Quantum principle of sensing gravitational waves: From the zero-point fluctuations to the cosmological stochastic background of spacetime

Diego A. Quiñones, Teodora Oniga, Benjamin T. H. Varcoe, and Charles H.-T. Wang

Phys. Rev. D 96, 044018 – Published 15 August 2017

Abstract

We carry out a theoretical investigation on the collective dynamics of an ensemble of correlated atoms, subject to both vacuum fluctuations of spacetime and stochastic gravitational waves. A general approach is taken with the derivation of a quantum master equation capable of describing arbitrary confined nonrelativistic matter systems in an open quantum gravitational environment. It enables us to relate the spectral function for gravitational waves and the distribution function for quantum gravitational fluctuations and to indeed introduce a new spectral function for the zero-point fluctuations of spacetime. The formulation is applied to two-level identical bosonic atoms in an off-resonant high- $Q$ cavity that effectively inhibits undesirable electromagnetic delays, leading to a gravitational transition mechanism through certain quadrupole moment operators. The overall relaxation rate before reaching equilibrium is found to generally scale collectively with the number $N$ of atoms. However, we are also able to identify certain states of which the decay and excitation rates with stochastic gravitational waves and vacuum spacetime fluctuations amplify more significantly with a factor of $N^{2}$ . Using such favorable states as a means of measuring both conventional stochastic gravitational waves and novel zero-point spacetime fluctuations, we determine the theoretical lower bounds for the respective spectral functions. Finally, we discuss the implications of our findings on future observations of gravitational waves of a wider spectral window than currently accessible. Especially, the possible sensing of the zero-point fluctuations of spacetime could provide an opportunity to generate initial evidence and further guidance of quantum gravity.

Received 13 February 2017

DOI:https://doi.org/10.1103/PhysRevD.96.044018

Physics Subject Headings (PhySH)

Gravitational waves Open quantum systems Quantum gravity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Diego A. Quiñones^1,§, Teodora Oniga^2,†, Benjamin T. H. Varcoe^1,‡, and Charles H.-T. Wang^2,*

¹School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom
²Department of Physics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

^*Corresponding author. c.wang@abdn.ac.uk
^†t.oniga@abdn.ac.uk
^‡b.varcoe@leeds.ac.uk
^§pydaqo@leeds.ac.uk

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Vol. 96, Iss. 4 — 15 August 2017

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Figure 1
Theoretical lower bounds for the spectral functions of stochastic gravitational waves $Ω (f)$ determined by Eq. (66) that could be detected using an ensemble of correlated two-level atoms contained in an off-resonant high- $Q$ cavity of volume $L^{3}$ and measurement time $τ$ . Specifically, we consider heliumlike atoms with atomic quantum numbers $n_{1} = n_{2} = 2$ , $l_{1} = l_{2} = 1$ , and $- m_{1} = m_{2} = 1$ , with a transition frequency induced by a Zeeman-type shift. For conventional stochastic gravitational waves, we have $Ω (f) = Ω_{gw} (f)$ with $τ = τ_{gw}$ , whereas for zero-point, i.e., vacuum, metric fluctuations, we have $Ω (f) = Ω_{vac} (f)$ with $τ = τ_{vac}$ introduced in this work by Eqs. (62) and (64). In both cases, we choose $τ = 1 \sec$ , min, and h as possible (future) transition times for the atoms with $L = 1$ and 10 m in generating the above diagram. It suggests, for example, measuring the zero-point fluctuations of spacetime would require a $1 m^{3}$ cavity with a measurement time of 1 sec at $f = 1 THz$ approximately. Note that regions where the values of $Ω_{gw} (f)$ are below that of $Ω_{vac} (f)$ would present a less stochastic gravitational wave signal to the vacuum fluctuations “noise” ratio and hence the detection of the latter would be more likely using the discussed quantum methods. For comparison, we have superimposed the expected spectral functions of stochastic gravitational waves from various sources such as inflation, up to frequencies dominated by spacetime fluctuations, that may be detected by different methods such as LIGO from Ref. [21] and its references.
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Figure 2
Simulations with $N$ atoms using the master equation (1) with the quantum dissipator (37) for the correlated two-level atoms and negligible matter interaction $H_{int}$ . From Eq. (45), the relaxation time is approximately given by $τ_{N} = (N Γ_{0})^{- 1}$ . The initial state of the forms (32) and (33) at $t = 0$ is given by $| 0, N ⟩ = | p = N + 1 ⟩$ with all $N$ atoms occupying the excited atomic level. During the dynamical evolution, the density matrix remains diagonal of the form (39) and relaxes into an inverse exponential profile represented by Eq. (48), yielding the asymptotic height of $ρ_{p = 1} = [N_{gw} (ω_{0}) (1 - e^{- N / N_{gw} (ω_{0})})]^{- 1}$ . The stochastic gravitational wave distribution function evaluated at the atomic transition frequency $ω_{0}$ is chosen to be $N_{gw} (ω_{0}) = 1$ . Accordingly, in plot (a) with $N = 100$ atoms, the initial state corresponding to the spike of height 1 at $(p, Γ_{0} t) = (101, 0)$ relaxes into an inverse exponential profile peaked at $(p, Γ_{0} t) = (1, 0.1)$ on a time scale of $τ_{N} = 0.01 / Γ_{0}$ . Similarly, in plot (b) with $N = 200$ atoms, the initial state corresponding to the spike of height 1 at $(p, Γ_{0} t) = (201, 0)$ relaxes into an inverse exponential profile similarly peaked at $(p, Γ_{0} t) = (1, 0.1)$ but on a halved time scale of $τ_{N} = 0.005 / Γ_{0}$ .
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Figure 3
Simulations of short-time collective spontaneous decays due to vacuum fluctuations of spacetime without additional stochastic gravitational waves so that $N_{gw} (ω_{0}) = 0$ . In plot (a) with $N = 100$ atoms, the initial state corresponds to the spike of height 1 at $(p, t) = (50, 0)$ . As shown, it decays into adjacent lower states with a relaxation time $τ_{vac} = 4 / (N^{2} Γ_{0}) = 0.0004 / Γ_{0}$ according to Eqs. (61) and (62). In plot (b) with $N = 200$ atoms, the initial state corresponds to the spike of height 1 at $(p, t) = (100, 0)$ . For a doubled $N$ , this state decays into adjacent lower states with a quarter of the preceding relaxation time with $τ_{vac} = 4 / (N^{2} Γ_{0}) = 0.0001 / Γ_{0}$ .
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Figure 4
Simulations of short-time collective transitions due to vacuum fluctuations of spacetime and additional stochastic gravitational waves with a chosen $N_{gw} (ω_{0}) = 1$ . In plot (a) with $N = 100$ atoms, the initial state corresponds to the spike of height 1 at $(p, t) = (50, 0)$ . While this state decays similarly to the descriptions in Fig. 4, its adjacent higher state with $p = 51$ is excited with an initial time scale of $τ_{gw} = 4 / (N^{2} Γ_{0} N_{gw} (ω_{0})) = 0.0004 / Γ_{0}$ according to Eqs. (60) and (62). In plot (b) with $N = 200$ atoms, the initial state corresponds to the spike of height 1 at $(p, t) = (100, 0)$ . For a doubled $N$ , its adjacent higher state with $p = 101$ is excited with a quarter of the preceding initial time scale with $τ_{gw} = 4 / (N^{2} Γ_{0} N_{gw} (ω_{0})) = 0.0001 / Γ_{0}$ .
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Physical Review D

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