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Register a new user$pi$-Corrected Heisenberg Limit
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Added by
Wojciech G\\'orecki
Publication date
2019
fields
Physics
and research's language is
English
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We consider the precision $Delta varphi$ with which the parameter $varphi$, appearing in the unitary map $U_varphi = e^{ i varphi Lambda}$ acting on some type of probe system, can be estimated when there is a finite amount of prior information about $varphi$. We show that, if $U_varphi$ acts $n$ times in total, then, asymptotically in $n$, there is a tight lower bound $Delta varphi geq frac{pi}{n (lambda_+ - lambda_-)}$, where $lambda_+$, $lambda_-$ are the extreme eigenvalues of the generator $Lambda$. This is greater by a factor of $pi$ than the conventional Heisenberg limit, derived from the properties of the quantum Fisher information. That is, the conventional bound is never saturable. Our result makes no assumptions on the measurement protocol, and is relevant not only in the noiseless case but also if noise can be eliminated using quantum error correction techniques.
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To quantify quantum optical coherence requires both the particle- and wave-natures of light. For an ideal laser beam [1,2,3], it can be thought of roughly as the number of photons emitted consecutively into the beam with the same phase. This number, $mathfrak{C}$, can be much larger than $mu$, the number of photons in the laser itself. The limit on $mathfrak{C}$ for an ideal laser was thought to be of order $mu^2$ [4,5]. Here, assuming nothing about the laser operation, only that it produces a beam with certain properties close to those of an ideal laser beam, and that it does not have external sources of coherence, we derive an upper bound: $mathfrak{C} = O(mu^4)$. Moreover, using the matrix product states (MPSs) method [6,7,8,9], we find a model that achieves this scaling, and show that it could in principle be realised using circuit quantum electrodynamics (QED) [10]. Thus $mathfrak{C} = O(mu^2)$ is only a standard quantum limit (SQL); the ultimate quantum limit, or Heisenberg limit, is quadratically better.
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We provide efficient and intuitive tools for deriving bounds on achievable precision in quantum enhanced metrology based on the geometry of quantum channels and semi-definite programming. We show that when decoherence is taken into account, the maximal possible quantum enhancement amounts generically to a constant factor rather than quadratic improvement. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: dephasing,depolarization, spontaneous emission and photon loss.
We propose an approach to quantum phase estimation that can attain precision near the Heisenberg limit without requiring single-particle-resolved state detection. We show that the one-axis twisting interaction, well known for generating spin squeezing in atomic ensembles, can also amplify the output signal of an entanglement-enhanced interferometer to facilitate readout. Applying this interaction-based readout to oversqueezed, non-Gaussian states yields a Heisenberg scaling in phase sensitivity, which persists in the presence of detection noise as large as the quantum projection noise of an unentangled ensemble. Even in dissipative implementations -- e.g., employing light-mediated interactions in an optical cavity or Rydberg dressing -- the method significantly relaxes the detection resolution required for spectroscopy beyond the standard quantum limit.
In this work an exactly solvable model of N two-level systems interacting with a single bosonic dephasing reservoir is considered to unravel the role played by collective non-Markovian dephasing. We show that phase estimation with entangled states for this model can exceed the standard quantum limit and demonstrate Heisenberg scaling with the number of atoms for an arbitrary temperature. For a certain class of reservoir densities of states decoherence can be suppressed in the limit of large number of atoms and the Heisenberg limit can be restored for arbitrary interrogation times. We identify the second class of densities when the Heisenberg scaling can be restored for any finite interrogation time. We also find the third class of densities when the standard quantum limit can be exceeded only on the initial stage of dynamics in the Zeno-regime.
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