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Quantum Mechanical Noise in a Michelson Interferometer with Nonclassical Inputs: Nonperturbative Treatment

S.‐K. King

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

The variances of the quantum-mechanical noise in a two-input-port Michelson interferometer within the framework of the Loudon-Ni model were solved exactly in two general cases: (1) one coherent state input and one squeezed state input, and (2) two photon number states inputs. Low intensity limit, exponential decaying signal and the noise due to mixing were discussed briefly.

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In the present study we extend and generalize previous results for coherent-state and squeezed-state Michelson interferometer quantum mechanical uncertainties (or fluctuations), which are commonly referred to as `quantum noise'. The calculation of photon counting (PC) fluctuations in the squeezed-state interferometer is extended to fourth-order correlation functions used as the measured signal. We also generalize a `unified model' for treating both PC and radiation pressure fluctuations in the coherent-state interferometer, by using mathematical methods which apply to Kerr-type interactions. The results are more general than those reported previously in two ways. First, we obtain exact expressions, which lead to previous results under certain approximations. Second, we deal with cases in which the responses of the two mirrors to radiation pressure are not equal.

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It is shown in the present Letter that the quantum noise due to high laser intensities in Michelson interferometer for gravitational waves detection can be reduced by sending squeezed vacuum states to the 'dark' port of the interferometer. The experimental details of such physical system have been described in a recent article by Barak and Ben-Aryeh (JOSA-B, 25, 361(2008)). In another very recent article by Voronov and Weyrauch (Phys. Rev.A 81, 053816 (2010)) they have followed our methods for treating the same physical system, and have pointed out an error in the sign of one of our expressions thus claiming for the elimination of our physical results. I show here a method by which the expectation value for the photon number operator is increased and at the same time the standard deviation is reduced. Although due to the mistake in sign in our expression the physical method for obtaining such result is different from our original study [1], the main physical effect can remain .

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We study the photon counting noise in optical interferometers used for gravitational wave detection. In order to reduce quantum noise a squeezed vacuum state is injected into the usually unused input port. Here, we specifically investigate the so-called “dark port case,” when the beam splitter is oriented close to 90° to the incoming laser beam, such that nearly all photons go to one output port of the interferometer, and only a small fraction of photons is seen in the other port (“dark port”). For this case it had been suggested that signal amplification is possible without concurrent noise amplification [R. Barak and Y. Ben-Aryeh, J. Opt. Soc. Am. B 25 , 361 (2008)]. We show that by injection of a squeezed vacuum state into the second input port, counting noise is reduced for large values of the squeezing factor, however the signal is not amplified. Signal strength only depends on the intensity of the laser beam.

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Quantum-mechanical noise in an interferometer: Intrinsic uncertainty versus measurement uncertainty

Wei-Tou Ni

In many quantum-mechanical measurements, the total uncertainties (\ensuremath{\sigma}) are greater than or equal to the intrinsic uncertainties (${\ensuremath{\sigma}}_{I}$) of the quantities to be measured; the differences are due to the particular measurement processes chosen. In each of these situations, a measurement uncertainty ${\ensuremath{\sigma}}_{M}$ can be defined as ${\ensuremath{\sigma}}_{M}$=(${\ensuremath{\sigma}}^{2}$-${\ensuremath{\sigma}}_{I}$${\mathrm{}}^{2}$${)}^{1/2}$. A question to ask is whether ${\ensuremath{\sigma}}_{M}$ is independent of the initial quantum state of the system to be measured, i.e., whether ${\ensuremath{\sigma}}_{M}$ is only dependent on the measurement process chosen. For a Michelson interferometer, we extend Loudon's model to show that in the model ${\ensuremath{\sigma}}_{M}$ is only dependent on the measurement process chosen for the case with small radiation-pressure fluctuation. When the intensity and radiation-pressure fluctuation are high, there are cross-correlated terms. Some of these terms are negative and give hope to the possibility of uncertainty reductions for appropriate schemes.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•May 15, 1987•7 citations
The variances of the quantum-mechanical noise in a two-input-port Michelson interferometer within the framework of the Loudon-Ni model were solved exactly in two general cases: (1) one coherent state input and one squeezed state input, and (2) two photon number states inputs. Low intensity limit, exponential decaying signal and the noise due to mixing were discussed briefly.
In the present study we extend and generalize previous results for coherent-state and squeezed-state Michelson interferometer quantum mechanical uncertainties (or fluctuations), which are commonly referred to as `quantum noise'. The calculation of photon counting (PC) fluctuations in the squeezed-state interferometer is extended to fourth-order correlation functions used as the measured signal. We also generalize a `unified model' for treating both PC and radiation pressure fluctuations in the coherent-state interferometer, by using mathematical methods which apply to Kerr-type interactions. The results are more general than those reported previously in two ways. First, we obtain exact expressions, which lead to previous results under certain approximations. Second, we deal with cases in which the responses of the two mirrors to radiation pressure are not equal.
The interferometers now being developed to detect gravitational waves work by measuring the relative positions of widely separated masses. Two fundamental sources of quantum-mechanical noise determine the sensitivity of such an interferometer: (i) fluctuations in number of output photons (photon-counting error) and (ii) fluctuations in radiation pressure on the masses (radiation-pressure error). Because of the low power of available continuous-wave lasers, the sensitivity of currently planned interferometers will be limited by photon-counting error. This paper presents an analysis of the two types of quantum-mechanical noise, and it proposes a new technique---the "squeezed-state" technique---that allows one to decrease the photon-counting error while increasing the radiation-pressure error, or vice versa. The key requirement of the squeezed-state technique is that the state of the light entering the interferometer's normally unused input port must be not the vacuum, as in a standard interferometer, but rather a "squeezed state"---a state whose uncertainties in the two quadrature phases are unequal. Squeezed states can be generated by a variety of nonlinear optical processes, including degenerate parametric amplification.
It is shown in the present Letter that the quantum noise due to high laser intensities in Michelson interferometer for gravitational waves detection can be reduced by sending squeezed vacuum states to the 'dark' port of the interferometer. The experimental details of such physical system have been described in a recent article by Barak and Ben-Aryeh (JOSA-B, 25, 361(2008)). In another very recent article by Voronov and Weyrauch (Phys. Rev.A 81, 053816 (2010)) they have followed our methods for treating the same physical system, and have pointed out an error in the sign of one of our expressions thus claiming for the elimination of our physical results. I show here a method by which the expectation value for the photon number operator is increased and at the same time the standard deviation is reduced. Although due to the mistake in sign in our expression the physical method for obtaining such result is different from our original study [1], the main physical effect can remain .
We study the photon counting noise in optical interferometers used for gravitational wave detection. In order to reduce quantum noise a squeezed vacuum state is injected into the usually unused input port. Here, we specifically investigate the so-called “dark port case,” when the beam splitter is oriented close to 90° to the incoming laser beam, such that nearly all photons go to one output port of the interferometer, and only a small fraction of photons is seen in the other port (“dark port”). For this case it had been suggested that signal amplification is possible without concurrent noise amplification [R. Barak and Y. Ben-Aryeh, J. Opt. Soc. Am. B 25 , 361 (2008)]. We show that by injection of a squeezed vacuum state into the second input port, counting noise is reduced for large values of the squeezing factor, however the signal is not amplified. Signal strength only depends on the intensity of the laser beam.
In many quantum-mechanical measurements, the total uncertainties (\ensuremath{\sigma}) are greater than or equal to the intrinsic uncertainties (${\ensuremath{\sigma}}_{I}$) of the quantities to be measured; the differences are due to the particular measurement processes chosen. In each of these situations, a measurement uncertainty ${\ensuremath{\sigma}}_{M}$ can be defined as ${\ensuremath{\sigma}}_{M}$=(${\ensuremath{\sigma}}^{2}$-${\ensuremath{\sigma}}_{I}$2{\mathrm{}}^{2}2)1/2{)}^{1/2})1/2. A question to ask is whether ${\ensuremath{\sigma}}_{M}$ is independent of the initial quantum state of the system to be measured, i.e., whether ${\ensuremath{\sigma}}_{M}$ is only dependent on the measurement process chosen. For a Michelson interferometer, we extend Loudon's model to show that in the model ${\ensuremath{\sigma}}_{M}$ is only dependent on the measurement process chosen for the case with small radiation-pressure fluctuation. When the intensity and radiation-pressure fluctuation are high, there are cross-correlated terms. Some of these terms are negative and give hope to the possibility of uncertainty reductions for appropriate schemes.