Quantum sensing limits#

Quantum limits of a phase measurement, and the acceleration sensitivity of an atom interferometer.

A sensor that reads a phase \(\varphi\) from \(N\) independent probes (atoms, photons) repeated \(\nu\) times cannot do better than the standard quantum limit,

\[\Delta\varphi_{\mathrm{SQL}} = \frac{1}{\sqrt{N\nu}} ,\]

while entangled probes can in principle reach the Heisenberg limit \(\Delta\varphi_{\mathrm{H}} = 1/(N\sqrt{\nu})\). Entangled states are fragile, so real instruments sit near the standard quantum limit.

A light-pulse atom interferometer measures a constant acceleration \(a\) as a phase \(\varphi = k_{\mathrm{eff}}\, a\, T^2\), with \(k_{\mathrm{eff}} = 4\pi/\lambda\) the effective two-photon wave number and \(T\) the time between pulses (Kasevich and Chu, 1992). Its single-shot acceleration sensitivity is therefore \(\Delta a = \Delta\varphi/(C\, k_{\mathrm{eff}} T^2)\) with \(C\) the fringe contrast.

Functions

phase_sensitivity

Smallest phase resolvable with n_probes probes repeated repeats times.

atom_interferometer_sensitivity

Single-shot acceleration sensitivity of a light-pulse atom interferometer.

Data

RB87_WAVELENGTH_M

Wavelength of the rubidium-87 D2 line, a common choice for atom interferometers, in metres.
phase_sensitivity(n_probes, repeats=1, scaling='sql')[source]#

Smallest phase resolvable with n_probes probes repeated repeats times.

Parameters:

  • n_probes (float) – Number of probes per shot (atoms or photons), at least 1.

  • repeats (float, optional) – Number of independent shots.

  • scaling ({'sql', 'heisenberg'}, optional) – Standard quantum limit \(1/\sqrt{N\nu}\) or Heisenberg limit \(1/(N\sqrt{\nu})\).

Returns:

float – Phase uncertainty in radians.

Raises:

ValueError – For fewer than one probe or one repeat, or an unknown scaling.

Examples

>>> from quantum_mind.sensing import phase_sensitivity
>>> round(phase_sensitivity(10_000), 4), round(phase_sensitivity(10_000, scaling='heisenberg'), 6)
(0.01, 0.0001)
atom_interferometer_sensitivity(n_atoms, interrogation_time, contrast=0.5, wavelength=7.8024e-07, scaling='sql')[source]#

Single-shot acceleration sensitivity of a light-pulse atom interferometer.

Parameters:

  • n_atoms (float) – Atoms detected per shot.

  • interrogation_time (float) – Time T between pulses in seconds.

  • contrast (float, optional) – Fringe contrast in (0, 1].

  • wavelength (float, optional) – Laser wavelength in metres (default: rubidium D2).

  • scaling ({'sql', 'heisenberg'}, optional)

Returns:

float – Acceleration uncertainty per shot in m/s^2.

Examples

>>> from quantum_mind.sensing import atom_interferometer_sensitivity
>>> a = atom_interferometer_sensitivity(1e6, 0.1)
>>> 1e-9 < a < 1e-7
True
RB87_WAVELENGTH_M#

Wavelength of the rubidium-87 D2 line, a common choice for atom interferometers, in metres.