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arXiv:cond-mat/0210247

Qubits as Spectrometers of Quantum Noise

R. J. Schoelkopf, A. A. Clerk, S. M. Girvin, Konrad W. Lehnert, Michel Devoret

Published Oct 10, 2002• 8 citations
arXivHTMLPDF

Abstract

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. In this paper 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 derive and discuss the quantum noise associated with both a resistor and a classical tunnel junction, and study the effect these noise sources will have on a Cooper-pair box. Finally, we describe a theoretical technique for calculating quantum noise which involves studying the noisy system of interest coupled to a two-level system. We use this technique to calculate and give a heuristic explanation of the quantum charge noise of a normal metal single electron transistor.

DOI
10.48550/arxiv.cond-mat/0210247

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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. In this paper 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 derive and discuss the quantum noise associated with both a resistor and a classical tunnel junction, and study the effect these noise sources will have on a Cooper-pair box. Finally, we describe a theoretical technique for calculating quantum noise which involves studying the noisy system of interest coupled to a two-level system. We use this technique to calculate and give a heuristic explanation of the quantum charge noise of a normal metal single electron transistor.
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.
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.
We calculate the full-frequency spectral density of voltage fluctuations in a single-electron transistor (SET), used as an electrometer biased above the Coulomb threshold so that the current through the SET is carried by sequential tunneling events. We consider both a normal-state SET and a superconducting SET. The whole spectrum, from low-frequency telegraph noise to quantum noise at frequencies comparable to the SET charging energy ${(E}_{C}/\ensuremath{\Elzxh})$ to high-frequency Nyquist noise, is described. We take the energy exchange between the SET and the measured system into account using a real-time diagrammatic Keldysh technique. The voltage fluctuations determine the backaction of the SET on the measured system, and we specifically discuss the case of superconducting charge qubit read-out and measuring the so-called Coulomb staircase of a single Cooper-pair box.
We examine the energy level structure of a Josephson charge qubit (Cooper pair box) coupled to its environment, utilising a backreaction technique originally proposed for characterising persistent current qubits. The system consists of a model charge qubit coupled to three electromagnetic modes: a low frequency bias field, a higher frequency microwave to excite the qubit into higher states, and a lossy reservoir representing the cavity containing the qubit and control fields. Specifically, the qubit is a small superconducting island connected to a bulk superconductor via a weak Josephson junction, allowing Cooper pairs to tunnel on and off the island coherently. The classical bias field experiences a backreaction from the qubit and the noise power increases when the qubit is operating near a transition.
Quantum mechanics can strongly influence the noise properties of mesoscopic devices. To probe this effect we have measured the current fluctuations at high-frequency (5-90G Hz) 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, non-symmetrized noise as demonstrated for a Josephson junction. The same scheme has been used to detect the current fluctuations arising from coherent charge oscillations in a two-level system, a superconducting charge qubit. A narrow band peak was observed in the spectral noise density at the frequency of the coherent charge oscillations.
Andreev reflection, which corresponds to the tunneling of two electrons from a metallic lead to a superconductor lead as a Cooper pair (or vice versa), can be exploited to measure high frequency noise. A detector is proposed, which consists of a normal lead--superconductor circuit, which is capacitively coupled to a mesoscopic circuit where noise is to be measured. We discuss two detector circuits: a single normal metal-superconductor tunnel junction and a normal metal separated from a superconductor by a quantum dot operating in the Coulomb blockade regime. A substantial dc current flows in the detector circuit when an appropriate photon is provided or absorbed by the mesoscopic circuit, which plays the role of an environment for the junction to which it couples. Results for the current can be cast in all cases in the form of a frequency integral of the excess noise of the environment weighted by a kernel, which is specific to the transport process (quasiparticle tunneling, Andreev reflection, etc) considered. We apply these ideas to the measurement of the excess noise of a quantum point contact, and we provide numerical estimates of the detector current.