H(t)=f(t)(e^{ⅈ t}a+e^{-ⅈ t}a^†)

Define a time-dependent Hamiltonian as H(t)=f(t)(e^{ⅈ t}a+e^{-ⅈ t}a^†) with[a,a^†]=1 where a (a^†) is the annihilation (creation) operator:

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The Hamiltonian above is expressed in the interaction picture; see Sec. 4.1.1 of the preprint (arXiv:0810.5488) for details. To avoid any conflict, we shall use formal symbol of t for the time.

Define the variables within the non-commutative algebra together with their corresponding commutation relations:

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Since in the Magnus series, there are many time-dependent terms that are only scalar variables, define corresponding non-commutative algebra and Gröbner basis for given n (where we needs to specify what are scalar variables for a given n):

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For scalar variables in above definitions, we use the same format as in the Magnus series defined in the previous section.

A function that simplifies and reduces the non-commutative polynomial of the annihilation and creation operators:

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We use Nest with ReleaseHold since the Magnus terms defined in the previous section are expressed in the hold commutator form.

As noted earlier, we chose a Hamiltonian that does not commute at different times. Show the Hamiltonian at different times do not commute:

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Show[H(t₁),[H(t₂),H(t₃)]]=0:

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Find commutation term [H(t₁),H(t₂)] in the second term of Magnus expansion :

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Find commutation term, [H(t₁),[H(t₂),H(t₃)]], in the third term of Magnus expansion :

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It is interesting that the third them and so on are all zero. Therefore, only first and 2nd terms contribute to the exponential: which can be rewritten as U(t)=Exp[-ⅈ (α a+α^*a^†)-ⅈ β] with and :

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H(t)=f₁(t) σ₁+f₂(t) σ₂+f₃(t) σ₃

Define a time-dependent Hamiltonian as H(t)=f₁(t) σ₁+f₂(t) σ₂+f₃(t) σ₃ with σ₁ the Pauli-X, σ₂ the Pauli-Y, and σ₃ the Pauli-Z:

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Define the variables within the non-commutative algebra together with their corresponding commutation relations:

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Since in the Magnus series, there are many time-dependent terms that are only scalar variables, define corresponding non-commutative algebra and Gröbner basis for given n (where we needs to specify what are scalar variables for a given n):

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A function that simplifies and reduces the non-commutative polynomial of Pauli matrices:

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As noted earlier, we chose a Hamiltonian that does not commute at different times. Show the Hamiltonian at different times do not commute:

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In above code, we have applied NonCommutativePolynomialReduce two times to take care of all reductions (note the difference in the order of variables):

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Define the Hamiltonian as :

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Define corresponding algebra and Groebner Basis:

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Define a function to simplify corresponding polynomials:

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Calculate {Ω₁(t),Ω₂(t),Ω₃(t),Ω₄(t)} and then collect them by Pauli matrix terms:

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For each term, find the coefficient for Pauli matrices (keep in mind that we will have only linear terms):

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Now let's repeat the same process, but this time assigning explicit values to Pauli matrics and doing the commutation/dot calculations:

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Ω1To4Manual=a₁σ₁+a₂σ₂+a₃σ₃

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Note a_i=Tr[σ_i.Ω1To4Manual]/2. One can calculate it and compare with the previous result:

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As expected, the coefficients in front of Pauli matrices are the same the ones we obtained from non - commutative algebraic calculations.

Some numerical exploration. To find approximation to unitary operator determined by H(t), calculate {-ⅈ Ω₁(t),(-ⅈ)²Ω₂(t),(-ⅈ)³Ω₃(t),(-ⅈ)⁴Ω₄(t)}:

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Calculate the unitary operator:

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Calculate {Ω₁(t),∑n=12Ω_n(t),∑n=13Ω_n(t),∑n=14Ω_n(t)}:

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Calculate ⅇ^∑_nΩ_n(t) and compare it with the solution :

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