Squeezing by a quantum conductorChristophe Mora

ENS, Paris

GDR Mesoscopic Quantum Physics, Aussois, 2016

U. Mendes, C. Altimiras, P. Joyez, F. Portier

Differential capacitanceMesoscopic QED

PrincetonENS, Paris

Saclay Darmouth collegeETH Zurich

Differential capacitanceParametric excitationofaswing

Child stands up at zero angle and squats at maximum angle

Frequency of pumping is twicethe oscillator frequency

In-phase noise is amplified, out-of-phase noise is reduced

02ww =P

Parametricexcitation

Squeezing of noise

Differential capacitanceSqueezing experiment

PRL 2013

Radiation of a tunnel junction

I. Squeezed light with tunnel junction

II. Dynamical Coulomb Blockade

Outline ofthetalk

Squeezed light with tunnel junction

Differential capacitanceHamiltonian vsdissipativesqueezing

Hamiltonian squeezing

Example: Josephson parametricAmplifier (JPA), limited to half-squeezingfor cavity mode

[ ] ( )..42

)2sin(12

2222

cheaiaaPtXH ti --»++= + wewew!

2/12 ³DX

Differential capacitanceJosephsonparametric amplifier(JPA)

Mallet et al., PRL 2011

Dissipative squeezingEnvironment noise engineered to squeezeExample: time-dependent damping rate

Differential capacitanceHamiltonian vsdissipativesqueezing

Hamiltonian squeezing

Example: Josephson parametricAmplifier (JPA), limited to half-squeezingfor cavity mode

[ ] ( )..42

)2sin(12

2222

cheaiaaPtXH ti --»++= + wewew!

2/12 ³DX

1

12

11

ll

+-

=DX

Didier, Qassemi, Blais, PRA 2014Kronwald, Marquardt, Clerk, PRA 2014

( )tic et wlkl 211)( +=

Differential capacitanceCoupling electrons andphotons

Hamiltonian for tunneling

..,

,, chcctHqk

qRkLT +L=å +

Dmytruk, Trif, Mora, Simon, PRB 2015

QQei =L=L F ],[

L transfers one electroniccharge through the junction

)( aai -=F +l

Gauge transform (cavity)

Cqk

qRkLT HchcctH ++=å +

,,, ..

( )RRLLC NNaaH ˆˆ)( ll ++= +

Gives a capacitive coupling

Udson, Mora PRB 2016

Differential capacitanceTunneljunction feeding aLCresonant circuit

(LC) Harmonic oscillator

Derivation of a Quantum Langevin equation (a, I are quantum operators)

The noise of the current I both damps and excites the LC resonator

)(22

22F=L+

F+= i

T eHLC

QH

[ ] !iQ =F,

Iaaia lkw =++20! [ ])()( 0000

2 wwlk --= SS

)( aai -=F +l

Fluctuation-dissipation theorem

Differential capacitanceEquilibrium noise

( ) ( ) )(2)( 211021 wwpdwww += SII

wb

ww !--=

eRS

T

eq

12)(0

absorptionemission

Iaaia lkw =++20!

At equilibrium (no voltage)

S0 is absorption noise for positive frequency and emission noise for negative

kwl )( 00

2 -=+ Saa

Solution of the Langevin equation

Differential capacitanceOut-of-equilibrium noise

( ) ( )å -+

=

nn nS

II

)2(2)( 0211

21

wwwpdw

ww

Iaaia lkw =++20!

0¹n Non-stationary terms

Quadrature Squeezing

kwl )( 01

2Saa =

Differential capacitanceOptimizing squeezing

)(1 aaiX -= +

Single tone

0

0

21

706.02

618.0

ww!

!

=

=

=D

eV

eVX

AC

Full agreement with Reulet’sexperiment

Squeezing depends on photo-assisted properties by the ac modulation or on emission/absorption probabilities cn

With harmonics

Differential capacitanceOptimalsqueezing foratunneljunction

Optimal squeezing is reached with the pulse shape

å ÷÷ø

öççè

æ-=

lopt

ltehtV

02)(

wpd

405.042

21 ==D

pX

Dynamical Coulomb blockade

Differential capacitanceInput-outputformalism,field emission

Input-output formalism describes how the conductor interacts with its electromagnetic circuit environment

)(winaincoming photon field(from coax line)

)(woutaoutgoing photon field(to coax line)

Photon radiation in the transmission line (impedances are not matched)

CSnSntPtPtP IBIB

inout 2)()()())(1()()()( 0000 wwww --+

=-=

Lesovik, Loosen, JETP 1997

Differential capacitanceDynamical Blockade foratunneljunction

Dynamical Coulomb blockade : transferred electric charge may excite environmental modes : reduction of current due to elastic processes

q=eq=-e

Z

Differential capacitanceDynamical Blockade foratunneljunction

QQei =L=L F ],[

Dynamical Coulomb blockade : transferred electric charge may excite environmental modes : reduction of current due to inelastic processes

q=eq=-e

Z

1002£L

Probability not to excite environment

0 Ground state (electromagnetic circuit)

Differential capacitanceDynamical Blockade foratunneljunction

QQei =L=L F ],[

Dynamical Coulomb blockade : transferred electric charge may excite environmental modes : reduction of current because of inelastic processes

q=eq=-e

Z

1002£L

Probability not to excite environment

0 Ground state (electromagnetic circuit)

Inelastic transport encoded in P(E) function

( )2// ehZ

CeEC 2/2=

Z

Differential capacitanceSqueezed radiation

We propose a circuit where dynamical Coulomb blockade and squeezed radiation readout are spatially separated

Strong DCB

2LClT ZZR =Impedance matching

www kdwk

wkdwdw I

iiRa

ia T

inoutˆ

2 0,, +-

+=

!No DCB

0wwdw -=

Differential capacitanceSqueezed radiationfrom Dynamical Coulomb

Strong DCB, reflected in the noise properties of the junction

( ) ( )

( )+-

+-+

+-++

---

+=+

wwwwww

wwww

w

kwkw

000

00

,,10

,,

2ˆ

2 inoutl

inout

aaCiYIZi

aiai

!

With DCB

LZCY l //2 0/ ±=-+k )()( 01/001/01/0 ww --µ SSY

Differential capacitanceSqueezed radiationfrom Dynamical Coulomb

Important temperature correctionsReflection coefficient is effectively strongly

modulated sysl

sysl

ZZZZ

r+

-=

ConclusionA tunnel junction is in principle able to produce squeezedlight in a resonator

Squeezing is improved with concentrated pulses of voltage

Non-linearities in a tunnel junction under strong Coulomb blockade could be used to achieve a competitivesqueezed radiation

Conclusions

Udson, Mora, NJP 2015 Mora, Altimiras, Joyez, Portier Arxiv 2015 Udson, Mora, PRB 2016