Correlation harvesting between particle detectors in uniform motion

Let us first review the correlation harvesting protocol. Consider two pointlike UDW detectors A and B with an energy gap $\Omega_{j},\,j\in\{\text{A},\text{B}\}$ between ground $\ket{g_{j}}$ and excited states $\ket{e_{j}}$. These detectors interact with the quantum Klein-Gordon field $\hat{\phi}$ along their trajectories $\mathsf{x}_{j}(\tau_{j})=(t(\tau_{j}),{\bm{x}}(\tau_{j}))$, where $\tau_{j}$ is the proper time of detector-$j$.

In the interaction picture, the interaction Hamiltonian (as a generator of time-translation with respect to $\tau_{j}$) describing the coupling between detector-$j$ and $\hat{\phi}$ is given by

$\displaystyle\hat{H}_{j}^{\tau_{j}}(\tau_{j})$

$\displaystyle=\lambda_{j}\chi_{j}(\tau_{j})\hat{\mu}_{j}(\tau_{j})\otimes\hat{\phi}(\mathsf{x}_{j}(\tau_{j}))\,,~j\in\{\text{A},\text{B}\}$

(2)

where $\lambda_{j}$ is a coupling constant and $\chi_{j}(\tau_{j})$ is the switching function that governs the time-dependence of the coupling. Here, $\hat{\mu}_{j}(\tau_{j})$ is the monopole moment given by

$\displaystyle\hat{\mu}_{j}(\tau_{j})$

$\displaystyle=\ket{e_{j}}\bra{g_{j}}e^{\mathsf{i}\Omega_{j}\tau_{j}}+\ket{g_{j}}\bra{e_{j}}e^{-\mathsf{i}\Omega_{j}\tau_{j}}\,,$

(3)

which describes each detector’s internal dynamics, and the field operator $\hat{\phi}(\mathsf{x}_{j}(\tau_{j}))$ is pulled back along the trajectory of detector-$j$. The superscript on $\hat{H}_{j}^{\tau_{j}}(\tau_{j})$ indicates the time-translation that the Hamiltonian is generating.

The total interaction Hamiltonian, $\hat{H}_{\text{I}}^{t}(t)$, can be written as a generator of time-translation with respect to the time $t$ that is common to both detectors:

$\displaystyle\hat{H}_{\text{I}}^{t}(t)$

$\displaystyle=\dfrac{\text{d}\tau_{\text{A}}}{\text{d}t}\hat{H}_{\text{A}}^{\tau_{\text{A}}}\big{(}\tau_{\text{A}}(t)\big{)}+\dfrac{\text{d}\tau_{\text{B}}}{\text{d}t}\hat{H}_{\text{B}}^{\tau_{\text{B}}}\big{(}\tau_{\text{B}}(t)\big{)}\,,$

(4)

Note that the proper times $\tau_{\text{A}}$ and $\tau_{\text{B}}$ are each now functions of $t$. From this Hamiltonian, one obtains the time-evolution operator $\hat{U}_{\text{I}}$ [55, 56]:

$\displaystyle\hat{U}_{\text{I}}$

$\displaystyle=\mathcal{T}_{t}\exp\left(-\mathsf{i}\int_{\mathbb{R}}\text{d}t\,\hat{H}_{\text{I}}^{t}(t)\right)\,,$

(5)

where $\mathcal{T}_{t}$ is a time-ordering symbol with respect to the common time $t$.

One can then use a perturbative analysis and obtain the final density matrix of a joint system $\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{B}}$, where $\mathcal{H}_{j}$ is a Hilbert space for detector-$j$. Assuming a small coupling strength, $\lambda\ll 1$, the Dyson series expansion of $\hat{U}_{\text{I}}$ reads

	$\displaystyle\hat{U}_{\text{I}}$	$\displaystyle=\mathds{1}+\hat{U}_{\text{I}}^{(1)}+\hat{U}_{\text{I}}^{(2)}+\mathcal{O}(\lambda^{3})\,,$		(6a)
	$\displaystyle\hat{U}_{\text{I}}^{(1)}$	$\displaystyle=-\mathsf{i}\int_{-\infty}^{\infty}\text{d}t\,\hat{H}_{\text{I}}^{t}(t)\,,$		(6b)
	$\displaystyle\hat{U}_{\text{I}}^{(2)}$	$\displaystyle=-\int_{-\infty}^{\infty}\text{d}t_{1}\int_{-\infty}^{t_{1}}\text{d}t_{2}\,\hat{H}_{\text{I}}^{t}(t_{1})\hat{H}_{\text{I}}^{t}(t_{2})\,.$		(6c)

By assuming that the initial state, $\rho_{0}$, of the detectors-field system is

$\displaystyle\rho_{0}$

$\displaystyle=\ket{g_{\text{A}}}\bra{g_{\text{A}}}\otimes\ket{g_{\text{B}}}\bra{g_{\text{B}}}\otimes\ket{0}\bra{0}\,,$

(7)

where $\ket{0}$ is the vacuum state of the field, one finds the final total density matrix $\rho_{\text{tot}}$ after the interaction to be

	$\displaystyle\rho_{\text{tot}}$	$\displaystyle=\hat{U}_{\text{I}}\rho_{0}\hat{U}_{\text{I}}^{\dagger}$
		$\displaystyle=\rho_{0}+\rho^{(1,1)}+\rho^{(2,0)}+\rho^{(0,2)}+\mathcal{O}(\lambda^{4})\,,$		(8)

where $\rho^{(i,j)}=\hat{U}^{(i)}\rho_{0}\hat{U}^{(j)\dagger}$ and all the odd-power terms of $\lambda$ vanish [7]. Then the final density matrix of the detectors, $\rho_{\text{AB}}$, is obtained by tracing out the field part: $\rho_{\text{AB}}=\operatorname{Tr}_{\phi}[\rho_{\text{tot}}]$. By employing the basis $\ket{g_{\text{A}}g_{\text{B}}}=[1,0,0,0]^{\top},\ket{g_{\text{A}}e_{\text{B}}}=[0,1,0,0]^{\top},\ket{e_{\text{A}}g_{\text{B}}}=[0,0,1,0]^{\top},\ket{e_{\text{A}}e_{\text{B}}}=[0,0,0,1]^{\top}$, the density matrix $\rho_{\text{AB}}$ reads

$\displaystyle\rho_{\text{AB}}$

$\displaystyle=\left[\begin{array}[]{cccc}1-\mathcal{L}_{\text{AA}}-\mathcal{L}_{\text{BB}}&0&0&\mathcal{M}^{*}\\ 0&\mathcal{L}_{\text{BB}}&\mathcal{L}_{\text{AB}}^{*}&0\\ 0&\mathcal{L}_{\text{AB}}&\mathcal{L}_{\text{AA}}&0\\ \mathcal{M}&0&0&0\end{array}\right]+\mathcal{O}(\lambda^{4})\,,$

(13)

where

	$\displaystyle\mathcal{L}_{ij}$	$\displaystyle=\lambda^{2}\int_{\mathbb{R}}\text{d}\tau_{i}\int_{\mathbb{R}}\text{d}\tau_{j}^{\prime}\,\chi_{i}(\tau_{i})\chi_{j}(\tau_{j}^{\prime})e^{-\mathsf{i}\Omega(\tau_{i}-\tau_{j}^{\prime})}$
		$\displaystyle\qquad\qquad\qquad\qquad\times W\big{(}\mathsf{x}_{i}(\tau_{i}),\mathsf{x}_{j}(\tau_{j}^{\prime})\big{)}\,,$		(14a)
	$\displaystyle\mathcal{M}$	$\displaystyle=-\lambda^{2}\int_{\mathbb{R}}\text{d}\tau_{\text{A}}\int_{\mathbb{R}}\text{d}\tau_{\text{B}}\,\chi_{\text{A}}(\tau_{\text{A}})\chi_{\text{B}}(\tau_{\text{B}})e^{-\mathsf{i}\Omega(\tau_{\text{A}}+\tau_{\text{B}})}$
		$\displaystyle\hskip 14.22636pt\times\big{[}\Theta\big{(}t(\tau_{\text{A}})-t(\tau_{\text{B}})\big{)}W\big{(}\mathsf{x}_{\text{A}}(\tau_{\text{A}}),\mathsf{x}_{\text{B}}(\tau_{\text{B}})\big{)}$
		$\displaystyle\hskip 28.45274pt+\Theta\big{(}t(\tau_{\text{B}})-t(\tau_{\text{A}})\big{)}W\big{(}\mathsf{x}_{\text{B}}(\tau_{\text{B}}),\mathsf{x}_{\text{A}}(\tau_{\text{A}})\big{)}\big{]}\,,$		(14b)

where $\Theta(t)$ is the Heaviside step function and $W(\mathsf{x},\mathsf{x}^{\prime})\coloneqq\bra{0}\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{x}^{\prime})\ket{0}$ is the vacuum Wightman function. In $(3+1)$-dimensional Minkowski spacetime, the Wightman function reads

$\displaystyle W(\mathsf{x},\mathsf{x}^{\prime})$

$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{(t-t^{\prime}-\mathsf{i}\epsilon)^{2}-({\bm{x}}-{\bm{x}}^{\prime})^{2}}\,,$

(15)

where $\epsilon$ is the UV cutoff. The elements $\mathcal{L}_{jj},\,j\in\{\text{A},\text{B}\}$ are the so-called transition probabilities (or response functions), which describe the probability of a detector transitioning from the ground to excited states, $\ket{g_{j}}\to\ket{e_{j}}$. The off-diagonal elements $\mathcal{M}$ and $\mathcal{L}_{\text{AB}}$ are responsible for harvesting entanglement and quantum mutual information, respectively, as we shall see in the next subsection.

Throughout this paper, we use a Gaussian switching function

$\displaystyle\chi_{j}(\tau_{j})$

$\displaystyle=e^{-\tau_{j}^{2}/2\sigma^{2}}\,,$

(16)

where $\sigma>0$ is the characteristic Gaussian width, which has the units of time. We will use $\sigma$ to make all quantities unitless (such as $\Omega\sigma$).

Let us introduce two measures for correlation: concurrence $\mathcal{C}_{\text{AB}}$ and quantum mutual information $I_{\text{AB}}$.

Concurrence is a measure of entanglement [57, 58]. Let $\rho_{\text{AB}}$ be the density matrix of a two-qubit system. We first define a matrix $\tilde{\rho}_{\text{AB}}$ as

$\displaystyle\tilde{\rho}_{\text{AB}}$

$\displaystyle\coloneqq(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\rho_{\text{AB}}^{*}(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\,,$

(17)

where $\hat{\sigma}_{y}$ is the Pauli-$y$ operator and $\rho_{\text{AB}}^{*}$ is the complex conjugate of $\rho_{\text{AB}}$. Then by denoting $w_{i}\in\mathbb{R}$, $(i=1,2,3,4)$ as eigenvalues of a Hermitian operator $\sqrt{\sqrt{\rho_{\text{AB}}}\tilde{\rho}_{\text{AB}}\sqrt{\rho_{\text{AB}}}}$, the concurrence is defined as follows.

	$\displaystyle\mathcal{C}_{\text{AB}}$	$\displaystyle\coloneqq\max\{0,\,w_{1}-w_{2}-w_{3}-w_{4}\}\,,$		(18)
		$\displaystyle\hskip 42.67912pt(w_{1}\geq w_{2}\geq w_{3}\geq w_{4})\,.$

The concurrence is zero if and only if the state $\rho_{\text{AB}}$ is separable. In the case of our density matrix (13), the concurrence is known to be

$\displaystyle\mathcal{C}_{\text{AB}}$

$\displaystyle=2\max\{0,\,|\mathcal{M}|-\sqrt{\mathcal{L}_{\text{AA}}\mathcal{L}_{\text{BB}}}\}+\mathcal{O}(\lambda^{4})\,.$

(19)

Quantum mutual information [59], on the other hand, quantifies the amount of total correlation, both classical and quantum. Quantum mutual information $I_{\text{AB}}$ between two qubits A and B up to second order in $\lambda$ is [7]

	$\displaystyle I_{\text{AB}}$	$\displaystyle=\mathcal{L}_{+}\ln\mathcal{L}_{+}+\mathcal{L}_{-}\ln\mathcal{L}_{-}$
		$\displaystyle\hskip 14.22636pt-\mathcal{L}_{\text{AA}}\ln\mathcal{L}_{\text{AA}}-\mathcal{L}_{\text{BB}}\ln\mathcal{L}_{\text{BB}}+\mathcal{O}(\lambda^{4})\,,$		(20)

where

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

$\displaystyle\coloneqq\dfrac{1}{2}\left(\mathcal{L}_{\text{AA}}+\mathcal{L}_{\text{BB}}\pm\sqrt{(\mathcal{L}_{\text{AA}}-\mathcal{L}_{\text{BB}})^{2}+4|\mathcal{L}_{\text{AB}}|^{2}}\right).$

(21)

Note that, while concurrence (19) vanishes when the “noise term” $\sqrt{\mathcal{L}_{\text{AA}}\mathcal{L}_{\text{BB}}}$ exceeds the nonlocal element $|\mathcal{M}|$, the mutual information becomes zero when $|\mathcal{L}_{\text{AB}}|=0$. In addition, if $\mathcal{C}_{\text{AB}}=0$ but the mutual information is nonvanishing, then the extracted correlation by the detectors is either classical correlation or nondistillable entanglement.

The most well-known trajectory for a uniformly accelerating (i.e., $a=const.$) pointlike particle is linear accelerated motion. However, Letaw pointed out that there are, in fact, five classes of uniformly accelerated trajectories, excluding the case where $a=0$. Along with the linear case, the other classes are circular, catenary, cusped, and helix [13]. Consider a trajectory in $(3+1)$-dimensional Minkowski spacetime. Such a trajectory can be characterized by three geometric invariants: the curvature $a(\tau)$, which represents the magnitude of proper acceleration, the first torsion $b(\tau)$, and the second torsion (also known as hypertorsion) $\nu(\tau)$ of the worldline. The torsions $b(\tau)$ and $\nu(\tau)$ correspond to the proper angular velocities in a given tetrad frame [13]. Assuming that these invariants are constants, the trajectory becomes stationary. In a nutshell, these motions are characterized by the following:

1. 1.

   linear: $a\neq 0,b=\nu=0$

2. 2.

   catenary: $a>b$, $\nu=0$

3. 3.

   cusped: $a=b,\nu=0$

4. 4.

   circular: $a<b,\nu=0$

5. 5.

   helix: $\nu\neq 0$

In this subsection, we review these trajectories and consider the corresponding vacuum Wightman functions. We will suppress the UV cutoff $\epsilon$ for readability.

We now consider two UDW detectors A and B, both undergoing uniform acceleration motion. In particular, we categorize the detector configurations into two classes: stationary (time-translation invariant) and nonstationary scenarios.

Let us begin by considering the transition probability $\mathcal{L}_{jj}$ for a uniformly accelerating detector. We are particularly interested in the cases of linear ($\bar{b}=0$), catenary ($0<\bar{b}<1$), cusped ($\bar{b}=1$), and circular ($\bar{b}>1$) motions, and their respective transition probabilities are given by (35). We consider $\mathcal{L}_{jj}/\lambda^{2}$ and write the parameters in units of $\sigma$, which makes the transition probability a function of three variables: $a\sigma,\bar{b}$, and $\Omega\sigma$. It is important to note that $\mathcal{L}_{\text{AA}}=\mathcal{L}_{\text{BB}}$, as we are assuming both detectors are identical.

Figure 3 depicts the transition probability $\mathcal{L}_{jj}/\lambda^{2}$ as a function of the magnitude of acceleration $a\sigma$ for fixed $\Omega$ (panel (a)) and $\log_{10}\bar{b}$ for different values of the acceleration (panels (b), (c)). In figure 3(a), the transition probabilities for a detector with $\Omega\sigma=2$ in linear ($\bar{b}=0$), catenary ($\bar{b}=0.5$), cusped ($\bar{b}=1$), and circular ($\bar{b}=2$) motions are shown. We find that in all these cases, $\mathcal{L}_{jj}/\lambda^{2}$ increases with the acceleration $a\sigma$.¹¹1$\mathcal{L}_{jj}/\lambda^{2}$ is not guaranteed to monotonically increase with $a\sigma$ for a finite interaction duration. For a detector in the linearly accelerated motion, such phenomenon is known as the (weak) anti-Unruh effect [60, 61].

However, the relationship between the transition probabilities of detectors in different uniform motions is highly nontrivial. For instance, when a detector has $\Omega\sigma=2$ and $a\sigma\lesssim 5$, as depicted in figure 3(a), a detector in circular motion with $\bar{b}=2$ shows the largest value of $\mathcal{L}_{jj}/\lambda^{2}$, whereas a detector in linear motion ($\bar{b}=0$) shows the smallest. This relation, however, flips for $a\sigma\gtrsim 5$. We will numerically demonstrate that such relationships depend on the interplay between $a\sigma$ and $\Omega\sigma$.

In figure 3(b), the magnitude of the acceleration is fixed at $a\sigma=1$, and $\mathcal{L}_{jj}/\lambda^{2}$ is plotted as a function of $\log_{10}\bar{b}$. Each curve in this figure corresponds to a different value of $\Omega\sigma$, with the curve for $\Omega\sigma=2$ corresponding to figure 3(a) at $a\sigma=1$. For each value of $\Omega\sigma$ in 3(b), the transition probability has a peak for $\log_{10}\bar{b}>0$ (i.e., $\bar{b}>1$), and then decreases with increasing $\log_{10}\bar{b}$, becoming smaller than the value for the linear case ($\log_{10}\bar{b}\to-\infty$). This means that $\mathcal{L}_{jj}/\lambda^{2}$ at $a\sigma=1$ in figure 3(a) increases with $\bar{b}$ until it reaches a maximum and then decreases. We note that the presence of the peak is contingent on larger values of $\Omega\sigma$ relative to $a\sigma$; In fact, the peak does not appear for smaller energy gaps, in which case the transition probability monotonically decreases with $\bar{b}$, as shown in figure 3(b). This trend is further illustrated in figure 3(c), where $a\sigma=6$ is chosen. In this scenario, the peak is nonexistent for $\Omega\sigma=1$ and 1.5 (as well as for $\Omega\sigma<1$), but becomes manifest when $\Omega\sigma\gtrsim 2$. Thus we infer that detectors with smaller energy gaps $\Omega\sigma$ compared to $a\sigma$ do not have a peak in $\mathcal{L}_{jj}(\bar{b})/\lambda^{2}$.

The behavior of $\mathcal{L}_{jj}/\lambda^{2}$ is related to the concept of the “effective temperature” perceived by a detector. For now, let us denote the transition probability as $\mathcal{L}_{jj}(\Omega,\sigma)$. The effective temperature, $T_{\text{eff}}$, is defined as

$\displaystyle T_{\text{eff}}^{-1}$

$\displaystyle\coloneqq\dfrac{1}{\Omega}\ln\dfrac{\mathcal{L}_{jj}(-\Omega,\sigma)/\lambda^{2}\sigma}{\mathcal{L}_{jj}(\Omega,\sigma)/\lambda^{2}\sigma}\,$

(49)

where this formula is derived in the Appendix. We divide $\mathcal{L}_{jj}(\Omega,\sigma)$ by $\sigma$ so that it is well defined in the long interaction limit, $\sigma\to\infty$ [62, 61]. Note that if the Wightman function obeys the Kubo-Martin-Schwinger (KMS) condition [63, 64], then the effective temperature converges to the KMS temperature (which is the temperature of the field formally defined in quantum field theory) in the limit $\sigma\to\infty$. However, in the case of finite interaction duration, the effective temperature is an estimator for the actual field temperature. For a detector in a uniform acceleration motion, the effective temperature for each scenario has been examined in, e.g., [14, 15, 16, 17, 18, 19, 20].

We plot the effective temperature $T_{\text{eff}}$ as a function of $\log_{10}\bar{b}$ when $\sigma=1$ and $a\sigma=1$ in figure 3(d), which corresponds to figure 3(b). We see that the locations of the peaks in $T_{\text{eff}}$ align with those of $\mathcal{L}_{jj}(\Omega)$ in 3(b). This suggests that, for a given acceleration and energy gap, a detector in circular motion within a certain range of $\log_{10}\bar{b}$ can register higher effective temperatures than those in other types of motion. However, as $\bar{b}\to\infty$, which corresponds to the speed of a detector in circular motion with $v_{\text{circ}}(=\bar{b}^{-1})\to 0$, the temperature becomes colder.

We now move on to the correlation harvesting protocol using two uniformly accelerating detectors, exploring both stationary and nonstationary configurations as described in section III.2.

We first examine the difference between the stationary and nonstationary configurations by plotting concurrence $\mathcal{C}_{\text{AB}}/\lambda^{2}$ and quantum mutual information $I_{\text{AB}}/\lambda^{2}$ as a function of $\bar{b}$ in figure 4. In these plots, we fix $\Omega\sigma=0.1$ and $L/\sigma=1$, and consider $a\sigma=1$ and $a\sigma=2$. We notice two characteristics: (i) In the vicinity of $\bar{b}\approx 0$, stationary detectors consistently harvest greater correlations than nonstationary detectors, for both concurrence and mutual information. (ii) As $\bar{b}$ becomes larger, both plots begin to oscillate with $\bar{b}$, and the curve representing correlations harvested by nonstationary detectors oscillates around the curve for the stationary case. The frequency of the oscillation increases as $a\sigma$ grows.

These observations can be traced back to the form of the Wightman functions (41) and (48). Let us recall that the denominators of these expressions contain $\sinh(x)$ when $\bar{b}\in[0,1)$ and transform into $\sin(x)$ when $\bar{b}>1$. Therefore, within the range $\bar{b}\in[0,1)$, the correlations are characterized by an exponential pattern, while for $\bar{b}>1$, an oscillatory behavior emerges. These traits explain the observation above. In particular, the suppression of correlations near $\bar{b}\approx 0$ for nonstationary detectors can be attributed to an additional term in the denominator of (48), which is absent in the stationary Wightman function (41). This extra term diminishes the amount of harvested correlations relative to the stationary scenario, and simultaneously gives rise to the oscillations noticed in the nonstationary case around the stationary one.