2164-2 Workshop on Nano-Opto-Electro-Mechanical Systems...

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2164-2

Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime

Tobias BRANDES

6 - 10 September 2010

Tech. Univ. Berlin Instit. fuer Theoretische Phys.

Hardenbergstr. 36, Sekr. EW-7, 10623 Berlin

GERMANY

Feedback Control of Quantum Transport

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Feedback Control of Quantum TransportTobias Brandes (Institut fur Theoretische Physik, TU Berlin)

Examples, basic idea.

Quantum transport.

Open questions.

Tobias Brandes (Berlin) Feedback Control Trieste workshop 2010 1 / 1

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Feedback Control: ExamplesJ. C. Maxwell (1868)

.Centrifugal Governor..

Stochastic input is stabilized.

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Feedback Control: ExamplesS. van der Meer (1972), Nobel Prize (1984) - discovery of W and Z bosons

.Stochastic cooling of particle collider beam..

Transverse kicks correct trajectory.

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Feedback Control: ExamplesS. Machida and Y. Yamamoto (1986)

.Negative Feedback Semiconductor Laser..

Photodetector signal corrects laser diode pump current.

Photon statistics changed into sub-Poissonian.

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Feedback Control: ExamplesA. Kubanek, M. Koch, C. Sames, A. Ourjoumtsev, P. W. H. Pinkse, K. Murr and G.Rempe (2009)

.Feedback control of a single-atom trajectory..

Potential switch conditioned on photon count.

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Quantum Feedback Control

.Basic Goals..

Feedback control at nanoscales.

Feedback control of quantum dynamics.

Microscopically justify classicalfeedback schemes.

Double quantum dot..Challenge for Quantum Systems..

Include feedback control into Schrodinger equations,Liouville-von-Neumann equations.

. . . . . .

.Basic Goals..

Double quantum dot..Challenge for Quantum Systems..

Measurement process.

Quantum noise.

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.Basic Goals..

Double quantum dot.Quantum feedback control.BelavkinMilburnWisemanDohertyKorotkovMabuchi

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.Basic Goals..

Double quantum dot.This work: feedback control for quantum transport.

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Quantum TransportElectron Statistics in Quantum Dots

.Full Counting Statistics..

Probability p(n, t) of nelectrons after time t.

C. Flindt, C. Fricke, F. Hohls, T.

Novotny, K. Netocny, T. Brandes, and

R. J. Haug; PNAS 106, 10116 (2009).

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Feedback Control of Electron Statistics

time tcharge n

.Aim: to suppress current fluctuations..

n electrons measured after time t, target current I0.

Charge error !qn(t) ! I0t " n.

Speed up (!qn(t) > 0) or slow down (!qn(t) < 0) tunneling.

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Quantum Transport ModelNo-Feedback Master Equation

Open system Hamiltonian.

H = HS +Hres +HT .

! HS system.! Hres reservoir.! HT system-reservoir coupling.

Reduced density matrix "(t), Liouvillian L, Born-Markovapproximation

"(t) = L"(t).

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Quantum Transport ModelMarkovian Master equation !(t) = L!(t): unravelling, quantum jumps

.L = L0 + J , J : jump super-operator

"(t) =!!

"n(t) !!!

0dtn...

c(t; tn, ..., t1)

"c(t; tn, ..., t1) ! eL0·(t"tn)J eL0·(tn"tn!1)J ...J eL0·t1"0

Non-unitary ‘free’ time-evolution, interrupted by n quantum jumps attimes ti (Carmichael; Zoller; Moelmer; Hegerfeldt;... 1980s).

Full counting statistics (FCS)

p(n, t) ! Tr"n(t).

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Feedback Master Equation

.Conditional density matrix !n(t) !TrresPn!total(t)Pn..

Partial trace, keeps track of reservoir charge n.

"n(t) = L0(t, n)"n(t) + J (t, n)"n"1(t).

Example: junction with bare tunnel rate !,

"L0(t, n) = J (t, n) = ! (1 + g!qn(t)) .

g # 0: feedback strength.

!qn(t) ! I0t " n error charge.

I0: target current, t: time.

time tcharge n

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Tunnel Junction ModelFull counting statistics p(n, t): numerical results

Feedback freezes in the counting statistics!

t=60 t=100 t=140 t=180t=30

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Tunnel Junction ModelSimulation of n-resolved Master equation

0 100 200 300 400 500 600 700 800

time t

g=0 g=0.1g=1.0

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Feedback in Coupled Quantum System

collector

emitter

Charge qubits, electron-phonondissipation.

G. Kießlich, E. Scholl, T. Brandes, F. Hohls, and R.J.

Haug; Phys. Rev. Lett. 99, 206602 (2007).

Molecular transport.

H. Hubener and T. Brandes, Phys.

Rev. Lett. 99, 247206 (2007).

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Mathematical Challenges

.Numerical stability..

Large ODE systems.

"n(t) = L0(t, n)"n(t) + J"(t, n)"

n"1(t) + J+(t, n)"n+1(t).

Conditional density matrix "n(t).

Partial trace, keeps track of reservoir charge n.

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Mathematical Challenges

.Analytical results..

Cumulants of feedback-frozen p(n, t $ %).

PDE systems.

Example: tunnel junction,

Generating Function "(#, t) !#

n ein!p(n, t).

"(#, t) = (e i! " 1)

$1 + g

$I0t "

%%"(#, t).

Cumulants C1(t) = !t,Cn#2(t $ %) = " 1

g& Bernoulli-Sekinumber.

T. Brandes, Phys. Rev. Lett. 105, 060602 (2010).

t=60 t=100 t=140 t=180t=30

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Relation between feedback and no-feedback distribution

no Feedback homogeneous Feedbackp(n, t) di!usive decay frozen

""t "(#, t) = L(#)"(#, t) L(#)f

&I0t " "

'"(#, t)

type ODE PDECGF %0(#)& t h(#) ! ln Tr "(#, t)" i#I0t

cumulants ''I n(( & t ! ("i)n%(n)0 (#)& t ("i)nh(n)(#)

.CGF h(") for homogeneous feedback (t $ %)..

%0(#)I0 = e"h(!)f

%eh(!).

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Relation between feedback and no-feedback distributionExplicit Formulas: First three cumulants

no FB linear Feedback exponential Feedback''I 1(( = I0 I0''I 2(( = I0 & 2gC2 I0 & 2gC2 + O(g2)''I 3(( = I0 &

(6g2C 2

2 + 3gC3)

I0 &(3g2C 2

2 + 3gC3 " 32g

2C4)+ O(g3)

.Fano Factor F (g = 0) from second frozen cumulant C2(g > 0)..

F = 2gC2 + O(g2), F ! ''I 2((''I 1((

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Relation between feedback and no-feedback distributionExample: Single level dot

Dot asymmetry

!R = !L1" a

1 + a, "1 ) a ) 1.

Feedback asymmetry

gR = gL1" b

1 + b, "1 ) b ) 1.

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Relation between feedback and no-feedback distribution

-1 -0.5 0 0.5 1

dot asymmetry parameter a

FB asymmetry b= 0.0FB asymmetry b= 0.5FB asymmetry b=-0.5

Fano factor (no FB)

Homogeneous FB b = 0 recovers Fano factor F = 2gC2 =12(1 + a2).

(g = 0.02 here)

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Relation between feedback and no-feedback distributionExample: double quantum dot

!L = 10!R , Tc = !R

0 0.5 1 1.5 2 2.5 3 3.5 4

F 2, F

F2 (homo)F3 (homo)

F2 (inhomo)F3 (inhomo)

F2/3 reconstructed Fano factor/ skewness.

inhomogenous FB: & not changed.

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Feedback control and internal states

How does feedback a"ect internal system state?

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Feedback control and internal statesExample: Single level dot, p1 stationary dot occupation.

-0.005

-1 -0.5 0 0.5 1

dot asymmetry parameter a

FB asymmetry b= 0.0FB asymmetry b= 0.5FB asymmetry b=-0.5

g/2(1-F)

b = 0: relation 1" p1(g > 0)/p1(g = 0) = g2 (1" F ).

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Open Questions

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Open Questions

Thermodynamic interpretation of frozenFCS: information gain.

t=60 t=100 t=140 t=180t=30

Related work: ”Generalized Jarzynski Equality under Nonequilibrium Feed-back Control”

'e"#(W"!F )"I ( = 1

T. Sagawa and M. Ueda, Phys. Rev. Lett. 104, 090602 (2010).

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Open Questions

E#ciency as an accurate charge transfer device.

To go beyond Markovian master equation.

Semiclassical limits.

Fully quantum feedback loops.

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Summary

Instantaneous feedback of I0t " n ! frozen FCS at large times.

PhD / PostDoc positionsavailable

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Feedback e!ciencyComparison with pump/turnstile

.Single electron turnstile model..

Single level dot, rates !L/R(t) = 'T#!

j=1 !(t " tj ,L/R).

Strong bias from left to right, cycles electrons in - out -in -out ...

Transfers on average 'n( = j tanh('T/2) electrons after j cycles.

Fluctuations c2 ! 'n2( " 'n(2 = 'n(/(2 cosh2('T/2))! grow with time.! surpass corresponding feedback system fluctuations after time

!CFB2 cosh2('T/2) =

2gIcosh2('T/2).

(I = !/2 current for symmetric dot, g FB coupling strength).

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Continuous operation for long times: feedback always wins!

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Feedback e!ciencyComparison with pump/turnstileHowever: the costs ....

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FB requires continuous monitoring of bath.

Cost of high-precision multiplication in feedback function I0t " n issuper-linear in t.

FB COST

FB BETTERAND CHEAPER

PUMP ERROR

PUMP COST

TIMEt*

FB ERROR

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Semiclassical Limit

Double quantum dot + single resonator mode.

Expansions around classical trajectories: limit cycles.

R. Hussein, A. Metelmann, P. Zedler, T. Brandes; arXiv:1006.2076 (2010).

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n-dependent rates: Milburn’s detector feedback model

G. J. Milburn, J. Mod. Opt 38 (10), 1973 (1991)

Random classical variable x = 0, 1: ’(not) transmitted’.

Generating function 'e"i$x( = 1 + p(e"i$ " 1) with p = 'x(.Now N independent transmissions: variables xl with 'xn( ! pn.

Variable XN !#N

l=1 xl , 'e"i$XN ( = $Nn=1

(1 + pn(e"i$ " 1)

Specific choice of the pn (e.g. saturation at large n) ! some controlof 'XN(, 'X 2

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Closed versus Open Loop Control

SYSTEM

Measurement Device

CLOSED LOOP (FEEDBACK)

System parameters are permanentlychanged, conditioned on measurementresult.

SYSTEM

Measurement Device

Open loop (no feedback)

’Design’ of system parameters.

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Some key players: Belavkin; Wiseman, Milburn (textbook 2010!); Doherty,Jacobs; Korotkov; Mabuchi;....

Measurement vs. coherent feedback control.

Markovian measurement feedback ! Lindblad Master equation.! Avoiding decoherence of cat states.! Purification of otherwise mixed qubit states: resonance fluorescent

atoms.

Solid state context: qubit coupled to detector, n-resolved masterequation.

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