Superconductivity

1. Superconductivity

2. Superconductivity The phonon-mediated attractive electron-electron interaction leads to the formation of Cooper-pairs which undergo a k-space condensation This condensate is a charged quantum liquid described by a macroscopic wave function The superconducting transition at produces a gap in the electronic excitation spectrum, thus removing all low energy excitations. Metal

3. Josephson junctions

4. Josephson junctions, are weak links connecting two superconducting leads/islands. They appear in various forms, e.g., as constrictions tunnel junctions Josephson relations describing the junction energy and the phase evolution

5. Two energy scales charge  0 phase current The particle number N = Q/2eand the phase are conjugate variables, i.e., we have a particle- phase duality [N,] = i (Anderson) Fock space fixed charge Q=2eN fixed phase 

6. Classical & quantum limits charge  0 phase current action Hamiltonian [N,] = i

7. H  2  Classical limit:gauge invariance and fluxoid quantization in a loop   unique gauge invariant phases  kinetic energy of currents () Free energy of a loop with inductance L: 2 junctions: 1 junction: In a small inductance loop, In a large inductance loop,

8. Quantum limit:Coulomb blockade and charge quantization on an island V Vn V2

9. Quantum limit:Coulomb blockade and charge quantization on an island  = 0 Vg V2 = 0 Account for the work done by the batteries when changing the island charge N

10. flux mixing by charge mixing by … in general

11. RCSJ model, adding dissipation additional shunt resistor, e.g., accounting for quasi-particle tunneling. Effective action describing the Resistively and Capacitively Shunted Josephson junction: ohmic dissipation (Caldeira-Leggett)

12. Schmid transition:T=0 quantum phase transition, driven by the environment H small inductance loop: two well potential  2 large inductance loop: particle in periodic potential weak dissipation limit with delocalized phase  strong dissipation limit with localized phase  -- superconducting junction (Leggett-Chakravarti)

13. classical computing

14. First `Programmer’ and The `full’ version of this machine was built in1991 by the Science Museum,London Ada Byron, Lady Lovelace 1815-1852 Enigma, cracked by Alan Turing with help of COLOSSUS Charles Babbage 1791-1871 Mechanics Inventor of the Difference Engine 1834

15. Electronics The ENIAC (Electronic Numerical Integrator and Computer) computer was built in 1946 Built at University of Pennsylvania, it included 18’000 tubes, weighed 30 tons, required 6 operators, and 160 m2 of space.

16. 1-bit gate: NOT Classical computer Hardware Vsd R Capacitors: input Vg Bits output 2-bit gate: AND Vsd Transistors: Gates Vg1 input Vg2 output Si-wafer R

17. Pentium Processor, 1997, Intel First Transistor, 1947 Bell Laboratories Bardeen, Brattain, & Shockley First Integrated Circuit, 1958 Jack Kilby, Texas Instruments Packed Device Transistors in 50 Years from 1 to 107 transistors

18. channel PTB gate 2 mm box Ultra-short channel Si-MOSFET, IBM 0.5 mm wide, 0.1 mm channel source drain Single Electron Transistor (SET), Al-Technology insulator gate Nanoscale Technology drain source Si-wafer Switch a MOSFET with 1000 electrons, while a SET requires only one!

19. Your track control in the car Applications Classical computers solve any computational task ….. Your bank account Your washing machine Your science Your agenda …. but some are really hard !

20. A modern computer can factor a 130-decimal-digits number (L = 300) in a few weeks - days; 1827365426354265930284950398726453672819048374987653426354857645283905612849667483920396069782635471628694637109586756325221365901 doubling L would take millions of years to carry out this calculation. Computational Complexity An input x is quantified via its information content L = log2 x. A calculation is characterized by the number s of steps (logical gates) involved. A problem is class P (efficiently solvable) if s is polynomial in L, A problem is deemed `hard’ (not in P) if s scales exponentially in L, A `classic’ hard problem is that of prime factorization: given a non-prime number N, find its factors; the best known algorithm scales as s ~ exp(2 L1/3 (lnL)2/3). A quantum computer would do the job within minutes

21. Encoding Decoding Public Key Encryption (Rivest, Shamir & Adleman, 1978) A quantum computer would crack this encryption scheme

22. Quantum computing

23. ``…nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical…” R. Feynman If one has N quantum two level systems (e.g. L spins) they can have different states. To describe such a system in classical computer one needs to have complex numbers, that requires exponentially large computational resources. Thus modeling even small quantum system on a classical computer is practically impossible task. But since Nature does it very efficiently one can try to use its ability to deal with quantum systems and to apply it also for computational problem.

24. Bits and Qubits spin language Aquantum bit (qubit) is the quantum mechanical generalization of a classical bit, a two-level system such as a spin, the polarization of a photon, or ring currents in a superconductor. Classical bit Quantum bit Physical realization via a quantum two-level system Physical realization via a charged/uncharged capacitor spins polarizations ring-currents

25. i f 0 1 1 0 E.g., with H = Hz we obtain the time evolution H = Hx we obtain the time evolution phase shifter amplitude shifter (a = 0) The possibilities to manipulate a classical bit are quite limited: The NOT-gate simply interchanges the two values 0 and 1 of the classical bit. Classical & quantum gates I On the other hand, manipulation of a quantum bit is much richer!For a spin / two-level system we can perform rotations around the x -, y -, and z - axis; placing the `spin’ S (with magnetic moment ) into a magnetic field H, the Hamiltonian produces the desired rotation. ….

26. i i f i i i i i f f f 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 Two-qubit gate: XOR (CNOT) Single-qubit gates: Rotations The target flips if the control is on 1 phase shifter amplitude shifter control Hadamard(basis change): H target irreversible Classical & quantum gates II NOT AND OR The combination of the classical gates allows us to construct all manipulations on classical bits. Is there a set of universal quantum gates ? How does such a set look like ?

27. Entangling two qubits put control qubit C into superposition state, then future gates act on two states simultaneously H C maximally entangled Bell state T i.e., target qubit T gets flipped AND non-flipped And subsequently: flipping a qubit in an entangled state modifies all its components

28. Quantum Algorithms

29. Grover’s Search algorithm (1997) searches unstructured database (e.g. telephone book) of N entries by steps. Quantum algorithms Shor’s Factorization algorithm (1994) finds prime factors in polynomial, rather than exponential time.

30. One can show, that no quantum algorithm can do better, than , thus it is optimal! It is not hard to prove, that classical algorithms can do no better, than just straight search through the list, requiring on average N/2 steps The `quantum speed-up’ ~ is greater than that achieved by Shor’s factorization algorithm (i) (ii) (iii) Although Grover’s algorithm doesn’t change complexity class it is not less fundamental, than Shor’s algorithm.

31. Task: Given an unstructured set of elements, find the one, that corresponds to the answer of some question. The goal is to to increase the amplitude of the component Grover’s Search algorithm The Algorithm Assume that the database contains N elements, N is some power of 2. Let there be only one solution, that we are looking for . We start with a homogeneous superposition of all basis states This can be achieved, e.g., by starting with n qubits in the state and applying the Hadamard transform

32. The algorithm will iterate the Grover rotation G a certain number of times to obtain state very close to . The Grover rotation consists of two parts: an oracle call and a reflection about . The oracle call is just quantum implementation of searching. There must exist unitary operator

33. Each Grover rotation rotates our state by an angle towards . We want Thus we should stop and measure after

34. Quantum Hardware

35. All hardware implementations of quantum computers have to deal with the conflicting requirements of controllability while minimizing the coupling to the environment in order to avoid decoherence. Physical implementation Solid state implementations enjoy good scalability & variability but require careful designs in order to avoid decoherence when trying to build Schrödinger cats

36. Parallel evolution providing the quantum speedup. Perturbations from the environment destroy the parallel evolution of the computation Network model of quantum computing (David Deutsch, 1985) initial state • each qubit can be prepared in some known state, • each qubit can be measured in a basis, • the qubits can be manipulated through quantum gates • the qubits are protected from decoherence final state

37. Physical implementations Quantum optics, NMR-schemes Good decoupling & precision: Solid state implementations Good scalability & variability: • trapped atoms (Cirac & Zoller) • photons in QED cavities (Monroe ea, Turchette ea) • molecular NMR (Gershenfeld & Chuang) • 31P in silicon (Kane) • spins on quantum dots (Loss & DiVincenzo) • 31P in silicon (Kane) • Josephson junctions, charge (Schön ea, Averin) • phase (Bocko ea, Mooij ea) All hardware implementations of quantum computers have to deal with the conflicting requirements of controllability while minimizing the coupling to the environment in order to avoid decoherence. Have to deal with individual atoms, photons, spins,…… Problems with control, interconnections, measurements. Have to deal with many degrees of freedom. Problems with decoherence.

38. not a problem up to Q ~ 104 2 charge qubits, interacting 4 9Be ions, deterministic Condensed Matter Condensed Matter Quantum optics Quantum optics Quantum optics 15 = 3 * 5 NMR Bell inequailty checks The rules of the game achievements • Find a system which emulates a spin / quantum two-level system and • which remains coherent • which can be manipulated(rotations) • which can be interconnected and entangled with other qubits • which can be projected (measured) • which carries out an algorithm(e.g., Shor’s prime factorization)

39. Superconducting quantum bits

40. Josephson junctions Superconducting qubits /2 Currents in a superconducting ring Charges on a superconducting island

41. Superconducting quantum bitsLoop vs Island In a superconducting ring, the wave function satisfies periodic boundary conditions. zero flux state A finite bias draws a Cooper-pair onto the island The macroscopic wave function winds once around the ring; the ring carries a current CP  flux one state These two states are degenerate at half-Cooper-pair frustration These two states are degenerate at half-flux frustration

42. Superconducting quantum bits Frustrate a ring with half-flux and obtain two degenerate flux states Frustrate an island with half-Cooper pair and obtain two degenerate charge states 0/2 CP/2 OR Produce a weak spot to flip between flux states: Josephson junction. Connect the box to allow charge hopping: Josephson junction. The Josephson junctions are key ingredients in any superconducting qubit design. Or, in other words, the Josephson junctions introduce the quantum dynamics into the superconducting structure. superconductor superconductor  0 phase

43. charge mixing by Charge Josephson Schönet al. Averin 1997 Three types Flux/Phase Ioffe et al. Orlando et al. 1999 superconductor  Bocko et al. 1997 0 phase  phase states V  

44. Manipulation

45.  charge mixing by Charge Phase Manipulation +ac-microwave voltage / current induces transitions across the gap OR + fast (non-adiabatic) switching induces (incomplete) Zener tunneling. flux mixing by tunneling gap

46. one-parameter (q) qubit two-parameter (q,Q) qubit Manipulation (general) ac-microwave induced transitions (NMR scheme) mixing Q trivial idle state phase shift decoupled states coupled states fast non-adiabatic switching, amplitude shift phase shift Q potential or dynamical About the NMR scheme: Otherqubits nearby are excited with probability and a precise addressing requires long times With Nqu qubits the distance between resonances is ~  / Nqu. The transition time of the k-th qubit is related to the ac-signal V via

47. SQUID- loop detector box 1m pulse gate Charge(Nakamura et al., 1999) Coherent devices detector Q e time fast gate voltage pulse Q

48. SQUID- loop detector box 1m pulse gate This time domain experiment shows coherent charge oscillations of 50 - 100 ps duration during a total coherence time of 2 ns. Charge(Nakamura et al., 1999) Coherent devices detector Q e time fast gate voltage pulse Q

49. Charge Balanced energy scales EJ ~ EC 104coherent charge oscillations observed via Ramsey fringes, Vion et al.,2002 Box Phase Second generation Ramsey interference free evolution ~100 coherent oscillations observed via Ramsey interference Chiorescu et al., 2003

50. Josephson-qubits Charge-qubits Qubit duet, entanglement Spectral analysis of interacting J-qubits Berkley et al., 2003 Spectral analysis of interacting q-qubits Pashkin et al., 2003