Consecutive measurements of photon number and quantum phase

Thomas Schürmann

Published Dec 20, 2006• 0 citations

Abstract

We introduce the conditional probability to consider consecutive measurements of photon number and quantum phase of a single mode. Let $P$ be the conditional probability to measure the phase $\\alpha$ with precision $\\Delta\\alpha$, given a previous measurement of $k$ photons with precision $\\Delta k$. Two upper bounds of the probability are derived. For arbitrary given precisions, these bounds refer to the variation of $k$, $\\alpha$, and the state vector $\\psi$ in Hilbert space. The first (weaker) bound is given by the inequality $P\\leq\\xi$, with $\\xi=\\frac{\\Delta\\alpha (\\Delta k+1)}{2\\pi}$. It is nontrivial for measurements with $\\xi<1$. As our main result the least upper bound of $P$ is determined. We obtain an analytical representation of this bound in the asymptotic limit $\\Delta k\\to\\infty$ and $\\Delta\\alpha\\to 0$ such that $\\xi>0$ is fixed. Finally, we present a rigorous prove that the well-known 'Heisenberg limit' in precision phase measurement can never be attained with measurement probabilities greater than $1/pi$.

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Consecutive measurements of photon number and quantum phase

Thomas Schürmann

Published Dec 20, 2006• 0 citations

OpenAlex

Abstract

We introduce the conditional probability to consider consecutive measurements of photon number and quantum phase of a single mode. Let

P

be the conditional probability to measure the phase

\\alpha

with precision

\\Delta\\alpha

, given a previous measurement of

k

photons with precision

\\Delta k

. Two upper bounds of the probability are derived. For arbitrary given precisions, these bounds refer to the variation of

k

\\alpha

, and the state vector

\\psi

in Hilbert space. The first (weaker) bound is given by the inequality

P\\leq\\xi

, with

\\xi=\\frac{\\Delta\\alpha (\\Delta k+1)}{2\\pi}

. It is nontrivial for measurements with

\\xi<1

. As our main result the least upper bound of

P

is determined. We obtain an analytical representation of this bound in the asymptotic limit

\\Delta k\\to\\infty

and

\\Delta\\alpha\\to 0

such that

\\xi>0

is fixed. Finally, we present a rigorous prove that the well-known 'Heisenberg limit' in precision phase measurement can never be attained with measurement probabilities greater than

1/pi

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Association between probability distribution of optical phase and relative phase detection in photonic quantum metrology

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The measurement of a phase difference between two optical modes in an interferometer has several important applications in metrology. This measurement can be useful in several contexts, such as in the detection of gravitational waves. The quantum nature of light imposes a limit on how accurately a phase difference can be measured, this limit being known as the Heisenberg limit (HL). On the other hand, photonic quantum metrology studies alternatives to reach the HL through entangled or squeezed quantum states. According to the theory of parameter estimation, the minimum uncertainty possible in the estimation of a phase $\theta$ considering classical light scales with $1/\sqrt{\Bar{N}}$, where $\Bar{N}$ is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with $1/{\Bar{N}}$, known as HL. In this thesis, starting from the relative phase probability distribution P( $\phi$ ) introduced by Luis and S\'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of $0,1\%$ . Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.The measurement of a phase difference between two optical modes in an interferometer has several important applications in metrology. This measurement can be useful in several contexts, such as in the detection of gravitational waves. The quantum nature of light imposes a limit on how accurately a phase difference can be measured, this limit being known as the Heisenberg limit (HL). On the other hand, photonic quantum metrology studies alternatives to reach the HL through entangled or squeezed quantum states. According to the theory of parameter estimation, the minimum uncertainty possible in the estimation of a phase $\theta$ considering classical light scales with $1/\sqrt{\Bar{N}}$, where $\Bar{N}$ is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with $1/{\Bar{N}}$, known as HL. In this thesis, starting from the relative phase probability distribution P($\phi$) introduced by Luis and S\'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of $0,1\%$. Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.

Nov 8, 20220 citations

We introduce the conditional probability to consider consecutive measurements of photon number and quantum phase of a single mode. Let

P

be the conditional probability to measure the phase

\\alpha

with precision

\\Delta\\alpha

, given a previous measurement of

k

photons with precision

\\Delta k

. Two upper bounds of the probability are derived. For arbitrary given precisions, these bounds refer to the variation of

k

\\alpha

, and the state vector

\\psi

in Hilbert space. The first (weaker) bound is given by the inequality

P\\leq\\xi

, with

\\xi=\\frac{\\Delta\\alpha (\\Delta k+1)}{2\\pi}

. It is nontrivial for measurements with

\\xi<1

. As our main result the least upper bound of

P

is determined. We obtain an analytical representation of this bound in the asymptotic limit

\\Delta k\\to\\infty

and

\\Delta\\alpha\\to 0

such that

\\xi>0

is fixed. Finally, we present a rigorous prove that the well-known 'Heisenberg limit' in precision phase measurement can never be attained with measurement probabilities greater than

1/pi

We consider a two-mode bosonic state with fixed photon number

n \in \mathbb {N}

, whose upper and lower modes pick up a phase

\phi

and

-\phi

, respectively. We compute the optimal Fock coefficients of the input state, such that the mean square error (MSE) for estimating

\phi

is minimized, while the minimum MSE is always attainable by a measurement. Our setting is Bayesian, i.e., we consider

\phi

to be a random variable that follows a prior probability distribution function (PDF). Initially, we consider the flat prior PDF and we discuss the well-known fact that the MSE is not an informative tool for estimating a phase when the variance of the prior PDF is large. Therefore, we move on to study truncated versions of the flat prior in both single-shot and adaptive approaches. For our adaptive technique, we consider

n=1

and truncated prior PDFs. Each subsequent step utilizes as prior PDF the posterior probability of the previous step, and at the same time we update the optimal state and optimal measurement.

The ultimate precision of phase estimation is limited by the Heisenberg scaling

Δϕ_0 = K/N

, where

K\sim1

is a numerical prefactor and

N

is the mean number of photons interacting with the phase shifting object(s). However, achieving this fundamental limit often comes at the cost of an extremely narrow high-sensitive range, rendering schemes impractical. We analyze the precision limits of phase measurements in single- and two-arm optical interferometers with input Gaussian states. We consider two detection methods: conventional homodyne measurement and non-Gaussian threshold detection that saturates the quantum Cramér-Rao bound. We characterize the performance by two complementary metrics: the peak sensitivity

Δϕ_0

and the width

δϕ

of the high-sensitivity range. We demonstrate that Heisenberg scaling is attainable in all configurations considered. However, we reveal that

δϕ

strongly depends on

K

. We derive an approximate analytic expression that describes this trade-off. We show also that the two-arm interferometer with antisymmetrically squeezed inputs exhibits exceptional performance, simultaneously achieving Heisenberg-limited sensitivity and a broad high-sensitivity range

δϕ=π/2

\theta

considering classical light scales with $1/\sqrt{\Bar{N}}$, where $\Bar{N}$ is the average number of photons. This dependence is known as the standard quantum limit. However, when considering quantum light sources, there is a significant improvement in the estimation of the phase parameter, as its minimum uncertainty can scale with $1/{\Bar{N}}$, known as HL. In this thesis, starting from the relative phase probability distribution P(

\phi

) introduced by Luis and S\'anchez-Soto in 1996, we obtained this distribution for several quantum states of light useful in photonic quantum metrology. We show that, within the numerical precision of our calculations, the Fisher information obtained via the relative phase distribution is equal to the quantum Fisher information for the considered states, such that the average differences between these quantities were on the order of

0,1\%

. Our results indicate that the relative phase distribution can be used to predict the minimum possible uncertainty in the phase detection process in quantum metrology, since this uncertainty depends on the quantum Fisher information, at least for pure states.