Sensing and learning in phase space
with squeezed light

Jonas S. Neergaard-Nielsen
QPIT/bigQ - Department of Physics, Technical University of Denmark

STORMYTUNE conference, Gaeta, 2024-06-19

Q. Computing

Q. Comm. + (semi)di protocols

Q. light sources

Q. sensing

Distributed sensing

Single phase estimation with squeezed vacuum

LEARNING A RANDOM DISPLACEMENT CHANNEL

interferometric
measurement

Sensing a phase shift

limited by quantum noise

increase probe power - if possible

or squeeze...

Amplitude squeezing

Phase squeezing

or squeeze...

Squeezed vacuum

or squeeze...

PPKTP linear

PPKTP bow-tie

PPLN waveguide

Renato
Domeneguetti

Michael
Stefszky

Distributed sensing

\phi_\mathrm{avg} = \frac{\phi_1 + \ldots + \phi_M}{M}

Estimate a global (distributed) parameter:

Different task

\phi_\mathrm{avg} = \frac{\phi_1 + \ldots + \phi_M}{M}

"Cheap" entanglement enhances estimation of a
global parameter of spatially separated systems

Estimate a global (distributed) parameter:

Different task

\phi_\mathrm{avg} = \frac{\phi_1 + \ldots + \phi_M}{M}

\phi_\mathrm{avg} = \frac{\phi_1 + \ldots + \phi_M}{M}

Xueshi Guo

Casper Breum

Johannes Borregaard

\phi_\mathrm{avg} = \frac{\phi_1 + \ldots + \phi_M}{M}

Estimate the average of multiple optical phase shifts

Homodyne detector has been pre-aligned close to the optimal phase by some rough (or adaptive) initial estimation.

For small \(\phi_i\), estimate with homodyne detection of phase quadrature:

\langle \hat{P}_\mathrm{avg}\rangle \approx \sqrt{2\eta}\alpha \phi_\mathrm{avg}

SETTING

\sigma_\mathrm{separable} = \frac{1}{2\sqrt{M}N\sqrt{1+\frac{1}{N}}}

\sigma_\mathrm{entangled} = \frac{1}{2MN\sqrt{1+\frac{1}{MN}}}

\sigma_\mathrm{SQL} = \frac{1}{2\sqrt{MN}}

sensitivity \(\sigma \equiv 1/SNR\)

- minimum resolvable phase shift

HEISENBERG SCALING IN PROBE ENERGY AND # OF SITES/SAMPLES

With losses,
Heisenberg scaling disappears but sensitivity gain remains

Realistic (lossy) situation

\sigma_\mathrm{separable} = \frac{1}{2\sqrt{M}N} \sqrt{\frac{N(1-\eta)+\frac{\eta}{2}(1+\sqrt{1+4N(1-\eta)})}{1+\eta/N}}

\sigma_\mathrm{entangled} = \frac{1}{2MN} \sqrt{\frac{MN(1-\eta)+\frac{\eta}{2}(1+\sqrt{1+4N(1-\eta)})}{1+\eta/N}}

For optimal balance between squeezed and coherent photons:

Experiment

\sigma \equiv \frac{1}{SNR} = \frac{\sqrt{\langle\Delta \hat P^2_\mathrm{avg} \rangle}}{|\partial \langle \hat P_\mathrm{avg}\rangle /\partial\phi_\mathrm{avg}|}

Measured sensitivities

\sigma \equiv \frac{1}{SNR} = \frac{\sqrt{\langle\Delta \hat P^2_\mathrm{avg} \rangle}}{|\partial \langle \hat P_\mathrm{avg}\rangle /\partial\phi_\mathrm{avg}|}

Measured sensitivities

\mu=\frac{N_\mathrm{sqz}}{N_\mathrm{total}}

Optimised photon number balance

single phase estimation with squeezed vacuum

Single phase sensing - improving sensitivity

\sigma = \frac{1}{2N} \sqrt{\frac{N}{1+N}}

N\sim\langle\hat n\rangle=\sinh^2r+|\alpha|^2

estimator:

\hat\phi\propto\langle \hat P\rangle

sensitivity:

photons per probe:

displaced
squeezing

squeezed
vacuum

\sigma = \infty

N\sim\langle\hat n\rangle=\sinh^2r

estimator:

\hat\phi\propto\langle \hat P\rangle

sensitivity:

photons per probe:

\hat\phi\propto\langle \hat P^2\rangle

\sigma = \frac{1}{\sqrt{8}\sqrt{N^2+N}}

Jens AH Nielsen

Comparison with Noon states

\frac{1}{\sqrt{2}}(|N0\rangle+|0N\rangle)

Comparison with Noon states

Experiment

Experiment

P(\phi|\{P_i\})

Bayesian updating of the likelihood of \(\phi\):

- Estimate is \(\hat\phi = \text{arg max }P(\phi|\{P_i\})\),

- Sensitivity is \(\sigma = \sqrt{\text{Var }P(\phi|\{P_i\})}\)

Scaling of the sensitivity

demo of the "deterministic" in the title

Applying a weak 3 kHz phase modulation,

we recover this signal in the recorded trace and spectrum

Learning a random displacement channel

(work in progress)

New scenario

\Lambda

\hat\rho

\Lambda(\hat\rho)

\Lambda(\hat\rho) = \int d^{2n}\alpha \ p(\alpha) \hat{D}(\alpha) \hat\rho \hat D^\dagger(\alpha)

\alpha\in\mathbb{C}^n

p(\alpha) = \delta(\alpha)

Channel:

Dynamics:

Goal:

Small phase shift (\(\approx\) p-displacement) on \(n\) modes

Static - can be probed repeatedly

Estimate average phase shift

Distributed sensing

x- AND p-displacements on \(n\) modes

Random - varies shot to shot, distribution \(p(\alpha)\)

Learn \(p(\alpha)\)

Channel:

Dynamics:

Goal:

Channel learning

\Lambda

\hat\rho

\Lambda(\hat\rho)

\Lambda(\hat\rho) = \int d^{2n}\alpha \ p(\alpha) \hat{D}(\alpha) \hat\rho \hat D^\dagger(\alpha)

\alpha\in\mathbb{C}^n

x- AND p-displacements on \(n\) modes

Random - varies shot to shot, distribution \(p(\alpha)\)

Learn \(p(\alpha)\)

Channel:

Dynamics:

Goal:

Channel learning

\alpha

\hat{x}

\hat{p}

EPR state

\hat{D}(\alpha)

← Squeezed probes are no longer very useful

Take inspiration from dense coding →

New scenario

Random displacement Channel learning

arXiv:2402.18809

Random displacement Channel learning

arXiv:2402.18809

\Lambda(\hat\rho) = \int d^{2n}\alpha \ p(\alpha) \hat{D}(\alpha) \hat\rho \hat D^\dagger(\alpha)

= \frac{1}{\pi^n}\int d^{2n}\beta \ \lambda(\beta) \text{Tr}[ \hat\rho \hat{D}(\beta)] \hat D^\dagger(\beta)

\lambda(\beta) = \int d^{2n}\alpha\ p(\alpha) e^{\alpha^\dagger\beta-\beta^\dagger\alpha}

Alternative description in terms of \(p\)'s characteristic function \(\lambda\):

Large \(\beta\) values ⇒ fast ripples in \(p(\alpha)\)

Random displacement Channel learning

arXiv:2402.18809

\lambda(\beta) = \int d^{2n}\alpha\ p(\alpha) e^{\alpha^\dagger\beta-\beta^\dagger\alpha}

Aim:

Sample the channel \(N\) times, obtaining \(n\)-mode samples \(\{\zeta_i\}\),

after which an estimate \(\tilde\lambda(\beta)\) can be obtained with sufficiently low error for any \(\beta\) within a range \(|\beta|^2\le \kappa n\).

For the entangled scheme, an unbiased estimate is

\zeta_1, \ldots, \zeta_N

\tilde\lambda(\beta) = e^{e^{-2r}|\beta|^2} \tfrac{1}{N} \sum_{i=1}^{N} e^{\zeta_i^\dagger\beta-\beta^\dagger\zeta_i}

\(e^{2e^{-2r}|\beta|^2} \approx 5\cdot10^8\) for \(|\beta|^2=10\) and \(r=0\), but \(\approx 7\) for \(r=1.15\) (10 dB squeezing)