Adiabatic “following” is a well-studied effect [1] but we believe that diabatic following has not previously been examined. Here we present the dynamical solutions of a model simple enough to allow both of these two kinds of following to occur and to be explained within a common framework. All of this is done entirely analytically using only sines and cosines, although two short movies are included for help in visualizing the limiting cases.

Adiabatic quantum states play a key role in many physical contexts, and were probably first used in understanding the transfer of probability during collisional excitation of an atom, as worked out independently by Landau [2] and Zener [3] in 1932. A key result is always the probability of remaining in the initial bare state, i.e., of avoiding adiabatic transfer. This probability approaches zero in the adiabatic limit. As a canonical example, if we call this the escape probability P_esc , then as the Landau-Zener (LZ) adiabaticity parameter ʌ becomes large: P_esc → e ^-πʌ → 0. [1] One says that the occupation probability of the initial state “follows” the adiabatic transformation of the ground state into the final excited state.

In this note we present a uniform exact solution to the two-level population transfer problem for long times and short times as well as for the adiabatic and diabatic limits. We can do this by adopting a particular model for the interaction Hamiltonian, and the exercise is motivated by elements of the result, e.g., in expression (1) below, that have not appeared before. Since the analytic expressions obtained are so straightforward, we might also claim that the model permits a clearer understanding of (even a clearer definition of) “following” itself by providing an unexpected result, namely that faithful following can also occur, in a sense to be explained, in the fully diabatic limit.

Let us begin by focusing on the “escape probability.” We will show below that our expression for P_esc is given by:

Here x is our rapidity parameter, to be defined below in (19). The super-slow limit x → 0 is the fully adiabatic limit in which 100% transfer occurs, i.e., no probability escapes from the adiabatically evolving initial state. The opposite limit for very large x is the extremely rapid diabatic case in which the escape probability is unity, meaning that the system does not follow at all but remains in the initial bare state. A graph of the expression for P_esc is shown in Fig. 1.

 

Figure 1. A plot of P_esc , given in (1), as a function of the rapidity parameter x, which is defined in (19). An expanded view of the approach to the singular oscillations near the adiabatic limit is given in the left graph.

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Our model is an extension of a “quarter-cycle” spin model apparently first used by Ramsey and Schwinger [4]. The model has an exact analog in matched pulse excitation of three-level systems, in which case explicit solutions have been worked out by Carroll and Hioe [5]. We have extended these earlier treatments to the long-time diabatic limit.

The specific physical context we adopt is near-resonant excitation of a two-level quantum system that has a transition frequency ω ₀ being irradiated by a laser field with amplitude ε and frequency ω_L . Spin-based magnetic analogies are obvious. We will start with the usual Bloch-vector equation for two-level evolution:

where S is the Bloch vector and Ω is the vector around which the Bloch vector precesses. In the ordinary rotating frame, having made the Rotating Wave Approximation (RWA) as usual (see, e.g., [6]), one finds that Ω has the three components

where the non-zero components are the on-resonance Rabi frequency and the detuning from resonance:

Here we consider the case in which Ω rotates uniformly, maintaining its length. This is guaranteed if Ω itself obeys the same kind of equation as S, namely

where A is a new constant “axis” vector. Thus we must satisfy (4) at the same time as the primary equation (2). It is not difficult to verify that a special solution that obeys both equations, apparently first reported by Güttinger in 1931 [7], can be written directly in terms of A and Ω:

where T_ad is a constant with the dimensions of time and constrained by S·S = 1, or

where A, the length of A, is Ω’s uniform rotation rate. When A ≪ Ω then Ω rotates slowly compared to the precession of S. Solution (5) for S(t) is sometimes referred to as the “spin-locking” solution.

Now we will go beyond spin-locking and find more general explicit solutions. These describe motion that can strongly depart from the adiabatic limit. We continue to focus on the idealized case (4) in which Ω simply rotates. The specific time-dependences of its components can be chosen to be:

which are consistent with:

The initial vector Ω ₀ may be oriented in any direction in the 1 – 3 plane. Since A is a constant vector, we can write the exact solution of (4) in terms of Ω ₀ and the unit vector  as

and this is easily checked. We’ll take Ω ₀ = -3̂Ω = {0, 0, -Ω}, for simplicity and because it corresponds to excitation by a laser field that is turned on at t = 0. Then the expression for Ω(t) reduces to

where $\Omega =\sqrt{r^2+{\Delta }^2}=\mathrm{const}$.

Since the anticipated result is that S will “follow” Ω, we will consider more “natural” components of S, those along three orthogonal (but time dependent) directions that include Ω̂, defined by the three unit vectors:

This choice is equivalent to working in a rotating frame (rotating with Ω̂(t) about the A axis).

We will write

and use the Bloch vector equation to determine the equations obeyed by α, β and γ. We can then use the equation for S to find the α, β, γ equations: dα/dt = Ωγ, dβ/dt = Aγ, and dγ/dt = -Aβ - Ωα. Here the initial conditions are α ₀ = γ ₀ = 0, and β ₀ = 1 if the atom starts in its ground state.

The equation for γ separates from the others in second order:

The solutions in agreement with the ground-state initial conditions are:

Probability conservation (i.e., fixed unit length of S) demands α ² + β ² + γ ² = 1, which is easily confirmed.

When the α, β, γ solutions are substituted into (12) this gives the exact solution for the Bloch vector:

or

The adiabatic limit is the one in which A ≪ Ω, i.e., Ω changes slowly compared to the precession of S around Ω. This means that the “rapidity” parameter x = A/Ω is very small:

In the limit x ≪ 1 the motion is easily visualized, and Movie #1 is given in Fig. 2 to show the Bloch vector faithfully following Ω, as expected. In the movie, x = 1/40, so the Bloch vector precesses 40 times around Ω while Ω makes a full revolution. To show the precession more clearly in the movie, a slightly different initial condition was chosen in which ${\alpha }_0^2$, =0, ${\beta }_0^2$ = 0.01 and ${\gamma }_0^2$ = 0.99.

 

Figure 2. A frame from a movie (352KB) showing adiabatic following of the Bloch vector as it precesses rapidly about Ω, which is rotating slowly with the relative rotation rate 1/40.

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The extreme non-adiabatic case is more interesting and we will describe it shortly. However, let us first recover formula (1). This gives the “escape probability,” the probability of failure to achieve inversion when the rotation speed of the axis vector Ω is too fast for S to follow. We must calculate the probability that at the end of a π rotation of Ω from -3 to +3, the Bloch vector has remained aligned with -3, i.e., the atom has been left in its ground state. Thus we must evaluate $P_{\mathit{esc}}=\frac{1}{2}\left(1-S_3\left(t\right)\right)$ for At = π, and (18) gives:

From this we easily obtain the escape probability in the form already given in (1). Obviously the half-rotation solution gives the two limiting cases:

with a clear singularity in the oscillation frequency that accompanies the approach to fully adiabatic transfer. This was already shown in Fig. 1.

Finally, it is interesting to explore the consequences of allowing the interaction to run beyond the At = π point. In the adiabatic case, as Movie #1 showed, the Bloch vector simply continues to follow. The diabatic case is more interesting. In the limit A ≫ Ω we may take AT_ad ≈ 1 and retain only terms of order ΩT_ad or higher in (18). Then we find the extremely simple expression:

Plainly, formula (22) contains several surprising predictions. Since the component along 2 is practically negligible compared to the other components, we see that the Bloch vector rotates smoothly in the 1 – 3 plane with full amplitude, eventually even achieving full inversion. Just as in the adiabatic limit, the rate of rotation is very slow but now the rate, 1/T_ad - A, is slow because A is large, not small:

This means that the vector ω sweeps rapidly past S on every revolution, but each kick it gives to S is coherent with the next kick when Ω encounters S again on its next revolution. Therefore, even in the extreme diabatic case, it could be said that S follows Ω, if only in the sense that it “remembersΩ Ω and accumulates its influence from one kick to the next.

Note an additional non-inutitive element of this “diabatic following”. Remarkably, S rotates in the opposite sense from Ω, making it even harder to visualize the nature of the kick-to-kick coherence. All of this is seen in Movie #2, given in Fig. 3. The movie parameters are AT_ad = 1 and ΩT_ad = 1/3, i.e., x is greater than unity, but is still far from the extreme diabatic limit, allowing the encounters of S and Ω to be seen as kicks, but still permitting full inversion.

 

Figure 3. A frame from a movie (570KB) showing diabatic following of the Bloch vector as it achieves inversion and then full 2π rotation as a consequence of successive “kicks” by Ω.

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We close with the remark that the model is probably impossible to implement in the pure two-level form so long as an optical interaction is insisted upon. The magnetic formulation is much less far-fetched, while the three-level two-pulse formulation may be possible to explore optically. In that case there is an interesting open question whether rapid sinusoidal intensity modulation (as opposed to modulation of Rabi frequencies) of the two driving fields would be sufficient to produce diabatic inversion.