# Presentation: 559484

## Mesoscopic Detectors and the Quantum Limit

**Q:What characterizes an “ideal” quantum detector?**

- (Phys. Rev. Lett. 89, 176804 (2002); Phys. Rev. B 67, 165324 (2003); cond-mat/0401103)

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

- Keck Foundation

**Notes:**
Constraints on measurement in a # of diff. Meso systems...

## Mesoscopic Detection Experiments

- Qubit measured by an SET

- Quantum dot measured by a quantum point contact

- Nanomechanical oscillator measured by an SET

*Lehnert et al, 2003*

*Buks et al, 1998*

*Knobel et al, 2003*

*LaHaye et al, 2004*

**Notes:**
Q of QL has direct relevance to a # of ongoing expts... expts where the Q nat. of det. Important.
Though they seem diff., there is a common way of thinking about the QL in each one...

## Generic Weakly-Coupled Detector

**Notes:**
Detector could in principle be quite complicated...
Fund thing. There is randomness in the detector... 
“its this noise which is fundamentally constrained”

## The Quantum Limit of Detection

- Quantum limit: the best you can do is measure as fast as you dephase:

- Measurement? Need                   ***distinguishable*** from

- What symmetries/properties must an ***arbitrary*** detector possess to reach the quantum limit?

**Notes:**
Now in a pos. to formulate the QL... not only must the meas give you some dephase, but the best you can do is...
Can ask some general, important Q’s at this state:

## Why care about the quantum limit?

**Notes:**
More basic Q...
“there are things you can only do with a quantum limited detector”  

## How to get to the Quantum Limit

- Now, we have:

- A.C., Girvin & Stone, Phys. Rev. B 67, 165324 (2003) Averin, cond-mat/0301524

**Notes:**
Return to the Q of what we need to do to reach the QL; 
Heur. Idea is that there should be no extran. Dof in det....can we make this more precise?
Can we even prove QL rig?

## 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)

- L

- R

## What does it mean?

- Mesoscopic Scattering Detector: (Pilgram & Buttiker; AC, Girvin & Stone)

- L

- R

- Transmission probability depends on qubit:

- To reach the quantum limit, there should be no unused information in the detector...

## The Proportionality Condition

- Need:

***Not usual symmetries!***

## Transmission Amplitude Condition

- Ensures that no information is lost when averaging over energy

***versus***

***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:

- Reaching quantum limit = no wasted information

- (Levinson)

## Information and Fluctuations (2)

- Can connect charge fluctuations to information in more complex cases:

- 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*

**Notes:**
Mention interactions a la Pilgram & Buttiker?

## Quantifying Information

- Once we choose a detector quantity (“y”) to measure, can think of our system as a ***noisy classical communication*** channel

- What is the maximum number of bits we can ***reliably*** send after *N *uses of the channel? (i.e. optimize over *all* codings)

-  *t*

## Mach-Zender Interferometer as Detector

**1-T**

**T**

- input

- 1/Nmeas = 2 02 T (1-T)

- Fock State Input: 1/N = 2 02 T (1-T)
- Coherent State Input: 1/N = 2 02 (1-T)

***Coherent state input misses the quantum limit due to “wasted” phase information at the output***

## Linear Response Position Detectors

***F***

***I***

*H**int* = -*A** ****x**** **¢** ****F***

- h ***I(t)*** i = A  h ***x(t)*** i

- Unlike qubit case, this is ***not*** a “non-demolition” setup

- A.C., cond-mat/0406536

- Though detector is not in equilibrium, can derive a Langevin equation for the oscillator:

**Notes:**
Now tell you how this approach to the QL applies to our system viewed as a lin. Amp, or say a pos. detector
At first glance, quite different from qubit problem...
First thing is describing osc... problem because detect. Out of equilib...

## Effective Temperature and Damping for DJQP

**Notes:**
Now turn to the detector...
Start by defining a QL det. As one which has ideal noise props (min SI SF product); in exact analogy to qubit;
Requires...
Also, want our detector to be a good amp the gain has to be large proper measure is the power gain...

## Ideal detectors & power gain

***F***

***I***

*H**int* = -*A** ****x**** **¢** ****F***

- h ***I(t)*** i = A  h ***x(t)*** i

- Want the power absorbed at the detector input to be much smaller than the power available at the detector ouput:

- Define a quantum-limited detector which has “ideal” noise properties (min. *S**I** S**F*), ***exactly*** the same as the qubit case...

***Large T******eff****** implies a large power gain!***

**Notes:**
Now turn to the detector...
Start by defining a QL det. As one which has ideal noise props (min SI SF product); in exact analogy to qubit;
Requires...
Also, want our detector to be a good amp the gain has to be large proper measure is the power gain...

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?
- What do we have to do to reach the quantum limit (not answered by the “standard” treatment of Caves!)

- A.C., cond-mat/0406536

***Three steps for reaching the quantum limit:***

**Notes:**
Now turn to the QL on the displacement sensitivity... what is the minimum possible noise added by the amp?  What do we need to do to get there?  Std. Treatment of Caves not good enough...
(“how” not answered; difficult to describe meso detector as a set of bosonic modes)

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?
- What do we have to do to reach the quantum limit (not answered by the “standard” treatment of Caves!)

- A.C., cond-mat/0406536

*Intrinsic output noise of detector*

*Effect of back-action force noise*

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

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?

- A.C., cond-mat/0406536

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

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?

- A.C., cond-mat/0406536

***Three steps for reaching the quantum limit:***
- Balance back action and intrinsic noise via tuning coupling A
-  ***Use a quantum-limited detector! *****h**** *****f | I | i***** ****i**** = ******** ****h**** *****f | F | i***** ****i**

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?

- A.C., cond-mat/0406536

***Three steps for reaching the quantum limit:***
- Balance back action and intrinsic noise via tuning coupling A
-  ***Use a quantum-limited detector! *****h**** *****f | I | i***** ****i**** = ******** ****h**** *****f | F | i***** ****i**

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?

- A.C., cond-mat/0406536

***Three steps for reaching the quantum limit:***
- Balance back action and intrinsic noise via tuning coupling A
-  ***Use a quantum-limited detector! *****h**** *****f | I | i***** ****i**** = ******** ****h**** *****f | F | i***** ****i**
- Tune the cross-correlator *S**IF*

## Minimum Displacement Sensitivity

- What is the minimum possible noise ***added*** by the detector?

- A.C., cond-mat/0406536

***Three steps for reaching the quantum limit:***
- Balance back action and intrinsic noise via tuning coupling A
-  ***Use a quantum-limited detector! *****h**** *****f | I | i***** ****i**** = ******** ****h**** *****f | F | i***** ****i**
- Tune the cross-correlator *S**IF*

## On resonance,  = 

- The condition for an optimal coupling takes a simple form:

***The detector dependent damping must be much weaker than the intrinsic damping of the oscillator!***

## The Single Electron Transistor (SET)

- Addition of interactions complicates the story...:

***Q***

- Without superconductivity?

- Sequential tunneling

- 2. Co-tunneling regime

- (Makhlin, Shnirman & Schön; Averin; Devoret & Schoelkopf)

- (Averin; Maassen van den Brink)

- Large signal (I / g), but
- = meas /  / g2

- = meas /  ' 1, but small signal (I / g2),

***g***

**Notes:**
Shift gears, discuss a slightly more complex class of detectors...
SETs, in widespread use in a number of diff. Expts.
Consist of a small metallic island w/ a large cap EC; attached via TJs to electrodes;
SD current very sensitive to pot of island, basis for use as a detector...
Unlike MSD, interactions important, can’t just assume coh. Transport...  CAN IT REACH QL?

SEEMS TO BE A TRADEOFF: CAN”T HAVE A LARGE SIGNAL AND REACH THE QL...

## Superconducting SET Detector

-  ***Large signal AND near quantum-limited! ***
-  Used in recent experiments:
-  *K. Lehnert et al., Phys. Rev. Lett., 2003*
*M. LaHaye et al., Science, 2004*

- A.C., Girvin, Nguyen & Stone, PRL 89, 176804 (2002)

- Use the high-signal “DJQP” process;
- Model using a density matrix approach.

*E**J** / ***

*BUT: Why doesn’t incoherence of transport matter?*

**Notes:**
Puzzle: how can this process be near-QL?  Not completely coherent...
General Q: is detector coherence necessary to reach QL

## Are Partially Coherent Detectors Bad?

- Consider a simple resonant-level model with a Lorentzian conductance resonance

- L

- Quantum Limited if (Averin, 2000):
*eV* small (no lost information due to energy-averaging)
-  L = R (no lost phase information)

***dI / dV***

***V******gate***

***What happens if we now add dephasing to the detector?***

- R

- A.C. and Stone, cond-mat/0401103

## Adding Detector Dephasing

- Consider different sources of dephasing:

- Dephasing due to escape into a voltage probe (Buttiker)

- Dephasing due an external, classical, fluctuating potential

- Coherent and  incoherent broadening: = L + R + 

- A.C. and Stone, cond-mat/0401103

***dI / dV***

***V***

**Notes:**
Discuss magnitude of dephasing here!!!!

## Adding Detector Dephasing

- Both the magnitude of dephasing *and* the nature of the dephasing source affect the quantum limit...

- For a “pure” voltage probe, possible to remain near quantum-limited even with ***complete*** detector incoherence

****

****

********** ****/ ********tot**

********** ****/ ********tot**

- A.C. and Stone, cond-mat/0401103

**Notes:**
Two slides perhaps, need to be a lot cleaner...

## Conclusions

- Reaching the quantum limit requires a detector with ***ideal noise properties***; this is equivalent to requiring that there be ***no wasted information*** in the detector
- Looking at information provides a new way to view mesoscopic systems (new “symmetries”)
- Both the magnitude and origin of detector incoherence affects the ability of a detector to reach the quantum limit
- A *generic *out-of-equilibrium detector described by a damping kernel and an effective temperature *T**eff*; *T**eff* sets the detector power gain.

## Measurement Rate for Phase Experiment

**t**

**r**

## Large Voltage, “Co-tunneling” Regime

## A Microscopic Calculation of Relevant Quantities...

- A ***Microscopic*** Calculation of Relevant Quantities...

- Each cycle transfers 3 electrons
-  Sub-Poissonian shot noise

- slow

- slow

-  e.g., consider the measurement rate

**Notes:**
What happens to shot noise off resonance?  Does single JQP also have sub-P shot noise?

## Normal-Superconducting Detector

- Make one reservoir superconducting
- consider Andreev transport at *eV* much smaller than the gap 

- N

- S

- electrons
- holes

- Can define Q via currents in normal regions; for *eV* ! 0:

- Same general form as normal state expression, but diff. phases
- Time reversal symmetry + inversion symmetry not enough!
- Turning ON superconductivity can increase dephasing (exp?)

## Normal-Superconducting Detector (2)

- Proportionality conditions between I and Q take the form:

- Many channels?  If no channel mixing, and:

- 	can derive a necessary form for T(E)...

- Larger *eV*: electron-hole structure makes it difficult to reach quantum limit, even if one has electron hole symmetry...

- T

- E

- NS

- N