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Two-Photon Interference Markus Perner Technische Universit¨ at M¨ unchen 08. Mai 2013 1 2 3 4
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Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

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Page 1: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Two-Photon Interference

Markus Perner

Technische Universitat Munchen

08. Mai 2013

1

2 3

4

Page 2: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Motivation

Why is two-photon interference so important?

Show quantum nature of light

Measurement of the length of a photon

Characterization of single-photon sources

Linear quantum information processing (QIP)

Page 3: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Outline

1 Theory

2 Hong-Ou-Mandel Interference [Hong, Ou & Mandel (1987)]

3 Quantum Beat of Two Single Photons [Legero et al. (2004)]

4 Summary

Page 4: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Outline

1 Theory

2 Hong-Ou-Mandel Interference [Hong, Ou & Mandel (1987)]

3 Quantum Beat of Two Single Photons [Legero et al. (2004)]

4 Summary

Page 5: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Theory Physical Description

Physical Description

When a photon enters a beam splitter (BS), there are two possibilities: it will either betransmitted or reflected

1 2

3

Consider two photons, one in each input mode of a 50:50 BS. There are four possibilitiesfor the photons to behave:

1 2 3 4

Page 6: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Theory Physical Description

The state of the system after interference is given by a superposition of all possibilitiesfor the photons to pass through the BS:

1 2 3 4

Where does the minus sign come from?

Reflection off bottom side =⇒ relative phase shift of π(i.e. reflection off the higher index medium)

Reflection off top side =⇒ no phase shift(i.e. reflection off the lower index medium).

Page 7: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Theory Physical Description

Now let’s assume that the two photons are identical in their physical properties (i.e.,polarization, spatio-temporal mode structure, and frequency):

1 2 3 4

cancelled out

The photons will always exit the same (but random) output port!

Page 8: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Theory Mathematical Description

Mathematical Description

Consider two identical photons input ports A and B of 50:50 BS

Quantum state given by applying photon-creation operators a†A,B on vacuum state |0〉

|Ψi 〉 = |1A1B〉 = a†Aa†B |0〉

Detection of photon in output port C or D evaluated by applyingphoton-annihilation operators aC ,D to |Ψi 〉Effect of 50:50 BS described by unitary transformation(

aAaB

)→ 1√

2

(1 11 −1

)(aCaD

)

Page 9: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Theory Mathematical Description

Initial state in terms of photons created in output ports:

|1A1B〉 = a†Aa†B |0〉

= 12(a†C +a†D)(a

†C −a†D) |0〉

= 12(a†2C −a†2D ) |0〉

= 1√2(|2C0D〉− |0C2D〉)

Remark: Since the operators a†C ,D act on different output ports their commutator

vanishes: [a†C ,a†D ] = 0

The photons will always exit the same (but random) output port!

Page 10: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Outline

1 Theory

2 Hong-Ou-Mandel Interference [Hong, Ou & Mandel (1987)]

3 Quantum Beat of Two Single Photons [Legero et al. (2004)]

4 Summary

Page 11: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Hong-Ou-Mandel Interference

Hong-Ou-Mandel Interference

Aim: Measurement of photon wave packet length δt

Problem: Time resolution � wave packet length

Page 12: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Hong-Ou-Mandel Interference Experimental setup

Experimental setup

UVKDP

UV3Filter

BS

M1

M2

IF1

IF2

D2

D1

Amp.&

Disc.

Amp.&

Disc.

Counter

Counter

CoincidenceCounter

PDP11/23+ω0

ω1

ω2

Photon source: Parametric down-conversion Type I (ω0 = ω1 +ω2)

Indistinguishability of photons:

Photon source ensures same polarization

Pass bands of interference filters (IF) ensure (at most) same frequency

BS is displaced (by ±cδτ) from symmetry position to ensure same time of arrival

Measurement of simultaneous detections N depending on BS displacement δτ

Page 13: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Hong-Ou-Mandel Interference Results

Hong-Ou-Mandel-Dip

Overlap controlled by BS displacement:no coincidences for perfect overlap

Photon length from FWHM = 16 µm = c · 50 fs

Width of dip connected toIF-bandwidth due to Fourier-limitedphotons

Page 14: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Outline

1 Theory

2 Hong-Ou-Mandel Interference [Hong, Ou & Mandel (1987)]

3 Quantum Beat of Two Single Photons [Legero et al. (2004)]

4 Summary

Page 15: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons

Quantum Beat of Two Single Photons

Now: Time resolution � wave packet lengthPhotons impinge simultaneous on BSDetection-time delay τ measured

1

2 3

4

Page 16: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Experimental Setup

Experimental Setup

Photon source: Atom-cavity system

Atom-cavity system emitsunpolarized single photons withwell determined frequency

Polarizing beam splitter (PBS)randomly directs them along twopaths

Photon travelling along path Agets delayed through fiber

Subsequent photon travelling along path B impinges on the BS simultaneously withdelayed photon from path A

A HWP along path B guarantees that both photons have the same polarization

Page 17: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Results 1

Results 1

Indistinguishable photons for parallel polarization: Correlation drops to minimumwith width of 460 ns

Width corresponds to inhomogenous broadening of photon spectrum ofδω/2π = 690 kHz

In general depth and width of minimum indicate initial purity of photon state, i.e.broadening of photon spectrum, frequency and emission-time jitter

Page 18: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Jitters

Jitters

No perfect single-photon source =⇒ Consider more realistic scenario, in which stream ofFourier-limited photons shows a jitter in its parameters:

Frequency jitter: Emission-time jitter:

expected shapereal shape

Page 19: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Results 2

Results 2

Introducing a frequency difference of ∆ω/2π = 3 MHz: Quantum beat visible

No coincidences for simultaneous detection

Fringes with a visibility of almost 100% (classical: 50%)

Page 20: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Theoretical Prediction of Quantum Beat

Theoretical Prediction of Quantum Beat

Initial state: |Ψi 〉 = |1A1B〉Detection of photon at time t0 in output C :

|ΨC (t0)〉 = aC |1A1B〉 = 1√2(|1A0B〉+ |0A1B〉)

Introduce frequency difference ∆ = ω2 −ω1. After time τ new state reads:

|ΨC (t0 + τ)〉 = 1√2(eiω1τ |1A0B〉+ eiω2τ |0A1B〉)

Introduce global phase |ΨC 〉 −→ e−iω1τ |ΨC 〉:

=⇒ |ΨC (t0 + τ)〉 = 1√2(|1A0B〉+ ei∆·τ |0A1B〉)

Probability of detecting the second photon with either C or D:

PCC = 〈ΨC | a†C aC |ΨC 〉 = 1

2[1 + cos(∆ · τ)]

PCD = 〈ΨC | a†D aC |ΨC 〉 = 1

2[1 − cos(∆ · τ)]

A frequency difference results in a quantum-beat signal

Page 21: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Theoretical Prediction of Quantum Beat

perpendicular polarization:no interference

parallel polarization:interference

interference of photons withdifferent frequencies

no simultaneous detections(τ = 0) for δτ = 0

even for distinguishablephotons no simultaneous

detections (τ = 0)

oscillation of coincidenceprobability for δτ ' 0

Comparison to HOM-Experiment:

No time resolved measurement =⇒ Integrate over detection-time difference τ:No difference between cases 1 and 3

Page 22: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Quantum Beat of Two Single Photons Visibility

Visibility

Classical fields: 50% visibility

QM two-photon intereference: 100% visibility

Fourier-limited photons Inhomogenous broadened photons

Page 23: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Outline

1 Theory

2 Hong-Ou-Mandel Interference [Hong, Ou & Mandel (1987)]

3 Quantum Beat of Two Single Photons [Legero et al. (2004)]

4 Summary

Page 24: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Summary

Summary

Hong-Ou-Mandel InterferencePhoton source: PDC

No time-resolved measurement possible

Width of dip connected to length of photon wave packet

Page 25: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Summary

Two-Photon InterferencePhoton source: Single Photon Emitter

Identical Frequencies

Quantum Beat

Time-resolved measurement possible

Photon source can be characterized from width and depth of dip (i.e. broadening ofphoton spectrum, frequency and emission-time jitter)

Frequency difference between photons induces quantum-beat with visibility of 100%

Page 26: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

Thank you for your attention

and I look forward for your questions on this topic!

Page 27: Markus Perner - Max Planck Society · Initial state in terms of photons created in output ports: j1 A1 Bi = ^a y A a^ y B j0i = 1 2 (a^y C +a^ y D)(a^ y-a^y)j0i = 1 2 (a^y2 C-a^ y2

References

C.K. Hong, Z.Y. Ou, L. Mandel: Phys. Rev. Lett. 59, 2044 (1987)http://prl.aps.org/abstract/PRL/v59/i18/p2044_1

T. Legero, T. Wilk, M. Hennrich, G. Rempe, A. Kuhn: Phys. Rev. Lett. 93, 070503(2004) http://prl.aps.org/abstract/PRL/v93/i7/e070503

T. Legero, T. Wilk, A. Kuhn, G. Rempe: Appl. Phys. B 77, 797-802 (2003)http://link.springer.com/article/10.1007%2Fs00340-003-1337-x

T. Legero, T. Wilk, A. Kuhn, G. Rempe: Adv. At. Mol. Opt. Phys. 53, 253 (2006)http://arxiv.org/abs/quant-ph/0512023

T. Legero: Ph.D. Thesis, Technical University Munich (2005)http://mediatum2.ub.tum.de/node?id=603072

H.P. Specht, J. Bochmann, M. Mucke, B.Weber, E. Figuerora, D.L. Moehring, G.Rempe: Nat. Photon. 3 Iss. 8, 469-472 (2009) http:

//www.nature.com/nphoton/journal/v3/n8/full/nphoton.2009.115.html

http://prl.aps.org/abstract/PRL/v59/i18/p2044_1
http://prl.aps.org/abstract/PRL/v93/i7/e070503
http://link.springer.com/article/10.1007%2Fs00340-003-1337-x
http://arxiv.org/abs/quant-ph/0512023
http://mediatum2.ub.tum.de/node?id=603072
http://www.nature.com/nphoton/journal/v3/n8/full/nphoton.2009.115.html
http://www.nature.com/nphoton/journal/v3/n8/full/nphoton.2009.115.html

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