Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Optical Society of America B

Generation of number-phase minimum-uncertainty states and number states

Y. Yamamoto, S. Machida, N. Imoto, Masahiro Kitagawa, Gunnar G. Bjork

Published Oct 1, 1987• 70 citations
Publisher

Abstract

The difference between the two nonclassical photon states, i.e., the squeezed state and the number-phase minimum-uncertainty state, is discussed. Four different generation principles for number-phase minimum-uncertainty states (and number states) are described. They are the following: (1) unitary evolution, using self-phase modulation and interference; (2) nonunitary state reduction, using either quantum nondemolition photon-number measurement or parametrically amplified idler-wave photon counting; (3) negative-feedback oscillators, incorporating a correlated photon-pair generator; and (4) pump-amplitude noise-suppressed lasers. A number-phase minimum-uncertainty state with photon-number noise reduced to 1 dB (20%) below the standard quantum limit in a frequency range over 100 MHz was produced by the fourth-generation principle.

DOI
10.1364/josab.4.001645

Related papers

Generation of amplitude-squeezed states by nondegenerate four-wave mixing with idler measurement feedback

Xi-Zeng Li, Zhenfeng Hu, Lixin Li

If the idler photon flux is continuously measured in a photon-counting detector and the photocurrent fluctuations are fed back to control the pump intensity, the output signal state will be an amplitude-squeezed state (near-number state) at low frequencies and high feedback loop gain. Electromagnetic field in number-phase minimum-uncertainty states (amplitude- squeezed states), and its limit (photon-number states) have potential applications in optical communication, precision measurement, and atomic spectroscopy, and gravitational wave detection. These states have been the focus of considerable attention.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Oct 28, 1992•0 citations

Realization of number-phase minimum-uncertainty states and number states by quantum nondemolition measurement

Masahiro Kitagawa, N. Imoto, Y. Yamamoto

Number-phase minimum uncertainty states and number states are shown to be realized by quantum nondemolition measurements of the photon number. Interphase modulation in a Kerr medium establishes a quantum-mechanical correlation between the signal and probe waves initially in coherent states. State reduction is realized by homodyne detection of a probe quadrature-phase amplitude. The number uncertainty can be arbitrarily reduced by increasing the probe intensity. The minimum uncertainty relation between the number and sine operators approximately holds until a near-number state is realized.

Physical Review A•Jun 1, 1987•48 citations

Number-phase minimum-uncertainty state with reduced number uncertainty in a Kerr nonlinear interferometer

Masahiro Kitagawa, Y. Yamamoto

The output state of a nonlinear Mach-Zehnder interferometer is shown to be an effective number-phase minimum-uncertainty state with reduced photon-number uncertainty. This interferometer includes an optical Kerr medium in one arm with a coherent-state input. Unusual ``crescent''-shaped squeezing which preserves photon number is revealed in the unitary evolution associated with a self-phase-modulation in the Kerr medium. Photon-number uncertainty 〈\ensuremath{\Delta}n${^}^{2}$〉 can be reduced by interference with a coherent-state reference wave. It can be minimized to 〈n^${〉}^{1/3}$, far below the limit 〈n^${〉}^{2/3}$ achieved by an ordinary squeezed state. The increased phase uncertainty due to self-phase-modulation and the reduced photon-number uncertainty still preserve the minimum-uncertainty product 〈\ensuremath{\Delta}n${^}^{2}$〉〈\ensuremath{\Delta}\ensuremath{\Phi}${^}^{2}$〉\ensuremath{\sim}(1/4). .AE

Physical Review A•Nov 1, 1986•404 citations

Generation and amplification of number states by nondegenerate parametric oscillators with idler-measurement feedback

Gunnar G. Bjork, Y. Yamamoto

Noise spectra of the in-phase and quadrature-phase components of the signal and idler fields of a nondegenerate optical parametric amplifier are calculated. It is shown that, as is suggested by the operator Manley-Rowe relations, the external in-phase (amplitude) fluctuations of the signal and idler fields are correlated in the frequency region lower than the signal and idler cavity bandwidths. If the idler photon flux is continuously measured and the photocurrent fluctuations are fed back to control the pump intensity, the signal output will feature an amplitude noise spectral density reduced to below the standard quantum limit (amplitude squeezing) in the frequency region below the feedback loop and signal and idler cavity bandwidths. An all-optical realization of such a negative-feedback parametric oscillator is presented. It is also shown that a nondegenerate parametric amplifier with idler measurement feedback can add a fixed number of photons to a photon-number state.

Physical Review A•Jan 1, 1988•51 citations

Quantum correlation and state reduction of photon twins produced by a parametric amplifier

Kimitaka Watanabe, Y. Yamamoto

Signal and idler waves produced by a nondegenerate parametric amplifier feature strong positive correlation between their photon numbers, or in-phase amplitudes. They feature equally strong negative correlation between their phases, or quadrature-phase amplitudes. These photon twins can produce arbitrarily squeezed states via state reduction by appropriate measurements of an idler wave. When the quadrature amplitude of an idler wave is measured, the signal wave is collapsed to a quadrature-amplitude squeezed state. When the two-quadrature amplitudes of an idler wave are measured simultaneously, the signal wave becomes a coherent state.

Physical Review A•Oct 1, 1988•43 citations

Preparation, measurement and information capacity of optical quantum states

Y. Yamamoto, H. A. Haus

The preparation, or generation of coherent states, squeezed states, and photon number states is discussed. The quantum noise is evaluated for various simultaneous measurements of two quadrature components: heterodyning, the beam splitter followed by two single quadrature measurements, the parametric amplifier, the (degenerate and/or nondegenerate) four-wave mixer, the Brillouin and Raman amplifiers, and the laser amplifier. A quantum nondemolition measurement followed by a measurement of the conjugate variable is also categorized as a simultaneous measurement. It is shown that, for all of these schemes, the minimum uncertainty product of the measured variables is exactly equal to that required for a simultaneous measurement of two noncommuting variables. On the other hand, measurements of a single quadrature component are noise-free. Such measurements are degenerate heterodyning, degenerate parametric amplification, and cavity degenerate four-wave mixing and photon counting by a photomultiplier or avalanche photodiode. The Heisenberg uncertainty principle and the quantum-mechanical channel capacity of Shannon are discussed to address the question "How much information can be transmitted by a single photon?" The quantum-mechanical channel capacity provides an upper bound on the achievable information capacity and is ideally realized by photon number states and photon counting detection. Its value is $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{(\mathrm{ln}2)kT}$ bit per photon. The use of coherent or squeezed states and a simultaneous measurement of two quadrature field components or the measurement of one single quadrature field component does not achieve the ultimate limit.

Reviews of Modern Physics•Oct 1, 1986•447 citations
The difference between the two nonclassical photon states, i.e., the squeezed state and the number-phase minimum-uncertainty state, is discussed. Four different generation principles for number-phase minimum-uncertainty states (and number states) are described. They are the following: (1) unitary evolution, using self-phase modulation and interference; (2) nonunitary state reduction, using either quantum nondemolition photon-number measurement or parametrically amplified idler-wave photon counting; (3) negative-feedback oscillators, incorporating a correlated photon-pair generator; and (4) pump-amplitude noise-suppressed lasers. A number-phase minimum-uncertainty state with photon-number noise reduced to 1 dB (20%) below the standard quantum limit in a frequency range over 100 MHz was produced by the fourth-generation principle.
If the idler photon flux is continuously measured in a photon-counting detector and the photocurrent fluctuations are fed back to control the pump intensity, the output signal state will be an amplitude-squeezed state (near-number state) at low frequencies and high feedback loop gain. Electromagnetic field in number-phase minimum-uncertainty states (amplitude- squeezed states), and its limit (photon-number states) have potential applications in optical communication, precision measurement, and atomic spectroscopy, and gravitational wave detection. These states have been the focus of considerable attention.
Number-phase minimum uncertainty states and number states are shown to be realized by quantum nondemolition measurements of the photon number. Interphase modulation in a Kerr medium establishes a quantum-mechanical correlation between the signal and probe waves initially in coherent states. State reduction is realized by homodyne detection of a probe quadrature-phase amplitude. The number uncertainty can be arbitrarily reduced by increasing the probe intensity. The minimum uncertainty relation between the number and sine operators approximately holds until a near-number state is realized.
The output state of a nonlinear Mach-Zehnder interferometer is shown to be an effective number-phase minimum-uncertainty state with reduced photon-number uncertainty. This interferometer includes an optical Kerr medium in one arm with a coherent-state input. Unusual ``crescent''-shaped squeezing which preserves photon number is revealed in the unitary evolution associated with a self-phase-modulation in the Kerr medium. Photon-number uncertainty 〈\ensuremath{\Delta}n${^}^{2}$〉 can be reduced by interference with a coherent-state reference wave. It can be minimized to 〈n^〉1/3{〉}^{1/3}〉1/3, far below the limit 〈n^〉2/3{〉}^{2/3}〉2/3 achieved by an ordinary squeezed state. The increased phase uncertainty due to self-phase-modulation and the reduced photon-number uncertainty still preserve the minimum-uncertainty product 〈\ensuremath{\Delta}n${^}^{2}$〉〈\ensuremath{\Delta}\ensuremath{\Phi}${^}^{2}$〉\ensuremath{\sim}(1/4). .AE
Noise spectra of the in-phase and quadrature-phase components of the signal and idler fields of a nondegenerate optical parametric amplifier are calculated. It is shown that, as is suggested by the operator Manley-Rowe relations, the external in-phase (amplitude) fluctuations of the signal and idler fields are correlated in the frequency region lower than the signal and idler cavity bandwidths. If the idler photon flux is continuously measured and the photocurrent fluctuations are fed back to control the pump intensity, the signal output will feature an amplitude noise spectral density reduced to below the standard quantum limit (amplitude squeezing) in the frequency region below the feedback loop and signal and idler cavity bandwidths. An all-optical realization of such a negative-feedback parametric oscillator is presented. It is also shown that a nondegenerate parametric amplifier with idler measurement feedback can add a fixed number of photons to a photon-number state.
Signal and idler waves produced by a nondegenerate parametric amplifier feature strong positive correlation between their photon numbers, or in-phase amplitudes. They feature equally strong negative correlation between their phases, or quadrature-phase amplitudes. These photon twins can produce arbitrarily squeezed states via state reduction by appropriate measurements of an idler wave. When the quadrature amplitude of an idler wave is measured, the signal wave is collapsed to a quadrature-amplitude squeezed state. When the two-quadrature amplitudes of an idler wave are measured simultaneously, the signal wave becomes a coherent state.
The preparation, or generation of coherent states, squeezed states, and photon number states is discussed. The quantum noise is evaluated for various simultaneous measurements of two quadrature components: heterodyning, the beam splitter followed by two single quadrature measurements, the parametric amplifier, the (degenerate and/or nondegenerate) four-wave mixer, the Brillouin and Raman amplifiers, and the laser amplifier. A quantum nondemolition measurement followed by a measurement of the conjugate variable is also categorized as a simultaneous measurement. It is shown that, for all of these schemes, the minimum uncertainty product of the measured variables is exactly equal to that required for a simultaneous measurement of two noncommuting variables. On the other hand, measurements of a single quadrature component are noise-free. Such measurements are degenerate heterodyning, degenerate parametric amplification, and cavity degenerate four-wave mixing and photon counting by a photomultiplier or avalanche photodiode. The Heisenberg uncertainty principle and the quantum-mechanical channel capacity of Shannon are discussed to address the question "How much information can be transmitted by a single photon?" The quantum-mechanical channel capacity provides an upper bound on the achievable information capacity and is ideally realized by photon number states and photon counting detection. Its value is $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{(\mathrm{ln}2)kT}$ bit per photon. The use of coherent or squeezed states and a simultaneous measurement of two quadrature field components or the measurement of one single quadrature field component does not achieve the ultimate limit.