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200422 Finalarxivversion.Pdf Classical and Nonclassical Time Dilation for Quantum Clocks A. J. Paige,1, ∗ A. D. K. Plato,1 and M. S. Kim1 1QOLS, Blackett Laboratory, Imperial College London, South Kensington, London, SW7 2AZ, UK. Proper time, ideal clocks, and boosts are well understood classically, but subtleties arise in quan- tum physics. We show that quantum clocks set in motion via momentum boosts do not witness classical time dilation. However, using velocity boosts we find the ideal behaviour in both cases where the quantum clock and classical observer are set in motion. Without internal state depen- dent forces additional effects arise. As such, we derive observed frequency shifts in ion trap atomic clocks, indicating a small additional shift, and also show the emergence of non-ideal behaviour in a theoretical clock model. Ideal clocks and proper time are key concepts in special observer and quantum clock being set in motion is cap- and general relativity [1]. Full understanding of the union tured by translation operators, and that it is the trans- between relativity and quantum mechanics, must include formation under translation operators that enables the how these ideas extend to the quantum realm. Recent velocity boost to describe both situations. We demon- work in this area can broadly be divided by whether the strate how these translation operators can be understood quantum clocks follow classical or quantum trajectories. via considering the placement of the origin for the re- Adopting the former approach [2{5] enables the utiliza- quired potentials. In addition, we show that considering tion of techniques from quantum field theory in curved the velocity kick as the classical observer changing frames spacetime. In particular this has allowed explorations immediately tells us that entanglement is frame depen- into consequences of the Unruh effect [6], and applica- dent and demonstrate consistency with the equivalence tions of techniques from relativistic quantum metrology principle. We highlight that without an internal state de- [7,8]. On the other hand, for quantum clocks following pendent force one should expect additional effects. We quantum mechanical trajectories [9{19], most progress demonstrate this for frequency shifts in ion traps, predict- has been made investigating connections between proper ing the already observed shifts and an additional smaller time and mass superpositions [20]. This has necessitated shift. We also analyse the effect on a Salecker-Wigner- the rejection of the Bargmann mass superselection rule Peres clock [24, 25] finding non-ideal clock behaviour. [21], on the grounds that our universe is not Galilean. Modified Hamiltonian | We begin by presenting the Notably this paradigm was used to investigate ideas for simplest argument for the modified Hamiltonian. The intrinsic time dilation decoherence caused by gravity [10]. modification was introduced and used to study quantum In this work we follow the second approach, where the mechanical proper time [9], and has also been shown to clock's motion is described quantum mechanically. We resolve paradoxes in quantum optics [26, 27]. For a free show that a quantum clock set into motion by a force composite particle of mass M; the non-relativistic Hamil- 2 that does not depend on the internal state is not wit- p tonian will consist of a kinetic energy term 2M ; and an nessing classical time dilation. This is because quantum internal energy term H0: However the internal energy clocks require coherence in some non-degenerate energy should contribute to the inertial mass since special rela- states [22, 23] but the inertial mass of this energy means tivity dictates that energy and inertial mass are equiva- 2 that assigning an identical momentum to each branch of lent. This leads to M = m + H0=c ; where we take m as the superposition does not correspond to a well defined the rest mass of the particle in its internal energy ground velocity. We therefore show that momentum boosts lead state j0i; and set H0j0i = 0: Thus we have to a nonclassical dilation due to the lack of a unique Lorentz factor. On the other hand, by suitably coupling p2 H = + H ; the motional and internal degrees of freedom, one can ap- 2M 0 (1) ply a velocity boost which exactly recovers the expected p2 p2 = + H 1 − : classical time dilation results for the \twin paradox," in 2m 0 2mMc2 both cases where the observer and the quantum clock are respectively the ones set in motion. To arrive at the second line we have used 1=(x + y) = We start from a Hamiltonian modified to account for 1=x − y=x(x + y): Note that since M is now operator the inertial mass of internal energy. We then consider valued, there is potential for ambiguity with operator or- sequences of appropriately centred boosts and evolution dering. However if the internal Hamiltonian commutes −1 operators to derive the different possible clock behaviours with the total momentum [H0; p] = 0; then [M ; p] = 0: in a twin paradox scenario. From the classical observer's Fully accounting for relativity would strictly imply that frame we show that the difference between the classical the internal degrees of freedom should be described by 2 a relativistic wave equation (or a quantum field theory). pbi: This represents the physical situation typically con- However the approach here is that regardless of the for- sidered for use in the laboratory [30], with no internal malism, the effect on the centre of mass dynamics should state dependence as per the potentials typically used to only be via a mass change, otherwise we could not claim move quantum systems (e.g. an ion moved via an Elec- that energy and inertial mass are equivalent. tromagnetic potential). 2 This Hamiltonian is often expanded in H0=mc ne- Using this boost, the translation operator T (pbt=m) = glecting higher order terms [10, 27]. It is then tempting e−ippbt=m~, and the free evolution under the Hamiltonian p2 of Eq. (1), we have to claim that (1 − 2m2c2 ) represents our familiar notion of time dilation, however, this is not correct as will be shown later. It is also important to emphasise that we Bp(pb)U(t)T (pbt=m)Bp(−2pb)T (−pbt=m)U(t)Bp(pb) shall always be working in the limit where the energy E = 2 p2 p2 − i2t p 2it b − 2it H (1− b ) p 2M 2m 0 2mMc2 p2c2 + M 2c4 is approximated by E = p2=2M + Mc2: = e ~ e ~ e ~ : (3) In words, one can consider the regime we utilise as one The first exponential term is the unaltered motional evo- where the mechanics can be treated in a \Newtonian" lution of the state that we would expect if we had not sense, but the rest mass of the internal energy is now ac- applied any of the boosts. This term has no p depen- counted for. On a technical level one must appreciate (as b dent relativistic corrections, which is not surprising since noted in [13]) that there are two relevant small quanti- the adopted formalism does not include the relevant rel- ties: H =mc2 and p2=m2c2; where the first relates to the 0 ativistic motional terms. The second term is a global internal degrees of freedom and the second can be viewed phase that is connected to the choice of the translation as a motional v2=c2 term (note p2=m2c2 > p2=M 2c2;). It operators, discussed in detail below. However, it is the is therefore not sufficient to merely think of approxima- final term that captures the effect on the evolution of the tions in terms of how many factors of 1=c2; are present. 2 internal state and therefore is our primary focus. One can consider the regime where H0=mc 1; but 2 2 2 2 The third exponential term in Eq. (3) has the internal H0=mc p =m c : With this in mind one arrives at Eq. 2 pb (1) (plus ground state rest mass energy) by expanding Hamiltonian multiplied by a factor (1 − 2mMc2 ) that is the full relativistic energy E = pM 2c4 + p2c2 = γMc2; less than unity. This is precisely the factor we would get neglecting terms of O(p4=m4c4) in the Lorentz factor by considering Eq.1 acting on a momentum eigenstate 2 jp i: We see that if we have some coherence in the in- whilst retaining the H0=mc terms and translating the b zero of energy by mc2: Note that in practise the order ternal state and are using it as a clock, then it appears 2 that the clock is running slower. Note it is not dilated of H0=mc terms kept would be dictated by the physical system under consideration and depends on the integer by a constant inverse Lorentz factor as in classical rela- 2 n 2 2 2 tivity. We can see why this is so by asking what Lorentz n for which (H0=mc ) p =m c : However, as will become apparent, it will prove more straightforward for factor we would expect. The clock has been given a mo- our initial theoretical study to work with the untruncated mentum pb but because our clock is in a superposition (1 + H =mc2)−1: We shall denote the unitary evolution of energies En, we also have a superposition of different 0 2 generated by Eq. (1) over time t as U(t): masses Mn = m + En=c : Hence we can write various Different boosts | We now explore the consequences velocities and thus Lorentz factors. For example, the N p 2 2 2 of the modified Hamiltonian. Time is a complex topic single shot values γn = 1 + pb =Mnc ; or the expec- in quantum mechanics [28, 29], but here we shall sim- tation value of an operator hγi: Furthermore, Eq.
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