Entanglement lab

 

 

Intro

We have to assemble the undegraduate experiment. The basic idea is published here.

Related articles:

• "Comparing measurements of g2(0) performed with different coincidence detection techniques"

The more information can be found in Berkley undergraduate lab , the student manual is available Some information can be found on .

Theory

Some practical formulas

Beam separation as a function of optical axis angle Theta and block length D: $d= \frac{D(n^{2}_{e}-n^{2}_{o})\tan \Theta}{n^{2}_{e} +n^{2}_{o} \tan^{2} \Theta}$

Type I for entanglement

As opposed to type II phase matching that produces orthogonally polarized photons in parametric down conversion (PDC), the type I PDC process produces identically polarized photons in the output signal and idler modes (labels s and i below).

Normally the output state from type I PDC is not entangled: to get the required phase matching in the nonlinear material, the pump polarization must be fixed. Both the PDC photons may then either be horizontally or vertically polarized. An often-used trick is to employ two similar nonlinear crystals (placed one after the other with their optic axes orthogonal) and sending a pump with a 45 degree polarization . If the crystals are thin enough to simultaneously lie inside the coherence length of the pump, and losses between the first and second crystal are negligible, then a pump photon is equally likely to excite the PDC process in either of the two crystals. In that case, the output state may be approximated as |Hs,Hi⟩+eiϕ|Vs,Vi⟩ which is an entangled state. The relative phase ϕ is a function of the phase matching, thickness of the crystals, etc.

Components

Electronics

Part name	Part number	Digikey	Ebay	Ebay price
FPGA	Cyclone FPGA	P0082-ND, 115.06 CAD
Arduino	DUE	1050-1049-ND, 49.89 CAD
Keypad		1528-1136-ND, 5.27 CAD
Display		1528-1536-ND, 26.61 CAD
AC-DC converter		945-2212-ND
USB connector		A111306CT-ND
Arduino
Arduino

Optomechanics

Missing: Beam displacer adaptor

Fibers

Part Number	Fiber Type	Connector Type	Diameter			Numerical Aperture
			Core	Cladding	Outter
SPCM-Q9	Multimode	FC/FC	100 um	140 um	2.5 mm	0.29

 

Missing: 2x2 Fiber coupler FC/FC, 850nm 62.5 um/125

62.5 um/125 FC/FC

Multimode Fused Fiber Spliter Oz-optics multimode Fiber fused coupler ( datasheet)

 

 

Fiber mechanics

• Fiber coupling stage option 1:

1. f = 11.0 mm, NA = 0.25, Mounted Geltech Aspheric Lens, AR: 600-1050 nm C220TMD-B $71.00

• Fiber coupling stage option 2:

1. Fiber Collimation Pkg., 780 nm, f = 11.07 mm, NA = 0.26 FC/PC F220FC-780 $148.00

• FC/PC to FC/PC Dual L-Bracket Mating Sleeve ADAFCB2 $54.50

SPCM

See the dedicated page

 

Lasers

Diode lasers are bought here.

• 808nm 10mW Laser Module, 12x35mm (AD808-101235)
• 405nm 150mw TTL TEC cooled OEM 12VDC (AIX-405-150T)

 

 

 

Crystals

The crystal picture of BBO for Type I SPDC. The crystal is cut for Type I (e-oo) phase matched SPDC. The crystal must be orientated as indicated in the diagram with the pump beam properly polarized and at normal incidence. The crystal can be tuned slightly with respect to the

Crystal	Information
PABBO5050-405(I)-HA3	Paired BBO crystals, size 5x5x0.5mm(each), cut for Type I SPDC pumped by 405nm with the half opening angle of 3 degrees. The two crystals mounted in a 1" holder with one crystal rotated by 90 degrees about the axis normal to the incidence face
NCBBO5050-405(I)-HA3	BBO crystal, Size 5x5x0.5mm, cut for Type I phase matched SPDC pumped by 405nm with the half opening angle of 3 degrees, AR coated, OD 1" mounted
NCBBO5300-405(I)-HA3	BBO crystal, Size 5x5x3.0mm, cut for Type I phase matched SPDC pumped by 405nm with the half opening angle of 3 degrees, AR coated, OD 1" mounted

Wavefront distortion:	less than λ/8 @ 633 nm
Clear aperture:	> 90% central area
Flatness:	λ/8 @ 633 nm
Scratch/Dig:	10/5 to MIL-O-13830A
Parallelism:	better than 20 arc seconds
Angle tolerance:	Δθ < +/-0.25o, Δφ < +/-0.25o

Coating: AR@405/810 nm for 405 nm pumped SPDC crystals

P-coating for pumps other than 405 nm unless coating is specified.

Experimental set-up

List of experiments

Hong-Ou-Mandel

Description of the HOM theory or here. If two photons are indistinguishable, we get regular N00N state with N=2. If we consider spectral bandwidth of a photon, we can take it in the form: $|1\!> = \int \phi(\omega)a^\dagger(\omega)|0\!> d\omega$, where $\phi(\omega)$ is spectral amplitude function. Before the BS we have next two-photon wavefunction with a delay in one arm: $\int \phi(\omega_1)a^\dagger(\omega_1)|0\!> d\omega_1 \int \phi(\omega_2)b^\dagger(\omega_2)e^{-i\omega_2\tau}|0\!> d\omega_2$ Following a regular BS transformation and applying measurement operator $\hat M_c=\int c^\dagger(\omega_c) |0><0| c(\omega_c) d \omega$. The probability to get photon in each arm to coincide is given by: $<\psi|M_a \otimes M_b| \psi> = \frac{1}{4}\int d \omega_a\int d \omega_b|\phi(\omega_a)\phi(\omega_b)|^2(2-e^{i\tau(\omega_a-\omega_b)}-e^{i\tau(\omega_b-\omega_a)})$

Visibility of HOM dip

Visibilty is related to density matrices of the photons and is equal to purity of photons: $V=\text{Tr}(\rho_a\rho_b)=\text{Tr}(\rho_a^2)=P$

Latest articles

The Hong-Ou-Mandel effect gives opportunity to measure attosecond delays and nanometer distances. "Lyons, Ashley, et al. "Attosecond-resolution Hong-Ou-Mandel interferometry." Science advances 4.5 (2018): eaap9416. "

[5],

Traditional Hong-Ou-Mandel interference scheme. Two identical photons arriving simultaneously at a balanced, broadband beam splitter (BS) will be conveyed along only one of the possible outgoing channels—and so, in contrast to the general case, no coincidence counts will be accumulated.

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The figure shows the complete alignment where the He–Ne laser helps to align the trajectories that the photons should follow. The use of the mirrors (M) helps to modify the difference in optical path using a stepper-motor (PM). A second beam splitter is placed where the beams intersect to generate the superposition and complete the IMZ interferometer. The balance of the optical path is made by using the interference produced from a white light (WL).

g2 function measurement

Degree of second-order coherence

$g^{(2)}( \mathbf{r}_1,t_1;\mathbf{r}_2,t_2)= \frac{\left \langle E^*(\mathbf{r}_1,t_1)E^*(\mathbf{r}_2,t_2)E(\mathbf{r}_1,t_1)E(\mathbf{r}_2,t_2) \right \rangle}{\left \langle\left | E(\mathbf{r}_1,t_1)\right |^2 \right \rangle \left \langle \left |E(\mathbf{r}_2,t_2)\right |^2 \right \rangle }$

If the electric fields are considered classical, we can reorder them to express $g^{(2)}$ in terms of intensities. A plane parallel wave in a stationary state will have

$g^{(2)}( \tau)= \frac{\left \langle I_{1}(t)I_{2}(t+\tau) \right \rangle}{\langle I(t)_{1} \rangle \langle I(t)_{2} \rangle }$

The above expression is even, $g^{(2)}(\tau)= g^{(2)}(-\tau)$.

violation of the Bell inequality

Bell inequality was violated, although each time new assumptions are made, which open loopholes in proof of an initial statement. At this moment a few experiments were performed to close locality and detection loopholes simultaneously. These experiments were made with photon pairs, NV vacancies (the first to close both loophholes) and with neutral atoms. (So called Event-ready Bell test)

1. "Significant-Loophole-Free Test of Bell’s theorem with Entangled Photons", PRL 115, 250401 (2015)
2. "Strong Loophole-Free Test of Local Realism", PRL 115, 250402 (2015)
3. PRL 119, 010402 (2017)

Assembly progress

May

• 17.05.17: Pat finished the rack above the optical table, optical table is on its legs
• 18.05.17: Sorted optical elements, Duncan finished his counting module
• 19.05.17: Duncan draws the plan. Crystals generate pairs on a specific angles determined by phase-matching conditions. All of the crystals have 3 degree an half opening angle.
• 23.05.17: Leveled a blue laser, collimated a red laser, mounted optics
• 24.05.17: sent beams of an alignment laser into fibers
• 25.05.17: installed crystal (long single crystal), saw counts on both SPCMs: 6 MHz of single counts, 1 MHz of coincidences
• 26.05.17: we have doubts regarding the coincidences. Possible check: misalignment of one of the fibers should lead to decrease of coincidences, although leave single counts similar.
• 30.05.17: we have more confidence that we see coincidences, which come from the same part of the cone. We decided to check entanglement between photons.
• 31.05.17: Duncan installed paired crystal, so we can observe polarization entanglement. We are basing our experiments on this article and this test.

June

• 1.06.17: Violated the Bell-inequality
• 2.06.17: Trying to find a way to interface available multi-mode fibers with a single-mode fiber beam splitter (TW805R5A2). Multi-mode fiber has a core 100um, fiber in the beam splitter 4.4um (uses this single-mode fiber); coupling efficiency with a use of ADAFCB2 is around 1% (multi-mode -> single-mode).
• 5.06.17: Found that only a small portion of photons emitted have Gaussian mode, so working with just single mode fibers is impossible. Single counts were 6*10^6 (multi-mode fiber) and 50*10^3 (single-mode fiber). Coincidence counts were negligible.
• 6.06.17: Alex thinks that multimode fiber coupler wouldn't work. We have seen some lab reports< where people used single-mode beam splitter. Don't have holders for beam displacers, planning to assemble regular interferometer () with mirrors and PBSs.
• 07.06.17: Alex doesn't want to try regular Mach-Zendner interferometer, asks us to change to interferometer based on beam displacers. Duncan build an interferometer, checked interference with a camera. No single photon interference.
• 12.06.17: We found that coherence length of the emitted photons scales as $l_c= \frac{\lambda^2}{\Delta \lambda}$, where ${\Delta \lambda}$ is the bandwidth of the filter in front of the multimode fiber. Usually people use 10 nm bandpass filter, we have only low-pass.
• 16.06.17: Got results on $g_2(0)$ for single photon and coherent cases

Milestones

Polarization entanglement

HWP for module #1: 0 degr

Pump HWP: 16 degr

Polarizer: 18 degr

Singles, in thousands, Module #1	Singles, in thousands, Module #2	Coincidences	HWP, Module #2
156	175	11100	0
162	198	6300	22.5
160	212	1400	45
159	195	6200	67.5
159	178	11240	90
155	196.5	6200	112.5
164	215	1200	135

 

 

 

 

Bell-inequality violation

A typical CH74 (single-channel) experiment

File:Single-channel Bell test.svg

Setup for a

Prior to 1982 all actual Bell tests used "single-channel" polarisers and variations on an inequality designed for this setup. The latter is described in Clauser, Horne, Shimony and Holt's much-cited 1969 article as being the one suitable for practical use. As with the CHSH test, there are four subexperiments in which each polariser takes one of two possible settings, but in addition there are other subexperiments in which one or other polariser or both are absent. Counts are taken as before and used to estimate the test statistic.

(3) S = (N(a, b) − N(a, b′) + N(a′, b) + N(a′, b′) − N(a′, ∞) − N(∞, b)) / N(∞, ∞),

where the symbol ∞ indicates absence of a polariser.

If S exceeds 0 then the experiment is declared to have infringed Bell's inequality and hence to have "refuted local realism". In order to derive (3), CHSH in their 1969 paper had to make an extra assumption, the so-called "fair sampling" assumption. This means that the probability of detection of a given photon, once it has passed the polarizer, is independent of the polarizer setting (including the 'absence' setting). If this assumption were violated, then in principle a local hidden variable (LHV) model could violate the CHSH inequality.

In a later 1974 article, Clauser and Horne replaced this assumption by a much weaker, "no enhancement" assumption, deriving a modified inequality, see the page on Clauser and Horne's 1974 Bell test.

31st of May

HWP, module #1	HWP, module #1	Coincidences	Singles, module #1	Singles, module #2
0	0	12300	162	190
0	45	1205	164.3	180
45	0	988	157	190
45	45	11300	152	179
45	none	12145	151	369
none	0	11500	318	171
none	none	25000	305	363

 

 

1st of June

N(a, b)	N(a, b')	N(a', b)	N(a', b')	N(a', ∞)	N(∞, b)	N(∞, ∞)	S
4271.3435114504	172	177.58	5177.4849785408	5362.9060402685	4581.5054945055	10000.768115942	0.48725

g₂ function measurement

Setup for a g2 function measurement

 

Single photons

N3, module #3, single counts	N1, module #1, single counts	N2, module #2, single counts	N13, mod. #1 & #3, coincidences	N23, mod. #2 & #3, coincidences	N123, mod. #1, #2 & #3, coincidences	Sampling time, s	Averaged over N	g2(0)
55261.6	14323.26	14344.026	1300.43	1403.29	2.1	1	121	0.064

$g_2(0) = \frac{\langle I_{1} I_{2}\rangle}{\langle I_{1} \rangle \langle I_{2} \rangle} = \frac{N_{123} \cdot N_3}{N_{13} \cdot N_{23}}$

Coherent state

Strongly attenuated red diode laser is sent through one of the paths, it comes to modules #1 and #2.

N1, module #1, single counts	N2, module #2, single counts	N12, mod. #1 & #2, coincidences	Sampling time, s	Averaged over N	g2(0)
49132.96	48341.5	23.6	1	100	0.99

$g_2(0) = \frac{\langle I_{1} I_{2}\rangle}{\langle I_{1} \rangle \langle I_{2} \rangle} = \frac{N_{12} }{N_1 \cdot N_2\cdot \tau_c}$ Which is normalized on the detection frequency (10 ns) rather than on the number of incident photons.

Possible extensions

• Quantum entanglement witnesses (Berkman), 4 detectors, FPGA Duncan
• Bell test with 4 SPDC
• Hong-Ou-Mandel (Mexican)-phys 598

Other educational experiments

• atomic clock (microwave source, amplifier, cavity 3.4 GHz)
  • Ramsey spectroscopy
  • radio-freq absorption
• optical pumping manual
• mode-locked laser
• harmonics generation
• titan-saphire

Questions:

1. do we have extra fiber coupling stage?
2. FPGA Duncan,adding 4th channel