WHAT IS A PHOTON

1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ ΓΓΡΑΦ ΊΣ πδεκδθωθά ηααέ ηαμ ΢χ δεΪ η ηΪμ ΨΆΞΙΜΟ Πκζδ δεά Απκλλά κυ Φυ δεά ωηα δ έωθ: Θ ωλέα ωθ χκλ υθ εαδ β φτ β βμ πλαΰηα δεσ β Ϊμ ηαμ Μ Πλαΰηα δεσ β α πΫλα απσ βθ τζβ · 31 httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html ε ηίλέκυ 2017 1ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ SHARE Labels Astronomy Big Bang cosmology dimensions nature Physics space Γα α, α α α α φυ α απ υ α π α α π α υ α απ α π α α π α α π α απ αυ υ π π α α υ . α φυ . string theory universe α α υπ α α απ υ υ α ππ υ α. httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html βή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ πα απ υ υ π υ α α α π αυ α. Κ α α , α απ π π α πα α α υ π , υ α φυ φα υ π α α αα α α α, π π υ υ α α π υ απ υ α υ π υ α α, α π α α α υ httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html α υ φ φυ α α. Ό α π π υ π π . α ,π π α α α α α υ φ υ α απ υ α υ . Αυ απ υ π φ α α απ α απ υ π α α α ππ υ π υπ α α απ υ υ π υ απ α υ α α π α α ππ φυ π α π π α α φυ υ α φα υ π α α α . φ π υ . γή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ α α α α υ α α α π π α α φ α φ α α α α. υ, α π απ , α α π , π . α α α α α π φα α π α φα α α α α α π υ α α π αυ αα α υ υ α απ αυ υ . αυ α α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html υ α υ . α απ α π α α. υ α α α . α π φυ π α α α α αα φ π υ α π α α α α α α φα α π υ π υ α υ α π υα π υ α υ π π υ π α α υ π α π απ υ α π α υ π α α απ υα φυ υ υ π α α α. υ α α π , π υ α α υ α, α υ . 4ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ Ό α αφ απ απ α π α Garry McKinnon π α α π α , π α α α α π υ , απ α. α φα α, π α , α απ α απ αυ α π υ α α φ α α υ α α π α π υ π α πα α .Κ π α α αυ π α α α α π α α Gary McKinnon α α α π α φ α α π α. Α φ α α, α α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html υ π α α υ αφ α α π α α Α α α α. π α α φυ π Leumeria, π π π π α υ φυ αυ υ φυ υπ υ 5ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ α . Απ α α υ φ π υ π υ π , α φ υ απ υ α α π υ απ υπ α ), α, , υ ,α α αυ α α α π π α α, 'αυ π α α π α α α π υ α α π υ α απ π φ π α α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html α ,α π α α α υπ α π α α υ α α , π υ α α φ φυ φυ α, υ π π α α π φ υ φυ α αα φυ υ α Γ π ,( υ υ υ α φ α. α. υ α α α φ α . π π α π α π υ , π υ π αυ υ α υ , φυ αφυ Φα π α α α α π φυ α π υ α α α π α. υ αυ α .Α υ 6ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ α α α φ . π α α α απ φυ π π α ,α π α α π α π α α α α α π φ α π υ αυ α .Α φυ π α α ,α π α α α α. α α α υ υ φυ υ υ α, α α α α 'αυ α φ απ φυ α .Α . π π π α α υ π α α α, httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html υ α α α α Θεωρία γρα α α υπ α αυ α π α α α α π π φ απ π π α α α απ α υ α. υ 'αυ α α α α φ α, π α α α α . ώ ιή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ Απ π υ π υ, π α Η α απ υπ π ππ , α υ π α α α α α α α π πα αυ φα π υ υ π α α α απ . αφ α π υ φ φ α, ,α α α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html π υ , α . α α φυ α . π υ α , π α υ π α π α α υ α α α. π α απ α Αυ φ φυ α α α. α πα α α α α απ αυ απ α.Η υ απ υ αυ α απ α π α υ . υ α α απ α π υ α απ α π α α φυ π α π α α υ α. απ π α κή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ π υ α Αυ π υ υ υ α (α π α α) π υ υ π α α π φ υ π , α α; υ υ π α φυ α ,α α υ , , α α α α υ αφ α α π . Απ υ α α π αφ α π υ πα α π α υ α α α α φα π αυ Αυ υ φ α α π α φ α α. π α α απ φυ α α υ α υπ π α π α α πα α, α π φ υ α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html α αυ υ . α υπ α πα α α, π α π α . α, α π α 9ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ υ α φ υ απ υ φ υ π .Η α α , α φ υ π υ α α α. α π π υ π . απ , α υ α ; α π υ α α α α υ α α ,π α υ . α απα α π α απ αυ π υ π υ υ α Η α. φ π α α ,α αα α α απ υπ απ α α απ αφ α α απ α υπ π α π .Η α π π υ α α υ π α α α υ υ α α α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html π α υ α υ α φυ α υ , α 1ίή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ π α α α, π π α α υ π αυ απ α α α Α υ υ π α α α π α υ α α απ α. υ α α α α. φυ α α υπ π α α α α , α π α α π α α υ . α α α α υ π υπ . φα α α, α υ α υ π υ α αφ φυ π υ α υ ; α . υ π α . υ πα υ υ α α . Κα π π α α πα α υπ π υ αυ π α httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html α α υ αυ α α υ α α π α α 11ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ υ α . Αυ π π α α α α α α π υ α. The video below shows that scientists are still trying to determine weather or not it would be physically possible to go through a wormhole. At the same time, I believe our black budget world has already developed the technology to do so. Like I mentioned earlier, this is why I believe there is a group on the planet that operates at a completely different level of understanding about science and technology, while the mainstream world is left to ponder what is already known. httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 1βή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ Michio Kaku Explains String Theory Sources: http://www.damtp.cam.ac.uk/research/gr/public/qg_ss .html httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 1γή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ http://plus.maths.org/content/string-theory-newtoneinstein-and-beyond http://www.dummies.com/how-to/content/the-basicelements-of-string-theory.html ΣΙΚΈΣ ΢: Α΢ΣΡΟΝΟΜ Α , BIG BANG , ΚΟ΢ΜΟΛΟΓ Α , ΣΗ΢ Φ ΢Η΢ , ΦΤ΢ΙΚ ΢ , Χ ΡΟ , Θ ΩΡ Α ΣΩΝ ΧΟΡ ΙΑ΢ΣΆ΢ Ι΢ , Ν , ΣΟ ΢ ΜΠΑΝ Μ ΡΊ ΙΟ ΢χσζδα Enter your comment... ΠλΫπ δ θα δαίΪ httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 14ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ έθαδ αζζκ απσμ πκυ πλκ παγ έ θα πδεκδθωθά δ η β Γβ; Πβΰά ωθ ζ υ αέωθ «αζζκ απυθ» βηΪ ωθ Ϋχ δ απκηαελτθ δ κυμ α λκθσηκυμ Μ Πλαΰηα δεσ β α πΫλα απσ βθ τζβ · 27 ε ηίλέκυ 2017 Μ ΡΊ ΙΟ Ι ΗΜΟ΢Ί Τ΢Η ΢ΧΟΛΊΟΤ Ά΢Σ Π ΡΙ΢΢ΌΣ Ρ Οδ πδ άηκθ μ απκεαζτπ κυθ πλυ μ httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 15ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ παλα βλβ δεΫμ απκ έι δμ σ δ κ τηπαθ ηαμ ά αθ εΪπκ Ϋθα κζσΰλαηηα Μ Πλαΰηα δεσ β α πΫλα απσ βθ τζβ · 26 ε ηίλέκυ 2017 Μ ΡΊ ΙΟ Ι 1 ΢ΧΌΛΙΟ Ά΢Σ Π ΡΙ΢΢ΌΣ Ρ Σκ ζδεσ σλδκ αχτ β αμ εαδ β πέ λα β κυ χλσθκυ Μ Πλαΰηα δεσ β α πΫλα απσ βθ τζβ · 04 Ικυζέκυ 2017 Μ ΡΊ ΙΟ Ι ΗΜΟ΢Ί Τ΢Η ΢ΧΟΛΊΟΤ Ά΢Σ Π ΡΙ΢΢ΌΣ Ρ httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 16ή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ Οδ φυ δεκέ Ϋζθκυθ ωηα έ δα φω σμ κ παλ ζγσθ, απκ δεθτκθ αμ σ δ κ αιέ δ κ χλσθκ έθαδ φδε σ! Μ Πλαΰηα δεσ β α πΫλα απσ βθ τζβ · 30 ε ηίλέκυ 2017 Μ ΡΊ ΙΟ Ι ΗΜΟ΢Ί Τ΢Η ΢ΧΟΛΊΟΤ Ά΢Σ Π ΡΙ΢΢ΌΣ Ρ Ο πδ άηκθαμ Brian Cox ζΫ δ σ δ ΰθωλέα δ ΰδα έ θ httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 1ιή1κ 1ή1ήβί1κ Φυ δεά ωηα δ έωθμ Θ ωλέα ωθ χολ υθ εαδ η φτ η ημ πλαΰηα δεσ η άμ ηαμ ίλάεαη αζζκ απκτμ Μ Πλαΰηα δεσ β α πΫλα απσ βθ τζβ · 28 ε ηίλέκυ 2017 Μ ΡΊ ΙΟ Ι 3 ΢ΧΌΛΙ Ά΢Σ Π ΡΙ΢΢ΌΣ Ρ Φοί λσ SPACE Powered by Blogger Θέματα από enot-poloskun httpμήήwww.realitybeyondmatter.comήβί1ιή1βήparticle-physics-string-theory-and.html 1κή1κ 

31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications λγλο α α α πυ π υ , Μα γαέομ . Broome, Casey R. Myers, Andrew G. White & Timothy C. Ralph Martin Ringbauer Nature Communications 5 , αλδγησμ αλξ έου: 4145 Λάοβ: 07 Νο ηβλέου 2013 πο (2014) Δβηο δ υηΫθο online: 19 Ιουθέου 2014 doi : 10.1038 / ncomms5145 Κα βΪ ε σ: 16 Μα ου 2014 βθ παλαποηπά Θ πλβ δεά φυ δεά Αφηρη η υ α π φυ , αυ α - απα α α π α. Η αυ α α α α α υ π φ υ αυ α απ υπ π υ, α π υ αφ υ α, α υ α α α httpsμήήwwwέnatureέcomήarticlesήncommsη14η α α α α π π π α α α υ υ α α α α υ α α π ξσζδα , α α α υ α πα α α . π π απ α α α .Ε qubit π υ α α α α α α α απ υ π υ α α α ο νέο χέ ιο. φά υ , π , πα π α υ Βλέπ α , αφ φ α υΑ . υ α π υ α πα α π υ α υ timelike. Αυ πα α φα α πα πα α π π α απ α φ π α α π απ . α α α απ υ α, π υ αυ 1ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications ππ απ α π α α α α π α α α απ φ , υ αα . Ει αγωγ Μαπ υ υ α π υ α π π α π υ π π υ π φ υ φα α υ π π α π α α α α α π π π α απ 8 α πα α πα α πα α α α Deutsch CTCs υ π , υ , α α υπ υ π 7 πα αφ 1991 α α α CTC. υ CTCs υπ α α, π α, α π α α υ υ π α φ α . Επ , πα α πα απ π α π α π υ 7 , 15 π υ π φ πα α α α πα α α υ α απ υ α π α α υ 9 α α απα α α π υ υ 10 , 11 α α α α NP-π π αφ α, α, π π α υ Heisenberg 12 . α π π α , α α α π υ π α α φα α. α απ υ α απ . υ α υ π υ α- α π υ π υ α υΑ α. π Η α α α π υ π π υ Hawking 5 , π , - υπ α α αφ α α α 1,2,3 υπ υ υ υ παππ α υ Deutsch 6 α (CTCs) - α υ Novikov 4 υ π υ α α απ αυ α π πα α α α α π - π αυ α υ φυ α α α α α φ α π α α CTC υ φα α α α π α , υ α υ π 13 , 14 . αυ , α υ π . α α α π υ πα httpsμήήwwwέnatureέcomήarticlesήncommsη14η π υ π π φ π α π υ π φ α α 2ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications υ .Α απ 16 , 17 , 18 α 20 , 21 , 22 , 23 π π φα υ α π π α α α α α π α . Αυ αφ α α π π φ υ α α φα υ α α υ α φυ υ α φ αυ - υ π , Απο ε ο ο α α α α υ α φ α α π π α α α | ψ >, α α qubit CTC- υ Deutsch υ6. απ υπ Bacon 8 α υ α υ π π U υ π α απ φ υ υ ,π απ α π π α απ αφ απ υ υ π υ . υ Brun et al. 9 υπ υα υ π π απ υπ , α α απ υα .7 α α π υπ α π α , α α α υ π π α α α α α 19 , φ αυ α α α α α υ π υ υα υ α π α α α α απ Deutsch CTC. υ π υ α, α απ CTC π υ π υ α traversing qubit φ α α αυ α α φ α α υ α α υ . . α α ο Deutsch While there has been some recent success on alternative models of CTCs based on post-selection23,24,25, we focus on the most prominent model for describing quantum mechanics in the presence of CTCs, introduced by Deutsch6. Here a quantum state |ψLj interacts unitarily with an older version of itself (Fig. 1). With the inclusion of an additional swap gate, this can equivalently be treated as a two-qubit system, where a chronology-respecting qubit interacts with a qubit ρCTC trapped in a CTC. The quantum state of ρCTC in this picture is determined by Deutsch’s consistency relation: httpsμήήwwwέnatureέcomήarticlesήncommsη14η 3ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Figure 1: Model of a quantum state |ψ 〉 interacting with an older version of itself. This situation can equivalently be interpreted as a chronology-respecting qubit interacting with a qubit trapped in a CTC. The CTC in general consists of a causal worldline with its past and future ends connected via a wormhole (indicated by black triangles). where U′ is the unitary U followed by a SWAP gate (Fig. 1). This condition ensures physical consistency—in the sense that the quantum state may not change inside the wormhole—and gives rise to the nonlinear evolution of the quantum state | ψLj. The state after this evolution is consequently given by ρOUT=Tr2[U′(|ψLjLJψ| ⊗ρCTC)U′†]. The illustration in Fig. 1 further shows that the requirement of physical consistency forces ρCTC to adapt to any changes in the surroundings, such as a different interaction unitary U or input state |ψLj. While equation (1) is formulated in terms of a pure input state |ψLj, it can be directly generalized to mixed inputs7. Simulating CTCs Our experimental simulation of a qubit in the (pure) state |ψLj traversing a CTC relies on the circuit diagram shown in Fig. 2a). A combination of single-qubit unitary gates before and after a controlled-Z gate allows for the implementation of a large set of controlled-unitary gates U. Using polarization-encoded single photons, arbitrary single-qubit unitaries can be realized using a combination of quarter-wave (QWP) and half-wave plates (HWP); additional SWAP gates before or after U are implemented as a physical mode swap. The controlled-Z gate is based httpsμήήwwwέnatureέcomήarticlesήncommsη14η 4ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications on non-classical (Hong-Ou-Mandel) interference of two single photons at a single partially polarizing beam splitter (PPBS) that has different transmittivities V=1/3 for vertical (V) and H=1 for horizontal (H) polarization26—a more detailed description of the implementation of the gate can be found in ref. 27. Conditioned on post-selection, it induces a π phase shift when the two interfering single-photon modes are vertically polarized, such that |VVLj →−|VVLj with respect to all other input states. Figure 2: Experimental details. (a) The circuit diagram for a general unitary interaction U between the state |ψLj and the CTC system. (b) The specific choice of unitary in the demonstration of the (i) nonlinear evolution and (ii) perfect discrimination of non-orthogonal states. (c) Experimental setup for case (ii). Two single photons, generated via spontaneous parametric down-conversion in a nonlinear β-barium-borate crystal, are coupled into two optical fibres (FC) and injected into the optical circuit. Arbitrary polarization states are prepared using a Glan–Taylor polarizer (POL), a quarter-wave (QWP) and a half-wave plate (HWP). Nonclassical interference occurs at the central partially polarizing beam splitter (PPBS) with reflectivities H=0 and V=2/3. Two avalanche photo-diodes (APDs) detect the single photons at the outputs. The states |ΨLj are chosen in the x–z plane of the Bloch sphere, parameterized by ϕ, and CUxz is the corresponding controlled unitary, characterized by the angle xz. The SWAP gate was realized via relabelling of the input modes. httpsμήήwwwέnatureέcomήarticlesήncommsη14η ηή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications One of the key features of a CTC is the inherently nonlinear evolution that a quantum state |ψLj undergoes when traversing it. This is a result of Deutsch’s consistency relation, which makes ρCTC dependent on the input state |ψLj. In order to simulate this nonlinear behavior using linear quantum mechanics we make use of the effective nonlinearity obtained from feeding extra information into the system. In our case we use the classical information about the preparation of the state |ψLj and the unitary U to prepare the CTC qubit in the appropriate state ρCTC as required by the consistency relation equation (1). After the evolution we perform full quantum state tomography on the CTC qubit in order to verify that the consistency relation is satisfied. Nonlinear evolution As a first experiment we investigate the nonlinearity by considering a Deutsch CTC with a CNOT interaction followed by a SWAP gate as illustrated in Fig. 2b(i). This circuit is well known for the specific form of nonlinear evolution: which has been shown to have important implications for complexity theory, allowing for the solution of NP-complete problems with polynomial resources8. According to Deutsch’s consistency relation (equation (1)) the state of the CTC qubit for this interaction is given by We investigate the nonlinear behaviour experimentally for 14 different quantum states , with and a variety of phases φ∈{0, 2π}, where the locally available information ϕ and φ is used to prepare ρCTC. In standard (linear) quantum mechanics no unitary evolution can introduce additional distinguishability between quantum states. To illustrate the nonlinearity in the system we thus employ two different distinguishability measures: the trace distance , where , and a single projective measurement with outcomes ‘+’ and ‘−’: httpsμήήwwwέnatureέcomήarticlesήncommsη14η θή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications While the metric is a commonly used distance measure, it does not have an operational interpretation and requires full quantum state tomography in order to be calculated experimentally. The measure in contrast is easily understood as the probability of obtaining different outcomes in minimum-error discrimination of the two states using a single projective measurement on each system. The operational interpretation and significance of is discussed in more detail in the Supplementary Note 1. Both and are calculated between the state |ψLj and the fixed reference state |HLj after being evolved through the circuit shown in Fig. 2b(i). The results are plotted and compared to standard quantum mechanics in Fig. 3. If the state |ψLj is not known then, based only on the knowledge of the reference state |HLj and the evolution in equation (2) it is natural and optimal to use the measure with a σz-measurement. httpsμήήwwwέnatureέcomήarticlesήncommsη14η ιή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Figure 3: Nonlinear evolution in a Deutsch CTC with SWAP.CNOT interaction. Both the trace distance and the σz-based distinguishability measure (equal to within experimental error in this case) of the evolved states ρOUT a er the interaction with the CTC are shown as yellow diamonds. The blue circles (red squares) represent the measure () between the input states |ψLj and |HLj in the case of standard quantum mechanics. Note that due to the phase independence of the evolution in equation (2), states that only di er by a phase collapse to a single data point. Crucially, the metric does not capture the e ect of the nonlinearity, while does, indicated by the red-shaded region. Error bars obtained from a Monte Carlo routine simulating the Poissonian counting statistics are too small to be visible on the scale of this plot. Inset: The dashed black lines with decreasing thickness represent theoretical expectations for and from 2, 3, 4 and 5 iterations of the circuit. We observe enhanced distinguishability for all states with an initial trace distance to |HLj smaller than (that is, ), as clearly demonstrated by the σz-based measure, see Fig. 3. Note, however, that this advantage over standard quantum mechanics is not captured by the metric (ρ1, ρ2) unless the nonlinearity is amplified by iterating the circuit on the respective output at least three times, see the inset of Fig. 3. This shows that the nonlinearity is not directly related to the distance between two quantum states. By testing states with various polar angles for each azimuthal angle on the Bloch sphere, we confirm that any phase information is erased during the evolution and that the httpsμήήwwwέnatureέcomήarticlesήncommsη14η κή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications evolved state ρOUT is indeed independent of φ, up to experimental error. We further confirm, with an average quantum state fidelity of F=0.998(2) between the input and output states of ρCTC in equation (3), that the consistency relation (1) is satisfied for all tested scenarios. Non-orthogonal state discrimination While it is the crucial feature, nonlinear state evolution is not unique to the SWAP.CNOT interaction, but rather a central property of all non-trivial CTC interactions. Similar circuits have been found to allow for perfect distinguishability of non-orthogonal quantum states9, leading to discomforting possibilities such as breaking of quantum cryptography9, perfect cloning of quantum states10,11 and violation of Heisenberg’s uncertainty principle12. In particular it has been shown that a set of N distinct quantum states in a space of dimension N can be perfectly distinguished using an N-dimensional CTC system. The algorithm proposed by Brun et al.9 relies on an initial SWAP operation between the input and the CTC system, followed by a series of controlled unitary operations, transforming the input states to an orthogonal set, which can then be distinguished. In our simulation of this effect we consider the qubit case N=2, which consequently would require two controlled unitary operations between the input state and the CTC system. We note, however, that without loss of generality the set of states to be discriminated can be rotated to the x–z plane of the Bloch sphere, such that |ψ0Lj=|HLj and for some angle ϕ. In this case, the first controlled unitary is the identity operation , while the second performs a controlled rotation of |ψ1Lj to |VLj as illustrated in Fig. 4a). In detail, the gate CUxz applies a π rotation to the target qubit conditional on the state of the control qubit, about an axis in the x–z plane defined by the angle For the optimal choice xz. the gate rotates the state |ψ1Lj to |VLj, orthogonal to |ψ0Lj, enabling perfect distinguishability by means of a projective σz measurement (see Fig. 4a). httpsμήήwwwέnatureέcomήarticlesήncommsη14η λή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Figure 4: Bloch-sphere evolution of states traversing a CTC. In the case of (a) local state preparation, the state |ψ0Lj=|HLj (blue) is una ected by CUxz, while |Ψ1Lj (green) undergoes a π rotation about the axis defined by xz. The axis is chosen as such that , which can then be perfectly distinguished from |ψ0Lj. (b) For non-local preparation of the initial states and the same choice of xz the controlled unitary maps both initial states to the maximally mixed state . The probability of distinguishing the two states is therefore 1/2—as good as randomly guessing. In practice the gate CUxz is decomposed into a controlled-Z gate between appropriate single-qubit rotations, defining the axis xz. The latter are realized by half-wave plates before and after the PPBS, set to an angle of xz/4 with respect to their optic axis (see Fig. 2c): httpsμήήwwwέnatureέcomήarticlesήncommsη14η 1ίή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Note that relation (1) requires that ρCTC=|HLjLJH|, whenever the input state is |HLj, independent of the gate CUxz. Crucially, this consistency relation ensures that any physical CTC system adapts to changes in ϕ and xz, parameterizing the input state and gate, respectively. In our simulation these two parameters are used to prepare the corresponding state ρCTC, as shown in Fig. 2c. In a valid experimental simulation the input and output states ρCTC have to match, that is, ρCTC has to satisfy relation (1). This has been verified for all following experiments with an average quantum state fidelity of =0.996(7). In the experiment, we prepared near-pure quantum states directly on single photons using a Glan–Taylor polarizer followed by a combination of a HWP and a QWP. We simulated CTC-aided perfect discrimination of non-orthogonal states for 32 distinct quantum states |ψ1Lj with ϕ∈[0, 2π). For each state we implemented CUxz with the optimal choice of . Furthermore, we tested the ability of this system to distinguish the set {|ψ0Lj, |ψ1Lj} given nonoptimal combinations of ϕ and xz. For this we either chose ϕ=3π/2 and varied the gate over the full range of , or chose CUxz as a controlled Hadamard (optimal for ϕ=3π/2) and varied the state |ψ1Lj over the full range of ϕ∈[0, 2π). The output state is characterized by quantum state tomography, which provides sufficient data to obtain for arbitrary measurement directions as well as for the calculation of the trace distance. Figure 5a illustrates the observed distinguishability for the above experiments and compares it to the expectation from standard quantum mechanics. In the latter case the measure is maximized by choosing the optimal projective measurement, based on the available information about the states |ψ0Lj and |ψ1Lj. Crucially, the optimized L is directly related to the trace distance as and therefore captures the same qualitative picture, without the requirement for full quantum state tomography. In the CTC case a σz-measurement is chosen, which is optimal when based on the knowledge of . Otherwise further optimization is possible xz (see Supplementary Note 2 and Supplementary Fig. 1 for more details). Furthermore, we note that the above scenario can also be httpsμήήwwwέnatureέcomήarticlesήncommsη14η 11ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications interpreted in a state-identification rather than state-discrimination picture, which is discussed in more detail in Supplementary Note 3 and Supplementary Fig. 2. Figure 5: Experimental results. Probability of state discrimination for (a) locally prepared and (b) non-locally prepared states |ψ0Lj =|HLj and as measured by . The surface represents the theoretically predicted probability depending on the state and gate parameters ϕ and xz, respectively. Solid, red (open, blue) data points indicate better (worse) performance than standard quantum mechanics. (c) Cross-sectional views of the combined plots a and b reveal the rich structure in the dependencies on the initial parameters for (top) a fixed state (ϕ=3π/2) and (bottom) a fixed gate ( xz=π/4). Here red squares (yellow diamonds) correspond to the CTC case with local (non-local) preparation and blue circles represent standard quantum mechanics. Error bars obtained from a Monte Carlo routine simulating the Poissonian counting statistics are too small to be visible on the scale of this plot. Local versus non-local state preparation Owing to the inherent nonlinearity in our simulated system, care must be taken when describing mixed input states ρin. In particular a distinction between proper and improper mixtures can arise, which is unobservable in standard (linear) quantum mechanics28. This ambiguity is resolved in ref. 7 by requiring the consistency condition to act shot-by-shot—that is, independently in every run of the experiment—on the reduced density operator of the input state. For proper mixtures this means that ρin is always taken as a pure state, albeit a different one shot-by-shot. For improper mixtures in contrast, ρin will always be mixed. A similar, but much more subtle and fascinating feature that has received less httpsμήήwwwέnatureέcomήarticlesήncommsη14η 12ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications attention in the literature so far occurs with respect to preparation of pure states29. While in standard quantum mechanics a pure state prepared directly (locally) on a single qubit is equivalent to one that has been prepared non-locally through space-like separated post-selection of an entangled resource state, this is not true under the influence of a CTC. The origin of this effect is not the nonlinear evolution, but rather the local absence of classical information about the post-selection outcome. The role of locally available classical information in entanglement-based preparation schemes is a matter of current debate and still to be clarified. A possible resource state for alternatively preparing |ψ0Lj and |ψ1Lj could be of the form , where projection of the first qubit onto the state |0Lj and |1Lj leaves the second qubit in the state |ψ0Lj and |ψ1Lj, respectively. From the point of view of ρCTC, however, there exists no information about the outcome of this projective measurement. Hence it ‘sees’ and adapts to the mixed state . The state of the CTC qubit is therefore different for local and non-local preparation. If this was not the case, it would enable superluminal signalling, which is in conflict with relativity29. Figure 4b) illustrates the evolution induced by CUxz, when the input states |HLj and |ψ1Lj are prepared using an entangled resource | Lj, rather than directly. The results of the previously discussed distinguishability experiments for this case are shown in Fig. 5b). In Fig. 5c) they are compared to the case of local preparation and to standard quantum mechanics for a fixed input state and a fixed gate, respectively. Again, consistency of our simulation is ensured by a quantum state fidelity of =0.9996(3) between the input and output states of ρCTC. In our simulation we find that the CTC system can indeed achieve perfect distinguishability of the (directly prepared) states |ψ0Lj and |ψ1Lj even for arbitrarily close states if the appropriate gate is implemented (see Fig. 5a). Furthermore, we show that the advantage over standard quantum mechanics persists for a wide range of non-optimal gate–state combinations, outside of which, however, the CTC system performs worse (blue points). Notably, we find that for non-locally prepared input states CTC-assisted state discrimination httpsμήήwwwέnatureέcomήarticlesήncommsη14η 13ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications never performs better than random guessing—a probability of (as shown in Fig. 5b). The predictions for standard quantum mechanics, in contrast are independent of the way the states |ψ0Lj and |ψ1Lj are prepared. Decoherence We further investigated the effect of two important decoherence mechanisms on the simulated CTC system (shown in Fig. 2a). The first is a single-qubit depolarizing channel acting on the input state |ψLj, which can be modelled as where (σx, σy, σz) are the three Pauli matrices and p∈[0, 1] quantifies the amount of decoherence. The second form of decoherence concerns the controlled unitary CUxz and is described as where ε∈[0, 1] is the probability of the gate to fail, describing the amount of decoherence that is present. For ε=0 the gate acts as an ideal controlled rotation CUxz, while it performs the identity operation for ε=1. We tested the robustness of the state-discrimination circuit in Fig. 2b(ii) against both forms of decoherence. For this test we chose CUxz as a controlled Hadamard (that is, xz=π/4) and the initial states |ψ0Lj=|HLj and (that is, ϕ=3π/2). Fig. 6 shows the distinguishability of the evolved states as a function of both decoherence mechanisms over the whole range of parameters p∈[0, 1] and ε∈[0, 1]. Note that the decoherence channel in equation (7) does not have an analogue in the standard quantum mechanics case (that is, without a CTC); hence only the channel in equation (6) is considered for comparison. It is further naturally assumed that the experimenter has no knowledge of the specific details of the decoherence and therefore implements the optimal measurements for the decoherence-free case. The physical validity httpsμήήwwwέnatureέcomήarticlesήncommsη14η 14ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications of the simulation is ensured by the consistency of ρCTC across the boundary of the wormhole with an average fidelity of =0.997(4). Figure 6: State discrimination as a function of gate and qubit decoherence for locally prepared states. Here ε quantifies the decoherence of the unitary interaction CUxz (with xz=π/4), which has no analogue in the standard quantum mechanics case, and p the single-qubit depolarization of the input qubits |HLj and |ψ1Lj (with ϕ=3π/2). The system demonstrates robustness against both forms of decoherence and the CTC advantage persists up to and , respectively. The semi-transparent blue surface represents the optimum in standard quantum mechanics. Error bars obtained from a Monte Carlo routine simulating the Poissonian counting statistics are too small to be visible on the scale of this plot. It is worth noting that the interpretation of decoherence effects in the circuit in Fig. 2a) is very different from the linear scenario without a CTC. In the case of httpsμήήwwwέnatureέcomήarticlesήncommsη14η 1ηή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications single-qubit depolarization the initially pure input state becomes mixed. In contrast to the linear case now an important distinction has to be made with respect to the origin of the decoherence. If it results from an interaction with the environment, which is the case considered here, then ρCTC ‘sees’ an improper mixture and adjusts to the mixed density matrix of the input state. If, however, the origin of the mixture is classical fluctuations in the preparation apparatus, then shot-by-shot pure states enter the circuit and the consistency relation holds accordingly shot-by-shot, resulting in a proper mixture at the output. This suggests that in the presence of a CTC it would be possible to identify the origin of the decoherence in an experimental setup. Furthermore, careful analysis of the decoherence of the unitary gate CUxz reveals parallels to the effects seen in non-local state preparation. The decoherence is assumed to arise from non-local coupling to the environment. Again, due to a lack of classical knowledge of the outcome of an eventual measurement of the environment, ρCTC ‘sees’ the mixed process in equation (7) in every run of the experiment. In the case of full decoherence the distinguishability is reduced to 0.5 as in standard quantum mechanics. The differences between local and nonlocal decoherence in their interpretation and effect is one of the key insights from our simulation. Discussion Quantum simulation is a versatile and powerful tool for investigating quantum systems that are hard or even impossible to access in practice20. Although no CTCs have been discovered to date, quantum simulation nonetheless enables us to study their unique properties and behaviour. Here we simulated the immediate adaption of ρCTC to changes in the CTC’s environment and in particular the effect of different forms of decoherence. We also show that the nonlinearity inherent in the system is in fact not uniform (as shown in Fig. 3), suggesting that nonlinear effects only become apparent in certain scenarios and for a specific set of measurements. httpsμήήwwwέnatureέcomήarticlesήncommsη14η 1θή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Moreover, we find intriguing differences with respect to nominally equivalent ways of pure state preparation. Although acknowledged in ref. 29, this feature has not been further investigated in the present literature. Importantly, this effect arises due to consistency with relativity, in contrast to the similar effect for mixed quantum states discussed earlier, which is a direct result of the nonlinearity of the system7. Our study of the Deutsch model provides insights into the role of causal structures and nonlinearities in quantum mechanics, which is essential for an eventual reconciliation with general relativity. Additional information How to cite this article: Ringbauer, M. et al. Experimental simulation of closed timelike curves. Nat. Commun. 5:4145 doi: 10.1038/ncomms5145 (2014). References 1. Morris, M. S., Thorne, K. S. & Yurtsever, U. Wormholes, time machines, and the weak energy condition. Phys. Rev. Lett. 61, 1446–1449 (1988). 2. Morris, M. S. & Thorne, K. S. Wormholes in spacetime and their use for interstellar travel: a tool for teaching general relativity. Am. J. Phys. 56, 395– 412 (1988). 3. Gott, J. R. Closed timelike curves produced by pairs of moving cosmic strings: exact solutions. Phys. Rev. Lett. 66, 1126–1129 (1991). 4. Novikov, I. Evolution of the Universe Cambridge University Press: Cambridge, England, (1983). httpsμήήwwwέnatureέcomήarticlesήncommsη14η 1ιή22 31ή12ή2ί1ι 5. ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Hawing, S. W. Chronology protection conjecture. Phys. Rev. D 46, 603–611 (1992). 6. Deutsch, D. Quantum mechanics near closed timelike lines. Phys. Rev. D 44, 3197–3217 (1991). 7. Ralph, T. C. & Myers, C. R. Information flow of quantum states interacting with closed timelike curves. Phys. Rev. A 82, 062330 (2010). 8. Bacon, D. Quantum computational complexity in the presence of closed timelike curves. Phys. Rev. A 70, 032309 (2004). 9. Brun, T. A., Harrington, J. & Wilde, M. M. Localized closed timelike curves can perfectly distinguish quantum states. Phys. Rev. Lett. 102, 210402 (2009). 10. Ahn, D., Myers, C. R., Ralph, T. C. & Mann, R. B. Quantum state cloning in the presence of a closed timelike curve. Phys. Rev. A 88, 022332 (2013). 11. Brun, T. A., Wilde, M. M. & Winter, A. Quantum state cloning using Deutschian closed timelike curves. Phys. Rev. Lett. 111, 190401 (2013). 12. Pienaar, J. L., Ralph, T. C. & Myers, C. R. Open timelike curves violate Heisenberg's uncertainty principle. Phys. Rev. Lett. 110, 060501 (2013). 13. Bennett, C., Leung, D., Smith, G. & Smolin, J. A. Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems? Phys. Rev. Lett. 103, 170502 (2009). 14. Kłobus, W., Grudka, A. & Wójcik, A. Comment on ‘Information flow of quantum states interacting with closed timelike curves’. Phys. Rev. A 84, 056301 (2011). 15. Ralph, T. C. & Myers, C. R. Reply to ‘Comment on ‘Information flow of quantum states interacting with closed timelike curves’’. Phys. Rev. A 84, 056302 (2011). httpsμήήwwwέnatureέcomήarticlesήncommsη14η 1κή22 31ή12ή2ί1ι 16. ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Kitagawa, T. et al. Observation of topologically protected bound states in photonic quantum walks. Nature Commun. 3, 882 (2012). 17. Ma, X.-S. et al. Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems. Nat. Phys. 7, 399–405 (2011). 18. Simon, J. et al. Quantum simulation of antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011). 19. Gerritsma, R. et al. Quantum simulation of the Dirac equation. Nature 463, 68–71 (2010). 20. Casanova, J. et al. Quantum simulation of the Majorana Equation and unphysical operations. Phys. Rev. X 1, 021018 (2011). 21. Philbin, T. G. et al. Fiber-optical analog of the event horizon. Science 319, 1367–1370 (2008). 22. Menicucci, N. C., Jay Olson, S. & Milburn, G. J. Simulating quantum effects of cosmological expansion using a static ion trap. New J. Phys. 12, 095019 (2010). 23. Lloyd, S. et al. Closed timelike curves via postselection: theory and experimental test of consistency. Phys. Rev. Lett. 106, 040403 (2011). 24. Lloyd, S., Maccone, L., Garcia-Patron, R., Giovannetti, V. & Shikano, Y. Quantum mechanics of time travel through post-selected teleportation. Phys. Rev. D 84, 025007 (2011). 25. Brun, T. A. & Wilde, M. M. Perfect state distinguishability and computational speedups with postselected closed timelike curves. Found. Phys. 42, 341–361 (2011). 26. Ralph, T. C., Langford, N., Bell, T. & White, A. G. Linear optical controlledNOT gate in the coincidence basis. Phys. Rev. A 65, 062324 (2002). httpsμήήwwwέnatureέcomήarticlesήncommsη14η 1λή22 31ή12ή2ί1ι 27. ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications Langford, N. K. et al. Demonstration of a simple entangling optical gate and its use in bell-state analysis. Phys. Rev. Lett. 95, 210504 (2005). 28. d'Espagnat, B. Conceptual Foundations of Quantum Mechanics 2nd edn Addison Wesley (1976). 29. Cavalcanti, E. G., Menicucci, N. C. & Pienaar, J. L. The preparation problem in nonlinear extensions of quantum theory, Preprint at http://arxiv.org/abs/1206.2725 (2012). Acknowledgements We thank Nathan Walk and Nicolas Menicucci for insightful discussions. We acknowledge financial support from the ARC Centres of Excellence for Engineered Quantum Systems (CE110001013) and Quantum Computation and Communication Technology (CE110001027). A.G.W. and T.C.R. acknowledge support from a UQ Vice-Chancellor's Senior Research Fellowship. Author information Affiliations Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia Martin Ringbauer, Matthew A. Broome, Casey R. Myers & Andrew G. White Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia Martin Ringbauer, Matthew A. Broome, Andrew G. White & Timothy C. Ralph Contributions M.R., M.A.B., C.R.M. and T.C.R. developed the concepts, designed the experiment, analysed the results and wrote the paper. M.R. performed the experiments and httpsμήήwwwέnatureέcomήarticlesήncommsη14η 2ίή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications analysed the data. T.C.R. and A.G.W. supervised the project and edited the manuscript. Competing interests The authors declare no competing financial interests. Corresponding author Correspondence to Martin Ringbauer. Supplementary information PDF files 1. Supplementary Information υ π α α α 1-2, υ π α 1-3 α υ π α α απ π χό ια π α α , υ φ α .Ε α , πα α α φ α α α Επ ο νωνίες φύσης ISSN 2041-1723 ( httpsμήήwwwέnatureέcomήarticlesήncommsη14η α υ υ α α απ υγ έαμ τθ υ Ό φ α υ υ α υ . β) 21ή22 31ή12ή2ί1ι ExperimentalΝsimulationΝofΝclosedΝtimelikeΝcurvesΝ|ΝσatureΝωommunications © 2017 Macmillan Publishers Limited, ηΫλομ ου Springer Nature. Οζα α δεαδυηα α δα βλοτθ αδ. httpsμήήwwwέnatureέcomήarticlesήncommsη14η 22ή22 

WHAT IS A PHOTON? Spontaneous emission: The need for quantum field theory In these notes I would like to try and give an introduction to the quantum mechanical theory of the photon. The treatment I give is in the spirit of a treatment you can find in Dirac’s quantum mechanics monograph, The Principles of Quantum Mechanics. I believe that Dirac was one of the first (if not the first) person to work out these ideas. Along the way, we will be generalizing the way we use quantum mechanics in a non-trivial way. More precisely, the way we model nature using the rules of quantum mechanics will change significantly, although the rules themselves will not actually change. Let us begin by setting the stage for this generalization. Consider the well-known processes of emission and absorption of photons by atoms. The processes of emission and absorption of photons by atoms cannot, ultimately, be accommodated in the usual quantum mechanical models based on particle mechanics. Instead, one must use a new class of models that go under the heading of quantum field theory. The reasons for this necessity are relatively simple if one focuses on spontaneous emission in atoms. This is where an atom in an excited state will spontaneously decay to a lower energy state (and emit one or more photons). First, we all know that the electron states we use to characterize atoms are the stationary states, which are energy eigenstates. But stationary states have the property that all their observable features are time independent. If an atomic electron occupying an atomic energy level were truly in a stationary state there could never be any spontaneous emission since a stationary state has no time dependent behavior. The only way out of this conundrum is to suppose that atomic energy levels are not really stationary states once you take into account the interaction of photons and electrons. But now consider emission of a photon by an atomic electron. The initial state of the system has an electron. The final state of the system has an electron and a photon. Now, in the usual quantum mechanical formalism for particles the number of particles is always fixed. Indeed, the normalization of the wave function for a particle (or for particles) can be viewed as saying that the particle (or particles) is (or are) always somewhere. Evidently, such a state of affairs will not allow us to treat a particle such as a photon that can appear and disappear. Moreover, it is possible to have atomic transitions in which more than one photon appears/disappears. Clearly we will not be able to describe such processes using the quantum mechanical models developed thus far. If this isn’t surprising enough, I remind you that there exist situations in which a photon may “transform” into an electron-positron pair and, conversely, in which an electron-positron can turn into a photon. So even electrons are not immune from the appearance/disappearance phenomenon. 1 Remarkably, it is possible to describe these multi-particle processes using the axioms of quantum theory, provided these axioms are used in a clever enough way. This new and improved use of quantum mechanics is usually called quantum field theory since it can be viewed as an application of the basic axioms of quantum mechanics to continuous systems (“field theories”) rather than mechanical systems. The picture that emerges is that the building blocks of matter and its interactions are neither particles nor waves, but a new kind of entity: a quantum field. Every type of elementary particle is described by a quantum field (although the groupings by type depend upon the sophistication of the model). There is an electron-positron field, a photon field, a neutrino field, and so forth. In this way of doing things, particles are elementary excitations of the quantum field. Quantum field theory (QFT) has lead to spectacular successes in describing the behavior of a wide variety of atomic and subatomic phenomena. The success is not just qualitative; some of the most precise measurements known involve minute properties of the spectra of atoms. These properties are predicted via quantum field theory and, so far, the agreement with experiment is perfect. We will only be able to give a very brief, very superficial, very crude introduction to some of the basic ideas. Our goal will be to show how QFT is used to describe photons. In a future discussion this could be the basis of an explanation of phenomena such as spontaneous emission. Harmonic Oscillators again Our first goal will be to describe the photon without considering its interaction with other (charged) particles. This is a quantum version of considering the EM field in the absence of sources. Indeed, our strategy for describing photons will be to extract them from a “quantization” of the source-free electromagnetic field. The key idea that allows this point of view is that the EM field can, in the absence of sources, be viewed as an infinite collection of coupled harmonic oscillators. We know how to describe harmonic oscillators quantum mechanically, and we can try to carry this information over to the EM field. First, let us quickly review the key properties of the harmonic oscillator. Recall that the energy of a classical oscillator with displacement x(t) is given by E= m 2 1 ẋ + mω 2 x2 . 2 2 In the quantum description, we view the energy in terms of coordinate and momentum operators, in which case the energy is known as the Hamiltonian: H= p2 1 + mω 2 x2 . 2m 2 2 Here we view x and p as operators on a Hilbert space obeying the commutation relation [x, p] = ih̄1, with “1” being the identity operator. Let us recall the definition of the “ladder operators”: r r p p mω mω † (x + i ), a = (x − i ), a= 2h̄ mω 2h̄ mω satisfying the commutation relations (exercise) [a, a† ] = 1. The Hamiltonian takes the form (exercise) 1 H = h̄ω(a† a + 1). 2 We can drop the the second term (with the “ 21 ”) if we want; it just defines the zero point of energy. The stationary states are labeled by a non-negative integer n, 1 En = (n + )h̄ω. 2 H|ni = En |ni, The ground state |0i satisfies a|0i = 0. Excited states are obtained via the identity (exercise) a† |ni = √ We also have a|ni = n + 1|n + 1i. √ n|n − 1i. It is easy to generalize this treatment to a system consisting of a number of uncoupled harmonic oscillators with displacements xi , i = 1, 2, . . ., momenta pi , masses mi and frequencies ωi . In particular, the Hamiltonian is (exercise) !  X1 X † p2i 1 1 2 2 H= + mω x = ai ai + 1 . 2 2mi 2 i i i 2 i i Even with a set of coupled oscillators, if the couplings are themselves harmonic, we can pass to the normal mode coordinates and momenta. In this case the Hamiltonian again takes the form given above. So, this description is quite general. 3 Fourier components of an EM field To find a quantum description of the EM field we need some elementary results from classical electromagnetic theory. To begin with, we introduce the EM potentials. Recall that the homogeneous subset of the Maxwell equations ∇×E+ 1 ∂B =0 c ∂t and ∇·B=0 are equivalent to the existence of a vector field, the vector potential A, and a scalar field, the scalar potential φ, such that E=− 1 ∂A − ∇φ, c ∂t B = ∇ × A. This is the general solution to the homogeneous subset of the Maxwell equations, so if we choose to work with the electromagnetic potentials we have eliminated half of the Maxwell equations. The potentials are far from uniquely defined. If (φ, A) are a set of potentials for a given EM field (E, B), then so are (exercise) φ′ = φ − 1 ∂Λ , c ∂t A′ = A + ∇Λ, where Λ is any (well-behaved) function of space and time. This transformation between two sets of potentials for the same EM field is called a gauge transformation, for historical reasons that we shall not go into. Physical quantities will be unchanged by gauge transformations. The notion of gauge invariance, which just seems like a technical detail in Maxwell theory, is actually pretty profound in physics. However, for our purposes, we just note that the freedom to redefine potentials via a gauge transformation means that we can try to make a convenient choice of potentials. Our choice will be always to put the potentials in the radiation gauge. What this means is as follows. Any electromagnetic field can be described by a set of potentials such that φ = 0, ∇ · A = 0. This should amuse you (at least a little). In electrostatics it is conventional to work in a gauge in which A = 0 and then the static electric field is (minus) the gradient of the scalar potential. This is certainly the most convenient way to analyze electrostatics, but one could opt to use a time-dependent (and curl-free) vector potential if so-desired (exercise). 4 The Hamiltonian of the electromagnetic field To use the harmonic oscillator paradigm to “quantize” the EM field, we first express the total energy of the field in terms of the potentials: "  # 2 Z Z 1 1 1 ∂A d3 x (E 2 + B 2 ) = d3 x 2 + (∇ × A)2 . H= 8π all space 8π all space ∂t c You can think of the first integral in the sum as the kinetic energy of the field and the second integral as the potential energy. This is something more than an analogy. It is possible to think of the electromagnetic field as a Hamiltonian dynamical system with the vector potential playing the role of (an infinite number of) generalized coordinate(s) and the electric field as its “canonically conjugate momentum”. In this interpretation the function H above is the Hamiltonian (expressed in terms of position and velocity). The behavior of any such dynamical system is determined by the Hamiltonian expressed as a function of canonical coordinates and momenta, therefore we focus all our attention on H. Next we make a Fourier decomposition of A: Z 1 d3 k Ak (t)eik·x . A(x, t) = 3/2 (2π) Note that (exercises) (Ak )∗ = A−k , ∇ · A = 0 ⇐⇒ k · Ak = 0, Z 1 ∂A d3 k Ȧk (t)eik·x , = 3/2 ∂t (2π) Z i d3 k [k × Ak (t)]eik·x . ∇×A= 3/2 (2π) Insert the Fourier expansion of the vector potential into the Hamiltonian. We get (exercise): Z n1 o H = d3 k 2 |Ȧk |2 + |k × Ak |2 c Using (1) a vector identity for the dot product of a pair of cross products, and (2) the radiation gauge condition, we get: Z o n1 H = d3 k 2 |Ȧk |2 + k 2 |Ak |2 . c Let’s compare this with the harmonic oscillator Hamiltonian. Think of the integrand as the energy of a single oscillator and the integral as a sum. Then we have (exercise) ω −→ ω(k) = kc, 5 and 1 m −→ 2 . c The frequency correspondence is perfectly reasonable; it describes the frequency-wave number dispersion relation of an EM wave. The mass analogy is okay mathematically, but shouldn’t be taken too literally in a physical sense; there is no particularly meaningful way to ascribe a rest mass to the electromagnetic field. Further confirmation of our interpretation of things comes by considering the remaining Maxwell equations,* 1 ∂E ∇×B− =0 c ∂t and ∇ · E = 0, which imply that each component of A satisfies the wave equation with propagation velocity c (and we still have the side condition ∇ · A = 0). This means that (exercise) Ak (t) = ck eiω(k)t + c∗−k e−iω(k)t , for some constants ck . Thus for each value of k the vector potential behaves like 3 harmonic oscillators with frequency kc. Thus you can think of the vector potential as a sort of generalized coordinate, and the electric field as the canonically conjugate momentum. The Hamiltonian is then that of a (continuous) collection of harmonic oscillators labeled by the wave vector of a Fourier decomposition. The fact that there is a continuous family of coordinates and momenta (one for each spatial point, or one for each wave vector) leads to the statement that the EM field has “an infinite number of degrees of freedom”. This is the principal feature distinguishing quantum field theory from quantum mechanics, and it is what allows one to describe processes in which particles are created and destroyed. Now let us massage our formula for the Hamiltonian into an even nicer form. We do this by taking account of the properties of the Fourier components of the vector potential. We have seen that the Fourier component of the vector potential with wave vector k is orthogonal to k, and that it satisfies a complex conjugation relation. We take both of these conditions into account (and introduce a convenient normalization) via the definition Ak = 2 X σ=1 r h̄c (a ǫ + a∗−k,σ ǫ−k,σ ), k k,σ k,σ * Recall that we consider the electromagnetic field in regions of space in which there are no charges or currents. 6 where σ labels the polarization and ǫk,σ are two orthonormal vectors orthogonal to k. The variables ak,σ carry the amplitude information about each polarization, and the polarization direction is determined by the choice of ǫk,σ . Note that we have used h̄ to define the new variables. From a purely classical EM point of view this is a bit strange, but it is convenient for the quantum treatment we will give. For now, just think of the use of h̄ as a convenient way of forming the amplitudes ak,σ , which are dimensionless (exercise). Using this form of Ak (t) it follows that, for solutions of the Maxwell equations, 2 X √ d (ak,σ ǫk,σ − a∗−k,σ ǫk,σ ). Ak = i h̄kc dt σ=1 Using this result we have (exercise) Z   X 3 ∗ H= d k h̄ω(k) ak,σ ak,σ . σ Hopefully, this very simple form for the energy justifies to you all the effort that went into deriving it. Up to a choice of zero point of energy, this is clearly a classical version of the Hamiltonian for a collection of oscillators labeled by k and σ. A similar computation with the Lagrangian for the Maxwell field, Z 1 L= d3 x(E 2 − B 2 ), 8π all space leads to a sum (really, integral) of harmonic oscillator Lagrangians. Thus, mathematically at least, the electromagnetic field (in the radiation gauge) is a continuous collection of harmonic oscillators. The quantization of the EM field The classical Hamiltonian for the electromagnetic field can be expressed as a continuous superposition over harmonic oscillator Hamiltonians: Z   X 3 ∗ H= d k h̄ω(k) ak,σ ak,σ . σ We thus view the quantum EM field as an infinite set of quantum oscillators. The oscillators’ degrees of freedom are labeled by the wave vector k and the polarization σ. We view † the ladder operators for each degree of freedom as ak,σ and ak,σ . In the context of quantum field theory, we call these operators annihilation and creation operators, respectively. We will see why in a moment. The quantum Hamiltonian is built from the creation and annihilation operators via Z   † X H = d3 k h̄ω(k) ak,σ ak,σ . σ 7 This is clearly just the sum of energies for each individual oscillator (with the “zero point energy” dropped). Incidentally, it is not too hard to compute the total momentum P of the electromagnetic field. It is obtained from the integral of the Poynting vector. (This means that the Cartesian components of the total momentum are integrals of the corresponding components of the Poynting vector field. In terms of the creation and annihilation operators we get (exercise) Z Z  †  X c 3 d x E × B = d3 k h̄k ak,σ ak,σ . P= 4π σ This is the same as the total energy except that the energy of each mode, h̄ω(k) has been replaced by the momentum of each mode h̄k. Do you recall the usual lore of photons? You know, the lore that says a photon with a definite energy and momentum will have E = h̄ω, and P = h̄k, where ω = kc. We see that we are in a position to model a photon with definite energy and momentum as a quantum normal mode of the EM field! To make detailed sense of all this we should spell out the meaning of all these ladder operators. The idea is to first use the harmonic oscillator point of view to understand the definition of the operators. Then we can reinterpret the mathematical set-up in terms of photon states. The vacuum To begin with, let us suppose that all the oscillators are in their ground state. This state of the EM field, denoted |0i, is called the vacuum state. You can easily see why. This state is an eigenstate of H and P with eigenvalue zero: H|0i = 0 = P|0i. This you can verify from the fact that all the annihilation (i.e., lowering) operators map the ground state to the zero vector: ak,σ |0i = 0. Thus the vacuum of the EM field (in the absence of any other interactions, which we are not modeling here) is a state in which the energy and momentum are known with probability one. It can be shown that this state is the state of lowest energy and momentum. By the way, have you ever encountered claims that, thanks to the uncertainty principle, there is an “infinite reservoir of zero point energy in the vacuum”? Perhaps you have even seen seemingly learned schemes to extract this energy for practical use. Now you are in 8 a position to be a little skeptical: the total energy-momentum of the vacuum is perfectly well-defined – it vanishes! Where do these wacky claims come from? As with most popular distortions of scientific results, there is a kernel of truth here. To uncover it, return to the case of a single harmonic oscillator. The ground state energy is perfectly well defined, but the energy and position and/or momentum operators are not compatible. This means that if you know the energy with probability one, you cannot know the position or momentum of the oscillator with probability one. Indeed, you may recall that in the ground state of the oscillator the probability distributions for position and momentum are Gaussians with zero average. The EM field has a similar behavior. Recall that the “position” is, essentially, the vector potential (hence the magnetic field is a “function of position”) and the “momentum” is, essentially, the electric field. In the EM vacuum state, the energy is minimized and sharply defined, but the EM fields themselves “fluctuate”. More precisely, the EM fields have a probability distribution with non-zero standard deviation; basically, each mode (and polarization) is a described by a Gaussian probability distribution in the vacuum state. (This fact is responsible for the “Casimir effect”.) Very roughly speaking, the uncertainty principle means that, while the total energy is known with certainty, the energy density is “uncertain”. I say “very roughly” since the notion of quantum energy density of an EM field is rather touchy; there is no well-defined quantum version that behaves much like the classical analog. Indeed, much of the zero point energy nonsense that appears in print is based upon trying to use classical ideas to describe a feature of the theory that is, well, not at all classical! Photon states With the vacuum state under control, we can now consider excited stationary states of the quantum EM field, which we will interpret as states with one or more photons. Think again about the EM field as a large collection of harmonic oscillators, labeled by wave number and polarization. Suppose we put one of these oscillators in its first excited state, † say, by applying ak,σ to the vacuum for some fixed choice of k and σ. We denote this state as † |1k,σ i = ak,σ |0i. It can now be shown that the resulting state is an eigenvector of H and P: H|1k,σ i = h̄ω(k)|1k,σ i, P|1k,σ i = h̄k|1k,σ i. This state has energy-momentum values defined with probability unity, which take the form appropriate for a single photon. We interpret this state as a 1 photon state with the indicated wave number, frequency, and polarization. Thus, the first excited stationary 9 state of the quantum electromagnetic field can be viewed as a single photon. In this sense photons are “quanta of the electromagnetic field”. Superpositions of photon states over momenta lead to photons that have probability distributions for their energy and momenta. Thus photons need not have a definite momentum or energy any more than, say, an electron must. More generally, we can build up a Hilbert space of states by repeatedly applying to the vacuum state the creation operators with various wave numbers and polarizations, taking † superpositions, etc. The result of each application of the creation operator ak,σ is to define a state with one more photon of the indicated type. Each application of the annihilation operator ak,σ results in a state with a photon of the indicated type removed. If the state doesn’t have such a photon, (e.g., the vacuum state won’t) then the result is the zero vector. The states we are describing have a particle interpretation, but the theory is richer than just a collection of particles since, e.g., one can superimpose the states described above to get states which encode the field like properties of the EM field. Of course, such states will not be stationary states. Also, recall our discussion of the “fluctuations” of E and B in the vacuum. Charges respond to the electromagnetic field strengths (think: Lorentz force law), and so the behavior of charged particles can be affected by EM phenomena, even when no photons are present! This is the idea behind the “Lamb shift” found in the spectra of atoms. And it is the key idea needed to explain spontaneous emission of photons from atoms. We see, then, how the “normal modes” of a field satisfying wave-type equations can be “quantized”. The resulting theory admits particle-like properties in its stationary states. To date, every known elementary particle has been successfully described by a quantum field in much the way we described photons using a quantized electromagnetic field. Interactions between particles (better: between quantum fields) have also been described with considerable success using the quantum field formalism. Indeed, it is reasonable to adopt the point of view that, in our current best understanding of nature, quantum fields are the stuff out of which everything is made! 10