Michael Mehring Physikalisches Institut, Univ. Stuttgart, Germany. Coworkers. A. Heidebrecht, J. Mende, W. Scherer

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1 Decoherence and Entanglement Tomography in Solids Michael Mehring Physikalisches Institut, Univ. Stuttgart, Germany Coworkers A. Heidebrecht, J. Mende, W. Scherer 5 N@C 60 and 3 P@C 60 Cooperation Boris Naydenov and Wolfgang Harneit FU Berlin

2 Richard Feynman was one of the first physicists, who contemplated about quantum computing already in 98 R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwart, J. Appl. Phys. 8, 49 (957).

3 The Quantum Bit (Qubit) Stern Gerlach Experiment (9)

4 NMR-Quantum-Computing a là Kane A silicon-based nuclear spin quantum computer B.E.Kane Nature 393, 33 (May 998)

5 Array of Quantum Dots with Spins Single spin in Q-dot F. H. L. Koppens et. al. Nature, 44, 766 (006) Daniel Loss proposal

6 Quantumstates of a Qubit (Spin ½) The Spin Density Matrix Population 0 Superposition 0 Phase X Phase Y 0

7 Basic density matrix algebra Time dependence Magnus expansion with Zeroth order term Average Hamiltonian U.Haeberlen amd J. S. Waugh, Phys. Rev. 85, 40 (969) M.M. and V.A. Weberuss, Object Oriented Magnetic Resonance, Academic Press, 00

8 The Single Qubit Basis:, or, or 0, z 0 0 ;, com plex e i i co s 0 e sin y Bloch sphere x Bloch s equations with R = /T and R = /(T )+/T 8:35

9 Single Qubit Gates 0 0 i e 0 0 e i universal gates Hadamard H 0 H H H 0 0 H 0

10 T Relaxation polarisation saturation recovery recovery

11 T Relaxation Decay of the superposition state Recovery of the equilibrium state (takes longer) Only transitions between eigenstates: T = T General case (phase flucutations): T << T

12 Distribution of local fields Spin echo E. L. Hahn, Phys. Rev. 80, 580 (950) spin echo 0

13 Random phasefluctuations of a qubit with N avg = N avg = 4 N avg = 5 N avg = 60 N avg = 50 N avg = 000

14 Decoherence caused by fluctuations T relaxation assuming and else results in

15 Spin lattice relaxation T

16 Two Qubit Gates Bell states (entangled states):

17 Decoherence of a two qubit entangled state

18 The Greek LAOCOON and Quantum Entanglement The EPR Pair Einstein, Podolsky and Rosen, Phys. Rev. 47, 777 (935) EPR H 0 Hadamard U CNOT

19 Quantencomputing in Hilbertspace 8 qubits span a 56 dimensional Hilbertspace Quantum evolution in Hilbertspace Uˆ 0 U t t 0 Quantumcomputing is: Preparation+Superposition+Entanglement+Projection

20 Proposals for Quantumcomputing with 60 Wolfgang Harneit, Phys. Rev. A 65, 03 (00) D. Suter and K. Lim, Phys. Rev. A. 65, (00) 9:0

21 The pseudo pure initial state M.M. and W. Scherer. Phys. Rev. Lett. 93, (004) W. Scherer and M. Mehring, J. Chem. Phys. 8, (008) 7 7 S 3 5 N@C 60 I I S 4 4 z ( 8 ) P P ( ) / ˆ I I I I I z z z z two qubit subspace

22 Pulses and unitary evolutions Hahn echo sequence free evolution window

23 Preparation and tomography of entangled states in 5 N@C 60 M.M. and W. Scherer. Phys. Rev. Lett. 93, (004) W. Scherer and M. Mehring, J. Chem. Phys. 8, (008)

24 Phase Encoding of Entangled States Application of phase rotations about the z-direction phase frequencies = MHz; = MHz = MHz e i 3 e i 3 = 6 MHz i 3 i 3 e e

25 Decoherence of the entangled state (+) phase frequencies =0 MHz; = 4 MHz 3 = 6 MHz with

26 Two Qubit Subsystems in 3 P@C 60 Cooperation with W. Harneit, FU Berlin Preliminary results (J. Mende, B. Naydenov) Reduced symmetry Lifting of degeneracies Planned experiments: Swap and qubit cloning, decoherence free subspaces IWEPNM 007, Kirchberg 9:0

27 Find the Prime Factors (Shor Algorithm) It is a simple task to build the product of two large prime numbers p und q. Calculating n = p x q is easy. But: =? 799 x 7389 is extremely demanding and requires log(n) steps with arbitrary large superpolynomial Factoring a 400 digit number would take 0 0 years with today's fastest computers P. Shor(994): Quantum computer requires only O[(ln n) 3 ] steps A quantum computer based on the current fastest clock rates would factor a 400 digit number in only about 3 years. 9:05

28 Implementing the Shor Algorithm on a Nuclear Spin Quantumcomputer L. M. K. Vandersypen et al., Nature 44, 883 (00) We need two quantum registers R(3 qubit) und R(4 qubit), which contain x and f(x): R> R>= x,f(x)> = n3,n,n> m4,m3,m,m> Superposition: 8 x 7 0 x x,7 m o d 5 8 0,, 7, 4 3,3 4, 5, 7 6, 4 7,3

29 Dynamic Decoupling CPMG sequence Carr, Purcell, Meiboom, Gill, Phys. Rev. A 94, 530 (954) Rev. Sc. Instr. 9, 688 (958) Phase encoded CPMG sequence. see number factoring

30 NMR experiment factors numbers with Gauss sums M M A l exp im N M m 0 N l M M C l cos m N M m 0 N l from number theory = M. M., K. Müller, W. Merkel, I. S. Averbukh, W. P. Schleich, Phys. Rev. Lett. 98, 050 (007)

31 H implementation of the Gauss sum algorithm with phase controlled pulse sequence M. M., K. Müller, W. Merkel, I. S. Averbukh, W. P. Schleich, Phys. Rev. Lett. 98, 050 (007) 30 th Anniversary Kernresonanzspektroskopie University Ulm 006

32 The S-Bus Concept M. M. and J. Mende Phys. Rev. A 73, (006)

33 Energy Ce:CaF single crystal: The qubyte+ m S Mims ENDOR m S 9:5

34 Implementation of the Collins version of the Deutsch-algorithm H U 00 H M. M. and J. Mende Phys. Rev. A 73, (006)

35 Two Qubit ( 9 F) Entanglement in the S-Bus Ce:CaF J. Mende, doctoral thesis, Univ. Stuttgart (005)

36 Tomografie of large qubit registers by selective phase encoding Q Q Q Q 3Q Q Q 3Q 4Q 0Q Q Q 0Q Q Q 3Q 0Q Q Q 3Q 4Q For more details see: M. M. and J. Mende Phys. Rev. A 73, (006)

37 Detection of multi-qubit correlations pulse prep ˆ ˆ ˆ ˆ ˆ I ˆ ˆ z zi I I I S z z 3 z 4 z z I I I x x 3 x 4 x Sˆ S ˆ Iˆ ˆ S ˆ 4 z Iˆ Iˆ Iˆ Iˆ 3 4 ;, 3;4 S ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ z I I I I I I I I S ˆ 4 z 3 4 Iˆ Iˆ Iˆ Iˆ Iˆ Iˆ Iˆ Iˆ ,,3; 4 0 Quantum Quantum 4 Quantum

38 Fighting decoherence in a hostile environment S-bus system with 9 F qubits (spins I = /) surrounding the master qubit Ce 3+ (spin S = /) 9 F FID (one site) 9 F rotary echos T * decoherence time decouple a particular 9 F from the rest time increments

39 Decoupling and selective recoupling A. Heidebrecht, J. Mende and M.M. Solid State Nucl. Magn. Reson. 9, 90 (006) Spin echo double resonance (SEDOR) M. Emschwiller, E. L. Hahn and D. Kaplan, Phys. Rev. 8, 44 (960) coupling constant 4.6 khz

40 Decoherence free subspaces? A. Heidebrecht, J. Mende and M.M. Solid State Nucl. Magn. Reson., 9:90--94, 006 SEDOR CPMG Spin-Locking

41 Fight Decoherence The 8 block evolution A. Heidebrecht, J. Mende and M.M. Solid State Nucl. Magn. Reson., 9, , 006

42 Global decoupling and selective recoupling Average Hamiltonian: Average cycle Hamiltonian:

43 Entanglement without direct spin-spin interaction? S I I Initial density matrix 4 I a I a I a I I z z z z CNOT evolution Entanglement + Indirect spin-spin interactions allow long distance entanglement Preliminary result A. Heidebrecht, doctoral thesis Univ. Stuttgart (007) Enrico fermi School, Varenna 008

44 J. Mende A. Heidebrecht W. Scherer