def transmon_hamiltonian(n_qubit,
n_cavity,
non_herm=False,
non_linear_decay=False):
"""Return the Transmon symbolic drift and control Hamiltonians. If
`non_herm` is True, the drift Hamiltonian will include non-Hermitian decay
terms for the spontaneous decay for the qubits and the cavity. This will be
the "standard" decay with a linear decay rate, or with an independent decay
rate for each level in the transmon and cavity if `non_linear_decay` is
True.
"""
HilQ1 = LocalSpace('1', dimension=n_qubit)
HilQ2 = LocalSpace('2', dimension=n_qubit)
HilCav = LocalSpace('c', dimension=n_cavity)
b1 = Destroy(identifier='b_1', hs=HilQ1)
b1_dag = b1.adjoint()
b2 = Destroy(identifier='b_2', hs=HilQ2)
b2_dag = b2.adjoint()
a = Destroy(hs=HilCav)
a_dag = a.adjoint()
δ1, δ2, Δ, α1, α2, g1, g2 = sympy.symbols(
r'delta_1, delta_2, Delta, alpha_1, alpha_2, g_1, g_2', real=True)
H0 = (δ1 * b1_dag * b1 + (α1 / 2) * b1_dag * b1_dag * b1 * b1 + g1 *
(b1_dag * a + b1 * a_dag) + δ2 * b2_dag * b2 +
(α2 / 2) * b2_dag * b2_dag * b2 * b2 + g2 *
(b2_dag * a + b2 * a_dag) + Δ * a_dag * a)
if non_herm:
if non_linear_decay:
for i in range(1, n_qubit):
γ = sympy.symbols(r'gamma_%d' % i, real=True)
H0 = H0 - sympy.I * γ * LocalSigma(i, i, hs=HilQ1) / 2
for j in range(1, n_qubit):
γ = sympy.symbols(r'gamma_%d' % j, real=True)
H0 = H0 - sympy.I * γ * LocalSigma(j, j, hs=HilQ2) / 2
for n in range(1, n_cavity):
κ = sympy.symbols(r'kappa_%d' % n, real=True)
H0 = H0 - sympy.I * κ * LocalSigma(n, n, hs=HilCav) / 2
else:
γ, κ = sympy.symbols(r'gamma, kappa', real=True)
H0 = H0 - sympy.I * γ * b1_dag * b1 / 2
H0 = H0 - sympy.I * γ * b2_dag * b2 / 2
H0 = H0 - sympy.I * κ * a_dag * a / 2
H1 = a / 2 # factor 2 to account for RWA
return H0, H1
def _toSLH(self):
a = Destroy(self.space)
a_d = a.adjoint()
S = identity_matrix(1)
if self.sub_index == 0:
# Include the Hamiltonian only with the first port of the kerr cavity circuit object
H = self.Delta * (a_d * a) + self.chi * (a_d * a_d * a * a)
L = Matrix([[sqrt(self.kappa_1) * a]])
else:
H = 0
L = Matrix([[sqrt(self.kappa_2) * a]])
return SLH(S, L, H)
def _toSLH(self):
a = Destroy(self.space)
a_d = a.adjoint()
S = identity_matrix(1)
if self.sub_index == 0:
# Include the Hamiltonian only with the first port of the kerr cavity circuit object
H = self.Delta * (a_d * a) + self.chi * (a_d * a_d * a * a)
L = Matrix([[sqrt(self.kappa_1) * a]])
else:
H = 0
L = Matrix([[sqrt(self.kappa_2) * a]])
return SLH(S, L, H)