Lasers & Spectroscopy
Work in progress
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The symmetries upon which we base our understanding of physics on tell us that matter and antimatter should be created and destroyed in equal quantities. However, since we observe that the universe is almost purely matter, we must conclude that these symmetries are broken at some level. At ALPHA, we compare the energy levels of antihydrogen to hydrogen using precision spectroscopy to search for subtle differences between matter and antimatter.
1. Spectroscopy in general
1.1. Principle quantum number n
The orbiting electron possesses orbital angular momentum, to which we assign the quantum number l. This can be thought of as the shape of the orbital in which the electron can be found, with n allowed shapes in each principal level. For example, the ground level (n=1) has only one allowed orbital (l=0), and the first excited level (n=2) has l=0,1. However, we rarely refer to the angular momentum numerically (0,1,2,3…), but rather by the historic notation of S (sharp), P (primary), D (diffuse), F (fundamental), and continuing alphabetically from F. Photons – the particle of light – possess one unit of angular momentum, meaning that electronic transitions are only allowed if l changes by ± 1. As we will see later, this selection rule can be bypassed if the atom absorbs or emits multiple photons simultaneously.
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[if gte vml 1]><v:shapetype id=”_x0000_t75″ coordsize=”21600,21600″ o:spt=”75″ o:preferrelative=”t” path=”m@4@5l@4@11@9@11@9@5xe” filled=”f” stroked=”f”> <v:stroke joinstyle=”miter”/> <v:formulas> <v:f eqn=”if lineDrawn pixelLineWidth 0″/> <v:f eqn=”sum @0 1 0″/> <v:f eqn=”sum 0 0 @1″/> <v:f eqn=”prod @2 1 2″/> <v:f eqn=”prod @3 21600 pixelWidth”/> <v:f eqn=”prod @3 21600 pixelHeight”/> <v:f eqn=”sum @0 0 1″/> <v:f eqn=”prod @6 1 2″/> <v:f eqn=”prod @7 21600 pixelWidth”/> <v:f eqn=”sum @8 21600 0″/> <v:f eqn=”prod @7 21600 pixelHeight”/> <v:f eqn=”sum @10 21600 0″/> </v:formulas> <v:path o:extrusionok=”f” gradientshapeok=”t” o:connecttype=”rect”/> <o:lock v:ext=”edit” aspectratio=”t”/> </v:shapetype><v:shape id=”Picture_x0020_4″ o:spid=”_x0000_s1026″ type=”#_x0000_t75″ style=’position:absolute;margin-left:0;margin-top:.6pt;width:165.75pt; height:113.9pt;z-index:251658240;visibility:visible;mso-wrap-style:square; mso-wrap-distance-left:9pt;mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; mso-wrap-distance-bottom:0;mso-position-horizontal:left; mso-position-horizontal-relative:margin;mso-position-vertical:absolute; mso-position-vertical-relative:text’> <v:imagedata src=”file:///C:/Users/weber/AppData/Local/Packages/oice_16_974fa576_32c1d314_9e2/AC/Temp/msohtmlclip1/01/clip_image001.png” o:title=””/> <w:wrap type=”square” anchorx=”margin”/> </v:shape><![endif][if !vml][endif]Figure 2: The energy levels for a given principal quantum number (n = 1, 2, 3) and orbital angular momentum (l = S, P, D) are shown in black. The dashed green lines show the allowed transitions following the l ± 1 selection rule. Since there is no level to decay to from 2S, this level is “metastable”. It has a lifetime of 0.12 s, compared to the 2P lifetime of just 1.6 ns. Level spacings are not to scale.
1.2. Orbital angular momentum l
1.3 Fine structure (l+s)
In addition to orbital angular momentum, the electron also possesses an intrinsic angular momentum characterised by the quantum number s = ± ½. We call this the “spin” of the electron, and its coupling to the orbital angular momentum leads to a small splitting of the energy levels. Since it is small, we call it the “fine structure”. If we define a new quantum number j = l + s to describe the total angular momentum, we see that each orbital l consists of 2j-1 levels. We label these levels as nlj, so the ground state of hydrogen is described as 1S1/2, and the first excited states are described as 2S1/2, 2P1/2, and 2P3/2. Whilst we might expect that the 2S1/2 and 2P1/2 levels have the same energy, it turns out that the S state has a slightly lower energy due to its greater overlap with the finite sized proton. We name the collection of QED effects that cause this the “Lamb shift”.
The proton has an intrinsic angular momentum associated with it too, and the interaction between the spin of the electron and the proton gives rise to an even smaller splitting of the energy levels, which we call the “hyperfine structure”. This can be described by the quantum number f = j ± ½.
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Figure 3: A schematic illustration of the energy level splitting of the hydrogen atom. The diagram is not to scale – the hyperfine structure is of order 1 GHz, the fine structure is of order 10 GHz, and the principal levels n=1 and n=2 are separated by 2.5 PHz. The Lamb shift between 2S1/2 and 2P1/2 is also 1 GHz.
So far, we have only considered the atom in a field-free environment. An external magnetic field will apply a torque to the internal dipoles of the atom, named the “Zeeman effect” after its discoverer. Each hyperfine state has a 2f+1 degeneracy associated with the magnetic quantum number mf, which is the projection of the total angular momentum along the axis of the applied field and can take the values mf = -f, -f+1…0…f-1, f. The interaction of the atom with an external magnetic field breaks this degeneracy, leading to a splitting of the energy levels. Since some of the levels increase in energy as the magnetic field increases, they exhibit diamagnetism, and can be trapped by constructing a magnetic minimum.
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Figure 4: The Zeeman splitting of the hydrogen 1S level. The blue states seek a lower magnetic field and are trappable, whilst the red states seek a higher magnetic field and, since it is not possible to make a magnetic maximum trap, are not trappable.