# The Quantum Wavefunction

Nagaganesh Jaladanki

###### Abstract

When most people think of physics, they think of what they learned in high
school physics: that the world is fundamentally predictable. Given the
position and velocity of a particle in space, it should be possible to predict
its position at any moment in the future—right? Though this was thought to be
true for thousands of years, recent developments in the field of physics have
shown that this isn’t actually true. Instead of being fundamentally
predictable, the universe is fundamentally unpredicable. However, this doesn’t
seem to make sense. What happened to the centuries of physics developed by
Newton, Bernoulli, and Lagrange? Well, as it turns out, they weren’t actually
wrong. Their equations were actually an approximation of a formula called the
wavefunction, which is the “Newton’s Laws” equivalent for modern physics. In
this paper, we’ll take a look at this peculiar wavefunction—and why our
intuition about the world is completely wrong.

This is Quantum Physics.

##  1 Introduction and Background

###  1.1 Quantum vs. Classical

Let’s think back to our example of a particle moving through space. This
particle would have (among many) two properties that we would be able to
measure: its position $\vec{x}$ in space, and its momentum $\vec{p}$.

.$\vec{x}$$\vec{p}$ Figure 1: Because we need a figure for everything.

In classical physics, it would have been possible to know both of these values
with exact precision. And we could then have coupled this information with the
(classical) laws of physics to predict the future of this particle at any
moment. If we think about this for a moment, it seems to make sense. If we
release a box in deep space at some velocity, we would expect it to stay at
that velocity for millions of years, as there is nothing that would block its
movement. As a result, we can use some simple equations to predict its exact
movement and position.

This, essentially, forms the backbone of classical physics. If we know the
state of the universe at a single moment in time, then we can use the laws of
physics to predict the future. This is known as determinism—the idea that
everything can be determined from an initial value. And this is exactly what
quantum physics breaks.

Here are the bare facts: quantum physics states that there is a limit to how
precisely we can measure the position and momentum of anything in the
universe. This is known as the Heisenberg Uncertainty Principle. It says that
the error of measurement of the position and momentum have to be greater than
a specific (though very small) value. In this case, the term “error of
measurement” doesn’t concern our ability to measure these properties—the
Uncertainty Principle is true no matter what device we use to measure position
and momentum. As a result, the more specifically we know position, the less
sure we can be for the particle’s momentum. Similarly, the more information we
have about the particle’s momentum, the more uncertain we are about its
position. You can see this in Figure 2.

The more we knowabout the position……the less we knowabout
momentum.$\vec{x}$$\vec{p}$

The less we knowabout the position……the more we knowabout
momentum.$\vec{x}$$\vec{p}$

Figure 2: The Uncertainty Principle

To understand the Uncertainty Principle, suppose that in a special deal, a
restaurant is offering an all-you-can-eat for one plate of food. There are two
separate piles of food, which correlate to the lists A and B. At the
restaurant, all of the food that a customer takes can be from pile A, or it
can be from pile B. This is comparable to knowing exactly where a particle is,
or knowing exactly its momentum. However, the customer can also choose to fill
half of his plate with items from pile A, and half with items from pile B. In
this case, the scientist would know with only some certainty on where the
particle is and some certainty on the particle’s momentum.

One could ask: why don’t we see this in everyday life? It doesn’t take much
effort to find the position and the momentum of a car on the highway. The
answer is that the objects that we handle every day are so big and full of
energy (compared to the quantum world) that the effects of quantum physics are
barely measurable. Where are they measurable? Let’s see in a small example.

Imagine the same scenario as Figure 1, but replace the ball with an electron.
And, being the scientists that we are, we decide to measure the electron’s
position and momentum. Because of the Uncertainty Principle, we can’t be
completely certain of both position and momentum. This is how we can visualize
it:

Figure 3: Electrons and Uncertainty

We can’t. This is because the electron is very, very, small, and the levels of
uncertainty are much bigger than the size of the electron. “Small” is a word
that quantum physics likes, as most of its effects are apparent at these
levels. Consequently, we can visualize a car or a box very easily, but not an
electron. However, this doesn’t mean that the electron doesn’t exist. It still
exists, but may be anywhere within the uncertainty bound—which will be very
high for an electron.

That’s a curious thing to think about: the fact that the electron, a particle,
can be anywhere in a wide region of space or have a range of values for
momentum. At this point, the electron doesn’t seem to act like a particle
anymore.

###  1.2 The Wave-Particle Duality

When we said that the particle could be anywhere within the bound of
uncertainty when we measured it, we were actually lying. The truth isn’t that
the particle could be anywhere within the bound. It is actually everywhere
within the bound. Yet at the same time, it’s still a particle. This idea is
what led to the creation of the Wave-Particle Duality.

The Wave-Particle Duality states that every particle acts as both a particle
and a wave. To understand this a little better, think of a particle as a
cylinder. There are two ways of looking at it. Viewed from the top, it appears
as a circle, while a side view paints a rectangular picture. Likewise, we can
think of a particle as having the properties of both a particle (intuitive)
and a wave (unintuitive).

So far, you’ve been taking the things that I’ve been saying as true. But we
need to be grounded in reality, so let’s take a look at an experiment that
provides a real life basis for all this theory we’ve been talking about.

The experiment is as follows: a stream of very small particles (such as
electrons) are fired at a small barrier with two narrow slits in it. The
particles that go through the slit are then recorded on a particle detector
that lies far from the detector. The experimental apparatus is shown in Figure
4.111This experiment is commonly known as Young’s experiment.

Figure 4: The Experimental Setup.

With this figure, it’s easy enough to see that the detector will have two
broad stripes corresponding for each of the slits. This is true for most
objects that we can see. For example, consider that we were hurling baseballs
to go between the slits. Most of the baseballs will bounce back after missing
the slit. However, the few that make it through will hit the detector in one
of the two big stripes.

However, once again, this doesn’t hold true for very small particles, such as
electrons or photons. Instead of two stripes, we see a very interesting
pattern, which you can see in Figure 10.

Figure 5: The Experiment for Waves

In Figure 10, the $y$-axis refers to the point on the detector which is
exactly between the two slits, and the $x$-axis lies along the
detector.222This graph is just an approximation to the actual, more
complicated data. However, the essential features have remained unchanged. In
addition, the crests represents areas where many particles hit the detector,
and the troughs show the areas which were hit relatively fewer times. Even at
first glance, we can tell that something is pretty wrong here. For example,
why are the particles hitting the detector at multiple spots? And how do they
hit the detector so far from the origin? Both of these questions are answered
by the Wave-Particle Duality.

As they say, a picture is worth a thousand words. Take a look at Figure 6 for
a visual. As you can see, the particles are behaving as waves when they pass
the slit. The waves then interfere, producing constructive and destructive
interference. By the time the particles reach the detector, they have
interfered greatly, producing the unintuitive pattern that we see in Figure 5.
However, that’s not all there is to it. It’s important to note that different
electrons are not interfering with each other—each electron is interfering
with itself.

Figure 6: Particles in the Experiment.

Each electron is producing a wave that passes through both slits, interferes
with itself, and lands on the detector in one spot. As a result, this pattern
would appear even if each particle was released separately. So far, we’ve been
dealing with the theory of quantum physics. Let’s make this a little more
concrete. But how do we do that?

Math is the answer. (It always is.) We can quantitatively describe what is
happening to the particles when they become waves by graphing their wave
function. How do we get ahold of this wavefunction?

Behold the greatest formula of our time. Schrödinger’s equation.

${E\Psi(x)=U\Psi(x)-\frac{\hbar}{2m}\frac{d^{2}\Psi(x)}{dx^{2}}}$

##  2 The Holy Wavefunction

###  2.1 What is the Wavefunction?

Though they seem similar, the wavefunction and Schrödinger’s equation are
separate things. The wavefunction describes the Wave-Particle action of a
system (or a particle), while the Schrödinger equation gives you a way to
generate the wavefunction of a particular system.

Before we take a look at why Schrödinger’s equation makes sense, let’s take a
look at what it’s trying to say.

${E\Psi(x)=U\Psi(x)-\frac{\hbar}{2m}\frac{d^{2}\Psi(x)}{dx^{2}}}$

$E$ = Energy in the system
$\Psi(x)$ = The Wave Function of the System
$U$ = The potential energy
$\hbar$ = The reduced Planck constant
$m$ = The mass of the system
$\frac{d^{2}\Psi(x)}{dx^{2}}$ = The second derivative of the wavefunction.

Figure 7: Understanding Schrödinger’s equation.

Most equations are like a map. Schrödinger’s equation is as well. You can use
the legend in Figure 7 to understand what the symbols in this equation mean.
As you may have seen, many of the symbols in this equation are simple symbols
from classical physics.

In this equation, the letter $\Psi$ (pronounced sigh and written as Psi)
represents the wavefunction.

Now, let’s take a look at how this equation is consistent with some previous
examples of physics, such as the conservation of energy and other properties.

###  2.2 The Phony Derivation

Disclaimer: This isn’t actually a derivation. Schrödinger’s equation is
actually a fundamental principle of physics, so it isn’t actually possible to
“derive” it using other principles. Instead, we will merely show that it is
compatible with the laws of physics.

First, let’s establish some basic principles that we will modify later on. To
begin with, the wavelength $\lambda$ of a particle (due to the Wave-Particle
Duality), is:

$\lambda=\frac{h}{p}$ (1)

where $h$ is Planck’s constant and $p$ the magnitude of the momentum.

We also know from the Wave-Particle duality that particles act like
waves—something like a pendulum moving back and forth. How do we represent a
wave mathematically? Using a differential equation:

$\frac{d^{2}\Psi(x)}{dx^{2}}=-k^{2}\Psi(x)$ (2)

For this wave, $\Psi$ represents the wavefunction (as we have seen that it
acts like a simple wave) and $k$ represents the wavenumber, or “how quickly
the wave moves”. The variable $k$ is defined to be:

$k=\frac{2\pi}{\lambda}$ (3)

Almost immediately, we can see that we can shift some variables around to get
a slightly diferent equation. Combining Equation 3 with Equation 1, we can
get:

$k=\frac{p}{\hbar}$ (4)

where $\hbar$ is just $\frac{h}{2\pi}$. We seem to be getting somehwere.
Replacing $k$ in Equation 2 and doing some rearranging,

$\frac{d^{2}\Psi(x)}{dx^{2}}=-\frac{p^{2}\Psi(x)}{\hbar^{2}}$ (5a)

$-\frac{d^{2}\Psi(x)}{dx^{2}}\hbar^{2}=p^{2}\Psi(x)$ (5b)

Our equation is taking form from the symbols. To get further, we decide to
divide both sides by $2m$.

$-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}=\frac{p^{2}}{2m}\Psi(x)$ (6)

Now, remember that:

$\frac{p^{2}}{2m}=K$ (7)

where $K$ is the kinetic energy in the system.333Assuming nonrelativistic
speeds. This leads to:

$-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}=K\Psi(x)$ (8)

Let’s relate this to the Principle of Conservation of Energy, which states
that the total energy in a system is equal to the kinetic energy added to the
potential energy.

$E=K+U$ (9a)

$K=E-U$ (9b)

Plugging this into Equation 8 and rearranging, we get:

$-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}=(E-U)\Psi(x)$ (10a)

$-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}+U\Psi(x)=E\Psi(x)$ (10b)

$E\Psi(x)=U\Psi(x)-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}$ (10c)

Voila! Our equation is identical to Schrödinger’s equation. Let’s take a look
back at what we’ve done. We’ve shown that Schrödinger’s equation is compatible
with Conservation of Energy and the Wave-Particle Duality. Nevertheless, this
is still a phony derivation. Schrödinger’s equation is a fundamental aspect of
nature, so it isn’t possible to derive it—you can only show that it’s
compatible with other fundamental principles of nature, which is what we’ve
shown. The version that we’ve just shown is true is known as the one-
dimensional time-independent Schrödinger equation.

We now know why Schrödinger’s equation is what it is. However, what does it
mean?

###  2.3 Remarks

Let’s take a few moments to understand what exactly $\Psi(x)$ is. We’ve
already established that it represents the wave that particle acts as.
However, we can ask: what does the wave displace? Water waves displace water,
and sound waves displace pressure. Unfortunately, the waves that $\Psi(x)$
represents don’t displace anything. In fact, they don’t even exist. The wave
function acts as a mathematical representation of what happens in reality. If
we base this off of experimental data, we find that the particles merely act
as waves. No one knows whether they are or aren’t.

Though the wave that the wavefunction represents doesn’t exist, the square of
the wavefunction has a physical meaning. The square of the wavefunction at a
particular location represents the probability that a particle will be at that
location.

This makes a little more sense in context. Here is an example of a probability
density function derived from the wavefunction:

Figure 8: The Experiment for Waves

It looks familiar, as it’s merely the square of the wave function for the
Wave-Particle Duality experiment. However, here’s an example of how it looks
when the experiment is actually done.

Figure 9: Experimental Results.

We can see that the wavefunction matches up with the experimental results,
which is good. However, let’s trace the path of one particle as it goes
through the experiment. As we’ve seen above, the particle acts as a wave until
it reaches the barrier. The instant it reaches the barrier, the wave picks a
position based on probability ($\Psi^{2}(x)$) and collapses into a particle,
which the detector records. This collapse is the reason why we can’t measure
the actual wave in a laboratory experiment—we can only see the effects that it
has.

##  3 Schrödinger’s Equation Applied

In this section, we’ll go over an example of Schrödinger’s equation in action.
In addition, we’ll also assume a little more math and physics knowledge to
speed up the process.

###  3.1 Particle in an Infinitely Deep Well

In this example, we’ll be analyzing an electron trapped in a potential well,
so that its potential energy function something looks like this:

Figure 10: Infinitely Deep Potential Well

We’re also given that the electron is trapped between the points 0 and $L$.
It’s important to note that the electron exists only in one dimension—it’s
just the potential energy that differs between various points in space.

Now that we’ve got the description out of the way, let’s try to to apply
Schrödinger’s Equation to this situation.

$E\Psi(x)=U\Psi(x)-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}$ (11)

We know that the potential energy at any point where the particle can exist
must equal zero, as shown by the graph. As a result,

$E\Psi(x)=-\frac{d^{2}\Psi(x)}{dx^{2}}\frac{\hbar^{2}}{2m}$ (12a)
$-\frac{2Em}{\hbar^{2}}\Psi(x)=\frac{d^{2}\Psi(x)}{dx^{2}}$ (12b)

This equation is really similar to Equation 2, and in fact shares the same
general solution.

$\Psi(x)=A\sin(kx)+B\cos(kx)$ (13)

where $A$ and $B$ are constants (that can be chosen
arbitrarily)444Technically, these constants can be chosen such that they are
imaginary numbers, but for the purposes of this discussion, we won’t be making
it too complex., and $k=\sqrt{\frac{2mE}{\hbar^{2}}}$. Now that it seems like
we have the makings of a solution, let’s think back to the original problem
statement to see if we can narrow down some more things.

We’re tasked with finding the wavefunction of an electron that’s trapped in
this infinitely deep potential well that lies between the points 0 and L. The
electron will never have enough energy to pass either point—there will be no
probability that we will find it outside that range. As a result, we know that
the wavefunction must come to zero at at points 0 and L. We can express this
as follows:

$\Psi(x)=0\>when\>x=0,L$ (14)

As a result, $B$ must be zero—if we had any nonzero constant in front of the
$\cos(kx)$ term, $\Psi(x)$ wouldn’t equal $0$ at the point $0$.

Now, let’s think about the other bound: at L. We need the wavefunction to come
to 0 at L, so:

$\sin(kx)\rightarrow k=\frac{n\pi}{L}$ (15)

where $n$ $\in$ $\mathbb{N}$. However, remember that we have already defined
the constant $k$ when we solved our differential equation. This leads to:

$k=\frac{n\pi}{L}$ (16a) $k=\sqrt{\frac{2mE}{\hbar^{2}}}$ (16b)
$\sqrt{\frac{2mE}{\hbar^{2}}}=\frac{n\pi}{L}$ (16c)
$E=\frac{\hbar^{2}}{2m}\frac{n\pi}{L}^{2}$ (16d)

Once we rearrange the equation so that all of the constants are on one side,
we find a strange result. When we plug in the numbers, we find that the energy
that the electron can have in the well is actually limited to certain values!
This is because everything except the $n$ is constant on the right side of the
equation. The $n$ term, however, can hold positive integer values. As a
result, the energies of the particle in the box can only have specific values.
Let’s try to calculate the first few for an electron trapped in a box 0.5
nanometers wide.

$E=\frac{\hbar^{2}}{2m}\left(\frac{n\pi}{L}\right)^{2}$ (17a)
$E_{1}=\frac{\hbar^{2}}{2(9.11\times 10^{-31})}\left(\frac{(1)(\pi)}{5\times
10^{-10}}\right)^{2}$ (17b) $E_{1}=\frac{(1.055\times
10^{-34})^{2}}{2(9.11\times 10^{-31})}\left(\frac{(1)(3.14}{5\times
10^{-10}}\right)^{2}$ (17c) $E_{1}=2.41\times 10^{-19}$ (17d)

Plugging all of this in, we get $2.41\times 10^{-19}$ joules. Converting in to
electron volts (eV), we get that our equation is approximately equivalent to
1.5 eV for the $n=1$ energy level. Following a similar approach, we get the
$n=2$ energy level to be approximately 4.5 eV. This is a very unintuitive
concept. It tells us that the amount of energy in a system only comes in
certain levels—you can’t have any more or less than that amount. In addition,
it also tells us that there is always at least some energy in a system, since
the smallest value for $n$ is $1$. This lowest energy value is called the
zero-point energy of the system. This idea is known as energy quantization.

Using the information above, we can finally get the general wave function for
this system:

$\Psi(x)=A\sin\left(\frac{n\pi}{L}x\right)$ (18)

You can take a look at the different wavefunctions for different energy levels
in Figures 12 and 12.555Keep in mind that the amplitude of the wave doesn’t
correspond the the energy of the wave—the potential energy well is just there
for reference.

Figure 11: $n$=1 next to the potential well.

Figure 12: $n$ = 2 next to the potential well.

As you can see, the wavefunction changes with each energy level. Because of
this, the probability of finding a particle also changes with each energy
level. Now, let’s use the wavefunction to help us find the probability
function of the particle in the box. The probability $P(x)=\Psi^{2}(x)$.
However, we have to make sure that the all of the probabilities summed
together produce 1. We can do this as follows:

$\int_{0}^{L}P(x)\,dx=1$ (19a) $\int_{0}^{L}\Psi(x)^{2}\,dx=1$ (19b)
$\int_{0}^{L}\left(A\sin\left(\frac{n\pi}{L}x\right)\right)^{2}\,dx=1$ (19c)
$\int_{0}^{L}A^{2}\sin^{2}\left(\frac{n\pi}{L}x\right)\,dx=1$ (19d)

A little bit of math leads to:

$A^{2}\frac{L}{2}=1$ (20)

As we still haven’t chosen the value of $A$, we can now set it such that the
left side of Equation 20 will equal 1.

$A=\sqrt{\frac{2}{L}}$ (21)

We now find that our probability function is:

$P(x)=\frac{2}{L}\sin^{2}\left(\frac{n\pi}{L}x\right)$ (22)

This process of finding the right constants for $A$ (and $B$) is known as
normalizing the wave function. You can find a visualization for the
probability in Figures 14 and 14.

Figure 13: P(x) for $n$ = 1.

Figure 14: P(x) for $n$ = 2.

Once again, we see some strange results. For the $n=1$ energy level, we find
that the particle is more likely to be found in the middle of the well rather
than the sides. Looking at the $n=2$ energy level, we find that it acts
strange in other ways. Namely, we can never find the particle in the middle of
the box—the probability there is zero.

Once again, we find that this makes no sense whatsoever intuitively. Why would
the predicted position change if we add just a little bit of energy to the
particle? No one knows. After all, this is quantum physics: it’s impossible to
get an intuition for.

##  4 Conclusion

It’s important to understand the position that math carries in the field of
quantum physics. Math gives us a window to glimpse the truth in this field,
but after all, it’s only a window. Suppose the Wave-Particle Duality, for
example. In an ordinary laboratory experiment, it’s possible to measure a
particle, but not a particle’s wave. How do we know it’s there, then? We carry
out another experiment which shows a different behavior than than which would
have been expected if the object had acted like a particle.

At this point, we can take the evidence of the discrepant behavior and see if
it matches up with anything else. And it does. By modeling the particle as a
mathematical wave, we can get the behavior that suits the results of the
experiment.

We now have the results that we want, but we’ve gotten them in a very
dangerous way. We’ve “modeled” the particle as a wave, but we have no
experimental evidence that this is true. At this point, we’re using math as a
crutch to support what we think is true about the world.

This isn’t a very safe place to be in for the subject of physics, which is, at
heart, an experimental science. However, as scientists get deeper and deeper
into the field of quantum physics, we will start to understand the true beauty
of the universe as it was, is, and always will be.

## References

  * [1] Ford, Kenneth W., _101 Quantum Question_. Harvard University Press, London, 1st edition, 2011\. Print.
  * [2] Griffiths, David J., _Introduction to Quantum Mechanics_. Prentice Hall, New Jersey, 2nd edition., 1995\. Print.
  * [3] Kopot, Andrey. “Time-Independent Schrodinger Equation.” AKLectures. YouTube 26 Feb. 2014, Web., 29 July 2014.
  * [4] Miller, David A.B., _Quantum Mechanics for Scientists and Engineers_. Cambridge University Press, New York, 1st edition, 2008\. Print.
  * [5] Munrose, Randall., “Purity” xkcd. http://xkcd.com/435/ xkcd 11 Jun. 2008. Web., 30 July 2014.
  * [6] Penrose, Roger, “The Quantum Particle” _The Road to Reality_. Random House, Great Britain, 1st edition., 2004\. Print.
  * [7] Ward, David W., and Sabine Volkmer. “How to Derive the Schrodinger Equation.” ArXiv. 2 Feb. 2008, Web., 27 July 2014.
  * [8] ”Wavefunction Properties” _The Wavefunction_. Georgia State University, 11 Nov. 2010, Web., 29 July 2014.
  * [9] “Schrodinger Equation - A Simple Derivation.” YouTube. YouTube 19 Jan. 2012. Web., 28 July 2014.


# A Spectroscopic Binary at the M/L Transition

Cullen H. Blake11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138<EMAIL_ADDRESS>55affiliation:
Harvard Origins of Life Initiative Fellow David Charbonneau11affiliation:
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138<EMAIL_ADDRESS>66affiliation: Alfred P. Sloan Research Fellow
Russel J. White22affiliation: University of Alabama in Huntsville, Physics
Department, 301 Sparkman Drive, 201B Optics Building, Huntsville, AL 35899
Guillermo Torres11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138<EMAIL_ADDRESS>Mark S.
Marley33affiliation: NASA Ames Research Center, MS 245-3, Moffett Field, CA
94035  Didier Saumon44affiliation: Los Alamos National Laboratory, PO Box
1663, MS F663, Los Alamos, NM 87545

###### Abstract

We report the discovery of a single-lined spectroscopic binary with an Ultra
Cool Dwarf (UCD) primary with a spectral type between M8 and L0.5. This system
was discovered during the course of an ongoing survey to monitor L dwarfs for
radial velocity variations and is the first known small separation ($a<1$ AU)
spectroscopic binary among dwarfs at the M/L transition. Based on radial-
velocity measurements with a typical precision of 300 m s-1, we estimate the
orbital parameters of this system to be $P=246.73\pm 0.49$ d,
$a_{1}\sin{i}=0.159\pm 0.003$ AU, $M_{2}\sin{i}=0.2062(M_{1}+M_{2})^{2/3}\pm
0.0034$ $M_{\sun}$. Assuming a primary mass of $M_{1}=0.08M_{\sun}$ (based on
spectral type), we estimate the secondary minimum mass to be
$M_{2}\sin{i}=0.054M_{\sun}$. With future photometric, spectroscopic, and
interferometric observations it may be possible to determine the dynamical
masses of both components directly, making this system one of the best
characterized UCD binaries known.

stars: low-mass, brown dwarfs, techniques: radial velocity

##  1 Introduction

Near infrared (NIR) surveys, such as 2MASS, DENIS, and SDSS (with its $z$-band
capability), have resulted in a rapid increase in our knowledge of the
properties of stars. This is particularly true for the late M, L and T
spectral types, collectively known as Utracool Dwarfs (UCDs, Kirkpatrick
2005). Today we know of more than 600 L and T
dwarfs111http://www.dwarfarchives.org. Despite these discoveries, we still do
not have a clear understanding of how molecular cloud material assembles
itself into such relatively low mass objects and fundamental properties such
as temperature, age and mass remain coarsely determined. Binary star systems
are a crucial tool for addressing both of these issues. UCD binaries allow us
to directly measure the masses, and possibly the radii, of these objects and
constrain theoretical models of the structure and emergent flux (i.e. Stassun
et al. 2006). Even without determining absolute ages, assuming that the
components of a binary have identical ages can provide constraints on
atmospheric models through estimates of the luminosity ratios. Different
formation scenarios make varied predictions for the statistical properties of
UCD binary systems (see Burgasser et al. 2007a for an overview). As a result,
studying the properties of UCD binaries may shed light on the formation
mechanism of the entire class of objects.

Although models of field L and T dwarfs have reached a high degree of
sophistication, model spectra, particularly for L dwarfs, are highly dependent
upon the assumed cloud model. Cushing et al. (2008), for example, found that
changes in the assumed degree of cloud sedimentation could alter the derived
effective temperature, $T_{\rm eff}$, by up to several hundred Kelvin and the
log of the surface gravity, $\log g$, by 0.5 dex. A spectroscopic binary
offers the prospect of constrained masses and coevality and would provide
excellent simultaneous constraints on the cloud model, object masses, and
effective temperatures (Marley & Leggett, 2008).

The binarity of UCDs has been studied both spectroscopically and with direct
imaging. Relative to main sequence stars (i.e. Duquennoy & Mayor 1991), the
UCD binaries are more rare, lie at closer separations, and are more likely to
have components with equal masses (Burgasser et al. 2007a and references
therein). Owing to the difficulties in detecting close, high contrast ratio
binaries, imaging surveys do not provide a clear picture of UCD binarity at
separations less than 1 AU (see Reid et al. 2008). Despite these difficulties,
Bouy et al. (2004), Lane et al. (2001), and Golimowski et al. (2007) have used
high-resolution imaging to determine the orbit of young L or M dwarf binaries.
The binaries with small separations are more readily detectable by searching
for the radial velocity signal due to the reflex motion of the primary star.
Spectroscopic searches for UCD binaries have identified several binaries and
candidate binaries (Joergens & Müller, 2007; Kenyon et al., 2005; Guenther &
Wuchterl, 2003; Basri & Reiners, 2006), though the total number of
observations of each object tends to be small. Recent work by Burgasser
(2007b) has demonstrated that low-resolution spectra alone can be used to
detect the faint companions by searching for the subtle signature of the
cooler object at the wavelengths of certain molecular features.

We presented results from a pilot study targeting nine field L dwarfs with the
Phoenix instrument on Gemini-S in Blake et al. (2007). Here, we report the
detection of a single-lined UCD spectroscopic binary (SB1) with an orbital
separation of approximately 0.4 AU. 2MASSJ03202839$-$0446358 (hereafter
2M0320$-$04) was included in the catalog of nearby, cool stars presented by
Cruz et al. (2003) who spectroscopically classified it as a possible M8 dwarf
based on optical spectra. Wilson et al. (2003) classified the object as an
L0.5 dwarf based on low resolution NIR spectroscopy. In §2 we describe the
high-resolution NIR spectroscopy of 2M0320$-$04, in §3 we describe the
modeling process used to extract radial velocities from these data, the fit of
a Keplerian orbit to the radial velocities, and the search for the spectral
lines of the secondary, and in §4 we describe the implications of this
discovery for future studies of UCDs.

##  2 Observations

We observed 2M0320$-$04 with the NIRSPEC spectrograph (McLean et al., 1998) on
the Keck telescope on 14 nights between September 2003 and January 2007 as
part of a program to monitor a large sample of L dwarfs for radial velocity
variations. In all cases, we used the same instrumental setup. This setup was
selected to cover the CO bandhead and R-branch features around 2.3$\micron$.
This spectral region is rich in telluric absorption features due to methane.
As described in the next section, these telluric features will serve as the
wavelength reference for our radial velocity measurements. We used a 3 pixel
(0.432$\arcsec$) slit, with the N7 blocking filter, to produce an approximate
scale of 0.3 Å pixel -1 and a resolution of $R=\lambda/{\Delta\lambda}\approx
25000$. The extracted spectra contain 1000 pixels and cover the approximate
spectral range 2.285$\micron$ to 2.318$\micron$. Exposures ranged from 900s to
1200s per nod position with adjustments made according to observing conditions
in order to maintain approximately constant S/N. The data were gathered in nod
pairs so as to facilitate the subtraction of sky emission lines. In total, we
collected 16 nod pairs.

##  3 Analysis

After subtraction of the nod pairs to remove sky emission lines, we extracted
the spectra following the optimal extraction procedures outlined in Horne
(1986). We modeled the extracted spectra following a procedure similar to that
described in Blake et al. (2007). Our model begins with two high-resolution
template spectra: one for the Earth’s atmosphere and one for the L dwarf. The
high-resolution ($5\times 10^{-6}\micron$ pixel-1) spectrum of the Earth’s
atmosphere is provided by Livingston & Wallace (1991). The high-resolution
synthetic spectra of L dwarfs were computed as described in Marley et al.
(2002), with a number of improvements to be described in a future publication.
The models apply the condensation cloud model of Ackerman & Marley (2001) with
a sedimentation parameter of $f_{\rm sed}=3$, corresponding to a moderate
amount of condensate settling. The models used here have solar metallicity
(Lodders, 2003), use the opacities described in Freedman et al. (2008), and a
fixed gravity of $\log g=5$ (cgs) and cover a range of $T_{\rm eff}$ from 1200
to 2400 K. The synthetic spectra provide monochromatic fluxes spaced
$4.2\times 10^{-6}\,\mu$m apart. We convolve and re-sample the product of the
telluric and L dwarf high-resolution spectra to generate the model that we
then fit to the extracted 1-D spectra. Our model has several free parameters.
The parameters related to the L dwarf are the projected rotation velocity of
the L dwarf ($V\sin{I}$) where $I$ indicates the inclination of the rotation
axis to the line of sight, its $T_{\rm{eff}}$, and its radial velocity. The
parameters related to the spectrograph are the PSF width, flux normalization,
and the wavelength solution (i.e. the mapping from wavelength to pixel
position).

For the parameters of the L dwarf we first determined a best initial fit with
$V\sin{I}=16.5\pm 0.5$ km s-1 using the L dwarf spectral template with
$T_{\rm{eff}}=2200$K using least squares fitting. This spectral template
provided the best overall fit to the data and is also consistent with the L0
spectral type (Basri et al., 2000; Golimowski et al., 2004). We modeled the
wavelength solution as a third-order polynomial, the overall flux
normalization as a third-order polynomial, and the spectrograph PSF as a
single Gaussian. The PSF is expected to be asymmetric at some level, and to
vary across the spectrograph order. Spectral models that accommodate these
subtleties may yield even higher precision than that achieved here. With the
$V\sin{I}$ and $T_{\rm{eff}}$ fixed the model has a total of ten free
parameters. For each spectrum, we determined the best fit values for each
parameter, including the radial velocity, using the AMOEBA algorithm. An
example of a NIRSPEC spectrum and best fit model is shown in Figure 1. The
algorithm failed to converge on a solution for one out of 28 spectra,
resulting in a total of 27 radial-velocity measurements. Visual inspection of
this spectrum indicated a strong flux gradient across the order, likely due to
a poor extraction. We used a bootstrap simulation to estimate the error on the
individual radial velocities, similar to the technique described in Blake et
al. (2007). We simulated NIRSPEC observations of L dwarfs with different S/N
and radial velocities and used our modeling procedure to estimate our ability
to recover the known radial velocity. Since the NIRSPEC data are of relatively
high S/N ($\approx 60$ per pixel), and the V$\sin{I}$ is a modest 16.5 km s-1,
the results of our simulation indicate that the statistical errors on the
radial velocities should be rather small ($\approx 60$ m s-1). The measured
radial velocities are listed in Table 1.

An initial search for periodicity in the radial velocities with the Lomb-
Scargle Periodogram revealed a clear signal with a period $P\approx 250$d,
prompting a more detailed analysis. The radial velocities covering nearly 5
cycles of the binary were fit with a Keplerian orbital model using standard
non-linear least-squares techniques. The six parameters of this model are the
period, $P$, the systemic velocity, $\gamma$, the radial velocity semi-
amplitude, $K_{1}$, the eccentricity, $e$, the time of periastron passage,
$T_{0}$, and the longitude of periastron, $\omega$. The scatter in the fit was
found to be significantly larger than the internal uncertainties, possibly as
a result of systematic errors. For the final fit we adjusted the internal
errors by adding 0.32 km s-1 in quadrature to the error estimates based on the
bootstrap simulations, so as to force a reduced $\chi^{2}$ value near unity.
The resulting orbital elements are listed in Table 2, wherein we also state
our estimate of the projected semi-major axis of the orbit of the primary,
$a_{1}\sin{i}$, and the observations and orbital fit are shown in Figure 2.
The eccentricity ($e=0.065\pm 0.016$) is small but significant at the
4$\sigma$ confidence level. The radial velocity semi-amplitude of the primary,
$K_{1}=7.02\pm 0.12$ km s-1, is 21 times the per-point measurement precision.

##  4 Discussion and Conclusions

Based on the estimated spectral type (M8 to L0.5) we can estimate the
effective temperature of the primary of 2M0320$-$04 if we assume that the
light from the secondary is negligible. Golimowski et al. (2004) present
estimates of $T_{\rm{eff}}$ as a function of spectral type and from this work
we estimate $2200<T_{\rm{eff}}<2400$K for an assumed age of 3 Gyr, in general
agreement with our fits to the L dwarf templates. Using the models of Baraffe
et al. (2003) to relate $T_{\rm{eff}}$ to mass at an age of 3 Gyr, we estimate
that the primary has a mass $0.075<M_{1}<0.081$M☉. We can use the mass
function derived from the radial-velocity fitting procedure to estimate the
mass of the secondary as a function of $\sin{i}$ where $i$ is the inclination
of the system to our line of sight. If we assume $M_{1}=0.08$M☉, and that
$\sin{i}\approx 1$, then the mass of the secondary is $M_{2}=0.054M_{\sun}$.
Using the models of Baraffe et al. (2003) and the work of Golimowski et al.
(2004), the secondary would have $T_{\rm{eff}}\approx 1350$K with a spectral
type of approximately L7 to T3 at the assumed age of 3 Gyr. As $\sin{i}$
decreases, the total mass of the system increases, resulting in an observed
spectrum that is not dominated by the L0 primary. This would be inconsistent
with observations without requiring that the more massive component of this
system be significantly under-luminous. In particular, values of $\sin{i}<0.7$
would result in a secondary star that is more massive, though less luminous,
than the primary. While it is possible that $\sin{i}\sim 1$ and the system
presents eclipses, the relatively small value of $(R_{1}+R_{2})/a\sim 10^{-3}$
makes eclipses unlikely. Given the constraint $\sin{i}>0.7$ the actual
probability of observable eclipses is likely somewhat higher.

While we have no direct measure of the age of the system, it is possible to
constrain the age from its kinematics. Using the distance estimate of
26.2$\pm$ 4.3 pc from Cruz et al. (2003), the proper motion of $0.678\pm
0.038\arcsec$ year-1 (Deacon et al., 2005), and our measurement of the radial
velocity, we can calculate the space velocity and U,V,W velocity components
following Johnson & Soderblom (1987). Since the parallax of this system is not
known, the large error on the distance estimate results in large errors on the
kinematic estimates. We find velocity components [U,V,W]=[$-$62.0$\pm
9.3$,$-$38.6$\pm 8.9$, $-$34.3$\pm 7.2$] km s-1 relative to the local standard
of rest. Using the age-velocity relation from Eqn. 8 of Wielen (1977) we
estimate an age based on the W velocity of $\tau>3.6$ Gyr (95$\%$ confidence).

It is interesting to consider the detectability of the spectral lines of the
secondary in our spectra, which would allow us to estimate directly the mass
ratio. Based on the models of Baraffe et al. (2003), the expected $K$-band
flux ratio for objects of these masses is $\approx 10$ at $\tau=1$Gyr and
$\approx 100$ at $\tau=5$Gyr. While detecting the secondary lines at the later
age would be challenging, methods like TODCOR (Zucker & Mazeh, 1994) have been
used to recover secondaries in systems with flux ratios $\approx 50$ (D.
Latham; private communication). Following a method similar to TODCOR we
searched for the spectral lines of the secondary as follows. Using the orbital
solution presented in Table 2, we searched a grid in two parameters; the mass
of the secondary and the $K$-band flux ratio. At each grid point, a second
theoretical template spectrum with $T_{\rm{eff}}=1400$K was added to the
fitting procedure described in Section 3. The V $\sin{I}$ of the secondary was
assumed to be the same as that of the primary. During this process the radial
velocity of the primary of 2M0320$-$04 was fixed to the value from the orbital
solution, the radial velocity of the secondary was also fixed based on the
orbital solution and the assumed value of $M_{2}$, and the $K$-band flux ratio
was fixed. At each grid point this modified modeling scheme was used to
determine the best fit of this model to the data. Significant improvements in
$\chi^{2}$ from the case of a flux ratio of 0.0 would indicate the detection
of the secondary component. We carried out this procedure for a subset of 12
of our spectra gathered near times of quadrature, when the velocity separation
of the primary and secondary would likely be greatest. We found no evidence
for the spectral signature of a secondary component with a flux ratio greater
than 0.1 in $K$ band. We note that the non-detection of the secondary spectral
lines also implies $\sin{i}\approx 1.0$. If $\sin{i}=0.8$ the expected flux
ratio at $\tau=5$ Gyr would be $\sim 0.3$ and the secondary would likely have
been detected. Detection of the spectral lines of the secondary and resolution
of the system with direct imaging would allow for the first direct measurement
of the mass of a field T dwarf. Since the orbital solution combined with
direct imaging provides a distance measurement, future observations of this
system could also provide important empirical tests of theoretical models for
old objects at such low masses.

This work demonstrates the importance of radial velocity searches for binary
UCDs with small ($a<1$ AU) separations as a complement to the direct imaging
searches for companions at larger separations. A more detailed analysis of our
data, including a more sophisticated model of the individual spectra, may lead
to the detection of the spectral lines of the secondary and an estimate of the
mass ratio. If the inclination can also be measured, then the masses of both
components may be determined. If we assume the models of Baraffe et al.
(2003), $\sin{i}=1.0$, and a distance of 26 pc (Cruz et al., 2003), then the
maximum angular separation of the pair is $\approx 16$ mas, below the
capabilities of the Keck Laser Guide Star AO system. While the flux ratio may
be large ($\Delta K\approx 5$ mag, depending on age), future interferometric
systems may be able to resolve both components and provide the measurement of
the inclination required to directly measure the masses of both components.

Note: During the completion of this Letter we became aware of work by
Burgasser et al. (2008) describing a tentative detection of the secondary in
the 2M0320$-$04 system using the spectral diagnostics described in Burgasser
(2007b).

CB acknowledges support from the Harvard Origins of Life Initiative. GT
acknowledges partial support from NSF grant AST-0708229. We thank the referee
for thoughtful comments that helped to improve this manuscript. Part of this
research was supported by a Spitzer Science Center Theory grant. The data
presented herein were obtained at the W.M. Keck Observatory, which is operated
as a scientific partnership among Caltech, the University of California, and
NASA. The Observatory was made possible by the generous financial support of
the W.M. Keck Foundation. The authors wish to recognize and acknowledge the
very significant cultural role and reverence that the summit of Mauna Kea has
always had within the indigenous Hawaiian community. We are most fortunate to
have the opportunity to conduct observations from this mountain. This research
has benefited from the M, L, and T dwarf compendium housed at
DwarfArchives.org and maintained by C. Gelino, D. Kirkpatrick, and A.
Burgasser. Facilities: Keck/NIRSPEC

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Table 1: RV Data for 2M0320$-$04

HJD-2400000 | RV | $\sigma_{RV}$
---|---|---
| km s-1 | km s-1
52921.0960 | 6.95 | 0.33
52921.1102 | 6.26 | 0.33
52922.1054 | 6.70 | 0.33
52922.1196 | 6.35 | 0.33
52957.0218 | 4.18 | 0.34
52957.0361 | 4.49 | 0.33
53272.1197 | -6.17 | 0.33
53272.1304 | -6.85 | 0.33
53273.0793 | -6.56 | 0.33
53273.0900 | -5.97 | 0.33
53328.8225 | -3.97 | 0.34
53328.8343 | -4.03 | 0.33
53421.7131 | 6.74 | 0.33
53421.7239 | 6.18 | 0.32
53669.8797 | 6.53 | 0.32
53669.8919 | 5.84 | 0.33
53670.8719 | 6.37 | 0.33
53670.8841 | 6.26 | 0.33
53686.8538 | 5.59 | 0.33
53686.8659 | 5.78 | 0.33
53742.8030 | -3.25 | 0.33
53742.8151 | -2.92 | 0.33
54023.9636 | -7.25 | 0.33
54023.9757 | -7.70 | 0.33
54100.7404 | 2.03 | 0.33
54100.7526 | 1.03 | 0.33
54101.7468 | 1.67 | 0.33

Note. — Individual radial-velocity measurements. Errors include the estimated
statistical errors added in quadrature with a 320 m s-1 systematic error.

Table 2: Orbital and System Parameters

Parameter | Value | Units
---|---|---
$P$ | 246.73$\pm 0.49$ | days
$\gamma$ | $-0.063\pm 0.078$ | km s-1
$K_{1}$ | $7.02\pm 0.12$ | km s-1
$e$ | $0.065\pm 0.016$ |
$\omega$ | $177\pm 17$ | ∘
$T_{0}$ | $2453537\pm 11$ | HJD
$a_{1}\sin{i}$ | $23.75\pm 0.41$ |  $10^{9}$ m
$M_{2}\sin{i}$ | $0.2062(M_{1}+M_{2})^{(2/3)}\pm 0.0034$ | $M_{\sun}$
V$\sin{i}$ | 16.5$\pm 0.5$ | km s-1
$J$ | 13.259$\pm 0.024$ |
$H$ | 12.535$\pm 0.023$ |
$K_{s}$ | 12.134$\pm 0.026$ |
RA | 03:20:28.39 | hh:mm:ss (J2000)
DEC |  $-$04:46:36.4 | dd:mm:ss (J2000)

Note. — Derived and observed parameters of the 2M0320$-$04 system.

Figure 1: Example of a NIRSPEC spectrum of 2M0320$-$04 (bottom) along with the
best fit model offset by a constant value (top). The spectra consist of
rotationally broadened CO and H2O features from the L dwarf along with narrow
telluric CH4 lines.  Figure 2: Radial-velocity measurements and orbital
solution for 2M0320$-$04 plotted as a function of time. Our observations span
approximately five orbital periods.


# Setting the photoelectron clock through molecular alignment

Andrea Trabattoni  Center for Free-Electron Laser Science, Deutsches
Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany  The
Hamburg Center for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Joss Wiese  Center for Free-Electron Laser
Science, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg,
Germany  Department of Chemistry, Universität Hamburg, Martin-Luther-King-
Platz 6, 20146 Hamburg, Germany  Umberto De Giovannini  Max Planck Institute
for the Structure and Dynamics of Matter and Center for Free-Electron Laser
Science, 22761 Hamburg, Germany  Dipartimento di Fisica e Chimica, Universitá
degli Studi di Palermo, Via Archirafi 36, I-90123, Palermo, Italy  Jean-
François Olivieri  Center for Free-Electron Laser Science, Deutsches
Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany  Terry
Mullins  Center for Free-Electron Laser Science, Deutsches Elektronen-
Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany  Jolijn Onvlee
Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY,
Notkestraße 85, 22607 Hamburg, Germany  Sang-Kil Son  Center for Free-Electron
Laser Science, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607
Hamburg, Germany  The Hamburg Center for Ultrafast Imaging, Universität
Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany  Biagio Frusteri
Dipartimento di Fisica e Chimica, Universitá degli Studi di Palermo, Via
Archirafi 36, I-90123, Palermo, Italy  Angel Rubio<EMAIL_ADDRESS>Max Planck Institute for the Structure and Dynamics of Matter and Center for
Free-Electron Laser Science, 22761 Hamburg, Germany  Center for Computational
Quantum Physics (CCQ), The Flatiron Institute, 162 Fifth Avenue, New York NY
10010  Department of Physics, Universität Hamburg, Luruper Chaussee 149, 22761
Hamburg, Germany  Sebastian Trippel  Center for Free-Electron Laser Science,
Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
The Hamburg Center for Ultrafast Imaging, Universität Hamburg, Luruper
Chaussee 149, 22761 Hamburg, Germany  Jochen Küpper<EMAIL_ADDRESS>https://www.controlled-molecule-imaging.org Center for Free-Electron Laser
Science, Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg,
Germany  The Hamburg Center for Ultrafast Imaging, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany  Department of Chemistry,
Universität Hamburg, Martin-Luther-King-Platz 6, 20146 Hamburg, Germany
Department of Physics, Universität Hamburg, Luruper Chaussee 149, 22761
Hamburg, Germany

###### Abstract

The interaction of strong laser fields with matter intrinsically provides
powerful tools to image transient dynamics with an extremely high
spatiotemporal resolution. In strong-field physics, the initial conditions of
this interaction are generally considered a weak perturbation. We investigated
strong-field ionisation of laser-aligned molecules and showed, for the first
time, a full real-time picture of the photoelectron dynamics in the combined
action of the laser field and the molecular interaction. We demonstrated that
the molecule defines the initial conditions of the photoelectron at birth and
has a dramatic impact on the overall strong-field recollision dynamics: it
sets the clock for the emission of electrons with a given rescattering kinetic
energy. This result represents a new benchmark for the seminal statements of
molecular-frame strong-field physics. Our findings have strong impact on the
interpretation of self-diffraction experiments, where the photoelectron
momentum distribution is used to retrieve molecular structures. Furthermore,
the resulting encoding of the time-energy relation in molecular-frame
photoelectron distributions shows the way of probing the molecular potential
in real-time and accessing a deeper understanding of electron transport during
strong-field interactions.

In the prototypical strong-field interaction, an intense driving field
extracts a valence electron from the target through tunnel ionisation,
accelerates the free electron in vacuum, and eventually drives it back to the
parent ion, predominantly resulting in rescattering or radiative recombination
Corkum (1993); Corkum  _et al._ (1997). The radiative recombination results in
the emission of high-energy photons by high-harmonic generation Corkum (1993)
and this is a powerful tool to investigate the electronic structure with
attosecond temporal resolution Calegari  _et al._ (2014); Lépine  _et al._
(2014); Kraus  _et al._ (2015). Alternatively, the rescattered portion of this
electron wavepacket is exploited in laser-induced electron diffraction (LIED)
Corkum and Krausz (2007) experiments as a coherent diffraction pattern of the
molecular target, potentially providing time-dependent images of the molecule
at sub-femtosecond and few-picometer resolution. Recently, corresponding
experimental results for the structure or dynamics of small or highly-
symmetric molecules were obtained Blaga  _et al._ (2012); Pullen  _et al._
(2015); Wolter  _et al._ (2016); Walt  _et al._ (2017); Fuest  _et al._
(2019); Amini  _et al._ (2019). At the same time, the initial conditions of
the strong-field interaction have attracted much attention for capturing the
intrinsic nature of strong-field physics. While pioneering attosecond
experiments and molecular-frame measurements revealed non-trivial
spatiotemporal features in electron tunnelling Eckle  _et al._ (2008); Meckel
_et al._ (2014), these initial conditions are still generally considered a
weak perturbation in strong-field physics. All the results obtained in LIED
experiments, for example, are interpreted in the framework of the strong-field
approximation, where the electron is considered to be born in the continuum
with a negligible initial momentum and to propagate as a plane wave Chen  _et
al._ (2009). Furthermore, the post-ionisation dynamics before rescattering are
assumed to be fully driven by the laser field, by neglecting, for example, the
Coulomb interaction with the ionised molecule. Common strategies to analyse
photoelectron momentum distributions rely on the quantitative rescattering
theory (QRS) Chen  _et al._ (2009), where angular dependence in the final
photoelectron wavepacket is introduced solely through rescattering. Within
this approach, diffraction patterns were analysed utilising the angular Blaga
_et al._ (2012); Pullen  _et al._ (2015) or radial Xu  _et al._ (2014)
photoelectron distribution. However, the relevance of the ionised molecular
orbital in the rescattered photoelectrons is still under discussion Schell
_et al._ (2018). So far, this was included by an overall weighting factor in
the rescattering probability Lein (2003); Niikura  _et al._ (2002), or as a
spatial phase or an angular feature in the rescattering electron wavepacket
Busuladžić  _et al._ (2008); Meckel  _et al._ (2014). Recently, the influence
of molecular alignment on molecular structure retrieval was discussed Xu  _et
al._ (2014); Pullen  _et al._ (2016). However, general predictions are still
extremely challenging with new models appearing Suárez  _et al._ (2016); Liu
_et al._ (2016).

Figure 1: Sketch of the experimental arrangement with alignment of OCS
molecules in the laboratory frame, parallel and perpendicular to the $Y$ axis.
The ionising laser is linearly polarised along the $Y$ axis and the detection
is in the $XY$ plane. The molecular-frame angle-resolved photoelectron spectra
are projected onto a 2D detector in a velocity-map-imaging spectrometer. The
spectra are displayed on a logarithmic intensity scale in units of
electrons/shot/bin.

Here, we experimentally and computationally studied molecular-frame
photoelectron spectroscopy from strongly aligned molecules in order to
investigate the relation between the molecular frame and the strong-field-
induced ultrafast electron dynamics. We demonstrated that the initial
conditions acquired by the electron in the molecular frame set a time-energy
relation governing the subsequent electron dynamics.

Fig. 1 depicts the experiment. An ultracold ensemble of carbonyl sulphide
(OCS) molecules Chang  _et al._ (2015) was adiabatically aligned in the
laboratory frame, with $\left<\cos^{2}\\!\theta_{\text{2D}}\right>=0.9$, by
using a linearly polarised, 500 ps laser pulse, centred at 800 nm Trippel  _et
al._ (2013, 2014), with a peak intensity $I=3\times
10^{11}~{}\text{W}/\text{cm}^{2}$. The molecules were aligned in two different
configurations, shown in Fig. 1 , with the molecular axis along the $Y$ and
$Z$ axes, named parallel and perpendicular alignment, respectively. A second
laser pulse, centred at $1300$ nm, with a duration of $65$ fs, and a peak
intensity $I=8\times 10^{13}~{}\text{W}/\text{cm}^{2}$, was used to singly
ionise the OCS molecules. For this intensity the ponderomotive energy of the
laser field is $U_{\text{p}}\approx 13$ eV and the ionisation occurred in the
tunnelling regime. The electric field of the ionising laser pulse,
$\mathbf{E}_{L}$ in Fig. 1 , was linearly polarised along the $Y$ axis
(ellipticity $\epsilon=I_{Z}/I_{Y}<0.005$). The produced molecular-frame
angle-resolved photoelectron spectra (MF-ARPES) were recorded in a velocity
map imaging (VMI) spectrometer Eppink and Parker (1997) with its detector
parallel to the $XY$ plane. It is important to note that the de Broglie
wavelength of rescattering electrons in the experiment was larger than $200$
pm. In this regime no diffraction feature is expected to appear in the
photoelectron distributions Xu  _et al._ (2014). Fig. 1 shows the MF-ARPES for
parallel (left) and perpendicular (right) alignment. The two distributions
show several differences. The spectrum for parallel alignment has a larger
width at small transverse momenta, $p_{X}<0.5~{}\text{a.\,u.}$ (atomic units),
while the spectrum for perpendicular alignment shows a number of angular
features for transverse momenta $p_{X}$ between $0.5~{}\text{a.\,u.}$ and
$1~{}\text{a.\,u.}$.

Figure 2: Molecular-frame angle-resolved photoelectron spectra of OCS obtained
a, b, c experimentally and computationally from d, e, f TDDFT and g, h, i ADK-
SM calculations. a, d, g: Split graphical representation as a comparison of
the photoelectron distributions for parallel and perpendicular alignment for
the experimental and computational results, respectively. b, e, h:
Corresponding projected energy distributions of photoelectrons along the $Y$
axis, angularly integrated within a cone of $\pm 20\,^{\circ}$, as well as the
ratio of the integral-normalised perpendicular and parallel distributions, on
logarithmic scales. Energies are reported in units of the ponderomotive energy
$U_{p}$. c, f, i: First derivatives of the photoelectron-energy distributions
to evaluate the high-energy cutoff for the two molecular-alignment cases. All
TDDFT computational results were obtained by averaging over different laser-
molecule orientations according to the experimental alignment distributions
and by adding a constant to account for the experimental background level. The
ADK-SM results refer to a single laser intensity and perfect alignment for
both cases. See _Methods_ for details.

These angular structures, which are much weaker in the spectrum for parallel
alignment, could be identified as forward-rescattering features Becker and
Milošević (2015). Focussing the attention on large longitudinal momenta
$p_{Y}$, the counts for parallel alignment drop around $2.5$ a. u.. In the
case of perpendicular alignment, however, the spectrum extends to larger
momenta, showing an appreciable amount of counts at $p_{Y}=3~{}\text{a.\,u.}$.
Following the strong-field approximation, the hard cutoff of photoelectron
momentum is expected to only depend on the properties of the laser field
Paulus  _et al._ (1994). Experimentally, this quantity is hard to measure.
Thus, the turning-point of the signal drop, i. e., the minimum of the first
derivative, at large longitudinal momenta is used instead. In the following,
we use the term cutoff in the latter sense. Surprisingly, in the current study
we found a clear dependence of the cutoff on the molecular frame.

Fig. 2 a shows a close comparison of the two experimental distributions for
the complete range of $p_{X}$ and $p_{Y}$, between $0$ and $4$ a. u.. Here,
the spectra were split along the $Y$ axis and the spectrum from parallel
alignment is shown on the left and the one from perpendicular alignment on the
right. Now, the differences at small momenta as well as at the cutoff are even
more evident. To perform a quantitative analysis of the cutoffs, the momentum
distributions were angularly integrated within a cone of $\pm 20\,^{\circ}$
with respect to the longitudinal axis ($Y$) and converted to an energy scale.
In Fig. 2 b the resulting photoelectron spectra are shown for parallel (blue)
and perpendicular (red) alignment, with energies in units of $U_{\text{p}}$.
The perpendicular/parallel ratio of the two area-normalised spectra (green)
shows a predominance of photoelectrons for perpendicular alignment in the
energy range between 2 $U_{\text{p}}$ and 10 $U_{\text{p}}$, where the
distribution is dominated by rescattered electrons Becker  _et al._ (2002).
Furthermore, the ratio increases with energy, reaching the maximum around the
cutoff. To evaluate the cutoffs, the first derivative of the energy
distributions are shown in Fig. 2 c and their minima were used to find the
edges of the distributions, which allowed us to analyse the cutoff region. The
first minimum represents the drop of direct electrons Becker  _et al._ (2002)
and it was around 2 $U_{\text{p}}$ for both alignment cases. This excluded any
significant alignment-dependent direct-electron-cutoff enhancement Smeenk  _et
al._ (2014). Surprisingly, the second minimum behaves differently for the two
alignments. While it is located around 10 $U_{\text{p}}$ for perpendicular
alignment, as expected from the well established above-threshold ionisation
theory Paulus  _et al._ (1994), the cutoff is shifted down to a value around
8.5 $U_{\text{p}}$ for parallel alignment.

To unravel the experimental observations, state-of-the-art calculations were
performed using both, time-dependent density-functional theory (TDDFT) Marques
_et al._ (2011) and a novel semiclassical molecular trajectory simulation
setup. Using TDDFT, the MF-ARPES probability was calculated by simulating the
complete dynamics of the many-body ionisation process in real-time and real-
space with the tSURFF method Tao and Scrinzi (2012); Wopperer  _et al._
(2017). With this technique the spectrum was obtained by computing the entire
time-dependent electron dynamics, including many-body electron interactions,
and collecting the flux of electrons through a closed surface surrounding the
molecule. Fig. 2 d, e, f report the same analysis of the numerical results as
performed for the experimental data in Fig. 2 a, b, c. The simulations capture
the principal features of experimental data very well. In particular, Fig. 2 f
shows that the calculations reproduce the experimental cutoff positions for
parallel and perpendicular alignment as well as the corresponding shift
between them very well. This result is strongly affected by the electron-
electron interaction and the interplay between different orbitals, see
_Methods_. Indeed, it is evident from the calculation that the molecule is
predominantly ionised from the highest-occupied molecular orbital (HOMO) for
both alignments. In the case of parallel alignment, nevertheless, a small
contribution of HOMO-1 to the yield of high-energy rescattered electrons is
observed. When the electron-electron interaction is artificially turned off
the HOMO-1 contribution becomes significant and in this scenario the reduced
cutoff observed in the experiment is not reproduced. Instead, in the case of
fully interacting electrons the yield of the rescattered electrons ionised
from HOMO-1 is suppressed, resulting in the really good agreement with the
experiment.

Furthermore, semiclassical trajectory simulations based on the Ammosov-Delone-
Kraǐnov (ADK) tunnelling theory Ammosov  _et al._ (1986) in conjunction with a
“simple man” propagation (SM) Gallagher (1988); Corkum  _et al._ (1989) were
conducted in order to track the molecular-frame electron dynamics during the
strong-field interaction, see _Methods_ for details. Based on the TDDFT
analysis of the different molecular orbitals contributing to the photoelectron
dynamics, the ionisation was assumed to occur solely from HOMO. In the
underlying model, the initial phase-space distribution of the electron
wavepacket in the continuum at birth was described by the quasistatic ADK
tunnelling theory, and the nodal structure of the HOMO was accounted for as
imprint onto this initial momentum distribution. Post-ionisation dynamics of
the electron wavepacket were evaluated in the combined interaction with the
laser electric field and the cation’s Coulomb field modelled as a point
charge, see _Methods_. To evaluate the accuracy of this semiclassical
description, the resulting MF-ARPES for parallel and perpendicular alignment
were calculated and analysed, see Fig. 2 g, h, i, and they show a really good
agreement, both, with the experimental data and the full TDDFT calculation,
reproducing the main features and cutoffs observed in the experiment very
well. In particular, as seen by the local maximum around 10 $U_{\text{p}}$ of
the ratio of the two alignment cases, Fig. 2 h, this semiclassical model
captures the enhanced yield of high-energy rescattered electrons for
perpendicular alignment with respect to parallel alignment. This result is
corroborated by the enhanced cutoff around 10.7 $U_{\text{p}}$ for
perpendicular alignment in Fig. 2 i, although a smaller yield at this energy
is present also for parallel alignment. In addition, a relevant minimum
appears around 9.3 $U_{\text{p}}$ for both alignments. These features of ADK-
SM, together with the pronounced yield along the centreline of Fig. 2 g, are
known to be mainly due to Coulomb focussing Daněk  _et al._ (2018), i. e., the
dynamics of a continuum electron wavepacket being focussed along a perfectly
linear laser polarisation axis. The relevance of this effect is discussed
further below.

Figure 3: a, b, c: Differential momentum distributions (parallel-
perpendicular) from a the experiment and the b TDDFT and c ADK-SM
calculations. To estimate the difference of the transverse momentum component
the signal is integrated along the $Y$ axis, shown by the black dashed lines.
d, e: Final absolute photoelectron momentum as a function of the recollision
phase (bottom) and revisit order (top) for d perpendicular and e parallel
alignment, calculated with ADK-SM. The colour scale maps the electron counts
at every momentum-phase point. The dashed black circles highlight the largest-
momentum electrons at the most probable revisit order for the two alignments.
A distance of $r<5$ atomic units between electron and point charge is
interpreted as a collision and only electrons with exactly one collision are
shown. The solid black line depicts the external electric field. The insets
give pictorial representations of molecular-frame electron trajectories, where
the cardinals represent the revisit order.

To obtain a more comprehensive picture of the alignment-dependent
photoelectron dynamics and, in particular, a glimpse at the initial electron
wavepacket, we performed a differential analysis by subtracting the
photoelectron distributions of the two alignment cases from each other. Fig. 3
a, b, c show the relative normalised differences, parallel minus
perpendicular, for the experiment, the TDDFT simulations, and the ADK-SM
calculations, respectively. The agreement between experimental data and both
models is excellent. Here, a strong depletion along the vertical axis and two
transversely offset broad lines of positive yield appear as main features,
with a really good agreement between experimental and both computational
results. The depletion along the centreline is due to the nodal structure of
the degenerate $\Pi$ HOMO of OCS, see _Methods_ : it represents a forbidden
direction of electron ejection Lein (2003); Holmegaard  _et al._ (2010).
Therefore, when the molecular axis was aligned along the polarisation axis of
the strong field, the electron preferentially acquired an initial transverse
momentum $p_{0X}$ that was much larger than in the case of perpendicular
alignment, shown by the red vertical ridges in Fig. 3 a, b, c.

The features observed in Fig. 3 a, b, c show the crucial impact of the
electronic structure on the initial conditions of the electron at birth.
However, a quantitative evaluation of the initial conditions of the electron
at tunnelling is challenging Pfeiffer  _et al._ (2012); Geng  _et al._ (2015);
Tian  _et al._ (2017). In general, they are defined by the tunnel-exit
position as well as by the temporal phase and the momentum acquired during the
ionisation with respect to the external field. Here, we demonstrate that the
molecular potential, i. e., the combination of the electrostatic potential and
the electronic structure of the molecule, has in fact not only a primary role
in setting the initial conditions for electron emission, but that it also
drives the whole photoelectron dynamics: it defines the cutoff of rescattered
electrons and it shapes the time-energy relation for electron recollision. To
investigate this, we exploited the ADK-SM calculations to analyse the final
absolute momentum acquired by the electron after photoionisation as a function
of the recollision phase, i. e., the difference between the rescattering phase
and the ionisation phase. The resulting momentum distributions are reported in
Fig. 3 d, e for perpendicular and parallel alignment, respectively. They
consist of broad peaks appearing every half cycle of the electric field at
phases $(k+1/2)\pi$, $k=1,2,3\ldots$, for which the electron collides with the
molecular cation when the laser field’s vector potential is maximum. The first
recollision event, i. e., the first peak in Fig. 3 d at a phase of $3\pi/2$,
allows the electron to reach the largest momentum as expected in the classical
theory Paulus  _et al._ (1994). This is close to the maximum asymptotic
kinetic energy, i. e., the 10 $U_{\text{p}}$ cutoff. The peaks appearing later
correspond to electrons that have initially missed and then revisited the ion
at later times. These subsequent rescattering events are expected to lead to
lower photoelectron energies Paulus  _et al._ (1994). At the same time, these
multiple revisits are possible only due to the Coulomb attraction of the
ionised molecule Paulus  _et al._ (1994). Since the current understanding and
analysis of strong-field self-diffraction experiments only consider the
photoelectron recollision on the first revisit Blaga  _et al._ (2012); Amini
_et al._ (2019), the relevance of Coulomb attraction is usually neglected.
However, our results demonstrate that it is a crucial ingredient to correctly
understand molecular-frame electron rescattering. Note that a small yield at
large momentum ($>3~{}\text{a.\,u.}$) is visible for both alignments, Fig. 3
d, e, at the fourth and the sixth revisit. These revivals, caused by Coulomb
focussing Daněk  _et al._ (2018), are expected to vanish for imperfect linear
polarisation of the laser field, as usually occurring in any experiment. This
explains why ADK-SM for parallel alignment has another cutoff around 10.7
$U_{\text{p}}$, as well as the more pronounced cutoff around 9.3
$U_{\text{p}}$ in Fig. 2 i. Due to the subtle conditions of Coulomb focussing,
this effect will not be further considered in the discussion below; it does
not contradict any of our general conclusions.

In this framework, the largest absolute momentum for perpendicular alignment
comes from the first rescattering event at a phase around $3\pi/2$, see Fig. 3
d and its inset, which yields the largest momentum
$\mathord{\sim}3.15~{}\text{a.\,u.}$. This momentum corresponds to an
asymptotic kinetic energy $\mathord{\sim}10.5~{}U_{\text{p}}$ and thus
explains the experimental observation of the 10 $U_{\text{p}}$ cutoff for
perpendicular alignment, see the red marker in Fig. 2 c. This rescattering
event is attenuated by the imprinting of the nodal plane perpendicular to the
molecular axis Schell  _et al._ (2018). Otherwise, this first peak would not
only correspond to the largest photoelectron momentum, but also to the most
probable recollision event. While this attenuation for perpendicular alignment
is responsible for the secondary cutoff at $\mathord{\sim}9~{}U_{\text{p}}$,
Fig. 2 i, it is not observed in the experiment nor in the TDDFT calculations,
Fig. 2 c, f. In the case of parallel alignment, instead, the first
rescattering event is strongly suppressed and most of the large-momentum
electrons come from the third revisit at a phase of $7\pi/2$, depicted in the
inset of Fig. 3 e; the fifth revisit also yields comparable momenta. As a
result, the momentum cutoff is smaller, i. e.,
$\mathord{\sim}2.9~{}\text{a.\,u.}$, corresponding to a final kinetic energy
of $\mathord{\sim}9~{}U_{\text{p}}$. This is in good agreement with the
experimentally observed reduced cutoff for parallel alignment, see the blue
marker in Fig. 2 c and Fig. 2 f, i. This dynamics is mainly driven by the
molecular potential: Here, the node of the HOMO along the laser polarisation
imprints an angle on the electron emission at tunnelling. For OCS this angle
was estimated to be $\mathord{\sim}30\,^{\circ}$ with respect to the
longitudinal $Y$ axis by the TDDFT calculations. This angle prevents the
electron from rescattering at the first revisit. However, then the Coulomb
attraction of the ionised molecule forces the electron to stay in the
interaction region and to recollide at later revisits. It is important to note
that the angle of emission and the rescattering at the $n$-th revisit are
strongly correlated. Indeed, larger emission angles lead to later revisits and
_vice versa_. As the consequence, the photoelectron cutoff carries a clear
signature of the electronic structure at tunnelling. This angular dependence
imprinted in the momentum distribution of the initial electron wavepacket
leads to the breakdown of the common product ansatz in QRS Chen  _et al._
(2009), where the initial and the rescattered parts of the wavepacket are
separated and only the latter contains angular dependence in the recollision
frame.

Furthermore, the photoelectron cutoff in the molecular frame carries crucial
time information: While the cutoff for perpendicular alignment is strongly
shaped by electrons recolliding $3/4$ of an optical cycle after ionisation, as
usually assumed, this is not true for parallel alignment: the cutoff is
dominated by electrons revisiting the molecule much later, namely one or
multiple optical cycles later. For a wavelength of $1300$ nm this corresponds,
at least, to a delay of $\mathord{\sim}4.3$ fs and it linearly increases with
the wavelength. From Fig. 3 d, e, apart from the aforementioned effects of
Coulomb focussing, it is also evident that even-numbered revisits yield lower
kinetic energies $<8~{}U_{\text{p}}$ Paulus  _et al._ (1994). Since the time
spent by the photoelectron before rescattering is usually exploited as the
elementary delay step for time-resolved self-diffraction experiments Blaga
_et al._ (2012), the use of this lower range of photoelectron energy Wolter
_et al._ (2016); Amini  _et al._ (2019) results in any time information being
smeared out on much longer timescales. Furthermore, the analysis performed
here demonstrated that this delay step strongly depends on the molecular-frame
alignment and that the molecular potential sets a complex time-energy encoding
in the electron dynamics. This molecular-frame clock for electron recollision
could clearly be exploited to disentangle the structural dynamics with few-fs
or even sub-fs temporal resolution. For instance, signals from specific
revisit orders could be selected in experiment with near-single-cycle laser
pulses.

We demonstrated, experimentally and computationally, that the molecular frame
determines the momentum distribution of high-energy rescattered electrons in
strong-field ionisation. The basic concept of molecular-frame strong-field
ionisation is captured by considering the initial conditions imposed by the
molecular potential in the dynamics of the photoelectron. Furthermore, from
the analysis of the rescattering trajectories it is evident that the molecular
interaction plays a crucial role in setting a clock for the emission and the
dynamics of high-energy electrons. It highlights that the molecular frame has
a strong impact on the relation between the photoelectron energy and the
rescattering time. This finding redefines the delay step of time-resolved
self-diffraction experiments and opens up a new perspective on time-resolved
diffraction experiments. These conclusions hold similarly for other
observables related to electron recollision, e. g., high-harmonic-generation
spectroscopy.

This result represents an important benchmark for any self-diffraction
measurement and represents a breakdown of the usual interpretation of LIED
experiments Blaga  _et al._ (2012); Wolter  _et al._ (2016); Amini  _et al._
(2019). It highlights the molecular-frame conditions as a crucial ingredient
of self-diffraction experiments. This new framework is general and can, in
principle, be extended to any molecular system. Furthermore, the molecular-
frame strong-field interaction was quantitatively modelled here, for the first
time, by a fully-interacting-electron TDDFT calculation and, in conjunction,
by a semiclassical single-active-electron theory. We exploited TDDFT to
evaluate the contribution of different molecular orbitals to the ionisation-
rescattering dynamics as a benchmark for the applicability of the
semiclassical approach. We expect this double-sided theoretical framework to
become more and more important with increasing molecular complexity, where
modelling the photoelectron dynamics may go beyond the capabilities of a
single-orbital picture. In general, this also opens the perspective to
investigate electron-correlation-driven phenomena in molecular strong-field
physics Lépine  _et al._ (2014). Furthermore, the earliest moments of a
strong-field interaction are intrinsically imprinted in the initial conditions
of the photoelectron and in the final energy distribution. Thus, molecular-
frame strong-field-ionisation experiments, in principle, allow one to achieve
a deeper understanding of electron tunnelling, for instance, regarding the
tunnelling time, and to track the molecular potential in real-time.

Acknowledgements
This work has been supported by the Clusters of Excellence “Center for
Ultrafast Imaging” (CUI, EXC 1074, ID 194651731) and “Advanced Imaging of
Matter” (AIM, EXC 2056, ID 390715994) of the Deutsche Forschungsgemeinschaft
(DFG), by the European Research Council under the European Union’s Seventh
Framework Programme (FP7/2007-2013) through the Consolidator Grant COMOTION
(ERC-Küpper-614507) and under the Horizon 2020 Research and Innovation through
Programme the Advanced Grant QSpec-NewMat (ERC-Rubio-694097), and by the
Helmholtz Association Initiative and Networking Fund. A.T. and J.O. gratefully
acknowledge fellowships by the Alexander von Humboldt Foundation.

Author contributions
A.T., S.T., and J.K. conceived the experiment. A.T., S.T., J.F.O., J.W., T.M.,
and J.O. performed the experiment. A.T. performed the data analysis. J.W. set
up and performed the ADK-SM simulations. U.D.G. and B.F. performed the TDDFT
calculations, and together with S.-K.S. and A.R. provided theoretical support.
A.T., S.T., U.D.G., J.W., A.R., and J.K. interpreted the results and prepared
the manuscript. All authors contributed to the discussion of the results and
commented on the manuscript.

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