$ ⁠, where once again the coherent state is given by $| α ⟩ = e^{- | α |^{2} / 2} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} | n ⟩$ ⁠. Considering the transformation of Eq. (58) in which the beam splitters are of ${\hat{J}}_{x}$ -type, the state after the first beam splitter is given by (see Appendix A 3)

| α ⟩_{a} | 0 ⟩_{b} \overset{BS 1}{\to} {| \frac{α}{\sqrt{2}} ⟩}_{a} | - \frac{i α}{\sqrt{2}} ⟩_{b},

(104)

and the state after the accumulated phase shift in the b-mode (the ${\hat{J}}_{z}$ operation introduces an anti-symmetric phase shift of $\pm ϕ / 2$ 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

| \frac{α}{\sqrt{2}} ⟩_{a} | \frac{i α}{\sqrt{2}} ⟩_{b} \overset{ϕ ¯ shift}{\to} | \frac{α}{\sqrt{2}} ⟩_{a} | - \frac{i α e^{i ϕ}}{\sqrt{2}} ⟩_{b} .

(105)

Finally, the state after the second beam splitter is

| \frac{α}{\sqrt{2}} ⟩_{a} | - \frac{i α e^{i ϕ}}{\sqrt{2}} ⟩_{b} \overset{BS 2}{\to} | \frac{α}{2} (1 + e^{i ϕ}) ⟩_{a} | \frac{i α}{2} (1 - e^{i ϕ}) ⟩_{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,⁸⁷ $I \propto ⟨ \hat{n} ⟩$ ⁠. Consequently, the difference between mode intensities at the output of the second beam splitter can be written as $δ I \propto ⟨ {\hat{a}}^{†} \hat{a} - {\hat{b}}^{†} \hat{b} ⟩$ ⁠. In terms of the SU(2) Lie algebra, this amounts to the expectation value of the operator $2 {\hat{J}}_{z}$ ⁠. Using Eq. (106), the mode intensities are given by $⟨ {\hat{n}}_{a (b)} ⟩ = \frac{{\bar{n}}_{0}}{2} (1 \pm cos ϕ)$ ⁠, where ${\bar{n}}_{0} = | α |^{2}$ (note that we are assuming a lossless interferometer such that ${\bar{n}}_{0} = ⟨ {\hat{n}}_{a} ⟩ + ⟨ {\hat{n}}_{b} ⟩$ is conserved), leading to the average value $⟨ 2 {\hat{J}}_{z} ⟩ = {\bar{n}}_{0} cos ϕ$ ⁠. It is also straight-forward to show that $⟨ {(2 {\hat{J}}_{z})}^{2} ⟩ = {\bar{n}}_{0} (1 + ⟨ 2 {\hat{J}}_{z} ⟩ cos ϕ)$ ⁠. Combining these expressions and Eq. (63) yields a phase uncertainty

Δ ϕ_{2 {\hat{J}}_{z}} = \frac{\sqrt{⟨ {\hat{J}}_{z}^{2} ⟩ - {⟨ {\hat{J}}_{z} ⟩}^{2}}}{| \partial_{ϕ} ⟨ {\hat{J}}_{z} ⟩ |} = \frac{1}{\sqrt{{\bar{n}}_{0}} | sin ϕ |},

(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 $sin ϕ$ with $cos ϕ$ in Eq. (107). It is worth pointing out that $Δ {\hat{J}}_{z}$ 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 ${\hat{Π}}_{b} = {(- 1)}^{{\hat{b}}^{†} \hat{b}}$ and likewise for the a-mode ${\hat{Π}}_{a} = {(- 1)}^{{\hat{a}}^{†} \hat{a}}$ ⁠. From this, the corresponding expectation values and their first derivatives are found to be⁸⁵

⟨ {\hat{Π}}_{a (b)} ⟩ = e^{- {\bar{n}}_{0} (1 \pm cos ϕ)},

(108)

\partial_{ϕ} ⟨{\hat{Π}}_{a (b)}⟩ = \pm {\bar{n}}_{0} sin ϕ ⟨{\hat{Π}}_{a (b)}⟩ .

(109)

Noting that $⟨ {\hat{Π}}_{a (b)}^{2} ⟩ \equiv 1$ ⁠, the phase uncertainty can immediately be found from the error propagation calculus as follows:

Δ ϕ_{{\hat{Π}}_{a}} = \frac{\sqrt{1 - {⟨ {\hat{Π}}_{a} ⟩}^{2}}}{| \partial_{ϕ} ⟨ {\hat{Π}}_{a} ⟩ |} = \frac{\sqrt{e^{4 {\bar{n}}_{0} {cos}^{2} ϕ / 2} - 1}}{{\bar{n}}_{0} | sin ϕ |},

(110)

Δ ϕ_{{\hat{Π}}_{b}} = \frac{\sqrt{1 - {⟨ {\hat{Π}}_{b} ⟩}^{2}}}{| \partial_{ϕ} ⟨ {\hat{Π}}_{b} ⟩ |} = \frac{\sqrt{e^{4 {\bar{n}}_{0} {sin}^{2} ϕ / 2} - 1}}{{\bar{n}}_{0} | sin ϕ |} .

(111)

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

Δ ϕ_{{\hat{Π}}_{a}} = \frac{1}{\sqrt{{\bar{n}}_{0}}} |_{ϕ \to π}, Δ ϕ_{{\hat{Π}}_{b}} = \frac{1}{\sqrt{{\bar{n}}_{0}}} |_{ϕ \to 0},

(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 $Δ ϕ_{min} = 1 / \sqrt{{\bar{n}}_{0}} = Δ ϕ_{SQL}$ ⁠.

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 $N 00 N$ states can be found in works by Dowling and collaborators⁸⁹ in which the projection operator ${\hat{Σ}}_{N}$ ⁠, 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 $λ \to λ / N$ (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 $N ϕ$ (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

Fig. 11.

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Comparison of measurement resolution obtained for an input state $| in ⟩ = | α ⟩_{a} \otimes | 0 ⟩_{b}$ between scaled intensity-difference $⟨ 2 {\hat{J}}_{z} ⟩ / {\bar{n}}_{0}$ (red, dashed) and parity $⟨ {\hat{Π}}_{b} ⟩$ with both $| α | = 4$ (blue, dotted) and $| α | = 8$ (purple, solid). Note that the output signal, when using parity-based measurements, narrows with the increasing average photon number. This does not occur when considering intensity-difference measurements.

G. Effects of losses and decoherence in optical interferometric schemes

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 $1 / \sqrt{\bar{n}}$ ⁠) 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 $N 00 N$ 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 ${\hat{Σ}}_{N}$ (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 $N 00 N$ state is worse than that of an equally attenuated separable N photon state. They conclude that for realistic experimental conditions, the $N 00 N$ 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 $N 00 N$ state with the form $| m : : m' ⟩ = \frac{1}{\sqrt{2}} (| m, m' ⟩ + | m', m ⟩)$ and average photon number $\bar{N} = m + m'$ ⁠. Defining the loss factor for each mode $L_{a (b)} = 1 - T_{a (b)}$ ⁠, where $T_{a (b)}$ is the transmission rate of the beam splitter modeling loss, the phase sensitivity can be expanded out for $L_{a} = L_{b} = L$ as

Δ ϕ = \{\begin{matrix} \frac{1}{Δ m} + \frac{(m + m')}{Δ m} csc (Δ m ϕ) L + O [L^{2}], & Δ m even \\ \frac{1}{Δ m} + \frac{(m + m')}{Δ m} sec (Δ m ϕ) L + O [L^{2}], & Δ m odd, \end{matrix})

(113)

where $Δ m = m - m', m > m'$ ⁠. In the limit of $L \to 0$ ⁠, Eq. 113 minimizes to $Δ ϕ = 1 / Δ m$ for the optimal value of phase,

ϕ \to \{\begin{matrix} \frac{(2 n - 1) π}{2 Δ m}, & Δ m even \\ \frac{n π}{Δ m}, & Δ m odd . \end{matrix})

(114)

Taking losses into account, the SQL and HL are given by $1 / \sqrt{\bar{N}'}$ and $1 / \bar{N}'$ ⁠, respectively, where $\bar{N}' = \bar{N} (1 - \frac{1}{2} L_{a} - \frac{1}{2} L_{b})$ ⁠. Consequently, the SQL is beaten provided that the criterion $Δ m > \sqrt{m + m'}$ 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 $| 6 : : 0 ⟩$ $N 00 N$ state and the $| 8 : : 2 ⟩$ entangled Fock state, assuming equal losses in both modes. For losses up to $\sim 10 %$ ⁠, 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 $N 00 N$ 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 $m m'$ -entangled Fock states outperform the $N 00 N$ 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 $\hat{O}, V_{det} = {⟨ \hat{O} ⟩}_{ϕ = 0}$ ⁠) in the face of losses when considering different permutations of the superposition state for a given $N = m - m'$ ⁠. 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 $N 00 N$ 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 $0.995 \leq T \leq 1$ (where T is the transmission rate of the beam splitter used to model losses). They also show that states of the form $| ψ ⟩ = N_{+} {(| α, 0 ⟩}_{1, 2} + | 0, α ⟩_{1, 2})$ ⁠, where $| α ⟩$ is a coherent state, outperform $N 00 N$ states of the same average photon number for nonlinear phase shifts described by $\hat{U} (k, ϕ) = exp [i {({\hat{b}}^{†} \hat{b})}^{k}]$ 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.

V. EXPERIMENTAL REALIZATIONS OF PARITY-BASED DETECTION

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-QED^105,106 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.

A. QND measures of atomic parity

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

{\hat{Π}}_{even} = \sum_{m = - j}^{j} {cos}^{2} [\frac{(j - m) π}{2}] | j, m ⟩ ⟨ j, m |,

(115)

{\hat{Π}}_{odd} = \sum_{m = - j}^{j} {sin}^{2} [\frac{(j - m) π}{2}] | j, m ⟩ ⟨ j, m |,

(116)

satisfying the POVM condition $\sum_{i} {\hat{Π}}_{i} = \hat{I}$ ⁠. Note that j + m represents the total number of atoms found in the excited state, while j – m represents the total number of atoms found in the ground state. Consequently, $j \pm m$ takes on only integer values such that for a given $j \pm m$ ⁠, only one of the projectors will be nonzero. These projection operators can be used to express the atomic parity operator as

\begin{matrix} \hat{Π} = \sum_{i} λ_{i} {\hat{Π}}_{i} = {\hat{Π}}_{even} - {\hat{Π}}_{odd} \\ = \sum_{m = - j}^{j} cos [(j - m) π] | j, m ⟩ ⟨ j, m | \end{matrix},

(117)

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

⟨ {\hat{Π}}_{even} ⟩ = \frac{1}{2} (1 + ⟨ \hat{Π} ⟩),

(118)

⟨ {\hat{Π}}_{odd} ⟩ = \frac{1}{2} (1 - ⟨ \hat{Π} ⟩),

(119)

where $⟨ {\hat{Π}}_{odd} ⟩ + ⟨ {\hat{Π}}_{even} ⟩ = 1$ ⁠. 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 $| τ, j_{b} ⟩_{b}$ and the coupling Hamiltonian and corresponding evolution operator,

\hat{H} = ℏ χ ({\hat{J}}_{0}^{a} + {\hat{J}}_{z}^{a}) ({\hat{J}}_{0}^{b} + {\hat{J}}_{z}^{b}) \to \hat{U} = e^{- i t \hat{H} / ℏ},

(120)

where ${\hat{J}}_{0} | j, m ⟩ = j | j, m ⟩$ (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

\begin{matrix} | initial ⟩ = | ψ ⟩_{a} \otimes | τ, j_{b} ⟩_{b} = \sum_{m = - j_{a}}^{j_{a}} C_{m}^{(j_{a})} | j_{a}, m ⟩_{a} \\ \otimes {(1 + | τ |^{2})}^{- j_{b}} \sum_{m' = - j_{b}}^{j_{b}} (\begin{matrix} 2 j_{b} \\ j_{b} + m' \end{matrix}) τ^{j_{b} + m'} | j_{b}, m' ⟩_{b} . \end{matrix}

(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,

\begin{matrix} | final ⟩ = e^{- i t \hat{H} / ℏ} | initial ⟩ \\ = \sum_{m = - j_{a}}^{j_{a}} C_{m}^{(j_{a})} | j_{a}, m ⟩_{a} \otimes | τ e^{- i t χ (j_{a} + m)}, j_{b} ⟩_{b} . \end{matrix}

(122)

For the choice of $χ t \to π$ ⁠, this becomes

\begin{matrix} | final ⟩ = | ψ_{even} ⟩_{a} | τ, j_{b} ⟩_{b} + | ψ_{odd} ⟩_{a} | - τ, j_{b} ⟩_{b} \\ = \sum_{m = - j_{a}}^{j_{a}} [{cos}^{2} (\frac{(j_{a} + m) π}{2}) | τ, j_{b} ⟩_{b} \\ + {sin}^{2} (\frac{(j_{a} + m) π}{2}) | - τ, j_{b} ⟩_{b}] C_{m}^{(j_{a})} | j_{a}, m ⟩ \\ = \sum_{m = - j_{a}}^{j_{a}} C_{m}^{(j_{a})} | j_{a}, m ⟩_{a} | Φ_{m} ⟩_{b} . \end{matrix}

(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,

e^{- i \frac{π}{2} \hat{J_{y}}} | j, - j ⟩ = | - 1, j ⟩ = | ψ_{-} ⟩^{\otimes N} = \frac{1}{2^{N / 2}} {(| g ⟩ - | e ⟩)}^{\otimes N} .

(124)

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

e^{- i \frac{π}{2} \hat{J_{y}}} | j, j ⟩ = | 1, j ⟩ = | ψ_{+} ⟩^{\otimes N} = \frac{1}{2^{N / 2}} {(| g ⟩ + | e ⟩)}^{\otimes N} .

(125)

With this, we can rewrite the state $| Φ_{m} ⟩_{b}$ ⁠, assuming $τ = - 1$ ⁠, as

\begin{matrix} | Φ_{m} ⟩_{b} = {cos}^{2} (\frac{(j_{a} + m) π}{2}) | - 1, j_{b} ⟩_{b} + {sin}^{2} (\frac{(j_{a} + m) π}{2}) | 1, j_{b} ⟩_{b} \\ = {cos}^{2} (\frac{(j_{a} + m) π}{2}) e^{- i \frac{π}{2} \hat{J_{y}^{b}}} | j_{b}, - j_{b} ⟩_{b} \\ + {sin}^{2} (\frac{(j_{a} + m) π}{2}) e^{- i \frac{π}{2} \hat{J_{y}^{b}}} | j_{b}, j_{b} ⟩_{b} . \end{matrix}

(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

e^{i \frac{π}{2} {\hat{J}}_{y}^{b}} | Φ_{m} ⟩_{b} = {\begin{matrix} {cos}^{2} (\frac{(j_{a} + m) π}{2}) | j_{b}, - j_{b} ⟩_{b}, & j_{a} + m even \\ {sin}^{2} (\frac{(j_{a} + m) π}{2}) | j_{b}, j_{b} ⟩_{b}, & j_{a} + m odd, \end{matrix}

(127)

where $j_{a} + m$ 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 $j_{a} + m$ ⁠. Note that this works for an arbitrary value of j_b; 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

\hat{H} = ℏ χ {\hat{b}}^{†} \hat{b} {\hat{J}}_{z} \to \hat{U} = e^{- i \frac{t}{ℏ} \hat{H}},

(128)

where ${\hat{b}, {\hat{b}}^{†}}$ are the usual boson annihilation and creation operators, respectively. Consider an ancillary field state, given by the usual coherent state

| α ⟩_{b} = e^{- \frac{1}{2} | α |^{2}} \sum_{n = 0}^{\infty} \frac{α^{n}}{\sqrt{n!}} | n ⟩_{b} .

(129)

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

| final ⟩ = \hat{U} | initial ⟩ = \sum_{m = - j}^{j} C_{m}^{(j)} | j, m ⟩_{a} \otimes | {(- 1)}^{m} α ⟩_{b} .

(130)

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

| final ⟩ = {\begin{matrix} | ψ_{odd} ⟩_{a} | - α ⟩_{b} + | ψ_{even} ⟩_{a} | α ⟩_{b}, & j = 0, 2, 4, \dots, N = 0, 4, 8, \dots \\ | ψ_{even} ⟩_{a} | - α ⟩_{b} + | ψ_{odd} ⟩_{a} | α ⟩_{b}, & j = 1, 3, 5, \dots, N = 2, 6, 10, \dots \\ | ψ_{even} ⟩_{a} | - i α ⟩_{b} + | ψ_{odd} ⟩_{a} | i α ⟩_{b}, & j = \frac{1}{2}, \frac{5}{2}, \frac{9}{2}, \dots, N = 1, 5, 9, \dots \\ | ψ_{odd} ⟩_{a} | - i α ⟩_{b} + | ψ_{even} ⟩_{a} | i α ⟩_{b}, & j = \frac{3}{2}, \frac{7}{2}, \frac{11}{2}, \dots, N = 3, 7, 11, \dots . \end{matrix}

(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 $N = 0, 4, 8, \dots$ ⁠. As per Eq. (131), the final state is given by $| final ⟩ = | ψ_{odd} ⟩_{a} | - α ⟩_{b} + | ψ_{even} ⟩_{a} | α ⟩_{b}$ ⁠. Mixing the field at a 50:50 beam splitter with an equal-amplitude phase-adjusted coherent state $| α ⟩_{c}$ ⁠. Assuming a $50 : 50$ ${\hat{J}}_{y}$ -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

\begin{matrix} | α ⟩_{b} | α ⟩_{c} \overset{BS 2}{\to} | \sqrt{2} α ⟩_{b} | 0 ⟩_{c}, \\ | - α ⟩_{b} | α ⟩_{c} \overset{BS 2}{\to} | 0 ⟩_{b} | \sqrt{2} α ⟩_{c} . \end{matrix}

(132)

The total state is now given by

| final ⟩_{BS 2} = | ψ_{odd} ⟩_{a} | 0 ⟩_{b} | \sqrt{2} α ⟩_{c} + | ψ_{even} ⟩_{a} | \sqrt{2} α ⟩_{b} | 0 ⟩_{c},

(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

| Ψ ⟩_{a} = \{\begin{matrix} | ψ_{even} ⟩_{a} & b - mode click \\ | ψ_{odd} ⟩_{a} & c - mode click . \end{matrix})

(134)

The optical fields $| α ⟩_{b}$ and $| α ⟩_{c}$ used in this procedure can be derived from the same beam if one starts with the state $| \sqrt{2} α ⟩_{c}$ and uses a $50 : 50$ ${\hat{J}}_{y}$ -type beam splitter of angle $+ π / 2$ such that

| \sqrt{2} α ⟩_{b} | 0 ⟩_{c} \overset{BS 1}{\to} | α ⟩_{b} | α ⟩_{c} .

(135)

After beam splitting, the b-mode coherent state can be coupled with the target atomic system before being photomixed with the coherent state $| α ⟩_{c}$ 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.

B. Detection of photon parity

As mentioned above, one way to obtain photon number parity is to perform photon number counts $n_{count}$ and raise –1 to that power: $Π_{measured} = {(- 1)}^{n_{count}}$ ⁠. 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., $\hat{a} | α ⟩ = α | α ⟩$ ⁠). 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 $⟨ {\hat{Π}}_{a} ⟩ = e^{- {\bar{n}}_{0} (1 + cos ϕ)} ≃ e^{- ({\bar{n}}_{0} / 2) {(ϕ - π)}^{2}}$ ⁠, where ${\bar{n}}_{0}$ is the total number of photons within the interferometer, which peaks at unity for $ϕ \to π$ ⁠. In this limit $Δ ϕ \to 1 / \sqrt{{\bar{n}}_{0}}$ ⁠, 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 $| 0 ⟩ ⟨ 0 |$ ⁠, or any number of photons detected but without resolution, $\sum_{n = 0}^{\infty} | n ⟩ ⟨ n | = I - | 0 ⟩ ⟨ 0 |$ ⁠. Defining $\hat{Z} = | 0 ⟩_{a} ⟨ 0 |$ ⁠, from the output state given in Eq. (106,) we find the probability of there being no photon detections to be

P_{0} (ϕ) = ⟨ \hat{Z} ⟩_{out} = e^{- ({\bar{n}}_{0} / 2) (1 + cos ϕ)} \overset{ϕ \to π}{≃} e^{- {(\bar{n}}_{0} / 4) {(ϕ - π)}^{2}} .

(136)

From the error propagation calculus, one easily finds that for $ϕ \to π$ ⁠, the phase uncertainty is given by $Δ ϕ_{a} \to 1 / \sqrt{{\bar{n}}_{0}}$ ⁠, 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

Fig. 12.

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A schematic of the experimental setup used by Cohen et al. The coherent states are produced by a Ti:sapphire laser with average photon numbers controlled by a calibrated neutral density filer (NDF), and the MZI is composed from two polarizers P at 45° and a phase shifter. What is defined in the text as the output a-mode is filtered spectrally by $3 nm$ band pass filter (BPF) and spatially by a single mode fiber (SMF). This mode is detected by the silicon photomultiplier. Reprinted with permission from Cohen et al., Opt. Express 22, 11945 (2014).⁹⁷ Copyright 2014 Optical Society of America.

The expectation values of the parity operator ${\hat{Π}}_{a}$ (in our notation) and $\hat{Z}$ 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 ${\bar{n}}_{0} = 200$ 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 $\frac{λ}{288}$ ⁠, where $λ = 780 nm$ 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 $\sqrt{2}$ ⁠, 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.

Fig. 13.

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Expectation values of the parity operator and the no-photons-detected projector $\hat{Z}$ ⁠. For each curve, the average photon numbers are 4.6 ± 0.2 (purple, square), 25 ± 1 (red, circle), 200 ± 8 (blue, triangle), 1190 ± 50 (pink, inverted triangle), and 4150 ± 150 (green, rhombus). The classic interference curves are presented in black-dashed lines on each plot. Errors are not shown as they are too small to be represented. Reprinted with permission from Cohen et al., Opt. Express 22, 11945 (2014).⁹⁷ Copyright 2014 Optical Society of America.

Fig. 14.

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Phase uncertainties obtained through experimental data for ${\bar{n}}_{0} = 200$ (blue, square) as a function of the phase $ϕ$ for both parity (left-most) and no-photons-detected probability P₀ (right-most). Fits based on the theory are represented by solid black lines and a correction made to account for imperfect visibility [see Eq. (9) in Cohen et al.⁹⁷] by orange-dashed lines. Errors representing the standard deviation in the phase uncertainty estimation process are shown when larger than their symbols. Reprinted with permission from Cohen et al., Opt. Express 22, 11945 (2014).⁹⁷ Copyright 2014 Optical Society of America.