Introduction

In mesoscopic physics, a lot of effort has been put in the measurement and understanding of current fluctuations1,2. Of particular interest are the deviations from the ubiquitous Gaussian noise, i.e. the study of high order cumulants. As a matter of fact, while the only parameter characterizing a Gaussian distribution, the variance or second cumulant of the fluctuations, contains some information about electron transport, much more could be learned by the statistical study of the fluctuations beyond their variance. These are characterized by cumulants of order three and higher, which are zero for a Gaussian distribution. For the simplest systems, such as a tunnel junction between normal metals or a quantum point contact, only the third cumulant has been measured until now3,4,5,6. Higher order cumulants have been experimentally accessible solely in quantum dots where electrons enter only one by one7,8,9. In all these measurements the system is driven out of equilibrium by a dc voltage bias. Here we address the statistics of photo-assisted shot noise, i.e. current fluctuations in the presence of an ac excitation. While the variance of such fluctuations has been well explored both theoretically10,11,12,13 and experimentally14,15,16,17,18,19, no experiment has been performed yet that reports the existence of higher order cumulants in such conditions.

In the following we present a link between the measurement of the correlation G2 = 〈P1P2〉 − 〈P1〉 〈P2〉 between the noise powers P1 and P2 at two different frequencies in the GHz range, f1 and f2. The source of shot noise is a tunnel junction that is dc biased and excited at frequency f0. Since power fluctuations of Gaussian noise are independent at all frequencies, our measurement probes only the non-Gaussian part of current fluctuations. This technique has been applied many times to the study of 1/f noise in glassy or disordered materials20,21,22,23 and has been proposed to measure the fourth cumulant of a quantum point contact without ac excitation24. In optics, G2 corresponds to intensity-intensity correlations, a usual probe of the statistical properties of light25. Here we show that G2 gives the fourth cumulant at frequencies (f1, f2, 0) of photo-assisted shot noise in the tunnel junction. In the optics community, current/voltage fluctuations, which are simply another point of view of a fluctuating electromagnetic field, are rather thought of as time dependent electric and magnetic fields. Thus, a noisy electronic device is also a light source and it is natural to apply tools developed in optics to analyze high frequency electronic noise24,26,27. This point of view will also be studied.

Results

Experimental principle

The experimental setup is depicted on Fig. 1. The sample, a tunnel junction described in the Methods section, is voltage biased by a dc source and connected to microwave signal generators below 4 GHz and above 8 GHz while the noise generated by the junction at point A is amplified by a cryogenic amplifier in the 4–8 GHz frequency range. This setup allows noise to be measured in the 4–8 GHz range while the excitation frequency f0 can take any value below 4 GHz or above 8 GHz. This insures that the amplifier never sees the ac excitation of the sample, which avoids spurious signals due to non-linearities in the amplifier at high excitation power. However, it prevents the use of an excitation frequency in the range 4–8 GHz.

Figure 1
figure 1

Experimental setup.

Diode symbols represent fast power detectors.

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The amplified noise is split into two branches: in branch 1 (respectively branch 2) the signal is bandpass filtered around f1 = 4.5 GHz with a bandwidth Δf1 = 0.72 GHz (resp. f2 = 7.1 GHz, Δf2 = 0.60 GHz). In each branch a fast power detector (diode symbol on Fig. 1, bandwidth ) measures the “instantaneous” (averaged over a few ns) noise power integrated in the corresponding bandwidth, i.e. with k = 1, 2. The dc voltage at point Bk is thus proportional to the noise spectral density, S(fk) = 〈|i(fk)|2〉 with i(f) the Fourier component of the instantaneous current: . A dc block (capacitor on Fig. 1) removes the dc part of vB to give . Finally, voltages at points C1 and C2 are simultaneously digitized with a 2-channel, 14-bit, 400 MS/s acquisition card and the correlator G2 = 〈δP1δP2〉 is computed in real time by a 12-core parallel computer, as are the autocorrelations of both channels .

The fourth cumulant of noise

In order to describe and understand the meaning of the observed correlator, let us first express the noise generated by the sample in Fourier space. We define the power fluctuations δPk(t) = Pk(t) − 〈Pk〉 so that G2 = 〈δP1(t)δP(2(t)〉. The average 〈.〉 is performed over time. Introducing the Fourier component of the power fluctuations δPk(ε) at frequency ε, one has: . Here ε is a low frequency limited by the output bandwidth of the power detectors. Note that δPk(ε = 0) = 0, while power fluctuations at finite frequency ε are related to current fluctuations by where i(f) is the fluctuating current's Fourier component at frequency f. This integral spans over the bandwidth of the bandpass filters, i.e. fk ± Δfk/2. Thus, G2 is proportional to the correlator between currents at four different frequencies which, by definition, is the time averaged fourth cumulant of current fluctuations taken at frequencies f1, f2 and ε (with ε → 0 but ε ≠ 0).

Noting ik(t) the current after bandpass filtering around ±fk, one has , the averaging being performed over the time t. Here 〈〈x4〉〉 = 〈x4〉 − 3 〈x2〉2 is the fourth cumulant of the random variable x with 〈x〉 = 0. For a Gaussian distribution, 〈〈x4〉〉 = 0, so there is no information in the fourth moment that is not contained in the variance. In contrast, for a non-Gaussian distribution, the fourth moment differs from 3 〈x2〉2, though only very slightly for current fluctuations involving many electrons, so the fourth cumulant is non-zero.

Frequency dependence

The first step in the observation of G2(Vdc, Vac, f0) is to determine the excitation frequency for which G2 ≠ 0. This is done by measuring G2 at fixed bias voltage (Vdc = 0 or Vdc = 2.4 mV) and fixed excitation amplitude (Vac = 1.1 mV) while varying the excitation frequency f0 from 10 MHz to 3.83 GHz and from 10 GHz to 14 GHz. These results are reported in Fig. 2, where G2 has been reduced to units of K2 as the measured noise spectral density S of a conductor of resistance R is often given in terms of equivalent noise temperature Tnoise = RS/2kB. We observe that G2 is large at low frequency for both bias voltages. This simply reflects that a slow oscillation of the bias voltage induces a slow modulation of the noise, i.e. a slow oscillation of both P1 and P2. The response disappears when the modulation frequency exceeds the output bandwidth of the power detectors.

Figure 2
figure 2

Reduced G2 vs excitation frequency f0 for two bias voltages: Vdc = 0 (in red) and Vdc = 2.24 mV (in blue).

The ac excitation amplitude is Vac = 1.1 mV for both measurements. The exact shape of the peaks is determined by the frequency response of the experimental setup.

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At much higher excitation frequencies, G2 strongly depends on Vdc. For Vdc = 2.4 mV, G2 shows peaks at f0 = 2.6 GHz and f0 = 11.6 GHz, which correspond to , where f1 and f2 are the signal observation frequencies. At zero bias, G2 peaks at f0 = 1.3 GHz, or f−/2. Choosing f0 = f+/2 was not possible with this experimental setup since it lies in the 4–8 GHz range.

To understand why G2 is nonzero at high excitation frequency, let us first consider the correlator 〈i(f)i(f′)〉. The latter is non-vanishing only for frequencies such that f + f′ = nf0 with n, an integer. The case n = 0 corresponds to photo-assisted noise whereas n ≠ 0 describes the noise dynamics, characterized by the correlator Xn(f, f0) = 〈i(f)i(nf0 − f)〉17,28,29. In order to detect the fourth cumulant and not the fourth moment of current fluctuations, it is crucial that the four frequencies involved in G2 (see Eq. 1 where ε ≠ 0) be different so that correlators 〈i(f)i(−f)〉 are never involved. Experimentally, the separation of the signal into two branches with non-overlapping bandwidths insures that f1 ≠ ± f2, while the presence of the dc blocks, by imposing ε ≠ 0, prevents all other possible occurrences of such correlators. In our experimental setup, relevant frequencies are close to ±f1 and ±f2, so this condition becomes f1 ± f2 = nf0, i.e. f0 = f±/n. In such cases, the fourth order correlator of Eq. (1) is dominated by the terms . This product is zero unless f1 ± f2 = nf0, as we observe on Fig. 2.

Voltage dependence

We now consider the variation of G2 as a function of the dc voltage for various ac excitation amplitudes at fixed excitation frequency f0 = f±/n. Data on Fig. 3 correspond to an excitation at f0 = f+. We observe that G2 is maximal at high dc bias and vanishes at Vdc = 0. We obtained identical results for f0 = f− (data not shown). Data on Fig. 4 correspond to f0 = f−/2. Here G2 peaks at 0 dc bias and decays when |Vdc| increases. Moreover, the maximum of G2(f0 = f±) for a given Vac is an order of magnitude larger than that of G2(f0 = f−/2).

Figure 3
figure 3

Reduced G2 vs dc bias voltage for various ac excitation amplitudes at frequency f0 = f+ = 11.6 GHz.

Symbols are experimental data and solid lines theoretical expectations of Eq. (2) and (3).

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Figure 4
figure 4

Reduced G2 vs dc bias voltage for various ac excitation amplitudes at frequency f0 = f−/2 = 1.3 GHz.

Symbols are experimental data and solid lines theoretical expectations of Eq. (2) and (3).

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The voltage dependence of the signal can be explained by expressing G2 in terms of the correlators Xn(f, f0):

with . The exact value of K involves integrals of the gain of the setup over the actual bandwidth of the filters as well as the coupling coefficient between the sample and the detection setup, which depends on the impedance of the sample. The correlators Xn have been calculated and measured in the quantum regime at very low temperature17,28,29. In the high temperature, classical regime that corresponds to the present experiment, Xn reduces to:

Here S0(V) = 2 eGV coth(eV/2kBT) is the noise of the junction at zero frequency in the absence of ac excitation. Eq. 3 is interpreted as follows: in the classical regime, the noise responds instantaneously to the time-dependent voltage V(t) = Vdc + Vac cos θ with θ = 2πf0t, so it oscillates at frequency f0 and its harmonics. Xn is the amplitude of the nth harmonics. Xn is independent of f and f0 as long as hf, , so depends only on |n|.

For small Vac and f0 = f±, the noise oscillates with an amplitude given by , so computes to zero at Vdc = 0 and is maximal at large V, which corresponds to the shape observed on Fig. 3. For f0 = f±/2, G2 is given by the amplitude of the noise that oscillates at 2f0, which is given by for small Vac. Therefore is maximal at Vdc = 0 and decays at finite dc bias, as observed on Fig. 4. Solid lines on Fig. 3 and 4 represent the theoretical predictions of Eqs. (2,3) and agree very well with the measurements.

Photon bunching

Current fluctuations generated by a tunnel junction are known to be non-Gaussian and thus should exhibit a non-zero fourth cumulant even in the absence of ac excitation30. This contribution, together with potential environmental effects31,32 are negligible as compared to the signals we report here. For example, at Vdc = 2 mV and Vac = 0, the correlator would correspond to G2 ~ 10−4 K2. Thus, by adding an ac excitation on the sample we have been able to boost the fourth cumulant by 5 orders of magnitude.

Furthermore, our experiment corresponds to intensity-intensity correlation, which is the usual way used to differentiate classical from quantum light by showing evidence of photon bunching or anti-bunching. More precisely, in order to make a connection with experiments performed in optics, let us define the dimensionless correlator:

which is usually referred to as the same-time two-mode second order correlator of the electromagnetic field radiated by the junction.

The variations of g2 as a function of Vdc are depicted on Fig. 5 for fixed Vac = 1.0 mV for both f0 = f+ (red triangles) and f0 = f−/2 (green circles). It is clear from these data that we always observe a positive correlation between the power fluctuations, i.e. photon bunching (g2 > 1). Each acquisition performed by the digitizer (integrated over τ = 2.5 ns) corresponds to an averaged number of 〈n2〉 = P2τ/hf2 ~ 21 photons at 7.2 GHz emitted by the sample at Vdc = Vac = 1 mV and f0 = f+, plus ~40 photons from the amplifier. Thus our experiment does not measure correlations at the single photon level, but is not very far from that limit, which can be reached by lowering the temperature and the ac power.

Figure 5
figure 5

Measured g2 vs dc bias voltage at Vac = 1.0 mV for an excitation frequency f0 = f+ (red triangles) or f0 = f−/2 (green circles).

The dashed line at g2 = 1 represents independent photons, i.e. chaotic light, while g2 > 1 corresponds to photon bunching.

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Discussion

It follows from the observed behaviour that by choosing the excitation frequency, dc bias and ac excitation, we can control how non-Gaussian the shot noise of the junction can be. The level of non-Gaussianity of the signal is characterised here by the fourth cumulant, which is directly linked to the measured correlator.

It should be noted that the power detector, which measures the square of the electric field, cannot differentiate absorption from emission of photons by the sample. This has to be taken into account when comparing the data with theories such as24,26,27 which consider photon detectors. In particular, the correlations between photon detectors will involve other current correlators33. Whereas G2 at excitation frequency f0 = f1 + f2 corresponds to absorption of one photon of frequency f0 and emission of two correlated photons, one at f1 and one at f2, the case of excitation at frequency f0 = f2 − f1 corresponds to two photons being absorbed, one at f0 and one at f1 while one photon at frequency f2 is emitted. We indeed observe no difference in G2 for f0 = f+ and f0 = f−.

The tunnel junction behaves as a source of white noise whose amplitude is instantaneously modulated by the bias voltage. This description holds only because the temperature is large in the present experiment. At very low temperature, the noise can no longer be considered as white and no longer responds adiabatically to an ac excitation, so Eq.(3) will no longer be valid. In particular, Xn depends on f and f0, so that excitations at f1 + f2 or f1 − f2 will no longer correspond to the same G2. Still the present analysis and in particular the link between G2 and Xn given by Eq.(2), will remain valid. Our measurements open the way to the study of the fourth cumulant of current fluctuations in the quantum regime at very low temperature, where the same setup can be used to detect correlations at the single photon level.

Methods

Sample

We have chosen to perform the measurement on the simplest system that exhibits well understood shot noise, the tunnel junction. The sample is a ~1 μm × 15 μm Al/Al oxide/Al tunnel junction made by photolithography, similar to that used for noise thermometry34, cooled at T = 3.0 K so the aluminum remains a normal metal. The resistance of the junction at that temperature, R = 22 Ω, is close enough to the 50 Ω impedance of the microwave circuitry to ensure a good coupling. The capacitance of the junction corresponds to an RC frequency cutoff of ~6 GHz, so it influences the amplitude of the noise we measure and the amplitude of the ac excitation experienced by the junction. Both effects are calibrated out as explained below.

Calibration

In order to have a quantitative measurement of G2, it is necessary to calibrate the ac excitation voltage reaching the sample and the overall gain of the detection. The calibration of the ac voltage across the sample is performed by measuring the usual photo-assisted noise, i.e. S vs Vdc, in the presence of a microwave excitation for various excitation voltages10,14. The temperature is large enough () to approximate the noise measured in either branch by its value at zero frequency. In the absence of ac excitation, the noise spectral density is given by S0(Vdc) = eGVdc coth(eVdc/2kBT) with G = 1/R, the sample's conductance. Since the excitation frequency f0 is always such that , the photo-assisted noise can be approximated by its time-average value as if the junction were responding instantaneously to the time-dependent voltage. To calibrate the gain of the setup, we consider the single channel autocorrelations . Those are also related to fourth order current correlators 〈i (f) i (ε − f) i (f′) i (−ε − f′)〉. However, here f and f′ belong to the same frequency band, so the correlator is dominated by terms f = −f′ and . The fourth cumulant G2 is only a very small correction to this. Thus, power correlations within the same frequency band are totally dominated by Gaussian fluctuations, as in35. Note that since the amplifier noise dominates 〈Pk〉, it also dominates , but only the fourth cumulant of the amplifier's noise contributes to G2 (here a very small contribution).