Quantum Impurity Physics with Microwave Photons Moshe Goldstein - - PowerPoint PPT Presentation

Quantum Impurity Physics with Microwave Photons

Moshe Goldstein (Simons Fellow @ Yale), Michel Devoret (Yale), Manuel Houzet (CEA, Grenoble), Leonid Glazman (Yale)

Outline

• Introduction

– Quantum impurities – Microwave photons

• System and relation to anisotropic Kondo
• AC conductance: Photon elastic scattering
• Photon inelastic scattering

Circuit QED

[Scholkoepf and Girvin, Nature ‘08]

• Quantum optics with microwave circuits:

– optical cavity ► microwave resonator – atom ►qubit

• Small “cavity”, large “atom” ► strong light-

matter interaction

Many Body Physics

[Koch et al., PRA ‘10]

Ultracold atoms in an optical lattice Microwave photons in a circuit:

[Bloch, Nature ‘08]

• Could we start with something simpler?
• Controllable simulators of many-body physics

Outline

• Introduction

– Quantum impurities – Microwave photons

• System and relation to anisotropic Kondo
• AC conductance: Photon elastic scattering
• Photon inelastic scattering

Quantum Impurities

• Small system coupled to quantum environment

– 2 level system in a bosonic bath (spin-boson) – Magnetic impurity in a Fermi sea (Kondo)

• Easy to study:

– Experimentally: nanophysics (QDs,nanograins,…) – Theoretically (RG,bosonization,CG,CFT,Bethe,NRG,DMRG,…)

• Teach us about:

– Strong correlations (asymptotic freedom, quantum phase

transitions, non Fermi liquid, ...)

– Nanophysics and quantum computation (qubits)

Example: Kondo

• Realizations

– Magnetic impurity

t

• Anderson impurity model:

– Quantum dot with odd electron number (spin qubit)

   

 

H.c. 2 2

env

                      

         

x c x c t n Un n B n B H H

z z

   

G=pn|t|2: level width







s k s k s k k

c c H

, , , env



  



k s k s

c , 

Impurity spin: Environment spin:

t

The Kondo Problem

• Problem: divergences [Kondo ’64]

– Example: susceptibility

   

                      T I T I T

2 2

ln ln 1 1 ~  n  n 

• Local moment regime [G<<+U,||] ► Kondo model:

Ixy=Iz=I~t2/U>0: exchange Bz: local magnetic field 0: bandwidth

 

z z z z z xy

S B s S I s S s S I H H     

   

) ( ) ( ) ( 2

env K

S 

 







' , , ' , ' , ' ' , ,

2 1

s s k k s k s s s k

c c s   

Kondo Physics

• RG equations (Bz=0)

[Anderson]:

 

2 2

ln d d

xy z

I I n  n  

 

xy z xy

I I I

2

ln d d n  n  

• Ferromagnetic Kondo:

– impurity decoupled – susceptibility: zz~c(I)/T+…

• Antiferromagnetic Kondo:

– impurity strongly-coupled (asymptotic freedom) – susceptibility: zz~1/TK+… (Fermi liquid) Kosterlitz- Thouless transition

TK: Kondo temperature

nIz nIxy

0: bandwidth n: local DoS

 

 

I TK n  / 1 exp  

Outline

• Introduction

– Quantum impurities – Microwave photons

• System and relation to anisotropic Kondo
• AC conductance: Photon elastic scattering
• Photon inelastic scattering

Superconducting Grain Array

[Manucharyan et al., Science ‘09]

Superconducting Grain Array

Velocity:

Admittance:

 



        

 i i i i

C Q J H 2 cos

2 1 lead

 

2Z R e JC π g

Q *

 

) ( ) ( x x i

x

 

C J J C C J C C C C C J J J C' C' C' C' C' C' J C'

   

 





   dx x g x g v H

x 2 1 2 lead

) ( ) ( 2 p  p

Assuming C>>C’

C/J ω v/a

0 



– Usually g>>1, but g~1 possible

[g<1: Glazman & Larkin, PRL ‘97]

– Waveguide for microwave photons



[Manucharyan et al., Science ‘09]

Adding a Quantum Impurity

• Motivations:

– Quantum optics ► many-body effects – Condensed matter ► bosons C J C C C C C J J J C' C' C' C' Quantum Impurity

?

• Artificial atom in

microwave waveguide

Adding a Quantum Impurity

• Transport Measurement:

– Charge ► conserved – Energy ► not conserved (dissipation)

• Where does energy go?

– Photons at different frequencies!

  

C J C C C C C J J J C' C' C' C' Quantum Impurity

Outline

• Introduction

– Quantum impurities – Microwave photons

• System and relation to anisotropic Kondo
• AC conductance: Photon elastic scattering
• Photon inelastic scattering

System

• Quantum impurity: two capacitively coupled grains,

weakly coupled (JL/R<<J) to the leads

– Only two charging states (nL/R=0,1 Cooper pairs)

L R

CLR C J JL CL CR JR C C C C C J J J C'L C' C' C'R C' C' Vg,L Vg,R

System

• Quantum impurity: two capacitively coupled grains,

weakly coupled (JL/R<<J) to the leads

– Only two charging states (nL/R=0,1 Cooper pairs)

   

 dx

x g x g v H

R L x



 

  

, 2 1 2 lead

) ( ) ( 2

  

p  p

 



  

  

R L i

e J n H

, ) ( imp lead

H.c. 1 ) ( U

      







R L LR R L

n n U n H   

 , imp   



L R

CLR JL CL CR JR C'L C'R Vg,L Vg,R L R

Relation with Anderson Impurity

• Generalized Anderson

impurity model:

R

R

t

LR

U

L

t

L

L



R



 /

t ↑/↓

 /



– Spin anisotropy – Luttinger liquid (g≠1) – Level-lead interaction spinless spinful refermionization

L R

CLR JL CL CR JR C'L C'R L R

Kondo Description

• In Coulomb blockade valley – “local moment”

regime {nL+nR1}:

– Singly occupied states – “spin” {Sz=(nL-nR)/2} – Equivalent to Kondo with noninteracting lead:

 

z z z z z xy

S B s S I s S s S I H H     

   

) ( ) ( ) ( 2

env K

 /

t ↑/↓

 /



CLR JL CL CR JR C'L C'R L R

R L 2e 2e







s k s k s k k

c c H

, , , env



Impurity spin: Environment spin:  







' , , ' , ' , ' ' , ,

2 1

s s k k s k s s s k

c c s   

S 

Kondo Parameters

 /

t ↑/↓

 /



• Schrieffer-Wolf (simplified expressions):

 

z z z z z xy

S B s S I s S s S I H H     

   

) ( ) ( ) ( 2

env K

CLR JL CL CR JR C'L C'R L R

R L 2e 2e

R g L g R L z

V V B

, , 

    

Kondo Parameters

 /

t ↑/↓

 /



CLR JL CL CR JR C'L C'R L R

• Schrieffer-Wolf (simplified expressions):

          

LR R L xy

U J J I 1 1 2    n

  2

R L

    

0 – bandwidth n – local DoS

 

z z z z z xy

S B s S I s S s S I H H     

   

) ( ) ( ) ( 2

env K

R L 2e 2e

R g L g R L z

V V B

, , 

    

Kondo Parameters

 /

t ↑/↓

 /



CLR JL CL CR JR C'L C'R L R

• Schrieffer-Wolf (simplified expressions):





             

R L LR R L Z

U J I

, 2 2 2

1 1 2 1

   

     n           

LR R L xy

U J J I 1 1 2    n

        v gU g p 

 

1 1

  2

R L

    

0 – bandwidth n – local DoS

 

z z z z z xy

S B s S I s S s S I H H     

   

) ( ) ( ) ( 2

env K

R L

2e 2e

R g L g R L z

V V B

, , 

    

nIz nIxy

Relevant Regime for SC

Kosterlitz- Thouless transition





             

R L LR R L Z

U J I

, 2 2 2

1 1 2 1

   

     n         v gU g p 

 

1 1           

LR R L xy

U J J I 1 1 2    n

• Typically: g>>1, UL/R>0 ► Iz>>Ixy, AFM Kondo
• g<1 possible [Glazman & Larkin, ‘97] ► Iz< – Ixy, FM Kondo

L R

CLR JL CL CR JR C'L C'R L R

Outline

• Introduction

– Quantum impurities – Microwave photons

• System and relation to anisotropic Kondo
• AC conductance: Photon elastic scattering
• Photon inelastic scattering

L R

CLR JL CL CR JR C'L C'R L R

• Current transmission coefficient:

AC conductance

) ( ) ( ) (

in trans

   I I TL 

Iin( Itrans( Iref( linear response

(xin<0, xout>0)

) ( ) ( x x i

x

 

) (

• ut

in

• ut

in

) ( ); ( ) ( ); ( ) ( ) (

 

  x i x i x i x i G G  

– Similarly for reflection RL( TL( V(

) (

ref trans in R L

n n dt d I I I    

• Elastic T matrix:

Elastic Scattering

          1 ) ( ) ( ) ( 1 ) ( ) ( ˆ el

| '

     p

R L R L

R T T R T i

 

– Generalization (finite temperature) of photon elastic scattering amplitudes:

) ; ' , ( ˆ ) ( ˆ ) ; , ( ˆ ) ; ' , ( ˆ ) ; ' , ( ˆ

) ( ph el ) ( ph ) ( ph ph

     x G T x G x x G x x G  

R L, ' ,   

L R

CLR JL CL CR JR C'L C'R L R

  

Scattering and Susceptibility

) ( ) 1 ( ) ( ˆ

' el | '

'

    

 zz

T

   



 

• T matrix Related to local spin susceptibility (by

equations of motion):

– Experimental probe for dynamic susceptibility!

        v gU g p 

 

1 1

L R

CLR JL CL CR JR C'L C'R L R Bz()

2e 2e

Iin( Itrans( Iref(

Kondo Susceptibility

      

K K zz

T i T  p  

2

1 1 ~ ) (

 

z

I K xy zz

T i I i

n

    n         

2

2 / ) ( ~ ) (

 

z

I xy K

I T

n

n 

/ 1

~

) 1 ( 2

2 2 2 z R L

I n       

 TK |TL|

• New low energy scale:
• For T, Bz<<TK:

– Low  [Shiba]: – High :

L R

CLR JL CL CR JR C'L C'R L R

Outline

• Introduction

– Quantum impurities – Microwave photons

• System and relation to anisotropic Kondo
• AC conductance: Photon elastic scattering
• Photon inelastic scattering

• Rate of energy loss / total inelastic scattering

probability for a photon at frequency 

Is Scattering Fully Elastic?

   

2 2

| | | | 1 ) (

  

R T  

 

 

2 2 2 2 2 2 2

) ( 4 ) ( Im 4       p    p

zz R L zz

 

 

• Zero for harmonic systems

– Kondo at <<TK : vanishes to O(2) [Nozieres, Shiba]

• Nonzero in general !

– Kondo at >>TK : O( [nIxy/2 ) – dominating over elastic transmission, O( [nIxy/4 )

L R

CLR JL CL CR JR C'L C'R L R Iin( Itrans( Iref(

L R

CLR JL CL CR JR C'L C'R L R

Inelastic Scattering

  

L R

CLR JL CL CR JR C'L C'R L R

Inelastic Scattering

 4 1  3 5

• Spectrum of outgoing photons (“Raman”)

) | ' (

| '

  

 

’  # of outgoing photons in lead ℓ’ at frequency interval [’, ’+d’] per each incoming photon in ℓ and 

Inelastic Spectrum

• Reducible to local correlators (Keldysh formalism):

              

                 

   p   

, , ' , ' , , ' , ' , , ' , ' 2 2 '|

2 ' cot ' ) | ' (

'

q z q z c z q z q z q z q z c z q z q z c z c z

S S S S S S S S T S S S S

 

 

• linear response in energy flux –

2nd order in charge current

) , ( ) , ( ) | ' (

) ( ; ; ; ; ; | '

'

      

 

  

r i i n r i i n

G G  

 

) ( ) (

' 2

    





z z z z

B B S S

) ( ) ( ) ' ( ) ' (

3

        

z z z z

B B B S

L R

CLR JL CL CR JR C'L C'R L R Bz()

• To lowest (2nd) order in IxyJLJR (>>TK):

    

' ) ( ) ( ;

Im ) ' ( ) ' ( ) ' ( 1 ) ' ( 1 ) ' (

   

        

 

     

i i B B B B

e e n n n n

Perturbative Regime (I)

  ' ) 4 / ( 4 ) | ' (

2 2 2 | '

'

 p  p    a I xy

 

 

{ } +

• Equivalent to Bolzmann equation:

 ’

}

   

    

           

 ' , ' ' ' ' 1 1 1 1 ' 2

' 1 1

1 ... 1 ... ' ' ... ' ' ... ! ' ! 2

N N N N N N N N xy

N N

n n n n d d d d N N a I dt dn

    

         p

   

) ' ' ... ( ) ' ' ... ( 1

' 1 1 ' 1 1 N N N N

n n            

 

           

Perturbative Regime (II)

 

z z

I K I xy

T I

n n

      p      n  p   

2 2 2 2 1 2 2 2 | '

' ' ' ' ' ) | ' (

' '

                  



   

 

• What happens for -’ or ’ small w.r.t. TK ?

’ 

• At T=0:

        ) ( ' ' ) | ' (

, ' '|    





 R L

d

– Obeys energy conservation:

Bosonic Kondo at Low Energy (I)

• Kondo Hamiltonian:
• Equivalent to spin-boson:

 

H.c. 4 2 ) (

) ( 2 env

    

 

s

i xy s x z z K

e S a I S I H H



p p 

x xy s x z z K SB

S a I S I H U H U H p p  2 2 ) (

env

     



) ( 2

s z

S i

e U





v I I

z z

p 2   

• At “strong coupling Toulouse point”, Iz/(4v)=p/2:

  z I

t

↑/↓

 /



– Spin in Sx eigenstate Iz/(4v) is a phase shift

 

 

 dx x v H

s x 2 env

) ( 4  p

Bosonic Kondo at Low Energy (II)

• Deviations from “Toulouse point”:

2 ) ( 2

env

p  p

s x z z x xy SB

S I S a I H H     

• At low energy, flipping Sx is not allowed. Thus:

 

2 2 env eff

) (

s x K

T v H H    

   

2 2 env

) (

s x xy z

I I a H H  p    

Kondo at Low Energy

• Low energy description for Fermions [Nozieres]:
• Bosonization:

– Lowest term allowed by symmetries (time reversal)

 

2 2 env eff

) (

s x K

T v H H    

   

   

 

  pn pn   

K s k k s k s k K k k s k s k s k k

T c c T c c H

2 , ' , , ' , ' , , , eff

1 2

 

 

 dx x v H

s x 2 env

) ( 4  p

t

↑/↓

 /



– No inelastic scattering of bosons to O(2)

Low Energy Inelastic Scattering

• Inelastic spectrum for <<TK:

’ 

 

6 2 2 2 2 | '

) ' ( ) ' ( ' ~ ) | ' (

'

K z

T B             

 

 

 

... ) (

4 3 2 1

4 3 2 1 4 3 4

  

 k k k k s x K

a a a a k k k k T v 

• Least-irrelevant many boson term:

– no magnetic field – with magnetic field

 

... ) (

3 2 1

3 2 1 3 3 3

  

 k k k s x K z

a a a k k k T v B 

Back to High Energies

• For >>TK (following considerations similar to <<TK):

z

I K

T

n

    

2

' ' '        

  

 

2 2 2

' ' '

K z

T B        

 

z

I K K

T T

n

     

2 2

' ' '        

’ 

2 ' 2 '|

) | ' (

   

 p   

` small ` small: at least two photons for Bz=0, one otherwise

TK TK

Conclusions

• Quantum impurity in circuit QED: Many

body physics with photons

– No dissipative elements, yet dissipative linear response of charge current – Missing energy? ► inelastic photon scattering – Scattering amplitudes ►local response functions

• Example: anisotropic Kondo for microwaves

– N leads ► SU(N) [Carmi et al., PRL’11]; 2 channel?