The Importance of The Difference in Adiabatic Phases in Non-Cyclic Evolution on Various Time Scales

Abstract

To understand the importance of the interference effect of non-physical non-cyclic phases difference on the expectation value of physical operators that do not commute with the time-dependent adiabatic Hamiltonian, we compare the value of $\langle \psi (t)| {\textbf{s}}_z| \psi (t)\rangle$ of the exact soluble two-level model with its adiabatic approximation in a non-cyclic evolution. This expectation value is calculated in different orders of magnitude of time periods. For time intervals of the same order as the adiabatic Hamiltonian’s period, the inability to account for the contribution of non-physical, non-cyclic adiabatic phases in the temporal evolution of this quantum system may introduce a percentage disagreement of up to $60\%$ with the precise result of measurable physical quantities. The results obtained in the article are independent of the Hamiltonian instantaneous eigenstate basis that we use to decompose the vector state in adiabatic evolution. For times greater than any integer number of periods $T_0$, in which the classical parameters return to their initial values, we only obtain the correct result of the expected value of the physical operators in state vectors with adiabatic evolution when we include in the vector state its Berry phase plus the non-cyclic phase acquired in the elapsed time interval after this integer number of periods $T_0$. This simple model emphasizes the relevance of the interference effects of physical or non-physical adiabatic phases in calculating the time evolution of the expectation value of physical operators in an adiabatically evolving quantum state with non-cyclic development.

Appendices

Appendix A: The Interference Effect Does not Contribute to the Expected Value of Operators Commuting with a Non-Degenerate Hamiltonian

Assume that at each instant t, a physical operator $\textbf{A}(t)$ commutes with the Hamiltonian ${\textbf{H}} (t)$ which has a non-degenerate energy spectrum,

$$\begin{aligned}{}[{\textbf{A}} (t), {\textbf{H}} (t)] = {\textbf{0}} . \end{aligned}$$

(28)

The null operator is ${\textbf{0}}$. The Hamiltonian ${\textbf{H}} (t)$ drives the quantum system. We also assume that the operator ${\textbf{A}}(t)$ has no time derivative operator in any order.

Consider a basis $\{|\phi _n; t \rangle , n = 1, 2, \cdots M \}$ of simultaneous eigenstates of the operators ${\textbf{H}} (t)$ and ${\textbf{A}} (t)$ at time t, that is,

$$\begin{aligned} {\textbf{A}}(t) \, |\phi _n; t \rangle = a_n(t) |\phi _n; t \rangle \hspace{0.3cm} \text{ and } \hspace{0.3cm} {\textbf{H}}(t) \, |\phi _n; t \rangle = E_n(t) |\phi _n; t \rangle , \end{aligned}$$

(29)

where $\langle \phi _n; t|\phi _m; t \rangle = \delta _{n m}$, $m, n = 1, 2, \cdots , M$. At moment t, $a_n (t)$ and $E_n (t)$ are the eigenvalues of the operators ${\textbf{A}}(t)$ and ${\textbf{H}}(t)$, respectively.

The vector state $|\psi (t) \rangle$, which is expanded on this basis, describes the quantum system at time t,

$$\begin{aligned} |\psi (t) \rangle = \sum _{n = 1}^M \, c_n(t) \, |\phi _n; t \rangle , \end{aligned}$$

(30)

and the coefficients $c_n (t) \in \mathbb {C}$, with $n = 1, 2, \cdots , M$.

Then we have

$$\begin{aligned} \langle \psi (t)| {\textbf{A}}(t) |\psi (t) \rangle = \sum _{n=1}^M \, a_n(t) \, |c_n (t)|^2 . \end{aligned}$$

(31)

From the result (31), we verify that the interference effect, due to the difference of phases from the evolution of distinct instantaneous eigenstates of a non-degenerate Hamiltonian, does not contribute to the physical object $\langle \psi (t)| {\textbf{A}}(t) |\psi (t) \rangle$ when the operator ${\textbf{A}} (t)$ commutes with this non-degenerate Hamiltonian ${\textbf{H}}(t)$ at every instant t.

Appendix B: The Exact Evolution of any Vector State of the Two-Level Model

The reference [11], gives the exact state vector $|\psi (t)\rangle _{ex}$ which is the solution of the Schrödinger equation of the two-level model with the initial vector state (7a), that is,

$$\begin{aligned} |\psi (t)\rangle _{ex} = \sum _{j = 1}^2 \, c_j (t) \, e^{- \frac{i E_j t}{\hbar }} \; |\phi _j; t\rangle . \end{aligned}$$

(32)

The vectors $|\phi _1; t\rangle$ and $|\phi _2; t\rangle$ are given by (4a) and (4b), respectively.

The expressions of the coefficients $c_1 (t)$ and $c_2 (t)$ are [11],

$$\begin{aligned} c_1 (t)= & {} e^{- \frac{i (\omega + \omega _0) t}{2}} \times \left\{ cos(\Gamma t) \, a_1 + \right. \nonumber \\+ & {} \left. i \frac{sin(\Gamma t)}{2 \Gamma } \, \left[ (\omega - \omega _0 \, cos(\theta )) \, a_1 - \omega _0 \, sin(\theta ) \, a_2 \right] \right\} , \end{aligned}$$

(33a)

$$\begin{aligned} c_2 (t)= & {} e^{ \frac{i (\omega - \omega _0) t}{2} } \times \left\{ cos(\Gamma t) \, a_2 + \right. \nonumber \\- & {} \left. i \frac{sin(\Gamma t)}{2 \Gamma } \left[ (\omega - \omega _0 \, cos(\theta )) \, a_2 + \omega _0 \, sin(\theta ) \, a_1 \right] \right\} . \end{aligned}$$

(33b)

The constants $a_1$ and $a_2$ in (7a) determine the initial vector state $|\psi (0)\rangle$. $|\psi (0)\rangle$ is written in terms of instantaneous eigenstates (4a) and (4b) of the Hamiltonian (2) at the initial time, t=0.

$\Gamma$ is the Rabi‘s frequency. It is equal to[11],

$$\begin{aligned} \Gamma = \frac{\omega }{2} \; \Big [ \big ( 1 - \frac{\omega _0}{\omega }\, cos(\theta )\big )^2 + \big (\frac{\omega _0}{\omega }\big )^2 \, sin^2 (\theta ) \Big ]^{\frac{1}{2}} . \end{aligned}$$

(33c)

We remember that $\omega _0$ is the angular frequency of the classical magnetic field $\mathbf {\textbf{B}}(t)$ that precesses around the z-axis. The angular frequency $\omega$, as defined in (5b), is the difference between the two energies of the two-level model under consideration, $E_1$ and $E_2$.

Appendix C: The General Initial Condition of the Initial vector State $|\psi (0)\rangle$

The coefficients $a_1$ and $a_2$ in the decomposition (7a) of the initial vector state $| \psi (0)\rangle$ are complex numbers in the general case. We can write them as:

$$\begin{aligned} a_1 \equiv a_1^{\, (R)} \, e^{\, i \varphi _1} \hspace{0.3cm} \text{ and } \hspace{0.3cm} a_2 \equiv a_2^{\, (R)} \, e^{\, i \varphi _2} . \end{aligned}$$

(34)

We have $a_1^{\, (R)}, a_2^{\, (R)} \in \mathbb {R}$ and the phases $\varphi _1$ and $\varphi _2$ are also real numbers.

The condition that the norm of the vector state $| \psi (0)\rangle$ is equal to 1 becomes:

$$\begin{aligned} |a_1|^2 + |a_2|^2 = 1 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \left( a_1^{\, (R)} \right) ^2 + \left( a_2^{\, (R)} \right) ^2 = 1 . \end{aligned}$$

(35)

In the general case of the initial coefficients $a_1$ and $a_2$, the condition (14) in the paper is replaced by the equality (35). The last equality allows us to use, in the general case, the parametrizations,

$$\begin{aligned} a_1^{\, (R)} \equiv sin(\beta ) \hspace{0.3cm} \text{ and } \hspace{0.3cm} a_2^{\, (R)} \equiv cos(\beta ) . \end{aligned}$$

(36)

For generic coefficients (34), the expectation value of the operator ${\textbf{s}}_z$ in the adiabatic vector state $| \psi (t); \alpha \rangle _{ad}$ is equal to:

$$\begin{aligned} S_z^{\, (ad)} (\theta , \beta , \omega _0, \varphi _1 - \varphi _2; t; \alpha )\equiv & {} \,\,_{ad}\langle \psi (t); \alpha | {\textbf{s}}_z | \psi (t); \alpha \rangle _{ad} \end{aligned}$$

(37a)

$$\begin{aligned}{} & {} \hspace{-3.5cm} = \frac{\hbar }{2} \; \Big \{ cos(\theta ) cos(2 \beta ) + \nonumber \\{} & {} \hspace{-6.5cm} - sin(\theta ) sin(2 \beta ) \times cos\big [ (\varphi _1 - \varphi _2) + \big ( 1 - \alpha \big ( \frac{\omega _0}{\omega }\big ) \, cos(\theta ) \big ) \, \omega t \big ] \Big \} . \end{aligned}$$

(37b)

The aim of this article is to investigate the importance of interference effect caused by differences in adiabatic phases in non-cyclic evolutions in measurable physical quantities. For this reason, in the main text of the paper, in the study of the two-level model, we decreased the number of the phase parameters by choosing the coefficients $a_1$ and $a_2$ to be real, that is, $\varphi _1 = \varphi _2 = 0$.