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CRC Press eBooks

Theory of Phase Measurement

Stephen M. Barnett, John A. Vaccaro

Published Apr 27, 2007• 0 citations
Publisher

Abstract

Here the angle θ , and hence the measured quadrature, is determined by the phase of the local oscillator. This quadrature is reminiscent of the product of the field amplitude and the cosineof thephase, cos(φ−θ), and indeed“measuredphaseoperators”havebeendefined in terms of these (Barnett and Pegg 1986 Paper 1.7). The quadratures are not phase operators, of course, but measuring them can provide a very good approximation to measurement of the phase (Vaccaro and Pegg 1994c Paper 5.2). Homodyne detection can also provide something close to a single-shot measurement of the phase by actively adjusting the local oscillator phase as the photocurrent is measured (Wiseman 1995 Paper 5.3). The idea is to use the accumulated photocurrent data, up to any instant during the measurement, to determine the new value selected for the local oscillator phase.

DOI
10.1201/b16006-8

Related papers

Adaptive single-shot phase measurements: The full quantum theory

Howard M. Wiseman, Rowan Killip

The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.

arXiv:quant-ph/9710056•Physical Review A•Mar 1, 1998•71 citations

Adaptive single-shot phase measurements: A semiclassical approach

Howard M. Wiseman, Rowan Killip

The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as ${N}^{\ensuremath{-}1}$, where $N$ is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus $\ensuremath{\pi}/2$, so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time $t$ is a functional of the photocurrent from time $0$ up to time $t$ in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as ${N}^{\ensuremath{-}3/2}$. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as ${N}^{\ensuremath{-}2}$. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.

Physical Review A•Jul 1, 1997•75 citations

Operational formulation of homodyne detection

Tomá Tyc, Barry C. Sanders

We obtain the standard quadrature-phase positive operator-valued measure (POVM) for homodyne detection directly and rigorously from the POVM for photon counting without directly employing the mean field approximation for the local oscillator. In addition, we obtain correction terms for the quadrature-phase POVM that are applicable for relatively weak local oscillator field strengths and typical signal states.

arXiv:quant-ph/0404090•Journal of Physics A Mathematical and General•Jul 8, 2004•47 citations

Operational formulation of homodyne detection

Tyc, T, Barry C. Sanders

We obtain the standard quadrature-phase positive operator-valued measure (POVM) for homodyne detection directly and rigorously from the POVM for direct photon counting. In addition we obtain correction terms for the quadrature-phase POVM that are applicable for relatively weak local oscillator field strengths and typical signal states.

•Jan 1, 2004•18 citations

The Phase Operator

Stephen M. Barnett, John A. Vaccaro

More particularly, the sets {|n〉 : n = 0, 1, · · · s} and {|θm〉 : m = 0, 1, · · · s} are two mutually unbiased bases for s as |〈n|θm〉|2 = 1/(s + 1) for n, m = 0, 1, · · · , s. The exponential phase operator is defined in terms of the phase state basis as ei φˆθ = s∑ m=0 eiθm |θm〉〈θm|, (2.4) or equivalently, in terms of the energy eigenstates as |n〉〈n + 1| + ei(s+1)θ0 |s〉〈0|. (2.5) The subscript θ serves as a reminder that the values of phase variable θm lie in the range from θ0 to θ0 + 2π . This construction ensures that the phase states are the eigenstates of ei φˆθ , i.e., ei φˆθ |θm〉 = eiθm |θm〉. (2.6) The exponential phase operator is unitary, because n=0 |n〉〈n| = 1ˆ s (2.7) where 1ˆ s is the identity operator on s . The Hermitian phase operator representing the phase angle itself was introduced in (Pegg and Barnett 1988b Paper 2.2) and (Barnett and Pegg 1989 Paper 2.3) as simply φˆθ = s∑ m=0 θm|θm〉〈θm|. (2.8) The operator on the left hand sides of Eqs. (2.4) and (2.5) is the exponential of i φˆθ .Moreover, the operator corresponding to an arbitrary function of the phase angle f (θ) was introduced in Barnett and Pegg (1992 Paper 3.3) and also discussed in Pegg and Barnett (1997a Paper 2.6) as f (φˆθ ) = s∑ m=0 f (θm)|θm〉〈θm|. (2.9) It follows that the Hermitian conjugate of ei φˆθ is the operator (ei φˆθ )† = s∑ m=0 e−iθm |θm〉〈θm| = e−i φˆθ . (2.10) This construction on the finite dimensional spaces elegantly sidesteps the problems faced with the infinite dimensional Hilbert space discussed in the previous chapter. Its simplicity is particularly appealing. The question remains, however, of how to deal with states such as | f 〉 = ∑∞n=0 cn|n〉 which reside in the infinite dimensional Hilbert space. For each value of s, there exists a phase operator defined on s ; this leads naturally to sequences of operators, such as e.g. φˆθ |s=S, φˆθ |s=S+1, φˆθ |s=S+2 · · · which are phase angle operators defined on the sequence of spaces S, S+1, S+2, · · · for arbitrarily large integer S. The space s contains a state | f 〉s which approximates | f 〉 to arbitrary precision for a sufficiently large value of s, such that the modulus of the overlap 〈 f | f 〉s is as close to unity as desired. Thus the issue of including states such as | f 〉 entails dealing with such sequences as s tends to infinity.

CRC Press eBooks•Apr 27, 2007•0 citations

Measurement of the Polarization State of a Weak Signal Field by Homodyne Detection

Sun-Hyun Youn

Information carried by an optical beam of light can usually be conveyed in the form of a temporal modulation of the intensity, phase, frequency or the polarization of the constituent mode(s). In that regard, homodyne detection is one of the most popular and standard method to measure the quantum mechanical properties of light Yuen & Shapiro (1978); Yuen et al. (1979); Yuen & Shapiro (1980); Yurke (1985). The quantum theory of homodyne detection originated principally from the works of Yuen and Shapiro et al. Yuen & Shapiro (1978); Yuen et al. (1979). In this method, a weak signal field is combined with a strong local oscillator field at a beam splitter, and the resulting signal is measured as a photocurrent. With the homodyne detection method, the quantum state of the signal field, such as the quadrature amplitude of the squeezed state, is easily measured Slusher et al. (1985); Wu et al. (1986 ); Polzik et al. (1992), and the quasiprobability distribution function can be measured by using the so-called optical homodyne tomography Smithey et al. (1993); Banaszek & W’odkiewicz (1996); Wallentowitz & Vogel (1996); Youn et al. (2001). Furthermore, optical homodyne detection is used to eavesdrop on the quantum key in quantum cryptography Hirano et al. (2000).

InTech eBooks•Mar 23, 2012•2 citations
Here the angle θ , and hence the measured quadrature, is determined by the phase of the local oscillator. This quadrature is reminiscent of the product of the field amplitude and the cosineof thephase, cos(φ−θ), and indeed“measuredphaseoperators”havebeendefined in terms of these (Barnett and Pegg 1986 Paper 1.7). The quadratures are not phase operators, of course, but measuring them can provide a very good approximation to measurement of the phase (Vaccaro and Pegg 1994c Paper 5.2). Homodyne detection can also provide something close to a single-shot measurement of the phase by actively adjusting the local oscillator phase as the photocurrent is measured (Wiseman 1995 Paper 5.3). The idea is to use the accumulated photocurrent data, up to any instant during the measurement, to determine the new value selected for the local oscillator phase.
The phase of a single-mode field can be measured in a single-shot measurement by interfering the field with an effectively classical local oscillator of known phase. The standard technique is to have the local oscillator detuned from the system (heterodyne detection) so that it is sometimes in phase and sometimes in quadrature with the system over the course of the measurement. This enables both quadratures of the system to be measured, from which the phase can be estimated. One of us [H. M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] has shown recently that it is possible to make a much better estimate of the phase by using an adaptive technique in which a resonant local oscillator has its phase adjusted by a feedback loop during the single-shot measurement. In a previous work [H. M. Wiseman and R. B. Killip, Phys. Rev. A 56, 944 (1997)] we presented a semiclassical analysis of a particular adaptive scheme, which yielded asymptotic results for the phase variance of strong fields. In this paper we present an exact quantum mechanical treatment. This is necessary for calculating the phase variance for fields with small photon numbers, and also for considering figures of merit other than the phase variance. Our results show that an adaptive scheme is always superior to heterodyne detection as far as the variance is concerned. However, the tails of the probability distribution are surprisingly high for this adaptive measurement, so that it does not always result in a smaller probability of error in phase-based optical communication.
The standard single-shot estimate for the phase of a single-mode pulse of light is the argument of the complex amplitude of the field. This complex amplitude can be measured by heterodyne detection, in which the local oscillator is detuned from the system so that all quadratures are sampled equally. Because different quadratures do not commute, such a measurement introduces noise into the phase estimate, with a variance scaling as ${N}^{\ensuremath{-}1}$, where NNN is the maximum photon number. This represents the shot-noise limit or standard quantum limit (SQL). Recently, one of us [H.M. Wiseman, Phys. Rev. Lett. 75, 4587 (1995)] proposed a way to improve upon this: a real-time feedback loop can control the local oscillator phase to be equal to the estimated system phase plus $\ensuremath{\pi}/2$, so that the phase quadrature of the system is measured preferentially. The phase estimate used in the feedback loop at time ttt is a functional of the photocurrent from time 000 up to time ttt in the single-shot measurement. In this paper we consider a very simple feedback scheme involving only linear electronic elements. Approaching the problem from semiclassical detection theory, we obtain analytical results for asymptotically large photon numbers. Specifically, we are able to show that the noise introduced by the measurement has a variance scaling as ${N}^{\ensuremath{-}3/2}$. This is much less than the SQL variance, but still much greater than the minimum intrinsic phase variance which scales as ${N}^{\ensuremath{-}2}$. We briefly discuss the effect of detector inefficiencies and delays in the feedback loop.
More particularly, the sets {|n〉 : n = 0, 1, · · · s} and {|θm〉 : m = 0, 1, · · · s} are two mutually unbiased bases for s as |〈n|θm〉|2 = 1/(s + 1) for n, m = 0, 1, · · · , s. The exponential phase operator is defined in terms of the phase state basis as ei φˆθ = s∑ m=0 eiθm |θm〉〈θm|, (2.4) or equivalently, in terms of the energy eigenstates as |n〉〈n + 1| + ei(s+1)θ0 |s〉〈0|. (2.5) The subscript θ serves as a reminder that the values of phase variable θm lie in the range from θ0 to θ0 + 2π . This construction ensures that the phase states are the eigenstates of ei φˆθ , i.e., ei φˆθ |θm〉 = eiθm |θm〉. (2.6) The exponential phase operator is unitary, because n=0 |n〉〈n| = 1ˆ s (2.7) where 1ˆ s is the identity operator on s . The Hermitian phase operator representing the phase angle itself was introduced in (Pegg and Barnett 1988b Paper 2.2) and (Barnett and Pegg 1989 Paper 2.3) as simply φˆθ = s∑ m=0 θm|θm〉〈θm|. (2.8) The operator on the left hand sides of Eqs. (2.4) and (2.5) is the exponential of i φˆθ .Moreover, the operator corresponding to an arbitrary function of the phase angle f (θ) was introduced in Barnett and Pegg (1992 Paper 3.3) and also discussed in Pegg and Barnett (1997a Paper 2.6) as f (φˆθ ) = s∑ m=0 f (θm)|θm〉〈θm|. (2.9) It follows that the Hermitian conjugate of ei φˆθ is the operator (ei φˆθ )† = s∑ m=0 e−iθm |θm〉〈θm| = e−i φˆθ . (2.10) This construction on the finite dimensional spaces elegantly sidesteps the problems faced with the infinite dimensional Hilbert space discussed in the previous chapter. Its simplicity is particularly appealing. The question remains, however, of how to deal with states such as | f 〉 = ∑∞n=0 cn|n〉 which reside in the infinite dimensional Hilbert space. For each value of s, there exists a phase operator defined on s ; this leads naturally to sequences of operators, such as e.g. φˆθ |s=S, φˆθ |s=S+1, φˆθ |s=S+2 · · · which are phase angle operators defined on the sequence of spaces S, S+1, S+2, · · · for arbitrarily large integer S. The space s contains a state | f 〉s which approximates | f 〉 to arbitrary precision for a sufficiently large value of s, such that the modulus of the overlap 〈 f | f 〉s is as close to unity as desired. Thus the issue of including states such as | f 〉 entails dealing with such sequences as s tends to infinity.
Information carried by an optical beam of light can usually be conveyed in the form of a temporal modulation of the intensity, phase, frequency or the polarization of the constituent mode(s). In that regard, homodyne detection is one of the most popular and standard method to measure the quantum mechanical properties of light Yuen & Shapiro (1978); Yuen et al. (1979); Yuen & Shapiro (1980); Yurke (1985). The quantum theory of homodyne detection originated principally from the works of Yuen and Shapiro et al. Yuen & Shapiro (1978); Yuen et al. (1979). In this method, a weak signal field is combined with a strong local oscillator field at a beam splitter, and the resulting signal is measured as a photocurrent. With the homodyne detection method, the quantum state of the signal field, such as the quadrature amplitude of the squeezed state, is easily measured Slusher et al. (1985); Wu et al. (1986 ); Polzik et al. (1992), and the quasiprobability distribution function can be measured by using the so-called optical homodyne tomography Smithey et al. (1993); Banaszek & W’odkiewicz (1996); Wallentowitz & Vogel (1996); Youn et al. (2001). Furthermore, optical homodyne detection is used to eavesdrop on the quantum key in quantum cryptography Hirano et al. (2000).