Coherent and incoherent evolution of qubits in semiconductor systems

Coherent and incoherent evolution of qubits in semiconductor systems Iain Chapman, Thornton Greenland, Sev Savory and Andrew Fisher Departments of Physics and Astronomy and Electrical Engineering and London Centre for Nanotechnology UCL

Overview • Some proposed and actual semiconductor qubits, and their principal sources of decoherence: • Spin states • Orbital states • Charge states • “The standard model” of decoherence – a reminder • Some consequences for fluctuations, dissipation and decoherence • Charge qubits in double quantum dots, surface acoustic waves and double defects • Spin qubits at defects and the use of control spins

Electron spin qubits – quantum dots Petta et al. Science 309 2180 (2005) Double quantum dot structures, e.g. Johnson et al. Nature 435 925 (2005) Spin echo and Rabi flopping on a single logical qubit Relaxation of spins in (1,1) charge state away from singlet state inhibits transfer to (0,2) (spin blockade) T2*=9ns Dominant decoherence mechanism is via nuclear spin bath, relaxation strongly suppressed by Bext. Spin-orbit dephasing will be important as an ultimate limit.

Orbital qubits – excitons in quantum dots Beats between x and y polarizations of excitons in a dot: or Large (band-gap) energy scales mean very rapid qubit evolution Monitor state of system by delayed probe pulse Bonadeo et al.Science282 1473 (1998)

Exciton decoherence Rapid initial decoherence (strongly pulse-area dependent) followed by long decay: Borri et al. PRL 87157401 (2001) and PRB 66 081306(R) (2002) Time-domain versus Energy-domain

Förstner et al Phys. Stat. Sol. B 238 419 (2003); Phys. Rev. Lett. 91 127401 (2003) Phonon dephasing Dominant mechanism appears to be coupling to acoustic phonons Long-time tail arises because phonon-induced processes cannot conserve energy in the long-time limit; rapid short-time decoherence comes from transient processes allowed by uncertainty principle (non-linear in optical field) Pulse-area dependent dephasing or Rabi oscillations: Reproduction of weak-field lineshape:

Coherent oscillations of electron charge states in double quantum dots Prepare electron in left dot, follow free evolution and detect probability it emerges on the right (“free-induction decay”) Double quantum-dot structure in which dots can be gated separately Gate-defined dot in GaAs with leads: Hayashi et al. PRL 91 226804 (2003); period ~0.3ns, T2 ~ 2ns SOI structure: Gorman et al. PRL 95 090502 (2005); period ~0.3 μs, T2~2 μs

Can be eliminated, at least in principle Charge qubits in quantum dots (contd) • Possible sources of decoherence: • Co-tunnelling (in experiments with leads) • Circuit noise • Phonons Micro-wave-irradiation produces absorption peaks as the electron is excited into the upper state of the two-level system. Relaxation of occupation gives T1=16ns; linewidths give T2* at least 0.4ns Petta et al.Phys. Rev. Lett. 93 186802 (2004) Intrinsic

The Lindblad master equation Lindblad (1966): most general form for Liouville equation in an open system that is Markovian (i.e. evolution depends only on current state) is Lindblad operators, cause incoherent “jumps” in the state of the system Hamiltonian (may be modified by environment)

The Born-Markov limit Evolution of full density matrix (interaction representation): Find formal solution and trace over environment to get system evolution • In Markovian limit (i.e. can replace ρ(s) by ρ(t) and extend limit on integral to -∞), and also assuming • First term is zero (achieve by redefining H0 if necessary) • Factorization of density matrix into “system” and “environment” parts at all times • we get

Correlation functions Explicit form for interaction Hamiltonian: Not necessarily evaluated in thermal equilibrium Environment operator System operator Correlation functions of environment operators: Write equation of motion in terms of these:

Good qubits: the Rotating Wave Approximation For “good” qubits (systems which can evolve through many cycles before damped by environment), decompose system operators in terms of transition frequencies of system: Neglecting rapidly-oscillating terms in ei(ω-ω’)t , equation of motion becomes Hamiltonian (coherent) evolution (Lamb shift) Quantum jumps where we define the Fourier transforms Note non-zero S requires spectral structure in J Causality (c.f. Kramers-Kronig):

A B E Coupling through the environment Environment in a typical solid-state implementation plays two roles: (1) Generates interactions between qubits (essential for multi-qubit gates) (2) Introduces decoherence (bad) Figure of merit: ~ # operations before decoherence sets in (c.f. Q-factor) No direct coupling between spatially separate qubits, since physics is ultimately local

Good news, bad news Good news: the two spins are coupled by an effective Hamiltonian Can generate time-evolution that entangles the spins, leads to a non-trivial 2-qubit gate Bad news: there is inevitably a corresponding contribution to the decoherence (provided the wek-coupling limit is appropriate), scaling like Destroys coherence of quantum evolution

Continuum environment Find standard Lindblad master equation for an open system (even though physics is partly non-Markovian): Effective Hamiltonian: Lindblad operators Both determined by same spectral functions

Coupling and decoherence Coupling  real part of susceptibility • Produces a link between coupling of qubits and decoherence introduced • Inter-qubit coupling and decoherence linked by Hilbert transform • Forms a generalisation of the fluctuation-dissipation theorem Fluctuations (and hence decoherence)  imaginary part of susceptibility Enables us to bound the figure of merit (Q-factor) of system by knowing only the spectrum of the correlations

Discrete environment Suppose environment has discrete spectral response: Effective Hamiltonian: Effective Lindblad operators Both expressed in terms of : (a) Choose operating frequencies far from environment frequencies Ωn Allows extra “engineering” possibilities: (b) Choose operating times to coincide with zeros of ψss’(t,Ωn) c.f. Ion-trap “warm” entanglement (Mølmer, Sørensen etc.)

No decoherence No entanglement Irreducible decoherence • This decoherence is irreducible because • There may be other mechanisms, contributing additional decoherence, which do not also contribute to the entanglement • A decoherence-free subspace affords no protection:

Morivation • Would like to • Understand processes operating in recent experiments • Assess the suitability of charge qubits over a wide range of length and time scales

σ σ x=-a x=+a Charge qubits - minimal model Single electron in double dot coupled to phonons Two-state electronic system coupled to phonons: Coupling depends on interaction

Types of coupling Two types of coupling to acoustic phonons important for low-frequency quantum-information processing: (1) Deformation potential: (2) Piezoelectric coupling: Direction-dependent coupling:

Relative importance of couplings Piezoelectric coupling dominates on distance scales above about 10nm provided do not exceed screening length Relative importance of piezoelectric term in GaAs very sensitive to assumptions about screening length q-10

Two approaches: Two treatments of decohering processes Rotating Wave Approximation: perturbative and Markovian (disregards memory of environment) TCL (time-convolutionless) projection operator technique – perturbative but explicitly non-Markovian Projector onto decoupled dots/phonons “TCL kernel”. To 2nd order in coupling: Results similar in this case.

Is the RWA OK?

Results (RWA, GaAs parameters) Piezo-electric (unscreened limit) Deformation-potential Assumes electron transfer decays with distance like exp(- a/σ) T=20 mK

Results as a function of operating frequency For the geometry of Hayashi et al: Dot size = 30nm Dot separation = 300nm T=20mK Typical frequency range of experiments

Aside – charge decoherence in the GHz range and the behaviour of trapped charges T=20 mK

Surface Acoustic Waves • What is the relative magnitude of the surface and bulk contributions to relaxation and decoherence in a double dot? b a Motivation: surface component readily easily controlled by surface cavities and transducers. Will this help?

Importance of anisotropy in piezoelectric terms when treating surface couplings Anisotropy in piezoelectric matrix elements now becomes important:

SAW contribution to relaxation Compare approximately 5ns lifetime from bulk terms, so SAW never dominates Incomplete “burial” of dot charge density (unphysical) Dot size = 30nm Separation = 300nm Splitting = 0.01 meV

Excited state (interaction present) Ground state (no interaction) Is there another way? • Would like to • Use (electron) spin qubits in order to avoid rapid phonon-induced decoherence as much as possible; • Control coupling of qubits without presence of nearby electrodes and associated electromagnetic fluctuations; • Avoid small excitation energies susceptible to decoherence at the lowest temperatures. Our proposal (Stoneham et al., J Phys Conden Matt 15 L447 (2003)): use real optical transitions in a localized state to drive an atomic-scale gate: Exploit properties of point defect systems conveniently occurring in Si, but concept also generalises to many other systems

Dopants Silicon The basic idea • Qubits are S=1/2 electron spins which must be controlled by one- and two-qubit gates • The spins are associated with dopants (desirable impurities) • Chosen so they do not ionise thermally at the working temperatures (“deep donors”) • The dopants are spaced 7-10nm to have negligible interactions in the “off” state

Dopants Silicon Basic Ideas (Continued)… • The new concept is to control the spins producing the A-gates and J-gates using laser pulses • Another new concept is separation of the storing of Quantum information from the control of Quantum interactions • Uniquely, the distribution of dopant atoms is disordered • A disordered distribution is desirable for system reasons • Dopants do not have to be placed at precise sites

ALL GATES OFF ONE GATE ON Controlling Spins Control gate by laser-induced electron transfer Gate addressed by combination of position and energy Silicon Donors carrying Qubit Spins Source of Control Electron

ALL GATES OFF ALL GATES OFF ONE GATE ON Many different charge transfer events possible Different laser wavelengths allow discrimination Controlling Spins Control gate by laser-induced electron transfer Gate addressed by combination of position and energy Silicon Donors carrying Qubit Spins Source of Control Electron

SFG M=815 N=904 H SFG M=1595 N=2137 H Qubit 0 H Laser on SFG M=1584 N=2177 H Qubit 1 Qubit 2 SFG M=815 N=904 H H The dynamics of optically-controlled gates Identify which data manipulations can be efficiently produced using optical excitation without interfering with (decohering) the qubits, as a function of the controlling parameters: Zero B-field Finite field Then analyse the chain of gates required to produce a demonstration quantum algorithm (3-qubit Deutsch-Jozsa), and estimate the overall accuracy (fidelity): Safe to turn laser off without damaging quantum information

Conclusions • Provided it may be safely applied, the weak-coupling approach to decoherence gives an attractively universal picture of incoherent processes and limits attainable figures of merit in many cases • For charge qubits in semiconductors the model (with no free parameters) suggests recent experiments are close to the attainable limits • Partly motivated by these considerations, we are investigating the optically-driven dynamics of defect spins in semiconductors

Acknowledgments • Thanks to • EPSRC, IRC in Nanotechnology, UK Research Councils Basic Technology Programme for support • Marshall Stoneham, Gabriel Aeppli, Wei Wu, Che Gannarelli

Coherent and incoherent evolution of qubits in semiconductor systems