Localisation problem for relativistic detectors in the Unruh-DeWitt model

Evan Gale^a, and Magdalena Zych^{a, b}

Physical Review D 107, 056023 (2023)

Detectors with quantised centre of mass

Considered by Unruh and Wald for a box detector [3]

$\hat{H}_I = \lambda \int d^3x \, \hat{\mu}(\vec{x}) \otimes \hat{\phi}(\vec{x})$

with a position-dependent monopole operator of the form

$\hat{\mu}(\vec{x}) \equiv \psi_g^*(\vec{x}) \psi_e(\vec{x}) | g \rangle \langle e | + \psi_e^*(\vec{x}) \psi_g(\vec{x}) | e \rangle \langle g |$

Detectors with quantised centre of mass

Alternative extension considered by Stritzelberger and Kempf [4]

$\hat{H}_I = \lambda \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D \otimes \hat{\phi}(\vec{x})$

Detector's free Hamiltonian now includes a kinetic term

$\hat{H}_D = \frac{\hat{\vec{p}}^2}{2 M} + E | e \rangle \langle e |$

Full atom-light interaction treated by Wilkens [5]

Relativistic second-quantised detector

Relativistic second-quantised detector model proposed by Unruh [1]

Replace energy levels of detector by fields

$| j \rangle \to \hat{\psi}_j$

$\hat{\mu}_\hat{\psi}(\vec{x}) = \hat{\psi}^\dagger_g(\vec{x}) \hat{\psi}_e(\vec{x}) + \hat{\psi}^\dagger_e(\vec{x}) \hat{\psi}_g(\vec{x})$

The interaction Hamiltonian takes the form

$\hat{H}_I^{(\mathrm{2nd})} = \lambda^{(\mathrm{2nd})} \int d^3x \, \hat{\mu}_\hat{\psi}(\vec{x}) \otimes \hat{\phi}(\vec{x})$

Coupling constants between models are dimensionally distinct!

Relativistic second-quantised detector

Giacomini and Kempf [6] simplified by restricting to one-particle sector

Interaction couples position and internal degrees of freedom

$\hat{H}_I^{(\mathrm{2nd})} \Big |_{\mathcal{H}^D_1} = \lambda^{(\mathrm{2nd})} \int d^3x \, \Big( | \vec{x}_g \rangle \langle \vec{x}_e |_D^{(\mathrm{2nd})} + | \vec{x}_e \rangle \langle \vec{x}_g |_D^{(\mathrm{2nd})} \Big) \otimes \hat{\phi}(\vec{x})$

Looks similar to previous non-relativistic model, except that

$| \vec{x}_j \rangle_D^{(\mathrm{2nd})} \equiv \frac{1}{(2\pi)^{3/2}} \int \frac{d^3p}{\sqrt{2 E_j(\vec{p})}} e^{-i \vec{p} \cdot \vec{x}} | \vec{p}, j \rangle_D$

Position states are not Fourier transforms of momentum eigenstates!

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

$\hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, .$

But we have to be a little careful!

Q. What time parameter does the Hamiltonian generate time translations for? [8]

A. Inertial detector. Generate time translations wrt coordinate time $t$

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

$\hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, .$

But we have to be a little careful!

Q. How do we define the position states $| \vec{x} \rangle_D^{(\mathrm{1st})}$ ?

A. Take states to be Fourier transform of momentum eigenstates

$| \vec{x} \rangle_D^{(\mathrm{1st})} \equiv \frac{1}{(2\pi)^{3/2}} \int d^3p \, e^{-i \vec{p} \cdot \vec{x}} | \vec{p} \rangle_D \, .$

A relativistic first-quantised model?

Interaction Hamiltonian should have the form

$\hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \hat{\mu} \otimes | \vec{x} \rangle \langle \vec{x} |_D^{(\mathrm{1st})} \otimes \hat{\phi}(\vec{x}) \, .$

But we have to be a little careful!

Q. How are the coupling constants between models related?

A. Take couplings to coincide for detector at rest

$\lambda^{\mathrm{(2nd)}} = \sqrt{2 (M_g^2 + M_e^2)} \, \lambda^{\mathrm{(1st)}} \, .$

Summary

First-quantised model

$\hat{H}_I^{(\mathrm{1st})} = \lambda^{(\mathrm{1st})} \int d^3x \, \Big( | \vec{x}_g \rangle \langle \vec{x}_e |_D^{(\mathrm{1st})} + | \vec{x}_e \rangle \langle \vec{x}_g |_D^{(\mathrm{1st})} \Big) \otimes \hat{\phi}(\vec{x}) \, .$

Second-quantised model

Different localisations are given by

$| \vec{x}_j \rangle_D^{(\mathrm{1st})} \equiv \frac{1}{(2\pi)^{3/2}} \int d^3p \, e^{-i \vec{p} \cdot \vec{x}} | \vec{p}, j \rangle_D \, , \quad | \vec{x}_j \rangle_D^{(\mathrm{2nd})} \equiv \frac{1}{(2\pi)^{3/2}} \int \frac{d^3p}{\sqrt{2 E_j(\vec{p})}} e^{-i \vec{p} \cdot \vec{x}} | \vec{p}, j \rangle_D \, .$

Transition rate for spontaneous emission

We idealise the atom-light interaction, finding transition from

$| \Psi_i \rangle = | \psi_i, e \rangle_D \otimes | 0 \rangle \, , \text{ with } \, | \psi_i \rangle = \int{d^3p \, \psi_i(\vec{p}; \vec{p}_0) | \vec{p}} \rangle \, ,$

to the final state via spontaneous emission of the atom-detector

$| \Psi_f \rangle = | \vec{p}_f, g \rangle_D \otimes \hat{a}^\dagger(\vec{k}) | 0 \rangle \, .$

Interaction picture: evolution wrt free Hamiltonian $\hat{H}_D + \hat{H}_F$

Weak coupling: find to first-order in perturbation theory

$\hat{U}(t_f, \, t_i) = 1 - i \int_{t_i}^{t_f} dt \, \hat{H}_{I}(t) + \mathcal{O}(\lambda^2) \, .$

Transition rate for spontaneous emission

Transition amplitude and probability found from

$\mathcal{A}_{| \vec{p}_i, e, 0 \rangle \to | \vec{p}_f, g, 1_{\vec{k}} \rangle} = \langle \Psi_f | \hat{U}(t_f, \, t_i) | \Psi_i \rangle \, , \\ P_{| \vec{p}_i, e, 0 \rangle \to | g \rangle} = \int d^3k \int d^3p_f \, \left| \mathcal{A}_{| \vec{p}_i, e, 0 \rangle \to | \vec{p}_f, g, 1_{\vec{k}} \rangle} \right|^2 \, .$

Transition rate given as a functional of the detector wave function (probability density function), convolved with a 'template function'

$\dot{P}[\psi_i] = \frac{\lambda^2}{2 \pi} \int d^3p \, |\psi_i(\vec{p}; \vec{p}_0)|^2 \, \mathcal{T}_{\mathrm{rel}}(\vec{p}) \, .$

Transition rate for spontaneous emission

Transition rate given as a functional of the detector wave function (probability density function), convolved with a 'template function'

$\dot{P}[\psi_i] = \frac{\lambda^2}{2 \pi} \int d^3p \, |\psi_i(\vec{p}; \vec{p}_0)|^2 \, \mathcal{T}_{\mathrm{rel}}(\vec{p}) \, .$

First-quantised model

$\hspace{-3em} \mathcal{T}_{\mathrm{rel}}^{\mathrm{(1st)}}(\vec{p}) = \frac{1}{4} \left( 1 - \frac{M_g^4}{M_e^4} \right) \sqrt{\vec{p}^2 + M_e^2} \, ,$

Second-quantised model

$\hspace{1em} \mathcal{T}_{\mathrm{rel}}^{\mathrm{(2nd)}}(\vec{p}) = \frac{1}{4} \left( 1 - \frac{M_g^4}{M_e^4} \right) \frac{M_e^2}{\sqrt{\vec{p}^2 + M_e^2}} \, .$

How should we interpret these results?

Localisation in RQM II

Philips [12] defined a Lorentz-invariant position operator

Superposition of localised systems is also localised.
Invariant wrt spatial rotations and reflections.
Localised states are invariant wrt Lorentz boosts.
The localised states are normalisable.
No subset of the set of localised states satisfies the above conditions.

In second-quantised formalism, Philips' position state is equivalent to the action of field operator on vacuum $| \vec{x}_p \rangle \equiv \hat{\psi}(\vec{x}) | 0 \rangle$

Localisation problem for relativistic detectors in the Unruh-DeWitt model Evan Gale a , and Magdalena Zych a, b a ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia b Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden Physical Review D 107 , 056023 (2023)

Localisation problem for relativistic detectors in the Unruh-DeWitt model

Evan Gale^a, and Magdalena Zych^{a, b}

Relativistic quantum physics

Unruh-DeWitt (UDW) detector model

Unruh-DeWitt (UDW) detector model

Detectors with quantised centre of mass

Detectors with quantised centre of mass

Relativistic centre of mass?

Relativistic centre of mass?

Relativistic second-quantised detector

Relativistic second-quantised detector

A relativistic first-quantised model?

A relativistic first-quantised model?

A relativistic first-quantised model?

A relativistic first-quantised model?

A relativistic first-quantised model?

Summary

Transition rate for spontaneous emission

Transition rate for spontaneous emission

Transition rate for spontaneous emission

Template functions for different localisations

Template functions for different localisations

Template functions for different localisations

Transition rate for detector at rest

Transition rate for detector at rest

Transition rate for detector at rest

Transition rate for highly localised detector

Open Questions

How should we interpret these results?

Localisation in RQM I

How should we interpret these results?

Localisation in RQM II

How should we interpret these results?

Localisation in RQM III

Two sides to relativistic localisation

Summary

Thank you!

Localisation problem for relativistic detectors in the Unruh-DeWitt model

Evan Galea, and Magdalena Zycha, b

Evan Gale^a, and Magdalena Zych^{a, b}