# Assessing Engineering Students’ Conceptual Understanding of Introductory Quantum Optics

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## Abstract

**:**

## 1. Introduction

#### 1.1. Educating the Future Quantum Workforce

- (1)
- The experimental and physical foundations of experiments with heralded photons may directly be leveraged to quantum technology applications within teaching scenarios, e.g., with regard to quantum computing [28], quantum metrology [29], or quantum information [30,31]. This content-specific argument is especially important with regard to undergraduate courses within study programmes for prospective quantum engineers.
- (2)
- Such experiments “provide the simplest method to date for demonstrating the essential mystery of quantum physics” [32] (p. 1), and “elegantly illustrate the fundamental concepts of quantum mechanics such as the wave-particle duality of a single photon, single-photon interference, and the probabilistic nature of quantum measurement” [33] (p. 1). Hence, such quantum optics-based approaches are likely to be conducive to circumvent widespread learning difficulties regarding quantum concepts, as has already been indicated by empirical results presented in Ref. [34].

#### 1.2. Aim of This Study

- (1)
- investigate as to how our concept inventory psychometrically functions on a sample of engineering students,
- (2)
- establish a difficulty scale regarding concepts covered by quantum optics experiments with heralded photons suggested by students’ scores on the concept inventory.

## 2. Research Background

#### 2.1. Assessment of Students’ Conceptual Understanding of Quantum Optics

#### 2.2. Students’ Conceptions on Quantum Optics Aspects

- a part of the learners interpret single-photon interference via photons that divide and then overlap with themselves [55] (p. 216);
- a part of the respondents equate photons with waves and conclude from this the necessity of observing interference phenomena [55] (p. 221);
- some learners claim that the photons’ localization would fail due to the small size of the photons [55] (p. 227);
- furthermore, a part of the participants believe that quantum anticorrelation at a beam splitter cube would be caused by photons behaving like haptical particles, being either reflected or transmitted [55] (p. 232).

## 3. The Quantum Optics Concept Inventory for Engineering Students

- (1)
- we updated the existing items (and the corresponding distractors) from Ref. [49], in terms of language, and
- (2)
- beyond the content domains covered in the test of Ref. [49], namely (a) the theoretical and (b) the experimental basics of quantum optics, we developed new items to incorporate the thematic area of the technical basics of quantum optics experiments into the instrument, since we believe this content domain to be relevant to future quantum engineers.

- (1)
- helps to minimize the effect of guessing, and hence, is “useful for gauging the quality of students’ understanding” [59] (p. 3); on the other hand,
- (2)
- it allows for the exploration of learning difficulties regarding the content area under investigation, namely by analyzing incorrect answers that were given confidently [62]. This point is postponed for the future research.

## 4. Research Questions

- RQ1:
- How does the Quantum Optics Concept Inventory function on a sample of engineering students?

- RQ2:
- What difficulty scale of quantum optics content aspects is suggested by students’ scores on the concept inventory?

## 5. Methods

#### 5.1. Test Administration and Sample

#### 5.2. Intervention

- (1)
- (2)

#### 5.3. Data Analysis

#### 5.3.1. Analysis Carried out to Answer RQ1

- the skewness and kurtosis of the items do not exceed the range of $-2$ to $+2$,
- the items are locally independent, and
- the uni-dimensionality of the construct can be assumed.

#### 5.3.2. Analysis Carried out to Answer RQ2

## 6. Results

#### 6.1. Psychometric Characterization

#### 6.2. Difficulty Scale of Quantum Optics Concepts

## 7. Discussion

#### 7.1. Discussion of RQ1

#### 7.2. Discussion of RQ2

- (1)
- The item difficulties of the items 12 ($1.647$ logits), and 13 ($0.745$ logits) both lay well above average—hence, single-photon interference obviously poses conceptual difficulties to the study participants. This observation is in accordance with prior research, and can be enriched by findings from various qualitative studies exploring student difficulties on the interference of single quanta [33,55].
- (2)
- Item 14, focusing on the anticorrelation effect of single-photon states at an optical beam splitter cube, has an item difficulty of $0.948$ logits. Hence, the concept that photons are either transmitted or reflected at a beam splitter cube seems to be difficult for students. This observation fits well with results presented in an earlier contribution [51], where the authors used a micro-intervention to explore students’ understanding of single-photons’ anticorrelation, using the technique of probing acceptance. Beyond that, this observation is particularly striking against the background of the finding described above regarding single-photon interference: while items 12 and 13 address the wave nature of photons, item 14 can be associated with a particle nature of photons. Hence, the three items represent one of the fundamental issues of quantum physics: wave-particle duality.

## 8. Conclusions

- aimed at engineering university students, and
- designed to introduce these students to quantum physics and modern quantum technologies via quantum optics experiments with heralded photons.

#### 8.1. Limitations

#### 8.1.1. Limitations of this Study

#### 8.1.2. Limitations Related to the Quantum Optics Concept Inventory

- (i)
- intelligible wording,
- (ii)
- coverage of overarching concepts of quantum optics in the heralded photon realm, rather than addressing all the details of a topic,
- (iii)
- subject-specific correctness.

#### 8.2. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Quantum Optics Concept Inventory

**Item 1.**A beam splitter

- (a)
- is employed in the Michelson interferometer, because it can be used to split an incident ray of light into two partial beams.
- (b)
- is made out of two merged prisms, where one of them is responsible for the transmitted beam and one for the reflected beam.
- *(c)
- separates incident rays of light or superimposes two rays of light.

**Item 2.**To precisely adjust a laser beam along a specific straight line on a breadboard in a quantum optics experiment,

- (a)
- two mirrors can be used to deflect the laser on this line.
- *(b)
- two apertures can be used to define two points laying on this line.
- (c)
- two lenses can be used to focus the laser beam on this line.

**Item 3.**If a non-linear crystal is irradiated by laser light, then,

- *(a)
- light is emitted by the crystal.
- (b)
- the laser beam is divided in two beams.
- (c)
- diffraction leads to laser light forming a cone after passing the crystal.

**Item 4.**Experiments with heralded photons differ from experiments with single electrons, because:

- (a)
- electrons are bigger than photons, leading to technical difficulties.
- (b)
- conducting the double-slit experiment one after the other with single electrons does not lead to an interference pattern.
- *(c)
- experiments with single electrons require a vacuum.

**Item 5.**The human eye may not be used as a detector for single photons, because:

- *(a)
- the intensity of the light falling on the eye in the single-photon regime ist too low.
- (b)
- the eye can only perceive light at specific wavelengths, but not those of single photons.
- (c)
- the photons are too small to be resolved with human eyes.

**Item 6.**If an electron transitions from an energy state ${E}_{1}$ to an energy state ${E}_{2}<{E}_{1}$, light is emitted. The bigger the energy difference, $\Delta E={E}_{1}-{E}_{2}$, (adopted from [43])

- (a)
- the more photons are emitted.
- (b)
- the longer the wavelength of the emitted light.
- *(c)
- the shorter the wavelength of the emitted light.

**Item 7.**When irradiated with laser light at a certain wavelength, parametric downconversion can be driven in a nonlinear crystal. This process leads to the emission of photon pairs. Both photons have

- (a)
- half the wavelength of the incident laser light.
- (b)
- the same wavelength of the incident light of the laser.
- *(c)
- double the wavelength of the incident light of the laser.

**Item 8.**Avalanche photodiodes used for single-photon detection

- (a)
- count the number of registered photons within a certain time interval.
- *(b)
- lead to electron avalanches when some kind of energy portion is registered.
- (c)
- point to the detection of a single-photon with each click.

**Item 9.**Coincident events measured at avalanche photodiodes in experiments with heralded photons

- *(a)
- mark simultaneous clicks at two or more detectors.
- (b)
- represent a measure of the detectors’ dark count rate.
- (c)
- can each be associated with the detection of a single-photon state.

**Item 10.**The coincidence technique is used in experiments with heralded photons

- (a)
- to show that photons are physical entities with finite size.
- (b)
- in order to experiment with two single-photons simultaneously.
- *(c)
- for single-photon detections.

**Item 11.**When the double-slit experiment is repeatedly performed with only one single-photon in the apparatus at a time,

- *(a)
- an interference pattern with minima and maxima can be observed.
- (b)
- two well-defined detection zones can be observed.
- (c)
- two well-defined detection zones and a zero-order maximum can be observed.

- (a)
- the single-photon follows a specific path, regardless of whether I observe this path or not.
- (b)
- the current position of a photon between source and detector is not indeterminate in principle, but unknown to the experimenter.
- *(c)
- the photon behaves like a particle and like a wave. It is none of them.

**Item 13.**Which statement about the behavior of a single-photon state in an interferometer is correct? (answer options adopted from [34,45])

- *(a)
- No one can say with certainty at which output port of the beam splitter cube a single-photon will be detected.
- (b)
- A single-photon state is divided at the beam splitter cube.
- (c)
- In the interferometer, the single-photon state is found in a superposition state of both, particle and wave.

**Item 14.**Anticorrelation of single-photon states can be observed at the outputs of a beam splitter cube because

- (a)
- more coincident events are detected between the outputs of the beam splitter than can be expected at random.
- *(b)
- a single-photon may only be detected once.
- (c)
- a single photon can be in both states, ’reflected’ and ’transmitted’, at the same time.

**Item 15.**For the preparation of single-photon states in experiments with heralded photons, one needs

- *(a)
- exactly two single-photon detectors.
- (b)
- at least one single-photon detector.
- (c)
- at least three single-photon detectors.

**Item 16.**Single-photons can be regarded as

- (a)
- spherical entities moving along a wavy path with the speed of light.
- (b)
- elementary energy portions of light surrounded by a wave that is responsible for interference.
- *(c)
- indivisible energy portions of light that are never detected on both output ports of a beam splitter cube simultaneously.

## Appendix B. Overview of the Four-Week Program on Introductory Quantum Optics

#### Appendix B.1. Part I: Foundations of Quantum Optics Experiments with Heralded Photons

- Second, the study participants were introduced to the properties of avalanche photodiodes operating above their breakdown voltage—also referred to as single-photon avalanche diode (SPAD) [95,96]: For example, quantum efficiency, dark count rate, and dead time [68] were covered. SPADs are binary detectors, which means that the “outcome of these APD’s is either ‘off’ (no photons detected) or ‘on,’ i.e., a ‘click,’ indicating the detection of one or more photons” [97] (p. 1).
- Third, the students were introduced to spontaneous parametric downconversion (PDC) [98], a quantum electrodynamic process in which “an incoming pump photon decays, under energy and momentum conservation, into a photon-pair” [99] (p. 351). Here, the students learned that a) PDC is driven by irradiating a nonlinear crystal (e.g., $\beta $-barium borate) with a pump beam, most often emitted by a laser, and that b) the “spectral properties of PDC states are governed by the phasematching properties of the nonlinear material, and this determines the frequencies of the downconverted photons” [100] (p. 3442). For an comprehensive overview of parametric downconversion, we refer the reader to [101].

#### Appendix B.2. Part II: Experiments with Heralded Photons

- First, the students learned about heralded single-photon state preparation, which is widely used in quantum optics research [102,103,104,105]). That is, one of the PDC photons is detected and heralds the second photon at a spatially separated detector—hence, simultaneous clicks at two detectors, also referred to as a coincident events, are “taken as preparation and detection of a single photon state” [37] (p. 348).
- Second, heralded single-photon states incident on a 50:50 beam splitter were investigated. In this experiment, the lack of coincident events at the output ports of the beam splitter, i.e., photon antibunching [106], was discussed. This anticorrelation effect is irreconcilable with any classical description of light: “A single photon can only be detected once” [67] (p. 173). These observations were further substantiated on a formal level a) by means of the second-order correlation function ${g}^{\left(2\right)}$, which allows for a judgement of the purity of single-photon states [94], and b) by highlighting the ideas of quantum superposition and quantum random, as has been achieved in [31].
- The investigation and quantum description of Grangier et al.’s experiment from 1986 [67] represents the last step of the course. In this experiment, the students realized that by using only one single-photon state, the anticorrelation at a 50:50 beam splitter appeared simultaneously to the single-photon interference observed in a spatially separated interferometer (e.g., a Michelson interferometer) using the same single-photon state in the same experimental set-up. The students experienced that the “quantum interference phenomenon shown experimentally is a consequence of the interplay of superposition and nonlocality” [64] (p. 17), while the idea of the photon as a localizable particle is not valid. Instead, in this course, the photon was introduced as an elementary field mode excitation in the sense of quantum electrodynamics [83,84,107,108].
- In the outlook, the engineering students were given first insights into quantum technologies 2.0, namely by applying an understanding of heralded photons experiments to the context of quantum cryptography.

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**Figure 1.**Infit MNSQ (crosses) and Outfit MNSQ (circles) for all items of the concept inventory. See text for details.

**Figure 2.**Item characteristic curves (ICC) for all items of the concept inventory. While the blue, light-blue, and light-green curves correspond to items that show good fit to the Rasch model, the outliers, namely items 4, 7, and 9, can also be identified. This observation is further substantiated analyzing the Wright Map presented in Figure 3.

**Figure 3.**Wright Map of our concept inventory. The left-hand side represent the frequency of respondents’ latent trait levels within the sample. The right-hand side represents a hierarchical order of all items along the logit scale.

**Figure 4.**Average difficulties of the quantum optics concepts, evaluated by our concept inventory, measured in logits.

**Table 1.**Overview of the content domains covered in the Quantum Optics Concept Inventory, and the topics addressed with the related items. For a didactically prepared overview of the subject-specific topics represented in the test, e.g., the anticorrelation of single-photon states (see [18,51]). The list of items is given in Appendix A.

Content Domain | No. of Items | Topics Covered (Related Items) |
---|---|---|

Technical-experimental foundations | 5 | Set-up/adjustment of quantum optical experiment (items 1, 2, and 4), Single-photon detection (items 5 and 8) |

Preparation of single-photon states | 6 | Parametric downconversion and energy conservation (items 3, 6, and 7), Coincidence technique (items 9, 10, and 15) |

Quantum effects | 5 | Interference of single quanta (items 11, 12, and 13), Anticorrelation of single-photon states (items 14 and 16) |

**Table 2.**Each item’s difficulty and point-biserial-coefficent, as well as the adjusted Cronbach’s $\stackrel{\u203e}{\alpha}$ if the respective item is dropped. See text for details.

Item No. | Difficulty | Point-Biserial | $\stackrel{\u203e}{\mathit{\alpha}}$ |
---|---|---|---|

1 | 0.15 | 0.35 | 0.73 |

2 | 0.37 | 0.30 | 0.73 |

3 | 0.24 | 0.46 | 0.72 |

4 | 0.91 | 0.22 | 0.74 |

5 | 0.43 | 0.48 | 0.72 |

6 | 0.63 | 0.39 | 0.73 |

7 | 0.82 | 0.33 | 0.73 |

8 | 0.37 | 0.29 | 0.74 |

9 | 0.80 | 0.21 | 0.74 |

10 | 0.59 | 0.20 | 0.75 |

11 | 0.54 | 0.34 | 0.73 |

12 | 0.20 | 0.34 | 0.73 |

13 | 0.35 | 0.44 | 0.72 |

14 | 0.32 | 0.31 | 0.73 |

15 | 0.35 | 0.34 | 0.73 |

16 | 0.61 | 0.36 | 0.73 |

**Table 3.**Overview of the relevant parameters for a dichotomous Rasch Model; “SE” stands for the standard error of item difficulty. See text for details.

Item No. | Skewness | Kurtosis | Difficulty | SE | Infit | Outfit |
---|---|---|---|---|---|---|

1 | 1.99 | 2.00 | 2.096 | 0.206 | 0.929 | 0.916 |

2 | 0.54 | −1.72 | 0.648 | 0.156 | 1.056 | 1.042 |

3 | 1.22 | −0.51 | 1.393 | 0.174 | 0.902 | 0.772 |

4 | −2.83 | 6.07 | −2.697 | 0.247 | 0.985 | 0.873 |

5 | 0.30 | −1.93 | 0.364 | 0.152 | 0.902 | 0.889 |

6 | −0.54 | −1.72 | −0.652 | 0.155 | 0.951 | 1.139 |

7 | −1.63 | 0.67 | −1.787 | 0.189 | 0.956 | 0.952 |

8 | 0.54 | −1.72 | 0.648 | 0.156 | 1.079 | 1.048 |

9 | −1.48 | 0.20 | −1.649 | 0.183 | 1.080 | 1.109 |

10 | −0.38 | −1.87 | −0.462 | 0.153 | 1.148 | 1.197 |

11 | −0.15 | −2.00 | −0.184 | 0.151 | 1.022 | 1.002 |

12 | 1.48 | 0.20 | 1.647 | 0.183 | 0.987 | 0.907 |

13 | 0.63 | −1.62 | 0.745 | 0.157 | 0.924 | 0.931 |

14 | 0.80 | −1.37 | 0.948 | 0.161 | 1.028 | 1.003 |

15 | 0.63 | −1.62 | 0.745 | 0.157 | 1.017 | 0.973 |

16 | −0.46 | −1.81 | −0.556 | 0.154 | 1.005 | 0.967 |

**Table 4.**Categorial judgment scheme and assignment rules for evaluating a concept inventory according to [70]. Values in parenthesis indicate the number of items that can fall outside of this recommendation. The judgement of our concept inventory is presented in the last column labeled QOCI.

Analysis | Excellent | Good | Average | Poor | QOCI | ||
---|---|---|---|---|---|---|---|

Classical Test theory | |||||||

Item Statistics | |||||||

Difficulty | $0.2$–$0.8$ | $0.2$–$0.8$ (3) | $0.1$–$0.9$ | $0.1$–$0.9$ (3) | good | ||

Discrimination | >0.2 | >0.1 | >0 | >−0.2 | excellent | ||

Total score reliability | |||||||

$\alpha $ of total score | >0.9 | >0.8 | >0.65 | >0.5 | average | ||

$\alpha $-with-item-deleted | All items less than overall $\alpha $ | (3) | (6) | (9) | good | ||

Item Response Theory | |||||||

Individual item measures | |||||||

Infit MNSQ | $0.7$–$1.3$ | $0.6$–$1.4$ | $0.5$–$1.5$ | – | excellent | ||

Outfit MNSQ | $0.7$–$1.3$ | $0.6$–$1.4$ | $0.5$–$1.5$ | – | excellent | ||

All items fit the model | (2) | (4) | (6) | (8) | excellent |

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**MDPI and ACS Style**

Bitzenbauer, P.; Veith, J.M.; Girnat, B.; Meyn, J.-P.
Assessing Engineering Students’ Conceptual Understanding of Introductory Quantum Optics. *Physics* **2022**, *4*, 1180-1201.
https://doi.org/10.3390/physics4040077

**AMA Style**

Bitzenbauer P, Veith JM, Girnat B, Meyn J-P.
Assessing Engineering Students’ Conceptual Understanding of Introductory Quantum Optics. *Physics*. 2022; 4(4):1180-1201.
https://doi.org/10.3390/physics4040077

**Chicago/Turabian Style**

Bitzenbauer, Philipp, Joaquin M. Veith, Boris Girnat, and Jan-Peter Meyn.
2022. "Assessing Engineering Students’ Conceptual Understanding of Introductory Quantum Optics" *Physics* 4, no. 4: 1180-1201.
https://doi.org/10.3390/physics4040077