Experiments with correlated photons: From advanced-lab projects to

dedicated laboratories

Advanced-Lab Conference2009

• Charles Holbrow • Lauren Heilig ‘01 • Naomi Courtemanche ’02• James Martin ‘03 • Matt Pysher ‘04, • Justin Spencer ’05• Mehul Malik ’06• Brad Melius ’06• Kyle Wilson ‘06• Bryce Gadway ‘07• Ben Reschovski ’07• Erik Johnson ’08• Ushnish Ray ’08• Laura Coyle ’09• Claire Watts ’10• Ashish Shah ‘10• Thanks to: Mark Beck, Joe Eberly, Paul Kwiat, Vic Mansfield, Beth

Parks, Anton Zeilinger, Bill Wooters.

Funded by CCLI grants DUE-9952626DUE-0442882

Colgate University, Hamilton, New YorkContributorsLiberal-arts college in CentralNew York – 2800 students

Quantum Agenda at Colgate

• 1st semester course on modern physics (20+ years old)– Text: Modern Introductory Physics, Holbrow et al– Covers main features of modern physics: wave-particle duality, relativity,

quantization, plus baby quantum mechanics (5 years old). Includes a lab on the quantum eraser.

– Rationale: teach the topics that excite physicists first.

• Junior-Senior quantum mechanics with lab (since 2005)– Dirac notation (text: Townsend)– Emphasize fundamentals (Superposition etc.-include Bell)– Lab has five experiments using single photons. – See Holbrow et al AJP 70, 260 (2002) ; Galvez et al AJP 73, 127 (2005).– See also http://departments.colgate.edu/physics/pql.htm– (Friday workshop Rm 4472).

• Capstone projects as advanced-lab requirement (poster 17)– Independent, table-top– Ulterior goal is to develop new labs for QM course– Very productive side-shows: 2 Phys Rev A’s, 1 J. Phys B, more to come.

• Main Apparatus• Quantum labs

– Quantum superposition: one photon at a time– The Eraser and distinguishability: quantum

interference without Heisenberg– Wavepackets and measurement– Biphotons: nonclassical path interference– Entanglement and reality

Agenda: new labs, but also new views

The photon source: photon pairs produced by spontaneous parametric down conversion

Parametric down conversion (DC) requires that the photon energy and momentum must be conserved:– Ep = EDC-signal + EDC-idler energy correlation– kp = kDC cos θs + kDC cos θi spatial correlation

BBO crystal

kp

kDC-signal

kDC-idler

θs

θi

Downconverted photons are produced simultaneously

Crystal:Beta-barium-borate 5x5x3 mm$0.5-1 k

Leonard Mandel

Pump laser

• SPDC produce photon pairs at half the energy, twice the wavelength. Visible lasers would produce DC in near-IR, where PMT are inefficient.• Best option are avalanche photodiodes APD.• Peak efficiency at 700 nm.• Best affordable source are blue diode lasers at 402 nm--410 nm.

GaN laser: 375nm (few mW), 405 nm (up to 200 mW) ($7k module)($2.5k no temp control)

If other gas lasers are at hand they also work:• Ar ion lasers 350 nm, 458 nm• HeCd 442 nm

Optical layout

• 2’x5’ optical breadboard• Standard optics for Near-IR• Low-height mounting hardware(pedestals are best).

• HeNe or fiber laser needed for alignment.

Avalanche Photodiode Detectors detectors

Bare or fiber-coupled (best): $4k each or $10k for four

Multimode fibers lenses

Bare (need to be boxed in)

Key to insure single-photon events: coincidence detection

Options: - NIM Electronics

- black box—does it all

(Mark Beck – Whitman CollegeBranning et al AJP 77, 667 (2009))

+ PC

Quantum superposition

A

Bl1

l2

Light going through an interferometer– moves in two directions (qubit). If the arms have the same length and no distinguishable features, the

path taken by the light from A to B is undefined. The state of the light is in a superposition of going through both

paths: C

x

yyixx

BS 221

1

+→

yxiM 2

12

+→

( ) ( ) yeexeei iiii

BS

2121

221

2δδδδ +−++→

interference

λπδ i

il2

=

( )( )21cos121 δδ −+=→BAP

• Outcomes of measurements are described in terms of probabilities P. The probability amplitude has both magnitude p and phase δ, and the probability of an outcome is the square of the probability amplitude for that outcome:

• When an event can occur in several alternative ways, with probability amplitudes (p1,δ1) and (p2,δ2) then the total probability amplitude is the (vector) sum of the individual probability amplitudes, and the probability is the square of the magnitude of the combination

For the interferometerSo, if the paths are indistinguishable then if we make• If the alternate paths are distinguishable

there is no interference

Feynman’s approach

M

M

BS

BS

2pP =

( )212122

21 cos2 δδ −++= ppppP

21

21 == pp( )δcos1

21

+=P21 δδδ −=

212

221 =+= ppP

RichardFeynman

The probability is P = ½ ( 1 + cos δ )

We change δ by changing the length of one of the arms (2πΔl/λ). “…the photon then only interferes with itself” P.A.M. Dirac

An experiment: single photons through an interferometer

Ingredients:• Heralded photon source• Interferometer• Single-photon detectors• Electronics/computer

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Classical vs. non-classical source

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fitdatactriplesg2*1000

The single photon must not split at a beam splitter: source mustpass the Hanbury-Brown-Twiss test of the degree of second order coherence

C

B

0)0()2( ==CB

BCquant PP

Pg 1)0()2( ≥=CB

CBclass II

IIg

Why not an attenuated source?

Experiment: measure triple coincidences at A, B and C

A

B

C

ACAB

ABCNN

Ng =exp)2( )0(

HWP=0°: polarization is not disturbed: indistinguishable paths. The probability is: P = ½ (1 + cos δ)

→ there is interference

HWP=45°: Rotates the polarization to horizontal. The probability is:

P = ½ no interference(possibilities are distinguishable)

A polarizer (at 45°) is added after the interferometer. The probability becomes:

P = ¼ (1 + cos δ) interference reappears(the distinguishing information is erased)

Polarizerat 45°APD

APDM

M

BS

BS

HWP (θ)The Eraser

( )

↔ = ( − )

↕ = ( + )

Note: photon is not disturbed: we do not need to appeal to Heisenberg to destroy the interference.

John Wheeler:“it is bit”

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HWP=0

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HWP=0HWP=45Pol in

The photons are in a coherent superposition of energy eigenstates. They form a wavepacket.

Δλ

λ

Δt = λ2/(cΔλ)

t

The length of the photon wavepacket

lc = c Δt

is the coherence length

Photon Wavepackets and post selection

M

M

BS

BSΔl < lc

Photons arrive at similar times: paths are indistinguishable.

Δl > lc

Photons arrive at distinguishable times: in principle we can determine the “which-way” information by timing the photon pulses.

When the paths are indistinguishable there is interference.

If the difference in length of the two arms is Δ l

Which way? skip

APD

APDM

M

BS

BSF

F

The experiment:1. We put 10-nm filters in front of

the detectors: the length of the photon wave packet is lc = 80 μm.

2. We align the laser to see fringes (as in previous experiment).

3. We quantify the degree of coherence with the visibility of the fringes V:

P = ½ (1 + V cos δ )

V = (Nmax – Nmin) / (Nmax + Mmin) V = 0.82 ± 0.05

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Increase the length of one of the arms

by 36 μm

by 72 μm

V = 0.31 ± 0.05

V = 0.10

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Increase the path length difference by 180 μm

V = 0.01 ± 0.10

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Note: a timing measurement of photon arrival times is not made. Interference disappears as soon as the path information is available, regardless of whether we measure it or not.

Post-selection

Put a 0.1-nm filter (in front of the idler).

This forces the detected photon wave packet to be ten times larger:

lc = 8 mm > Δl , making the which-way information unavailable, and thus the paths indistinguishable.

APD

APDM

M

BS

BSF

F

F=0.1 nm

V = 0.59 ± 0.04

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State of the light is determined by post-selection: a collapse of the two-photon wave function.

Two collinear photons enter an interferometer.

A

B

C

PAB = (1/4) (1 + cos δ )2

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δ/2π

P = (1/2) (1 + cos δ )

Pxx = (1/4) (1 + cos 2δ )

Two Photons or biphotons? An example of nonclassical path interference

Possibilities for one-photon: 2 +

Possibilities for 2 photons leaving through the same port: 4 + + +

Possibilities for two-photons leaving through separate ports : 8 + + +

+ + +

Displaced for sake of clarity

Glauber: “it is amplitudes that interfere”

MZ irisfilter

opticalfiber

BBO crystal

polarizers

piezo+stage

A

B

CApparatus

The result can be shown analytically to be due to the symmetry of the bosonic wave function.

Galvez & Beck, Proceedings of ETOP, 2007

The two photons are in a correlated state. They act as a single quantum: the biphoton

If the source crystal is indistinguishable then the light is in a superposition:

P

P

H

H

WD

D

Entanglement and Reality

1st crystal produces

)(2

12121 bb±↔↔=Φ±

21↔↔

2nd crystal produces 21bb

Entangled state cannot be decomposed into a product of single-particle states; measurement on one photon determines the state of the other. Two qubits

Einstein: “spukhafte fernwirkung” or“spooky action at a distance”.

ErwinSchrödinger

1

2

θ1

θ2Polarizers

Experiment:P1 fixed ( ), P2 turned

Photon pairs are correlated (parallel) regardless of the orientation:

P = ½ cos2(θ1 − θ2).

If θ1 = π/4 P = ¼ ( 1 + sin θ2 )

If photons are in a mixed state (i.e., half the time in 1 2 and the other half in 1 2), then the results are different: when θ1 = π/4, P = ¼

Polarization Correlations

( 1 2 + 1 2 )

Φ+ = ( 1 2 + 1 2 )

This can best be treated analytically with the density matrix.

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P1=45 ent

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Bell Inequalities

The Clauser-Horne-Shimony Holt tests against reality and localityCorrelation parameter:

),(),(),(),(),( βαβαβαβαβα ⊥⊥⊥⊥ −−+= PPPPE

1±=E0=E

Realistic view predicts:

),(),(),(),( βαβαβαβα ′′+′+′−= EEEES

In the lab students get S = 2.39 ± 0.09: violation!

α, β angles, α⊥ = α + π/2 , β ⊥ = β + π/2. For an entangled state , and for a mixed state If we define

2

• Realistic view: a reality exists independent of the observation• Quantum view: observables do not have preexisting values• Other quantum tenets: indeterminism, nonlocality, contextuality

From lab write-up…

(see also Dehlinger & Mitchell AJP 70, 903 (2002))

≤S

John Bell

The quantum labs

Advanced lab

20072009

To conclude…

• Experiments with single photons directly probe quantum mechanics—they are explained by the quantum mechanics of a single quantum (“on your face quantum mechanics”).

• Experiments are table-top, feasible, and reproducible.

• Student feedback is very positive.

• Experiments provoke discussion and debate about fundamental questions, plus they are fun and spooky…

Webpage: http://departments.colgate.edu/physics/pql.htmsee also http://people.whitman.edu/~beckmk/QM

Density Matrix

Crash course: states are represented by matrices ψψρψ =ˆ

If ⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

2

12211 a

aaa φφψ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∗∗

∗∗

2221

2111ˆaaaaaaaa

ψρthen

Thus 1)ˆ( =ρTr (conservation of probability)

The probability of state |Ψ> being in state |ϕ> is

)ˆˆ(ˆ ψϕψ ρρϕρϕ TrP ==

An incoherent mixture of basis states |φ1> and |φ2> can be represented by

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

2

12211 0

0P

PPPm φφφφρ ϕφ Mixed state does not have

off-diagonal elements.

Measuring correlations

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

θθ

θθθsincos

sincos 111 VH

1

2

θ

π/4Polarizers

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

=⊗=

θθθθ

θθ

sinsincoscos

21

21 DD

( )θθρθ 2sin121ˆ +=+Φ

DD

21ˆ =DD m θρθ

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P2=45 Phi+P2=45 mix

⎟⎟⎠

⎞⎜⎜⎝

⎛=+=

11

21

21

21

222 VHD

⎟⎟⎠

⎞⎜⎜⎝

⎛=

01

HBasis ⎟⎟⎠

⎞⎜⎜⎝

⎛=

10

V

Photon 1pol at angle θ

Photon 2pol at 45º

Two-photon state:

Entangled state projected onto :Dθ

Entangled state projected onto Dθ

ret