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When textbooks in QM give example of finite dimensional Hilbert spaces they give examples of photon polarizations or of 2-states systems and sometimes they mention how one can achieve superposition in such cases experimentally.

On the other hand when they talk about simple potentials like particle in an infinite potential well, and talk about superposition of the stationary states of this problem they never mention how such superposition can be achieved experimentally (despite the fact that we are in the era of nanotechnology and scientists can "engineer" effectively such potentials and many others). Typical example that may appear in textbooks can be as simple as $\Psi(x)=\frac{4}{5}\phi_1(x)+\frac{3}{5}\phi_5(x)$, where {$\phi_n(x)$} are the normalized eigenfunctions of the particle in a box Hamiltonian. Other more general superpositions could be between infinite numbers of stationary states via $\Psi=\sum a_n\phi_n(x)$.

That makes me wonder, is there a fundamental reason that prevents us from engineering such superposition in case of particle in a box and the like, or that we just do not know how to do it yet? why is it possible with spin and seem to be hard with particle in a box? or is it something related to the energy eignestates in the position representation?

It is kind of frustrating to study for long hours/read/solve problems/HW on all kinds of potentials and on superpositions without knowing how/if they can be realized in experiment.

If someone knows references in which this issue is discussed it would be greatly appreciated.

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+1 on the pythagorean amplitudes :) –  Emilio Pisanty Aug 18 '12 at 20:34
    
There are some papers about ways of constructing any superposition in certain experimental systems. I don't have time to locate them now, but hopefully somebody can do that. –  Peter Shor Aug 18 '12 at 22:25
    
Beware of question like, "Can ... be realized experimentally?" Do you mean that, theoretically, can an experiment realize it? I would argue that's not the same as, "Can an experiment realize ..."? (More below on my answer.) –  emarti Jan 13 '13 at 19:59
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4 Answers 4

Theoretically, any superposition can be experimentally realized. Experimentally, most can't. The fundamental reason is that a system must be decoupled from its environment so as not to decohere, yet still coupled strongly to an extremely well-calibrated apparatus to generate the superposition. I would guess the 'advanced information' for the 2012 Nobel Prize would be a good starting point, since this issue was so central for both Haroche and Wineland.

As a very rough experimental summary, superpositions of two-level systems has been seen in a huge number of systems, from charge or flux states in qubits to electronic states of semiconductor impurities. A general superpositions on the order of up to maybe half a dozen states has been realized in a few systems. The systems that leap to mind are nuclear spins in a molecule, internal spins in an atom, the spin/electronic state of a chain of ions, momentum state of a free-falling aotm, or the polarization and spatial mode of one (or several) photons.

For a large number of particles, very particular superpositions can be made. For example, spin squeezing has been observed for probably hunderds to thousands of atoms, but a general superposition state of so many atoms is beyond current experiments. That, in a word, is why it's hard to build a quantum computer.

To answer the broader question, superpositions are very useful to know about and understand, even if they're very challenging to make.

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It all depends on exactly what you want to do - what system you're handling, what state you want to engineer, and what you plan to do with it. (Note that "I just want to make it" is definitely a perfectly legitimate purpose, but then you also have to think about how you're going to detect it and make sure you've got it!)

For the specific example you pose, creating a superposition of two particle-in-a-box states, you first have to make the box. This is now doable using quantum dots (semiconductor islands in a different semiconductor, possibly with an electron-donating impurity inside) with the right geometry. You also need to make sure that your well is deep enough to accommodate the states you want without shifting their energies too much. After that, though, it's a piece of cake (relatively), since the $\propto n^2$ dependence of the energy levels makes all the transition frequencies distinct. Then you just have to shine a laser pulse at the right frequency and you effectively eliminate all the other levels to get a two-level system interacting with a laser field - a Rabi problem - and you just need to drive a Rabi cycle long enough to get the superposition you need.

However, not all systems are as easily manipulated, and the creation of specific states can be quite challenging. For example, for a harmonic oscillator, all the transition frequencies are the same, and you cannot do this kind of trick, so that making states with a well-defined number of photons/quanta can be very difficult (but doable!). For example, creating superpositions of different coherent states (i.e. "cat states") in light is currently only possible in certain geometries, as I found out on this question. Number states, coherent states, squeezed states, superpositions, entangled states, and so on, have been realized to some degree or other in light beams, mechanical oscillators, atoms and ions, circuit QED, and so on. Again, it depends on what you want your "weird quantum state" to do.

A word of warning, though, on your more general infinite superposition $\Psi=\sum_n a_n \phi_n$. While in principle this is (more-or-less) doable, depending on the state, you also have to bear in mind that one can only ever do a finite number of measurements on the state and therefore you can only ever confirm with certainty a finite number of the coefficients $a_n$. This is another way of saying that you can only ever do stuff with some finite precision. Thus all you can create is a finite sum like $$\Psi=\sum_n^N a_n \phi_n+\textrm{ some amount of noise.}$$ Other than that, it again depends on what system you have and what state you want and it's up to your experimental ingenuity do design a procedure that will take you there.

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I do not quite understand the part on shining the laser. Regarding the finite sum of stationary states to reach an approximate $\Psi$, I am wondering how well the stationary states, $\phi_n$, are prepared? are they approximate as well? –  Revo Aug 19 '12 at 21:36
    
The $\phi_n$ are basically in your head, so they can be approximate or exact depending on what you need. Ultimately you need to refer to exactly what you are measuring. –  Emilio Pisanty Aug 20 '12 at 0:02
    
About the laser: when you shine EM radiation of angular frequency $\omega$ on a system, it can only excite transitions at energy $\sim\Delta E=\hbar\omega$. If the system is on, say, the ground state, then states not at this energy difference will not be populated and can therefore be ignored. This is effectively the Bohr principle that atoms absorb and emit radiation at frequencies corresponding to the energy differences of transitions within the different energy levels. –  Emilio Pisanty Aug 20 '12 at 0:44
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Comment to Shor (apologies for the answer, I cannot yet write comments):

Maybe you are referring to Quantum controllability theorems.

Basically quantum controllability tells you what are the requirements needed for any state of the system to be accessible from any other state by means of an external electromagnetic field at a finite time. The problems are of course related to degeneracies in the spectra of many Hamiltonians. The first papers addressing this problem are J. Math. Phys. 24, 2608, (1983) and Phys. Rev. A, 51, 960 (1995). There are many works after this, particularly due to its importance in Quantum Control and its connection with Quantum Computation.

To Emilio Pisanty: By the way, the harmonic oscillator is a well known uncontrollable system. However any truncation of the Hamiltonian makes the system controllable again.

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Yes, for any finite-dimensional Hilbert space that effectively describes some physical system, it is possible to design a procedure that prepares the physical system in any state $|\psi\rangle \in {\mathcal H}$.

The precise description of the procedure or gadget is an awkward task because the wording inevitably depends on the physical interpretation of the degrees of freedom and their interactions with various macroscopic fields we have.

However, let me just pick a simple example. $\phi_1$ and $\phi_5$ may be interpreted as the states $|up\rangle$ and $|down\rangle$. Then any linear combination of them, including your combination $0.8 \phi_1+ 0.6 \phi_5$ (incidentally, I also love to use this Pythagorean combination as an example), may be prepared as the state "electron up" with respect to a particular axis that is calculable. I could do the calculation of the axis for you if you want, it's trivial. Of course, the state's phase will be undetermined; the state vector's overall phase is always unphysical (unless you can compare it with another phase of the same system, like in Berry's phase or the Aharonov-Bohm effect etc.).

Quantum computation is a systematic "industry" that is able to perform many elementary operations on the Hilbert space, like exchanging $\phi_2$ and $\phi_5$ if you found it natural. Typically, only several operations – like rotations by preferred angles – are allowed operations on a quantum computer. However, it's straightforward to extend the basic operations of a quantum computer so that you may compose them to any unitary matrix you want. That's also enough to prepare any complex linear superposition of any basis vectors.

It's also possible to "remap" qubits encoded e.g. in many electrons' spins to the amplitudes for any other states even though the detailed "diagram" of the apparatus that achieves such a goal will tend to depend on the precise technical implementation. But in principle, such a "remapping" is analogous to copying the classical information (in bits) from a CD to a Flash memory. The same things may be done at the quantum level but the bits are not copied; the original has to be destroyed.

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But this does not answer how to superimpose experimentally 2 eigenstates or more of the particle in a box Hamiltonian. –  Revo Aug 18 '12 at 19:53
    
OK, just shoot the particle into the box using a double-slit-like gadget where the kinetic energy is ready for the first level in the first arm, and accelerated to the 5th level in the second arm. Let the electrons in the two arms interfere before they're trapped in the box. The relative size of the slits will determine the ratio of the normalizations of the two amplitudes, fine adjustments of the arms' length will adjust the relative phase. If I can't shoot the particles inside, e.g. because you want the particle to be inside the box, you would have to tell me what I can do to manipulate it. –  Luboš Motl Aug 18 '12 at 19:58
    
Let me mention that this technical description sounds very different than a quantum computer or others and it shouldn't be surprising. Your task "prepare a physical system in a particular quantum state" is the quantum counterpart of "prepare a classical system in a particular state". This is an extremely universal task that may mean anything - cook a soup or deliver a rover to Mars. All these things are states of a physical system, even classically. The procedures to achieve them of course depend on the physical system but with enough tools and interactions, it can always be done in principle. –  Luboš Motl Aug 18 '12 at 20:01
    
Sorry, but I do not understand what you mean by "arm" let alone "the kinetic energy is ready", "accelerated", and "the arms length" to control the relative phases. Could you please elaborate on that mechanism. Are you saying that we have 2 holes, they act like filters, one will let only particles in the quantum state 1 to pass, and the other will allow only particles in the quantum state 5 to pass, then we let them interfere then collect them in a box? (I guess that would destroy the interference pattern so seems my understanding is wrong) –  Revo Aug 19 '12 at 21:49
    
Yes, you can definitely construct holes that act as filters. A hole always is a filter, after all. It corresponds to a projection operator that only keeps the particle if its position belongs to a set, the hole. By accelerating the particle or doing something else with it along the path, the position information may be converted to another quantum number such as the momentum. If the initial state of the particle is the same (independent of the history/slit) and the final state is the same up to the final position along the would-be interference pattern, there is always interference. –  Luboš Motl Aug 20 '12 at 5:46
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