[Paper Review] Shaping a time-dependent excitation to control the electron distribution function: noise minimization in a tunnel junction

Julien Gabelli, Bertrand Reulet|arXiv (Cornell University)|May 16, 2012

Quantum and electron transport phenomena3 citations

TL;DR

This paper demonstrates that shaping a bi-harmonic ac excitation—comprising frequencies ν and 2ν—with controlled amplitude and relative phase enables dynamic control of the non-equilibrium electron distribution function in a tunnel junction, leading to a reduction in shot noise below the level achieved by single-frequency excitation. The key result is that coherent interference between photon absorption processes at ν and 2ν can suppress noise, with optimal noise minimization achieved at non-zero dc bias when the phase and amplitude are tuned appropriately.

ABSTRACT

We report measurements of shot noise in a tunnel junction under bi-harmonic illumination, Vac(t) = Vac1 cos(2πft) + Vac2 cos(4πft+ ϕ). The experiment is performed in the quantum regime, hf >> k_BT at low temperature T = 70 mK and high frequency f = 10 GHz. From the measurement of noise at low frequency, we show that we can infer and control the non-equilibrium 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 2f with the proper amplitude and phase can reduce the noise of the junction excited at frequency f only.

Motivation & Objective

• To explore how time-dependent ac excitation shapes the non-equilibrium electron distribution function in mesoscopic conductors.
• To investigate whether coherent photon-assisted processes can be engineered to minimize current noise in tunnel junctions.
• To demonstrate that noise can be reduced below the minimum achievable with single-frequency excitation by adding a second harmonic component at 2ν.
• To establish a method for inferring and controlling the electron distribution function through low-frequency noise measurements under tailored ac excitation.

Proposed method

• The experiment uses a bi-harmonic ac voltage excitation: $ V_{ac}(t) = V_{ac1}\cos(2\pi\nu t) + V_{ac2}\cos(4\pi\nu t + \varphi) $, with independently tunable amplitudes $ V_{ac1}, V_{ac2} $ and phase $ \varphi $.
• Low-frequency shot noise is measured using a cryogenic amplifier and power detector, with the noise spectral density $ S_2 $ extracted via integration over a 0.5–1.8 GHz bandwidth.
• The electron distribution function is inferred from noise measurements by fitting to theoretical models of photon-assisted tunneling in a tunnel junction.
• Theoretical modeling uses the formalism of periodic, time-dependent excitation to calculate the non-equilibrium stationary distribution function and predict noise behavior.
• The system is operated at $ T = 70\,\text{mK} $, $ \nu = 10\,\text{GHz} $, and $ h\nu \gg k_B T $, ensuring the quantum regime is accessible.
• A directional coupler and bias tee ensure impedance matching and allow superposition of dc and ac voltages on the tunnel junction.

Experimental results

Research questions

• RQ1 Can a bi-harmonic ac excitation at frequencies ν and 2ν be used to shape the non-equilibrium electron distribution function in a tunnel junction?
• RQ2 Does coherent interference between photon absorption processes at ν and 2ν lead to a reduction in shot noise below the level of single-frequency excitation?
• RQ3 At finite temperature and non-integer $ eV_{dc}/h\nu $, does the noise minimum still occur at quantized values of $ V_{dc} $, or is it shifted?
• RQ4 Can the noise be minimized at non-zero dc bias by tuning the amplitude and phase of the 2ν component?

Key findings

• The addition of a coherent excitation at 2ν with amplitude $ eV_{ac2} = 2.4\,h\nu $ and phase $ \varphi = 0 $ reduces the shot noise below the minimum observed under single-frequency excitation at $ V_{ac2} = 0 $.
• The noise minimum shifts from $ V_{dc} = 0 $ to $ eV_{dc} = \pm 2.3\,h\nu $ when $ \varphi = 0 $ or $ \pi $, demonstrating phase-dependent control of the noise minimum.
• For $ \varphi = \pi/2 $, the noise minimum remains at $ V_{dc} = 0 $, confirming the symmetry dependence of the noise profile.
• The difference in noise $ \Delta S_{2,ac}(V_{dc}) = S_2(V_{dc}, V_{ac1}, V_{ac2}) - S_2(V_{dc}, V_{ac1}, V_{ac2} = 0) $ is negative over a range of $ V_{dc} $, proving noise reduction via destructive interference.
• The optimal waveform for noise minimization at finite temperature is not Lorentzian but resembles the first two harmonics of a Lorentzian with a dc offset, indicating departure from idealized zero-temperature assumptions.
• The result is a purely quantum effect: noise suppression occurs due to coherent interference between two-photon processes at ν and one-photon processes at 2ν, not classical averaging.