C c V Det. V Emi. C Self R S,E

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1 Z( ) ( ) S I,Em (A /Hz) S V,Det (V /Hz) SUPPLEMENTARY INFORMATION: DETECTING NOISE WITH SHOT NOISE USING ON-CHIP PHOTON DETECTOR SUPPLEMENTARY FIGURES (a) V Emi C c V Det S I,Emi R E S I,SE C Self R S,E C Self R S,D R D r c r c r c r c (b) (c) 000 3,0x0-4,0x0-7,0x0-500,0x0-7,0x Frequency (Hz) 0,0 0, Frequency (Hz) Supplementary Figure : Experimental set up and transimpedance. (a) Schematic representation of the experimental set up for the PASN measurement including contact resistances. The coupling capacitance was designed to be C C ff. Ohmic contact resistances r c are 700 Ω. C self is the self capacitance to the ground. V Emi is the bias of the emitted line and V Det for the detected one. R E (R D) is the emitter (detector) resistance and R S,E (R S,D) is the series resistance in the emitter (detector) line. The noise source of the emitter (series resistance) is noted S I,Emi (S I,SE) (b) Calculated transimpedance as a function of the frequency considering the following parameters: R E = h/e, R SE = h/e, R D = h/e, R SD = h/4e, C C = ff, C self = 3 ff. (c) S V,Det(ω) and S I,Emi(ω) as a function of the frequency for R E = h/e, T e = 300 mk and V ds = mv.

2 Detection C SD T SD T det R SD V det PC D Supplementary Figure : Detection set up. Simplified electrical circuit in the high frequency limit 0 GHz seen by the detector. V det depends on the emitted shot noise. The detection part is surrounded by the dashed line. C SD is the self capacitance to the ground of the detector line. We note T det the effective temperature of the detector PC and T SD the one for the series resistance of the detector line.

3 I ph (pa) D D Supplementary Figure 3: Photocurrent measurement. In red dots, photocurrent measurement as a function of D D (series PC tuned on the first plateau) for D E tuned at 0.78 (series PC tuned on the second plateau). The emitter excitation is a sine function V amp. mv. The photocurrent measurement is realized before the PASN measurement and is used as a calibration to extract the different experimental parameters. Theoretical prediction is represented by a black solid line.

4 4 PC Contact = R PCI T e = R PCI Contact x Supplementary Figure 4: Schematic representation of the heat transport in a single PC. When a bias is applied on the PC, a power R P C I is injected on both side of the mesa. The heat equation can be solved using the Wiedemann-Franz law, assuming the contact being at the fridge temperature ridge. We note T e the electronic temperature at the PC.

5 5 PC serie PC Det Contact = R PC,LI T e,left = R PC,LI = R PC,RI T e,right = R PC,RI Contact T 0 Inner ohmic contact Supplementary Figure 5: Schematic representation of the heat transport in the detector line. The series PC is now tuned at transmission T =. In between, a floating ohmic contact with an unknown contact resistance assumed to be at the fridge temperature ridge. Regarding the detector PC we will consider the left (resp. right) electronic temperature noted T e,left (resp. T e,right ). The temperature inside the cavity is noted T 0.

6 6 SUPPLEMENTARY DISCUSSIONS Supplementary Discussion : transimpedance coupling the Emitter and Detector DEG circuits The transimpedance Z(ω) coupling the two DEG circuits is defined as S V,Det = Z(ω) S I,Emi with S V,Det being the voltage fluctuations seen by the detector line and S I,Emi the shot noise generated by the emitter PC. Z(ω) is equal to: Z(ω) = Z E(ω) + Z D(ω)+ jccω + jc CωZ D(ω) with C c the coupling capacitance, R E (R D ) the emitter (detector) resistance and Z D (ω), Z E (ω) (see in Fig. (a)) defined by: Z D (ω) = jc self ω + R SD+r c + R D +r c () Z E (ω) = jc self ω + R SE+r c + R E+r c (3) with R SE (R SD ) the series resistance in the emitter (detector) line and C self the self capacitance. The frequency dependence of the detection is therefore related to the frequency dependence of the transimpedance. Fig. (b) shows Z(ω) as a function of the frequency: current fluctuations up to 0 GHz can be detected. In the low temperature limit (k B T ω), the excess current noise spectral density generated by the PC emitter S I,Emi (V ds, ω) is : 4e h + ev ds + ω D E,n ( D E,n )( e ((ev ds+ ω)/k BT e) n ev ds + ω e (( ev ds+ ω)/k BT e) ω e ( ω)/kbte) ) (4) where T e is the electronic temperature and D E,n the transmission of the n-electronic mode of the PC emitter. Fig. (c) shows S V,Det (ω) and S I,Emi (ω) as a function of the frequency. Clearly, the bandwidth of the detection is limited by the transimpedance and from the emission point of view, we are in the low frequency limit ( ω ev ds ). At high bias, we also have to consider S I,SE (ω) due to the heating effect of the series resistance. () Supplementary Discussion : the distribution probability P(E) In the continuous limit (i.e. non periodic excitation) which is relevant for random potential fluctuations, the probability to create an electron hole pair of energy E is: P (E) = π + dτe ieτ/ e i(φ(τ) φ(0)) (5) where φ(τ) = e τ dt V det (t ). Since e i(φ(τ) φ(0)) e (φ(τ) φ(0)), we have to solve (φ(τ) φ(0)) = e τ 0 dt τ 0 dt (V det (t )V det (t ). In the high frequency regime ( 0 GHz), the electrical circuit can be simplified as represented in Fig.. We introduce a characteristic time τ c : τ c = R Eq C self (6) with R Eq composed of R SD and R D in parallel. A good approximation of the environment noise temperature T E seen by the detector is: T E Z(ω 0) (S I,Emi + S I,SE ) R eq 4k B (7)

7 with ω 0 the resonant frequency of Z(ω). We are interested in the voltage fluctuations V det (t) seen by the PC detector. We have the relation V det (t)v det (0) = kbt E C self e t/τc. Then, we can show that e i(φ(τ) φ(0)) is equal to: e i(φ(τ) φ(0)) = e e k B T E C τ self c ( τ τc +e τ /τc ) To discuss this expression, we need to introduce the parameter λ= e k BT E C self τc. When λ >, one recovers the classical voltage fluctuation probability for a RC circuit at temperature TE : P (E) = πkb TE e /(C self ) e while for the lower temperature λ one enters a quantum regime, with: P (E) = 7 (8) E k B T E e /(C self ) (9) (πk B T α) + (E/k B T α ) (0) where T α = π ReqT E h/e. Typically in this experiment λ. The classical limit being easier to treat and still a good approximation, it will be considered in the following. Supplementary Discussion 3: the photocurrent In the continuous limit, the photocurrent is given by: I ph = e dɛ( f h ɛ )( D D ɛ ) For P(E) in the classical limit, we get the expression: E P (E) () I ph = e h ( D D ɛ ) E F k B T E e C self () To obtain ( DD ɛ ) EF, one considers the saddle point model of a PC where the transmission can be written as D D,n (V g ) = /(+e π(v0,n Vg)/Vg,n ) where V g,n defines the shape of the saddle potential []. The lever arm ( = ɛ/ V g ) is extracted from transconductance (dg/dv g ) measurements 0.0e. We finally obtain for I ph : I ph = e h k B T E e C self n π V g,n D D,n ( D D,n ) (3) In figure 3, we compare this theoretical prediction to photocurrent measurements. C c 0.9 ff and C self 3 ff being fixed by the geometry of the device, we extract 0.04e (independent measurements extracted from differential PC conductance versus gate and bias voltages gives 0.0e; for these measurements, the series PC are opened and the electrostatic environment is slightly different). Supplementary Discussion 4: Photon assisted shot noise The detected PASN low frequency shot noise S PASN I S PASN I is given by: = e h [4k BT e DD + D D ( D D ) EP (E) coth E k B T e ] (4) where P(E) is given by (9). To calculate S PASN I, we need to develop in Laurent series coth E k BT e (E/k B T e ) + E/(6k B T e ). We obtain the following expression: S PASN I = e h (4k BT e D D ) e T E 4e h D D( D D ) (5) (C self ) 6T e

8 8 Our measurements are presented in terms of the excess noise SI PASN off to the noise with V ds on. It finally gives: obtained by subtracting the noise with V ds S PASN I e T E = 4e h D D( D D ) (6) (C self ) 6T e Supplementary Discussion 5: heating effect single PC configuration At low temperature, electrons in the two dimensional electron gas are hardly thermalized by the phonons since typical sample lengths are very small compared to electron-phonon temperature relaxation length. A temperature gradient between the PC and the contacts (assumed to be thermalized at the base temperature of the fridge ridge ) will therefore appear. Combining the Wiedemann-Franz law and the Joule heating, the problem can be exactly solved. We note T (x) the electronic temperature at position x. The injected power by the PC on each side of the mesa is equal to = RPCI (see Fig. 4). The heat equation can be written: and T (x) j (x) = κ x (x) x (7) = ρ mesa I (8) with κ the thermal conductivity of the mesa, R PC the resistance of the PC and ρ mesa the linear resistance of the mesa. The Wiedemann-Franz law enables us to relate κ, the conductivity of the mesa σ and T : κ σt = π 3 [k B e ] (9) We introduce T e the electronic temperature at the PC. Integrating over the length of the right mesa we get: T e T f = 4 G PC R m π VDS[ + R mg PC ] (0) where R m is the total resistance of the mesa + contact resistance. We assume the contact resistance to be the same on the right and left of the PC. Therefore the total conductance of the right and left mesa + contact in parallel is equal to G m = 4/R m. We finally obtain: Since G PC G m, we have: with L = π k B 3e. T e = T f + 4 π G PC T e = T f G m [ + G PC ]( ev DS ) () G m k B + G PC VDS () L G m Series PC configuration We now discuss the electron heating in our experiment, where two PCs are in series with in between an inner ohmic contact. As depicted in Fig. 5, because of the series PC, we introduce the left electronic temperature of the PC noted T e,left, the right one T e,right, in between the two PCs T 0 and the inner ohmic contact at. A similar approach to the one described in the single PC configuration gives: L (Te,right ) = G PC,RVR (R m,r ) (3)

9 with G PC,R the conductance of the detector PC, V R the applied bias on the right PC and R m,r the resistance of the right mesa+contact resistance. For T e,left : 9 L (Te,left ) = G PC,LVL (R m,l ) (4) with G PC,L the conductance of the serie PC (tuned on a plateau), V L the applied bias on the left PC and R m,l the resistance of the left mesa+contact resistance. For T 0 the temperature in between the PCs, we write: L (T0 Tin ) = G PC,LVR + G PC,RVL (R c,in ) (5) with R c,in the contact resistance of the inner ohmic contact. The electronic temperature of the left PC is given by the average of the left temperature T e,left and the temperature in between the two PCs T 0, (T e,left +T 0 )/. From our measurements we extract R m,l =R m,r = 400 Ω and R c,in = 500 Ω. The inner ohmic contact resistance being much smaller this difference is not surprising. [] Y.M. Blanter and M. Buttiker. Phys. Rep., 336: 66, 000. [] M. Büttiker, Phys. Rev. B 4, 7906(R), 990.