Evolution of Quantum Systems with a Discrete Energy Spectrum in an Adiabatically Varying External Field

Abstract

We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.

1. Introduction

The behavior of quantum systems with a discrete energy spectrum in varying external fields is of interest and relevant today, just as it was many years ago. In particular, this interest is due to the multitude of applications in quantum technologies. The problem of solving the Schrödinger equation when the Hamiltonian depends on adiabatically varying parameters was solved at an early stage of the development of quantum mechanics [1,2,3,4,5]. Adiabatic quantum gates based on the well-known Landau–Zener–Majorana–Stückelberg transitions play an important role in quantum information processing (see, for example [6,7,8,9,10,11,12]). Subsequently, it was shown that in the adiabatic approximation, there is a Berry phase in a nondegenerate quantum state [13,14,15,16] which depends on adiabatically varying parameters, which also plays an important role for quantum technologies (for example, see [17]). Adiabatic approximation using time-dependent basis sets, where the state vectors depend on time due to changes in time of some parameters of the Hamiltonian, can be successfully applied in many problems, in particular to describe the adiabatic transfer of a population between molecular states [18].

The standard adiabatic approximation assumes that the solution of the stationary Schrödinger equation is known at any given time and that this approximation is applicable, as a rule, for a nondegenerate energy spectrum and when energy levels are not close to each other. Such a situation often does not exist in real systems of interest for tasks of quantum technologies. Specifically, it is inapplicable for describing the evolution of entangled states in complex systems. In addition, the exact solution is unfortunately usually unknown or unsuitable for executing appropriate measurements of quantum states. Thus, the adiabatic approximation should be modified for all cases of interest when considering the evolution of quantum systems in varying external fields. This generalization and modification can be successful if we introduce the so-called adiabatic representation to solve the Schrödinger equation in a time-varying set of base states while introducing an additional effective term in the Hamiltonian. Recently, this adiabatic representation was introduced to solve the problem of two interacting 1/2 spins in an external magnetic field that is varying in time [19,20].

The rest of this article is organized as follows. In Section 2, the Schrödinger equation is written in the adiabatic representation and the evolution operator is obtained. Section 3.1 shows the evolution of a two-level system in the adiabatic representation, and Section 3.2 describes the evolution of qubit states in multilevel systems. Lastly, Section 4 discusses the application of the adiabatic representation for systems with two interacting spins in an adiabatically varying magnetic field.

2. Materials and Methods

In this section, we introduce an adiabatic representation for solving the Schrödinger equation with a time-dependent Hamiltonian in a nonstationary basis.

When the Hamiltonian of a system depends on some parameters ${\xi }_t$ that change slowly over time $\widehat{H}\left({\xi }_t\right)$, an adiabatic representation can usually be used. For real physical systems, the role of the parameter ${\xi }_t$ is usually played by an external magnetic field for spin systems or an electric field for quantum dots, etc. In the case of a discrete energy spectrum, the problem of solving the stationary Schrödinger equation can be represented as

$$\widehat{H}\left({\xi }_t\right)|{\psi }_n\left({\xi }_t\right)\rangle =E_n\left({\xi }_t\right)|{\psi }_n\left({\xi }_t\right)\rangle .$$

(1)

Here $|{\psi }_n\left({\xi }_t\right)\rangle$ are the eigenstates of the Hamiltonian at each moment of time t, time-dependent via the parameter ${\xi }_t$, and accordingly are orthogonal to each other $\langle {\psi }_n\left({\xi }_t\right)|{\psi }_m\left({\xi }_t\right)\rangle ={\delta }_{nm}$ at any given time. For the standard adiabatic representation, the discrete spectrum $E_n\left({\xi }_t\right)$ depending on the parameters ${\xi }_t$ should be nondegenerate. The evolution operator is written as a T-exponent:

$$U(t,t_0)=\widehat{T}exp\left(-i\underset{t_0}{\overset{t}{\int }}\widehat{H}\left({\xi }_{t^{\prime }}\right)d{t}^{\prime }\right)$$

(2)

and if the initial state coincides with one of the eigenstates (1) $|{\psi }_n\left({\xi }_0\right)\rangle$, we can write

$$|\Psi (t,{\xi }_t)\rangle =\widehat{T}exp\left(-i\underset{t_0}{\overset{t}{\int }}\widehat{H}\left({\xi }_{t^{\prime }}\right)d{t}^{\prime }\right)|{\psi }_n\left({\xi }_0\right)\rangle \approx exp\left(-i\underset{t_0}{\overset{t}{\int }}{E}_n\left({\xi }_{t^{\prime }}\right)d{t}^{\prime }\right)|{\psi }_n\left({\xi }_t\right)\rangle .$$

(3)

Here and in what follows, we assume $\hslash =1$.

How can we solve the usually stationary Schrödinger equation in the case of a finite number of states? We write the Hamiltonian in a suitable basic set in the form of a matrix and solve a system of linear algebraic equations. The resulting solution can be represented as a unitary transformation $\widehat{\Theta }$ diagonalizing the nondiagonal initial Hamiltonian matrix in the chosen basis $\langle n^{\prime }\left|\widehat{H}\right|n\rangle$. Thus, if we introduce the transformed state vector $|\phi \rangle ={\widehat{\Theta }}^{†}|\psi \rangle$ into the stationary Schrodinger equation $\widehat{H}|\psi \rangle =E|\psi \rangle$, we obtain for it an equation with a diagonal Hamiltonian matrix in the chosen basis $|n\rangle$:

$$\widehat{H}|\psi \rangle =E|\psi \rangle ⟶{\widehat{\Theta }}^{†}\widehat{H}\widehat{\Theta }|\phi \rangle =E|\phi \rangle .$$

(4)

It is obvious that the matrix elements of the unitary transformation matrix $\widehat{\Theta }$ are determined by the matrix elements of the Hamiltonian matrix. If the Hamiltonian does not depend on time, then the Schrödinger equation is invariant for this transformation:

$$i{\displaystyle \frac{\partial }{\partial t}}|\phi \left(t\right)\rangle ={\widehat{\Theta }}^{†}\widehat{H}\widehat{\Theta }|\phi \left(t\right)\rangle ,|\phi \left(0\right)\rangle ={\widehat{\Theta }}^{†}|\psi \left(0\right)\rangle .$$

(5)

The time-evolution of the state $\left|\phi \right(t)\rangle$ is determined by a simple exponential dependence.

The situation changes when the Hamiltonian depends on time, as in our case. The unitary transformation matrix also depends on time, as $\widehat{\Theta }\left({\xi }_t\right)$; thus, the diagonalizing procedure should be considered as “instantaneous” and transformation can be treated as ‘quasi-interaction’ representation [19], which in what follows we call adiabatic representation.

$$|\psi \left(t\right)\rangle =\widehat{\Theta }\left(t\right)|\phi \left(t\right)\rangle ,|\phi \left(0\right)\rangle ={\widehat{\Theta }}^{†}\left(0\right)|\psi \left(0\right)\rangle .$$

(6)

The second Equation (6) is necessary because $\widehat{\Theta }\left(0\right)\ne \widehat{1}$. The insertion of this transformation into Equation (5) provides

$$i\dot{\widehat{\Theta }}\left(t\right)|\phi \left(t\right)\rangle +i\widehat{\Theta }\left(t\right){\displaystyle \frac{\partial }{\partial t}}|\phi \left(t\right)\rangle =\widehat{H}\widehat{\Theta }\left(t\right)|\phi \left(t\right)\rangle ,$$

and by multiplying both sides of this equation by ${\widehat{\Theta }}^{†}\left(t\right)$, we obtain

$$i{\displaystyle \frac{\partial }{\partial t}}|\phi \left(t\right)\rangle =\widehat{\tilde{H}}|\phi \left(t\right)\rangle -i{\widehat{\Theta }}^{†}\left(t\right)\dot{\widehat{\Theta }}\left(t\right)|\phi \left(t\right)\rangle ,$$

(7)

where $\widehat{\tilde{H}}={\widehat{\Theta }}^{†}\left(t\right)\widehat{H}\widehat{\Theta }\left(t\right)$. The second term on the right side of the Equation (7) looks like a kinematic effect, and can be considered as an effective pseudo-interaction.

It is worth noting that while the procedure can be applied independently of the rate of field change, and consequently independently of the applicability of the adiabatic approximation, the time-dependent eigenstates and energy eigenvalues retain a clear physical meaning as evolutes of the initial ones only when the adiabatic approximation is applicable. That is, if the system is prepared in the eigenstate of the Hamiltonian at the initial moment of time, then the system remains in the evolved eigenstate of the Hamiltonian at time t only if the change in the parameter $\xi \left(t\right)$, for example, the external magnetic field, is slow enough to satisfy the requirements of the adiabatic approximation, i.e.,

$${\displaystyle \frac{\dot{\xi }\left(t\right)}{\xi \left(t\right)}}\ll {\omega }_{nk},\mathrm{where}{\omega }_{nk}=|H_{nn}-H_{kk}|.$$

(8)

Here, $H_{nn}$ and $H_{kk}$ are the corresponding diagonal matrix elements of the Hamiltonian matrix in the chosen basis. If the inequality in (8) is violated for some states, then the problem is usually reduced to a solution for a two-level system when $H_{nk}\ne 0$. In the opposite case, $H_{nk}=0$, the transition cannot be observed. Thus, we can accept the inequality in (8) as a sufficient requirement for adiabatic approximation. An example of a two-level system will be examined below.

For adiabatically slowly varying parameters $\xi \left(t\right)$, the last term in (7) can be considered as a small perturbation and the standard approach of time-dependent perturbation theory can be applied. It is usually convenient to use the interaction representation to solve the problem when the unperturbed Hamiltonian does not depend on time. In our case, the transformed Hamiltonian $\widehat{\tilde{H}}$ should play the role of an unperturbed Hamiltonian, and an additional term in the Equation (7) can be considered as a perturbation. Despite the fact that the unperturbed Hamiltonian depends on time, the evolution operator in a time-dependent basis has a simple view when it acts on the basis states of $|{\phi }_n\left(t\right)\rangle$:

$$|{\phi }_n\left(t\right)\rangle ={\tilde{U}}_0(t,t_0)|{\phi }_n\left(0\right)\rangle =\widehat{T}exp\left(-i\underset{t_0}{\overset{t}{\int }}\widehat{\tilde{H}}\left(t^{\prime }\right)d{t}^{\prime }\right)|{\phi }_n\left(0\right)\rangle =e^{-i\underset{t_0}{\overset{t}{\int }}{E}_n\left(t^{\prime }\right)d{t}^{\prime }}|{\phi }_n\left(0\right)\rangle .$$

(9)

Now and in what follows, we write $\widehat{\tilde{H}}\left({\xi }_t\right)\equiv \widehat{\tilde{H}}\left(t\right)$ for simplicity, omitting the symbol ${\xi }_t$ of time-dependent parameters.

Taking into account that the additional term in the Equation (7) appears due to the kinematic effect, we can introduce an adiabatic representation, after which the Schrödinger equation looks like

$$i{\displaystyle \frac{\partial }{\partial t}}|{\phi }_I\left(t\right)\rangle ={\widehat{V}}_{eff}\left(t\right)|{\phi }_I\left(t\right)\rangle ,\mathrm{where}{\widehat{V}}_{eff}\left(t\right)=-i{\tilde{U}}_0^{†}\left(t\right){\widehat{\Theta }}^{†}\left(t\right)\dot{\widehat{\Theta }}\left(t\right){\tilde{U}}_0\left(t\right),$$

(10)

and $|\phi \left(t\right)\rangle ={\tilde{U}}_0\left(t\right)|{\phi }_I\left(t\right)\rangle$.

Correspondingly, the evolution operator in (2) should be written as follows:

$$U(t,t_0)=\widehat{\Theta }\left(t\right){\tilde{U}}_0(t,t_0){\tilde{U}}_{ad}(t,t_0){\widehat{\Theta }}^{†}\left(0\right),$$

(11)

where

$${\tilde{U}}_{ad}(t,0)=\widehat{T}exp\left(-i\underset{t_0}{\overset{t}{\int }}{\widehat{V}}_{eff}\left(t^{\prime }\right)d{t}^{\prime }\right).$$

Note that Equation (11) is a solution to the problem in an arbitrary case, as in order to obtain the evolution operator only the existence of a solution to the stationary Schrödinger Equation (1) is necessary. Unfortunately, an analytical solution can be obtained approximately for an adiabatically slowly varying parameter $\xi \left(t\right)$.

In the following, we illustrate the application of the adiabatic representation to describe the evolution of quantum systems with a nondegenerate energy spectrum that depend on some time-varying parameter when two levels become close to each other, as well as the evolution of entangled states in a system of two coupled subsystems. As is well known, in the first case the problem can be described in terms of effective spin 1/2. In the second case, we consider the main features of the evolution of entangled states in the simplest case of two interacting particles with spin 1/2.

3. Results

In this section, we present a solution to the problem for two special cases that have wide application in various problems.

3.1. Two-Level System

Consider a situation where, for a multilevel system in the chosen basis, the inequality $|H_{nn}-H_{kk}|\lesssim |H_{nk}|$ is satisfied for two states $|n\rangle$ and $|k\rangle$. In this case, these two states can be described in terms of effective spin 1/2. As is well known, a solution for a two-level system can be obtained in the general case. Recently, a solution has been obtained for nonstationary two-level systems [21]. Based on these reasons, a two-level system can be considered as a basic one; thus, we first of all illustrate its description in the adiabatic representation. In general, the Hamiltonian of any two-level system can be written as the scalar product of some vector $\mathbf{b}\left(t\right)$ by a vector of three Pauli matrices: ${\widehat{H}}_2=\widehat{a}+\left(\mathbf{b}\left(t\right)\sigma \right)$.

In the standard basis, the Hamiltonian matrix has the form

$${\widehat{H}}_2=\left(\begin{array}{cc}a+b_3& b_{-}\\ b_{+}& a-b_3\end{array}\right)\mathrm{where}{b}_{\pm }=b_1\pm i{b}_2.$$

(12)

The Hamiltonian matrix in (12) is diagonalized by the matrix of the unitary transformation

$$\widehat{\Theta }=\left(\begin{array}{cc}cos\vartheta & -sin\vartheta \\ sin\vartheta & cos\vartheta \end{array}\right),\mathrm{where}tan2\vartheta e^{i\phi }={\displaystyle \frac{b_{-}}{b_3}},\phi =\arg b_{-}.$$

(13)

The corresponding instantaneous eigenstates and numbers are equal:

$$|{\chi }_1\rangle =cos\vartheta |+\rangle +sin\vartheta |-\rangle ,|{\chi }_2\rangle =-sin\vartheta |+\rangle +cos\vartheta |-\rangle ,{\epsilon }_{1,2}=a\pm \sqrt{b_3^2+{\left|b_{-}\right|}^2}.$$

(14)

Now, we need to introduce the transformed Pauli operators in the form ${\widehat{\tau }}_i={\widehat{\Theta }}^{†}{\widehat{\sigma }}_i\widehat{\Theta }$, where $i=x,y,z$. In this case, the transformed operators pose the following properties:

$${\widehat{\tau }}_z|{\chi }_{1,2}\rangle =\pm |{\chi }_{1,2}\rangle ,{\widehat{\tau }}_x|{\chi }_{1,2}\rangle =|{\chi }_{2,1}\rangle .$$

The transformed diagonal Hamiltonian matrix is equal to

$${\widehat{\tilde{H}}}_2=a\widehat{1}+\sqrt{b_3^2+{\left|b_{-}\right|}^2}{\widehat{\tau }}_z.$$

(15)

Effective interaction in this case has a very simple form:

$$-i{\widehat{\Theta }}^{†}\left(t\right)\dot{\widehat{\Theta }}\left(t\right)=-\left(\begin{array}{cc}0& -i\dot{\vartheta }\\ i\dot{\vartheta }& 0\end{array}\right)=-\hslash \dot{\vartheta }{\widehat{\tau }}_y.$$

(16)

The evolution operator of an “unperturbed” system is equal to

$${\tilde{U}}_0\left(t\right)=e^{-iat}exp\left(-i\underset{t_0}{\overset{t}{\int }}{\epsilon }_0\left(t^{\prime }\right)d{t}^{\prime }{\widehat{\tau }}_z\right),$$

(17)

where ${\epsilon }_0\left(t\right)=\sqrt{b_3^2+{\left|b_{-}\right|}^2}$.

For a two-level system, Equation (7) in the adiabatic representation takes the form

$$|\chi \left(t\right)\rangle =U_0\left(t\right)|{\chi }_I\left(t\right)\rangle ,{\displaystyle \frac{\partial }{\partial t}}|{\chi }_I\left(t\right)\rangle =i\dot{\vartheta }\left(cos2\Phi \left(t\right)\widehat{{\tau }_y}+sin2\Phi \left(t\right)\widehat{{\tau }_x}\right)|{\chi }_I\left(t\right)\rangle ,$$

(18)

where $\Phi \left(t\right)=\underset{t_0}{\overset{t}{\int }}{\epsilon }_0\left(t^{\prime }\right)d{t}^{\prime }$.

Correspondingly, the evolution operator for the interaction representation in (11) is equal to

$${\tilde{U}}_{ad}(t,t_0)=\widehat{T}exp\left\{i\underset{t_0}{\overset{t}{\int }}\dot{\vartheta }\left(t^{\prime }\right)\left(cos2\Phi \left(t^{\prime }\right)\widehat{{\tau }_y}+sin2\Phi \left(t^{\prime }\right)\widehat{{\tau }_x}\right)d{t}^{\prime }\right\}.$$

(19)

If we take into account that all time dependence is related to the time-dependent parameter ${\xi }_t$, then we can replace time integration with parameter integration:

$${\tilde{U}}_{ad}(t,t_0)=\widehat{T}exp\left\{i\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}\left(cos2\Phi \left({\xi }_{t^{\prime }}\right)\widehat{{\tau }_y}+sin2\Phi \left({\xi }_{t^{\prime }}\right)\widehat{{\tau }_x}\right)d{\xi }_{t^{\prime }}\right\}.$$

(20)

In the adiabatic approximation, we can assume that $|\dot{\vartheta }\left(t\right)|\ll \dot{\Phi }\left(t\right)$ and limit ourselves only to the first order of decomposition of the operator (19):

$${\tilde{U}}_{ad}(t,0)\approx 1+i\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}\left(cos2\Phi \left({\xi }_{t^{\prime }}\right)\widehat{{\tau }_y}+sin2\Phi \left({\xi }_{t^{\prime }}\right)\widehat{{\tau }_x}\right)d{\xi }_{t^{\prime }}.$$

(21)

In the first-order of perturbation theory, we can calculate the evolution of a quantum system for two initial states: $|\chi \left(0\right)\rangle =|{\chi }_2\rangle$ and $|\chi \left(0\right)\rangle =|{\psi }_1\rangle$. The first case corresponds to a situation where two states are considered as a qubit subsystem in the ground state. The second case corresponds to a system, for example, of interacting spins 1/2, when the system is far from the region of close of energy levels.

If the initial state corresponds to the “ground” state of the qubit, then the evolution of the state in the first order of perturbation is described by the relation

$$|\chi \left(t\right)\rangle =e^{i\Phi \left(t\right)}|{\chi }_2\rangle +e^{-i\Phi \left(t\right)}\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}{e}^{i2\Phi \left({\xi }_{t^{\prime }}\right)}d{\xi }_{t^{\prime }}|{\chi }_1\rangle .$$

(22)

The probability of observing a system in an “excited” state $|{\chi }_1\rangle$ at time t, that is, the transition probability, is determined as follows:

$$W_{21}\left(t\right)={\left|\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}{e}^{i2\Phi \left({\xi }_{t^{\prime }}\right)}d{\xi }_{t^{\prime }}\right|}^2.$$

(23)

Note that for a two-level system the expression for the evolution operator in (20) is valid for arbitrary dependence of the parameter $\xi \left(t\right)$ on time and that the inequality in (8) is important for applying the perturbation approach.

It can be seen that the transition probability is small, but not exponentially small. Let us now make an estimation of the expression in (23). If we remove the slowly varying function $\dot{\vartheta }\left({\xi }_{t^{\prime }}\right)$ from under the integral at the initial moment of time and leave only the rapidly varying exponential function, we obtain the following result:

$$W_{21}\approx {\left|\dot{\vartheta }\left({\xi }_0\right)\underset{t_0}{\overset{t}{\int }}{e}^{i2\Phi \left({\xi }_{t^{\prime }}\right)}d{t}^{\prime }\right|}^2\sim {\left|{\displaystyle \frac{\dot{\vartheta }\left({\xi }_0\right)}{2\dot{\Phi }\left({\xi }_t\right){|}_{t=t_0}}}\right|}^2\sim {\left|{\displaystyle \frac{\dot{\xi }}{\xi \omega }}\right|}^2\ll 1$$

where $\omega =2{\epsilon }_0$ is the transition frequency. This estimate shows that for the adiabatically varying parameter $\xi$, the expression in (23) for the transition probability provides a good approximation for an arbitrary point in time.

Let the initial state correspond to one of the base states $|{\psi }_2\rangle$, which after changing the external parameters becomes close to another base state $|{\psi }_1\rangle$. In this case, these two states effectively form two qubit states $|{\chi }_1\rangle$ and $|{\chi }_2\rangle$. For the formed effective qubit, the initial state is

$$|\chi \left(0\right)\rangle =|{\psi }_2\rangle =sin{\vartheta }_0|{\chi }_1\rangle +cos{\vartheta }_0|{\chi }_2\rangle .$$

(24)

Here, ${\vartheta }_0$ is the transformation parameter at the initial time $t=0$.

At time t, the qubit state in the first order of perturbation is

$$\begin{array}{cc}\hfill \left|\chi \right(t)\rangle =& e^{-i\Phi \left(t\right)}\left(sin{\vartheta }_0+cos{\vartheta }_0\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}{e}^{i2\Phi \left({\xi }_{t^{\prime }}\right)}d{\xi }_{t^{\prime }}\right)|{\chi }_1\rangle \hfill \\ \hfill +& e^{i\Phi \left(t\right)}\left(cos{\vartheta }_0-sin{\vartheta }_0\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}{e}^{-i2\Phi \left({\xi }_{t^{\prime }}\right)}d{\xi }_{t^{\prime }}\right)|{\chi }_2\rangle .\hfill \end{array}$$

(25)

To determine the state of the system at time t in the initial basis, we need to apply the expression in (11) for the evolution operator. After some transformations, we obtain

$$\begin{array}{cc}\hfill |\Psi (t)\rangle =& \left[-sin\vartheta \left(t\right)+cos\vartheta \left(t\right)\Re F\left(t\right)+icos\left(\vartheta \left(t\right)+2{\vartheta }_0\right)\Im F\left(t\right)\right]|{\psi }_1\rangle \hfill \\ \hfill +& \left[cos\vartheta \left(t\right)+sin\vartheta \left(t\right)\Re F\left(t\right)+isin\left(\vartheta \left(t\right)+2{\vartheta }_0\right)\Im F\left(t\right)\right]|{\psi }_2\rangle ,\hfill \end{array}$$

(26)

where $\Re F\left(t\right)$ and $\Im F\left(t\right)$ are the real and imaginary parts of the function

$$F\left(t\right)=e^{-i\Phi \left(t\right)}\underset{{\xi }_0}{\overset{{\xi }_t}{\int }}{\displaystyle \frac{d\vartheta \left({\xi }_{t^{\prime }}\right)}{d{\xi }_{t^{\prime }}}}{e}^{i2\Phi \left({\xi }_{t^{\prime }}\right)}d{\xi }_{t^{\prime }}.$$

If the transformation parameter ${\vartheta }_0=0$ and $\vartheta \left(t\right)\to \pi /2$ at a finite time, then Equation (25) provides a well known result of stationary perturbation theory. Now, however, we can describe it as the evolution of a quantum system. In addition, we observe oscillations in the transitions between the basic states in the intermediate region of the value of the parameter $\xi$ when the two energy levels become close to each other.

3.2. Qubit States in a Multilevel System

In this section, we consider the case of a quantum system with a discrete nondegenerate energy spectrum when the system has well-defined base states at initial time $t_0$$|{\psi }_n^{\left(0\right)}\rangle$ for some values of the parameter $\xi \left(t\right)$ (the external field). In other words, all nondiagonal matrix elements of the Hamiltonian matrix satisfy the inequality $|H_{kk}-H_{nn}|\ll |H_{kn}|$ with the exception of two states, which we denote as $|{\psi }_1^{\left(0\right)}\rangle$ and $|{\psi }_2^{\left(0\right)}\rangle$. These two states form qubit states that strongly depend on the parameter and that can be described using the adiabatic representation discussed in the previous sections, while all the others can be approximated as eigenstates of the Hamiltonian. Thus, we can assume that all other states change exponentially little for an adiabatically varying parameter, in accordance with the standard theory of adiabatic perturbations.

Following our approach, we need to perform a unitary transformation, which in the initial basis is described by the matrix

$$\Theta \left(t\right)=\left(\begin{array}{cccccc}cos\vartheta & -sin\vartheta & 0& 0& \cdots & 0\\ sin\vartheta & cos\vartheta & 0& 0& \cdots & 0\\ 0& 0& 1& 0& \cdots & 0\\ 0& 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& 0& \cdots & 1\end{array}\right),\mathrm{where}tan2\vartheta \left(t\right)={\displaystyle \frac{2|H_{12}\left(t\right)|}{H_{11}\left(t\right)-H_{22}\left(t\right)}}.$$

(27)

We represent the transformed Hamiltonian matrix as the sum of the diagonal ‘unperturbed’ Hamiltonian matrix and the nondiagonal matrix, which can be considered as a small perturbation:

$$\widehat{\tilde{H}}={\widehat{\Theta }}^{†}\widehat{H}\widehat{\Theta }={\widehat{\tilde{H}}}_0+\widehat{\tilde{V}}.$$

(28)

The matrix elements of the Hamiltonian matrix ${\widehat{\tilde{H}}}_0$ are equal to energy levels in the zero approximation:

$${\epsilon }_{1,2}^{\left(0\right)}={\displaystyle \frac{1}{2}}\left(H_{11}+H_{22}\pm \sqrt{{(H_{11}-H_{22})}^2+4{\left|H_{12}\right|}^2}\right),\mathrm{and}{\epsilon }_k^{\left(0\right)}=H_{kk}\mathrm{for}k\ne 1,2.$$

(29)

Now, we need to change the initial base states $|{\psi }_1^{\left(0\right)}\rangle$ and $|{\psi }_2^{\left(0\right)}\rangle$ to the states $|{\phi }_{1,2}^{\left(0\right)}\rangle$, as in Equation (14):

$$|{\phi }_1\rangle =cos\vartheta |{\psi }_1^{\left(0\right)}\rangle +sin\vartheta |{\psi }_2^{\left(0\right)}\rangle ,|{\phi }_2\rangle =-sin\vartheta |{\psi }_1^{\left(0\right)}\rangle +cos\vartheta |{\psi }_2^{\left(0\right)}\rangle .$$

(30)

After this transformation, the transition probabilities from the states in (30) to all other states $|{\psi }_k\rangle ,k\ne 1,2$ should be exponentially small, and we can consider the states $|{\phi }_{1,2}^{\left(0\right)}\rangle$ like two states of a qubit.

In the perturbation matrix, only the first two columns and first two rows change after the transformation:

$${\tilde{V}}_{1k}={\tilde{V}}_{k1}^{\ast }=cos\vartheta \left(t\right)H_{1k}+sin\vartheta \left(t\right)H_{2k},{\tilde{V}}_{2k}={\tilde{V}}_{k2}^{\ast }=-sin\vartheta \left(t\right)H_{1k}+cos\vartheta \left(t\right)H_{2k},$$

(31)

where $k\ne 1,2$. All other elements coincide with the nondiagonal matrix elements of the initial Hamiltonian matrix.

The evolution of qubit states in the zero approximation is described by the relations obtained in Section 3.1 for a two-level system, except that now we need to take into account the existence of a nonzero perturbation. Taking this perturbation into account is necessary because the transition probabilities between qubit states have a power dependence, as shown by the relation in (23).

To solve a nonstationary problem within the framework of perturbation theory, it is advisable to use the interaction representation. Because the transformed unperturbed Hamiltonian ${\widehat{\tilde{H}}}_0$ is represented by a diagonal matrix in the transformed basis, the evolution operator is defined by Equation (9) with new energy levels (29) for transformed states (30). All nondiagonal matrix elements of the perturbation operator obtain an additional phase multiplier:

$${\tilde{V}}_{I,ik}\left(t\right)={\tilde{V}}_{ik}{e}^{-i\underset{t_0}{\overset{t}{\int }}{\omega }_{ik}\left({\xi }_{t^{\prime }}\right)d{t}^{\prime }}$$

(32)

where ${\omega }_{ik}\left(\xi \right)={\epsilon }_i^{\left(0\right)}\left(\xi \right)-{\epsilon }_k^{\left(0\right)}\left(\xi \right)$. Here, ${\epsilon }_k^{\left(0\right)}\left(\xi \right))=H_{kk}$ for $k\ne 1,2$ and ${\epsilon }_i^{\left(0\right)}\left(\xi \right))$ is equal to (29) for $i=1$ and 2.

In the interaction representation in the zero approximation, we see that all states are $|{\psi }_{I,k}^{\left(0\right)}\left(t\right)\rangle =|{\psi }_{I,k}^{\left(0\right)}\left(0\right)\rangle$ for $k\ne 1,2$, while $|{\phi }_{I,i}^{\left(0\right)}\left(t\right)\rangle$ for $i=1,2$ are described by the relations in (19)–(21) obtained for a two-level system. In the following, we need to obtain corrections for the states of the qubit $|{\phi }_{I,i}^{\left(0\right)}\left(t\right)\rangle$, as the nondiagonal matrix elements of the perturbation are not small for them.

In our case, after choosing the basic states, the Schrödinger equation is represented as a system of linear equations. Therefore, it is convenient to write a set of states in the form of a vector-column ${\overrightarrow{\phi }}_I\left(t\right)$, elements of which are equal to state vectors $|{\phi }_{I,n}\left(t\right)\rangle$. In these notations, the Schrödinger equation in the interaction representation has the form

$$i{\displaystyle \frac{\partial }{\partial t}}{\overrightarrow{\phi }}_I\left(t\right)={\tilde{V}}_I\left(t\right){\overrightarrow{\phi }}_I\left(t\right),$$

(33)

where ${\tilde{V}}_I\left(t\right)$ is a square matrix defined by Equations (31) and (32).

We can write down the corrected states which are components of the vector ${\overrightarrow{\phi }}_I\left(t\right)$, as in standard perturbation theory, taking into account our results:

$${\overrightarrow{\phi }}_I\left(t\right)=\left(1+A\left(t\right)\right){\overrightarrow{\phi }}_I^{\left(0\right)}\left(0\right)$$

(34)

where $A\left(t\right)$ is a square Hermitian matrix with elements $a_{ik}\left(t\right)$ satisfying inequalities $|a_{ik}\left(t\right)|\ll 1$ at any given time. We take into account that $|{\psi }_{I,k}^{\left(0\right)}\left(t\right)\rangle =|{\psi }_{I,k}^{\left(0\right)}\left(0\right)\rangle$. Substituting these into the Schrödinger Equation (33), we obtain

$$i\dot{A}\left(t\right){\overrightarrow{\phi }}_I^{\left(0\right)}\left(0\right)={\tilde{V}}_I\left(t\right)\left(1+A\left(t\right)\right){\overrightarrow{\phi }}_I^{\left(0\right)}\left(0\right).$$

(35)

We have now obtained a system of inhomogeneous equations, the solution of which is the sum of the general solution of a homogeneous equation and a partial solution of an inhomogeneous one. In the first order of perturbation, we need to limit ourselves to only a partial solution. Thus, for each component of the vector equation in (35), we can write the following equations:

$$i\sum _k\left({\dot{a}}_{ik}-{\tilde{V}}_{I,ik}\left(t\right)\right)|{\phi }_k^{\left(0\right)}\rangle =0.$$

(36)

All vectors $|{\phi }_k^{\left(0\right)}\rangle$ are independent, and we have simple solutions for the elements of the matrix $A\left(t\right)$:

$$a_{ik}\left(t\right)=a_{ik}\left(0\right)-i\underset{t_0}{\overset{t}{\int }}{\tilde{V}}_{I,ik}\left(t^{\prime }\right)d{t}^{\prime }=a_{ik}\left(0\right)-i\underset{t_0}{\overset{t}{\int }}{\tilde{V}}_{ik}\left({\xi }_{t^{\prime }}\right)e^{-i\underset{t_0}{\overset{t^{\prime }}{\int }}{\omega }_{ik}\left({\xi }_{t^{\prime \prime }}\right)d{t}^{\prime \prime }}d{t}^{\prime }.$$

(37)

The matrix elements at the initial moment of time $a_{ik}\left(0\right)$ are found in the same way as in the stationary perturbation theory:

$$a_{ik}\left(0\right)={\displaystyle \frac{{\tilde{V}}_{I,ik}\left({\xi }_0\right)}{{\epsilon }_i^{\left(0\right)}-{\epsilon }_k^{\left(0\right)}}}={\displaystyle \frac{{\tilde{V}}_{ik}\left({\xi }_0\right)e^{i{\omega }_{ik}\left({\xi }_0\right)t_0}}{{\epsilon }_i^{\left(0\right)}-{\epsilon }_k^{\left(0\right)}}}.$$

(38)

The integral function in the second term of the expression in (37) is the product of slowly and rapidly varying functions. In this case, the slowly varying function is averaged over another frequently oscillating function. Thus, as in the estimation of Equation (23), we can extract a slowly varying function from under the integral at the initial moment of time. After this approximation, we obtain the following expression:

$$\begin{array}{cc}\hfill -i\underset{t_0}{\overset{t}{\int }}{\tilde{V}}_{ik}\left({\xi }_{t^{\prime }}\right)e^{-i\underset{t_0}{\overset{t^{\prime }}{\int }}{\omega }_{ik}\left({\xi }_{t^{\prime \prime }}\right)d{t}^{\prime \prime }}d{t}^{\prime }\approx & -i{\tilde{V}}_{ik}\left({\xi }_0\right)\underset{t_0}{\overset{t}{\int }}{e}^{-i\underset{t_0}{\overset{t^{\prime }}{\int }}{\omega }_{ik}\left({\xi }_{t^{\prime \prime }}\right)d{t}^{\prime \prime }}d{t}^{\prime }\hfill \\ \hfill \approx & {\tilde{V}}_{ik}\left({\xi }_0\right)\left({\displaystyle \frac{e^{-i{\omega }_{ik}\left({\xi }_t\right)t}}{{\omega }_{ik}\left({\xi }_t\right)}}-{\displaystyle \frac{e^{-i{\omega }_{ik}\left({\xi }_0\right)t_0}}{{\omega }_{ik}\left({\xi }_0\right)}}\right).\hfill \end{array}$$

(39)

Here, we have taken into account that the transition frequencies are slowly varying functions of the parameter $\xi$. It can be seen that the second term in Equation (39) is reduced with the first term in Equation (37), and finally we have

$$a_{ik}\left(t\right)\approx {\displaystyle \frac{{\tilde{V}}_{ik}\left({\xi }_0\right)e^{-i{\omega }_{ik}\left({\xi }_0\right)t}}{{\epsilon }_i^{\left(0\right)}\left({\xi }_t\right)-{\epsilon }_k^{\left(0\right)}\left({\xi }_t\right)}}$$

(40)

The results obtained here are in full agreement with the results of perturbation theory, and confirm the results for an adiabatically varying perturbation in the case of a nondegenerate spectrum. We can consider the expression in (40) as a condition for determining qubit states in multilevel systems if $|a_{ik}\left(t\right)|\ll 1$.

4. Discussion

The introduced adiabatic representation can be successfully applied to describe the evolution of a multilevel system in which the avoided level-crossing effect is observed. In particular, this is a common situation for spin systems. The problem of two interacting 1/2 spins in an adiabatically varying external magnetic field was considered in the adiabatic representation in [19,20]. For two equivalent spins, the problem was formulated as the evolution of the pseudo-qutrit [22]. The spin Hamiltonian of these systems is equivalent to the spin Hamiltonian of a system with axially symmetric hyperfine interaction (HFI), and can be written as follows (see, for example [23,24,25]):

$$\widehat{H}=-{\displaystyle \frac{1}{2}}\omega \left(t\right){\widehat{\sigma }}_{1z}-{\displaystyle \frac{\zeta }{2}}\omega \left(t\right){\widehat{\sigma }}_{2z}+A_{\perp }\left({\sigma }_1{\sigma }_2\right)+(A_{\parallel }-A_{\perp })\left({\sigma }_1\mathbf{n}\right)\left({\sigma }_2\mathbf{n}\right)$$

(41)

where ${\omega }_1=g_1{\mu }_{0,1}B\left(t\right)\equiv \omega$ and ${\omega }_2=g_2{\mu }_{0,2}B\left(t\right),\mathbf{B}\left(t\right)\parallel z$ is the external magnetic field, while ${\mu }_{0,1}$ and ${\mu }_{0,2}$ are particle magnetons. We introduce the ratio $g_2{\mu }_{0,2}/g_1{\mu }_{0,1}=\zeta$ and assume scalar g-factors, $g_1>g_2$. The HFI interaction tensor is written as $A_{ik}=A_i{\delta }_{ik}$ with respect to the principal axes $\mathbf{n}$, and is determined by the two constants $A_{\parallel }\parallel \mathbf{n}$ and $A_{\perp }\perp \mathbf{n}$, while ${\sigma }_1$ and ${\sigma }_2$ are the Pauli matrices of the two spins. As usual, we express the Hamiltonian in frequency units by taking $\hslash =1$. In this problem, the external field, and accordingly the frequency $\omega \left(t\right)$, plays the role of the ${\xi }_t$ parameter.

In the standard basis set for a two-spin system

$$|{\chi }_1\rangle =|+\rangle |+\rangle ,|{\chi }_2\rangle =|+\rangle |-\rangle ,|{\chi }_3\rangle =|-\rangle |+\rangle ,|{\chi }_4\rangle =|-\rangle |-\rangle ,$$

(42)

the Hamiltonian matrix of the system can written as follows:

$$\widehat{H}=\left(\begin{array}{cccc}A-\omega (1+\zeta )/2\hfill & \Delta A_1\hfill & \Delta A_1\hfill & \Delta A_2\hfill \\ \Delta A_1\hfill & -A-\omega (1-\zeta )/2\hfill & 2A+\Delta A_2\hfill & -\Delta A_1\hfill \\ \Delta A_1\hfill & 2A+\Delta A_2\hfill & -A+\omega (1-\zeta )/2\hfill & -\Delta A_1\hfill \\ \Delta A_2\hfill & -\Delta A_1\hfill & -\Delta A_1\hfill & A+\omega (1+\zeta )/2\hfill \end{array}\right)$$

(43)

where $A=A_{\parallel }{cos}^2\theta +A_{\perp }{sin}^2\theta ,\Delta A_1=(A_{\parallel }-A_{\perp })sin\theta cos\theta ,\Delta A_2=(A_{\parallel }-A_{\perp }){sin}^2\theta$, and $\theta$ is the angle between the magnetic field $\mathbf{B}$ and the axes of symmetry $\mathbf{n}$.

Consider the case of a strong magnetic field $\omega \gg A$ and $g_2\ll g_1$, that is, $\zeta \ll 1$. In an arbitrary case, all nondiagonal matrix elements of the Hamiltonian matrix in (43) are much smaller than the difference between any two diagonal elements. Therefore, diagonal elements of the matrix in (43) could be considered as energy levels in the zero approximation. However, it is easy to see that ${\tilde{H}}_{11}\approx {\tilde{H}}_{22}$ or ${\tilde{H}}_{33}\approx {\tilde{H}}_{44}$ for $\omega \approx 2A/\left|\zeta \right|$. In this case, the difference between the diagonal elements of the Hamiltonian matrix in (43) may be small ($|H_{11}-H_{22}|\ll |H_{12}|$ or $|H_{33}-H_{44}|\ll |H_{34}|$) and it is necessary to perform diagonalization (13) for the corresponding close energy levels of the zero approximation. In our case, we can perform the necessary unitary transformation for two signs of the $\zeta$ ratio simultaneously using the following matrix [24,25]:

$$\Theta \left(\theta \right)=\left(\begin{array}{cccc}cos{\vartheta }_1& -sin{\vartheta }_1& 0& 0\\ sin{\vartheta }_1& cos{\vartheta }_1& 0& 0\\ 0& 0& cos{\vartheta }_2& -sin{\vartheta }_2\\ 0& 0& sin{\vartheta }_2& cos{\vartheta }_2\end{array}\right)$$

(44)

where

$$tan\left(2{\vartheta }_1\right)={\displaystyle \frac{\Delta A_1}{A-\zeta \omega /2}};tan\left(2{\vartheta }_2\right)={\displaystyle \frac{\Delta A_1}{A+\zeta \omega /2}}.$$

The approximate energy levels are equal to

$$\begin{array}{cc}\hfill {\epsilon }_{1,2}^{\left(0\right)}=& -{\displaystyle \frac{1}{2}}\omega \mp \sqrt{\Delta A_1^2+{(A+\zeta \omega /2)}^2},{\epsilon }_1^{\left(0\right)}\approx {\tilde{H}}_{11},{\epsilon }_2^{\left(0\right)}\approx {\tilde{H}}_{22},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\epsilon }_{3,4}^{\left(0\right)}=& {\displaystyle \frac{1}{2}}\omega \mp \sqrt{\Delta A_1^2+{(A-\zeta \omega /2)}^2},{\epsilon }_3^{\left(0\right)}\approx {\tilde{H}}_{33},{\epsilon }_4^{\left(0\right)}\approx {\tilde{H}}_{44}.\hfill \end{array}$$

(46)

We can see that energy levels are not crossed now.

Finally, we define the derivatives of the transformation parameters:

$${\dot{\vartheta }}_1=-{\displaystyle \frac{\Delta A_1\zeta \dot{\omega }}{4{(\Delta A_1)}^2+{(2A-\zeta \omega )}^2}}=-{\displaystyle \frac{\Delta A_1\zeta }{{\omega }_{12}^2}}\dot{\omega },{\dot{\vartheta }}_2={\displaystyle \frac{\Delta A_1\zeta }{{\omega }_{34}^2}}\dot{\omega }.$$

(47)

The qubit states in this problem are the entangled states of two interacting spin subsystems.

If one of the interacting spins is greater than 1/2, then the problem will be more difficult but more interesting due to the presence of a quadrupole moment. Let us consider magnetic interactions in a system of two nuclei with spins $s=1/2$ and $I\ne 1/2$. Such systems can exist in dielectric crystals, for example diamond and silicon in crystals with a diamond structure. The nuclei of the main isotopes${}^{12}$C and ${}^{28}$Si have spins equal to 0, while the nuclei of the isotopes ${}^{13}$C and ${}^{29}$Si have spins $s=1/2$. The main impurities in diamond are nitrogen and boron, and the nucleus of the ${}^{14}$N isotope has spin $I=1$. In silicon crystals, the isotopes of the impurities ${}^{27}$Al and ${}^{33}$S have spins $I=5/2$ and 3/2, respectively.

The spin Hamiltonian of the problem can be written as follows:

$$\widehat{H}=-{\omega }_1\left(t\right){\widehat{s}}_z-{\omega }_2\left(t\right){\widehat{I}}_z+{\widehat{V}}_Q+{\widehat{V}}_{dd}$$

(48)

where the quadrupole interaction has the form

$${\widehat{V}}_Q={\Omega }_Q\left(3\left(\widehat{\mathbf{I}}\mathbf{n}\right)\left(\widehat{\mathbf{I}}\mathbf{n}\right)-I(I+1)\right)$$

(49)

and the magnetic dipole–dipole interaction is equal to

$${\widehat{V}}_{dd}={\Omega }_d\left(3\left(\widehat{\mathbf{I}}\mathbf{n}\right)\left(\widehat{\mathbf{s}}\mathbf{n}\right)-\left(\widehat{\mathbf{s}}\widehat{\mathbf{I}}\right)\right).$$

(50)

The Zeeman frequencies are ${\omega }_{1,2}=g_{1,2}{\mu }_{N}B$, where ${\mu }_N$ is a nuclear magneton and $g_1$ and $g_2$ are the corresponding g-factors of the nuclei. The constant ${\Omega }_Q$ is determined by the nuclear quadrupole moment and the gradient of the local electric field. As a rule, this constant corresponds to the precession frequency of the magnetic moment in a magnetic field of the order of 100 Gs. We suppose that the main axes of the electric field gradient tensor are determined by the unit vector $\mathbf{n}$ between two interacting nuclei. The frequency ${\Omega }_d$ is determined as usual:

$${\Omega }_d={\displaystyle \frac{g_1{g}_2{\mu }_N^2}{R_{12}^3}},$$

where $R_{12}$ is the distance between nuclei.

It is easy to see that ${\Omega }_Q\gg {\Omega }_d$; in addition, in an external magnetic field $B\gtrsim 1$ kGs, the Zeeman part of (48) can be considered as an unperturbed Hamiltonian. Let us consider the case of strong magnetic fields when both quadrupole and magnetic dipole–dipole interactions can be considered as small perturbations.

As in the previous example, well-defined states of the system in a strong magnetic field have certain projections of nuclear spins:

$$|{\psi }_n\rangle ⟶|m_s,M_I\rangle =|m_s\rangle |M_I\rangle .$$

(51)

The states in (51) represent the basic states for the problem when the diagonal matrix elements of the Hamiltonian matrix in (48) provide energy levels in the zero approximation and are equal to

$${\epsilon }_{m_s,M_I}^{\left(0\right)}=-{\omega }_1\left(t\right)(m_s+\zeta M_I)+{\Omega }_Q\left(3M_I^2{cos}^2\theta -I(I+1)\right)+{\Omega }_d{m}_s{M}_I(3{cos}^2\theta -1),$$

(52)

where, as before, $\zeta ={\omega }_2/{\omega }_1$.

It can be seen that the energy levels of the zero approximation in (52) can cross at some values of the external field. The number of intersections depends on the value of I and the value of the ratio $\zeta$, taking into account that the inequality $\omega \gg {\Omega }_Q$ should be fulfilled. Let us consider the simplest example for the case of $I=1$. We enumerate the basic states as follows:

$$\begin{array}{cc}\hfill |{\psi }_1\rangle =& |+\rangle |+1\rangle ,|{\psi }_2\rangle =|-\rangle |+1\rangle ,|{\psi }_3\rangle =|+\rangle |0\rangle ,\hfill \\ \hfill |{\psi }_4\rangle =& |-\rangle |0\rangle ,|{\psi }_5\rangle =|+\rangle |-1\rangle ,|{\psi }_6\rangle =|-\rangle |-1\rangle .\hfill \end{array}$$

(53)

If we write down the transition frequencies, we can see that only two ‘levels’ can intersect in the limit of a strong magnetic field in the case of $\zeta \ll 1$:

$${\omega }_{13}^{\left(0\right)}=-\zeta {\omega }_1+{\displaystyle \frac{1}{2}}(3{\Omega }_Q+{\Omega }_d)(3{cos}^2\theta -1),{\omega }_{24}^{\left(0\right)}=-\zeta {\omega }_1+{\displaystyle \frac{1}{2}}(3{\Omega }_Q-{\Omega }_d)(3{cos}^2\theta -1).$$

(54)

The corresponding nondiagonal elements of the matrix are equal

$$V_{13}={\displaystyle \frac{3\sqrt{3}}{4}}\left({\Omega }_Q+{\displaystyle \frac{1}{2}}{\Omega }_d\right)sin2\theta e^{-i\phi },V_{24}={\displaystyle \frac{3\sqrt{3}}{4}}\left({\Omega }_Q-{\displaystyle \frac{1}{2}}{\Omega }_d\right)sin2\theta e^{-i\phi },$$

(55)

where $\phi$ is the azimuthal angle of the unit vector $\mathbf{n}$.

If the angle $\theta =0$ ($\mathbf{B}\left(t\right)\parallel \mathbf{n}$), then the problem is solved exactly and the states $|{\psi }_1\rangle$ and $|{\psi }_3\rangle$ or $|{\psi }_2\rangle$ and $|{\psi }_4\rangle$ are not entangled. In other cases, the evolution of the system will be determined by the appropriate entangled states. The transformation parameters are found from the relations

$$tan2\vartheta ={\displaystyle \frac{3\sqrt{3}\left({\Omega }_Q\pm \frac{1}{2}{\Omega }_d\right)sin2\theta }{(3{\Omega }_Q\pm {\Omega }_d)(3{cos}^2\theta -1)-2\zeta {\omega }_1}}.$$

The expression for the derivative of the transformation parameter is rather cumbersome:

$$\dot{\vartheta }={\displaystyle \frac{3\sqrt{3}\left({\Omega }_Q\pm \frac{1}{2}{\Omega }_d\right)sin2\theta }{{\left(3\sqrt{3}\left({\Omega }_Q\pm \frac{1}{2}{\Omega }_d\right)sin2\theta \right)}^2+{\left((3{\Omega }_Q\pm {\Omega }_d)(3{cos}^2\theta -1)-2\zeta {\omega }_1\right)}^2}}\zeta {\dot{\omega }}_1.$$

Unfortunately, the considered example can be implemented for a nuclear system only in rare cases, for example, the parameter $\zeta \approx 0.1$ for hydrogen and nitrogen. For most nuclei, this parameter is not as small and it is not possible to observe an ideal picture. In particular, for the isotopes ${}^{13}$C and ${}^{14}$N, this ratio is $\zeta \approx 1/3$. Nevertheless, this approach qualitatively describes the evolution of states in an adiabatically varying external magnetic field. In particular, the situation becomes more interesting for paramagnetic centers interacting with nuclear magnetic moments in the lattice when the ratio $\zeta \sim {10}^{-3}$. However, it is necessary to take into account the possible existence of more significant interactions.

5. Conclusions

In this article, we have considered the adiabatic representation for describing the evolution of states of a quantum system in slowly varying external fields. We have shown that this representation allows us to clearly describe the time evolution of the states of multilevel systems when the energy levels of the system are close to each other at certain values of the external field. We have illustrated the application of the adiabatic representation using two simple but clear examples of two interacting spin systems. An exact analytical solution for the evolution operator $U(t,t_0)$ of spin systems in time-dependent magnetic fields is relevant for solving the Liouville equation for the density matrix. Our results continue the latest research on obtaining exact solutions for the SU(2) evolutionary operator [21,26,27,28,29,30,31]. Note that the adiabatic representation is very suitable if we cannot use the standard interaction representation, when the Hamiltonian has the form of the sum $\widehat{H}={\widehat{H}}_0+\widehat{V}\left(t\right)$ and the unperturbed Hamiltonian ${\widehat{H}}_0$ does not depend on time. In this case, the introduced representation reduces the problem to solving the stationary Schrödinger equation and finding a simpler evolution operator of the modified equation. The adiabatic representation using a time-dependent basis can be applied to non-adiabatically varying external fields or other parameters, similar to the perturbation decomposition provided in [18]. It will be interesting to apply the introduced representation to solve another urgent problem of shortcuts to adiabaticity (STA) [8,32,33,34,35], which uses the time scaling transformation of the Hamiltonian. Obviously, in this case the basic Equation (7) should be changed for the appropriate transformation.

Notably, our results are not only applicable to spin systems. The introduced representation can be successfully applied to quantum systems consisting of several subsystems when it is convenient to describe the states of the system in a separable basis; for example, spin Hamiltonian models can describe the quantum dynamics of two electron spins in a double quantum dot [36,37,38,39,40] and in a double quantum well [41,42]. The adiabatic representation also makes it possible to clearly explain and describe the formation and evolution of entangled states in complex systems when external fields or the environment change. Our results can also be useful for describing the evolution of the states of trapped ions. Currently, the use of trapped ions is being actively studied for the implementation of quantum algorithms with processors based on trapped ions ( for example, see [43,44]).

The obtained evolution operator allows us to calculate the evolution of the density matrix of the system and obtain all the necessary average values. The evolution operator is also necessary when it is necessary to take relaxation processes into account.