![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg34.png\")
<\/p>\n\n\n\nBeam Amplitudes Spot intensity on screen<\/p>\n\n\n\n
Left Right as SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
is slowly turned o\ufb00<\/p>\n\n\n\n
L R L R Final (SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
o\ufb00)<\/p>\n\n\n\n
+ +<\/p>\n\n\n\n
+ \u2212 no spot<\/p>\n\n\n\n
\u2212 + no spot<\/p>\n\n\n\n
\u2212 \u2212<\/p>\n\n\n\n
In the language of optics, we can say that the middle two beams, having<\/p>\n\n\n\n
<\/p>\n\n\n\n
26 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
opposite phases, destructively interfere so that the corresponding intensity is<\/p>\n\n\n\n
zero.<\/p>\n\n\n\n
This was an example of interference between just two states. In more com-<\/p>\n\n\n\n
plex situations where multiple states exist, each state must be associated with<\/p>\n\n\n\n
a phase that is in general complex. It is this phenomenon of interference in<\/p>\n\n\n\n
quantum mechanics that calls for description of states with complex ampli-<\/p>\n\n\n\n
tudes. In mathematical language, each state is associated with a complex<\/p>\n\n\n\n
vector, one that has a magnitude as well as a phase.<\/p>\n\n\n\n
The Stern\u2013Gerlach setup we have described in this chapter serves multiple<\/p>\n\n\n\n
purposes for us. First, it demonstrates the quantum property of spin of an<\/p>\n\n\n\n
electron as a prototypical two-state system that can be used as a qubit. Second,<\/p>\n\n\n\n
we can use the setup to prepare a quantum system in a prede\ufb01ned state:<\/p>\n\n\n\n
initializing it to |0i or |1i by \ufb01ltering out one of the outputs. Third, the setup<\/p>\n\n\n\n
can be used as a detector to measure the state of the input beam.<\/p>\n\n\n\n
Exercise 2.2. Suppose that one of the four beams output from the middle SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
were blocked (in Figure 2.9). What would be the intensities of the various<\/p>\n\n\n\n
output beams?<\/p>\n\n\n\n
Box 2.1: Polarization States of Light<\/p>\n\n\n\n
The quantum spin described in this chapter is novel and has no classical<\/p>\n\n\n\n
analog. However, the same picture of a 2-dimensional Hilbert space emerges<\/p>\n\n\n\n
from considering the polarization states of light. This example is worth con-<\/p>\n\n\n\n
sidering, as it will be particularly useful later when we use light for quantum<\/p>\n\n\n\n
information processing. The analogy with spin is also complete, with a classi-<\/p>\n\n\n\n
cal picture to peg our understanding on.<\/p>\n\n\n\n
Classically, light is electromagnetic radiation, with oscillating electric and<\/p>\n\n\n\n
magnetic \ufb01elds. The form of the \ufb01elds comes from solutions to Maxwell\u2019s<\/p>\n\n\n\n
equations. It is easier to detect the electric \ufb01eld, so we will describe light by<\/p>\n\n\n\n
its electric \ufb01eld vector. The important parameters that describe a monochro-<\/p>\n\n\n\n
matic light wave are its wavelength \u03bb and angular frequency (color) \u03c9 and the<\/p>\n\n\n\n
wave vector<\/p>\n\n\n\n
~<\/p>\n\n\n\n
k =<\/p>\n\n\n\n
2\u03c0<\/p>\n\n\n\n
\u03bb<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
k giving the direction of propagation. The direction<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
k is<\/p>\n\n\n\n
conventionally taken to be \u02c6z, just as SG<\/p>\n\n\n\n
z<\/p>\n\n\n\n
is the standard for the spin system.<\/p>\n\n\n\n
An important property of the electromagnetic wave in free space is transver-<\/p>\n\n\n\n
sality: the electric \ufb01eld vector always lies in a plane perpendicular to<\/p>\n\n\n\n
\u02c6<\/p>\n\n\n\n
k. The<\/p>\n\n\n\n
direction of the electric \ufb01eld vector is known as its polarization. This direction<\/p>\n\n\n\n
could be constant, as in linearly polarized light, or rotate in the polarization<\/p>\n\n\n\n
plane, as in circularly polarized light.<\/p>\n\n\n\n
<\/p>\n\n\n\n
A Simple Quantum System 27<\/p>\n\n\n\n
FIGURE 2.10: Linearly polarized light wave.<\/p>\n\n\n\n
Let\u2019s \ufb01rst look at linearly polarized light with the<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
E \ufb01eld oscillating along<\/p>\n\n\n\n
a direction we\u2019ll call \u02c6\ue00f (Figure 2.10). It is possible to produce such light by<\/p>\n\n\n\n
passing unpolarized light from a monochromatic source through a polaroid<\/p>\n\n\n\n
\ufb01lter. Such a \ufb01lter has a \u201cpass axis\u201d that allows polarizations parallel to this<\/p>\n\n\n\n
axis alone to be transmitted through. The \ufb01eld is described by<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
E = E<\/p>\n\n\n\n
0<\/p>\n\n\n\n
\u02c6\ue00f cos(kz \u2212\u03c9t). (2.11)<\/p>\n\n\n\n
The intensity of light is given by the magnitude square of the electric \ufb01eld.<\/p>\n\n\n\n
If a second polarizer, with its pass-axis at an angle \u03b4 to the \ufb01rst, is placed in<\/p>\n\n\n\n
the light path, then only the component of the<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
E \ufb01eld along this angle is<\/p>\n\n\n\n
passed through. So the electric \ufb01eld of the transmitted light is E cos \u03b4 in a<\/p>\n\n\n\n
direction parallel to the new pass axis. The intensity of the light falls by a<\/p>\n\n\n\n
factor cos<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b4 (Figure 2.11).<\/p>\n\n\n\n
FIGURE 2.11: E\ufb00ect of a linear polarizer on unpolarized light; subsequent<\/p>\n\n\n\n
polarizer allows only a component \u221d cos<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u03b4 through.<\/p>\n\n\n\n
More generally, the electric \ufb01eld could have components oscillating along the<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bg35.png\")
<\/p>\n\n\n\n
28 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
\u02c6x and the \u02c6y directions with di\ufb00erent amplitudes and even di\ufb00erent phases:<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
E = E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u02c6xe<\/p>\n\n\n\n
i(kz\u2212\u03c9t)<\/p>\n\n\n\n
+ E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
\u02c6ye<\/p>\n\n\n\n
i(kz\u2212\u03c9t+\u03c6)<\/p>\n\n\n\n
. (2.12)<\/p>\n\n\n\n
FIGURE 2.12: Light with elliptic polarization: the electric \ufb01eld vector traces<\/p>\n\n\n\n
out an ellipse in the x-y plane<\/p>\n\n\n\n
One can de\ufb01ne the polarization vector<\/p>\n\n\n\n
\u02c6\ue00f =<\/p>\n\n\n\n
E<\/p>\n\n\n\n
1<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
E |<\/p>\n\n\n\n
\u02c6x +<\/p>\n\n\n\n
E<\/p>\n\n\n\n
2<\/p>\n\n\n\n
|<\/p>\n\n\n\n
\u2212\u2192<\/p>\n\n\n\n
E |<\/p>\n\n\n\n
e<\/p>\n\n\n\n
i\u03c6<\/p>\n\n\n\n
\u02c6y.<\/p>\n\n\n\n
This is in general elliptical polarization (Figure 2.12). The special case \u03c6 = \u03c0\/2<\/p>\n\n\n\n
corresponds to circular polarization while \u03c6 = 0 is linear polarization.<\/p>\n\n\n\n
It is easy to see that if \u02c6x-polarized light is incident on a \u02c6y-polarizer (a polaroid<\/p>\n\n\n\n
\ufb01lter with its pass axis along the \u02c6y direction), no light passes through. We<\/p>\n\n\n\n
can thus de\ufb01ne two orthogonal polarization states of light corresponding to<\/p>\n\n\n\n
the vertical (\u02c6y) and horizontal (\u02c6x) directions. This experiment is analogous<\/p>\n\n\n\n
to the Stern\u2013Gerlach-z machine, with up and down ports being analogous to<\/p>\n\n\n\n
the vertical and horizontal polarizations. We can thus draw analogy between<\/p>\n\n\n\n
the vector space of light polarizations and the spin Hilbert space:<\/p>\n\n\n\n
|0i \u2194 l<\/p>\n\n\n\n
|1i \u2194 \u2194 .<\/p>\n\n\n\n
<\/p>\n\n\n\n
A Simple Quantum System 29<\/p>\n\n\n\n
To create the optical analogue of the states produced by SG<\/p>\n\n\n\n
x<\/p>\n\n\n\n
\ufb01lters, we will<\/p>\n\n\n\n
need to use polarizers that are rotated by 45<\/p>\n\n\n\n
\u25e6<\/p>\n\n\n\n
with respect to the earlier ones.<\/p>\n\n\n\n
Light produced by these polarizers can be in polarization states<\/p>\n\n\n\n
l<\/p>\n\n\n\n
and<\/p>\n\n\n\n
l<\/p>\n\n\n\n
de\ufb01ned by the 45<\/p>\n\n\n\n
\u25e6<\/p>\n\n\n\n
orthogonal directions:<\/p>\n\n\n\n
l<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(\u02c6x + \u02c6y)<\/p>\n\n\n\n
l<\/p>\n\n\n\n
=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(\u02c6x \u2212 \u02c6y).<\/p>\n\n\n\n
It is easy to see that if<\/p>\n\n\n\n
l<\/p>\n\n\n\n
light is incident on an \u02c6x or \u02c6y polarizer then 50% of<\/p>\n\n\n\n
the incident beam passes through. Similarly for<\/p>\n\n\n\n
l<\/p>\n\n\n\n
.<\/p>\n\n\n\n
The analogy of the SG<\/p>\n\n\n\n
y<\/p>\n\n\n\n
basis is with right and left circular polarization. From<\/p>\n\n\n\n
Equation 2.12, we see that light with electric \ufb01eld rotating in the plane of<\/p>\n\n\n\n
polarization arises due to a phase di\ufb00erence of \u03c0\/2 between the x- and y-<\/p>\n\n\n\n
components. The complex notation is most suitable for expressing this phase<\/p>\n\n\n\n
relationship (using i = e<\/p>\n\n\n\n
i\u03c0\/2<\/p>\n\n\n\n
):<\/p>\n\n\n\n
|\u2191<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i \u2194 \ue009=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(\u02c6x + i\u02c6y) (2.13)<\/p>\n\n\n\n
|\u2193<\/p>\n\n\n\n
y<\/p>\n\n\n\n
i \u2194 \ue008=<\/p>\n\n\n\n
1<\/p>\n\n\n\n
\u221a<\/p>\n\n\n\n
2<\/p>\n\n\n\n
(\u02c6x \u2212 i\u02c6y) (2.14)<\/p>\n\n\n\n
The necessity of complex probability amplitudes becomes clear now, due to<\/p>\n\n\n\n
considerations of phase being unavoidable.<\/p>\n","protected":false},"excerpt":{"rendered":"
Beam Amplitudes Spot intensity on screen Left Right as SG x is slowly turned o\ufb00 L R L R Final (SG x o\ufb00) + + + \u2212 no spot \u2212 + no spot \u2212 \u2212 In the language of optics, we can say that the middle two beams, having 26 Introduction to Quantum Physics and […]<\/p>\n","protected":false},"author":1,"featured_media":3940,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[488],"tags":[],"class_list":["post-3972","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-3-the-quantum-postulates"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/science-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3972","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3972"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3972\/revisions"}],"predecessor-version":[{"id":4552,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3972\/revisions\/4552"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3940"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3972"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3972"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}