Minkowski Vacuum State Research Articles

Relativistic quantum Otto engine: instant work extraction from a quantum field

In this study, we carry out a non-perturbative approach to a quantum Otto engine, employing an Unruh-DeWitt particle detector to extract work from a quantum Klein-Gordon field in an arbitrary globally hyperbolic curved spacetime. We broaden the scope by considering the field in any quasi-free state, which includes vacuum, thermal, and squeezed states. A key aspect of our method is the instantaneous interaction between the detector and the field, which enables a thorough non-perturbative analysis. We demonstrate that the detector can successfully extract positive work from the quantum Otto cycle, even when two isochoric processes occur instantaneously, provided the detector in the second isochoric process receives a signal from the first interaction. This signaling allows the detector to release heat into the field, thereby the thermodynamic cycle is completed. As a demonstration, we consider a detector at rest in flat spacetime and compute the work extracted from the Minkowski vacuum state.

Polarization of the Conformal Vacuum in the Milne Universe

The vacuum expectation values of the field squared and energy-momentum tensor for a massless scalar field are investigated in the Milne universe with general number of spatial dimensions. The vacuum state depends on the choice of the mode functions in the canonical quantization procedure and we assume that the field is prepared in the conformal vacuum. As the first step an integral representation for the difference of the Wightman functions corresponding to the conformal and Minkowski vacua is derived. The mean field squared and energy-momentum tensor are obtained in the coincidence limit. It is shown that the Minkowski vacuum state is interpreted as a thermal one with respect to the conformal vacuum. The thermal factor is of the Bose-Einstein type in odd dimensional space and of the Fermi-Dirac type in even number of spatial dimensions.

Well-tempered Minkowski solutions in teleparallel Horndeski theory

Well-tempering stands among the few classical methods of screening vacuum energy to deliver a late-time, low energy vacuum state. We build on the class of Horndeski models that admit a Minkowski vacuum state despite the presence of an arbitrarily large vacuum energy to obtain a much larger family of models in teleparallel Horndeski theory. We set up the routine for obtaining these models and present a variety of cases, all of which are able to screen a natural particle physics scale vacuum energy using degeneracy in the field equations. We establish that well-tempering is the unique method of utilizing degeneracy in Horndeski scalar-tensor gravity—and its teleparallel generalisation—that can accommodate self-tuned flat Minkowski solutions, when the explicit scalar field dependence in the action is minimal (a tadpole and a conformal coupling to the Ricci scalar). Finally, we study the dynamics of the well-tempered teleparallel Galileon. We generate its phase portraits and assess the attractor nature of the Minkowski vacuum under linear perturbations and through a phase transition of vacuum energy. ‘The effort to understand the Universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy.’-Steven Weinberg

Space and time averaged quantum stress tensor fluctuations

We extend previous work on the numerical diagonalization of quantum stress tensor operators in the Minkowski vacuum state, which considered operators averaged in a finite time interval, to operators averaged in a finite spacetime region. Since real experiments occur over finite volumes and durations, physically meaningful fluctuations may be obtained from stress tensor operators averaged by compactly supported sampling functions in space and time. The direct diagonalization, via a Bogoliubov transformation, gives the eigenvalues and the probabilities of measuring those eigenvalues in the vacuum state, from which the underlying probability distribution can be constructed. For the normal-ordered square of the time derivative of a massless scalar field in a spherical cavity with finite degrees of freedom, analysis of the tails of these distributions confirms previous results based on the analytical treatment of the high moments. We find that the probability of large vacuum fluctuations is reduced when spatial averaging is included, but the tail still decreases more slowly than exponentially as the magnitude of the measured eigenvalues increases, suggesting vacuum fluctuations may not always be subdominant to thermal fluctuations and opening up the possibility of experimental observation under the right conditions.

Entanglement of the vacuum between left, right, future, and past: Dirac spinor in Rindler and Kasner spaces

We study the relations of the positive frequency mode functions of Dirac field in 4-dimensional Minkowski spacetime covered with Rindler and Kasner coordinates, and describe the explicit form of the Minkowski vacuum state with the quantum states in Kasner and Rindler regions, and analytically continue the solutions. As a result, we obtain the correspondence of the positive frequency mode functions in Kasner region and Rindler region in a unified manner which derives vacuum entanglement.

Physical observers, T-vacuum and Unruh like effect in the radiation dominated early universe

We report the existence of an Unruh like effect, for physical observers (cosmo- logical and comoving observers) in the radiation dominated early universe, which is possible due to the discovery of a new vacuum state (referred here as the T−vacuum). Both the comoving and the cosmological observers, who are crucial in our understanding of cosmology, observe this T−vacuum as a particle excited state and are able to detect radiation due to particle creation. We draw a robust analogy with the Unruh effect, whereby — (i) the physical observers here are closely analogous to the accelerated (Rindler) observers in the flat spacetime, and (ii) the T−vacuum plays the role of the Minkowski vacuum state which contains particles when viewed from the physical observers frame. Our analogy is further supported by a proof of well-defined (hadamard) behavior of the T−vacuum in the entire spacetime. All our analysis of the particle creation process is done here within a two dimensional set up.

Continuous-variable quantum teleportation with vacuum-entangled Rindler modes

We consider a continuous-variable quantum teleportation protocol between a uniformly accelerated sender in the right Rindler wedge, a conformal receiver restricted to the future light cone, and an inertial observer in the Minkowski vacuum. Using a nonperturbative quantum circuit model, the accelerated observer interacts unitarily with the Rindler modes of the field, thereby accessing entanglement of the vacuum as a resource. We find that a Rindler-displaced Minkowski vacuum state prepared and teleported by the accelerated observer appears mixed according to the inertial observer, despite a reduction of the quadrature variances below classical limits. This is a surprising result, since the same state transmitted directly from the accelerated observer appears as a pure coherent state to the inertial observer. The decoherence of the state is caused by an interplay of opposing effects as the acceleration increases: the reduction of vacuum noise in the output state for a stronger entanglement resource, constrained by the amplification of thermal noise due to the presence of Unruh radiation.

Vacuum quantum stress tensor fluctuations: A diagonalization approach

Large vacuum fluctuations of a quantum stress tensor operator can be described by the asymptotic behavior of the probability distribution of the time or spacetime averaged operator. Here we focus on the case of stress tensor operators averaged with a sampling function in time. The Minkowski vacuum state is not an eigenstate of the time-averaged operator, but can be expanded in terms of its eigenstates. We calculate the probability distribution and the cumulative probability distribution for obtaining a given value in a measurement of the time-averaged operator taken in the vacuum state. In these calculations, we use the normal ordered square of the time derivative of a massless scalar field in Minkowski spacetime as an example of a stress tensor operator. We analyze the rate of decrease of the tail of the probability distribution for different temporal sampling functions, such as compactly supported functions and the Lorentzian function. We find that the tails decrease relatively slowly, as exponentials of fractional powers, in agreement with previous work using the moments of the distribution. Our results lead additional support to the conclusion that large vacuum stress tensor fluctuations are more probable than large thermal fluctuations, and may have observable effects.

Entanglement of the vacuum between left, right, future, and past: The origin of entanglement-induced quantum radiation

The Minkowski vacuum state is expressed as an entangled state between the left and right Rindler wedges when it is constructed on the Rindler vacuum. In this paper, we further examine the entanglement structure and extend the expression to the future (expanding) and past (shrinking) Kasner spacetimes. This clarifies the origin of the quantum radiation produced by an Unruh--DeWitt detector in uniformly accelerated motion in the four-dimensional Minkowski spacetime. We also investigate the two-dimensional massless case where the quantum radiation vanishes but the same entanglement structure exists.

Entanglement-induced quantum radiation

Quantum entanglement of the Minkowski vacuum state between left and right Rindler wedges generates thermal behavior in the right Rindler wedge, which is known as the Unruh effect. In this letter, we show that there is another consequence of this entanglement, namely entanglement-induced quantum radiation emanating from a uniformly accelerated object. We clarify why it is in agreement with our intuition that incoming and outgoing energy fluxes should cancel each other out in a thermalized state.

Quantum radiation produced by the entanglement of quantum fields

We investigate the quantum radiation produced by an Unruh-De Witt detector in a uniformly accelerating motion coupled to the vacuum fluctuations. Quantum radiation is nonvanishing, which is consistent with the previous calculation by Lin and Hu [Phys. Rev. D 73, 124018 (2006)]. We infer that this quantum radiation from the Unruh-De Witt detector is generated by the nonlocal correlation of the Minkowski vacuum state, which has its origin in the entanglement of the state between the left and the right Rindler wedges.

Radiative processes of uniformly accelerated entangled atoms

We study radiative processes of uniformly accelerated entangled atoms, interacting with an electromagnetic field prepared in the Minkowski vacuum state. We discuss the structure of the rate of variation of the atomic energy for two atoms travelling in different hyperbolic world lines. We identify the contributions of vacuum fluctuations and radiation reaction to the generation of entanglement as well as to the decay of entangled states. Our results resemble the situation in which two inertial atoms are coupled individually to two spatially separated cavities at different temperatures. In addition, for equal accelerations we obtain that one of the maximally entangled antisymmetric Bell state is a decoherence-free state.

Spacetime diamonds

We show that the particle-number distribution of diamond modes, modes that are localized in a finite spacetime region, are thermal for the Minkowski vacuum state of a massless scalar field, an analogue to the Unruh effect. The temperature of the diamond is inversely proportional to its size. An inertial observer can detect this thermal radiation by coupling to the diamond modes using an appropriate energy-scaled detector. We further investigate the correlations between various diamonds and find that entanglement between adjacent diamonds dominates.

Quantum field theory in the Rindler-Rindler spacetime

It is well known that Minkowski vacuum appears as a thermal bath in the Rindler spacetime when the modes on the left wedge are traced out. We introduce the concept of a Rindler-Rindler spacetime, obtained by a further coordinate transformation from the Rindler spacetime, in a manner similar to the transformation from inertial to Rindler frame. We show that the Rindler vacuum appears as a thermal state in the Rindler-Rindler frame. Further, the spectrum of particles seen by the Rindler-Rindler observers in the original Minkowski vacuum state is shown to be identical to that seen by a detector accelerating through a real thermal bath. Thus the Davies-Unruh effect acts as a proxy for a real thermal bath, for a certain class of observers in the Rindler-Rindler spacetime. We interpret this similarity as indicating further evidence of the indistinguishability between thermal and quantum fluctuations along the lines of the recent work in Kolekar and Padmanabhan [arXiv:1308.6289]. The implications are briefly discussed.

Entropy of a Rindler Observer

We compute the entropy of a Rindler particle-detector (observer) in the presence of a quantum field in the Minkowski vacuum state; due to the Unruh effect, the observer is immersed in a thermal bath at a temperature proportional to its proper acceleration.

Fully renormalized stress tensor correlator in flat space

We present a general procedure to renormalize the stress tensor two-point correlation function on a Minkowski background in position space. The method is shown in detail for the case of a free massive scalar field in the standard Minkowski vacuum state, and explicit expressions are given for the counterterms and finite parts, which are in full accordance with earlier results for the massless case. For the general case in position space, only regularized --- but not renormalized --- results have been obtained previously. After a Fourier transformation to momentum space, we also check agreement with a previous calculation there. We generalize our results to general Hadamard states. Furthermore, the proposed procedure can presumably be generalized to the important case of an inflationary spacetime background, where the transition to momentum space is in general not possible.

A ground state for the causal diamond in 2 dimensions

We apply a recent proposal for a distinguished ground state of a quantum field in a globally hyperbolic spacetime to the free massless scalar field in a causal diamond in two-dimensional Minkowski space. We investigate the two limits in which the Wightman function is evaluated (i) for pairs of points that lie in the centre of the diamond (i.e. far from the boundaries), and (ii) for pairs of points that are close to the left or right corner. We find that in the centre, the Minkowski vacuum state is recovered, with a definite value of the infrared cutoff. Interestingly, the ground state is not the Rindler vacuum in the corner of the diamond, as might have been expected, but is instead the vacuum of a flat space in the presence of a static mirror on that corner. We confirm these results by numerically evaluating the Wightman function of a massless scalar field on a causal set corresponding to the continuum causal diamond.

Entanglement entropy of two black holes and entanglement entropic force

We study the entanglement entropy ${S}_{C}$ of the massless free scalar field on the outside region $C$ of two black holes $A$ and $B$ whose radii are ${R}_{1}$ and ${R}_{2}$ and how it depends on the distance $r(\ensuremath{\gg}{R}_{1},{R}_{2})$ between two black holes. If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force acting on the two black holes from the $r$ dependence of ${S}_{C}$. We develop the computational method based on that of Bombelli et al. to obtain the $r$ dependence of ${S}_{C}$ of scalar fields whose Lagrangian is quadratic with respect to the scalar fields. First, we study ${S}_{C}$ in ($d+1$)-dimensional Minkowski spacetime. In this case the state of the massless free scalar field is the Minkowski vacuum state, and we replace two black holes by two imaginary spheres and take the trace over the degrees of freedom residing in the imaginary spheres. We obtain the leading term of ${S}_{C}$ with respect to $1/r$. The result is ${S}_{C}={S}_{A}+{S}_{B}+\frac{1}{{r}^{2d\ensuremath{-}2}}G({R}_{1},{R}_{2})$, where ${S}_{A}$ and ${S}_{B}$ are the entanglement entropy on the inside region of $A$ and $B$, respectively, and $G({R}_{1},{R}_{2})\ensuremath{\le}0$. We do not calculate $G({R}_{1},{R}_{2})$ in detail, but we show how to calculate it. In the black hole case we use the method used in the Minkowski spacetime case with some modifications. We show that ${S}_{C}$ can be expected to be the same form as that in the Minkowski spacetime case. But in the black hole case, ${S}_{A}$ and ${S}_{B}$ depend on $r$, so we do not fully obtain the $r$ dependence of ${S}_{C}$. Finally, we assume that the entanglement entropy can be regarded as thermodynamic entropy and consider the entropic force acting on two black holes. We argue how to separate the entanglement entropic force from other forces and how to cancel ${S}_{A}$ and ${S}_{B}$ whose $r$ dependences are not obtained. Then we obtain the physical prediction, which can be tested experimentally in principle.

Universal viscosity to entropy density ratio from entanglement

We present evidence that the universal Kovtun-Son-Starinets shear viscosity to entropy density ratio of 1/4\pi can be associated with a Rindler causal horizon in flat spacetime. Since there is no known holographic (gauge/gravity) duality for this spacetime, a natural microscopic explanation for this viscosity is in the peculiar properties of quantum entanglement. In particular, it is well-known that the Minkowski vacuum state is a thermal state and carries an area entanglement entropy density in the Rindler spacetime. Based on the fluctuation-dissipation theorem, we expect a similar notion of viscosity arising from vacuum fluctuations. Therefore, we propose a holographic Kubo formula in terms of a two-point function of the stress tensor of matter fields in the bulk. We calculate this viscosity assuming a minimally coupled scalar field theory and find that the ratio with respect to the entanglement entropy density is exactly 1/4\pi in four dimensions. The issues that arise in extending this result to non-minimally coupled scalar fields, higher spins, and higher dimensions provide interesting hints about the relationship between entanglement entropy and black hole entropy.

Acceleration radiation and the Planck scale

A uniformly accelerating observer perceives the Minkowski vacuum state as a thermal bath of radiation. We point out that this field-theory effect can be derived, for any dimension higher than two, without actually invoking very high energy physics. This supports the view that this phenomenon is robust against Planck-scale physics and, therefore, should be compatible with any underlying microscopic theory.