Abstract
Quantum measurement has challenged physicists for almost a century. Classically, there is no lower bound on the noise a measurement may add. Quantum mechanically, however, measuring a system necessarily perturbs it. When applied to electrical amplifiers, this means that improved sensitivity requires increased backaction that itself contributes noise. The result is a strict quantum limit on added amplifier noise^{1,2,3,4,5,6}. To approach this limit, a quantumlimited amplifier must possess an ideal balance between sensitivity and backaction; furthermore, its noise must dominate that of subsequent classical amplifiers^{7}. Here, we report the first complete and quantitative measurement of the quantum noise of a superconducting singleelectron transistor (SSET) near a double Cooperpair resonance predicted to have the right combination of sensitivity and backaction^{8}. A simultaneous measurement of our SSET’s charge sensitivity indicates that it operates within a factor of 3.6 of the quantum limit, a fourfold improvement over the nearest comparable results^{9}.
Main
The two mesoscopic devices most commonly used to electrically measure spin and chargebased quantum systems are the singleelectron transistor (SET) and quantum point contact (QPC). These devices operate according to the same scheme: the electrometer is biased by a source–drain voltage V_{sd} and the current I through it is measured. Motion of charges near the electrometer causes its differential conductance G_{d}to change, resulting in changes in I. The ultimate sensitivity of an electrometer operated in this way is therefore set by the nonequilibrium 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 S_{I}^{sym}(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 this^{10}. The transition rate for the qubit to be promoted to its excited state is proportional to S_{I}(−ω_{0}), where is the unsymmetrized quantum noise spectrum of the electrometer current and ℏω_{0} 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 S_{I}(+ω_{0}). To make a complete measurement of the quantum noise of an electrometer, one must obtain information regarding both S_{I}(+ω_{0}) and S_{I}(−ω_{0}).
Rather than couple our SSET electrometer to a twolevel system to carry out our quantum noise measurements, we instead couple it to another canonical quantum system, namely a harmonic oscillator consisting of an onchip superconducting L C resonator^{11} as shown in Fig. 1a. This resonator serves both to impedance match the SSET to the impedance Z_{0}=50 Ω of the measurement electronics^{12}, and also to amplify its current noise so that it can be detected by a subsequent cryogenic amplifier.
In our SSET/resonator system, the unsymmetrized shot noise of the SSET 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 SSET 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 SSET. 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 resonator^{13,14,15}. 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 ω_{0} the SSET can be viewed as an effective thermal bath, as illustrated in Fig. 1b, characterized by an effective temperature T_{SET} and a damping rate γ_{SET}.
To make a complete noise measurement, it is not necessary to measure S_{I}(+ω_{0}) or S_{I}(−ω_{0}) separately, as has been done in other systems^{16,17,18}. 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 ℏω_{0}sufficiently small compared with k_{B}T_{SET})
where C_{p} is the resonator capacitance. Note that k_{B}T_{SET} can be significantly smaller than the energy of either the SSET’s physical temperature k_{B}T or its bias voltage e V_{sd}. Furthermore, it can be either positive or negative, as can γ_{SET}, depending on whether absorption or emission, respectively, dominates the quantum noise.
Measuring the total symmetrized noise the SSET injects into the resonator, while simultaneously measuring the rate γ_{SET} at which it damps the resonator modes therefore enables a complete measurement of the SSET quantum noise without a separate measurement of either S_{I}(+ω_{0}) or S_{I}(−ω_{0}). A similar approach was used to investigate the backaction of an SSET capacitively coupled to a nanomechanical resonator^{9}. Our approach, in which there is a direct electrical connection between the SSET and a superconducting L C resonator, is simpler to implement and enables a more accurate measurement of effective temperature and damping. Furthermore, the SSET 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 γ_{SET}=G_{d}/C_{p} by means of the SSET’s differential conductance G_{d}. Although G_{d} is usually measured near d.c., extensive measurements of the reflection coefficient for waves incident on the resonator Γ_{in} show^{11} that G_{d} accurately predicts Γ_{in} and therefore γ_{SET} at ω_{0}. A plot of our measurements of differential conductance G_{d} versus V_{sd} and island charge number n_{g}=V_{g}C_{g}/e, where V_{g} is the gate voltage is shown in Fig. 2a. Interestingly, there are several points in the V_{sd}–n_{g} plane at which G_{d}<0. At these points, the SSET exhibits negative differential conductivity (NDC). NDC is also clearly visible in Fig. 2b as decreasing current with increasing bias just past the doubleJosephson quasiparticle (DJQP) current maximum. The NDC regions are associated with Cooperpair resonances, occurring on the highbias side of both the supercurrent and the DJQP features. In the DJQP subgap transport cycle, current flows by means of a combination of Cooperpair and quasiparticle tunnelling^{19}. This cycle appears as a peak in current near the intersection of two Cooperpair resonances^{8,20}, one for each junction in the SSET, at V_{sd}=2E_{c}/e as in Fig. 2b, where E_{c}=e^{2}/2C_{Σ} is the SSET charging energy.
When the SSET is biased above the DJQP resonance (blue detuning), Cooper pairs must emit energy to tunnel. Similarly, when the SSET is biased below the resonance (red detuning), Cooper pairs must absorb energy. As illustrated in Fig. 2d, because the SSET’s electromagnetic environment is dominated by the L C resonator, most absorption (emission) will take the form of photon exchange with the tank circuit^{21}. In terms of the picture of resonator damping given above, if G_{d}<0 we expect both γ_{SET}<0 and Γ_{in}>1. Physically, this negative damping corresponds to net emission of energy into the resonator by the SSET. For the total symmetrized noise of the SSET to remain positive (as it must), the SSET effective temperature T_{SET} must also be negative in this region.
The second step in characterizing the total quantum noise of the SSET is a measurement of T_{SET}. 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 SSET operated near the DJQP resonance has been predicted to possess the ideal balance of sensitivity and backaction needed to approach the quantum limit^{8}. Second, near this cycle, the SSET’s quantum noise properties are expected to depend strongly on the SET bias in V_{sd} and n_{g} with respect to this intersection^{8,14,15}. Last, the charge sensitivity δ q of the SSET 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 normalstate SET biased near threshold is expected to operate far from the quantum limit^{22}.
Our measurements of the SSET noise characteristics show excellent correspondence with photon emission and absorption by the SSET. We show this correspondence by measuring the reflection coefficient Γ_{in} of the tank circuit over the same range of V_{sd} and n_{g}, as in Fig. 3b. For most values of V_{sd} and n_{g}, we found Γ_{in}<1, indicating net absorption by the SSET. However, when the SSET is blue detuned, there is a region for which Γ_{in}>1, indicating emission. Here the SSET provides negative damping, returning more power to the resonator than is delivered by the radiofrequency excitation. Remarkably, therefore, as we measure Γ_{in}>1, we are directly measuring photon emission by Cooper pairs as they tunnel. Comparing this to the SSET 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 γ_{SET} and Γ_{in}, and with the more sophisticated quantum noise viewpoint of equations (2) and (3).
Having measured G_{d} and , we now proceed to completely and quantitatively determine the quantum noise of our SSET. As indicated above, we treat the SSET as a thermal bath with conductance G_{d} and available noise power k_{B}T_{SET}. It can be shown that (where γ_{T} is the total damping rate of the resonator—see the Methods section), so that T_{SET} can be found from measurements of , whereas γ_{SET} can be determined directly from G_{d} through the relation γ_{SET}=G_{d}/C_{p}. The resulting values of γ_{SET} and T_{SET} at ω_{0} versus V_{sd} and n_{g} near the DJQP for 300 mK are shown in Fig. 4a,b. The tendency of the SSET to either emit or absorb (as measured by γ_{SET}) and its degree of asymmetry (as measured by T_{SET}∝(S_{I}(ω_{0})+S_{I}(−ω_{0}))/(S_{I}(ω_{0})−S_{I}(−ω_{0}))) vary strongly with V_{sd} and n_{g}. For blue detuning where Cooper pairs must give off energy, we observe both negative damping and a negative effective temperature. Although T_{SET} is large in some areas, for most bias points T_{SET}≲1 K, making it smaller than e V_{sd}/k_{B} but still large enough that our assumption k_{B}T_{SET}≫ℏω_{0} in equations (2) and (3) is still valid. For red detuning, where the SSET is strongly absorbing, T_{SET} can be as low as 100±40 mK, less than the ambient temperature and indicating that the SSET is capable of refrigeration. Although theoretical expressions for γ_{SET} and T_{SET} near the DJQP exist^{14,15}, they assume capacitive coupling of the SSET to a resonator rather than our direct electrical connection, and also ignore higherorder tunnelling processes known^{9,20} to be important for our relatively lowresistance SSETs. Nonetheless, theory predicts a minimum T_{SET}≈250 mK for an SSET with our parameters, in reasonable agreement with our results. Finally, we prefer T_{SET} and γ_{SET} as a description of the SSET quantum noise over the Fano factor because the latter is due only to fluctuations of the number of tunnelling electrons^{23}. In our experiment, variations in arising from electron number fluctuations are indistinguishable from those due to emission/absorption of photons.
We now estimate the measurement capability of our SSET relative to the quantum limit. We imagine coupling the SSET to some external device such as a quantum dot. The ratio of the time it takes the SSET 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, S_{Q}^{sym} and S_{I}^{sym} are the symmetrized zerofrequency spectral densities of charge fluctuations on and current through the SSET, E_{int} describes its interaction with the measured system and ΔI is the change in SSET current corresponding to a change in the system charge state^{3} (see Supplementary Information). Using the current I through the SSET to estimate S_{Q}^{sym}≈3e^{3}/8I, we find independent of the specifics of the dot and its coupling to the SSET. 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 highelectronmobility 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 SSET (ref. 9), and a nearoptimal normalstate 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 higherorder tunnelling processes that could bring the SSET closer to the quantum limit^{25}. In addition to its inherent interest in terms of quantum measurement, an SSET 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 ^{3}He 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 V_{sd} and small a.c. voltage v_{ac} were filtered and introduced to the highfrequency 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 onchip spiral^{11} for which internal losses are negligible, and can be fully described by its inductance L≈169 nH and its parasitic capacitance to ground C_{p}=0.14 pF, giving a resonant frequency ω_{0}=1.04 GHz. Its total damping rate γ_{T} is given by γ_{T}=γ_{0}+γ_{SET}, where γ_{0}=Z_{0}/L is the damping due to the coupling to the feedline. The reflection coefficient Γ_{in} for waves incident on the resonator can be written in terms of γ_{0} and γ_{SET} as Γ_{in}=(γ_{SET}−γ_{0})/(γ_{SET}+γ_{0}). On the basis of the simple model for the SSET/resonator circuit in Fig. 1b, we expect γ_{SET}=G_{d}/C_{p}. This relation, verified by extensive measurements of Γ_{in} versus G_{d}, agrees with the expectation that ω_{0} is still in the lowfrequency limit for the SSET, because tunnelling events typically occur at a much higher rate (tens of gigahertz). To measure Γ_{in}, we applied a very small carrier wave (−149 dB m), measured the reflected power, and after accounting for the HEMT and circulator, computed Γ_{in}. To find the integrated noise we started by applying a d.c. current I and measuring the total output noise power P_{n} at ω_{0} in a bandwidth Δf=5 MHz at the output of the amplifier chain, as shown in Fig. 1a,d. The total output noise P_{n} includes contributions from the SSET, HEMT amplifier and circulator: P_{n}=A(k_{B}T_{HEMT}+Γ_{in}^{2}k_{B}T_{circ}+P_{SET}(I))Δf, where T_{HEMT} and T_{circ} are the HEMT and circulator noise temperatures, P_{SET}(I) is the spectral noise density of the SET referred to the HEMT input and A is the total gain of the amplifier chain^{26}. We use our noise power data to determine A=61 dB, T_{HEMT}=9.5 K and T_{circ}≈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 P_{SET} versus source–drain bias V_{sd}and island charge number n_{g}. The integrated SET noise referred to the input of the HEMT is given by .
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Acknowledgements
This work was supported by the ARO under Agreement No. W911NF0610312, by the NSF under Grant Nos DMR0804488 and DMR0804477 and by the NSA, LPS and ARO under Agreement No. W911NF0810482. We thank T. J. Gilheart, M. Bal and F. Chen for experimental assistance.
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A.J.R. planned the experiment. W.W.X. and Z.J. fabricated the samples. W.W.X. carried out the measurements with assistance from Z.J., F.P. and J.S. M.P.B. proposed the method of analysis. W.W.X. and A.J.R. analysed the data. A.J.R. and J.S. wrote the paper with input from W.W.X. and M.P.B.
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Xue, W., Ji, Z., Pan, F. et al. Measurement of quantum noise in a singleelectron transistor near the quantum limit. Nature Phys 5, 660–664 (2009). https://doi.org/10.1038/nphys1339
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DOI: https://doi.org/10.1038/nphys1339
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