Abstract
The lownoise 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 phasepreserving 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 fluxdriven 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 phasepreserving 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 phasepreserving amplifier needs to add at least half a noise photon in the highgain 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. Quantumlimited amplification of quantum microwave states has been achieved with superconducting Josephson parametric amplifiers (JPAs)^{27,28,29}, but also by employing Josephson travelingwave 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 multiqubit 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 fluxdriven JPA and is complimentary to the conventional phasepreserving nondegenerate (narrowband and quantumlimited) or phasesensitive 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 phasedependent or phaseindependent amplification regimes, respectively. One could also note that the nondegenerate amplifier allows to realize a twomode squeezing operation between the input signal and input idler modes^{39,40}. Here, we experimentally demonstrate a JPA for the phaseindependent 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 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
Parametrically driving the JPA results in amplification of the incoming signal and creation of an additional phaseconjugated 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 highgain 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 threewave 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 downconversion 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 output \(\hat{c}(\omega )\) results from a linear combination of amplified incoming signals \(\hat{a}(\tilde{\omega })\) and phaseconjugated idler signals \({\hat{a}}^{{\dagger} }({\tilde{\omega }}_{{{{\rm{i}}}}})\) as well as from an additive noise mode \(\hat{f}(\omega )\)^{26}
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 ω_{s} and has a singleside bandwidth b_{s} (total bandwidth 2b_{s}). 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} = G_{n}(ω) and ∣L(ω_{i})∣^{2} = G_{n}(ω) − 1, where G_{n}(ω) is the power gain. The strength of the additive noise is described with the noise power spectral density S_{f}(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
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: b_{1} = 2Δ − B and b_{2} = 2Δ + B. For b_{s} ≤ b_{1}, Eq. (6) 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 fluxdriven JPA serially connected to a cryogenic highelectronmobility 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 referencestate 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 \({G}_{{{{\rm{n}}}}}(\omega )=1+{G}_{0}{b}_{{{{\rm{J}}}}}^{2}/({b}_{{{{\rm{J}}}}}^{2}+{(\omega {\omega }_{0})}^{2})\), 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 \(\tau \equiv {b}_{{{{\rm{J}}}}}\sqrt{{G}_{0}}\) denotes the gainbandwidth product^{36} (see Supplemental Material). The corresponding limit η_{ql} of the quantum efficiency η can be calculated with Eq. (1). The case b_{s} ≤ B is not considered in Eq. (8) as it is only of technical relevance since we can always achieve b_{s} = 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 η_{ql} = 1/2 for coherent input signals, approximately reproducing Eq. (3), whereas we expect η_{ql} = 1 for broadband signals.
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 phasesensitive JPAs^{36,46} and is notably higher than the quantum efficiency of 0.32 reached with the phasepreserving 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 highgain 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} by
where α_{0} 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 gaindependent JPA noise n_{J}(G) can be approximated by
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 fluxdriven 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 noncommensurable 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 phasesensitive 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 lownoise amplification is a key prerequisite. For instance, it can be used for highefficiency 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}.
Methods
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 \(m,n\in {{\mathbb{N}}}_{0}\), 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, \(\tilde{G}\) is the gain of the HEMT and the remaining amplification chain and κ denotes the photonnumberconversion 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 twopulsed 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 \({G}_{{{{\rm{b}}}}}\tilde{G}\) 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 referencestate 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 twopulsed 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 gaindependent 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 \({n}_{{{{\rm{J,b}}}}}^{\prime}\) 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
implying that we use
as a fit function for this case, where we treat \({n}_{{{{\rm{J}}}},{{{\rm{n}}}}}^{\prime}\) and ϵ_{n} as fit parameters (see Supplemental Material).
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We acknowledge support by the German Research Foundation through the Munich Center for Quantum Science and Technology (MCQST), Elite Network of Bavaria through the program ExQM, EU Flagship project QMiCS (Grant No. 820505), JST ERATO (Grant No. JPMJER1601).
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K.G.F. and F.D. planned the experiment. M.R., S.P., and K.G.F. performed the measurements and analyzed the data. M.R. and K.G.F. developed the theory. Q.C., Y.N., and M.P. contributed to development of the measurement software and experimental setup. K.I. and Y.N. provided the JPA samples. F.D., A.M., and R.G. supervised the experimental part of this work. M.R. and K.G.F. wrote the manuscript. All authors contributed to discussions and proofreading of the manuscript.
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Renger, M., Pogorzalek, S., Chen, Q. et al. Beyond the standard quantum limit for parametric amplification of broadband signals. npj Quantum Inf 7, 160 (2021). https://doi.org/10.1038/s4153402100495y
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DOI: https://doi.org/10.1038/s4153402100495y
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