Semiconductors: Applications in spintronics and quantum computation. Tatiana G. Rappoport Advanced Summer School Cinvestav 2005

Size: px

Start display at page:

Download "Semiconductors: Applications in spintronics and quantum computation. Tatiana G. Rappoport Advanced Summer School Cinvestav 2005"

Magdalen Goodman
5 years ago
Views:

1 Semiconductors: Applications in spintronics and quantum computation Advanced Summer School 1

2 I. Background II. Spintronics Spin generation (magnetic semiconductors) Spin detection III. Spintronics - electron transport in QD Spin injection Spin relaxation Coulomb blockade in quantum dots IV. Quantum computation with spins Introduction to quantum computation Using quantum dots for qc. 2

3 A short introduction to quantum computation 3

4 What is computation? Wikipedia: Computation can be defined as finding a solution to a problem from given inputs by means of an algorithm. Turing Machine: Church-Turing Thesis: Every 'function which would naturally be regarded as computable' can be computed by a Turing machine. 4

5 Feynman s computation Fact: Many quantum problems are not efficiently solved by a Turing machine (requires exponentially large amount of resources). Feynman in 1982: Use quantum computers to solve quantum problems! and.. some traditional problems, like factorization of big numbers, are also not efficiently solved by a Turing machine. What physics has to do with that? Turing Machine is very physical! So, let s use physics to define a new powerful machine (Deutsch 1985). 5

6 Factorization Classical Computer: deterministic and sequential Factorization of: x = x x = (x 0,x 1,x 2, x N ) Solution: Try all primes from 2 to x 1/2 2 N/2 =e N ln(2)/2 Quantum Computer: probabilistic and non-sequential For the same problem, there is Shor s algorithm: N 3 6

7 Classical bits Wikipedia: bit is the most basic information unit used in computing and information theory. A single bit is a one or a zero, a true or a false, a "flag" which is "on" or "off", or in general, the quantity of information required to distinguish two mutually exclusive states from each other Bits: = x

8 Quantum bits (qubits) qubits are two-level quantum systems Photons (left and right polarization) Spins (up and down) and many others. The basic requirement is: E ΔE 2» ΔE 1 ΔE 1 ΔE

9 Qubit: Formal representation! = " 0 + # 1 State vector in a 2D Hilbert state space! 2 + " 2 = 1 Bloch sphere representation:! = e i" cos(# /2) 0 + e i$ sin(# /2)1 z :! = 0 " 0 ( ) x :! = # / 2,$ = 0 " 1 ( ) y :! = # / 2,$ = # / 2 " 1 ( i1 ) 9

10 Classical logical gates For two bits 10

11 Quantum logical gates Example: Quantum gate Input qubit X Output qubit X 0 = 1 ; X 1 = 0! 0 + " 1 #! 1 + " 0! x = 0 1 $ # " 1 0 & % Matrix representation: 0 1! 0 = 1 $ # " 0 & %! 1 = 0 $ # " 1 & % 11

12 More on quantum gates One qubits quantum gates are rotations in the Bloch sphere General dynamics of a closed quantum system (including logic gates) can be represented as a unitary matrix:!' = U!,!' =! U U=UU =1 12

13 Pauli matrices are quantum gates X gate: Y gate: Z gate: X Y Z! x = 0 1 $ # " 1 0 & % or ' x " y = 0!i % $ # i 0 ' & or ( y " z = 1 0 % $ # 0!1 ' & or ( z 13

14 Measuring qubits A measurement can be done by a projection of ψ in the basis states (for example, 0 and 1 ). A Measurement can be done in any orthonormal and linear combination of basis states 0 & 1. Ex:! = " 0 + # 1 Using the orthonormal basis + 1 ( )! 1 ( 2 0! 1 ) P + = 1 2! + " 2 P # = 1 2! # " 2 14

15 More on measurement A measurement disturbs the system leaving it in a defined quantum state determined by the outcome! M Probabilistic Classical Bit Probabilistic Classical Bit 15

16 Multiple qubit systems The state space of n qubits can be represented by Tensor Product in Hilbert space with 2 n orthonormal base vectors ( Ψ = ψ 1 ψ 2 ) Ex: 0 0 (also represented by 00 or 0,0 But entangled states cannot be represented by a tensor product ( Ψ ψ 1 ψ 2 ) Ex:! = 1 2 ( ) States represented my tensor products can be separated and the measurement of one do not affect the other. Entangled states cannot be separated! 16

17 Multiple qubit gates 2-qubits states! = " " " " qubits gates: 00 =! 1$ # 0& # 0& # & " 0% 01 =! 0$ # 1& # 0& # & " 0% 10 =! 0$ # 0& # 1& # & " 0% 11 =! 0$ # 0& # 0& # & " 1% U CNOT =! $ # & # & # & " % 17

18 18

19 Experimental realizations Using quantum dots 19

20 Experimental requirements Scalable: large number of qubits Preparation of the state Operation time << Decoherence time (isolation) Universal set of gate operations Single-quantum measurements (readout) D. P. DiVincenzo, quant-ph/

21 Experimental realizations From micro To macro Quantum but hard to scale Easy to scale but hard to get quantum and MNR 21

22 Solid state realizations Single particle states in semiconductor structures Nuclear spins of P impurities in Si Electron spins in quantum dots QHE edge states Global quantum states of superconducting josephson circuit 22

Spin qubits Loss & DiVicenzo, PRA 57, 120 (1998) Qubit defined by the Zeeman-split levels of an electron in a quantum dot 1-qubit control using: Electron spin resonance

23 Spin qubits Loss & DiVicenzo, PRA 57, 120 (1998) Qubit defined by the Zeeman-split levels of an electron in a quantum dot 1-qubit control using: Electron spin resonance Modulated effective g-factor Local variation of magnetic field 2-qubits coupling: Exchange interaction between dots Read out through charge Experimental realization: DELFT Group 23

24 Using lateral quantum dots 24

25 Ground and excited states 25

26 Detecting Zeeman splitting E ΔE ΔE Z Magnetic field parallel to the 2DEG! R. Hanson et al. PRL

27 Coulomb blockade with spin 27

28 Zeeman splitting 20 μev per Tesla! 28

29 1 qubit gate Not done yet. One of the big issues Some proposals Electron spin resonance 29

30 Read-out Coupling charge and spin Pulsed variation of V Real time monitor of tunneling 30

31 Zeeman energy Elzerman et al.nature 2004 Spin dependent tunneling rate Hanson et al.prl

32 32

33 Two qubits Two qubits gates: Exchange interaction between spins controled by tunnel barrier H = JS " S = J (!! +!! +!! 2) 1 2 x1 x2 y1 y2 z1 z 33

34 Transport in coupled dots 34

2.0 Basic Elements of a Quantum Information Processor. 2.1 Classical information processing The carrier of information

QSIT09.L03 Page 1 2.0 Basic Elements of a Quantum Information Processor 2.1 Classical information processing 2.1.1 The carrier of information - binary representation of information as bits (Binary digits).