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) 4 2 I(t) = δ 2 v(t) C ( 0 ) I ( t ) 0 2 -2 -4 100x10 3 Time average 0 20 40 60 80 time τ = δ + τ δ Correlation function: C ( V , ) I ( t ) I ( t ) 2 ω = δ ω δ − ω S ( V , ) I ( ) I ( ) Noise spectral density: 2
Current in a tunnel junction V = 0 I(t) Γ + time e - e - Γ - I eV >> k B T I(t) time V
Current fluctuations in a tunnel junction at low frequency eV δ = 2 I 2 eIB coth B=bandwidth 2 k T B << Equilibrium (Johnson) noise: 4 k TG if eV k T B B macroscopic, fluctuation- = S 2 dissipation theorem >> 2 eI if eV k T B Shot noise: discreteness of Noise spectral density in A 2 /Hz charge
Experiment S 2 ( ω =0,T=4.2K) Tunnel junction made by L. Spietz at Yale 250 T = 4.2 K ∆ f ~ 1 GHz 200 ) nA = eV k T ( Shot noise 150 B f -6 ∆ >> = x10 e S ( eV k T ) eI / 2 B 100 2 I δ 50 0 -200 -100 0 100 200 < Ι > ( μ A) Equilibrium noise -6 x10 = = S ( V 0 ) 2 k TG 2 B What about finite frequencies ?
Current noise / electromagnetic radiation Current / voltage fluctuations = fluctuating electromagnetic field = white light ! Average power in a bandwidth Δ f ~ intensity of light: 𝑄 = 𝑆 𝜀𝜀 2 = 𝑆𝑇 2 𝑔 Δ𝑔 = [ 𝑜 𝑔 + 1 2 ] ℎ𝑔 Noise = average photon number At equilibrium: Thermal (Johnson) noise = blackbody radiation !
How to measure S 2 ( ω ) ? 1) « Classical » detection with a linear amplifier δ V(t) band pass ω [ ] 2 ω = ω 2 = ω − ω + − ω ω ˆ ˆ ˆ ˆ S ( ) I ( ) I ( ) I ( ) I ( ) I ( ) / 2 2) « Quantum » detection with a photo-detector ω = − ω ω ˆ ˆ S em ( ) I ( ) I ( ) 2 δ V(t) band pass Photo-multiplier: absorbs photons ω
S 2 in the quantum regime ħω >k B T,eV Tunnel junction R=50 Ω T phonons = 22 mK T electrons = 27 mK 7.30 f = 5.5 - 6.5 GHz No photon emitted: 7.25 hf/k B = 290 mK Zero point fluctuations Ghf/e = 0.50 μ A S 2 (K) 7.20 eV= ħω 7.15 It is not possible to 7.10 separate the noise 7.05 of the amplifier from the ZPF ! -1.0 0.0 1.0 current (µA)
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? Quadratures? Optics Electronics
Experimental set-up Quantum regime :
Result ω 0 = 2 ω
Result ω 0 = ω
Generation of photon pairs ? Hamiltonian responsible for squeezing: 𝐼 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑏 2 + 𝑏 +2 Creation / destruction of pairs of photons Statistics of emitted photons ? Correlation between photons of different colors ?
Fluctuations (noise !) of light intensity (power) ? 𝑜 ~ 𝜀𝜀 2 𝜀𝑜 2 ~ 𝜀𝜀 4 Photon noise (fluctuations of the light intensity) = fourth moment of current fluctuations (noise of the noise)
Measurement of power fluctuations 𝜀𝑄 band pass f 𝜀𝑄 2 δ I(t) 𝑄 Square law detector 2 ) V out = low pass filter (V in
In terms of current 𝜀𝑄 𝜁 = � 𝜀𝜀 𝑔 1 𝜀𝜀 𝜁 − 𝑔 1 𝑒𝑔 1 band pass f 𝜀𝑄 2 δ I(t) 𝑄 = � 𝑒𝑔 2 𝜀𝜀 𝑔 2 𝜀𝜀 ( −𝑔 2 ) 𝜀𝑄 −𝜁 = � 𝜀𝜀 𝑔 2 𝜀𝜀 −𝜁 − 𝑔 2 𝑒𝑔 2
What is measured: 𝜀𝑄 2 ≈ 𝜀𝜀 𝑔 1 𝜀𝜀 𝜁 − 𝑔 1 𝜀𝜀 𝑔 2 𝜀𝜀 −𝜁 − 𝑔 2 ≈ 𝜀𝜀 𝑔 1 𝜀𝜀 −𝑔 𝜀𝜀 𝜁 − 𝑔 1 𝜀𝜀 −𝜁 + 𝑔 𝑔 2 = −𝑔 1 1 1 ≈ 𝑄 2 𝜀𝜀 4 = 3 𝜀𝜀 2 2 Dominated by gaussian contribution. 𝜀𝑄 2 = 𝑄 2 + 𝑄 One obtains for Gaussian noise: Identical to chaotic (thermal) light !
Correlation of power fluctuations at different frequencies: photon-photon correlations ? S 2 (f 1 ) f 1 𝜀𝑄 1 𝜀𝑄 2 δ I(t) f 2 S 2 (f 2 )
Experimental setup: 𝐻 2 = 𝜀𝑄 1 𝜀𝑄 2 + PHOTO-EXCITATION f 0 f 1 =4.3 GHz f 2 =7.3 GHz
G 2 vs excitation frequency f 1 -f 2 f 1 +f 2 (f 1 -f 2 )/2 f 0
G 2 vs. V dc , excitation at f 1 ±f 2
𝑄 1 𝑄 2 Normalized correlation: 2 = 𝑄 1 𝑄 2
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 = 𝑄 2 𝑄 2 = 1 + 𝜀𝑄 2 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 = 𝑜 2 𝑜 2 = 𝑞 2 /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 f 1 +f 2 : absorption of one photon at frequency f 1 +f 2 , emission of a pair (f 1 ,f 2 ), i.e. three wave mixing - Excitation at (f 1 +f 2 )/2: absorption of two photons, emission of a pair (f 1 ,f 2 ), i.e. four wave mixing - Excitation at f 1 -f 2 : absorption of one photon at frequency f 1 -f 2 , emission of a photon f 1 , absorption of a photon f 2
Excitation at f 1 +f 2
Photon-photon correlation
Interpretation: noise modulation • Noise at frequency f 1 modulated at f 2 -f 1 gives a sideband at f 2 and vice-versa • Noise at frequency f 1 modulated at f 2 +f 1 gives a sideband at -f 2 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 ?
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