Radiation pressure induced EPR paradox
0 Marcello Cini, Dipartimento di Fisica, Universita La Sapienza, Roma, Italy Istituto Nazionale di Fisica della Materia , Sez. di Roma
he debate between Einstein and Bohr about the interpretation T of Quantum Mechanics, initiated at the end of the twenties and formalized in the Einstein, Podolski and Rosen paper of 1935 [1], left both contenders with his own opinion. As is well known, Einstein and coworkers showed that, in a system formed by two spatially separated but physically correlated particles A and B, the possibility of determining indirectly (at the experimenter's choice) the value of one or the other of two canonically conjugate variables (e.g. x and p) ofB by measuring the correspondingvariable of A implied one of these two alternatives: (a) the two incompatible (according to QM) variables of each particle, contrary to the fundamental uncertainty established by the Heisenberg principle, possess simultaneously well defined, even if unknowable, values; (b) the result of the measurement performed on A is instantaneously transmitted to B whose corresponding variable, uncertain until that moment, acquires the value required by the correlation's constraint. Einstein rejected (b) and deduced from (a) that QM was an incomplete theory, while Bohr eliminated the problem by denying the possibility of speaking of the values of the variables of B not directly measured. Since no means seemed to exist to decide who was right, because the issue was a counterfactual statement, the question remained open for almost thirty years. It was John Bell [2] who, in 1964, found a way to test whether Einstein was right or wrong. Instead of discussing about the unknowable values of unmeasured quantities he suggested to compare the values of measured quantities. Bell showed in fact that, when the variables of the two particles are only partially correlated, their experimentally measurable correlation coefficient is different if the incompatible variables of each particle actually do possess well defined simultaneous values, as Einstein believed, or not. In the former case a set of inequalities were satisfied; in the latter, violated. The first reliable evidence that the correlation coefficient between the polarizations of two photons is indeed inconsistent with Einstein's choice was performed by Aspect, Granger and Roger in 1981. This introduction, which will sound trivial to the readers who have already read these things many times, is useful not only in order to explain to newcomers the essence of this still open fundamental issue but also to provide the context in which the paper Radiation Pressure Induced EinsteinPodolskiRosen Paradox by V. Giovannetti, S. Mancini and P. Tombesi (GMT) recently published in EPL [3] should be placed. The line of research to which the paper discussed here belongs, in fact, is not the same as Bell's. Its purpose is not to test once again whether Einstein was right or wrong. It is rather to· deepen our knowledge of the various aspects of the fundamental, but counterintuitive, quantum property called entanglement, which is an essential ingredient of the EPR paradox. Very briefly I recall its definition. Suppose that 'ljJA' (r=1,2,..) is the wave function of A in a state labelled by the eigenvalue r of a variable 0 Aand 'ljJBr is the wave function of B in the corresponding state r of the correlated variable OB. Then the wave function W for the total system, describing a state in which eitherA and B are both in state r or A and B are both in an other state s, is given by: This quantum state does not describe a statistical ensemble of N pairs AB in which Nr=Nlcrl2 pairs are in state r and Ns=Nlcsl2

pairs are in state s as Einstein maintained. It describes an ensem
ble in which each pair has probabilities Pr=ICrI2 and PFlesl2
(Pr+PFI)for the two alternatives. The difference between these
two ensembles arises when one considers the measurement of
variables "CA and "CB which are incompatible with a Aand aB respec
tively. The expression for the mean value of "CA"CB contains in fact
non vanishing interference terms of the form
In conclusion, the proposed scheme achieves the goal of
obtaining the entanglement of radiation fields with a macroscopic
number of photons, by means of a classical ponderomotive force
on a macroscopic object. This work offers therefore new prospects
for the investigation of the tricky borderline between the quan
tum and the classical world.
(
2
)
The presence of these terms is at the origin of all the strange prop
erties of two (or more) components systems [4, 5, 6] usually
attributed to what is called the spooky actionatadistance of QM.
The proposal of GMT follows the avenue suggested by Reid and
Drummond [7]. These authors consider two distinct light modes
A and B in a cavity spatially separated but strongly correlated by a
nondegenerate optical parametric oscillator. Each mode is char
acterized by two conjugate quadratures whose phase differs by
rr.12. These quadratures XA(O),XA(rr.12) andXB(0), XB(rr./2) play the
role of the conjugate variables x and p of the two particles A and B
of EPR. Since each mode ofA is correlated with the corresponding
mode of B one can infer the values of XA(O), XA(rr.12) by measur
ing XB(O) or XB(rr./2). The errors of these inferences can be
quantified by the variances L12infXA(0) and L12infXA(rr./2) between
the "true" values and the "inferred" values of XA(0), XA(rr.12). The
EPR paradox arises if the correlation is high enough that one can
reduce the values of the variances to an extent that
[4] I only mention here the still unsettled discussion on the so called
wavefunction collapse during a measurement of a variable of a quan
tum object by means of a macroscopic instrument, and all the
connected questions of decoherence, macroscopic quantum coher
ence, and so on.
(
3
)
(4)
because, according to quantum mechanics, the two conjugate
variables must satisfy the Heinsenberg principle
Of course QM predicts that, as a result of entanglement, both
inequalities should be satisfied. They are not contradictory. The
variances appearing in inequality (
2
) in fact refer to values of
XA(O), XA(rr./2) inferred by different measurements (XB(O) or
XB(rr./2) performed on B. The prediction (
3
) has been experi
mentally realized [8).
The interest of the GMT paper arises from the fact, up to now,
the production of entangled states has been generally considered
as the result of quantum dynamics at the microscopic level. These
authors, instead, suggest that "entanglement can be obtained via a
classical force acting on a macroscopic object:' To this purpose,
they"consider the radiation field having a macroscopic number of
photons and impinging on a completely reflecting and oscillating
mirror in an optical cavity". By studying the properties of the out
put field they demonstrate that the state of the radiation field can
become nonclassical, giving rise to the appearance of the EPR
paradox on its continuous variables.
In the proposed experiment, in fact, the two input modes in the
cavity interact by means of the radiation pressure force by which
each of them acts on the mirror. The entanglement of the two out
put field quadratures arises from the reaction of the mirror
displacement quantum fluctuations. The Hamiltonian assumed to
describe the system allows the calculation of the left hand side of
(
2
) as a function of the input power and the temperature. Its value
turns out to be about 0.7 for values of these parameters within the
possibilities of the available technology.
from =_J""""'""'...j';;::::"J
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[1] A. Einstein et al., Phys. Rev: 47 , 777 (1935
[2] J. Bell , Physics 1 , 195 ( 1964 )
[3] V. Giovannetti , S. Mancini , and P. Tombesi , EPL
[5] M. Brune , E. Hagley , J. Dreyer , X. Maitre , A. Maali , C. Wunderlich , J.M. Raimond and S. Haroche , Phys. Rev: Lett. 77 , 4887 ( 1996 )
[6] C.J. Myatt , RE. King, Q.A. Turchette , C.A. Sackett , D. Kielpinski , WM. Itano, C. Monroe , and D.J. Wineland , Nature (London) 403 , 269 ( 2000 ).
[7] M.D. Reid and P.D. Drummond , Phys Rev Lett . 60 , 2731 ( 1988 ) ; M.D. Reid , Phys. Rev. A 40 , 913 ( 1989 ) ; M.D. Reid and P.D. Drummond , Phys. Rev: A 41 , 3930 ( 1990 )
[8] Z.Y. Ou et al. Phys. Rev: Lett. 68 , 3663 ( 1992 ).