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Exploring Quantum Superconducting Qubits in Electrical Circuits

Dive into the world of quantum physics with Josephson circuits, from discrete energy levels to quantum information on chips. Learn about superconducting qubits, readout methods, and processor architectures. Discover the Hamiltonian of electrical circuits, Cooper-pair boxes, and quantum processor design. Delve into real atom quantization with examples like the hydrogen atom and harmonic oscillators. Understand the Josephson junction and its role in building superconducting qubits. Explore the Hamiltonian of arbitrary circuits and how to identify independent variables. Uncover different types of qubits like Cooper-pair boxes, flux qubits, and charge qubits. Join the journey to understand the quantum behavior of "man-made" circuits!

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Exploring Quantum Superconducting Qubits in Electrical Circuits

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  1. Josephson qubits P. Bertet SPEC, CEA Saclay (France), Quantronics group

  2. Introduction : Josephson circuits for quantum physics From a fundamental question (25 yearsago) …. CAN MACROSCOPIC « MAN-MADE » ELECTRICAL CIRCUITS BEHAVE QUANTUM-MECHANICALLY ???? YES THEY CAN Discreteenergylevels M.H. Devoret, J.M. Martinis and J. Clarke, PRL85, 1908 (1985) M.H. Devoret, J.M. Martinis and J. Clarke, PRL85, 1543 (1985) … to genuine quantum information and quantum opticson a chip … ANDMUCH MORE TO COME !! M. Hofheinz et al., Nature (2009) Q. state engineering and Tomography M. Neeley et al., Nature (2011) 3-qubitentanglement

  3. Outline Lecture 1: Basics of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics Lecture 3: 2-qubit gates and quantum processor architectures

  4. Outline Lecture 1: Basics of superconducting qubits 1) Introduction: Hamiltonian of an electrical circuit 2) The Cooper-pair box 3) Decoherence of superconducting qubits Lecture 2: Qubitreadout and circuit quantum electrodynamics Lecture 3: 2-qubitgates and quantum processor architectures

  5. Real atoms quantization Hydrogen atom |3> |2> |1> E01=E1-E0=hn01 « two-level atom » |0> I.1) Introduction

  6. Hydrogen atom Optical Bloch equations + spontaneous emission G laser Spectroscopy (weak field) Rabi oscillations (short pulses, strong field at n=n01) |0> |0> n01 1 FWHM =G/2p P(1) P(1) (a.u) |1> 0 Time (a.u) Frequency n (a.u) Real atoms quantization I.1) Introduction

  7. Electrical harmonic oscillator f +q k x -q m |4> E |3> Quantum regime ?? |2> |1> |0> X,f I.1) Introduction

  8. 2 conditions : |4> E |3> |2> Typic : |1> |0> At T=30mK : X,f |4> E |3> |2> |1> OK if dissipation negligible |0> Superconductors at T<<Tc X,f LC oscillator in the quantum regime ? I.1) Introduction

  9. Microwave superconducting resonators Q=4 1010 at 51GHz and at 1K Tc(Nb)=9.2K Quantum regime S. Kuhr et al., APL90, 164101 (2006) T<<Tc : dissipation negligble at GHz frequencies I.1) Introduction

  10. |4> E |3> |2> |1> |0> X,f Necessity of anharmonicity f +q -q How to prepare |1> ? Need non-linear and non dissipative element : Josephson junction I.1) Introduction

  11. Josephson DC relation : Josephson AC relation : B. Josephson, Phys. Lett.1, 251 (1962) P.W. Anderson & J.M. Rowell, Phys. Rev.10, 230 (1963) S. Shapiro, Phys. Rev.11, 80 (1963) Basics of the Josephson junction The building block of superconducting qubits I.1) Introduction

  12. Josephson DC relation : Classical variables ?? Josephson AC relation : NON-LINEAR INDUCTANCE POTENTIAL ENERGY Basics of the Josephson junction The building block of superconducting qubits I.1) Introduction

  13. Hamiltonian of an arbitrary circuit = = Correct procedure described in : HAMILTONIAN ??? M. H. Devoret, p. 351 in Quantum fluctuations (Les Houches 1995) G. Burkard et al., Phys. Rev. B 69, 064503 (2004) G. Wendin and V. Shumeiko, cond-mat/0508729 M.H. Devoret, lectures at Collège de France (2008) accessible online

  14. Hamiltonian of an arbitrary circuit = = HAMILTONIAN ??? 1) Identify the relevant independent circuit variables 2) Write the circuit Lagrangian 3) Determine the canonical conjugate variables and the Hamiltonian

  15. Hamiltonian of an arbitrary circuit branch node = = Identifying the relevant independent circuit variables 1) Choosereferencenode (ground)

  16. Hamiltonian of an arbitrary circuit = = Identifying the relevant independent circuit variables 1) Choosereferencenode (ground) 2) Choose « spanningtree » (no loop)

  17. Hamiltonian of an arbitrary circuit Fd Fc = Fb Fe Fa = Identifying the relevant independent circuit variables 1) Choosereferencenode (ground) 2) Choose « spanningtree » (no loop) 3) Define « treebranch fluxes »

  18. Hamiltonian of an arbitrary circuit Fd Fc F4= Fb+Fd F3 F2 = Fb Fe Fa F1 F5 = Identifying the relevant independent circuit variables 1) Choosereferencenode (ground) 2) Choose « spanningtree » (no loop) 3) Define « treebranch fluxes » 4) Definenode fluxes = sum of branch fluxes fromground

  19. Hamiltonian of an arbitrary circuit Fd Fc F4= Fb+Fd F3 F2 = Fb Fe Fa F1 F5 = WriteLagrangian takingintoaccountconstraintsimposed by externalbiases (fluxes or charges) F2 F1 Fext

  20. Hamiltonian of an arbitrary circuit Fd Fc F4= Fb+Fd F3 F2 = Fb Fe Fa F1 F5 = Conjugate variables : With Quantum Hamiltonian ClassicalHamiltonian or with

  21. NIST Santa Barbara Different types of qubits Cooper-pair boxes Flux qubits Phase qubits TU Delft MIT Berkeley NEC Saclay Chalmers NEC Yale ETH Zurich Charge qubit/Quantronium/Transmon Junctions sizes 100mm2 .04mm2 .01 to 0.04 mm2 Shape Of the Potential Energy I.2) Cooper-Pair Box

  22. NIST Santa Barbara Different types of qubits Cooper-pair boxes Flux qubits Phase qubits TU Delft MIT Berkeley NEC Saclay Chalmers NEC Yale ETH Zurich Charge qubit/Quantronium/Transmon Junctions sizes 100mm2 .04mm2 .01 to 0.04 mm2 Shape Of the Potential Energy I.2) Cooper-Pair Box

  23. The Cooper-Pair Box 1 degree of freedom 1 knob « chargingenergy »

  24. small or The split CPB inductance 2 d° of freedom 2 knobs

  25. The split CPB 1 d° of freedom 2 knobs tunable

  26. Energy levels of the CPB ( ) Solve either in charge basis |N> Diagonalize I.2) Cooper-Pair Box

  27. Energy levels of the CPB ) … or in phasebasis |q> ( Solve Mathieu equation I.2) Cooper-Pair Box

  28. Two simple limits : (1) (charge regime) EJ/Ec=0 Energy |N=0> |N=1> |N=2> Ng q N N q

  29. Two simple limits : (1) (charge regime) EJ/Ec=0.1 E2(Ng) Energy E1(Ng) QUBIT E0(Ng) Ng=0.01 Ng

  30. Two simple limits : (1) (charge regime) EJ/Ec=0.1 E2(Ng) Energy E1(Ng) QUBIT E0(Ng) Ng=0.5 Ng

  31. From to EJ/Ec=0.5 EJ/Ec=2 Energy Ng Ng EJ/Ec=10 EJ/Ec=5 Energy Ng Ng STILL A QUBIT !

  32. Two simple limits : (2) (phase regime) J. Koch et al., PRA (2008) EJ/Ec=10 Ng I.2) Cooper-Pair Box

  33. Experimentalspectrum of a transmon J. Schreier et al., PRB (2008)

  34. One-qubit gates Ng(t)=DNgcoswt F TRANSMON QUBIT transmon drive Two-level approximation I.2) Cooper-Pair Box

  35. One-qubit gates w=w01 f0 F Rotation : X TRANSMON QUBIT |0> Z X Y |1> I.2) Cooper-Pair Box

  36. One-qubit gates w=w01 f0 f0+f F Xf Rotation : X TRANSMON QUBIT |0> Z X Y f Xf |1> I.2) Cooper-Pair Box

  37. One-qubit gates w=w01 f0 f0+f F+dF Xf Rotation : X TRANSMON QUBIT |0> Z Z rotation X Y f Xf All rotations on Bloch sphere Fidelity ? J.M. Chow et al., PRL 102, 090502 (2009) |1> I.2) Cooper-Pair Box

  38. Decoherence Ng+dNg(t) F+dF(t) Noise in Hamiltonian parameters DECOHERENCE MAJOR OBSTACLE TO QUANTUM COMPUTING I.3) Decoherence

  39. Pure dephasing li(t) t Low-frequency noise Decoherence in superconducting qubits (Ithier et al., PRB 72, 134519, 2005) Relaxation (Spontaneous emission) environmental density of modes at qubit frequency I.3) Decoherence

  40. Decoherence Ng+dNg(t) F+dF(t) Noise in Hamiltonian parameters DECOHERENCE Origin of the noise ??? I.3) Decoherence

  41. Decoherence Ng+dNg(t) F+dF(t) Noise in Hamiltonian parameters R DECOHERENCE R Origin of the noise ??? 1) ELECTROMAGNETIC Low-frequency : Johnson-Nyquist due to thermal noise High-frequency : spontaneous emission (quantum noise) Under control I.3) Decoherence

  42. Decoherence e- Spin flips Ng+dNg(t) F+dF(t) Noise in Hamiltonian parameters R DECOHERENCE R Origin of the noise ??? Flux noise Charge noise 2) MICROSCOPIC Low-frequency noise well studied : High-frequency (GHz) microscopic noise TOTALLY UNKNOWN !! I.3) Decoherence

  43. Decoherence e- Ng+dNg(t) Spin flips F+dF(t) Noise in Hamiltonian parameters R DECOHERENCE R Origin of the noise ??? Flux noise Charge noise 2) MICROSCOPIC Low-frequency noise well studied : CPB in charge regime High-frequency (GHz) microscopic noise TOTALLY UNKNOWN !! Transmon I.3) Decoherence

  44. State-of-the-art coherence times T1=1-2ms T2=1-3ms Schreier et al., PRB 77, 180502 (2008) I.3) Decoherence

  45. Very recent breakthrough: transmon in 3D cavity H. Paik et al., quant-ph (2011) T1=60ms T2=15ms ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET I.3) Decoherence

  46. Fabrication techniques Al/Al2O3/Al junctions O2 small junctions e-beam lithography 1) e-beam patterning 2) development 3) first evaporation 4) oxidation 5) second evap. 6) lift-off 7) electrical test PMMA PMMA-MAA SiO2 small junctions Multi angle shadow evaporation I.3) Decoherence

  47. gate 160 x160 nm QUANTRONIUM (Saclay group) I.3) Decoherence

  48. FLUX-QUBIT (Delft group) I.3) Decoherence

  49. TRANSMON QUBIT (Saclay group) 40mm I.3) Decoherence 2mm

  50. END OF FIRST LECTURE

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