Abstract
Landau levels and states of electrons in a magnetic field are fundamental quantum entities underlying the quantum Hall and related effects in condensed matter physics. However, the realspace properties and observation of Landau wave functions remain elusive. Here we report the realspace observation of Landau states and the internal rotational dynamics of free electrons. States with different quantum numbers are produced using nanometresized electron vortex beams, with a radius chosen to match the waist of the Landau states, in a quasiuniform magnetic field. Scanning the beams along the propagation direction, we reconstruct the rotational dynamics of the Landau wave functions with angular frequency ~100 GHz. We observe that Landau modes with different azimuthal quantum numbers belong to three classes, which are characterized by rotations with zero, Larmor and cyclotron frequencies, respectively. This is in sharp contrast to the uniform cyclotron rotation of classical electrons, and in perfect agreement with recent theoretical predictions.
Introduction
Classical electrons in a uniform magnetic field propagate freely along the field and form confined circular orbits in the plane perpendicular to the field. The angular velocity of such orbiting is constant and is known as the cyclotron frequency. Accordingly, quantummechanical eigenstates of a scalar electron in a uniform magnetic field are localized in the transverse plane and are characterized by two quantum numbers. Excluding the longitudinal motion of the electron (as, for example, in twodimensional (2D) condensedmatter systems), this leads to the quantization of the energy levels, which are degenerate and are characterized by a single quantum number. Quantum electron states and their corresponding energy levels in a magnetic field were described by Fock^{1}, Landau^{2} and Darwin^{3} in the early days of quantum theory, and are commonly referred to as Landau states and Landau levels.
Landau eigenstates play a key role in various solidstate phenomena, such as the diamagnetism of metals, as well as quantum Hall, Shubnikov–De Haas and De Haas–van Alphen effects^{4,5,6}. Landau energy levels reveal themselves in quantumHall conductance plateaus^{6}; they are measured spectroscopically^{7}; and recently they attracted enormous attention in relation to graphene systems^{8,9,10,11,12}. However, Landau levels are highly degenerate and do not provide information about the actual state and spatial distribution of the electron. Moreover, the drift of the states in an external random potential blurs the picture in condensedmatter systems^{13,14}, and it is impossible to observe the fast rotational dynamics of electrons in such systems. Although considerable progress was achieved recently in Fourier analysis of Landau modes^{15} (based on a single radial quantum number), their realspace properties remain elusive. Thus, the observation of spatially resolved Landau eigenstates and their internal dynamics remains a challenging problem.
Landau states can appear not only in condensedmatter systems but also for free electrons in a uniform magnetic field. Recently, we argued^{16} that, allowing free propagation along the magnetic field, the Landau states represent nondiffracting versions of the socalled electron vortex beams^{17,18,19,20}. Furthermore, both the radial and azimuthal (vortex) quantum numbers of Landau modes crucially determine their properties and evolution. Electron vortex beams were predicted^{17} and recently generated using transmission electron microscopes (TEM)^{18,19,20,21,22,23,24}, and they promise applications in various areas of both fundamental and applied physics^{24,25,26,27,28,29,30,31,32,33,34} (for a review, see ref. 35). Here we report the realspace observation of individual Landau eigenstates, which are formed by freeelectron vortex beams in a uniform magnetic field inside a TEM. We measure the fast rotational dynamics of electrons within different states and reveal their unusual nonclassical behaviour. Instead of cyclotron orbiting, we observe that Landau electrons rotate with three different angular velocities, determined by the vortex quantum number.
Results
Rotational dynamics of electrons in quantum Landau states
The Landau states of an electron in a zdirected homogeneous magnetic field B>0 and uniform gauge with azimuthal vector potential A_{ϕ}=Br/2 are described by cylindrical vortex wave functions^{1,3,16}
Here (r, ϕ, z) are cylindrical coordinates, m=0, ±1, ±2, ... and n=0, 1, 2, ... are the azimuthal and radial quantum numbers, respectively, is the magnetic length parameter, are the generalized Laguerre polynomials and k_{z} is the wave number of the free longitudinal electron motion. The transverse energy of the electron is quantized according to Landau levels^{1,3,16}
is the principal Landau quantum number and Ω=eB/2μ_{e} is the Larmor frequency corresponding to the electron charge e=−e and mass μ_{e}. Although the classical electron dynamics is determined by the cyclotron frequency ω_{c}=2Ω, it is the Larmor frequency that is fundamental in the quantum evolution of electrons^{16,36}.
The Landau states (1) resemble zpropagating freespace vortex beams^{17,18,19,20}, which are characterized by the phase factor exp(im ϕ), circulating mdependent azimuthal current and kinetic orbital angular momentum L_{z}=ℏm per electron. However, the kinetic orbital angular momentum of the Landau states differs significantly from that of freespace beams and is determined by the principal quantum number: L_{z}=E_{⊥}/Ω=ℏ(2N+1)>0 (ref. 16). It is this positive angular momentum and the corresponding negative magnetic moment M_{z}=(e/2μ_{e})L_{z}<0 that are responsible for the diamagnetism of electrons predicted by Landau^{2,3}. The nontrivial angular momentum of the Landau states appears because the gaugeinvariant probability current is modified by the presence of the vector potential A (ref. 37): . According to this, the current in the Landau states (1) becomes^{16}
where and are the unit vectors of the corresponding coordinate axes. Here the mdependent azimuthal term originates from the freespace vortex current, whereas the second azimuthal term describes the contribution from the vector potential A_{ϕ}=Br/2.
Equation (3) describes the spiralling of the Landau electron about the magneticfield direction in the sense of Bohmian trajectories, that is, streamlines of the probability current^{38,39,40,41}. The expectation value of the electron’s angular velocity, ω(r)=v_{ϕ}(r)/r (where v=j/ψ^{2} is the local Bohmian velocity), can be obtained from equations (1) and (3):
where we used , for m≠0. Equation (4) represents a highly surprising result. It shows that the rotation of electrons in a magnetic field in the quantum picture is drastically different from the uniform classical orbiting. Instead of rotation with a single cyclotron frequency ω_{c}=2Ω, states with negative, zero and positive azimuthal indices rotate with zero, Larmor and cyclotron rates, respectively. This result is independent of the radial index n and is also valid for any superposition of modes with different n and the same m as long as . The probabilitydensity distributions and internal Bohmian trajectories in the Landau states (1) with n=0 and m=−1, 0, 1 (which correspond to the degenerate Landau levels (2) with N=0, 0, 1) are shown in Fig. 1. One can see that in the m=0 mode the trajectory rotation is uniform and corresponds to the Larmor frequency Ω. At the same time, the rotations inside the m≠0 modes depend on the radius r and coincide with the averaged values ‹ω› (equation (4)) at the maximalintensity radii . As was recently demonstrated for photons^{40,41}, Bohmian trajectories can be measured experimentally using statistical averaging over many identical singleparticle events without interparticle interactions. The same conditions are realized in electronoptical measurements in electron microscopy^{42}.
Experimental measurements
The main goal of our experiment is twofold: first, the creation of freeelectron Landau states (equation (1)) and, second, the observation of their extraordinary internal rotational dynamics (equation (4)). To produce the freeelectron Landau states, we use the fact that they represent nondiffracting versions of the freespace Laguerre–Gaussian vortex beams^{16,17}. Such electron vortex beams are generated in a TEM using a holographic fork mask (a diffraction grating with a dislocation)^{19,20}. The mask shown in Fig. 2a has a bar/slit ratio of 1, and it produces beams with different azimuthal indices m=0, ±1, ±3, ±5, ... for different diffraction orders^{19}. (For other bar/slit ratios, even values of m can also be produced^{20}.) The vortex beams are then focused with a magnetic lens, which has a region of a quasiuniform zdirected strong magnetic field. Figure 2 shows a schematic diagram of our experimental setup with the converging vortex beams. We tune the parameters of the system, such that in the zregion of a quasiuniform magnetic field the beam radius w(z) comes close to the magnetic radius (Fig. 2b). Thus, the vortex beams approximate Landau states (1) with n=0 and different azimuthal indices m. Then, the observation of the peculiar rotational dynamics described by equation (4) would verify that the beams indeed acquire properties of the Landau modes. The more the beam radius deviates from , the more ‹r^{−2}› deviates from , changing the electron angular velocity given by equation (4).
To observe the internal rotational dynamics of equation (4) and spiralling Bohmian trajectories (Fig. 1) inside the cylindrically symmetric beams, we borrow a technique successfully employed in optics^{43,44,45} and also recently demonstrated for electrons^{46}. Namely, we obstruct half of the beam with an opaque knife edge stop and trace the spatial rotation of the visible part of the beam when the knife edge is moved along the z axis (Fig. 2a). Although such truncation of the beam breaks the cylindrical symmetry of the initial vortex state, it does not perturb significantly the probability currents in the visible part of the beam, so that the truncated beam approximately follows the internal Bohmian trajectories of the initial cylindrical state. Note that in condensedmatter systems the Larmor rotation of electrons is fast and cannot be observed as compared with the slow motion of the centre of mass in an external potential^{13,14,15}. In contrast, for paraxial electrons in a TEM, the transverse Larmor dynamics is slow as compared with the relativistic longitudinal velocity of electrons: . This allows mapping of the internal dynamics on the z axis with extremely high resolution (in our experiment, the Larmor time scale Ω^{−1}≃8 ps corresponded to the propagation distance z_{L}=v/Ω≃1.7 mm (ref. 16)).
The experiment was performed according to the above approach in a FEI TECNAI F20 TEM at 200 kV acceleration voltage (v≃0.7c). The focusing lens produced the maximal longitudinal field B≃1.9T, which corresponds to the Larmor frequency Ω≃120 GHz (using the relativistic mass μ_{e}=γμ_{e0}) and the magnetic radius w_{B}≃26 nm. Choosing z=0 as the observation plane, the region of interest was zε(−80, −30) μm (Fig. 2). In this region the magnetic field was uniform up to negligible variations ~10^{−2}B in the longitudinal component and ~10^{−6}B in the radial component. The knife edge was made from a Si crystal. Its position z_{k} was varied in the region of interest (that is, the propagation distance to the observation plane was varied by Δz≃50 μm) to measure the rotations of the images of the cut beam (the corresponding Larmor angle is Δϕ=Δz/z_{L}≃30 mrad, see Fig. 3). Note that the focal plane of the beams was set at z≃8 μm, that is, few Rayleigh ranges below the observation plane, in order to reduce the diffractive Gouyphase rotation of the images^{46,47} and to improve the accuracy of the measurements using sufficiently large vortex radii (Fig. 2a).
Figures 3 and 4 show the results of the experimental measurements of the cut vortex modes at different positions z_{k} of the knife edge. In Fig. 3a the images of the modes with m=−5, −3, −1, 0, 1, 3, 5 are shown for three values of z_{k}. Note that the cut edges of the beams with opposite m have opposite inclination with respect to the line joining the beams. This is the residual Gouyphase rotation^{46,47} visible at the observation plane. At the same time, a slow rotation of the m>0 states as a function of z_{k} can be detected visually, while the m<0 modes do not experience visible rotation. A quantitative analysis of the differential rotations of modes with different m is depicted in Fig. 3b. One can clearly see three different rates of rotations for modes with m<0, m=0 and m>0, in precise agreement with the prediction of equation (4) and in sharp contrast to the classical cyclotron orbiting. This confirms that electron vortices form Landau states and acquire their peculiar properties in the region of interest. We repeated measurements of the mode rotations with z_{k} with slightly different defocus values and holographic masks (including those producing m=±2 modes). For each of such experiments we determined the average value of the rotational velocities ‹ω›=v‹dϕ/dz_{k}› for different m. The results are shown in Fig. 4. One can clearly see that rotational frequencies in different measurements fluctuate around the theoretical values of equation (4), and their averages over all measurements are in very good agreements with the theoretical Landaustate behaviour. Note that variations of the rotational frequencies in the m≠0 modes and robustness of such frequencies for the m=0 mode can be related to the radial dependence (independence) of the rotation in the m≠0 (m=0) Landau states (see Fig. 1).
Discussion
To summarize, the extraordinary mdependent rotational dynamics, which is impossible in classical electron propagation, reveals the peculiar behaviour of quantum Landau states. Although it is commonly believed that the cyclotron rotation of electrons underpins Landau states, we have shown that electrons can also rotate in the quantum Bohmian picture with either zero or Larmor frequencies. (It is worth remarking that the observed mdependent rotation of electrons can be related to the Aharonov–Bohm effect^{48}. Indeed, it is the asymmetry of the azimuthal currents in the m>0 and m<0 modes, caused by the presence of the vector potential, that is responsible for the Aharonov–Bohm phenomenon^{16,48}.) We emphasize several striking features and the fundamental importance of these results. First, we demonstrated the appearance of quantum Landau states within freeelectron optics, rather than in a condensedmatter system. Second, in contrast to condensedmatter experiments and analyses, we separated modes with different azimuthal quantum numbers m (some of which correspond to the same Landau levels) and showed that this index is crucially important for the electron rotation in a magnetic field. Third, while the observation of the fast Larmor dynamics is currently impossible in condensedmatter systems, this rotation becomes measurable for free relativistic electrons. In our setup, this allowed the detection of rotations with frequency Ω~100 GHz, corresponding to the energy difference ℏΩ~100 μeV. Thus, our results provide new insights into the fundamental properties of Landau states and pave the way towards detailed investigations of their otherwise hidden characteristics.
Additional information
How to cite this article: Schattschneider, P. et al. Imaging the dynamics of freeelectron Landau states. Nat. Commun. 5:4586 doi: 10.1038/ncomms5586 (2014).
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Acknowledgements
This work was supported by the Austrian Science Fund (FWF; grant no. I543N20), RIKEN iTHES Project, MURI Center for Dynamic MagnetoOptics, JSPSRFBR contract no. 120292100, GrantinAid for Scientific Research(S), MEXT Kakenhi on Quantum Cybernetics and the JSPS via its FIRST programme.
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P.S. designed the experiment, wrote the paper and contributed to the theoretical analysis; T.S. and S.L. performed the experiment and processed the data, M.S.P. contributed to the experiment; A.S.T. made the holographic mask; K.Y.B. performed the theoretical analysis and wrote the paper; F.N. wrote the paper.
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Schattschneider, P., Schachinger, T., StögerPollach, M. et al. Imaging the dynamics of freeelectron Landau states. Nat Commun 5, 4586 (2014). https://doi.org/10.1038/ncomms5586
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