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The two mesoscopic devices most commonly used to electrically measure spin- and charge-based quantum systems are the single-electron transistor (SET) and quantum point contact (QPC). These devices operate according to the same scheme: the electrometer is biased by a source–drain voltage Vsd and the current I through it is measured. Motion of charges near the electrometer causes its differential conductance Gdto change, resulting in changes in I. The ultimate sensitivity of an electrometer operated in this way is therefore set by the non-equilibrium current noise (shot noise) present in I(t). The same current fluctuations also determine its backaction, and, therefore, its proximity to the quantum limit.

Classically, current noise is described by a spectral density SIsym(ω) that is symmetric in frequency ω. Quantum mechanically, however, we must distinguish between positive frequency noise, which transfers energy from a measured system to the electrometer, and negative frequency noise, which transfers energy from the electrometer to the measured system. A simple Fermi’s golden rule calculation of, for example, an electrometer coupled to a qubit prepared in its ground state shows this10. The transition rate for the qubit to be promoted to its excited state is proportional to SI(−ω0), where is the unsymmetrized quantum noise spectrum of the electrometer current and ℏω0 is the separation in energy between the ground and excited states. Similarly, the rate at which a system prepared in its excited state decays to the ground state is given by SI(+ω0). To make a complete measurement of the quantum noise of an electrometer, one must obtain information regarding both SI(+ω0) and SI(−ω0).

Rather than couple our S-SET electrometer to a two-level system to carry out our quantum noise measurements, we instead couple it to another canonical quantum system, namely a harmonic oscillator consisting of an on-chip superconducting L C resonator11 as shown in Fig. 1a. This resonator serves both to impedance match the S-SET to the impedance Z0=50 Ω of the measurement electronics12, and also to amplify its current noise so that it can be detected by a subsequent cryogenic amplifier.

Figure 1: Experimental method for quantum noise measurements.
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a, Measurement circuit, showing the sample and radiofrequency electronics, including a bias tee, circulator and cryogenic HEMT amplifier. b, Model for the S-SET/resonator, showing the S-SET as an effective bath with temperature TSET and damping rate γSET. The asymmetric current noise SI(+ω0) and SI(−ω0) is related to the probability of the S-SET absorbing or emitting a photon, as illustrated. c, Electron micrograph of a typical S-SET. d, Noise power Pn at the output of the amplifier chain versus SET current I.

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In our S-SET/resonator system, the unsymmetrized shot noise of the S-SET at is related to its probability to either emit energy to or absorb energy from the resonator. This enables a complete characterization of the noise. To see this, consider the Hamiltonian for the L C resonator and S-SET given by

where is the flux in the inductor, is the charge on the capacitor and is the operator describing the noisy current flowing through the S-SET. This Hamiltonian is formally equivalent (see Supplementary Information) to one recently explored in the context of measuring the backaction of a charge detector on a nanomechanical resonator13,14,15. Assuming a large separation of timescales between fluctuations in and the response time of the L C resonator, it can be shown rigorously that at the resonant frequency ω0 the S-SET can be viewed as an effective thermal bath, as illustrated in Fig. 1b, characterized by an effective temperature TSET and a damping rate γSET.

To make a complete noise measurement, it is not necessary to measure SI(+ω0) or SI(−ω0) separately, as has been done in other systems16,17,18. As long as two linearly independent combinations can be measured, complete noise information is obtained. This is how we proceed. Using equation (1) and the approach of refs 13, 14, 15, it is simple to show that (for ℏω0sufficiently small compared with kBTSET)

where Cp is the resonator capacitance. Note that kBTSET can be significantly smaller than the energy of either the S-SET’s physical temperature kBT or its bias voltage e Vsd. Furthermore, it can be either positive or negative, as can γSET, depending on whether absorption or emission, respectively, dominates the quantum noise.

Measuring the total symmetrized noise the S-SET injects into the resonator, while simultaneously measuring the rate γSET at which it damps the resonator modes therefore enables a complete measurement of the S-SET quantum noise without a separate measurement of either SI(+ω0) or SI(−ω0). A similar approach was used to investigate the backaction of an S-SET capacitively coupled to a nanomechanical resonator9. Our approach, in which there is a direct electrical connection between the S-SET and a superconducting L C resonator, is simpler to implement and enables a more accurate measurement of effective temperature and damping. Furthermore, the S-SET can easily be replaced by some other nanostructure such as graphene with interesting noise properties.

Our first step in characterizing the total quantum noise is to measure γSET=Gd/Cp by means of the S-SET’s differential conductance Gd. Although Gd is usually measured near d.c., extensive measurements of the reflection coefficient for waves incident on the resonator Γin show11 that Gd accurately predicts Γin and therefore γSET at ω0. A plot of our measurements of differential conductance Gd versus Vsd and island charge number ng=VgCg/e, where Vg is the gate voltage is shown in Fig. 2a. Interestingly, there are several points in the Vsd–ng plane at which Gd<0. At these points, the S-SET exhibits negative differential conductivity (NDC). NDC is also clearly visible in Fig. 2b as decreasing current with increasing bias just past the double-Josephson quasiparticle (DJQP) current maximum. The NDC regions are associated with Cooper-pair resonances, occurring on the high-bias side of both the supercurrent and the DJQP features. In the DJQP subgap transport cycle, current flows by means of a combination of Cooper-pair and quasiparticle tunnelling19. This cycle appears as a peak in current near the intersection of two Cooper-pair resonances8,20, one for each junction in the S-SET, at Vsd=2Ec/e as in Fig. 2b, where Ec=e2/2CΣ is the S-SET charging energy.

Figure 2: Subgap transport in the S-SET.
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a, Gd for the S-SET versus Vsd and ng. NDC is visible for Vsd and ng in the vicinity of the supercurrent and the DJQP cycle. Cooper-pair resonances and are shown as the dashed lines; the DJQP cycle occurs at their intersection. b, I–V characteristics of the S-SET for ng≈0.5, emphasizing the presence of NDC (arrow) on the high-bias side of the DJQP resonance where current decreases with increasing bias. c, Amplitude-modulated reflected power for a charge modulation of 0.01e at 100 kHz. The lower curve is the noise floor of the amplifier chain for I=0. d, DJQP cycle. When the S-SET is biased in ng and Vsd so that Cooper pairs do not have enough energy to tunnel on or off the island (that is, the S-SET is biased to the left of both Cooper-pair resonance lines in a), a photon must be absorbed from the resonator for tunnelling to occur. Similarly, when the S-SET is biased so that Cooper pairs have excess energy (to the right of both resonances in a), a photon must be emitted during tunnelling.

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When the S-SET is biased above the DJQP resonance (blue detuning), Cooper pairs must emit energy to tunnel. Similarly, when the S-SET is biased below the resonance (red detuning), Cooper pairs must absorb energy. As illustrated in Fig. 2d, because the S-SET’s electromagnetic environment is dominated by the L C resonator, most absorption (emission) will take the form of photon exchange with the tank circuit21. In terms of the picture of resonator damping given above, if Gd<0 we expect both γSET<0 and |Γin|>1. Physically, this negative damping corresponds to net emission of energy into the resonator by the S-SET. For the total symmetrized noise of the S-SET to remain positive (as it must), the S-SET effective temperature TSET must also be negative in this region.

The second step in characterizing the total quantum noise of the S-SET is a measurement of TSET. This in turn first requires a measurement of the integrated SET shot noise , which has not previously been measured in the subgap regime. at 300 mK in the vicinity of the DJQP resonance is shown in Fig. 3a on a logarithmic scale. The noise is minimal for red detuning with respect to the DJQP, and maximal for blue detuning. We focused on the DJQP region for several reasons. First and foremost, an S-SET operated near the DJQP resonance has been predicted to possess the ideal balance of sensitivity and backaction needed to approach the quantum limit8. Second, near this cycle, the S-SET’s quantum noise properties are expected to depend strongly on the SET bias in Vsd and ng with respect to this intersection8,14,15. Last, the charge sensitivity δ q of the S-SET is typically excellent here; charge sensitivity measurements as in Fig. 2c gave for operation as a radiofrequency SET (ref. 12). It is interesting to note that in contrast, a normal-state SET biased near threshold is expected to operate far from the quantum limit22.

Figure 3: Noise and reflected power measurements.
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a,  at 300 mK. Cooper-pair resonances are shown by the dashed lines, and the centre of the DJQP occurs at their intersection. Noise is maximal for blue detuning and minimal for red detuning. b, |Γin|(Vsd,ng) at 300 mK. A small region for which |Γin|>1 exists for blue detuning. c, At 900 mK, is smaller in the blue-detuned region (in agreement with a lessening of NDC there for higher temperature). The reduction of in the red-detuned region is more pronounced, and tracks nearly exactly the Cooper-pair resonance lines. d, Gd(Vsd,ng) at 300 mK. The region of NDC corresponds nearly exactly to that for which |Γin|>1 in b.

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Our measurements of the S-SET noise characteristics show excellent correspondence with photon emission and absorption by the S-SET. We show this correspondence by measuring the reflection coefficient |Γin| of the tank circuit over the same range of Vsd and ng, as in Fig. 3b. For most values of Vsd and ng, we found |Γin|<1, indicating net absorption by the S-SET. However, when the S-SET is blue detuned, there is a region for which |Γin|>1, indicating emission. Here the S-SET provides negative damping, returning more power to the resonator than is delivered by the radiofrequency excitation. Remarkably, therefore, as we measure |Γin|>1, we are directly measuring photon emission by Cooper pairs as they tunnel. Comparing this to the S-SET conductance in the same region as shown in Fig. 3d, we again see excellent correspondence. The region of negative damping corresponds exactly to the region of NDC. This is in accord with our expectation based both on the forms of γSET and Γin, and with the more sophisticated quantum noise viewpoint of equations (2) and (3).

Having measured Gd and , we now proceed to completely and quantitatively determine the quantum noise of our S-SET. As indicated above, we treat the S-SET as a thermal bath with conductance Gd and available noise power kBTSET. It can be shown that (where γT is the total damping rate of the resonator—see the Methods section), so that TSET can be found from measurements of , whereas γSET can be determined directly from Gd through the relation γSET=Gd/Cp. The resulting values of γSET and TSET at ω0 versus Vsd and ng near the DJQP for 300 mK are shown in Fig. 4a,b. The tendency of the S-SET to either emit or absorb (as measured by γSET) and its degree of asymmetry (as measured by TSET∝(SI(ω0)+SI(−ω0))/(SI(ω0)−SI(−ω0))) vary strongly with Vsd and ng. For blue detuning where Cooper pairs must give off energy, we observe both negative damping and a negative effective temperature. Although TSET is large in some areas, for most bias points TSET≲1 K, making it smaller than e Vsd/kB but still large enough that our assumption kBTSET≫ℏω0 in equations (2) and (3) is still valid. For red detuning, where the S-SET is strongly absorbing, TSET can be as low as 100±40 mK, less than the ambient temperature and indicating that the S-SET is capable of refrigeration. Although theoretical expressions for γSET and TSET near the DJQP exist14,15, they assume capacitive coupling of the S-SET to a resonator rather than our direct electrical connection, and also ignore higher-order tunnelling processes known9,20 to be important for our relatively low-resistance S-SETs. Nonetheless, theory predicts a minimum TSET≈250 mK for an S-SET with our parameters, in reasonable agreement with our results. Finally, we prefer TSET and γSET as a description of the S-SET quantum noise over the Fano factor because the latter is due only to fluctuations of the number of tunnelling electrons23. In our experiment, variations in  arising from electron number fluctuations are indistinguishable from those due to emission/absorption of photons.

Figure 4: Quantum noise of the S-SET.
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a, S-SET damping rate γSET. b, S-SET effective temperature TSET at ω0. Together, these give a complete and quantitative description of the S-SET quantum noise.

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We now estimate the measurement capability of our S-SET relative to the quantum limit. We imagine coupling the S-SET to some external device such as a quantum dot. The ratio of the time it takes the S-SET to measure the dot’s charge state to the time it takes to dephase it must be greater than one. Quantitatively we express this condition in terms of the square root χ of this ratio given by , where equality corresponds to the quantum limit. Here, SQsym and SIsym are the symmetrized zero-frequency spectral densities of charge fluctuations on and current through the S-SET, Eint describes its interaction with the measured system and ΔI is the change in S-SET current corresponding to a change in the system charge state3 (see Supplementary Information). Using the current I through the S-SET to estimate SQsym≈3e3/8I, we find independent of the specifics of the dot and its coupling to the S-SET. This is as it should be: an amplifier’s proximity to the quantum limit is an intrinsic property of the amplifier and does not depend on properties of the measured system. For typical currents I≈5 nA near the DJQP and neglecting the noise of the high-electron-mobility transistor (HEMT) amplifier, we find χ≈3.6, indicating that our radiofrequency SET operates near the quantum limit. If amplifier noise is included, we find χ≈8. These estimates each represent a fourfold improvement in χ over other results for both the S-SET (ref. 9), and a near-optimal normal-state SET (ref. 24). For the latter, we estimate an intrinsic χ≈20. Note also that this approach probably overestimates χ, because it ignores the presence of higher-order tunnelling processes that could bring the S-SET closer to the quantum limit25. In addition to its inherent interest in terms of quantum measurement, an S-SET charge sensor operating in the vicinity of the quantum limit has potentially broad implications in terms of its ability to measure a host of quantum systems.

Methods

All measurements were carried out in a 3He refrigerator at its base temperature of 290 mK. The circulator and HEMT amplifier were at a temperature of 2.9 K. A d.c. source–drain bias Vsd and small a.c. voltage vac were filtered and introduced to the high-frequency circuit by means of a bias tee. The input coaxial line included attenuation of 34 dB. The data presented are for a representative sample. In all, five samples were measured, each producing similar results. (See Supplementary Information for details on sample parameters.) The resonator is a superconducting on-chip spiral11 for which internal losses are negligible, and can be fully described by its inductance L≈169 nH and its parasitic capacitance to ground Cp=0.14 pF, giving a resonant frequency ω0=1.04 GHz. Its total damping rate γT is given by γT=γ0+γSET, where γ0=Z0/L is the damping due to the coupling to the feedline. The reflection coefficient Γin for waves incident on the resonator can be written in terms of γ0 and γSET as Γin=(γSET−γ0)/(γSET+γ0). On the basis of the simple model for the S-SET/resonator circuit in Fig. 1b, we expect γSET=Gd/Cp. This relation, verified by extensive measurements of Γin versus Gd, agrees with the expectation that ω0 is still in the low-frequency limit for the S-SET, because tunnelling events typically occur at a much higher rate (tens of gigahertz). To measure Γin, we applied a very small carrier wave (−149 dB m), measured the reflected power, and after accounting for the HEMT and circulator, computed Γin. To find the integrated noise we started by applying a d.c. current I and measuring the total output noise power Pn at ω0 in a bandwidth Δf=5 MHz at the output of the amplifier chain, as shown in Fig. 1a,d. The total output noise Pn includes contributions from the S-SET, HEMT amplifier and circulator: Pn=A(kBTHEMT+|Γin|2kBTcirc+PSET(I))Δf, where THEMT and Tcirc are the HEMT and circulator noise temperatures, PSET(I) is the spectral noise density of the SET referred to the HEMT input and A is the total gain of the amplifier chain26. We use our noise power data to determine A=61 dB, THEMT=9.5 K and Tcirc≈2.9 K, the last of which is in excellent agreement with the circulator’s physical temperature (see Supplementary Information for details). This information enables us to extract PSET versus source–drain bias Vsdand island charge number ng. The integrated SET noise referred to the input of the HEMT is given by .