Quantop > Quantum Optics Lab > Research > Mesoscopic cold atomic ensembles > Projection Noise

Projection Noise

Even when $\nu_{clock}$ and $\nu_{osc}$ are exactly identical sometimes we find more atoms in $\uparrow$ (and sometimes more in $\downarrow$).

If $\Delta \nu=0$ after all the microwave pulses every single one of the $N_a=100000$ atoms is in the quantum state $$|\psi\rangle = \frac{1}{\sqrt{2}} \Bigl( |\downarrow\rangle + |\uparrow\rangle \Bigr) $$ so that for each atom there is an equal chance of finding it in $\uparrow$ or $\downarrow$.

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 $\textrm{var}(N_\uparrow-N_\downarrow) = N_a$. 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 $\varphi$ is $\delta \varphi = \frac{1}{\sqrt{N_a}}$.

Since $\varphi = \Delta \nu \cdot \tau$, for an interrogation time $\tau$ this means that we can measure the frequency with a precision $\delta \nu = \frac{1}{\tau \sqrt{N_a}}$.

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