1. Macroscopic quantum dynamics in superconducting nanocircuits Jens Siewert Institut für Theoretische Physik, Universität Regensburg, Germany Imperial College, London, 23 January 2007

2. Outline 1. What is macroscopic quantum coherence ? 2. Simple examples for macroscopic quantum dynamics 3. Towards quantum control in complex circuits Is there any macroscopic quantum dynamics at all? ‚Ingrediences‘ for superconducting nanocircuits that exhibit macroscopic quantum dynamics Cooper-pair box & Rabi oscillations Berry phases adiabatic passage more complex circuits circuit QED

3. macroscopic quantum coherence consider a collective variable -- correlated motion of ~10 10...10 20 degrees of freedom quantum dynamics of that variable Anderson 1964 Leggett 1966 examples: superconducting order parameter superfluid density magnetization in ferromagnets dipole moment in ferroelectrics ⋮

4. Reminder superconductivity / Josephson junctions superconducting phase transition collective variable tunnel junction between two superconductors   EJEJ I phase difference     classical dynamics of phase EJEJ R C corresponds to particle in a tilted washboard potential  U(  Josephson relations

5. Elements of superconducting nanocircuits small superconducting islands Josephson tunnel junction charge Q = 2e n capacitance C  1 fF charging energy E c = (2e) 2 / 2C phase difference  critical current I C Josephson energy E J = (ħ/2e) I C

6. a simple Hamilton function with these elements: required for quantum dynamics Macroscopic quantum dynamics H = E c ( n – n x ) 2 – E J cos  [ n,  ] = - i Anderson 1964 Ambegaokar, Eckern, Schön 1982

7. Flavors of superconducting qubits charge qubit flux qubit phase qubit E C >> E J E C << E J circuit computational states     Nakamura et al Tsukuba Delsing et al Gothenburg Wallraff et al Yale + ETH [Esteve et al Saclay] Martinis et al UCSBMooij et al Delft Clarke et al Berkeley island charge states persistent current states phase eigenstates in potential well x x x  I

8. Why is it so difficult to observe macroscopic quantum behavior? system of interest is always coupled to other degrees of freedom ( ``environment‘‘, in solids, e.g., electronic quasiparticles, phonons, eigenmodes of electric circuits, motion of defects...) environment ``measures‘‘ the system: coherence is destroyed e.g. that is X X decoherence classical mixture of states

9. Description of noise – (at least) two types of noise high-frequency noise, ohmic environments (ohmic parts of impedances, incomplete filtering, etc) low-frequency noise, 1/f noise (background charges, critical-current noise, noise caused by spurious two-level systems)

10. theory Some historical remarks AO Caldeira & AJ Leggett 1983 U Weiss 2000 J Clarke  1985... 1990 (SQUID experiments) M Devoret J Martinis macroscopic quantum tunneling C Tesche... macroscopic quantum coherence ? V Bouchiat et al 1997 superpositions Y Nakamura et al (charge qubit) Y Nakamura et al 1999 Rabi oscillations J Friedman et al 2000 C vdWal et al superpositions (flux qubit) Chiorescu et al 2003 Rabi oscillations A Wallraff et al 2004 circuit QED experiment

11. Important recent experimental results Vion, Esteve et al 2002 + charge-phase qubit as analogue of two-level atom (protocols like Ramsey fringes, spin echo) A Wallraff et al 2004 circuit QED setup 2005 enhancement visibility of Rabi oscillations (2006) resolving photon number states J Clarke et al (2006) flux qubits with tunable coupling, manipulations of two-qubit states J Martinis et al 2006 generation of Bell states with two phase qubits, two-qubit state tomography

12. 2. Simple examples

13. Cooper-pair box H = E c ( n – n x ) 2 – E J cos    n,   = -i Büttiker 1987 Bouchiat et al 1997  – phase difference across junction n – number of Cooper pairs n x =C g V – induced gate charge CgCg

14. typical energy scales H = E c ( n – n x ) 2 – E J cos     n,  = -i  – phase difference n – number of Cooper pairs n x =C g V – induced gate charge  – superconducting gap ``charge regime‘‘ 4  > E c > E J > kT E c  500  eV  5 K  (2  ) 100 GHz E J  100  eV kT  30 mK,   200  eV

15. operators written in the basis of charge eigenstates charging energy Josephson energy write use Fourier transform

16. Cooper-pair box as two-state system ``charge regime´´ E c > E J > k T | 0  - some reference state | 1  - charge of reference state + 1 Cooper-pair charge leakage is negligible Fazio, Palma, Siewert 1999 H = E c ( n – n x ) 2 – E J cos  effective description as a two-state system is possibl``charge qubit´´ Shnirman, Schön, Hermon 1997

17. Cooper-pair box as two-state system ``charge regime´´ E c > E J > k T | 0  - some reference state | 1  - charge of reference state + 1 Cooper-pair charge leakage is negligible Fazio, Palma, Siewert 1999 H = E c ( n – n x ) 2 – E J cos  effective description as a two-state system is possibl``charge qubit´´ Shnirman, Schön, Hermon 1997

18. Cooper-pair box as two-state system ``charge regime´´ E c > E J > k T | 0  - some reference state | 1  - charge of reference state + 1 Cooper-pair CHARGE leakage is negligible Fazio, Palma, Siewert 1999 H = E c ( n – n x ) 2 – E J cos  effective description as a two-state system is possibl``charge qubit´´ Shnirman, Schön, Hermon 1997

19. Artificial spin in a fictitious magnetic field  z   z  choose a basis: ByBy BxBx BzBz H = E c (n x -1/2)  z - E J /2  x this corresponds to H = –  B B    direction of spin island charge

20. Rabi oscillations in a Cooper-pair box Y Nakamura, Yu Pashkin, JS Tsai 1999 control of charging energy difference E c (1)- E c (0) by switching the gate voltage

21. Berry phases with a Cooper-pair box Berry phase for spin ½ = ½  solid angle  B(t) 

22. Berry phases with a Cooper-pair box Berry phase for spin ½ = ½  solid angle  B(t)  H = E c (n x -1/2)  z - E J /2 [ cos  (  )  x + sin  (  )  y ] realized with an asymmetric SQUID

23. Berry phases with a Cooper-pair box Berry phase for spin ½ = ½  solid angle  B(t)  H = E c (n x -1/2)  z - E J /2 [ cos  (  )  x + sin  (  )  y ] realized with an asymmetric SQUID Falci, Fazio, Palma, Siewert, Vedral 2000 protocol:spin flip charge measurement total Berry phase 2 

24. Adiabatic passage in a three-state system “  configuration” |0  |1  |e  K Bergmann, H Theuer, BW Shore 1998

25. Cooper-pair box driven by microwave field eigenstates are superpositions of charge states Vion et al: operate quantronium as a two-state system (Rabi oscillations, Ramsey fringes...) “quantronium” device H = E C ( n – n g (t)) 2 – E J cos  Vion et al 2002

26. STIRAP with Cooper-pair box important: avoid symmetry points (selection rules!) Nori et al 2005 rf couples diagonally in the basis of charge states  setup still works now: operation as a three-state system apply two microwave frequencies

27. action of decoherence study time evolution of density matrix with decoherence population transfer is robust against action of decoherence  should work with existing technology relaxation time ~ 300 ns dephasing time ~ 50 ns STIRAP with quantronium device

28. action of decoherence (II) decoherence is peculiar for solid-state devices we have used standard description (master equation) includes relaxation and dephasing relaxation time ~300 ns dephasing time ~ 50 ns

29. action of decoherence (II) low frequency components of noise (background charge fluctuations) may cause decoherence is peculiar for solid-state devices we have used standard description (master equation) includes relaxation and dephasing relaxation time ~300 ns dephasing time ~ 50 ns fluctuations with equal detunings fluctuations with different detunings  level crossings  described by effective dephasings ??

30. 3.Towards quantum control in complex circuits

31. Alternative circuit for adiabatic passage one excess Cooper pair in the system small superconducting islands effective Josephson couplings J 0, J 1 can be controlled by local magnetic fields possible application:  charge pumping ``tool kit´´ applied to three-level system ``  configuration ´´ Siewert, Brandes 2004 Siewert, Brandes, Falci 2006

32. Coupling two-state systems to harmonic oscillators Cooper-pair box + electromagnetic resonator  circuit QED Wallraff et al 2004 Cooper-pair box + nanomechanical resonator Armour, Blencowe, Schwab 2002 possible applications: - quantum communication - distance sensoring...

33. Circuit QED with electrical cavity Cooper-pair box + electromagnetic resonator  Jaynes-Cummings model coupling strength g ~ 10 -2  circuit QED Wallraff et al 2004

34. Cooper-pair box coupled to nanomechanical oscillator important: box-resonator coupling can be controlled with gate voltage Armour, Blencowe, Schwab 2002

35. Fock state generation in nanomechanical oscillator single-phonon generation process works with realistic parameters for circuit, resonator & decoherence rates resonator frequency ~ 1.5 GHz quality factor ~ 5000 relaxation time ~300 ns dephasing time ~ 50 ns quantum optics: Parkins, Marte, Zoller, Kimble 1993 expt: Hennrich, Kuhn, Rempe 2000 Siewert, Brandes, Falci, cond-mat/0509735

36. Summary macroscopic quantum coherence is possible and has been demonstrated new area of research at the border between quantum optics and solid-state physics many possibilities for tailor-made quantum systems - few-level systems (  few-level atoms) - coupling to resonators (  cavity QED) -... need to overcome system-specific decoherence

37. Superconducting nanocircuit for adiabatic passage vary effective Josephson couplings J 0, J 1 corresponds to rotation of ``dark state’’ effective Hamiltonian for charge states lowest in energy choose gate voltages such that

38. Geometrical phases Josephson junction as asymmetric SQUID ring scalar geometrical phases (Berry phases) Falci, Fazio, Palma, Siewert, Vedral 2000 four-level system (tripod scheme)  two dark states non-Abelian holonomies (Wilczek-Zee phases) Unanyan 1999 Faoro, Siewert, Fazio 2003 possible applications - ``holonomic quantum computation‘‘ Zanardi & Rasetti1999 - charge pumping  metrology

39. example I : two-island device choose working point, e.g.

40. example I : two-island device without decoherence  perfect population transfer possible during transfer, state is distributed over many charge states Siewert, Brandes, Falci 2006