2164-2 workshop on nano-opto-electro-mechanical systems...

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..

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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..

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Transverse kicks correct trajectory.


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

.Negative Feedback Semiconductor Laser..

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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..

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Potential switch conditioned on photon count.


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Quantum Feedback Control

.Basic Goals..

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Feedback control at nanoscales.

Feedback control of quantum dynamics.

Microscopically justify classicalfeedback schemes.

Double quantum dot..Challenge for Quantum Systems..

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


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.Basic Goals..

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Double quantum dot..Challenge for Quantum Systems..

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Measurement process.

Quantum noise.


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.Basic Goals..

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Double quantum dot.Quantum feedback control.BelavkinMilburnWisemanDohertyKorotkovMabuchi

...


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.Basic Goals..

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


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Quantum TransportElectron Statistics in Quantum Dots

.Full Counting Statistics..

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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..

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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

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"(t) =!!

n=0

"n(t) !!!

n=0

" t

0dtn...

" t2

0dt1"

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..

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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

t=10


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Tunnel Junction ModelSimulation of n-resolved Master equation

0.9

0.95

1

1.05

1.1

0 100 200 300 400 500 600 700 800

n(t)/

t

time t

g=0 g=0.1g=1.0


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Feedback in Coupled Quantum System

Γe

collector

QD1

QD2

Γc

emitter

curr

en

t

Ω

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..

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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..

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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 "

$

$i#

%%"(#, 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

t=10


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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 " "

"i!

'"(#, 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 $ %)..

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i#

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

$" $

$i#

%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)..

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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

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

2g C

2

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

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3 3.5 4

F 2, F

3

ε

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.

-0.01

-0.005

0

0.005

0.01

-1 -0.5 0 0.5 1

1-p 1

(g>0

)/p1(

g=0)

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

t=10

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..

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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

t! =4

!CFB2 cosh2('T/2) =

1

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

N(.


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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.


2164-2 workshop on nano-opto-electro-mechanical systems...

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