Quantum State Orthogonalization and a Toolset for Quantum Optomechanical Phonon Control

We introduce a method that can orthogonalize any pure continuous variable quantum state, i.e., generate a state $|{\psi }_{\perp }⟩$ from $|\psi ⟩$ where $⟨\psi |{\psi }_{\perp }⟩=0$, which does not require significant a priori knowledge of the input state. We illustrate how to achieve orthogonalization using the Jaynes-Cummings or beamsplitter interaction, which permits realization in a number of physical systems. Furthermore, we demonstrate how to orthogonalize the motional state of a mechanical oscillator in a cavity optomechanics context by developing a set of coherent phonon level operations. As the mechanical oscillator is a stationary system, such operations can be performed at multiple times providing considerable versatility for quantum state engineering applications. Utilizing this, we additionally introduce a method how to transform any known pure state into any desired target state.

Figure 1

A continuous variable pure state can be orthogonalized by coupling with a qubit via the Jaynes-Cummings (a) or the beamsplitter (b) interaction and then measurement of the qubit. Alternatively, simultaneously using the beamsplitter and two-mode-squeezing interactions can be used for state orthogonalization. This can be realized with cavity optomechanics to coherently manipulate the quantum state of motion of a mechanical oscillator (c). (PBS, polarizing beam splitter; FR, Faraday rotator). One of the drive fields is blue detuned and gives rise to a phonon-number-increasing process, whereas the other is red detuned and gives rise to a phonon-number-reducing process. This is shown in (d); a truncated energy level diagram of the optomechanical system where the left kets describe the intracavity photon number and the right kets describe the mechanical phonon number. Each drive generates a sideband at cavity resonance, which is shown in (e), an example optomechanical spectrum. Thus, after erasure of the polarization information, photon detection at the cavity resonance frequency causes the mechanical element to undergo a coherent superposition of phonon addition and subtraction.

Figure 2

An equally weighted superposition of quanta addition and subtraction can orthogonalize any pure quantum state. (a) The orthogonalizer ${{\rm Y}}_{\perp }$ is a quadrature perpendicular to the angle $\vartheta$ made by the input state’s mean in phase space. The Wigner function (blue-cyan, positive; red-yellow, negative; larger ticks mark the origin and they increment by unity) of a displaced squeezed state (b), which has been orthogonalized (c). A superposition of an initial state with an orthogonal state may be prepared to create a qubit from any initial pure state. In (d) such a superposition is shown by action with ${{\rm Y}}_{\perp }+|\mu |e^{-i\pi /2}/\sqrt{2}$, where $|\mu |=r$.