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How can somebody demonstrate the quantum superposition of states directly by other means than the double slit experiment?

And why can't macroscopic objects like a pen be in superpostion of states? Will it ever be possible to have an object like a pen to be in superposition of more than one state?

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Others have given good answers, but it all depends on what you mean by "directly." You simply can't "see" a wavefunction. There are many experiments showing quantum superpositions, but they all rely to some degree on interference since that is how you measure the relative phase of different components of the wavefunction. – Michael Brown Jan 11 '13 at 9:54
So, the reason why macroscopic objects like pen can't be in superposition of different because of decoherence. But will it be ever possible to "remove" decoherence and demonstrate the quantum of superposition of large objects? – user774025 Jan 13 '13 at 13:06

This reference describes an experiment in which an ion is prepared in a superposition state for which the relative phase between the component states is controllable, and ulimately measurable.

The obstruction to having objects like pens in superposition states is the inability to isolate its parts from the environment. Interactions between the environment and the system (the atoms in the pen) act through a process called decoherence, which has the effect of suppressing these relative phase terms incredibly quickly.

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Observation of a kilogram-scale oscillator near its quantum ground state Abbott, B. et al.

Quantum Upsizing Aspelmeyer, Schwab, Zeilinger.

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There are experiments with cold atom gases in optical traps that deal with matter's essential quantum nature. Cold atom gases' is actually quite a hot field (no pun intended). In an attempt to answer your question, I will try to explain how I understand quantum superposition experiments work in the most general possible way I can.

A superposition state $\left|\Psi\right\rangle$, under this most general description possible, is a linear superposition (that is precisely the reason why they are called that way) of the different quantum states that the system may have: $$\left|\Psi\right\rangle=\sum_i c_i\left|\tilde{\Psi}_i\right\rangle$$ Where the sum is performed over all possible $\left|\tilde{\Psi}_i\right\rangle$ states in the chosen basis, each one of them weighed by a complex factor $c_i$. The common idea in every experiment dealing with quantum superposition is simple: to have a system whose initial state, which is given by the $c_i$'s, can be prepared and controlled (that is called a coherent state), and in which a certain observable quantity can be measured (associated with an operator branded $\hat{A}$). At the end of the day, the result of every observable measurement can be understood in terms of how the operator $\hat{A}$ acts over the different states in the basis, how each one of these states in the particular basis of choice evolve in time and which are the weights $c_i$.

What happens in the double slit experiment, trying to explain it in these terms without delving into further details, is that the different time evolutions of the (initially coherent) electrons moving through the experimental setup introduce phase factors that keep adding up to the $c_i$'s. When the position of the electrons is recorded on a screen (which constitutes the measurement), the mean value of the position observable has these different phase factors interfering with each other, giving rise to an interference pattern in the screen as a result.

The key idea in these experiments is always the following: To start with a coherent state (or at least as coherent as it can be managed in practice), to let interactions and time evolution make their respective jobs, and later on to see what happens when the system is looked at.

What happens with a macroscopic system? The abridged answer is: 'Way too many things'. For starters, the number of components would be of the order of $N_A=6'022\cdot10^{23}$ when not greater, so the number of states in our basis for a complete, many-body description would be absolutely humongous. On top of that, atoms and electrons are subject to many interactions on a microscopic level, so any coherence that may be momentaneously achieved will be destroyed in a quick, uncontrolled way before any measurement may be performed...

... Unless what we are dealing with is a system of a very particular kind, either with not too many particles (cold atom gases) or some specific interaction which favours the appearance of a coherent state even on a $N_A$-level macroscopic scale (as in conventional superconductivity and superfluidity).

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In my opinion, the concept behind the famous Schroedinger's cat is exactly this: to Schroedinger, it appeared paradoxical that a macroscopic object -- even a living being -- could be in a superposition of states. However, cases in which the radiation inside a cavity was prepared in a superposition of two macroscopically distinguishable states have already been observed experimentally. For example, this was one of the result for which the Nobel Prize this year was awarded, see here. In the case cited, decoherence was also observed. Even better: it was observed a slow, gradual process of decoherence.

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Others have covered the usual microscopic systems that are currently amenable to controlled quantum manipulations, but there are in fact "macroscopic" objects (to some definitions of the word) that can be placed in quantum superpositions. These fall broadly within a field known usually as cavity optomechanics, which has a reasonable wikipedia page. The essential idea is to couple the oscillations of light to those of one of the mirrors of a cavity; this allows 'quantumness' in the state of the light to translate into the state of the mirror.

In this and usually all macroscopic examples, it is only one degree of freedom - the centre-of-mass position in this case - that gets put in a superposition; all other degrees of freedom are left in classical, often thermal, states. This is nevertheless quite enough to get quantum behaviour. For example, you can obtain double-slit interference patterns using buckyballs (C$_{60}$) using their centre-of-mass positions, while maintaining the rotational and vibrational degrees of freedom (which are merely a better way of accounting for all motional degrees of freedom apart from the average) in (fairly cold) thermal, classical states.

Another quantum superposition of a macroscopic object - (just) visible to the naked eye! - is the placing of a microwave oscillator in a superposition state using other microwave sources en lieu of laser beams on an atom. This is fairly well explained by Aaron O'Connell in his TED talk. For more formal references try the UCSB Cleland group page; good popular science articles are listed in O'Connell's wikipedia page.

Finally, I would try macroscopic quantum phenomena" in wikipedia.

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