Understanding spin-qubit decoherence: From models to reality

Transcription

1 Understanding spin-qubit decoherence: From models to reality Bill Coish IQC, University of Waterloo, Canada and (starting in Sept.) McGill University, Montreal, QC, Canada Collaborators: Basel: Jan Fischer, Daniel Klauser, Daniel Loss Waterloo: Farzad Qassemi, Frank Wilhelm, Jonathan Baugh Oslo: Joakim Bergli

2 Goal: Quantum Computation Initialization Arbitrary unitary Readout j0i 0 j0i 1 j0i N 1 j0i N U jªouti h0j ª out i = 0 1 h0j ª out i = 1 N h0j ª out i = 1

3 Goal: Quantum Computation Initialization Arbitrary unitary Readout j0i 0 j0i 1 j0i N 1 j0i N U jªouti h0j ª out i = 0 1 h0j ª out i = 1 N h0j ª out i = 1 Physical Implementation: ½ U = T exp i Z t 0 ¾ dt 0 H(t 0 ) H 2 H S

4 Reality: Imperfections Initialization Arbitrary unitary Readout j0i 0 j0i 1 j0i N 1 j0i N U jªouti h0j ª out i = 0 1 h0j ª out i = 1 N h0j ª out i = 1 ~U = T exp ½ i Z t 0 ¾ dt 0 (H(t 0 ) + ±H(t 0 )) ±H 2 H S H E

5 hsx(t)i Types of error In addition to initialization/readout error Gate error (free-induction decay)» t g =T FID 2 t=t hs x (t)i / e 2 FID Error-correction threshold < c» t g

6 hsx(t)i hsx(t)i Types of error In addition to initialization/readout error t g Gate error (free-induction decay)» t g =T FID 2 t=t hs x (t)i / e 2 FID Memory error (echo/dynamical decoupling) Error-correction threshold < c» » t=t2 ECHO ECHO t=t» e 2» T FID 2 t

7 hsx(t)i hsx(t)i Types of error In addition to initialization/readout error t g Gate error (free-induction decay)» t g =T FID 2 t=t hs x (t)i / e 2 FID Memory error (echo/dynamical decoupling) Error-correction threshold < c» » t=t2 ECHO ECHO t=t» e 2» T FID 2 Even for single spin: T2 ECHO 6= T2 FID 'Intrinsic' decay time is a myth! t

8 hsx(t)i Types of error In addition to initialization/readout error Gate error (free-induction decay)» t g =T FID 2 t=t hs x (t)i / e 2 FID Error-correction threshold < c» t g Focus on reducing gate error (increasing FID time): T FID 2 = T 2 Caveat: Gating and decay not always independent (should really determine the gate fidelity). e.g.: F = Tr n U y ~U o

9 Nuclear spins are (almost) everywhere NV centers in diamond Quantum dots 60 Phosphorus donors

10 Coherence Problem: One spin sees many N» 10 6 nuclei WAC and J. Baugh, `Nuclear spins in nanostructures', Phys. Stat. Solidi B (2009)

11 Theory of everything for spins in the solid state ~I H contact = 8¼ 3 S I ±(r)s I ~B ~E + H dip: = S I 3(n S)(n I) S I r 3 ~S - H LI = S I L I r 3

12 Confined electron H e ' hã 0 j H jã 0 i hr jã 0 i ' u(r)ã 0 (r)

13 Interactions: s vs. p s-state (electron) p-state (hole) u c (r) + Ze Z e + u v (r) hã s orbj H contact jã s orbi 6= 0 hã p orb j H contact jã p orb i = 0 hãs orbj H dip: jã s orbi = 0 hã p orb j H dip: jã p orb i 6= 0 hã s orbj H LI jã s orbi = 0 hã p orb j H LI jã p orb i 6= 0

14 Interactions: s vs. p s-state (electron) p-state (hole) u c (r) + Ze Z e + u v (r) hã s orbj H contact jã s orbi 6= 0 hã p orb j H contact jã p orb i = 0 hãs orbj H dip: jã s orbi = 0 hã p orb j H dip: jã p orb i 6= 0 hã s orbj H LI jã s orbi = 0 hã p orb j H LI jã p orb i 6= 0 Anything else: NV Center, Nanotubes, graphene,...combination

15 Interactions: s vs. p s-state p-state + Ze Z e + Project onto m J = 3 2 ) s z = 1 2 H e s = A s S I H e p = A p s z I z For 4s, 4p Hydrogen-like atomic orbitals (valence states of Ga, As): 3 A p A s = 1 5 µ Ze (4p) Z e (4s) The two coupling strengths are comparable!

16 In a 2D GaAs system A h A e ¼ 0:15 Large! H = (b + h z )s z Ising: no flip-flops Fischer, WAC, Bulaev, Loss, PRB (2008)

17 Hole-spin Decoherence H = h z s z + bs x z b ±h z Transverse fluctuations suppressed for large in-plane b hs z i t ' cos bt arctan (t= )» p 1 2 [1 + (t= ) 2 ] 1=4 t Predicted: = b (±h z ) 2 ' 1 ¹s D. Brunner et al., Science (2009) Measured: & 1 ¹s

18 Electrons: Hyperfine Hamiltonian H hf = bs z + h S +... h = X k A k I k Electron Zeeman energy A = X k A k Coupling to nuclear field Neglect: Nuclear dipole-dipole, Quadrupolar splitting, Spin-orbit, Phonons,...

19 Hyperfine Hamiltonian H hf = bs z + h S Electron Zeeman energy Coupling to nuclear field h = X k A = X k A k I k A k h S = h z S z S h + S + h S + I V does not conserve energy for large b S I

20 Hyperfine Hamiltonian H hf = bs z + h S Electron Zeeman energy Coupling to nuclear field h = X k A = X k A k I k A k h S = h z S z S h + S + h S + I V does not conserve energy for large b S I Perturbation theory in A b 1 b=g ¹ B & 3:5 T (GaAs)

21 Nuclear-spin bath preparation h z B h ) hs x i t / e (t= )2» ns measurement B h ) hs x i t / e i!t (narrowed state) Theory: WAC and Loss, PRB (2004), Klauser, WAC and Loss, PRB (2006,2008), Stepanenko et al., PRL (2006), Giedke et al., PRA (2006), Ribeiro and Burkard, PRL (2009), Expt.: Greilich et al., Science (2006), (2007), Reilly et al., Science (2008), Xu et al., Nature (2009), Vink et al., Nat. Phys. (2009), Latta et al., Nat. Phys. (2009)

22 After Narrowing... Dynamics in nuclear-spin system lead to decay B e» B + B N (t) B V V 2 hs x i t / e t=t 2

23 Nuclear-spin dynamics D. Klauser, WAC, D. Loss, PRB (2008) Short time: Ã hh z (t)i ' hh z (0)i 1 µ t n 2 + O(t 3 )! n» N 3=2 b A 2» 10 4 s b e» A 2 =N 3=2 b '

24 Nuclear-spin dynamics D. Klauser, WAC, D. Loss, PRB (2008) Short time: hh z (t)i ' hh z (0)i à 1 µ t Beyond short time (generalized master equation): n 2 + O(t 3 )! n» N 3=2 b A 2» 10 4 s hh z (t)i» 1 N hh z(0)i (nearly uniform polarization) c» N A» 10 6 s di & 1 10 s (Nuclear correlation time) (Dipolar spin diffusion)

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26 Not an 'easy' problem!

27 New approach: A general theory of coherent quantum dynamics Relevant observables fo g All observables Von Neumann: _½ = i [H; ½] ho i t = Tr fo½(t)g

28 New approach: A general theory of coherent quantum dynamics Relevant observables fo g All observables Von Neumann: _½ = i [H; ½] ho i t = Tr fo½(t)g Nakajima-Zwanzig Generalized Master Equation D _ O E t = i X! ho i t i X Z t 0 dt 0 (t t 0 ) ho i t 0 H = H 0 + V (t) = X n (n) (t) (n) (t) = O (V n )

29 Free-induction decay: history Khaetskii, Loss, Glazman, PRL (2002), PRB (2003) WAC and Loss, PRB (2004) Exact solution for p=1, Generalized Master Equation for p<1.

30 Free-induction decay: history Equation-of-motion method. Deng and Hu, PRB (2006), (2008)

31 Free-induction decay: history Effective Hamiltonian, pair correlation approximation. Yao, Liu, Sham, PRB (2006), NJP (2007)

32 Free-induction decay: history Effective Hamiltonian, Born-Markov approximation. WAC, Fischer, Loss, PRB (2008)

33 Free-induction decay: history Effective Hamiltonian, High-order resummation, low b-field. Cywinski, Witzel, Das Sarma, PRL (2009), PRB (2009)

34 Free-induction decay: history Generalized Master Equation, Higher order. WAC, Fischer, Loss, PRB (2010)

35 Free-induction decay: history WAC, Fischer, Loss, PRB (2010) Envelope modulations

36 Free-induction decay: history WAC, Fischer, Loss, PRB (2010) Non-monotonic decay rate for p 1 1 T 2 b Envelope modulations

37 Solve the problem in two ways: hoi t = hã(0)j e iht Oe iht jã(0)i H = H 0 + V (1) Effective Hamiltonian ~Ã(0) E Expand in powers of ~H = e S He S = H 0 + V e + O V 3 = e S jã(0)i = jã(0)i + O (V ) V e» O µ V 2 A» O b neglected (2) Work directly with the 'real' Hamiltonian Expand in powers of V

38 Initial conditions Fast initialization: ½(0) = ½ S (0) ½ I (0) Sufficient condition: init. 1=A ' 50 ps Narrowed bath: ½ I (0) = X i ½ ii jn i i hn i j! jn i i =! n jn i i

39 Generalized Master Equation (GME) Rotating frame: x t = 2e i(! n+!)t hs + i t GME: _x t = i!x t i Z t 0 dt 0 ~ (t t 0 )x t 0 Lamb shift:! = Re Z 1 0 dt ~ (t) Markov: Z 1 1 = Im T 2 0 dt ~ (t) x t ' x 0 e t=t 2

40 Direct expansion vs. effective H Expanding in V (s) = ~ ' ~ (2) + ~ (4) + O V 6! ' Re~ (2) (s = 0 + ) = O(V 2 ) 1 T 2 ' Im ~ (4) (s = 0 + ) Z 1 0 e st (t) V e» V 2 ~ e = ~ (2) e + O V 8 Expanding in! e ' Re~ (2) e (s = 0+ ) = O(V 4 ) 1 T 2 ' Im~ (2) e (s = 0+ ) For one isotope: Multiple isotopes: ~ (4) = ~ (2) e ~ (4) 6= ~ (2) e (with 1/N corrections)

41 Non-monotonic decay rate 1 ' Im T ~ (4) (s = 0 + ) / 1 X 2 b 2 k;k 0 A 2 ka 2 k 0 ±(A k A k 0!)! / 1 b A k A=N Qualitative behavior (maximum) is controlled by 1 p 2 A b < 1

42 Full Non-Markovian time dependence x(s) = x 0 s i! i ~ (s) 1 x t = lim!0 2¼i Z +i1 i1 e st x(s) = X i Res[e st x(s); x = s i ] X (t) Exponential decay or sustained oscillations Power-law decay

43 Envelope Modulations t À 1! Re [x t ]» C cos (!t + Á) t 2 Higher-order corrections needed

44 Envelope modulations: A collective quantum effect z Conventional envelope modulations (semiclassical); the spin remains on the surface of the Bloch sphere. x y z New quantum modulations due to collective excitations of the nuclear spin bath. x y

45 Initialization and readout Spin Charge

46 Spin dynamics in transport Ono and Tarucha, PRL (2004) Koppens et al., PRL (2007) Reilly et al., Science (2008) Foletti et al., Nature Phys. (2009)

47 The problem: Identify multiple relaxation rates F. Qassemi, WAC, J. Bergli, F. K. Wilhelm (in preparation) E E ? I(t)

48 Solution?: Frequency-dependent current noise S II (!) = 1 2 Z 1 1 dte i!t hf±i(t); ±I(0)gi ±I(t) = I(t) hii I(t) 2 >! > 1 : I(t) 3 >! > 2 : ¹ne» 1= 1 t t I(t)! > j :» 1= 2 e t

49 Dynamical Channel Blockade W. Belzig, PRB (2005) F. Qassemi, ¹n = 1 WAC, J. Bergli, F. K. Wilhelm (in preparation) F (0) = 2¹n 1 F (!) = S II(!) P B e hii Probability to be blocked

50 Dynamical Channel Blockade W. Belzig, PRB (2005) F. Qassemi, WAC, J. Bergli, F. K. Wilhelm (in preparation) ¹n = 1 P B F (0) = 2¹n 1 F (!) = S II(!) e hii Probability to be blocked Spin diode: _½ = M½ F (!) = 1 + X j j : Eigenvalues of M 2j F j! 2 + 2j F j : From right/left eigenvectors

51 Pauli spin blockade Blocked jt + i = j""i jt 0 i = 1 p 2 (j"#i + j#"i) jt i = j##i jsi = 1 p 2 (j"#i j#"i)

52 Fano factor W T+ W T0 jt + i W T+ W T0 jt 0 i jsi jt i F = 2¹n 1 ¹n = 1 P B F (!) ' W 2 T + 4! 2 + W 2 T W 2 T 0 3(9! 2 + 4W 2 T 0 )! <<

53 Conclusions Electron spin dynamics depend on the nuclear environment (new quantum effects); Gate errors. Frequency-dependent noise as a probe of spin relaxation; Initialization and readout.