Shaping a time-dependent excitation to minimize the shot noise in a tunnel junction

We report measurements of shot noise in a tunnel junction under biharmonic illumination, $V_{ac}\left(t\right)=V_{ac1}\cos \left(2\pi \nu t\right)+V_{ac2}\cos (4\pi \nu t+\phi )$. The experiment is performed in the quantum regime, $h\nu \gg k_{B}T$ at low temperature $T=70\mathrm{mK}$ and high frequency $\nu =10\mathrm{GHz}$. From the measurement of noise at low frequency, we show that we can infer and control the nonequilibrium electronic distribution function by adjusting the amplitudes and phase of the excitation, thus modeling its shape. In particular, we observe that the noise depends not only on the amplitude of the two sine waves but also on their relative phase, due to coherent emission/absorption of photons at different frequencies. By shaping the excitation we can minimize the noise of the junction, which no longer reaches its minimum at zero dc bias. We show that adding an excitation at frequency $2\nu$ with the proper amplitude and phase can reduce the noise of the junction excited at frequency $\nu$ only.

Figure 1

Experimental setup for the measurement of the photon-assisted noise in a tunnel junction under biharmonic excitation. Inset B: $T$-periodic sequence of biharmonic excitation (blue line) with $e{V}_{dc}=e{V}_{ac1}=2e{V}_{ac2}=h\nu$ and $\phi =0$, approximating Lorentzian pulses of width $\tau =\ln 2/\left(2\pi T\right)$ and height $N=1$ (red dashed line). Inset C: normalized differential noise spectral density with (blue line) and without (black dotted line) microwave excitation vs normalized dc bias. The power of the generator, the variable attenuator, and the phase shifter are tuned to obtain $e{V}_{ac1}=2e{V}_{ac2}=5.4h\nu$ and $\phi =0$.

Figure 2

Nonequilibrium distribution functions obtained from numerical deconvolution of the measured differential noise. (a) Noise is measured for $e{V}_{ac1}=2e{V}_{ac2}=5.4h\nu$, for phase shifts $\phi =0$ (blue), $\phi =\pi /2$ (green), and $\phi =\pi$ (red). (b) [resp. (c)] Noise is measured for various amplitudes of excitation $V_{ac1}$ and $V_{ac2}$ keeping $V_{ac1}=2V_{ac2}$, for $\phi =0$ (resp. $\phi =\pi /2$).

Figure 3

Calculated (a) and measured (b) second derivative of the biharmonic photon-assisted noise ${\partial }^2{S}_{2,ac}/\partial V_{dc}^2$ as a function of normalized dc bias and phase shift. In both cases $e{V}_{ac1}=2e{V}_{ac2}=5.4h\nu$ with $\nu =10$ GHz, and the temperature is $T_{el}=0.14h\nu /k_B=70\mathrm{mK}$. Red curves correspond to the calculated (a) and measured (b) minimum of the photon-assisted noise, $\partial S_{2,ac}/\partial V_{dc}=0$. Dashed lines correspond to phase shifts that are used in Fig. 4.

Figure 4

(a) Shape of the ac excitation with the different phase shifts: $\phi =0$ (blue), $\phi =\pi /2$ (green) and $\phi =\pi$ (red). Dashed line: monoharmonic signal. (b) Normalized biharmonic photon-assisted noise $S_{2,ac}/Gh\nu$ vs normalized dc bias for $e{V}_{ac1}=5.4h\nu$. Blue square, green circle, red triangle symbols: data for $e{V}_{ac2}=2.7h\nu$ and phase shifts $\phi =0,\pi /2,\pi$. Black circles: data for $V_{ac2}=0$, i.e., pure sine wave excitation. Cross symbols ($\times$): data for $V_{ac1}=V_{ac2}=0$ i.e., shot noise without any ac excitation. Solid lines: theoretical predictions, Eqs. (5) and (6). (c) Difference between biharmonic and monoharmonic photon-assisted noise $\Delta S_{2,ac}\left(V_{dc}\right)=S_{2,ac}(V_{dc},V_{ac1},V_{ac2})-S_{2,ac}(V_{dc},V_{ac1},V_{ac2}=0)$.

Figure 5

(a) Optimal value of the reduced dc voltage $e{V}_{dc}/h\nu$. Optimal value of the reduced amplitude $e{V}_{ac2}/h\nu$ at frequency $2\nu$. (c): Noise reduction $R=S_2(V_{dc}=0,V_{ac1},V_{ac2}=0)-S_2(V_{dc}^{☆},V_{ac1},V_{ac2}^{☆})$ between the monoharmonic photon-assisted noise at zero dc bias and the optimal biharmonic one with the same ac amplitude $V_{ac1}$ at frequency $\nu$. (a), (b), and (c) are plotted as a function of the reduced amplitude $e{V}_{ac1}/h\nu$ at frequency $\nu$, for various reduced temperature $k_B{T}_{el}/h\nu$ ranging from 0 (blue line) to 0.65 by steps of 0.05. Inset: Temperature above which the noise cannot be reduced by biharmonic excitation.

Figure 6

(a) $T$-periodic sequence of Lorentzian pulses of width $\tau =\ln 2/2\pi T$ [red line, $V_L\left(t\right)$] and its harmonic [green line, $V_1\left(t\right)$] and biharmonic [blue line, $V_2\left(t\right)$] approximations. (b) Noise difference $S_2-S_{2,dc}$ for the different waveforms. Dashed lines correspond to zero temperature whereas solid lines correspond to our experimental temperature $k_{B}T=0.14h\nu$.