Jul 22 2012

## Quantum dynamics of single trapped ions, Sections I-III

Mostly a collection of notes to myself. I'm reading through a review article called Quantum Dynamics of single trapped ions, (Leibfreid, et al, Rev. Mod. Phys. 75, 281). It seems to be a pretty good description of the basic physics involved in ion traps, or in Paul traps at any rate. What follows is some highlights that I thought were interesting.

#### Sec. II

• Sec. II starts with the classical solution of an ion in an oscillating rf potential with some additional static potential. We start with a potential of the form
$\Phi(x,y,z,t) = \frac{U}{2}( \alpha x^2 + \beta y^2 + \gamma z^2) + \frac{\tilde{U}}{2}\cos(\omega_{rf} t) (\alpha' x^2 + \beta' y^2 + \gamma' z^2)$
and then try to write down the equations of motion. If you write down the classical EOM, and make the following substitutions
$\xi = \omega_{rf}t/2$, $a_x = \frac{4Z|e|U\alpha}{m \omega_{rf}^2}$, $q_x = \frac{2 Z |e| \tilde{U} \alpha'}{m \omega{rf}^2}$
then the equation of motion can be transformed into the Mathieu equation, which is (sort of) nicely solvable.
• The stable solutions of the Mathieu equation are periodic and have the form
$x(\xi) = A e^{i \beta_x \xi}\sum_{n = - \infty}^{\infty} C_{2n}e^{i 2n \xi} + Be^{-i\beta_x \xi}\sum_n D_{2n}e^{- i 2n \xi}$.
In general one obtains continued fractions for the coefficients and $\beta_x$. I won't go into the details, but I need to get the notation started. One important result: a trapped particle is stable in all three dimensions if $0 \leq \beta_i \leq 1$ for all $i$.
• It's easiest to understand the dynamics in the lowest order case, where $(|a_x|, q_x^2)\ll 1$. We get:
• $\beta_x \approx \sqrt{a_x + q_x^2/2}$
• $x(t) \approx 2 A C_0 \cos(\beta_x \frac{\omega_{rf}}{2} t)(1 - \frac{q_x}{2}\cos(\omega_{rf}t))$
• In the above: we see motion at the secular frequency of $\beta_x \omega_{rf}/2$ (denoted $\nu$ by the authors), modulated by micromotion at the rf frequency.
• The quantum solution isn't really much different (which is not exactly surprising). The authors follow a fully quantum mechanical treatment due to Glauber. You basically use the eigenstates of a static potential oscillator of frequency $\nu$ to build a basis for the time-dependent problem. These states are not actually energy eigenstates of the full time-dependent potential, because their energy is periodically altered through micromotion.

#### Sec. III

• We start with the usual (and instructive) two-level atom approximation with the states $|g\rangle$ and $|e\rangle$.
• With some re-definitions, we can use the following Hamiltonian to account for dipole, quadrupole, and Raman transitions:
$H^{i} = \hbar \Omega/2(\sigma^+ + \sigma^-) ( e^{i(k x - \omega t + \phi)} + e^{-i(kx - \omega t + \phi)})$
• Then you go into an interaction picture to get rid of the motion and electronic terms in the Hamiltonian, and apply the rotating wave approximation. We get:
$H_{int}(t) = \hbar \Omega /2 \sigma^+ \exp(i(\phi + \eta[ a u^*(t) + a^\dagger u^(t)] - \delta t) + \rm{H.c.}$
$\eta = k x_0 = k \sqrt{\hbar/(2 m \nu)}$
$u(t)$ is a periodic function that basically accounts for driven motion, and is discussed in detail in Sec. II. $\eta$ is the Lamb-Dicke parameter.
• Plugging in the exact solution for $u(t)$, you can see that there are resonant terms at multiples of the secular and micromotion frequencies and their combinations. If you look only at the secular sidebands, and make the usual assumptions of $\eta \ll 1$ and $(a_x, q_x^2) \ll 1$, then the interaction Hamiltonian becomes
$H_{int}(t) = \hbar \Omega_0 /2 \sigma^+ \exp [ i \eta (a e^{-i \nu t} + a^\dagger e^{i \nu t})]e^{i(\phi - \delta t)} + \rm{H.c.}$
$\Omega_0 = \Omega(1 + q_x/2)$ is a rescaled interaction strength.
• The Lamb-Dicke regime is where the ion's wave function is much smaller than $1/k$. (i.e. the light wave does not vary much over the extent of the ion's wavefunction.) In this regime, the interaction contains three resonant terms: the carrier resonance, and sidebands at $\pm$ the secular frequency. The Rabi frequencies are different:
• $\Omega_{n,n-1} = \Omega_0 \sqrt{n} \eta$
• $\Omega_{n,n+1} = \Omega_0 \sqrt{n+1}\eta$.
• Higher order sidebands are suppressed by successive powers of the Lamb-Dicke parameter.

May 18 2012

## Property rights are a bad foundation for your free market philosophy

I'm all for markets. Markets work great a lot of the time. They're efficient, they demonstrate self-organizing behavior, and they have built-in feedback mechanisms to keep them flowing efficiently.

But they don't always work as advertised. Conservatives, in the mold of Ayn Rand (say), defend capitalism from charges of market failure by means of a moral argument. They say that capitalism is the only moral way to allocate capital. They say that taxation is theft (or slavery). They say that to it is immoral for the state to take a man's property without his consent.

But this rather misses the point. The concept of property exists because it's useful in organizing society, not because you have some "natural right" to own land.

For instance, most of us recognize that there's a certain amount of value to society in having someone extract oil from the earth. Before it's extracted, it seems like a dubious claim to say it "belongs" to anyone--if anything, it's held in common. But since most of us don't feel like putting the effort into drilling it, we come up with a deal: if you want to drill it up for us, we'll grant you certain rights to the oil you drill, such as the right to bring it to market and sell it to us at market rates. But this right exists only as a result of this tacit agreement. You have the right to sell it under the conditions of this agreement, not under any conditions you set. If you decide that you're going to collude with the three other guys who are drilling oil to drive the price far above some rate the rest of us are willing to pay, and we decide to impose price controls or nationalize your refineries, we're not infringing on some fundamental right you have. Any entitlement you had existed only within the context of an agreement--an agreement which no longer exists.

Markets. They're good policy (usually), but bad philosophy (always).

Mar 29 2012

## More fun with FRED: analyzing data and reaching dubious conclusions edition

A quick followup to yesterday's post, I thought it might be interesting to Fourier transform the FRED data on national average regular unleaded gasoline prices. The data only go back to 1990, which makes it a bit difficult to draw conclusions about fluctuations on a timescale of less than about a year, but we do what we can. On the bright side, we get lots of information about high frequencies since the data are sampled weekly.

Here's what it looks like at frequencies up to about 5/yr.

And here's what it looks like zoomed in to low frequency fluctuations, the highest being about 1/yr.

Like I said, the data don't go back far enough to get convincing data on fluctuations happening at a few years, but it looks like there could be some peaks corresponding to fluctuations with periods 2 years and 4 years. Maybe elections noticeably change gas prices.

Mar 27 2012

## Fun With FRED: Lying with statistics about gasoline prices edition

I've just started playing around with the St. Louis Fed's FRED tool for easily browsing economic statistics. I highly recommend it: it's easy to get statistics on more or less any economic data you want, and lets you play around with the data with ease. For the inaugural post of my playing with FRED, I thought I'd pick some low hanging fruit and point out people lying with numbers.

Take, for instance, this moron at WorldNetDaily:

Do you know what a gallon of gasoline cost on the day Barack Obama was sworn into office Jan. 20, 2009?

$1.84 Why is the price of gasoline the day a president is sworn in a meaningful value to compare to? Answer: because it's the most convenient. Of course, gasoline was over$4 in 2008.

Jan 14 2012

## Science can determine moral values?

I've been listening to a few of Sam Harris's talks about his Moral Landscape. The full disclaimer that I have not yet read his book applies here, and I could be grossly mischaracterizing his argument. But I've heard a few of his discussions at length on the topic, so I think I have some kind of general idea of the claims and the standard objections he raises.

I don't know how I feel about it. I know, I know, I'm violating the first commandment of blogging by not having a strong opinion on the argument, but that's the only way I know how to respond to a book which claims to solve a branch of philosophy that's thousands of years old.

First, I think I should say that I'm sympathetic to the idea that science can determine moral truths. Understanding morality in terms of the "well being of conscious creatures" seems eminently sensible, and in many cases that's a subject science can say something about.

But I'm still not entirely convinced.

I'm fine with using well being as a foundation of morality. And Sam makes lots of references to the moral atrocities of human history: slavery, genocide, and so on, and rightly would argue that we've made moral progress insofar as we have disabused ourselves of the idea that these practices are somehow acceptable ways to behave.

Problem is, these are the easy questions. You don't need science to figure out that beating your children is a perhaps somewhat suboptimal way to improve human happiness. All the cases of moral progress to which Sam makes reference weren't accomplished by scientific inquiry.

There might be an exception to this in the case of mental health. The scientific treatment of psychology has allowed us to understand types of suffering that didn't conceptually exist in the past. (I believe Sam has made this point actually.) Things like depression couldn't even enter into the moral calculus two hundred years ago. It also seems likely that as our understanding of consciousness grows, we will learn more about what kinds of happiness and suffering are available to nonhuman animals, and this will inform the ethical discussion of how we ought to treat them.

So it's not immediately obvious to me that science can't help us determine what we ought to value morally. But this still seems to fall short of a prescriptive ethics. If there's a weakness in the argument, it probably lies in how we are to measure happiness. Do all humans enter with equal weight into a moral calculation? Should I value the happiness of strangers as a discount to my own? The families of strangers to one's own family? Our moral intuitions seem to favor self sacrifice, assigning a moral value to the lack of consideration of one's own well being. I'm not sure there's an obvious empirical way to handle these questions.

Dec 29 2011

## There are no agnostics

At least, not in the popular sense of being entirely undecided about the existence of god(s). Ok, fine. Maybe there's a select few who are able to maintain a completely open mind about the existence of deities, but it seems like such a worldview is about as difficult to hold as it is to play a chaotic neutral character in D&D--and for roughly the same reasons.

Since I'm feeling lazy, I'll just argue this one by analogy. Suppose somebody makes a dire prediction. It doesn't really matter what it is, but let's just say he tells you that you're going to die in some horrifying way this very night. Your interlocutor, however, helpful fellow that he is, also informs you that you can avoid this gruesome fate if you pay him five hundred dollars and performing some complicated ritual in your backyard.

The gauntlet has been thrown. You have to play the game. Either you decide to pay off your prognosticator of doom or you don't. If you think that there's even some reasonable chance that he's right, then you're going to follow his instructions. After all, even if you think there's only, say, a 25% chance that he's telling the truth it's quite a bargain. We spend quite a lot more to deal with threats that are much less likely than one in four. Only in the event that you think he's completely bonkers do you not.

So it is with religions and their promises of eternal damnation or bliss. Either you think it's reasonably likely that there's a judging god, in which case you'd be religious, or you don't--in which case you aren't. The sheer immensity of religions' claims make it impossible to be agnostic about them. And yes, I know the argument doesn't work for deistic gods or other supernatural agents who don't make moral judgments. But I'm not really interested in discussing gods that don't interact with the world.

Nov 27 2011

## Real life superheroes?

So apparently there's a "real life superhero" movement. Why? I don't know. But I won't let that stop me from armchair psychoanalyzing. Now, before I get started, all the usual disclaimers apply: I'm sure all of the "real life superheroes" are socially conscious and responsible citizens. They're not really (not in particular, anyway) what I'm trying to get at.

The question I'm more interested in is whether comic book superheroes should serve as role models for proactive citizenship. And I don't think it should. It's a bit hard to put my finger on exactly what bothers me about it, but I'll give it a shot.

There seems to be something deeply right-wing about superheroism. It's this... sort of pathological obsession with crime. For one thing, in order to endorse the need for superheroes, you need to believe that the police are basically failing in their policing. And not just in need of a few improvements, but really failing in a fundamental way. Like, crime running rampant in the streets failing. And I don't think there's any evidence for that.

Second, of all the problems that are in dire need of solving in either your community or the world at large, is street crime really the most important? This point is a bit of a slippery slope and so I should clarify a bit. I spend a lot of time trying to build quantum computers, and the case could probably be made that my efforts would be better spent working on, say, sustainable farming. So I guess the issue really gets at the psychology of the whole business. It seems like holding up vigilantism as exemplary of concerned, responsible citizenship reflects an underlying fear about the world that is largely unjustified. And maybe more importantly, it's easy for a culture of vigilantism to take on violent, nationalistic undertones.

Jul 20 2011

## Now I'm no economist...

... but some conservatives' claims about the economy just don't make sense. Robert Reich recounts an encounter with a "conservative Republican," who repeated the line I must have heard Avagadro's number of times by now:

“The Depression ended because of World War Two,” [the Republican] pronounced, as if government had played no part in it.

Wait. What? No, really. What?

What was World War Two if not a giant government spending program? While I could be wrong, I'm pretty sure it wasn't the private sector buying all those B52's they built...

Do people seriously consider this a plausible argument against government stimulus?

Jul 12 2011

## In which Randal Rauser is wrong about everything

Randal Rauser, over at the Tentative Apologist, attempts to demolish the following claim from a reader of his:

Intelligent Design could only be a valid scientific explanation if you were proposing an intelligent entity that is bound by the laws of physics.

Randal goes on for several paragraphs, mostly repeating the same idea over and over again (emphasis Randal's):

The problem is that we don't understand the laws of physics as they ultimately are.

Uhm. No.

When quantum mechanics was first developed, it was not necessary that our understanding of atoms fit with previously known classical physics. When developing a theory of atoms, scientists didn't proceed under classical assumptions. They did, however, proceed under the assumption that however atoms did work, they worked in a consistent and repeatable way, and their behavior could in principle be described by a set of laws.

We didn't have to know what those laws were to make that claim. But without the assumption that atoms operate by a set of laws, there's no science.

Science rests on a set of assumptions. One of them is that empirical observation can yield knowledge about reality. It's conceivable, perhaps, that physics works one way while someone is doing an experiment, and an entirely different way when a scientist isn't looking at them. You could write a theory about this. You could even have equations if you wanted. But it wouldn't be a scientific theory.

If you want to both postulate an intelligent designer of the universe and write down a scientific theory about it, then you've got to insist that your intelligent designer operates according to some set of rules. This designer must interact with the world in a consistent way. So far, my impression is that the intelligent design crowd wants their designer to be somehow apart from physics; a designer that cannot even in principle be bound by physical laws.

It's not even that hard. You could, for instance, propose that we're living in a simulation. (How exactly you'd test this is hard. But I can imagine it could be done. If you think we're living in a classical simulation, you could try to build a quantum computer.) But you can't propose that whoever wrote the simulation--and even the simulation itself--is not bound by any consistent physics.

Jul 9 2011

## Clever title

Jerry asks for a coherent definition of free will. I'm not a sophisticated philosopher, but to be honest I don't really understand the trouble here. It seems to depend entirely on what it is that you want free will to "do."

Jerry's definition of free will seems to be something along the lines of the following: suppose we take a series of snapshot of an individual's brain before and up to immediately after a "choice" appears to have been made. At each stage in the slideshow, their neurons are in some configuration that somehow encodes their state of consciousness. Each configuration should follow, in a physical sense, from the one immediately prior. Presumably, if a contra-causal choice occurs somewhere in this process, there should be a change in neuron configuration that doesn't follow from the previous one. I tend to think of these kind of choices as sort of discontinuities in neural configuration.

This definition, while I suppose perfectly fine as it goes, suffers the problem that at some stage there must be a non-physical. intervention in the brain's path through time. Somewhere, an apparently uncaused physical effect has to occur. Such a notion seems unlikely, given what we know about how the universe works.

If we accept that no neural state is physically uncaused, then we're forced to accept that our actions are dictated by physics. You can no more have chosen chocolate ice cream instead of vanilla than fly by jumping out a window. Fine. What of it?

To me, the entire issue comes down to what you want from this concept of free will. Do you want to hold people responsible for their choices? (Actions, perhaps?) You don't need contra-causal free will to do that.

In my view, an action can be called freely chosen if it can be explained by processes within a system (perhaps in response to external stimuli). But, I hear you object, (or I would, if anyone read this blog), that encompasses many things. Computers, for instance.

Fine; it includes computers. But there's no point in thinking about computers having free will. The concept is useless insofar as our interactions with them are concerned. If computers don't do what we want, we can reprogram them. We can give them new rules to operate under.

Humans, on the other hand, we can't (yet?) reprogram. Our basic neural wiring is, presently at least, unchangeable in any predictable way. It's useful to think about classifying human minds as evil or immoral precisely because we can't change them. (And certainly we can call destructive minds evil just we can cancer.) Perhaps one day brains will be as programmable as computers. Perhaps one day we'll understand how consciousness works physically. Then, perhaps, free will won't be a useful concept.

Until then, I don't think I understand why determinism is such a problem.