Deviation between analytical and numerical solutions

[Or Golan]

Hi,

I'm trying to simulate the Hamiltonian

H = g1 Z1 (b+bdagger) + g2 Z2 (b + bdagger) + ϵ bdagger*b

where Z1, Z2 are the Pauli Z operators on qubits 1,2, and b is the destroy operator on some QHO (say a 3 level system). And I want to find the expectation value of b over time.

This can be solved analytically in the rotating frame to obtain

<b> = b + (1-exp(-iϵt))(g1 Z1 + g2 Z2)/ϵ

Which results in a circle when we plot the imaginary/real parts.

When I try to find the expectation value numerically, by evolving in time a set of states (either specific states or the eigenstates of the Hamiltonian), I get a dramatically different result. Here is the code:

-------------------

import qutip
import numpy as np

import matplotlib.pyplot as plt

g1 = 5.6

g2 = 6.0

z1 = qutip.tensor(qutip.sigmaz(), qutip.qeye(3), qutip.qeye(2))
z2 = qutip.tensor(qutip.qeye(2), qutip.qeye(3), qutip.sigmaz())
b = qutip.tensor(qutip.qeye(2), qutip.destroy(3), qutip.qeye(2))

H = g1 * z1 * (b.dag() + b) + g2 * z2 * (b.dag() + b) + ϵ*b.dag()*b

energies, states = H.eigenstates()
# states = [qutip.tensor(qutip.basis(2, i), qutip.basis(3, 0), qutip.basis(2, j)) for i in range(2) for j in range(2)]

for s in states:
    solve = qutip.sesolve(H, s, np.linspace(0, tau/ϵ))
    p = [qutip.expect(b, x) for x in solve.states]
    plt.plot(np.real(p), np.imag(p))
    plt.show()

--------------

Any idea what I'm doing wrong?

[Paul Menczel]

Hi,

I am not exactly sure how you got to that analytical solution, but you say it is in a rotating frame. Are you maybe missing a transformation into that rotating frame?

Also, I think 3 might be a bit low for the Fock space cutoff.

Best

Paul

[Or Golan]

Thanks,

To solve analytically:

We have

Going to the Magnus expansion, we see that the 3rd order is already 0, so the analytical solution is just the second order Magnus expansion.

Returning to the lab frame will just be a e^{-I\epsilon t} factor.

I went for 3 as a cutoff since I’m not populating the QHO at any time.

Cheers,

Or

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[Or Golan]

You were right - a higher cutoff fixed the problem. Thanks!