Abstract
After the extensive discussion of bosonic and fermionic structures, we now dig deeper into the dynamical properties of these particles. More specifically, we focus on the concept of true indistinguishability and the additional interference effects arising because of it—even in the absence of physical interaction between the particles (Tichy 2011; Tichy et al. 2010, 2012). These many-body interferences are still rather poorly understood and not too much research has been done on them. The reason is that there are few many-particle systems with sufficient control that allow to treat non-Gaussian states (e.g. number states), which are required to see such fundamental quantum effects. For example, in well-controlled settings as photonics it is very difficult to generate number states, since one usually starts out from coherent or thermal light sources (Aspect et al. 1982; Ates et al. 2009; Eibl et al. 2003; Grice et al. 1998; Grice and Walmsley 1997; Huang et al. 2011; Kwiat et al. 1995; Metcalf et al. 2013; Mosley et al. 2008; Ou et al. 1999; Ra et al. 2013a; Santori et al. 2001, 2002; Tichy et al. 2011).
“O brave new world!”
John in “Brave new World” (Huxley 1932)
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Notes
- 1.
Later in Chap. 9 we explore particle transport properties in open systems where the number of particles is allowed to fluctuate.
- 2.
Because the creation and annihilation operators are generators of the algebra of observables, this requirement is logical.
- 3.
Since the derivations in this section hold both for fermions and bosons, we omit, for the time being, the index f / b in the second quantisation operators \(\Gamma \), originally defined in (7.89).
- 4.
“Normally ordered” (Davies 1977a) refers to monomials where all creation operators are ordered on the left, whereas all annihilation operators are on the right.
- 5.
Note that \(a^{\#}\) is a collective term for \(a^{\dag }\) and a, i.e. (8.10) holds for, both, creation and annihilation operators.
- 6.
In the Schrödinger picture.
- 7.
Which is actually not an observable, but rather a coherence (between the particles in modes \(\psi _1\) and \(\psi _2\)).
- 8.
- 9.
We were also confronted with this issue in Chap. 4, see e.g. p. 79.
- 10.
More specifically (Reck et al. 1994) shows that this amount is \(N(N-1)/2\) when one wishes to describe an \(N \times N\) unitary operation.
- 11.
Thus the concept of “exchange interaction” e.g. in the structure theory of helium (Madroñero 2004).
- 12.
One can simply use that all spaces of equal, finite dimension are isomorphic (Igodt and Veys 2011).
- 13.
This makes sense, because they can be told apart by the input direction they are following.
- 14.
Note that in principle the normalisation factor \({\mathcal N}'\) depends only on \(\xi _1, \dots \xi _n\) in this case, because \(\left\langle f_i,f_j\right\rangle = \delta _{ij}\).
- 15.
The proof relies on the “permanent-of-Gaussians conjecture” and on the “permanent anticoncentration conjecture” (see Aaronson and Arkhipov (2013) for details). Therefore it is formally not completely correct to refer to the result as a closed proof.
- 16.
In computational complexity theory jargon the result states that the existence of an efficient simulation scheme for boson sampling implies a collapse of the polynomial hierarchy up to the third order. For details on this statement, we refer the reader to Aaronson and Arkhipov (2013), Moore and Mertens (2011). For the context of this dissertation, it suffices to know that this is a complexity theorist’s way of stating that it is highly unlikely to be possible.
- 17.
In random matrix theory (RMT), the set of unitary matrices equipped with its Haar measure is also referred to as the Circular Unitary Ensemble (Akemann et al. 2011).
- 18.
- 19.
Even though these algorithms still have an unfavourable scaling with increasing numbers of particles, they are much more capable of simulating small systems. One can, for example, apply them to simulate boson sampling with several dozens of particles on a normal laptop.
- 20.
Experimentally, such a counter is rather difficult to construct. Standard photodetectors just click upon detection of one or more photons, e.g. the single-photon avalanche diodes (Cova et al. 1996) that are commonly used cannot resolve the photon number at a given time. The number of photons in a single mode is much harder to resolve, but efforts are being made (Humphreys et al. 2015). We are somewhat ambitious in the text and assume that full counting is feasible.
- 21.
More formal proofs can be constructed in various ways. An appealing construction exploits the equivalence of a bosonic system with a tensor product of harmonic oscillators (see Sect. 7.3.2), to explicitly construct eigenstates of a number operator acting on one of these oscillators.
- 22.
In principle, for each eigenvalue n, we obtain an enormous eigenspace on which to project. This implies that we must consider all wave functions which have a non-zero component in the n-particle sector. However, because we consider a projector on the whole eigenspace, we ultimately find that the only wave functions which are relevant in the sum (8.105) are those which are n-particle wave functions.
- 23.
And it is obviously unreasonable to hope to do so.
- 24.
Multisets are very similar to sets, but are an extension in that sense that elements can occur several times. For example \(\{1,2,3\}\) is a set, but \(\{1,1,2,3\}\) is a multiset. For an overview of the theoretical framework, see Blizard (1988)
- 25.
To give a simple example, when we consider the set \(\mathcal {A} = \{1,1,2,3\}\), then we find that \(\mathcal {A} \cap \mathbb {N} = \{1,2,3\}\).
- 26.
Were this not the case, we would find a row of zeroes in \(G^{V\setminus \mathcal {V},V\setminus \mathcal {V}'}\).
- 27.
Which actually implies finding benchmarks of many-particle interference.
- 28.
\(d < m\) is an integer which depends on the specific unitary scattering matrix.
- 29.
Or any other correlation function that we derived for that matter.
- 30.
- 31.
To avoid indices of indices of indices, we simply write j rather than \(i_j\) in the following step.
- 32.
As an example, consider the bunching event where all particles are detected in a single mode. There are only m such events which can occur for a total of \(m!/n!(m-n)!\) possible outcomes. Even with an enhanced probability, a lot of sampling is required to probe the relative frequency of such events.
- 33.
“Low” order is an order which is significantly smaller than the number of particles. In this Dissertation, we focus on second and third order.
- 34.
Keep in mind that these are multivariate analog of cumulants. They also occurs under the name joint cumulants, but we will follow the terminology of the quantum statistical mechanics community (Bratteli and Robinson 1997).
- 35.
The decrease in order is seen by the term
$$\sum _{k=1}^n U_{r i_k}U_{s i_k}\overline{U}_{r i_k}\overline{U}_{s i_k}$$which appears in all truncated correlation functions \(C^{b,f,d}_{rs}\). This summation only contains n terms, which is a considerable decrease compared to the \(n(n-1)\) terms which appear in
$$\sum _{\begin{array}{c} k,l = 1\\ k \ne l \end{array}}^n U_{r i_k}U_{s i_l}\overline{U}_{r i_k}\overline{U}_{s i_l},$$for the correlation functions \(c^{b,f,d}_{rs}\).
- 36.
The full single-particle Hilbert space is described by \(\mathcal {H}\cong \mathbb {C}^m \otimes \mathcal {H}_\mathrm{add}\). To observe interference patterns of indistinguishable bosons in the interferometer, each particle much be described by the same single-particle wave function in \(\mathcal {H}_\mathrm{add}\). For an extended discussion, see Sect. 8.3.4.
- 37.
In the sense that we can obtain them from a simulation of a circuit described by a random unitary matrix U.
- 38.
This notation was introduced by J.-D. Urbina and J. Kuipers in private communication.
- 39.
The cycle notation of permutations is commonly used in abstract algebra and discrete mathematics (Biggs 1989). A cycle of length n \((i_1, \dots , i_n)\) represents a cyclic permutation \(i_1 \rightarrow i_2 \rightarrow \dots \rightarrow i_n \rightarrow i_1\). It turns out that any permutation can be interpreted as combination of several cycles. As an example, we may consider the set of three elements \(\{a,b,c\}\) and consider the permutation \(\{a,b,c\} \mapsto \{a,b,c\}\), which can be written as a combination of three cycles of length one: (a)(b)(c). As a second example, we consider \(\{a,b,c\} \mapsto \{c,b,a\}\) which is denoted by (b)(a, c) in cycle nations. As a final example, \(\{a,b,c\} \mapsto \{c,a,b\}\) is written as (a, b, c) in cycle notation. In representation theory, it is common to classify permutations based on cycle lengths (Hamermesh 1989). To use the examples of before, we could characterise (a)(b)(c) as a permutation with cycle lengths (1, 1, 1), whereas (b)(a, c) has cycle lengths (1,2), and finally (a, b, c) has cycle length (3).
- 40.
Let us consider the difference between \(K_{l,1}\) and \(K_{l,2}\) in more depth to emphasise the idea: Both \(K_{l,1}\) and \(K_{l,2}\) start from the index set \(\{i_k,i_k,i_l\}\), however to represent the permutation of k and l, there are two possible choices for k. \(K_{l,1}\) represents the permutation \(\{i_k,i_k,i_l\} \rightarrow \{i_k,i_l,i_k\}\), whereas \(K_{l,2}\) represents \(\{i_k,i_k,i_l\} \rightarrow \{i_l,i_k,i_k\}\).
- 41.
For the Fourier matrix, the normalised means of both \(C_{rs}\) and \(C_{qrs}\) are completely independent of the chosen input state (8.177). The physical interpretation of this result is currently unclear to us.
- 42.
Mathematically, \(J=J'\) simply follows because the factor associated with the factor related to the additional degrees of freedom is of the type \(\left|\left\langle \chi _k,\chi _k\right\rangle \right|^2\left\langle \chi _l,\chi _l\right\rangle \), which is equal to one due to normalisation.
- 43.
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Walschaers, M. (2018). Many-Particle Interference. In: Statistical Benchmarks for Quantum Transport in Complex Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93151-7_8
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