sign in sign up
<<
Home , Problem of time, Quantum clock, Quantum entanglement, World line

Downloaded by guest on September 27, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1616427114 a Ruiz Castro through Esteban of Entanglement ehnclefcsmdf u ocpino pc n ieand and quantum of how question conception open our an modify is effects (4). It mechanical 6). mod- intervals (5, in gravity relevant space–time quantum also of of are els time measurability of measurability the the to of investigat- Limitations for limits framework the important ing an provides context tivistic behave physical, relativity. being general and the- clocks, physical fundamental quantum the most ories: our that of principles assume the consistency, to to of according sake natural a the is in For This, (3). it etc. inconsistent...” point, is material sense, things, the certain other field, all (2) electromagnetic clocks, physi- that the and of fact] e.g., rods kinds measuring the (1) two [by i.e., introduces struck things, relativity]... cal is special “One [of Einstein: theory of arti- and the is words clocks therefore the and treat footing In equal not ficial. on the does systems with physical assumption of interact This rest not the . do the that objects of external rest are clocks the far-away that a it, time to measures next to who located not observer next but an sitting , observer. call given location. be will a We his/her to to so. at according need do clocks to not clock the does then the by observer observer recorded The the occurred. data Importantly, the time the the out the when and reads with event clock the them this to label by nearest of read and clock region the events a of the over coordinate spatial record locally, space– clocks clocks of places These split time, space. particular and a space events with into locate observer, to time An clocks of (2). latticework space–time a in uses frame picture reference common a A of (1). the hypothesis” clocks by way predicted “clock that a time so-called such proper believed theory—the the in with field is coincide metric readings It the their to lines. that correlate world lines world In along terms these clock. in time along locally a proper specified as is the acts time of relativity, which general sys- entity, of the context physical the between another correlation and a itself tem establishing means this tionally, A clocks quantum clocks. of limit classical relativistic the a in general recovered by the is However, measured time defined. as of well time notion not the Hence, is clock. clock loss single single entan- a a get to of leads necessarily eventually clocks which of back- effect, the dilation hold that metric time principles through show gled nonfixed both we that situation, a assumption this to the in on leads only eigenstates Based superpo- ground. mechanical energy quantum of the sition whereas rela- clocks, general a the interaction the most, between gravitational that at implies equivalence show is, mass–energy general picture we tivistic of Specifically, a principles such fiction. the mechanics, convenient obeys quantum that and clock to nearby relativity physical pointer along according a a clocks as time operationally, of of defined position of presence is time the flow if However, due by the lines. affected effects world and not gravitational is ignored clock ideal, ideal are one Being an clocks line. 2016) assigns these 4, world October space–time to review each for of (received to picture 2017 30, clock the January approved relativity, and PA, general Park, In University University, State Pennsylvania The Ashtekar, V. Abhay by Austria Edited Vienna, A-1090 Sciences, of Academy Austrian Information, Quantum and inaCne o unu cec n ehooy aut fPyis nvriyo ina -00Ven,Asra and Austria; Vienna, A-1090 Vienna, of University , of Faculty Technology, and Science Quantum for Center Vienna ngnrl h td fcok sqatmssesi rela- a in systems quantum as clocks of study the general, In considered conventionally is it picture, latticework clock the In o fsseswt epc otepsaeo ie Opera- time. of passage the to respect with behav- systems the describe of to is ior theory physical any of aspect crucial | entanglement a,b,1 lmnaGiacomini Flaminia , | gravity | lsia limit classical a,b and , alvBrukner Caslav ˇ 1073/pnas.1616427114/-/DCSupplemental at online information supporting contains article This option. access open 1 PNAS the through online available Freely Submission. Direct PNAS a is article This interest. of conflict and F.G. no from declare input authors with The paper the wrote E.C.R. and research; formed oei rae.Fnly ercvrtecasclnto ftime of notion black classical a the before recover magnitude we of Finally, orders created. regime will is a We Moreover, hole in too. holes. that effect, significant this black argue is of of independent 6) it is creation (5, effect our the space–time that from on show to arises has due limitation itself time (4). of clock the fixed measurability the be of effects to lack the assumed the the is regarding than metric works space–time Other the different which a in of ones is considerations, the relativistic by eral recorded as time of measurability the clocks. fundamen- are to there limitations that entangle- tal implying number to clocks, the leads nearby effect which between the in ment Moreover, limit, in conserved. velocities is already slow of is and effect gravity world this weak nearby that the along show evolving ill-defined.We of clocks is consequence of a lines dilation as time that, the prove fact, We this superposition. space–time a the in therefore eigenstates, is and energy metric, vicinity, of field. its in superposition gravitational field a gravitational different in the a runs to clock eigen- the corresponds energy Because clock of each the limit fact, of In the . state in time nonnegligible of become state precision the effects high to These corresponding clock. energies the the of the from arise between gravita- effects equivalence, difference tional mass–energy the energy to one, the Due orthogonal (7–11). super- to an eigenstates a into proportional the evolves in inversely state as system the is understood a which precision, in is Its time clock minimal eigenstates. quantum energy a of feature, position lines. world general nearby along a clocks joint As by the given to as limits time fundamental of put measurability clocks the of properties tional quantum neglected. where be limit can the effects in obtained mechanical is conception usual the how uhrcnrbtos ... .. and F.G., E.C.R., contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To ftm ngnrlrltvt mre rmti iuto nthe in situation this from limit. notion emerges classical the relativity how general in show time We of equivalence. principle superposition mass–energy the the clock, of the and consequence of necessary constitution a particular that is the sense and on the clock depend in the not fundamental, of does is an effect it location for This the vicinity. possibilities its at in intervals the and time between define trade-off to a observer to equiv- leads mass–energy the alence eigenstates, a energy in be of must mechanics superposition clock quantum quantum any between Because relativity. general interplay and the from trajecto- joint arising space–time the neighboring ries, to along limitations time fundamental of exist measurability there that find We Significance h iiain tmigfo unu ehncladgen- and mechanical quantum from stemming limitation, The gravita- and mechanical quantum that show we work, this In a,b PNAS | ulse nieMrh7 2017 7, March online Published ..dsge eerh ... .. and F.G., E.C.R., research; designed C.B. ˇ . b nttt o unu Optics Quantum for Institute www.pnas.org/lookup/suppl/doi:10. C.B. ˘ | E2303–E2309 ..per- C.B. ˇ

PHYSICS PNAS PLUS measurement in the limit where the clocks are increasingly of this section, and hence we will treat here only the two-level large quantum systems and the measurement precision is coarse case. This two-level clock model does not aim to describe all of enough not to reveal the quantum features of the system. In this the features involved in time , like, for example, way, we show how the (classical) general relativistic notion of the reconstruction of the “flow of time” from repetitions of mea- emerges from our model in terms of the average surements (4, 14). Our intention in this section is to point out mass–energy of a gravitating quantum system. the minimal requirements for a system to be a clock, i.e., that From a methodological point of view, we propose a gedanken the system must be in a superposition of energy eigenstates. It experiment where both general relativistic time dilation effects follows from these requirements that the orthogonalization time and quantum superpositions of space– play significant is inversely proportional to the energy gap of the clock. A more roles. Our intention, as is the case for gedanken experiments, is elaborate model of a clock, that addresses the issue of repetitive to take distinctive features from known physical theories (quan- measurements, will be considered in Clocks in the Classical Limit, tum mechanics and , in this case) and explore when studying how the general relativistic notion of time dilation their mutual consistency in a particular physical scenario. We emerges in the classical limit. believe, based on the role gedanken experiments played in the The gravitational effects due to the energies involved are to early days of and relativity, that such consid- be expected at a fundamental level. In particular, for a given erations can shed light on regimes for which there is no complete energy of the clock, there is a time dilation effect in its surround- physical theory and can provide useful insights into the physical ings, due to the mass–energy equivalence. However, because effects to be expected at regimes that are not within the reach of the mass–energy corresponding to the amplitude of |0i is dif- current experimental capabilities. ferent than that corresponding to |1i, the time dilation in the vicinity of the clock in the state given by Eq. 2 is uncertain Clock Model (Fig. 1). Consider a second clock localized at a coordinate dis- Any system that is in a superposition of energy eigenstates can be tance x from the first clock (in the reference frame of the before- used as a reference clock with respect to which one defines time mentioned observer). Due to time dilation, this clock would run evolution. The simplest possible case is that in which the clock is as in flat space–time for the amplitude corresponding to |0i, a with an internal degree of freedom that forms a two- and it would run (to second-order approximation in c−2) as level system. In the following, we assume the clock to follow a t→t + ∆t = t 1 + G∆E/(c4x), for the amplitude correspond- semiclassical that is approximately static, that is, it has ing to the |1i. Here, G denotes the gravitational (approximately) zero velocity with respect to the observer who constant, and t can be operationally defined as the uses the clock to define operationally his/her reference frame, in of the observer, who is sufficiently far away from the mass-energy the sense stated in the Introduction. In this way, special relativis- distribution so that the effects of the different gravitational fields tic effects can be ignored. We stress the fact that the observer originating from the two states of the clocks are indistinguish- does not need to be located next to the clock. He/she can per- able at his/her location. This observer ensures that the coordi- form measurements on it by sending a probe quantum system to nate distance x between the first observer and the clocks is kept interact with the clock and then measuring the probe in his/her fixed. In SI Appendix, Analysis of the Coordinate t, we quantify location. In the following, we focus only on the clock’s inter- the minimum distance between the observer and the clocks such nal degrees of freedom, which are the only ones relevant to our that he/she cannot operationally distinguish between the differ- model. The internal Hamiltonian of the particle in its rest refer- ent gravitational fields. ence frame,

Hint = E0|0ih0| + E1|1ih1|, [1] generates the evolution of the clock. For convenience, we choose the origin of the energy scale so that E0 = 0, and we define ∆E = E1 − E0 = E1. An operational meaning of the “passage of a unit of time,” in which, by definition, the system goes through a noticeable change from an initial state to a final state, can be given in terms of the “orthogonalization time” of the clock, that is, the time it takes for the initial state to become orthogonal to itself. For a two- level system, the orthogonalization time is equal to t⊥ = }π/∆E (12). Note that t⊥ quantifies the precision of the clock, and it is, in this sense, a measure of time uncertainty. The optimal initial state of the clock is one with an equal superposition of energies, which we choose to be 1 |ψin i = √ (|0i + |1i). [2] 2 Fig. 1. Pictorial representation of the fundamental trade-off between uncertainty of time measurement by a given clock and uncertainty of time For this state, the optimal measurement to determine the pas-√ measurement by nearby clocks. The clock at the frontal plane of the picture sage of time is given by projectors in the |±i = (|0i ± |1i)/ 2 has a relatively high accuracy, depicted by its sharply defined hands. The . It is important to stress that the relation between orthogo- uncertainty of time reading for this clock is inversely proportional to the nalization time and energy difference is fundamental: Any clock energy gap ∆E of the internal degree of freedom that constitutes the clock model has a precision limited by the difference of energies (Clock Model). By the mass–energy equivalence, the energy of the clock will involved in the time measuring process. This fact was already produce gravitational time dilation effects on nearby clocks. Because the noticed in earlier works (4, 10). It is this feature, also shared by energy is not well defined but has an uncertainty ∆E, nearby clocks will have an uncertainty in their time dilation with respect to the main clock, as more-detailed clock models (13), that plays a fundamental role depicted by the “fuzzy” hands in a superposition. There exists, therefore, in this work. The fact that the clock can return periodically to its a limitation to the possibility of defining time accurately at nearby points, initial state and therefore give ambiguous time readings can be given by the joint effects of quantum mechanics (superposition principle) dealt with by choosing a more elaborate clock model, e.g., a sys- and general relativity (gravitational time dilation). This effect is fundamen- tem with more energy levels. This fact is irrelevant for the result tal and independent of the energy gap ∆E of the clock, as stated in Eq. 3.

E2304 | www.pnas.org/cgi/doi/10.1073/pnas.1616427114 Castro Ruiz et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 n n ou nyo h fetdet h nenldgesof degrees internal is the system two-clock to the due for effect Hamiltonian dynamical the the the Thus, on with freedom. only compared focus small and negligibly one is mass static equiva- the mass–energy the be using and can lence: operators interaction to particle The masses each the probed. promoting of by is terms mechanical system quantum in the described energy the which of matter with a scale effectively, is, view, distinction interaction of of their and and mass point constituents energy, between relativistic difference the a no conceptually of from is fact, there interaction In the particle. composite from our arises and freedom of nature, degrees internal the H of energy the gravitational mass to the rest corresponding static to both contribution including mass–energy field, whole the mean particles of labels masses The manner. be same can the corrections in solu- post-Newtonian analyzed the However, to metric. approximation the first-order for the tion on only focus we by work, described energy is gravitational Newtonian interaction the the of gravitational solution the the equations, to approximation Einstein lowest a To observer). away by labeled B clocks, interacting gravitationally two Consider between Clocks Two interaction gravitational of terms clocks. in quantum effect above the In nonoperational. and classical been has constitution. particular its quan- or clock both the of to gap indepen- energy due holds the It of arises effects. dently relativistic that general relation and mechanical uncertainty tum an is which atoRi tal. et Ruiz Castro is observer far-away the to lated: H for freedom. (18), of dynam- Isham degrees the and fundamental from more emerges (17) of itself Rovelli time ics that of suggested works is it the the- example, In . fundamental of a ory require back- no would with space–time physics the ground gravi- of description from here full arising a clocks presented interaction, of tational methods entanglement as the the describe equivalence, to Although suffice mass–energy Approach. the Theory of of Field sector use two-particle the in sketch the the- and we field to (16) a restriction field from the the obtained by be perspective can ory Hamiltonian same The nian. Eq. equivalence, of mass–energy the discussion see and heuristic principle a superposition For the 15. on is and relativity 12 freedom general refs. perturbative of in in given degree evolves internal particle quantum the a how of regarding derivation full A at dilation the time by described of succinctly uncertainty be can the relation trade-off and This the points. clock at nearby the time measuring of of accuracy location the between trade-off mental ˆ ˆ int B eaae yacodnt distance coordinate a by separated , ofr u ramn ftm iaini h iiiyo h clock the of vicinity the in dilation time of treatment our far, So e sasm htteeege fbt Hamiltonians both of energies the that assume us Let funda- a is there considerations, these of consequence a As IApni,HuitcDrvto fteToCokHamilto- Two-Clock the of Derivation Heuristic Appendix, SI r qa n htteiiilsaeo h lcsi uncorre- is clocks the of state initial the that and equal are hsnto fdnmclms so ueyrelativistic purely a of is mass dynamical of notion This . | m ψ in →m i = IApni,ToCokHmloinfo Quantum from Hamiltonian Two-Clock Appendix, SI | + ψ  i H (|0i ˆ = int H ˆ /c √ + A = 1 2 2 |1i)/ and H o esn fsmlct,w suethat assume we simplicity, of reasons For . ˆ  t A |0i|ϕ ⊥ + ∆ B √ t H epciey yteemse,we masses, these By respectively. , 2 ˆ 0 =  B i ⊗2 e + − π h tt ttime at state The . c }Gt U 4 c − G x m 4 (x it x } x , ∆E n h yaia mass, dynamical the and H = ) ˆ i h rm ftefar- the of frame the (in m eexplain we Clocks, Two A | A H 1i|ϕ ˆ −Gm B and . 1 i m A  m , B B t ee othe to refer /x according H 4 nthis In . ˆ A A based and and [3] [5] [4] form for between argument symmetric is ation in |ψ order first of to value time expectation proper is the the clocks ing of both derivative of clock the state for initial equation the tion that assume |ψ We value. clock clock tion of To of evolution gravity. the influence semiclassical describe the in us a done let as is point, this as clocks expec- explain energy, the the the account of of into value taking energy tation just internal of instead the , consider quantum we that crucial eeaiaino h neatn aitna of write Hamiltonian consider and interacting clocks the can of of we generalization pair a effect, any between the distance on coordinate a largest bound other of the lower functioning of a presence the the give how by affected of is question clock the single the in discussed ask sense We the in Introduction. frame reference distance a constitutes coordinate clocks the by characterized are space there suppose Now N for consider clocks, next interacting we tationally effect, becomes the holes strengthen black To of relevant. formation before long occurs we that As regime 6). (5, in formation hole see measurability black will to the due of intervals limitations space–time of of independent concerning is argument It limit. usual weak-field the the mechanics in quantum relativity from general only and follows that time of measurement the of unitarity the system. of composite consequence the a of is evolution fact This again. increase clocks. in of see limit will clock classical We the one clock. of Limit other dilation Classical the the time of in energy overall entan- mean an the get is to not result due do only clocks the the and approach, gled, semiclassical the in that, labels the with but H etbetween ment H contributes that terms the Hamiltonian Specifically, the clocks. between of entanglement the part to any a in of enter of value the not presence the would for alter the it not clock that would Hamiltonian the Note the in answers. “ask mass random static we only when get because, func- we clock, to time,” able proper longer a no is as clock tion the and state mixed maximally the approach as expressed is time this units, t Planck in variables dimensionless time the Using for state entangled The clock. maximally other the gets of energy the clock of one value the in to runs correlated time is which at rate The interaction: gravitational e ∆E where ˆ ˆ P −i A B in i ti motn opitotta,frti fett rs,i is it arise, to effect this for that, out point to important is It h fetpeetdhr a udmna nuneo the on influence fundamental a has here presented effect The after that, Note esefo Eq. from see We Clocks 1 + = B , t ⊗ /E ∆E i ~ H l ˆ exp = = |ψ P H A ˆ P ( /c | i 1− B and , ψ ⊗ = t |ϕ in stePac ie and time, Planck the is does. G mix h feti infiati parameter a in significant is effect the Clocks, 1 + N m |ψ ∆E i 0 −it A i B ξ i h eue tt fayo h lcsapproaches clocks the of any of state reduced the , / A A (|0i = A and 1, c |ψ ⊗ = 4 and  n ban fe vlto,ajitsaeo the of state joint a evolution, after obtain, and x |ψ ⊗ H ˆ x ) + 1 c |1i]/ /l −2 A = t in mix P B htti iuto sefcieyrcvrdin recovered effectively is situation this that e + i and hrfr,tesaeof state the Therefore, . hH i 5 where , m a X B n ontchange not do and , B u with but A, N =0 √ PNAS τ h uiyo h eue ytmwill system reduced the of purity the , where , A httecok e nage through entangled get clocks the that olwn es 2ad1,teevolu- the 15, and 12 refs. Following . − A mix B N 2. 1 i/(c B H it } ˆ 1 + ∆E is ⊗ a = necagd hnw aeshown have we Then interchanged. | A H − l 4 P i}∂ m πξ/ε | A ulse nieMrh7 2017 7, March online Published x 1i)/ lcscnandi einof region a in contained clocks and |ψ N = c B ehave we , E ) G 4 t  1 i P /} |ψ p  x A τ 2 H √ ontcet n entangle- any create not do = B nPac ieuis swe As units. time Planck in , a X i  }G a h aefr of form same the has 1. A ihrsetto respect with and 2, B

PHYSICS PNAS PLUS where the indices a and b label each of the individual clocks. Planck scale in the possibility of defining time, see refs. 5, 19, In this part, we concentrate only on the interacting part of the and 20). Fig. 2 shows the decoherence time td as a function of Hamiltonian, because we wish to analyze the loss of coherence of the energy gap ∆E and the distance x for a macroscopic num- the reduced state of a single clock. We therefore analyze the evo- ber of particles N = 1023. Despite the fact that the effect is very lution in the . For an initial state of the form small with respect to the regimes of current atomic clocks, it is  √ ⊗N +1 |ψin i = (|0i + |1i)/ 2 , the reduced state of the zeroth important to analyze the order of magnitude of the limitations clock is from a conceptual point of view. For instance, for a distance x ≈ 10−13    2 N  cm (the order of magnitude of the charge radius of −i τε 1 1 1 + e ξ a proton), an energy gap of ∆E ≈ 10 GeV, which is compara- 1  2  ρ0 = . [7] ble to, for example, the energy of the nuclear bound state of a   2 N  − 4 2  1 i τε  K particle in He (21), and a macroscopic number of parti- 1 + e ξ 1 23 2 cles N ≈ 10 , we find td ≈ 80 s. The important point is that the regime of these parameters is several orders of magnitude away Interestingly, the time for maximal mixing is independent of N from the Planck scale. It is also important to note that, for these and is equal to τmix from the two-clock case. However, coherence 2 values of ∆E, N , the Schwarzschild radius r = 2GM /c [where can be significantly reduced for times earlier than τmix . To quan- 2 M = E/c and E = N ∆E is the total energy] is of the order of tify this, we use the visibility V , defined by (twice) the absolute −29 10 m, so that the effect we predict is orders of magnitude value of the nondiagonal element of the density operator. away from the regime where a is formed. In our case, To end this section, we note that, despite the fact that this  1  τε2 N effect is not large enough to be measured with the current exper- V = 2|(ρ ) | = 1 + cos 0 12 2 ξ imental capabilities, it might be possible to perform experiments on analog systems to test this effect. Specifically, in ref. 22, the √ 2  √ 2 2  N τε2  − N τε authors consider an atom traversing an oscillating quantum ref- ≈ 1 − ≈ e 2ξ , 2ξ erence frame, and show that the phase of the of the √ atom has an uncertainty that can be related to the uncertainty in 2 for τ  2ξ/( N ε ). From here, we identify a decoherence time the atom’s elapsed proper time. By the , it that is, back in the initial units, is possible to interpret the acceleration that the oscillating refer- 4 2}c x ence frame induces on the atom as the gravitational effect that td = √ . [8] one clock suffers as a consequence of the presence of another NG(∆E)2 nearby clock. This characterizes the fundamental limit on the time after which quantum clocks lose their ability to measure time when their Clocks in the Classical Limit gravitational effects are taken into account. We now give an estimate of the parameter regime where deco- Given the ill-definedness of time measured by a single clock herence is significant. The calculations are done ignoring all when it is in the presence of other clocks, how does the classi- effects external to our model and should be understood in terms cal notion of a clock, including relativistic time dilation effects, of a gedanken experiment. The intention is to contrast the pre- arise? In what follows, we answer this question by considering dictions given by our model with the usual predictions given by the classical limit of our model. The that is clos- quantum gravity models, which do not expect limitations due to est to the classical state of a clock is a or atomic coher- the combined effects of quantum mechanics and general relativ- ent state. In general, spin coherent states can be defined as iϕ ⊗2j ity before the Planck scale. (For a discussion of the role of the |ϑ, ϕ, j i = cosϑ/2|0i + e sinϑ/2|1i and can be understood

1. 1016

1. 1011

1. 106

1. 101

Fig. 2. Clock decoherence time td of Eq. 8 as a function of the clocks’ energy gap (∆E) and the separation between clocks (x) for a macroscopic number of particles N = 1023. The dotted lines show three different decoherence time regimes for different scales of ∆E and x: 1017 s (the order of the age of the universe), 107 s (the order of 1 year), and 100 s. Note that the blue region, showing relatively short decoherence times, corresponds to energies and distances far from the Planck scale regime, suggesting a breakdown of the measurability of time at larger distance and lower energy scales.

E2306 | www.pnas.org/cgi/doi/10.1073/pnas.1616427114 Castro Ruiz et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 h lcseeg ept h oreeso h measurement. the of coarseness the despite emerge clocks clock the the of energy average the by these defined is to measurements corresponding (POVM) sys- measure the operator-valued resolution measure- experimental itive of time the features coarse-grained by consider is characterized quantum we ments that case, the our resolution reveal In (23). experimental to tem an not on enough based coarse is mechanics tum the of equivalence. mass–energy nonnegativity the considering the when ensuring mass nonnegative, is Hamiltonian where is Hamiltonian ∆ momen- the angular in the operator of tum terms In above. presented Hamiltonian angular to extension atoRi tal. et Ruiz Castro For (B) of spread. functions Husimi average resolution the The experimental evolve, functions. they finite Husimi As the different point. at represents the same function bins of the Husimi at separation the centered the of the of are probability R representing and size functions the clock circle Husimi finite gives the the the The that of of of POVM circle. All motion measurement color condition: the average time the divide the as a darker limit: that grows consider clock classical The We bins the the location. rate. the that to dilation of at contribute one time pointer that in different clock effects the state a finding clock at of the precesses probability finding mixture the the higher the in location, state one Each evolves. it as (Eq. distribution binomial spins a with by clocks state coherent Two surements. 3. Fig. such bin, particular a in clock the behaves of clock state mea- the the coarse-grained finds and a that of after regime, surement that, nature this also quantum in Note classically. the visible effectively not to are due Con- system fluctuations negligible. the the becomes bin of one all than more sequently, in found be to clock R to proportional is tion matrix density the of state Q coherent spin yields is bin a sphere measurement for Bloch a that the probability probability dividing the as the POVM along this picture into can We state: coherent spin a the in that is assume clock We the [(|0i representation. of sphere state Bloch initial a admits angles a polar of terms in  ρ E  n fteapoce otecascllmtfo ihnquan- within from limit classical the to approaches the of One the is clock this of state the evolves that Hamiltonian The = ϕ) (ϑ, j ( B −1/2 1 + j / R 2π/ p −1/2 21 k M |1i)/ mrec ftecasclnto fgnrlrltvsi iedlto rmteasmto fcok nchrn ttsadcas-rie mea- coarse-grained and states coherent in clocks of assumption the from dilation time relativistic general of notion classical the of Z ˆ l ftechrn ttsleisd n i n o“unu utain”ocrwe esrn ie The time. measuring when occur fluctuations” “quantum no and bin one inside lie states coherent the of all , TrM = − k = = drcin nteeutro h pee Then, sphere. the of equator the in ϕ-direction, u still but Z ,ϕ hϑ, ˆ P √ bn. h oeetsaeeovsb moving by evolves state coherent The “bins.” hrfr,fraspin- a for Therefore, ). ϑ 2j 2] −j j k j j 4π A and ⊗2j ρ 1 + | hra h fetv eaaino h oeetsae spootoa to proportional is states coherent the of separation effective the whereas , ρ m pnpitn ntedrcingvnb the by given direction the in pointing spin = | ,ϕi ϑ, j = |m 2j H Z  hspcuei ovnetbcueit because convenient is picture This ϕ. z ˆ 4π j 0 |ϑ h hrceitcwdho hsfunc- this of width characteristic The ρ. +1 free ih −1/2 π direction, ABC h rbblt o h one fthe of pointer the for probability the 1, m = dϑ steHsm ucin or function, Husimi the is R .TeHsm ucino ahchrn tt srpeetdb lecrl htpeessaogtebakcircumference black the along precesses that circle blue a by represented is state coherent each of function Husimi The 12). ∆E = oeta h pcrmo this of spectrum the that Note |. 0 π π ( A. n,teeoe ntergm where regime the in therefore, and, , {M / sin R ϕ 2, ( C kR k k For ) −1)R ϑ } (j 0 = k 2π/ Z Z ˆ =1 j 1 ( t Eq. , k kR j , R − −1)R dϑ ftetolvl(j two-level the of > j ytm h corresponding the system, with , i. Z t ˆ dϕsinϑ ∗ ), j ifrn uiifntosocp ifrn is n h fet fteqatmetnlmn between entanglement quantum the of effects the and bins, different occupy functions Husimi different , 1 A ρ ,ϕihdϕ|ϑ, and switnas written is ob nthe in be to k j Q B nt,ie,the i.e., units, neatgaiainly h eue tt o the for state reduced The gravitationally. interact ρ where ϕ), (ϑ, ,ϕ|. ϑ, R Q h pos- The . function, |ψ 1 = H ˆ in int [10] k /2) i [9] th = = ihec te.Frfl eeaiy espoeta the spin that total suppose with we system a generality, is gravitationally full clock interacting For and other. state each coherent with a in initially being ino h eincrepnigt h i ilntatrthe alter not (24). continue will measurement effectively after bin will behavior classical the clock its the to Therefore, corresponding significantly. region state the on lies size that tion of it of bin part the the because outside nonperturbed, effectively is state a where the for nFg .Frt hr ste“vlto ftecoka a as clock the of “evolution clock the of the phase average is the depicted of there movement is ϕ the First, situation is, that The 3. whole,” limit. Fig. classical the in defines that of eters evolution the to tributions Eq. phases of mixture the in states 12 coherent the interaction, This of bin. clocks presence one noninteracting inside for only also zero applies from different Eq. significantly of is state POVM coarse-grained on the depends by When states measured coherent is these them state of of The width each typical states, The phase coherent tion. of a sum with a evolving of consists It equation. |ψ where ato h vrg hs spootoa to proportional is dilation time phase the average of evolution the the that of hand, ill- part one to the therefore leading on effects, and two note, other, these we state quantify each To reduced measurements. clock the time from of of the definedness spread mixing of to a to energy tend evolves eventually states average detection, coherent the for ond, probability to highest due the time-dilated with pointer the j A in osdrnwtocok,lbldby labeled clocks, two now Consider hr r w fet,dfeeti aue hs eaiecon- relative whose nature, in different effects, two are There ρ vlewt ifrn iedlto atr,gvnb aho the of each by given factors, dilation time different with evolve B = i = = − j ϕ B ϕ B |ϑ H ˆ t k k 4 ∆E slreand large is A ~ 1 lc ttime at clock . j = = A π − X k (1 ( 2j / p =0 H ˆ ϕ 2, A t j − ∆E B A ~

aeteform the have ) H lcsi h lsia Limit Classical the in (Clocks ˆ Gj 0 = 2j (1 k A = A R ∆E − ! , H j  ˆ t PNAS R A ϕ

/ Gk A ϑ B is c i⊗ k 4 j sngiil n,teeoe projec- a therefore, and, negligible is lc sasmo oeetsae modulated states coherent of sum a is clock + ouae yabnma distribu- binomial a by modulated , B = ∆E x −1/2 hspae hc orsod to corresponds which phase, This ).

H | ϑ ˆ / π 2 c ρ B = ulse nieMrh7 2017 7, March online Published ϕ , 4 ti xetdta ahcoherent each that expected is it , j B A x − π e saayecoeythis closely analyze us Let ). 2 k B ieu h eieo param- of regime the us give (j ϕ , , c t lc stm-iae codn to according time-dilated is clock B j G 4  B .Tefl aitna is Hamiltonian full The ). x o h nta state initial the For 9. A 0 = ED t H ∗ ˆ A and .In ). A = ϑ , j H and ˆ B }c = B edpc h initial the depict we A, B

4 h eue state reduced the , , x π 2 j oee,i the in However, . /[G B A ϕ , nteother the On . aho them of each , hr r two are There R. p k , 2j j A B (∆E

, Sec- A. | A ) 2 E2307 ] [12] [11] (B and 10. j B ) .

PHYSICS PNAS PLUS hand, despite the fact that the angle separation of the coher- heuristic arguments based on the superposition principle and 2 2GjA(∆E) t 4 ent states ∆ϕ = ϕ2jA − ϕ0 = /c x} is also proportional gravitational time dilation to our treatment of interacting to jA, not all of the terms in 12 contribute significantly to the clocks in the classical limit. Consider the two-clock sce- −j 2j A A nario at the beginning of this section with jA = 1/2 and state, due to the binomial distribution p(k) = 4 k . Indeed, for large jA, p(k) can be approximated by a Gaussian distribu- jB  1. We will analyze the limit in which the time dila- 2 q 1  k−j  tion of clock B due to clock A is significant [that is, tion, that is, p(k) ≈ exp √ A , which has a characteris- 4 πjA jA G∆EA/(c x) is nonnegligible], but the time dilation effect √ 4 tic width proportional to jA. This means that the√ effective angle on A due to B can be neglected, i.e., GjB ∆EB /(c x)  1. separation between coherent states grows with jA, rather than Let us focus first on clock B. Its reduced state after evo- √ π G 2j (∆E)2t 1 2 π 2 A lution is given by ρB = / (|ϑ = 2 , ϕ0, jB ihϑ = / ,ϕ0, jB | + with jA; that is, ∆ϕeff = 4 . Therefore, for times much (~c x) π π t∆EB |ϑ = /2,ϕ1, jB ihϑ = /2,ϕ1, jB |), with ϕk = − (1− smaller than a characteristic time 4 ~ Gk∆EA/c x), k = 0, 1. We now define the operator

4 Z π Z 2π c x j (2jB + 1) ∗ } T B = } dθ sin θ dφφ |θ, φ, j ihθ, φ, j |. [15] t = √ 2 , [13] B B G 2jA(∆E) 4π∆EB 0 0 Physically, this operator represents the pointer position of clock ∗ say t = Γt , where Γ  1, the angle separation will grow as B and has physical of time. In the limit jB →∞, 4 q t ∗ c x jA spin coherent states are orthonormal, and, therefore, the state ∆ϕeff = /t = Γ, but we will have ϕj = − √ Γ + Γ, A G∆E 2jA 2 π jB √ ϑ = 2 , ϕk , jB becomes an eigenstate of T with eigenvalue which grows as jA for large values of jA. Therefore, for this scal- }ϕk /∆EB , for ϕk ∈ (0, 2π) and k = 0, 1. Using this fact, it is ing with respect to jA and in the limit where jA  1, we reach the easy to show that, in this limit, the variance of the operator T jB regime where the classical limit of clocks holds, because entan- is given by glement is negligible at these scales. If, apart from this char- ∗ t j G∆EAt acteristic time , we have a coarse enough measurement, i.e., ∆T B = } (ϕ − ϕ ) = . [16] −1/2 2∆E 1 0 2c4x R  jB , measurements of time will detect time dilation, as B classical general relativity predicts, but with no “quantum fluc- On the other hand, the probability of measuring one unit of tuations.” Significantly, the time dilation factor corresponds to time on clock A is given by the operator TA = }/∆E |−ih−|. A A the average energy of clock , consistent with the semiclassical Operationally, the time it takes for the average of TA to approximation to gravity in the quantum domain. change significantly is given by dTA = ∆TA/(|dhTAi/dt|), The evolution of the reduced state ρB can also be studied in where the bars denote absolute value. We can terms of its master equation. Following ref. 15, where a full treat- now compute dTA for the reduced state of clock A, ment of the master equation for systems of particles evolving ED 1 P2jB 2jB  π π ρA = ϑ = /2,ϕk , 1/2 ϑ = /2, ϕk , 1/2 , with in the presence of relativistic time dilation is given, the master 4jB k=0 k t∆EA Gk∆E 4 equation in this case can be written as ϕk = − (1 − B/c x). Because, by assumption, the time ~ dilation effect of clock B on clock A is negligible, we take into     account only the ϕ0 contribution to dTA, yielding the result dρB i ˆ GjA∆E = HB 1 + 4 , ρB dTA = /∆EA. Putting the pieces together, we get dt } c x } r !2 j Gt j G∆E Z t h h i i dT ∆T B = } , [17] A ˆ ˆ A 4 − 4 ds HB , HB , ρB , [14] 2c x 2 2c x 0 s which coincides with Eq. 3 up to a factor of π/2.

h i ˆ h i ˆ −is/ HB is/ HB Discussion where HˆB , ρB = e } HˆB , ρB e } . We note that the s In the (classical) picture of a reference frame given by gen- first term, corresponding to the unitary part of the evolution, has eral relativity, an observer sets an array of clocks over a region a time dilation factor (1 + GjA∆E/c4x) that corresponds to the of a spacial hypersurface. These clocks trace world lines and mean energy jA∆E of the A clock. On the other hand, the sec- ˆ tick according to the value of the along their tra- ond term, responsible for decoherence and quadratic in HB , is jectory. Here we have shown that, under an operational defi- 2 proportional to the square of the variance, jA(∆E) /2. We then nition of time, this picture is untenable. The reason does not see that, in a state of clock A where the variance of the energy only lie in the limitation of the accuracy of time measurement is negligible, clock B evolves unitarily with a time-dilation fac- by a single clock, coming from the usual quantum gravity argu- tor given by the average energy of clock A, just as expected for ment in which a black hole is formed when the energy density a quantum state of matter in the semiclassical limit, where its used to probe space–time lies inside the Schwarzschild radius energy–momentum tensor operator is replaced by its average for that energy. Rather, the effect we predict here comes from value. For completeness, we derive Eq. 14 in SI Appendix, Deriva- the interaction between nearby clocks, given by the mass–energy tion of the Master Equation, following closely ref. 15. We show equivalence, the validity of the Einstein equations, and the that the derivation of the master Eq. 14 holds, in general, for any linearity of . We have shown that clocks inter- quantum system and any form of the Hamiltonians HˆA and HˆB . acting gravitationally get entangled due to gravitational time This fact implies that, as long as the initial state of the clocks is dilation: The rate at which a single clock ticks depends on the not in an energy eigenstate (a condition needed for the system energy of the surrounding clocks. This interaction produces a to be a clock), the second term in Eq. 14 will be nonzero, as the mixing of the reduced state of a single clock, with a character- variance of the energy will not vanish; implying that, irrespective istic decoherence time after which the system is no longer able of the nature of the clocks, they will get entangled. to work as a clock. Although the regime of energies and dis- Finally, in the light of the analysis of the present section, let tances in which this effect is considerable is still far away from us now return to Eq. 3, obtained via a heuristic semiclassical the current experimental capabilities, the effect is significant at argument in Clock Model, and show that it can also be derived energy scales that exist naturally in bound from the classical limit of two interacting clocks, connecting the states.

E2308 | www.pnas.org/cgi/doi/10.1073/pnas.1616427114 Castro Ruiz et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 0 hrnvY ta.(98 esrmn ftm farvli unu mechanics. quantum in arrival of time of Measurement (1998) al. et Y, Aharonov 10. 3 Bu 13. Brukner I, Pikovski F, Costa M, Zych 12. (2007) eds I, Egusquiza MR, Sala G, Muga 11. ia ii fgaiyi h unu eie nwihteenergy– the semiclas- as which the in clocks regime, with quantum of the consistent in limit is gravity of appropriate limit limit sical This the states. in coherent relativity character- spin dilation general time shown of classical the istic have reproduces we model our Moreover, that subjected measurements. systems accuracy-limited obtained quantum high-dimensional is to of structure measurability limit causal time the in well-defined indefinite only of with notion scenarios here The discussed physical (25). situations to the lead that expect may can space-like, of one measurability are the intervals, events on time the depend whether separated, time-like to events, or as light-like, between question is distance the lines space–time hence world and the nearby Because along intervals defined. question time well to of us notion lead intervals the may pre- conclusion whether time arbitrary This principle. with relevant, in measured even be are cision, cannot lines clocks world the nearby along of effects gravitational atoRi tal. et Ruiz Castro .Mrou ,LvtnL 19)Temxmmsedo yaia evolution. dynamical of speed maximum The (1998) LB Levitin N, Margolus states. nonstationary of 9. evolution the on in bound time unitarity A and (1973) energy GN Fleming between relation 8. uncertainty The the (1945) and I decoherence, Tamm universal L, clocks, Mandelstam Realistic 7. (2004) J Pullin RA, (the Porto in R, distances Gambini of 6. measurability the on Limits (1994) spacetime G of Amelino-Camelia measurement the 5. of limitations Quantum (1958) EP Wigner H, Salecker 4. (1979) A Einstein 3. (2000) JA mean- Wheeler EF, the Taylor and relativity, general 2. in clocks and rods of behaviour The (2009) HR Brown 1. hs eut ugs ht nteacrc eieweethe where regime accuracy the in that, suggest results These 2210. A Rev 120:188–195. A Cimento mechanics. quantum non-relativistic paradox. information hole black gravity. quantum of) approximation semiclassical distances. Chicago). Court, York). New Longman, Wesley (Addison arXiv:0911.4440. field. metric the of ing ins fgnrlrltvsi rprtime. proper relativistic general of witness 1. Vol York), e ,DraR asrS(99 pia unu clocks. quantum Optimal (1999) S Massar R, Derka V, zek ˇ 57(6):4130–4139. hsRev Phys 16:232–240. rn dShlpP (Open PA Schilpp ed trans Notes. Autobiographical Einstein: Albert 109:571–577. xlrn lc oe:Itouto oGnrlRelativity General to Introduction Holes: Black Exploring hsRvLett Rev Phys 21)Qatmitreoercvsblt sa as visibility interferometric Quantum (2011) C Phys J ˇ iei unu Mechanics Quantum in Time a Commun Nat 9:249–254. 93:240401. o hsLt A Lett Phys Mod 2:505. hsRvLett Rev Phys 9:3415–3422. Srne,New (Springer, 82(10):2207– hsc D Physica Nuovo Phys 1 kih ,Tsiis 20)NcerKbudsae nlgtnuclei. light in states bound K Nuclear (2002) Y Toshimitsu Y, Akaishi events spacetime 21. of definition operational the on Limitations (1987) measurement. T Padmanabhan spacetime to 20. Limit (1994) H Dam Van YJ, Ng 19. time. of problem the hypothesis. and gravity An quantum Canonical gravity: (1993) CJ quantum Isham in 18. Time (1991) equations. C Newton–Schroedinger the Rovelli with 17. Problems (2014) BL Hu C, Anastopoulos 16. Brukner F, Costa M, Zych I, Pikovski 15. Rankovi 14. 5 rskvO ot ,Bunr, Brukner F, Costa O, Oreshkov 25. Brukner J, Kofler 24. Brukner J, Kofler quan- as resonators 23. mechanical Mesoscopic (2015) KC Schwab MP, Blencowe BN, Katz 22. ne rjc 11-2,adIdvda rjc 24621. Project Individual Sci- and (CoQuS) Systems” W1210-N25, Quantum Quantum Project of “Complex under the program Applications doctoral through the and (FWF) (FoQuS), with “Foundations Fund ence” Science Interface program Austrian Gravity research the and special and Quantum (TURIS), “Testing ” Structures,” platform Single Causal research the “Quantum from the 60609, support from acknowledge Project We Foundation, discussions. Templeton interesting John for Zych M. and ACKNOWLEDGMENTS. effects regimes relativistic at general relevant. expected and are mechanical new be quantum to to both lead phenomena where might the objects concerning real idealized as insights of clocks instead considering systems that suggest physical it from obtained quences within understood be cannot despite effect the value, approximation. general, this expectation in its that, by fact the replaced is tensor momentum h prtoa prahpeetdhr n h conse- the and here presented approach operational The n unu gravity. quantum and MA Rodrguez 340. LA, Ibort eds Theories, 157–287. Field pp York), Quantum New (Springer, and Groups, Quantum tems, 456. Phys J New dilation. time tional arXiv:1506.01373. . ticks nate a Commun Nat measurements. Lett Rev Phys coarse-grained of restriction 180403. the under frames. reference noninertial tum 65:044005. ,LagYC enrRQatmcok n hi ycrnsto h alter- the – synchronisation their and clocks Quantum R Renner Y-C, Liang S, c ´ 16:085007. 101:090403. 3:1092. 20)Cniin o unu ilto fmcocpcrealism. macroscopic of violation quantum for Conditions (2008) C ˇ a Phys Nat 20)Casclwrdaiigoto unu physics quantum of out arising world Classical (2007) C ˇ ls unu Grav Quantum Class etakF ot,A ex .Hen .Wieland, W. Hoehn, P. Feix, A. Costa, F. thank We PNAS 11:668–672. 21)Qatmcreain ihn aslorder. causal no with correlations Quantum (2012) C ˇ hsRvA Rev Phys | 21)Uiesldchrnedet gravita- to due decoherence Universal (2015) C ˇ ulse nieMrh7 2017 7, March online Published 4:L107–L113. 92:042104. o hsLt A Lett Phys Mod hsRvD Rev Phys hsRvLett Rev Phys nerbeSys- Integrable hsRvC Rev Phys | 3442– 43 E2309 9:335– 99:

PHYSICS PNAS PLUS