Introduction

In quantum technology, resolving low power quantum signals in a noisy environment is essential for an efficient readout of quantum states1. Linear phase-preserving amplifiers are a central tool to accomplish this task by simultaneously increasing the amplitude in both signal quadratures without altering the signal phase2. Due to the low energy of microwave photons, this amplification process is of particular relevance for the tomography of quantum microwave states3. State tomography is crucial for experimental protocols with quantum propagating microwaves such as entanglement generation4,5,6,7,8, secure quantum remote state preparation9, quantum teleportation10,11,12, quantum illumination13, or quantum state transfer14. Furthermore, a conventional dispersive readout of superconducting qubits15,16,17,18,19,20,21,22 relies on the amplification of microwave signals which carry information about the qubit and consist only of a few photons23. This approach has proven to be extremely successful and has led to the realization of important milestones in superconducting quantum information processing24,25. However, fundamental laws of quantum physics imply that any phase-preserving amplifier needs to add at least half a noise photon in the high-gain limit 26. This bound is known as the standard quantum limit (SQL) and results from the bosonic commutation relations of input and output fields constituting the original and amplified signals, respectively. Quantum-limited amplification of quantum microwave states has been achieved with superconducting Josephson parametric amplifiers (JPAs)27,28,29, but also by employing Josephson traveling-wave parametric amplifiers (JTWPAs)30. With JTWPAs, a noise performance of 2.1 times the SQL has been reached31. Nevertheless, alternative ways to realize noiseless amplification are important for a large variety of quantum applications which rely on the efficient detection of signal amplitudes such as the parity measurements in multi-qubit systems32, quantum amplitude sensing33, detection of dark matter axions34, or the detection of the cosmic microwave background35, among others.

In this work, we investigate the nondegenerate parametric amplification of broadband microwave signals and derive conditions under which noiseless amplification is possible. This broadband nondegenerate regime is realized by employing a flux-driven JPA and is complimentary to the conventional phase-preserving nondegenerate (narrowband and quantum-limited) or phase-sensitive degenerate (narrowband and potentially noiseless) regimes36,37,38. In the current context, the terms ‘narrowband’ and ‘broadband’ relate to the bandwidth of input signals and not to the bandwidth of the JPA itself. It is important to emphasize that, in contrast to the degenerate regime, signal and idler are separate frequency modes here. Their phases can be, in principle, either correlated or uncorrelated, which leads to the phase-dependent or phase-independent amplification regimes, respectively. One could also note that the nondegenerate amplifier allows to realize a two-mode squeezing operation between the input signal and input idler modes39,40. Here, we experimentally demonstrate a JPA for the phase-independent linear amplification of broadband signals with performance beyond the SQL.

Results

Quantum limits on quantum efficiency

An ideal linear amplifier increases the signal photon number ns by the power gain factor Gn2. The fluctuations in the output signal consist of the amplified vacuum fluctuations of the input signal and the noise photons nf, added by the amplifier, where nf is referred to the amplifier input. We use the quantum efficiency η to characterize the noise performance of our amplifiers41. The quantum efficiency is defined as the ratio between vacuum fluctuations in the input signal and fluctuations in the output signal. Thus, η can be expressed as

$$\eta =\frac{1}{1+2{n}_{{{{\rm{f}}}}}}.$$
(1)

Parametrically driving the JPA results in amplification of the incoming signal and creation of an additional phase-conjugated idler mode. This idler mode consists of, at least, vacuum fluctuations, implying that it carries ni ≥ 1/2 photons. As a result of the narrowband parametric signal amplification, the idler adds42

$${n}_{{{{\rm{f}}}}}={n}_{{{{\rm{i}}}}}\frac{{G}_{{{{\rm{n}}}}}-1}{{G}_{{{{\rm{n}}}}}}\ge \frac{1}{2}\left(1-\frac{1}{{G}_{{{{\rm{n}}}}}}\right)$$
(2)

noise photons to the signal, referred to the amplifier input, as schematically depicted in Fig. 1a. Equation (2) implies that η is bounded by

$$\eta \le \frac{{G}_{{{{\rm{n}}}}}}{2{G}_{{{{\rm{n}}}}}-1},$$
(3)

reaching 1/2 in the high-gain limit, Gn ≫ 1. Figure 1b illustrates the amplification process for broadband signals, where the input signal bandwidth is large enough to cover both the signal and idler modes of the parametric amplifier. As a result, the idler mode does not add any noise but contributes to the amplified output signal. Thus, we expect that a quantum efficiency of η = 1 can be reached in this case.

Fig. 1: Phase space representation of nondegenerate parametric amplification and experimental setup.
figure 1

a Top: Phase space transformation for the amplification of narrowband input signals. Colored circles depict the respective variances of the input signal (orange), output signal (blue), and idler mode (green). Bottom: Spectrum of parametric amplification of narrowband signals with bandwidth bs. The purple solid line shows a Lorentzian gain function. Blue-shaded and green-shaded regions represent measurement bands with full bandwidth 2B around the signal and idler modes, respectively. For input signals with bs ≤ b1, the idler adds at least vacuum fluctuations to the output. b Top: Phase space transformation for amplification of broadband input signals. The idler does no longer act as a noise port and the signal is amplified with gain Gb = 2Gn − 1. Bottom: Spectrum of parametric amplification process for broadband signals. If bs ≥ b2, each mode in the signal bandwidth corresponds to a correlated input mode on the idler side, resulting in amplification with the total gain Gb and the absence of the SQL. c Illustration of the experimental setup. The amplification chain consists of a nondegenerate Josephson parametric amplifier (JPA) and a cryogenic HEMT amplifier. Narrowband coherent states can be applied via a microwave source and a heatable attenuator enables the generation of broadband thermal states. The red dot labels the signal reconstruction point.

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Nondegenerate Josephson parametric amplifier

Parametric amplification can be realized by driving a nonlinear electromagnetic resonator with a strong coherent field (pump) at ωp = 2ω0, where ω0 is the resonance frequency37,42. In the resulting three-wave mixing process, a pump photon splits into a signal photon at frequency ωs = ωp/2 + Δ and a corresponding idler photon at ωi = ωp/2 − Δ. Here, Δ denotes the detuning of the signal reconstruction frequency ωs from ω0. In the nondegenerate regime, we have Δ ≠ 0, leading to spectrally separated signal and idler modes43. This parametric down-conversion process results in a Lorentzian spectral gain function, which is depicted by the purple solid line in Fig. 1a, bottom. The input signal as well as the amplified output signal is detected within the measurement bandwidth 2B around ωs. For a narrowband state, the idler necessarily adds broadband noise to the signal, leading to broadening of the output variances (see Fig. 1a, top). In contrast, if the signal bandwidth bs is large enough to cover the idler input modes, as shown in Fig. 1b, the amplified output signal consists of contributions from both the signal and idler modes. In this case, the idler no longer serves as a noise port but rather as an additional signal port and the part of the input signal at the idler frequency is mixed into the measurement bandwidth. In the case where the input signal phase and the input idler phase are uncorrelated, we obtain a total broadband gain (see Supplemental Material).

$${G}_{{{{\rm{b}}}}}=2{G}_{{{{\rm{n}}}}}-1.$$
(4)

Standard quantum limit for multimode input signals

The SQL fundamentally results from the fact that the amplifier input and output signals have to fulfill the bosonic commutation relations. We assume that the JPA output \(\hat{c}(\omega )\) results from a linear combination of amplified incoming signals \(\hat{a}(\tilde{\omega })\) and phase-conjugated idler signals \({\hat{a}}^{{\dagger} }({\tilde{\omega }}_{{{{\rm{i}}}}})\) as well as from an additive noise mode \(\hat{f}(\omega )\)26

$$\hat{c}(\omega )={\int_{\mathcal{I}}}d\tilde{\omega }\left[M(\omega ,\tilde{\omega })\hat{a}(\tilde{\omega })+L(\omega ,\tilde{\omega }){\hat{a}}^{{\dagger} }(\tilde{\omega })\right]+\hat{f}(\omega ),$$
(5)

where the integral is taken over all input modes \({{{\mathcal{I}}}}\). Frequencies \(\tilde{\omega }\) denote input modes, whereas output modes are described by ω. We assume that the input signal is centered around the frequency ωs and has a single-side bandwidth bs (total bandwidth 2bs). The signal amplitude gain \(M(\omega ,\tilde{\omega })\) and the idler amplitude gain \(L(\omega ,\tilde{\omega })\) satisfy \(M(\omega ,\tilde{\omega })=M(\tilde{\omega })\delta (\omega -\tilde{\omega })\) and \(L(\omega ,\tilde{\omega })=L({\tilde{\omega }}_{{{{\rm{i}}}}})\delta (\omega -{\tilde{\omega }}_{{{{\rm{i}}}}})\), respectively, where \({\tilde{\omega }}_{{{{\rm{i}}}}}=2{\omega }_{0}-\tilde{\omega }\). Furthermore, ∣M(ω)∣2 = Gn(ω) and ∣L(ωi)∣2 = Gn(ω) − 1, where Gn(ω) is the power gain. The strength of the additive noise is described with the noise power spectral density Sf(ω) of the bosonic mode \(\hat{f}(\omega )\), corresponding to the number of noise photons per mode. We evaluate the integral in Eq. (5) and calculate the commutator \([\hat{c}(\omega ),{\hat{c}}^{{\dagger} }({\omega }^{\prime})]\). Next, we calculate the spectral autocorrelation function of the fluctuations. This allows us to use the bosonic continuum commutation relations and the Heisenberg uncertainty principle to show that

$$\begin{array}{ll}\int\nolimits_{{\omega }_{{{{\rm{s}}}}}-{b}_{{{{\rm{s}}}}}}^{{\omega }_{{{{\rm{s}}}}}+{b}_{{{{\rm{s}}}}}}{S}_{{{{\rm{f}}}}}(\omega )d\omega \equiv 2{b}_{{{{\rm{s}}}}}{n}_{{{{\rm{f}}}}}\ge 2{b}_{{{{\rm{s}}}}}{n}_{{{{\rm{ql}}}}}\\ \qquad\qquad\qquad\,\,\,=\frac{1}{2}\int\nolimits_{{\omega }_{{{{\rm{s}}}}}-{b}_{{{{\rm{s}}}}}}^{{\omega }_{{{{\rm{s}}}}}+{b}_{{{{\rm{s}}}}}}\left[1-{{\mathbb{1}}}_{{{{\rm{i}}}}}(\omega )+\frac{1}{{G}_{{{{\rm{n}}}}}(\omega )}({{\mathbb{1}}}_{{{{\rm{i}}}}}(\omega )-1)\right]d\omega ,\end{array}$$
(6)

where nql is the quantum limit for the number of additive noise photons (see Supplemental Material). We define

$${{\mathbb{1}}}_{{{{\rm{i}}}}}(\omega )={{\Theta }}(\omega -2{\omega }_{0}+{\omega }_{{{{\rm{s}}}}}+{b}_{{{{\rm{s}}}}})-{{\Theta }}(\omega -2{\omega }_{0}+{\omega }_{{{{\rm{s}}}}}-{b}_{{{{\rm{s}}}}}),$$
(7)

where Θ is the Heaviside step function. In Eq. (6), the range of integration over ω is limited by bs, if bs < B, otherwise it is set by B. We observe that there are two threshold values of the bandwidth: b1 = 2Δ − B and b2 = 2Δ + B. For bs ≤ b1, Eq. (6) reproduces the SQL, whereas for b1 ≤bs ≤ b2, the input signal starts to overlap with idler modes and the lower bound for the additive noise decreases. In the broadband case bs ≥ b2, the signal covers all idler modes and Eq. (6) reduces to nf ≥ 0, implying that there is no fundamental lower limit for the added noise.

Experimental setup

Our experimental setup is schematically shown in Fig. 1c and consists of a flux-driven JPA serially connected to a cryogenic high-electron-mobility transistor (HEMT) amplifier with a gain of GH = 41 dB. The JPA is operated in the nondegenerate regime, which is realized by detuning the signal frequency ωs by Δ/2π = 300 kHz from half the pump frequency ωp/2 = ω0. A circulator at the JPA input separates the resonator input and output fields. The moments of the output signal are reconstructed with a bandwidth B/2π = 200 kHz using the reference-state reconstruction method at the reconstruction point indicated by the red circle in Fig. 1c7,44. The experiment is performed with two distinct JPAs, labelled JPA 1 and JPA 2, which are operated at different flux spots to check for reproducibility of our results. For JPA 1 (JPA 2), we reconstruct the signal at ωs/2π = 5.500 GHz (5.435 GHz). A continuous coherent tone can be applied via a microwave input line and a heatable 30 dB attenuator allows us to generate thermal states as broadband input signals.

Bandwidth dependence of the limits on quantum efficiency

We solve Eq. (6) for the Lorentzian JPA gain function \({G}_{{{{\rm{n}}}}}(\omega )=1+{G}_{0}{b}_{{{{\rm{J}}}}}^{2}/({b}_{{{{\rm{J}}}}}^{2}+{(\omega -{\omega }_{0})}^{2})\), where G0 denotes the maximal JPA gain and bJ is the half width at half maximum JPA bandwidth43. Then, assuming G0 ≫ 1, the quantum limit nql for the number of additive noise photons is given by

$${n}_{{{{\rm{ql}}}}}=\frac{1}{4\beta }\left\{\begin{array}{ll}\arctan \left(\frac{2\beta }{1+{\delta }^{2}-{\beta }^{2}}\right)\quad &B\le {b}_{{{{\rm{s}}}}}\le {b}_{1},\\ \arctan \left(\frac{2(\delta +\beta -{\beta }_{{{{\rm{s}}}}})}{1+\beta {\beta }_{{{{\rm{s}}}}}+\delta ({\beta }_{{{{\rm{s}}}}}-\beta )-{\delta }^{2}}\right)\quad &{b}_{1}\le {b}_{{{{\rm{s}}}}}\le {b}_{2},\\ 0\quad &{b}_{{{{\rm{s}}}}}\ge {b}_{2},\end{array}\right.$$
(8)

with β = B/τ, βs = bs/τ, and δ = Δ/τ, where \(\tau \equiv {b}_{{{{\rm{J}}}}}\sqrt{{G}_{0}}\) denotes the gain-bandwidth product36 (see Supplemental Material). The corresponding limit ηql of the quantum efficiency η can be calculated with Eq. (1). The case bs ≤ B is not considered in Eq. (8) as it is only of technical relevance since we can always achieve bs = B by adopting B. A discussion of this case is included in the supplement (see Supplemental Material). The solution Eq. (8) allows us to distinguish quantitatively between broadband and narrowband regimes and is plotted for τ/2π = 15 MHz in Fig. 2a for varying Δ with B/2π = 30 kHz and in Fig. 2b for varying B with Δ/2π = 37.5 kHz. According to Eq. (8), we obtain ηql = 1/2 for coherent input signals, approximately reproducing Eq. (3), whereas we expect ηql = 1 for broadband signals.

Fig. 2: Theoretical limits on quantum efficiency.
figure 2

Limit ηql of the quantum efficiency η as a function of the signal bandwidth bs for a varying detuning Δ at a fixed measurement bandwidth B/2π = 30 kHz and b for varying measurement bandwidth B at a fixed detuning Δ/2π = 37.5 kHz, according to Eq. (8). For bs ≥ b2, amplification with η = 1 is possible. The case bs ≤ B is only of technical interest and is discussed in the supplement (see Supplemental Material).

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Experimental determination of the quantum efficiency

We experimentally extract the quantum efficiency by measuring the total noise photon number of the amplification chain. To achieve this goal, we vary the temperature of the heatable 30 dB attenuator from 40 mK to 600 mK and perform Planck spectroscopy of the amplification chain45. As a result, we detect the photon number nb at the reconstruction point for varying broadband gain Gb and show the result of this measurement in Fig. 3a for JPA 2. For each value of Gb, the experimentally determined outcomes for nb (dots) are fitted with corresponding Planck distributions (cyan solid lines) and the respective noise photon number is extracted from the offset. The quantum efficiency for narrowband signals is determined in a similar experiment by amplifying a coherent input tone with varying input photon number nin for different JPA gains Gn. For each value of Gn, the amplifier response nn at the reconstruction point is linearly fitted, which allows us to extract the noise photons from the respective offset. This procedure also proves that the JPA acts as a linear amplifier here. Figure 3b shows a logarithmic plot of the experimental results (dots) for JPA 2 as well as the respective linear fits (orange lines). The dependence of the broadband gain Gb on Gn is depicted in Fig. 3c. The results are in agreement with Eq. (4).

Fig. 3: Experimental determination of quantum efficiency.
figure 3

a Planck spectroscopy for varying broadband gain Gb. The average photon number nb of the amplified signal (dots) is indicated at the reconstruction point. The lines are fits to the corresponding Planck distributions. b Average photon number nn of the amplified signal (dots) at the reconstruction point for the amplification of a continuous coherent signal for different values of the narrowband gain Gn. The average photon number of the input signal is labeled with nin. From the offsets of the linear fits (lines), we can calculate the number of added noise photons. c Experimentally determined broadband gain Gb vs. narrowband gain Gn of the JPA.

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In Fig. 4a, we plot the measured quantum efficiencies for JPA 1 for the amplification of thermal states (cyan dots) and coherent states (purple dots), respectively. The red dashed line depicts the SQL determined by Eq. (3). Figure 4b shows the result for the same experiment with JPA 2 instead of JPA 1. For both JPAs, we find a gain region where we clearly exceed the SQL for the amplification of broadband states.

Fig. 4: Quantum efficiency for nondegenerate broadband and narrowband amplification.
figure 4

Experimental quantum efficiency of the flux-driven JPAs for broadband (cyan dots) and narrowband (purple dots) input signals. Panel a shows the data for JPA 1 and panel b the data for JPA 2. The red dashed line represents the SQL. The gain dependence of the quantum efficiencies for the narrowband and broadband cases is fitted using Eq. (10).

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Experimental limitations on quantum efficiency

Importantly, Fig. 4b shows that we can achieve a maximal quantum efficiency η = 0.69 ± 0.02 with our setup, which substantially exceeds the SQL. This value is comparable to quantum efficiencies reached with degenerate phase-sensitive JPAs36,46 and is notably higher than the quantum efficiency of 0.32 reached with the phase-preserving JTWPA in ref. 31. The deviation from the theoretically achievable quantum efficiency of unity can be explained by noise in the pump signal, as discussed below. Furthermore, we observe that the experimentally determined dependence of quantum efficiency on the gain in Fig. 4 reaches a maximum and decreases for higher gains for the narrowband and broadband case. For low JPA gains, the noise photons nH = 11.3, which are added by the HEMT, limit the quantum efficiency. This contribution becomes irrelevant in the high-gain limit, since its influence decreases with 1/G, where G = Gn (G = Gb) for narrowband (broadband) amplification. Since the parametric gain depends on the pump power, fluctuations in the pump photon number imply additional noise in the signal mode47. We describe the noisy pump signal in the frame rotating at a pump frequency ωp by

$$\hat{A}(t)=({\alpha }_{0}+{\hat{f}}_{{{{\rm{p}}}}}(t)){e}^{-i{\omega }_{{{{\rm{p}}}}}t},$$
(9)

where α0 denotes the amplitude of the coherent pump signal and the operator \({\hat{f}}_{{{{\rm{p}}}}}(t)\) represents thermal noise. We use the Wiener–Khinchin theorem to calculate the variance of the corresponding power fluctuations (see Supplemental Material)48. The experimentally determined dependence of the parametric gain on the pump power can be fitted by an exponential function (see Supplemental Material). Thus, the gain-dependent JPA noise nJ(G) can be approximated by

$${n}_{{{{\rm{J}}}}}(G)={n}_{{{{\rm{J}}}}}^{\prime}{(G-1)}^{\epsilon },$$
(10)

where ϵ depends on JPA parameters and \({n}_{{{{\rm{J}}}}}^{\prime}\) is a constant prefactor (see Supplemental Material). We use Eq. (10) to fit the measured quantum efficiencies in Fig. 4a, b and treat \({n}_{{{{\rm{J}}}}}^{\prime}\) and ϵ as fitting parameters. In Fig. 4a, the last data point for broadband amplification is not considered for the fit, as JPA 1 starts entering a nonlinear compression regime. The fit is depicted by the solid lines in Fig. 4a, b for both JPAs and successfully reproduces the maximum as well as the behavior for low gain values.

Discussion

In conclusion, we have investigated a nondegenerate linear parametric amplification of broadband signals and have derived a quantitative criterion for an input signal bandwidth under which a quantum efficiency of η = 1 can be achieved. We have used a superconducting flux-driven JPA to experimentally determine the quantum efficiencies for amplification of broadband thermal states and demonstrated η = 0.69 ± 0.02 which significantly exceeds the SQL ηql = 0.5. Thus, we have verified that for the parametric amplification of broadband input states, an idler mode may also carry signal information and does not add extra noise to the output. However, the SQL violation in our experiment comes not from the phase interference but rather from the fact that average amplitudes for both the signal and idler modes encode the original signal information (see Supplemental Material). Since it is difficult to define a phase for a general broadband signal, which results from the technical difficulty to stabilize the relative phase of signal and idler in case of non-commensurable frequencies, encoding information in the photon number is a more natural choice. The absence of the SQL is furthermore reflected by the fact that the idler mode does not simply lead to a constant power offset, but alters the amplifier gain. This experimental observation is in stark contrast with the conventional phase-sensitive amplification regime, where the SQL violation relies on the phase interference. In our case, the latter is impossible due to the absence of any phase correlations in broadband thermal signals used for amplification. However, although a relative phase between signal and idler does not have an impact on the quantum limit for the noise, the amplifier output itself can depend on such a phase relationship. Furthermore, we have shown that the gain dependence of η can be explained by the photon number fluctuations in the pump tone. One can exploit quantum efficiencies above the SQL η > 0.5 in experiments where ultra low-noise amplification is a key prerequisite. For instance, it can be used for high-efficiency parity detection of entangled superconducting qubits via readout of dispersively coupled resonators20. Assuming that these resonators are probed at the respective signal and idler frequencies of a readout JPA, one can amplify the combined resonator response with quantum efficiency beyond the SQL. Another useful application could be a direct broadband dispersive qubit readout with weak thermal states generated artificially or naturally occurring due to finite temperatures of the cryogenic environment49.

Methods

Extracting the quantum efficiency for broadband signals

To measure the noise added by the amplification of broadband input signals, we vary the temperature Ta of the heatable 30 dB attenuator from 40 mK to 600 mK using a PID control architecture. We detect the quadrature moments 〈ImQn〉 with \(m,n\in {{\mathbb{N}}}_{0}\), m + n ≤ 4 from the digitized and filtered output signal. The detected power P(Ta) is determined by the sum 〈I2〉 + 〈Q2〉 of the second order moments and follows a Planck curve45

$$\begin{array}{ll}P({T}_{{{{\rm{a}}}}})\,=\,\frac{\langle {I}^{2}\rangle +\langle {Q}^{2}\rangle }{R}\\ \qquad\quad=\,\frac{\kappa {G}_{{{{\rm{b}}}}}\tilde{G}}{R}\left[\frac{1}{2}\coth \left(\frac{\hslash {\omega }_{{{{\rm{s}}}}}}{2{k}_{{{{\rm{B}}}}}{T}_{{{{\rm{a}}}}}}\right)+{n}_{{{{\rm{f,b}}}}}\right],\end{array}$$
(11)

where nf,b is the total noise added by the amplification chain referred to the input, R = 50 Ω is the line impedance, Gb is the broadband JPA gain, \(\tilde{G}\) is the gain of the HEMT and the remaining amplification chain and κ denotes the photon-number-conversion factor (PNCF). The gain dependence of nf,b can be determined by repeating the temperature sweep for varying values of Gb and fitting a Planck curve to each of the results. To be able to experimentally control Gb, we use a vector network analyzer to measure the pump power dependence of the narrowband parametric gain Gn. We then expect Gb = Gn + 3 dB and measure Planck curves for expected Gb ranging from 6 dB to 27 dB in steps of 3 dB and fit Eq. (11) to each experimental outcome. Since we measure with a two-pulsed scheme, where the JPA pump signal is only switched on during the second pulse, Gb can be extracted directly from the measurement by calculating the ratio of the prefactors \({G}_{{{{\rm{b}}}}}\tilde{G}\) of the Planck curves corresponding to the second pulse and first pulse. We calculate the broadband quantum efficiencies ηb = 1/(1 + 2nf,b) from the fit parameters. The error bars for the quantum efficiency are determined from the fit error Δnf,b by error propagation.

Extracting the quantum efficiency for narrowband signals

To calibrate the photon number in a coherent input signal, we switch the JPA pump off and tune the JPA resonance frequency out of the measurement bandwidth such that the JPA does not have any impact on the calibration procedure. We then perform Planck spectroscopy to determine the PNCF for the signal reconstruction point45. Following that, we vary the power Pcoh of the coherent input signal and determine the photon number ncoh at the reconstruction point using the reference-state reconstruction method for each value of Pcoh (see Supplemental Material). This data can be linearly fitted according to

$${n}_{{{{\rm{coh}}}}}({P}_{{{{\rm{coh}}}}})={k}_{1}{P}_{{{{\rm{coh}}}}}+{k}_{2},$$
(12)

where k1 and k2 are fitting parameters. To measure the additive noise number, we tune the JPA into resonance and measure in the two-pulsed scheme. We vary the coherent input photon number nin and measure the output photon number nout for the case in which the JPA pump is switched off (first pulse). We repeat the measurement and detect the signal power nJPA when the JPA pump is switched on. Both results are fitted linearly according to

$${n}_{{{{\rm{out}}}}}={k}_{3}{n}_{{{{\rm{in}}}}}+{k}_{4},$$
(13)
$${n}_{{{{\rm{JPA}}}}}={k}_{5}{n}_{{{{\rm{in}}}}}+{k}_{6}$$
(14)

with fitting constants k3, k4, k5, k6. From this, we extract the narrowband gain Gn = k5/k3 and the total number of added noise photons nf,n = k6/Gn, referred to the input which allows us to calculate the narrowband quantum efficiency ηn. The error bars for ηn are determined from the fit by error propagation.

Fitting the measured quantum efficiencies

The total additive noise nf, referred to the input of the amplification chain, can be related to the JPA noise nJ and the HEMT noise nH with the Friis equation50

$${n}_{{{{\rm{f}}}}}={n}_{{{{\rm{J}}}}}+\frac{{n}_{{{{\rm{H}}}}}}{G},$$
(15)

where G denotes the JPA gain (either narrowband or broadband). Thus, we can express the quantum efficiency η as

$$\eta =\frac{G}{G+2G{n}_{{{{\rm{J}}}}}+2{n}_{{{{\rm{H}}}}}}.$$
(16)

The JPA noise nJ is dependent on G. For G = 1, i.e., when the JPA is switched off, we expect nJ = 0. We assume that the gain-dependent noise nJ,b(Gb) for amplification of broadband signals mainly results from pump induced fluctuations and fit the measured quantum efficiencies for the broadband case using

$${\eta }_{{{{\rm{b}}}}}({G}_{{{{\rm{b}}}}})=\frac{{G}_{{{{\rm{b}}}}}}{{G}_{{{{\rm{b}}}}}+2{G}_{{{{\rm{b}}}}}{n}_{{{{\rm{J,b}}}}}^{\prime}{({G}_{{{{\rm{b}}}}}-1)}^{{\epsilon }_{{{{\rm{b}}}}}}+2{n}_{{{{\rm{H}}}}}},$$
(17)

where we treat \({n}_{{{{\rm{J,b}}}}}^{\prime}\) and ϵb as fitting parameters and insert nH = 11.3 (see Supplemental Material). For the narrowband quantum efficiency ηn, we assume that the idler mode adds additional vacuum fluctuations to the signal. Thus, we assume for the noise photons

$$n_{{\mathrm{J}},{\mathrm{n}}}(G_{\mathrm{n}}) = \underbrace{n_{{\mathrm{J}},{\mathrm{n}}}^{\prime} \left(G_{\mathrm{n}}-1\right)^{\epsilon_{\mathrm{n}}}}_{{{\mathrm{Pump}}-{\mathrm{induced}}}} + \underbrace{\frac{1}{2}\left(1-\frac{1}{G_{\mathrm{n}}}\right)}_{{{\mathrm{SQL}}}},$$
(18)

implying that we use

$${\eta }_{{{{\rm{n}}}}}({G}_{{{{\rm{n}}}}})=\frac{{G}_{{{{\rm{n}}}}}}{2{G}_{{{{\rm{n}}}}}-1+2{G}_{{{{\rm{n}}}}}{n}_{{{{\rm{J}}}},{{{\rm{n}}}}}^{\prime}{\left({G}_{{{{\rm{n}}}}}-1\right)}^{{\epsilon }_{{{{\rm{n}}}}}}+2{n}_{{{{\rm{H}}}}}}$$
(19)

as a fit function for this case, where we treat \({n}_{{{{\rm{J}}}},{{{\rm{n}}}}}^{\prime}\) and ϵn as fit parameters (see Supplemental Material).