Electron shot noise is quantum light Gabriel Gasse, Jean-Charles - - PowerPoint PPT Presentation

Electron shot noise is quantum light

Gabriel Gasse, Jean-Charles Forgues, Fatou Bintou Sane, Christian Lupien & BR, Sherbrooke, Canada Julien Gabelli, Orsay, France Samples: Lafe Spietz, NIST, Boulder; Karl Thibault, Sherbrooke

Fluctuations (noise)

) ( ) ( ) , ( ) ( ) ( ) , (

2 2

ω δ ω δ ω δ τ δ τ − = + = I I V S t I t I V C

Correlation function: Noise spectral density:

Time average

4 2

• 2
• 4

v(t) 100x103 80 60 40 20 time

I(t)

) ( ) (

2 2

t I C δ =

Current in a tunnel junction

e-

Γ+ Γ-

e-

V I

time I(t) time I(t)

V = 0 eV >> kBT

Current fluctuations in a tunnel junction at low frequency

        = T k eV eIB I

B

2 coth 2

2

δ      >> << = T k eV eI T k eV TG k S

B B B

if 2 if 4

2

Equilibrium (Johnson) noise: macroscopic, fluctuation- dissipation theorem Shot noise: discreteness of charge Noise spectral density in A2/Hz B=bandwidth

Experiment S2(ω=0,T=4.2K)

250 200 150 100 50 x10

• 6
• 200
• 100

100 200 x10

• 6

T = 4.2 K ∆f ~ 1 GHz

< Ι > (μA) ) nA ( /

2

f e I ∆ δ

Tunnel junction made by L. Spietz at Yale

T k eV

B

=

Equilibrium noise

TG k V S

B

2 ) (

2

= =

What about finite frequencies ?

eI T k eV S

B

= >> ) (

2

Shot noise

Current noise / electromagnetic radiation

Current / voltage fluctuations = fluctuating electromagnetic field = white light ! Average power in a bandwidth Δf ~ intensity of light:

𝑄 = 𝑆 𝜀𝜀2 = 𝑆𝑇2 𝑔 Δ𝑔 = [𝑜 𝑔 + 1

2]ℎ𝑔

Noise = average photon number

At equilibrium: Thermal (Johnson) noise = blackbody radiation !

How to measure S2(ω) ?

δV(t)

band pass ω

1) « Classical » detection with a linear amplifier 2) « Quantum » detection with a photo-detector δV(t)

band pass ω

Photo-multiplier: absorbs photons

[ ] 2

/ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ˆ ) ( ) (

2 2

ω ω ω ω ω ω I I I I I S − + − = = ) ( ˆ ) ( ˆ ) (

2

ω ω ω I I S em − =

7.30 7.25 7.20 7.15 7.10 7.05 S2 (K)

• 1.0

0.0 1.0 current (µA)

S2 in the quantum regime ħω>kBT,eV

eV=ħω Tunnel junction R=50Ω No photon emitted: Zero point fluctuations Tphonons= 22 mK T

electrons= 27 mK

f = 5.5 - 6.5 GHz hf/kB = 290 mK Ghf/e = 0.50μA

It is not possible to separate the noise

• f the amplifier from

the ZPF !

Can one generate a quantum state of light with a circuit ?

• A classical current generates a coherent state (Glauber)
• A classical current in a non-linear medium can generate

a quantum state with the help of a non-linear component Ex: parametric amplifier made of Josephson junction(s)

• What about a quantum current ?

Ex: a mesoscopic sample placed at very low temperature

Source of non-linearity= electron charge !

Squeezing ?

Heisenberg principle :

Current squeezing?

Optics Electronics Quadratures?

Experimental set-up

Quantum regime :

Result ω0 = 2ω

Result ω0 = ω

Generation of photon pairs ?

Hamiltonian responsible for squeezing:

𝐼𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑏2 + 𝑏+2

Creation / destruction of pairs of photons Statistics of emitted photons ? Correlation between photons of different colors ?

Fluctuations (noise !) of light intensity (power) ? Photon noise (fluctuations of the light intensity) = fourth moment of current fluctuations (noise of the noise) 𝑜 ~ 𝜀𝜀2 𝜀𝑜2 ~ 𝜀𝜀4

Measurement of power fluctuations

δI(t)

band pass f

𝜀𝑄2 𝑄 𝜀𝑄

Square law detector Vout= low pass filter (Vin

2)

In terms of current

δI(t)

band pass f

𝜀𝑄2

𝜀𝑄 𝜁 = 𝜀𝜀 𝑔

1 𝜀𝜀 𝜁 − 𝑔 1 𝑒𝑔 1

𝑄 = 𝑒𝑔

2 𝜀𝜀 𝑔 2 𝜀𝜀(−𝑔 2)

𝜀𝑄 −𝜁 = 𝜀𝜀 𝑔

2 𝜀𝜀 −𝜁 − 𝑔 2 𝑒𝑔 2

What is measured:

𝜀𝑄2 ≈ 𝜀𝜀 𝑔

1 𝜀𝜀 𝜁 − 𝑔 1 𝜀𝜀 𝑔 2 𝜀𝜀 −𝜁 − 𝑔 2

≈ 𝜀𝜀 𝑔

1 𝜀𝜀 −𝑔 1

𝜀𝜀 𝜁 − 𝑔

1 𝜀𝜀 −𝜁 + 𝑔 1

≈ 𝑄2

𝑔

2 = −𝑔 1

Dominated by gaussian contribution. One obtains for Gaussian noise:

𝜀𝜀4 = 3 𝜀𝜀2 2 𝜀𝑄2 = 𝑄2 + 𝑄 Identical to chaotic (thermal) light !

Correlation of power fluctuations at different frequencies: photon-photon correlations ?

δI(t) S2(f2)

f1

𝜀𝑄

1𝜀𝑄2

S2(f1)

f2

Experimental setup: 𝐻2= 𝜀𝑄

1𝜀𝑄2

f1=4.3 GHz f2=7.3 GHz + PHOTO-EXCITATION f0

G2 vs excitation frequency f1+f2 f1-f2 (f1-f2)/2

f0

G2 vs. Vdc, excitation at f1±f2

Normalized correlation: 𝑕2 =

𝑄

1𝑄2

𝑄

1 𝑄2

Classical, correlated fluctuations

• vs. photon pairs

Toy model: a light bulb that flickers: perfect correlation between any two colors 𝑄 𝑢 = 𝑄 + 𝜀𝑄(𝑢) 𝑕2 = 𝑄2 𝑄 2 = 1 + 𝜀𝑄2 𝑄 2 = 1 + 𝜁 For ON/OFF with 50% duty cycle: 𝑕2 = 2 If we attenuate the light down to a less than 1 photon p=probabilty to detect 1 photon when ON 𝑕2 = 𝑜2 𝑜 2 = 𝑞2/2 (𝑞/2)2 = 2 For ON/OFF with x% duty cycle: 𝑕2 = 1/𝑦

The quantum regime: emission of photon pairs ?

• What happens when the mean number of photons

per measurement is smaller than one ?

• Excitation at f1+f2: absorption of one photon at

frequency f1+f2, emission of a pair (f1,f2), i.e. three wave mixing

• Excitation at (f1+f2)/2: absorption of two photons,

emission of a pair (f1,f2), i.e. four wave mixing

• Excitation at f1-f2: absorption of one photon at

frequency f1-f2, emission of a photon f1, absorption

• f a photon f2

Excitation at f1+f2

Photon-photon correlation

Interpretation: noise modulation

• Noise at frequency f1 modulated at f2-f1

gives a sideband at f2 and vice-versa

• Noise at frequency f1 modulated at f2+f1

gives a sideband at -f2 and vice-versa

• This works even at the single photon level !
• Squeezing: modulation of the zero point

fluctuations !

Conclusions

• Current fluctuations in a conductor can

generate quantum electromagnetic fields: photon pairs, squeezing

• Can we use this as an interesting source ?
• Entanglement ? Bell-like inequalities ?
• Can one trigger the emission of photons ?