A. A. Clerk, S. M. Girvin, and A. D. StoneDepartments of Applied Physics and Physics,

Yale University

Q:What characterizes an “ideal” quantum detector?

(cond-mat/0211001)

(and many discussions with M. Devoret & R. Schoelkopf)

Mesoscopic Detectors and Mesoscopic Detectors and the Quantum Limitthe Quantum Limit

Generic Weakly-Coupled Detector

QI

“gain”

1. Measurement Rate: How quickly can we distinguish the two qubit states?

0

m

P(m

,t)

SQQ ´ 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:

QI

•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 Vnoise?

SI

Q Iz

• 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)

QI

A.C., Girvin & Stone, cond-mat/0211001Averin, cond-mat/0301524

•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)

QI

L R

L

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)

QI

L R

L

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….

L R

Q I

Not usual symmetries!

Transmission Amplitude Condition

Ensures that no information is lost when averaging over energy

versus

Q IL R

L

R

L

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)

4 2 2 4

0.2

0.4

0.6

0.8

1 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 IL 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 IL 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

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…

0.2 0.4 0.6 0.8 1

1.2

1.4

1.6

1.8

2

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1Charge Noise (SQ)

Conclusions

QI

• 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