Quantum interferometric visibility as a witness of general relativistic proper time

Abstract

Current attempts to probe general relativistic effects in quantum mechanics focus on precision measurements of phase shifts in matter-wave interferometry. Yet, phase shifts can always be explained as arising because of an Aharonov-Bohm effect, where a particle in a flat space-time is subject to an effective potential. Here we propose a quantum effect that cannot be explained without the general relativistic notion of proper time. We consider interference of a 'clock'--a particle with evolving internal degrees of freedom--that will not only display a phase shift, but also reduce the visibility of the interference pattern. According to general relativity, proper time flows at different rates in different regions of space-time. Therefore, because of quantum complementarity, the visibility will drop to the extent to which the path information becomes available from reading out the proper time from the 'clock'. Such a gravitationally induced decoherence would provide the first test of the genuine general relativistic notion of proper time in quantum mechanics.

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Received 13 Jun 2011 | Accepted 5 Sep 2011 | Published 18 Oct 2011 DOI: 10.1038/ncomms1498

Current attempts to probe general relativistic effects in quantum mechanics focus on precision measurements of phase shifts in matterwave interferometry. Yet, phase shifts can always be explained as arising because of an AharonovBohm effect, where a particle in a at spacetime is subject to an effective potential. Here we propose a quantum effect that cannot be explained without the general relativistic notion of proper time. We consider interference of a clocka particle with evolving internal degrees of freedomthat will not only display a phase shift, but also reduce the visibility of the interference pattern. According to general relativity, proper time ows at different rates in different regions of spacetime. Therefore, because of quantum complementarity, the visibility will drop to the extent to which the path information becomes available from reading out the proper time from the clock. Such a gravitationally induced decoherence would provide the rst test of the genuine general relativistic notion of proper time in quantum mechanics.

Quantum interferometric visibility as a witness of general relativistic proper time

Magdalena Zych1, Fabio Costa1, Igor Pikovski1 &aslav Brukner1,2

1 Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. 2 Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. Correspondence and requests for materials should be addressed to M.Z. (email: [email protected]).

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In the theory of general relativity, time is not a global background parameter but ows at dierent rates depending on the space time geometry. Although veried to high precision in various

experiments1, this prediction (as well as any other general relativistic eect) has never been tested in the regime where quantum eects become relevant. There is, in general, a fundamental interest in probing the interplay between gravity and quantum mechanics2. The reason is that the two theories are grounded on seemingly dierent premises and, although consistent predictions can be extrapolated for a large range of phenomena, a unied framework is still missing and fundamentally new physics is expected to appear at some scale.

One of the promising experimental directions is to reveal, through interferometric measurements, the phase acquired by a particle moving in a gravitational potential3,4. Typically considered is a MachZehnder type interferometer (Fig. 1), placed in the Earths gravitational eld, where a particle travels in a coherent superposition along the two interferometric paths 1, 2 that have dierent proper lengths. The two amplitudes in the superposition acquire different, trajectory-dependent phases i, i = 1, 2. In addition, the particle acquires a controllable relative phase shi . Taking into account the action of the rst beam splitter and denoting by |ri the mode associated with the respective path i, the state inside the Mach Zehnder setup |MZ, just before it is recombined, can be written as

| = 12 | | .

1 1 2 2

D

x

BS

1

2

D+

g

PS

BS

y

Figure 1 | MachZehnder interferometer in the gravitational eld.

The setup considered in this work consists of two beam splitters (BS), a phase shifter (PS) and two detectors D . The PS gives a controllable phase difference between the two trajectories 1 and 2, which both lie in the x y plane. A homogeneous gravitational eld (g) is oriented antiparallel to thex direction. The separation between the paths in the direction of the eld is h. General relativity predicts that the amount of the elapsed proper time is different along the two paths. In our approach, we will consider interference of a particle (which is not in free fall) that has an evolving internal degreeof freedom that acts as a clock. Such an interference experiment will therefore not only display a phase shift, but also reduce the visibility of the interference pattern to the extent to which the path information becomes available from reading out the proper time of the clock.

MZ i i i

ie r e r

+

( )

Here we predict a quantum eect that cannot be explained without the general relativistic notion of proper time and thus show how it is possible to unambiguously distinguish between the two interpretations discussed above. We consider a MachZehnder interferometer placed in the gravitational potential and with a clock used as an interfering particle. By clock we mean some evolving internal degree of freedom of the particle. If there is a dierence in proper time elapsed along the two trajectories, the clock will evolve into dierent quantum states for the two paths of the interferometer. Because of quantum complementarity between interference and which-path information the interferometric visibility will decrease by an amount given by the which-way information accessible from the nal state of the clock1012. Such a reduction in the visibility is a direct consequence of the general relativistic time dilation, which follows from the Einstein equivalence principle. Seeing the Einstein equivalence principle as a corner stone of general relativity, observation of the predicted loss of the interference contrast would be the rst conrmation of a genuine general relativistic eect in quantum mechanics.

One might sustain the view that the interference observed with particles without evolving degrees of freedom is a manifestation of some intrinsic oscillations associated with the particle and that such oscillations can still be seen as the ticking of a clock that keeps track of the particles time. If any operational meaning was to be attributed to this clock, it would imply that which-way information is, in principle, accessible. One should then either assume that proper time is a quantum degree of freedom, in which case, there should be a drop in the interferometric visibility, or that the quantum complementarity relation (between which-path information and interferometric visibility) would be violated when general relativistic eects become relevant. Our proposed experiment allows to test these possibilities. The hypothesis that proper time is a degree of freedom has indeed been considered in various works1315.

The above considerations are also relevant in the context of the debate over ref. 16 (determination of the gravitational redshi by reinterpreting interferometric experiment9 that measured the acceleration of free fall). It was pointed out in refs 1720 that only states non-trivially evolving in time can be referred to as clocks. In ref. 18, the interference in such a case was discussed, however, the role of the interferometric visibility as a witness of proper time in quantum mechanics and as a tool to test new hypotheses has not been previously considered.

+ j

(1)(1)

(2)(2)

Finally, the particle can be registered by one of the two detectors D with corresponding probabilities P :

P +

= 12

1

2 ,

cos j

where := 1 2. The phase i is proportional to the action along the corresponding (semiclassical) trajectory i on which the particle moves. For a free particle on an arbitrary spacetime background, the action can be written in terms of the proper time that elapsed during the travel, S mc

i i

= 2

g t

( )

d . This might suggest that the measurement of is an experimental demonstration of the general relativistic time dilation.

There is, however, a conceptual issue in interpreting experiments measuring a gravitationally induced phase shi as tests of the relativistic time dilation. The action Si above can be written in terms of an eective gravitational potential on a at spacetime. Thus, all the eects resulting from such an action are fully described by the Schdinger equation with the corresponding gravitational potential and where the time evolution is given with respect to the global time. Note that a particle in a eld of arbitrary nature is subject to a Hamiltonian where the potential energy is proportional to the elds charge and a position-dependent potential. Therefore, even in a homogeneous eld, the particle acquires a trajectory-dependent phase although the force acting on it is the same at any pointthe phase arises only because of the potential. For a homogeneous electric eld, this relative phase is known as the electric AharonovBohm eect5. The case of Newtonian gravity is directly analogousthe role of the particles electric charge and of the Coulomb potential are taken by the particles mass and the Newtonian gravitational potential, respectively 6.

All quantum interferometric experiments performed to date (see for example, refs 79) are fully explainable by this gravitational analogue of the electric AharonovBohm eect. Moreover, even if one includes non-Newtonian terms in the Hamiltonian, this dichotomy of interpretations is still present. Again, one can interpret the phase shi as a type of an AharanovBohm phase, which a particle moving in a at spacetime acquires because of an eective, non-Newtonian, gravitational potential (at least for an eective gravitational potential arising from the typically considered Kerr or Schwarzschild spacetimes).

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In the present paper, we discuss an interferometric experiment in the gravitational eld where the interfering particle can be operationally treated as a clock. We predict that as a result of the quantum complementarity between interference and which-path information the general relativistic time dilation will cause the decrease in the interferometric visibility. The observation of such a reduction in the visibility would be the rst conrmation of a genuinely general relativistic eect in quantum mechanics, in particular, it would unambiguously probe proper time as predicted by general relativity. The proposed experiment can also lead to a conclusive test of theories in which proper time is treated as a quantum degree of freedom.

ResultsWhich-way information from proper time. Consider an interferometric experiment with the setup as in Fig. 1, but in a situation where the particle in superposition has some internal degree of freedom that can evolve in time. In such a case, state (1) is no longer the full description of the system. Moreover, if this degree of freedom can be considered as a clock, according to the general relativistic notion of proper time it should evolve dierently along the two arms of the interferometer in the presence of gravity. For a trajectory i, let us call |i the corresponding state of the clock. The superposition (1) inside the interferometer now reads

| = 12 | | | | .

1 1 1 2 2 2

the clock. Moreover, when the information about the time elapsed is not physically accessible, the drop in the visibility will not occur. This indicates that the eect unambiguously arises because of the proper time as predicted by general relativity, in contrast to measurements of the phase shi alone. The gravitational phase shi occurs independently of whether the system can or cannot be operationally treated as a clock, just as the phase shi acquired by a system in the electromagnetic potential. Therefore, the notion of proper time is not probed in such experiments.

Massive quantum clock in an external gravitational eld. In the next paragraphs, we present how the above idea can be realized when the clock degrees of freedom are implemented in internal states of a massive particle (neglecting the nite-size eects). Let H be the Hamiltonian that describes the internal evolution. In the rest reference frame, the time coordinate corresponds to the proper time , and the evolution of the internal states is given by i H

( )

t = . Changing coordinates to the laboratory frame, the evolution is given by i t H

( )

= t , where

t t

= d dt describes how fast the proper time ows with respect to the coordinate time. For a general metric g, it is given by

t mn

m n

= g x x , where we use the signature ( + + + ) and summation over repeated indices is understood. The energymomentum tensor of a massive particle described by the action S can be dened as the functional derivative of S with respect to the metric, that is, T S g

mn

d d

:= (see, for example, ref. 21). Since the particles energy E is dened as the T00 component, it reads E = g0 g0T. In the case of a free evolution

in a spacetime with a stationary metric (in coordinates such that g0j = 0 for j = 1, 2, 3), we have

E mc gg x x

= ,

mn

MZ i i i

ie r e r

+

( )

+

t t

j

(3)(3)

(4)(4)

(5)(5)

In general, the state (3) is entangled and according to quantum mechanics interference in the path degrees of freedom should correspondingly be washed away. The reason is that one could measure the clock degrees of freedom and in that way read out the accessible which-path information. Tracing out the clock states in equation (3) gives the detection probabilities

P + +

mn

m n

2 00

(6)(6)

= 12

1

2 | | | ,

1 2

( )

where m is the mass of the particle. Spacetime geometry in the vicinity of Earth can be described by the Schwarzschild metric. In isotropic coordinates (x, , ) and with d d d

2 2 2 2

+

q q J

sin it

takes the form21

t t a j

cos

where 1|2 = |1|2|ei. When the ancillary phase shi is varied, the probabilities P oscillate with the amplitude V , called the visibility (contrast) of the interference pattern. Formally

V := Max P Min P

Max P Min P

(1 ( )

2 )

= (1 ( ) 2 )

c

2 2 2

+

f

f

x cx c

2 c t xc x x

2 2

f

d d d

t

1 ( ) 2

+ dd2

( ),

4 2 2

2

2

+ .

Whereas without the clock the expected contrast is always maximal (equation (2) yields V =1), in the case of equation (4) it reads

V =| | |.

1 2

2

j j

j j

where f( ) =

x GM x

is the Earths gravitational potential (G denotes the gravitational constant and M is the mass of Earth). We consider the limit of a weak eld and of slowly moving particles. In the nal result, we therefore keep up to quadratic terms in the kinetic and potential energy. In this approximation, the metric components read21

g x c

x c

t t

The distinguishability D of the trajectories is the probability to correctly guess which path was taken in the two-way interferometer by measuring the degrees of freedom that serve as a which-way detector12 (in mathematical terms it is the trace norm distance between the nal states of the detectors associated with dierent paths). In our case, these are the clock degrees of freedom and we obtain D = 1 | | |

1 2

2

+ +

1 2 ( ) 2 ( )

f f , g c

x c

1 1 2 ( )

2 2

d f

00 2

2

4

ij ij

,

so that

t f f f

1 2 ( ) 2 ( ) 1 2 ( )

2

t t . The amount of the which-way information that is potentially available sets an absolute upper bound on the fringe visibility and we recover the well-known duality relation1012 in the form V D

2 2 =1

+ +

2

4

2

x c

x c

x c

xc .

2

+ , as expected for pure states. The above result demonstrates that general relativistic eects in quantum interferometric experiments can go beyond previously predicted corrections to the non-relativistic phase shi. When proper time is treated operationally we anticipate the gravitational time dilation to result in the reduction of the fringe contrast. This drop in the visibility is expected independently of how the proper time is measured and which system and interaction are used for

The total Hamiltonian in the laboratory frame is given by H H H

Lab = 0 +

t , where the operator H0 describes the dynamics of the external degrees of freedom of the particle and is obtained by canonically quantizing the energy (6), that is, the particles coordinate x and kinematic momentum p mx

= become operators satisfying the canonical commutation relation ( )

[ , ]=

x p i . Thus, approximating up to the second order also in the internal energy,

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HLab reads

H mc H E xc mc H E

k

P+ P

1

1

GR GR Lab corr

2 2

2

+ + + + +

f ( )

( ) ,

(7)(7)

where

E pm

pmc mc H

2 2

2

T (s)

k

GR = 2 1 3 2

1

+

and

E m x pm

GRcorr = 12 ( ) 3 2

2

f .

We consider a semiclassical approximation of the particles motion in the interferometer. Therefore, all terms in HLab, apart from the internal Hamiltonian H , appear as purely numerical functions dened along the xed trajectories.

In a setup as in Figure 1, the particle follows in superposition two xed non-geodesic paths 1, 2 in the homogeneous gravitational eld. The acceleration and deceleration, which the particle undergoes in the x direction, is assumed to be the same for both trajectories, as well as the constant velocity along the y axis. This assures that the trajectories have dierent proper length, but there will be no time dilation between the paths stemming from special relativistic eects. The particle inside the interferometer will thus

be described by the superposition | = 12 | |

1 2

Figure 2 | Visibility of the interference pattern and the phase shift in the cases with and without the clock. The plot of the difference betweenthe probabilities P (, m, E, V, T), equation (12), to nd the particlein the output path of the MachZehnder interferometer as a functionof the time T for which the particle travels in a superposition of two trajectories at constant heights (this corresponds to changing the length of the interferometric arms). The term proportional to the particles mass is the phase originating from the Newtonian potential energy mV. General relativistic corrections stemming from external degrees of freedom are given by EGRcorr , see for example, ref. 3. Without the clock degrees of freedom, only these terms are present in the result (dashed, black linein the plot). In the situation with the clock (blue line), we expect twonew effects: the change of the interferometric visibility given by the absolute value of the rst cosine (thick red line) and an extra phase shift proportional to the average internal energy of the clock. The values for the energy gap E and the gravitational potential difference V between the interferometric paths are chosen such that

E V c

2 = 1

j ,

where the states |i associated with the two paths i are given by applying the Hamiltonian (7) to the initial state, which we denote by |xin| in. Up to an overall phase, these states read

| = | | .

( ) 2

MZ i ei

+

( )

Hz . Whereas the phase shift alone can always be understood as an AharonovBohm phase of an effective potential, the notion of general relativistic proper time is necessary to explain the decrease of the visibility.

2

+ +

i

d corr in in

For a small size of the interferometer, the central gravitational potential (x) can be approximated to linear terms in the distance h between the paths:

f f

( ) = ( ) ( ),

2

t x c

mc H E

e x

( )

2

GR

(8)(8)

(9)(9)

(10)(10)

(11)(11)

(12)(12)

the corrections EGRcorr from equation (7) averaged over the two trajectories and E: = E1 E0. The expectation value H is taken with respect to the state (11). The corresponding visibility (5) is

V = 2 .

2

i

g

f

i

t

R h R g h h

+ + +

O

where g GM R

= 2 denotes the value of the Earths gravitational acceleration in the origin of the laboratory frame, which is at distance R from the centre of Earth.

For a particle having two internal states |0, |1 with corresponding energies E0, E1, the rest frame Hamiltonian of the internal degrees of freedom can be written as

H E E

= |0 0 | |1 1|

0 1

+

and if we choose the initial state of this internal degrees to be

| = 12 (|0 |1 )

cos

E V T c

(13)(13)

The introduction of the clock degrees of freedom results in two new quantum eects that cannot be explained without including general relativity: the change of the interferometric visibility and the extra phase shi proportional to the average internal energy (Fig. 2; equation (12)). The drop in the visibility is a consequence of a direct coupling of the particles internal degrees of freedom to the potential in the eective Hamiltonian (7). Such a coupling is never found in Newtonian gravity, and it is the mathematical expression of the prediction that the clock ticks at dierent rates when placed in dierent gravitational potentials. This coupling can directly be obtained from the Einstein equivalence principle. Recall that the latter postulates that accelerated reference frames are physically equivalent to those in the gravitational eld of massive objects. When applied within special relativity, this exactly results in the prediction that initially synchronized clocks subject to dierent gravitational potentials will show dierent times when brought together. The proposed experiment probes the presence of such a gravitational time dilation eect for a quantum systemit directly shows whether the clock would tick at dierent rates when taken along the two possible trajectories in the interferometer. On the other hand, to obtain the correct phase shi, it is sufficient to consider a semiclassical coupling of the average total energy of the system to the gravitational potential. With such a coupling, the time displayed by the clock used in

tin +

the detection probabilities read

P m E V T E V Tc mc H

E

( , , , , ) = 12

1

2 2 2

j

cos cos

+

(

2

+

)+ +

corrr

GR V T c

2 j ,

where T is the time (as measured in the laboratory frame) for which the particle travels in the interferometer in a superposition of two trajectories at constant heights, V: = gh is the dierence in the gravitational potential between the paths, EGRcorr represents

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the experiment will not depend on the path taken. This means that a gravitationally induced phase shi can probe general relativistic corrections to the Newtonian gravitational potential but is always consistent with having an operationally well-dened notion of global time, that is, with a at spacetime.

The eect described in our work follows directly from the Einstein equivalence principle, which is itself crucial for the formulation of general relativity as a metric theory22. Thus, the drop in the fringe contrast is not only genuinely quantum mechanical but also a genuine general relativistic eect that in particular unambiguously probes the general relativistic notion of proper time.

General clocks and gravitational elds. Let us call t the orthogonalization time of a quantum system, that is, the minimal time needed for a quantum state to evolve under a given Hamiltonian into an orthogonal one23,24. For the initial state (11) subject to the rest frame Hamiltonian H given by equation (10) we obtain

t E

1 1

t H Egr

1 2 ( )

a a a

p ,

> 0 (provided the initial state is in the domain of (H Egr))

where H denotes the internal Hamiltonian and Egr the energy of its ground state.

Discussion

Current approaches to test general relativistic eects in quantum mechanics mainly focus on high precision measurements of the phase induced by the gravitational potential. Although such experiments would probe the potential and thus could verify non-Newtonian corrections in the Hamiltonian, they would not constitute an unambiguous proof of the gravitational time dilation, because they are also explainable without this concept by the AharonovBohm eect: a trajectory-dependent phase acquired by a particle moving in a at spacetime in the presence of a position-dependent potential.

In our proposed experiment, the eects arising from general relativistic proper time can be separated and probed independently from the AharonovBohm type of eects. Unlike the phase shi, which occurs independently of whether the interfering particle can be treated as a clock, the change of the interferometric visibility (equation (13)) is a quantum eect that arises if and only if general relativistic proper time has a well dened operational meaning. Indeed, if one prepares the initial state |in as an eigenstate of the internal energy Hamiltonian H , only the phase of such a state would change during the time evolution and, according to equation (16), interferometric visibility would be maximal. This clock would not tick (it has orthogonalization time t = ) so the concept of proper time would have no operational meaning in this case. Moreover, reasoning that any (even just an abstract) frequency which can be ascribed to the particle allows considering proper time as a physical quantity would imply that interference should always be lost, as the which-path information is stored somewhere. This once again shows that, in quantum mechanics, it makes no sense to speak about quantities without specifying how they are measured.

The interferometric experiment proposed in this work can also be used to test whether proper time is a new quantum degree of freedom. This idea was discussed in the context of, for example, the equivalence principle in refs 13,14 and a massproper time uncertainty relation15. The equations of motion for proper time treated dynamically, as put forward in refs 1315, are in agreement with general relativity. Therefore, the predictions of equation (5) would also be valid, if the states |i, introduced in equation (3), stand for this new degree of freedom. Already performed experiments, like

= p

.

(14)(14)

(15)(15)

(16)(16)

A system with nite t can be seen as a clock that ticks at a rate proportional to t1. Thus, the orthogonalization time gives also the precision of a considered clock. From the expression for

t in the approximation (9), it follows that the total time dilation between the trajectories is

t = .

2

V T c

We can, therefore, phrase the interferometric visibility V solely in terms of t and :

V =

2

cos t p

t

.

The total time dilation is a parameter capturing the relevant information about the paths, and t grasps pertinent features of the clock. It is only their ratio that matters for the fringe visibility. Equation (16) is a generalization of the result (13) to the case of an arbitrary initial state, clock Hamiltonian and a non-homogeneous gravitational eld: whenever the time dilation between the two trajectories through the MachZehnder interferometer is equal to the orthogonalization time t of the quantum mechanical system that is sent through the setup, the physically accessible proper time dierence will result in the full loss of fringe contrast. There are several bounds on the orthogonalization time based on energy distribution moments23,25,26.

Such bounds can through equation (16) give some estimates on the gravity-induced decoherence rates in more general situations. As an example, for mixed states one generally has26:

Table 1 | Discussion of possible outcomes of the proposed interferometric experiment.

Experimental visibility Possible explanation Current experimental status

Vm = 0 Proper time: quantum d.o.f., sharply dened Disproved in, for example, refs 7,9

0 V V

< <

m QM Proper time: quantum d.o.f. with uncertainty Consistent with current data for st t

> | | 8 (1 )

ln V

= Proper time: not a quantum d.o.f. or has a very broad uncertainty Consistent with current data

V V

m QM

> Quantum interferometric complementarity does not hold when

general relativistic effects become relevant

V V

m QM

Not tested

The measured visibility Vm is compared with the quantum mechanical prediction VQM given by equation (13). Depending on their relation, different conclusions can be drawn regarding the possibility that proper time is a quantum degree of freedom (d.o.f.). Assuming that the distribution of the proper time d.o.f. is a Gaussian of the width , current interferometric experiments give bounds on

possible in terms of the proper time difference between the paths and the experimental error V of the visibility measurement.

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Table 2 | Comparison of different systems for the experimental observation of the reduced interferometric visibility.

System Clock (Hz) DhDT (ms) achieved DhDT (ms) required

Atoms Hyperne states 1015 10 5 10

Electrons Spin precession 1013 10 6 103

Molecules Vibrational modes 1012 10 8 104

Neutrons Spin precession 1010 10 6 106

Several possible systems are compared on the basis of theoretically required and already experimentally achieved parameters, which are relevant for our proposed experiment. For a clock with a frequency w = E , the required value of the parameter hT (h being the separation between the interferometers arms and T the time for which the particle travels in superposition at constant heights) for the full loss of the fringe visibility (see equation (13)), is given in the rightmost column. In our estimations, we assumed a constant gravitational acceleration g m/s

= 10( )

2 . See section

Methods for further discussion on possible experimental implementations.

in refs 7,16, which measured a gravitational phase shi, immediately rule out the possibility that the state of proper time was sharply dened in those tests, in the sense of 1|2 = (1 2). However, such experiments can put a nite bound on the possible uncertainty in the state of proper time. The phase shi measured in those experiments can be phrased in terms of the dierence in the proper time between the paths. Denote by V the experimental error with which the visibility of the interference pattern was measured in those tests. As a result, a Gaussian state of the proper time degree of freedom of width such that s t

t >| | 8 (1 )

/ ln

V , is consistent with the experimental data. An estimate of the proper time uncertainty can be based on the Heisenberg uncertainty principle for canonical variables and the equation of motion for the proper time. In such an analysis, the rest mass m can be considered as a canonically conjugated momentum to the proper time variable , that is, one assumes [ , ]=

2

t mc i1315. In Table 1, we discuss what

can be inferred about proper time as a quantum degree of freedom from an experiment in which the measured visibility would be Vm

and where VQM is the visibility predicted by quantum mechanics, as given by equation (13).

In conclusion, we predicted a quantum eect in interferometric experiments that, for the rst time, allows probing general relativistic proper time in an unambiguous way. In the presence of a gravitational potential, we showed that a loss in the interferometric visibility occurs, if the time dilation is physically accessible from the state of the interfered particle. This requires that the particle is a clock measuring proper time along the trajectories, therefore revealing the which-way information. Our predictions can be experimentally veried by implementing the clock in some internal degrees of freedom of the particle (see Methods). The proposed experiment can also lead to a conclusive test of theories in which proper time is treated as a quantum degree of freedom. As a nal remark, we note that decoherence due to the gravitational time dilation may have further importance in considering the quantum to classical transition and in attempts to observe collective quantum phenomena in extended, complex quantum systems because the orthogonalization time may become small enough in such situations to make the predicted decoherence eect prominent.

Methods

Systems for the implementation of the interferometric setup. Here we briey discuss various systems for the possible implementation of the interferometric setup. Interferometry with many dierent massive quantum systems has been achieved, for example, with neutrons7,8, atoms16,27, electrons28,29 and molecules30,31. In our framework, further access to an internal degree of freedom is paramount, as to initialize the clock which measures the proper time along the interferometric path. Therefore, the experimental requirements are more challenging. To observe full loss of the interferometric visibility, the proper time dierence in the two interferometric arms needs to be = t. For a two level system, the revival of the visibility due to the indistinguishability of the proper time in the two arms occurs when = 2t.

The best current atomic clocks operate at optical frequencies around 1015 Hz. For such systems, we have t = /, and one would therefore require an atomic superposition with hT~10 ms to see full disappearance of the interferometric visibility. For example, the spatial separation would need to be of the order of 1 m, maintained for about 10 s. Achieving and maintaining such large superpositions of

atoms still remains a challenge, but recent rapid experimental progress indicates that this interferometric setup could be conceivable in the near future. For neutrons, a separation of 10 2 m with a coherence time of t~10 4 s has been achieved8. To implement our clock in neutron interferometry, one can use spin precession in a strong, homogeneous magnetic eld. However, such a clock could reach frequencies up to ~109 Hz (for a magnetic eld strength of order of 10T (ref. 32)), which is still a few orders of magnitude lower than necessary for the observation of full decoherence owing to a proper time dierence. Improvements in the coherence time and the size of the interferometer would still be necessary. Other systems, such as molecules, could be used as well and Table 2 summarizes the requirements for various setups (note again that the particles are assumed to travel at xed height during the time T).

The eect we predict can be measured even without achieving full orthogonalization of the clocks. Note that even for t the small reduction of visibility can already be sufficient to prove the accessibility of which-path information due to the proper time dierence. With current parameters in atom interferometry,an accuracy of the measurement of the visibility of V = 10 6 would have to be achieved for the experimental conrmation of our predictions. A very good precision measurement of the interferometric visibility and a precise knowledge about other decoherence eects would therefore make the requirements for the other parameters less stringent.

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Acknowledgements

We thank M. Arndt, B. Dakic, S. Gerlich, D. M. Greenberger, H. Mller, S. Nimmrichter, A. Peters, and P. Wolf for insightful discussions. The research was funded by the Austrian Science Fund (FWF) projects: W1210, P19570-N16 and SFB-FOQUS, the Foundational Questions Institute (FQXi) and the European Commission Project Q-ESSENCE (No. 248095). F.C., I.P. and M.Z. are members of the FWF Doctoral Program CoQuS.

Author contributions

M.Z., F.C., I.P. and .B. contributed to all aspects of the research with the leading input from M.Z.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Zych, M. et al. Quantum interferometric visibility as a witness of general relativistic proper time. Nat. Commun. 2:505 doi: 10.1038/ncomms1498 (2011).

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