Quantum States of Light

110 Pages·2015·11.09 MB·English

by  Akira Furusawa (auth.)

SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 10
Akira Furusawa
Quantum States
of Light
123
SpringerBriefs in Mathematical Physics
Volume 10
Series editors
Nathanaël Berestycki, Cambridge, UK
Mihalis Dafermos, Cambridge, UK
Tohru Eguchi, Tokyo, Japan
Atsuo Kuniba, Tokyo, Japan
Matilde Marcolli, Pasadena, USA
Bruno Nachtergaele, Davis, USA
More information about this series at http://www.springer.com/series/11953
Akira Furusawa
Quantum States of Light
123
AkiraFurusawa
Department ofApplied Physics
TheUniversity of Tokyo
Tokyo
Japan
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
ISBN978-4-431-55958-0 ISBN978-4-431-55960-3 (eBook)
DOI 10.1007/978-4-431-55960-3
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Preface
We learn about the properties of quantized optical fields in quantum optics.
Althoughthismaysoundoldandtraditional,itisnot.Inreality,weassumedoptical
fields as classical fields until very recently. We did not have to quantize the field
because our light source was only a laser and whose state, a coherent state, can be
regarded as a classical field.
Wehavetousequantumopticsnowadays,ofcourse.Itisbecausesqueezedlight
is easily created these days and we have to handle it. Squeezed light is a pure
quantum mechanical state, which cannot be described without quantum optics. In
thatsense,the“phasetransition”occurredwhenSlusheretal.createdthesqueezed
light for the very first time in 1985. After the “phase transition,” various “pure”
quantum states were created, which include superposition of a vacuum and a
single-photon state, a Schrödinger’s cat state, and so on.
In this book, we explain the definition and the way to create these “modern”
quantum states of light. For that purpose we use many figures to visualize the
quantum states to help the readers’ understanding, because the quantum states
sometimes look very counterintuitive when one only looks at equations.
Tokyo Akira Furusawa
v
Acknowledgments
All the experimental results presented in this book come from experiments carried
out by the members of Furusawa group at the Department of Applied Physics of
The University of Tokyo, including Nobuyuki Takei, Hidehiro Yonezawa, Yuishi
Takeno, Jun-ichi Yoshikawa, Noriyuki Lee, Mitsuyoshi Yukawa, Yoshichika
Miwa, Kazunori Miyata, and Maria Fuwa. The author would like to thank all
membersofFurusawagroup.TheauthoralsoacknowledgesIlianHäggmarkforthe
language review.
This book was originally written in Japanese and published by Uchida
Roukakuho in 2013.
Tokyo Akira Furusawa
vii
Contents
1 Quantum States of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Quantization of Optical Fields. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Coherent States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Balanced Homodyne Measurement. . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Beam Splitters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Balanced Homodyne Measurement. . . . . . . . . . . . . . . . . 13
1.3.3 Eigenstates of Quadrature Amplitude Operators
and Marginal Distributions . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Single-Photon States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Marginal Distribution of a Single-Photon State . . . . . . . . 21
1.5 Photon-Number States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Superposition States of a Vacuum and a Single-Photon State . . . . 26
1.7 Coherent States and Schrödinger Cat States . . . . . . . . . . . . . . . . 29
1.8 The Wigner Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.9 Superposition States of a Vacuum and a Two-Photon State. . . . . . 45
1.10 Squeezed States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.11 Squeezing Operation and Squeezed States. . . . . . . . . . . . . . . . . . 52
1.12 Quantum Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Creation of Quantum States of Light . . . . . . . . . . . . . . . . . . . . . . . 69
2.1 Creation of Coherent States of Light . . . . . . . . . . . . . . . . . . . . . 69
2.2 Creation of a Squeezed Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 74
2.3 Creation of a Single-Photon State . . . . . . . . . . . . . . . . . . . . . . . 80
2.4 Creation of a Minus Cat State. . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.5 Creation of a Superposition of Photon-Number States . . . . . . . . . 86
2.6 Creation of Quantum Entanglement . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
ix
Chapter 1
Quantum States of Light
1.1 Quantization ofOpticalFields
Inthissection,wepresentanintuitivedescriptionofquantumoptics.
Accordingtothequantumfieldtheoryavectorpotentialoperatorofopticalfields
Aˆ(r,t)canbedescribedas
Aˆ(r,t)=A ei(k·r−ωt)aˆ +A∗e−i(k·r−ωt)aˆ†. (1.1)
0 0
Herethespatialmodeisaplanewavewhichpropagatesinthedirectionofthewave
vectork,A denotesacomplexvectorpotentialorthogonaltothewavevector,ωis
0
theangularfrequencyoftheopticalfield,aˆ andaˆ† aretheannihilationandcreation
operators,respectively,andnˆ =aˆ†aˆ isthenumberoperator.
aˆ andaˆ†actoneigenstatesofthenumberoperator(Fockstates)|n(cid:3)(nˆ|n(cid:3)=n|n(cid:3))
as
√
aˆ|n(cid:3)= n|n−1(cid:3),
√
aˆ†|n(cid:3)= n+1|n+1(cid:3), (1.2)
nˆ|n(cid:3)=n|n(cid:3).
We can create an electrical-field operator Eˆ(r,t) and a magnetic-flux density
operatorBˆ(r,t)oftheopticalfieldbyusingthefollowingequationsandEq.(1.1).
Namely,byusing
∂A(r,t)
E(r,t)=− ,
∂t (1.3)
B(r,t)=∇×A(r,t)
andEq.(1.1),wecanget
©TheAuthor(s)2015 1
A.Furusawa,QuantumStatesofLight,SpringerBriefs
inMathematicalPhysics,DOI10.1007/978-4-431-55960-3_1
2 1 QuantumStatesofLight
Eˆ(r,t)=iω(A ei(k·r−ωt)aˆ −A∗e−i(k·r−ωt)aˆ†), (1.4)
0 0
Bˆ(r,t)=ik×(A ei(k·r−ωt)aˆ −A∗e−i(k·r−ωt)aˆ†). (1.5)
0 0
Moreover,fromk·A =0and|A |=A ,wecangettheHamiltonianHˆ which
0 0 0
correspondstothefieldenergyas
(cid:2) (cid:3) (cid:4)
Hˆ = 1(cid:4) Eˆ(r,t)·Eˆ(r,t)+ 1 Bˆ(r,t)·Bˆ(r,t) dr
2 0 2μ
(cid:3) (cid:4)(cid:2) 0
1 1
= (cid:4) ω2+ |k|2 A2dr(aˆaˆ†+aˆ†aˆ)
2 0 μ 0
0
(cid:2)ω
= (aˆaˆ†+aˆ†aˆ)
2 (cid:3) (cid:4)
1
=(cid:2)ω nˆ + , (1.6)
2
w(cid:5)here(cid:4)0 isthepermittiv(cid:5)ityofvacuum,μ0 isthemagneticpermeabilityofvacuum,
A2dr=(cid:2)/2(cid:4) ω,and e±2ik·rdr=0.
0 0
Now let the optical field operators evolve in time. When the Hamiltonian does
notchangeintime,theHeisenbergequationofmotionofanoperatorAˆ(t)becomes
i(cid:2)dAˆ(t) =[Aˆ(t),Hˆ]. (1.7)
dt
Byusingthisequation,wecanget
Aˆ(t)=eiH(cid:2)ˆtAˆ(0)e−iH(cid:2)ˆt. (1.8)
So the time evolution of an electrical-field operator Eˆ(r,t) of an optical field
shouldobey
Eˆ(r,t)=eiH(cid:2)ˆtEˆ(r,0)e−iH(cid:2)ˆt. (1.9)
By using Eqs.(1.6) and (1.9) we can check Eq.(1.4) from the view point of time
evolutionofoperators.Notethatweusedthefollowingequationhere:
ei(cid:2)ω(nˆ(cid:2)+1/2)taˆe−i(cid:2)ω(nˆ(cid:2)+1/2)t =aˆe−iωt. (1.10)
Similarly we can check Eq.(1.5) for the magnetic-flux density operator Bˆ(r,t)
ofanopticalfieldfromtheviewpointoftimeevolutionwithEqs.(1.6)and(1.9).
Althougheverythingis“peacefulandquiet”sofar,wedomoreinquantumoptics.
Inquantumopticswethinkthatanannihilationoperatoraˆ evolvesaccordingto
Eq.(1.10). Namely we set aˆ(t)=aˆe−iωt. It is a misunderstanding in some sense,
because an annihilation operator is a field operator and should not evolve in time.
However,ifwesetitlikethis,itbecomesveryconvenient.Soinquantumopticswe