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Quantum Fluctuations in Linear Systems

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Quantum Limits on Measurement and Control of a Mechanical Oscillator

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

The need to observe a system precludes it form being in perfect isolation. Typical physical systems also exist in equilibrium with some thermal environment. Here we provide a unified description of a system in contact with a thermal environment, and a measurement device.

Here are some words which have no place in a formulation with any pretension to physical precision: system, apparatus, environment, microscopic, macroscopic, reversible, irreversible, observable, information, measurement.

John Bell

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Notes

  1. 1.

    It turns out that, on technical grounds, the framework of Hilbert space is too restrictive to realize the flexibility of Dirac’s formulation of quantum mechanics [3, 4]; here, we will however be satisfied with using the Dirac formalism rather than justifying each step of the usage rigorously.

  2. 2.

    A peculiarity of quantum mechanics is that although the value taken by each observable, for a fixed state, can be assumed to be drawn from a classical probability distribution (exhibited in Appendix A), there is generally no joint probability distribution for the values of a set of operators [6].

  3. 3.

    Attempts to formulate inequalities applicable to sequential measurements [13,14,15,16] give results very different from Eq. (2.1.2).

  4. 4.

    For positive real numbers x, y, it is true that \(x+y \ge 2\sqrt{xy}\); this follows from the identity, \((\sqrt{x}-\sqrt{y})^2 \ge 0\).

  5. 5.

    There exists two, progressively finer, levels of description of the evolution of a quantum system in contact with an environment. The coarse description concerns itself with the time evolution of observables, and some of its statistics. The finer description addresses the question of how the quantum state itself changes. The former is subsumed by the latter in a variety of equivalent ways [19,20,21,22,23].

  6. 6.

    The normalisation warrants clarification: if the integrand were a classical Brownian process, its root-mean-square diverges as the square root of the observation window, i.e. as \(T^{1/2}\), which is checked by the normalisation. For a wide class of classical stochastic processes, a theorem due to Donsker [24] guarantees that the integral limits to a Brownian process (a “functional central limit theorem”)—the \(T^{-1/2}\) normalisation is necessary. This result from classical probability theory suffices to justify the normalisation.

  7. 7.

    Short for “symmetrised power spectral density”, by abuse of terminology.

  8. 8.

    Note that the notion of a continuous observable, as introduced here, is very different from that of a continuous variable used in the context of quantum information [35, 36]. The latter refers to hermitian operators (i.e. observables) whose eigenspectrum is continuous. The former, as used here, refers to time-dependent observables (with a continuous, or discrete, eigenspectrum) which can (in principle) be continuously monitored in time.

  9. 9.

    An outline of a proof is as follows (see [41] for the setup required to justify some of the steps). Assume then that there is some state \(\hat{\rho }\) for which all operators (not just observables) of the system satisfy Eq. (2.2.3); i.e. \(\langle {\hat{A}(t)\hat{B}(0)}\rangle =\langle {\hat{B}(0)\hat{A}(t+i\hbar \beta )}\rangle \), for all operators \(\hat{A},\hat{B}\). Time-translation invariance means that only the case \(t=0\) need to be considered, i.e., \(\langle {\hat{A}(0)\hat{B}(0)}\rangle =\langle {\hat{B}(0)\hat{A}(i\hbar \beta )}\rangle \,\, \forall \,\, \hat{A},\hat{B}\). Dropping the time argument and writing this out with the unknown state \(\hat{\rho }\) explicitly,

    $$\begin{aligned} \mathrm {Tr}[\hat{\rho }\hat{A}\hat{B}] =\mathrm {Tr}[\hat{\rho } \hat{B} e^{-\beta \hat{H}_0} \hat{A} e^{\beta \hat{H}_0}] \qquad \forall \, \hat{A},\hat{B}. \end{aligned}$$

    This can be expressed in two different ways. Firstly, since it applies for any \(\hat{A}\), it must also apply for \(\hat{A}=e^{\beta \hat{H}_0}\); in this case, \(\mathrm {Tr}[\hat{\rho }e^{\beta \hat{H}_0} \hat{B}] = \mathrm {Tr}[e^{\beta \hat{H}_0} \hat{\rho } \hat{B}]\,\, \forall \,\, \hat{B}\), implying that,

    $$\begin{aligned} \hat{\rho }e^{\beta \hat{H}_0} = e^{\beta \hat{H}_0}\hat{\rho }. \end{aligned}$$

    Secondly, permuting within the trace gives the alternate form, \(\mathrm {Tr}[\hat{B}\hat{\rho }\hat{A}] = \mathrm {Tr}[e^{\beta \hat{H}_0} \hat{\rho } \hat{B}e^{-\beta \hat{H}_0} \hat{A}]\,\, \forall \,\, \hat{A},\hat{B}\), implying that,

    $$\begin{aligned} \begin{aligned} \hat{B}\hat{\rho }&= e^{\beta \hat{H}_0} \hat{\rho } \hat{B}e^{-\beta \hat{H}_0} \\ \text{ i.e., }\quad \hat{B}\hat{\rho }e^{\beta \hat{H}_0}&= e^{\beta \hat{H}_0} \hat{\rho } \hat{B}\qquad \forall \, \hat{B}. \end{aligned} \end{aligned}$$

    Combining the results from the two forms gives,

    $$\begin{aligned} \hat{B}e^{\beta \hat{H}_0}\hat{\rho } = e^{\beta \hat{H}_0} \hat{\rho } \hat{B}\qquad \forall \, \hat{B}, \end{aligned}$$

    i.e. the operator \(e^{\beta \hat{H}_0} \hat{\rho }\) commutes with every operator in the Hilbert space. This means that it must be proportional to the identity operator, i.e. \(e^{\beta \hat{H}_0} \hat{\rho } \propto {1}\), or, \(\hat{\rho }\propto e^{-\beta \hat{H}_0}\). The normalization of the state fixes the proportionality constant.

  10. 10.

    This is because the meter is expected to output a classical record of the system observable being measured; this can only be arranged for if the states of the meter corresponding to the various values taken by the system observable are macroscopically distinguishable [50].

  11. 11.

    The most general linear relationship is of the form \(\hat{Y}(t)=\int f(t)\hat{X}(t-t')\, \mathrm {d}t'\), corresponding to a filtered version of the observable. However, without loss of generality, the filtering may be considered as happening on the classical measurement record, after the detector.

  12. 12.

    On the other hand, if it can be arranged that the observable \(\hat{X}\) already satisfies \([\hat{X}(t),\hat{X}(t')]= 0\), i.e. it is a continuous observable in the sense defined in Eq. (2.1.18), then there is in principle no additional contamination.

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    Vivishek Sudhir

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Sudhir, V. (2018). Quantum Fluctuations in Linear Systems. In: Quantum Limits on Measurement and Control of a Mechanical Oscillator. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-69431-3_2

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