Bohmian trajectory gravityin the BMV quantum gravity experiment

Thomas C Andersen nSCIr Canada tandersennscirca

The recent experimental proposals by Bose et al and Marletto et al (BMV) outline a way to test for the quantumnature of gravity by measuring gravitationally induced phase accumulation over the paths of two entangled sim 10minus14kgmasses This work predicts the outcome of the BMV experiment in Bohmian trajectory gravity - where classical gravityis assumed to couple to the particle configuration in each Bohmian path Bohmian trajectory gravity predicts thatthere will quantum entanglement This is surprising as the gravitational field is treated classically

The BMV experimentIn the BMV experimental proposal Bose[1] and Marletto and Vedral[2] propose tomeasure the expected behaviour of quantum systems when particles interact onlyvia mutual gravitational attraction Christodoulou and Rovelli [3]

detecting the effect counts as evidence that the gravitational field can bein a superposition of two macroscopically distinct classical fields and sincethe gravitational field is the geometry of spacetime (measured by rods andclocks) the BMV effect counts as evidence that quantum superposition ofdifferent spacetime geometries is possible can be achieved[3]

Figure 1 Particles 1 and 2 start off in the spin up |u〉 (defined as +z) directionand are spatially split into superposed |L〉 + |R〉 states by a Stern-Gerlachmechanism The particles fly through the experiment at the same time and thenthrough a second Stern-Gerlach where they are brought back into updown state

Christodoulou and Rovelli[3] point out that the effect can be understood as thetime dilation effects of one mass on the other Although the time dilation effect istiny the experimental effect is amplified as the experimenters can take advantageof the extremely high Compton frequency of the massive (in the quantum particlesense) particle The experimenters only need to measure the difference in phaseaccumulation along different paths and once that phase difference is sim π theexperimenters will detect that the state has become quantum entangled

Figure 2 Left (entanglement) In quantum gravity particle 1 on the right branchfeels gravity from both 2L and 2R in superposition Right (no entanglement) InRosenfeld style semi-classical gravity particle 1R feels the same (negligible) gravityno matter which branch particle 2 is on

Initial StateEach particle is passed through a typical Stern-Gerlach device where L and Rrepresent spins in the plusmnx direction

|u1〉 =|L1〉 + |R1〉radic

2 |u2〉 =

|L2〉 + |R2〉radic2

(1)

andu dσxσyσz (2)

take on their usual meanings The state at t1 is thus

|Ψt1〉 =12

983059|LR〉 + |RL〉 + |LL〉 + |RR〉

983060 (3)

Measuring quantum entanglementWe wish to quantify the entanglement of the final state in the quantum gravitymodels considered here A well chosen entanglement witness is a good measureWe use a witness similar to that of Bose et al[1] Witness

W = 〈σ(1)x otimes σ(2)

z 〉 + 〈σ(1)y otimes σ(2)

y 〉 (4)which shows entanglement (ρ is the density matrix)

W(ρ) le 0 (5)See for instance[4] The witness W is in the range [-2 2] Witness operatorsoperate on density matrix representations of final statesmixturesFor quantum gravity if the experiment is arranged so that a phase change of πoccurs on the R1L2 branch we have

|Ψt4〉qg =12

983059|LR〉 minus |RL〉 + |LL〉 + |RR〉

983060 (6)

and our entanglement witness showsW(ρqg) = Tr(ρqgW) = minus2 (quantum gravity) (7)

For semi-classical gravity its worthwhile to recall the physical modelRmicroν minus

12

gmicroνR =8πG

c4 〈Ψ|Tmicroν|Ψ〉 (8)We see that Einsteins gravity connects to the expectation value of thematter-energy configuration T The state at time t4 Ψt4 = Ψt0 (excepting overallphase)

and our entanglement witness shows

W(ρsc) = Tr(ρscW) = 0 (semi minus classical gravity) (9)

Bohmian trajectory predictionde Broglie Bohm mechanics[5] uses non local hidden variables to create aninterpretation of quantum mechanics where particles have well defined trajectoriesStruyve writes[6]

Bohmian mechanics solves the measurement problem by introducing an ac-tual configuration (particle positions in the non-relativistic domain particlepositions or fields in the relativistic domain) that evolves under the influ-ence of the wave function According to this approach instead of couplingclassical gravity to the wave function it is natural to couple it to the actualmatter configuration

In equation form comparing to the semi-classical form (8) above we have

Rmicroν minus12

gmicroνR =8πG

c4 Tmicroν(ϕB g) (10)

Where ϕB is the trajectory of one member of the ensemble of Bohmiantrajectories and Tmicroν(ϕB g) is the stress energy tensor as determined by thespecific trajectory in combination with perhaps other (eg earths gravity) sourcesIn order to interpret what happens in the BMV experiment assuming Bohmiantrajectory gravity governs the quantum - gravity coupling considerSuperposition since gravity couples directly to each particle position gravity is

not in a superposition instead the configuration of the gravitational fieldchanges for each run of the experiment with 4 possible configurations ofparticle trajectories

Measurement There is no measurement taking place so there is no collapseThe effect on the quantum state when the particle trajectories take the 1R and2L paths is for the wave function to be altered in the same way as in thequantized gravity solution

The end result is that at t4 there are two possible states we have a mixed staterepresented with a density matrix ρbt34 of the time there will be no gravitational interaction and the final state will bethe same as the starting state at t1 while 14 of the time there is gravitationaltime dilation and the final state will be the same as that for quantized gravity

75 |Ψt4〉bt =12

983059|LR〉 + |RL〉 + |LL〉 + |RR〉

983060(11)

25 |Ψt4〉bt =12

983059|LR〉 minus |RL〉 + |LL〉 + |RR〉

983060(12)

Figure 3 Four runs of the experiment showing Bohmian trajectories Only in thesecond run from the left will there be gravitational interaction between thetrajectories

Evaluating the witness

W(ρbt) = Tr(ρbtW) = minus12(Bohmian trajectory) (13)

The state is entangled The witness shows a result 14 of that of the maximallyentangled quantized gravity caseThus we have created entanglement with a classical single valued field

DiscussionThe BMV (and related) experiments provide a good probe of the nature of theinteraction of gravity and quantum mechanics The experiment proposed by BMVwhile very challenging to perform is simple to study theoretically

Of course some form of quantized gravity is the most popular prediction for theBMV experiment[3] with an entangled final state showing an entanglement witnessW sim minus2 The semi-classical model does not show any entanglement The resultprediction for Bohmian trajectory gravity shows entanglement with a witness ofW sim minus1

2Semi-classical gravity is perhaps more complicated than Bohmian trajectory

gravity In semi-classical gravity the gravitational field has to somehow integratethe entire position space of the wave function (a non local entity) in real time (viathe Schroumldinger - Newton equation) in order to continuously use the expectationvalue as a source for the gravitational field In Bohmian mechanics thegravitational field connects directly to an existing hidden particle position which isconceptually simpler

AcknowledgementsThe author wishes to thank Hans-Thomas Elze and the other organizers and attendees of DICE2018 Spacetime - Matter - Quantum Mechanics Ninth International Workshopin Castiglioncello Italy for the chance to speak on the matter presented in this here[1] Bose S Mazumdar A Morley G W Ulbricht H Toroš M Paternostro M Geraci A A Barker P F Kim M S and Milburn G 2017 Phys Rev Lett 119 240401

[2] Marletto C and Vedral V 2017 Phys Rev Lett 119 240402 (Preprint ONarXiv170706036v2)

[3] Christodoulou M and Rovelli C Preprint arXiv 180805842[4] Buchleitner A Viviescas C and Tiersch M 2008 Entanglement and decoherence foundations and modern trends vol 768 (Springer Science amp Business Media)

[5] Bohm D 1952 Phys Rev 85 166--179

[6] Sudarsky D 2017 Quantum Origin of Cosmological Structure and Dynamical Reduction Theories The Philosophy Of Cosmology ed Chamcham K Silk J Barrow J D and Saunders S (Cambridge CambridgeUniversity Press) pp 330--355 ISBN 9781316535783