Hensgens et al.: Quantum simulation of a Fermi - Hubbard model using a semiconductor quantum dot array. QIP II: Implementations,

Transcription

1 Hensgens et al.: simulation of a Fermi - using a semiconductor quantum dot array Kiper Fabian QIP II: Implementations,

2 Outline

3 simulation: using a quantum system to a Hamiltonian. digital: using a quantum computer with qubits and gates that are applied sequentially universal, but requires large number of highly controlled qubits. analog: no gates, system designed to emulate the Hamiltonian less versatile, but easier and more controlled. Fermi - : describes interacting particles (here: electrons) on a lattice

4 Fermi - Intuitive explanation Approximate of solid state: Fixed lattice (here: quantum ) filled with fermions (here: electrons). Illustration of [von Stecher, 2011]. Electrons are able to tunnel to neighbouring. Electrons interact with other electrons. How?

5 H = ɛ i n i i }{{} energy of each dot <i,j>,spin ɛ i : Energy of i-th dot t ij (c iσ c jσ + h.c.) + i } {{ } Electron hopping/tunneling Fermi - Hamiltonian U i 2 n i(n i 1) } {{ } Coulomb repulsion in same dot c iσ : Creation operator of electron on dot i with spin σ n i = σ c iσ c iσ t ij : Tunnel coupling between i-th and j-th dot + i,j U: Sets strength of coulomb repulsion between electrons in same dot V ij n i n j }{{} Coulomb repulsion between different V : Sets strength of coulomb repulsion between electrons in different

6 H = ɛ i n i i }{{} energy of each dot <i,j>,spin t ij (c iσ c jσ + h.c.) + i } {{ } Electron hopping/tunneling Fermi - Hamiltonian U i 2 n i(n i 1) } {{ } Coulomb repulsion in same dot + i,j V ij n i n j }{{} Coulomb repulsion between different Illustration of [von Stecher, 2011].

7 Fermi - Motivation: Why bother simulating? is base for many exotic phases of matter This Hamiltonian predicts Mott insulators: Materials which should be conductors according to band theory but are insulators [Biswas, 2016]

8 Fermi - Mott insulator (a) Mott insulator according to Band theory. (b) of Mott insulator. [Biswas, 2016]

9 Fermi - Mott insulator If U is big enough, electrons cannot hop anymore = Insulator! [Greiner, 2016]

10 Fermi - Mott insulator If U is big enough, electrons cannot hop anymore = Insulator! [von Stecher, 2011]

11 Fermi - Mott transition How to get from insulating to conducting behaviour? Increase tunnel coupling t!

12 Dot Basic principle Image courtesy of Jann Hinnerk Ungerer

13 setup [Hensgens et al., 2017] linear triple-quantum-dot array 3 plunger gates (P 1,2,3 ) 4 barrier gates (B 1L,12,23,3R ) sensing dot channel 4 Fermi reservoirs (2 each for QD array and sensing dot channel)

14 Dot As a Charge Sensor QIP II lecture slides (Andreas Wallraff, 2017)

15 setup [Hensgens et al., 2017]

16 setup Charge Stability Diagram [Hensgens et al., 2017]

17 Eliminating cross-talk through the definition of virtual gates The value of the energy offset of each dot ɛ i depends linearly on the gate values up to several tens of millivolts Small changes in energy offsets are linear combination of voltage changes on the 7 gates: ( ) ( ) ɛ1 α11 α 12 α 13 α 14 α 15 α 16 α 17 δ ɛ 2 ɛ 3 = α 21 α 22 α 23 α 24 α 25 α 26 α 27 α 31 α 32 α 33 α 34 α 35 α 36 α 37 α ii : describes coupling of plunger P i to offset ɛ i α ij, i j: cross-talk δ(p 1, P 2, P 3, B 1L, B 12, B 23, B 3R ) T

18 Eliminating cross-talk through the definition of virtual gates The values α ij can be related to the α ii s via the slope of the charge addition lines. δb 12 δp 1 = α 11 α 14 T. Hensgens et al., simulation of a Fermi using a semiconductor quantum dot array, Nature volume 548, (2017)

19 Eliminating cross-talk through the definition of virtual gates The relative weights of the α ii s are determined by the slope of the polarization lines α 11 = α 21 + δp 2 δp 1 (α 22 α 12 ) [Hensgens et al., 2017] The absolute value of α 22 (called the lever arm) can be measured using photon-assisted tunnelling experiments.

20 Eliminating cross-talk through the definition of virtual gates All matrix elements known ɛ i s can be deterministically changed Virtual gates are defined to simplify the tuning process. For example for ɛ 1 : δɛ 1 α 11 α 12 α 13 P 1 P 1 α 11 α 12 α 13 0 = α 21 α 22 α 23 δ P 2 = δ P 2 = α 21 α 22 α 23 0 α 31 α 32 α 33 P 3 P 3 α 31 α 32 α 33 similarly for the barrier gates. δɛ

21 Collective Coulomb Blockade Physics Goal: Want to see transition from Mott insulator to metal.: Coulomb Blockade Collective Coulomb Blockade What do we expect from a triple quantum dot filled with electrons? [Hensgens et al., 2017]

22 Collective Coulomb Blockade Physics For small tunnel coupling: H = ɛ i n i t ij i }{{} (c iσ c jσ + h.c.) + U ( i 2 n ) i(n i 1) + V ij n i n j <i,j>,spin i i,j }{{}}{{}}{{} energy of Electron hopping/tunneling Coulomb repulsion Coulomb repulsion each dot in same dot between different = System behave as isolated

23 Collective Coulomb Blockade Physics [Hensgens et al., 2017]. = Coulomb Bockade! (Insulating behaviour)

24 Collective Coulomb Blockade Physics [Hensgens et al., 2017] Energy eigenstates become non-degenerate.

25 Collective Coulomb Blockade Physics [Hensgens et al., 2017] = Collective Coulomb Blockade! (Conducting behaviour) Electrons are completely delocalized

26 Collective Coulomb Blockade Physics [Hensgens et al., 2017]

27 Collective Coulomb Blockade Physics [Hensgens et al., 2017]

28 Next step: More in different architectures difficult calibration of virtual gates Gallium arsenide is not the future Silicon CMOS technology as potential candidate Combination of charge and spin degrees of freedom

29 Abhijit Biswas. Metalinsulator transitions and non-fermi liquid behaviors in 5d perovskite iridates, M. Greiner. Beyond mean eld physics with bose-einstein condensates in optical lattices, T. Hensgens, T. Fujita, L. Janssen, Xiao Li, C. J. Van Diepen, C. Reichl, W. Wegscheider, S. Das Sarma, and L. M. K. Vandersypen. simulation of a fermi hubbard using a semiconductor quantum dot array Javier von Stecher. Strongly interacting atoms in optical lattices, 2011.