Quantum Noise and Measurement

Rob Schoelkopf Applied Physics Yale University

Gurus: Michel Devoret, Steve Girvin, Aash Clerk

And many discussions with D. Prober, K. Lehnert, D. Esteve, L. Kouwenhoven, B. Yurke, L. Levitov, K. Likharev, …

Thanks for slides: L. Kouwenhoven, K. Schwab, K. Lehnert,…

Noise and Quantum Measurement 1 R. Schoelkopf And God said: [,aa† ]=1

“Go forth, be fruitful, and multiply (but don’t commute)”

And there was light, and quantum noise…

Noise and Quantum Measurement 2 R. Schoelkopf Manifestations of Quantum Noise

Spontaneous emission Casimir effect Well-known: Lamb shift g-2 of electron

Mesoscopic and solid-state examples (less usual?): Shot noise Minimum noise temperature of an amplifier Measurement induced dephasing of qubit Environmental destruction of Coulomb blockade Quasiparticle renormalization of SET’s capacitance …

Noise and Quantum Measurement 3 R. Schoelkopf Overview of Lectures

Lecture 1: Equilibrium and Non-equilibrium Quantum Noise in Circuits Reference: “Quantum Fluctuations in Electrical Circuits,” M. Devoret Les Houches notes.

Lecture 2: Quantum Spectrometers of Electrical Noise Reference: “Qubits as Spectrometers of Quantum Noise,” R. Schoelkopf et al., cond-mat/0210247 Lecture 3: Quantum Limits on Measurement References: “Amplifying Quantum Signals with the Single-Electron Transistor,” M. Devoret and RS, Nature 2000. “Quantum-limited Measurement and Information in Mesoscopic Detectors,” A.Clerk, S. Girvin, D. Stone PRB 2003. And see also upcoming RMP by Clerk, Girvin, Devoret, & RS. Noise and Quantum Measurement 4 R. Schoelkopf Outline of Lecture 1

• Quantum circuit intro and toolbox

• Electrical quantum noise of a harmonic oscillator (L-C)

• How to make a quantum resistor (= the vacuum!)

• Noise of a resistor: the quantum Fluctuation-Dissipation Theorem (FDT)

• Experiments on the zero point noise in circuits

• Shot noise and the nonequilibrium FDT (time permitting)

Noise and Quantum Measurement 5 R. Schoelkopf Quantum Circuit Toolbox

L-C Resonator Cooper-Pair Box Single Electron Transistor

Vg Cg Cge Vds

Vge Two-level Harmonic oscillator system Voltage/Charge (qubit) amplifier Superconductors: quality factor 106 or greater – levels sharp ω > kT 1 GHz = 50 mK, very few levels populated

Noise and Quantum Measurement 6 R. Schoelkopf The Electrical Harmonic Oscillator 1 iLω Z == HO 1/iLωω+ iC 2 1/− ()ω ω0 t φ()tL==I()t ∫ V(ττ)d −∞ 11 =−Cφ22φ ω0 =1 LC 22L L C ⇔ mass 1/ L ⇔ spring constant Z0 = C QC= φ ⇔ momentum

2 Thermal equilibrium: Qk~ TC

Noise and Quantum Measurement 7 R. Schoelkopf The Quantum Electrical Oscillator

22 Q φ ⎛⎞† 1 Ha= +=ω0 ⎜⎟a+ 22CL ⎝⎠2 “p” “x”

1 † Z Qa=−()aφ =+0 ()aa† iZ2 0 2

† [Qi,,φ]= −=⎣⎡⎤aa⎦ −i [QH,0]≠ Q and φ are not constants of motion! []φ,0H ≠ [Qt(),Q(0)]≠ 0 At()= eiHt //A(0)e−iHt []φφ()t , (0) ≠ 0 Noise and Quantum Measurement 8 R. Schoelkopf Noise of Quantum Oscillator What about correlation functions of φ and Q ? e.g. for thermal equilibrium !? Z00⎡ ⎛⎞ω ⎤ φφ(tt) (0) =−⎢coth⎜⎟cos()ω00isin()ωt⎥ 22⎣ ⎝⎠kT ⎦ 1) Correlator not real, how to define/interpret a spectral density? 2) Non-zero variances even at T=0

Z00⎛ ω ⎞ φφ(0) (0) = coth⎜ ⎟ 22⎝ kT ⎠

⎛ ω0 ⎞ QQ(0) (0) = coth⎜ ⎟ 22Z0 ⎝ kT ⎠

Noise and Quantum Measurement 9 R. Schoelkopf Quantum Fluctuations of Charge

2 ⎛⎞ω00hωω⎛0⎞ Qk==coth ⎜⎟TCcoth⎜⎟ 22Z0 ⎝⎠kT 2kT ⎝2kT ⎠

xx⎛ ⎞ Quantum: coth ⎜ ⎟ ω 22⎝ ⎠ Q2 ~ 0 Qk2 / TC 2C

2 Thermal: Qk~ TC

ω0 kT Noise and Quantum Measurement 10 R. Schoelkopf Noise and Spectral Densities Classically Vt() Random variable Vt() t

Auto-correlation function CtVV ()− t′′= V(t)V(t)

1 T /2 Fourier transform Vd()ω = teitω V(t) T ∫−T /2 ∞ Spectral density Sd()ω ==teitω V(t)V(t′) V()ωωV(−) VV ∫−∞ Since Vt() is classical and real, Vt()Vt(′) = Vt(′)Vt()

And so: SSVV ()ω = VV (−ω)

Noise and Quantum Measurement 11 R. Schoelkopf Spectral Density of Classical Oscillator for mechanical harmonic oscillator in thermal equil.: 1 mass, m xt()=+x(0c)os()ωωt p()0 sin()t 00mω resonant 0 2 freq, ω0 p()tp=−(0c)os()ω00tx()0mωωsin()0t position correlation function: 0 in equil. 1 Ctxx ()==x()tx(00) x()x()0cos()ω00t+p()0x()0sin()ω t mω0 11 1 equipartition thm: kx22==mω x2kT 220 2 kT Ctxx ()= 2 cos()ω0t F.T. mω0 kT Sxx ()ω =−πδ2 ⎡ ()ωω00+δ(ω+ω)⎤ symmetric in ω! ⎣ Noise and Quantum Measurement⎦ 12 mω0 R. Schoelkopf Spectral Density of Quantum Oscillator - I 1 xtˆˆ()=+x(0c)os()ωω00t pˆ()0 sin()t mω0 2 pˆˆ()tp=−(0c)os()ω00txˆ()0mωωsin()0t

1 Ctxx ()=+xˆˆ()00x()cos()ω00t pˆ()0xˆ()0 sin()ω t mω0

but because ⎣⎡xp(0,) (0)⎦⎤=i

xp(00) ( ) − p(0)x(0) = i and px(00) ( ) = −≠i /20

Noise and Quantum Measurement 13 R. Schoelkopf Spectral Density of Quantum Oscillator - II

† using xxˆ = RMS (a+ a)

22 with xxRMS ==00ˆ 2mω0 at†/()= eiHt a†(0)e−iHt /= eitω0 a†(0)

2†itωω00−it † Ctxx ( )==xˆˆ(t)x(0() xRMS ea0)a(0)+e a(0)a(0)

2 +−itωω00it Ctxx ( )=+xRMS (nBE (ωω00)e ⎣⎦⎡⎤nBE ( )+1 e ) 1 where nBE ()ω0 = eω0 kT −1 is the Bose-Einstein occupation Sxω =+21π 2 ⎡⎤nω δω ω +⎡⎤nω +δω−ω xx ( ) RMS ⎣⎦BE ( 00) ( ) ⎣⎦BE ( 0) ( 0)

Noiseasymmetric and Quantum Measurement in frequency!! 14 R. Schoelkopf How to Make a Resistor - 1

Caldeira-Legget prescription: “Sum infinite number of oscillators to make continuum”

Caldeira and Leggett, Ann. Phys. 149, 374 (1983).

Noise and Quantum Measurement 15 R. Schoelkopf How to Make a Resistor - 2

Admittance = parallel sum of series resonances

L’s and C’s chosen to give dense comb of frequencies and the correct value of impedance/admittance

Noise and Quantum Measurement 16 R. Schoelkopf How to Make a Resistor - 3 Transmission line = infinite LC ladder

L and C are constants (all same) = to the inductance and capacitance per unit length, calculated from electro/magneto-statics of the particular transmission line the “vacuum” or a perfect blackbody! Line needs to be infinite – no reflections/memory and infinite number of d.o.f. to make reservoir Noise and Quantum Measurement 17 R. Schoelkopf Quantum Noise of an Impedance

SV (ω ) 2Rω e[Z ] T ≠ 0 ω = kT T = 0

2RkT e[Z ]

0 ω The quantum fluctuation- Three limiting cases: dissipation relation: ω kT SkV = 2RTe[Z] 2Rω e[Z ] S ()ω = ω kT SZV = 2Rω e[ ] V 1− e−ω / kT ω −kT SV = 0 Noise and Quantum Measurement 18 R. Schoelkopf Symmetrized and Antisymmetrized Noise 2ωR 1 S ()ω = n = V 1− e−ω / kT e ω / kT −1 stim. emission spont. emission ω > 0: SnV (+ω) = 2Rω ( +1) absorption ω < 0: SnV (−ω) = 2Rω Symmetrized noise spectrum: S SSVV()ω = (+ωω)+−SV( )∼22ω( n+1)R SkS ()ω = 2Rωωcoth⎡ /2T⎤ Callen and Welton, V ⎣ ⎦ Phys. Rev. 83, 34 (1951) Anti-Symmetrized noise spectrum: A SSVV()ω = (+ωω)−−SV( )∼2Rωdissipation (T indep.!) Noise and Quantum Measurement 19 R. Schoelkopf Symmetrized (One-Sided) Johnson Noise

SRV = 2ω

S SkV /4 TR 2Rω coth⎣⎡ ω /2kT ⎦⎤

SkV = 4 TR

ω0 kT S S ()ω → 0 S S (ω →∞) ω TT==V T ==V RJ 4kR Q 42kR k “energy per mode = ½ photon” Noise and Quantum Measurement 20 R. Schoelkopf Experiments On Quantum Johnson Noise

Method: measure low-freq. noise of resistively-shunted JJ With zero-point 10-21

I = I sinφ

C /Hz 2 A

10-22 w/out zero-point Inferred Johnson noise

1010 Frequency 1012 Rectified noise from ω = 2/eVDC JJ “mixes down” noise from THz frequencies to audio Koch, van Harlingen, and Clarke, PRL 47, 1216 (1981)

Noise and Quantum Measurement 21 R. Schoelkopf Experiments On Quantum Johnson Noise Work by Bernie Yurke et al. at Bell Labs Josephson parametric amplifier: observed zero-point part 19 GHz and 30 mK of waveguide’s noise (and then squeezed it!) 4.5

fit to coth! Noise power

2 Tamp = 0.45K = hν/2k quantum limited amplifier! 01Temperature K Noise and Quantum Measurement 22 Movshovich et al.,R. Schoelkopf PRL 65, 1419 (1990) Shot Noise – “Classically”

what’s up here? n D I ∼ qDn

Incident “current” of particles Poisson-distributed fluctuations

Barrier w/ finite “white” noise with trans. probability

SqI = 2 I

Noise and Quantum Measurement 23 R. Schoelkopf Shot Noise is Quantum Noise Einstein, 1909: Energy fluctuations of thermal radiation “Zur gegenwartigen Stand des Strahlungsproblems,” Phys. Zs. 10 185 (1909) 23 2 ⎡⎤π c 2 ()∆=EV⎢⎥ωρω()+2 ρ()ω dω ⎣ ω ⎦ particle term = shot noise! wave term first appearance of wave-particle complementarity?

† Can show that “particle term” is a consequence of ⎣⎡aa,1⎦⎤ = (see Milloni, “The Quantum Vacuum,” Academic Press, 1994) 2 −1 ∆=na2†aa†a−a†a nn==(eω / kT −1) n+1 †† †2 n =+aa(1a )a−aa Pnn =+/1(n)

†† † † 2 †† 2 =+aaaa aa −aa aaaa =−∑n(1n )Pn =2n 22 ∆=nn+n Noise and Quantum Measurement 24 R. Schoelkopf Conduction in Tunnel Junctions I

V

G I =−ff(1 )dE LR→ e ∫ L R G I =−ff(1 )dE RL→ e ∫ R L Difference gives current:

Fermi functions I = IILR→→−=RL GV Assume: Tunneling amplitudes and D.O.S. independent of energy Conductance (G) Noise and Quantum Measurement is constant 25 Fermi distributionR. Schoelkopf of electrons Non-Equilibrium Noise of a Tunnel Junction

(Zero-frequency limit) Sum gives noise:

SfI ()= 2e(ILR→→+ IRL)

⎛⎞eV SfI ()= 2eIcoth⎜⎟ ⎝⎠2kTB I = VR/

*D. Rogovin and D.J. Scalpino, Ann Phys. 86,1 (1974)

Noise and Quantum Measurement 26 R. Schoelkopf Non-Equilibrium Fluctuation Dissipation Theorem

2eI Shot Noise

Transition Region 4kBTJohnson Noise eV~kBT R

⎛⎞eV SfI ()= 2eV/Rcoth⎜⎟ ⎝⎠2kTB

Noise and Quantum Measurement 27 R. Schoelkopf Noise Measurement of a Tunnel Junction

P 5µ SEM

Al-Al2O3-Al Junction

Measure symmetrized noise spectrum at ω < kT

Noise and Quantum Measurement 28 R. Schoelkopf Seeing is Believing

δ P = 1 P Bτ

High bandwidth measurements of noise 8 δ P B ~10 Hz, τ = 1 second =10−4 P Noise and Quantum Measurement 29 R. Schoelkopf Test of Nonequilibrium FDT Agreement over four decades in temperature

To 4 digits of precision

Noise and Quantum Measurement 30 R.L. Schoelkopf Spietz et al., Science 300, 1929 (2003) Comparison to Secondary Thermometers

Noise and Quantum Measurement 31 R. Schoelkopf Two-sided Shot Noise Spectrum (Quantum, non-equilibrium FDT) (ω + eV )/R ( ω − eV )/R S ()ω = + I 11− ee−+()ωωeV //kT − −−()eV kT

SI (ω)

V eV / 2/ω R

eI T = 0 −eV / ω = 0 ω Aguado & Kouwenhoven, PRL 84, 1986 (2000). Noise and Quantum Measurement 32 R. Schoelkopf Finite Frequency Shot Noise

Symmetrized Noise: SSsym = ()++ω S(−ω) don’t add powers! Shot noise

Quantum noise

Noise and Quantum Measurement 33 R. Schoelkopf Measurement of Shot Noise Spectrum Theory Expt.

Schoelkopf et al., PRL 78, 3370 (1998)

Noise and Quantum Measurement 34 R. Schoelkopf Shot Noise at 10 mK and 450 MHz

hkν /2T=

L. Spietz, in prep.

Noise and Quantum Measurement 35 R. Schoelkopf With An Ideal Amplifier and T=0

SI 2eI

ShI = 2 νG Quantum noise from source

Quantum noise Sh= 2 νG I added by amplifier V eV=-hν eV=hν

Noise and Quantum Measurement 36 R. Schoelkopf Summary – Lecture 1

• Quantizing an oscillator leads to quantum fluctuations present even at zero temperature.

• This noise has built in correlations that make it very different from any type of classical fluctuations, and these cannot be represented by a traditional spectral density- requires a “two-sided” spectral density.

• Quantum systems coupled to a non-classical noise source can distinguish classical and quantum noise, and allow us to measure the full density – next lecture!

Noise and Quantum Measurement 37 R. Schoelkopf