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"much of our discussion has been directly relevant to the measurement of mechanical nanoresonators, a topic attracting considerable recent attention. These nanoresonators are typically studied by coupling them either to electrical often superconducting
circuits or to optical cavities. A key goal is to achieve quantum-limited continuous position detection set by the quantum limit is in principle independent from being at low temperatures, it becomes interesting only when the systems are near their ground state; one could then, e.g., monitor the oscillator’s zero-point fluctuations ... Other important
directions in nanomechanics include the possibility of detecting quantum jumps in the state of a mechanical resonator via QND measurement of its energy ... Back-action evasion using a
microwave cavity detector coupled to a nanomechanical resonator was recently reported Hertzberg et al., 2010. Another area distinct from nanomechanics where rapid progress is being made is the readout of solid state qubits using microwave signals sent through cavities
whose transmission properties are controlled by the qubit. At the moment, one is close to achieving good fidelity single-shot QND readout, which is a prerequisite for a large number of applications in quantum information processing. The gradually growing information
about the qubit state is extracted from the measured noisy microwave signal trace, leading to a corresponding collapse of the qubit state. This process can also be described by conditional quantum evolution and quantum trajectories.

A promising method for superconducting qubit readout currently employed is a so-called latching measurement, where the hysteretic behavior of a strongly driven
anharmonic system e.g., a Josephson junction is exploited to toggle between two states depending on the qubit state Siddiqi et al., 2004; Lupas¸cu et al., 2006.
Although this is then no longer a linear measurement scheme and is therefore distinct from what was discussed discussed in this review, it can be turned into a linear amplifier
for a sufficiently weak input signal. An interesting and important open question is whether such a setup can reach the quantum limit on linear amplification.

Both qubit detection and mechanical measurements in electrical circuits would benefit from quantum-limited on-chip amplifiers. Such amplifiers are now being developed using the tools of circuit quantum electrodynamics, employing Josephson junctions or SQUIDs coupled to microwave transmission line cavities Bergeal et al., 2008; Castellanos-Beltran et al., 2008. Such an amplifier has already been used to perform continuous position
detection with a measurement imprecision below the SQL level Teufel et al., 2009."

Introduction to quantum noise, measurement, and amplification

A. A. Clerk*
Department of Physics, McGill University, 3600 rue University Montréal, Quebec, Canada
H3A 2T8
M. H. Devoret
Department of Applied Physics, Yale University, P.O. Box 208284, New Haven,
Connecticut 06520-8284, USA
S. M. Girvin
Department of Physics, Yale University, P.O. Box 208120, New Haven,
Connecticut 06520-8120, USA
Florian Marquardt
Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for
Theoretical Physics, Ludwig-Maximilians-Universität Mu?nchen, Theresienstrasse 37,
D-80333 Mu?nchen, Germany
R. J. Schoelkopf
Department of Applied Physics, Yale University, P.O. Box 208284, New Haven,
Connecticut 06520-8284, USA
Published 15 April 2010
"The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and atomic, molecular, optical–quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, the basics of weak continuous measurements are described. Particular attention is given to the treatment of the standard quantum limit on linear amplifiers and position detectors within a general linear-response framework. This
approach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quantum optics and its application to the case of electrical circuits is illustrated, including mesoscopic detectors and resonant cavity detectors.
DOI: 10.1103/RevModPhys.82.1155 PACS numbers: 72.70.m
CONTENTS
I. Introduction 1156
II. Quantum Noise Spectra 1159
A. Introduction to quantum noise 1159
B. Quantum spectrum analyzers 1161
III. Quantum Measurements 1162
A. Weak continuous measurements 1164
B. Measurement with a parametrically coupled
resonant cavity 1164
1. QND measurement of the state of a qubit
using a resonant cavity 1167
2. Quantum limit relation for QND qubit state
detection 1168
3. Measurement of oscillator position using a
resonant cavity 1169
IV. General Linear-Response Theory 1173
A. Quantum constraints on noise 1173
1. Heuristic weak-measurement noise
constraints 1173
2. Generic linear-response detector 1174
3. Quantum constraint on noise 1175
4. Evading the detector quantum noise
inequality 1176
B. Quantum limit on QND detection of a qubit 1177
V. Quantum Limit on Linear Amplifiers and Position
Detectors 1178
A. Preliminaries on amplification 1178
B. Standard Haus-Caves derivation of the quantum
limit on a bosonic amplifier 1179
C. Nondegenerate parametric amplifier 1181
1. Gain and added noise 1181
2. Bandwidth-gain trade-off 1182
3. Effective temperature 1182
D. Scattering versus op-amp modes of operation 1183
E. Linear-response description of a position detector 1185
1. Detector back-action 1185
2. Total output noise 1186
3. Detector power gain 1186
*clerk@physics.mcgill.ca 4. Simplifications for a quantum-ideal detector 1188

"having a near-quantum-limited detector would allow one to continuously monitor the
quantum zero-point fluctuations of a mechanical resonator. It is also necessary to have a quantum-limited detector is for such tasks as single-spin NMR detection.... as well as gravitational wave detection ... Particular attention is given to the quantum mechanics of transmission lines and driven electromagnetic cavities, topics that are especially relevant given recent experiments making use of microwave stripline resonators. ...

TABLE II. Contents of online appendix material. Page numbers
refer to the supplementary material.
Section Page
A. Basics of classical and quantum noise 1
B. Quantum spectrum analyzers: further details 4
C. Modes, transmission lines and classical
input-output theory
8
D. Quantum modes and noise of a transmission line 15
E. Back-action and input-output theory for driven
damped cavities
18
F. Information theory and measurement rate 29
G. Number phase uncertainty 30
H. Using feedback to reach the quantum limit 31
I. Additional technical details 34