Beyond the standard quantum limit for parametric amplification of broadband signals

The low-noise amplification of weak microwave signals is crucial for countless protocols in quantum information processing. Quantum mechanics sets an ultimate lower limit of half a photon to the added input noise for phase-preserving amplification of narrowband signals, also known as the standard quantum limit (SQL). This limit, which is equivalent to a maximum quantum efficiency of 0.5, can be overcome by employing nondegenerate parametric amplification of broadband signals. We show that, in principle, a maximum quantum efficiency of unity can be reached. Experimentally, we find a quantum efficiency of 0.69 ± 0.02, well beyond the SQL, by employing a flux-driven Josephson parametric amplifier and broadband thermal signals. We expect that our results allow for fundamental improvements in the detection of ultraweak microwave signals.


INTRODUCTION
In quantum technology, resolving low power quantum signals in a noisy environment is essential for an efficient readout of quantum states 1 . 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 phase 2 . Due to the low energy of microwave photons, this amplification process is of particular relevance for the tomography of quantum microwave states 3 . State tomography is crucial for experimental protocols with quantum propagating microwaves such as entanglement generation [4][5][6][7][8] , secure quantum remote state preparation 9 , quantum teleportation [10][11][12] , quantum illumination 13 , or quantum state transfer 14 . Furthermore, a conventional dispersive readout of superconducting qubits [15][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 photons 23 . This approach has proven to be extremely successful and has led to the realization of important milestones in superconducting quantum information processing 24,25 . 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 reached 31 . 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 systems 32 , quantum amplitude sensing 33 , detection of dark matter axions 34 , or the detection of the cosmic microwave background 35 , 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 phasepreserving nondegenerate (narrowband and quantum-limited) or phase-sensitive degenerate (narrowband and potentially noiseless) regimes [36][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 modes 39,40 . Here, we experimentally demonstrate a JPA for the phase-independent linear amplification of broadband signals with performance beyond the SQL.

Quantum limits on quantum efficiency
An ideal linear amplifier increases the signal photon number n s by the power gain factor G n 2 . The fluctuations in the output signal consist of the amplified vacuum fluctuations of the input signal and the noise photons n f , added by the amplifier, where n f is referred to the amplifier input. We use the quantum efficiency η to characterize the noise performance of our amplifiers 41 . 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 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 n i ≥ 1/2 photons. As a result of the narrowband parametric signal amplification, the idler adds 42 noise photons to the signal, referred to the amplifier input, as schematically depicted in Fig. 1a. Equation (2) implies that η is bounded by reaching 1/2 in the high-gain limit, G n ≫ 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.

Nondegenerate Josephson parametric amplifier
Parametric amplification can be realized by driving a nonlinear electromagnetic resonator with a strong coherent field (pump) at ω p = 2ω 0 , where ω 0 is the resonance frequency 37,42 . In the resulting three-wave mixing process, a pump photon splits into a signal photon at frequency ω s = ω p /2 + Δ and a corresponding idler photon at ω i = ω p /2 − Δ. Here, Δ denotes the detuning of the signal reconstruction frequency ω s from ω 0 . In the nondegenerate regime, we have Δ ≠ 0, leading to spectrally separated signal and idler modes 43 . 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 ω s . 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 b s 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).

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 outputĉðωÞ results from a linear combination of amplified incoming signalsâðωÞ and phaseconjugated idler signalsâ y ðω i Þ as well as from an additive noise modef where the integral is taken over all input modes I . Frequenciesω denote input modes, whereas output modes are described by ω.
We assume that the input signal is centered around the frequency ω s and has a single-side bandwidth b s (total bandwidth 2b s ). The signal amplitude gain Mðω;ωÞ and the idler amplitude gain Lðω;ωÞ satisfy Mðω;ωÞ ¼ MðωÞδðω ÀωÞ and Lðω;ωÞ The strength of the additive noise is described with the noise power spectral density S f (ω) of the bosonic modef ðωÞ, corresponding to the number of noise photons per mode. We evaluate the integral in Eq. (5) and calculate the commutator ½ĉðωÞ;ĉ y ðω 0 Þ. 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 where n ql is the quantum limit for the number of additive noise photons (see Supplemental Material). We define where Θ is the Heaviside step function. In Eq. (6), the range of integration over ω is limited by b s , if b s < B, otherwise it is set by B. We observe that there are two threshold values of the bandwidth reproduces the SQL, whereas for b 1 ≤b s ≤ b 2 , the input signal starts to overlap with idler modes and the lower bound for the additive noise decreases. In the broadband case b s ≥ b 2 , the signal covers all idler modes and Eq. (6) reduces to n f ≥ 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-electronmobility transistor (HEMT) amplifier with a gain of G H = 41 dB. The JPA is operated in the nondegenerate regime, which is realized by detuning the signal frequency ω s by Δ/2π = 300 kHz from half the pump frequency ω p /2 = ω 0 . 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. 1c 7,44 . 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 ω s /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 where G 0 denotes the maximal JPA gain and b J is the half width at half maximum JPA bandwidth 43 . Then, assuming G 0 ≫ 1, the quantum limit n ql for the number of additive noise photons is given by with β = B/τ, β s = b s /τ, and δ = Δ/τ, where τ b J ffiffiffiffiffi G 0 p denotes the gain-bandwidth product 36  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 chain 45 . As a result, we detect the photon number n b at the reconstruction point for varying broadband gain G b and show the result of this measurement in Fig. 3a for JPA 2. For each value of G b , the experimentally determined outcomes for n b (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 n in for different JPA gains G n . For each value of G n , the amplifier response n n 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 G b on G n is depicted in Fig. 3c. The results are in agreement with Eq. (4).
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.

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 JPAs 36,46 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 n H = 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 = G n (G = G b ) 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 mode 47 . We describe the noisy pump signal in the frame rotating at a pump frequency ω p bŷ where α 0 denotes the amplitude of the coherent pump signal and the operatorf p ðtÞ 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 n J (G) can be approximated by where ϵ depends on JPA parameters and n 0 J is a constant prefactor (see Supplemental Material). We use Eq. (10) to fit the measured quantum efficiencies in Fig. 4a, b and treat n 0 J 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 η ql = 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 resonators 20 . 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 environment 49 .

Extracting the quantum efficiency for broadband signals
To measure the noise added by the amplification of broadband input signals, we vary the temperature T a of the heatable 30 dB attenuator from 40 mK to 600 mK using a PID control architecture. We detect the quadrature moments 〈I m Q n 〉 with m; n 2 N 0 , m + n ≤ 4 from the digitized and filtered output signal. The detected power P(T a ) is determined by the sum 〈I 2 〉 + 〈Q 2 〉 of the second order moments and follows a Planck curve 45 where n f,b is the total noise added by the amplification chain referred to the input, R = 50 Ω is the line impedance, G b is the broadband JPA gain, 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 n f,b can be determined by repeating the temperature sweep for varying values of G b and fitting a Planck curve to each of the results. To be able to experimentally control G b , we use a vector network analyzer to measure the pump power dependence of the narrowband parametric gain G n . We then expect G b = G n + 3 dB and measure Planck curves for expected G b 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, G b can be extracted directly from the measurement by calculating the ratio of the prefactors G bG of the Planck curves corresponding to the second pulse and first pulse. We calculate the broadband quantum efficiencies η b = 1/(1 + 2n f,b ) from the fit parameters. The error bars for the quantum efficiency are determined from the fit error Δn f,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 point 45 . Following that, we vary the power P coh of the coherent input signal and determine the photon number n coh at the reconstruction point using the reference-state reconstruction method for each value of P coh (see Supplemental Material). This data can be linearly fitted according to where k 1 and k 2 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 n in and measure the output photon number n out for the case in which the JPA pump is switched off (first pulse). We repeat the measurement and detect the signal power n JPA when the JPA pump is switched on. Both results are fitted linearly according to with fitting constants k 3 , k 4 , k 5 , k 6 . From this, we extract the narrowband gain G n = k 5 /k 3 and the total number of added noise photons n f,n = k 6 /G n , referred to the input which allows us to calculate the narrowband quantum efficiency η n . The error bars for η n are determined from the fit by error propagation.

Fitting the measured quantum efficiencies
The total additive noise n f , referred to the input of the amplification chain, can be related to the JPA noise n J and the HEMT noise n H with the Friis equation 50 where G denotes the JPA gain (either narrowband or broadband). Thus, we can express the quantum efficiency η as The JPA noise n J is dependent on G. For G = 1, i.e., when the JPA is switched off, we expect n J = 0. We assume that the gain-dependent noise n J,b (G b ) for amplification of broadband signals mainly results from pump induced fluctuations and fit the measured quantum efficiencies for the broadband case using where we treat n 0 J;b and ϵ b as fitting parameters and insert n H = 11.3 (see Supplemental Material). For the narrowband quantum efficiency η n , we assume that the idler mode adds additional vacuum fluctuations to the signal. Thus, we assume for the noise photons n J;n ðG n Þ ¼ n 0 J;n G n À 1 ð Þ ϵn |fflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflffl ffl} implying that we use η n ðG n Þ ¼ G n 2G n À 1 þ 2G n n 0 J;n G n À 1 ð Þ ϵn þ 2n H (19) as a fit function for this case, where we treat n 0 J;n and ϵ n as fit parameters (see Supplemental Material).