Entangled states dynamics of moving two-level atoms in a thermal field bath

We consider a massless scalar quantum field $\hat{\phi}(\mathsf{x})$ in the (3+1)-dimensional Minkowski spacetime with metric $\eta_{\mu\nu}$. The scalar field satisfies the Klein-Gordon equation $\square\hat{\phi}(\mathsf{x})=0$, where $\square=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}=-\partial_{t}^{2}+\nabla^{2}$ is the d'Alembert operator. In terms of plane wave solutions to the Klein-Gordon equation the field can be expressed as

where $\hat{a}_{\boldsymbol{k}}$ and $\hat{a}_{\boldsymbol{k}}^{\dagger}$ are the annihilation and creation operators of the field mode with momentum $\boldsymbol{k}$. They satisfy the canonical commutation relations

The free Hamiltonian of the field, after subtracting the infinite zero-point energy, reads

We consider an Unruh-DeWitt (UDW) particle detector [14, 15, 17, 16] modeled as a pointlike two-level atom (qubit) with energy gap $\omega$ and Hamiltonian

where $\hat{\sigma}_{3}$ is the usual Pauli operator. The detector moves along a worldline $\mathsf{x}(\tau)=(t(\tau),\mathbf{x}(\tau))$ parametrized by its proper time $\tau$, while interacting with the scalar field through the interaction Hamiltonian

where $\lambda(\tau)$ describes how the coupling between the detector and the field is switched on and off,

is the detector's monopole moment operator expressed in terms of the SU(2) ladder operators $\hat{\sigma}_{\pm}$, $\hat{\phi}(\tau):=\hat{\phi}(\mathsf{x}(\tau))$ is the scalar field operator evaluated along the detector's trajectory, and $p$ is a non-negative integer. We will next consider a sudden switching $\lambda(\tau)=\lambda\theta(\tau)$, where $\lambda$ specifies the detector-field coupling strength and $\theta(\tau)$ is the Heaviside step function.

In this work, we will focus on the cases: (i) $p=0$, which refers to the standard UDW detector model, and (ii) $p=1$, which corresponds to a derivative coupling detector model [41, 42, 43, 44, 45] that linearly couples the atom with the proper-time derivative of the quantum field. The derivative coupling model closely resembles the dipole interaction $\hat{\mathbf{d}}\cdot\hat{\mathbf{E}}$ between an atom with dipole moment $\hat{\mathbf{d}}$ and an external electromagnetic field $\hat{\mathbf{E}}(t,\boldsymbol{x})=-\partial_{t}\hat{\mathbf{A}}(t,\boldsymbol{x})$ expressed in terms of the vector potential $\hat{\mathbf{A}}$ (in the Coulomb gauge) [47].

In the followings, we will consider that the detector is moving with a constant velocity $\upsilon$ along the x-axis, i.e, following the trajectory [17]

where $\gamma=1/\sqrt{1-\upsilon^{2}}$ is the Lorentz factor.

The interaction Hamiltonian (5) constitutes a special case of the spin-boson model Hamiltonian [46]. This implies considering the UDW detector as an open quantum system with the scalar field playing the role of the environment that induces dissipation and decoherence. We suppose that initially the field is in a thermal state

at temperature $T=\beta^{-1}$, and that no correlations between the detector and the field environment are present, i.e., $\hat{\rho}_{\text{tot}}(0)=\hat{\rho}(0)\otimes\hat{\rho}_{\beta}$.

The dynamics of the density operator of the combined detector-field system in the interaction picture is described by the Liouville-von Neumann equation

where $[\cdot,\cdot]$ stands for the commutator. We can formally solve Eq. (9) by integration to obtain

Inserting then the integral form (10) back into (9) and taking the partial trace over the field bath degrees of freedom yields an integro-differential equation for the reduced density matrix of the detector

We next successively employ: (i) the Born approximation, that is we assume a weak coupling between the detector and the field so that the combined state $\hat{\rho}_{\text{tot}}(s)\simeq\hat{\rho}(s)\otimes\hat{\rho}_{\beta}$ remains approximately factorized at all times, and (ii) the Markov approximation, by which we replace $\hat{\rho}(s)$ by $\hat{\rho}(\tau)$ in the integral in (11) and extend the upper limit of integration to infinity. The Markov approximation is valid provided that the detector's time evolution is much slower than that of the field bath [48]. After performing a change of variables $s\to\tau-s$ we obtain the Markovian master equation

In terms of the interaction Hamiltonian (5) the above master equation takes the form [49, 50]

where

is the Wightman two-point correlation function of the field evaluated along the detector's trajectory; for convenience we have set $\hat{\varphi}(\tau)=\frac{d^{p}}{d\tau^{p}}\hat{\phi}(\tau)$. We note that when the state of the field is stationary and the detector follows a stationary spacetime trajectory [51], such is the one with constant velocity, the Wightman function depends only on the proper time deference $\tau-\tau^{\prime}$ between the two points on the detector's worldline and it can be expressed as $\mathcal{W}(\tau,\tau^{\prime})=\mathcal{W}(\tau-\tau^{\prime})$.

In the case of the standard UDW coupling, the Wightman function pull-backed to the inertial trajectory (7) followed by the detector takes the form (see Appendix A for details)

where $n_{k}=(e^{\beta|\boldsymbol{k}|}-1)^{-1}$ denotes the Planck distribution, and the limit $\epsilon\to 0^{+}$ is understood in the distributional sense. The Wightman function (II.2) is stationary. The first term is independent of the temperature and corresponds to the vacuum two-point correlation function of the field. The second term incorporates the thermal effects of the field environment. On the other hand, in the case of the time-derivative coupling the Wightman function is

Markov approximation implies that the correlation function of the environment decays rapidly with respect to some characteristic timescale $\tau_{c}$. For an atom at rest, with worldline $\mathsf{x}(\tau)=(\tau,0,0,0)$, and interacting with the thermal field bath, this timescale is determined by $\tau_{c}=T^{-1}$. The correlation function is expressed as (see Appendix A)

exhibiting a sharp peak around $s=0$ and decaying exponentially as $s/\tau_{c}>>1$, i.e., $\mathcal{W}(s)\sim e^{-s/\tau_{c}}$. Thus, as commonly understood [46, 48], the Markov approximation fails to hold at low temperatures of the environment. Besides, as demonstrated in [52], an atom moving inertially at a constant speed $\upsilon$ and interacting with the thermal field bath, experiences a continuum of thermal baths with temperatures $T^{\prime}$ within the range

Consequently, in the ultra-relativistic limit $\upsilon\to 1$, the moving atom can perceive the field bath as if it were in its vacuum (zero-temperature) state. This observation is also supported by the correlation function Eq. (II.2) (or equivalently Eq. (A)). For a fixed temperature, the correlation function $\mathcal{W}(s)$ remains sharply peaked around $s=0$ and exponentially decays with respect to the temperature. In the ultra-relativistic limit $\upsilon\to 1$, the term blue-shifted by the Doppler factor $\sqrt{\frac{1+\upsilon}{1-\upsilon}}$ exponentially decays as $\sim\exp^{-T\sqrt{\frac{1+\upsilon}{1-\upsilon}}s}$, but instead the term red-shifted by the factor $\sqrt{\frac{1-\upsilon}{1+\upsilon}}$ behaves as $\sim\frac{1}{T\sqrt{\frac{1-\upsilon}{1+\upsilon}}s}$. Therefore, we can conclude that in the ultra-relativistic regime the Markov approximation ceases to hold and a more thorough investigation that takes into account memory effects is required.

Calculating the Fourier transform time integrals in (13), employing the (post-trace) rotating wave approximation (RWA) [53, 54] by which the rapidly oscillating terms of the form $\hat{\sigma}_{+}\hat{\rho}\hat{\sigma}_{+}e^{2i\omega\tau}$ and $\hat{\sigma}_{-}\hat{\rho}\hat{\sigma}_{-}e^{-2i\omega\tau}$ are ignored, and lastly transforming back to the Schrödinger picture, we can write (see Ref. [52] for a detailed derivation) the master equation in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) [55, 56] form

where $\{\cdot,\cdot\}$ stands for the anticommutator, we have defined $\Omega\equiv\omega+\Delta\omega$ with $\Delta\omega$ being the frequency (Lamb) shift induced by the field environment, and $\Gamma$ is the decay constant of the qubit detector in the vacuum. We note that the evolution map $\Phi:\rho(0)\mapsto\rho(\tau)$ that the GKSL master equation generates is a completely positive and trace-preserving (CPTP) map [46].

In the case of the UDW coupling the decay coefficient reads

whereas when considering the derivative coupling it takes the form

Note that when the detector is static ($\upsilon=0$), the UDW coupling and derivative coupling cases produce equivalent expressions for the decay constant $\Gamma$ (with the exception of an additional factor of $\omega^{2}$ arising from the derivative nature of the latter coupling). Nonetheless, this is not the case when the detector is in constant motion ($\upsilon\neq 0$). Accordingly, the coefficient $N$ is equal to

in the case of the standard UDW coupling, and equal to

in the case of the derivative coupling, where the function

is expressed in terms of the polylogarithm function [57]

which appears in the Bose-Einstein integral

Note that in the static detector limit $\upsilon\to 0$ the usual thermal Planck distribution

that describes the mean number of field modes is recovered in both coupling cases. This implies that the atom's motion changes the way it experiences the Planckian spectrum of the background thermal environment, causing it to respond differently to the black body spectrum.

We do not report here the expressions for the frequency shift $\Delta\omega$, as its exact form is not involved in our later calculations and results (see, e.g., Eq. (36)). The analytic expressions for the Lamb shift in both coupling cases can be found in [52]. For a more detailed discussion regarding the different outcomes produced by the two couplings in the context of moving detectors in a thermal field bath we refer the reader to [52].

To quantify the amount of entanglement present in an arbitrary two-qubit state $\rho$ we employ the concurrence [59, 60]

where $\lambda_{i}$ are, in decreasing order, the square roots of eigenvalues of the matrix $R=\rho(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$. Here, $\rho^{*}$ denotes the complex conjugate of $\rho$ and $\sigma_{y}$ is the usual Pauli matrix. Concurrence ranges from $C=0$ for a state with no entanglement to $C=1$ for a maximally entangled state.

We note that in the case of the so-called X-states [61, 62], which are characterized by the fact that their density matrix has non-zero elements along the main diagonal and anti-diagonal, i.e.,

the concurrence is shown to be given by

It can be easily shown that the state (III) is a X-state of the form (34). As a result, we can directly calculate concurrence by employing Eq. (35). We find that

The concurrence (36) is plotted in Fig. 1 as a function of time, in the case of the standard UDW (plots (a)-(c)) and derivative (plots (d)-(f)) detector-field coupling, for various velocities $\upsilon$ of the moving detector and temperatures $\beta^{-1}$ of the field bath. As may be expected, the initial amount of entanglement present in the shared state between the moving atom and its auxiliary partner monotonically decreases with time. In all cases, the rate of decrease becomes greater with the increase of the field bath temperature.

Despite this fact, we observe that in the UDW coupling case and for temperatures $T\geq\omega$, entanglement loss is hindered when the atom is moving at a constant speed. This is a direct consequence of the way the moving atom responds to the perceived thermal spectrum $N$ of the environment. In Figs. 2 and 3, we plot $N_{\text{\tiny{UDW}}}(\beta,\upsilon)$ (Eq. (22)) as a function of the atom's speed, for high and low temperatures of the field bath respectively. Concurrence approximately decreases exponentially as $C\sim e^{-\frac{\Gamma}{2}(2N+1)}$. Besides, for higher temperatures of the environment, the coefficient $N$ behaves as

This means that $N_{\text{\tiny{UDW}}}$ decreases as a function of the atom's speed (see, also, Fig. 2), consequently reducing the rate at which concurrence decreases. In contrast, in the low-temperature regime, $N$ exponentially increases with the atom's speed (see, Fig. 3), behaving as

This leads to a greater loss of an initial amount of entanglement for a moving atom. In summary, a greater delay of the entanglement degradation can be achieved with an increase of the atom's speed. Although the higher the reservoir's temperature, the greater the loss of any initial amount of entanglement, the latter can be delayed by allowing the detector to move along a worldline of constant velocity.

Instead, this conclusion does not hold true for the derivative coupling case, where as the atom's speed increases, there is a substantial decrease in the concurrence. As it is evident from Figs. 2 and 3$,N_{\text{\tiny{TD}}}$ monotonically decreases with the atom's speed in both the low and high-temperature regimes. However, unlike the UDW case, where the decay coefficient $\Gamma_{\text{\tiny{UDW}}}$ is independent of the detector's speed, in the derivative coupling case, $\Gamma_{\text{\tiny{TD}}}$ exponentially increases with the atom's speed (see, Eq. (21)). As a consequence, entanglement rapidly decoheres as the atom's speed increases.

The different environments are classified according to the form of their spectral density $J(\omega)$, which usually follows the power-law $J(\omega)\propto\omega^{\alpha}$¹¹1Physically, the spectral density should vanish for very high values of frequencies. The pointlike detector case implies the introduction of a high frequency cut-off $\omega_{c}=\epsilon^{-1}$ as an exponential factor $e^{-\epsilon\omega}$ in the spectral density to regularize the high frequency behavior [63, 64]. [46]. The most common choice for the exponent $\alpha$ is $\alpha=1$, in which case the the environment is characterized as Ohmic. If $\alpha<1$ the environment is characterized as subohmic, and if $\alpha>1$ as supraohmic respectively. We note that the massless scalar field bath $\hat{\phi}(\mathsf{x})$ represents an Ohmic environment (from Eq. (A) we have that $J(|\boldsymbol{k}|)\propto|\boldsymbol{k}|$), that is an environment whose spectral density is linear in frequency. Instead, when considering the derivative coupling case, the field bath $\dot{\hat{\phi}}(\mathsf{x})$ acts as a supraohmic environment (from Eq. (A) we have that $J(|\boldsymbol{k}|)\propto|\boldsymbol{k}|^{3}$). Thus, for higher values ($\beta\leq\omega^{-1}$) of temperature of an Ohmic environment, we conclude that an atom's motion can slow down the rate at which entanglement is lost, resulting in the preservation of entanglement for longer time. However, this is not true for supraohmic heat reservoirs: any interaction of moving atoms with them is always disastrous for entanglement.

Let us note finally that the time point at which entanglement completely disappears (i.e., $C(\rho)=0$) is given by the expression