PRL 114, 126801 (2015)
PHYSICAL
REVIEW
week ending 27 MARCH 2015
LETTERS
Fluctuation-Dissipation Relations of a Tunnel Junction Driven by a Quantum Circuit O. Parlavecchio,1 C. Altimiras,1' J.-R. Souquet,2' ’ P. Simon,2 I. Safi,2 P. Joyez,1 D. Vion,1 P. Roche,1 D. Esteve,1 and F. Portier1* 'SPEC (UMR 3680 CEA-CNRS), CEA Saclay, 91191 Gif-sur-Yvette, France ~Laboratoire de Physique des Solides, Universite Paris-Sud, 91405 Orsay, France (Received 28 October 2014; published 25 March 2015) We derive fluctuation-dissipation relations for a tunnel junction driven through a resonator displaying strong quantum fluctuations. We find that the fluctuation-dissipation relations derived for classical external drives hold, provided the effect of the circuit’s quantum fluctuations is incorporated into the modified nonlinear current voltage characteristics. We also demonstrate that all quantities measured under a time dependent bias can be reconstructed from their values measured under a dc bias using photoassisted tunneling relations. We confirm these predictions by implementing the circuit and measuring the dc current through the junction, its high frequency admittance, and its current noise at the frequency of the resonator. DOI: 10.1103/PhysRevLett. 114.126801
PACS numbers: 73.23.-b, 05.40.Ca, 42.50.Lc, 72.70.+m
The fluctuations of any physical system held at equilib rium are proportional to its dissipative linear response [1-4], This universal fluctuation-dissipation theorem (FDT) relates the fluctuations (noise) of any passive system to an easier linear response measurement. Moreover, measuring both quantities implements a primary thermometer, reaching metrological accuracy [5], The FDT provides a physical picture on the origin of macroscopic irreversibility: it shows that dissipation within a Hamiltonian system corresponds to the system’s ability to dissolve an incoming excitation within its internal degrees of freedom through the system’s fluctuations of both quantum and thermal origin. Turned into a rigorous formalism as in Ref. [6], such a picture provides an efficient way to model dissipation in macroscopic quantum systems. Despite these successes, the standard FDT [3] relates the fluctuations of a quantum system to its response to a classical drive, ignoring the quantum fluctua tions of the driving forces. In the particular case of a quantum electrical conductor connected to an electromagnetic envi ronment, these quantum fluctuations are known to trigger inelastic electron tunneling, yielding the nonlinear dc current-voltage characteristics I(Vic), an effect known as environmental or dynamical Coulomb blockade (DCB) [7], This raises the question of the existence of FDT relations for a quantum conductor coupled to an environment displaying strong quantum fluctuations. Such relations have been derived [8-10], but the bias was described as a timedependent classical voltage across the junction. Here instead, we explicitly include the quantum fluctuations of the voltage across a normal tunnel junction biased through the elementary building block of the description of a linear circuit, i.e., a harmonic oscillator, driven in a coherent state. We find that in this case, the junction’s response and current fluctuations can be recast in terms of the nonlinear 7(Vdc) curve. We thus extend the validity of expressions found for tunnel conductors driven by classical fields [8-17], We 0031-9007/15/114(12)/126801 (5)
probe these predictions by embedding a tunnel junction in a high impedance microwave resonator. The junction’s dc conductance, its finite frequency admittance, and its current fluctuations are found in good agreement with predictions. We consider [see Fig. 1(a)] a tunnel junction of tunnel conductance GT embedded at temperature T in an LC circuit with resonant frequency z/0 = 1/(2ns/LC) and characteristic impedance Z c = yjL/C, where C is the oscillator’s capacitance, and L its inductance. Harmonic oscillators have been considered as detectors for the current fluctuations of quantum conductors [18-21], but neglecting the backaction due to their quantum voltage fluctuations. We assume that the quantum average of the voltage across the junction reads Fdc + Vac cos(2nu(jt). We describe the resonator field by a “thermal coherent state” density matrix p — D(ia/2r)pTD ( ia /2 r y [22], where pT is the equilib rium density matrix at temperature T, D(ia/2r) = exp [ia(a + aT)/2r\ is the displacement operator corre sponding to an amplitude ia/2r, with a = eVac/ hv() and r = (nZc/ R K) (R k = h / e 1 = 25.8 k£2) characterizing the coupling between the oscillator and the tunnel junction. We evaluate [23] the time-dependent quantum average of the current I and current fluctuations’ spectral density Sr to lowest order in the tunnel coupling. Their time averages, 7 and Sr, are obtained as copies of the same quantity measured under dc bias, translated by the different har monics of u0 and weighted by Bessel functions, following photoassisted tunneling relations [9,11,15-17]:
126801-1
/(Vdc, a) = ; > > ( a ) 2/(V dc ~ khv0/e,0),
(1)
k
Si(v, Vdc, a) = J ^ J k(a)2S,(is, Vdc - khv0/e, 0). (2) k
© 2015 American Physical Society
PHYSICAL
PRL 114, 126801 (2015)
REVIEW
(a)
l
*0— I I — 1 c-cl »c
'G t
Vdc Vac =
week ending
27 MARCH 2015
density we recover the noise susceptibility derived in Ref. [32] using the Landauer-Biittiker formalism. For a dc bias a = 0, the current [7] and current noise [33] read
I
—omrv>-
LETTERS
a !f
(b)
/(V *) = ~ - [ Y * P(eVdc) - y * P ( - e V dc)}, Si(v> y dc) = 2Gj[y * P(eVdc - h v ) + y * P{ -h v - eVdc)], where y * P(E) = f de'y(e')P(E - e') with P(e) the prob ability density for a tunneling electron to emit the energy e in the form of photons into the impedance [7], with y{e) = j d e ’f { e ’)[ 1 - f ( e ’ + e)] = e/( \ - e~Elk*T), and / the Fermi function. Combining these expressions with Eqs. (l)-(3), we obtain a Kubo-like relation [4,8]: S[(-v, Vdc,a) - Sj(v, Vdc,a) = 2huReY(v, Vdc,a). The detailed balance property of y{E) and P(E) yields Vfr. V -r) e~I(Vi c - h u , a ) eI{Vic + hv,a) Vdc, a) - i _ e_fiieVic_hv) + e^ (_eVdc_to) _ j • (4)
FIG. 1 (color online), (a) Model system: a small conductance tunnel junction GT is embedded in a resonator of frequency vo = l /\ [ L C and characteristic impedance Zc = y /L /C . The system is connected to a dc (rf) voltage source Vdc (Vac) through a large inductance (capacitance), (b) Experimental setup: A normal tunnel junction, cooled at 20 mK by a dilution refrigerator, is connected to a 50 £2 line through a high-impedance 2/4resonator, whose inner conductor consists in a serial SQUID array. The resonator is connected to a bias-Tee, whose inductive port allows us to dc bias the junction and to measure its low frequency conductance. The rf port allows us to shine microwaves onto the resonator, and to measure the microwave signals emitted or reflected by the sample.
The time dependence of 7 and S, can also be retained to calculate their Fourier transform, allowing the derivation of the junction’s admittance Y(v, Vdc, a) from the current response to an infinitesimal drive in a small impedance additional fictitious mode at an arbitrary frequency v [23]. We find that Re[Y(i/, Vdc, a)] obeys a photoassisted tunnel ing formula analog to Eqs. (1) and (2) and keeps a structure known for classical drives [9,11,13,14]:
Re[?(z/,Vdc,a)]
7(Vdc + hv/e,a) - 7(Vdc - hv/e,a) 6 2hv ’ (3)
whereas its imaginary part follows from Kramers-Kronig relations. From a similar calculation for the noise spectral
Equation (4) is the main prediction we probe exper imentally: The results derived for classically dc biased tunnel elements [13,14,34] can be extended to quantum biasing circuits, provided one incorporates the effect of their quantum fluctuations into a “renormalized” nonlinear 7(Vdc) curve [10], even in the presence of a time-dependent drive [9] since it modifies equally both the current and noise. The crucial assumptions of our derivation are (i) that the quantum conductor is in the tunnel regime with (ii) a tunnel conductance small enough to have negligible impact on the density matrix of the system, which (iii) follows a detailed balance. Note that in Refs. [10,12-14], only the symmetrized spectral density of the current fluctuations [5/(1/, Vdc) + Sj(—v, Vdc)]/2 was considered having a sim ilar yet different expression. To probe the above predictions, we implement a circuit equivalent to Fig. 1(a) by embedding a high impedance tunnel junction in a microwave resonator with an imped ance high enough to significantly modify the tunnel junction transport properties [33,35]. We then measure the dc conductance, the finite frequency admittance and the finite frequency current noise of the junction as a function of its dc voltage bias, in the presence of an additional microwave tone inducing a coherent state in the resonator, using a cryogenic low noise amplifier. Our setup only gives access to the emission noise spectral density of the sample [33], unlike on-chip quantum detectors [36-38] such as Superconductor-Insulator-Superconductor detectors [39] which allow the measurement of the emission and absorp tion noise of their linear environment [40], Coupling two such devices via a low impedance circuits, displaying negligible quantum fluctuations, was used to measure their
126801-2
PRL 114, 126801 (2015)
PHYSICAL
REVIEW
emission noise [40], found in agreement with Eq. (4), and to probe their absorption noise [34] and admittance [41], Our experimental setup is represented in Fig. 1(b): a l OOxl OOnm2 tunnel junction with tunnel resistance G j { = 270 k£2 is embedded in a 2/4 coplanar resonator whose inner conductor is made of an array of identical and equally spaced A1/A10X/A1 SQUIDs. The SQUIDs’ Josephson inductance can be increased by an external magnetic flux, increasing the resonator’s characteristic impedance Zc from 1.2 kfl to 1.8 k£2, while decreasing its resonant frequency v0 from 6 GHz to 4 GHz [33,35], A 30 x 50 x 0.3 /tm3 gold patch is inserted between the tunnel junction and the SQUID array to evacuate the Joule power generated at the tunnel junction via electronphonon coupling. The chip is connected to a comme rcial 50 Q. matched bias tee. The low frequency path is used to bias the sample through a cold 13 MQ resistor, and to measure the tunnel junction dc conductance G(Vic,a) — d I /d V dc. The rf path is connected to a 20 dB directional coupler, which allows us to (i) shine microwave power on the tunnel junction through the weakly coupled port while (ii) measuring the microwave signals emitted or reflected by the sample via the well transmitted port, itself connected to a cryogenic amplifier with a ~2.5 K noise temperature in the 4-8 GHz band width. Two cascaded circulators (only one being repre sented) divert the backaction noise of the amplifier onto thermalized 50 £2 matched loads. The low temperature kBT
PHYSICAL
REVIEW
week ending 27 MARCH 2015
LETTERS
Fluctuation-Dissipation Relations of a Tunnel Junction Driven by a Quantum Circuit O. Parlavecchio,1 C. Altimiras,1' J.-R. Souquet,2' ’ P. Simon,2 I. Safi,2 P. Joyez,1 D. Vion,1 P. Roche,1 D. Esteve,1 and F. Portier1* 'SPEC (UMR 3680 CEA-CNRS), CEA Saclay, 91191 Gif-sur-Yvette, France ~Laboratoire de Physique des Solides, Universite Paris-Sud, 91405 Orsay, France (Received 28 October 2014; published 25 March 2015) We derive fluctuation-dissipation relations for a tunnel junction driven through a resonator displaying strong quantum fluctuations. We find that the fluctuation-dissipation relations derived for classical external drives hold, provided the effect of the circuit’s quantum fluctuations is incorporated into the modified nonlinear current voltage characteristics. We also demonstrate that all quantities measured under a time dependent bias can be reconstructed from their values measured under a dc bias using photoassisted tunneling relations. We confirm these predictions by implementing the circuit and measuring the dc current through the junction, its high frequency admittance, and its current noise at the frequency of the resonator. DOI: 10.1103/PhysRevLett. 114.126801
PACS numbers: 73.23.-b, 05.40.Ca, 42.50.Lc, 72.70.+m
The fluctuations of any physical system held at equilib rium are proportional to its dissipative linear response [1-4], This universal fluctuation-dissipation theorem (FDT) relates the fluctuations (noise) of any passive system to an easier linear response measurement. Moreover, measuring both quantities implements a primary thermometer, reaching metrological accuracy [5], The FDT provides a physical picture on the origin of macroscopic irreversibility: it shows that dissipation within a Hamiltonian system corresponds to the system’s ability to dissolve an incoming excitation within its internal degrees of freedom through the system’s fluctuations of both quantum and thermal origin. Turned into a rigorous formalism as in Ref. [6], such a picture provides an efficient way to model dissipation in macroscopic quantum systems. Despite these successes, the standard FDT [3] relates the fluctuations of a quantum system to its response to a classical drive, ignoring the quantum fluctua tions of the driving forces. In the particular case of a quantum electrical conductor connected to an electromagnetic envi ronment, these quantum fluctuations are known to trigger inelastic electron tunneling, yielding the nonlinear dc current-voltage characteristics I(Vic), an effect known as environmental or dynamical Coulomb blockade (DCB) [7], This raises the question of the existence of FDT relations for a quantum conductor coupled to an environment displaying strong quantum fluctuations. Such relations have been derived [8-10], but the bias was described as a timedependent classical voltage across the junction. Here instead, we explicitly include the quantum fluctuations of the voltage across a normal tunnel junction biased through the elementary building block of the description of a linear circuit, i.e., a harmonic oscillator, driven in a coherent state. We find that in this case, the junction’s response and current fluctuations can be recast in terms of the nonlinear 7(Vdc) curve. We thus extend the validity of expressions found for tunnel conductors driven by classical fields [8-17], We 0031-9007/15/114(12)/126801 (5)
probe these predictions by embedding a tunnel junction in a high impedance microwave resonator. The junction’s dc conductance, its finite frequency admittance, and its current fluctuations are found in good agreement with predictions. We consider [see Fig. 1(a)] a tunnel junction of tunnel conductance GT embedded at temperature T in an LC circuit with resonant frequency z/0 = 1/(2ns/LC) and characteristic impedance Z c = yjL/C, where C is the oscillator’s capacitance, and L its inductance. Harmonic oscillators have been considered as detectors for the current fluctuations of quantum conductors [18-21], but neglecting the backaction due to their quantum voltage fluctuations. We assume that the quantum average of the voltage across the junction reads Fdc + Vac cos(2nu(jt). We describe the resonator field by a “thermal coherent state” density matrix p — D(ia/2r)pTD ( ia /2 r y [22], where pT is the equilib rium density matrix at temperature T, D(ia/2r) = exp [ia(a + aT)/2r\ is the displacement operator corre sponding to an amplitude ia/2r, with a = eVac/ hv() and r = (nZc/ R K) (R k = h / e 1 = 25.8 k£2) characterizing the coupling between the oscillator and the tunnel junction. We evaluate [23] the time-dependent quantum average of the current I and current fluctuations’ spectral density Sr to lowest order in the tunnel coupling. Their time averages, 7 and Sr, are obtained as copies of the same quantity measured under dc bias, translated by the different har monics of u0 and weighted by Bessel functions, following photoassisted tunneling relations [9,11,15-17]:
126801-1
/(Vdc, a) = ; > > ( a ) 2/(V dc ~ khv0/e,0),
(1)
k
Si(v, Vdc, a) = J ^ J k(a)2S,(is, Vdc - khv0/e, 0). (2) k
© 2015 American Physical Society
PHYSICAL
PRL 114, 126801 (2015)
REVIEW
(a)
l
*0— I I — 1 c-cl »c
'G t
Vdc Vac =
week ending
27 MARCH 2015
density we recover the noise susceptibility derived in Ref. [32] using the Landauer-Biittiker formalism. For a dc bias a = 0, the current [7] and current noise [33] read
I
—omrv>-
LETTERS
a !f
(b)
/(V *) = ~ - [ Y * P(eVdc) - y * P ( - e V dc)}, Si(v> y dc) = 2Gj[y * P(eVdc - h v ) + y * P{ -h v - eVdc)], where y * P(E) = f de'y(e')P(E - e') with P(e) the prob ability density for a tunneling electron to emit the energy e in the form of photons into the impedance [7], with y{e) = j d e ’f { e ’)[ 1 - f ( e ’ + e)] = e/( \ - e~Elk*T), and / the Fermi function. Combining these expressions with Eqs. (l)-(3), we obtain a Kubo-like relation [4,8]: S[(-v, Vdc,a) - Sj(v, Vdc,a) = 2huReY(v, Vdc,a). The detailed balance property of y{E) and P(E) yields Vfr. V -r) e~I(Vi c - h u , a ) eI{Vic + hv,a) Vdc, a) - i _ e_fiieVic_hv) + e^ (_eVdc_to) _ j • (4)
FIG. 1 (color online), (a) Model system: a small conductance tunnel junction GT is embedded in a resonator of frequency vo = l /\ [ L C and characteristic impedance Zc = y /L /C . The system is connected to a dc (rf) voltage source Vdc (Vac) through a large inductance (capacitance), (b) Experimental setup: A normal tunnel junction, cooled at 20 mK by a dilution refrigerator, is connected to a 50 £2 line through a high-impedance 2/4resonator, whose inner conductor consists in a serial SQUID array. The resonator is connected to a bias-Tee, whose inductive port allows us to dc bias the junction and to measure its low frequency conductance. The rf port allows us to shine microwaves onto the resonator, and to measure the microwave signals emitted or reflected by the sample.
The time dependence of 7 and S, can also be retained to calculate their Fourier transform, allowing the derivation of the junction’s admittance Y(v, Vdc, a) from the current response to an infinitesimal drive in a small impedance additional fictitious mode at an arbitrary frequency v [23]. We find that Re[Y(i/, Vdc, a)] obeys a photoassisted tunnel ing formula analog to Eqs. (1) and (2) and keeps a structure known for classical drives [9,11,13,14]:
Re[?(z/,Vdc,a)]
7(Vdc + hv/e,a) - 7(Vdc - hv/e,a) 6 2hv ’ (3)
whereas its imaginary part follows from Kramers-Kronig relations. From a similar calculation for the noise spectral
Equation (4) is the main prediction we probe exper imentally: The results derived for classically dc biased tunnel elements [13,14,34] can be extended to quantum biasing circuits, provided one incorporates the effect of their quantum fluctuations into a “renormalized” nonlinear 7(Vdc) curve [10], even in the presence of a time-dependent drive [9] since it modifies equally both the current and noise. The crucial assumptions of our derivation are (i) that the quantum conductor is in the tunnel regime with (ii) a tunnel conductance small enough to have negligible impact on the density matrix of the system, which (iii) follows a detailed balance. Note that in Refs. [10,12-14], only the symmetrized spectral density of the current fluctuations [5/(1/, Vdc) + Sj(—v, Vdc)]/2 was considered having a sim ilar yet different expression. To probe the above predictions, we implement a circuit equivalent to Fig. 1(a) by embedding a high impedance tunnel junction in a microwave resonator with an imped ance high enough to significantly modify the tunnel junction transport properties [33,35]. We then measure the dc conductance, the finite frequency admittance and the finite frequency current noise of the junction as a function of its dc voltage bias, in the presence of an additional microwave tone inducing a coherent state in the resonator, using a cryogenic low noise amplifier. Our setup only gives access to the emission noise spectral density of the sample [33], unlike on-chip quantum detectors [36-38] such as Superconductor-Insulator-Superconductor detectors [39] which allow the measurement of the emission and absorp tion noise of their linear environment [40], Coupling two such devices via a low impedance circuits, displaying negligible quantum fluctuations, was used to measure their
126801-2
PRL 114, 126801 (2015)
PHYSICAL
REVIEW
emission noise [40], found in agreement with Eq. (4), and to probe their absorption noise [34] and admittance [41], Our experimental setup is represented in Fig. 1(b): a l OOxl OOnm2 tunnel junction with tunnel resistance G j { = 270 k£2 is embedded in a 2/4 coplanar resonator whose inner conductor is made of an array of identical and equally spaced A1/A10X/A1 SQUIDs. The SQUIDs’ Josephson inductance can be increased by an external magnetic flux, increasing the resonator’s characteristic impedance Zc from 1.2 kfl to 1.8 k£2, while decreasing its resonant frequency v0 from 6 GHz to 4 GHz [33,35], A 30 x 50 x 0.3 /tm3 gold patch is inserted between the tunnel junction and the SQUID array to evacuate the Joule power generated at the tunnel junction via electronphonon coupling. The chip is connected to a comme rcial 50 Q. matched bias tee. The low frequency path is used to bias the sample through a cold 13 MQ resistor, and to measure the tunnel junction dc conductance G(Vic,a) — d I /d V dc. The rf path is connected to a 20 dB directional coupler, which allows us to (i) shine microwave power on the tunnel junction through the weakly coupled port while (ii) measuring the microwave signals emitted or reflected by the sample via the well transmitted port, itself connected to a cryogenic amplifier with a ~2.5 K noise temperature in the 4-8 GHz band width. Two cascaded circulators (only one being repre sented) divert the backaction noise of the amplifier onto thermalized 50 £2 matched loads. The low temperature kBT
