Standard quantum limit for diffusion

I just posted my newest paper: “Decoherence from classically undetectable sources: A standard quantum limit for diffusion” (arXiv:1504.03250). [Edit: Now published as PRA 92, 010101(R) (2015).] The basic idea is to prove a standard quantum limit (SQL) that shows that some particles can be detected through the anomalous decoherence they induce even though they cannot be detected with any classical experiment. Hopefully, this is more evidence that people should think of big spatial superpositions as sensitive detectors, not just neat curiosities.

Here’s the abstract:

In the pursuit of speculative new particles, forces, and dimensions with vanishingly small influence on normal matter, understanding the ultimate physical limits of experimental sensitivity is essential. Here, I show that quantum decoherence offers a window into otherwise inaccessible realms. There is a standard quantum limit for diffusion that restricts some entanglement-generating phenomena, like soft collisions with new particle species, from having appreciable classical influence on normal matter. Such phenomena are classically undetectable but can be revealed by the anomalous decoherence they induce on non-classical superpositions with long-range coherence in phase space. This gives strong, novel motivation for the construction of matter interferometers and other experimental sources of large superpositions, which recently have seen rapid progress. Decoherence is always at least second order in the coupling strength, so such searches are best suited for soft, but not weak, interactions.

Here’s Figure 2:


Standard quantum limit for forces and momentum diffusion. A test mass is initially placed in a minimal uncertainty wavepacket with a Wigner distribution W(x,p) over phase space (top) that contains the bulk of its mass within a 2\sigma-contour of a Gaussian distribution (dashed black line).
[continue reading]

How to think about Quantum Mechanics—Part 4: Quantum indeterminism as an anomaly

[Other parts in this series: 1,2,3,4,5,6,7,8.]

I am firmly of the view…that all the sciences are compatible and that detailed links can be, and are being, forged between them. But of course the links are subtle… a mathematical aspect of theory reduction that I regard as central, but which cannot be captured by the purely verbal arguments commonly employed in philosophical discussions of reduction. My contention here will be that many difficulties associated with reduction arise because they involve singular limits….What nonclassical phenomena emerge as h 0? This sounds like nonsense, and indeed if the limit were not singular the answer would be: no such phenomena.Michael Berry

One of the great crimes against humanity occurs each year in introductory quantum mechanics courses when students are introduced to an \hbar \to 0 limit, sometimes decorated with words involving “the correspondence principle”. The problem isn’t with the content per se, but with the suggestion that this somehow gives a satisfying answer to why quantum mechanics looks like classical mechanics on large scales.

Sometimes this limit takes the form of a path integral, where the transition probability for a particle to move from position x_1 to x_2 in a time T is

(1)   \begin{align*} P_{x_1 \to x_2} &= \langle x_1 \vert e^{-i H T} \vert x_2 \rangle \\ &\propto \int_{x_1,x_2} \mathcal{D}[x(t)] e^{-i S[x(t),x'(t)]/\hbar} = \int_{x_1,x_2} \mathcal{D}[x(t)] e^{-i \int_0^T \mathrm{d}t L(x(t),x'(t))/\hbar} \end{align*}

where \int_{x_1,x_2} \mathcal{D}[x(t)] is the integral over all paths from x_1 to x_2, and S[x(t),x'(t)]= \int_0^T \mathrm{d}t L(x(t),x'(t)) is the action for that path (L being the Lagrangian corresponding to the Hamiltonian H). As \hbar \to 0, the exponent containing the action spins wildly and averages to zero for all paths not in the immediate vicinity of the classical path that make the action stationary.

Other times this takes the form of Ehrenfest’s theorem, which shows that the expectation values of functions of position and momentum follow the classical equations of motion.… [continue reading]

Decoherence detection and micromechanical resonators

In this post I want to lay out why I am a bit pessimistic about using quantum micromechanical resonators, usually of the optomechanical variety, for decoherence detection. I will need to rely on some simple ideas from 3-4 papers I have “in the pipeline” (read: partially written TeX files) that seek to make precise the sense in which decoherence detection allows us to detect classical undetectable phenomena, and to figure out exactly what sort of phenomena we should apply it to. So this post will sound vague without that supporting material. Hopefully it will still be useful, at least for the author.

The overarching idea is that decoherence detection is only particularly useful when the experimental probe can be placed in a superposition with respect to a probe’s natural pointer basis. Recall that the pointer basis is the basis in which the density matrix of the probe is normally restricted to be approximately diagonal by the interaction with the natural environment. Classically detectable phenomena are those which cause transitions within the pointer basis, i.e. driving the system from one pointer state to another. Classically undetectable phenomena are those which cause pure decoherence with respect to this basis, i.e. they add a decoherence factor to off-diagonal terms in this basis, but preserve on-diagonal terms.

The thing that makes this tricky to think about is that the pointer basis is overcomplete for most physically interesting situations, in particular for any continuous degree of freedom like the position of a molecule or a silicon nanoparticle. It’s impossible to perfectly localize a particle, and the part of the Hamiltonian that fights you on this, p^2/2m, causes a smearing effect that leads to the overcompleteness.… [continue reading]

Records decomposition talk

Last month Scott Aaronson was kind enough to invite me out to MIT to give a seminar to the quantum information group. I presented a small uniqueness theorem which I think is an important intermediary result on the way to solving the set selection problem (or, equivalently, to obtaining an algorithm for breaking the wavefunction of the universe up into branches). I’m not sure when I’ll have a chance to write this up formally, so for now I’m just making the slides available here.

Screen Shot 2015-01-26 at 7.42.02 AM

Scott’s a fantastic, thoughtful host, and I got a lot of great questions from the audience. Thanks to everyone there for having me.… [continue reading]

Ambiguity and a catalog of the actions

I had to brush up on my Hamilton-Jacobi mechanics to referee a paper. I’d like to share, from this Physics.StackExchange answer, Qmechanic’ clear catalog of the conceptually distinct functions all called “the action” in classical mechanics, taking care to specify their functional dependence:

At least three different quantities in physics are customary called an action and denoted with the letter S.

  1. The (off-shell) action

    (1)   \[S[q]~:=~ \int_{t_i}^{t_f}\! dt \ L(q(t),\dot{q}(t),t)\]

    is a functional of the full position curve/path q^i:[t_i,t_f] \to \mathbb{R} for all times t in the interval [t_i,t_f]. See also this question. (Here the words on-shell and off-shell refer to whether the equations of motion (eom) are satisfied or not.)

  2. If the variational problem (1) with well-posed boundary conditions, e.g. Dirichlet boundary conditions

    (2)   \[ q(t_i)~=~q_i\quad\text{and}\quad q(t_f)~=~q_i,\]

    has a unique extremal/classical path q_{\rm cl}^i:[t_i,t_f] \to \mathbb{R}, it makes sense to define an on-shell action

    (3)   \[ S(q_f;t_f;q_i,t_i) ~:=~ S[q_{\rm cl}],\]

    which is a function of the boundary values. See e.g. MTW Section 21.1.

  3. The Hamilton’s principal function S(q,\alpha, t) in Hamilton-Jacobi equation is a function of the position coordinates q^i, integration constants \alpha_i, and time t, see e.g. H. Goldstein, Classical Mechanics, chapter 10.
    The total time derivative

    (4)   \[ \frac{dS}{dt}~=~ \dot{q}^i \frac{\partial S}{\partial q^i}+ \frac{\partial S}{\partial t}\]

    is equal to the Lagrangian L on-shell, as explained here. As a consequence, the Hamilton’s principal function S(q,\alpha, t) can be interpreted as an action on-shell.

These sorts of distinctions are constantly swept under the rug in classical mechanics courses and textbooks (even good books like Goldstein). This leads to serious confusion on the part of the student and, more insidiously, it leads the student to think that this sort of confusion is normal. Ambiguity is baked into the notation! This is a special case of what I conjecture is a common phenomena in physics:

  • Original researcher thinks deeply, discovers a theory, and writes it down.
[continue reading]

Approach to equilibrium in a pure-state universe

(This post is vague, and sheer speculation.)

Following a great conversation with Miles Stoudenmire here at PI, I went back and read a paper I forgot about: “Entanglement and the foundations of statistical mechanics” by Popescu et al.S. Popescu, A. Short, and A. Winter, “Entanglement and the foundations of statistical mechanics” Nature Physics 2, 754 – 758 (2006) [Free PDF].a  . This is one of those papers that has a great simple idea, where you’re not sure if it’s profound or trivial, and whether it’s well known or it’s novel. (They cite references 3-6 as “Significant results along similar lines”; let me know if you’ve read any of these and think they’re more useful.) Anyways, here’s some background on how I think about this.

If a pure quantum state \vert \psi \rangle is drawn at random (according to the Haar measure) from a d_S d_E-dimensional vector space \mathcal{H}, then the entanglement entropy

    \[S(\rho_S) = \mathrm{Tr}[\rho_S \mathrm{log} \rho_S], \qquad \rho_S = \mathrm{Tr}_E[\vert \psi \rangle \langle \psi \vert]\]

across a tensor decomposition into system \mathcal{S} and environment \mathcal{E} is highly likely to be almost the maximum

    \[S_{\mathrm{max}} = \mathrm{log}_2(\mathrm{min}(d_S,d_E)) \,\, \mathrm{bits},\]

for any such choice of decomposition \mathcal{H} = \mathcal{S} \otimes \mathcal{E}. More precisely, if we fix d_S/d_E and let d_S\to \infty, then the fraction of the Haar volume of states that have entanglement entropy more than an exponentially small (in d_S) amount away from the maximum is suppressed exponentially (in d_S). This was known as Page’s conjectureD. Page, Average entropy of a subsystem.b  , and was later provedS. Foong and S. Kanno, Proof of Page’s conjecture on the average entropy of a subsystem.c  J. Sánchez-Ruiz, Simple proof of Page’s conjecture on the average entropy of a subsystem.d  ; it is a straightforward consequence of the concentration of measure phenomenon.… [continue reading]

Undetected photon imaging

Lemos et al. have a relatively recent letterG. Lemos, V. Borish, G. Cole, S. Ramelow, R. Lapkiewicz, and A. Zeilinger, “Quantum imaging with undetected photons”, Nature 512, 409 (2014) [ arXiv:1401.4318 ].a   in Nature where they describe a method of imaging with undetected photons. (An experiment with the same essential quantum features was performed by Zou et al.X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference”, Phys. Rev. Lett. 67, 318 (1991) [ PDF ].b   way back in 1991, but Lemos et al. have emphasized its implications for imaging.) The idea is conceptually related to decoherence detection, and I want to map one onto the other to flesh out the connection. Their figure 1 gives a schematic of the experiment, and is copied below.


Figure 1 from Lemos et al.: ''Schematic of the experiment. Laser light (green) splits at beam splitter BS1 into modes a and b. Beam a pumps nonlinear crystal NL1, where collinear down-conversion may produce a pair of photons of different wavelengths called signal (yellow) and idler (red). After passing through the object O, the idler reflects at dichroic mirror D2 to align with the idler produced in NL2, such that the final emerging idler f does not contain any information about which crystal produced the photon pair. Therefore, signals c and e combined at beam splitter BS2 interfere. Consequently, signal beams g and h reveal idler transmission properties of object O.''

The first two paragraphs of the letter contain all the meat, encrypted and condensed into an opaque nugget of the kind that Nature loves; it stands as a good example of the lamentable way many quantum experimental articles are written.… [continue reading]

Quantum Brownian motion: Definition

In this post I’m going to give a clean definition of idealized quantum Brownian motion and give a few entry points into the literature surrounding its abstract formulation. A follow-up post will give an interpretation to the components in the corresponding dynamical equation, and some discussion of how the model can be generalized to take into account the ways the idealization may break down in the real world.

I needed to learn this background for a paper I am working on, and I was motivated to compile it here because the idiosyncratic results returned by Google searches, and especially this MathOverflow question (which I’ve answered), made it clear that a bird’s eye view is not easy to find. All of the material below is available in the work of other authors, but not logically developed in the way I would prefer.

Preliminaries

Quantum Brownian motion (QBM) is a prototypical and idealized case of a quantum system \mathcal{S}, consisting of a continuous degree of freedom, that is interacting with a large multi-partite environment \mathcal{E}, in general leading to varying degrees of dissipation, dispersion, and decoherence of the system. Intuitively, the distinguishing characteristics of QBM is Markovian dynamics induced by the cumulative effect of an environment with many independent, individually weak, and (crucially) “phase-space local” components. We will defined QBM as a particular class of ways that a density matrix may evolve, which may be realized (or approximately realized) by many possible system-environment models. There is a more-or-less precise sense in which QBM is the simplest quantum model capable of reproducing classical Brownian motion in a \hbar \to 0 limit.

In words to be explained: QBM is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the evolution is specified by a quadratic Hamiltonian and linear Lindblad operators.… [continue reading]