Input-output theory of Coulomb Blockade

Quantum Properties of the radiation emitted by a conductor in the Coulomb Blockade Regime C. Mora,1 C. Altimiras,2 P. Joyez,2 and F. Portier2 ´ Laboratoire Pierre Aigrain, Ecole Normale Sup´erieure-PSL Research University, CNRS, Universit´e Pierre et Marie Curie-Sorbonne Universit´es, Universit´e Paris Diderot-Sorbonne Paris Cit´e, 24 rue Lhomond, 75231 Paris Cedex 05, France 2 Service de Physique de l’Etat Condens´e, CEA, CNRS, Universit´e Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France (Dated: March 30, 2017)

arXiv:1512.05812v2 [cond-mat.mes-hall] 28 Mar 2017

1

We present an input-output formalism describing a tunnel junction strongly coupled to its electromagnetic environment. We exploit it in order to investigate the dynamics of the radiation being emitted and scattered by the junction. We find that the non-linearity imprinted in the electronic transport by a properly designed environment generates strongly squeezed radiation. Our results show that the interaction between a quantum conductor and electromagnetic fields can be exploited as a resource to design simple sources of non-classical radiation. PACS numbers: 73.23.b, 72.70.+m, 73.23.Hk, 42.50.Lc, 42.50.Dv

I.

INTRODUCTION

Circuit quantum electrodynamics (cQED) describes at a quantum level the interaction between electromagnetic fields and artificial atoms implemented by quantum conductors such as Josephson junctions1 or quantum dots2–4 . This new architecture has triggered a number of pioneering experiments5–8 . However quantum conductors can also be continuously driven out-of-equilibrium by dc biases, giving rise to situations having no evident counterpart in atomic physics. Recent predictions and experiments relevant to these situations already point to interesting quantum electrodynamics effects. To cite a few, dc driven quantum conductors can be used as stochastic amplifying media giving rise to lasing (or masing) transitions in the field stored in RF cavities9–11 , as sources of sub-poissonian12–16 and squeezed radiation17–26 . Conversely, non-classical features of an incoming field may be revealed in the I(V ) curves of the conductor27 . The common underlying mechanism for these effects is the probabilistic transfer of discrete charge carriers through the quantum conductors. The resulting current fluctuations excite the surrounding electromagnetic environment. This coupling not only results in photon emission, but also modifies the transport properties of the conductor itself28–37 , an effect known as Dynamical Coulomb Blockade (DCB)38 . By increasing the impedance of the electromagnetic environment to values comparable to the resistance quantum RK = h/e2 ' 25.9 kΩ, the resulting strong coupling suppresses the transport at low voltage and low temperature for a normal conductor28–30,35–37 . So far most DCB studies focused on the conductor’s transport properties at low frequencies39,40 , or at higher frequencies41–43 but without describing photon radiation. Recent progress in microwave techniques opens new perspectives for investigating the quantum properties of the emitted radiation11,23,34,44 . Such radiative properties lie

out of the scope of the standard DCB approach which focus primarily on electrons. Yet one may expect the emission of non-classical radiation in a strong coupling regime as DCB generally induces important non-linearities. In the case of a Josephson junction, it has indeed been predicted45,46 that the emitted photons are strongly antibunched. Focusing on photons rather than on electronic variables is also a well-suited point of view if one is interested in what is actually measured in a quantum circuit experiment at GHz frequencies. The classical and quantum backaction of the measurement channels are thus treated equally with the system. The question of modeling measurement was initially addressed by Lesovik and Loosen47,48 by coupling a quantum conductor to LC resonator representing the measurement apparatus. The general point of view considered here is that the system is ultimately connected to a transmission line carrying the photon radiation and that the measurements are realized on these output photons. Recently, the standard inputoutput theory of quantum optics49,50 has been adapted to describe the field response of quantum conductors20,25,25 . In this paper, we consider the case of a normal tunnel junction arbitrarily coupled to radiation. As a first step, we reconsider the model of Lesovik and Loosen and extend it to include measurement backaction, i.e. DCB induced by the LC resonator on the quantum conductor. We find that the energy transfer between the conductor and the resonator conserves the same structure as in the absence of DCB, a combination of emission and absorption noises, but with current correlators simply dressed by DCB inelastic processes. Next, we develop a Hamiltonian approach which considers not only the electronic transport through the conductor but also the associated radiative dynamics via an input-output description49,50 . Considering the Lesovik&Loosen geometry complemented by a transmission line, we find that the energy transfer can be read out directly in the power of the output field. Finally, we exploit the Hamilto-

2

F

(a)

Q

Iqp

ID

F

(b)

FA Q

Iqp

IL

IL aout

ID

(c)

FB QB

F Iqp

FA Q ID

ain

ain

IL aout

conductor towards to the detector then accurately describes finite-frequency noise measurement. For a highimpendance and therefore strongly coupled LC resonator, the measurement backaction must be included in the formalism. We therefore consider a tunnel junction element shunted by an LC circuit of resonant frequency ω0 = p √ 1/ LC and characteristic impedance ZLC = L/C. We treat it in the standard DCB formulation38 assuming the LC detector to be always in a thermal state. Describing the relaxation dynamics of the detector will be the subject of the next section. The electrodynamic coupling gives rise to inelastic tunneling events, modifying the charge transfer dynamics of the junction. This physics is described by the Hamiltonian H0 = Hqp + HT + HLC , with X † X Hqp = l cl cl + r c†r cr (1) r

l

FIG. 1: Schematics of the considered circuits: (a) A tunnel junction with quasiparticle current Iˆqp is embedded in an LC circuit described by conjugated fields at the depicted node: the inductive magnetic flux Φ, and the capacitive influence charge Q. (b) A transmission line in series with the inductor damps the circuit by radiating outgoing modes aout , which can can detected with a matched detection chain (not shown). (c) A high resistance (R > RK ) RC circuit is connected to the junction. The resulting quantum flux fluctuations ΦB (t) are responsible for strong dynamical Coulomb blockade modifying the quasiparticle current Iˆqp , which can be efficiently collected in a matched detection band of the transmission line.

describing the left and right electrodes of the junction, HLC =

Q2 Φ2 + 2C 2L

(2)

the energy stored in the LC circuit, and HT = T + T † P with T = l,r τl,r c†l cr eieΦ/~ the tunnel coupling which simultaneously transfers quasiparticles from the right to the left electrode with amplitude τl,r , while displacing the influence charge of the capacitance by the electron charge eieΦ/~ Qe−ieΦ/~ = Q − e,

(3)

nian approach and propose a well-suited circuit geometry (Fig. 1(c)) in which a normal tunnel junction in the strong DCB regime can efficiently squeeze radiation. Squeezing in this scheme is induced by the dissipative bath provided by the tunnel junction51,52 , together with direct parametric down-conversion resulting from the ac modulation of the field reflection coefficient. The outline of the paper is the following: in Sec. II, we first discuss the DCB effect of a high-impedance LC circuit on a tunnel junction, and evaluate the power emitted in the LC circuit. In Sec. III, in order to account for energy dissipation in the system, we add a coupling to a dissipation line and set the stage for an input-output description of the junction-resonator circuit. In Sec. IV, extending the input-output analysis to a particular circuit where the tunnel junction is both strongly coupled to a DCB resistive circuit and weakly coupled to a transmission line for readout via a LC resonator, we show how strong squeezing emerges under a parametric ac driving.

HT is the minimal coupling of the junction to its circuit, neglecting the intrinsic electrodynamics of the electrodes15,53,54 beyond the mean-field approximation encompassed in the shunting capacitance C and the inductance L. From this Hamiltonian we obtain the quasiparticle current X † ie (5) Iˆqp = d(e (cl cl ))/dt = (T † − T ), ~

II.

and the inductive current IˆL = Φ/L which correctly compensate at the circuit node Iˆqp = IˆD + IˆL as required by gauge invariance. We now consider the radiative properties in the presence of both dc and ac bias, by evaluating the power emitted in the LC circuit

POWER EMITTED WITH STANDARD DCB

As discussed in the introduction, the measurement of finite-frequency current fluctuation can be accounted for by the weak coupling to a LC resonator modeling the detector47,48 . The power emitted by the quantum

corresponding to the node commutation relation [Φ, Q] = i~.

(4)

l

the displacement current ie IˆD = dQ/dt = (T † − T ) − Φ/L, ~

PLC = dHLC /dt =

Iˆqp Q + QIˆqp 2C

(6)

(7)

3 and computing its expectation value to lowest order in the tunnel coupling HT . To do so, we take the uncoupled boundary condition for the density matrix ρ = ρqp ⊗ρLC . The electrodes are initially at thermal equilibrium ρqp = e−βHqp /Zqp . The LC circuit is set in a displaced thermal state55,56

III.

CIRCUIT MODEL FOR DISSIPATION

We now go beyond the standard DCB approach and include a dissipative channel in the model, see Fig. 1(b), by adding a semi-infinite transmission line49,58 characterized by the impedance Z` . This not only provides (i) a precise mechanism for the damping of the LC circuit, but (8) ρLC = D[γ]e−βHLC /ZLC D† [γ] also (ii) a way to compute the properties of the radiation emitted by the junction into a linear detection circuit ieVac , with r = where the displacement vector γ = 2r~ω using an input-output approach. Our analysis thus ex0 p tends previous works26,59 by considering both quasipartiπe2 ZLC /h, gives the deterministic voltage cles and strong (DCB) backaction. We find in particular hV (t)i = T r(ρLC dΦ/dt) = Vac cos(ω0 t) + Vdc . (9) that the standard DCB formulation used to describe the circuit of Fig. 1(a) is justified in the limit case where The average power reads57 : the LC resonator leaks photons in the transmission line much faster than it exchanges photons with the tunnel junction, so that a separation of time scales occurs. This (1 + nB (~ω0 ))SIqp (ω0 , t) − nB (~ω0 )SIqp (−ω0 , t) corresponds to an impedance mismatch to the readout hPLC (t)i = 2 2C circuit RT  ZLC /Z` , wherepRT denotes the junction’s − hIˆqp (t)iVac cos(ω0 t), (10) tunnel resistance and Z` = `/c the transmission line characteristic impedance, with ` and c standing respecwhere nB (~ω0 ) is the bosonic thermal population of the tively for the lineic inductance and capacitance LC mode, The energy stored in the LC mode now reads Z Q2 (Φ − ΦA )2 HLC = + . (13) SIqp (ω, t) = dτ e−iωτ hIˆqp (t + τ )Iˆqp (t)i (11) 2C 2L is the power spectral density of quasiparticle current fluctuations56 (emission noise being here at positive frequency), and hIˆqp (t)i is the average quasiparticle current. In this expression both SIqp (ω0 , t) and hIqp (t)i have an explicit time-dependence due to the breaking of timetranslational invariance by the ac bias. The first term in Eq. (10) describes the power being emitted/absorbed by the junction via its emission/absorption current fluctuations. We recover the same structure as for a weakly coupled LC detector in the absence of AC driving47,48 , the important difference being that the tunneling dynamics encompassed in SIqp (ω0 , t) take into account both DCB and photonassisted tunneling effects. The second term describes the Joule power dissipated in the junction via its mean current response in phase with the ac excitation. Indeed, computing the power injected in the electrodes Pqp = dHqp /dt57 confirms that its average value is equal to the electrical power delivered by the dc source, minus that carried away by the LC circuit which, here, acts both as a power source and sink: hPqp (t)i = hIˆqp (t)iVdc − hPLC (t)i.

(12)

This perturbative approach, valid in the high tunneling resistance limit, considers flux (voltage) fluctuations arising only from the external circuit dynamics. Moreover, it implicitly assumes the presence of additional mechanisms not specified in the Hamiltonian H0 , restoring the initial state of the full system in between every tunneling event. In the following we explicitly consider such mechanism.

The dynamics of the transmission line is decribed by the Hamiltonian49 : # 2 Z +∞ "  1 ∂Φline (x) qline (x)2 Hline = dx , (14) + 2` ∂x 2c 0 where we introduced the conjugated variables Φline (x) and qline (x) describing the flux and charge density in the line, x being the position along the line. The bosonic operators describing the input ain,ω and output aout,ω fields enter the mode decomposition of Φline , r Z  ~Z` +∞ dω  √ ain,ω e−ikx + aout,ω eikx + h.c. . Φline (x) = 2 8π 0 ω (15) The input-output theory is obtained by considering the time-evolution equations in the interaction representation and imposing the coupling between the line and the LC oscillator via ΦA ≡ Φline (0). In the Heisenberg picture, the equations of motion for the LC circuit variables are ∂t Φ = ∂Q H = Q/C and ∂t Q = −∂Φ H. Combined, they couple the fields of the line, the flux of the LC circuit, C∂t2 Φ =

ΦA − Φ ˆH + Iqp , L

(16)

H and the current operator Iˆqp = eiHt Iˆqp e−iHt defined in Eq. (5) in the Heisenberg representation. No charge can accumulate on the node A, and we obtain a second equation

1 ∂Φline (0) ΦA − Φ = , ` ∂x L

(17)

4 from ∂ΦA H = 0. In a second step, the current operator is expanded in the linear response regime i H Iˆqp (t) = Iˆqp (t) + ~

Z

t

dt0 [HT (t0 ), Iˆqp (t)],

(18)

−∞

where Iˆqp (t) denotes the current evolved with the Hamiltonian unperturbed by HT . Solving for equations (16) and (17) in frequency domain, one arrives at aout,ω

∆∗ (ω) = ain,ω − iω02 ∆(ω)

r

H (ω) 2Z` Iˆqp , ~ω ∆(ω)

(19)

with ∆(ω) = ω 2 − ω02 + iωκ. κ = ZL` is the LC resonator damping rate due to the transmission line. The first term corresponds to the input field reflected by the resonator with a phase shift (time delay). The second term is the field emitted by the tunnel junction itself and carrying current noise fluctuations. As such Eq. (19) does not fully solve the circuit dynamics since the output field still H in the flux Φ dressing the tunneling enters the current Iˆqp operator T , calling for a self-consistent solution. ˜ = pHowever, writing the rescaled flux Φ(ω) 4 2ω/(~Z` ω0 )Φ(ω) as ˜ Φ(ω) =−

2ain,ω +

i ω

+

1 κ


q 2ωZ` ~

H (ω) Iˆqp

∆(ω)

,

(20)

we find that the flux fluctuations at frequency ω0 arising from the second term, which calls for the self-consistency, are negligible in the case of strong impedance mismatch SIqp (ω0 )L2 ω0 Z2 ∼ LC  1. ~Z` RT Z `

(21)

This can also be formulated in the time domain by inspecting the dynamics of the LC resonator: photons will leak much faster to the transmission line rather than to the tunnel junction25,27 , when κ  (ZLC /~)[SIqp (−ω0 )− SIqp (ω0 )] ∼ (ZLC /RT ) ω0 , yielding the same small parameter as Eq. (21). For circuits having this separation H of time scales, the approximation Iˆqp = 0 in Eq. (20) then reproduces standard DCB expressions37,38 (see also appendix A) for phase fluctuations across the tunnel junction characterized by the impedance seen by the junction ReZt (ω) = Z` ω04 /|∆(ω)|2 . The resulting flux Φ is finally H substituted in the current Iˆqp in Eq. (19), enabling the calculation of spectral properties of the emitted light even for very strong DCB backaction. The net power carried out by the transmission line reads PT L (t) =

hA†out (t)Aout (t)i R +∞

dω 2π

√

−

hA†in (t)Ain (t)i, −iωt

As a result, in the high quality factor limit ω0  κ, the expression of PT L (t) agrees precisely with the mean power hPLC (t)i derived in the previous section, see Eq. (10), with the same form as in the absence of DCB. It is worth noting that the products of the two terms in Eq. (19) mix the field and the junction dynamics giving rise to the Bose factors in Eq. (10), which vanish at low temperature ~ω0  kB T , and to the Joule power dissipated in the junction.

IV.

SQUEEZED RADIATION

We continue with the Hamiltonian approach and input-output framework to analyze a specific circuit, illustrated in Fig. 1(c), for which we will demonstrate that efficient squeezing in the output radiation can be realized. As strong DCB is responsible for non-linearities in transport, it is expected to also favor squeezing in the field emitted by the tunnel junction. The circuit of Fig. 1(b) is however not adapted to this effect. For strong impedance mismatch, the incoming mode is almost perfectly reflected by the junction, polluting the outgoing field with unsqueezed fluctuations. The impedancematched junction23,26 on the other hand shunts environment fluctuations, thereby reducing non-linearities and squeezing efficiency. We thus consider Fig. 1(c) where DCB and readout are spatially separated: on one side, the tunnel junction is coupled to a resistive circuit producing strong DCB, on the other side, a weakly coupled (i.e. low impedance) resonant circuit is used to probe the radiation emitted by the junction, providing a good impedance matching to the junction over a narrow bandwidth. In the following, we consider a situation where the classical bias at the junction consists in a dc voltage superimposed to an ac modulation at twice the resonator’s frequency: Vcl (t) = Vdc + Vac cos 2ω0 t.

A.

Noise and linear response

We assume the following hierarchy of resistances RT  R  RK  Z` , ZLC . The high resistance R imposes strong fluctuations for ΦB at the tunnel junction. An even larger RT is necessary to avoid shunting those fluctuations. In contrast to that, the resonant circuit produces weak flux fluctuations giving a negligible contribution to DCB effects and the flux Φ can be expanded to second order yielding the inductive coupling H ' H u − Φ Iˆqp − (eΦ/~)2 HTu /2

(23)

(22)

where Ain/out (t) = 0 ~ωain/out,ω e . The details of its calculation are given in appendix A taking the incoming field to be described by a displaced thermal state.

in the Hamiltonian in which the tunnel coupling T u is dressed by the flux ΦB only, HTu = T u + T u† ,

Tu =

X l,r

τl,r c†l cr e−ieΦB /~ .

(24)

5 H u governs the uncoupled evolutions of the DCB tunnel junction and weakly damped LC resonator. Hence, dynamics of the tunnel junction and DCB resistive circuit have been isolated, only weakly probed by the readout circuit. The input-output theory is constructed similarly to Ref.26 : time evolution is still described by Eqs. (16) and (17), the current operator is expanded in the flux Φ and in the linear reponse regime (tunnel limit) with Hamiltonian (23), H Iˆqp (t)

Z

B.

n

in the presence of the ac bias with frequency 2ω0 , where ˙ we introduced the quantum voltage V (t) = Φ(t). The admittances are related to the quasiparticles shot noise via a Kubo-like relation Z δSn (ω1 ) ω dω1 Yn (ω) = i . (27) 2π ~(ω − 2nω0 )(ω − ω1− )ω1+ Here

+

= ω1 ± i0 and δSn (ω1 ) = Sn (−ω1 ) − Sn (ω1 − 2nω0 ),

where the quasiparticle current noise spectral power of Eq. (11) is expanded in Fourier components SIqp (ω, t) =

X

Sn (ω)e−2inω0 t .

(28)

n

For a tunnel junction in the absence of DCB, the identity Sn (−ω1 ) = Sn (ω1 − 2nω0 ) for n 6= 0 implies that all admittances Yn6=0 vanish, indicating an absence of current rectification, as well as Y0 (ω) = 1/RT , recovering the tunnel junction bare resistance. In the general case, the imaginary parts of the admittances Yn (ω) all vanish for ω → 0 as there can be no phase shift with respect to the applied bias in the dc regime. The evaluation of the functions Sn (ω), and thus Yn (ω) is straightforward for a tunnel junction, using the standard P (E) theory42,56 . The resulting expressions are given in appendix B. The effect of the very strong DCB assumed here is encoded in the function38 P (E) = √

2 1 e−(E−Ec ) /(4Ec kB T ) , 4πEc kB T

in terms of the finite-temperature noise power spectrum of a simple tunnel junction 2 ~ω . RT e~ω/kB T − 1

(31)

(25)

This double expansion is in fact justified by two small parameters: ZLC /RK  1 ensures weak flux fluctuations for Φ and RK /RT  1 controls the linear regime for the tunneling current. Eq. (25) can be rewritten in a more suggestive linear response form X H Iˆqp (ω) = Iˆqp (ω) − Yn (ω)V (ω − 2nω0 ) (26)

ω1±

−∞

(0) Seq (ω) =

t

i dt0 Φ(t0 )[Iˆqp (t0 ), Iˆqp (t)] = Iˆqp (t) − ~ −∞ Z t  e 2 i + Φ(t) dt0 [HTu (t0 ), HTu (t)] . ~ ~ −∞

which gives the probability of the environment to absorb an energy E from a tunneling electron. Ec = e2 /(2CB ) is the charging energy. The equilibrium noise is expressed as a convolution product Z +∞ (0) dE P (E)Seq (ω + E/~), (30) Seq (ω) =

(29)

Photon correlators

The current response (26) is the missing piece needed to complete the input-output calculation with Eqs. (16) and (17). Assuming κ  ω0 , we obtain the boundary equation relating the input and output fields (ω + iκ+ ) aout,ω+ω0 = (ω + iκ− ) ain,ω+ω0 r  (32) ω0 Z` ˆ iY1  † Iqp,ω+ω0 − −i aout,ω0 −ω − a†in,ω0 −ω 2~ 2C where 2κ± = Y0 /C ± κ, with the notation Yn ≡ Yn (ω0 ). Y0 /C and κ are respectively the damping rates of the LC resonator to the tunnel junction and transmission line. One can check that in the absence of DCB, Y0 (ω) = 1/RT and Y1 vanishes, so that only dissipative squeezing occurs25,26 . On resonance and for impedance matched junction and resonator71 2 ZLC QZLC = = 1, RT Z` RT

(33)

or κ− = 0, where Q = ZLC /Z` is the quality factor of the resonator, aout,ω0 ∝ Iˆqp,ω0 and squeezing properties in the noise fluctuations of the junction are imprinted in the output field23 . With DCB and higher non-linearities, Y1 6= 0 and the last term in Eq. (32) introduces a parametric down-conversion mechanism on the input field, similarly to a parametric amplifier60,61 . In the general case, squeezing of the output field thus results from an interplay between these two mechanisms: squeezed radiation from the junction and direct parametric downconversion. Note that the first r.h.s. term in Eq. (32) corresponds to the reflected part of the input field, detrimental to squeezing. Eq. (32) is linear and can be easily inverted to express the output field aout,ω0 +ω in terms of the fields ain,ω0 +ω , a†in,ω0 −ω , Iˆqp,ω+ω0 and Iˆqp,ω−ω0 , in order to compute the output field correlations. The details of the calculation and the precise coefficients are given in appendix C. Squeezing is characterized by the power spectrum Sθ (ω) of the quadrature Xθ,ω = e−iθ aout,ω+ω0 + eiθ a†out,ω0 −ω , rotated by the angle θ compared to the quadrature in phase with the ac excitation, defined as h{Xθ,ω , Xθ,ω0 }i = 2Sθ (ω)2πδ(ω + ω 0 )

(34)

6

In summary, we formulated a general input-output theory that captures at the same level strong dynamical Coulomb blockade physics and the quantum properties of the emitted light. We showed how strong blockade amplifies quadrature squeezing in the emitted field under parametric excitation. We gave specific results for the case of a tunnel junction but the generality of our approach makes it applicable to other conductors. The cases of quantum dots2–4 and hybrid systems such as SIS junctions68 seem particularly appealing for the purpose of squeezing efficiency. The extension of our approach to cotunneling processes, where two electrons may cooperate to emit a photon69,70 , is another promising direction. We gratefully acknowledge discussions with the Quantronics Group, P. Roche and G. Johansson. The research leading to these results has received funding from the European Research Council under the European Union’s Programme for Research and Innovation (Horizon 2020) / ERC grant agreement number [639039], and support from the ANR AnPhoTeQ research contract.

24

Ec/ ℏω0

30

20

15

16 0

2

0

150

12

300

8 1

Vac Vdc

0 0

50

100

150

200

250

e Vdc/ ℏω0

e Vac/ ℏω0

3

4 0 300

(ZLC/RT) Q 40

3 2.5

30

2 20

1.5

θ

Ec/ ℏω0

At large QZLC /RT  1, the down-conversion mechanism dominates and a simple physical picture explains squeezing of the output field in analogy with flux-driven parametric amplifiers62,63 . For strong DCB, the 2ω0 ac modulation of the classical bias drives the junction from a nearly insulating state with a very large impedance to a conducting state with an impedance near RT . For QZLC  RT , the corresponding microwave reflection coefficient between the LC circuit and the junction oscillates between its extremal values, thereby implementing a parametric drive of the resonator. Microwave resonators with characteristic impedances in the range of a few kilo-Ohms have been reported using either kinetic64,65 or electromagnetic inductance66 have been reported, with quality factor Q well exceeding 105 (Ref. 65), allowing to reach values as high as a few 103 for the ratio QZLC /RT , while keeping RT in the 100 kΩ range to ensure that RT  RK . The optimum charging energy can be seen to increase with increasing QZLC /RT , but remain within realistic boundaries: Junctions with nanoscale cross-section67 can implement charging energies as large as Ec /h = 4 THz. Promising squeezing levels, well above 10 dB thus seem within experimental reach.

4

1

10

Ec

0.5

θ 0 0

50

100

150

200

0 300

250

(ZLC/RT) Q

T=0 kB T/ℏω0 = 0.025

0.6

kB T/ℏω0 = 0.05 SX1

which can be measured with RF heterodyning scheme e.g.11,24 . For each value of QZLC /RT , we determine numerically the optimal θ, Ec , Vdc and Vac which optimize squeezing. Sθ (ω) is always minimum at ω = 0, with a bandwidth of the order of κ. Results for the squeezed quadrature at ω = 0, noted SX1 , are displayed in Fig.2 at different temperatures. At Ec /~ω0 = 0, only dissipative squeezing survives and we recover the values in the absence of DCB23,25,26 , namely SX1 = 0.618 at zero temperature. Squeezing then improves with the ratio QZLC /RT , showing that DCB effects can significantly improve the squeezing efficiency.

0.4

0.2

0 0

50

100

150

200

250

300

(ZLC/RT) Q

FIG. 2: Lower Panel: From Eq. (32), squeezing of the output quadrature Xθ,ω = e−iθ aout,ω+ω0 + eiθ a†out,ω0 −ω , at the resonant frequency ω0 , as function of the impedance-matching parameter QZLC /RT for very strong resistive DCB. The continuous line corresponds to the zero temperature limit, whereas the dashed and dotted curves correspond to temperatures of respectively 7 mK and 14 mK for a resonance at ω0 /2π= 6 GHz. The charging energy Ec , the quadrature angle θ and the bias voltage at the junction are chosen to optimize (minimize) SX1 for each QZLC /RT . The middle and upper panels give the associated variations of EC , θ, and of Vdc and Vac at 7 mK.

7 Appendix A: DCB power radiated in the transmission line

We consider the closed form (19) as the starting point for evaluating the output field correlations. Neglecting H Iˆqp and using a thermal distribution for the input field, ha†in,ω ain,ω0 i = 2πnB (~ω)δ(ω − ω 0 ), we recover standard DCB expressions37,38 for phase fluctuations across the tunnel junction heieΦ(t)/~ e−ieΦ(0)/~ i ≡ eJ(t) +∞ dω ReZt (ω) × J(t) = 2 ω RK 0     βω (cos ωt − 1) − i sin ωt , × coth 2

(A1a)

in agreement with the prefactor in Eq. (10). SIqp (ω0 ) is interpreted as the emission noise corresponding to the power emitted by current fluctuations in the tunnel junction, even in the absence of the input field. It takes into account the influence of DCB on transport and can be written as a convolution SIqp (ω) = R the noise dεP (ε)SI0qp (ω − ε/~) between the energy distribution R +∞ function P (E) = h1 −∞ dt eJ(t)+iEt/~ and the noise in absence of DCB effect SI0qp (ω) =

Z

(A1b)

where β = 1/(kB T ) is the inverse temperature of the input field and ReZt (ω) = Z0 ω04 /|∆(ω)|2 is the real part of the impedance seen by the junction. Let us consider first the absence of an ac bias voltage. The power injected in the output field is expressed as Z +∞ hA†out (t)Aout (t)i = dω ~ωfout (ω) (A2)

1 X ~ω ± eVdc . RT ± eβ(~ω±eVdc ) − 1

(A7)

The second and third terms in Eq. (A3) are complex conjugate to each other. In contrast to the last term in Eq. (A3), it is now the second term in the expansion of Eq. (18) which contributes to the calculation. The first term in Eq. (18) creates electron-hole excitations across the junction and has a vanishing expectation. In the calculation, we make use of the following identity ha†in (t)[HT (t1 ),Iˆqp (t2 )]i = h[Iˆqp (t1 ), Iˆqp (t2 )]i   (A8) × ha†in (t)Φ(t2 )i − ha†in (t)Φ(t1 )i ,

0

where we introduced the photon-flux density from ha†out,ω aout,ω0 i = 2πfout (ω)δ(ω − ω 0 ). Defining the scat∗

(ω) tering phase eiθω = ∆∆(ω) and the normalization factor q ω02 0 Nω = i∆(ω) 2Z ~ω , the decomposition Eq. (19) of the input field produces four terms in the calculation of fout (ω)

ha†out,ω aout,ω0 i = ei(θω0 −θω ) ha†in,ω ain,ω0 i H H + Nω0 e−iθω ha†in,ω Iˆqp (ω 0 )i + Nω∗ eiθω0 hIˆqp (−ω)ain (ω 0 )i H H + Nω Nω∗0 hIˆqp (−ω)Iˆqp (ω 0 )i.

(A3) The first term is readily calculated ei(θω0 −θω ) ha†in,ω ain,ω0 i = 2πδ(ω − ω 0 )nB (ω).

(A4)

It is equal to the photon-flux ha†in,ω ain,ω0 i of the incoming field ain,ω . This term is subtracted in the net output power PLT defined in Eq. (22). The last term is written in terms of the power spectral density of quasiparticle current fluctuations SIqp (ω). It takes the form H H Nω Nω∗0 hIˆqp (−ω)Iˆqp (ω 0 )i = |Nω |2 SIqp (ω)2πδ(ω − ω 0 ), (A5) where only the first term is kept in the expansion (18) H of Iˆqp . If we use this result in the expression of the radiated power Eq. (A2), we arrive at the contribution SIqp (ω0 )/(2C) under the assumption of a sharp resonance κ  ω0 and the integral Z +∞ 1 Z0 ω02 = , (A6) dω ~ω|Nω |2 ' 2κ 2C 0

which is valid because the field ain , and therefore Φ(ω) = −i~Nω ain (ω), have Gaussian distributions. After a tedious but straightforward calculation, we find the additional contribution nB (ω)(SIqp (ω) − SIqp (−ω))|Nω |2 to fout (ω). This term vanishes at zero temperature. We then integrate over frequencies using the integral Eq. (A6) and obtain the correction to the power in the output field   nB (ω) SIqp (ω) − SIqp (−ω) . (A9) 2C To summarize, adding all contributions from Eq. (A3), the calculation of the net output power PLT in the inputoutput formalism coincides with the power hPLC (t)i received by the LC resonator in the standard DCB approach of Sec. II. The presence of an ac voltage can be included rigorously in the quantum formalism thanks to the displacement operator D acting on both frequencies ω0 and −ω0 . Quantum averages are then taken with respect to the displaced density operator ρ = De−βH D† /Z and the action on the input field is given by r Vac Z0 † D ain,ω D = ain,ω + (A10) 2ZLC 2~ω0 × [2πδ(ω − ω0 ) − 2πδ(ω + ω0 )] . Using the expression (20) of Φ, we obtain the shift in the flux D† Φ(ω)D = Φ(ω) +

iVac [2πδ(ω − ω0 ) − 2πδ(ω + ω0 )] , 2ω0 (A11)

8 ∂Φ(t) + Vac cos(ω0 t). Instead leading to D† ∂Φ(t) ∂t D = ∂t of dressing the density operator with D, it is possible to work with the undisplaced density operator e−βH /Z while all operators of the theory are dressed by D and D† . The input-output relation (19) is then transformed to r ∆∗ (ω) Vac Z0 aout,ω = ain,ω − ∆(ω) 2ZLC 2~ω0 r H (ω) 2Z0 Iˆqp 2 × [2πδ(ω − ω0 ) + 2πδ(ω + ω0 )] − iω0 , ~ω ∆(ω) (A12) H where the flux in the current Iˆqp (t) contains the clasVac sical evolution ω0 sin(ω0 t). Inserting this result into the expression of the net injected power PT L (t) retrieves Eq. (10) of Sec. II.

introducing the Bessel functions Jn , is used to derive the photo-assisted noise, or current-current correlators

     1 X eVac eVac Sn (ω) = Jm Jm+n 2 2~ω0 2~ω0 m∈Z   eVac × Seq (ω − eVdc /~ − 2mω0 ) + Jm 2~ω0    eVac Seq (ω + eVdc /~ + 2mω0 ) . ×Jm−n 2~ω0

(B2)

In particular, the effect of DCB fluctuations factorizes and is entirely encoded in the equilibrium noise function Seq (ω), see Eq. (30).

Appendix B: Current correlators

We consider a classical sinusoidal ac-bias applied to the tunnel junction Vcl (t) = Vdc + Vac cos 2ω0 t. The standard Fourier decomposition   X eVac eVac i 2~ω sin(2ω0 t) 0 e = Jm e2imω0 t , (B1) 2~ω0

Appendix C: Solution of the input-output equations

Injecting the linear response current expression (26) in Eqs. (16) and (17), we arrive at the coupled equation

m∈Z

      Z` Z` 1 − Z` Y0 (ω) Y0 (ω) Y0 (ω) 1 + Z` Y0 (ω) ω2 − − iω − + iω + ain,ω − ω 2 − aout,ω LC L C LC L C r r   X 2Z` ˆ 1 2nω0 ω − 2nω0 i Iqp (ω) − 1− Z` Yn (ω) +i = ain,ω−2nω0 + LC ~ω ω LC Z` C ω n∗ < 2ω 0 r      X ω − 2nω0 2nω0 1 2nω0 − ω † 1 −i − 1Z` Yn (ω) −i aout,ω−2nω0 + ain,2nω0 −ω LC Z` C ω LC Z` C ω n∗ > 2ω 0    1 2nω0 − ω † + +i aout,2nω0 −ω , LC Z` C

valid in the general case, where n∗ stands for n 6= 0. This lengthy expression can nevertheless be simplified in the limit of a high quality factor κ  ω0 . First, Eq. (17) can be written as Φ = − L` ∂x Φline (0) + ΦA where the second term is much smaller than the first one. Moreover, only the n = 0, 1 terms matter in Eq. (C1), and other values of n are suppressed in the limit κ/ω0 → 0, filtered by the LC resonator. n = 1 is the standard parametric term which couples frequencies ω0 and −ω0 . After a few straigthforward algebraic manipulations assuming κ  ω0 , we obtain the relation (32) given in the main text. It

(C1)

can also be written in matrix form,  M+

aout,ω+ω0 a†out,ω0 −ω



  ain,ω+ω0 = M− † ain,ω−ω0 r   ω0 Z` Iˆqp,ω+ω0 −i , 2~ −Iˆqp,ω−ω0

(C2)

with  M± =

 ω + iκ± iY1 /2C . −iY1∗ /2C −ω − iκ±

(C3)

9 e2iθ A∗2 (ω) with

The solution is aout,ω+ω0 = λω ain,ω+ω0 + µω a†in,ω0 −ω + αω Iˆqp,ω+ω0 + βω Iˆqp,ω−ω0 ,

(C4)


A1 (ω) = |λω |2 + |µω |2 [1 + 2nB (~ω)] + (|αω |2 + |βω |2 ) [S0 (ω0 ) + S0 (−ω0 )]
∗ ∗ + 2S1 (ω0 )Re αω βω + α−ω β−ω ,

with the frequency-dependent coefficients (κ+ − iω)(κ− − iω) − |Y1 /2C|2 Y1 κ µω = Dω 2C Dω r r ω0 Z` κ+ − iω ω0 Z` Y1 αω = − βω = , 2~ Dω 2~ 2CDω (C5)

λω =

and Dω = (κ+ −iω)2 −|Y1 /2C|2 . They satisfy the unitary identity |λω |2 − |µω |2 + (|αω |2 − |βω |2 ) 2~ω0 Y0 = 1 to preserve the commutation relation of the output field [aout,ω , a†out,ω0 ] = 2πδ(ω − ω 0 ). The quadrature power spectrum Sθ (ω) introduced in Eq. (34) indicates squeezing if Sθ (ω) < 1. Using Eq. (C4), we find Sθ (ω) = A1 (ω) + e−2iθ A2 (ω) +

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

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(C6a)

A2 (ω) = (λω µ−ω + λ−ω µω )[1/2 + nB (~ω)] 1 + (αω β−ω + α−ω βω ) [S0 (ω0 ) + S0 (−ω0 )] 2
+ |αω |2 + βω β−ω S1 (ω0 ).

(C6b)

Using the polar representation A2 = |A2 |eiϕ , one sees that the most squeezed quadrature is given by the angle θ = ϕ/2 + π/2, and Sθ=ϕ/2+π/2 = A1 − 2|A2 |.

(C7)

This is in particular the angle chosen in Fig. 2.

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11 Supplementary information for the article: Quantum Properties of the radiation emitted by a conductor in the Coulomb Blockade Regime

These supplementary information provides the full calculations allowing us to derive the expressions present in the main article. Cited equations not preceded by S− refer to the main text.

Appendix A: Standard P (E) approach 1.

Electromagnetic power

The power emitted into the LC circuit is defined by the power operator  1 ˆ Iqp Q + QIˆqp . 2C

PLC = dHLC /dt =

Here we compute the mean value of this operator up to lowest order in the coupling Hamiltonian HT . Making use 0 of the interaction picture of the power operator PLC (t) with respect to the uncoupled evolution Hqp + Henv , its time evolution up to first order in the tunnel coupling reads (here-after, the time-dependence of unlabeled operators are meant to be taken in the interaction picture): i ~

1 0 PLC (t) =PLC (t) +

Z

0 0 [HT (t + τ ), PLC (t)]dτ

−∞

Its quantum average over the initial states described in the article simplifies in: 1 hPLC (t)i

1 = − Re C

Z

0

−∞

2e hT (t + τ )T † (t)Q(t) − T † (t + τ )T (t)Q(t)i − hIˆqp (t + τ )Iˆqp (t)idτ, ~2

(A1)

2 where we identified hIˆqp (t + τ )Iˆqp (t)i = ~e 2 hT (t + τ )T † (t) + T † (t + τ )T (t)i. Equation (A1) contains the real part of the already known quasiparticle current time-correlator, and two new correlation functions which we will now compute. Since the initial states are uncoupled, the correlation functions factorize in terms of quasiparticle and environment P correlation functions. The quasiparticle correlation Θ(t + τ )Θ† (t), and Θ† (t + τ )Θ(t), where Θ = l,r τl,r c†l cr is the quasiparticle tunneling operator, are already well known. In the case of a particle-hole symmetric systems (which is the case of metallic tunnel junctions probed in the relevant range of energies much smaller than the barrier height and Fermi energy), one has: ! ~GT d π2 −2 πτ † † θ(τ ) = hΘ(t + τ )Θ (t)i = hΘ (t + τ )Θ(t)i = iπ~ δ(τ ) − 2 sinh ( ) , 2πe2 dt β ~β

where GT is the tunneling conductance, and β the inverse temperature. Therefore, we only need to compute the environment correlation functions: he±ieΦ(t+τ )/~ e∓ieΦ(t)/~ Q(t)i. For the sake of clarity, we first compute them for a dc bias, and then discuss how an ac bias modifies this first result.

a.

dc bias

q πZLC † The magnetic flux operator reads: Φ(t) = Vdc t + δΦ(t), with δΦ(t) = ~r e (a(t) + a (t)), where r = RQ with q L 2 † ZLC = C the mode impedance and RQ = h/e ' 25.8 kΩ the resistance quantum, and where a(t) and a (t) are correspondingly the mode annihilation and creation operators of the LC in the interaction picture. Therefore the operator e±ieδΦ(t)/~ and the products e±ieδΦ(t+τ )/~ e∓ieδΦ(t)/~ can be recast with displacement operators D[α] = † ∗ eαa −α a : e±ieδΦ(t)/~ = D[±ireiω0 t ], e±ieδΦ(t+τ )/~ e∓ieδΦ(t)/~ = e−ir

2

sin(ω0 τ )

D[±ireiω0 t (eiω0 τ − 1)].

12 From this, we can compute the correlation function eJ(τ ) = he±ieδΦ(t+τ )/~ e∓ieδΦ(t)/~ i appearing in P (E) theory, being its inverse Fourier transform: he

±ieδΦ(t+τ )/~ ∓ieδΦ(t)/~

e

i=

e−ir

2

sin(ω0 τ ) − 12 β~ω0 −r 2 (1−cos(ω0 τ ))

e

e

ZLC

X

e−βn~ω0 L0n (2r2 (1 − cos(ω0 τ )))

n

= eJ(τ ) where Lm n (x) are generalized Laguerre polynomials of order n, from which we recover the well known expression for a single mode    β~ω 
0 J(τ ) = r2 cos(ω0 τ ) − 1 coth − i sin(ω0 τ ) . 2 A similar calculation gives the result: e he±ieΦ(t+τ )/~ e∓ieΦ(t)/~ Q(t)i = ± e±ieVdc τ /~ 2

1 − eiω0 τ + eβ~ω0 e


−iω0 τ

ieβ~ω0 eJ(τ ) eJ(τ ) − ω0 r 2


0 ! .

(A2)

Inserting Eq. (A2) back in Eq. (A1), together with the already known expression for quasiparticle current fluctua2 J(τ ) cos(eVdc τ /~), the average electromagnetic power reads: tions hIˆqp (t + τ )Iˆqp (t)i = 2e ~2 θ(τ )e "
!# Z 0 β~ω0 J(τ ) 0 2
e ie 2e 1 dτ hPLC (eVdc )i = − 2 Re cos(eVdc τ /~)θ(τ ) eβ~ω0 e−iω0 τ − eiω0 τ eJ(τ ) − ~ C ω0 r2 −∞ 
1  1 + nB (~ω0 ) SIqp (Vdc , ω0 ) − nB (~ω0 )SIqp (Vdc , −ω0 ) = (A3) 2C which is the article Equation (1) specialized to the case of a dc bias. It can also take the following form: 1 hPLC (eVdc )i

Z

+∞

= −∞

SV (−ω)SIqp (Vdc , ω) dω , ~ω 2π

where we introduced the spectral density of voltage fluctuations of the LC circuit: Z +∞ ˙ + τ )Φ(t)i ˙ ˙ + τ )ihΦ(t)i. ˙ SV (ω) = dτ e−iωτ hΦ(t − hΦ(t −∞

This expression is more symmetric in the sense that the power emitted (absorbed) from the tunnel junction via its current fluctuations is proportional to the spectral density of emission (absorption) current fluctuations of the junction multiplied by the spectral density of absorption (emission) voltage fluctuations of the load electromagnetic environment. Note however that this expression is not fully symmetric: while the voltage fluctuations appear via their closed cumulant, the current fluctuations appear via their raw moment.

b.

ac bias

We now take the boundary condition described in the article, ρLC (t → −∞) = D[γ]

e−βHLC † D [γ], ZLC

p describing a thermal field being displaced by a ”classical” source. The displacement vector γ = iVac C/(2~ω0 ) gives
˙ rise to a deterministic time-dependent ac voltage: hV (t)i = T r ρLC (t → −∞)Φ(t) = Vdc + Vac cos(ω0 t), without perturbing the quantum and thermal voltage fluctuations. Again, we exploit the properties of displacement operators in order to get:  eV τ
 2e2 eVac dc hIˆqp (t + τ )Iˆqp (t)i = 2 θ(τ )eJ(τ ) cos + sin(ω0 (t + τ ) − sin(ω0 t) ~ ~ ~ω0

13 and heieΦ(t+τ )/~ e−ieΦ(t)/~ Q(t) − e−ieΦ(t+τ )/~ eieΦ(t)/~ Q(t)i =
0 !  eV τ

J(τ ) ieβ~ω0 eJ(τ ) eVac dc iω0 τ β~ω0 −iω0 τ e e cos + sin(ω0 (t + τ ) − sin(ω0 t) 1−e +e e − ~ ~ω0 ω0 r2  eV τ
 eVac dc + 2i sin + sin(ω0 (t + τ ) − sin(ω0 t) θ(τ )eJ(τ ) CVac cos(ω0 t), ~ ~ω0 From which we obtain the first equation of the article: 
1  1 1 + nB (~ω0 ) SI (ω0 , t) − nB (~ω0 )SI (−ω0 , t) − hIˆqp (t)iVac cos(ω0 t) 2C Z +∞ SV (−ω)SIqp (ω, t) dω 1 − hIˆqp (t)iVac cos(ω0 t). = ~ω 2π −∞

1 hPLC (t)i =

Contrary to the stationary case, where power is exchanged only via the current and voltage fluctuations of the circuit, now the junction can also dissipate some energy initially contained in the LC circuit via the average time-dependent current response, as stressed by the last equality.

2.

Joule power

We define now the power injected within the electrodes: Pqp =

iX i ∗ † [H0 , Hqp ] = − (l − r )τl,r c†l cr eieΦ/~ + (r − l )τl,r cr cl e−ieΦ/~ , ~ ~ l,r

and we expand its time evolution to first order in the tunnel coupling: 1 0 Pqp (t) = Pqp (t) +

2 Re ~2

Z

0

X T † (t + τ )(l − r )τl,r c†l cr ei(l −r )t/~ eieΦ(t)/~

−∞ l,r

∗ † − T (t + τ )(l − r )τl,r cr cl e−i(l −r )t/~ e−ieΦ(t)/~ dτ.

Again, the evaluation of this operator with the uncoupled boundary conditions factorizes in quasiparticle and environment correlation functions. The
environment correlation functions are just the standard he±ieΦ(t+τ )/~ e∓ieΦ(t)/~ i = e±i

eVdc τ ~

ac + eV ~ω (sin(ω0 (t+τ ))−sin(ω0 t)) 0

hΘ(t + τ )

X

eJ(τ ) . Then we only need to compute the new quasiparticle correlation functions:

∗ † ˙ ), cr cl e−i(l −r )t/~ i = −i~θ(τ (l − r )τl,r

l,r

= −hΘ† (t + τ )

X (l − r )τl,r c†l cr ei(l −r )t/~ i l,r

where the last equality holds for the particle-hole symmetric junctions. Picking up the terms we obtain for a dc bias: 1 hPqp (eVdc )i =

 2π  θ() ∗ P ()(eVdc ) + θ() ∗ P ()(−eVdc ) , ~

while for an ac bias we get: 1 hPqp (t)i =

2 Re ~2

Z

0

dτ 2 cos( −∞

eVdc τ eVac ˙ )eJ(τ ) . + (sin(ω0 (t + τ )) − sin(ω0 t)))i~θ(τ ~ ~ω0

14 3.

Power balances a.

dc bias

Since there is neither a dc voltage drop across the inductance, nor a dc displacement current, the average electrical power which is supplied by the voltage source is directly: Z 2π dE(eVdc − E)θ(E)P (eVdc − E) + (−eVdc − E)θ(E)P (−eVdc − E) hIˆqp (eVdc )iVdc = ~ Z 2π + dEEθ(E)P (eVdc − E) + Eθ(E)P (−eVdc − E), ~ where we identify the power absorbed in the quasiparticles Z 2π dEEθ(E)P (eVdc − E) + Eθ(E)P (−eVdc − E). Pqp (eVdc ) = ~ The other term can be worked out to match the average power emitted into the environment: Z 2π 1 dE(eVdc − E)θ(E)P (eVdc − E) + (−eVdc − E)θ(E)P (−eVdc − E) = PLC (eVdc ) ~ With these identifications we obtain the stationary power balance: PDC (eVdc ) = Pqp (eVdc ) + PLC (eVdc ). b.

ac bias

In the presence of the ac bias, the average power delivered by the dc source reads: hIˆqp (t)iVdc =

X k

Jk2

 eV  ac Iqp (eVdc + k~ω0 )Vdc , ~ω0

where we exploited the Jacobi-Angers expansion of exponentials having trigonometric arguments. On the other hand, the average dissipative ac response reads:  eV  V   eV   eV  ac ac ac ac Jk+1 + Jk−1 Iqp (eVdc + k~ω0 ) ~ω0 2 ~ω0 ~ω0 k X k~ω0  eVac  = Jk2 Iqp (eVdc + k~ω0 ). e ~ω0

hIˆqp (t)iVac cos(ω0 t) =

X

Jk

k

Combining the two expression with the results for a dc bias, we get the power balance of the circuit: hIˆqp (t)iVdc = Pqp (t) + PLC (t).