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Superconductivity Superconductivity Vk ' k † H k ck† ck Vk ' k ck† 'ck c c ' k k k , F k The phonon-mediated attractive electron-electron interaction leads to the formation of Cooper-pairs h D V n 1 3 = which undergo a k-space condensation This condensate is a charged quantum liquid described by a macroscopic wave function i r ,t r, t e . The superconducting transition at 1 N0V Tc 1.13 h D e k produces a gap 1.764 Tc in the electronic excitation spectrum, thus removing all low energy excitations. Metal k F kF k Josephson junctions Josephson junctions, are weak links connecting two superconducting leads/islands. They appear in various forms, e.g., as constrictions l g l r g r V tunnel junctions V Josephson relations describing the junction energy and the phase evolution i l l e l r r eir energy E ( ) Const EJ r l l r dynamic phase d / dt 2eV 1 cos E 1 cos J r l Two energy scales e2 charging (capacitive) energy EC 2C charge Q Q phase 0 current I I c 0 current (inductive) energy EJ 2 c The particle number N = Q/2e and the phase are conjugate variables, i.e., we have a particle phase duality [N, ] = i (Anderson) classical limit EJ EC fixed phase Fock space exp i exp i N 2 N 1 N quantum limit EJ EC N 1 N 2 fixed charge Q=2eN Classical & quantum limits e2 charging (capacitive) energy EC 2C charge Q Q phase 0 current I I c 0 current (inductive) energy EJ 2 c Einductive EJ 1 cos action L Ecapacitive C V 2 2 2 2 C 2 4e 2 2 4e2 C 2 E 1 cos J 2 Einductive : I c sin 2 Hamiltonian 2 H EC d 2 EJ 1 cos d ~ EJ 2 [N, ] = i N i dd Classical limit: gauge invariance and fluxoid quantization in a loop j s 20 A , H ext self hence 2 unique gauge invariant phases 0 2 and A 20 in the leads, A ds 20 ds leads A ds A ds junctions 20 2n 1 2 ... junctions n 0 1 2 ..... kinetic energy of currents () Free energy of a loop with inductance L: 1 .... F E1 1 cos E2 1 cos 2 .... 2L ext 1 2 2 . 2 junctions: 1 junction: In a small inductance loop, F example : ext 0 2 i ext i In a large inductance loop, 0 n 0 Quantum limit: Coulomb blockade and charge quantization on an island 2 1 electrostatic energy U Ci Visl Vi 2 i C1 V Cn C2 V C Ne C C i Vn Visl i i i i V2 Visl U 0 & add N electrons U 1 Ci C j Vi Vj 2C j i 2 ( Ne) 2 2C Quantum limit: Coulomb blockade and charge quantization on an island = 2 1 electrostatic energy U Ci Visl Vi 2 i C1 0 Cg C2 V C Ne C C i Visl Vg Visl U 0 & add N electrons i i i i V2 = U 1 Ci C j Vi Vj 2C j i 0 2 ( Ne) 2 2C Account for the work done by the batteries when changing the island charge N 2 1 E C V Ne const. 2C g g E N 2 EC e 2C 0 Vg N 0 N 1 Vg … in general E F EC 0 EJ 8 EJ EC EC E H EC N ext Q 2e 2J Q Q . 2 E F 14 Vg 0 EJ ext 0 2 p EJ EC e N 1 0 EC 0 2 H EC dd 2 EJ 1 cos N flux mixing by E C EC charge mixing by E J N N 1 EJ EC 0 0 Qext 2e 2 Vg RCSJ model, adding dissipation RQ C CRQ R 2Ice sin 2Ie , with the quantum resistance Ic I RQ 4e2 additional shunt resistor, e.g., accounting for quasi-particle tunneling. R Effective action describing the Resistively and Capacitively Shunted Josephson junction: 2 ' RQ 2 2 S d 16 E EJ 1 cos 4 R d ' 2 C ' ohmic dissipation (Caldeira-Leggett) . Schmid transition: T=0 quantum phase transition, driven by the environment H 0 2 F small inductance loop: two well potential 2 0 2 RQ R = 1, weak dissipation limit F large inductance loop: particle in periodic potential with delocalized phase 0 2 RQ R 1, strong dissipation limit with localized phase -- superconducting junction F F (Leggett-Chakravarti) 0 0 classical computing Mechanics First `Programmer’ and Inventor of the Difference Engine 1834 Ada Byron, Lady Lovelace 1815-1852 The `full’ version of this machine was built in1991 by the Science Museum,London Enigma, cracked by Alan Turing with help of COLOSSUS Charles Babbage 1791-1871 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. Classical computer Hardware Capacitors: Bits 0 : V 0, 1-bit gate: NOT Vsd R input Vg output 1 : V 0. 2-bit gate: AND Transistors: Gates Vg 0, closed, Vg 0, open. Vsd Vg1 input Vg2 output Si-wafer R Pentium Processor, 1997, Intel Transistors in 50 Years from 1 to 107 transistors First Integrated Circuit, 1958 Jack Kilby, Texas Instruments First Transistor, 1947 Bell Laboratories Bardeen, Brattain, & Shockley Packed Device Nanoscale Technology V insulator gate drain source Si-wafer PTB channel gate 2 mm box Ultra-short channel Si-MOSFET, IBM 0.5 mm wide, 0.1 mm channel Switch a MOSFET with 1000 electrons, while a SET requires only one! source drain Single Electron Transistor (SET), Al Applications Your track control in the car Classical computers solve any computational task ….. Your bank account Your washing machine Your agenda Your science …. but some are really hard ! 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, m s~L s ~ exp 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 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. A quantum computer would do the job within minutes Public Key Encryption (Rivest, Shamir & Adleman, 1978) Encoding Decoding M message s, N p q, (non-) public key s E M mod N t s , p ,q ME mod N ts 1 mod( p 1)(q 1) A quantum computer would crack this encryption scheme Quantum computing ``…nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical…” R. Feynman L If one has N quantum two level systems (e.g. L spins) they can have 2 different states. To describe such a system in classical computer one needs to have 2 L 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. Bits and Qubits A quantum bit (qubit) is the quantum mechanical generalization spin language 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 i a, 0 a e 0, 1 0 spins polarizations ring-currents Q V V 1 1 a Physical realization via a quantum two-level system Physical realization via a charged/uncharged capacitor Q 1 2 Classical & quantum gates I 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. i f 0 1 1 0 On the other hand, manipulation of a quantum bit is much richer! For a spin / twolevel system we can perform rotations around the x -, y -, and z - axis; placing the `spin’ S (with magnetic moment m) into a magnetic field H, the Hamiltonian m S H with produces the desired rotation. E.g.,…. H = Hz we obtain the time evolution e im H z t / 2 0 U , im H z t / 2 e 0 t eim H z t / 2 0 a eim H zt / 2 1 phase shifter 1 a2 . H = Hx we obtain the time evolution cos m H x t i sin m H xt U , i sin m H t cos m H t x x t cos m H x t 0 amplitude shifter (a = 0) i tan m H x t 1 . Classical & quantum gates II NOT The combination of the classical gates allows us to construct all manipulations on classical bits. i f 0 1 1 0 Is there a set of universal quantum gates ? How does such a set look like ? AND i 0 0 1 1 i 0 1 0 1 cos 2 iei sin 2 U i , cos 2 ie sin 2 Hadamard (basis change): H 1 1 2 1 1 : 1 f 0 0 0 1 i 0 0 1 1 i 0 1 0 1 f 0 1 1 1 The target flips if the control is on 1 1 0 U 0 0 phase shifter 0 a ei 1 OR Twoqubit gate: XOR (CNOT) Singlequbit gates: Rotations amplitude shifter irreversible 1 a2 . H 0 1 0 1 0 1 control 2, 2. target 0 0 0 1 0 0 0 0 1 0 1 0 i 0 0 1 1 i 0 1 0 1 f 0 1 1 0 Entangling two qubits C 0 H 0 0 1 put control qubit C into superposition state, then future gates act on two states simultaneously H 00 11 T 2 maximally entangled Bell state 0 XOR H 0 0 00 11 2 i.e., target qubit T gets flipped AND non-flipped And subsequently: flipping a qubit in an entangled state modifies all its components Quantum Algorithms Quantum algorithms Shor’s Factorization algorithm (1994) finds prime factors in polynomial, rather than exponential time. Grover’s Search algorithm (1997) searches unstructured database (e.g. telephone book) of N entries by N steps. Although Grover’s algorithm doesn’t change complexity class it is not less fundamental, than Shor’s algorithm. (i) 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 (ii) The `quantum speed-up’ ~ Shor’s factorization algorithm N is greater than that achieved by ~ exp(2(ln N )1/3) (iii) One can show, that no quantum algorithm can do better, than O thus it is optimal! N , Grover’s Search algorithm Task: Given an unstructured set of elements, find the one, that corresponds to the answer of some question. 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 x0. We start with a homogeneous superposition of all basis states N 1 x N x0 s 1 This can be achieved, e.g., by starting with n qubits in the state Hadamard transform Hn 0 H H 0 s 1 2 n 0 1 The goal is to to increase the amplitude of the n s x0 component 0 and applying the H 1 1 2 1 1 : 1 The algorithm will iterate the Grover rotation G a certain number of times to obtain state very close to 0 . The Grover rotation consists of two parts: an oracle call and a reflection about s . x The oracle call is just quantum implementation of searching. There must exist unitary operator O x0 x0 , O x x for x x0 x0 s O s Each Grover rotation rotates our state by an angle 2 towards x0 . x0 G s 2 s O s We want Gt s x0 , Thus we should stop and measure after sin( 2t ) 1 N 1 t 4 1 steps ( ) 2 N Quantum Hardware Physical implementation 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. Solid state implementations enjoy good scalability & variability but require careful designs in order to avoid decoherence when trying to build Schrödinger cats Network model of quantum computing (David Deutsch, 1985) initial state • each qubit can be prepared in some known state, 00000K 0000 . • each qubit can be measured in a basis, 01100K 1010 . • the qubits can be manipulated through quantum gates final state • the qubits are protected from decoherence f i Parallel evolution providing the quantum speedup. Perturbations from the environment destroy the parallel evolution of the computation Physical implementations Quantum optics, NMR-schemes Good decoupling & precision: • trapped atoms (Cirac & Zoller) • photons in QED cavities (Monroe ea, Turchette ea) • molecular NMR (Gershenfeld & Chuang) • 31P in silicon (Kane) Solid state implementations Good scalability & variability: • 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. The rules of the game achievements Find a system which emulates a spin / quantum two-level system and • which remains coherent Quantum optics not a problem Condensed Matter up to Q ~ 104 • which can be manipulated (rotations) • which can be interconnected and entangled with other qubits Condensed Matter 2 charge qubits, interacting Quantum optics 4 9Be ions, deterministic • which can be projected (measured) Quantum optics • which carries out an algorithm (e.g., Shor’s prime factorization) Bell inequailty checks NMR 15 = 3 * 5 Superconducting quantum bits Superconducting qubits Josephson junctions /2 Currents in a superconducting ring Charges on a superconducting island V Superconducting quantum bits Loop vs Island In a superconducting ring, the wave function x x exp i x Im satisfies periodic boundary conditions. 0 zero flux state Re A finite bias draws a Cooper-pair onto the island The macroscopic wave function winds once around the ring; the ring carries a current V CP h * x cc j x ~ 2mi flux one state These two states are degenerate at half-flux frustration These two states are degenerate at half-Cooper-pair frustration Superconducting quantum bits Frustrate a ring with half-flux and obtain two degenerate flux states OR Produce a weak spot to flip between flux states: Josephson junction. EJ Frustrate an island with half-Cooper pair and obtain two degenerate charge states CP/2 0/2 Connect the box to allow charge hopping: Josephson junction. EC EC EJ 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. e2 EC 2C superconductor phase L 2 16EC 0 2 EJ 1 cos superconductor EJ Ic 0 2c 2 H EC d 2 EJ 1 cos d Three types Charge Schön et al. Averin 1997 EC Flux/Phase C Bocko et al. 1997 EJ Vg EC n 1 n 0 2 EJ C Ioffe et al. Orlando et al. 1999 Ig n n Josephson superconductor phase 0 EC charge mixing by E J phase states V 2E J EJ EC 0 Vg 2 EJ 0 2 1/ 4 ~ p EC EJ exp EJ EC h p EJ p 8EJ EC Manipulation Manipulation Charge V Vg Phase Ig C EC EJ V 0 N N EJ I J , CJ n EC Vg C E 2E J E N EC N 1 charge mixing by E J EJ + ac-microwave voltage / current induces transitions across the gap OR + fast (non-adiabatic) 0 Vg phase amplitude shifter shifter switching induces (incomplete) Zener tunneling. 2 Ig 2 0 flux mixing by E C tunneling gap ~ p EC EJ 1/ 4 e 8 EJ EC one-parameter (q) qubit Manipulation (general) ac-microwave E induced transitions (NMR scheme) two-parameter (q,Q) qubit E mixing Q trivial idle state phase shift q q phase shift fast non-adiabatic switching, amplitude shift decoupled states q Q coupled states potential or dynamical About the NMR scheme: 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 top ~ Vk . Other qubits nearby are excited with 2 2 probability 2 Vk Vk and a precise addressing requires long times top N qu . Coherent devices Charge (Nakamura et al., 1999) SQUIDloop detector detector box 1mm pulse gate 0 2 Q Q t fast gate voltage pulse time Coherent devices Charge (Nakamura et al., 1999) SQUIDloop detector detector box 1mm pulse gate 0 2 Q Q t This time domain time fast gate experiment shows voltage pulse coherent charge oscillations of 50 100 ps duration during a total coherence time of 2 ns. Second generation Charge pulse 2 Balanced energy scales EJ ~ E C Phase 104 coherent charge oscillations observed via Ramsey fringes, Vion et al.,2002 Ramsey interference Box V free evolution Im pulse 2 ~100 coherent oscillations observed via Ramsey interference Chiorescu et al., 2003 Charge-qubits Qubit duet, entanglement Spectral analysis of interacting J-qubits Berkley et al., 2003 q0 Spectral analysis of interacting q-qubits Pashkin et al., 2003 q1 q Josephson-qubits Two-qubit gate CNOT (XOR) Device Result 1 0 U 0 0 0 0 0 1 0 0 0 0 1 0 1 0 Puls sequence Yamamoto et al., 2003 The End