A Quantum Noise Approach to Quantum NEMS

Aashish Clerk McGill University

S. Bennett (graduate student) Collaborators: K. Schwab, S. Girvin, A. Armour, M. Blencowe

• Quantum noise as a route to understanding quantum NEMS… Quantum NEMS: Interesting Issues

• How does the conductor affect the oscillator? • Non-trivial dissipative quantum mechanics… the oscillator sees a non-Gaussian, non-equilibrium “bath” ! Systematic way of describing non-equilibrium systems as effective equilibrium baths • Non-equilibrium cooling • strong-feedback “lasing” instabilities,… Quantum NEMS: Interesting Issues

• How does the oscillator affect the conductor? • Possibility for quantum limited position detection… opens door to doing quantum control ! Rigorous bound on quantum noise of a detector ! Reaching the quantum limit requires a conductor (detector) with ideal quantum noise properties Non-equilibrium baths?

Problem: the “bath” produced by the conductor is non-equilibrium and non-Gaussian! However: for several systems, theory shows that the conductor acts as an effective equilibrium bath! Point contact: Mozyrsky & Martin, 02; Smirnov, Mourokh & Horing, 03; !cond A.C. & Girvin, 04 Normal-metal SET: Armour & Blencowe, 04 kBTeff " eVDS Quantum dot: Mozyrsky, Martin & Hastings, 04

• Generic way of seeing this?

• What in general determines Teff? Effective bath descriptions • If oscillator-conductor coupling is weak, we can rigorously derive a Langevin equation!

damping kernel random force • Langevin equation determined by the quantum noise of the conductor:

n~F ">0: absorption of h" by bath "<0: emission of h" by bath

A.C., Phys. Rev. B 70 (2004) (see also J. Schwinger, J. Math Phys. 2 (1960); Mozyrsky, Martin & Hastings, PRL 92 (2004)) Effective temperature

n~F ">0: absorption of h" by bath "<0: emission of h" by bath • Not in equilibrium ! no fluctuation dissipation theorem! Effective temperature

n~F ">0: absorption of h" by bath "<0: emission of h" by bath • Use FDT to define the effective temperature of the detector Effective temperature

n~F ">0: absorption of h" by bath "<0: emission of h" by bath • Use FDT to define the effective temperature of the detector

Teff in a non-equilibrium system set by the asymmetry between absorption and emission Sequential Tunneling • Incoherent tunneling on and off the island: F~n #+ #-

•Form of #’s?

V • depend on what is tunneling (electrons, DS quasiparticles, ...) and where (metallic SET, quantum dot, ...).

• “effective bath” determined by sensitivity of tunnel rates to the addition or subtraction of energy Sequential Tunneling (2) #+ #-

VDS

Normal Metal SET #(E)

E More energy, more available states for tunneling….

Superconducting SET #(E)

E Sharp turn-on of rates… Cooper Pairs as an Effective Bath (Bennett, AC, NJP, 2005; Imbers, Blencowe & Armour, NJP, 2005) • Josephson-quasiparticle process in a superconducting SET

EJ #

$ = Efinal - Einitial

# # Vosc b a

$ = detuning of Cooper-pair resonance n Cooper Pairs as an Effective Bath (2) • Force from conductor? V Number of electrons n on the SET island osc • Calculate the quantum noise in n (A.C., Girvin, Nguyen & Stone, PRL 02) n

EJ #

• Teff can be as small as h#a/4!!

#b #a • Teff and ! can be negative!

$ = E2 - E0 Negative Damping & Temperature?

($ = E2 - E0) #b #a • $ > 0? • Cooper pairs move closer to resonance E if they absorb energy 2 • Net tendency to absorb energy from

oscillator… %, Teff > 0 E0 • $ < 0 • Cooper pairs move closer to resonance E0 if they emit energy • Net tendancy to emit energy to the

oscillator… negative damping and Teff! E2 (First discussed for qubits: % Unstable regime!! A.C., Girvin, Nguyen & Stone, PRL 02) See also Blencowe, Imbers & Armour 2005: master equation approach Ground state cooling using Cooper Pairs?

Vosc

#b #a n

• Calculate Teff as a function of frequency… Ground state cooling using Cooper Pairs?

Vosc

#b #a n

• Calculate Teff as a function of frequency… Ground state cooling using Cooper Pairs

Vosc

#b #a n

• Calculate Teff as a function of frequency…

• If $ = h&, h& >> #?

• In principle, ground-state cooling possible (reality: need to worry about cotunneling processes) (see also Martin, Shnirman & Tian, 2004) Equivalence to Optical Cavity Cooling

• Cooper-pair cooling identical in many ways to cavity cooling:

• Teff(",$) same. • Response time (as a function of $) same • BUT: • Driven cavity is a many-state (figure courtesy Jack Harris, Yale) system • SSET is a driven 3-state system

In our system, Cooper-pairs play the role of photons… Strong Feedback Effects (AC & Bennet, NJP 2005; Bennett & AC, PRB(R) 2006) • What happens when the total oscillator damping is negative? !weak coupling theory is unstable!

small x large x • Oscillator motion becomes large ! Strongly modifies behaviour of the SET Strong Feedback Effects (AC & Bennet, NJP 2005; Bennett & AC, PRB(R) 2006) • What happens when the total oscillator damping is negative? !weak coupling theory is unstable!

small x large x • Oscillator motion becomes large ! Strongly modifies behaviour of the SET • Result? Amplitude-dependent damping… !(E) Similar to a laser! 0 Tunneling Cooper-pairs ' act as the gain medium… Strong Feedback Effects (AC & Bennet, NJP 2005; Bennett & AC, PRB(R) 2006)

small x large x • Make this picture rigorous using the fact that the SET is much faster than the oscillator • At each instant, the properties of the effective bath depend on the oscillator’s position

• Langevin & Fokker-Planck equation • Weak damping limit: average over phase (Kramers) Strong Feedback Effects (AC & Bennet, NJP 2005; Bennett & AC, PRB(R) 2006)

small x large x

• Get an energy diffusion equation for the oscillator:

Energy-dependent damping and diffusion!

• Similar physics in a normal SET: Blanter, Usmani & Nazarov, PRL 2006 • Negative damping & non-linear friction: Dykman, 87 Stationary State - Bistability

small x large x • Stationary state of the oscillator determined by an “energy- dependent” temperature! Optimal Coupling & Scaling (Bennett & AC, PRB(R) 2006)

small x large x • Tuning coupling voltage ' tuning atom-field coupling… • There is an optimal coupling to maximize effect…

• ( & #? • Can get number squeezing! (Rodrigues, Imbers & Armour, 2006) Enhanced Shot Noise

small x large x • Slows dynamics in oscillator leads to long-time correlations in Cooper pair tunneling… enhanced current noise. SI / 2e Imax

See also: Blanter, Usmani & Nazarov, PRL 2006 Critical Slowing Down (Bennett & AC, PRB(R) 2006)

small x large x • Current noise also sensitive to critical slowing down at the “lasing” bifurcation… From Quantum Noise to the Quantum Limit Ideal:

Reality:

Added noise S ("): SI(") x,add

Quantum limit?

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

• Reaching quantum limit requires a detector with “ideal” quantum noise:

• A detector where the product SI SF reaches a minimum.

• For such a detector, the power gain GP is simply Teff! • Heuristic interpretation: no wasted information… Ideality of Cooper-Pair Tunneling

Other “ideal” detectors: • Tunnel junction • Adiabatic QPC • Normal SET in cotunneling regime Conclusions

• For weak coupling, the conductor in a NEMS system acts as an “effective thermal bath”

• Damping and Teff determined by conductor’s ability to emit and absorb energy • Tunneling Cooper pairs as a bath • Non-equilibrium cooling • analogy to optical cavity cooling… Cooper pairs ' cavity photons! • Unstable negative damping regime characterized by strong feedback (like a laser!)

A. Clerk & S. Bennett, New. J. Phys. (2005) A. Naik et. al, Nature (2006) S. Bennett & A. Clerk, cond-mat/0609329 Future Challenges

• What can we do with a quantum limited detector? • Quantum control? Feedback cooling? Squeezing? • Only a few proposals (Korotkov; Jacobs & Habib) • Need more thought here! • Strongly-coupled NEMS? • Can we get into a regime where the non-Gaussian nature of the conductor’s noise is important? • Quantum oscillators? • Reaching quantum limit on position sensitivity has nothing do to with the quantum nature of the oscillator….