Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons

Kim, Taesoo; Pfister, Olivier; Holland, Murray; Noh, Jaewoo; Hall, John L.

doi:10.1103/physreva.57.4004

Cited by 163 publications

(147 citation statements)

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“…The phase sensitivity of interferometers is believed to be limited by quantum fluctuations [12], and the phase sensitivity of various interferometers has been explored for different types of input states, such as squeezed states [12,13], and number states [14,15,16,17,18,19,20,21,22]. In all the above cases, the phase sensitivity ∆φ has been discussed in terms of two limits, known as the standard limit, ∆φ SL = 1/ √ N , and the Heisenberg limit [23], ∆φ HL = 1/N , where N is the number of particles that enter the interferometer during each measurement cycle.…”

Section: Introductionmentioning

confidence: 99%

Fidelity of quantum interferometers

Bahder

Lopata

2006

Phys. Rev. A

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For a generic interferometer, the conditional probability density distribution, p(φ|m), for the phase φ given measurement outcome m, will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as ∆φ ∼ 1/ √ N (the standard limit), or ∆φ ∼ 1/N (the Heisenberg limit). We propose an alternative measure of phase sensitivity-the fidelity of an interferometer-defined as the Shannon mutual information between the phase shift φ and the measurement outcomes m. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. We find the surprising result that an entangled N00N state input leads to a lower fidelity than a Fock state input, for the same photon number. Introduction.Phase sensitivity of interferometers has been a topic of research for many years because of interest in the fundamental limitations of measurement [1, 2], gravitational-wave detection [3], and optical [4,5], atom [6], and Bose-Einstein condensate(BEC)-based gyroscopes [7,8,9]. Recently, applications to sensors are being explored [10,11].The phase sensitivity of interferometers is believed to be limited by quantum fluctuations [12], and the phase sensitivity of various interferometers has been explored for different types of input states, such as squeezed states [12,13], and number states [14,15,16,17,18,19,20,21,22]. In all the above cases, the phase sensitivity ∆φ has been discussed in terms of two limits, known as the standard limit, ∆φ SL = 1/ √ N , and the Heisenberg limit[23], ∆φ HL = 1/N , where N is the number of particles that enter the interferometer during each measurement cycle. These arguments are based on results of standard estimation theory [24] which connects an ensemble of measurement outcomes, m i , i = 1, 2, · · · , M , with corresponding phases, φ i , through a theoretical relation m = m(φ). An example of the theoretical relation associated with some quantum observable is m(φ) = φ| m|φ , where the state is parameterized by a single parameter φ. Standard estimation theory predicts that the standard deviation, ∆φ, of the probability distribution for the phase φ, is related to the standard deviation in the measurements, ∆m, by [24]

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Section: Introductionmentioning

confidence: 99%

Fidelity of quantum interferometers

Bahder

Lopata

2006

Phys. Rev. A

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“…Not all detection schemes are capable of exploiting the full potential of non-classical light to be super-sensitive and super-resolving. For example, intensity difference measurement, which is standard for optical interferometry with coherent light, is not phase sensitive at all if TMSV input is used [9]. In our work, we consider parity detection for our measuring scheme.…”

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Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit

et al. 2010

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We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum (n photons on average). We show that super-resolution and subHeisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolvesn times faster than in traditional schemes, and uncertainty in the phase estimation is better than 1/n.PACS numbers: 07.60. Ly, 95.75.kK, 42.50.St Different physical mechanisms contribute to phase measurement. Thus, measuring phase provides insight into a number of physical processes. Therefore, improved phase estimation benefits multiple areas of scientific research, such as quantum metrology, imaging, sensing, and information processing. Consequently, enormous efforts have been devoted to improve the resolution and sensitivity of interferometers. Sensitivity is a measure of the uncertainty in the phase estimation, while resolution is rate at which signal changes with changing phase.In what follows, we direct our attention to quantum interferometry. The benchmark that quantum interferometry is compared against is one with coherent light input and intensity difference measurement at the output of a Mach-Zehnder interferometer (MZI). In general, phase sensitivity of this benchmark is shot-noise limited, namely ∆ϕ = 1/ √n , wheren is the average number of photons. However, better sensitivity is possible if nonlinear interaction between photons in the MZI takes place [1]. In what follows, we only consider phase accumulation due to linear processes.In 1981, Caves pointed out that by using coherent light and squeezed vacuum one could beat the shot-noise limit ∆ϕ < 1/ √n (super-sensitivity) [2]. In the work of Boto et al., it was shown that by exploiting quantum states of light, such as N00N states, it is possible to beat the Rayleigh diffraction limit in imaging and lithography (super resolution) while also beating the shot-noise limit in phase estimation [3,4,5,6]. Finally, it was shown in Ref.[7] that input state entanglement is important in order to achieve super-sensitivity in a linear interferometer.non-classical light Experimental realization of these predictions have been hindered by the fact that entangled states of light, with large numbers of photons, are difficult to obtain. Therefore we turn our attention to the brightest (experimentally available) nonclassical light -two-mode squeezed vacuum (TMSV). A state of TMSV is a superposition of twin Fock states |ψn = ∞ n=0 p n (n) |n, n , where the probability of a twin Fock state |n, n = |n A |n B to be present de- pends on average number of photons in both modes of TMSV,n, in the following way p n (n) = (1 − tn)t n n with tn = 1/ (1 + 2/n) [8].Light entering a MZI in TMSV state exits a lossless interferometer in the state |ψ f =Û MZI |ψn , where the MZI is described by the unitary transformation U MZI (Fig. 1). This transformation, in terms of the field operators for the optical modesâ andb, isÛ MZI =ÛP ϕÛ = exp ϕ â †b −b †â /2 , wherê P ϕ =exp −...

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“…More recent publications describing schemes that theoretically reach the Heisenberg limit have mostly considered quantum states which are very hard to synthesize [8,9,10,12,13,15,16,17] and suggest to use unrealistically high non-linearities to guide the light through the interferometer [10] or detectors which have single photon resolution even when dealing with very many photons [4,5,8,9,12,13,14,15,16,17].…”

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“…Present optical interferometers typically operate at the shot noise resolution limit δϕ ∼ 1/ N . Interest in reaching the Heisenberg-limit is great because it presents a fundamental limit and overcomes the shot-noise limit leading to potential applications in high resolution distance measurements, for instance, to detect gravitational waves [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].…”

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Approaching the Heisenberg limit with two-mode squeezed states

Steuernagel¹,

Scheel²

2004

J. Opt. B: Quantum Semiclass. Opt.

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Two mode squeezed states can be used to achieve Heisenberg limit scaling in interferometry: a phase shift of δϕ ≈ 2.76/ N can be resolved. The proposed scheme relies on balanced homodyne detection and can be implemented with current technology. The most important experimental imperfections are studied and their impact quantified.PACS numbers: 42.50. Dv, 42.87.Bg, 03.75.Dg The best possible phase resolution for an interferometer is given by the Heisenberg limit for the minimum detectable phase shift δϕ = 1/ N ; here N is the average intensity (number of photons or other bosons). Present optical interferometers typically operate at the shot noise resolution limit δϕ ∼ 1/ N . Interest in reaching the Heisenberg-limit is great because it presents a fundamental limit and overcomes the shot-noise limit leading to potential applications in high resolution distance measurements, for instance, to detect gravitational waves [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].Known, feasible schemes use degenerate squeezed vacuum combined with Glauber-coherent light to increase the phase sensitivity achieving sub-shot noise resolution, but do not reach the Heisenberg limit [6,7]. Indeed, no practical scheme has been found that shows scaling like the Heisenberg limit δϕ = κ/ N for large intensities (and preferably a small constant κ).More recent publications describing schemes that theoretically reach the Heisenberg limit have mostly considered quantum states which are very hard to synthesize [8,9,10,12,13,15,16,17] and suggest to use unrealistically high non-linearities to guide the light through the interferometer [10] or detectors which have single photon resolution even when dealing with very many photons [4,5,8,9,12,13,14,15,16,17].This Letter proposes to use a standard linear twopath interferometer fed with two mode squeezed vacuum states degenerate in energy and polarization [5,9], see FIG. 1. But rather than measuring photon numbers (intensities) we want to measure the product of the output ports' quadrature components, i.e. perform balanced homodyne detection [1,18]. The only non-linearities used in the setup proposed here are those of the crystal for parametric down-conversion to generate the two mode squeezed vacuum state. It turns out that modest squeezing, i.e. low intensities, suffice to reach interferometric resolution at approximately three times the Heisenberg limit The use of balanced homodyne detection removes the detection problems mentioned above. Because only well established technology is required [18,19,20] a proof-ofprinciple experiment will be immediately possible.In order to derive our main result (1) we follow the conventions of reference [21]: In the Heisenberg picture the action of the parametric amplifier is described by photon operator transformationsâ 1 = Uâ 0 + Vb † 0 andb 1 = Ub 0 + Vâ † 0 where U = cosh G and V = −i exp(iξ) sinh G with the single pass gain G = g|E p |L and a relative phase ξ which we will assume to be zero. L is the interaction path length, E p the pump laser's amplitude, and ...

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Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons

Cited by 163 publications

References 19 publications

Fidelity of quantum interferometers

Fidelity of quantum interferometers

Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit

Approaching the Heisenberg limit with two-mode squeezed states

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