Experiment

The basic concept of our method for broadband homodyne detection is illustrated in Fig. 1, showing in Fig. 1a the standard homodyne method and in Fig. 1b the parametric homodyne detection as realized by a broadband parametric amplifier acting on the quadratures of the light. To describe the effect of the parametric amplifier in Fig. 1b, we use the common expression for the optical field at the output of a parametric amplifier (based on either three-wave or four-wave mixing (FWM) optical nonlinearity: \(a_{{\mathrm{out}}} = a_{{\mathrm{in}}}{\kern 1pt} {\mathrm{cosh}}(g) + a_{{\mathrm{in}}}^\dagger {\kern 1pt} {\mathrm{sinh}}(g)\) = \(x_{{\mathrm{in}}}{{e}}^g + y_{{\mathrm{in}}}{{e}}^{ - g}\), where \(a_{{\mathrm{in}}},a_{{\mathrm{in}}}^\dagger\) are the input field operators, x in , y in are the input quadratures, and g is the parametric gain. Hence, the parametric amplification amplifies one input quadrature (x in eg) while attenuating the other (y in e−g), indicating that for sufficient amplification, the output field reflects one quadrature of the input primarily without adding noise to the measured quadrature, thus offering a quadrature selective quantum measurement. The amplification process responds instantaneously to time variations of the quadrature amplitudes x(t), y(t) and the amplification bandwidth is limited only by the phase matching conditions in the nonlinear medium, which can easily span an optical bandwidth of 10–100 THz (implications of the time dependence are deferred to a later discussion). In our experiment, we measured the spectrally resolved intensity of the chosen input quadrature x†(ω)x(ω) simultaneously across the entire bandwidth by detecting the output spectrum of a parametric amplifier with an input of broadband squeezed vacuum.

We note that the parametric amplifier used in the measurement need not be ideal. Specifically, since the attenuated quadrature is not measured, it is not necessarily required to be squeezed below vacuum, only to be sufficiently suppressed compared to the amplified quadrature. Consequently, restrictions on the measurement amplifier are considerably relaxed compared to sources of squeezed light, allowing it to operate with much higher gain.

The common source for squeezed light or squeezed vacuum, in our experiment, is also a parametric amplifier. If the amplification is spontaneous (vacuum input), the amplifier attenuates one of the quadratures of the vacuum input state, squeezing its quantum uncertainty. For measuring the squeezing, we exploit the same nonlinearity and the same pump that generates the squeezed state in the first place, thus guaranteeing a bandwidth match of the homodyne measurement to the squeezing process. The quadrature information over a broad frequency range is obtained simultaneously by measuring the spectrum of the light at the output of the detection parametric amplifier. With a single LO—the pump, each individual frequency component is measured independently, and the number of accessible Q-modes (or Q-bits) that could be utilized simultaneously would be multiplied by N (the number of resolved frequency bins) rather than log 2 N. As will be explained later, a single-frequency component of the quadrature is actually a combination of two-frequency modes, commonly termed signal ω s and idler ω i , symmetrically separated around the main carrier frequency Ω.

The experimental demonstration of broadband parametric homodyne consists of two parts (see Fig. 2): first, generation of broadband squeezed vacuum, and second, parametric homodyne detection of the generated squeezing. We generate broadband squeezed vacuum by collinear FWM in a photonic crystal fiber (PCF) that is pumped by narrowband picosecond pulses near the zero dispersion wavelength of the PCF. To measure the generated squeezing, we couple the light generated by the FWM process together with the pump into another PCF, which acts as the measurement parametric amplifier (in the experiment this was the same PCF in the backward direction). After this second (measurement) pass we record the parametric output spectrum to extract the quadrature information (see Fig. 3a).

Fig. 2 Experimental schematic of the parametric homodyne. The experiment consists of two parts: (1) generation of broadband squeezed light and (2) homodyne measurement of the generated squeezing. Broadband two-mode squeezed light is generated via spontaneous four-wave mixing (FWM) in a photonic crystal fiber (PCF) pumped by 12 ps laser pulses (786 nm). After generation, the pump is replaced by an appropriately delayed copy of the original pump light, via a narrowband filter, which allows independent intensity and phase control, to tune the parametric gain and to select the specific quadrature to be measured. Then, the new pump and the FWM light enter the second PCF for the homodyne measurement. After this second (measurement) pass through the amplifier, the pump is separated from the FWM light by a narrowband filter and the FWM light is measured by a spectrometer Full size image

Fig. 3 The procedure of parametric homodyne. Measurement of the quadratures includes three stages: a raw output measurement, b calibration, and c quadrature extraction. a Raw output measurements: In the most general case of arbitrary parametric gain, two measurements are needed to extract the quadrature information: (1) amplifying one quadrature (black); and (2) amplifying the orthogonal quadrature (purple). The specific quadrature to be amplified is defined by tuning the pump phase. The reduction of the raw output beneath the vacuum input level (dashed green) directly indicates squeezing. The inset shows the effect of loss at the input FWM light on the parametrically amplified output. As the loss is increased, the squeezing is reduced and the observed fringe minima rise towards the vacuum level (vertical arrows) even though the total input intensity is considerably decreased (a non-classical signature). b Calibration: To calibrate the parametric amplifier, the output response is measured for a set of three known inputs: (1) idler-input only (blocked signal, I zs —solid blue), (2) vacuum input (blocked entire FWM—both signal and idler, I zsi —dashed green), and (3) zero amplification (blocked pump, I zp —dotted red). c Extracted quadratures (black and purple in accordance with a)—with the analysis detailed in the Methods, quadrature information is extracted. Quadrature squeezing is evident across the entire 55 THz spectrum down to \(\left\langle {x^\dagger x} \right\rangle \approx 0.68\), 32% below the vacuum level (~1.7 dB) Full size image

Since squeezed vacuum is a gaussian state, its quadrature distribution is completely defined by the second moment. We therefore measure the average spectral intensity (with averaging times of a few 10 ms) and reconstruct the average quadrature fluctuations \(\left\langle {x^\dagger x} \right\rangle\),\(\left\langle {y^\dagger y} \right\rangle\). Measurement of the instantaneous intensity distribution is possible with a shorter integration time, but not necessary for squeezed vacuum.

Fringes appear across the output spectrum of the measurement parametric amplifier due to chromatic dispersion in the optical components (filters, windows, etc.), which introduces a varying spectral phase with respect to the pump across the FWM spectrum. Thus, for some frequencies the stretched quadrature is amplified (bright fringes) while for others the squeezed quadrature is amplified (relatively dark fringes), as seen in Fig. 3a. The specific quadrature to be amplified can be controlled by the pump phase (see Methods for more details on the experiment). The broadband squeezing is evident already from the raw output spectrum, shown in Fig. 3a, where reduction of the parametric output below the vacuum noise level (the parametric output when the input is blocked) is observed across the entire 55 THz. To verify this, we varied the squeezing by varying the loss of the input FWM field before the measurement (second) pass through the PCF. As the loss is increased, the squeezing slowly vanishes, and even though the total power entering the fiber is diminished, the minimum fringes at the output of the measurement amplifier rise towards the vacuum input level, as shown in the inset of Fig. 3a.

The extraction of the quadrature information from the measured parametric output assumes knowledge of the parametric gain. The calibration of the parametric amplifier is simple, performed by recording the output spectrum for a set of known inputs (Fig. 3b), when blocking various input fields (signal, idler, or pump). For example, the vacuum level of the parametric amplifier is observed when both the signal and the idler-input fields are blocked (I zsi —zero signal idler). Also, the average number of photons at the input is given by the ratio of the measured output when the signal is blocked (idler only, I zs —zero signal) to the vacuum input level \(\left\langle {N_{\mathrm{i}}} \right\rangle = \frac{{I_{{\mathrm{zs}}}}}{{I_{{\mathrm{zsi}}}}} - 1\). This calibration process is fully described in the Methods. After calibration, we obtain the parametric homodyne results of Fig. 3c, which show ~1.7 dB squeezing across the entire 55 THz bandwidth.

The observed squeezing in our experiment is far from ideal, primarily due to the fact that the pump is pulsed, which induces an undesirable time dependence of both the magnitude and phase of the parametric gain in the squeezing process, as well as in the parametric homodyne detection via self-phase and cross-phase modulation—SPM and XPM. Since our pump pulses are relatively long, their time dependence can be regarded as adiabatic, indicating that the instantaneous squeezing (source) and parametric amplification (measurement) are ideal, but the quadrature axis, squeezing level, and gain of the two amplifiers vary with time, not necessarily at the same rate. Thus, the measured spectrum, which represents a temporal average of the light intensity over the entire pulse, diminishes somewhat the expected squeezing (see illustration in the Methods).

Even with a pulsed pump, however, the various homodyne and calibration measurements are consistent and unequivocal for weak enough pump intensity (see Methods for further details on the pulse-averaging effects). With a pure CW pump, as is generally used in squeezing applications, this pulse-averaging limitation would not exist. Another limitation in our measurements is the need to re-couple the FWM back into the PCF, which introduces an inevitable loss of 30% and reduces the observed squeezing. This “known” loss can either be avoided completely in other experimental configurations or can be calibrated out to estimate the “bare” squeezing level of the measured light source (see Methods).

We verified the properties of the parametric homodyne detection in several ways. We measured the squeezed quadrature \(\left\langle {x^\dagger x} \right\rangle\), and the uncertainty area, \(\left\langle {x^\dagger x} \right\rangle \times \left\langle {y^\dagger y} \right\rangle\), of the squeezed state. Ideally, the generated squeezed light should be a minimum uncertainly state of \(\left\langle {x^\dagger x} \right\rangle \times \left\langle {y^\dagger y} \right\rangle = 1\), independent of the generation gain; and the average intensity of the squeezed quadrature should exponentially decrease with the gain. The results are presented in the Methods, showing a clear reduction of the normalized squeezed quadrature intensity down to \(\left\langle {x^\dagger x} \right\rangle \approx 0.68\) (32% below the vacuum level), and the uncertainty area remains nearly ideal at \(\left\langle {x^\dagger x} \right\rangle \times \left\langle {y^\dagger y} \right\rangle < 1.3\), up to a pump power of 60 mW. Further increase of the pump does not improve the measured squeezing due to pulse effects, and the minimum uncertainty property deteriorates. Based on the measured squeezing, the instantaneous squeezed quadrature at the peak of the pulse was estimated to be >3 dB (see Methods). Additional verification measurements of the broadband squeezing are presented and illustrated in the Methods.

Two-mode quadratures

The fundamental quadrature oscillation—a single-frequency component of a quadrature amplitude x(ω), y(ω), is a two-mode combination of frequencies ω s = Ω + ω and ω i = Ω − ω—the signal and idler25,26,27. Using the field operators of the signal a s = a(ω) and the idler a i = a(−ω), the quantum operators of the quadratures x(ω), y(ω) are (see Methods for an intuitive reasoning)

$$\left\{ \begin{array}{l}x(\omega ) = a_{\mathrm{s}} + a_{\mathrm{i}}^\dagger \\ y(\omega ) = i\left( {a_{\mathrm{s}}^\dagger - a_{\mathrm{i}}} \right)\end{array} \right..$$ (1)

This definition preserves the commutation relation [x, y] = 2i and reduces in the monochromatic case to the single-mode quadratures x = a + a†, y = i(a† − a).

The generalization of the standard quadratures to two-mode quadratures requires some attention. As opposed to the standard quadrature operators, which are hermitian and represent the time-independent real amplitude of the cosine (sine) oscillation, the two-mode quadrature operators of Eq. (1) are non-hermitian x†(ω) = x(−ω) ≠ x(ω) and represent a time-dependent beat between the signal and idler modes with an envelope frequency ω, carried by a cosine (sine) wave at frequency Ω (see Methods for some intuitive reasoning). The quadrature operators x(ω), x†(ω) represent the beat envelope, which has an amplitude and phase, in some similarity to the field operators a, a† that represent the amplitude and phase of the carrier oscillation. Yet, the two-mode quadrature x is an observable quantity (in contrast to the field operator a). Since x commutes with its conjugate [x(ω), x†(ω)] = 0 (as opposed to [a, a†] = 1), it is possible to simultaneously measure both the real and imaginary part of the quadrature envelope, and thereby obtain complete information on both amplitude and phase of the single quadrature:

$$\left\{ \begin{array}{l}{\mathrm{Re}}[x] = x + x^\dagger = X_{\mathrm{s}} + X_{\mathrm{i}}\\ {\mathrm{Im}}[x] = i\left( {x - x^\dagger } \right) = Y_{\mathrm{s}} - Y_{\mathrm{i}},\end{array} \right.$$ (2)

where X s,i , X s,i are the standard single-mode quadratures of the signal and idler modes. Our experiment measured x†(ω)x(ω).

Since the phase of the two-mode quadrature relates to commuting observables (as opposed to the carrier phase), it does not reflect a non-classical property of the quantum light field, but rather defines the classical temporal mode in which the field is measured. Specifically, the temporal mode of measurement is the two-frequency beat pattern of frequency ω (see Methods for an illustration), where the envelope phase defines the temporal offset of the beat. This offset, along with other mode parameters, such as polarization, spatial mode, carrier frequency, and so on define the mode of the LO. Of course, quantum entanglement is possible between the two envelope modes (cosine or sine) in direct equivalence to entanglement of a single photon (or photon pair, or cat state) between polarization modes, which is widely used for quantum information. However, this “quantumness” between modes is different and additional to the intra-mode quantum state, which is described by the quadratures x, y.

Due to the bandwidth limitation of standard homodyne measurement, the commonly used expression to interpret two-mode quadratures of optical frequency separation ω does not rely on Eq. (1), but rather on Eq. (2). Two independent homodyne measurements of the signal and idler quadratures, x s,i , y s,i need to be made relative to two correlated LOs at their respective frequencies ω s , ω i so that the two output homodyne signals are within the electrical bandwidth. Thus, the standard procedure to measure just a single-frequency component of the two-mode quadrature x(ω) (and its squeezing) requires two separate homodyne measurements of the independent quadratures of both the signal and the idler using a pair of phase-correlated LOs17,28. For a broadband spectrum, standard two-mode homodyne requires a dense set of correlated pairs of LOs for each frequency component of the measurement. As we have shown, however, in our experiment above, a single LO is sufficient to simultaneously extract a specific quadrature across the entire optical bandwidth, just as a single pump laser can simultaneously generate the entire bandwidth of quadrature squeezed mode pairs.

Quantum derivation of the parametric amplified output

To model quantum mechanically the parametric homodyne process, we derive an expression for the parametric output intensity (photon-number) operator of the signal (or idler) mode, \(N_{\mathrm{s}}(g) = a_{\mathrm{s}}^\dagger (g)a_{\mathrm{s}}(g)\) (g is the parametric gain) in terms of the input complex quadratures x(ω), y(ω). Mathematically, our method relies on the similarity between the quadrature operators of interest (Eq. (1)) \(x(\omega ) = a_{\mathrm{s}} + a_{\mathrm{i}}^\dagger\), \(iy^\dagger = a_{\mathrm{s}} - a_{\mathrm{i}}^\dagger\) and the field operator at the output of a parametric amplifier:

$$a_{\mathrm{s}}(g) = a_{\mathrm{s}}{\kern 1pt} {\mathrm{cosh}}(g) + e^{i\varphi }a_{\mathrm{i}}^\dagger {\kern 1pt} {\mathrm{sinh}}(g) \equiv Ca_{\mathrm{s}} + Da_{\mathrm{i}}^\dagger ,$$ (3)

where the coefficients C and D are generally complex. Since field operators must fulfill \(\left[ {a_{\mathrm{s}}^\dagger (g),a_{\mathrm{s}}(g)} \right] = 1\), the two coefficients C and D must obey \(\left| C \right|^2 - \left| D \right|^2 = 1\), which leads to the common description of C = cosh g and D = eiφ sinh g. However, the attributed phase of the parametric process φ, which is determined by the pump phase and the phase matching conditions in the nonlinear medium, can also be expressed explicitly, leaving the two coefficients C, D real and positive (rather than complex), using \(a_{\mathrm{s}}(g,\theta ) = \left( {Ca_{\mathrm{s}}e^{i\theta } + Da_{\mathrm{i}}^\dagger e^{ - i\theta }} \right)e^{i\theta _0}\). Since the overall phase θ 0 does not affect the photon-number calculations, we may discard it as θ 0 = 0. In this expression we account for the phase of the pump as a rotation of the input quadrature axis—a s,i → a s,i eiθ. Accordingly, the rotated complex quadrature operators (Eq. (1)) become \(x_\theta (\omega ) = a_{\mathrm{s}}e^{i\theta } + a_{\mathrm{i}}^\dagger e^{ - i\theta }\) and \(y_\theta (\omega ) = i\left( {a_{\mathrm{s}}^\dagger e^{ - i\theta } - a_{\mathrm{i}}e^{i\theta }} \right)\).

Parametric amplification directly amplifies one quadrature of the input and attenuates the other, as evident by expressing the field operators a s (g) at the output using the quadrature operators x,y of the input:

$$a_{\mathrm{s}}(g) = \frac{{C + D}}{2}x + i\frac{{C - D}}{2}y^\dagger = \frac{{e^g}}{2}x + \frac{{e^{ - g}}}{2}iy^\dagger ,$$ (4)

where the index θ was dropped for brevity. Finally, the parametric photon-number operator at the output is

$$\begin{array}{*{20}{l}} {N_{\mathrm{s}}(g)} \hfill & = \hfill & {a_{\mathrm{s}}^\dagger (g)a_{\mathrm{s}}(g) = \frac{{N_{\mathrm{s}} - N_{\mathrm{i}} - 1}}{2} } \hfill \\ {} \hfill & {} \hfill & { + \left( {\frac{{C + D}}{2}} \right)^2x^\dagger x + \left( {\frac{{C - D}}{2}} \right)^2y^\dagger y} \hfill \\ {} \hfill & = \hfill & {\frac{{N_{\mathrm{s}} - N_{\mathrm{i}} - 1}}{2} + \frac{{e^{2g}}}{4}x^\dagger x + \frac{{e^{ - 2g}}}{4}y^\dagger y,} \hfill \end{array}$$ (5)

where \(N_{{\mathrm{s,i}}} = a_{{\mathrm{s,i}}}^\dagger a_{{\mathrm{s,i}}}\) represent the input photon numbers (intensities) of the signal and idler. When access is available simultaneously to the intensities of both the signal and the idler, their sum of intensities provides the cleanest measurement of the quadrature intensities

$$N_{\mathrm{s}}(g) + N_{i}(g) = \frac{{e^{2g}}}{2}x^\dagger x + \frac{{e^{ - 2g}}}{2}y^\dagger y - 1.$$ (6)

Note that N s (g) − N i (g) = N s − N i is a constant of the amplification, independent of the parametric gain.

With sufficient parametric gain, any given x quadrature at the input can be amplified above the vacuum noise to a “classical level”, even if it was originally squeezed, which allows complete freedom in measurement since vacuum fluctuations are no longer the limiting noise. If the measurement gain considerably exceeds the generation gain, such that e2gx†x ≫ e−2gy†y, the amplified quadrature will dominate the intensity of the output light allowing to neglect the intensity of the attenuated orthogonal y quadrature, and the measurement of the light intensity spectrum at the output will directly reflect (after calibration, see Methods) the single-shot value of the input quadrature intensity x†(ω)x(ω), just like the standard measurement of the electrical spectrum at the output of standard homodyne.

Although the concept of parametric homodyne is conveniently understood in the limit of large gain, where the quadrature of interest dominates the output light field, parametric homodyne is equally effective with almost any finite gain. When the measurement gain is not large enough and the attenuated quadrature cannot be neglected, the two quadrature intensities can be easily extracted using a pair of measurements; setting the pump phase to amplify one quadrature (θ = 0) and then to amplify the other (θ = π/2), as illustrated in Fig. 3. Indeed, the output intensity in this case will not directly reflect the quadrature intensity, but it still provides equivalent information about the quadrature at any finite gain, since two light intensity measurements along orthogonal axes uniquely infer the two quadrature intensities at any finite gain, indicating that the information content of a measurement of the output intensity is the same as that of the quadrature intensity. An analytic derivation of this equivalence is provided in the Methods.

Applicability to quantum tomography

Quantum state tomography is a major application of homodyne measurement. It allows reconstruction of an arbitrary quantum state (or its density matrix or Wigner function) from a set of quadrature measurements along varying quadrature axes7. Unique reconstruction requires a complete measurement of the quadrature distribution function, which necessitates single-shot measurements of the instantaneous quadrature value, not just its average. Although both standard two-mode homodyne and parametric homodyne provide incomplete quadrature information in a single shot (in somewhat different ways), they still allow reconstruction of the quantum state under some assumptions. Hereon we review the different limitations of both methods and their implications to quantum tomography, leading to a conclusion that a combination of parametric homodyne followed by standard homodyne alleviates all the limitations and allows unambiguous reconstruction of arbitrary states.

Standard two-mode homodyne cannot provide a complete measurement of x(ω) in a single shot since standard homodyne is a destructive measurement. Specifically, observation of Re[x(ω)] = X s + X i requires a standard homodyne measurement of both frequency modes, which inevitably destroys the quantum state by photo-detection and prevents a consecutive measurement of Im[x(ω)] = Y s − Y i . Splitting the state into two measurement channels is impossible since such a splitting will inevitably introduce additional vacuum noise. Thus, although Re[x(ω)] and Im[x(ω)] commute, standard two-mode homodyne can evaluate only one of them in a single shot. In analogy to light polarization, standard homodyne acts as an absorptive polarizer that detects one polarization but absorbs the other, preventing complete analysis of the polarization state.

Our current realization of parametric homodyne suffers from a different ambiguity in a single shot (envelope phase). Since parametric homodyne measures only the instantaneous intensity of the quadrature x†x (across a wide spectrum), but not its phase, only the probability distribution of the intensity P(x†x) can be measured.

Let us analyze the ambiguity that is introduced to the reconstruction of a two-mode quantum state by the incomplete measurement, for both standard homodyne (only real part) and parametric homodyne (only intensity). For standard homodyne, the interpretation of a null result is ambiguous: a zero measurement can arise either from a true null of the measured quadrature or from a wrong selection of the envelope phase. Thus, standard homodyne can reconstruct a two-mode quantum state only if the envelope phase is fixed and known a priori. For two-mode squeezed vacuum, however, which is the major two-mode quantum state that is experimentally accessible, the envelope phase is random, indicating that standard homodyne can provide only the average fluctuations \(\left\langle {x^\dagger x} \right\rangle = \left\langle {\left( {X_{\mathrm{s}} + X_{\mathrm{i}}} \right)^2} \right\rangle + \left\langle {\left( {Y_{\mathrm{s}} - Y_{\mathrm{i}}} \right)^2} \right\rangle\), but not the single shot value of the quadrature (or its intensity).

For parametric homodyne, where the quadrature intensity is measured, null (or any intensity) is unambiguously interpreted for any envelope phase, but the sign of the measured quadrature is ambiguous. Thus, complete reconstruction is possible (for any envelope phase) only if the symmetry of the quadratures is known, which is relevant to a large set of important quantum states. For example, photon-number states or squeezed states6 that are known to be symmetric can be reconstructed, and indeed the non-classicality of a single photon state is directly manifested by the fact that the probability to measure a null intensity vanishes \(P\left( {x_\theta ^\dagger x_\theta = 0} \right) = 0\) for any quadrature axis θ, which inevitably indicates negativity of the Wigner function at zero field. Yet, a two-mode coherent state \(\left| { \pm \alpha } \right\rangle\) and cat states like \(\left| \alpha \right\rangle \pm \left| { - \alpha } \right\rangle\)8 can be differentiated only if the symmetry of the state is assumed a priori. For broadband squeezed vacuum, where the envelope phase is inherently random, this measurement is ideal.

Clearly, the two methods complete each other in their capabilities, indicating that a combination of parametric homodyne with interferometric detection is the perfect solution to a complete measurement, as illustrated in Fig. 4. Specifically, parametric gain is a non-demolition process (contrary to standard homodyne) that provides a light output and allows extraction of the complete quadrature information in a single shot, including the phase. Thus, if the measured quadrature is amplified sufficiently above the vacuum, this quadrature becomes insensitive to loss, even for moderate gain values. The parametric output light can thus be split to two homodyne channels that will measure both Re[x(ω)] and Im[x(ω)] simultaneously (see Fig. 4). The splitting does not hamper the measurement (contrary to standard homodyne) since the added vacuum affects primarily the attenuated quadrature, which is not measured.

Fig. 4 A Complete broadband homodyne scheme. Parametric homodyne allows to fully measure the two-mode quadrature amplitude; once the quadrature of interest x(ω) is sufficiently amplified above the vacuum level by the parametric amplifier, this quadrature becomes insensitive to additional vacuum noise, which allows splitting of the light into two standard homodyne channels in order to measure simultaneously both Rex(ω) and Imx(ω) Full size image

In the literature, the possibility to add a parametric amplifier before electronic detection was analyzed in several different contexts: already the seminal paper of Caves from 1981 that introduced squeezed vacuum to the unused port of an interferometer for sub-shot noise interferometric measurement, suggested to include a parametric amplifier in the detection arm to overcome the quantum inefficiency of photo-detectors29, Leonhardt and Paul30 later suggested a similar use of parametric amplification for quantum tomography that is insensitive to loss, Ralph31 suggested it for teleportation and Davis et al.32 for the analysis of atomic spin-squeezing. Most recently, this concept was experimentally implemented for atomic spin measurements in33 enabling phase detection down to 20 dB below the standard quantum limit with inefficient detectors.

Comparison to standard homodyne

It is illuminating to examine on equal footing standard homodyne measurement and the parametric homodyne method. After all, the balanced detection in standard homodyne produces a down-converted RF field at the difference-frequency of the two optical inputs (LO and signal), similar to optical down-conversion, which is the core of parametric amplification. In that view, the well-known homodyne gain of balanced detection (proportional to the LO field) produces an amplified electronic version of the input quantum quadrature, directly analogous to the parametric gain (proportional to the pump amplitude), which optically amplifies a single input quadrature. Thus, both the standard homodyne gain and the optical parametric gain serve the same homodyne purpose—to amplify the quantum input of interest (the optical quadrature) to a classically detectable output level34, which is sufficiently above the measurement noise (the electronic noise for standard homodyne or the optical vacuum noise for parametric homodyne). Consequently, standard and parametric homodyne are two faces of the same concept.

The difference between the two schemes is both technical and conceptual. On the technical level, the gain of standard homodyne is generally very large, allowing to a priori neglect any effect of the unmeasured quadrature on the electrical output, whereas the optical parametric gain may not be sufficient to justify such an a priori assumption and may require more careful analysis of the output with finite gain, as we described earlier. On the conceptual level, parametric homodyne provides an optical output, as opposed to standard homodyne that destroys the optical fields. Since the optical parametric output can be sufficiently “classical” (amplified above the vacuum level), it is far less sensitive to additional vacuum noise from optical loss or detector inefficiency. Consequently, parametric homodyne does not only preserve the optical bandwidth across the quantum-classical transition (see Fig. 1), but can also allow complete reconstruction of the two-mode quadrature in a single shot, as was explained in the previous sub-section. Hence, adding a layer of optical parametric gain before the electronic photo-detection, be it intensity detection or homodyne provides an important freedom to quantum measurement beyond the ability to preserve the optical bandwidth.

Beyond the pure two-mode field

Last, let us briefly consider broadband time-dependent states of light beyond the single-frequency two-mode state. Any classical wave packet with spectral envelope \(f(\omega ) = \left| {f(\omega )} \right|{e}^{i\varphi (\omega )}\) around the carrier frequency Ω (normalized to \({\int} \mathrm{d}\omega \left| {f(\omega )} \right|^2 = 1\)) can be regarded as an electromagnetic mode with associated quantum field operators

$$\left\{ \begin{array}{l}a_f(t) = {\int} \mathrm{d}\omega f(\omega )a(\omega )e^{ - i\omega t}\\ a_f^\dagger (t) = {\int} \mathrm{d}\omega f^ \star (\omega )a^\dagger (\omega )e^{i\omega t},\end{array} \right.$$ (7)

and associated temporal quadratures

$$\left\{ \begin{array}{l}x_f(t) = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left[ {f(\omega )a(\omega ) + f^ \star ( - \omega )a^\dagger ( - \omega )} \right]} \\ y_f(t) = i{\int} {\mathrm{d}\omega e^{ - i\omega t}\left[ {f^ \star ( - \omega )a^\dagger ( - \omega ) - f(\omega )a(\omega )} \right]} ,\end{array} \right.$$ (8)

which is just the Fourier transform of Eq. (1) (see also Eq. (11) in the Methods).

We can express the temporal quadrature x f (t) in terms of the two-mode quadratures x(ω), y(ω) as

$$\left\{ \begin{array}{l}x_f(t) = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left[ {\frac{{f(\omega ) + f^ \star ( - \omega )}}{2}x(\omega ) + i\frac{{f(\omega ) - f^ \star ( - \omega )}}{2}y^\dagger (\omega )} \right]} \\ y_f(t) = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left[ {\frac{{f(\omega ) + f^ \star ( - \omega )}}{2}y^\dagger (\omega ) - i\frac{{f(\omega ) - f^ \star ( - \omega )}}{2}x(\omega )} \right],} \end{array} \right.$$ (9)

where the symmetric and antisymmetric parts of the wave packet \(\frac{{f(\omega ) + f^ \star ( - \omega )}}{2},\frac{{f(\omega ) - f^ \star ( - \omega )}}{2}\) are the Fourier transforms of Ref(t), Imf(t) the real and imaginary parts of the field envelope in time.

Equation (9) can be simplified considerably when the spectrum of the wave packet is symmetric \(\left| {f(\omega )} \right| = \left| {f( - \omega )} \right|\), which is the major situation to employ a quadrature representation to begin with. The temporal quadrature x f (t) is then simply a superposition of many two-mode components x θ (ω) with a spectrally varying axis θ(ω) and envelope phase δ(ω)

$$\left\{ \begin{array}{l}x_f(t) = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left| {f(\omega )} \right|e^{i\delta (\omega )}x_{\theta (\omega )}(\omega ) } \\ = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left| {f(\omega )} \right|e^{i\delta (\omega )}\left[ {x(\omega ){\kern 1pt} {\mathrm{cos}}{\kern 1pt} \theta (\omega ) + y^\dagger (\omega ){\kern 1pt} {\mathrm{sin}}{\kern 1pt} \theta (\omega )} \right]} \\ y_f(t) = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left| {f(\omega )} \right|e^{i\delta (\omega )}y_{\theta (\omega )}^\dagger (\omega ) } \\ = {\int} {\mathrm{d}\omega e^{ - i\omega t}\left| {f(\omega )} \right|e^{i\delta (\omega )}\left[ {y^\dagger (\omega ){\kern 1pt} {\mathrm{cos}}{\kern 1pt} \theta (\omega ) - x(\omega ){\kern 1pt} {\mathrm{sin}}{\kern 1pt} \theta (\omega )} \right]} .\end{array} \right.$$ (10)

The quadrature axis of each two-mode component is dictated by its carrier phase \(\theta (\omega ) = \frac{{\varphi (\omega ) + \varphi ( - \omega )}}{2}\)—the symmetric part of the spectral phase of the wave packet φ(ω); and the two-mode envelope phase \(\delta (\omega ) = \frac{{\varphi (\omega ) - \varphi ( - \omega )}}{2}\) relates to the antisymmetric part of φ(ω). Thus, for a transform limited mode, where φ(ω) = 0, both the envelope phase and the quadrature axis are constant across the spectrum δ(ω) = 0, θ(ω) = 0. An antisymmetric phase variation, (φ(ω) = −φ(−ω)), will affect only the envelope phase, but keep the quadrature axis constant θ(ω) = 0, as is the case for down-converted light. A purely symmetric phase φ(ω) = φ(−ω), as due to material dispersion, will affect only the quadrature axis, but keep the envelope constant δ(ω) = 0.

Therefore, measurement of an arbitrary generalized quadrature of broadband light requires measurement (or knowledge) of two spectral degrees of freedom—the quadrature axis θ(ω) and the envelope phase δ(ω). Parametric homodyne with intensity measurement provides complete information of the quadrature axis θ(ω) (by measuring the output spectrum for varying pump phase), but is insensitive to δ(ω). It therefore allows measurement if δ(ω) is either unimportant (down-conversion) or known a priori (transform limit or well-defined pulse), which is relevant to all current sources of broadband quantum light in spite of the limitations. The combination of parametric gain followed by standard homodyne allows complete arbitrary measurement, as explained above.