Observation of edge magnetoplasmon squeezing in a quantum Hall conductor H. Bartolomei1 R. Bisognin1 H. Kamata1 J.-M. Berroir1 E. Bocquillon12 G. M enard1 B. Plac ais1 A. Cavanna3 U. Gennser3 Y . Jin3 P. Degiovanni4 C. Mora5 and G. F eve1

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Observation of edge magnetoplasmon squeezing in a quantum Hall conductor

H. Bartolomei1, R. Bisognin1, H. Kamata1, J.-M. Berroir1, E. Bocquillon1,2, G. M´

enard1, B.

Plac¸ais1, A. Cavanna3, U. Gennser3, Y. Jin3, P. Degiovanni4, C. Mora5, and G. F`

eve1∗

1Laboratoire de Physique de l’Ecole normale sup´

erieure, ENS, Universit´

PSL, CNRS, Sorbonne Universit´

e, Universit´

e Paris Cit´

e, F-75005 Paris, France

2II. Physikalisches Institut, Universit¨

at zu K¨

oln, Z¨

ulpicher Str. 77, 50937 K¨

oln

3Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Universit´

e Paris-Saclay, 91120 Palaiseau, France.

4Univ Lyon, Ens de Lyon, Universit´

e Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France

5Universit´

e de Paris, Laboratoire Mat´

eriaux et Ph´

enom`

enes Quantiques, CNRS, F-75013 Paris, France.

∗To whom correspondence should be addressed; E-mail: gwendal.feve@ens.fr.

Squeezing of the quadratures of the electromagnetic ﬁeld has been extensively studied in optics and mi-

crowaves. However, previous works focused on the generation of squeezed states in a low impedance

(Z0≈50Ω) environment. We report here on the demonstration of the squeezing of bosonic edge magnetoplas-

mon modes in a quantum Hall conductor whose characteristic impedance is set by the quantum of resistance

(RK≈25kΩ), offering the possibility of an enhanced coupling to low-dimensional quantum conductors. By

applying a combination of dc and ac drives to a quantum point contact, we demonstrate squeezing and observe

a noise reduction 18% below the vacuum ﬂuctuations. This level of squeezing can be improved by using more

complex conductors, such as ac driven quantum dots or mesoscopic capacitors.

In quantum Hall conductors, charge excitations propagate

ballistically along one dimensional chiral channels. This bal-

listic propagation has been exploited in electron quantum op-

tics experiments1,2 focusing on the generation and manipula-

tion of elementary electron and hole excitations of the Fermi

sea. These are particle-like fermionic excitations, but the dy-

namics of charge propagation along one-dimensional edge

channels can be equivalently described in terms of collec-

tive bosonic excitations called edge magnetoplasmons (EMP),

which consist of coherent superpositions of electron-hole

pairs on top of the Fermi sea.

EMP have been largely investigated in the past by study-

ing the propagation of the time dependent electrical current

in the time3–8 or frequency domain9–12. Experiments have

highlighted the dependence of EMP propagation speed on the

magnetic ﬁeld and on the screening by nearby electrostatic

gates. All these studies are based on a classical description

of charge propagation along the edge channels which can be

modeled as transmission lines11. However, chiral edge chan-

nels have three important differences with respect to standard

50 Ohms coaxial cables. Firstly, the chirality results in the

separation between forward and backward propagating waves.

Secondly, the speed of the EMP8is of the order of 105m.s−1,

three orders of magnitude smaller than the speed of light, re-

sulting in wavelengths in the µm range at GHz frequencies

compared to the cm range in standard coaxial cables. EMPs

would thus allow for more compact circuits. Finally, their

characteristic impedance is of the order of the resistance quan-

tum, RK≈25 kΩ, much larger than the 50 Ohms standard,

offering the possibility of a strong coupling to low dimen-

sional quantum conductors of high impedance13.

These speciﬁcities motivated recent theoretical and

experimental1415 studies of EMP transmission lines for efﬁ-

cient coupling to on-chip high impedance quantum devices,

such as charge or spin qubits, for the study of Coulomb inter-

action effects in one-dimensional edge channels16, or for the

realization of on-chip microwave circulators17. So far, these

studies have focused on the classical regime, where EMP

states can be described as coherent states. However, as for

other bosonic modes, quantum EMP states can also be gener-

ated. In the last years there has been a strong interest for the

generation of quantum radiation by quantum conductors18–20

and in particular of squeezed states21–23. So far, it has been

limited to the study of low impedance (50 Ohms) transmis-

sion lines coupled to superconducting circuits24–26 or tunnel

junctions27. We report here on the generation of squeezed

EMP states at the output of a quantum point contact used as

an electronic beam-splitter in a GaAs quantum Hall conduc-

tor, as discussed in Ref. [23]. Using two-particle interfer-

ence processes2occurring between electron and hole excita-

tions colliding on the splitter, we generate a squeezed EMP

vacuum state at frequency f=Ω

2π= 7.75 GHz at the splitter

output with a noise minimum 18% below the vacuum ﬂuctu-

ations. The non-linear EMP scattering at the splitter breaks a

2fpump signal into coherent photon pairs, thereby achieving

squeezing28.

Squeezed EMP states could be used for quantum enhanced

measurements in EMP interferometers29, or to extend the

study of low dimensional quantum conductors in the regime

where they are driven by quantum voltage sources30, exploit-

ing the strong coupling of high impedance transmission lines

to high impedance low-dimensional quantum circuits.

In the bosonic description of charge propagation, the charge

density ρ(x, t)carried by a single edge channel can be ex-

pressed as a function of a chiral bosonic ﬁeld Φ(x, t)with

ρ(x, t) = −e

√π∂xΦ(x, t). The relation between the electri-

cal current and the ﬁeld can then be deduced directly from

charge conservation: i(x, t) = e

√π∂tΦ(x, t). At low fre-

quency (typically a few GHz), dispersion effects can be ne-

glected, such that Φ(x, t)can be decomposed in terms of ele-

mentary plasmon excitations at pulsation ωpropagating with

arXiv:2210.04279v1 [cond-mat.mes-hall] 9 Oct 2022

constant speed velocity v:

Φ(x, t) = −i

√4πX

ωr2π

ωTmeas

[bωeiω(x/v−t)−h.c.](1)

with [bω, b†

ω0] = δω,ω0.(2)

b†

ωis the operator which creates a single plasmon of energy ~ω

and obeys the usual bosonic commutation relations. The long

measurement time Tmeas sets the discretization of the plasmon

modes by steps of 2π/Tmeas. In order to address the squeezing

of EMP modes, it is useful to introduce the quadratures of

the bosonic ﬁeld at a given pulsation Ωdeﬁned for a phase

0≤ϕ≤π:

XΩ,ϕ =bΩeiϕ +b†

Ωe−iϕ

√2(3)

Their ﬂuctuations h∆X2

Ω,ϕican be decomposed into an

isotropic h∆X2

Ω,isoiand an anisotropic, ϕdependent, part

h∆X2

Ω,anigiven by:

h∆X2

Ω,isoi=hb†

ΩbΩi−hb†

ΩihbΩi+1

2(4)

h∆X2

Ω,ani=<(hb2

Ωi−hbΩi2)e2iϕ(5)

For classical coherent states, h∆X2

Ω,ϕiis isotropic

(h∆X2

Ω,ani= 0) and given by h∆X2

Ω,ϕi= 1/2which

are called vacuum ﬂuctuations. For squeezed states, the

minimum value of the noise, obtained for a certain value

ϕ=ϕ0of the angle, goes below the vacuum ﬂuctuations,

h∆X2

Ω,ϕ0i<1/2. As imposed by Heisenberg’s uncertainty

principle, the orthogonal quadrature then exhibits larger

ﬂuctuations: h∆X2

Ω,ϕ0+π/2i>1/2.

The quadratures of the ﬁeld and their ﬂuctuations can be

experimentally accessed from the measurements of the elec-

trical current i(t)and its ﬂuctuations at high frequency. More

precisely, deﬁning iΩ,ϕ(t) = cos(Ωt+ϕ)i(t), one has:

hiΩ,ϕ(t)Tmeas i=−2erΩ

πTmeas hXΩ,ϕi(6)

SΩ,ϕ = 2 ZdτhδiΩ,ϕ(t+τ/2)δiΩ,ϕ(t−τ /2)iTmeas

(7)

=e2Ω

2πh∆X2

Ω,ϕi(8)

where iΩ,ϕ(t)Tmeas denotes the average of iΩ,ϕ(t)over the

measurement time Tmeas. Classical states are thus deﬁned by

current ﬂuctuations SΩ,ϕ =e2Ω

4π=e2f

2and squeezed states

by SΩ,ϕ0<e2f

2. Experimentally, one measures the noise in

excess of the equilibrium ﬂuctuations ∆SΩ,ϕ =SΩ,ϕ −e2f

and squeezing occurs when ∆SΩ,ϕ0<031.

The principle of the experiment is represented in Figure

1.A. A quantum point contact (QPC) is used as a beam-splitter

for electronic excitations of transmission probability T(and

reﬂection probability R= 1 −T). By plugging two elec-

tronic sources at inputs 1 and 2 of the QPC, collisions between

FIG. 1. A. Principle of the experiment: a QPC is used as an elec-

tronic beam-splitter of transmission Tfor the collision of electron

and hole excitations generated at inputs 1 and 2. Input 1 is connected

to a dc source, which shifts the chemical potential of the edge chan-

nel by −eVdc. Input 2 is connected to an ac sinusoidal source of

amplitude Vac and frequency 2f. EMP squeezing is characterized at

output 4 by measuring the correlations SΩ,ϕ of the current iΩ,ϕ. For

a squeezed vacuum, SΩ,ϕ goes below the vacuum ﬂuctuations for

ϕ= 0 and above them for ϕ=π/2.B. Experimental setup: the low

frequency noise ∆S0is measured at output 3 and the high frequency

noise ∆SΩ,ϕ at output 4. The EMP current i(t)is weakly transmit-

ted to a Z0= 50 Ω coaxial cable, ampliﬁed using two cryogenic

ampliﬁers in a double balanced conﬁguration34,37 and multiplied to

the local oscillator Vl(t) = Vlcos (2πft +ϕ)using high frequency

mixers. The noise power is then integrated using a diode in a 800

MHz bandwidth set by a low pass ﬁlter. The noise is measured via

a lock-in detection by applying a square modulation at a frequency

fm= 234 Hz to the dc voltage Vdc. The resulting output excess

noise, ∆Sout

Ω,ϕ is proportional to ∆SΩ,ϕ with a proportionality factor

G(Ω) that needs to be calibrated.

electron and hole excitations32,33 emitted by each source oc-

cur at the beam-splitter. Previous implementations of high

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摘要：

ObservationofedgemagnetoplasmonsqueezinginaquantumHallconductorH.Bartolomei1,R.Bisognin1,H.Kamata1,J.-M.Berroir1,E.Bocquillon1;2,G.M´enard1,B.Plac¸ais1,A.Cavanna3,U.Gennser3,Y.Jin3,P.Degiovanni4,C.Mora5,andG.Feve11LaboratoiredePhysiquedel'Ecolenormalesup´erieure,ENS,Universit´ePSL,CNRS,SorbonneUni...

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Observation of edge magnetoplasmon squeezing in a quantum Hall conductor H. Bartolomei1 R. Bisognin1 H. Kamata1 J.-M. Berroir1 E. Bocquillon12 G. M enard1 B. Plac ais1 A. Cavanna3 U. Gennser3 Y . Jin3 P. Degiovanni4 C. Mora5 and G. F eve1.pdf

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Observation of edge magnetoplasmon squeezing in a quantum Hall conductor H. Bartolomei1 R. Bisognin1 H. Kamata1 J.-M. Berroir1 E. Bocquillon12 G. M enard1 B. Plac ais1 A. Cavanna3 U. Gennser3 Y . Jin3 P. Degiovanni4 C. Mora5 and G. F eve1

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