-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathqwChern.html
264 lines (246 loc) · 17.5 KB
/
qwChern.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
<!DOCTYPE html>
<html lang="en">
<head>
<!-- Required meta tags always come first -->
<meta charset="utf-8">
<meta http-equiv="x-ua-compatible" content="ie=edge">
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<title>Nonlinear disordered quantum walk | Random physics
</title>
<link rel="canonical" href="/qwChern.html">
<link rel="stylesheet" href="/theme/css/bootstrapr.min.css">
<link rel="stylesheet" href="/theme/css/font-awesome.min.css">
<link rel="stylesheet" href="/theme/css/pygments/autumn.min.css">
<link rel="stylesheet" href="/theme/css/style.css">
<link rel="icon" type="image/png" href="/extras/rphys.png" sizes="64x64">
<meta name="description" content="Effects of disorder and interaction on a quantum walk with nontrivial topology.">
</head>
<body>
<header class="header">
<div class="container">
<div class="row">
<div class="col-sm-12">
<h1 class="title"><a href="/">Random physics</a></h1>
<p class="text-muted">Alberto Verga, research notebook</p>
<ul class="list-inline">
<li class="list-inline-item"><a href="/">Blog</a></li>
<li class="list-inline-item text-muted">|</li>
<li class="list-inline-item"><a href="/pages/about.html">About</a></li>
<li class="list-inline-item"><a href="/pages/lectures.html">Lectures</a></li>
</ul>
</div>
</div>
</div>
</header>
<div class="main">
<div class="container">
<article class="article">
<header>
<ul class="list-inline">
<li class="list-inline-item text-muted" title="2016-05-02T12:00:00+02:00">
<i class="fa fa-clock-o"></i>
lun. 02 mai 2016
</li>
<li class="list-inline-item">
<i class="fa fa-folder-open-o"></i>
<a href="/category/blog.html">Blog</a>
</li>
<li class="list-inline-item">
<i class="fa fa-files-o"></i>
<a href="/tag/research.html">#research</a> </li>
</ul>
</header>
<div class="content">
<p><span class="math">\(\newcommand{\I}{\mathrm{i}}
\newcommand{\E}{\mathrm{e}}
\newcommand{\D}{\mathop{}\!\mathrm{d}}\)</span></p>
<blockquote>
<p>Work in progress to investigate the edge states of a quantum walk at the interface between two topologically distinct regions, when spatial disorder is present. <a href="http://dx.doi.org/10.1140/epjb/e2017-70433-1">Published in <span class="caps">EPJB</span>, 2017.</a></p>
</blockquote>
<p>Topological phases of matter appeared with the discovery of the <a href="http://dx.doi.org/10.1103/PhysRevLett.45.494">quantum Hall effect</a>. A quantum walk having the same topological properties as the anomalous quantum Hall system, was discussed by <a href="http://dx.doi.org/10.1103/PhysRevA.82.033429">Kitagawa et al</a> recently. It is defined by the product
</p>
<div class="math">\begin{equation}
\label{e:tr}
U = T_x R(\theta) T_y R(\alpha) T_{x+y} R(\theta)
\end{equation}</div>
<p>
of shift operators <span class="math">\(T_i\)</span>, which move the walker’s position a single step <span class="math">\(\pm1\)</span> in the direction <span class="math">\(i=x,y\)</span> of a square lattice according to their spin state, and rotation operators
</p>
<div class="math">\begin{equation}
R(\theta) = \E^{-\I \theta \sigma_y/2} = \begin{pmatrix}
\cos \theta/2 & -\sin \theta/2 \\
\sin \theta/2 & \cos \theta/2
\end{pmatrix} \,,
\end{equation}</div>
<p>
which rotate the <span class="math">\(1/2\)</span> particle’s spin by <span class="math">\(\theta\)</span> around the <span class="math">\(y\)</span>-axis (<span class="math">\(\boldsymbol \sigma\)</span> are Pauli matrices). Therefore, the quantum walk evolution is defined by the unitary operator <span class="math">\(U\)</span>, which advances the particle’s state a time step:
</p>
<div class="math">\begin{equation}
|\psi(t+1) \rangle = U \, |\psi(t)\rangle\,,
\end{equation}</div>
<p>
where <span class="math">\(|\psi(t)\rangle\)</span> is the quantum state of the walker at time <span class="math">\(t\)</span>; initially
</p>
<div class="math">\begin{equation}
|\psi(0)\rangle =
|0\rangle \otimes
\frac{|\uparrow\rangle + \I |\downarrow\rangle}{\sqrt{2}}\,,
\end{equation}</div>
<p>
the particle is at the origin in a superposition of spin up and down with equal probability. The first movie shows the propagation of the position probability:
</p>
<div class="math">\begin{equation}
P(x,y,t) = |\psi_\uparrow(x,y,t)|^2 + |\psi_\downarrow(x,y,t)|^2 \,,
\end{equation}</div>
<p>
where <span class="math">\(\psi_{\uparrow\downarrow}\)</span> are the spinor components. As other realizations of quantum walks, the information propagates in a ballistic fashion. </p>
<p>As in the quantum Hall effect, one may compute the <a href="http://arxiv.org/abs/1509.02295">Chern number</a>. Indeed, we can associate an effective Hamiltonian <span class="math">\(H\)</span> to the unitary operator:
</p>
<div class="math">\begin{equation}
U = \E^{-\I H}, \quad
H \equiv \I\,\log U\,,
\end{equation}</div>
<p>
which allows the study of the topological properties of the random walk using the same tools of condensed matter systems. According to the value of the angles <span class="math">\((\theta,\alpha)\)</span>, one finds that the Chern number takes one of the values <span class="math">\(C \in \{-1,0,1\}\)</span>. </p>
<video width="300" height="300" poster="/images/prob_02.png" preload="auto" controls > <source src="/images/prob_02.mp4" type="video/mp4" /> </video>
<video width="300" height="300" poster="/images/prob_06.png" preload="auto" controls > <source src="/images/prob_06.mp4" type="video/mp4" /> </video>
<h3>Topological properties</h3>
<p>We consider a system with an interface separating a <span class="math">\(C=-1\)</span> left region (for <span class="math">\(x < 0\)</span>) and a <span class="math">\(C=1\)</span> right region (for <span class="math">\(x \gt 0\)</span>). In the figure we show the two representative points in the <span class="math">\((\theta,\alpha)\)</span> phase plane, corresponding to each topological domain: red for the left and blue for the right. At the edge, localized modes exist that, in the case of the Hall system, are responsible for the conduction. These <a href="http://dx.doi.org/10.1038/ncomms1872">topologically protected bound states</a> were experimentally observed in the case of a one dimensional quantum walk.</p>
<p><img src="/images/c-phases.png" alt="quantum walk topology" style="height: 200px;"/>
<img src="/images/c-impurities.png" alt="spatial disorder" style="height: 200px;"/></p>
<p>In the case of the two dimensional quantum walk <span class="math">\(\ref{e:tr}\)</span>, we numerically computed the time evolution (second movie) and found that the position probability concentrates at the interface and spreads along the <span class="math">\(y\)</span> axis, in agreement with the existence of an edge state.</p>
<h3>Effect of disorder</h3>
<p>One important property of the edge states associated with the topology of the effective Hamiltonian (in analogy with the electronic bands of an insulator), is that they are protected against random perturbations. As long as the perturbation do not change the topology, for instance closing the gap or adding new states in the gap, the transport properties of the system are preserved. </p>
<p>We are interested in investigating the behavior of the edge states in the case of the two-dimensional quantum walk <span class="math">\(\ref{e:tr}\)</span>, when it is perturbed by <em>spatial disorder</em>. We introduce, at random sites uniformly distributed over the square lattice, a set of impurities <span class="math">\(I\)</span> (as shown in the figure, where we compare two sets with occupation probabilities <span class="math">\(p=0.01\)</span> and <span class="math">\(p=0.1\)</span>). These impurities will change the spin state of the walker.</p>
<video width="300" height="300" poster="/images/m_06.png" preload="auto" controls > <source src="/images/m_06.mp4" type="video/mp4"> </video>
<video width="300" height="300" poster="/images/m_05.png" preload="auto" controls > <source src="/images/m_05.mp4" type="video/mp4"> </video>
<p>In the presence of disorder, we consider various modifications of the coin operator <span class="math">\(R\)</span>:</p>
<ul>
<li>
<p>the rotation angle becomes random, <span class="math">\(\theta \rightarrow \theta + \delta\theta\)</span>,
<div class="math">$$R_J(\theta) = \E^{-\I \sigma_y (\theta + J\delta\theta(x))/2},\;
x \in I\,,$$</div>
where <span class="math">\(J\)</span> is a coupling constant and <span class="math">\(\delta\theta\)</span> is uniformly distributed on the circle (quenched “rotation” disorder);</p>
</li>
<li>
<p>the walker’s spin is changed by the impurity spin, polarized in the <span class="math">\(z\)</span> direction,
<div class="math">$$S_J(\phi) = \E^{-\I J \sigma_z \phi(x,t)},\;
x \in I\,,$$</div>
where the phase <span class="math">\(\phi(x,t)\)</span> can be randomly distributed on the circle (quench “spin” disorder, independent of time),</p>
</li>
<li>
<p>or dependent on the particle’s state (self-consistent disorder),
<div class="math">$$\phi(x,t) = |\psi_\uparrow|^2 - |\psi_\downarrow|^2$$</div>
or
<div class="math">$$\phi(x,t) = \frac{|\psi_\uparrow|^2 - |\psi_\downarrow|^2}{
|\psi_\uparrow|^2 + |\psi_\downarrow|^2}\,.
$$</div>
In the first case the impurity phase is proportional to the <span class="math">\(z\)</span> component of the particle spin, and in the second case, it is in addition normalized to the local particle density; this second form is invariant with respect to scale transformations of the spinor. For the self-consistent disorder the coin operator writes:
<div class="math">$$R_J(\theta,\phi) = S_J(\phi)R(\theta)\,,$$</div>
and the quantum walk becomes <a href="https://tel.archives-ouvertes.fr/tel-01230891">nonlinear</a>. Nonlinear quantum walks are studied in relation to the Dirac equation; in particular <a href="http://dx.doi.org/10.1103/PhysRevA.92.052336">solitons</a> and <a href="http://dx.doi.org/10.1103/PhysRevA.93.022329">attractors</a>, were recently predicted.</p>
</li>
</ul>
<p>The two videos compare the probability density at the interface of the free (without disorder) and the random (with spatial disorder) quantum walks. The two cases show ballistic propagation on the edge state. The following two figures show the width of the probability at <span class="math">\(x=0\)</span> (spreading in the <span class="math">\(y\)</span> direction), for the free walk and the nonlinear walk with <span class="math">\(p=0.1\)</span> and <span class="math">\(J=100\)</span>; we choose a large value of the coupling constant to strength the remarkable behavior in this case.</p>
<p><img src="/images/chern_w0_00.png" alt="free walk on the edge" style="height: 200px;"/>
<img src="/images/chern_w0_09.png" alt="random walk on the edge" style="height: 200px;"/></p>
<p>Indeed, it its worth noting that the presence of the nonlinear disorder do not prevent the walker to spread at a rate proportional to the time. This is in contrast with the usual behavior of a quantum walk, which localizes or diffuse as a result of disorder. </p>
<p><img src="/images/chern_p3d_00.png" alt="free walk on the edge" style="height: 300px;"/>
<img src="/images/chern_p3d_09.png" alt="random walk on the edge" style="height: 300px;"/></p>
<p>We also observe, as shown in the figures above, that in the random case the probability density distribution is slightly larger than in the free case near the interface. This is probably an effect due to disorder induced (anisotropic) localization, the density is confined in a region around its initial position. However, the presence of the edge state, wider in the case of disorder, prevents the localization along the <span class="math">\(y\)</span> axis.</p>
<h3>Noise and nonlinearity</h3>
<p>The edge state is protected against disorder. If the amplitude of the noise in the rotation angle is not large enough to change the topology of the system, the edge state persists. Strong disorder may destroy it.</p>
<p><img src="/images/chern_12b.png" alt="weak rotation noise" style="height: 200px;"/>
<img src="/images/chern_13d.png" alt="strong rotation noise" style="height: 200px;"/></p>
<p>Figures above compare the spreading of the walker in the weak (<span class="math">\(p=0.05\)</span>) and strong (<span class="math">\(p=0.2\)</span>) disorder cases.</p>
<p>The nonlinear walk restore the edge state even in the presence of strong noise.</p>
<blockquote></blockquote>
<table>
<tr style="text-align:center">
<td rowspan = "2"><img src="/images/chern_0809.png" alt="08 and 17" style="height: 260px;"/></td>
<td><img src="/images/chern_p3d_17.png" alt="17" style="height: 130px;"/></td>
</tr>
<tr style="text-align:center">
<td><img src="/images/chern_p3d_08.png" alt="08" style="height: 130px;"/></td>
</tr>
<tr style="text-align:center">
<td colspan = "2">Spreading along the interface in the case of nonlinear (gray) and linear $z$-phase noise (red) walks for the same parameters. Stronger noise (below panels) do not destroy the coherence of the *nonlinear* walk on the edge channel.</td>
</tr>
</table>
<script type="text/javascript">if (!document.getElementById('mathjaxscript_pelican_#%@#$@#')) {
var align = "center",
indent = "0em",
linebreak = "true";
if (true) {
align = (screen.width < 700) ? "left" : align;
indent = (screen.width < 700) ? "0em" : indent;
linebreak = (screen.width < 700) ? 'true' : linebreak;
}
var mathjaxscript = document.createElement('script');
mathjaxscript.id = 'mathjaxscript_pelican_#%@#$@#';
mathjaxscript.type = 'text/javascript';
mathjaxscript.src = 'https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.3/latest.js?config=TeX-AMS-MML_HTMLorMML';
var configscript = document.createElement('script');
configscript.type = 'text/x-mathjax-config';
configscript[(window.opera ? "innerHTML" : "text")] =
"MathJax.Hub.Config({" +
" config: ['MMLorHTML.js']," +
" TeX: { extensions: ['AMSmath.js','AMSsymbols.js','noErrors.js','noUndefined.js'], equationNumbers: { autoNumber: 'AMS' } }," +
" jax: ['input/TeX','input/MathML','output/HTML-CSS']," +
" extensions: ['tex2jax.js','mml2jax.js','MathMenu.js','MathZoom.js']," +
" displayAlign: '"+ align +"'," +
" displayIndent: '"+ indent +"'," +
" showMathMenu: true," +
" messageStyle: 'normal'," +
" tex2jax: { " +
" inlineMath: [ ['\\\\(','\\\\)'] ], " +
" displayMath: [ ['$$','$$'] ]," +
" processEscapes: true," +
" preview: 'TeX'," +
" }, " +
" 'HTML-CSS': { " +
" availableFonts: ['STIX', 'TeX']," +
" preferredFont: 'STIX'," +
" styles: { '.MathJax_Display, .MathJax .mo, .MathJax .mi, .MathJax .mn': {color: 'inherit ! important'} }," +
" linebreaks: { automatic: "+ linebreak +", width: '90% container' }," +
" }, " +
"}); " +
"if ('default' !== 'default') {" +
"MathJax.Hub.Register.StartupHook('HTML-CSS Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax['HTML-CSS'].FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
"VARIANT['normal'].fonts.unshift('MathJax_default');" +
"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
"});" +
"}";
(document.body || document.getElementsByTagName('head')[0]).appendChild(configscript);
(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
}
</script>
</div>
</article>
</div>
</div>
<footer class="footer">
<div class="container">
<div class="row">
<ul class="col-sm-6 list-inline">
<li class="list-inline-item"><a href="/archives.html">Archives</a></li>
<li class="list-inline-item"><a href="/tags.html">Tags</a></li>
<li>©Alberto Verga (2025)</li>
</ul>
<p class="col-sm-6 text-sm-right text-muted">
<a href="https://github.com/getpelican/pelican" target="_blank">Pelican</a> / <a href="https://getbootstrap.com" target="_blank"><img alt="Bootstrap" src="/theme/css/bootstrap-solid.svg" style="height: 18px;"/></a> / <a rel="license" href="http://creativecommons.org/licenses/by-nc/4.0/"><img alt="Creative Commons License Non-Commercial 4.0" style="border-width:0" src="https://i.creativecommons.org/l/by-nc/4.0/88x31.png" /></a> CC 4.0
</p>
</div>
</div>
</footer>
</body>
</html>