The Magic U box; Bimodal Hong-Ou-Mandel Interference in an InterferometricOptical System

Deepika Sundarraman, Cody C. LearyPhysics Department, The College of Wooster, Wooster, Ohio 44691, USA

(Dated: July 20, 2012)

We investigate interference of light in both classical and quantum optics. We discuss a generalapproach to predicting bimodal Hong Ou Mandel Interference (HOMI) in interferometric opticalsystems, which have applications to quantum information processing. This two-photon interferenceeffect was found in two of the investigated optical systems and the output quantum state wascontrasted with the corresponding classical prediction. A Mach Zehnder Interferometer with anextra mirror and a relative phase shift exhibits HOMI for identical input photons whereas a Sagnacinterferometer exhibits HOMI for non-identical input photons.

I. INTRODUCTION

One of the phenomenon of linear optics useful for quan-tum information processing is Hong Ou Mandel Interfer-ence. Hong, Ou and Mandel proposed this theory in 1987that involves two photon interference in a beam splitterresulting in both photons exiting out of the same outputport of the beam splitter by virtue of quantum indistin-guishability. This theory can be exploited in more com-plex optical systems involving interferometers. The the-oretical models of classical situations such as laser beamswere studied by us in the first stages of this project. Thenthe knowledge of quantum optics is applied to the clas-sical theory for the same optical system to build up thequantum case for photons. The results from various endsof our predictions draw a comparison between the out-puts in the case of ramming classical beams versus pho-tons which helps us gather intuition for their behavior.Here we investigated two interferometers, the Sagnac

and the Mach Zehnder for conditions of HOMI. The pre-dictions made through this are a result of both manualcomputation and Mathematica, both of which have beenuseful to us at various stages of development of our re-sults.

II. THEORY

Consider a general optical system comprising two inputports a and b each of which has a certain photonic stateor mode given by

cosθ

2| 0+ e

iΦ sinθ

2| 1, (1)

where the state | 0 is the even Hermite Gaussian modeFig.1 and the | 1 is the odd Hermite Gaussian mode Fig.2. These two modes are described by their electrical fieldintensity plots such that they form an orthogonal basisfor a qubit system in quantum computing.This system can be conveniently described using a

Bloch sphere Fig. 3 in such a manner that every pointon the sphere is a representation of Eq.1 or a mode with

FIG. 1: The | 0 state commonly referred to as the ‘even’mode.

FIG. 2: The | 1 state commonly referred to as the ‘odd’mode.

a certain value of θ and Φ. The Bloch sphere is an or-thogonal system(the even-odd basis) and the superposi-tion of these two independent modes can create variousmodes at different points on the Bloch sphere. Some ofthe modes on the equator of the Bloch sphere have givenunique results in our predictions and thus are importantto understand Fig. 3.

FIG. 3: The Bloch sphere showing the orthogonal basis withthe | 0 and | 1 state with the other significant modes on theequator of the sphere.

2

A. General Approach

The input ports a and b are traced through a setupwith an interferometer such that the inputs and the out-puts are given using linear optics

ec0

ec1

ed0

ed1

= U

ea0

ea1

eb0

eb1

, (2)

where the ec0 depicts the electric field out of port ‘c’ inthe | 0 or even state, ea1 depicts the electric field intoport ‘a’ in the | 1 or odd state and thus ec1, ed0, ed1and ea0, eb0 ,eb1 can be understood respectively. Theconjugate transpose of the same leads us to obtainingthe input fields in terms of the outputs

ea0

ea1

eb0

eb1

= U †

ec0

ec1

ed0

ed1

. (3)

Inverting the matrix is essential to carrying out all thequantum computations that are addressed later.

The quantization of the field leads us with the following

a†0

a†1

b†0

b†1

= U†

c†0

c†1

d†0

d†1

, (4)

where a†0 is the operator that creates a photon in port

a of the ‘even’ kind and similar interpretations can bemade regarding a

†1, b

†0, b

†1, c

†0, c

†1, d

†0 and d

†1 respectively.

The operators a† and b

† can be introduced here as alinear combination of the even and odd creation operatorsgiven by

a† = a0a

†0 + a1a

†1, (5)

and

b† = b0b

†0 + b1b

†1, (6)

where a0 = cos θA, b0 = cos θB , a1 = eıΦA sin θA and

b1 = eıΦB sin θB are the coefficients from Eq.1. The cre-

ation operator creates one photon in the specific port forexample

d†0 | vacc0 c1 d0 d1 = d

†0 | 0000c0 c1 d0 d1 =| 0010c0 c1 d0 d1

(7)

B. Understanding the U matrix

The U matrix can be computed depending on the var-ious elements Fig. 4 of the interferometric system. In ageneral way

U =

σz i

i σz

U+ 00 U−

σz i

i σz

, (8)

where σz is the Pauli’s matrix

1 00 −1

, i translates to

a matrix

i 00 i

and U+ and U− are determined by the

extra optical elements in the system discussed in the nextsection [1]. On computation of Eq.8 matrix we get

U =

σzU+σz − U− i(σzU+ + U−σz)

i(U+σz + σzU−) −(U+ − σzU−σz)

. (9)

FIG. 4: The linear optics matrices for the various opticalelements of our interferometers

C. U+ and U−

The U+ and U− matrices depend on the ‘special’ ele-ments we add to the interferometer to control the inputsat various stages.

1. Mach Zehnder Interferometer

The glass plate and an extra mirror are the key ele-ments introduced in the arms of the Mach Zehnder in-terferometer Fig. 5. Physically twisting the glass platehas a direct dependence on φ which changes the phasedifference between the beams through ports c and d, Theglass plate is thus represented by a pure phase,

eiφ 00 1

= e

iφ2

eiφ2 0

0 e−iφ

2

, (10)

where eiφ2 and e

−iφ2 are multiples of the identity matrix

each being a 2× 2 matrix.

φ± = eiφ2

e±iφ

2 0

0 e±iφ

2

. (11)

3

FIG. 5: The Mach Zehnder Interferometer with an extra mir-ror and glassplate.

FIG. 6: The parity flips caused by mirrors.

The extra mirror only causes a parity flip in one armFig. 6 and thus,

σz

2 00 σz

, (12)

reducing to1 00 σz

. (13)

The Mach Zehnder interferometer with a glass plate hasa simple U± associated with it that is given by,

U+ = eiφ2

eiφ2 0

0 eiφ2

1 00 1

. (14)

= eiφ2

eiφ2 0

0 eiφ2

(15)

and

U− = eiφ2

e−iφ

2 0

0 e−iφ

2

σz 00 σz

. (16)

= eiφ2

e−iφ

2 σz 0

0 e−iφ

2 σz

. (17)

2. Sagnac Interferometer

The Sagnac interferometer Fig. 7 has a simple rotationmatrix and by function it rotates one arm by θ and theother by −θ,

FIG. 7: The Sagnac Interferometer with an rotator and glass-plate.

R(θ) 00 R(−θ)

, (18)

where R(θ) =

cos θ − sin θsin θ cos θ

. In addition it also has a

glass plate given as above that can be visualized as in theearlier case Eq.11 The Sagnac has thus a more complexmatrix,

U± = e±iφ

2

cosΩ ∓ sinΩ± sinΩ cosΩ

. (19)

D. U†; Computing the inputs

As in 11 we note that following the computations of theU matrix, our next step is to find its transpose conjugateU† to be able to use it for all the quantum computations.Hence for the Mach Zehnder we have the following matrixwhich is given in its transpose conjugate Eq.21.

U = eiφ2

i sin φ2 0 −i cos φ

2 00 cos φ

2 0 − sin φ2

−i cos φ2 0 −i sin φ

2 00 − sin φ

2 0 − cos φ2

, (20)

4

U† = e−iφ

2

sin φ2 0 cos φ

2 00 cos φ

2 0 − sin φ2 0

cos φ2 0 − sin φ

2 00 − sin φ

2 0 − cos φ2 0

, (21)

Additionally, for computations and simplicity we can alsorewrite the above switching basis,

a†0

b†0

a†1

b†1

= U†

c†0

d†0

c†1

d†1

, (22)

U† =

i sin φ2 −i cos φ

2 0 0−i cos φ

2 −i sin φ2 0 0

0 0 cos φ2 − sin φ

2

0 0 − sin φ2 − cos φ

2

, (23)

where the symbols have their usual meanings.

For the Sagnac we have

U = eiφ2

cosΩ sin φ2 sinΩ sin φ

2 cosΩ cos φ2 − sinΩ cos φ

2

− sinΩ sin φ2 − cosΩ sin φ

2 − sinΩ cos φ2 − cosΩ cos φ

2

cosΩ cos φ2 sinΩ cos φ

2 − cosΩ sin φ2 sinΩ sin φ

2

sinΩ cos φ2 − cosΩ cos φ

2 − sinΩ sin φ2 − cosΩ sin φ

2

, (24)

and thus

U† = e−iφ

2

cosΩ sin φ2 − sinΩ sin φ

2 cosΩ cos φ2 sinΩ cos φ

2

sinΩ sin φ2 − cosΩ sin φ

2 sinΩ cos φ2 − cosΩ cos φ

2

cosΩ cos φ2 − sinΩ− cos φ

2 − cosΩ sin φ2 − sinΩ sin φ

2

− sinΩ cos φ2 − cosΩ cos φ

2 sinΩ sin φ2 − cosΩ sin φ

2

. (25)

In a special scenario when φ = π2 and changing the basis

as shown earlier Eq.22 we get a matrix with symmetryas

U† =

cosΩ cosΩ sinΩ − sinΩcosΩ − cosΩ sinΩ sinΩ− sinΩ − sinΩ cosΩ − cosΩsinΩ − sinΩ − cosΩ − cosΩ

(26)

E. Mach Zehnder with Glass Plate

To find the conditions for Hong-Ou-Mandel Interfer-ence we performed computations using the U† to findthe outputs in terms of the inputs. Thus, we use 23 toget four consequent linear equations.

a†0 = i(c†0 sin

φ

2− d

†0 cos

φ

2) (27)

b†0 = −i(c†0 cos

φ

2+ d

†0 sin

φ

2) (28)

a†1 = c

†1 cos

φ

2− d

†1 sin

φ

2(29)

b†1 = −(c†1 sin

φ

2+ d

†1 cos

φ

2) (30)

Now assuming two photons enter, one from each porta and b.

a†b† | vac = (a0a

†0 + a1a

†1)(b0b

†0 + b1b

†1) (31)

Replacing the operators a†0, a†1, b

†0 and b

†1 with the out-

put operators and the coefficients a0, b0, a1, b1 using 5and 6 we get an output state that resembles:

5

| ψ = 1

2cos

θA

2cos

θB

2sinφ[| 2000− | 0002]− 1

2sin

θA

2sin

θB

2sinφei(ΦA+ΦB)[| 0200− | 0020]

− i(sinθA

2cos

θB

2cos2

φ

2eiΦA + cos

θA

2sin2

φ

2sin

θB

2eiΦB ) | 1100

+ (cosθA

2sin

θB

2cos2

φ

2eiΦB + sin

θA

2cos

θB

2sin2

φ

2eiΦA) | 0011

− cosφ(cosθA

2cos

θB

2| 1010+ sin

θA

2sin

θB

2ei(ΦA+ΦB)) | 0101

+ i sinφ(sinθA

2cos

θB

2eiΦA − cos

θA

2sin

θB

2eıΦB )[| 1001− | 0110]

(32)

From the above should see that there are six two-photonstates that are indicative of HOMI. These are |1100,|0011, |0200, |0020, |0002 and |2000. Thus we wantto find conditions that can eliminate the other states.In doing so we find that HOMI occurs only under thefollowing conditions

φ =π

2,

θA = θB ≡ θo,

ΦA = ΦB ≡ Φo,

for which we get an output wavefunction as

| ψ = 1

2cos2

θ

2[| 2000− | 0002]− 1

2sin2

θ

2e2iΦ[| 0200− | 0020]− i

2sin θeiΦ[| 1100+ | 0011] (33)

FIG. 8: The classical laser beam case for the diagonal-anti-diagonal mode basis for comparison.

The equation Eq.33 tells us that in every case thatthe given conditions are true we always get two pho-tons exiting from the same port and each event has anequal probability of occurring. We also investigate thesame in the diagonal-antidiagonal mode basis Fig.9 tounderstand and compare our outputs from the classi-cal laser beam case Fig. 8 that we obtained by cre-ating a program in Mathematica titled ‘1dinterferom-eter2inputsblochspheres’. [3] This is important in notonly helping us gather intuition to review the case for

slamming photons instead of laser beams but also as itcan give us an idea of the differences in behavior of lightbeams and photons.

FIG. 9: The diagonal-antidiagonal mode basis.

The conversion from even-odd to the diagonal basisinvolves,

c†a

c†dd†a

d†d

=

1 1 0 01 −1 0 00 0 1 10 0 1 −1

c†0

c†1

d†0

d†1

, (34)

6

and thus we obtain a simpler relation for the followingconditions

Φ =π

2,

θo =π

2,

ψ =1

2[| 2000cacddadd− | 0002cacddadd ]. (35)

Thus each state | 2000cacddadd and | 0002cacddadd has aprobability of 50%.

The same result for

Φ = −π

2, ,

and the same values of θ we get

ψ =1

2[| 0200cacddadd− | 0020cacddadd ]. (36)

This result can be easily interpreted in the Fig. 10 andcompared with the classical result we obtained for thesame inputs. Similarly while switching to a circular mode

FIG. 10: The quantum case for the diagonal-anti-diagonalmode basis.

basis as shown in Fig. 11 we obtain another useful result.To do so we use the following relationship

c†L

c†R

d†L

d†R

=

1 1 0 0i −i 0 00 0 1 10 0 i −i

c†0

c†1

d†0

d†1

, (37)

Under special conditions, we can use the equations above

FIG. 11: The right-left handed circular mode basis for com-parison.

to simplify further as done in the diagonal mode basis.For

Φ = 0,

θo =π

2,

ψ =1

2[| 2000cLcRdLdR− | 0002cLcRdLdR ] (38)

This can be also clearly understood from Fig. 12. Simi-larly we find another similar condition i.e.

Φ = π,

ψ =1

2[| 0200cLcRdLdR− | 0020cLcRdLdR ] (39)

The classical comparison for the same case also results in

FIG. 12: The quantum case for the right-left handed circularmode basis for comparison.

getting a donut LG+10 mode out of the C port only Fig.

13. These four results Eq.35, Eq.36 and Eq.38, Eq.39yield us an interpretation of the symmetry between thediagonal modes and the circular modes such that in thecase of Hong Ou Mandel interference each gives rise tothe other.

7

FIG. 13: The classical case for the right-left handed circularmode basis for comparison.

1. Interpreting the output states

To understand the output states from the circularmode basis we interpret the outputs in terms of the in-puts of the circular mode basis.

e−iφ

2

√2

ec0

ed0

ec1

ed1

=

−i i 0 0i i 0 00 0 1 −10 0 −1 −1

ea0

eb0

ea1

eb1

(40)

Thus from the above we get the following equations

ec0 =ie

−iφ2

√2

(ea0 + eb0) (41)

ec1 =e−iφ

2

√2(ea1 − eb1) (42)

ed0 =ie

−iφ2

√2

(ea0 − eb0) (43)

ed1 = −e−iφ

2

√2(ea1 + eb1) (44)

Since for HOMI θA = θB ≡ θo and ΦA = ΦB ≡ Φo

ec = ec0 | 0+ ec1 | 1 (45)

From Eq. 1 we thus have,

ec = 0 (46)Next we determined the output electric field for the ed

port using,

ed = ed0 | 0+ ed1 | 1. (47)

The output field ed is given by,

√2(i cos

θo

2| 0 − sin

θo

2eiΦo | 1). (48)

Thus,

i

√2(cos

θo

2| 0+ i sin

θo

2eiΦo | 1). (49)

Thus, when θ = π2 and φ = 0, our output becomes,

ed =1√2| 0+ i | 1, (50)

which is the LG+10 mode.

The other cases can also be studied in a similar mannerand the output states can be determined from our inputs.

F. Sagnac with Rotator

Using Eq.25 we obtain the following four equations.

a†0 = i sin

φ

2cosΩ c

†0 − cos

φ

2sinΩ c

†1 + i cos

φ

2cosΩ d

†0 − sin

φ

2sinΩ d

†1, (51)

a†1 = cos

φ

2sinΩ c

†0 + i sin

φ

2cosΩ c

†1 − sin

φ

2sinΩ d

†0 − cos

φ

2cosΩ d

†1, (52)

b†0 = i cos

φ

2cosΩ c

†0 − sin

φ

2sinΩ c

†1 − i sin

φ

2cosΩ d

†0 + cos

φ

2sinΩ d

†1, (53)

b†1 = − sin

φ

2sinΩ c

†0 − i cos

φ

2cosΩ c

†1 − cos

φ

2sinΩ d

†0 − i sin

φ

2cosΩ d

†1. (54)

At this stage I used a Mathematica program ‘Glass-plateandrotator’ [3] to carry out the computations as our

answer for the output was more complex than the Mach

8

Zehnder case. Again replacing the input creation op-erators with the output operators we get a result .seeappendix. that simplifies for one of the investigated con-ditions for Hong-Ou-Mandel Interference

ΦA = ΦB + π ≡ Φo,

θA = θB ≡ θo,

Ω =π

4,

φ =π

2.

The resultant two photon state after simplification isgiven by

| ψ =1

4[

−e

2iΦo sin2θo

2+

1

2eiΦo sin θo − cos2

θo

2

|2000

+

e2iΦo sin2

θo

2+

1

2eiΦo sin θo + cos2

θo

2

|0002

+

e2iΦo sin2

θo

2− 1

2eiΦo sin θo + cos2

θo

2

|0020

+

−e

2iΦo sin2θo

2− 1

2eiΦo sin θo − cos2

θo

2

|0200

−cos2

θo

2− e

2iΦo sin2θo

2

(|1100+ |0011)]

(55)

Thus we obtain a two photon state with equal probabil-ities of |1100, |0011, |0200,|0020, |0002 and |2000events.

III. RESULTS AND CONCLUSIONS

We obtain two photon interference in the Mach Zehn-der Interferometer when φ = π

2 and the modes that are

fired into the interferometer are identical. The two out-put photons in this case either have the probability to exitout of ports c or d such that states |0200 and |0020, and|0002 and |2000 are quantum entangled states.When the input modes are diagonal-antidiagonal

(HG+45

10 and HG−45

10 ) the output modes are either theLG

+10 or the LG

−10 donut modes. Similarly when the

LG+1

0 or the LG−1

0 modes are fired in then either the

HG+45

10 or the HG−45

10 exit out of the ports. There isthus an inherent symmetry between these modes. Thephoton case was also compared to the classical modescase when the same two modes are sent into the inter-ferometer but now in the form of classical laser beamsthat give out an output from only one port, Fig. 8 ,specifically the even port of the interferometer.In the next investigated case of the Sagnac we found

that one of the conditions for bimodal HOMI is when φ =π2 and Ω = π

4 , the inputs are both on the same latitudeof the sphere but on opposite ends( ΦA = ΦB + π).Thus the inputs are non-identical for HOMI in com-

parison to the Mach Zehnder case where they had to beidentical.This proves to us that different inputs can create

HOMI for indistinguishability in the outputs of the pho-ton state versus in the inputs.

IV. APPLICATIONS

The applications of HOMI span across various disci-plines. In molecular biology it can be used for the vari-able control over the spatial mode profile of optical tweez-ers, which have been used to impart orbital angular mo-mentum to microscopic particles, and probe the mechan-ics of RNA transcription.[2]. The investigated two pho-ton entangled states can be used for enhanced resolutionof light-sensitive microscopic objects via quantum imag-ing techniques.The two photon HOMI state is essential for quantum

information processing as discussed earlier and ties thefields of quantum information and optics.

[1] R. Loudon, The Quantum Theory of Light(Oxford SciencePublishing), 89, (2000).

[2] Abbondanzieri, E.A. et al., Nature 438, 460-465, (2005).

[3] Dropbox/REU/Deepika Sundarraman