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Time and Matter in the Interaction Between Gravity and Quantum

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Time and Matter in the Interaction Between Gravity and Quantum Time and Matter in the Interaction between Gravity and Quantum Fluids: Are there Macroscopic Quantum Transducers between Gravitational and Electromagnetic waves? Raymond Y. Chiao∗ and Walter J. Fitelson† April 3, 2003 Abstract Measurements of the tunneling time are briefly reviewed. Next, time and matter in general relativity and quantum mechanics is examined. In particular, the question arises: How does gravitational radiation inter- act with a coherent quantum many-body system (a “quantum fluid”)? A minimal coupling rule for the coupling of the electron spin to curved spacetime in general relativity implies the possibility of a coupling be- tween electromagnetic (EM) and gravitational (GR) radiation mediated by a quantum Hall fluid. This suggests that quantum transducers between these two kinds of radiation fields might exist. We report here on a first attempt at a Hertz-type experiment, in which a high-Tc superconductor (YBCO) was the material used as a quantum transducer to convert EM into GR microwaves, and a second piece of YBCO in a separate apparatus was used to back-convert GR into EM microwaves. An upper limit on the conversion efficiency of YBCO was measured to be 1.6 10−5. × arXiv:gr-qc/0303089v3 3 Apr 2003 1 Introduction In this conference in Venice on “Time and Matter,” one of us (RYC), was invited to speak on the tunneling time problem: How quickly does a particle traverse a barrier in the quantum process of tunneling? A. M. Steinberg, P. G. Kwiat, and RYC have used a photon-pair emission light source (spontaneous parametric down-conversion) for measuring the single-photon tunneling time, using the “click” of a Geiger counter as the registration of when one photon, which had succeeded in tunneling ∗Department of Physics, University of California at Berkeley, CA 94720-7300, USA (E-mail: [email protected]) †University of California, Space Sciences Laboratory, Berkeley, CA 94720-7450, USA (E- mail: [email protected]) 1 through the barrier, reached the detector, relative to a second, vacuum- traversing photon, which was born at the same time as the first photon (hence its “twin”). The arrival time of the tunneled photon was measured with respect to that of its twin, which had traversed a distance equal to the tunnel barrier thickness, but in the vacuum, by means of the difference in the two “click” times of two Geiger counters. These two Geiger coun- ters were used in the coincidence detection of the two photons, with one counter placed behind the tunnel barrier, and the other counter placed behind the vacuum, in conjunction with a Hong-Ou-Mandel interferome- ter. By means of this two-photon interferometer, we achieved the sub- picosecond time resolution necessary for measuring the tunneling time of a photon relative to the vacuum-traversal time of its twin. The result was that the Wigner theory of tunneling time was confirmed to be the one that applied to our experiment. The surprising result was that when a photon succeeded in tunneling (which is rare), it arrived earlier than its twin which had traversed the vacuum, as indicated by the fact that the “clicks” of the Geiger counter registering the arrival of the tunneling photons occurred earlier on the average than the Geiger counter “clicks” registering the arrival of the vacuum-traversing twin photons, as if the tunneling photons had traversed the tunnel barrier superluminally. The effective group velocity of the tunneling single-photon wavepacket was measured to be 1.7 0.2 times the vacuum speed of light. ± Since our tunneling-time work has already been adequately reviewed,[1] here we shall concentrate instead on a different question concerning “Time and Matter,” namely, the role of time in the interaction of gravity, in par- ticular, of gravitational radiation, with matter in the form of quantum fluids, i.e., many-body systems which exhibit off-diagonal long-range or- der (ODLRO), such as superconductors, superfluids, atomic Bose-Einstein condensates (BECs), and quantum Hall fluids.[2] As we shall see, under the proper circumstances, there results the interesting possibility that such quantum fluids could in principle mediate the conversion of EM into GR waves, and vice versa. 2 Quantum fluids as antennas for gravi- tational radiation Can quantum fluids circumvent the problem of the tiny rigidity of classical matter, such as that of the normal metals used in Weber bars, in their feeble responses to gravitational radiation? One consequence of the tiny rigidity of classical matter is the fact that the speed of sound in a Weber bar is typically five orders of magnitude less than the speed of light. In order to transfer energy coherently from a gravitational wave by classical means, for example, by acoustical modes inside the bar to some local detector, e.g., a piezoelectric crystal glued to the middle of the bar, the length scale L of the Weber bar is limited to a distance scale on the order of the speed of sound times the period of the gravitational wave, i.e., an acoustical wavelength λsound, which is typically five orders of magnitude 2 smaller than the gravitational radiation wavelength λ to be detected. This makes the Weber bar, which is thereby limited in its length to L λsound, ≃ much too short an antenna to couple efficiently to free space. However, rigid quantum objects, such as a two-dimensional electron gas in a strong magnetic field which exhibits the quantum Hall effect, in what Laughlin has called an “incompressible quantum fluid”,[3] are not limited by these classical considerations, but can have macroscopic quantum phase coherence on a length scale L on the same order as (or even much greater than) the gravitational radiation wavelength λ. The origin of this rigidity is that the phase of the wavefunction must remain rigidly single-valued everywhere inside the quantum fluid, whenever the many-body system is perturbed by gravity waves whose time variations are slow compared to the time scale of the gap time ¯h/Egap, where Egap is the energy gap separating the ground state from all excited states. Then the wavefunction will remain adiabatically, and hence rigidly, in its ground state during these time variations. Since the radiation efficiency of a quadrupole antenna scales as the length of the antenna L to the fourth power when L << λ, such quantum antennas should be much more efficient in coupling to free space than classical ones like the Weber 4 bar by a factor of (λ/λsound) . Weinberg gives a measure of the radiative coupling efficiency ηrad of a Weber bar of mass M, length L, and velocity of sound vsound, in terms of a branching ratio for the emission of gravitational radiation by the Weber bar, relative to the emission of heat, i.e., the ratio of the rate of emission of gravitational radiation Γgrav relative to the rate of the decay of the acoustical oscillations into heat Γheat, which is given by[4] 4 Γgrav 64GMvsound −34 ηrad = 2 5 3 10 , (1) ≡ Γheat 15L c Γheat ≃ × where G is Newton’s constant. The quartic power dependence of the efficiency ηrad on the velocity of sound vsound arises from the quartic dependence of the coupling efficiency to free space of a quadrupole antenna upon its length L, when L<<λ. The long-range quantum phase coherence of a quantum fluid allows the typical size L of a quantum antenna to be comparable to the wavelength λ. Thus the phase rigidity of the quantum fluid allows us in principle to replace the velocity of sound vsound by the speed of light c. Therefore, 4 quantum fluids can be more efficient than Weber bars, based on the vsound factor alone, by twenty orders of magnitude, i.e., 4 c 20 10 . (2) vsound ≃ Hence quantum fluids could be much more efficient receivers of this radi- ation than Weber bars for detecting astrophysical sources of gravitational radiation. This has previously been suggested to be the case for superflu- ids and superconductors.[5],[6] Another important property of quantum fluids lies in the fact that they can possess an extremely low dissipation coefficient Γheat, as can be inferred, for example, by the existence of persistent currents in superfluids 3 that can last for indefinitely long periods of time. Thus the impedance matching of the quantum antenna to free space,[7] or equivalently, the branching ratio ηrad can be much larger than that calculated above for the classical Weber bar. Since it is difficult to calculate Γheat, we need to measure ηrad experimentally. 3 Minimal-coupling rule for a quantum Hall fluid The electron, which possesses charge e, rest mass m, and spin s = 1/2, obeys the Dirac equation. The nonrelativistic, interacting, fermionic many-body system, such as that in the quantum Hall fluid, should obey the minimal-coupling rule which originates from the covariant-derivative coupling of the Dirac electron to curved spacetime, viz. (using the Einstein summation convention),[4],[8] 1 AB pµ pµ eAµ ΣAB ω (3) → − − 2 µ where pµ is the electron’s four-momentum, Aµ is the electromagnetic four- potential, ΣAB are the Dirac γ matrices in curved spacetime with tetrad AB (or vierbein) A, B indices, and ωµ are the components of the spin con- nection AB Aν B ω = e µ e ν (4) µ ∇ Aν B where e and e ν are tetrad four-vectors, which are sets of four or- thogonal unit vectors of spacetime, such as those corresponding to a local inertial frame. The vector potential Aµ leads to a quantum interference effect, in which the gauge-invariant Aharonov-Bohm phase becomes observable. AB Similarly, the spin connection ωµ , in its Abelian holonomy, should also lead to a quantum interference effect, in which the gauge-invariant Berry phase[9] becomes observable.
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