2.1.

Lab. 1. Entanglement and Bell’s inequalities

Entanglement is the most exciting and mysterious property of some quantum mechanical systems when property of one particle depends on the property of the other. Measurements performed on a first particle would change the state of the second particle, no matter how far apart they may be. This nonlocal character is a key in entanglement and it is important that no propagation of information occurs. Entanglement can be on some physical value, e.g., polarization, energy, momentum, time, etc. Mathematically we can say that in quantum mechanics, particles are called entangled if their state cannot be factored into single-particle states:

Figure 2 shows a cartoon made by A.K. Jha and L. Elgin showing “entangled particles”. Applications of quantum entanglement are quantum computers, quantum communication, and quantum teleportation (teleportation of a state, but not a particle).

Figure 2.

Cartoon showing “entangled particles”.

The idea of entanglement was introduced into physics in 1935 by Einstein, Podolsky and Rosen (EPR) [2], as “spooky action at a distance” (entanglement), in which authors (EPR) did not believe, because of information cannot propagate with the speed greater than the speed of light. EPR also did not like probabilities in description of particle behavior. They suggested, that quantum mechanics is incomplete theory, and future, complete theory of quantum mechanics will be something like statistical mechanics with some additional variables which later have been named “hidden”. Shortly after the EPR paper, Schrödinger coined the word entanglement (verschränkung in German) and further developed this concept [3]. Bohr replied to the EPR paradox that in a such quantum state one could not speak about the individual properties of each of the particles, even if they were distant from one another.

In the mid-sixties of the last century it was realized that the nonlocality of nature is a testable hypothesis. 1964, John Bell showed that the “locality hypothesis” with “hidden” variables leads to a conflict with quantum mechanics [4,5]. He proposed a mathematical theorem containing certain inequalities. An experimental violation of his inequalities would suggest the states obeying the quantum mechanics with nonlocality. I would like to emphasize that Bell’s inequalities (there are many types of Bell’s inequalities, some of them were obtained later by other researchers) are classical relations [6], and they are violated only in quantum mechanics and only for some values of parameters (e.g., under some polarizer angles if we have entanglement in polarization). In many values of parameters both classical physics and quantum mechanics give the same results without violation of Bell’s inequalities. Quantum correlation is a very rare event, and it is not easy to find it. The most popular form of Bell’s inequality for the experimentalists is a CHSH inequality described by Clauser, Horne, Shimony and Holt, in a widely-cited paper [7] published in 1969. The CHSH inequality for polarization entanglement can be obtained from a trivial relation that modulus of sum is less or equal to sum of moduli. And in a quantum world this classical relation can be violated. It is impossible to understand with a “common sense” mind, that inequality |a + b + c| ≤ |a|+|b|+|c| can be violated, but in this lab students violate this inequality.

The most popular way to obtain entangled photons is to use a spontaneous parametric down conversion (SPDC) process. This paper describes the lab on photons entangled in polarization. The first experiment with the same type of setup was made by Kwiat [8]. To learn how to build a similar setup see also papers [9-17], books [18, 19] and websites [1, 20, 21]. To learn about entanglement and quantum optics in general see books [22-27], and about SPDC – book of Klyshko [28].

In UR teaching lab two polarization entangled photons are produced through a SPDC in two type I Beta Barium Borate (BBO) crystals pumped by a laser. We used three experimental setups with diode lasers (405 and 408 nm wavelengths) and an argon ion laser (363.8 nm wavelength) with an intracavity etalon inside. In this process a single pump photon spontaneously splits into “signal” and “idler” photons with a longer wavelengths inside a nonlinear crystal. The efficiency of this process is only ~10^-10. Because of spontaneous process, the angles of emitted photons and their wavelength may have any values obeying the conservation of energy and momentum, so that downconverted photons, like in a rainbow, have various wavelengths. To create an entangled state at 2λ wavelength (e.g., 727.6 nm for an ion-argon laser), we select definite angles of signal and idler photons’ propagation and narrow-bandwidth interference filters (10 nm bandwidth). The signal and idler photons with 2λ wavelength are emitted in a cone (see Figure 3).

Figure 3.

Experimental setup for entanglement and Bell’s inequality violation. LEFT: Schematics of a SPDC process with conservation of energy and momentum; CENTER: Top - Type I SPDC: down-conversion of a horizontal photon (for this orientation of the crystal, vertical photons pass strait through) and down-conversion of a vertical photon (for this orientation of the crystal horizontal photons pass straight through). RIGHT (top): cross-section of a SPDC cone (727.6 nm wavelength) from a type I BBO crystal (measurements were made by an EM-CCD-camera).

Type I downconversion means that the signal and idler photons have the same polarization, which is opposite to that of the pump photon (Figure 3, top center). Our source of entangled photons uses two identical BBO crystals, with one rotated 90° from the other about the beam propagation direction. In this arrangement each crystal can support downconversion of one pump polarization, the other polarization simply passes through the transparent crystal. A 45° polarized pump photon can downconvert in either crystal, producing a polarization entangled pair of photons (Figure 3, center bottom).

Mathematically it can be represented as |H〉 + |V〉 →|V_sV_i〉 + exp(iΔ)|H_sH_i〉 (2). Here V and H represent a horizontal- or a vertical-polarized photon. Signal and idler photons are denoted by subscripts “s” and “i” respectively. Δ is a phase difference after two crystals due to different path length for different down-converted polarizations in a birefringent BBO crystal. Downconverted photons with horizontal and vertical polarizations are produced in different crystals, so they are independent from each other, and two overlapping cones of downconverted photons (Figure 3, center bottom) with vertical and horizontal polarizations give cone of unpolarized photons, although we will measure correlations in polarizations. Cross-section of a cone of downconverted photons of one 727.6 nm wavelength selected through a 10-nm transmission filter recorded by an EM-CCD camera is shown in the top of Figure 3, right (it was obtained with an argon ion laser).

Figure 4 shows the schematics of the experimental setup with a 405 nm diode laser. 10-nm bandwidth interference filters are used for selection of definite wavelength of downconverted photons (810 nm in this setup) propagating at definite cone angle. Two single-photon counting avalanche photodiode modules (APDs) are used as detectors A and B. Collection system (microscope objectives and optical fibers) to APDs is located at the opposite ends of a downconverted cone diameter. See also Figure 5, left with collection system, APDs and two linear polarizers mounted on rotating mounts. The crystal set mounting for its precise alignment is shown in Figure 5, right.

Figure 4

(prepared by R. Lopez-Rioz) shows the schematics of experimental setup of polarization entangled photons with two BBO crystals and a diode laser. We did not use a quartz plate for compensation a phase Δ to obtain entangled photons in a last version of a setup with a 405 nm laser, although in an OPT 253 class students investigated its effect on results by its rotation around two axes.

Figure 5.

LEFT: Collection system with a 3-D adjustable mount with two polarizers and APD detectors. RIGHT: BBO crystal set with a 3-D rotation mounting (an iris diaphragm serves for alignment).

Using polarizers rotated to angles α and β in the signal and idler paths, respectively, one measures the polarization correlation of the downconverted photons. It is very important that response from two single-photon counting avalanche photodiode modules (APDs) was processed with a counter-timer computer card that permit to record not only signals from each APDs (singles), but also their simultaneous response (coincidences).

The probability P of coincidence detection for the case of 45° incident polarization on a BBO crystal set depends only on the relative polarizer angle β-α:

It should be noted that APD singles’ count does not depend on polarizers’ rotation angles (overlapping of two cones of downconverted light from two independent polarized sources created in two different crystals results in an unpolarized light). The correlations described by equation (3) can be recorded only in coincidence count using a counter-timer board (time window of 25 ns).

The pictures of two entanglements setups with a diode (405 nm, 120 mW) and an ion-argon (363.8 nm, 100 mW, single-longitudinal mode) lasers are shown below in Figure 6. Two fiber-connected APDs are used in both setup. BBO crystal set were cut at different angles for each setup.

Figure 6.

Two entanglement setups for the teaching labs of the Institute of Optics, University of Rochester (different views of the same optical table).

Figure 7 shows experimental polarization correlation for fixed angles of a polarizer A α = 45° and 135° and rotation of a polarizer B only. The experimental data are in a good agreement with relation (3): coincidence counts reach a maximum, when polarizers are parallel (α = β). Fringe visibility of 0.9 was observed that is greater than 0.71 that indicates a possibility of entanglement.

Figure 7.

Polarization correlations: dependence of coincidence count on relative polarizer angle.

To prove an entanglement in our system Bell’s inequality should be violated. The CHSH Bell inequality [7] constrains the degree of polarization correlation under measurements at different polarizer angles. The proof involves two measures of correlations, Ε(α,β) and S (P_VV and P_HH are probabilities of both photons to have vertical or horizontal polarizations, P_VH and P_HV are probabilities of photons to have opposite polarizations), Ν(α,β) is measured coincidence counts at polarizers angles α and β. Violation of CHSH Bell inequality occurs if S > 2. S does not have a clear physical meaning.

The above calculation of S requires a total of 16 coincidence measurements (N), at definite polarization angles a and b (see the Table 1 bellow with experimental coincidence counts). Polarization angles are selected with maximum value of S from CHSH Bell’s inequality to violate it. At many other angles S is less than 2 from a CHSH theory, so to prove entanglement we need to select polarizer angles with max value of S. Calculation of Bell’s inequality in the CHSH form shows its violation (S~2.65 > 2).

Table 1.

LEFT: Table 1. Data of 16 measurements of coincidence counts at 16 combinations of polarizer angle with maximum value of S. In this case S > 2. At some angles, the value of S can be less than 2. RIGHT: Figure 8. Dependence of PVV on relative polarizer angles for quantum mechanical case (solid line) and for classical physics (dashed line) [10]. For some values of relative polarizer angle quantum mechanics and classical physics give the same value of PVV as well as S.

2.2.

Lab 2. Single photon interference (Young’s double slit experiment and Mach-Zehnder interferometer)

In this lab, wave-particle duality (complementarity) on example of single photons was demonstrated. For this lab an attenuated laser beam (Poissonian light source) is a good approximation for a source of single photons, although photon antibunching (separation of all photons in time) cannot be achieved in a such a source. Interference between single photons was observed in both Young’s double slit and Mach-Zehnder interferometers. In a Mach-Zehnder interferometer experiment, the effect of “which path” information was also shown, since we could only acquire an inference pattern when that information was hidden by using a quantum eraser (45° linear polarizer). This laboratory provides a visual demonstration of the appearance and disappearance of interference fringes, both at high light and single-photon levels, by carefully destroying and restoring “which-path” information using a quantum eraser in a Mach-Zehnder interferometer (see also Ref. [15, 29]).

Measurements are made using a He-Ne laser beam, attenuated to a single photon level with neutral density filters. On advanced lab level (OPT 253 class), cooled electron multiplying (EM) CCD camera iXon of Andor Technologies, sensitive to single-photons, is used. It was also employed with MCC students’ groups and freshman research projects (OPT 101). In the OPT 204 class with 14 students’ groups working at different time on the same setup (49 students total in Spring 2017), a conventional CCD camera was utilized for recording interference fringes from faint laser light attenuated for a single photon level (~ one photon per meter or kilometer), but at a longer exposure time (several to tens of seconds).

Figures 9 and 10 show the schematics of two experiments of this lab: Young’s double slit (Figure 9) and Mach-Zehnder interferometers (Figure 10). As a double slit we initially used 10 μm width slits with 90 μm slit separation fabricated by a lithographic deposition of the metal on a glass substrate. This type of a double slit provides interference pattern as a result of interference of light diffracted by the slit and reflected light from reflective surface of a substrate. In this case a fine structure appears in the interference maxima of a double slit interference (see insert in Figure 9, right). Although it does not influence a wave-particle duality, we transferred recently to the slit with opening in the air. The third part of the lab (even in a 3-hour version) is alignment of a Mach-Zehnder interferometer. A teaching assistant (TA) destroys the alignment by removing the mirror mounts from the post holders. After that students should realign an interferometer.

Figure 9

(prepared by R. Lopez-Rios). Setup of a single photon Young’s double slit experiment. ND: Neutral density filters; EMCCD: electron multiplying CCD camera; He-Ne: Helium-Neon laser. The insert at the right shows double-slit interference with “lithographic” slit. A fine structure is seen inside the maxima (see explanation in the text).

Figure 10

(prepared by R. Lopez-Rios). Setup for a Mach-Zehnder Interferometer. PBS: polarizing beam splitter; NPBS: non-polarizing beam splitter. At each point where the system state changes the polarization vector is shown.

Figure 11 shows the results on wave-particle duality using single-photon interference pattern in a Mach-Zehnder interferometer recorded by an EM-CCD camera with a single, 200 ms exposure time and with accumulation of 50 images. It also shows dependence of fringe visibility of an interference pattern on the angle of quantum eraser. Figure 12 shows a single-photon interference lab setup in different views with two CCD cameras (conventional and EM-CCD). See also Ref. [23] with a similar teaching lab experiment.

Figure 11

(from a M. Dupuis and B. Maas report, Fall 2016, OPT 453 class). LEFT: EM-CCD images of the same interference pattern at a single photon level with a 200 ms single exposure time (top) and with accumulation of 50 images with a 200 ms exposure time. RIGHT: calculation of fringe visibility of accumulated images versus a linear polarizer (quantum eraser) angle. A zero on a mount scale of a linear polarizer (quantum eraser) had s small shift relative a polarizer axis. Maximum fringe visibility should be at a 45° angle.

Figure 12.

Experimental setup for single-photon interference lab at different viewing angles. TOP: The whole view including a He-Ne laser from a Young’s double slit interferometer side; CENTER: View from a Mach-Zehnder interferometer side. A conventional CCD camera without cooling and amplification of the signal is used on top and center images. BOTTOM: A Young’s double slit interferometer setup with an EM-CCD camera and a slit set with air slit opening (a He-Ne laser is not shown).

2.3.

Lab 3. Single photon source: Confocal microscope imaging of single-emitter fluorescence

One strength of the OPT 253 course is the students’ immersion in a real research environment, working on state-of-the-art, fragile, and expensive equipment that modern quantum-optics research uses around the globe. Every student understands the cost of each piece of equipment he/she is entrusted with. In addition, in the Labs 3 and 4, a class time was reserved for addressing “real” research questions on actual research samples or, time permitting, have students prepare their own samples with single emitters that nobody else before them had ever investigated. This intentional blurring of the dividing line between “education” and “research” strongly boosts student interest. Occasionally students obtained results that were reported at the professional conferences, e.g. QELS, IQEC or Frontiers in Optics. Some undergraduates felt sufficiently motivated to focus their senior research projects on work in Lukishova’s research lab after this class.

Labs 3 and 4 are devoted to single (antibunched) photon source (SPS) [30-31], a key hardware element in quantum cryptography. To create single photons a laser beam should be focused on a single emitter that emits single photons at a time (for reviews see [32-33]. Single colloidal semiconductor nanocrystal quantum dots and color centers in nanodiamonds were used as single emitters in these lab. A confocal fluorescence microscope (Figure 13) with excitation by the laser light with different wavelengths was used in a Lab 3 for imaging of single emitter fluorescence and measuring their spectra.

Figure 13

(prepared by R. Lopez Rios). A schematics of a confocal fluorescence microscope. Top-left inset shows oil immersion microscope objective.

Figure 14 shows confocal microscope fluorescence images of NV-color centers nanodiamonds of 40 and 20 nm diameter (left two images) and photoluminescence of gold bowtie nanoantenna arrays (insert shows nanoantenna shape with 75 nm arms and 30-60 nm gaps) (center image). Right picture shows a wide-field view of a sample area in a white light. Numbers identify the positions of nanoantenna arrays with different gaps. Students learned how to find a specific array for imaging.

Figure 14.

LEFT: Confocal microscope imaging of single NV-centers fluorescence in 40-nm and 20 nm size nanodiamonds. Right figure of two shows blinking in fluorescence (horizontal stripes) of a single color center in a 20-nm-nanodiamond (2 μm × 2 μm raster scan). CENTER: Confocal microscope imaging of gold photoluminescence from bow-tie nanoantenna arrays. Insert shows the bow-tie shape of nanoantenna (SEM micrograph). RIGHT: A wide-field sample view recorded by a CCD-camera showing a position of different nanoantenna arrays (numbers 30nm-50 nm are the gaps of nanoantennas located close to these numbers).

Figure 15 shows the experimental setup of Labs 3 and 4 in different viewing angles. Bottom pictures show sample table and a vial of 40 nm size NV nanodiamonds dispersed in a deionized water.

Figure 15.

Pictures of a Labs 3 and 4 experimental setup at different view. TOP: A 532-nm excitation green laser is shown on the picture. CENTER: A spectrometer with a low-light level EM-CCD camera is shown. Insert shows students of the OPT 253 class. BOTTOM: A sample table with a piezo-electric scanning stage for a raster scan of the sample (left) and a vial with dispersion in a deionized water of NV-color-center nanodiamonds. For comparison a jewelry with color center rough diamond powder is shown as well.

After fluorescence imaging of nanodiamonds students of the OPT 253 class obtain micrographs of nanodiamond topography (Figure 16, left top micrographs) using a compact atomic force microscope (Figure 16, top right). Bottom left picture shows Fall 2016 students of OPT 253 working on an AFM.

Figure 16.

TOP LEFT: Raw data of AFM imaging of topography of nanodiamond clusters obtained by OPT 253 students; TOP RIGHT: A compact AFM (Easy Scan 2, Nanosurf Inc.) on a vibration isolation platform and a AFM controller. BOTTOM LEFT: Undergraduate students of a Fall 2016 OPT 253 class working on an AFM. BOTTOM RIGHT: Students of a Fall 2016 OPT 253/OPT 453 class on the lecture.

Weekly 1 hour lectures about the labs, their main concepts, devices and optical components, lab materials and overview of modern advances of quantum nanophotonics are very important in the OPT 253 class (see Figure 16, bottom right). In addition to lab report submission and quizzes before and after each lab, students have two exams (MidTerm and Final). Graduate students must submit also individual essays about single and entangled photon sources and make a Power Point presentation at a special meeting with TAs (for 20 students in the OPT 253/OPT 453 class two TAs were involved).

2.4.

Lab. 4. Single-photon source: Hanbury Brown and Twiss setup. Fluorescence antibunching

To prove a single photon nature of single-emitter fluorescence (separation of all photons in time from the emitter (antibunching)), in a Lab 4 students measure time interval between consecutive photons. Antibunching first was obtained at the University of Rochester in 1977 by Mandel, Kimble and Dagenais [30-31]. See also book [32] and review [33].

In the case of antibunching should not be any photons at all at zero interphoton time. For such measurements a Hanbury Brown and Twiss correlator [34] is used (Figure 17). It consists on a beamsplitter and two single-photon counting avalanche photodiode modules APDs. Electronics part in our measurements consists of a time-correlated single photon counting PCI card with a start and stop inputs connected to APDs. This card (Time Harp 200, Picoquant) and a computer program permit to build a histogram of interphoton times. In the case of antibunching a dip appears at zero interphoton time.

Figure 17.

LEFT: Schematics of a Hanbury Brown and Twiss correlator at the ouput of a confocal fluorescence microscope (prepared by L. Bissell). RIGHT: Raw data histogram of interphoton times with a dip at zero time interval indicating photon antibunching (SPS). Fluorescence of NQD inside a gap of a bow-tie nanoantenna was studied. Insert shows a time trace of this NQD.

Figure 17, right shows user interface of the program used for building the histogram with a dip at zero photon time indicating antibunching. In these measurements in class we used a CdSeTe colloidal nanocrystal quantum dot (NQD) within a gold bow-tie nanoantenna. These 2015 OPT 253 class results were reported at 2016 Frontiers in Optics conference [35]. During collection data for the histogram, on another computer with imaging software students recorded time traces of changing intensities of single emitters (NQD) in time showing blinking of a single NQD (see Figure 17, insert).

3 hours from this 15-hour lab (five 3 hour-classes for each group of 3-4 students) of the OPT 253/OPT 453 class were devoted to electronics. Students measured on an oscilloscope TTL pulses from APDs, they also calibrated a PCI correlator by finding a zero-time position on a histogram using an electrical delay line. BNC and PCI card connectors and cables and impedance matching are discussed as well. Detailed discussion was included on computer cards used in this lab.

4.1.

Monroe Community College

From 2009 to 2015 years, 142 MCC students carried out labs at the UR Quantum and Nano-Optics Lab facility. 52 MCC students from them also carried out the lab on photolithography in the UR Integrated Nanosystems Center (see paper [36] of this proceedings issue). MCC students carried out 3-hour labs on (1) entanglement and Bell’s Inequalities; (2) singlephoton interference with single-photon counting EM-CCD camera; and (3) atomic force microscopy of nanodiamonds. Prof. P. D’Alessandris (MCC) was trained to work with an EM-CCD camera to teach his MCC students a single-photon interference lab at the UR (Figure 20). Prof. S. Lukishova taught MCC students other two labs. Within this collaboration supported by 3 NSF grants, lab manuals were specifically rewritten for MCC students [Lukishova (UR), D’Alessandris (MCC)]. Methods of evaluation of MCC students’ knowledge were developed using quizzes with participation of an external evaluator (Prof. Zawicki, Buffalo State College).

Figure 20.

Two groups of MCC students with their professor (right) in the UR lab on single-photon interference.

4.2.

Rochester Institute of Technology

Rochester Institute of Technology (RIT) was a collaborator of the UR on one of the NSF-supported projects on developing teaching experiments on photon quantum mechanics using photon counting instrumentation. During Fall 2009, Prof. R. Jodoin from RIT spent his sabbatical in UR teaching labs. Later RIT established their own quantum-optics teaching lab, strengthening the Upstate New York future-optics roots. Prof. S. Preble established quantum optics teaching lab on entanglement learning UR experience (see Figure 21).

Figure 21.

LEFT: S. Preble and R. Jodoin - RIT collaborators of UR. RIGHT: S. Preble at the UR facility learning alignment of entanglement and Bell’s inequalities setup.

4.3.

UR participation at ALPhA’s Laboratory Immersions Program

In August 2011, UR participated in the Immersion Program of the Advanced Laboratory Physics Association (ALPhA). Lukishova hosted six visitors from different universities for 3 days, familiarizing them with UR lab-course experience on photon quantum mechanics (http://www.advlab.org/imm_singlephotonr2.html).

Prof. M. Braunstein (Central Washington University), Prof. J. Buchholz (California Baptist University and University of California, Riverside), Prof. T. Perera (Illinois Wesleyan University), Prof. M.C. Sullivan (Ithaca College), Prof. W.F Smith (Haverford College) and D. Dominguez (graduate student responsible for teaching Labs, Texas Tech University) carried out all four labs of the OPT 253 class and at the end made final presentations about these labs (see Figure 22).

Figure 22.

Six participants of the ALPhA’s immersion program at the UR and a TA J. Winkler (right).

4.4.

Adelphi University students’ visit for Quantum Lab experience

During two days (October 2012), 5 students of Adelphi University carried out 4 labs at the UR Quantum/Nano-Optics Lab facility. They also prepared 1D-photonics bandgap materials for SPS based on cholesteric liquid crystals [37] (Figure 23).

Figure 23.

Adelphi university students and their professor at the UR Quantum and Nano-Optics Lab facility. LEFT: During their experiment on SPS (single-molecule confocal fluorescence microscopy of colloidal nanocrystal quantum dots). Prof. S. Bentley - at right. RIGHT: During their preparation of 1-D photonic band gap structures for SPS with a cholesteric liquid crystal microcavity [37].

4.5.

Dissemination of results in some other universities

One graduate student of Prof. P. Verma (University of Oklahoma (UO), Tulsa) was trained on a Rochester lab facility in October 2010. Earlier (in September of 2010) S. Lukishova was invited to Tulsa to share her experience on quantum optics labs with this University. Figure 24, left shows UO audience at the Tulsa Schusterman Center Seminar during this lecture. Insert shows P. Verma with graduate student and S. Lukishova.

Figure 24.

LEFT: Dissemination of results at the UO, Tulsa: lecture of S. Lukishova at the Tulsa Schusterman Center Seminar. Insert shows (from right to left) P. Verma, S. Lukishova and Verma’s graduate student. RIGHT: Invited lecture of S. Lukishova at the ITMO University, St. Petersburg, Russia.

S. Lukishova was also invited to the Young Scientists Workshop at ITMO University in June 2013 (Saint Petersburg, Russia). See Figure24, right.

In Spring 2012, 6 graduate students and Prof. S.K. Sundaram (SUNY Alfred) visited UR Quantum and Nano-Optics Lab facility and attended lecture-demonstrations (Figure 25).

Figure 25.

Students of Alfred University and their professor S.K. Sundaram at single-photon interference lab of the UR.

During several years students of Colgate University (Prof. E. Galvez) attended similar lecture demonstrations at the UR.