Symbolic Analysis and Simulation

Symbolic Analysis of the Pseudo NAND gate and the Pseudo NAND SR-Latch

Pseudo NAND gate

In[1]:

from qnet.algebra.circuit_algebra import *

In[2]:

from qnet.circuit_components import pseudo_nand_cc as nand

# real parameters
kappa = symbols('kappa', positive = True)
Delta, chi, phi, theta = symbols('Delta, chi, phi, theta', real = True)

# complex parameters
A, B, beta = symbols('A, B, beta')

N = nand.PseudoNAND('N', kappa=kappa, Delta=Delta, chi=chi, phi=phi, theta=theta, beta=beta)
N

Out[2]:

\[{\rm N}\]

Circuit Analysis of Pseudo NAND gate

In[3]:

N.creduce()

Out[3]:

\[\begin{split}\left(\mathbf{1}_{1} \boxplus (\left(\mathbf{1}_{1} \boxplus (\left({\rm N.P} \boxplus \mathbf{1}_{1}\right) \lhd {{\rm N.BS2}} \lhd \left({W(\beta)} \boxplus \mathbf{1}_{1}\right))\right) \lhd {\mathbf{P}_\sigma \begin{pmatrix} 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}} \lhd \left({\rm N.K} \boxplus \mathbf{1}_{1}\right))\right) \lhd \left({\rm N.BS1} \boxplus \mathbf{1}_{2}\right) \lhd {\mathbf{P}_\sigma \begin{pmatrix} 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 2 \end{pmatrix}}\end{split}\]

In[4]:

# yields a single block
N.show()

# decompose into sub components
N.creduce().show()

SLH model

In[5]:

NSLH = N.coherent_input(A, B, 0, 0).toSLH()
NSLH

Out[5]:

\[\begin{split}\left( \begin{pmatrix} \frac{1}{2} \sqrt{2} & - \frac{1}{2} \sqrt{2} & 0 & 0 \\ \frac{1}{2} \sqrt{2} & \frac{1}{2} \sqrt{2} & 0 & 0 \\ 0 & 0 & e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right) & - e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) \\ 0 & 0 & \operatorname{sin}\left(\theta\right) & \operatorname{cos}\left(\theta\right)\end{pmatrix}, \begin{pmatrix} \frac{1}{2} \sqrt{2} A - \frac{1}{2} \sqrt{2} B \\ \left(\frac{1}{2} \sqrt{2} A + \frac{1}{2} \sqrt{2} B\right) + \sqrt{\kappa} {a_{{{\rm N.K}}}} \\ \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right) - \sqrt{\kappa} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N.K}}}} \\ \beta \operatorname{sin}\left(\theta\right) + \sqrt{\kappa} \operatorname{cos}\left(\theta\right) {a_{{{\rm N.K}}}}\end{pmatrix}, \frac{1}{2} \mathbf{\imath} \left( - \left(\frac{1}{2} \sqrt{2} A \sqrt{\kappa} + \frac{1}{2} \sqrt{2} B \sqrt{\kappa}\right) {a_{{{\rm N.K}}}^\dagger} + \left(\frac{1}{2} \sqrt{2} \sqrt{\kappa} \overline{A} + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \overline{B}\right) {a_{{{\rm N.K}}}}\right) + \chi {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} {a_{{{\rm N.K}}}} + \Delta {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} \right)\end{split}\]

Heisenberg equation of motion of the mode operator \(a\)

In[6]:

s = N.space
a = Destroy(s)
a

Out[6]:

\[{a_{{{\rm N.K}}}}\]

In[7]:

NSLH.symbolic_heisenberg_eom(a).expand().simplify_scalar()

Out[7]:

\[\frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- A - B\right) - \left(\mathbf{\imath} \Delta + \kappa\right) {a_{{{\rm N.K}}}} - 2 \mathbf{\imath} \chi {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} {a_{{{\rm N.K}}}}\]

Super operator algebra: The system’s liouvillian and a re-derivation of the eom for \(a\) via the super-operator adjoint of the liouvillian.

In[8]:

LLN = NSLH.symbolic_liouvillian().expand().simplify_scalar()
LLN

Out[8]:

\[\frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(A + B\right) {\rm spost}\left[{a_{{{\rm N.K}}}^\dagger}\right] + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- \overline{A} - \overline{B}\right) {\rm spost}\left[{a_{{{\rm N.K}}}}\right] + \mathbf{\imath} \chi {\rm spost}\left[{a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} {a_{{{\rm N.K}}}}\right] + \left(\mathbf{\imath} \Delta - \kappa\right) {\rm spost}\left[{a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}}\right] + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- A - B\right) {\rm spre}\left[{a_{{{\rm N.K}}}^\dagger}\right] + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(\overline{A} + \overline{B}\right) {\rm spre}\left[{a_{{{\rm N.K}}}}\right] - \mathbf{\imath} \chi {\rm spre}\left[{a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} {a_{{{\rm N.K}}}}\right] - \left(\mathbf{\imath} \Delta + \kappa\right) {\rm spre}\left[{a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}}\right] + 2 \kappa {\rm spre}\left[{a_{{{\rm N.K}}}}\right] {\rm spost}\left[{a_{{{\rm N.K}}}^\dagger}\right]\]

In[9]:

(LLN.superadjoint() * a).expand().simplify_scalar()

Out[9]:

\[\frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- A - B\right) - \left(\mathbf{\imath} \Delta + \kappa\right) {a_{{{\rm N.K}}}} - 2 \mathbf{\imath} \chi {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} {a_{{{\rm N.K}}}}\]

A full Pseudo-NAND SR-Latch

In[10]:

N1 = nand.PseudoNAND('N_1', kappa=kappa, Delta=Delta, chi=chi, phi=phi, theta=theta, beta=beta)
N2 = nand.PseudoNAND('N_2', kappa=kappa, Delta=Delta, chi=chi, phi=phi, theta=theta, beta=beta)

# NAND gates in mutual feedback configuration
NL = (N1 + N2).feedback(2, 4).feedback(5, 0).coherent_input(A, 0, 0, B, 0, 0)
NL

Out[10]:

\[\begin{split}{\left\lfloor\left(\mathbf{1}_{3} \boxplus {\rm N_2}\right) \lhd \left(({\mathbf{P}_\sigma \begin{pmatrix} 0 & 1 & 2 & 3 \\ 0 & 1 & 3 & 2 \end{pmatrix}} \lhd {{\rm N_1}}) \boxplus \mathbf{1}_{3}\right)\right\rfloor_{5\to0}} \lhd \left({W(A)} \boxplus \mathbf{1}_{2} \boxplus {W(B)} \boxplus \mathbf{1}_{2}\right)\end{split}\]

The circuit algebra simplification rules have already eliminated one of the two feedback operations in favor or a series product.

In[11]:

NL.show()
NL.creduce().show()
NL.creduce().creduce().show()

SLH model

In[12]:

NLSLH = NL.toSLH().expand().simplify_scalar()
NLSLH

Out[12]:

\[\begin{split}\left( \begin{pmatrix} - \frac{1}{2} \sqrt{2} & 0 & 0 & 0 & \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right) & - \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) \\ \frac{1}{2} \sqrt{2} & 0 & 0 & 0 & \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right) & - \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) \\ 0 & \operatorname{sin}\left(\theta\right) & \operatorname{cos}\left(\theta\right) & 0 & 0 & 0 \\ 0 & \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right) & - \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) & - \frac{1}{2} \sqrt{2} & 0 & 0 \\ 0 & \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right) & - \frac{1}{2} \sqrt{2} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) & \frac{1}{2} \sqrt{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \operatorname{sin}\left(\theta\right) & \operatorname{cos}\left(\theta\right)\end{pmatrix}, \begin{pmatrix} \frac{1}{2} \sqrt{2} \left(- A + \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) - \frac{1}{2} \sqrt{2} \sqrt{\kappa} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N_2.K}}}} \\ \frac{1}{2} \sqrt{2} \left(A + \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) + \sqrt{\kappa} {a_{{{\rm N_1.K}}}} - \frac{1}{2} \sqrt{2} \sqrt{\kappa} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N_2.K}}}} \\ \beta \operatorname{sin}\left(\theta\right) + \sqrt{\kappa} \operatorname{cos}\left(\theta\right) {a_{{{\rm N_1.K}}}} \\ \frac{1}{2} \sqrt{2} \left(- B + \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) - \frac{1}{2} \sqrt{2} \sqrt{\kappa} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N_1.K}}}} \\ \frac{1}{2} \sqrt{2} \left(B + \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) - \frac{1}{2} \sqrt{2} \sqrt{\kappa} e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N_1.K}}}} + \sqrt{\kappa} {a_{{{\rm N_2.K}}}} \\ \beta \operatorname{sin}\left(\theta\right) + \sqrt{\kappa} \operatorname{cos}\left(\theta\right) {a_{{{\rm N_2.K}}}}\end{pmatrix}, \frac{1}{4} \sqrt{2} \mathbf{\imath} \sqrt{\kappa} \left(- A - \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) {a_{{{\rm N_1.K}}}^\dagger} + \frac{1}{4} \sqrt{2} \mathbf{\imath} \sqrt{\kappa} \left(- B - \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) {a_{{{\rm N_2.K}}}^\dagger} + \frac{\sqrt{2} \mathbf{\imath} \sqrt{\kappa} \left(e^{\mathbf{\imath} \phi} \overline{A} + \operatorname{cos}\left(\theta\right) \overline{\beta}\right)}{4 e^{\mathbf{\imath} \phi}} {a_{{{\rm N_1.K}}}} + \frac{\sqrt{2} \mathbf{\imath} \sqrt{\kappa} \left(e^{\mathbf{\imath} \phi} \overline{B} + \operatorname{cos}\left(\theta\right) \overline{\beta}\right)}{4 e^{\mathbf{\imath} \phi}} {a_{{{\rm N_2.K}}}} + \chi {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_1.K}}}} + \Delta {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} + \frac{\sqrt{2} \mathbf{\imath} \kappa \left(e^{2 \mathbf{\imath} \phi} -1\right) \operatorname{sin}\left(\theta\right)}{4 e^{\mathbf{\imath} \phi}} {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_2.K}}}} + \chi {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} {a_{{{\rm N_2.K}}}} + \Delta {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} + \frac{\sqrt{2} \mathbf{\imath} \kappa \left(e^{2 \mathbf{\imath} \phi} -1\right) \operatorname{sin}\left(\theta\right)}{4 e^{\mathbf{\imath} \phi}} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_2.K}}}^\dagger} \right)\end{split}\]

Heisenberg equations of motion for the mode operators

In[13]:

NL.space

Out[13]:

\[{{\rm N_1.K}} \otimes {{\rm N_2.K}}\]

In[14]:

s1, s2 = NL.space.operands
a1 = Destroy(s1)
a2 = Destroy(s2)

In[15]:

da1dt = NLSLH.symbolic_heisenberg_eom(a1).expand().simplify_scalar()
da1dt

Out[15]:

\[\frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- A - \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) - \left(\mathbf{\imath} \Delta + \kappa\right) {a_{{{\rm N_1.K}}}} + \frac{1}{2} \sqrt{2} \kappa e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N_2.K}}}} - 2 \mathbf{\imath} \chi {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_1.K}}}}\]

In[16]:

da2dt = NLSLH.symbolic_heisenberg_eom(a2).expand().simplify_scalar()
da2dt

Out[16]:

\[\frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- B - \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) + \frac{1}{2} \sqrt{2} \kappa e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {a_{{{\rm N_1.K}}}} - \left(\mathbf{\imath} \Delta + \kappa\right) {a_{{{\rm N_2.K}}}} - 2 \mathbf{\imath} \chi {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} {a_{{{\rm N_2.K}}}}\]

Show Exchange-Symmetry of the Pseudo NAND latch Liouvillian super operator

Simultaneously exchanging the degrees of freedom and the coherent input amplitudes leaves the liouvillian unchanged.

In[17]:

C = symbols('C')
LLNL = NLSLH.symbolic_liouvillian().expand().simplify_scalar()
LLNL

Out[17]:

\[\frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(A + \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) {\rm spost}\left[{a_{{{\rm N_1.K}}}^\dagger}\right] + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(B + \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) {\rm spost}\left[{a_{{{\rm N_2.K}}}^\dagger}\right] + \frac{\sqrt{2} \sqrt{\kappa} \left(- e^{\mathbf{\imath} \phi} \overline{A} - \operatorname{cos}\left(\theta\right) \overline{\beta}\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spost}\left[{a_{{{\rm N_1.K}}}}\right] + \frac{\sqrt{2} \sqrt{\kappa} \left(- e^{\mathbf{\imath} \phi} \overline{B} - \operatorname{cos}\left(\theta\right) \overline{\beta}\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spost}\left[{a_{{{\rm N_2.K}}}}\right] + \mathbf{\imath} \chi {\rm spost}\left[{a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_1.K}}}}\right] + \left(\mathbf{\imath} \Delta - \kappa\right) {\rm spost}\left[{a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}}\right] + \frac{\sqrt{2} \kappa \operatorname{sin}\left(\theta\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spost}\left[{a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_2.K}}}}\right] + \mathbf{\imath} \chi {\rm spost}\left[{a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} {a_{{{\rm N_2.K}}}}\right] + \left(\mathbf{\imath} \Delta - \kappa\right) {\rm spost}\left[{a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}}\right] + \frac{\sqrt{2} \kappa \operatorname{sin}\left(\theta\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spost}\left[{a_{{{\rm N_1.K}}}} {a_{{{\rm N_2.K}}}^\dagger}\right] + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- A - \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) {\rm spre}\left[{a_{{{\rm N_1.K}}}^\dagger}\right] + \frac{1}{2} \sqrt{2} \sqrt{\kappa} \left(- B - \beta e^{\mathbf{\imath} \phi} \operatorname{cos}\left(\theta\right)\right) {\rm spre}\left[{a_{{{\rm N_2.K}}}^\dagger}\right] + \frac{\sqrt{2} \sqrt{\kappa} \left(e^{\mathbf{\imath} \phi} \overline{A} + \operatorname{cos}\left(\theta\right) \overline{\beta}\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spre}\left[{a_{{{\rm N_1.K}}}}\right] + \frac{\sqrt{2} \sqrt{\kappa} \left(e^{\mathbf{\imath} \phi} \overline{B} + \operatorname{cos}\left(\theta\right) \overline{\beta}\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spre}\left[{a_{{{\rm N_2.K}}}}\right] - \mathbf{\imath} \chi {\rm spre}\left[{a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_1.K}}}}\right] - \left(\mathbf{\imath} \Delta + \kappa\right) {\rm spre}\left[{a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}}\right] + \frac{1}{2} \sqrt{2} \kappa e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {\rm spre}\left[{a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_2.K}}}}\right] - \mathbf{\imath} \chi {\rm spre}\left[{a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} {a_{{{\rm N_2.K}}}}\right] - \left(\mathbf{\imath} \Delta + \kappa\right) {\rm spre}\left[{a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}}\right] + \frac{1}{2} \sqrt{2} \kappa e^{\mathbf{\imath} \phi} \operatorname{sin}\left(\theta\right) {\rm spre}\left[{a_{{{\rm N_1.K}}}} {a_{{{\rm N_2.K}}}^\dagger}\right] + 2 \kappa {\rm spre}\left[{a_{{{\rm N_1.K}}}}\right] {\rm spost}\left[{a_{{{\rm N_1.K}}}^\dagger}\right] + \frac{\sqrt{2} \kappa \left(- e^{2 \mathbf{\imath} \phi} -1\right) \operatorname{sin}\left(\theta\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spre}\left[{a_{{{\rm N_1.K}}}}\right] {\rm spost}\left[{a_{{{\rm N_2.K}}}^\dagger}\right] + \frac{\sqrt{2} \kappa \left(- e^{2 \mathbf{\imath} \phi} -1\right) \operatorname{sin}\left(\theta\right)}{2 e^{\mathbf{\imath} \phi}} {\rm spre}\left[{a_{{{\rm N_2.K}}}}\right] {\rm spost}\left[{a_{{{\rm N_1.K}}}^\dagger}\right] + 2 \kappa {\rm spre}\left[{a_{{{\rm N_2.K}}}}\right] {\rm spost}\left[{a_{{{\rm N_2.K}}}^\dagger}\right]\]

In[18]:

C = symbols('C')
(LLNL.substitute({A:C}).substitute({B:A}).substitute({C:B}) - LLNL.substitute({s1:s2,s2:s1}).expand().simplify_scalar()).expand().simplify_scalar()

Out[18]:

\[\hat{0}\]

Numerical Analysis via QuTiP

Input-Output Logic of the Pseudo-NAND Gate

In[19]:

NSLH.space

Out[19]:

\[{{\rm N.K}}\]

In[20]:

NSLH.space.dimension = 75

Numerical parameters taken from

Mabuchi, H. (2011). Nonlinear interferometry approach to photonic sequential logic. Appl. Phys. Lett. 99, 153103 (2011)

In[21]:

# numerical values for simulation

alpha = 22.6274                              # logical 'one' amplitude

numerical_vals = {
                  beta: -34.289-11.909j,     # bias input for pseudo-nands
                  kappa: 25.,                # Kerr-Cavity mirror couplings
                  Delta: 50.,                # Kerr-Cavity Detuning
                  chi : -50./60.,            # Kerr-Non-Linear coupling coefficient
                  theta: 0.891,              # pseudo-nand beamsplitter mixing angle
                  phi: 2.546,                # pseudo-nand corrective phase
    }

In[22]:

NSLHN = NSLH.substitute(numerical_vals)
NSLHN

Out[22]:

\[\begin{split}\left( \begin{pmatrix} \frac{1}{2} \sqrt{2} & - \frac{1}{2} \sqrt{2} & 0 & 0 \\ \frac{1}{2} \sqrt{2} & \frac{1}{2} \sqrt{2} & 0 & 0 \\ 0 & 0 & 0.628634640249695 e^{2.546 \mathbf{\imath}} & - 0.777700770912654 e^{2.546 \mathbf{\imath}} \\ 0 & 0 & 0.777700770912654 & 0.628634640249695\end{pmatrix}, \begin{pmatrix} \frac{1}{2} \sqrt{2} A - \frac{1}{2} \sqrt{2} B \\ \left(\frac{1}{2} \sqrt{2} A + \frac{1}{2} \sqrt{2} B\right) + 5.0 {a_{{{\rm N.K}}}} \\ 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}} - 3.88850385456327 e^{2.546 \mathbf{\imath}} {a_{{{\rm N.K}}}} \\ - \left(26.666581733824 + 9.2616384807988 \mathbf{\imath}\right) + 3.14317320124847 {a_{{{\rm N.K}}}}\end{pmatrix}, \frac{1}{2} \mathbf{\imath} \left( - \left(2.5 \sqrt{2} A + 2.5 \sqrt{2} B\right) {a_{{{\rm N.K}}}^\dagger} + \left(2.5 \sqrt{2} \overline{A} + 2.5 \sqrt{2} \overline{B}\right) {a_{{{\rm N.K}}}}\right) - 0.833333333333333 {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} {a_{{{\rm N.K}}}} + 50.0 {a_{{{\rm N.K}}}^\dagger} {a_{{{\rm N.K}}}} \right)\end{split}\]

In[23]:

input_configs = [
            (0,0),
            (1, 0),
            (0, 1),
            (1, 1)
          ]

In[24]:

Lout = NSLHN.L[2,0]
Loutqt = Lout.to_qutip()
times = arange(0, 1., 0.01)
psi0 = qutip.basis(N.space.dimension, 0)
datasets = {}
for ic in input_configs:
    H, Ls = NSLHN.substitute({A: ic[0]*alpha, B: ic[1]*alpha}).HL_to_qutip()
    data = qutip.mcsolve(H, psi0, times, Ls, [Loutqt], ntraj = 1)
    datasets[ic] = data.expect[0]

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In[25]:

figure(figsize=(10, 8))
for ic in input_configs:
    plot(times, real(datasets[ic])/alpha, '-', label = str(ic) + ", real")
    plot(times, imag(datasets[ic])/alpha, '--', label = str(ic) + ", imag")
legend()
xlabel('Time $t$', size = 20)
ylabel(r'$\langle L_out \rangle$ in logic level units', size = 20)
title('Pseudo NAND logic, stochastically simulated time \n dependent output amplitudes for different inputs.', size = 20)

Out[25]:

<matplotlib.text.Text at 0x1100b7dd0>

Pseudo NAND latch memory effect

In[26]:

NLSLH.space

Out[26]:

\[{{\rm N_1.K}} \otimes {{\rm N_2.K}}\]

In[27]:

s1, s2 = NLSLH.space.operands
s1.dimension = 75
s2.dimension = 75
NLSLH.space.dimension

Out[27]:

5625

In[28]:

NLSLHN = NLSLH.substitute(numerical_vals)
NLSLHN

Out[28]:

\[\begin{split}\left( \begin{pmatrix} - \frac{1}{2} \sqrt{2} & 0 & 0 & 0 & 0.314317320124847 \sqrt{2} e^{2.546 \mathbf{\imath}} & - 0.388850385456327 \sqrt{2} e^{2.546 \mathbf{\imath}} \\ \frac{1}{2} \sqrt{2} & 0 & 0 & 0 & 0.314317320124847 \sqrt{2} e^{2.546 \mathbf{\imath}} & - 0.388850385456327 \sqrt{2} e^{2.546 \mathbf{\imath}} \\ 0 & 0.777700770912654 & 0.628634640249695 & 0 & 0 & 0 \\ 0 & 0.314317320124847 \sqrt{2} e^{2.546 \mathbf{\imath}} & - 0.388850385456327 \sqrt{2} e^{2.546 \mathbf{\imath}} & - \frac{1}{2} \sqrt{2} & 0 & 0 \\ 0 & 0.314317320124847 \sqrt{2} e^{2.546 \mathbf{\imath}} & - 0.388850385456327 \sqrt{2} e^{2.546 \mathbf{\imath}} & \frac{1}{2} \sqrt{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.777700770912654 & 0.628634640249695\end{pmatrix}, \begin{pmatrix} \frac{1}{2} \sqrt{2} \left(- A + 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}}\right) - 1.94425192728164 \sqrt{2} e^{2.546 \mathbf{\imath}} {a_{{{\rm N_2.K}}}} \\ \frac{1}{2} \sqrt{2} \left(A + 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}}\right) + 5.0 {a_{{{\rm N_1.K}}}} - 1.94425192728164 \sqrt{2} e^{2.546 \mathbf{\imath}} {a_{{{\rm N_2.K}}}} \\ - \left(26.666581733824 + 9.2616384807988 \mathbf{\imath}\right) + 3.14317320124847 {a_{{{\rm N_1.K}}}} \\ \frac{1}{2} \sqrt{2} \left(- B + 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}}\right) - 1.94425192728164 \sqrt{2} e^{2.546 \mathbf{\imath}} {a_{{{\rm N_1.K}}}} \\ \frac{1}{2} \sqrt{2} \left(B + 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}}\right) - 1.94425192728164 \sqrt{2} e^{2.546 \mathbf{\imath}} {a_{{{\rm N_1.K}}}} + 5.0 {a_{{{\rm N_2.K}}}} \\ - \left(26.666581733824 + 9.2616384807988 \mathbf{\imath}\right) + 3.14317320124847 {a_{{{\rm N_2.K}}}}\end{pmatrix}, 1.25 \sqrt{2} \mathbf{\imath} \left(- A - 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}}\right) {a_{{{\rm N_1.K}}}^\dagger} + 1.25 \sqrt{2} \mathbf{\imath} \left(- B - 0.628634640249695 \left(-34.289 - 11.909 \mathbf{\imath}\right) e^{2.546 \mathbf{\imath}}\right) {a_{{{\rm N_2.K}}}^\dagger} + 1.25 \frac{\sqrt{2} \mathbf{\imath} \left(e^{2.546 \mathbf{\imath}} \overline{A} -21.5552531795218 + 7.48640993073362 \mathbf{\imath}\right)}{e^{2.546 \mathbf{\imath}}} {a_{{{\rm N_1.K}}}} + 1.25 \frac{\sqrt{2} \mathbf{\imath} \left(e^{2.546 \mathbf{\imath}} \overline{B} -21.5552531795218 + 7.48640993073362 \mathbf{\imath}\right)}{e^{2.546 \mathbf{\imath}}} {a_{{{\rm N_2.K}}}} - 0.833333333333333 {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_1.K}}}} + 50.0 {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_1.K}}}} + 4.86062981820409 \frac{\sqrt{2} \mathbf{\imath} \left(-1 + e^{5.092 \mathbf{\imath}}\right)}{e^{2.546 \mathbf{\imath}}} {a_{{{\rm N_1.K}}}^\dagger} {a_{{{\rm N_2.K}}}} - 0.833333333333333 {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} {a_{{{\rm N_2.K}}}} + 50.0 {a_{{{\rm N_2.K}}}^\dagger} {a_{{{\rm N_2.K}}}} + 4.86062981820409 \frac{\sqrt{2} \mathbf{\imath} \left(-1 + e^{5.092 \mathbf{\imath}}\right)}{e^{2.546 \mathbf{\imath}}} {a_{{{\rm N_1.K}}}} {a_{{{\rm N_2.K}}}^\dagger} \right)\end{split}\]

In[29]:

input_configs = {
            "SET": (1, 0),
            "RESET": (0, 1),
            "HOLD": (1, 1)
          }

models = {k: NLSLHN.substitute({A:v[0]*alpha, B:v[1]*alpha}).HL_to_qutip() for k, v in input_configs.items()}

In[30]:

a1, a2 = Destroy(s1), Destroy(s2)
observables = [a1.dag()*a1, a2.dag()*a2]
observables_qt = [o.to_qutip(full_space = NLSLH.space) for o in observables]

In[31]:

def model_sequence_single_trajectory(models, durations, initial_state, dt):
    """
    Solve a sequence of constant QuTiP open system models (H_i, [L_1_i, L_2_i, ...])
    via Quantum Monte-Carlo. Each model is valid for a duration deltaT_i and the initial state for
    is given by the previous model's final state.
    The function returns an array with the times and an array with the states at each time.

    :param models: Sequence of models given as tuples: (H_j, [L1j,L2j,...])
    :type models: Sequence of tuples
    :param durations: Sequence of times
    :type durations: Sequence of float
    :param initial_state: Overall initial state
    :type initial_state: qutip.Qobj
    :param dt: Sampling interval
    :type dt: float
    :return: times, states
    :rtype: tuple((numpy.ndarray, numpy.ndarray)
    """
    totalT = 0
    totalTimes = array([])
    totalStates = array([])
    current_state = initial_state

    for j, (model, deltaT) in enumerate(zip(models, durations)):
        print "Solving step {}/{} of model sequence".format(j + 1, len(models))
        HQobj, LQObjs = model
        times = arange(0, deltaT, dt)
        data = qutip.mcsolve(HQobj, current_state, times, LQObjs, [], ntraj = 1, options = qutip.Odeoptions(gui = False))

        # concatenate states
        totalStates = np.hstack((totalStates,data.states.flatten()))
        current_state = data.states.flatten()[-1]
        # concatenate times
        totalTimes = np.hstack((totalTimes, times + totalT))
        totalT += times[-1]

    return totalTimes, totalStates

In[32]:

durations = [.5, 1., .5, 1.]
model_sequence = [models[v] for v in ['SET', 'HOLD', 'RESET', 'HOLD']]
initial_state = qutip.tensor(qutip.basis(s1.dimension, 0), qutip.basis(s2.dimension, 0))

In[33]:

times, data = model_sequence_single_trajectory(model_sequence, durations, initial_state, 5e-3)

Solving step 1/4 of model sequence

100.0%  (1/1)  Est. time remaining: 00:00:00:00
Solving step 2/4 of model sequence

100.0%  (1/1)  Est. time remaining: 00:00:00:00
Solving step 3/4 of model sequence

100.0%  (1/1)  Est. time remaining: 00:00:00:00
Solving step 4/4 of model sequence

100.0%  (1/1)  Est. time remaining: 00:00:00:00

In[34]:

datan1 = qutip.expect(observables_qt[0], data)
datan2 = qutip.expect(observables_qt[1], data)

In[36]:

figsize(10,6)
plot(times, datan1)
plot(times, datan2)
for t in cumsum(durations):
    axvline(t, color = "r")
xlabel("Time $t$", size = 20)
ylabel("Intra-cavity Photon Numbers", size = 20)
legend((r"$\langle n_1 \rangle $", r"$\langle n_2 \rangle $"), loc = 'lower right')
title("SET - HOLD - RESET - HOLD sequence for $\overline{SR}$-latch", size = 20)

Out[36]:

<matplotlib.text.Text at 0x1121d8990>