A. A. Clerk, S. M. Girvin, and A. D. Stone Departments of Applied Physics and Physics, Yale University Q:What characterizes an “ideal” quantum detector? (cond-mat/ ) (and many discussions with M. Devoret & R. Schoelkopf) Mesoscopic Detectors and the Quantum Limit

Generic Weakly-Coupled Detector Q I “gain” 1.Measurement Rate: How quickly can we distinguish the two qubit states? 0 m P(m,t) S QQ ´ 2 s dt h  Q(t)  Q(0) i 2.Dephasing Rate: How quickly does the measurement decohere the qubit?

The Quantum Limit of Detection Quantum limit: the best you can do is measure as fast as you dephase: Q I Dephasing? Need orthogonal to Measurement? Need distinguishable from What symmetries/properties must an arbitrary detector possess to reach the quantum limit?

Why care about the quantum limit? Minimum Noise Energy in Amplifiers: (Caves; Clarke; Devoret & Schoelkopf) Minimum power associated with V noise ? SISI Q I zz  Detecting coherent qubit oscillations (Averin & Korotkov)

How to get to the Quantum Limit Now, we have: λ’ is the “reverse gain”: IQ λ’ vanishes (monitoring output does not further dephase) Q I A.C., Girvin & Stone, cond-mat/ Averin, cond-mat/ Quantum limit requires: (i.e. no extra degrees of freedom)

What does it mean? To reach the quantum limit, there should be no unused information in the detector… Mesoscopic Scattering Detector: (Pilgram & Buttiker; AC, Girvin & Stone) Q I LR LL RR

What does it mean? To reach the quantum limit, there should be no unused information in the detector… Mesoscopic Scattering Detector: (Pilgram & Buttiker; AC, Girvin & Stone) Q I LR LL RR Transmission probability depends on qubit:

The Proportionality Condition Need: Phase condition? Qubit cannot alter relative phase between reflection and transmission No “lost” information that could have been gained in an interference experiment…. LR Q I Not usual symmetries!

Transmission Amplitude Condition Ensures that no information is lost when averaging over energy versus Q I L R LL RR LL RR 1) 2)

The Ideal Transmission Amplitude Necessary energy dependence to be at the quantum limit Corresponds to a real system-- the adiabatic quantum point contact! (Glazman, Lesovik, Khmelnitskii & Shekhter, 1988) T  -  0

Information and Fluctuations No information lost when energy averaging: Look at charge fluctuations: No information lost in phase changes:  meas for current experiment  meas for phase experiment Q I L R Reaching quantum limit = no wasted information

Measurement Rate for Phase Experiment  meas for current experiment  meas for phase experiment t r

Information and Fluctuations (2) Can connect charge fluctuations to information in more complex cases: Q I L R Reaching quantum limit = no wasted information  meas for current experiment  meas for phase experiment 2. Normal-Superconducting Detector 1. Multiple Channels Extra terms due to channel structure

Partially Coherent Detectors What is the effect of adding dephasing to the mesoscopic scattering detector? Look at a resonant-level model… LL RR Symmetric coupling to leads  no information in relative phase L R  I  = 0 Assume dephasing due to an additional voltage probe (Buttiker)

Partially Coherent Detectors Reducing the coherence of the detector enhances charge fluctuations… total accessible information is increased A resulting departure from the quantum limit… Charge Noise (S Q )

Conclusions Q I Reaching the quantum limit requires that there be no wasted information in the detector; can make this condition precise. Looking at information provides a new way to look at mesoscopic systems: New symmetry conditions New way to view fluctuations Reducing detector coherence enhances charge fluctuations, leads to a departure from the quantum limit