Introduction Resonant Cooling of Nuclear Spins in Quantum Dots Mark Rudner Massachusetts Institute of Technology For related details see: M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99, 036602 (2007); arxiv:0705.2177; Phys. Rev. Lett. 99, 246602 (2007).
Introduction The Plan I. Quantum dots and spin blockaded transport II. Post -diction: nuclear spin pumping III. Prediction: resonant suppression of fluctuations Summary IV. Bonus Topic: Reverse Overhauser Effect (?)
Introduction Part I: Quantum dots and spin-blockade Goal: Understand electron transport in double quantum dots 1) relevant experimental context 2) connection to nuclear spins
Introduction What is a quantum dot? Quantum dots as artificial atoms Confinement leads to discrete energy levels Energy scale for level separation: L
Introduction Double quantum dots Source L R Drain Artificial molecule Exceptional degree of control and tunability
Spin Blockade Notation and the relevant part of the Hilbert space L Source R Occupancy/spin labels Drain Shorthand notation
Spin Blockade A simple model based on sequential tunneling Source L When the source populates R Drain, transport proceeds unimpeded
Spin Blockade A simple model based on sequential tunneling Source L R Drain x? When the source populates, transport is blocked
Experimental Observations Experimentally observed suppression of current Koppens et al. Science 309,1346-1350 (2005); see also Ono et al. Science 297, 1313-1317 (2002).
Experimental Observations Nuclear-spin-induced oscillatory current in spin-blockaded quantum dots K. Ono and S. Tarucha Phys. Rev. Lett. 92, 256803 (2004).
Intermission Part II: Nuclear spin pumping Goal: Understand feedback and nuclear spin dynamics
Spin Blockade Electrons escape from via two channels: (1) x (2) (1) cotunneling/coupling to source/drain (2) spin-flip due to interaction with nuclei
Hyperfine Interaction Interaction between electron and nuclear spins allows mutual flip-flop Electron and nuclear spins interact via: Flip-flop possible if energy is conserved
Energy Levels Zero-field electron energy levels divided into 1 triplet and 2 singlets E
Magnetic Field Dependence Triplet degeneracy is lifted by magnetic field E B B
Magnetic Field Dependence Spin flip decay rate is enhanced when energy levels align Decay only by cotunneling decays by spin flip decays by spin flip
Nuclear Spin Pumping Polarization feeds back into spin flip rate to produce instability
Blockaded Transport Electrons must pass through system one by one No polarization can develop if each incoming spin must flip
Spin Blockade Electrons escape from via two channels: (1) x (2) (1) cotunneling/coupling to source/drain (2) spin-flip due to interaction with nuclei
Nuclear Spin Pumping Average rate for flipping spins up/down Assume on average 1 out of every 4 electrons enters Hyperfine rate depends on polarization through Overhauser shift
Nuclear Spin Pumping What about relaxation? Low T: direct relaxation of nuclear spin is slow (minutes) Polarization diffuses out via dipole-dipole flips (seconds)
Nuclear Spin Pumping Rate (107 s-1) Nuclear spin flip rate at finite magnetic field Fixed point jumps to finite polarization Polarization
Nuclear Spin Pumping Rate (107 s-1) Nuclear spin flip rate at zero magnetic field Relaxation stabilizes unpolarized state Polarization
Nuclear Spin Pumping Stability Diagram: Polarization vs Magnetic Field Stable fixed point + Low B: unpolarized state is stable Unstable fixed point
Nuclear Spin Pumping Current hysteresis in magnetic field sweep
Relation to Experiments Model accounts well for observed instabilities and hysteresis Basic features of Delft and Tokyo data described by our model Time-dependent oscillations remain a mystery Coming up: analysis of nuclear spin fluctuations
Intermission Part III: Resonant suppression of fluctuations Goal: Understand dynamics of nuclear spin distribution
Nuclear Spin Fluctuations How does the nuclear spin distribution evolve? Distribution is broad due to fluctuations (finite system) Goal: drive nuclear spin to a precise value of polarization Relevance: electron spin dephasing via hyperfine interaction
Nuclear Spin Fluctuations Feedback suppresses fluctuations at small detuning
Nuclear Spin Fluctuations Diffusive dynamics of the nuclear spin distribution Polarization performs 1-D directed random walk Distribution evolves according to Fokker-Planck equation with P 0-2 P0 P 0+ 2 P
Nuclear Spin Fluctuations Steady state distribution In the steady state, the distribution takes this form: Linearization gives Gaussian of width
Squeezing Estimate of squeezing efficiency High temperature state: V depends on P through (P/N), yielding With identical resonances of width dependence : Greater than 10-fold reduction for!
Squeezing Dependence on detuning (fixed magnetic field) Polarization Fixed Points RMS Fluctuations vs Detuning
Squeezing Dependence on magnetic field (fixed detuning)
Experimental Signatures How can the squeezing effect be observed? ESR measurements look for change in the line width T2* measurement in pulsed gate setup (?) Other ideas?
Summary What have we learned? 1) The spin-blockade regime contains rich physics and the possibility for controlling nuclear spins by electric current 2) Competition between resonant spin-flips and indirect tunneling leads to polarization and hysteretic behavior 3) Dynamics in the resonant transport regime possess a squeezing effect that can be used to suppress nuclear spin fluctuations well below the thermal level
Electron Spin Resonance Electrically Induced Hyperfine-Mediated Electron Spin Resonance Time varying local electron density stimulates inelastic hyperfine transitions E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P. Hanson, A. C. Gossard, arxiv:0707.0557.
Electron Spin Resonance Reverse Overhauser Pumping of Nuclear Spins Overhauser Reverse Overhauser
Electron Spin Resonance Hysteresis and Spatial Distribution of Nuclear Polarization Overhauser Reverse Overhauser
Electron Spin Resonance Experimentally observed direction of pumping Reverse Overhauser E. A. Laird et al., arxiv:0707.0557.
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Electron Spin Resonance Electrically Induced Hyperfine-Mediated Electron Spin Resonance E. A. Laird, C. Barthel, E. I. Rashba, C. M. Marcus, M. P. Hanson, A. C. Gossard, arxiv:0707.0557.
Fokker-Planck Parameters Drift velocity: expected change of polarization per time step Diffusion constant: mean square change of polarization per time step P
Squeezing Thermal/Environmental Broadening Noise opposes squeezing, restores random distribution Noise acts by randomly flipping spins with microscopic rate Account for noise by making the replacement:
Microscopic Rates Formulas Fermi's Golden Rule transition rates: Singlet density of states: Equilibrium condition:
Details Time Scales Electron precession in polarized nuclear field: 10 ps Electron precession in random nuclear field: 10 ns Nuclear precession in electron effective field: 10 μs Nuclear dipole-dipole flip time: 100 μs Nuclear relaxation time: > 1 s
Nonlinear Dynamical Behavior Can we explain Ono and Tarucha's oscillations? Simplest approach: rate equation for nuclear polarization P A first order dynamical system in one variable cannot oscillate x o x o x
Spin Blockade Coherent manipulation of electron spins; pumping of nuclear spins Petta et al. Science 309,2180-2184 (2005).
Spin Blockade Current switching and hysteresis in resonant transport F. H. L. Koppens et al. Science 309,1346-1350 (2005).
Dynamical Instability Instability and oscillations in the absence of periodic driving Tacoma Narrows Bridge: torsional feedback into aerodynamic torque