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units.lyx
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#LyX 1.6.6 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
\textclass revtex4
\options prl, twocolumn
\use_default_options true
\language english
\inputencoding auto
\font_roman default
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\font_sf_scale 100
\font_tt_scale 100
\graphics default
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\spacing single
\use_hyperref false
\papersize default
\use_geometry false
\use_amsmath 1
\use_esint 1
\cite_engine basic
\use_bibtopic false
\paperorientation portrait
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\papercolumns 1
\papersides 1
\paperpagestyle default
\tracking_changes false
\output_changes false
\author ""
\author ""
\end_header
\begin_body
\begin_layout Title
System of units for Phase Qubit design
\end_layout
\begin_layout Author
Daniel Sank
\end_layout
\begin_layout Date
28 November 2010
\end_layout
\begin_layout Section
Goal
\end_layout
\begin_layout Standard
It's really useful to work with a system of units in which quantities defining
qubit design have values near 1.
In particular we want to choose units suitable for the physical circuit
elements we use, ie.
inductors measured in picohenries, etc.
\end_layout
\begin_layout Section
Choice of Fundamental Dimensions
\end_layout
\begin_layout Standard
There are four fundamental physical dimensions:
\end_layout
\begin_layout Itemize
Frequency,
\begin_inset Formula $f$
\end_inset
\end_layout
\begin_layout Itemize
Length,
\begin_inset Formula $l$
\end_inset
\end_layout
\begin_layout Itemize
Mass,
\begin_inset Formula $M$
\end_inset
\end_layout
\begin_layout Itemize
Charge,
\begin_inset Formula $Q$
\end_inset
\end_layout
\begin_layout Standard
In circuit design we don't think about lengths, charges, or masses, so we'd
like a new set of four dimensions that trade these three for dimensions
like capacitance, inductance, and resistance.
In other words, we want to recombine the four usual dimensions into four
\begin_inset Quotes eld
\end_inset
electronic
\begin_inset Quotes erd
\end_inset
dimensions.
To do this we look at how electronic quantities are made out of the four
fundamental ones,
\end_layout
\begin_layout Itemize
Inductance:
\begin_inset Formula $Ml^{2}/Q^{2}$
\end_inset
\end_layout
\begin_layout Itemize
Flux:
\begin_inset Formula $Ml^{2}f/Q$
\end_inset
\end_layout
\begin_layout Itemize
Current:
\begin_inset Formula $fQ$
\end_inset
\end_layout
\begin_layout Itemize
Voltage:
\begin_inset Formula $Ml^{2}f^{2}/Q$
\end_inset
\end_layout
\begin_layout Itemize
Capacitance:
\begin_inset Formula $Q^{2}/Ml^{2}f^{2}$
\end_inset
\end_layout
\begin_layout Itemize
Action:
\begin_inset Formula $Ml^{2}f^{2}$
\end_inset
\end_layout
\begin_layout Standard
Frequency is a useful unit in circuit design so we keep that one.
When designing devices we think in terms of junction current and circuit
loop inductances.
We choose current as a fundamental dimension.
Since current is frequency*charge our choices of frequency and current
is equivalent to a choice of charge.
We have therefore locked down two of the four dimensions, frequency and
charge, and made no use of the other two, mass and length.
We can lock down the product
\begin_inset Formula $Ml^{2}$
\end_inset
by choosing inductance to be a fundamental dimension.
This leaves us to choose either a unit of mass or length and we choose
length.
\end_layout
\begin_layout Section
Unit System
\end_layout
\begin_layout Standard
We now choose standard units for our four fundamental dimensions.
Qubit frequencies are in the GHz range so we measure frequency in GHz.
For current and inductance we use uA and pH respectively.
Length is unimportant for our work but we choose um.
These four choices completely determine our system of units.
The complete system, which I call the
\begin_inset Formula $\textbf{Circuit Designer}$
\end_inset
system (CD), is summarized in Table
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:unitSummary"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="12" columns="3">
<features>
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Dimension
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
symbol
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
unit
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Frequency
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $f$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
GHz
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Current
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $I$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
uA
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Inductance
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $L$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
pH
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Capacitance
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $C$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
uF
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Resistance
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $R$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
m
\begin_inset Formula $\Omega$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Flux
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\Phi$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $10^{-18}$
\end_inset
W
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Charge
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $Q$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $10^{-15}$
\end_inset
C
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Action
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $S$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $10^{-33}$
\end_inset
Js
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Energy
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $E$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $10^{-24}$
\end_inset
J
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Mass
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $M$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $10^{-30}$
\end_inset
Kg
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Length
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $l$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
um
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\begin_inset Caption
\begin_layout Plain Layout
Complete
\begin_inset Formula $\textrm{\textbf{C}ircuit}$
\end_inset
\begin_inset Formula $\textrm{\textbf{D}esigner}$
\end_inset
(CD) unit system
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:unitSummary"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
We need to know the values of physical constants in this system.
These values are given in Table
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:contants"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="4">
<features>
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Constant
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
symbol
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
SI value
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
CD value
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Planck's constant
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $h$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $6.626*10^{-34}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $0.6626$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
flux Quantum
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\Phi_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2.0678*10^{-15}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $2067.8$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
electron charge
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $e$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.6*10^{-19}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $1.6*10^{-4}$
\end_inset
\end_layout
\end_inset
</cell>
</row>
</lyxtabular>
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption
\begin_layout Plain Layout
Physical constants in the SI and CD unit systems
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Flo:contants"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Finally we give the important qubit relations in Table
\begin_inset CommandInset ref
LatexCommand ref
reference "Flo:qubitRelations"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Tabular
<lyxtabular version="3" rows="4" columns="3">
<features>
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Quantity
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Formula
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Value
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $L_{J_{0}}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\Phi_{0}/2\pi I_{0}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $318/I_{0}$
\end_inset
\end_layout
\end_inset
</cell>
</row>