# Wavepacket spreading produces force sensitivity

I’m still trying to decide if I understand this correctly, but it looks like coherent wavepacket spreading is sufficient to produce states of a test-mass that are highly sensitive to weak forces. The Wigner function of a coherent wavepacket is sheared horizontally in phase space (see hand-drawn figure). A force that perturbs it slightly with a small momentum shift will still produce an orthogonal state of the test mass.

The Gaussian wavepacket of a test mass (left) will be sheared horizontally in phase space by the free-particle evolution governed by . A small vertical (i.e. momentum) shift by a weak force can then produce an orthogonal state of the test mass, while it would not for the unsheared state. However, discriminating between the shifted and unshifted wavepackets requires a momentum-like measurement; position measurements would not suffice.

Of course, we could simply start with a wavepacket with a very wide spatial width and narrow momentum width. Back when this was being discussed by Caves and others in the ’80s, they recognized that these states would have such sensitivity. However, they pointed out, this couldn’t really be exploited because of the difficulty in making true momentum measurements. Rather, we usually measure momentum indirectly by allowing the normal free-particle () evolution carry the state to different points in space, and then measuring position. But this doesn’t work under the condition in which we’re interested: when the time between measurements is limited.The original motivation was for detecting gravitational waves, which transmit zero net momentum when averaged over the time interval on which the wave interacts with the test mass. The only way to notice the wave is to measure it in the act since the momentum transfer can be finite for intermediate times. a   (If we can take as much time as we want, we could always just allow the test-masses to evolve away from each other, and there’d be no limit to how sensitive we could be to weak forces.)

The one advantage of letting a wavepacket shear in phase-space (as opposed to just preparing a wavepacket that’s narrow in momentum) is that you don’t need extremely wide spatial traps. It’s conceivable that cooling a given test-mass to its ground state in a low-frequency trap (which we take to be a harmonic oscillator) is too difficult because the ground state energy is far below the experimentally achievable temperatures. But if you use a tighter trap, whose ground state can still be reached for much higher temperatures, then you can still achieve the momentum-shift sensitivity by allowing free expansion of the state.

Edit 2015-4-23: I think this concept might be related to this (new?) method for creating macroscopic quantum states that are sensitive to decoherence by kicking a mirror with an extremely short laser pulse: arXiv:1504.00790.

### Footnotes

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1. The original motivation was for detecting gravitational waves, which transmit zero net momentum when averaged over the time interval on which the wave interacts with the test mass. The only way to notice the wave is to measure it in the act since the momentum transfer can be finite for intermediate times.
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