1. Journal Club 2012. február 16. Tóvári Endre Resonance-hybrid states in a triple quantum dot PHYSICAL REVIEW B 85, 081301(R) (2012) Using QDs as building blocks: exploring quantum effects seen in real molecules and solids (but with tunable parameters, # of electrons, arrangement of QDs in an arbitrary structure, even lattice of artificial atoms) QD arrays (flat band ferromagnetism, GMR, superconductivity calculations) quantum information processors (seperating entangled electrons, topological quantum computation for fault-tolerant quantum computers) modelling chemical reactions quantum simulations Here: resonance-hybrid states in a few-electron TQD, exploring the origin of the hybrid bond stability focusing on spin Model: 3-site Hubbard model Phys. Rev.B 65, 085324 (2002)Phys. Rev. Lett. 90, 166803 (2003)

2. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 2 Valence bond theory; multiple contributing structures (bonds) the bonding cannot be expressed by one single Lewis formula delocalized electrons (or superposition of wavefunctions) lower energy  hybrids are more stable than any of the contributing structures Resonance hybrid molecules http://en.wikipedia.org/wiki/Resonance_(chemistry)#Resonance_hybrids Valence bond theory; multiple contributing structures (bonds) the bonding cannot be expressed by one single Lewis formula delocalized electrons (or superposition of wavefunctions)

3. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 3 DC current from S to D, QDs in parallel V g1 V g3 V g2 V g2’ =V g2 Al 0.3 Ga 0.7 As/GaAs double-barrier resonant tunneling structure Size: adjusted to attain the few- electron regime, 100 mK DC current, B=0, V g2 =V g2’

4. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 4 Determining charge configurations: from the slope (ΔV g1 /ΔV g2 ) of each Coulomb oscillation line away from the anticrossing regions separation between, and “rounding” of lines at anticrossing regions (X,Y,Z): interdot Coulomb interaction and tunnel coupling V sd =300μV V g1 V g3 V g2 V g2’ =V g2 the levels in dots 1 and 3 are aligned near Z... On increasing Vsd, the Coulomb oscillation lines broaden into current stripes and excited states within the energy window eV sd become accessible V sd =1mV 3-site Hubbard model: U i intradot Coulomb-energies V ij interdot Coulomb-en. t ij interdot tunnel coupling E i lowest single-e - level: E 1 =0.5ε=-E 3, E 2 =δ ε energy detuning between QD1 and QD3 side view top view drain

5. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 5 110: ground state, excited state 011: ground state, excited state Near Z: ε→0, E 1 =E 3 S 12 and S 23, T 12 and T 23 become resonant, (1,1,0) and (0,1,1) hybridize, neither is dominant, the energy separation between S and T (ground and excited) levels increases: μ g (2) < μ e (2) * hybridization: first-order (direct) tunneling: (1,1,0) → (0,1,1) second-order tunneling via intermediate virtual states (important if ΔE(δ) between intermediate and initial states is small): 110→101→011 and 110→020→011 E i lowest single-e - level: E 1 = 0.5ε = - E 3, E 2 =δ ε energy detuning between QD1 and QD3 S-S hybridization is stronger  weaker curvature, more stable resonance μ g (2) μ e (2) μ e (3) μ g (3) CALCULATION (3-site Hubbard) |ε|>0: 110 or 011 is dominant, energy ~ -| ε| * μ(N) is the energy of the N electron state minus the energy of the N −1 electron ground state N=3 doublet states 2 levels, S tot =1/2 D 1, D 0 doublet states μ D1 = μ g (3) < μ D0 = μ e (3) μ e (3) excited state D 0 : S’=S 1 +S 3 =0 μ g (3) ground state D1: S’=1

6. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 6 Tuning E 2 =δ (and thus t 31 and the resonance) with V g3 ↑ 010 N=1: 010 becomes more stabilized (the 1st line shifts ), δ decreases N=2: the separation between the N=2 singlet and N=2 triplet levels increases due to stronger tunneling and hybridization, the former’s curvature weakens further μ g (2) μ e (2) μ e (3) μ g (3)

7. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 7 Tuning E 2 =δ (and thus t 31 and the resonance) with V g3 ↑ 010 μ g (2) μ e (2) μ e (3) μ g (3) N=3: the sign of the curvature of level μ g (3) changes from + to -, while the separation of the doublet levels at ε=0 remains small

8. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 8 μ g (2) μ e (2) Calculated charge state contributions in the ground state: N=2 QD1 QD2 QD3 δ is reduced → the weight of 020 increases, stronger hybridization and resonance, so μ g (2) flattens δ=-1.9meVδ=-2.2meVδ=-2.5meV stronger resonance-hybrid bond between the 110 and 011 singlet states compared to the hybridized triplet states (the former can hybridize with tunneling through 020, which is promoted by lowering E 2 =δ) → |μ e (2) – μ g (2)| increases

9. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 9 δ is reduced: (n,2,m) configurations more and more preferable μ e (3) μ g (3) δ P < δ: positive curvature of μ g (3) because: 111 is still dominant (its energy is independent of ε), and μ g (2) varies as ε or –ε (N=2 dominant config.: 110 or 011)* δ Q < δ < δ P : μ g (3) flattens because: 020 gains weight and the 111-energy is ε-independent δ < δ Q : μ g (3) has negative curvature because: 120 and 021 gain weight, their energy varies as ε or –ε * * μ(N) is the energy of the N electron state minus the energy of the N −1 electron ground state Calculated charge state contributions in the ground state: N=3

10. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 10 if 111 is dominant: doublet and quadruplet states: S tot =1/2 - D 1, D 0 doublet states μ e (3) excited state D 0 : S’=S 1 +S 3 =0 μ g (3) ground state D1: S’=1 S tot =3/2 - Q quadruplet doublet states symmetric and asymmetric states of 120 and 021 120 and 021 hybridize at δ Q < δ: D 1, D 0, Q N=3 μ e (3) μ g (3) π phase gain no phase gain the separation between two doublet states remains small (ε=0): both doublet states are stabilized (Q is not) geometrical phase from the single electron in QD2 One might expect that D 1 should hybridize with the S state from the permutation process of electrons in dots 1 and 3, but this is not so due to additional geometrical phase hybridization of D 1 and AS, and D 0 and S→ the separation between two doublet states remains small

11. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 11 doublet states symmetric and asymmetric states of 120 and 021 N=3 μ e (3) μ g (3) A quantum computation aspect: Changing δ adiabatically, using the level crossing: going from a charge qubit to a spin qubit (S→D 1 or AS →D 0 ) hybridization of D 1 and AS, and D 0 and S→ the separation between two doublet states remains small

12. 2012.02.16. JC: Resonance-hybrid states in a triple quantum dot 12 Conclusions Enhanced stability of the 110 ↔ 011 singlet resonance over the triplet resonance was observed due to the difference in accessibility of the (0,2,0) intermediate state Evolution of the three-electron ground and excited-state energies: from the accessibility of (1,2,0) and (0,2,1) intermediate states with the resonance-hybrid picture and geometrical phase in the electron hopping process