Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
New Phys.: Sae Mulli 2021; 71: 1096-1104
Published online December 31, 2021 https://doi.org/10.3938/NPSM.71.1096
Copyright © New Physics: Sae Mulli.
Changsuk NOH*
Department of Physics, Kyungpook National University, Daegu 41566, Korea
Correspondence to:cnoh@knu.ac.kr
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The one-dimensional variable-mass Dirac equation is connected to various models used throughout many branches of physics. An analog simulation of the equation in a spinor slow light system allows experimental realizations of such models. This work concentrates on an interesting model of historical importance that led to a prediction of charge fractionalization, which in turn occurs due to the presence of a topologically protected zero-energy mode. After describing how the model can be realized in a spinor slow light system, the current work explains how the presence of the zero-energy mode can be verified from the dynamics of the spinor slow light.
Keywords: Dirac equation, Quantum optics, Dark state polaritons
The Dirac equation was the first successful attempt at merging quantum mechanics and relativity [1]. The relativistic wave equation predicted paradoxical effects such as
An analog simulation, or an emulation, of the Dirac equation was first proposed in a trapped ion system in 2007 [2]. The one-dimensional Dirac equation was implemented experimentally soon after, and both
Usefulness of the Dirac equation is not limited to the description of a fermionic particle. By allowing the mass term to vary in space (or even in time) the model’s applicability extends to many other areas of physics including condensed matter physics and nuclear physics. Pointing out such a connection is one of the aims of this work. For simplicity and for the ease of experimental implementation, we stick to the one-dimensional Dirac equation, which, despite the limitation, gives rise to plenty of interesting physics as we will see. Only one-dimensional models will be considered in the rest this work and the Dirac equation will always be in one-dimension.
A convenient starting point for a derivation of the onedimensional Dirac equation is the covariant form
where {
with a two-component spinor
This work provides brief descriptions of various models associated with the variable-mass Dirac equation and show how some of their predictions can be emulated using spinor slow light (SSL). Section II gives a gentle introduction to the so-called dark state polaritons (see Ref. [18] for a short pedagogical introduction), concentrating on how they can be coerced to form an SSL, which follows the dynamics of the Dirac equation. It also explains the preparation, evolution, and measurement stages of the emulation process and sets the stage for the rest of the work. Section III introduces the Jackiw- Rebbi model, whose implementation in an SSL system has already been proposed in [19]. Whereas the latter work concentrated on a transmission-and-reflection scenario, this work concentrates on a loading-and-releasing scenario. Section IV discusses three other models related to the variable mass Dirac equation, namely polyacetylene, the random mass Dirac model, and the Lorentz scalar potential.
That the so-called dark state polaritons (DSPs) can be manipulated to simulate the physics of the Dirac equation has been noticed in Refs. [10,20] and experimentally realized in [21]. The basic setup consists of atoms coupled to an effectively one-dimensional waveguide as illustrated schematically in Fig. 1(a). Atomic levels have a double-tripod structure as shown in Fig. 1(b). In order to understand this setup, let us first consider a simpler atomic structure of the lambda-type configuration depicted in Fig. 2(a). When a probe field resonant with the atomic transition |
The situation changes completely when a strong control field, driving the transition |
By introducing a counter-propagating control fields Ω_{1,2} to the above system as shown in Fig. 2(b), two counter propagating DSPs,
To obtain a spinor slow light, consisting of twocomponents, additional ground (or metastable) states are needed [10,20]. The double tripod setup proposed in [20] is shown in Fig. 1(b). Such a configuration yields two DSPs, Ψ_{1,2}, that obey an effective equation
where Ψ = (Ψ_{1},Ψ_{2})^{T} ,
In its essence, the Jackiw-Rebbi (JR) model is described by the Dirac equation with a specific type of position-dependent mass profile [25]. It was used by the authors to predict charge fractionalization, before fractional quantum Hall effect was discovered. Essentially the same effect has been found independently in studying the electron-phonon coupling in polyacetylene [26]. Before we delve into the details of the model, let us start with a brief remark on charge fractionalization, the details of which can be found in a review article [27].
The JR model arises as an approximation to the quantum field theory for an electron coupled to bosonic fields. The equations of motion for the bosonic fields allow soliton solutions when treated classically, which, in the approximate Dirac equation, acts as a position-dependent mass term with the soliton profile. A fractionalization of charge occurs in this model because it allows a zeroenergy mode, or a zero-mode for short. The ground state of the quantum field can either contain the zero-mode or not, and the charge conjugation symmetry dictates that these two degenerate ground states must have opposite charges. Now, because only one electron may occupy the zero-mode, the latter must have a fractional charge of
As noted above, charge fractionalization is ultimately connected to the presence of the zero-mode, which is topologically protected by global properties of the soliton field. While the polariton emulator cannot exhibit fractionalization explicitly because the DSP is not a fermionic field, the existence of the zero-mode and its topological robustness can be probed as shown in Ref. [19]. This section explains the basic theory of the zero-mode and describes how it can be observed in the spinor slow light system. While the original proposal has concentrated on optical transmission measurements [19], this work concentrates on the dynamics of the spinor.
In the JR model, a Dirac particle is coupled to a real scalar field
where
The ground state solutions are
which is localized around
where
Before we move on to the topic of emulation using SSL, a discussions on a few important aspects of the zero-energy solution is in order. Firstly, the zero-mode solution only depends on the value of
Due to the similarity between Eqs. (4) and (5), the SSL system follows the dynamics of the Dirac equation with an effective speed of light
In the first scheme, one utilizes the intrinsic nonequilibrium property of the quantum optical systems by driving the waveguide with a laser field [28] and observing the transmission spectrum [19]. Transmission spectrum shows a peak at zero energy, revealing the presence of the zero-mode.
The second scheme, which has been only briefly touched upon in [19], focuses on the time evolution of a spinor. To understand this scheme, let us assume that the zero-energy state has been prepared at time
Under time evolution, the initial Gaussian state exhibits completely different behavior depending on whether the mass term has a soliton profile or not. In Fig. 5(a) the mass term
Actually, this is not the whole story and Fig. 5(a) can be deceiving. For example, for
An experiment to verify the presence of the zero-mode can be run as follows. First, a Gaussian state is prepared initially. This can be achieved either by using the EIT technique described earlier to store two counter propagating light pulses as atomic coherences or by using Raman transitions to create the coherences directly. Second, all four control fields Ω_{ij} with
Third, after waiting for a certain period of time
There can be many complications in an actual experiment, one of which is that the total intensity decays in time [21,29]. This occurs for many reasons, such as couplings to other states in an actual atom and instability of the states |
where Γ is a 2 by 2 matrix with diagonal components γ_{1} and γ_{2}, corresponding to the decay rates of Ψ_{1} and Ψ_{2}, respectively. This causes the total intensities of the individual components to decay, limiting the evolution time, but the effects of the zero-mode can still be observed from the initial time evolution. This is illustrated in Fig. 7, where an asymmetrical choice γ_{1} = 0.1 and
Lastly, the topological character of the zero-mode can be verified by varying the details of the spatial mass profile. For example, upon changing the value of
The variable-mass Dirac model is not only useful in high-energy physics, but also in other branches of physics such as condensed matter and nuclear physics. The purpose of this section is to point out three such cases: Polyacetylene, random mass Dirac model, and Lorenz scalar potential. We limit ourselves to the most basic aspects of each topic and briefly discuss how the relevant physics can be emulated in a SSL setup.
Polycetylene can be modeled as electrons confined to move in a one-dimensional chain [27]. Phonon modes of the atoms are coupled to the electrons because the hopping strength depends on the distance between the sites. In such a system, the configuration of equally spaced sites does not support the ground state. Instead, it supports two degenerate ground state configurations with broken reflection symmetry. This is called the Peierls instability [30]. A soliton is formed when the two configurations coexist in a single chain, at the location at which the configuration changes from one to the other. A localized electron state forms around the site, whose energy lies between the gap created by the electron-phonon coupling.
The situation is analogous to the Jackiw-Rebbi model, where a zero-mode forms between the mass gap of the Dirac electrons. This can be established formally by taking the continuum limit of the electron hopping model and using the Jordan-Wigner transformation, which turns out to yield a slightly more generalized version of Eq. (5). Therefore, analogous physics to the JR model occur in polyacetylene, which can be observed in the same SSL setup described above. One complication that arises in polyacetylene is the spin of the electron, which is absent in the one-dimensional Dirac particle. In short, the soliton is associated with a charge
Random mass Dirac model is a variant of the Jackiw- Rebbi model, where solitons are assumed to take the form of a step-function (a kink) and are randomly distributed in space. The mass term in the Dirac equation thus randomly flips its sign. This is called the telegraph signal and is described by
where sgn denotes the sign function and
The spin-Peierls model describes an antiferromagnetic spin-1/2 chain with an additional dimerization term [32]:
Δ_{n} determines the strength of the phonon-mediated dimerization and is related to the mass term in the continuum limit. When impurities are introduced, domain walls are formed at their locations, affecting Δ_{n}, which in turn introduces a kink in the mass term. The lowenergy physics of this model is described by the random mass Dirac model.
One of the characteristics of the model is a long-range correlation in the mid-gap (zero-mode) state [32]. Because the localization length of a state is proportional to ln(1/
at length scales greater than the mean free path.
An emulation in terms of our polaritonic setup starts from noting that the Gaussian distribution of the mass term, instead of the telegraph signal described earlier, exhibits the same scaling in the correlation function [32]. Note that the mass term does not follow a Gaussian profile in space, but has a random value picked out from a Gaussian distribution. In an emulation, the two-photon detuning should be varied randomly in space according to a Gaussian distribution. After loading and evolving an initial state following the same procedure introduced in the previous section, the intensity correlation function
We note that the random mass Dirac model can be realized in the SSL system was already noted by Unanyan et al. [10] and a related model with a random vector potential has been studied in the setting of cold atoms [33]. Even an experimental demonstration was performed in a waveguide arrays setup [7].
As another application of the variable-mass Dirac equation, imagine a Dirac field coupled to a Lorentzscalar potential. Such a case was first studied as a spinoff of the MIT bag model, devised as a phenomenological model for quark confinement [34, 35]. Yet another area that uses similar models is nuclear physics (see e.g. [36]). The Hamiltonian describing a Dirac field interacting with both a scalar potential
The vector potential is known to exhibit Klein’s paradox when it is strong enough to create a particle and an antiparticle [1]. This type of behaviour occurs because the positive and negative energy states see different potentials and does not occur for a scalar potential.
In the phenomenological study of quarkonium, an equal mixture of a vector and a scalar confining potential is used to obtain a best fit for the spin-orbit splitting [37]. One widely studied potential is a linearly increasing potential,
The vector potential term can be introduced by slightly modifying the double tripod scheme in Fig. 1(b). Instead of choosing the equal and opposite two photon detunings as shown in the figure, choose different detunings
In conclusion, a connection between the onedimensional Dirac equation and a spinor slow light system has been pointed out. The latter system was shown to be able to emulate a variable-mass Dirac equation which in turn is related to models in diverse branches of physics such as high-energy, condensed matter, and nuclear physics. In particular, the SSL setup realizing the Jackiw-Rebbi model was considered in detail, along with a possible experimental procedure to observed the presence of the topologically protected zero-mode.
This research was supported by Kyungpook National University Research Fund, 2019.