On Decoherence in Solid-State Qubits Josephson charge qubits Classification of noise, relaxation/decoherence Josephson qubits as noise spectrometers Decoherence due to quadratic 1/ f noise Decoherence of spin qubits due to spin-orbit coupling Gerd SchönKarlsruhe work with: Alexander ShnirmanKarlsruhe Yuriy MakhlinLandau Institute Pablo San JoséKarlsruhe Gergely ZarandBudapest and Karlsruhe Universität Karlsruhe (TH)

2 energy scales E C, E J charging energy, Josephson coupling 2 degrees of freedom charge and phase 2 control fields: V g and  x gate voltage, flux VgVg xx n tunable 2 states only, e.g. for E C » E J VgVg g x /0x /0 C g V g /2e Shnirman, G.S., Hermon (97) 1. Josephson charge qubits

Observation of coherent oscillations Nakamura, Pashkin, and Tsai, ‘99  op ≈ 100 psec,   ≈ 5 nsec Q g /e major source of decoherence: background charge fluctuations

Quantronium (Saclay) Operation at saddle point: to minimize noise effects - voltage fluctuations couple transverse - flux fluctuations couple quadratically Charge-phase qubit EC ≈ EJEC ≈ EJ gate C g V g /2e x /0x /0

Decay of Ramsey fringes at optimal point  /2 Vion et al., Science 02, …

Experiments Vion et al. Gaussian noise SS  1/  4MHz S Ng  1/  0.5MHz Coherence times (ns)  x   Free decay Spin echo |N g -1/2|

Sources of noise - noise from control and measurement circuit, Z(  ) - background charge fluctuations - … Properties of noise - spectrum: Ohmic (white), 1/ f, …. - Gaussian or non-Gaussian coupling: 2. Models for noise and classification longitudinal – transverse – quadratic (longitudinal) …

Ohmic Spin bath 1/ f (Gaussian ) model noise Bosonic bath

Relaxation (T 1 ) and dephasing (T 2 ) Bloch (46,57), Redfield (57) Dephasing due to 1/ f noise, T=0, nonlinear coupling ? Golden rule: exponential decay law For linear coupling, regular spectra, T ≠ 0 pure dephasing: *   Example: Nyquist noise due to R (fluctuation-dissipation theorem)

1/ f noise, longitudinal linear coupling Cottet et al. (01) non-exponential decay of coherence  time scale for decay

eigenbasis of qubit Josephson qubit + dominant background charge fluctuations 3. Noise Spectroscopy via JJ Qubits probed in exp’s transverse component of noise  relaxation longitudinal component of noise  dephasing 1/ f noise  Astafiev et al. (NEC) Martinis et al., …

Relaxation (Astafiev et al. 04) data confirm expected dependence on  extract T 2 dependence of 1/ f spectrum observed earlier by F. Wellstood, J. Clarke et al. Low-frequency noise and dephasing E 1/f

same strength for low- and high-frequency noise Relation between high- and low-frequency noise

Qubit used to probe fluctuations X(t) each TLS is coupled (weakly) to thermal bath H bath. j at T and/or other TLS  weak relaxation and decoherence Source of X(t) : ensemble of ‘coherent’ two-level systems (TLS) High- and low-frequency noise from coherent two-level systems qubit TLS bath inter- action

Spectrum of noise felt by qubit distribution of TLS-parameters, choose exponential dependence on barrier height for 1/ f for linear  -dependence overall factor One ensemble of ‘coherent’ TLS Plausible distribution of parameters produces: - Ohmic high-frequency ( f ) noise → relaxation - 1/ f noise → decoherence - both with same strength a - strength of 1/ f noise scaling as T 2 - upper frequency cut-off for 1/ f noise Shnirman, GS, Martin, Makhlin (PRL 05) low  : random telegraph noise large  : absorption and emission

4. At symmetry point: Quadratic longitudinal 1/ f noise Paladino et al., 04 Averin et al., 03 static noise 1/ f spectrum “quasi-static” Shnirman, Makhlin (PRL 03)

Fitting the experiment G. Ithier, E. Collin, P. Joyez, P.J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnirman, Y. Makhlin, J. Schriefl, G.S., Phys. Rev. B 2005

5. Decoherence of Spin Qubits in Quantum Dots or Donor Levels with Spin-Orbit Coupling Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots Petta et al., Science, 2005

Generic Hamiltonian = strength of s-o interaction direction depends on asymmetries published work concerned with large, → vanishing decoherence for (Nazarov et al., Loss et al., Fabian et al., …) We find: the combination of s-o and X  x and Z  z leads to decoherence, based on a random Berry phase. spin + ≥ 2 orbital states + spin-orbit coupling noise coupling to orbital degrees of freedom dot 2 orbital states noise 2 independent fluct. fields coupling to orbital degrees of freedom spin-orbit spin

Rashba + Dresselhaus + cubic Dresselhaus Specific physical system: Electron spin in double quantum dot Phonons with 2 indep. polarizations Charge fluctuators near quantum dot Spectrum:  + Z(t) X(t) 2-state approximation: Fluctuations

= natural quantization axis for spin For two projections ± of the spin along For each spin projection ± we consider orbital ground state Ground (and excited) states 2-fold degenerate due to spin (Kramers’ degeneracy)    x y z b -b

   x y z In subspace of 2 orbital ground states for + and - spin state: Instantaneous diagonalization introduces extra term in Hamiltonian Gives rise to Berry phase random Berry phase  dephasing

X(t) and Z(t) small, independent, Gaussian distributed  effective power spectrum and dephasing rate Small for phonons (high power of  and T ) Estimate for 1/ f  – noise or 1/ f ↔ f noise Nonvanishing dephasing for zero magnetic field due to geometric origin (random Berry phase) measurable by comparing  1 and   for different initial spins

Conclusions Progress with solid-state qubits Josephson junction qubits spins in quantum dots Crucial: understanding and control of decoherence optimum point strategy for JJ qubits:    1  sec >>  op ≈ 1…10 nsec origin and properties of noise sources (1/ f, …) mechanisms for decoherence of spin qubits Application of Josephson qubits: as spectrum analyzer of noise