Optical and electrical parallel molecular and nanoscale computing Françoise Remacle Theoretical Physical Chemistry, UR Molsys, University of Liège FP7 FET projects H2020 FET open project COPAC , starts Nov 1st, 2017 Collaboration : R. D. Levine (HUJI), B. Fresch (UNIPD) Experiments : E. Collini (UNIPD), S. Rogge (UNSW, Sydney) CD08 Mons Aug 23-25, 2017 1
Objectives Depart from the switching paradigm and Boolean logic Develop a new approach to logic and computing that takes advantage of the atomic and molecular scale by exploiting the dynamics of the transitions between molecular states Design and implement at the level of proof concept : • massively parallel computations • multivalued computations • inbuilt memory (Finite State Machines) • reconfigurable/programmable logic machines CD08 Mons Aug 23-25, 2017 2
Quasi classical approach • Boolean or multi-valued variables are encoded in the discrete charge, energy or conformational states of confined molecular or nano systems. • The inputs are provided as a selective perturbation that induces specific transitions between these states. The inputs can be optical, electrical and chemical. • Information processing is performed by the quantum dynamics of the multi state system. • The outputs are a probe of the final states. Different from quantum computing : no information is encoded into the phase of the wave function. CD08 Mons Aug 23-25, 2017 3
Beyond miniaturization of electronic devices : Unconventional computing at the molecular scale Exploits the dynamical response of molecular systems to implement logic Molecular systems possess many states that can be selectively addressed and undergo time-dependent transitions, opening the way to multivalued and parallel logics and programmability Our approach belongs to the broader field of 'rebooting computing' (http://rebootingcomputing.ieee.org/) CD08 Mons Aug 23-25, 2017 4
Two kinds of physical systems Optical addressing of the electronic states of multichromophoric rhodamine supramolecular complexes in solution for implementing molecular decision trees Elisabetta Collini, Padua Light induced transitions between states Molecular decision tree operated in parallel pulse 1 pulse 3 pulse 2 x 1 x 3 x 2 CD08 Mons Aug 23-25, 2017 5
Electrical addressing of the electronic states of dopant molecules in silicon Interacting P atom embedded in Si Sven Rogge School of Physics & CQC 2 T University of New South Wales Australia The pulse of voltages induces transition between states of the coupled dopant atoms. These can be mapped to multivalued decision trees. The logic is processed in parallel. CD08 Mons Aug 23-25, 2017 6
Assignment of the electronic states of the As dopant atom from the stability maps of a FinFET transistor dI/dV b ( S) 13G14 V b = 40 mV 40 30 0.6 I SD (nA) 30 20 20 10 0.4 V b (mV) 10 0 320 340 360 380 V G (mV) 0 -10 0.2 -20 -30 0 -40 E C = 31 meV 300 320 340 360 V G (mV) Excited electronic states of a given charge As dopant Lansberger et al, Nature Physics, 4 , 656 (2008) CD08 Mons Aug 23-25, 2017 7
Physical realization of a cascade of two full binary adders 1 . SAT : arithmetic addition A i + B i in base 4 2. CIB : FET + load resistor : V i is the arithmetic sum A i + B i + C i-1 3. A SET decodes the arithmetic sum to a Boolean variable • multivalued logic scheme • CMOS compatible • scalable • each full addition requires 4 transistors reduces the number of switches by a factor 7 J.A. Mol, J. Verduijn, R. D. Levine, F. Remacle and S. Rogge, PNAS, 108 , 13969 (2011). M. Klein, G. P. Lansbergen, J. A. Mol, S. Rogge, R. D. Levine, and F. Remacle, ChemPhysChem. 10 : 162-173, 2009. CD08 Mons Aug 23-25, 2017 8
Arithmetic addition performed by the SAT • A and B are encoded as two value of V b and V g • depending on these values, input (0,0) :zero level contribute input (0,1) and (1,0) : one level to the input (1,1) : two levels current half addition CIB SET decoder 8 CD08 Mons Aug 23-25, 2017
CIB and decoder to Boolean V i = A i + B i + V i-1 SET i converts the a.s. to a Boolean signal. The conversion is implemented using the periodicity of the SET. V i is applied to the V i gate of the SET and gives a current that corresponds to S i a.s is a four valued signal CIB : carry in buffer , made of • a load resistor (converts I i-1 to V i-1 ) • a carry in FET opened if the arithmetic sum, V i-1 , is larger than 1 (C i = 1). • provides gain CD08 Mons Aug 23-25, 2017 10
Finite state machines • Logic circuits whose present output depends on the inputs the present state stored in the memory unit outputs inputs logic memory The internal states of the machine provide the storage unit ( memory) . • internal states: implemented by the quantum states of the molecule • parallel logic : mapping of the multi state dynamics. CD08 Mons Aug 23-25, 2017 11
Computing by observables for parallel logic Allows to increase the number of state variables ( ) Ψ t The state of the system, , is a coherent System with N stationary time-dependent superposition of the N states: quantum states N ( ) = ( ) Ψ i Apply a perturbation ∑ c i t Ψ t which induces i = 1 transitions between . . states The coefficients c i ( t ) are complex numbers. . ( ) 2 In the superposition, the quantum states have a population c i t Since the state of the system is a superposition, there are also coherences between pairs of states, i and j , that evolve as the product of the coefficients of the two states : c i ( t ) c j ( t ). The state of the system, that is the coherences and the populations, can be described by the density matrix operator (von Neumann) ( ) = ψ t ( ) ψ t ( ) ρ t ˆ CD08 Mons Aug 23-25, 2017 12
Computing by observables for parallel logic Allows to increase the number of state variables N stationary N ( ) = ( ) Ψ i ∑ c i t Ψ t quantum states i = 1 N N ( ) = ψ t ( ) ψ t ( ) = ( ) c j t ( ) Ψ i * t ∑ ∑ c i ρ t Ψ j ˆ i = 1 j = 1 . . . ( ) ( ) ( ) ⎛ ⎞ ρ 11 t ρ 12 t ρ 1 N t ! ⎜ ⎟ ( ) ( ) ( ) ⎜ ⎟ ρ 21 t ρ 22 t ρ 2 N t ! ( ) = Matrix form in the basis of the N ρ t ˆ ⎜ ⎟ stationary quantum states is N x N ⎜ ⎟ " " " " ( ) ( ) ( ) ⎜ ⎟ ρ N 1 t ρ N 2 t ρ NN t ! ⎝ ⎠ The observables are the matrix elements of the density matrix ( ) , i = 1, ..., N ρ ii t populations of the quantum states : diagonal elements, real values (probabilities) ( ) = ρ ji ( ) N ( N -1) * t ρ ij t coherences between the quantum states : complex numbers N 2 observables ( N 2 -1) if normalisation is known CD08 Mons Aug 23-25, 2017 13
Dimer of two dyes (two coupled two level systems) Four quantum states : N =4 N x N = 16 observables: i = ˆ ρ ii = i ˆ E ii i =0,1,2,3 populations j = ˆ coherences ˆ ρ ij = i E ij i , j =0,1,2,3 i ≠ j ˆ ⎡ E ij , ˆ ˆ ⎤ ⎦ = δ jk ˆ E il − δ li ˆ E ij E kl E kj The 16 operators are close under commutation : ⎣ They are the generators of the Lie Group SU(4) ( ) ⋅ ˆ ˆ H = ε 0 0 0 + ε 1 1 1 + ε 2 2 2 + ε 3 3 3 − E t µ Hamiltonian µ = µ ij ˆ ∑ ˆ E ij dipole operator i , j E ( t ) is the oscillating electric field N N ( ) . ∑ ε i ˆ ∑ µ ij ˆ ˆ of the electromagnetic field H = E ii − E t E ij ˆ µ The dipole operator, , induces i ≠ j i the transitions between the states. ( t ) ˆ ρ = ρ ij ∑ ˆ E ij i , j CD08 Mons Aug 23-25, 2017 14
Multi state dynamics Dynamics of the density matrix of a multi level system in Liouville space ( ) ρ t i ! d ˆ ( ) ˆ ⎡ ⎤ = ρ t H , ˆ dissipation can be included ⎣ ⎦ dt ( ) ∑ δ jk ˆ E il − δ li ˆ ⎡ µ , ˆ ρ ⎦ = ⎤ µ ij ˆ ρ kl ˆ E kj ⎣ ijkl Heisenberg equation of motion for an observable ( ) d ˆ E ij t ( ) ⎡ H , ˆ ˆ ⎤ = i ! E ij t ⎣ ⎦ dt Instead of the populations of the N states for encoding logic variables and performing parallel logic operations, we therefore have in principle N 2 state variables on which we can operate in parallel . B. Fresch, D. Hiluf, E. Collini, R. D. Levine, and F. Remacle, Molecular decision trees realized by ultrafast electronic spectroscopy, Proc. Natl. Acad. Sci. USA 110 , 17183 (2013). CD08 Mons Aug 23-25, 2017 15
Computing by observables Hamiltonian is a linear combination of operators that belong to the close set of observables k ˆ ⎡ O i , ˆ ˆ ⎤ ∑ ⎦ = c ij ˆ h l ˆ O j O k H = O l ⎣ l Using information theory and the Heisenberg equation of motion of an observable, Alhassid and Levine (1978) showed that the time evolution of the density matrix is equivalent to a set of N 2 coupled equations of the mean values of observables. ( ) ˆ d O i t ( ) ∑ ˆ = i ! g ij O j t i , j =0,..., N 2 - 1 dt j Technically, the observables are the generators of a Lie group. These relations are verified for both the model Hamiltonians used for describing the dynamics of chromophoric systems and the dopant systems. Basis for the implementation of parallel logic schemes and molecular decision trees Y. Alhassid, and R. D. Levine, Connection between maximal entropy and scattering theoretic analyses of collision processes, Phys. Rev. A 18 , 89 (1978). CD08 Mons Aug 23-25, 2017 16
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