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Classicalization and the Macroscopicity of Quantum Superposition States

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Macroscopic Matter Wave Interferometry

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Abstract

In this chapter I take a broader perspective and turn the question of the classical limitation to heavy-particle interference into the question of the macroscopicity of quantum superposition states in general. The central result of this chapter will be the definition of an objective measure of macroscopicity. It allows us to quantify how macroscopic various quantum superposition states are, which have been or may soon be observed in past and future experiments on mechanical systems.

Under the ideal measure of values there lurks the hard cash.—Karl Marx

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Notes

  1. 1.

    The concept of macroscopicity presented here only applies to quantum systems with a direct classical correspondence in nonrelativistic mechanics. This excludes, by definition, systems of genuinely quantum degrees of freedom such as spins.

  2. 2.

    The normalization \(\text {tr}\left( \rho \right) =1\) implies that the state is a norm-bounded operator. The set of bounded operators \( \mathcal {B} \left( \mathcal {H} \right) \) on a separable Hilbert space \( \mathcal {H} \) is also a Hilbert space.

  3. 3.

    An additional technical assumption is required apart from the semigroup property and the LCPT nature of the one-parameter family \(\left\{ \varPhi _{t}\,|\,\, t\ge 0\right\} \): Expectation values must evolve continuously in time, that is, the function \(\left\langle {\mathsf {A}} \left( t\right) \right\rangle =\text {tr}\left( {\mathsf {A}} \varPhi _{t}\left( \rho \right) \right) \) must be continuous in \(t\) for any state \(\rho \) and any bounded operator \( {\mathsf {A}} \in \mathcal {B} \left( \mathcal {H} \right) \).

  4. 4.

    The unitary representation of time translations in a system with Hamiltonian \( {\mathsf {H}} \) is given by the time evolution operators \( {\mathsf {U}} (t_{g})=\exp \left[ -i {\mathsf {H}} t_{g}/\hbar \right] \). The time translation-covariant Lindblad generator of a modification of the coherent time evolution would require the Lindblad operators to commute with \( {\mathsf {U}} \left( t_{g}\right) \) up to a complex phase.

  5. 5.

    An operator exponential can be split according to \(e^{ {\mathsf {X}} + {\mathsf {Y}} }=e^{ {\mathsf {X}} }e^{ {\mathsf {Y}} }e^{-\left[ {\mathsf {X}} , {\mathsf {Y}} \right] /2}\) when the commutator is a \(\mathbb {C}\)-number. From this follows also the identity \(e^{ {\mathsf {X}} } {\mathsf {Y}} e^{- {\mathsf {X}} }= {\mathsf {Y}} +\left[ {\mathsf {X}} , {\mathsf {Y}} \right] \).

  6. 6.

    Point particles with \(f\left( \varvec{r}\right) =m\delta \left( \varvec{r}\right) \) are not allowed in this model because they would lead to a diverging modification term. One must assume a finite mass density, or finite size, for every particle, even including electrons, for instance.

  7. 7.

    Although plane waves are improper states, i.e. not normalizable, they serve as a helpful idealization in many concrete situations.

  8. 8.

    The coefficient functions could actually share a common phase factor \(\exp \left[ i\theta \left( \varvec{s},\varvec{w}\right) \right] \) which needs not be isotropic. Such a global phase factor in the Lindblad operator \( {\mathsf {L}} ^{(\mathrm S)}\left( \varvec{s},\varvec{w}\right) \) may, however, always be omitted without loss of generality.

  9. 9.

    This corresponds to the \(j\)-summation in the general Galilei-covariant form of the modification (4.21), which is otherwise omitted.

  10. 10.

    It was noted in Sect. 4.1.3.2 that the single-particle modification is characterized by a minimal length scale where it affects superpositions. Smaller-scale coherences are hardly influenced, and the internal dynamics of a sufficiently small compound object should therefore remain unaffected under the same conditions.

  11. 11.

    Invariance with respect to particle exchange is fulfilled if the Lindblad operators \( {\mathsf {L}} ^{(\mathrm S)}\left( \varvec{s},\varvec{w}\right) \) of the modification commute with the exchange of any two out of \(N\) identical particles \(n,m\) up to a complex phase factor \(\exp \left[ i\phi _{nm}\left( \varvec{s},\varvec{w}\right) \right] \). Since this must generally hold for any \(N\ge 2\), all relative phases \(\theta _{n}\) between indistinguishable particles in (4.50) must be set to zero. This makes the Lindblad operators exchange-symmetric sums of single-particle operators.

  12. 12.

    Evidently, I assume that there exist (besides elementary point particles) point-like compound particles whose internal structure remains practically unaffected by the classicalizing effect. This is certainly a weaker prerequisite than establishing a fixed reference set of elementary particles with individual classicalization rates and distribution functions, and it can be justified a posteriori by limiting the strength of the classicalization effect in compliance with experimental observations on the microscopic scale. In practice one should describe electrons, protons, neutrons and nuclei as point particles, or otherwise even femtometer-sized superpositions of elementary particles would be affected. In this case one could keep microscopic systems unaffected only by limiting the classicalization rate of macroscopic systems to unsatisfactorily low values, as was discussed in the case of spontaneous localization models in [44].

  13. 13.

    Spins are finite-dimensional degrees of freedom and their basis states are thus labelled by a single running index \(\sigma \). The classicalizing modification only acts on mechanical degrees of freedom, and the internal dynamics of spin states remains unaffected.

  14. 14.

    Note that the modulated distribution function in (4.75) is not normalized; a renormalization to unity leads to an effective center-of-mass time parameter \(1/\tau _{\mathrm{cm}}<4/\tau _{0}\). Note also that we have to substitute the integration variable \(\varvec{s}\) by \(\varvec{s}/2\) to arrive at the single-particle form (4.33).

  15. 15.

    The approximation ceases to be valid sufficiently close to the origin \(\varvec{s}=\varvec{q}=0\), but we neglect this contribution to the integral over the phase space distribution \(g_{0}\left( s,q\right) \).

  16. 16.

    The \(N\) vectors \(\overline{\varvec{r}}_{n}\) are linearly dependent. They sum up to \(\sum _{n}m_{n}\overline{\varvec{r}}_{n}=0\).

  17. 17.

    This resembles the continuum approximation in [20, 21] for the specific case of spontaneous collapse models.

  18. 18.

    Note that, in particular, the classicalizing modification can only make physical sense if the distribution \(g\left( \varvec{s},\varvec{q}\right) \) is a well-behaved function with finite moments.

  19. 19.

    The thermal state \(\rho =\exp \left( - {\mathbf {\mathsf{{p}}}}^{2}/2mk_{B}T\right) /\left( 2\pi mk_{B}T\right) ^{3/2}\) of a free particle corresponds to the characteristic function \(\chi \left( \varvec{r},\varvec{p}\right) =\delta \left( \varvec{p}\right) \exp \left( -mk_{B}T {\mathbf {\mathsf{{r}}}}^{2}/2\hbar ^{2}\right) \). It is improperly normalized because the spatial coordinate extends over the whole coordinate space \(\mathbb {R}^{3}\). Regardless of this issue, which could be fixed by restricting to a finite volume, with growing temperature \(T\) the peak of the characteristic function around the origin gets more narrow. The same happens to the state under the influence of the classicalizing modification with growing time, which relates to the classicalization-induced diffusion heating discussed before.

  20. 20.

    To see the small-argument limit explicitly, expand the Fourier transform \(\widetilde{g}\left( \varvec{r},\varvec{p}\right) \) of the distribution function to the lowest non-vanishing order around the origin, \(\widetilde{g}\left( \varvec{r},\varvec{p}\right) \approx 1- \mathcal {E} \left[ s^{2} \right] p^{2}/2\hbar ^{2}- \mathcal {E} \left[ q^{2} \right] r^{2}/2\hbar ^{2}\). This leads to

    $$ \mathcal {L} _{1}\left[ \chi \left( \varvec{r},\varvec{p}\right) \right] \approx -\left( \frac{ \mathcal {E} \left[ s^{2} \right] }{2\hbar ^{2}\tau }p^{2}+\frac{ \mathcal {E} \left[ q^{2} \right] }{2\hbar ^{2}\tau }r^{2}\right) \chi \left( \varvec{r},\varvec{p}\right) , $$

    which is equivalent to the diffusion expression (4.101).

  21. 21.

    This case might be appealing as it also conserves the energy of a free particle, i.e. the invariance under time translations.

  22. 22.

    Controllable nonlinear interactions are the crucial ingredient in interference experiments with squeezed multi-component BECs to achieve longer coherence times and better phase sensitivity [54–56].

  23. 23.

    The effective Heisenberg equation of motion for an observable \( {\mathsf {B}} \) is obtained by taking the time derivative of the expectation value, \(\left\langle \partial _{t} {\mathsf {B}} \right\rangle \equiv \partial _{t}\left\langle {\mathsf {B}} \right\rangle =\partial _{t}\text {tr}\left( \rho {\mathsf {B}} \right) \), and using the cyclic property of the trace to shift the master equation \(\partial _{t}\rho =-i\left[ {\mathsf {H}} ,\rho \right] /\hbar + \mathcal {L} \left( \rho \right) \) to \( {\mathsf {B}} \). This results in the conjugate equation \(\partial _{t} {\mathsf {B}} =i\left[ {\mathsf {H}} , {\mathsf {B}} \right] /\hbar + \mathcal {L} ^{{\dagger }}\left( {\mathsf {B}} \right) \), where \( \mathcal {L} ^{{\dagger }}\left( {\mathsf {B}} \right) = \mathcal {L} \left( {\mathsf {B}} \right) \) holds due to the isotropy of the classicalizing modification (4.117).

  24. 24.

    The law of induction tells us that any change of magnetic flux through an open loop induces an electric potential difference \(u=-\mathrm{d}\varPhi /\mathrm{d}t\). Here, the voltage \(u\) must build up across the tunneling junction, which can be regarded as a capacitor \(C\). This is in accordance with the Josephson equation \(\mathrm{d}\varDelta \phi /\mathrm{d}t=2eu/\hbar =2\pi u/\varPhi _{0}\) that generally relates the phase difference with the voltage across a Josephson junction [51]. The voltage gives rise to a charge imbalance \(Q=Cu\) and to a capacitive energy \(E_{C}=Cu^{2}/2\).

  25. 25.

    We restrict our view to s-type superconductors with electrons pairing in a spin singlet state.

  26. 26.

    The momentum spread of the superposition is much smaller than the Fermi wave number \(k_{F}\) as only the momentum states close to the Fermi surface are accessible by electron-phonon scattering. The spatial extension of the superposition is therefore much larger than the de Broglie wavelength \(2\pi /k_{F}\) of the Fermi electrons.

  27. 27.

    Strictly speaking, the BCS ground state does not describe a fixed number of electrons, but rather a distribution of numbers around the expectation value \(N=\langle \mathrm{BCS}| {\mathsf {N}} |\mathrm{BCS}\rangle \). The effect of this uncertain number of electrons can be neglected as \(N\) is very large in most practical cases.

  28. 28.

    Discretized momenta must be used, as introduced in Sect. 4.1.4.6. The Weyl operator \( {\mathsf {W}} \left( \varvec{r},\varvec{v}\right) \) describes a translation of both the position and the momentum coordinate, which transforms the momentum creation operator to \( {\mathsf {W}} \left( \varvec{r},\varvec{v}\right) {\mathsf {c}} _{\sigma }^{{\dagger }}\left( \varvec{k}\right) {\mathsf {W}} ^{{\dagger }}\left( \varvec{r},\varvec{v}\right) \,{=}\,\exp \left( i\varvec{k}\cdot \varvec{r}-m_{e}\varvec{v}\cdot \varvec{r}/2\hbar \right) {\mathsf {c}} _{\sigma }^{{\dagger }}\left( \varvec{k}-m_{e}\varvec{v}/\hbar \right) \), as can be checked in a straightforward calculation using the CCR (4.64).

  29. 29.

    Recall that the observable consequences of the CSL model [20, 21] are described by a specific classicalizing modification of the present form with a Gaussian momentum distribution and no position diffusion, as discussed in Sect. 4.1.3.2.

  30. 30.

    The atom mass is mainly concentrated in the femtometer-sized nucleus, and the small contribution of the electrons is negligible. It can therefore be regarded as a point particle in the present context.

  31. 31.

    One observes the same behaviour in the case of thermal decoherence at low temperatures [69], where the average photon wavelength is greater than the path separation.

  32. 32.

    We must only include those parts of the setup where a loss of coherence in the particle state affects the interference visibility. This excludes, for instance, anything that happens before the first collimation slit in a double-slit experiment representing the point source. In a Talbot-Lau setup we must only consider the passage from the first to the second and from the second to the third grating.

  33. 33.

    The presence of a constant acceleration would only contribute to the phase difference.

  34. 34.

    The kernel is similar to the diffraction kernels used in Chap. 3 to describe near-field interference in the Wigner function representation, except that the integration is over the center position here.

  35. 35.

    The phase sensitivity and the coherence time of the BEC interferometer were increased in a later experiment [52]. However, I estimated a smaller macroscopicity in this case, \(\mu \approx 8.3\), since only \(f=15\,\%\) visibility was found after \(200\,\)ms time of flight.

  36. 36.

    This aspect prevents many-body product states of the form \(|\psi _{N}\rangle \propto \left( |A\rangle +|B\rangle \right) ^{\otimes N}\) from being considered more macroscopic than the single-particle state \(|\psi _{1}\rangle \propto |A\rangle +|B\rangle \). On the other hand, the disconnectivity of a GHZ-type state, \(|\psi \rangle \propto |A\rangle ^{\otimes N}+|B\rangle ^{\otimes N}\), would increase with the number of particles \(N\).

  37. 37.

    This triggered an interesting and illustrative debate on the macroscopicity of many-body quantum phenomena. Leggett’s prime example of a macroscopic quantum effect was the superposition of persistent currents in a superconducting Josephson loop, first observed in [6, 7]. He claimed that the disconnectivity of the measured state was given by the total number of Cooper-paired electrons, more than a billion to be precise, flowing either clockwise or anticlockwise through the loop. A proper many-body analysis, however, reveals that the state is given by a superposition of two slightly displaced but largely overlapping Fermi spheres, as found by Korsbakken et al. [62, 63]. Hence, the vast majority of indistinguishable electrons occupies the same momentum space region in both branches of the superposition, and only a few hundred to thousand are actually disconnected, i.e. found in different states.

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    Stefan Nimmrichter

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Nimmrichter, S. (2014). Classicalization and the Macroscopicity of Quantum Superposition States. In: Macroscopic Matter Wave Interferometry. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-07097-1_4

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