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arXiv:cond-mat/0311325 · Europhysics Letters (EPL)

Output spectrum of a measuring device at arbitrary voltage and temperature

Alexander Shnirman, Dmitry Mozyrsky, Ivar Martin

Published Aug 25, 2004• 25 citations
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Abstract

We calculate the noise spectrum of the electrical current in a quantum point contact which is used for continuous measurements of a two-level system (qubit). We generalize the previous results obtained for the regime of high transport voltages (when V is much larger than the qubit's energy level splitting B (we put e = ℏ = 1)) to the case of arbitrary voltages and temperatures. When V ∼ B , the background output spectrum is essentially asymmetric in frequency, i.e. , it is no longer classical. Yet, the spectrum of the amplified signal, i.e. , the two coherent peaks at ω = ± B , is still symmetric. In the emission (negative frequency) part of the spectrum the coherent peak can be 8 times higher than the background pedestal. Alternatively, this ratio can be seen in the directly measureable excess noise. For V < B and T = 0 the coherent peaks do not appear at all. We relate these results to the properties of linear amplifiers.

DOI
10.1209/epl/i2003-10309-6

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We calculate the noise spectrum of the electrical current in a quantum point contact which is used for continuous measurements of a two-level system (qubit). We generalize the previous results obtained for the regime of high transport voltages (when V is much larger than the qubit's energy level splitting B (we put e = ℏ = 1)) to the case of arbitrary voltages and temperatures. When V ∼ B , the background output spectrum is essentially asymmetric in frequency, i.e. , it is no longer classical. Yet, the spectrum of the amplified signal, i.e. , the two coherent peaks at ω = ± B , is still symmetric. In the emission (negative frequency) part of the spectrum the coherent peak can be 8 times higher than the background pedestal. Alternatively, this ratio can be seen in the directly measureable excess noise. For V < B and T = 0 the coherent peaks do not appear at all. We relate these results to the properties of linear amplifiers.
We calculate the noise spectrum of the output signal of a quantum detector during continuous measurement of a two-level system (qubit). We generalize the previous results obtained for the regime of high voltages (when eVeVeV is much larger than the qubit's energy level splitting ΔΔΔ) to the case of arbitrary voltages and temperatures. When V∼ΔV \sim ΔV∼Δ the output spectrum is essentially asymmetric in frequency, i.e., the output signal is no longer classical. In the emission (negative frequency) part of the spectrum the peak due to the qubit's coherent oscillations can be 8 times higher that the background pedestal. For V<ΔV < ΔV<Δ and T=0 the coherent peaks do not appear at all.
We employ a quantum Langevin-equation approach to establish Bloch-type dynamical equations, on a fully microscopic basis, to investigate the measurement of the state of a coupled quantum dot qubit by a nearby quantum point contact. The ensuing Bloch equations allow us to analyze qubit relaxation and decoherence induced by measurement, and also to determine the noise spectrum of the meter output current with the help of a quantum regression theorem, at arbitrary bias voltage and temperature. Our analyses attempt to provide a clear resolution of a recent debate concerning the occurrence of a quantum oscillation peak in the noise spectrum.
Electrical engineers and physicists are naturally very interested in noise in circuits, amplifiers and detectors. With the advent of quantum computation and other high frequency electronics operating at low temperatures, we have entered a regime where quantum noise and quantum-limited detectors are important. Here we describe the general concept of a two-level system as a quantum spectrum analyzer and apply it to a simple superconducting qubit, the Cooper-pair box. We then discuss the coupling of a Cooper-pair box to its electromagnetic environment, whose noise leads to a finite polarization and excited-state lifetime of the qubit. Finally, we describe a theoretical technique for treating a qubit coupled to a measurement system, which allows one to calculate the full quantum noise of the measurement device. We present results for such a calculation for the case of a normal SET.
Abstract We employ a microscopic quantum Heisenberg‐Langevin equation approach to establish a set of quantum Bloch equations for a two‐level system (coupled quantum dots) capacitively coupled to a quantum point contact (QPC). The resulting Bloch equations facilitate our analysis of qubit relaxation and decoherence in coupled quantum dots induced by measurement processes at arbitrary bias‐voltage and temperature. We also examine the noise spectrum of the meter output current for a symmetric qubit. These results help resolve a recent debate about a quantum oscillation peak in the noise spectrum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The electrical noise of mesoscopic devices can be strongly influenced by the quantum motion of electrons. To probe this effect, we have measured the current fluctuations at high frequency (5 to 90 gigahertz) using a superconductor-insulator-superconductor tunnel junction as an on-chip spectrum analyzer. By coupling this frequency-resolved noise detector to a quantum device, we can measure the high-frequency, nonsymmetrized noise as demonstrated for a Josephson junction. The same scheme is used to detect the current fluctuations arising from coherent charge oscillations in a two-level system, a superconducting charge qubit. A narrow band peak is observed in the spectral noise density at the frequency of the coherent charge oscillations.
In this work we revisit the problem of a Majorana box qubit (MBQ) readout by a point-contact (PC) detector. The logic states of the MBQ are associated with the combined fermion parities of the MBQ and its tunnel-coupled quantum dot, which is measured by a PC detector. Beyond the existing studies on limiting bias voltage regimes, we analyze the steady-state current and the current power spectrum across all bias voltages. Our results indicate that the MBQ readout via the parity-dependent detector current is effective only at low bias voltage regime and requires the dot energy level to be off-resonance with the Majorana qubit. In contrast, the current power spectrum allows MBQ readout through the parity-dependent Rabi oscillation peak signals for arbitrary bias voltages, without restrictions on the dot energy level. Particularly, with focus on the MBQ measurement visibility, we analyze the peak-to-pedestal ratio for each characteristic peak (associated with each logic state of the qubit) and the signal-to-noise ratio of the two peaks. By examining these two metrics, we identify the optimal bias voltage window for the PC detector at low temperature limit.