How do you rotate a Fock state qubit?

Question

I read that a qubit can be encoded in a Fock state, such as the presence or absence of a photon. How do you perform single qubit rotations on Fock states?

Answer 1

Superpositions in Fock space--and rotations in Fock space--are absolutely ubiquitous.

• It is important to note that all classical states of the electromagnetic field are superpositions of many different photon-number eigenstates.

• The entire discipline of quantum field theory (approximately) concerns which rotations within certain physically-motivated Fock spaces are allowed, and with what amplitudes they actually occur.

• The experimental paradigms of circuit and cavity QED--which validate exquisitely the predictions of that now 70-year-old theory--explicitly deals with operations on photon number states (in particular "the presence or absence of a single photon" as DaftWullie put it), and are cornerstones of atomic, molecular and optical physics. Circuit QED is the essential theory underpinning superconducting flux qubits, which devices have been shown to display coherent quantum effects beyond any reasonable or unreasonable doubt. Serge Haroche was awarded the 2012 Nobel Prize in physics for his work on cavity QED, in which he went on happily creating, controlling and measuring superpositions of small numbers of microwave photons. Lots of experimentalists do this every day.

• It has long been suggested that a single harmonic mode be used to represent one or more qubits in a practical quantum computer, in which logical states are encoded as superpositions of states of different occupation number. For a few ideas on how to do this as well a few reasons why it might not be the best idea, see Nielsen and Chuang, section 7.2.

There's no shortage of literature on ways to perform these kinds of operations. In fact, a nontrivial fraction of modern physics is concerned with exactly that. I can't imagine where or how you would get the opposite idea.

Answer 2

The short answer is that you can't. There's something called a "particle number superselection rule" which postulates that you can't create a superposition of different numbers of particles. So, if you prepare a Fock state, you can perform phase gates, and bit flips, but you cannot perform arbitrary rotations that create superpositions of different particle number.

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The longer answer is that sometimes you can make superpositions, if you have the right reference frame available. There's a good discussion of this stuff here. This is the reason why states such as the coherent states, which are a superposition of different numbers of photons, can be created (and they get used for quantum computation, but that's an entirely different question). But I believe that this can't work with small photon numbers (e.g. the presence or absence of a single photon). The only thing you can do in that context is create a superposition of a single photon being in one of two places.