Interference Visibility and Wave-Particle Duality in Multi-Path Interference

TL;DR

This paper introduces a new wave-particle duality relation for multi-path interference that accounts for complex visibility behaviors and links to coherence measures, resolving longstanding issues in the field.

Contribution

It presents a novel duality relation based on sum of pairwise visibilities, applicable even when traditional complementarity fails, and connects it to coherence measurement.

Findings

Sum of pairwise visibilities always respects the duality relation.

The relation offers a new method to measure coherence experimentally.

It resolves issues where interference visibility increases with path information.

Abstract

Wave-particle duality in multi-path interference is fraught with issues despite substantial progress in recent years. It was experimentally shown that in certain specific conditions, getting path information in a multi-path experiment can actually increase the visibility of interference. As a result, it was argued that in multi-path interference experiments, visibility of interference and 'which-path' information are not always complementary observables. In the present work, a new wave-particle duality relation is presented, based on a sum of visibilities of interference from individual pairs of path. This relation is always respected, even in the kind of specific situations mentioned above. This sum of visibilities turns out to be related to a recently introduced measure of coherence. As one of the consequences, it provides a novel way of experimentally measuring coherence in a multi-path interference experiment. As another consequence, this relation suggests a simple way of measuring path-distinguishability in multi-path interference. In addition, it resolves several outstanding issues concerning wave-particle duality in multi-path interference.

Equations

$$\frac{2}{n(n-1)}\sum _{\text{pairs}}{{\mathcal{D}}_Q}_{ij}+\frac{2}{n(n-1)}\sum _{\text{pairs}}{\mathcal{V}}_{ij}\le 1,$$

$$|\Psi ⟩=c_1|{\psi }_1⟩+c_2|{\psi }_2⟩+\cdots +c_n|{\psi }_n⟩,$$

$$|\Psi ⟩=c_1|{\psi }_1⟩|d_1⟩+c_2|{\psi }_2⟩|d_2⟩+\cdots +c_n|{\psi }_n⟩|d_n⟩,$$

$$\rho =T{r}_d[|\Psi ⟩⟨\Psi |]=\sum _{i=1}^n\sum _{j=1}^n{c}_i{c}_j^{\ast }|{\psi }_i⟩⟨{\psi }_j|⟨d_j|d_i⟩$$

$$\rho =\sum _{i=1}^n\sum _{j=1}^n{\rho }_{ij}|{\psi }_i⟩⟨{\psi }_j|⟨d_j|d_i⟩.$$

$${\rho }^{(2)}=\frac{1}{{\rho }_{ii}+{\rho }_{jj}}\left(\begin{array}{cc}{\rho }_{ii}& {\rho }_{ij}\\ {\rho }_{ji}& {\rho }_{jj}\end{array}\right),$$

$${\mathcal{V}}_{ij}=\frac{2|{\rho }_{ij}||⟨d_j|d_i⟩|}{{\rho }_{ii}+{\rho }_{jj}}.$$

$${{\mathcal{D}}_Q}_{ij}=1-2\frac{\sqrt{{\rho }_{ii}{\rho }_{jj}}}{{\rho }_{ii}+{\rho }_{jj}}|⟨d_j|d_i⟩|.$$

$${{\mathcal{V}}_{ij}+{\mathcal{D}}_Q}_{ij}+2\frac{\sqrt{{\rho }_{ii}{\rho }_{jj}}-|{\rho }_{ij}|}{{\rho }_{ii}+{\rho }_{jj}}|⟨d_j|d_i⟩|=1.$$

$${{\mathcal{D}}_Q}_{ij}+{\mathcal{V}}_{ij}\le 1.$$

$${{\mathcal{D}}_Q}_{ij}+{\mathcal{V}}_{ij}=1.$$

$$\sum _{\text{pairs}}{{\mathcal{D}}_Q}_{ij}+\sum _{\text{pairs}}{\mathcal{V}}_{ij}\le \frac{n(n-1)}{2},$$

$$\frac{2}{n(n-1)}\sum _{\text{pairs}}{{\mathcal{D}}_Q}_{ij}+\frac{2}{n(n-1)}\sum _{\text{pairs}}{\mathcal{V}}_{ij}\le 1,$$

$${{\mathcal{D}}_Q}_{ij}$$

$${\mathcal{V}}_{ij}$$

$$1-\frac{1}{n-1}\sum _{i\ne j}\sqrt{{\rho }_{ii}{\rho }_{jj}}|⟨d_j|d_i⟩|)+\frac{1}{n-1}\sum _{i\ne j}|{\rho }_{ij}||⟨d_j|d_i⟩|\le 1,$$

$${\mathcal{D}}_Q$$

$$\mathcal{C}$$

$${\mathcal{D}}_Q+\mathcal{C}\le 1,$$

$${\mathcal{D}}_Q=\frac{2}{n(n-1)}\sum _{\text{pairs}}{{\mathcal{D}}_Q}_{ij},\mathcal{C}=\frac{2}{n(n-1)}\sum _{\text{pairs}}{\mathcal{V}}_{ij}.$$

$$\sum _{i\ne j}({\rho }_{ii}+{\rho }_{jj}){{\mathcal{D}}_Q}_{ij}+\sum _{i\ne j}({\rho }_{ii}+{\rho }_{jj}){\mathcal{V}}_{ij}\le \sum _{i\ne j}{\rho }_{ii}+{\rho }_{jj},$$

$$\frac{1}{(n-1)}\sum _{\text{pairs}}({\rho }_{ii}+{\rho }_{jj}){{\mathcal{D}}_Q}_{ij}+\frac{1}{(n-1)}\sum _{\text{pairs}}({\rho }_{ii}+{\rho }_{jj}){\mathcal{V}}_{ij}\le 1.$$

$${\mathcal{D}}_Q$$

$$\mathcal{C}$$

Full text

Wave-particle duality; Complementarity; Visibility; Coherence

I Introduction

Last two decades have seen lot of research activity in the area of complementarity or wave-particle duality in multi-path interference durr ; bimonte ; mei ; luis ; bimonte1 ; englertmb ; prillwitz ; 3slit ; cd ; nslit ; predict ; coles1 ; bagan ; biswas ; tania ; anu ; misba . After Englert derived a duality relation $\mathcal{D}^{2}+\mathcal{V}^{2}\leq 1$, for a two-path interference, which puts a bound on how much path information can be obtained from a quanton and the sharpness of interference it can show englert , it was natural to look for a similar duality relation for multi-slit interference. Breakthrough came with the derivation of the duality relation $\mathcal{D}_{Q}+\mathcal{C}\leq 1$ cd , between a new path-distinguishability $\mathcal{D}_{Q}$ based on unambiguous quantum state discrimination (UQSD) uqsd , and a new quantity quantum coherence $\mathcal{C}$, based on quantification of coherence by Baumgratz, Cramer and Plenio coherence .

Despite this tremendous progress, several issues still remained. One was, how coherence $\mathcal{C}$, which is just based on the $l_{1}$-norm of the off-diagonal elements of the density matrix of the quanton, can be measured in an experiment. A way to measure $\mathcal{C}$ from interference has been suggested tania , but that does not work in all scenarios, especially the kind described in the following. Mei and Weitz carried out multi-path interference experiments where a controlled decoherence was introduced only in selected paths mei . In addition, the phase of one of the paths was flipped by $\pi$. In such a situation they saw that increasing decoherence, which could also amount to getting path information, actually increased the visibility or contrast of the interference. The visibility $\mathcal{V}$, it may be recalled, is simply Michelson’s expression for fringe contrast $\mathcal{V}=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$, where $I_{max},I_{min}$ refer to the maximum and minimum intensities of interference, respectively bornw . This is in clear contradiction with the spirit of the Bohr’s principle of complementarity bohr . Based on this result, several authors argued that the interference visibility is not a good measure of interference or wave nature bimonte1 ; luis . It was even argued that there exist path measurements which do not degrade interference luis . In this kind of a scenario, coherence $\mathcal{C}$ can be shown to always decrease with increasing decoherence, and appears to capture the wave nature of a quanton well. However, in such a scenario $\mathcal{C}$ cannot be measured from interference by the method suggested in Ref. tania . Thus, it remains an open question whether a measure of the wave nature can be gotten from interference in a multi-path experiment coh-rev . The main result of this paper is the following duality relation for an unbiased n-path interference

[TABLE]

where $\mathcal{V}_{ij}$ is the interference visibility if only the i’th and j’th slits are open, and the rest are blocked, and ${\mathcal{D}_{Q}}_{ij}$ is the maximum probability of unambiguously distinguishing between the i’th and j’th paths in such a scenario. It may be useful to recall that $n(n-1)/2$ is the total number of slit pairs, making the two terms, average of two-path distinguishability, and average of two-path visibility, with the average taken over all path-pairs. It will be shown that this inequality will hold in all situations, even the one described by Mei and Weitz’s experiments mei . There are several extremely useful consequences of this result which will also be discussed in the following.

II Interference visibility from pairs of paths

We begin by writing a general pure state of a quanton passing through a n-slit or a n-path interferometer. If $|\psi_{k}\rangle$ represent the state corresponding to the quanton taking the k’th path, the general state is given by

[TABLE]

where $|c_{k}|^{2}$ represents the probability of the quanton taking the k’th path. The states $\{|\psi_{i}\rangle\}$ can be assumed to form an ortho-normal set, without loss of generality. If we are talking about an experiment in which a path-detector is in place, which attempts to know which path the quanton followed, the basic requirement of the theory of quantum measurement is that certain path detector states should get entangled with the states $\{|\psi_{i}\rangle\}$:

[TABLE]

where $\{|d_{i}\rangle\}$ represent certain normalized states of the path-detector which may not necessarily be orthogonal to each other. In case they are orthogonal to each other, measuring an observable of the path-detector, which they are eigenstates of, will reveal which path the particle followed, e.g., $|\Psi\rangle\xrightarrow{\text{measurement}}|\psi_{k}\rangle|d_{k}\rangle$ (say). The density matrix for the above entangled state, after tracing over the path-detector states, can be written as

[TABLE]

If the quanton were in a mixed state, for some reason, before encountering the path-detector, a general form of the state would be given by

[TABLE]

In the subsequent discussion we will assume the above to be the general form of the density operator, and will specify $\rho_{ij}=c_{i}c_{j}^{*}$ for a pure quanton state.

Let us suppose that we block all the paths except the paths $i,j$. Then the effective density matrix of the quanton part will look like

[TABLE]

where the prefactor has been introduced to renormalize this 2x2 matrix. The actual density matrix of the quanton will additionally have $\langle d_{j}|d_{i}\rangle$ in the off-diagonal elements. For a two-slit interference, it is well known that the fringe visibility is given by twice the absolute value of the off-diagonal matrix elements. Hence we can write the visibility of interference from slits $i,j$ as

[TABLE]

Since $|d_{i}\rangle,|d_{j}\rangle$ are not in general orthogonal, one can do a UQSD measurement uqsd to determine whether the path-detector state is $|d_{i}\rangle$ or $|d_{j}\rangle$. The specific aspect of UQSD measurements is that if the method succeeds, one can tell for sure if the state is $|d_{i}\rangle$ or $|d_{j}\rangle$. But sometimes the method fails, giving no result. If two states $|d_{i}\rangle,|d_{j}\rangle$ occur with probabilities $p_{1},p_{2}$, respectively, the optimal probability of a successful distinguishing between the two is given by $P_{max}=1-2\sqrt{p_{1}p_{2}}|\langle d_{j}|d_{i}\rangle|$ uqsd . In our two-slit interference, the probability of the state $|d_{i}\rangle,|d_{j}\rangle$ occurring is $\frac{\rho_{ii}}{\rho_{ii}+\rho_{jj}},\frac{\rho_{jj}}{\rho_{ii}+\rho_{jj}}$, respectively. So the optimal probability of successfully distinguishing between the two path-detector states is $P_{max}=1-\frac{2\sqrt{\rho_{ii}\rho_{jj}}}{\rho_{ii}+\rho_{jj}}|\langle d_{j}|d_{i}\rangle|$. Consequently this is also the optimal probability of successfully telling whether the quanton followed path $i$ or $j$. This optimal probability is what we define our path-distinguishability as. Thus, the path-distinguishability for this two-slit interference is given by

[TABLE]

Using the (7) and (8) we can write

[TABLE]

Since the density matrix given by (6) is positive semi-definite, one can write $\sqrt{\rho_{ii}\rho_{jj}}-|\rho_{ij}|\geq 0$. Thus the above equation implies

[TABLE]

This is a wave-particle duality relation for two-path interference awpd . For a pure quanton state, $\rho_{ij}=c_{i}c_{j}^{*}$ leads to $\sqrt{\rho_{ii}\rho_{jj}}-|\rho_{ij}|=0$, and the duality relation saturates to an equality

[TABLE]

The result is that if all but two slits are closed, the effectively two-slit interference follows a tight duality relation (10), which saturates for the pure case.

This same procedure can be followed for all pairs of slits, thus yielding $\mathcal{V}_{ij}$ and ${\mathcal{D}_{Q}}_{ij}$ for all pairs $i,j$. Adding (10) for all pairs of slits, we get

[TABLE]

because for $n$ slits, there are $\frac{n(n-1)}{2}$ pairs. Dividing both sides by $\frac{n(n-1)}{2}$ we get the required duality relation

[TABLE]

which, for pure quanton state, will reduce to an equality. A skeptic may be excused for asking if the above relation, obtained by selectively opening only one pair of paths at a time, has anything to do with genuine multi-path interference. After all, we know that even for a three-slit experiment, the three-slit interference pattern cannot be obtained simply as a sum of the interference patterns from various pairs of slits. To address this criticism we delve deeper into (13), to understand its meaning.

III Interference visibility and coherence

We first consider the case where all the paths are equally probably, which implies that $\rho_{ii}=\frac{1}{n},~{}i=1,n$. Two-path distinguishability and visibility, in this situation, are given by

[TABLE]

We substitute these expressions for ${\mathcal{D}_{Q}}_{ij}$ and $\mathcal{V}_{ij}$ into (13) to get

[TABLE]

where we have used the fact that $\sum_{i\neq j}=2\sum_{\text{pairs}}$. From an earlier study of wave-particle duality in n-path interference, we recall cd

[TABLE]

where $\mathcal{D}_{Q}$ is path-distinguishability defined earlier for n-path interference, and $\mathcal{C}$ is the coherence defined again for n-path interference. Using (16), the duality relation (13) assumes the form

[TABLE]

which is exactly the duality relation derived in Ref. cd . So, for symmetric multi-path interference, we get a very elegant connection of the path-distinguishability and coherence of n-path interference with the path-distinguishability and visibility of two-path interference of pairs of slits or paths:

[TABLE]

The immense usefulness of this connection will become clear in the following analysis, which also applies to the case of unequal intensities in different paths, which is discussed later.

What (18) implies for coherence is that in an n-path interference with equal intensities in all beams, coherence $\mathcal{C}$ can be obtained simply by opening only a pair of path at a time and measuring visibility by the conventional method, and then averging this visibility over all the pairs of paths. Thus, (18) provides a simple way of directly obtaining coherence from the interference pattern, although by the special procedure mentioned above. The other important consequence of (18) is that in the kind of experiment hooked up by Mei and Weitz mei , flipping the phase of one path by $\pi$ will have no effect on the visibility of interference from any two paths, if all other paths are blocked. Thus n-path coherence can be measured as easily in Mei and Weitz’s experiment, as in any normal multi-path interference. This method then provides good measure of wave nature of a quanton, which can be obtained from the interference from various path pairs. Not only is coherence $\mathcal{C}$ a good measure of wave nature, it can be obtained from the interference in all situations, contrary to the pessimistic view taken by some authors luis ; bimonte1 .

In earlier studies on wave-particle duality in multi-path interference 3slit ; nslit , the distinguishability $\mathcal{D}_{Q}$ is defined as an upper bound on the probability with which the $n$ paths can be unambiguously distinguished from each other. One problem with this upper bound is that it is not the optimal probability, meaning there is no guaranty that this limit will be achievable for a give set of states $\{|d_{i}\rangle\}$. The second problem with $\mathcal{D}_{Q}$ is that UQSD for more than two states works only for a linearly independent set $\{|d_{i}\rangle\}$. If the states are linearly dependent, UQSD cannot be used, and there is no meaning one can assign to the expression (16) for $\mathcal{D}_{Q}$. The relations (18) solve this problem. Even if the states are linearly dependent, (18) gives a well defined meaning to $\mathcal{D}_{Q}$, in terms of the sum distinguishabilities of different pairs of paths. Two-path distinguishability is based on UQSD involving only two states, and is the optimal probability of distinguishing the two states. Therefore, $\mathcal{D}_{Q}$ as defined by (18) is always experimentally attainable. The third problem is that UQSD has only been experimentally demonstrated for two states uqsdexpt . No one knows how to implement it for more than two states. Since the present method represents the distinguishability in terms of two-path UQSD, it can be experimentally implemented.

Next we look at the more general case where all paths may not be equally probable. Here, instead of summing the two-path distinguishabilities and visibilities as done in (13), we multiply the duality relation (10) for each path-pair with the sum of probabilities of the two paths involved, and then sum over all $i,j~{}(i\neq j)$:

[TABLE]

A new duality relation for the asymmetric multi-path interference can then be written from the above as

[TABLE]

Substituting (7) and (8) in the above, we again get the known duality relation (17). So we see that the new duality relation (20), for asymmetric n-path interference, is the same as (17), with the following connection

[TABLE]

As a consistency check, for all equally probable paths, $(\rho_{ii}+\rho_{jj})=\frac{2}{n}$, and (20) reduces to (13).

It is clear from the preceding analysis that in a general multi-path interference, where the paths may not be equally probable, the multi-path distinguishability and multi-path coherence can be experimentally measured by carrying out a series of experiments where only one pair of paths is open, and the visibility of interference is measured. However, here one also needs to measure the relative intensity of each path in the multi-path experiment. One can then use (21) to get the coherence $\mathcal{C}$. Similarly, if one is able to set up an experiment to measure path-distinguishability of a pair of paths, one can use (21) to get the path distinguishability $\mathcal{D}_{Q}$ for the multi-slit interference.

IV Conclusion

To summarize, we have introduced a new way of studying wave-particle duality in multi-path interference, by opening only one pair of paths at a time, and measuring conventional visibility and using UQSD to measure the distinguishability ${\mathcal{D}_{Q}}_{ij}$. The multi-slit path distinguishability $\mathcal{D}_{Q}$ and multi-path coherence $\mathcal{C}$ (for symmetric paths) can then be obtained as average of ${\mathcal{D}_{Q}}_{ij}$ and average of ${\mathcal{V}}_{ij}$ over all path pairs, respectively. For a multi-path interference where the paths may not be equally probable, the same method works, but the average has to be taken with each term weighted with the total intensity from the two paths of the pair. This method resolves various outstanding issues in wave-particle duality in multi-path interference, which are listed below.

(1) A way of measuring coherence in multi-path interference is provided which works even for the experiment of Mei and Weitz mei , where interference visibility was shown to increase with increasing path knowledge.

(2) The method shows that wave-nature can always be characterized using interference, and that it is complementary to path information, even in multi-path interference, contrary to existing belief luis ; bimonte1 .

(3) Multi-path coherence has been given a new meaning in terms of interference visibilities of path pairs.

(4) Path-distinguishability in multi-path interference is given a new meaning in terms of path distinguishability for a pair of paths.

(5) Path-distinguishability in multi-path interference continues to hold even in the situation when path-detector states form a linearly dependent set.

(6) There was no known way to measure path-distinguishability $\mathcal{D}_{Q}$ in a multi-path interference. A way is provided here.