Quantum information processing in semiconductors

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1 FIRST Quantum information processing in semiconductors Yasuhiro Tokura (University of Tsukuba, NTT BRL) Part I August 14, afternoon I Part II August 15, morning I Part III August 15, morning II Summar School 2012 FIRST 1

2 1985 Peierls transition Monte Carlo 1985 NTT CNT* 2002 Pauli (ICORP) 2012 *CNT: Carbon nanotube Summar School 2012 FIRST 2

3 Plan of this lecture Part I (Aug. 14, afternoon I) Basics of semiconductor system +CNT, Graphene Quantum dots, Double quantum dots: Hubbard model Quantum point contacts: charge detection Charge qubits Charge detection Part II (Aug. 15, morning I) Which path detector, continuous weak measurement Spin detection - Spin to charge conversion Exchange based (only) qubits Single qubit manipulations, Hybrid qubit Two-qubit interaction Part III (Aug. 15, morning II) Single spin qubits Single spin manipulations: magnetic, electric (µ-magnet, SOI, etc) Two or more qubit manipulation Hyperfine interaction and material issues Coupling remote qubits (Resonator coupling, Flying qubits) Prospective Summar School 2012 FIRST 3

4 Part I Semiconductor Quantum Dots Quantum Point Contacts Summar School 2012 FIRST 4

Criteria of realizing quantum computers D. P. DiVincenzo Fortschr. Phys. (2000). 1. A scalable physical system with well characterized qubits ( Electrically controlled spin qubits 2.

5 Criteria of realizing quantum computers D. P. DiVincenzo Fortschr. Phys. (2000). 1. A scalable physical system with well characterized qubits ( Electrically controlled spin qubits 2. The ability to initialize the state of the qubits to a simple fiducial state 3. Long relevant decoherence times, much longer than the gate operation time 4. A universal set of quantum gates 5. A qubit-specific measurement capability Summar School 2012 FIRST 5

6 Main subject of this talk How can we realize coherent system in semiconductor? What is the current status of the research? (How good, how many ) To which direction can we go? Can we see straightforward milestones? Or do we need another big breakthrough? Summar School 2012 FIRST 6

7 One sheet summary of semiconductor We can enjoy the variety of material features and their combinations. Band gap E gap, - Important for optical interface Effective mass m* - scales `Quantum confinement, zero - metallic CNT/Graphene Multi-valley (Silicon, CNT, Graphene) additional quantum index? Lande g-factor g* - magnetic coupling of spin, electrically tunable Spin-orbit interaction (SOI) α,β enabling electric control of spin / topological states, Majorana Hyperfine coupling A enemy of spin coherence, isotope engineering Deformation/Piezoelectric Phonon Ξ, h 14 another source of decoherence Summar School 2012 FIRST 7

8 Carbon nanotube and Graphene Massless Dirac Fermion K and K valleys B. J. LeRoy, et al., Nature 432, 371 (2004). A. K. Geim, et al., Science 324, 1530 (2009). Summar School 2012 FIRST 8

Nano-technology ʼ # of transistors 10 9 10 8 Size in

nano-technology for the semiconductor devices may help

9 Nano-technology ʼ # of transistors Size in micron <32nm Fin-FET Potentially, the developed nano-technology for the semiconductor devices may help also to realize scalable quantum system. Summar School 2012 FIRST 9

10 Isolation of single charge and spin In contrast to naturally well-isolated systems like cold-atoms, ions, and photons, forming quantum two-level systems (qubits) in condensed matter is not a easy task. Controlling single charge one-by-one had been achieved in metallic small grains, but these systems cannot be a candidate of qubits, except for the superconducting states, where finite gap is formed and macroscopic quantum coherence is maintained. Therefore, isolation of single electron (artificial atom) is an important milestone to realize well-defined qubits in condensed matter. Summar School 2012 FIRST 10

Quantum dots (QDs) Environment in thermal equilibrium k B T µ Total C capacitance N Tunnel barrier Quantized levels ε i Stability condition of N electrons in the QD: Total energy of N electrons E(N)

11 Quantum dots (QDs) Environment in thermal equilibrium k B T µ Total C capacitance N Tunnel barrier Quantized levels ε i Stability condition of N electrons in the QD: Total energy of N electrons E(N) N ε i + N C 2 U i=1 Constant interaction model: U e2 2C No addition µ k BT E(N + 1) E(N) ~ UN+ε N+1 No escape U + ε N+1 ε N k B T Addition energy µ 1 2 k BT E(N) E(N 1) ~ U(N-1)+ε N Coulomb blockade for very low temperatures, small capacitance, large quantization energy Summar School 2012 FIRST 11

Fabrication of QDs Typical top-down approach, starting

hetero-interface, and depleting selective areas by the

Advent of one-electron single QDs Tarucha et al.

12 Fabrication of QDs Typical top-down approach, starting from two-dimensional (2D) electron gas formed at the hetero-interface, and depleting selective areas by the surface metallic gates negatively biased. Advent of one-electron single QDs Tarucha et al. PRL 96 Jung et al. APL05 Ciorga et al. PRB 02 Summar School 2012 FIRST 12

13 Spins in a QD N=1 a b D + D - N=2 Simple But, how can we probe these? Summar School 2012 FIRST 13

14 QDs coupled to the leads V SD C g gate capacitance I V g Conductance G=I/V SD is peaked when ( V SD << k B T) µ = E(N) E(N 1) U(N 1) + ε N C g C ev g Conductance 0 Coulomb blockade region N=0 N=1 N=2 N=3 Gate voltage V g Summar School 2012 FIRST 14

15 Tunneling spectroscopy U+ε 2 -ε 1 U+ε 3 -ε 2 V SD <<k B T Conductance N=0 N=1 N=2 N=3 V SD >> k B T 0 Conductance ε 2 Gate voltage V g ε 2 N=1 N=2 av SD ev SD Energy ε 2 ε 2 *a: lever-arm factor 0 Gate voltage V g V g Summar School 2012 FIRST 15

44 (bulk GaAs) g = 0.44 g =0.23 (AlGaAs/GaAs/ AlGaAs QD) g = 0.

16 g-factor in Quantum Dot: Excitation spectroscopy! di/dv Zeeman splitting µev) ( g =0.44 (bulk GaAs) g = 0.44 g =0.23 (AlGaAs/GaAs/ AlGaAs QD) g = B (T) Kodera et al. Physica E (04) Zeeman splitting! gµ B B" >! >! 16

17 Relaxation time (T 1 ) Energy relaxation time without changing spin state is very fast ~ 1ps (typical), by electron-phonon scatterings. Spin flip relaxation is forbidden in the lowest order, but is possible by spin-orbit interaction, hyperfine coupling etc. S T +,T - T ± = T ± + SOI δe ST S, S = S SOI δe ST T ± Fermi s golden rule Λ x q S H e p T ± Summar School 2012 FIRST 17

18 Determining spin T 1 Spin relaxation time is evaluated by pump-and-probe method T 1 > 200µs T. Fujisawa et al., Nature 419, 278 (2002). T 1 ~ 1s S. Amasha, et al., Phys. Rev. Lett. 100, (2008). Summar School 2012 FIRST 18

Charging energy U Coupled quantum dots Quantum mechanical tunneling t Charging energy U Energy levels ε Li Energy levels ε Ri V Inter-dot Coulomb interaction Minimum

19 Charging energy U Coupled quantum dots Quantum mechanical tunneling t Charging energy U Energy levels ε Li Energy levels ε Ri V Inter-dot Coulomb interaction Minimum realization of Hubbard model: ˆn µ,σ â µ,σ ˆn µ,σ ˆn µ ˆn µ,σ σ H DQD = µ=l,r ε µ â µ,σâ µ,σ t(â L,σâR,σ + H.c.) σ +U µ=l,r ˆn µ, ˆn µ, + V ˆn Lˆn R Summar School 2012 FIRST 19

ouble QDs holiding few electrons Fabrication of two QDs is

realizing tunable coupling between the two QDs and going

nanotube Advent of two-electron double QDs Mason et al.

20 ouble QDs holiding few electrons Fabrication of two QDs is straightforward extension in top-down approach, but realizing tunable coupling between the two QDs and going into few electron regime is not a simple task. nanotube Advent of two-electron double QDs Mason et al. Science 04 Hatano et al. Science 05 Petta et al. Science 04 Summar School 2012 FIRST 20

21 Charge qubits Effectively one electron in coupled QDs is simple two level system: charge qubit. H DQD = µ=l,r ε µ â µâ µ t(â LâR + H.c.) L R Detuning energy ε ε L ε R (n L,n R ) represents n L and n R electrons in the left and right QDs, resp. Summar School 2012 FIRST 21

22 Charge qubit experiments T* Rabi ~1 ns Origin: Cotunneling Phonon T. Hyashi, et al., Phys. Rev. Lett. 91, (2002). K. D. Petersson, et al., Phys. Rev. Lett. 105, (2010). Y. Dovzhenko, et al., Phys. Rev. B 84, (2011). Ramsey T* 2 ~60ps Summar School 2012 FIRST 22

23 Coupled charge qubits Mutual coherent osc. Conditional coherent osc. H 2DQD = 1 2 i (ε i σ (i) z t i σ x (i) )+ J 4 σ(1) z σ z (2) G. Shinkai, et al., Phys. Rev. Lett. 103, (2009). Summar School 2012 FIRST 23

24 Atomic limit of two electron in DQD Two electron problem in double dot system: Heisenberg Hamiltonian that enable swap operations J L R This is from an ideal situation of two identical dots in the atomic limit (t<<u). Here, we generalize the argument to finite detuning energy ε. H DQD = ε 2 (ˆn L ˆn R ) t(â L,σâR,σ + H.c.) +U ˆn µ, ˆn µ, + V ˆn Lˆn R µ=l,r Summar School 2012 FIRST 24

25 Two electron basis functions There are six two electron basis functions. S(1, 1) = 1 2 (a L a R a L a R 0, S(2, 0) = a L a L 0, Spin singlets S(0, 2) = a R a R 0 T 1 = a L a R 0, T 0 = 1 2 (a L a R + a L a R 0, T 1 = a L a R 0 Spin triplets Summar School 2012 FIRST 25

26 Eigen energies Singlet energies are the eigenvalues of the 3x3 matrix in the basis of ( S(1,1)>, S(2,0)>, S(0,2)>). 0 2t 2t H S = 2t U V + ε 0 2t 0 U V ε Triplet energies are degenerate up to the Zeeman energy, -#./0 () * 12345, V/U=0.67 E T = V 6 () * E S3. * () t/u=0.1 * 1237 Exchange energy is defined by the difference of energy of spin triplet to spin singlet ground states J 4t2 U for ε << U-V (E!" Sn # -E $% T & )/U '() * E S1 Summar School 2012 FIRST E S2 8() * 12 E T +() ε/u * J/U 9 Important region at large offset : 2 ;(: 3/2. t/u * 26

27 Region of large offset: Pauli blockade The very slow relaxation process from spin triplet to spin singlet is the origin of Pauli blockade. Result of exact diagonalization ε K. Ono et al., Science 297, 1313 (2001). Y. Tokura unpublished (2001). Summar School 2012 FIRST 27

28 Quantum point contact (QPC) Quantum point contact (QPC) is a vary short and narrow constriction. Landscape near QPC is the saddle point potential. 37 (2002) 363. Summar School 2012 FIRST 28

Condactance quantization B.J.van Wees, et al, Phys. Rev. B 43, 12431 (1991).

29 Condactance quantization B.J.van Wees, et al, Phys. Rev. B 43, (1991). K.J.Thomas, et al, Phys. Rev. Lett. 77, 135 (1996). Summar School 2012 FIRST 29

30 Adiabatic transport - no channel mixing Or carbon nanotube, graphene nanoribbon, E F Conductance Landauer s formula G = 2e2 h n T n Transmission probability of mode n T n T n =1 Noiseless mode Summar School 2012 FIRST 30

31 Statistics of transmitted current Input flux from degenerate Fermi sea with bias voltage V SD : J = ev F (ev SD )ρ F Here we used the density of states at the Fermi energy ρ F = 2 2π spin = e2 V SD π k E = 1 πv F the Fermi velocity v F Summar School 2012 FIRST 31

For N inputs (in other words, for a finite time τj), one possible series of events is

32 Counting statistics For each input electron, there are only two possible events, Transmitted (+) and Reflected (-). For N inputs (in other words, for a finite time τj), one possible series of events is (+,+,+,-,-,+,-,+,-,-,-,-). N # of + is Q The probability of this particular series, where Q out of N electrons are transmitted, is given by Binomial distribution function P N (Q) = N! Q!(N Q)! T Q (1 T ) N Q Summar School 2012 FIRST 32

Characteristic function It is very useful to introduce the characteristic function, C N (λ), with a counting field λ: N C N (λ) = P N (Q)e iλq Q=0 = (1 T ) N + N(T e iλ )(1 T ) N 1 N(N 1) + (T e iλ )

33 Characteristic function It is very useful to introduce the characteristic function, C N (λ), with a counting field λ: N C N (λ) = P N (Q)e iλq Q=0 = (1 T ) N + N(T e iλ )(1 T ) N 1 N(N 1) + (T e iλ ) 2 (1 T ) N 2 + +(T e iλ ) N 2! = (1 T + T e iλ ) N. With this function, we can obtain l-th cumulant Q = i d dλ ln C N(λ) λ=0 For example, the average and its variance are Q = NT Zero at T=1! Q 2 V = NT (1 T ) = Q 2 Q 2 Summar School 2012 FIRST 33

34 Dependence on statistics Marcus Kindermann, Ph.D thesis Flux of Fermion is regulated because of Pauli exclusion principle. Summar School 2012 FIRST 34

35 Shot noise Zero frequency current noise for ev SD >> k B T is given by the variance for large τ N = τj V = NT (1 T ) S V τ = JT (1 T )=I(1 T ) Average current is given by Landauer s formla: L. S. Levitov and G. B. Lesovik, JETP Lett. 58, 230 (1993). I = JT Summar School 2012 FIRST 35

Poisson distribution It is well-known that the Binomial distribution becomes Poisson distribution by letting N large while keeping average <Q>=NT finite: C N (λ) = (1 T + T e iλ ) N Then

36 Poisson distribution It is well-known that the Binomial distribution becomes Poisson distribution by letting N large while keeping average <Q>=NT finite: C N (λ) = (1 T + T e iλ ) N Then all the cumulants are identical. = (1 + (NT (e iλ 1) N N Q = i d dλ [NT (e iλ 1)] λ=0 = NT i d e iλ = NT e NT (e iλ 1) C P (λ) dλ ) N S I Walter Schottky (1918) Summar School 2012 FIRST 36

37 QPC Charge detection QPC is frequently used as a sensitive charge detector since the current changes with the potential barrier. M. Field, et al., Phys. Rev. Lett. 70, 1311 (1993). Necessary condition to the time required to distinguish the change of the QPC current by the change of transmission. t d ev SD π T Change of transferred charge t d ev SD π T (1 T ) Fluctuation 1 ev SD t d h ( T ) 2 T (1 T ) I. L. Aleiner, et al., Phys. Rev. Lett. 79, 3740 (1997). Summar School 2012 FIRST 37

38 Counting electrons T. Fujisawa, et al., Science 312, 1634 (2006). Summar School 2012 FIRST 38

39 Radio-frequency(rf)-SET R. J. Schoelkopf, et al., Science 280, 1238 (1998). Wei Lu, et al., Nature 425, 422 (2003). T. Fujisawa, et al., Appl. Phys. Lett. 84, 2343 (2003). Summar School 2012 FIRST 39

40 Comparison of QPC/QD detectors The sensitivity of QD charge detector is superior to that of QPC. C. Barthel, et al., Phys. Rev. B 81, (2010). Summar School 2012 FIRST 40

41 End of Part I Summar School 2012 FIRST 41

Lecture 8, April 12, 2017

Lecture 8, April 12, 2017 This week (part 2): Semiconductor quantum dots for QIP Introduction to QDs Single spins for qubits Initialization Read-Out Single qubit gates Book on basics: Thomas Ihn, Semiconductor