Analytic Solutions to Coherent Control of the Dirac Equation

TL;DR

This paper introduces an analytic framework for controlling Dirac spinors, enabling precise manipulation of quantum states and electromagnetic fields, with potential applications in quantum technology development.

Contribution

It presents a novel, simple parametrization of Dirac spinors that allows for analytic construction of electromagnetic fields to achieve desired quantum evolutions.

Findings

Achieves dispersionless rotation and translation of wave packets.

Reveals flexibility of the Dirac equation for quantum control.

Enables control interactions beyond electromagnetic fields.

Abstract

A simple framework for Dirac spinors is developed that parametrizes admissible quantum dynamics and also analytically constructs electromagnetic fields, obeying Maxwell's equations, which yield a desired evolution. In particular, we show how to achieve dispersionless rotation and translation of wave packets. Additionally, this formalism can handle control interactions beyond electromagnetic. This work reveals unexpected flexibility of the Dirac equation for control applications, which may open new prospects for quantum technologies.

Equations

$${\gamma }^{\mu }[ic\hslash {\partial }_{\mu }-ce{A}_{\mu }]\psi =m{c}^2\psi ,$$

$$\psi =\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_4\end{array}\right)⟺\{\genfrac{}{}{0ex}{}{\Psi =\left(\begin{array}{cc}{\psi }_1+{\psi }_3& -{\psi }_2^{\ast }+{\psi }_4^{\ast }\\ {\psi }_2+{\psi }_4& {\psi }_1^{\ast }-{\psi }_3^{\ast }\end{array}\right),}{\bar{\Psi }=\left(\begin{array}{cc}{\psi }_1^{\ast }-{\psi }_3^{\ast }& {\psi }_2^{\ast }-{\psi }_4^{\ast }\\ -{\psi }_2-{\psi }_4& {\psi }_1+{\psi }_3\end{array}\right).}$$

$$ic\hslash \bar{\partial }\Psi {\sigma }_3-ce\bar{A}\Psi -m{c}^2{\bar{\Psi }}^{†}=0,$$

$$ce\bar{A}=\left(ic\hslash \bar{\partial }\Psi {\sigma }_3-m{c}^2{\bar{\Psi }}^{†}\right){\Psi }^{-1}.$$

$$\Psi =\sqrt{\rho }BR{e}^{i\beta /2}.$$

$$B=B(\mathbf{u})=\frac{u^{\mu }{\sigma }_{\mu }+1}{\sqrt{2(1+u^0)}},$$

$$R=R(\mathbf{\theta })=\exp \left(-i{\theta }^k{\sigma }_k/2\right).$$

$$\Psi$$

$$\hslash {\omega }_0=m{c}^2-\sqrt{(m{c}^2{)}^2+2e{B}_0{c}^2\hslash },$$

$$\left|\mathbf{E}-\underset{\hslash \to 0}{\lim }\mathbf{E}\right|=\frac{\gamma r_0{\omega }^3}{e}\frac{\hslash }{2c^2}$$

$$\Psi$$

$$e{E}_1=\left(\underset{\hslash \to 0}{\lim }e{E}_1\right)-\frac{\hslash }{2c^2}\frac{d}{dt}\left(\gamma \stackrel{..}{Y}\right),$$

$$\Psi =e^{i\arcsin [f^{\prime }(z)]/2}{e}^{-\frac{e{B}_0(x^2+y^2)}{4\hslash }-mcf(z)/\hslash }{e}^{-iϵt{\sigma }_3/\hslash },$$

$$e{A}_0=\frac{2mc(f^{\prime }(z{)}^2-1)-\hslash f^{\prime \prime }(z)}{2\sqrt{1-f^{\prime }(z{)}^2}},$$

$$e{A}_0=-\frac{\xi mc}{\sqrt{{\xi }^2+z^2}}-\frac{\xi \hslash }{2({\xi }^2+z^2)}.$$

$$c{\gamma }^{\mu }{\hat{p}}_{\mu }\psi =(m{c}^2+V+\kappa |\psi {|}^2)\psi .$$

$$\Psi =e^{i\pi /4}{e}^{-mcz/\hslash }{e}^{-mc{z}^2/(\xi \hslash )}{e}^{-iϵt/\hslash {\sigma }_3},$$

$$\psi =\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_4\end{array}\right)⟺\Psi =\left(\begin{array}{cccc}{\psi }_1& -{\psi }_2^{\ast }& {\psi }_3& {\psi }_4^{\ast }\\ {\psi }_2& {\psi }_1^{\ast }& {\psi }_4& -{\psi }_3^{\ast }\\ {\psi }_3& {\psi }_4^{\ast }& {\psi }_1& -{\psi }_2^{\ast }\\ {\psi }_4& -{\psi }_3^{\ast }& {\psi }_2& {\psi }_1^{\ast }\end{array}\right),$$

$$\hslash c\partial /\Psi {\gamma }^2{\gamma }^1-ceA/\Psi -m{c}^2\Psi {\gamma }^0=0.$$

$$\frac{1}{2}({\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu })$$

$$({\gamma }^0{)}^{†}$$

$$({\gamma }^k{)}^{†}$$

$${\gamma }^{\mu }$$

$${\gamma }^0$$

$${\gamma }^j$$

$${\gamma }^0$$

$${\gamma }^j$$

$$\Psi =BR\in Spi{n}_{+}(1,3),$$

$$\Psi =\exp \left(-i{\gamma }^5\beta /2\right)RB\in Spin(1,3),$$

$$\Psi =\sqrt{\rho }\exp \left(-i{\gamma }^5\beta /2\right)RB.$$

$$\psi =\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\\ {\psi }_4\end{array}\right)⟺\Psi =\left(\begin{array}{cc}{\psi }_1+{\psi }_3& -{\psi }_2^{\ast }+{\psi }_4^{\ast }\\ {\psi }_2+{\psi }_4& {\psi }_1^{\ast }-{\psi }_3^{\ast }\end{array}\right),$$

$$ic\hslash \bar{\partial }\Psi {\sigma }_3-ce\bar{A}\Psi -m{c}^2{\bar{\Psi }}^{†}=0,$$

Full text

Introduction. The common aim of quantum control is to find a tailored external electromagnetic field to steer the ensuing dynamics in a desired fashion Brif et al. (2010). This capability, in particular, is enabling quantum technologies with the prospect of revolutionizing metrology, information processing, and matter manipulation. However, little is known about the control of the Dirac equation in spite of its modern applications reaching into nearly every domain of physics, going far beyond its original intention Greiner (2000); Bagrov and Gitman (2014). For example, lasers have already reached intensities where light-matter interactions must be described within the Dirac theory Di Piazza et al. (2012). Studies of the properties of heavy elements led to the establishment of relativistic quantum chemistry Grant (2007); Autschbach (2012); Schwerdtfeger et al. (2015); Pašteka et al. (2017) based on the Dirac equation. Moreover, there is a growing list of low energy systems emulating Dirac fermions in solids Novoselov et al. (2005); Katsnelson et al. (2006); Hasan and Kane (2010), optics Otterbach et al. (2009); Ahrens et al. (2015), cold atoms Boada et al. (2011); Suchet et al. (2016), trapped ions Gerritsma et al. (2010); Blatt and Roos (2012), and circuit quantum electrodynamics Pedernales et al. (2013).

The Dirac equation is commonly expressed as Greiner (2000)

[TABLE]

where the summation over repeated indices is adopted, $\psi$ is a four-component complex spinor, $m$ is the mass, $c$ is the speed of light, $\gamma^{\mu}$ are the $4\times 4$ so-called gamma matrices, $A_{\mu}$ is the four-vector potential and $\mu=0,1,2,3$.

The Dirac equation (1) can be viewed as a “first quantization” approximation to QED. The solutions of Eq. (1) exclude effects such as radiation reaction and particle creation/annihilation prominent at ultrarelativistic energies. Nevertheless, Eq. (1) provides a mean-field description of relativistic effects at low and moderate energies. A moving Dirac electron generates the current $J_{D}^{\mu}=\psi^{\dagger}\gamma^{0}\gamma^{\mu}\psi$ that emits secondary radiation, which is not accounted for by Eq. (1). Therefore, a solution of the Dirac equation is physical if the energy loss due to the secondary radiation is much smaller than the electron kinetic energy. This criterion should be satisfied in the applications of the Dirac equation to quantum control.

In this Letter we present the framework of Relativistic Dynamical Inversion (RDI) opening up a new route to coherent control for the Dirac dynamics: Given a desired wave packet evolution, we analytically design electromagnetic control fields obeying Maxwell equations. This should be compared with other techniques such as shortcuts to adiabaticity Deffner (2015); Song et al. (2016); Deffner et al. (2014) analytically constructing interactions that often go beyond electromagnetic fields.

The purpose of the current work is to solve the following problem: Given an arbitrary (desired) spinorial spacetime wave packet $\psi$, find an electromagnetic field $A_{\mu}$ such that Eq. (1) is satisfied. This is accomplished by RDI in two steps: First, we verify the attainability of the given evolution $\psi$ by assessing the existence of the underlying $A_{\mu}$ leading to valid Maxwell equations; second, if it exists, an explicit form of $A_{\mu}$ is obtained. Moreover, the method can also be used to assess for attainable dynamics.

The task of constructing the control field yielding the desired dynamics at all times and positions is one of the most important and challenging problems in quantum control. In particular, transporting coherent wave packets without disturbance is a required building block in quantum technologies. RDI allows for finding analytic solutions not feasible by other current methods. This is possible due to unique properties of the Dirac equation.

Exact solutions of Eq. (1), a system of four partial differential equations, are rare. The vast majority of them are for highly symmetric stationary systems Thaller (2013); Bagrov and Gitman (2014); Eleuch et al. (2012). Furthermore, finding exact solutions with probability densities having finite integrals over the whole three dimensional space is a formidable task. Only a handful of solutions for time dependent dynamics exist Varró (2013); Bialynicki-Birula (2004); Oertel and Schützhold (2015); Kaminer et al. (2015); Hayrapetyan et al. (2014); Bialynicki-Birula and Bialynicka-Birula (2017); Barnett (2017); Heinzl and Ilderton (2017). Most of the investigations call for either semiclassical methods Lazur et al. (2005) or numerical calculations Braun et al. (1999); Mocken and Keitel (2008); Bauke and Keitel (2011); Fillion-Gourdeau et al. (2012, 2016); Lv et al. (2016); Cabrera et al. (2016). In addition to being computationally demanding, commonly used numerical schemes are plagued by unphysical artifacts at the fundamental level Hammer and Pötz (2014); Hammer et al. (2014); thus, there is a need for systematic construction of analytic solutions. RDI fulfills all these needs by providing stationary as well as time-dependent exact solutions integrable in two and three dimensions.

RDI simultaneously seeks the state $\psi$ and the vector potential $A_{\mu}$ describing physically admissible dynamics. Considering that Eq. (1) is * bilinear* with respect to both $\psi$ and $A_{\mu}$, it may seem that the proposed approach is even more challenging than solving the linear Dirac equation for $\psi$. Nevertheless, the following four elements make RDI much simpler than the traditional methods: (i) The Dirac equation is written in the form where both $\psi$ and $A_{\mu}$ are $2\times 2$ complex matrices Baylis (1992, 1996). (ii) The cross term responsible for the bilinearity is eliminated by expressing the vector potential as an explicit function of the state. (iii) The physical consistency of the state is accomplished by demanding the Hermiticity of the vector potential expressed in matrix form. (iv) Enforcing the Lorentz covariance by decomposing the state into spacetime rotations as well as a transformation of the internal degrees of freedom significantly reduces the complexity of the analytic derivations.

Methodology of Relativistic Dynamical Inversion. The Dirac equation (1) can be written in different forms emphasizing the geometry of the Lorentz group Hestenes (1967, 1973, 1975, 2003, 2010); Lounesto (2001); Doran and Lasenby (2003); Baylis (1992, 1996). Here, we employ the Baylis formulation Baylis (1992, 1996); Baylis and Yao (1999); Baylis (1999); Baylis et al. (2010) (see also Sec. I of the Appendix) where the state $\psi$ in Eq. (1) is represented by the matrix $\Psi$ and its Clifford conjugate $\bar{\Psi}$,

[TABLE]

obeying the Dirac equation in the matrix form Baylis (1992, 1996)

[TABLE]

where $\bar{A}=\sigma_{\mu}A_{\mu}$, $\bar{\partial}=\sigma_{\mu}\partial_{\mu}$, $\sigma_{0}=\boldsymbol{1}$ is an identity matrix, $\sigma_{1,2,3}$ are Pauli matrices. Note that $\bar{A}$ must be a Hermitian matrix by construction. According to Ref. Lounesto (2001), $\det\Psi=0$ for the Majorana and Weyl fermions as well as for the flag-dipole spinors, whereas $\det\Psi\neq 0$ for electrons/positrons. Thus, in the latter case, the vector potential may be expressed as a function of the state

[TABLE]

A crucial insight is the spinor factorization for electrons/positrons: $\Psi=\sqrt{\rho}L$, where $\rho$ is a non-negative scalar function modulating the probability density 111 Stationary solutions of the Dirac equation rarely have nodes; as a result, they cannot be used to classify the eigensolutions. For example, the hydrogen atom eigenstates have no zeros in $\rho$ (except at the origin) Grant (2007); likewise, nodes in the Landau levels for the Dirac equation are hard to come across.

and $L$ is an invertible matrix representing a Lorentz group element Hestenes (1967, 1973, 1975).

Considering that a member $L$ of the special Lorentz group Hestenes (1967, 1973, 1975) is composed of spatial rotations $R$, a boost $B$ and a transformation of internal degrees of freedom generated by the Yvon-Takabayashi angle $\beta$ Yvon (1940); Takabayasi (1957), the state can be factorized as Hestenes (1967, 1973, 1975); Baylis (1996)

[TABLE]

The boost $B$ is parametrized by the velocity components $c\mathbf{u}=c(u^{1},u^{2},u^{3})$ (bold symbols denote three dimensional vectors throughout)

[TABLE]

with $u^{0}=\sqrt{1+\mathbf{u}^{2}}$; whereas, the spatial rotations are parametrized by the angles $\boldsymbol{\theta}=(\theta^{1},\theta^{2},\theta^{3})$

[TABLE]

Note that the density $\rho$, velocity $\mathbf{u}$, rotation angle $\boldsymbol{\theta}$, and Yvon-Takabayashi angle $\beta$ are in general functions of space and time.

RDI is performed in the following way: Spacetime functions $\rho$, $\mathbf{u}$, $\boldsymbol{\theta}$, and $\beta$ are initially selected to describe a desired dynamics of the Dirac state $\Psi$. The constructed factorization (3) is substituted in Eq. (2) to obtain the vector potential in the matrix form $\bar{A}$.

If $\bar{A}$ is not Hermitian, the proposed dynamics is not reachable with physical fields, and the parametrization $\rho$, $\mathbf{u}$, $\boldsymbol{\theta}$, and $\beta$ needs to be modified.

If $\bar{A}$ is Hermitian, then the procedure is completed: The obtained vector potential $A_{\mu}={\rm Tr}\,(\bar{A}\sigma_{\mu})/2$ enables to recover the electromagnetic fields $F^{\mu\nu}=c\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right)$ and the source $J^{\nu}=\partial_{\mu}F^{\mu\nu}/(\varepsilon_{0}c)$ generating them. Provided the current $J^{\nu}$, the obtained fields $F^{\mu\nu}$ necessarily satisfy Maxwell’s equations. Note that $J^{\nu}$ differs from the current $J_{D}^{\mu}={\rm Tr}\,(\Psi\Psi^{\dagger}\sigma_{\mu})=\psi^{\dagger}\gamma^{0}\gamma^{\mu}\psi$ emanating from the Dirac equation.

RDI is a trial-and-error procedure to find a suitable parametrization $\rho$, $\mathbf{u}$, $\boldsymbol{\theta}$, $\beta$ of the desired dynamics to yield a pair $A_{\mu}$, $\Psi$ analytically satisfying the Dirac equation. In a general case, the obtained $A_{\mu}$ may have a complicated temporal and special profile hard to implement experimentally.

Furthermore, RDI has a very general foundation, which is applicable to interactions beyond electromagnetic, e.g., non-linear Dirac equations and scalar interactions coupling through the mass ($mc^{2}\rightarrow mc^{2}+V$) as shown below. The inversion procedures in Refs. Krüger (1993); Oertel and Schützhold (2015) can be viewed as specialized cases of RDI.

Dispersionless rotation. We now find an electromagnetic field that moves a Gaussian wave packet along a circular trajectory in the $x-y$ plane without distortion. Since the center of the wave packet should follow the trajectory $\mathbf{r}(t)=r_{0}(\cos\omega t,\sin\omega t,0)$, the desired state evolution is

[TABLE]

where $\mathbf{u}=\dot{\mathbf{r}}/\sqrt{1-(\dot{\mathbf{r}}/c)^{2}}$ and the values of $r_{0}$ and $\omega$ must be selected such that $r_{0}\omega<c$ to avoid superluminal propagation. According to RDI, the vector potential generating the dynamics consists of a constant homogeneous magnetic field $B_{0}$ perpendicular to a planar electric field with a spatial and temporal profile. However, for the frequency

[TABLE]

the electric field acquires a fixed spatial configuration rotating in time. This expression for $\omega_{0}$ can be regarded as the cyclotron frequency corrected for quantum effects (see Sec. III of the Appendix).

In Fig. 1, the crossed circles represent the homogeneous magnetic field perpendicular to the plane, and the electric field at the initial time $t=0$ is displayed as blue arrows in the $x-y$ plane. According to Sec. III of the Appendix, these electromagnetic fields satisfy Maxwell’s equations with an electric current but without free charges. The black diffused circle (centered at $x=2\mu\mbox{m}$ and $y=0$) depicts the initial Gaussian [Eq. (55)] state whose shape is preserved during the rotation along the gray circular arrow.

The nonrelativistic limit $c\to\infty$ of the driving controls consist of the homogeneous magnetic field $B_{0}$ and the circularly polarized electric field: $-(r_{0}\omega/e)\left(\left(eB_{0}+m\omega\right)\cos\omega t,\left(eB_{0}+m\omega\right)\sin\omega t\right)$. This setup can be shown to preserve the Gaussian shape within the Schrödinger equation.

As shown in Appendix III C, the magnetic field is unaltered in the classical limit $\hbar\rightarrow 0$; whereas, the vector norm difference between the exact electric field $\mathbf{E}$ and its classical limit reads

[TABLE]

where $\gamma=[1-(r_{0}\omega/c)^{2}]^{-1/2}$ is the Lorentz factor. This reveals that quantum effects are enhanced by relativistic dynamics. The spatial inhomogeneity in the exact electric field depicted in Fig. 1 is due to spin-orbit coupling, which is simultaneously a relativistic and quantum effect. Note that this dynamics can be observed at experimentally available values of $B_{0}=0.35$T and $|\mathbf{E}|\sim 0.3$V/m employed in Fig. 1. In such a regime, the synchrotron radiation energy loss per cycle is infinitesimally (i.e., $11$ orders of magnitude) smaller than the electron’s kinetic energy. Therefore, the obtained solutions satisfy the physicality criterion.

Dispersionless translation. We now apply RDI to achieve a spatial translation of a wave packet without changing its initial shape. For example, consider the translation along the $y$ axis with the trajectory $Y(t)$. Calculating the proper velocity $u$ from $\mathbf{r}(t)=(0,Y(t),0)$, we apply RDI to the dynamics $\Psi=e^{-\frac{eB_{0}x^{2}}{4\hbar}}g(t,y)B(\mathbf{u}).$ It turns out that physical fields exist only if $g(t,y)=G(y-Y(t))/\sqrt{u^{0}(t)}$ for an arbitrary function $G(y)$. In particular, the translation of the Gaussian

[TABLE]

results in the electromagnetic field composed of a time dependent homogeneous magnetic field and an electric field with temporal and spatial dependence given in Sec. IV of the Appendix. For the specific trajectory $Y(t)=(L/2)[1+\sin(\pi(t-T/2)/T)]$ for $0\leq t\leq T$, Fig. 2 displays two snapshots of the electric field at the beginning of motion [Fig. 2(a)] and at the middle [Fig. 2(b)].

In the nonrelativistic limit $c\to\infty$ the driving control is made of a constant magnetic field $B_{0}$ along $z$ and a time dependent electric field exclusively directed along the trajectory as dictated by Newton’s law $eE_{2}=mY^{\prime\prime}(t)$.

As elaborated in Sec. IV of the Appendix, the classical limit $\hbar\to 0$ affects neither the magnetic field nor the electric field along the direction of motion. However, the exact component of the electric field perpendicular to the direction of motion can be written as

[TABLE]

where $\gamma=[1-(\dot{Y}/c)^{2}]^{-1/2}$ is the Lorentz factor. This quantum correction resembles the Abraham-Lorentz force describing the interaction of a charged particle with its own electromagnetic field. Similar to the dispersionless rotation discussed above, quantum effects are enhanced by the relativistic dynamics. The counterintuitive temporal and spatial structure of the control shown in Fig. 2(b) is a manifestation of strong relativistic spin effects even at weak electric ($|\mathbf{E}|\sim 10^{6}$ V/m) and magnetic ($B_{0}\sim 1$T) fields. In this case, the bremsstrahlung energy loss is negligible compare to the electron’s kinetic energy.

Integrable three dimensional solutions. Having demonstrated the RDI’s ability to synthesize dynamics in two spatial dimensions, we now turn to a challenging three dimensional case. For the following confined stationary state

[TABLE]

RDI uncovers the underlying constant homogeneous magnetic field $B_{0}$ along the $z$ direction and the static electric potential

[TABLE]

where the energy of the state (11) is set to $\epsilon=0$, and $f(z)$ is an arbitrary real function. The obtained potential has no nonrelativistic limit.

A noteworthy feature of the state (11) is the spatial dependence of the Yvon-Takabayashi angle $\beta=\arcsin f^{\prime}(z)$, which is a signature of antiparticles represented by negative energy components in a wave packet (see, e.g., page 275 of Ref. Baylis (1996)). The values of $\beta$ lie between $-\pi$ and $+\pi$, where particles (i.e., positive energy) and antiparticles are associated with $\beta=0$ and $\beta=\pm\pi$, respectively. From the point of view of Lorentz transformations, the Yvon-Takabayashi angle is a degree of freedom corresponding to the CPT conjugation Lounesto (2001) that includes the time inversion $t\to-t$; hence, $\beta$ is a parameter in the special Lorentz group not available in the restricted Lorentz group. Moreover, this degree of freedom is absent from the nonrelativistic Pauli-Schrödinger theory. Since $f(z)$ controls the density of the state in Eq. (11), the tighter the confinement along the $z$ axis, the higher the contribution of antiparticles.

In the particular case of $f(z)=\sqrt{\xi^{2}+z^{2}}$, where $\xi$ determines the density spreading in $z$, the confining static electric potential is the sum of soft-core Coulomb and short range potentials

[TABLE]

In Sec. V of the Appendix, the space and time dependent electromagnetic fields are obtained by RDI to yield the dispersionless rotation of the state (11).

Exact solutions beyond electromagnetic interactions. RDI is not restricted to the electromagnetic interactions. The Dirac equation describing the scalar field $V$ coupled to the mass is $c\gamma^{\mu}\hat{p}_{\mu}\psi=(mc^{2}+V)\psi$. This equation describes a Fermion in gravitational fields Jentschura and Noble (2014), topological materials Shen (2013), and quark models Chodos et al. (1974); Ru-keng and Yuhong (1984). Another generalization of the Dirac equation involves nonlinear interactions Thirring (1958); Soler (1970), which can also be used to model Bose-Einstein condensates Merkl et al. (2010).

Let us consider the following nonlinear interaction with unspecified $V$

[TABLE]

Applying RDI to the following state

[TABLE]

we find the scalar interaction $V=2mc^{2}z/\xi-\kappa\sqrt{\frac{2mc}{\pi\xi\hbar}}e^{-mc(2z+\xi)^{2}/(2\xi\hbar)}$ by demanding the absence of electromagnetic fields. Note that the state $\psi$ is confined in the potential $V$ unbounded from above and below and, even more surprisingly, in the presence of an additional repulsive force emanating from the nonlinear term. This is not possible in the nonrelativistic limit. Another example is presented in Sec. VI of the Appendix.

These cases extend a rather short list of analytic solutions of the Dirac equation with scalar interaction Hiller (2002); de Castro (2003); de Castro and Hott (2005). Further explorations reveal that RDI becomes more flexible by utilizing both scalar and electromagnetic interactions, opening new possibilities for controlling quantum dynamics.

Outlook. We have developed RDI, a new framework for analytically constructing electromagnetic fields controlling the dynamics of the Dirac equation. RDI has also been shown to be a flexible tool for discovering novel exact solutions. In particular, we have shown how relativistic coherent states could be constructed experimentally. A scalar interaction coupled to the mass has been incorporated into RDI. This opens up prospects for quantum technologies in new realms of physics and may further expand the scope of control landscape analysis Rabitz et al. (2004).

Since RDI relies on the matrix representation of the dynamical group generated by an equation of motion, the developed methodology may also be adaptable to other dynamical equations Yepez (2016). In a similar fashion, RDI may be used to yield exact solutions for non-Abelian fermions in the standard model Trayling and Baylis (2001) as well as curved spaces Rodrigues and de Oliveira (2007); Fabbri (2017) that are currently intractable.

Acknowledgments. We thank two anonymous referees for a number of valuable suggestions. A.G.C. acknowledges financial support from NSF CHE 1464569, D.I.B. from DOE DE-FG02-02-ER-15344, R. C. from ARO W911NF-16-1-0014 and H.R. from Templeton Foundation 52265. A.G.C. was also supported by the Fulbright Foundation. D.I.B. is also supported by AFOSR Young Investigator Research Program (FA9550-16-1-0254).

A.G.C. and R.C. contributed equally to this work.

Appendix A I: Matrix form of the Dirac equation

The traditional Dirac column spinor $\psi$ can be expressed in terms of $4\times 4$ complex matrices $\Psi$. In particular, employing the Dirac matrix representation we have

[TABLE]

where $\Psi$ belongs to the group $Spin(1,3)$, as the double cover of the Special Lorentz group (at each point in the spacetime). The spinor operator $\Psi$ obeys the Dirac-Hestenes equation Hestenes (1967, 1973, 1975, 2003, 2010); Lounesto (2001); Doran and Lasenby (2003)

[TABLE]

The Feynman slash notation is employed $A\!\!\!/=A^{\mu}\gamma_{\mu}$, $\partial\!\!\!/=\gamma^{\mu}\partial_{\mu}$ ($\mu,\nu=0,1,2,3$), where the gamma matrices $\gamma^{\mu}$ are constructed to contain the Minkowski metric $g^{\mu\nu}$ according to

[TABLE]

Other important properties of the gamma matrices are

[TABLE]

Among the infinite possibilities, the Dirac matrix representation is built in terms of the Kronecker product of Pauli matrices $\sigma_{\mu}$

[TABLE]

where $\sigma_{0}=\boldsymbol{1}$ and $j=1,2,3$. The Weyl representation is another possibility

[TABLE]

The group $Spin_{+}(1,3)$ is defined as the double cover of the restricted Lorentz group and is characterized by preserving the direction of time while avoiding spatial reflections. A group element can be decomposed as

[TABLE]

where $R$ and $B$ are unitary and Hermitian matrices, respectively. The matrix $R$ induces spatial rotations while $B$ performs Lorentz boosts.

The double cover of the special Lorentz group (allowing for $CPT$ conjugation) contains the additional factor $\exp\left(-i\gamma^{5}\beta/2\right)$

[TABLE]

where $\beta$ is the Yvon-Takabayashi angle Yvon (1940); Takabayasi (1957). The solutions of the Dirac equation (16) are spacetime modulations of $Spin(1,3)$ carried out by a non-negative scalar function $\rho$ according to

[TABLE]

Alternative to the Dirac-Hestenes formulation, Dirac spinors can be expressed in terms of $2\times 2$ complex matrices

[TABLE]

obeying the Dirac-Baylis equation Baylis (1992, 1996)

[TABLE]

where $\sigma_{3}$ is a Pauli matrix that defines the arbitrary initial spin direction and the overbar is the Clifford conjugation that applied to a matrix $L$ gives

[TABLE]

The gradient operator $\bar{\partial}$ and the vector potential $\bar{A}$ are

[TABLE]

where the physical components of the vector potential are given in contravariant indexes $A^{\mu}$ such that $A^{0}=A_{0}$ and $A^{k}=-A_{k}$.

Both Dirac-Hestenes and Dirac-Baylis approaches are completely equivalent.

A delocalized free particle with zero mean momentum and spin along $z$ can be represented as

[TABLE]

where the spinor subscript indicates an specific reference frame that is taken at rest.

The spacetime position $x$ is

[TABLE]

The unitless proper velocity $u$ is defined as

[TABLE]

where $\tau$ is the proper time (time in the rest frame attached to the particle). Correspondingly, the proper velocity can be expressed as

[TABLE]

where the shell mass condition must be imposed by writing $u^{0}=\sqrt{1+(u^{1})^{2}+(u^{3})^{2}+(u^{3})^{2}}=\sqrt{1+\mathbf{u}^{2}}$, effectively reducing the degrees of freedom of the proper velocity to three. A state with a net velocity can be obtained through an active Lorentz transformation carried out by a Lorentz boost given by

[TABLE]

which can be written as

[TABLE]

For example, the spinor operator corresponding to a Lorentz boost along the $z$ direction can be expressed as

[TABLE]

with $u^{0}=\sqrt{1+(u^{3})^{2}}$. The active Lorentz transformation of the coordinates is carried out by the following formula

[TABLE]

leading to

[TABLE]

A state $\Psi(t,z)$ is actively boosted to $\Psi^{\prime}(t^{\prime},z^{\prime})$ according the spinorial transformation law that in the particular case reads

[TABLE]

where we must emphasize that the final transformed spinor $\Psi^{\prime}$ must be expressed in terms of the boosted spacetime coordinates $t^{\prime}$ and $z^{\prime}$ according to Eq. (41). Thus, the state in Eq. (33) boosted along $z$ is

[TABLE]

Considering that the Dirac equation is covariant under (homogeneous) restricted Lorentz transformations, $\Psi^{\prime}$ must satisfy the Dirac equation.

Appendix B II: Time-dependent boost of a Gaussian state

Let us initially consider the Landau ground state for a constant homogeneous magnetic field $\mathbf{B}=(0,0,B_{0})$

[TABLE]

and a constant Lorentz boost along the $y$ direction with

[TABLE]

The ground state is boosted to

[TABLE]

Applying RDI we obtain the fields

[TABLE]

Independently, we can verify that the Lorentz transformations of the fields are

[TABLE]

Thus, demonstrating the covariance of the Dirac equation under (homogeneous) Lorentz transformations.

The Dirac equation is covariant for homogeneous Lorentz transformations that do not depend on time or position. Nevertheless, a non-homogeneous Lorentz transformed spinor can still statisfy the Dirac equation but for a properly designed electromagnetic field. Let us illustrate this situation by analyzing the action of a time dependent Lorentz transform that corresponds to the action of a constant electric field $E_{0}$

[TABLE]

That induces the following transformation of coordinates to the rest frame

[TABLE]

Applying this Lorentz boost to the Gaussian spinor at rest in Eq. (45) we get

[TABLE]

Ignoring the primes in the variables, RDI provides

[TABLE]

with $\text{Q}=c^{6}m^{4}+c^{4}{eE_{0}}^{2}m^{2}t^{2}+c^{2}{eE_{0}}^{4}t^{4}-2c{eE_{0}}^{3}t^{2}y\sqrt{c^{2}m^{2}+{eE_{0}}^{2}t^{2}}+{eE_{0}}^{4}t^{2}y^{2}$ and $R=\sqrt{(mc)^{2}+(eEt)^{2}}$. The configuration of this vector potential is evidently not the Lorentz transform of the initial reference vector potential. This becomes more obvious by observing that the complicated vector potential depends on the internal degrees of freedom of the state and on $\hbar$.

Appendix C III: Dispersionless rotation: Gaussian state in 2D

The state that describes a dispersionaless rotation of a Gaussian state is [Eq. (6) in the main text]

[TABLE]

According to RDI, the electromagnetic vector potential is

[TABLE]

The corresponding electric field is (see Fig. 4)

[TABLE]

The magnetic field is

[TABLE]

The obtained electromagnetic field obeys Maxwell’s equations

[TABLE]

with the current $\mathbf{J}$

[TABLE]

The current amplitude

[TABLE]

becomes time independent if the condition $2c(eB_{0}+cm\omega)-\omega^{2}\hbar=0$ is employed. The frequency accomplishing this is

[TABLE]

The following series expansion indicates that this contains quantum corrections to the proper cyclotron frequency $\omega_{c}=eB_{0}/m$

[TABLE]

In classical mechanics, the cyclotron frequency $\omega$ and the proper cyclotron frequency $\omega_{c}$ are related by $\omega=\omega_{c}/\gamma$, thus

[TABLE]

The low energy expansion of the current at the resonant frequency (70) is

[TABLE]

The non-relativistic limit of the electromagnetic field (56)-(61) is

[TABLE]

The classical limit $\hbar\rightarrow 0$ of the electric field (56)-(58) reads

[TABLE]

The difference between the exact (56)-(61) and its classical limit (78)-(80) is

[TABLE]

The leading term in $\hbar$ can be expressed in terms of the resonant frequency (70)

[TABLE]

where $\omega_{1}=\frac{\omega_{c}^{2}}{2mc^{2}}\hbar$ is the first quantum correction to the cyclotron frequency in Eq. (71). Note that Eq. (82) can be interpreted as a Lorentz force $e\mathbf{v}\times\mathbf{B}.$

The current constructed from the Dirac spinors is $J_{D}=\Psi\Psi^{\dagger}$, whose components $J_{D}^{\mu}=Tr(\Psi\Psi^{\dagger}\sigma_{\mu})$ are

[TABLE]

The velocity associated with the current $J_{D}$ is obatined as $v^{k}=cJ_{D}^{k}/J_{D}^{0}$

[TABLE]

with the magnitude given by $|v|=r_{0}\omega$. Thus, superluminal propagation is avoided if $r_{0}\omega<c$. Furthermore, the latter inequality is also obtained from the RDI consistency condition for the electromagnetic fields as described in the main text.

The current $\mathbf{J}$ (66)-(68) entering Maxwell’s equations (62)-(65) is related to the current $\mathbf{J}_{D}$ (83)-(86) constructed from the Dirac spinor in the following way: The Maxwell current $\mathbf{J}$ creates the electromagnetic field (56)-(61) steering the Dirac wave packet (55). Moving along a circular trajectory, a Dirac electron yields the current $\mathbf{J}_{D}$ emitting synchrotron radiation. If the radiation losses are large, the proposed control protocol may not work. Therefore, the control electromagnetic fields are physically meaningful if the electron kinetic energy is much larger than the energy emitted via synchrotron radiation. The dispersionless rotation shown in Fig. 1 of the main text obeys well this criterion because the radiative energy loss per period is $\propto 10^{-32}$J, whereas the electron kinetic energy is $\propto 10^{-21}$J.

Appendix D IV: Dispersionless translation

The state undergoing dispersionless translation is [Eq. (9) of the main text]

[TABLE]

where $c\mathbf{u}={\{}0,cu^{2},0{\}}={\{}0,\dot{Y}/\sqrt{1-(\dot{Y}/c)^{2}},0{\}}$ and $u^{0}(t)=1/\sqrt{1-(\dot{Y}/c)^{2}}$. RDI leads to the following vector potential components

[TABLE]

The corresponding electromagnetic fields are

[TABLE]

where $\gamma$ is the Lorentz factor.

The nonrelativistic limit $c\to\infty$ is

[TABLE]

The classical limit $\hbar\to 0$ is

[TABLE]

This means that only the perpendicular component of the electric field includes quantum corrections. Moreover, it can be shown that

[TABLE]

The electric field for the trajectory employed in the main text is evaluated at the origin as a function of time and shown in Fig. 5.

The electromagnetic fields (90)-(93) obey Maxwell’s equations (62)-(65) with the following electric current:

[TABLE]

where $Q=\sqrt{c^{2}-(\dot{Y})^{2}}$. The leading terms of the electric current at low energies are homogenous in the space

[TABLE]

Figure 6 depicts parametrically the current (100)-(103) creating the fields (90)-(93) that realize the dispersionless translation shown in Fig. 2 of the main text.

An electron moving linearly with an acceleration emits bremsstrahlung. Similarly to the rotational dynamics discussed in Sec. C, the control electromagnetic fields (90)-(93) are physically meaningful if the electron kinetic energy is much larger than the bremsstrahlung energy loss. The dispersionless translation depicted in Fig. 2 of the main text satisfies well this criterion since the radiative energy loss does not exceed $10^{-38}$J while the electron kinetic energy $\propto 10^{-24}$J.

Appendix E V: An integrable three-dimensional solution

The rotation of the three-dimensional state in Eq. (11) of the main text state is carried out by

[TABLE]

where $B(\mathbf{u})$ is the same boost as in Eq. (55).

The vector potential according to RDI is

[TABLE]

with $W=c\sqrt{c^{2}-r_{0}^{2}\omega^{2}}+c^{2}-r_{0}^{2}\omega^{2}$.

Appendix F VI: Additional example with a scalar potential interaction

RDI is not restricted to the construction of electromagnetic fields as shown in the main text. In particular, we saw how an unbounded scalar potential can hold a stationary state. In this section we show another example for a scalar interaction bounded from below with the absence of electromagnetic interactions. Supplying the stationary state with the predefined Yvon-Takabayashi angle,

[TABLE]

the requirement of $A_{0}=0$ specifies both the scalar potential and an arbitrary function $f(z)$

[TABLE]

A plot of the potential (107) along with the state (106) is shown in Fig. 8.