Is Quantum Linear Superposition an Exact Principle of Nature?

Abstract

The principle of linear superposition is a hallmark of quantum theory. It has been confirmed experimentally for photons, electrons, neutrons, atoms, for molecules having masses up to ten thousand amu, and also in collective states such as SQUIDs and Bose-Einstein condensates. However, the principle does not seem to hold for positions of large objects! Why for instance, a table is never found to be in two places at the same time? One possible explanation for the absence of macroscopic superpositions is that quantum theory is an approximation to a stochastic nonlinear theory. This hypothesis may have its fundamental origins in gravitational physics, and is being put to test by modern ongoing experiments on matter wave interferometry.

This essay received the Fourth Prize in the FQXi Essay Contest, 2012.

Technical Endnotes (For Details See [5])

The Physics of Continuous Spontaneous Localization

The essential physics of the CSL model can be described by a simpler model, known as QMUPL (Quantum Mechanics with Universal Position Localization), whose dynamics is given by the following stochastic nonlinear Schrödinger equation

$$\begin{aligned} d \psi _t = \left[ -\frac{i}{\hbar } H dt + \sqrt{\lambda } (q - \langle q \rangle _t) dW_t - \frac{\lambda }{2} (q - \langle q \rangle _t)^2 dt \right] \psi _t, \end{aligned}$$

(10.1)

where $q$ is the position operator of the particle, $\langle q \rangle _t \equiv \langle \psi _t | q | \psi _t \rangle$ is the quantum expectation, and $W_t$ is a standard Wiener process which encodes the stochastic effect. Evidently, the stochastic term is nonlinear and also nonunitary. The collapse constant $\lambda$ sets the strength of the collapse mechanics, and it is chosen proportional to the mass $m$ of the particle according to the formula $\lambda = \frac{m}{m_0}\; \lambda _0,$ where $m_0$ is the nucleon’s mass and $\lambda _0$ measures the collapse strength. If we take $\lambda _0\simeq 10^{-2}$ m$^{-2}$ s$^{-1}$ the strength of the collapse model corresponds to the CSL model in the appropriate limit.

The above dynamical equation can be used to prove position localization; consider for simplicity a free particle $(H=p^2/2m)$ in the gaussian state (analysis can be generalized to other cases):

$$\begin{aligned} \psi _{t}(x) = \text {exp}\left[ - a_{t} (x - \overline{x}_{t})^2 + i \overline{k}_{t}x + \gamma _{t}\right] . \end{aligned}$$

(10.2)

By substituting this in the stochastic equation it can be proved that the spreads in position and momentum

$$\begin{aligned} \sigma _{q}(t) \equiv \frac{1}{2}\sqrt{\frac{1}{a_{t}^{{\tiny \mathrm{{R}}}}}};\qquad \sigma _{p}(t) \equiv \hbar \,\sqrt{\frac{(a_{t}^{{\tiny \mathrm{{R}}}})^2 + (a_{t}^{{\tiny \mathrm{{I}}}})^2}{a_{t}^{{\tiny \mathrm{{R}}}}}}, \end{aligned}$$

(10.3)

do not increase indefinitely but reach asymptotic values given by

(10.4)

such that: $\sigma _{q}(\infty )\, \sigma _{p}(\infty ) \; {=} \; {\hbar }/{\sqrt{2}}$ which corresponds to almost the minimum allowed by Heisenberg’s uncertainty relations. Here, $\omega \; = \; 2\,\sqrt{{\hbar \lambda _{0}}/{m_{0}}} \; \simeq \; 10^{-5} \; \mathrm{{s}}^{-1}.$

Evidently, the spread in position does not increase indefinitely, but stabilizes to a finite value, which is a compromise between the Schrödinger’s dynamics, which spreads the wave function out in space, and the collapse dynamics, which shrinks it in space. For microscopic systems, this value is still relatively large ($\sigma _{q}(\infty ){\sim }1$ m for an electron, and ${\sim }1$ mm for a $C_{60}$ molecule containing some 1,000 nucleons), such as to guarantee that in all standard experiments—in particular, diffraction experiments—one observes interference effects. For macroscopic objects instead, the spread is very small ($\sigma _{q}(\infty ){\sim }3\times 10^{-14}$ m, for a 1 g object), so small that for all practical purposes the wave function behaves like a point-like system. This is how localization models are able to accommodate both the “wavy” nature of quantum systems and the “particle” nature of classical objects, within one single dynamical framework.

The same stochastic differential equation solves the quantum measurement problem and explains the Born probability rule without any additional assumptions. For illustration, consider a two state microscopic quantum system $\mathcal{S}$ described by the initial state

$$\begin{aligned} c_{+}|+\rangle + c_{-} |-\rangle \end{aligned}$$

(10.5)

interacting with a measuring apparatus $\mathcal{A}$ described by the position of a pointer which is initially in a ‘ready’ state $\phi _{0}$ and which measures some observable $O$, say spin, associated with the initial quantum state of $\mathcal{S}$. As we have seen above, the pointer being macroscopic (for definiteness assume its mass to be 1 g), is localized in a gaussian state $\phi ^{G}$, so that the initial composite state of the system and apparatus is given by

$$\begin{aligned} \Psi _{0} = \left[ c_{+} |+\rangle + c_{-} |+\rangle \right] \otimes \phi ^{\mathrm {G}}. \end{aligned}$$

(10.6)

According to standard quantum theory, the interaction leads to the following type of evolution:

$$\begin{aligned} {\left[ c_{+} |+\rangle + c_{-} |-\rangle \right] \otimes \phi ^{G}} \qquad \mapsto \qquad c_{+} |+\rangle \otimes \phi _{+} + c_{-} |-\rangle \otimes \phi _{-}, \end{aligned}$$

(10.7)

where $\phi _{+}$ and $\phi _{-}$ are the final pointer states corresponding to the system being in the collapsed state $|+\rangle$ or $|-\rangle$ respectively. While quantum theory explains the transition from the entangled state (10.7) to one of the collapsed alternatives by invoking a new interpretation or reformulation, the same is achieved dynamically by the stochastic nonlinear theory given by (10.1).

It can be proved from (10.1) that the initial state (10.6) evolves, at late times, to

$$\begin{aligned} \psi _{t} = \frac{|+\rangle \otimes \phi _{+} + \epsilon _{t}|-\rangle \otimes \phi _{-}}{\sqrt{1+ \epsilon _t^2}}. \end{aligned}$$

(10.8)

The evolution of the stochastic quantity $\epsilon _t$ is determined dynamically by the stochastic equation: it either goes to $\epsilon _t \ll 1$, with a probability $|c_{+}|^2$, or to $\epsilon _t \gg 1$, with a probability $|c_{-}|^2$. In the former case, one can say with great accuracy that the state vector has ‘collapsed’ to the definite outcome $|+\rangle \otimes \phi _{+}$ with a probability $|c_{+}|^2$. Similarly, in the latter case one concludes that the state vector has collapsed to $|-\rangle \otimes \phi _{-}$ with a probability $|c_{-}|^2$. This is how collapse during a quantum measurement is explained dynamically, and random outcomes over repeated measurements are shown to occur in accordance with the Born probability rule. The time-scale over which $\epsilon _t$ reaches its asymptotic value and the collapse occurs can also be computed dynamically. In the present example, for a pointer mass of 1 g, the collapse time turns out to be about $10^{-4}$ s.

Lastly, we can understand how the modified stochastic dynamics causes the outcome of a diffraction experiment in matter wave-interferometry to be different from that in quantum theory. Starting from the fundamental Eq.  (10.1) it can be shown that the statistical operator $\rho _t = \mathbb {E}[|\psi _t\rangle \langle \psi _t|]$ for a system of $N$ identical particles evolves as

$$\begin{aligned} \rho _t(x,y) = \rho _0(x,y) e^{- \lambda N (x-y)^2 t/2}. \end{aligned}$$

(10.9)

Experiments look for a decay in the density matrix by increasing the number of the particles $N$ in an object, by increasing the slit separation $|x-y|$, and by increasing the time of travel $t$ from the grating to the collecting surface. The detection of an interference pattern sets an upper bound on $\lambda$. The absence of an interference pattern would confirm the theory and determine a specific value for $\lambda$ (provided all sources of noise such as decoherence are ruled out.)

A detailed review of the CSL model and its experimental tests and possible underlying theories can be found in [5].