Quantum Noise of Resonant Cooper Pair Tunneling

1
Mesoscopic Detectors and the Quantum Limit
(Phys. Rev. Lett. 89, 176804 (2002) Phys. Rev. B
67, 165324 (2003) cond-mat/0401103)
A. A. Clerk, S. M. Girvin, and A. D.
StoneDepartments of Applied Physics and
Physics,Yale University
(and discussions with M. Devoret R. Schoelkopf)
QWhat characterizes an ideal quantum detector?
Keck Foundation
2
Mesoscopic Detection Experiments
Quantum dot measured by a quantum point contact
Qubit measured by an SET
Lehnert et al, 2003
Buks et al, 1998
Nanomechanical oscillator measured by an SET
Knobel et al, 2003
LaHaye et al, 2004
3
Generic Weakly-Coupled Detector
4
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?

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Why care about the quantum limit?
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How to get to the Quantum Limit
A.C., Girvin Stone, Phys. Rev. B 67, 165324
(2003) Averin, cond-mat/0301524

• Now, we have

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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)
mL
mR
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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)
mL
mR
Transmission probability depends on qubit
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The Proportionality Condition

• Need

Not usual symmetries!
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Transmission Amplitude Condition
Ensures that no information is lost when
averaging over energy
1)
versus
2)
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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
1
0.8
0.6
0.4
0.2
e - e0
-
4
-
2
2
4
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Information and Fluctuations
Reaching quantum limit no wasted information

• No information lost in phase changes

• No information lost when energy averaging

Look at charge fluctuations
(Levinson)
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Information and Fluctuations (2)
Reaching quantum limit no wasted information
Can connect charge fluctuations to information in
more complex cases
1. Multiple Channels
Extra terms due to channel structure
2. Normal-Superconducting Detector
Gmeas for phase experiment
Gmeas for current experiment
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Quantifying Information

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

t

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

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Mach-Zender Interferometer as Detector
1-T
T
input
1/Nmeas 2 q02 T (1-T)
Fock State Input 1/Nf 2 q02 T (1-T) Coherent
State Input 1/Nf 2 q02 (1-T)
Coherent state input misses the quantum limit due
to wasted phase information at the output
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Linear Response Position Detectors
A.C., cond-mat/0406536
Hint -A x F
F
I
h I(t) i A l h x(t) i

• Unlike qubit case, this is not a non-demolition
  setup

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

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Effective Temperature and Damping for DJQP
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Ideal detectors power gain
Hint -A x F
F
I
h I(t) i A l h x(t) i

• Define a quantum-limited detector which has
  ideal noise properties (min. SI SF), exactly
  the same as the qubit case

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

• Large Teff implies a large power gain!

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Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• 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!)

Three steps for reaching the quantum limit
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Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• 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!)

Effect of back-action force noise
Intrinsic output noise of detector

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

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Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• What is the minimum possible noise added by the
  detector?

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

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Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• What is the minimum possible noise added by the
  detector?

• 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
  a h f F i i

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Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• What is the minimum possible noise added by the
  detector?

• 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
  a h f F i i

24
Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• What is the minimum possible noise added by the
  detector?

• 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
  a h f F i i
• Tune the cross-correlator SIF

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Minimum Displacement Sensitivity
A.C., cond-mat/0406536

• What is the minimum possible noise added by the
  detector?

• 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
  a h f F i i
• Tune the cross-correlator SIF

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On resonance, w W

• 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!

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The Single Electron Transistor (SET)
g
Addition of interactions complicates the story
Q
Without superconductivity?

• Sequential tunneling

Large signal (I / g), but c Gmeas / Gf / g2
(Makhlin, Shnirman Schön Averin Devoret
Schoelkopf)
2. Co-tunneling regime
c Gmeas / Gf ' 1, but small signal (I / g2),
(Averin Maassen van den Brink)
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Superconducting SET Detector
A.C., Girvin, Nguyen Stone, PRL 89, 176804
(2002)
Use the high-signal DJQP process Model using
a density matrix approach.

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

BUT Why doesnt incoherence of transport matter?
EJ / G
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Are Partially Coherent Detectors Bad?
A.C. and Stone, cond-mat/0401103

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

gL
dI / dV
gR
Vgate

• Quantum Limited if (Averin, 2000)
• eV small (no lost information due to
  energy-averaging)
• gL gR (no lost phase information)

What happens if we now add dephasing to the
detector?
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Adding Detector Dephasing
A.C. and Stone, cond-mat/0401103

• Consider different sources of dephasing

Dephasing due to escape into a voltage probe
(Buttiker)
Dephasing due an external, classical, fluctuating
potential
dI / dV
Coherent and incoherent broadeningg gL gR
gf
V
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Adding Detector Dephasing
A.C. and Stone, cond-mat/0401103

• Both the magnitude of dephasing and the nature of
  the dephasing source affect the quantum limit

c
c
gf / gtot
gf / gtot
For a pure voltage probe, possible to remain
near quantum-limited even with complete detector
incoherence
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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
  Teff Teff sets the detector power gain.

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Measurement Rate for Phase Experiment
t
r
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Large Voltage, Co-tunneling Regime
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A Microscopic Calculation of Relevant Quantities

• e.g., consider the measurement rate

slow

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

slow
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Normal-Superconducting Detector

• Make one reservoir superconducting
• consider Andreev transport at eV much smaller
  than the gap D

S
N
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?)

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Normal-Superconducting Detector (2)

• Proportionality conditions between I and Q take
  the form

T

• Many channels? If no channel mixing, and

NS
N
can derive a necessary form for T(E)
E

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