Christophe Mora ENS, Paris · Christophe Mora ENS, Paris GDR Mesoscopic Quantum Physics, Aussois,...
Transcript of Christophe Mora ENS, Paris · Christophe Mora ENS, Paris GDR Mesoscopic Quantum Physics, Aussois,...
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