Quadrature and phase space Coherent displacements Coherent state

① Quadrature and phase space ② Coherent displacements ③ Coherent state properties

II- Coherent states

1 - Definition and properties Universe of all possible states of the SHO • Minimize the Heisenberg uncertainty principle RMS deviation of p • Simultaneously minimize the spread in both position and momentum operators: more classical state possible Coherent states Schrödinger, 1926 Glauber, 1963 • Minimize in one direction at the expense of increasing the uncertainty for the other conjugate variable: squeezed coherent state.

1. 1 Mathematical definition Nearly classical character: searching for pure states of the SHO with mean energy equal to the classical energy Quantum mechanical oscillator energy Zero of energy shifted (normally ordered) Ehrenfest’s theorem: motion of the center of the quantum wave packet obeys law of classical mechanics Classical energy Factorization condition States satisfying this condition are coherent states!

1. 2 Some properties of coherent states • Coherent states are eigenstates of the annihilation operator is not hermitian • They are normalized • Expansion in the number state basis and satisfy the factorization condition ? Consecutive use of Projection of the coherent state on the vacuum?

• Evaluation of ? • Coherent state projected on the vacuum state • In terms of the vacuum • Baker-Hausdorff relation More symmetric but is it allowed? if Displacement operator Coherent state is the vacuum displaced!

2 - Quadratures and phase space position momentum Uncertainty principle: the state at a particular time can’t be specified by a single point in the phase space Quantum state: distribution of points in phase space centered at • Vacuum state with and Classically Dynamics mapped out with deterministic trajectories Circular distribution centered at the origin with standard deviation in any direction ½

2. 1 Coherent displacements • Adding energy to the system in a coherent fashion without increasing the noise Displacement operator coherent amplitude We want to calculate and in the coherent state. , We thus need to know how the displacement operator acts on and • Some properties of the displacement operator Hermitian conjugate operation Inverse operation Displacement of opposite magnitude Transformation of annihilation and creation operators 1) 2) ?

• Quadrature in the coherent state The state is now centered at in the complex plane with the same noise properties as the vacuum Energy of the coherent state Average number of photons added to the vacuum without noise

Summary of coherent states’ properties Coherent states are: • eigenstates of the annihilation operator • not eigenstates of the creation operator! • minimum uncertainty states More classical states • superposition of number states • not orthogonal! becomes increasingly orthogonal if differs sufficiently from • Resolution of the identity Any state can be represented as a superposition of coherent states • Expectation values of normally ordered operator products