Super-Adiabatic States

• Super-adiabatic states are quantum approximations that optimize adiabatic expansion by truncating its divergent series to eliminate spurious oscillations.
• They use Berry’s smoothing of the Stokes phenomenon to produce smooth time evolution and reveal meaningful quantum interference effects.
• This framework enhances physical interpretations in contexts like scalar QED and cosmological particle production by controlling time derivative corrections.

A super-adiabatic state is defined as a quantum state or, more generally, a basis or class of approximations that incorporates corrections beyond standard leading-order adiabatic theory, typically by optimally truncating an (asymptotic and divergent) adiabatic expansion in time derivatives. This construction eliminates or minimizes non-physical artifacts—such as spurious oscillations or sharp transitions—in the adiabatic approximation, produces smooth time evolution across non-analyticities (Stokes lines), and can reveal physically meaningful features such as universal scaling and quantum interference effects associated with particle production, transport, or driven dynamics.

1. Adiabatic Expansion and Basis Dependence

In dynamical quantum systems subject to time-dependent backgrounds (e.g., nonstationary electric fields or curved spacetimes), the instantaneous particle number is often defined by decomposing the solution into positive- and negative-frequency modes. A generic prescription uses an adiabatic basis, with "mode functions" chosen in analogy to free particle solutions. For a time-dependent oscillator equation,

$\frac{d^2}{dt^2}f_k(t) + \omega_k^2(t) f_k(t) = 0,$

the standard ansatz is

$$f_k(t) = \frac{1}{\sqrt{2W_k(t)}}\exp\left(-i\int^t W_k(t')dt'\right),$$

where $W_k(t)$ is determined iteratively via

$$W_k^2(t) = \omega_k^2(t) - \left[\frac{1}{2W_k}\ddot{W}_k - \frac{3}{4}\left(\frac{\dot{W}_k}{W_k}\right)^2\right],$$

with the leading order $W_k^{(0)}(t) = \omega_k(t)$. Higher orders introduce increasingly complex time derivative corrections. The adiabatic particle number at intermediate times, defined via the Bogoliubov coefficient $\tilde{\mathcal{N}}_k(t) = |\beta_k(t)|^2$, can display large, non-physical oscillations depending on the specific order and how the basis is chosen. Asymptotically (as $t \rightarrow +\infty$), the final particle number is independent of basis, but the intermediate time dependence is not.

2. Super-Adiabatic Truncation and Optimal Order

The full adiabatic expansion in time derivatives is divergent. Truncation at finite order does not, in general, yield a controlled approximation throughout dynamical events—oscillations can worsen at high order. However, by analyzing the large-order structure one finds a "universal" form for the asymptotic series (Dingle and Berry): $$\varphi_k^{(2l+2)} \sim -\frac{(2l+1)!}{\pi F_k^{2l+2}},$$ with $F_k(t)$ the "singulant," defined by

$F_k(t) = 2i\int_{t_p}^t \omega_k(t')dt',$

where $t_p$ is the nearest complex turning point (where $\omega_k(t_p) = 0$). The optimal truncation order is

$$j \approx \left\lfloor \frac{1}{2}(|F_k^{(0)}| - 1) \right\rfloor,$$

at which the remainder of the truncated asymptotic series is minimized. Resumming only up to this order, the time evolution of the Bogoliubov coefficient and thus the particle number follows a universal "smooth" profile—rather than the discontinuous behavior predicted by low-order expansions.

3. Berry’s Smoothing of the Stokes Phenomenon

The Stokes phenomenon refers to the switching-on (or off) of exponentially subdominant solutions across certain lines in the complex time plane (Stokes lines), inherent in WKB and adiabatic approximations. Berry demonstrated that truncating the asymptotic series at optimal order yields a replacement of the abrupt Stokes transition by a smooth error-function profile: $$\beta_k(t) \approx \frac{i}{2} \operatorname{Erfc}(-\sigma_k(t)) \exp(-F_k^{(0)}),$$ with

$$\sigma_k(t) = \frac{\operatorname{Im}F_k(t)}{\sqrt{2\,\operatorname{Re}F_k(t)}}.$$

Consequently, the super-adiabatic particle number becomes

$$\tilde{\mathcal{N}}_k(t) \approx \frac{1}{4}\left|\operatorname{Erfc}(-\sigma_k(t)) \exp(-F_k^{(0)})\right|^2,$$

yielding a smooth, physical time dependence across temporal regions separated by Stokes lines. The key point is that the "birth" of particles (or analogous events) corresponds to the passage of $\sigma_k(t)=0$ across the real time axis.

4. Quantum Interference in Super-Adiabatic Formalism

The super-adiabatic approach reveals quantum interference phenomena in particle production processes. When backgrounds feature multiple pulses or periodically varying fields, each pulse is associated with complex-conjugate turning points. The total amplitude is a coherent sum over such contributions: $$N_k \equiv \tilde{\mathcal{N}}_k(+\infty) \approx \left| \sum_{t_p} \exp(2i\theta_k^{(p)}) \exp(-F_{k,t_p}^{(0)}) \right|^2,$$ where

$$F_{k,t_p}^{(0)} = i \int_{t_p^*}^{t_p} \omega_k(t)dt, \quad \theta_k^{(p)} = \int_{s_1}^{s_p} \omega_k(t)dt.$$

Constructive and destructive interference between different turning points leads to momentum-dependent amplification (such as $n^2$ enhancement for $n$ pulses) or suppression (even complete cancellation) of produced particles. For example, in global de Sitter space, interference is constructive in even dimensions (resulting in nonzero particle number) and destructive in odd dimensions (suppressing net production).

5. Physical Interpretation and Applications

The super-adiabatic framework not only provides a prescription for a smoother, more physically meaningful particle number, but also illuminates the mechanism of "multi-slit" quantum interference in dynamical particle production. The universality of the error-function profile ties the behavior of highly non-trivial quantum evolution in time-dependent backgrounds to the underlying complex analytic structure (turning points and Stokes lines) of the problem. In the context of scalar QED (Schwinger effect) or cosmological particle production (de Sitter space), these techniques identify physically observable consequences, such as constructive enhancement due to coherent field pulses or the dimensional dependence of pair creation in curved spacetime.

6. Summary Table: Key Constructs and Results

Concept	Mathematical Expression	Physical Interpretation
Adiabatic expansion (recursion)	$W_k^2(t) = \omega_k^2(t) - \cdots$	Iterative determination of auxiliary frequency
Super-adiabatic truncation	$$j \approx \lfloor \frac{1}{2}(|F_k^{(0)}| - 1) \rfloor$$	Optimal adiabatic order for smooth evolution
Smoothing (Berry’s formula)	$$\tilde{\mathcal{N}}_k(t)\approx\frac{1}{4}|\operatorname{Erfc}(-\sigma_k(t))\exp(-F_k^{(0)})|^2$$	Universal smoothing of Stokes jumps
Quantum interference	$$N_k \approx |\sum_{t_p} e^{2i\theta_k^{(p)}} e^{-F_{k,t_p}^{(0)}}|^2$$	Enhancement/suppression from multiple event paths

The adoption of super-adiabatic states addresses both technical and conceptual obstacles in describing non-equilibrium quantum processes in strong-field and cosmological physics. The approach replaces the basis ambiguity and oscillatory artifacts of naive adiabatic expansions with a mathematically controlled, smooth, and physically interpretable framework in which both universal features (such as the Stokes smoothing) and problem-dependent details (such as interference patterns and dimensionality dependence) can be quantitatively understood.