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https://doi.org/10.1038/s41467-020-18264-4 OPEN Quantum clocks observe classical and quantum dilation ✉ ✉ Alexander R. H. Smith 1,2 & Mehdi Ahmadi 3

At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper . We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one

1234567890():,; clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of- mass of these clocks move in localized wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time- energy/mass uncertainty relation.

1 Department of , Saint Anselm College, Manchester, NH 03102, USA. 2 Department of Physics and Astronomy, Dartmouth College, Hanover, NH ✉ 03755, USA. 3 Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053, USA. email: [email protected]; [email protected]

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’ 〉 〉 ′ 〉 hat allowed Einstein to transcend Newton s absolute ⎟ pB ⎟ pA +⎟ pA time was his insistence that time is what is shown by a W 1 clock : “[Time is] considered measurable by a clock (ideal periodic process) of negligible spatial extent. The time of an event taking place at a point is then defined as the time shown on the clock simultaneous with the event.” Bridgman highlighted the significance of this definition of time2: “Einstein, in seizing on the act of the observer as the essence of the situation, is actually adopting a new point of view as to what the concepts of physics should be, namely, the operational view.” Extending the operational view to quantum theory, one is led to define time through measurements of quantum systems serving as ===B = clocks3. Such descriptions of quantum clocks have been developed A in the context of quantum metrology4–7. In this regard, time observables are identified with positive-operator valued measures Fig. 1 Quantum time dilation. Two clocks are depicted as moving in (POVMs) that transform covariantly with respect to the group of Minkowski space. Clock B is moving in a localized momentum wave packet 8,9 p A time translations acting on the employed clock system .This with average momentum B, while clock is moving in a superposition of p p0 covariance property ensures that these time observables give the localized momentum wave packets with average momentum A and A. optimal estimate of the time experienced by the clock; that is, they Clock A experiences a quantum contribution to the time dilation it observes saturate the Cramer–Rao bound10. Furthermore, covariant time relative to clock B due to its nonclassical state of motion. observables allow for a rigorous formulation of the time-energy uncertainty relation4–7, circumvent Pauli’s infamous objection to espoused earlier, we employ the Page–Wootters formulation of the construction of a time operator11,12, and play an important quantum dynamics in which time enters like any other quantum 13–17 Ψ role in relational quantum dynamics . observable. One considers a state j ii2Hphys of a clock C and a Given that clocks are ultimately quantum systems, they too are system S that lives in the physical Hilbert space Hphys. This subject to the superposition principle. In a relativistic context, this Hilbert space is defined as the Cauchy completion of the set of leads to the possibility of clocks experiencing a superposition of solutions to the constraint equation: proper times. Such scenarios have been investigated in the context of Ψ Ψ ; relativistic clock interferometry18, in which two branches of a matter- CHj ii¼ðÞHC þ HS j ii¼0 ð1Þ wave interferometer experience different proper times on account of 2 LðH Þ 2 LðH Þ 19–26 where HC C and HS S denote the Hamiltonians of either special or general relativistic time dilation .Suchasetup C and S. One then associates with C a time observable defined as a leads to a signature of matter experiencing a superposition of proper POVM: times through a decrease in interferometric visibility. Other work has Z focused on quantum variants of the twin-paradox27,28 and exhibiting : ; 29–36 TC ¼ ECðtÞ8t 2 G j IC ¼ ECðdtÞ ð2Þ nonclassical effects in relativistic scenarios . G We introduce a proper time observable defined as a covariant where E ðtÞ¼jit hjt is a positive operator on H known as an POVM on the internal degrees of freedom of a relativistic particle C C effect operator, G the group generated by H , and jit will be moving through curved spacetime. This allows us to consider two C referred to as a clock state associated with a measurement of the relativistic quantum clocks, A and B, and construct the probability clock yielding the time t. What makes T a time observable is that that A reads a particular proper time conditioned on B reading a C the effect operators transform covariantly with respect to the different proper time. To compute this probability distribution we – group generated by H 5 7,9 extend the Page–Wootters approach37,38 to relational quantum C 0 0 y 0 dynamics to the case of a relativistic particle with internal degrees Eðt þ t Þ¼UCðt Þ EðtÞ UCðt Þ ð3Þ of freedom. We then consider two clocks prepared in localized : ÀiHCt momentum wave packets and demonstrate that they observe on where UCðtÞ ¼ e . This covariance condition implies that 0 0 fi average classical time dilation in accordance with special relativity. ji¼t þ t UCðt Þjit . One then de nes a state of S by conditioning jΨii on C reading the time t We then illustrate a quantum time dilation effect that occurs when one clock moves in a superposition of two localized momentum ψ ðtÞ :¼ hj t I jiΨi : ð4Þ wave packets: On average, the proper time of a clock moving in a S S coherent superposition of momenta is distinct from that of the It then follows from Eqs. (1) and (3), corresponding classical mixture, see Fig. 1. We describe the d average quantum correction to the classical time dilation observed i ψ ðtÞ ¼ H ψ ðtÞ ; ð5Þ dt S S S by such a superposed clock. In addition, our description of proper time as a covariant POVM allows for both proper time and par- which describes the evolution of S relative to C. ticle mass to be treated as dynamical quantum observables, leading As described in the “Methods” section, a relativistic particle to a time-energy/mass uncertainty relation. with an internal clock degree of freedom can be described by the 2 R Hilbert space Ht Hcm Hclock, where Ht ’ L ð Þ, 2 R3 Hcm ’ L ð Þ, and Hclock are Hilbert spaces associated respec- Results tively with the temporal, center-of-mass, and internal clock Page–Wootters description of a relativistic particle with an degrees of freedom of the particle. When the relativistic internal degree of freedom. In adhering to the operational view particle has positive energy, the physical state satisfies Eq. (1)

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= with HC Pt equal to the momentum operator on Ht and fundamental limit on the variance of the proper time measured sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by the clock P2 H 2 2 clock ; ð6Þ 2 1 HS ¼ mc 2 2 þ 1 þ 2 hðΔT Þ i ≥ ; m c mc clock ρ Δ 2 ð8Þ 4hð HclockÞ iρ 2 = η i j η where P : ijP P , ij the Minkowski metric, and m the rest mass Δ 2 fi of the particle. Equation (5) then becomes the relativistic where hð HclockÞ iρ is the variance of Hclock on the ducial state ρ Schrödinger equation and the state jψ ðtÞi may be interpreted . Equation (8) is a time-energy uncertainty relation between the S ’ as the state of the center-of-mass and internal clock of the particle proper time estimated by Tclock and a measurement of the clock s fi at the time t, interpreted as the time of an inertial frame observing energy Hclock. Now consider the related mass observable de ned = + 2 19 the particle with respect to which the center-of-mass degrees of by the self-adjoint operator Mclock: m Hclock/c (e.g., ). From freedom are defined. With this identification, the dynamics Eq. (8), an uncertainty relation between this mass observable and implied by the Page–Wootters formalism is in agreement with proper time follows previous descriptions of a relativistic particle with internal 1 19,31,39 ΔM ΔT ≥ ; ð9Þ degrees of freedom . clock clock 2c2 We note that in the “Methods” section the above analysis in the 2 1=2 Page–Wootters formalism is generalized to the case of a where ΔA :¼hðΔAÞ iρ denotes the standard deviation of the stationary curved spacetime and in Supplementary Note 1 the observable A. This inequality gives the ultimate bound on the Klein–Gordon equation is recovered. Further justification of the precision of any measurement of proper time. Page–Wootters formalism is provided in Supplementary Note 2. The time-energy/mass inequality above can be saturated using the optimal proper time observable provided that the effect τ fi “ ” Proper time observables. We now make precise how the internal operators Eclock( )de ning Tclock are proportional to projection degrees of freedom of the relativistic particle introduced in the operators previous section constitute a clock by introducing a proper time E ðτÞ¼μτjihjτ ; ð10Þ observable. We define a clock to be the quadruple: clock μ R ∫ τ = ; ρ; ; ; for 2 such that G Eclock(d ) Iclock, where Hclock is the fgHclock Hclock Tclock ð7Þ generator of proper time translations, and jiτ is the clock state fi corresponding to the proper time τ. Here the motivation is that the elements of which are the clock Hilbert space Hclock,a ducial state ρ, Hamiltonian H , and time observable T . Similar to measurements not described by one-dimensional projectors have clock clock 6 τ the definition of T above, T is defined as a POVM that less resolution , however, note that the clock states jiare not C clock τ τ0 ≠ transforms covariantly with respect to the group action necessarily orthogonal, h j i 0. τ U ðτÞ¼eÀiHclock . The physical significance of the covariance It turns out that covariant observables satisfying Eq. (10) clock τ condition in Eq. (3) is that it implies the time observable satisfies constitute an optimal measurement to estimate the parameter ρ τ = τ ρ τ † the following two physical properties commonly associated with a unitarily encoded in the state ( ): Uclock( ) Uclock( ) , provided that the fiducial state is pure ρ ¼jψ ihψ j and clock, which we state in a theorem. Z clock clock

Theorem 1 (Desiderata of physical clocks) Let Tclock be a τ ψ τψ τ i hHclockiρ τ ; covariant time observable relative to the group generated by clock ¼ d clockð Þ e ji ð11Þ ρ fi G Hclock, be a ducial state such that hTclockiρ ¼ 0, and pffiffiffi where ψ ðτÞ :¼ μhτjψ i and ψ ðτÞ is a real function ρðτÞ :¼ U ðτÞ ρ Uy ðτÞ. The following two physical proper- clock clock clock clock clock of τ such that hT i ¼ 06. Such a proper time observable T ties of such a time observable follow: clock ρ clock is optimal in the sense that it maximizes the so-called Fisher τ 10 1. Tclock is an unbiased estimator of the parameter such that information , which quantifies how well two slightly different τ hTclockiρðτÞ ¼ . values of proper time can be distinguished given a particular 2. The variance of the time observable is independent of the quantum measurement. For the effect operatrors Eclock and the τ Δ 2 Δ 2 fi parameter , i.e., hð TclockÞ iρðτÞ ¼hð TclockÞ iρ. ducial state in Eq. (11), we have Proof Statements 1 and 2 follow directly from the covariance τ; ρ τ Δ 2 : F½ ð ފ ¼ 4hð HclockÞ iρ ð12Þ property of Tclock; see Supplementary Note 3. fi τ τ0 τ0 τ This theorem justi es interpreting Tclock as a time observable: The covariance condition, jiþ ¼ Uclockð Þji, ensures that when a time observable is measured on a quantum clock, we the Fisher information is independent of τ. expect on average that it estimates the elapsed time τ unitarily Let us point out a connection between our above construction encoded in ρ(τ). Also, the variance of this measurement should be of a properp timeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi observable and quantum speed limits. From the independent of the time τ being estimated. fact that F½Šτ; ρðτÞ =2 ≤ ΔH40, together with Eq. (12), we can Taking this notion of a clock and applying it to the relativistic conclude that the covariant proper time observable in fact particle model introduced in the previous section, we may saturates the so-called Mandelstam and Tamm inequality. That is construct a proper time observable that transforms covariantly π π with respect to the internal clock Hamiltonian H of the τ ¼ ΔT ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; clock ? clock τ; ρ τ 2ΔM ð13Þ particle. As explained in the “Methods” section, such a F½Šð Þ clock Hamiltonian generates a unitary evolution of the internal clock where τ⊥ is the time that passes before the initial state of a system degrees of freedom of the particle, and thus a time observable evolves under the Hamiltonian Hclock into an orthogonal state. Tclock that transforms covariantly with respect to the group We remark that in this construction both proper time and ’ generated by Hclock will measure the particle s proper time. mass are treated as genuine quantum observables; the former as a covariant POVM Tclock and the latter as a self-adjoint operator Proper time-energy/mass uncertainty relation. For an unbiased Mclock. Such a formulation of proper time and mass in the regime estimator, like the proper time observable Tclock introduced in the of relativistic has been argued as necessary previous section, the Helstrom–Holevo lower bound4,5 places the by Greenberger41,42.

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Classical and quantum time dilation. Let us now consider two and the variance in such a measurement is  relativistic particles A and B, each with an internal degree of hHcmiÀhHcmi clock clock clock Δ 2 A B σ2: freedom serving as a clock, fH ; ψ ; H ; T g and hð TAÞ i¼ 1 À ð18Þ A A A A mc2 clock; ψclock ; clock; fHB B HB TBg. Suppose these clocks move through Ψ As might have been anticipated, the variance in a measurement of Minkowski space and are described by the physical state j ii, σ2 fi fi TA is proportional to , which quanti es the spread in the which satis es two copies of Eq. (1), one for each clock, as fi detailed in the “Methods” section. To probe time dilation effects ducial clock state. between these clocks we consider the probability that clock A Now suppose that the center-of-mass of both clocks are τ prepared in a Gaussian state localized around an average reads the proper time A conditioned on clock B reading the  Δ τ momentum pn with spread n >0, proper time B. This conditional probability is evaluated using the Z  2 Ψ ðpÀpnÞ physical state j ii and the Born rule as follows cm 1 À Δ2 ψ ffiffiffiffiffiffi 2 n :  ; ð19Þ n ¼ 1=4p dp e jipn ¼ jipn τ & τ π Δ prob½ŠTA ¼ A TB ¼ B n prob½Š¼T ¼ τ j T ¼ τ τ A A B B prob½ŠTB ¼ B 2 Δ2 : cm pn n Ψ τ τ Ψ ð14Þ for which hH i¼ þ . It follows that the observed average hh jEAð AÞEBð BÞj ii n 2m 4m ¼ Ψ τ Ψ hh jEBð BÞjii time dilation between two such clocks is  p2 À p2 þ 1 ðΔ2 À Δ2 Þ To evaluate this probability distribution note that the clock hT i¼ 1 À A B 2 A B τ : ð20Þ fi A 2 2 B states de ned below Eq. (2) form a basis dense in Ht, and thus a 2m c physical state may be expanded as Z If instead the two clocks were classical, moving with momenta p and p corresponding to the average velocity of the Ψ ψ A B j ii¼ dttji SðtÞ momentum wave packets of the clocks just considered, then to Z τ O ð15Þ leading relativistic order the proper time A read by A given that τ ¼ dttji U ðtÞ ψcm ψclock ; B reads the proper time B is n2fA;Bg n n n  γ 2 2 τ B τ pA À pB τ ; A ¼ B ¼ 1 À B ð21Þ where U ðtÞ :¼ eÀiHnt, H is the Hamiltonian given in Eq. (6), γ 2m2c2 n n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA and in writing Eq. (15) we have supposed that the conditional γ : 2 = 2 2 = ψ ψ ψ where n ¼ 1 þ pn m c . Therefore, upon comparison with state at t 0 is unentangled, Sð0Þ ¼ A B . Further Δ = Δ Eq. (20) and supposing that A B, quantum clocks whose suppose that the center-of-mass and internal clock degrees of center-of-mass are prepared in Gaussian wave packets localized ψ ψcm ψclock freedom of both particles are unentangled, n ¼ n n , around a particular momentum agree on average with classical ψcm cm where n 2Hn is the initial state of the center-of-mass of the time dilation described by special relativity. clock 2 R particle. Suppose that Hn ’ L ð Þ, so that we may consider It is natural to now ask: Does a quantum contribution to the clock time dilation observed by these clocks arise if the center-of-mass an ideal clock such that Hn ¼ Pn and Tn are the momentum clock of one of the clocks moves in a superposition of momenta? To and position operators on Hn . Such clocks represent a commonly used idealization in which the time observable is answer this question, suppose that the center-of-mass state of A sharp, that is, the clock states are orthogonal hτ jτ0i¼δðτ À τ0Þ begins in a superposition of two Gaussian wave packets with i i average momenta p and p0 , and so outcomes of differentÂà clock measurements are perfectly A A ϕ distinguishable. Note that T ; Hclock ¼ i, from which it follows ψcm  cos θjiþp ei sin θ p0 ; ð22Þ n n A A A that the effect operators satisfy the covariance relation where θ ∈ [0, π/2), ϕ ∈ [0, π], and jip and p0 are defined ji¼τ þ τ0 U ðτ0Þjiτ . We employ such clocks for their A A clock in Eq. (19). Further, suppose that the center-of-mass mathematical simplicity in illustrating the quantum time dilation degree of freedom of clock B is again prepared in a Gaussian effect, however we stress that for any covariant time observable, wave packet with average momentum p as in Eq. (19). on account of Eq. (14), a quantum time dilation effect is expected B Using Eq. (17), the average time read by A conditioned on B (e.g., see ref. 43). reading τ is By substituting Eq. (15) into Eq. (14), the probability that A B  τ τ reads A conditioned on B reading B can be evaluated to leading hT i¼ γÀ1 þ γÀ1 τ ; ð23Þ 〈 〉 A C Q B relativistic order in the center-of-mass momentum Pn/mc and cm= 2 internal clock energy hHn mc i where 2 2θ 02 2θ 2 Δ2 Δ2 τ τ À1 pAcos þ pA sin À pB A À B prob½ŠTA ¼ A j TB ¼ B γ :¼ 1 À À ; ð24Þ C 2m2c2 4m2c2 ðτ À τ Þ2  À A B cm cm 2 2 ð16Þ e 2σ2 hH iÀhH i τ À τ leads to the classical time dilation expected by a clock moving pffiffiffiffiffi A B A B ; ¼ 1 þ 2 1 À 2  0 2πσ 2mc σ in a statistical mixture of momenta pA and pA with probabilities cos2θ and sin2θ,and hi where hHcmi :¼ ψcm P2 ψcm =2m is the average nonrelativistic ÀÁÀÁ n n n n ϕ θ 0  2 02 2 θ cos sin 2 p À pA À 2 p À p cos 2 kinetic energy of the nth particle and we have assumed the γÀ1 : A A A ; clock Q ¼ 2 fiducial states of the clocks ψ to be Gaussian wave packets p0 Àp n 2 2 ϕ θ ðÞA A τ = σ 8m c cos sin 2 þ exp Δ2 centered at 0 and have a width in the clock state (i.e., 4 A position) basis. As described by this probability distribution, the ð Þ average proper time read by clock A conditioned on clock B 25 fi indicating the time τB is which quanti es the quantum contribution to the time dilation  between A and B. As expected, if either θ 2f0; πg or p ¼ p0 , cm cm 2 A A hHA iÀhHB i τ ; then the quantum contribution vanishes, γÀ1 ¼ 0. This is hTAi¼ 1 À B ð17Þ Q mc2 expected given that in these cases the center-of-mass of the

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a packets popt; as the total average momentum of the wave 4 0  = γÀ1 packets pA þ pA mc increases, the magnitude of Q and popt increase. γÀ1 θ Figure 2b is a plot of Q as a function of quantifying the 2 weight of each momentum wave packet comprising the super- –5 position in Eq. (22) for a fixed valueÀÁ of the difference in average momentum of each wave packet p0 À p =mc. It is observed × 10 ÀÁ A A

–1 0 À1 Q   = γ θ  0 that when pA þ pA mc ¼ 0, Q is positive for all values of and reaches its maximum value at θ = π/4. As the total average γÀ1 θ π momentum increases, Q decreases for 0 < < /4 and increases –2 for π/4 < θ < π/2 with the largest negative value at θ ≈ π/8 and largest positive value at θ ≈ 3π/8. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 ′ (p A – pA)/mc Quantum time dilation in experiment. Consider the two clock particles to be 87Rb atoms, which have a mass of m = 1.4 × 10−25 b kg and atomic radius of r = ℏ/Δ = 2.5 × 10−10 m. Suppose 4 A these clock particles can be prepared in aÀÁ superposition of 0  = momentumÀÁ wave packets such that pA þ pA mc ¼ p0 À p =mc ¼ 6:7 ´ 10À8, corresponding to each branch of the 2 A A

–5 momentum superposition moving at average velocities of  −1 0 −1 vA ¼ 5ms and vA ¼ 15 m s . We note that (classical) special

× 10 relativistic time dilation has been observed with atomic clocks –1 Q  0 moving at these velocities44,45, and perhaps the momentum superposition can be prepared by a momentum beam splitter realized using coherent momentum exchange between atoms and –2 light46,47. Supposing that θ = 3π/4 and ϕ = 0 results in γÀ1 À15 Q  10 . Assuming that the resolution of the clock formed by 0   3  the internal degrees of freedom of the 87Rb atoms is 10−14 s, 8 4 8 2 87 48  corresponding to the resolution of Rb atomic clocks , it is seen from Eq. (17) that the coherence time of the momentum super- p¯′ +p¯ p¯′ +p¯ p¯′ +p¯ A A A A = –0.03 A A = 0 position must be on the order of 10 s to observe a quantum time mc = –0.06 mc mc dilation effect. We note that the required coherence time is p¯′ +p¯ p¯′ +p¯ A A = 0.03 A A mc mc = 0.06 comparable to coherence times of the superpositions created in the experiments of Kasevich et al.49, which are on the order of Fig. 2 Magnitude of quantum time dilation. The strength of the quantum seconds. γÀ1 time dilation effect Q is plotted in a as a function of the momenta Concretely, one might imagine observing quantum time ðp0 À p Þ=mc  ¼ðp ; ; Þ 0 ¼ðp0 ; ; Þ difference A A , where pA A 0 0 and pA A 0 0 denote dilation in a spectroscopic experiment using the width of an the average momentum of the wave packets comprising the superposition emission line, which is inversely proportional to the lifetime of the state in Eq. (22) for θ = π/8, and in b as a function of superposition weight associated excited state, as a quantum clock. Indeed, it has recently θ ðp0 À p Þ=mc ¼ : for A A 0 17. Different values of the average total momentum been shown that the lifetime of an excited hydrogen-like atom ðp0 þ p Þ=mc Δ mc = A A are shown and A/ 0.01 in all cases. The thin black line moving in a superposition of relativistic momenta experiences p in plot a traces the trajectoryÀÁ of the optimal momentum difference opt for quantum time dilation in accordance with Eq. (25)43. Alterna- p0 þ p =mc 24 different total momentum A A . tively, Bushev et al. have proposed to use the precession of a single electron in a Penning trap as a clock to observe nonclassical relativistic time dilation effects, and it is conceivable that such a clock particle is no longer a superposition of momentum wave clock might be able to witness the quantum time dilation effect packets; see Eq. (22). From Eq. (24) it is clear that choosing Δ = Δ 2 2 2θ 02 2θ γÀ1 discussed here. Similar remarks apply to the ion trap atomic clock A B and pB ¼ pAcos þ pA sin results in C ¼ 0, and discussed in the ref. 32. thus any observed time dilation between clocks A and B would γÀ1 be a result of the quantum contribution Q . To illustrate the behavior of quantum time dilation stemming Discussion from the nonclassicality of the center-of-mass state of clock We considered the internal degrees of freedom of relativistic A, the quantity γÀ1 is plotted in Fig. 2.Forsimplicitytheone- particles to function as clocks that track their proper time. In Q doing so we constructed an optimal covariant proper time dimensional case is exhibited by supposing p ¼ðp ; 0; 0Þ and A A observable which gives an unbiased estimate of the clock’s proper p0 ¼ð0 ; ; Þ 0  A pA 0 0 with pA > pA.FigureÀÁ2a shows the behavior of time. It was shown that the Helstrom–Holevo lower bound4,5 γÀ1 0  = Q as a function of the difference pA À pA mc in the average implies a time-energy uncertainty relation between the proper momentum of each wave packet comprising the momentum time read by such a clock and a measurement of its energy. From superpositionÀÁ in Eq. (22) for different values of their total this relation, we derived an uncertainty relation between proper 0  = momentum pA þ pA mc. It is seen that quantum time time and mass, which provided the ultimate bound on the pre- dilation can be either positive or negative, corresponding to cision of any measurement of proper time. This yielded a con- increasing or decreasing the total time dilation experienced by sistent treatment of mass and proper time as quantum the clock compared to an equivalent clock moving in a classical observables related by an uncertainty relation, resolving past mixture of the same momenta wave packets. Further, there is an issues with such an approach41,42. The approach adopted here optimal difference in the average momentum of the two wave differs in that we construct a proper time observable Tclock in

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However, when the state of their center-of-mass is in a superposition of such localized momentum states, we demonstrated that a quantum time dilation effect occurs. We exhibited how this quantum time n dilation depends on the parameters defining the momentum superposition and gave an order of magnitude estimate for the size of this effect. We conclude that quantum time dilation may be observable with present day technology, but note that the Spatial/momentum degrees of freedom experimental feasibility of observing this effect remains to be i i (x n , P n) explored. It should be noted that the conditional probability distribution Fig. 3 Degrees of freedom of nth particle. The temporal degrees of ðx0; P0Þ ðxi ; Pi Þ n in Eq. (14) associated with clocks reading different times was a freedom n n and center-of-mass degrees of freedom n n of the th nonperturbative expression for clocks in arbitrary nonclassical relativistic particle are depicted. This particle carries an internal degree of freedom ðq ; P Þ that is used to construct a clock which measures the nth states in a curved spacetime. It thus remains to investigate the n qn effect of other nonclassical features of the clock particles such as particle’s proper time. shared entanglement among the clocks and spatial super- τ positions. In regard to the latter, it will be interesting to recover is the Lagrangian associated with the nth particle, n and mn denote respectively this particle’s proper time and rest mass, and Hclock ¼ Hclock ðq ; P Þ is the Hamiltonian previous relativistic time dilation effects in quantum systems n n n qn related to particles prepared in spatial superpositions and each governing its internal degrees of freedom. We use these internal degrees of freedom as ’ fi clock branch in the superposition experiencing a different proper time a clock tracking the nth particle s proper time. Note that Eq. (26)speci es that Hn – generates an evolution of the internal degrees of freedom of the nth particle with due to gravitational time dilation18 20,34,50. We emphasize that μ respect to its proper time. Let xn denote the spacetime position of the nth particle’s the quantum time dilation effect described here differs from these center-of-mass relative to an inertial observer, see Fig. 3. τ ’ μ results in that it is a consequence of a momentum superposition The differential proper time d n along the nth particle s world line xnðtnÞ, rather than gravitational time dilation. Nonetheless, it will be parametrized in terms of an arbitrary parameter tn,is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi interesting to examine such gravitational time dilation effects in μ ν dτ ¼ Àg x_ x_ =c2 dt ¼ Àx_ 2 dt ; ð27Þ the framework developed above and make connections with n μν n n n n n previous literature on quantum aspects of the equivalence prin- where the over dot denotes differentiation with respect to tn and we have used the 51–53 _ 2 : _ μ _ ν = 2 ciple . We also note that while we exhibited the quantum dimensionless shorthand xn ¼ gμν xnxn c . In terms of the parameters tn the time dilation effect for a specific clock model, based on the pre- action takes the form ceding analysis in terms covariant time observables it is expected X Z qffiffiffiffiffiffiffiffiffi ¼ À_ 2 ð Þ: that any clock will witness quantum time dilation. Given this, it S dtn xn Ln tn ð28Þ n will be fruitful to examine our results in relation to other models – This action is invariant under changes of the world line parameters tn, as long as 3,32,54 57 τ of quantum clocks that have been considered and there is a one-to-one correspondence between tn and n. This invariance allows for establish whether quantum time dilation is universal, affecting all the action to instead be parameterized in terms of a single parameter t, which is clocks in the same way, like its classical counterpart. connected to the nth particle’s proper time through a monotonically increasing τ = 73 Another avenue of exploration is the construction of relativistic function fn( n): t. Expressed in terms of the single parameter t , the action in Eq. (28)isS = ∫ dtL(t), where quantum reference frames from the relativistic clock particles 0 1 15,58–61 fi qffiffiffiffiffiffiffiffiffi considered here . In particular, one might de ne relational X B P q_ C : _ 2 @ 2 qqffiffiffiffiffiffiffiffiffin n clockA: coordinates with respect to a reference particle and examine the LðtÞ ¼ Àxn Àmnc þ À Hn ð29Þ n Àx_ 2 corresponding relational quantum theory and the possibility of n 62–69 changing between different reference frames . Related is the This Lagrangian treats the temporal, spatial, and internal degrees of freedom as perspective-neutral interpretation of the Hamiltonian constraint dynamical variables on equal footing described by an extended phase space in terms of which a formalism for changing clock reference sys- interpreted as the description of the particles with respect to an inertial observer. tems has recently been developed35,70–72. The Hamiltonian associated with L(t) is constructed by a Legendre transform of Eq. (29), which yields Xhi μ ν H :¼ gμνP x_ þ P q_ À LðtÞ Methods n n qn n Constraint description of relativistic particles with internal degrees of free- n  X qffiffiffiffiffiffiffiffiffiÀÁ ð30Þ dom. We present a Hamiltonian constraint formulation of N relativistic particles μ _ ν _ 2 2 clock ; ¼ gμν Pnxn þ Àxn mnc þ Hn with internal degrees of freedom. A complementary approach has been taken in n ref. 31. μ ’ μ Consider a system of N free relativistic particles each carrying a set of internal where Pn is the momentum conjugate to the nth particle s spacetime position xn fi fi degrees of freedom, labeled collectively by the con guration variables qn and their de ned as = … conjugate momentum Pq (n 1, , N), and suppose these particles are moving ∂ _ μ n μ : μν LðtÞ qxffiffiffiffiffiffiffiffiffin ; through a curved spacetime described by the metric gμν. The action describing such Pn ¼ g ∂_ ν ¼ Mn ð31Þ = ∑ ∫ τ τ xn _ 2 a system is S n d n Ln( n), where Àxn dq fi : clock= 2 L ðτ Þ :¼Àm c2 þ P n À Hclock; ð26Þ where we have de ned the mass function Mn ¼ mn þ Hn c , comprised of the n n n qn dτ n clock= 2 n nondynamic rest mass mn and the dynamic mass Hn c implied by mass-energy

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〈ψ ∣ψ 〉 = R equivalence. Upon substituting Eq. (31) into the Hamiltonian H, we see that each We demand that this state is normalized S(t) S(t) 1 for all t 2 . This term in Eq. (30) is constrained to vanish implies that the physical states are normalized with respect to the inner product14 qffiffiffiffiffiffiffiffiffi _ μ _ ν ÀÁ gμν xn xn 2 Ψ Ψ : Ψ Π Ψ ψ ψ ; H ¼ pffiffiffiffiffiffi M þ Àx_ m c2 þ Hclock hh j iiPW ¼hh j t ISj ii¼h SðtÞj SðtÞi ¼ 1 ð39Þ n Àx_ 2 n n n n n ÂÃÀÁÀÁ R x_ 2 ð32Þ for all t 2 . ¼ pffiffiffiffiffiffin m c2 þ Hclock À m c2 þ Hclock _ 2 n n n n Note that the set fgΠ ; 8 t 2 R constitutes a projective valued measure (PVM) Àxn Nt 0 0 δ 0 ∫ Π = 0 0  0; on the HilbertN space nHn, since htjt i¼ ðt À t Þ and dt t I , where I is the identity on H0 . Given this observation and the definition of the conditional ≈ 74 n n where means Hn vanishes as a constraint . Furthermore, using Eq. (31), the N state in Eq. (38), it is seen that the physical state jΨii is entangled relative to constraints in Eq. (32) can be expressed as cm clock Hn Hn , : μ ν 2 2 4 : Z Z CH ¼ gμν PnPnc þ Mnc  0 ð33Þ n jΨii¼ dt Π I jΨii¼ dttjiψ ðtÞ : ð40Þ This is a collection of primary first class constraints, which are quadratic in the t S S particles’ momentum and are a manifestation of the Lorentz invariance of the fi We emphasize that this entanglement is with respect to a partitioning of the action de ned by Eq. (26). 16 Similar to70,71, each of these constraints may be factorized as C ¼ CþCÀ, kinematical Hilbert space K, and thus is not physical (i.e., not gauge invariant) . Hn n n ± fi We consider physical states that satisfy where Cn is de ned as Âà þ Ψ Ψ ; ± : ; Cn j ii¼ ðPnÞ0 þ hn j ii¼0 ð41Þ Cn ¼ðPnÞ0 ± hn ð34Þ and for all n, where hn is the operator equivalent of Eq. (35). This amounts to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi demanding that the conditional state of the system has positive energy as measured h :¼ g Pi Pj c2 þ M2 c4: ð35Þ by hn. n ij n n n ψ fi fi We now show that the conditional state SðtÞ ,de ned in Eq. (38), satis es the 0 ; In Eq. (34) we have assumed the time-space components of the metric vanish, Schrödinger equation in the parameter t. Recall that ½xn ðPnÞ0Š¼i, and hence the g = 0. Such an assumption is not necessary, however, to illustrate the quantum 0 0 0i operators Pn generate translations in xn, time dilation effect we consider clocks in Minkowski space for which this is the case 0 0 Àiðt ÀtÞðPn Þ0 ; and hence we make this assumption for simplicity. tn ¼ e jitn ð42Þ By construction, the momentum conjugate to the nth particle’s spacetime μ ν μν 0 0 ; δ δ where jitn and tn are eigenkets of the operator xn with respective eigenvalues t coordinates satisfy the canonical Poisson relations fxm Png¼ mn. This implies μ and t0. Now consider how ψ ðtÞ changes with the parameter t: that the canonical momentum Pn generates translations in the spacetime S μ ± coordinate xn. Moreover, if it is the case that C  0, it follows that ðP Þ ¼ ±h , n n 0 n d ψ d Ψ which is the generator of translations in the nth particle’s time coordinate. Said i SðtÞ ¼ i ðÞ nhjtn IS j ii dt dt ! another way, ±hn is the Hamiltonian for both the center-of-mass and internal X ð43Þ degrees of freedom of the nth particle, generating an evolution of these degrees of 0 Ψ ; 0 ¼ hjðt PmÞ0 I:m IS jii freedom with respect to the time xn ¼ t, interpreted as the time measured by an m inertial observer employing the coordinate system xμ. 74–76 0 0 In what follows, we will employ Dirac’s canonical quantization scheme .We where I:m denotes the identity operator on all of the Hilbert spaces Hn for which promote the phase space variables of the nth particle to operators acting on n ≠ m and the derivative with respect to t was evaluated using Eq. (42). The 0 constraint Cþji¼Ψi 0 can be rewritten as appropriate Hilbert spaces: xn and ðPnÞ0 become canonically conjugate self-adjoint n operators acting on the Hilbert space H0 ’ L2ðÞR associated with the nth n ðP Þ I0 I ji¼ÀΨi I0 h I jiΨi ; ð44Þ ’ i m 0 :m S m S:m particle s temporal degree of freedom; xn and ðPÀÁnÞi become canonically conjugate operators acting on the Hilbert space Hcm ’ L2 R3 associated with the nth cm clock n for all m, where hm is the operatorN equivalent of Eq. (35) acting on Hm Hm ’ cm clock particle s center-of-mass degrees of freedom; and qn and Pq become canonically and I is the identity on ≠ H H . Substituting Eq. (44) into Eq. (43) n S:m n m n n clock conjugate operators acting on the Hilbert space Hn associated with the nth yields particle’s internal degrees of freedom. The Hilbert space describing the nth particle ! X 0 cm clock d is thus Hn ’Hn Hn Hn . i ψ ðtÞ ¼ hj t h I jiΨi The constraint functions in Eqs. (32) and (33) become operators C and C ± dt S m S:m Hn n mZ X acting on Hn. The quantum analog of the constraints is to demand that physical ¼ dt0 htjt0ih I ψ ðt0Þ ð45Þ states of the theory are annihilated by these constraint operators m S:m S Xm C ji¼Ψi CþCÀ ji¼Ψi 0; 8 n; ð36Þ Hn n n ¼ h I ψ ðtÞ ; m S:m S Ψ m where ji2Hi phys is a physical state that is an element of the physical Hilbert 77,78 space Hphys . The physical Hilbert space is introduced because the spectrum of where the second equality is obtained using Eq. (40). Equation (45) asserts that the C is continuous around zero, which implies solutions to Eq. (36) are not conditional state satisfies the Schrödinger equation Hn N normalizable in the kinematical Hilbert space K’ nHn. To fully specify Hphys a d fi i ψ ðtÞ ¼ H ψ ðtÞ ; ð46Þ physical inner product must be de ned, which is done in Eq. (39). Note that dt S S S because ½Cþ; CÀŠ¼0, it follows C jΨii¼0 if either Cþ jΨii¼0or P n n Hn n where H :¼ h I is the relativistic of all N particles. À Ψ S m m S:m Cn ji¼i 0. Leading relativistic expansion of the conditional time probability distribution. The Page–Wootters formulation of N relativistic particles. In this subsection, The Hamiltonian in Eq. (35) can be expressed as we recover the standard formulation of relativistic quantum mechanics with respect to a center-of-mass (coordinate) time using the Page–Wootters formalism. clock cm int; hn ¼ Hn þ Hn þ Hn ð47Þ To do so, the physical state jiΨi of N particles is normalized on a spatial hyper- surface by projecting a physical state jΨii onto a subspace in which the temporal where we have specialized to Minkowski space, dropped an overall constant mc2, 0 fi cm : 2 = degree of freedom of each particle is in an eigenstate state jitn of the operator xn and de ned the center-of-mass Hamiltonian Hn ¼ Pn 2m and the leading order R 0 0 relativistic contribution associated with the eigenvalue t 2 in the spectrum of xn, xnjitn ¼ ttjin . Expli- citly, ÀÁ 1 2 Hint :¼À Hcm Hclock þ Hcm ; ð48Þ Π Ψ ψ ; n 2 n n n t ISj ii¼jit SðtÞ ð37Þ mc N clock= 2 cm clock Π : which is derived by expanding Eq. (35) in both Pn/mc and H mc . where IS denotes the identity on H ’ H H , ¼ jit hjt is a projector n S n n n t Let us define the free evolution of the center-of-mass and internal clock degrees onto the subspace of H in which theN temporal degree of freedom of each particle is fi : of freedom respectively as in a de nite temporal state jit ¼ njitn associated with the eigenvalue t. Equa- fi cm cm tion (36)de nes the conditional state ρcm : ÀiHn t ρcm iHn t ; n ðtÞ ¼ e n e ð49Þ ψ : Ψ ; SðtÞ ¼ hjt ISj ii2HS ð38Þ and which describes the state of the center-of-mass and internal degrees of freedom of À clock clock ρ ðtÞ :¼ e iHn t ρ eiHn t ; ð50Þ N particles conditioned on their temporal degree of freedom being in the state jitn . n n

NATURE COMMUNICATIONS | (2020) 11:5360 | https://doi.org/10.1038/s41467-020-18264-4 | www.nature.com/naturecommunications 7 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-18264-4 for n ∈ {A, B}. Then the reduced state of the clock to leading relativistic order is equality in Eq. (53) and the results that follow. Nonetheless, a similar analysis  fi int int should lead to an analogous quantum time dilation effect that will be modi ed by ρ ÀiHn tρcm ρ iHn t nðtÞ¼trcm e n ðtÞ nðtÞe the specific details of the clock. The details of clocks described by discrete spectrum ð51Þ Hamiltonians and the associated covariant time observables have recently been hHcmi ÂÃ ¼ ρ ðtÞþit n Hclock; ρ ðtÞ : discussed in a related context16. n mc2 n n Using Eq. (51) the integrands defining the conditional probability distribution in Eq. (14) may be evaluated perturbatively Data availability Data sharing not applicable to this article as no datasets were generated or analysed. cm ÂÃ τ ρ τ ρ hHn i τ clock; ρ : trðEnð nÞ nðtÞÞ ¼ trðEnð nÞ nðtÞÞ þ it trðEnð nÞ Hn nðtÞ Þ ð52Þ mc2 fi ψclock clock Received: 20 December 2019; Accepted: 12 August 2020; Suppose that the ducial state n 2Hn of the clock is Gaussian with a spread σ, then the first term in Eq. (52)is ÀÁ clock 2 τ ρ τ ÀiHn t ψclock tr Enð nÞ nðtÞ ¼ hjn e n

2 2 Z ðτ0 ÀtÞ À e 2σ2 ¼ dτ0hτ jτ0i ffiffiffi References n π1 pσ ð53Þ R 4 1. Einstein, A. Out of My Later Years 39–46 (Wings Books, New York, 1996).

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