Quantum Noise andQuantum Measurement(APS Tutorial on Quantum Measurement)!F(t)Aashish Clerk
McGill Universityt(With thanks to S. Girvin, F. Marquardt, M. Devoret)• Use quantum noise to understand quantum measurement…

Quantum Measurement &Mesoscopic Physics• Quantum measurement relevant to many recent expts…
Quantum electromechanical systems…
(K. Schwab)(K. Lehnert)

Qubit + readout
experiments…
(C. Marcus, J. Petta)(R. Schoelkopf)

Quantum-limited &back-action evading
amplifiers…
(M. Devoret, I. Siddiqi)(K. Lehnert)

Quantum Measurement &Mesoscopic Physics• Quantum measurement relevant to many recent expts…

•Issues?
1. How do we describe the “back-action” of a detector?
• Detector is quantum and out-of-equilibrium
2. What is the “quantum limit”?
• How do we reach this ideal limit?
3. Conditional evolution?
• What is the state of the measured system given a
particular measurement record?

Weak Continuous Measurements• Information only acquired gradually in time…
• Need to average to reduce the effects of noise• e.g. oscillator measured by a single-electron transistor:n e-’s onisland(K. Schwab group, Cornell)

"(t)

t

Weak Continuous Measurements• Not measuring instantaneous x(t);rather “quadrature amplitudes”:

n e-’s onisland(K. Schwab group, Cornell)

"(t)

t

Quantum Limits?
• Naïve: for a more precise measurement, just increase
coupling, hence !....
• BUT: back-action puts a limit to how much you can do this!
Heisenberg microscope•Measure x " disturb p•This messes up meas. of xat later times

Quantum Limits?
• Naïve: for a more precise measurement, just increase
coupling, hence !....
• BUT: back-action puts a limit to how much you can do this!
Heisenberg microscope

SET position detector?
n e-’s onisland•Measure x " disturb p•This messes up meas. of x
at later times•A#n is a fluctuating force on
oscillator… will cause x to fluc.

Quantum Limits?
• Naïve: for a more precise measurement, just increase
coupling, hence !....
• BUT: back-action puts a limit to how much you can do this!
Heisenberg microscope

SET position detector?
n e-’s onisland•Measure x " disturb p•This messes up meas. of x
at later times•A#n is a fluctuating force onoscillator… will cause x to fluc.

How do we describe back-action? Is it “ideal” or not?

Quantum Noise Approach• What does back-action do to the oscillator?
• Is it as small as allowed by quantum mechanics?
• Need to understand the (quantum) noiseproperties of the detector’s back-action force..

• Force exerted by detector described by an operator F

F ~ cavity photon number

F ~ charge in QPCF~nBasics of Classical Noise• Start by thinking of the noisy force F(t) classically…

!F(t)
0t
• Power spectral density: how big is the noise at a given
frequency?

• Stationary noise? Autocorrelation only depends on time difference• Gaussian noise? Full probability distribution set by SF($)

What about a noisy quantum force?
n e-’s onisland• Back-action force is a quantum operator; also describedby a spectral density…

Heisenberg-picture operators
Expectation value is with respect
to the density matrix describing thedetector’s state…

What is so quantum about quantum noise?
1. Zero-point fluctuations• Noise does not vanish at zero temperature• At high frequencies, h$ > kBT; noise will be muchbigger than the classical prediction
2. Positive and negative frequencies not the same!
••

Classical: %F(-$) = %F($)*, thus SF($) = SF(-$)
Quantum: F(t) and F(0) do not commute!

3. Heisenberg-like quantum constraints on noise!
• The uncertainty principle places a rigorous lower
bound on SF

Effects of the back-action force?
• Classical case: use a Langevinequation:

damping kernelrandom force• Quantum case: Langevin equation still holds if the detector isin equilibrium and has Gaussian noise…

• But: any reasonable detector is NOT in equilibrium!
What is the “effective temperature” of the detector?

Positive versus Negative Frequency Noise?
•Instructive to write SF($) in terms of the exact eigenstatesof our “bath”:

Bath density matrix•

Bath energy eigenstates

Just the Golden Rule expression for a transition rate!
$>0: absorption of h$ by bath$<0: emission of h$ by bath

Effective Temperature$>0: absorption of h$ by bath$<0: emission of h$ by bath••In equilibrium, quantum noise directly tied to temperature…
Consider the rate at which the detector makes transitions
between states with energy E and E+h#...
E+h#

E

Thus:

|b &|a&Effective Temperature$>0: absorption of h$ by bath$<0: emission of h$ by bath••In equilibrium, quantum noise directly tied to temperature…
Consider the rate at which the detector makes transitions
between states with energy E and E+h#...
E+h#

E

Thus:

|b &|a&Effective Temperature$>0: absorption of h$ by bath$<0: emission of h$ by bath•In equilibrium, ratio between positive and negative noise
set by temperature:••

BUT: What if bath is not in equilibrium?
Can use this ratio to define an effective temperature….

Effective Temperature$>0: absorption of h$ by bath$<0: emission of h$ by bath

• Teff in a non-equilibrium system?
• a measure of the asymmetry betweenemission and absorption• Teff is frequency-dependent?
• the price we pay for being out-ofequilibrium!
Still… how does this relate to more usual notions of temperature?

Effective bath descriptions• For weak coupling, can rigorously derive a Langevin equation
A.C., Phys. Rev. B 70 (2004); ( also J. Schwinger, J. Math Phys. 2 (1960);Mozyrsky, Martin & Hastings, PRL 92 (2004))

damping kernelrandom force• Generic approach:• To understand how the detector acts as a “bath”, needto know its SF($)…
• Teff?
• Energy scale characterizing the difference between
energy absorption and emission

Applications of this Approach?
KEY: If we understand the quantum noise properties of our detector, weunderstand how it acts as a bath…

• Back-action cooling with Cooper Pairs:
EJ$n~F% = Efinal - Einitial

Teff NOT set by bias voltage
Rather, by lifetime of a resonance!
(expt: VDS ~ 5K, Teff ~ 200 mK)
Theory: AC, Girvin & Stone, 02; AC & Bennett, 05; Blencowe, Armour & Imbers, 05
Expt:
Naik et al, 2006

Applications of this Approach?
KEY: If we understand the quantum noise properties of our conductor, weunderstand how it acts as a bath…

• Back-action cooling with Cooper Pairs:
EJ$n~F% = Efinal - Einitial•What if oscillator frequency is not small?

• If ! = h#, h# >> $?
(AC, unpublished)Applications of this Approach?
KEY: If we understand the quantum noise properties of our conductor, weunderstand how it acts as a bath…

• Back-action cooling with photons:•Same expression as for
Cooper pair cooling!
•Can reach ground state forlarge $m / '…
Quantum theory: Marquardt, Chen, AC, Girvin, 07; Wilson-Rae, Nooshi, Zwerger & Kippenberg 07
Expts:
Hohberger-Metzger et al., 04; Arcizet et al., 06; Gigan et al. 06; Schliesser et al. 06; Corbitt et al07; Thompson et al. 08

Towards the Quantum Limit• How small can we makethe added noise?
Two parts to the noise:“Intrinsic” output noise:• Present even without coupling to oscillator (e.g. shot noise)• Make it smaller by increasing coupling strength…

Towards the Quantum Limit• How small can we makethe added noise?
Two parts to the noise:
Back-action noise:• Measuring x must disturb p in a random way;
this leads to uncertainty in x at later times.
• Make it smaller by decreasing coupling strength…

Amplifier Quantum Limit• How small can we makethe added noise?
Two parts to the noise:
Quantum Limit• If our detector has a “large” gain, then "˜ (t) cannot be
arbitrarily small• The smallest it can be is the size of the oscillator zero-point
motion…
!

A Precise Statement of the QL
If there were no noise:
Including noise added by detector:(Ignore correlation for the moment!)

Added noise Sx,add($):SI($)

Quantum limit?
MHzA loophole?
If there were no noise:
Including noise added by detector:(Ignore correlation for the moment!)

Added noise Sx,add($):
WAIT: what if back-action force and “shotnoise” anti-correlated?
Could in principle have back-action, yet still
have no added noise!

Why must there be added noise?
Before Amplification:pAfter Amplification:pxx• Liouville Theorem: phase-space volume can’t expand!
• Way out: there must be extra degrees of freedom• Quantum: these extra degrees of freedom must havesome noise (at the very least, zero-point noise)(can use this to derive amplifier quantum limt: Haus & Mullen, 62; Caves 82)

Aside: Noise-Free Amplification?
Before Amplification:pAfter Amplification:px• Can amplify one quadrature without any added noise:• Can realize this in many ways
e.g. driven cavity coupled to osc. (AC, Marquardt, Jacobs, 08)x

Detector Noisen e-’s onisland

Noise characterized by symmetrized spectral densities:

Quantum Constraint on Noise
AC, Girvin & Stone, PRB 67 (2003)
Averin, cond-mat/031524

Important aspect of quantum noise:
There are quantum constraints on noise that have noclassical analogue.

• If we have gain, we MUST in general have noise.
• To simplify inequality, have assumed:• No reverse gain (if you couple to I, F is not affected)• ! is real

Origin of Quantum Noise Constraint• Usual Heisenberg Uncertainty relation:• To have gain, I(t) and F(0) can’t commute for all times t!

• This non-commutation at different times leads directly toour quantum constraint on the noiseQuantum Constraint on Noise
AC, Girvin & Stone, PRB 67 (2003)
Averin, cond-mat/031524• A detector with “quantum ideal” noise?
• One where the product SI SF reaches a minimum.
• Reaching the quantum limit on the added requires adetector with “quantum ideal” noise….

Power Gain
• Only have a quantum limit if our detector truly amplifies• Need dimensionless measure of power gain….

PinPoutdrive

Need a large power gain!
Otherwise, we can’t ignore the added noise of thenext stage of amplification!

Power Gain
• Only expect a quantum limit if our detector truly amplifies• Need to introduce the notion of a dimensionless power
gain….

PinPoutdrive• If detector has “ideal” quantum noise:Power gain set by effective
temperature!

• If also GP >> 1:

SIF must be real!
(correlations can’t help beat QL!)

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Consider a large power gain… cross-correlator SIF is real

Intrinsic output
noise of detector

Effect of backaction force noise

Three steps for reaching the quantum limit:

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Assume the limit of a large power gain " SIF / ! is real

Intrinsic output
noise of detector

Effect of backaction force noise

Three steps for reaching the quantum limit:
1. Balance back action and intrinsic noise via tuning coupling A.

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Assume the limit of a large power gain " SIF / ! is realThree steps for reaching the quantum limit:
1. Balance back action and intrinsic noise via tuning coupling A.

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Assume the limit of a large power gain " SIF / ! is realThree steps for reaching the quantum limit:
1. Balance back action and intrinsic noise via tuning coupling A
2. Use a quantum-limited detector!

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Assume the limit of a large power gain " SIF / ! is realThree steps for reaching the quantum limit:
1. Balance back action and intrinsic noise via tuning coupling A
2. Use a quantum-limited detector!

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Assume the limit of a large power gain " SIF / ! is realThree steps for reaching the quantum limit:
1. Balance back action and intrinsic noise via tuning coupling
2. Use a quantum-limited detector!
3. Tune the cross-correlator SIF

Minimum Added Noise• Quantum noise constraint leads to the quantum limit…
• Assume the limit of a large power gain " SIF / ! is realSame as zero point noise!
Three steps for reaching the quantum limit:
1. Balance back action and intrinsic noise via tuning coupling A
2. Use a quantum-limited detector!
3. Tune the cross-correlator SIF

On resonance, $ = (• The condition for an optimal coupling takes a simple form:• At the quantum limit, the amplifier-oscillator coupling
has to be weak enough to offset the large Teff of theamplifierDetecting Zero Point Motion?
• Pick coupling tominimize added
noise on resonance• Oscillator peak is 4
times noisebackground…
Experiments:

Naik, Schwab et al. 06:

SET detector,15)QL (but…)
using actual output noise? 525 ) QL
Flowers-Jacobs, Lehnert et al. 07: APC detector, (1700±400)*QL (measured!)Meaning of “ideal noise”?
• Key point: need to have a detector with “ideal” quantum
noise to reach the quantum limit.

• What does this mean?
• Detector cannot be in a thermal equilibrium state;• More concrete: “no wasted information”
e.g. generalized QPC detector• scattering matrix must satisfy constraints related to“wasted information” (Pilgram & Buttiker; A.C., Stone & Girvin)Mesoscopic Scattering Detector(AC, Girvin & Stone 03)

LQ

T depends on oscillator:

IR
F=Q!

Insisting on “ideal” noise puts
constraints on s-matrix:µLµR

Wasted Information?
(AC, Girvin & Stone 03)

Phase Info:

Info in T(%):• try to learn more by doing
an interference expt.

• try to learn more by using theenergy-dependence of T
µL
µR

LR

versusµLµR

Conclusions• Teff of a non-equilibrium system:• Defined by the detector’s quantum noise spectrum• Characterizes asymmetry between absorption andemission of energy• In general, is frequency dependent• Quantum Limit• There are quantum constraints on noise• Reaching the quantum limit requires a detector
with “ideal” noise
Clerk, Phys. Rev. B 70, 245306 (2004)
Clerk & Bennett, New. J. Phys. 7, 238 (2005)
Clerk, Girvin, Marquardt, Devoret & Schoelkopf, RMP (soon!)