## QuanTechi

This page contains circuits, techical manuals, sortware, and other little pieces of wisdom we developed and found useful in our experimental work.

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Apart from usual laboratory hazards, such as high voltage, toxic chemicals, or dumb co-workers, there is a specific one relevant to optics: lasers. The power of typical laboratory lasers ranges up to 20 Watt which is probably less than your reading light. Which means that the horrors you’ve seen in sci-fi movies are greatly exaggerated and the worst a laser beam can do to your skin is a minor burn you may even wish to proudly show to your friends. However, there is one part of your body that is especially vulnerable: your eyes. If a direct laser beam of more than a few milliwatt power hits your eye, **your eye will get permanently damaged.**The main rule is therefore

**Never look directly into the laser beam!**

To make following this rule easier, it is strongly recommended that you **wear safety goggles **whenever a laser is on. Now, since you are not going to wear them anyway, at least try to follow the suggestions below:

– Familiarize yourself with the type of lasers used in your room: continuous or pulsed, visible or invisible, repetition rate, pulse energy, power, color, whatever is relevant.

– Avoid positioning your eyes at the level of the laser beam. If you drop something, close your eyes or look away from the table while bending.

Avoid or be well aware of laser beams flying around above their usual height or shooting straight up. Keep visitors out when a hazardous beam configuration is present.

– Remember, by hurting your eyes you will not just damage your health, but also your colleagues may have to suffer a shutdown of experimental activities and a lengthy investigation.

Every little optic in your lab costs a few hundred dollars. If it doesn’t, it probably costs a few thousand dollars. So, exercising proper care is essential.

Avoid touching optical surfaces with your fingers. If you do happen to touch something, however, this is not a tragedy. Just clean the optic immediately with acetone using one of the standard techniques. Much worse if you don’t notice / forget / neglect to clean it, your body fat residue will enter a chemical reaction with the glass surface and damage it permanently in a few hours.

Familiarize yourself with the turn-on and shut-down procedures. Best of all, hang them on the wall and check yourself every night before leaving. You don’t want to come back next day and discover your setup burnt or flooded.

Use latex gloves to handle unmounted optics, e.g. to place an optic into a mount or remove it. Store all unmounted optics in original boxes.

Avoid burning objects with the laser beam (sounds like a fun idea doesn’t it?). The smoke will condense on your optics and fog them. In particular, be careful about blocking laser beams with black carton.

If you remove an optic and lay it on the table, don’t leave any optical surface facing upwards, even for a few minutes. You’ll be surprised to see how much damage even a brief exposure to dust can do.

Do not eat, drink, or smoke in the lab.

Don’t be ashamed of your errors. A furious boss won’t kick you out for a scratched mirror! It is normal for beginners to make mistakes, and the cost of the equipment you will damage is already planned into your lab’s budget. It would be much worse if you hide your errors and your team spends days, weeks, months trying to figure out what’s wrong with the setup.

Mark all optics, both the element itself (pencil on the cylindrical side surface) and the mount (sticker). The information you write should be sufficient to track the item down to the original purchase order, but also to identify it quickly in the lab.

Decide consciously whether to use high-end or cheap optics/hardware for a particular location. Avoid combining high-end mounts with low-end posts.

Always close the door to the lab. Air movements and temperature gradients disturb your alignment.

Make sure your laser beam is well centered on all optics. Never use partial beam blocking for attenuation.

Attach your optics carefully with screws or magnetic mounts. Do not leave any items in use unattached, even temporarily. Do not use excessive force when screwing, unless exceptional stability is required.

Don’t lean on or against the optical table. Holding on to the overhead rack is ok.

If you are aligning an optic, let go of a knob after turning it. Holding a knob displaces a mount a bit.

You may want to keep the lights off when aligning. Darkness will inspire your creativity and also enhance the sensitivity of your eyes.

Do not track your beam with chewing gum wraps and old ripped newspapers. Use library cards and carton screens instead.

Light travels a foot in an nanosecond. Easy to remember.

If you lose alignment of a laser cavity, the main rule is – don’t panic! If you start turning all the knobs randomly, you’ll only make it worse. At best take a half an hour brake and ponder a safe recovery procedure.

Keep good records of your activities – unless your name is Hendrick Schön or you envy his fame (he comes from Konstanz too, by the way)! Record not only the data you take but also all major changes of the setup, procedures developed, problems encountered and the ways found to solve them. Spend some fifteen minutes a day doing this. It is a good investment: your notebooks will cost a few thousand dollars each once you win a Nobel prize!

Most of the data acquisition nowadays is computerized. Which means (a) hardware failures happen when you are least prepared so frequent backups are imperative and (b) you have to store your data in a way that would allow one to find a necessary file quickly. By experience, it is better to avoid file names like *P-t-InP-001-2_1-surf-MBE-circ.dat. *Instead, use a date-based file nomenclature and record the conditions of each dataset acquisition in your notebook.

Approach your supervisor with healthy skepticism. Remember that he is what *you *will be in a few years. He might be more experienced, but not any smarter! Besides, you are much more familiar with your setup and don’t have three more projects and a lecture course to take care of. Although it might be a good idea to listen what he/she has to say, always evaluate his ideas critically. Remember that it’s your, not his way into science that you are paving!

How can one determine a quantum state’s Wigner function? Although it cannot be measured directly as a probability density, all its marginal distributions can. Once we know all the marginal distributions associated with different quadratures – i.e. the Wigner function projections upon various vertical planes – we can reconstruct the Wigner function. This reconstruction procedure is very similar to that used by medical doctors in computer tomography – where one reconstructs a three-dimensional picture of a tissue anomaly from a set of two-dimensional X-ray images acquired from different angles.

Fig. 1

So our goal is to determine the marginal distributions for all different “points of view”. Suppose we can measure the coordinate of our oscillator (for example, by taking its photograph at a given time). Repeating this procedure many times for a set of identical quantum states we obtain the marginal distribution pr(X) for quadrature X. This is, however, insufficient for tomographic reconstruction of the Wigner function: we need distributions

for all values of q. In order to understand how to obtain these distributions we need to recall that our oscillator, well, oscillates. That is, our Wigner function rotates around the phase space origin with the oscillation frequency w. The marginal distribution pr(X) measured at the moment t = q / w is the same as the marginal distribution pr(Xq) measured at the moment t = 0. Therefore, instead of measuring the marginal distributions at a particular moment for all “view angles” we can measure marginal distribution associated with one view angle, but at different moments [Fig. 1].

Fig. 2

Let us now remember that we work with light, not with mechanical oscillators. As we know from classical physics, these two physical systems are described by the same equations of motion. The role of the mechanical coordinate in the electromagnetic wave is played by the electric field. If we had an “electroscope” able to perform phase-sensitive measurements of the electric field in an electromagnetic mode, we would be able to reconstruct the “quantum portrait” of the mode’s quantum state – its Wigner function.

Unfortunately, such an “electroscope” does not exist. Typical oscillation frequencies of the light fields are hundreds of terahertz, and no one can build a detector that can follow such fast changes. Yet one can use a trick that allows to do phase-sensitive measurements of the electric field using regular, “slow” detectors. This trick is called **balanced homodyne detection** – here is its main idea.

Quantum objects are usually small, and electromagnetic fields that exhibit quantum properties are usually weak. Let us overlap the mode whose quantum properties we want to measure (the “signal” mode) with a relatively strong laser beam (the “local oscillator”) on a beam splitter [fig. 2]. If the electric fields in the two modes are Es and ELO, the fields emerging in the two beam splitter output ports are given by

Let us now have both beam coming out of the beam splitter hit high-efficiency photodiodes. A photodiode is a device which generates electric current proportional to the intensity (not the amplitude) of the electromagnetic field incident on it. Our two photodiodes produce photocurrents that we subtract from each other. The photocurrent difference is given by

since ELO >> Es, we neglect the quantum noise of the local oscillator field and find out that the photocurrent difference is proportional to the amplitude of the signal field – exactly the quantity we are looking for! By changing the relative optical phase of the local oscillator and signal waves, we measure the electric field at different phases. At each phase, we perform a multitude of electric field measurements (each time preparing an identical quantum state in the signal channel) thus obtaining the marginal distribution. A set of marginal distributions for various q’s will provide us with full information about the quantum state – and allows us to reconstruct its Wigner function and the density matrix.

An excellent introduction into nonclassical light and methods of its characterization is given in the textbook by U. Leonhardt “Measuring the quantum state of light”, CambridgeUniversity Press, 1997.

Our review paper on homodyne tomography

Gerd Breitenbach’s homodyne tomography page

This is the simplest quantum state altogether: no light! But even when we switch off the light, the uncertainty principle holds valid, so both position and momentum exhibit some fluctuations. These are the vacuum fluctuations and they are the same for X and P – so the Wigner function is a round hill centered at the origin of phase space.

This state is emitted by a laser. Its Wigner function is the same as for the vacuum state, but displaced in the phase space so there is a nonzero average for the electric field.

This is the state of radiation a heated black body emits. It has no phase and its photon statistics is that of Bose-Einstein. By room temperature the thermal state is to a good approximation vacuum.

The first nonclassical state on this page. The uncertainty in one quadrature is reduced at the other quadrature’s expense. The product of the two is however the same as for the vacuum: this is a minimum uncertainty state.

See our experimental work on generating squeezed light.

The name says it all – a state containing exactly one photon. We did an experimental project on this one, too!

This is a coherent superposition of two coherent states: |a> and |-a>. It is hard to generate experimentally, but the Wigner function is instructive to look at. We see two round hills at the top and at the bottom, associated with each coherent state involved. If our ensemble were an incoherent, statistical mixture of these two states, its Wigner function would just feature these two hills. But because the superposition is coherent, the Wigner function exhibits an additional, highly nonclassical feature in the middle: a fine structured interference pattern with negative regions.

This state is an analog of the squeezed state, but is generated (in theoreticians’ notebooks) by means of 3-photon, rather than 2-photon, down-conversion.

A limiting case of the “odd” Schrödinger cat state |α> – |-α> for small α.

Gerd Breitenbach’s page featuring another gallery of quantum states.

This tutorial gives a technical description of the light storage phenomenon. The physical description – i.e. not what happens, but why it happens, is much more involved and is hard to describe in the semi-popular language.

Consider a glass cell filled with a gas of so-called Lambda-type atoms – a three-level system with two ground and one excited states:

We assume the gas to be optically dense on the transition between levels 1 and 3, that is, a laser beam of the transition wavelength will be absorbed by the gas. If we now, at the same time, apply an additional, strong laser field to the transition 2 – 3 to exhibit a narrow transparency window in the absorption spectrum. Moreover, if the blue laser is pulsed, the pulse will propagate through the cell at a speed much less than the vacuum speed of light – possibly slower than a bicycle! This phenomenon is called **electromagnetically-induced transparency (EIT)**.

Let us now turn the control (red) laser off slowly while the signal (blue) pulse is entirely within the sample making its way through. As we reduce the control field, the signal pulse will propagate slower and slower and will eventually stop. The quantum information contained in the pulse will be transferred to the atoms – more precisely, to the coherent superposition of the two ground levels 1 and 2. Because both are low energy states, their superposition is long-lived, that is – subject to little decay. It can remain unchanged for hundreds of microseconds – an epoch on the atomic time scale!

Theoreticians say, this storage procedure is completely reversible. When we wish to read our memory, we slowly turn the control field back on. The signal pulse will be reemitted with its original shape, phase and in its original quantum state – thus completing the storage/release procedure.

Here is a little movie showing the process of light storage. Notice how the EIT absorption and dispersion, as well as the group velocity of the light pulse change as a function of the control field.

Consider a classical harmonic oscillator. Its motion can be completely described by a point in the phase space – the two-dimensional space with the particle’s coordinate X and momentum P as dimensions (**quadratures**) [Fig. 1(a)]. For a large number of identical classical oscillators, one can talk about the **phase-space probability distribution** – a function W(X, P) which indicates the probability of finding a particle at a certain point in the phase space [Fig. 1(b)]. This function must, of course, be non-negative and normalized: its integral over the entire phase space must be equal to one.
This classical probability distribution has another important property. Consider a series of measurements in which we only measure the oscillator’s coordinate but not the momentum. After a large number of such measurements, one obtains the probability distribution associated with the coordinate – we call this a **marginal distribution** pr(X). This distribution is related to the phase-space probability density in the following way:

In other words, a marginal distribution is just a density projection of W(X, P) onto a plane associated with the given quadrature [Fig 1(d)].

In the quantum world [Fig. 1(c)], the notion of a “certain point in the phase space” does not make sense because the position and the momentum cannot be measured simultaneously (Heisenberg’s uncertainty principle). Neither does the phase-space probability density. However, even in the quantum domain one can perform quantum measurements of a single quadrature – be it X, P, or their linear combination. A multiple measurement of a quadrature on a set of identical quantum states will yield a probability density associated with this quadrature, i.e. a marginal distribution. Is there any connection between marginal distributions for different quadratures?

In the classical world this connection exists – through the phase-space probability density as discussed above. The amazing fact is that even in the quantum domain there exists so called phase space quasiprobability density – called the**Wigner function** – with exactly the same property. A marginal distribution associated with a particular quantum state and a particular quadrature is a projection of the state’s Wigner function upon the relevant vertical plane.

The Wigner function of a given state can be calculated from its density matrix. For each quantum ensemble there exists a Wigner function. Just as the classical phase-space probability density, it is real and normalized. However, the Wigned function has one very important difference from its classical analogue. Because by itself it does not have a meaning of a probability density, it does not have to be positive definite. An example is provided by so-called Fock states of harmonic oscillators – the states of definite energy. No matter what the energy is, the phase space has regions where the Wigner function takes on negative values.

In other words, a marginal distribution is just a density projection of W(X, P) onto a plane associated with the given quadrature [Fig 1(d)].

In the quantum world [Fig. 1(c)], the notion of a “certain point in the phase space” does not make sense because the position and the momentum cannot be measured simultaneously (Heisenberg’s uncertainty principle). Neither does the phase-space probability density. However, even in the quantum domain one can perform quantum measurements of a single quadrature – be it X, P, or their linear combination. A multiple measurement of a quadrature on a set of identical quantum states will yield a probability density associated with this quadrature, i.e. a marginal distribution. Is there any connection between marginal distributions for different quadratures?

In the classical world this connection exists – through the phase-space probability density as discussed above. The amazing fact is that even in the quantum domain there exists so called phase space quasiprobability density – called the

The Wigner function of a given state can be calculated from its density matrix. For each quantum ensemble there exists a Wigner function. Just as the classical phase-space probability density, it is real and normalized. However, the Wigned function has one very important difference from its classical analogue. Because by itself it does not have a meaning of a probability density, it does not have to be positive definite. An example is provided by so-called Fock states of harmonic oscillators – the states of definite energy. No matter what the energy is, the phase space has regions where the Wigner function takes on negative values.