QC 4 QuantumBrain

This week

We have two separate laboratories and the journey from one to the other is about ~100m. Up to now we have used only the old lab, now we have a new experiment to set up and we require a more stable temperature environment - the new lab does the trick apparently (but we haven’t tested the temperature fluctuations with the equipment actually in the lab yet!)

So first thing is first - move one of the work tables from the old lab to the new. Sounds easy, it’s not. We call in ‘Team Hector the Director’, I ask the stoutest member of the moving team, ‘What sort of things do you usually move?’, to which he answers in a strong Australian accent, ‘Anything heavy’. Good we have the right people in for this job as the table is 12ft x 4ft and weighs close to 1 ton. Credit where credit is due, they did a superb job and in under 90 minutes!

Today was our turn. All we had to do was move a laser. Again sounds easy, it’s not. It took three physicists and a rickety old trolly over 3 hours all in all. 

Film Review - Twlight

Made in America, Bill Bryson

The ultimate combination of social and sporting activities.

Step 1.

Take the scenic route around campus to the City Cat ferry port. Along the lakeside walk, through hidden gardens and over little bridges passing and enjoying the wild turkeys, possums and lizards on the way.

Step 2.

Board the high speed river ferry service ($5.00 for all zones) to the other side of town. Watch the sunset as you meander down the river and the speed induced breeze plays that whooshing sound which enhances the thrill of the ride.

Step 3.

Arrive at Kangaroo Point, a once abandoned quarry now an active climbing center. The face sits 25m or so from the river bank and stretches for about 250m along it. During the day the wall is not so busy, the heat and humidity render the sport practically unbearable for most climbers, at night however, the site is transformed. Fifty, maybe more, climbers gather along its foot to enjoy the floodlit wall from dusk till dawn. Myself and Bojan were looking for the University climbing club but were unsuccessful. Not to worry, a local offered us his rope and we climbed with that.

Oh, and it’s free of charge. Always.

Step 4.

Climb. The climbs number over 200 and range from very easy to insanely hard (so I’m told) so something to do for everyone. Get to the top and you can take in that beautiful river view. Across the river is the central business district, which at night is lit up a treat and looks superb!

Step 5.

Meet random people. A few guys slacklining invited (well I sort of invited myself) us to join in. Slacklining in the warm Australian evening with a river view to top it off - golden.

Step 6.

Play the didgeridoo. No night is complete without a practice on an ancient aboriginal musical instrument… is it?

Home.

Why so much gender confusion throughout the lyrics of Black Kids songs?

Bearing in mind these lyrics are sung by a guy…

1. I’m Not Gonna Teach Your Boyfriend How To Dance With You

Opening line;

You are the girl that I’ve been dreaming
of ever since I was a little girl.

2. Hurricane Jane

20 seconds in;

…Oh, you’ve been playing nice,
but I can see it in your eyes,
you’re thinking, “Christ.
He’s everybody’s girl.”

Perhaps I’m missing something here, a hidden joke maybe?

:-p

The Bunching Effect

I know I talked about beam splitters in a previous post, but I want to touch on a really cool effect that I have recently learned about.

Let us start from the quantum operation of a beam splitter again. I am going to use the bra-ket notation here as it is the most widely used in the field of quantum optics and in particular quantum information.

Schematic diagram of a beam splitter

We can represent the prepared input and output modes of the beam splitter in the following way

 \left|10\right>

The first and second elements in the bra-ket unit represent photons in spatial modes traced out by ‘a’ and ‘b’ respectively (see diagram above). The above state represents one photon in the ‘a’ spatial mode and no photons in the ‘b’ spatial mode. This is effectively a two qubit system where the qubit state is represented by either the presence of a photon in the mode, or not, this is known as the spatial mode degree of freedom. There are two spatial modes thus making it a two qubit system.

By defining the reflectivity of the beam splitter as R and the transmitivity as 1-R it is easy to convince yourself that the following equations represent zero, single and two photon events at the beam splitter.

 \hat{U}\left|00\right>=\left|00\right>

 \hat{U}\left|10\right>=i\sqrt{R}\left|01\right> + \sqrt{1-R}\left|10\right>

 \hat{U}\left|01\right>=i\sqrt{R}\left|10\right> + \sqrt{1-R}\left|01\right>

 \hat{U}\left|11\right>=i(2R-1)\left|11\right> + \sqrt{2R(1-R)}(\left|20\right>+\left|02\right>)

The first equation is the most trivial and of least interest, it represents no photons entering either mode of the beam splitter. The next two equations map the process of a single photon in one mode and no photon in the other. Both equations are equivalent and there is a phase difference of 180 degrees between the reflected and transmitted photon, equivalent to multiplying the reflected mode by i.

The last equation shows the consequence of having a 50/50 beam splitter. That is to say, R = 0.5 will always result in both photons leaving via the same mode (you do the math!). This effect is known as the bunching effect.

It is interesting to note that the two qubits (represented by the spatial modes) are now quantum mechanically entangled. Knowing information about one spatial mode will give you information about the other instantly. Entanglement lies at the heart of quantum information and quantum computation architecture and whilst this example is nice it by no means introduces entanglement in its full glory.

There, the bunching effect. If you find flaws in the math please tell me. I’m not entirely sure it is correct from my working. However, it gives the idea of the bunching effect. The phases are chosen arbitrarily, here I chose a phase change upon reflection, which is standard practice in linear optics.

So if I carry on my membership with the Institute of Physics as an associate member they will forward my monthly subscription of Physics World to my home address in Brisbane, Australia.

Sweeeeeeeeeeet.

Roger red hat.

Something just occured to me…

I’m working intensly with bosonic creation and annihilation operators in the field of quantum optics and whilst doing some simple calculations I realised the brilliance that are these two operators, it can be summed up in the expression:

[\hat{a}^\dagger,\hat{b}^\dagger] = \hat{a}^\dagger\hat{b}^\dagger - \hat{b}^\dagger\hat{a}^\dagger = 0

They commute! Of course I already knew this, but it took me a while to realize how nice this is and how thankful I am that I’m not working with stupid fermionic systems.

I apologize for all the geek speak in this post, but I had to tell someone. Now I have it off my chest I can live the rest of my days.

Optical components II

Beam splitters - A quantum mechanical approach

Schematic diagram of a beam splitter

A beam splitter can be represented by a quantum operator. From the diagram, the beam splitter maps the input states to the output states as;

 \hat{a}^\dagger_{out} \rightarrow r \hat{b}^\dagger_{in} + t \hat{a}^\dagger_{in}

 \hat{b}^\dagger_{out} \rightarrow r' \hat{a}^\dagger_{in} + t' \hat{b}^\dagger_{in}

For a 50/50 beam splitter, i.e. one that transmits 50% and reflects 50%, |r|=|t| and r=r’ and t=t’. The operation of the beam splitter is thus written as,

\begin{bmatrix}
\hat{a}^\dagger_{out} \\ \hat{b}^\dagger_{out}
\end{bmatrix} =
\hat{U}
\begin{bmatrix}
\hat{a}^\dagger_{in} \\ \hat{b}^\dagger_{in}
\end{bmatrix} =
\begin{bmatrix}
t & r\\
r & t
\end{bmatrix}
\begin{bmatrix}
\hat{a}^\dagger_{in} \\ \hat{b}^\dagger_{in}
\end{bmatrix}

An electromagnetic wave experiences a phase change upon reflection and transmission and so the operator looks like,

 \hat{U} = \frac{1}{\sqrt{2}}\begin{bmatrix}
e^{i\phi_{t}} & e^{i\phi_{r}}\\
e^{i\phi_{r}} & e^{i\phi_{t}}
\end{bmatrix}

By absorbing a global phase and defining,

\phi = \phi_{r} - \phi_{t}

 \hat{U} = \frac{1}{\sqrt{2}}\begin{bmatrix}
e^{i\phi} & 1\\
1 & e^{i\phi_}
\end{bmatrix}

For the beam splitter to conserve photon number its operator must be unitary, that is

 \hat{U}^\dagger \hat{U} = I

Therefore

 \frac{1}{2}\begin{bmatrix}
e^{-i\phi} & 1\\
1 & e^{-i\phi_}
\end{bmatrix}
\begin{bmatrix}
e^{i\phi} & 1\\
1 & e^{i\phi_}
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}

\Rightarrow \frac{1}{2}
\begin{bmatrix}
2 & (e^{i\phi}+e^{-i\phi})\\
(e^{i\phi}+e^{-i\phi}) & 2
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}

\Rightarrow
\begin{bmatrix}
1 & \cos \phi\\
\cos \phi & 1
\end{bmatrix} =
\begin{bmatrix}
1 & 0\\
0 & 1
\end{bmatrix}

\Rightarrow \cos\phi = 0

\Rightarrow \phi = \frac{\pi}{2}+n\pi

In order to conserve photon number there must be a phase difference of 90 degrees between the transmitted and reflected waves.

The first in what may be the ‘What I’ve leared archive‘…

Optical components

A mathematical description of some of the basic building blocks of linear optics.

Single-mode phase shift

A phase shifter has an index of refraction that is different to that of free space.

The operation of the single mode phase shifter can be represented by the creation and annihilation operators defined as

 \hat{a}^\dagger\left|n\right>=\sqrt{n+1}\left|n+1\right>

 \hat{a}\left|n\right>=\sqrt{n}\left|n-1\right>

respectively. Where the operation of the single-mode phase shift is given by

 \hat{a}^\dagger_{out}=e^{i\phi\hat{a}^\dagger_{in}\hat{a}_{in}}a^\dagger_{in}e^{-i\phi\hat{a}^\dagger_{in}\hat{a}_{in}}=e^{i\phi}\hat{a}^\dagger_{in}

The interaction Hamiltonian is

 H_{\phi}=\phi\hat{a}^\dagger_{in}\hat{a}_{in}

Where

\hbar=1

Since the Hamiltonian is proportional to the number operator

\hat{n}=\hat{a}^\dagger\hat{a}

this means the number of photons is conserved.

Beam splitters

Beam splitters are fun, and far more complex. I shall return to them in the near future!

Having been in Brisbane for less than 24 hours I don’t have much to say! I can tell you a few things though.

1. It’s hot, very hot. So warm here that it is not cool (hehe) to stay out in the early afternoon for any length of time - but it can be pleasant in the evening. Apparently yesterday was exceptionally warm, even for Brisbane. The day ended with torrential thunder storm  - ha, some guy opposite where I am living left a window open, fun times.
2. I’ve been down to campus, clean and big are two words that come to mind - sorry I’m not the most poetic descriptor.
3. Ummmm watched Australian Idol last night! Funny because they have an equivalent of ‘Ant and Dec’, but no way near as good - their jobs are safe for the time being, that is before James and I hit the stage.
4. I need to start exercise. Everyone here is fitness mad and I’m will try to blend in that way.

That’s all folks. I’m off to sort out my life today, perhaps I may even go to the lab for a moosh around.

See ya.

My last post from the UK for a long time.

Having torn my way through the labyrinth of Australian immigration and University bureaucracy I stand at the final hurdle - getting over there.

I find it remarkable that in less that 24 hours (actual time) I can be transported from one side of the globe to the other. A feat of human kind that must not be over looked as a trivial one, but is so often done so. At this moment I want to say something profound, something different, something that will make us all think a little more about our own achievements in life, what they mean to us and those around the world. But with fear of this being too heavy an undertaking for such a light mind I will say just this:

Without the benefit of true foresight it is not possible to predict with certainty the ramifications of our actions, all we can go with is what feels right and true at the time. Success, prosperity, triumph and achievement form the consequences of the decisions we make and the paths we choose in life. However, just like the great feats of human kind, our own achievements could at some point be someone’s great misfortune.

I believe by accepting this cost to others, by taking the time to think about how best to minimise it, it is natural that more love be shed onto the world.

We make a living by what we get. We make a life by what we give.

- Winston Churchill