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Two-level Rabi model

Statement

Let $σ_{x}, σ_{y}, σ_{z}$ be the Pauli matrices of Finite spin matrices (e), acting on $C^{2}$ , and let $ℏ > 0$ , $δ, Ω \in R$ . Put $H : = \frac{ℏ}{2} (δ σ_{z} + Ω σ_{x}), b : = \sqrt{δ^{2} + Ω^{2}} .$ Then:

(a) $H^{2} = \frac{ℏ^{2} b^{2}}{4} I$ and the unitary group generated by $- i H / ℏ$ is $U (t) = \cos ⁣ (\frac{b t}{2}) I - \frac{i}{b} \sin ⁣ (\frac{b t}{2}) (δ σ_{z} + Ω σ_{x}),$ which solves $i ℏ U^{'} (t) = H U (t)$ , $U (0) = I$ ; if $b = 0$ the formula is read as $U (t) = I$ .

(b) Squared transition amplitude. Put $e_{↑} = (1, 0)^{T}$ and $e_{↓} = (0, 1)^{T}$ , and use the inner product linear in its first argument from the Pauli supplier. Define the squared transition amplitude by $p_{↓} (t) = ∣ ⟨ e_{↓}, U (t) e_{↑} ⟩ ∣^{2} = \frac{Ω^{2}}{b^{2}} \sin^{2} ⁣ (\frac{b t}{2}) (b > 0) .$ For $b = 0$ it is $p_{↓} (t) = 0$ for every real $t$ .

(c) Periodicity. For $b > 0$ one has $U (t + 2 π / b) = - U (t)$ ; the propagator is $2 π / b$ -periodic up to the central sign $- I$ , and every squared transition amplitude is $2 π / b$ -periodic.

These are the exact mathematical dynamics of the stipulated finite matrix model; no physical identification of $δ, Ω$ with a field or detuning and no measurement statistics is asserted.

Facts & Assumptions

Given: the Pauli matrices, $ℏ > 0$ , $δ, Ω \in R$ , $b = \sqrt{δ^{2} + Ω^{2}}$ , $H$ as above.

[F1]

Pauli algebra: $σ_{i}^{2} = I$ , $σ_{i} σ_{j} = - σ_{j} σ_{i}$ for $i \neq j$ , $σ_{x} σ_{z} = - i σ_{y}$ and the products $σ_{x} σ_{z} + σ_{z} σ_{x} = 0$ ; hence $(δ σ_{z} + Ω σ_{x})^{2} = (δ^{2} + Ω^{2}) I$ (Finite spin matrices (e), algebra).

[F2]

For a finite matrix $A$ , the series $e^{- i t A / ℏ} = \sum_{k \geq 0} (- i t A / ℏ)^{k} / k!$ and its derivative series converge uniformly in operator norm on compact time intervals by scalar exponential majorants. Termwise differentiation gives $i ℏ U^{'} = A U$ and $U (0) = I$ . For Hermitian $A$ , $(U^{*} U)^{'} = 0$ , so $U^{*} U = I$ ; finite-dimensional injectivity gives surjectivity and unitarity. The difference $D$ of two solutions with the same initial value has $(D^{*} D)^{'} = 0$ and $D (0) = 0$ , so $D = 0$ . Uniqueness gives the group law. This local argument uses the norm conventions of The operator norm as the least bound and as the unit-sphere or unit-ball supremum and A bounded linear operator between normed spaces, with no Choice or spectral theorem premise.

[F3]

The eigenvectors: $e_{↑} = (1, 0)$ , $e_{↓} = (0, 1)$ satisfy $σ_{z} e_{↑} = e_{↑}$ , $σ_{z} e_{↓} = - e_{↓}$ , $σ_{x} e_{↑} = e_{↓}$ , $σ_{x} e_{↓} = e_{↑}$ (Finite spin matrices (e)).

Proof

1.1F1F2algebra

[Exponential form] By [F1], $H$ is Hermitian and $H^{2} = ℏ^{2} b^{2} I / 4$ . For $b > 0$ put $S = (δ σ_{z} + Ω σ_{x}) / b$ , so $S^{2} = I$ . The even terms in the series of [F2] sum to $\cos (b t / 2) I$ , and its odd terms sum to $- i S \sum_{m \geq 0} (- 1)^{m} (b t / 2)^{2 m + 1} / (2 m + 1)! = - i S \sin (b t / 2)$ . This gives the displayed $U$ and its unique unitary group and equation by [F2]. If $b = 0$ , realness of $δ, Ω$ gives $δ = Ω = 0$ , so $H = 0$ and $U = I$ .

2.1step 1.1F1F3algebra

[The transition amplitude] By [F3] and step 1.1, $U (t) e_{↑} = [\cos (b t / 2) - i δ \sin (b t / 2) / b] e_{↑} - i Ω \sin (b t / 2) e_{↓} / b$ when $b > 0$ . Conjugate linearity in the second argument gives $⟨ e_{↓}, U (t) e_{↑} ⟩ = + i Ω \sin (b t / 2) / b$ , whose squared modulus is $Ω^{2} \sin^{2} (b t / 2) / b^{2}$ . If $b = 0$ , $U = I$ and orthogonality gives $p_{↓} = 0$ .

3.1step 1.1step 2.1algebra∎

[Periodicity] For $b > 0$ , replacing $t$ by $t + 2 π / b$ in the closed form changes $\cos, \sin$ by their negatives; hence $U (t + 2 π / b) = - U (t)$ , and all squared transition amplitudes are $2 π / b$ -periodic.

Depends on

Dependency tree · 2 levels

Sources

R. Fitzpatrick, Quantum Mechanics (University of Texas at Austin, author-hosted course text), Two-State System pp. 153-155 and Spin Magnetic Resonance pp. 178-180 (exact two-level evolution and Rabi oscillations)
C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics I, Chapter IV (two-level systems and the Rabi formula; standard textbook reference)