Schrödinger’s cat for de Sitter spacetime

Despite numerous attempts over the last century, a unified theory of quantum gravity remains elusive. Since the spacetime metric in general relativity is dynamical, standard approaches to quantisation cannot be applied to the gravitational field. Attempts at unification such as string theory Gubser et al. (1998); Seiberg and Witten (1999); Polchinski (1998); Witten (1995) and loop quantum gravity Rovelli (1998); Rovelli and Vidotto (2014); Thiemann (2003) have addressed the problem from a top-down perspective. However, the well-known difficulties within these approaches have led to suggestions that spacetime could be fundamentally classical, or that the quantisation of gravity is an ill-posed question to begin with Dyson (2013). For pertinent discussions on the necessity of quantising the gravitational field, see Garay (1995); Carlip (2008) and references therein.

Any expected theory combining spacetime with the indeterminacy of quantum theory should admit such notions as a superposition of semiclassical geometries. Christodoulou and Rovelli Christodoulou and Rovelli (2019) have argued that a recent experimental proposal, involving the gravitationally-induced entangling of two mesoscopic particles (each in a spatial superposition state) Bose et al. (2017); Marletto and Vedral (2017), would provide evidence for a system described by a superposition of spacetime geometries. So far the focus has been on effects produced by spatial superpositions of massive objects. However, the resulting metrics only differ by a global coordinate transformation and are thus diffeomorphic. Such scenarios can equivalently be described as arising in a single classical spacetime where quantum systems are prepared and measured in expropriate quantum states. Quantum systems residing in background metrics defined by parameters in superposition, or even superpositions of different classes of background metrics, has not been studied.

In this paper, we propose a new phenomenological description of a spacetime metric in a genuine quantum superposition. Rather than offering a fully quantum-gravitational origin for this phenomenon, we work within the paradigm of quantum field theory in curved spacetime (QFT-CS), using a theoretical tool known as the Unruh-deWitt (UdW) detector Unruh and Wald (1984) to probe such superpositions. UdW detectors have been utilized widely in relativistic and gravitational settings, for example in the study of quantum fields and their entanglement structure in black hole spacetimes Ng et al. (2014, 2017, 2018); Henderson et al. (2020); Hodgkinson and Louko (2012a) and higher-dimensional topologies Hodgkinson and Louko (2012b). We adopt a recent framework Foo et al. (2020a); Barbado et al. (2020) which extends the model to include quantum-controlled superpositions of semiclassical detector trajectories. As suggested in Foo et al. (2020a), this would allow (for example) the detector to reside in a superposition state of different radial distances from a black hole horizon, in which it would be expected to perceive a superposition of redshifts induced by the local curvature, and respond to Hawking radiation as if it were in a coherent ‘superposition’ of thermal states.

In the first application of this novel idea, we study the experience of UdW detectors residing on a background de Sitter spacetime in superposition. We beging by modelling the detector as traveling in a superposition of spatially translated worldlines in the static patch of de Sitter spacetime. We demonstrate that this scenario is diffeomorphic to that in which the detector resides on a manifold in a quantum superposition of spatial translations Zych et al. (2018). Our main result is the application of the model to phenomenologically describe de Sitter spacetime in a superposition of curvatures. The detector interacts with quantum fields defined on a background spacetime described by a superposition of states, each associated with a different value (i.e. superposed) of the de Sitter curvature. This novel description bears an analogy to the superposition of ‘semiclassical’ coherent states in quantum information. Unlike superpositions of spatial degrees of freedom, there is no diffeomorphism that maps the metric in superposition to a single metric with a classical value of curvature.

In each scenario, we calculate the response of the detector as it interacts with a conformally coupled massless scalar field in static de Sitter spacetime. To leading order in perturbation theory, we discover particular scenarios where the detector response corresponds with that produced by superpositions of uniformly accelerated trajectories in Minkowski spacetime. Due to the well-known conformal relationship between the Rindler and de Sitter geometries, it is on one hand unsurprising that such a correspondence exists. On the other hand, this raises the interesting question of how one can operationally distinguish between genuine superpositions of spacetime metrics from mere superpositions of trajectory states of an UdW detector.

We organise this paper as follows: in Sec. II, we review the UdW detector model with the inclusion of a control degree of freedom, which allows the detector to travel in quantum superpositions of trajectories, or equally initialises the background spacetime in an arbitrary superposition of quantum states. In Sec. III, we demonstrate the diffeomorphic invariance of a superposition of detector trajectory states on a classical metric compared with a classical detector trajectory on a background in a superposition of spatial states. We apply the model to static de Sitter spacetime in Sec. V and VI, studying operationally the effect of metrics in superpositions of spatial and curvature states on the behaviour of quantum detectors. We discuss the question of how quantum probes may differentiate genuine superpositions of spacetime metrics from flat spacetime trajectories in Sec. VII, before offering some conclusions in Sec. VIII. Throughout, we utilise natural units, $\hslash=k_{B}=c=G=1$.

In the original formulation of the UdW detector model with quantum control, the detector is modelled as interacting with the quantum field on a classical spacetime background in a superposition of trajectory states. Such a system can be described in the tensor product Hilbert space, $\mathcal{H}=\mathcal{H}_{T}\otimes\mathcal{H}_{\text{UdW}}\otimes\mathcal{H}_{F}$ where $\mathcal{H}_{T}$, $\mathcal{H}_{\text{UdW}}$ and $\mathcal{H}_{F}$ are associated with the trajectory, detector, and field degrees of freedom respectively. We are interested here in describing the spacetime itself with quantised degrees of freedom. Thus, the Hilbert space can be decomposed as $\mathcal{H}=\mathcal{H}_{S}\otimes\mathcal{H}_{\text{UdW}}\otimes\mathcal{H}_{F}$, where $\mathcal{H}_{S}$, $\mathcal{H}_{\text{UdW}}$ and $\mathcal{H}_{F}$ are now associated with the spacetime, detector and field degrees of freedom. As we demonstrate in Sec. III, the two representations are essentially equivalent for spatial superpositions. Finally, we note that in our analysis in Sec. VI.1, both the spacetime and the trajectories are assigned quantum states.

The standard UdW model considers a point-like two-level system whose internal states $|g\rangle,|e\rangle$ couple to the massless, scalar field $\hat{\Phi}(\mathsf{x}(\tau))$. We assume that the detector is initially in its ground state, $|g\rangle$, and that the field is in the conformally coupled vacuum state $|0\rangle_{F}$ of de Sitter spacetime Birrell and Davies (1984), that is

$\displaystyle|\Psi\rangle_{FD}$

$\displaystyle=|g\rangle\otimes|0\rangle_{F}.$

(1)

The conformal vacuum is a natural choice, since it is a coordinate-independent vacuum state bearing a close analogy to the Minkowski vacuum in flat spacetime Birrell and Davies (1984). We introduce a control degree of freedom, $|\chi\rangle$, to the detector-field-spacetime system, leaving the initial state to be

$\displaystyle|\Psi\rangle_{CFD}$

$\displaystyle=|\chi\rangle\otimes|g\rangle\otimes|0\rangle_{F}$

(2)

where

$\displaystyle|\chi\rangle$

$\displaystyle=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}|i\rangle_{C}$

(3)

and $|i\rangle_{C}$ are orthogonal states. As alluded to, one may interpret the control as being attached to the detector itself, in which case it governs the trajectories which the detector traverses in superposition. However we are motivated by the possibility of describing the detector travelling on a single trajectory with the spacetime in a superposition state. Without postulating the (quantum-gravitational) mechanism by which one prepares a superposition of a gravitational source of curvature, we show in this article that one can equally take $|\chi\rangle$ as controlling the quantum state of the background spacetime. In such a scenario, the amplitudes of the superposition could correspond to the orthogonal ‘curvature states’ of an expanding spacetime, or the ‘quantised mass states’ of a black hole Arrasmith et al. (2019); Wood and Zych (2020).

Returning now to the UdW detector model with a quantum control, we introduce the interaction Hamiltonian,

$\displaystyle\hat{H}_{\text{int.}}(\tau)$

$\displaystyle=\sum_{i=1}^{N}\hat{\mathcal{H}}_{i}(\tau)\otimes|i\rangle\langle i|_{C}$

(4)

where

$\displaystyle\hat{\mathcal{H}}_{i}(\tau)$

$\displaystyle=\lambda\hat{\sigma}(\tau)\eta_{i}(\tau)\hat{\Phi}\big{(}\mathsf{x}_{i}(\tau)\big{)}$

(5)

governs the interaction along the worldline $\mathsf{x}_{i}(\tau)$ of the superposition. In Eq. (5), $\lambda$ is a weak coupling constant, $\eta_{i}(\tau)$ are switching functions, $\hat{\sigma}(\tau)=|e\rangle\langle g|e^{i\Omega\tau}+\text{H.c}$ is the interaction picture ladder operator between the states $|g\rangle,|e\rangle$ with energy gap $\Omega$. The evolution of the system in the interaction picture is given by

$\displaystyle\hat{U}|\Psi\rangle_{CFD}=\mathcal{T}\exp\bigg{\{}-i\int\mathrm{d}t\>\Big{(}\frac{\mathrm{d}\tau}{\mathrm{d}t}H_{\text{int.}}(\tau)\Big{)}\bigg{\}}|\Psi\rangle_{CFD}$

(6)

where $\mathcal{T}$ denotes time-ordering. The time-evolution operator takes the form,

$\displaystyle\hat{U}$

$\displaystyle=\sum_{i=1}^{N}\hat{U}_{i}\otimes|i\rangle\langle i|_{C}$

(7)

where, to leading order in perturbation theory

$\displaystyle\hat{U}_{i}$

$\displaystyle=1-i\lambda\int\mathrm{d}\tau\>\mathcal{\hat{H}}_{i}(\tau)+\mathcal{O}(\lambda^{2}).$

(8)

Using Eq. (7), we can time-evolve the initial state to obtain

$\displaystyle\hat{U}|\Psi\rangle_{CFD}$

$\displaystyle=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\hat{U}_{i}|i\rangle_{C}|0\rangle_{F}|g\rangle.$

(9)

Following Foo et al. (2020a, b), we are interested in the conditional state of the detector given the control is measured in some fixed state, which we take to be its initial state, $|\chi\rangle$. This yields

$\displaystyle\langle\chi|\hat{U}|\Psi\rangle_{CFD}\equiv|\Psi^{\prime}\rangle_{FD}$

$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\hat{U}_{i}|0\rangle_{F}|g\rangle.$

(10)

From this, one can obtain the final state of the detector by tracing out the field degrees of freedom, which to leading order in the coupling constant $\lambda$, is given by the density matrix

$\displaystyle\hat{\rho}_{D}$

$\displaystyle=\begin{pmatrix}1-\mathcal{P}_{D}&0\\ 0&\mathcal{P}_{D}\end{pmatrix}+\mathcal{O}(\lambda^{4})$

(11)

where

$\displaystyle\mathcal{P}_{D}$

$\displaystyle=\frac{\lambda^{2}}{N^{2}}\sum_{i,j=1}^{N}\mathcal{P}_{ij,D}=\frac{\lambda^{2}}{N^{2}}\left\{\sum_{i=j}^{N}\mathcal{P}_{ij,D}+\sum_{i\neq j}\mathcal{P}_{ij,D}\right\}$

(12)

is the transition probability of the detector (or fraction of excited detectors within an identically prepared ensemble) conditioned on the measurement of the control system in the asymptotic future. The individual contributions take the form

$\displaystyle\mathcal{P}_{ij,D}$

$\displaystyle=\int\mathrm{d}\tau\int\mathrm{d}\tau^{\prime}\chi_{i}(\tau)\overline{\chi}_{j}(\tau^{\prime})\mathcal{W}^{ji}\big{(}\mathsf{x}_{i},\mathsf{x}_{j}^{\prime}\big{)}$

(13)

where we have defined $\chi(\tau)=\eta(\tau)e^{-i\Omega\tau}$, while

$\displaystyle\mathcal{W}^{ji}\big{(}\mathsf{x}_{i},\mathsf{x}_{j}^{\prime}\big{)}:=\langle 0|\hat{\Phi}(\mathsf{x}_{i})\hat{\Phi}(\mathsf{x}_{j}^{\prime})|0\rangle_{F}$

(14)

are field correlation (Wightman) functions pulled back to the trajectories $\mathsf{x}_{i}(\tau),\mathsf{x}_{j}^{\prime}(\tau^{\prime})$ Birrell and Davies (1984), associated with the $i$-$j$th amplitude of the superposition. The $i$th trajectory is associated with the $|i\rangle_{C}$ state of the control system. Unlike a detector travelling along a single, classical trajectory, $\mathcal{P}_{D}$ now features ‘nonlocal’ correlation functions between each respective pair of amplitudes in the superposition, $i\neq j$.

For the switching function, we consider a Gaussian, $\eta(\tau)=\exp(-\tau^{2}/2\sigma^{2})$, where $\sigma$ is a characteristic timescale of the interaction. For stationary trajectories, the Wightman functions only depend on $s=\tau-\tau^{\prime}$ and the outer integral of Eq. (13) can be evaluated exactly, yielding the simplified expression,

$\displaystyle\mathcal{P}_{D}$

$\displaystyle=\frac{\lambda^{2}\sqrt{\pi}\sigma}{N^{2}}\sum_{i,j=1}^{N}\int\mathrm{d}s\>e^{-s^{2}/4\sigma^{2}}e^{-i\Omega s}\mathcal{W}^{ji}(s).$

(15)

In these time-independent scenarios, it will be useful to work with the normalised transition probability

$\displaystyle\mathcal{F}(\Omega)$

$\displaystyle=\frac{\mathcal{P}_{D}}{\lambda^{2}\sqrt{\pi}\sigma},$

(16)

which we refer to as the response function. In the infinite-interaction time limit, $\sigma\to\infty$, the response function becomes

$\displaystyle\mathcal{F}(\Omega)$

$\displaystyle=\frac{\lambda^{2}}{N^{2}}\sum_{i,j=1}^{N}\int_{-\infty}^{\infty}\mathrm{d}s\>e^{-i\Omega s}\mathcal{W}^{ji}(s).$

(17)

For non-stationary trajectories where analytic solutions are intractable, we can calculate the instantaneous transition rate of the detector, measured at the proper time $\tau$ while the interaction is still on. Following Foo et al. (2020a, b), and taking the infinite interaction-time limit $(\sigma\to\infty$), one obtains for the instantaneous transition rate,

$\displaystyle\dot{\mathcal{P}}_{D}$

$\displaystyle=\frac{2\lambda^{2}}{N^{2}}\text{Re}\sum_{i,j=1}^{N}\int_{0}^{\infty}\mathrm{d}s\>e^{-i\Omega s}\mathcal{W}^{ji}(\tau,\tau-s)$

(18)

where $\dot{\mathcal{P}}_{D}=\mathrm{d}\mathcal{P}_{D}/\mathrm{d}\tau$.

Equations (17) and (18) characterise the detector’s response as it travels along a given trajectory or superposition of trajectories in spacetime. These are essentially Fourier transforms of the Wightman functions, Eq. (14), which encode the field correlations between $\hat{\Phi}\big{(}\mathsf{x}_{i}(\tau)\big{)}$ and $\hat{\Phi}\big{(}\mathsf{x}_{j}(\tau)\big{)}$. In the language of superposed UdW detectors traveling on a classical background spacetime, it is natural to view $\mathsf{x}_{i}(\tau)$ and $\mathsf{x}_{j}(\tau)$ as the superposed trajectories of the detector itself. Alternatively, such a scenario can be mapped via a global coordinate transformation so that the detector travels along a single classical trajectory, with the spacetime in a superposition of spatial translations Zych et al. (2018).

To illustrate this, let us consider a detector in a superposition of two trajectories. One can associate the basis states $|i\rangle_{C}$ in the control superposition with the physical trajectories which the detector traverses. For example, we take

	$\displaystyle|1\rangle_{C}$	$\displaystyle\Rightarrow|\xi\rangle\vphantom{\frac{1}{2}}$		(19)
	$\displaystyle|2\rangle_{C}$	$\displaystyle\Rightarrow|\xi+\mathcal{L}\rangle$		(20)

which we refer to as the trajectory states, where $\xi\equiv\xi(\tau)$ denotes the worldline of the first trajectory and $\mathcal{L}\equiv\mathcal{L}(\tau)$ is some time-dependent function that relates the coordinates of the trajectory basis states. For fixed $\mathcal{L}$, this merely describes a constant spatial translation, which we assume for simplicity here.

A coordinate transformation relating the trajectory basis states can be enacted via a unitary transformation $\hat{\mathcal{T}}(\mathcal{L})$

	$\displaystyle|\xi+\mathcal{L}\rangle\otimes|0\rangle_{F}$	$\displaystyle=\hat{\mathcal{T}}(\mathcal{L})|\xi\rangle\otimes|0\rangle_{F}\vphantom{\frac{1}{2}}$		(21)
		$\displaystyle=\hat{\mathcal{T}}_{\xi}(\mathcal{L})|\xi\rangle\otimes\hat{\mathcal{T}}_{\phi}(\mathcal{L})|0\rangle_{F}$		(22)

where $\hat{\mathcal{T}}(\mathcal{L}):=\hat{\mathcal{T}}_{\phi}(\mathcal{L})\otimes\hat{\mathcal{T}}_{\xi}(\mathcal{L})$, i.e. the unitary $\hat{\mathcal{T}}(\mathcal{L})$ acts on the trajectory state (generating spatial translations through $\hat{\mathcal{T}}_{\xi}(\mathcal{L})$) and also on the field degrees of freedom, since they possess a dependence on the spacetime coordinates. When calculating transition amplitudes, $\hat{\mathcal{T}}_{\phi}(\mathcal{L})$ generates unitary coordinate transformations at the level of the field operator,

$\displaystyle\hat{\Phi}(\xi+\mathcal{L})$

$\displaystyle:=\hat{\mathcal{T}}_{\phi}(\mathcal{L})^{\dagger}\hat{\Phi}(\xi)\hat{\mathcal{T}}_{\phi}(\mathcal{L}).$

(23)

Due to the symmetries of general relativity, any arbitrary coordinate transformation of the spacetime manifold has an associated unitary representation that transforms quantum states defined on that manifold in a corresponding way. Our goal here is to demonstrate that the dynamical evolution of the detector-field-spacetime system (and hence observables measured by the detector) is diffeomorphic invariant under these unitary transformations of the field operators.

Using Eq. (21), the initial state of the system can be expressed as

	$\displaystyle|\Psi\rangle_{TFD}$	$\displaystyle=\frac{1}{\sqrt{2}}\big{(}|\xi\rangle+|\xi+\mathcal{L}\rangle\big{)}|0\rangle_{F}|g\rangle$		(24)
		$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\mathds{1}+\hat{\mathcal{T}}(\mathcal{L})\big{)}|\xi\rangle|0\rangle_{F}|g\rangle.$		(25)

Thus, the time-evolution of the state is given by

$\displaystyle\hat{U}|\Psi\rangle_{TFD}$

$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\hat{U}+\hat{U}\hat{\mathcal{T}}(\mathcal{L})\big{)}|\xi\rangle|0\rangle_{F}|g\rangle.$

(26)

From Eq. (26), we see that the detector motion has been mapped to a single trajectory, $\xi$, and the detector-field dynamics are now enacted via a superposition of unitaries. It is now the field operators contained in $\hat{U}$ which carry the superposition dynamics of the system.

Measuring the trajectory state in the modified basis $|\chi\rangle=(\mathds{1}+\hat{\mathcal{T}}(\mathcal{L}))|\xi\rangle/\sqrt{2}$ yields the conditional detector-field state

	$\displaystyle|\Psi\rangle_{FD}$	$\displaystyle=\frac{1}{2}\Big{(}\langle\xi|\hat{U}|\xi\rangle+\langle\xi|\hat{U}\hat{\mathcal{T}}(\mathcal{L})|\xi\rangle+\langle\xi|\hat{\mathcal{T}}(\mathcal{L})^{\dagger}\hat{U}|\xi\rangle$
		$\displaystyle+\langle\xi|\hat{\mathcal{T}}(\mathcal{L})^{\dagger}\hat{U}\hat{\mathcal{T}}(\mathcal{L})|\xi\rangle\Big{)}\otimes|0\rangle_{F}\otimes|g\rangle\vphantom{\frac{1}{2}}.$		(27)

Previously, we have considered the time-evolution operator as having components acting on all of the trajectory states of the detector,

$\displaystyle\hat{U}\equiv\sum_{\text{$\zeta=$ \text{paths}}}\hat{U}(\zeta)\otimes|\zeta\rangle\langle\zeta|.$

(28)

However now we only have the trajectory $|\xi\rangle$, so

$\displaystyle\hat{U}$

$\displaystyle=\hat{U}(\xi)\otimes|\xi\rangle\langle\xi|.$

(29)

where $\hat{U}(\xi)$ acts on the detector-field Hilbert space. Using Eq. (29), it is straightforward to show that $\langle\xi|\hat{U}\hat{\mathcal{T}}(\mathcal{L})|\xi\rangle=\langle\xi|\hat{\mathcal{T}}(\mathcal{L})^{\dagger}\hat{U}|\xi\rangle=0$ while

	$\displaystyle\langle\xi|\hat{U}|\xi\rangle$	$\displaystyle=\hat{U}(\xi)$		(30)
	$\displaystyle\langle\xi|\hat{\mathcal{T}}(\mathcal{L})^{\dagger}\hat{U}\hat{\mathcal{T}}(\mathcal{L})|\xi\rangle$	$\displaystyle=\hat{\mathcal{T}}_{\phi}(\mathcal{L})^{\dagger}\hat{U}(\xi)\hat{\mathcal{T}}_{\phi}(\mathcal{L}),$		(31)

where Eq. (31) is simply equal to $\hat{U}(\xi+\mathcal{L})$. This leaves the detector-field system in the state

$\displaystyle|\Psi\rangle_{FD}$

$\displaystyle=\frac{1}{2}\big{(}\hat{U}(\xi)+\hat{U}(\xi+\mathcal{L})\big{)}|0\rangle_{F}|g\rangle.$

(32)

We see clearly that the detector, now traversing the single classical trajectory $\xi\equiv\xi(\tau)$, undergoes dynamics in which the field operators are enacted in a superposition state of spatial translations (i.e. evaluated along two different trajectories). Of course, Eq. (32) is exactly Eq. (10), viewed from a different quantum reference frame Giacomini et al. (2019).

For completeness, we can write explicitly the expressions for the time-evolution operators, namely

	$\displaystyle\hat{U}(\xi)$	$\displaystyle=1-i\int\mathrm{d}\tau\>\hat{\mathcal{H}}_{i}(\xi)+\mathcal{O}(\lambda^{2})$		(33)
	$\displaystyle\hat{U}(\xi+\mathcal{L})$	$\displaystyle=1-i\int\mathrm{d}\tau\>\hat{\mathcal{H}}_{i}(\xi+\mathcal{L})+\mathcal{O}(\lambda^{2}).$		(34)

Applying these to the detector-field state in Eq. (32) yields, after tracing out the field states, the conditional transition probability

$\displaystyle\mathcal{P}_{D}$

$\displaystyle=\frac{\lambda^{2}}{4}\left\{\sum_{i=j}\mathcal{P}_{ij,D}+\sum_{i\neq j}\mathcal{P}_{ij,D}\right\}$

(35)

where

$\displaystyle\mathcal{P}_{ij,D}$

$\displaystyle=\int\mathrm{d}\tau\int\mathrm{d}\tau^{\prime}\chi(\tau)\overline{\chi}(\tau^{\prime})\langle 0|\hat{\Phi}(i)\hat{\Phi}(j^{\prime})|0\rangle_{F}$

(36)

are the individual contributions evaluated between the trajectories $i=\xi,\xi+\mathcal{L}$ and $j^{\prime}=\xi^{\prime},\xi^{\prime}+\mathcal{L}$.

Comparing Eq. (35) with Eq. (12), it is clearly seen that the transition probability is equivalent between the two perspectives. More generally, for a detector prepared in a superposition of trajectory states on a fixed background spacetime (i.e. in a quantum reference frame), there always exists a complementary scenario in which the detector traverses a single worldline (i.e. a classical reference frame) and the spacetime is in a quantum superposition of spatial states. Such a scenario is visualised in Fig. 1(b) for a superposition of angular rotations in the 5-dimensional embedding space. Observables like the transition probability of the detector are diffeomorphic invariant when transforming between these perspectives. As we discuss in Sec. VI, this invariance does not apply to situations involving superpositions of curvature, since two unique solutions to Einstein’s equations in superposition cannot be mapped to a single classical background. This scenario, visualised in Fig. 1(c), unambiguously represents a genuine quantum superposition of spacetime metrics that has no dual representation in terms of a superposition of detector trajectories.

Before studying the specific detector-de Sitter spacetime superpositions alluded to previously, we review the necessary geometric and field-theoretic details for our calculations. In this paper we study superpositions of static de Sitter spacetime De Sitter (1917a, b), which is particularly convenient for attaining simple analytic results due to its constant curvature. Throughout, we also make comparisons to analogous scenarios for accelerated trajectories in flat spacetime, and so we also briefly review the geometric details of Rindler space below.

We now apply the quantum-controlled UdW detector model to spatially translated superpositions in the static patch of de Sitter spacetime. The inclusion of the control degree of freedom allows us to model the detector as traveling in a quantum superposition of two spatially translated trajectories – equivalently, to reside in a background spacetime in a quantum superposition of spatial translations.

We consider first translations within the 5-dimensional Minkowski embedding space of the de Sitter spacetime (see Fig. 1 for a visualisation). Note that this is not a limitation of our model; the translations occur within the embedding space according to our choice of the superposed coordinate $\theta$. Alternatively we could equally superpose the radial coordinate of the detector (which we do in Sec. VI), which would be a translation on the de Sitter manifold itself. The trajectories associated with the quantum states in superposition differ by the angular separation $\theta_{S}=\theta_{A}-\theta_{B}$; these are parametrised by the coordinates

	$\displaystyle\textbf{Z}^{(A)}$	$\displaystyle=\big{(}Z_{0}^{(A)},Z_{i}^{(A)}\big{)}$		(71)
	$\displaystyle\textbf{Z}^{(B)}$	$\displaystyle=\big{(}Z_{0}^{(B)},Z_{i}^{(B)}\big{)},$		(72)

(the superscripts denoting the $i$th worldline of the superposition) where

$\displaystyle\begin{split}Z_{0}^{(A)}&=\sqrt{l^{-2}-R_{D}^{2}}\sinh(lt)\\ Z_{1}^{(A)}&=\sqrt{l^{-2}-R_{D}^{2}}\cosh(lt)\\ Z_{2}^{(A)}&=R_{D}\cos\theta_{A}\vphantom{\sqrt{l^{-2}-r^{2}}}\\ Z_{3}^{(A)}&=R_{D}\sin\theta_{A}\cos\phi\vphantom{\sqrt{l^{-2}-R_{D}^{2}}}\\ Z_{4}^{(A)}&=R_{D}\sin\theta_{A}\sin\phi\vphantom{\sqrt{l^{-2}-R_{D}^{2}}}\end{split}\begin{split}Z_{0}^{(B)}&=\sqrt{l^{-2}-R_{D}^{2}}\sinh(lt)\\ Z_{1}^{(B)}&=\sqrt{l^{-2}-R_{D}^{2}}\cosh(lt)\\ Z_{2}^{(B)}&=R_{D}\cos\theta_{B}\vphantom{\sqrt{l^{-2}-r^{2}}}\\ Z_{3}^{(B)}&=R_{D}\sin\theta_{B}\cos\phi\vphantom{\sqrt{l^{-2}-R_{D}^{2}}}\\ Z_{4}^{(B)}&=R_{D}\sin\theta_{B}\sin\phi\vphantom{\sqrt{l^{-2}-R_{D}^{2}}}.\end{split}$

(73)

This means that the superposed trajectories, each static at constant $(R_{D},\theta_{i},\phi)$, are translated (in the embedding space) by the Euclidean distance

$\displaystyle\mathcal{L}_{S}$

$\displaystyle=2R_{D}\sin\left(\frac{\theta_{S}}{2}\right).$

(74)

We refer to $\mathcal{L}_{S}$ as the superposition distance. The translation occurs along an axis orthogonal to the radial direction from the horizon, so the individual amplitudes of the superposition experience equal cosmological redshifts and hence, equal temperatures, $\kappa\equiv\kappa_{A}=\kappa_{B}$. The translation takes on values in the finite domain $\mathcal{L}_{S}\in(0,2/l)$. In the infinite interaction-time limit, the response function (35) takes the form

$\displaystyle\mathcal{F}_{\mathcal{L}_{S}}^{\sigma\to\infty}(\Omega)$

$\displaystyle=-\frac{\kappa^{2}}{32\pi^{2}}\int_{-\infty}^{\infty}\mathrm{d}s\>e^{-i\Omega s}\left\{\frac{1}{\sinh^{2}\big{(}\kappa s/2-i\varepsilon\big{)}}+\frac{1}{\sinh^{2}\big{(}\kappa s/2-i\varepsilon\big{)}-\big{(}\kappa\mathcal{L}_{S}/2\big{)}^{2}}\right\}.$

(75)

The first integral of Eq. (75) gives the thermal response function of a single static trajectory while the second can be evaluated using the residue theorem (see Appendix A). Performing this calculation yields

$\displaystyle\mathcal{F}_{\mathcal{L}_{S}}^{\sigma\to\infty}(\Omega)$

$\displaystyle=\frac{\Omega}{4\pi}\frac{1}{e^{2\pi\Omega/\kappa}-1}\left\{1+\frac{\sin\big{(}2\Omega\kappa^{-1}\operatorname{arcsinh}\big{(}\mathcal{L}_{S}\kappa/2\big{)}\big{)}}{\mathcal{L}_{S}\Omega\sqrt{1+\big{(}\mathcal{L}_{S}\kappa/2\big{)}^{2}}}\right\}.$

(76)

In Eq. (76), the response function has a local contribution from the individual trajectories of the superposition (yielding the familiar Planck distribution) and a nonlocal interference term between the trajectories. In the limit where the superposition distance vanishes, the interference term smoothly approaches the local term,

$\displaystyle\lim_{\mathcal{L}_{S}\to 0}\mathcal{F}_{\mathcal{L}_{S}}^{\sigma\to\infty}(\Omega)$

$\displaystyle=\frac{\Omega}{2\pi}\frac{1}{e^{2\pi\Omega/\kappa}-1}.$

(77)

As $R_{D}$ approaches the cosmological horizon, the acceleration grows unbounded and the interference term vanishes. In this limit the Gibbons-Hawking temperature diverges, and the response function grows unbounded with the temperature of the field,

$\displaystyle\lim_{R_{D}\to 1/l}\mathcal{F}^{\sigma\to\infty}_{\mathcal{L}_{S}}(\Omega)$

$\displaystyle=\frac{\kappa}{8\pi^{2}}.$

(78)

Even though the trajectories are separated by a finite Euclidean distance, any interference effects between them are washed out by the amplification of thermal particle production as the detector approaches the horizon. This leaves the response function in a classical mixture of contributions from the individual amplitudes of the superposition. In Fig. 2, the detector response is plotted as a function of the energy gap, for various angular separations. For $\theta_{S}=0$, we recover the Planckian spectrum of Eq. (77), which grows linearly for increasing negative energy gaps (corresponding to a detector prepared in its excited state). As one introduces the spatial superposition, the response function oscillates with $\Omega/l$.

In Fig. 3 and 4, we have plotted the detector response as a function of the angular separation of the superposition, for negative and positive energy gaps respectively. Since nonlocal correlations generally decrease with distance, we find likewise that the total response of the detector decays with the superposition distance.

The response function also satisfies the KMS detailed balance condition at temperature $T_{\text{dS}}=\kappa(2\pi)^{-1}$ for all $\mathcal{L}_{S}$,

$\displaystyle\frac{\mathcal{F}_{\mathcal{L}_{S}}(\Omega)}{\mathcal{F}_{\mathcal{L}_{S}}(-\Omega)}=e^{-2\pi\Omega/\kappa}.$

(79)

This contrasts with results found in Foo et al. (2020a), where a UdW detector was superposed along accelerated trajectories translated in the plane of motion, and the response was found in general to be nonthermal. This was due to the asymmetric causal structure between the trajectories, leading to time-dependent interference effects that perturbed the quantum field away from thermalisation. For static worldlines in de Sitter spacetime, the causal symmetry of the trajectories nullifies these effects, allowing the detector to thermalise exactly.

Interestingly, there is a functional equivalence between Eq. (76) and the response of a detector traveling in a superposition of accelerated trajectories in Minkowski spacetime, where the translation occurs along an axis orthogonal to the direction of motion. That is, for superposed worldlines in Minkowski spacetime defined by

$\displaystyle\begin{split}Z_{0}^{(A)}&=\kappa_{M}^{-1}\sinh(\kappa_{M}\tau)\\ Z_{1}^{(A)}&=\kappa_{M}^{-1}\cosh(\kappa_{M}\tau)\\ Z_{2}^{(A)}&=0\vphantom{\kappa_{M}^{-1}}\end{split}\begin{split}Z_{0}^{(B)}&=\kappa_{M}^{-1}\sinh(\kappa_{M}\tau^{\prime})\\ Z_{1}^{(B)}&=\kappa_{M}^{-1}\cosh(\kappa_{M}\tau^{\prime})\\ Z_{2}^{(B)}&=\mathcal{L}_{M}\vphantom{\kappa_{M}^{-1}}\end{split}$

(80)

and all other spatial coordinates equal to zero, the response function is given by (using similar techniques as the de Sitter case)

$\displaystyle\mathcal{F}^{\sigma\to\infty}_{R}(\Omega)$

$\displaystyle=\frac{\Omega}{4\pi}\frac{1}{e^{2\pi\Omega/\kappa_{M}}-1}\left\{1+\frac{\sin\big{(}2\Omega\kappa_{M}^{-1}\operatorname{arcsinh}\big{(}\mathcal{L}_{M}\kappa_{M}/2\big{)}\big{)}}{\mathcal{L}_{M}\Omega\sqrt{1+\big{(}\mathcal{L}_{M}\kappa_{M}/2\big{)}^{2}}}\right\}.$

(81)

The only difference between the response functions is that $\kappa_{M}$ is now the proper acceleration of the superposed detector trajectories in flat spacetime while $\mathcal{L}_{M}$ is still the Euclidean distance separating the trajectories. This means that the two spacetimes are operationally indistinguishable given the replacement $\kappa\Rightarrow\kappa_{M}$ and $\mathcal{L}_{S}\Rightarrow\mathcal{L}_{M}$. This correspondence reflects the conformal relationship between the Rindler wedge and static de Sitter spacetime, recalling that detectors on classical trajectories respond identically to the conformally coupled vacuum in static de Sitter space compared with uniformly accelerated detectors in the Minkowski vacuum.

The main difference between the two cases is that in Eq. (81), the Euclidean distance between the accelerated trajectories, $\mathcal{L}_{M}$, takes on values on the full real line, so the interference vanishes when the trajectories are infinitely separated. This implies what we refer to as a quantitative equivalence between the experience of detectors in these spacetimes. That is, for a particular subset of the $(l,R_{D},\theta_{S},\mathcal{L}_{M})$ parameter space, the constraints

	$\displaystyle\mathcal{\kappa}$	$\displaystyle=\frac{1}{\sqrt{l^{-2}-R_{D}^{2}}}=\kappa_{M}$		(82)
	$\displaystyle\mathcal{L}_{S}$	$\displaystyle=2R_{D}\sin\left(\frac{\theta_{S}}{2}\right)=\mathcal{L}_{M}$		(83)

are satisfied exactly. In this case, the two spacetimes induce identical physical effects (i.e. the detected rate of particle production from the respective vacua) within UdW detectors. Such a correspondence can be considered to be a special case of the functional equivalence explained previously. A similar equivalence has been pointed out for the entanglement properties of two comoving detectors in de Sitter spacelike and two uniformly accelerating detectors Salton et al. (2015).

Next, we consider the detector traveling in a background spacetime in a quantum superposition of de Sitter curvatures. Our approach is purely phenomenological, in that our results are obtained by simply calculating the Wightman functions with respect to fields quantised on spacetime manifolds with two different (superposed) values of the de Sitter curvature. In our interpretation, there is still only one spacetime, now in a superposition of semiclassical parameters (i.e. the de Sitter curvature). Hence we expect our method for calculating detector observables in such spacetimes, which reduces to calculating the Wightman functions pulled back to the trajectories on these two manifolds, is not limited to scenarios involving a higher-dimensional embedding space. A conceptually helpful example is a black hole in a superposition of masses; even though the individual branches of the superposition correspond to unique solutions to Einstein’s field equations, the detector still resides in a single spacetime, despite perceiving the Riemannian curvature to be superposed at each point in space. Finally, we do not posit the mechanism for how these kinds of superpositions are generated; however it is expected that a theory of quantum gravity should admit such solutions.