# Projection Noise

Even when ν clock   and ν osc   are exactly identical sometimes we find more atoms in ↑  (and sometimes more in ↓  ).

IfΔν=0  after all the microwave pulses every single one of theN a =100000  atoms is in the quantum state

|ψ⟩=1 2  √   (|↓⟩+|↑⟩)

so that for each atom there is an equal chance of finding it in↑  or↓  .

But as in tossing a coin 100000 times only on average we find 50000 heads and 50000 tails. From statistics we know that in this case var(N ↑ −N ↓ )=N a   . This is the projection noise.

If we add the Bloch-vectors of every atom to form a big vector with length N a   the total vector has an associated uncertainty which can be represented as probability distribution on the sphere.

Using this picture the whole Ramsey-spectroscopy sequence can be summed up in one picture: The uncertainty with which we can estimate the precession angle φ  is δφ=1 N a   √     .

Since φ=Δν⋅τ  , for an interrogation time τ  this means that we can measure the frequency with a precision δν=1 τN a   √     .

• Using more atoms (performing more measurements simultaneously) is one way to increase the precision.
• A longer interrogation time τ  increases the precision of frequency measurements (but not of phase measurements).
• We can use entanglement to reduce the phase fluctuations.