Optical and electrical parallel molecular and nanoscale computing - - PowerPoint PPT Presentation
Optical and electrical parallel molecular and nanoscale computing Franoise Remacle Theoretical Physical Chemistry, UR Molsys, University of Lige FP7 FET projects H2020 FET open project COPAC , starts Nov 1st, 2017 Collaboration : R. D.
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Collaboration : R. D. Levine (HUJI), B. Fresch (UNIPD) Experiments : E. Collini (UNIPD), S. Rogge (UNSW, Sydney)
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(http://rebootingcomputing.ieee.org/)
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Optical addressing of the electronic states of multichromophoric rhodamine supramolecular complexes in solution for implementing molecular decision trees
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Light induced transitions between states pulse 1 x1 pulse 2 x2 pulse 3 x3 Molecular decision tree operated in parallel Elisabetta Collini, Padua
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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. Interacting P atom embedded in Si Sven Rogge School of Physics & CQC2T University of New South Wales Australia
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Lansberger et al, Nature Physics, 4 , 656 (2008)
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V G (mV) V b (mV)
300 320 340 360 40 30 20 10
0.2 0.4 0.6
dI/dVb ( S)
13G14
EC = 31 meV
Vb = 40 mV 320 340 360 380 10 20 30 ISD (nA) VG (mV)
Excited electronic states of a given charge As dopant
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J.A. Mol, J. Verduijn, R. D. Levine, F. Remacle and S. Rogge, PNAS, 108, 13969 (2011).
reduces the number of switches by a factor 7 1 . SAT : arithmetic addition Ai + Bi in base 4
Vi is the arithmetic sum Ai + Bi + Ci-1
Boolean variable
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input (0,0) :zero level input (0,1) and (1,0) : one level input (1,1) : two levels contribute to the current half addition CIB SET decoder
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CIB : carry in buffer , made of
arithmetic sum, Vi-1, is larger than 1 (Ci = 1).
SETi converts the a.s. to a Boolean signal. The conversion is implemented using the periodicity of the SET. Vi is applied to the gate of the SET and gives a current that corresponds to Si Vi a.s is a four valued signal Vi = Ai + Bi + Vi-1
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logic memory inputs
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Allows to increase the number of state variables
. . . System with N stationary quantum states The state of the system, , is a coherent time-dependent superposition of the N states:
Ψ t
( )
Apply a perturbation which induces transitions between states
Ψ t
i=1 N
The coefficients ci(t) are complex numbers. The state of the system, that is the coherences and the populations, can be described by the density matrix operator (von Neumann) In the superposition, the quantum states have a population ci t
2
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 : ci (t) cj (t).
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Allows to increase the number of state variables
ˆ ρ t
( ) =
ρ11 t
( )
ρ12 t
( )
! ρ1N t
( )
ρ21 t
( )
ρ22 t
( )
! ρ2N t
( )
" " " " ρN1 t
( )
ρN 2 t
( )
! ρNN t
( )
⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟
populations of the quantum states :
ρii t
coherences between the quantum states : diagonal elements, real values (probabilities) . . .
Ψ t
i=1 N
ˆ ρ t
i=1 N
j=1 N
* t
Ψ j
N stationary quantum states Matrix form in the basis of the N stationary quantum states is N x N The observables are the matrix elements of the density matrix
ρij t
* t
complex numbers
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Four quantum states : N =4
N x N = 16 observables: i =0,1,2,3 populations coherences ˆ
i ≠ j Hamiltonian ˆ H = ε0 0 0 + ε1 1 1 + ε2 2 2 + ε3 3 3 − E t
µ i, j=0,1,2,3 The 16 operators are close under commutation :
ˆ Eij
ˆ Eij, ˆ Ekl ⎡ ⎣ ⎤ ⎦ = δ jk ˆ Eil −δ li ˆ Ekj
i, j
ˆ ρ = ρij
i, j
∑ (t) ˆ Eij
ˆ H =
i N
Eii − E t
i≠ j N
Eij
They are the generators of the Lie Group SU(4) dipole operator E(t) is the oscillating electric field
The dipole operator, , induces the transitions between the states.
ˆ µ
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Dynamics of the density matrix of a multi level system in Liouville space
i! d ˆ ρ t
dt = ˆ H, ˆ ρ t
⎡ ⎣ ⎤ ⎦
dissipation can be included Heisenberg equation of motion for an observable
i! d ˆ Eij t
dt = ˆ H, ˆ Eij t
⎡ ⎣ ⎤ ⎦
Instead of the populations of the N states for encoding logic variables and performing parallel logic operations, we therefore have in principle N2 state variables on which we can operate in parallel.
ˆ µ, ˆ ρ ⎡ ⎣ ⎤ ⎦ = µij ˆ ρkl
ijkl
δ jk ˆ Eil −δli ˆ Ekj
realized by ultrafast electronic spectroscopy, Proc. Natl. Acad. Sci. USA 110, 17183 (2013).
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Hamiltonian is a linear combination of operators that belong to the close set of observables
ˆ H =
l
hl ˆ Ol
k ˆ
i! d ˆ Oi t
dt =
j
gij ˆ Oj t
i,j=0,..., N2 - 1 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 Technically, the observables are the generators of a Lie group. 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 N2 coupled equations of the mean values of observables.
theoretic analyses of collision processes, Phys. Rev. A 18, 89 (1978).
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Physical realization of a molecular decision tree on the TAMRA-DNA complex
Multivalued Logic by the Two-Dimensional Photon-Echo Response of a Rhodamine–DNA Complex, J. Phys. Chem. Lett. 6, 1714 (2015). 2D-PE set up : addressing by a sequence
DNA-TAMRA Model level structure : GS (g) S1 v=0 and v=1 (e0 and e1) S2 v=1 (f) all transitions are within the laser band Weak fields, impulsive limit third order density matrix ˆ ρ
3
( ) τ,T,t
i ! ⎛ ⎝ ⎜ ⎞ ⎠ ⎟
3
ˆ U τ +T + t
µ τ +T
µ τ
µ 0
ρ 0
⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ˆ U
† τ +T + t
c ji t,T,τ
Eij
ij
All the operators can be written in terms of the 16 observables close under commutation.
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experiment computed response For a broad bandwidth, the Liouville pathways are followed in parallel The third order polarization along τ and t for given values of T :
P
3
( ) t
( ) = Tr ˆ
µ ˆ ρ
3
( ) t
( )
The 2D spectra are obtained as a double FT of :
P
3
( ) t
SkS ω1,T,ω 2
( ) =
dt dτ
e
−iω1τe iω2tiP kS 3
( ) τ,T,t
( )
i! d ˆ ρ t
dt = ˆ H, ˆ ρ t
⎡ ⎣ ⎤ ⎦
ˆ µ, ˆ ρ ⎡ ⎣ ⎤ ⎦ = µij ˆ ρkl
ijkl
δ jk ˆ Eil −δli ˆ Ekj
15fs 600fs
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Apply Shannon decomposition recursively
To build the tree So that the end nodes (leaf) correspond to the MIN term expansion of the function
XOR Truth Table
Decision tree
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A Liouville pathway corresponds to one of the branches of a molecular decision tree. Assignment of the three variables x1, x2 and x3. x1,2=the vibrational mode is excited by the interaction with the first (second) laser pulse. x3=the interaction with the laser stimulates the system emission Several pathways correspond to the same position on the map. This defines the class of functions that can be implemented. For the 6 positions can be reliably characterized and related to the dynamics, we can implement any of the 64 functions of 3 Boolean variables. x1 x2 x3 pulse 1 pulse 2 pulse 3
f 15 fs
( ) = x1x2x3 + x1x2x3 + x1x2x3 + x1x2x3
f 600 fs
( ) = x1x2x3 + x1x2x3
The MIN term that corresponds to (0,1,0,...,1) is
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Boolean functions : a decision tree provides the input for all possible sets of inputs. The spectral decomposition corresponds to a Sum of Products (SOP), also called MIN term expansion. f x1,...,xn
Function of n Boolean variables
j=1 2n
1,..,in
i
1,...,in
is a MIN term : the AND product of the values of the inputs xj, negated when they are zeros. aj is the Boolean output for the set of inputs
x1,..,xn
( ) j
x1x2x3...xn
There are
2n
( )
functions of n Boolean variables.
SOP
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The spectral decomposition is carried out on a basis of Hadamard products
Stankovic, R. S.; Astola, J. T.; Moraga, C., Representation of Multiple-Valued Logic Functions. Morgan and Claypool Publishers: San Rafael, 2012; Vol. 37.
SOP for a function of n logic variables of radix r :
f x1,..,xn
( ) =
j=1 rn
a j Ji1 x1
( ),Ji2 x2 ( )..,Jin xn ( )
( ) j
aj is the r valued output,
Ji1 x1
( ),Ji2 x2 ( )..,Jin xn ( )
( ) j is the Hadamard product and the analog of the MIN term
The functions
Jij x j
are called characteristic functions. They are defined such as
Jij x j
( ) = 1
if ij=xj and 0 otherwise r = 5 balanced notation, the Hadamard product is an element of the basis that identifies the input sequence (-2,1,0)
J−2 x1
( )J1 x2 ( )J0 x3 ( )
SOP decompositions of discrete functions of multivalued discrete variables are the analog to integral transforms of functions of continuous variables. Such a representation is important tools for functional analysis, i.e., for comparing two logic functions or for logic synthesis.
Example on a function of 2 base 3 variables
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Total number of functions : 3
32
( ) = 19683
Number of pairs of inputs : 32 = 9 balanced ternary (-1,0,1) Subset of functions that have a zero output for (0,0), (0,-1), (0,+1), (+1,0) and (-1, 0): 39-5=34 Mimic optical selection rules : Mimic output detection : {(1,1) , (-1,-1)} and {(-1,1) , (1,-1)} lead to the same output which can be -1,0,1. Subset of functions that can be computed is 32=9 functions. 6 characteristic functions :
(x1,x2) f1 f2 f3 f4 f5 f6 f7 f8 f9 00 0-1 0+1 +10
11 1
1
1
1
1
1
1
1
1 1-1 1
1
1
Ji x j
( ) =
1 if x j = i 0 otherwise ⎧ ⎨ ⎪ ⎩ ⎪
Ji x1
represented as vectors of 9 dimensions (because there are 9 possible pairs of inputs)
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f x1,x2
⊕ f −10
f 10
is the sum modulo 3
(x1,x2) f1 f2 f3 f4 f5 f6 f7 f8 f9 00 0-1 0+1 +10
11 1
1
1
1
1
1
1
1
1 1-1 1
1
1
f 8 x1,x2
⊕J−1 x1
⊕(−1)⋅ J−1 x1
⊕(−1)⋅ J1 x1
⊕
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Experimental setup (macroscopic view) Ensemble Microscopic view of polarizations
detector emission incident pulses
Phase matching conditions of the total polarization of the ensemble P(t,r)
l
l t
The phase matching conditions, kl, define the macroscopic directions of emission.
n
l ≡ l1,..,li,...
Number of partial polarizations, Pl (t), necessary to expand P(t,r) depends on the strength of the field and the number of pulses in the sequence. PE The partial polarizations Pl (t) are the outputs of a multivariable function for a given set
direction
l ≡ l1,..,li,...
kl =
n
lnkn
kn is the wave vector of the pulse n.
T.-M. Yan, B. Fresch, R. D. Levine, and F. Remacle, Information processing in parallel through directionally resolved molecular polarization components in coherent multidimensional spectroscopy, J. Chem. Phys. 143, 064106 (2015).
Spectral decomposition
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their wavelength, and polarization, also be changing the molecule.
Sl ≡ Sl τ,T
( ) =
dt
∞
∫
P
l τ,T,t
( )
The radix r of the variables is determined by the strength of the laser pulses.
l1,l2,l3
( )
Three pulses :
i
3
∑
li = n
determines the order of the process. For the pulse strengths used in Padua, first, third (fifth) orders are necessary to describe P(t,r) 6 directions at first order, 38 at third order and 102 at fifth order. Optical selection rules : Some directions can lead to a zero output, outputs in several directions can be the same.
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1 2 3 4
(a) (b) k1 k2 k3
60 fs 140 fs
k1 k2 k3
100 fs 100 fs
1 2
1 2
Five level structure (cf dimer)
1 2 3 4 5 6
(a) τ = T = 100 fs
(3,0,0) (-3,0,0) (0,3,0) (0,-3,0) (0,0,3) (0,0,-3) (2,1,0) (-2,-1,0) (2,-1,0) (-2,1,0) (2,0,1) (-2,0,-1) (2,0,-1) (-2,0,1) (1,2,0) (-1,-2,0) (1,-2,0) (-1,2,0) (0,2,1) (0,-2,-1) (0,2,-1) (0,-2,1) (0,1,2) (0,-1,-2) (0,1,-2) (0,-1,2) (1,0,2) (-1,0,-2) (1,0,-2) (-1,0,2) (1,1,-1) (-1,-1,1) (1,-1,1) (-1,1,-1) (-1,1,1) (1,-1,-1) (1,1,1) (-1,-1,-1)
1 2
Directions {l}3 (third order) Logic values (b)
we use r =5 32 triplets of inputs in base 5
i
3
∑
li = n
third order 3 pulses, 3 base 5 variables 32 phase matching directions selection rules :
P
−l j
( ) t
( ) ≡ P
−l1,−l2,−l3 j
( )
t
( ) = P
l1,l2,l3 j
( )
t
( )
⎡ ⎣ ⎤ ⎦
*
2 out of the subset of the 516 functions that will have a different
matching directions.
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!
dopants are unequally spaced to control
Observables : charge on the dots, and bond
Potentially 8 observables (N2-1). Only the real part of the bond order is measured. linear geometry : only 2 bond orders 4 observables (2 populations, 2 bond
Model Hamiltonian : SU3 Lie group Differences with the optical case : non perturbative regime, no optical selection rules The 4 observables are simultaneously directly addressed and read by the tip. Intrinsically multivalued Implementation of the spectral decomposition of multivalued logic functions in parallel by electrical addressing
ˆ H t
( ) =
i=0 2
( ) i
i +
i, j≠i 2
j
Vg t
( ) = V0 −
i
ΔVi 2 tanh fi t − ti
( )
( )
αi(t) = aiVg t
Single device is read, need for averaging over several cycles of Vg(t). In the optical case, the averaging is performed by the ensemble and provides directly a macroscopic output.
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Non perturbative regime : all the observables are coupled to one another each time the gate voltage varies. There are therefore a maximum of 9 possible transitions for a 3 dopant molecule. We define the radix r of the logic variable by the maximum number of possible transitions r = 9.
k m
The number of variables is fixed by the number of steps in the gate voltage profile, Vg (t). x1 x2 t Encoding of the value, p, of the logic variables in balanced notation :
Transition p varies from +4 ( ) to -4 ( )
0 0 → 2 2 2 2 → 0 0
For a M dopant system, there are M-1 independent populations that are real. There are only M (M -1)/2 real coherence terms that correspond bond orders. Linear dopant molecule, only two independent populations and two bond orders can be measured, so there are 94= 6561functions that can be implemented by varying the profile of the gate voltage and/or the structure of the dopant molecule.
k k
The number of functions that can be implemented is restricted to = 59049.
9(27−22)
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1 2 1 3 4
2
3
f 0,0
( )
f 1,−1
( )
f 2,−2
( )
f 3,−3
( )
f 4,−4
( )
f 0,1
( )
f 1,0
( )
f 2,−1
( )
f 3,−2
( )
f 4,−3
( )
f 0,2
( )
f 1,1
( )
f 2,0
( )
f 3,−1
( )
f 4,−2
( )
f 0,3
( )
f 1,2
( )
f 2,1
( )
f 3,0
( )
f 4,−1
( )
f 0,4
( )
f 1,3
( )
f 2,2
( )
f 3,1
( )
f 4,0
( )
...
Implementation of 3 different functions by varying the profile of gate voltage all sets of inputs that lead to the same output Implementation
parallel by electrical addressing a molecule of three dopants in Silicon, ChemPhysChem 18, 1790 (2017).
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functions
molecular dynamics responses that lead to the outputs.
functions
Funding : FP7 FET MULTI Moldynlogic, MOLOC