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 • Quantum optics with microwave circuits: [Scholkoepf and Girvin, Nature ‘ 08] – optical cavity ► microwave resonator – atom ► qubit • Small “cavity”, large “atom” ► strong light- • matter interaction

Many Body Physics • Controllable simulators of many-body physics Ultracold atoms in Microwave photons an optical lattice in a circuit : [Koch et al., PRA ‘ 10] [Bloch, Nature ‘08] • Could we start with something simpler?

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 – Magnetic impurity in a – 2 level system in a Fermi sea ( Kondo ) bosonic bath ( spin-boson ) • 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 – Quantum dot with odd electron number ( spin qubit ) t • Anderson impurity model :         B B                        0 z z 0 0 H.c. H H n n Un n t c x c x         env     2 2        G = pn |t| 2 : level width   0 H c c 0 c , env , , k k s k s s k s , k s k

The Kondo Problem t • Local moment regime [ G <<  +U,|  | ] ► Kondo model : I        0 xy ( 0 ) ( 0 ) ( 0 ) H H S s S s I S s B S     K env z z z z z 2  I xy =I z =I~t 2 /U>0 : exchange Impurity spin: S   B z : local magnetic field 1       0 s c c Environment spin: , , ' ' , ' k s s s k s 2 , ' , , ' k k s s • Problem: divergences [Kondo ’ 64] – Example: susceptibility         1       n  n    2   2 0 0  ~ 1 ln ln   I I  0 : bandwidth       T T T

Kondo Physics n I xy • RG equations ( B z =0 ) [Anderson] :   n d I   n 2 2 z I  xy d ln  0 : bandwidth 0   n: local DoS n d I   n xy 2 I I  z xy d ln 0 • Ferromagnetic Kondo : – impurity decoupled – susceptibility:  zz ~c(I)/T+… n I z Kosterlitz- • Antiferromagnetic Kondo : Thouless – impurity strongly-coupled ( asymptotic freedom ) transition – susceptibility:  zz ~1/T K +… ( Fermi liquid )        n exp 1 / T K I T K : Kondo temperature 0

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 [Manucharyan et al., Science ‘ 09] C' C' C' C' C' C' C' Assuming C>>C’   2    Q J J J J J J J        i cos H J  lead 1 i i  2  C C C C C C C C C i        v       p 2 2 1 ( ) ( ) H g x g x dx p lead x 2   x  ( ) ( ) i x x – Waveguide for microwave photons – Usually g>>1 , but g~1 possible  0  ω Velocity: v/a C/J π R JC [ g<1 : Glazman & Larkin, PRL ‘ 97 ]   Q g Admittance: * e 2Z

Adding a Quantum Impurity C' C' C' C' Quantum J J J J Impurity C C C C C C • Artificial atom in • ? microwave waveguide • Motivations: – Quantum optics ► many-body effects – Condensed matter ► bosons

Adding a Quantum Impurity C' C' C' C'  Quantum J J J J Impurity   C C C C C C • Transport Measurement : – Charge ► conserved – Energy ► not conserved (dissipation) • Where does energy go? – Photons at different frequencies!

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 (J L/R <<J) to the leads – Only two charging states (n L/R =0,1 Cooper pairs ) C' C' C' L C' R C' C' C LR L R J L J R J J J J C L C R C C C C C C V g,R V g,L

System • Quantum impurity: two capacitively coupled grains, weakly coupled (J L/R <<J) to the leads – Only two charging states (n L/R =0,1 Cooper pairs ) C' L C' R C LR L R J L J R L R C L C R V g,R V g,L   dx     v       p 2 2 1 ( ) ( ) H g x g x p   lead x 2   , L R     H n U n n imp   LR L R  ,  L R          ( 0 ) i U ( 0 ) 1 0 H.c. H n J e         lead imp   , L R

Relation with Anderson Impurity C' L C' R C LR L R L R J L J R C L C R refermionization U t t   LR L R R L L R spinless t   /  ↑ / ↓  /  spinful – Spin anisotropy • Generalized Anderson – Luttinger liquid ( g≠ 1 ) impurity model : – Level-lead interaction

Kondo Description C' L C' R C LR L R 2e R L 2e J L J R C L C R t   /  ↑ / ↓  /  • In Coulomb blockade valley – “ local moment ” regime { n L +n R  1 }: – Singly occupied states – “ spin ” { S z =(n L -n R )/2 } – Equivalent to Kondo with noninteracting lead : I        xy 0 ( 0 ) ( 0 ) ( 0 ) H H S s S s I S s B S     K env z z z z z 2   Impurity spin: S      1 Environment spin:   0  H c c    0 s c c env , , k k s k s k , s s , s ' k ' , s ' 2 , k s , ' , , ' k k s s

Kondo Parameters C' L C' R C LR L R 2e L R 2e J L J R C L C R t   /  ↑ / ↓  /  I        xy 0 ( 0 ) ( 0 ) ( 0 ) H H S s S s I S s B S     K env z z z z z 2 • Schrieffer-Wolf (simplified expressions):      ,  B V V , z L R g L g R

Kondo Parameters C' L C' R C LR L R 2e L R 2e J L J R C L C R t   /  ↑ / ↓  /  I        xy 0 ( 0 ) ( 0 ) ( 0 ) H H S s S s I S s B S     K env z z z z z 2 • Schrieffer-Wolf (simplified expressions):      ,  B V V   2      , z L R g L g R 0 L R   1 1  0 – bandwidth J J   n   L R 2 I   n – local DoS     xy U   0 0 0 LR

Kondo Parameters C' L C' R C LR L R 2e 2e L R J L J R C L C R t   /  ↑ / ↓  /  I        xy 0 ( 0 ) ( 0 ) ( 0 ) H H S s S s I S s B S     K env z z z z z 2 • Schrieffer-Wolf (simplified expressions):      ,  B V V   2      , z L R g L g R 0 L R   1 1  0 – bandwidth J J   n   L R 2 I   n – local DoS     xy U   0 0 0 LR      2 2 2 1 1    J  1 gU   n           L R 1 1 I    p       Z v g 2 U     , L R   0 LR

Relevant Regime for SC C' L C' R C LR L R L R J L J R n I xy C L C R Kosterlitz-   1 1 J J   n   Thouless L R 2 I       xy U   transition 0 0 0 LR      2 2 2 1 1 J    n      L R 1 I       Z 2  U    L , R   0 LR    1 gU      1  p   v g n I z • Typically: g>>1 , U L/R >0 ► I z >>I xy , AFM Kondo • g<1 possible [Glazman & Larkin, ‘ 97] ► I z < – I xy , FM Kondo

Outline • Introduction – Quantum impurities – Microwave photons • System and relation to anisotropic Kondo • AC conductance : Photon elastic scattering • Photon inelastic scattering