Broadband pulsed quadrature measurements with calorimeters

Abstract

A general one-dimensional quantum optical mode is described by a shape in the time or frequency domain. A fundamental problem is to measure a quadrature operator of such a mode. If the shape is narrow in frequency this can be done by pulsed homodyne detection, in which the mode and a matched local oscillator (LO) interfere on a beamsplitter, whose output ports are monitored by photo-detectors. The quadrature value is proportional to the difference between the photo-detectors’ signals. When the shape of the mode is broad in frequency, the lack of uniform response of the detectors across the spectrum prevents direct application of this technique. We show that pulsed homodyne detection can be generalized to broadband pulsed (BBP) homodyne detection setups with detectors such as calorimeters that detect total energy instead of total number of photons. This generalization has applications in quantum measurements of femtosecond pulses, and, speculatively, measurements of Rindler modes to verify the temperature of Unruh radiation. Like pulsed homodyne detection, BBP homodyne detection requires choosing the LO pulse such that the subtracted signal approaches the desired quadrature measurement for large LO amplitudes. A distinctive feature of the technique is that the LO pulse does not belong to the mode of the quadrature being measured. We analyze how the implemented measurement approaches an ideal quadrature measurement with growing LO amplitude. We prove that the moments of the measurement converge to the moments of the quadrature and that the measurement distributions converge weakly.

1 Introduction

Homodyne detection is an indispensable part of many quantum optics experiments. In homodyne detection one measures quadrature operators of an optical signal of interest. It is widely used in both the frequency and the time domains; for classic examples see [1, 2]. In homodyne detection one interferes the input signal with a high-amplitude, mode-matched local oscillator (LO) on a balanced beamsplitter and measures the light exiting the two output ports of the beamsplitter with high-efficiency detectors such as photodiodes (Fig. 1). The quadrature measurement outcome is obtained by scaling the difference between the detector outputs. The measurement outcome approaches that of an ideal quadrature measurement in the limit of large LO amplitude. Homodyne detection can be performed with an always-on LO (continuous wave homodyne) or with a pulsed LO (pulsed homodyne). In continuous wave homodyne, detectors continuously record intensity. To determine a specific quadrature measurement outcome, the intensity difference is integrated against the shape in time defining the quadrature, as long as this shape’s spectrum is within the bandwidth of the detector. Many quadratures can be simultaneously measured in this scheme. In pulsed homodyne, the LO is pulsed with a shape matching that of the quadrature of interest. Only the quadrature matching the LO is measured, but the detectors need not be time-resolving and the LO power can be concentrated where it is needed. It has the benefits of simplicity and is suitable for the case where the quadrature of interest is known ahead of time. Pulsed homodyne is required when using slow detectors such as calorimeters.

Standard pulsed homodyne detection requires the detector outputs to be proportional to the total number of photons arriving during the pulse’s time interval. This restricts the bandwidth of the LO pulse and of the optical signal during the pulse’s time interval to a spectral band over which the detectors have nearly uniform efficiency. To approach a single-shot quadrature measurement, the efficiency needs to be close to unity over the spectral band. There are no detectors with a wide enough spectral band to measure quadratures of optical signals with octave-spanning bandwidth such as those associated with femtosecond pulses. Such octave-spanning quadratures are also of interest in studies of the Unruh effect [3], which arises for accelerating observers due to the non-zero temperature of Rindler modes. Rindler modes are naturally extremely broadband in an inertial frame.

One type of detector with the potential for high-efficiency detection over a broad frequency band is a calorimeter. Rather than counting photons, calorimeters record total energy. Calorimeters are widely used and can achieve near-unit efficiency for measuring energy across the spectrum. For example transition edge sensors (TES) directly detect the change of temperature due to incident light on a superconducting island at the transition temperature [4, 5] and can be designed to have high efficiency over a wide bandwidth. In 1998, Ref. [6] demonstrated broadband detection over a $3.5$ octave bandwidth from $0.3\;\textrm{eV}$ to $3.5\;\textrm{eV}$ with an energy resolution of about $0.15\;\textrm{eV}$. Some of these parameters have since been improved. For example, Ref. [7] achieved $60\;\textrm{\%}$ efficiency with an energy resolution of $0.067\;\textrm{eV}$ in a band around $0.8\;\textrm{eV}$.

To enable the use of detectors such as calorimeters for measuring specific quadratures of broadband modes, we introduce broadband pulsed (BBP) homodyne detection. BBP homodyne is based on detectors that record observables, such as total energy, that are expressed as linear combinations of photon numbers in the modes of an orthogonal mode basis. As in standard pulsed homodyne, the BBP homodyne measurement outcome is the scaled difference between the detector outputs. We show that arbitrary quadratures expressible in this mode basis can be measured with a high-amplitude LO whose pulse shape is determined by the quadrature of interest. A feature of BBP homodyne is that the mode of the LO is typically different from the mode whose quadrature one wishes to measure. BBP homodyne is distinguished from methods that allow for the parallel measurement of narrow-band quadratures across a broad spectral band, such as those described in Ref. [8]. Such parallel quadrature measurements can in principle be combined in software to obtain measurement outcomes for specific broadband modes of interest, at the cost of the additional resources and high LO energy required for the simultaneous measurements of many narrow-band quadratures.

Theoretical treatments of pulsed homodyne detection argue that the homodyne measurement outcomes correspond to an observable that differs from the quadrature to be measured by an operator proportional to the inverse of the LO amplitude. For well-behaved states, such as those with bounded quadrature moments, this implies that the moments of the homodyne measurement outcome converge to those of the quadrature. We show that for BBP homodyne, the moments of the homodyne measurement outcomes also converge to those of the quadrature. For homodyne detection in general, the difference between a moment of the homodyne measurement outcomes and that of the quadrature depends on the properties of the state and the order of the moment, assuming the LO amplitude is finite. This difference can be arbitrarily large at any finite LO amplitude. Therefore, the convergence of the measurement outcome distribution is not addressed by considerations of the moments alone. An early study of the finite-LO amplitude behavior and the convergence of the outcome distribution is Ref. [9], where these outcome distributions are derived and compared to quadrature outcome distributions for superpositions of two coherent states. Several more detailed studies investigating the finite-LO amplitude measurement operators in the context of operational homodyne detection followed [10, 11]. Beyond convergence of moments, it desirable to have weak convergence of measurement distributions, which can be defined as convergence of expectation values of continuous bounded functions of the measurement outcomes. A general theory relating the convergence of moments to weak convergence was developed and applied by J. Kiukas and P. Lahti [12, 13]. Here, we apply this theory to show that weak convergence holds also for BBP homodyne.

To present our results, we begin with a review of standard homodyne detection in Sect. 2. We introduce BBP homodyne detection in Sect. 3. We then investigate the convergence of BBP homodyne measurements to ideal quadrature measurements. Convergence of moments and weak convergence of the measurement distribution is established in Sect. 4. We conclude with a discussion of BBP and our results in Sect. 5.

2 Homodyne detection

Standard pulsed homodyne detection is a way to measure a quadrature of a mode of a field. Typical applications involve electromagnetic fields whose excitations are photons, and we use terminology accordingly. Such a field may be described by annihilation and creation operator fields in momentum space, see [14] for a pedagogical treatment. We denote generic modes by $a,b,\ldots$ and the corresponding mode (annihilation) operators by $\hat{a},\hat{b},\ldots$ with or without indices. Mode operators satisfy $\left[\hat{a},\hat{a}^{\dagger}\right]=1$. In the physical field’s state space, $\hat{a}$ annihilates the vacuum, and $\hat{a}^{\dagger}$ applied to the vacuum creates a photon in mode $a$ with a particular wavefunction in momentum space. The number operator for generic mode $a$ is $\hat{n}_{a}=\hat{a}^{\dagger}\hat{a}$ and counts the number of photons in this mode. We denote the vacuum state of a mode by $\ket{0}$. The state space of a mode is the Hilbert space spanned by the number states $\ket{n}=\left(\hat{a}^{\dagger}\right)^{n}\ket{0}/\sqrt{n!}$.

We use the convention that generalized quadratures of mode $a$ are defined by operators $\hat{q}_{a,\alpha}=-i(\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a})$ where $\alpha$ is a complex number. Conventional, canonically conjugate $\hat{x}$ (“position”) and $\hat{p}$ (“momentum”) operators associated with the mode when viewed as a quantum harmonic oscillator are $\hat{x}=\hat{q}_{a,i/\sqrt{2}}$ and $\hat{p}=\hat{q}_{a,-1/\sqrt{2}}$. A coherent state is a normalized state $\ket{\beta}$ that satisfies $\hat{a}\ket{\beta}=\beta\ket{\beta}$. This identity determines $\ket{\beta}$ up to a global phase. Having fixed the normalized vacuum state $\ket{0}$, we fix the phase by requiring that $\bra{\beta}\ket{0}$ is positive real. This is equivalent to requiring that $\ket{\beta}$ has real, positive coefficient on $\ket{0}$ when expressed in the number basis. This coefficient is necessarily $e^{-|\beta|^{2}/2}$. The overlap formula for coherent states is then given by

$$\innerproduct{\beta}{\alpha} =e^{-(|\beta|^{2}+|\alpha|^{2}-2\alpha\beta^{*})/2}.$$ (1)

Displacement operators for mode $a$ are given by $\hat{D}_{\beta}=e^{i\hat{q}_{a,\beta}}$, and we say that $\hat{q}_{a,\beta}$ generates the displacement $\hat{D}_{\beta}$. They satisfy $\hat{D}_{-\beta}=\hat{D}_{\beta}^{\dagger}$ and transform the mode operator according to

$$\hat{D}_{-\beta}\hat{a}\hat{D}_{\beta} =\hat{a}+\beta.$$ (2)

They displace coherent states as follows:

$$\hat{D}_{\beta}\ket{\alpha} =e^{i\Im{\alpha^{*}\beta}}\ket{\alpha+\beta}.$$ (3)

The phase on the right-hand side can be calculated from $\hat{D}_{-\alpha-\beta}\hat{D}_{\beta}\hat{D}_{\alpha}=e^{i\Im{\alpha^{*}\beta}}$ and is given by twice the signed area of the triangular path followed in going from $0$ to $\alpha$ to $\alpha+\beta$ and back to $0$. For example, see Eq. (2) of Ref. [15].

Two modes $a$ and $b$ are orthogonal if $[\hat{a},\hat{b}^{\dagger}]=0$. This implies that the two associated photon wavefunctions are orthogonal and that polynomials of $\hat{a}$ and $\hat{a}^{\dagger}$ commute with polynomials of $\hat{b}$ and $\hat{b}^{\dagger}$. The joint state space of the two modes can be represented as the tensor product of the state spaces of each mode. For our analysis, we consider finitely many modes at a time. A family of orthogonal modes $\{a_{k}\}_{k=1}^{N}$ is associated with a vector of mode operators $\hat{\bm{a}}=(\hat{a}_{1},\ldots,\hat{a}_{N})$ satisfying $[\hat{a}_{k},\hat{a}_{l}^{\dagger}]=\delta_{kl}$. The joint vacuum of the modes is denoted by $\ket{\bm{0}}$, and its density matrix is $\bm{0}$. The joint state space of the modes is spanned by Fock states $\ket{\bm{n}}$, where $\bm{n}=(n_{1},\ldots,n_{N})$ is the vector of mode occupation numbers. That is, $n_{k}$ is the number of photons in mode $k$.

General mode operators in the system defined by the modes $\bm{a}$ can be associated with complex, normalized amplitude vectors $\bm{\beta}=(\beta_{1},\ldots,\beta_{N})$, $\sum_{k=1}^{N}|\beta_{k}|^{2}=1$. The operator $\hat{a}_{\bm{\beta}}=\sum_{k=1}^{N}\beta_{k}\hat{a}_{k}$ satisfies the defining properties of mode annihilation operators, namely $\hat{a}_{\bm{\beta}}\ket{\bm{0}}=0$ and $\left[\hat{a}_{\bm{\beta}},\hat{a}_{\bm{\beta}}^{\dagger}\right]=1$. As a result, generalized quadratures, number operators and states of mode $a_{\bm{\beta}}$ are well-defined.

Generalized quadratures of the modes $\bm{a}$ are defined as

$$\hat{q}_{\bm{a},\bm{\alpha}} =\sum_{k=1}^{N}-i(\alpha_{k}\hat{a}_{k}^{\dagger}-\alpha_{k}^{*}\hat{a}_{k}).$$ (4)

The quadrature is normalized if $\sum_{k}|\alpha_{k}|^{2}=1/2$. The corresponding displacement operators are

$$\hat{D}_{\bm{a},\bm{\alpha}} =e^{i\hat{q}_{\bm{a},\bm{\alpha}}}$$

$$=\prod_{k}e^{i\hat{q}_{a_{k},\alpha_{k}}}.$$ (5)

The coherent states of modes $\bm{a}$ are

$$\ket{\bm{\alpha}} =\otimes_{k=1}^{N}\ket{\alpha_{k}}_{a_{k}}$$

$$=\hat{D}_{\bm{a},\bm{\alpha}}\ket{\bm{0}},$$ (6)

where we use labels on the kets to specify the subsystem in a tensor product that the state belongs to. The algebraic properties of coherent states and displacement operators generalize accordingly.

Figure 1: Multi-mode homodyne measurement configuration. The signal modes enter from the left on modes ${\bm{a}}$ with mode operators $\hat{\bm{a}}$. The LO modes enter from the bottom on modes ${\bm{b}}$. To simplify calculations, the LO modes are initially in vacuum, and the LO coherent state is explicitly prepared by the displacement operator $\hat{D}_{{\bm{b}},R\beta}$. Modes $\bm{{a}}$ and ${\bm{b}}$ are combined on a balanced beam splitter (BS). The beam splitter’s outgoing modes are ${\bm{c}}$ and ${\bm{d}}$. They are measured with photon counters (standard pulsed homodyne) or calorimeters (BBP pulsed homodyne). In both cases the observable associated with the detector is of the form $\hat{E}=\sum_{k}\omega_{k}\hat{f}_{k}^{\dagger}\hat{f}_{k}$ with $\bm{f}=\bm{c}$ or $\bm{d}$. For standard pulsed homodyne, $\omega_{k}=1$ for all $k$. For BBP homodyne, $\omega_{k}$ is the energy of photons in the $k$’th mode. The homodyne measurement result is determined by subtracting one detector’s output from the other and rescaling the result by a factor of $1/R$. In general, $\omega_{k}$ can be arbitrary positive weights.

The configuration for pulsed homodyne is shown in Fig. 1, where for standard pulsed homodyne, the dectors count total number of photons. The full state space of the signal is that of $N$ modes $\bm{a}$ with mode operators $\hat{\bm{a}}$. The goal is to measure the generalized quadrature $\hat{q}_{\bm{a},\bm{\beta}}$. We refer to the quadrature to be measured as the target quadrature. For this purpose an LO is introduced on a family of modes $\bm{b}$, each of which is matched to the corresponding signal mode. To approximately measure a generalized quadrature of the modes, the LO state is the coherent state $\ket{R\bm{\beta}}_{\bm{b}}$, where $R$ is a large real number. It is convenient to explicitly introduce the displacement operator $\hat{D}_{\bm{b},R\bm{\beta}}$ that makes this state from vacuum, as shown in the figure. The initial state consists of the signal state $\rho_{\bm{a}}$ on the signal modes $\bm{a}$, and vacuum $\bm{0}_{\bm{b}}$ on the LO modes $\bm{b}$. The signal modes and the matching LO modes are combined on a balanced beamsplitter. The outgoing modes are labeled $\bm{c}$ and $\bm{d}$. With Heisenberg evolution, we can express the outgoing mode operators in terms of the original signal and the pre-displacement LO modes as follows:

$$\hat{\bm{c}} =\frac{1}{\sqrt{2}}\left(\hat{\bm{a}}+\hat{\bm{b}}+R\bm{\beta}\right),$$

$$\hat{\bm{d}} =\frac{1}{\sqrt{2}}\left(\hat{\bm{a}}-\hat{\bm{b}}-R\bm{\beta}\right).$$ (7)

Here we made a particular sign and phase choice for the balanced beamsplitter. The homodyne configuration is completed by photon counting on each of the two outgoing arms of the beamsplitter, subtracting the counts obtained, and dividing by $R$. To describe this measurement, we subtract the scaled total photon number operators for the arms to obtain the measurement operator

$$\hat{h} =\frac{1}{R}\left(\hat{n}_{\bm{c}}-\hat{n}_{\bm{d}}\right)$$

$$=\frac{1}{R}\left(\hat{\bm{c}}^{\dagger}\cdot\hat{\bm{c}}-\hat{\bm{d}}^{\dagger}\cdot\hat{\bm{d}}\right)$$

$$=\bm{\beta}\cdot\hat{\bm{a}}^{\dagger}+\bm{\beta}^{*}\cdot\hat{\bm{a}}+\frac{1}{R}\left(\hat{\bm{a}}^{\dagger}\cdot\hat{\bm{b}}+\hat{\bm{b}}^{\dagger}\cdot\hat{\bm{a}}\right).$$ (8)

Here, the inner product between two vectors is denoted by “$\cdot$” and complex conjugation and dagger is applied element-wise. Intuitively, for large $R$ the summand with a factor of $\frac{1}{R}$ can be neglected, so $\hat{h}$ approaches a measurement of the target quadrature $\hat{q}_{\bm{a},i\bm{\beta}}$.

Standard pulsed homodyne requires two ideal photon counters. In practice, noisy, high-efficiency photon counters are used. If the noise increases at a sublinear rate with photon number, a good quadrature measurement can still be obtained by increasing the LO amplitude $R$. High-efficiency photo-diodes have this property even though they are unable to resolve photon numbers. Another effect that needs to be considered is that the deviation of $\hat{h}$ from the target quadrature measurement is affected by the state of photons in modes orthogonal to that of the target quadrature. The amplitude $R$ needs to be increased to make the signal from such photons negligible. Finally, the efficiency of the photon counters needs to be uniformly high for all the relevant modes. Most photon counters have high efficiency in a relatively narrow frequency band, which limits the range of energies associated with the modes $a_{k}$. Calorimeters have the potential to overcome this limitation but measure the total energy $\sum_{k=1}^{N}\omega_{k}\hat{n}_{a_{k}}$ rather than the total photon number.

3 Quadrature measurement with calorimeters

An ideal calorimeter measures the total energy of the light. If we consider $N$ modes $\bm{a}$, then a calorimeter measures the energy operator

$$\hat{E}=\sum_{k=1}^{N}\omega_{k}\hat{a}_{k}^{\dagger}\hat{a}_{k}.$$ (9)

where $\omega_{k}$ is the energy of a photon in mode $k$. One can think of a calorimeter as a device performing measurement of a weighted sum of photon numbers in a family of modes, where the weights are the energies of the modes. Our treatment does not require $\omega_{k}$ to be the energy of a mode $k$. It can be an arbitrary real, positive weight. If $\omega_{k}=1$ for all $k$, $\hat{E}$ measures the total photon number. We therefore refer to the vector $\bm{\omega}$ as mode weights.

Consider the homodyne setup with calorimeters instead of photon counters at the outgoing arms of the balanced beamsplitter. The the measurement operator corresponding to subtracting the calorimeter measurement results and dividing by $R$ is

$$\frac{1}{R}\Delta\hat{E} =\frac{1}{R}\sum_{k=1}^{N}\omega_{k}\big(\hat{c}_{k}^{\dagger}\hat{c}_{k}-\hat{d}^{\dagger}_{k}\hat{d}_{k}\big)$$

$$=\sum_{k=1}^{N}\omega_{k}(\hat{a}_{k}^{\dagger}\beta_{k}+\beta_{k}^{*}\hat{a}_{k})+\frac{1}{R}\sum_{k=1}^{N}\omega_{k}(\hat{a}_{k}^{\dagger}\hat{b}_{k}+\hat{b}_{k}^{\dagger}\hat{a}_{k}).$$ (10)

Here all operators represent incoming mode operators according to Heisenberg evolution. In particular $\Delta\hat{E}$ when so expressed depends on the LO displacement and therefore on $R$. Assuming that the contribution from terms multiplied by $\frac{1}{R}$ is negligible, this approaches a measurement of the target generalized quadrature $\hat{q}_{\bm{a},i(\bm{\omega}*\bm{\beta})}$, where “*” denotes the element-wise product defined by $(\bm{\gamma}*\bm{\gamma^{\prime}})_{k}=\gamma_{k}\gamma^{\prime}_{k}$. From now on, we omit the mode label on operators such as $\hat{q}$ when the modes are clear from context. For example, we write $\hat{q}_{\bm{\alpha}}$ for $\hat{q}_{\bm{a},\bm{\alpha}}$. If we wish to measure the generalized coordinate $\hat{q}_{\bm{\alpha}}$, we choose $\bm{\beta}=-i\bm{\alpha}*(1/\bm{\omega})$, where $1/\bm{\omega}$ is the vector with $k$’th entry $1/\omega_{k}$. The required LO amplitude is $R\bm{\beta}=-i\bm{\alpha}*(R/\bm{\omega})$.

In our treatment, quadrature expectations and coherent state amplitudes are unitless and scaled so that the vacuum expectation of the square of a normalized quadrature is $1/2$. The units of the weights $\bm{\omega}$ and of the scale $R$ are therefore identical. In the case of calorimeters, they both have the same energy units. For the purposes of BBP homodyne, quantities of interest depend on $R$ and $\bm{\omega_{k}}$ only through ratios such as $R/\omega_{k}$.

For analyzing the behavior of BBP homodyne, we define $\delta=1/R$ and introduce the BBP measurement operator

$$\hat{q}_{\bm{\alpha},\delta} =\delta\Delta\hat{E}$$

$$=\hat{q}_{\bm{a},\bm{\alpha}}+\delta\quantity( \qty(\bm{\omega}*\hat{\bm{a}}^{\dagger}\vphantom{\hat{\bm{b}}^{\dagger}})\cdot\hat{\bm{b}} + \qty(\bm{\omega}*\hat{\bm{b}}^{\dagger})\cdot\hat{\bm{a}} ).$$ (11)

As noted previously $\Delta\hat{E}$ is defined as an operator on the incoming modes and therefore depends on $\delta$ through the displacement on the LO modes. The associated LO amplitudes are

$$\bm{\beta}_{\textrm{LO},\bm{\alpha},\delta} =-i\bm{\alpha}*\frac{1}{\bm{\omega}\delta}.$$ (12)

We define $\hat{q}_{\bm{\alpha},0}=\hat{q}_{\bm{a},\bm{\alpha}}$. When $\bm{\alpha}$ is clear from context, we abbreviate $\hat{q}_{\delta}=\hat{q}_{\bm{\alpha},\delta}$ and $\hat{q}=\hat{q}_{\bm{\alpha},0}$.

The BPP measurement operator $\hat{q}_{\delta}$ is by design proportional to a difference of two commuting total energy operators at the outgoing modes, so the spectrum of $\hat{q}_{\delta}$ is discrete. The outgoing mode Fock states that diagonalize the energy operators correspond to displaced Fock states in the incoming modes, where the displacement diverges to infinity as $\delta$ goes to zero. In contrast, $\hat{q}$ has continuous spectrum and improper eigenspaces associated with its spectral measure. This immediately suggests that the convergence of $q_{\delta}$ to the target quadrature $q_{0}$ is not straightforward. The BBP measurement operator is unbounded, as is the quadrature measurement. As a result, there are states in the Hilbert space not in the domains of these operators, and for such states convergence is impossible. Thus proofs and quantification of convergence are necessarily state-dependent.

The BBP homodyne configuration, as described above, has perfect calorimeters. However, as with the standard homodyne configuration, noisy but high-efficiency calorimeters can be used, provided that the noise scales sublinearly with total energy. We do not include such noise in our analysis.

4 Weak convergence to a quadrature measurement

Our first convergence result establishes that the moments for BBP homodyne measurement outcome distributions converge to the moments of the target quadrature. This result is well known for standard homodyne, for example, see Ref. [11], Eq. (3.2) and following, or Ref. [12] Prop. 8. The proofs for standard homodyne generalize with little modification to BBP homodyne. We use the angle-bracket notation $\langle\cdot\rangle_{\rho}$ for expectations of operators with respect to state $\rho$. We omit the subscript when the state with respect to which expectations are computed is clear from context.

Theorem 4.1.

Suppose the state $\rho_{\bm{a}}$ on the signal modes $\bm{a}$ has well-defined expectations for all polynomials in the mode operators and their adjoints up to degree $n$, and the joint initial state is $\rho_{\bm{a}}\otimes\bm{0}_{\bm{b}}$. Then the moments of $\hat{q}_{\delta}$ and $\hat{q}$ satisfy

$$\langle\hat{q}_{\delta}^{n}\rangle =\langle\hat{q}^{n}\rangle+O(\delta^{2}),$$ (13)

where the constant in the order notation depends on $\rho_{\bm{a}}$.

Proof.

Define the operator $\hat{C}=(\hat{q}_{\delta}-\hat{q})/\delta=\quantity(\bm{\omega}*\hat{\bm{a}}^{\dagger}\vphantom{\hat{\bm{b}}^{\dagger}})\cdot\hat{\bm{b}}+\quantity(\bm{\omega}*\hat{\bm{b}}^{\dagger})\cdot\hat{\bm{a}}$, which does not depend on $\delta$. We expand $\hat{q}_{\delta}^{n}=(\hat{q}+\delta\;\hat{C})^{n}$ as a sum of $2^{n}$ monomials expressed as ordered products of $\hat{q}$ and $\delta\;\hat{C}$. We can order the monomials by the power of $\delta$ that multiplies them. Let $\hat{Q}_{k}$ be the sum of the monomials that are multiplied by $\delta^{k}$, so $\hat{q}_{\delta}^{n}=\sum_{k=0}^{n}\hat{Q}_{k}$. Then $\hat{Q}_{0}=\hat{q}^{n}$, so $\langle\hat{Q}_{0}\rangle=\langle\hat{q}^{n}\rangle$ contributes the first term on the right-hand side of Eq. (13). We can express $\hat{Q}_{1}$ as

$$\hat{Q}_{1} =\delta\sum_{k=0}^{n-1}\hat{q}^{k}\hat{C}\hat{q}^{n-k-1}.$$ (14)

The factors of $\hat{q}$ in the sum act only on the signal modes $\bm{a}$, and $\hat{C}$ is linear in the LO mode operators. Since $\rho$ is vacuum on the pre-displacement LO modes, the expectations of the summands of $\hat{Q}_{1}$ are zero. Consequently $\langle\hat{Q}_{1}\rangle=0$. The remaining terms in the expansion are multiplied by a factor of order $\delta^{2}$ or smaller. Their expectations may be expressed as expectations of monomials of degree at most $n$ in the signal mode operators and their adjoints, and of degree at most $n$ in the LO mode operators and their adjoints. Taking account of the vacuum state in the LO mode, these expectations reduce to expectations of $\rho_{\bm{a}}$ of monomials of degree at most $n$ in the signal mode operators. The expectations of these monomials are finite by assumption and do not depend on $\delta$. Thus

$$\langle\hat{q}_{\delta}^{n}\rangle =\sum_{k=0}^{n}\langle\hat{Q}_{k}\rangle$$

$$=\langle\hat{Q}_{0}\rangle+\langle\hat{Q}_{1}\rangle+O(\delta^{2})$$

$$=\langle\hat{q}^{n}\rangle+O(\delta^{2}).$$ (15)

∎

We explicitly evaluate the differences between the moments of $\hat{q}_{\delta}$ and $\hat{q}$ for the first and second moment. The first moment does not depend on $\delta$ and is identical to the target quadrature’s. Because the pre-displacement LO modes are in vacuum,

$$\langle\hat{q}_{\delta}-\hat{q}\rangle =\delta\sum_{k=1}^{N}\omega_{k}\langle\hat{a}^{\dagger}_{k}\hat{b}_{k}+\hat{a}_{k}\hat{b}^{\dagger}_{k}\rangle$$

$$=0.$$ (16)

For the second moment we have

$$\langle\hat{q}_{\delta}^{2}-\hat{q}^{2}\rangle =\delta^{2}\sum_{k,k^{\prime}=1}^{N}\omega_{k}\omega_{k^{\prime}}\left\langle(\hat{a}^{\dagger}_{k}\hat{b}_{k}+\hat{a}_{k}\hat{b}^{\dagger}_{k})(\hat{a}^{\dagger}_{k^{\prime}}\hat{b}_{k^{\prime}}+\hat{a}_{k^{\prime}}\hat{b}^{\dagger}_{k^{\prime}})\right\rangle$$

$$\phantom{=\ }-\delta\sum_{k=1}^{N}\omega_{k}\left\langle(\hat{a}^{\dagger}_{k}\hat{b}_{k}+\hat{a}_{k}\hat{b}^{\dagger}_{k})\hat{q}\right\rangle-\delta\sum_{k=1}^{N}\omega_{k}\left\langle\hat{q}(\hat{a}^{\dagger}_{k}\hat{b}_{k}+\hat{a}_{k}\hat{b}^{\dagger}_{k})\right\rangle$$

$$=\delta^{2}\sum_{k}\omega^{2}_{k}\langle\hat{n}_{a_{k}}\rangle,$$ (17)

where all terms other than those with factors of the form $\hat{b}_{k}\hat{b}_{k}^{\dagger}$ are zero because the pre-displacement LO modes are in vacuum. If it is known that the signal state is Gaussian, then it is sufficient to measure the first and second moments of its quadratures. The above shows that the second moments of $\hat{q}_{\delta}$ are biased high by a term of order $\delta^{2}$ with a coefficient that can be estimated knowing only bounds on the expected photon numbers.

Theorem 4.1 applies specifically to states with well-defined expectations for all polynomials of mode operators. A dense linear space of pure states with such well-defined expectations is the set $\mathcal{D}_{S}$ of Schwartz states, defined as states whose Wigner functions decay superpolynomially [16]. This set of states is preserved by all polynomials of the mode operators and their adjoints. It includes number states and coherent states and their finite linear combinations.

Next we show that the moment convergence for a restricted family of signal states in Thm. 4.1 implies that for every bounded continuous function $f$ of the reals and every signal state, the expectations of $f(\hat{q}_{\delta})$ with respect to the BBP homodyne outcome distributions converge to the expectation of $f(\hat{q})$. This property is equivalent to weak convergence in the sense of probabilities of the positive operator valued measures (POVMs) realized by BBP homodyne to the spectral measure of $\hat{q}$ (see [12] Prop. 3).

Theorem 4.2.

For every continuous complex-valued bounded function $f$ on the reals and for every state $\rho=\rho_{\bm{a}}\otimes\bm{0}_{\bm{b}}$, we have

$$\lim_{\delta\rightarrow 0}\langle f(\hat{q}_{\delta})\rangle =\langle f(\hat{q})\rangle.$$ (18)

Proof.

To prove the theorem we implement a version of the sequence of steps given in Sect. 4 of Ref. [12] for establishing weak convergence for measurement schemes. In this reference, the steps are implemented to prove weak convergence of standard homodyne with one mode. The first step is to identify a dense subspace of states on which the target quadrature’s measurement outcome distribution is determined by its moments (Ref. [12] Def. 2). A probability distribution $\mu$ on the reals is determined by its moments if for every probability distribution $\nu$ whose moments are the same as those of $\mu$, we have $\mu=\nu$. The set $\mathcal{D}_{\text{coh}}\subseteq\mathcal{D}_{S}$ of finite linear combinations of coherent states suffices for this purpose. For one mode, this is a consequence of Ref. [12] Lemma 2. Because $\hat{q}$ is the quadrature of one mode, this Lem. 2 suffices for BBP homodyne. See also the discussion after Proposition 2 in the reference. The next step is to verify that for states in $\mathcal{D}_{\text{coh}}$, the positive-operator-valued measures (POVMs) associated with $\hat{q}_{\delta}$ have moments converging to those of $\hat{q}$. For BBP homodyne, since $\mathcal{D}_{\text{coh}}$ is a subset of the set of Schwartz states, this is a consequence of Thm. 4.1. For the purpose of applying the results of Ref. [12], the POVMs associated with $\hat{q}_{\delta}$ are the POVMs on the signal modes obtained from the projection-valued measures of the operators $\hat{q}_{\delta}$ by fixing the pre-displacement state of the LO modes to be vacuum. The POVM for $\hat{q}$ is projection-valued, but the POVMs for $\hat{q}_{\delta}$ are not, they are positive-operator valued and referred to as “semispectral measures” in Ref. [12]. With this, the conditions of Ref. [12] Prop. 5 are satisfied. That is, with the definitions of this reference, because of moment convergence, there is a POVM that is a moment limit of the POVMs associated with $\hat{q}_{\delta}$. One such moment limit is the spectral measure of $\hat{q}$. Since the latter is determined by its moments, this moment limit is unique. The conclusion from the reference’s Prop. 5 is that the POVMs associated with $\hat{q}_{\delta}$ converge weakly in the sense of probabilities to the spectral measure of $\hat{q}$. This is equivalent to the conclusion of our theorem. ∎

There are many equivalent definitions for weak convergence of the measures associated with $\hat{q}_{\delta}$ to the measures of $\hat{q}$. The version given in Thm. 4.2 corresponds to Ref. [12] Prop. 3 (iv), which is also equivalent to the convergence of overlaps as expressed in the following corollary.

Corollary 4.3.

Let $\bm{g}$ be any family of modes or other quantum systems that are not involved in the BBP homodyne measurements, and $f$ a continuous complex-valued bounded function on the reals. For all joint pure states $\ket{\phi}$ and $\ket{\psi}$ of $\bm{g}$, the signal, and the LO modes, if $\ket{\phi}$ and $\ket{\psi}$ are vacuum on the LO modes, then we have

$$\lim_{\delta\rightarrow 0}\bra{\phi}f(\hat{q}_{\delta})\ket{\psi} =\bra{\phi}f(\hat{q})\ket{\psi}.$$ (19)

Proof.

For any complex, bilinear form $\bra{\phi}\hat{A}\ket{\psi}$ with bounded operator $\hat{A}$,

$$\bra{\phi}\hat{A}\ket{\psi} =\frac{1}{4}\sum_{j=0,1,2,3}(-i)^{j}(\bra{\phi}+(-i)^{j}\bra{\psi})\hat{A}(\ket{\phi}+i^{j}\ket{\psi})$$

$$=\frac{1}{4}\sum_{j=0,1,2,3}(-i)^{j}\bra{\rho_{j}}\hat{A}\ket{\rho_{j}},$$ (20)

where $\ket{\rho_{j}}=\ket{\phi}+i^{j}\ket{\psi}$. We can re-express $\bra{\rho_{j}}\hat{A}\ket{\rho_{j}}=\tr(\ketbra{\rho_{j}}\hat{A})$. After substituting $\hat{A}=f(\hat{q}_{\delta})$ and tracing out systems other than the signal and LO modes in the state $\outerproduct{\rho_{j}}{\rho_{j}}$, we can apply Thm. 4.2 to complete the proof of the corollary. The identity in Eq. (20) is an instance of the polarization identity, a textbook method used to reconstruct an inner product from the associated norm, for instance see [17, 18]. ∎

5 Discussion

BBP homodyne can be seen to be a generalization of standard pulsed homodyne by setting all the weights to $\omega_{k}=1$. The well-known properties of standard pulsed homodyne are preserved. In particular, the moments of the BBP homodyne observables converge to the moments of the target quadrature and the distributions of BBP measurement outcomes converge weakly to that of the target quadrature.

The main motivation for introducing BBP homodyne is as a way of taking advantage of calorimeters to measure quadratures of broadband modes such as those present in optical femtosecond pulses or modes of interest in quantum field theory such as Rindler modes. The BBP homodyne configuration preserves the simplicity of the standard homodyne experimental configuration with only two detectors after one beamsplitter. The detectors need not resolve time and can be slow. An alternative approach to homodyne measurements of broadband modes is to use a device such as a Bragg grating to split the incoming modes according to their wavelengths, then combine standard, narrowband homodyne measurements at each wavelength. A version of this technique is proposed in Ref. [8]. This technique has the advantage of being able to simultaneously measure multiple quadratures across the resolved spectrum, at the cost of a more complicated experimental configuration.

We have not yet considered the effect of energy-dependent inefficiency on the performance of BBP homodyne. For standard pulsed homodyne, detector inefficiency can be taken into account with an operational theory of homodyne [10]. It is also of interest to directly determine and exploit the effective POVMs at finite LO amplitudes as done, for example in Ref. [11]. Going further, when the calorimeters perform well for low incident energy, at small LO amplitude and signal energies, one can directly take advantage of each calorimeter’s measurement outcome, generalizing or bypassing the subtraction, as done for weak-field homodyne in Refs. [19, 20].

For applications of BBP homodyne, it is desirable to quantify the convergence of the measurement outcome distribution to the ideal one for the target quadrature. In many applications, the measurement outcome is used in conditional operations such as displacements of unmeasured modes. Examples include CV quantum teleportation [21, 22] and CV quantum computing [23]. For these applications, it will be helpful to quantify the fidelity of the conditional operations.

For the application of BBP homodyne to characterizing the state of Rindler modes and verifying the thermal states of these modes, it is necessary to determine how to realize the local oscillator and beamsplitter while maintaining the compatibility with the relativistic quantum field under investigation. Because it is impossible to measure standard, single-frequency Rindler modes, it is also necessary to determine how to reveal the desired quadrature information from smeared such modes. Relevant suggestions have been offered for different detection systems in [24].

6 Acknowledgment

We thank Adriana Lita for help with identifying and describing the performance of broadband TES detectors. We also thank Zachary Levine and Michael Mazurek for assistance with reviewing the paper before submission. E. S and J. R. v. M. acknowledge support from the Professional Research Experience Program (PREP) operated jointly by NIST and the University of Colorado. This work includes contributions of the National Institute of Standards and Technology, which are not subject to U.S. copyright.