Chapter 1. Quantum interference 1.1 Single photon interference

Transcription

1 Chapter. Quantum interference. Single photon interference b a Classical picture Quantum picture Two real physical waves consisting of independent energy quanta (photons) are mutually coherent and so they interfere. Each individual photon simultaneously exists in the two arms with finite probability amplitudes, which interfere. input state: after st B.S.: in B.S. unitary matrix basis vector a 0 b ket vector Two states are uncorrelated. U ˆ BS in 0 0 a b a b 0 0 a b 0 0 a b Û U BS Chapter- linear superposition state Two mutually exclusive possibilities coexist.

2 after phase shifter: U ˆ PS 0 a b ei 0 a b after nd B.S.: out U ˆ BS 0 a 0 b 0 e i 0 a b ei Probability of obtaining a photon at the output port a P a out ˆ n a out 4 ei cos a 0 b e i 0 a b n ˆ a n nn aa n ˆ I b self-adjoint operator A ˆ A ˆ real eigenvalue Hermitian operator represents a dynamical variable (observable). Probability of obtaining a photon at the output port b P b out n ˆ b out 4 ei cos e i b 0 a e i b a 0 n ˆ b I ˆ a n nn bb n bra bector out out P Probability interpretation a a n b 0 out of QM P b a 0 b n out Chapter- Schrödinger wavefunction c-number carries the amplitude & phase information simultaneously

3 Probability P a P b QM is simply silent for an single event. The connection between the theory & experiment is only via statistics of many, many measurement events. 0 Phase shift What interfere with each other are the two probability amplitudes of the linear superposition state, a 0 b and 0 a b. Origin of interference is the lack of information for which path a photon takes before it is detected. One photon interference does not distinguish a quantum picture from a classical picture based on two real physical waves. In order to see a truly quantum mechanical interference effect for which a classical picture fails, we need to study a multi-photon interference effect.. Symmetrization postulate and quantum indistinguishability C. Cohen-Tannoudji et al., Quantum Mechanics (John Wiley and Sons, New York, 977).. statement of the postulate The physical state of a system including several identical quantum particles are completely symmetric or anti-symmetric with respect to permutation of these particles. or (?) Identical quantum particles are indistinguishable. cf. c c c c if 0 Chapter-3 : boson : fermion (orthogonal) does not correspond to a physical state even though it is a mathematically legal state.

4 .. Collision of two identical quantum particles Spinless particles L R in R L L R ˆ U BS R 0 L 0 L or (?) B.S. R L L (direct term) L R out R R L L R L L R : boson L L R R : fermion R L L R L L Only states with constructive interference between direct (ex. (exchange and exchange (ex. term) R L L L L R L R L L ) terms are selected out. ) output particle flux boson fermion L R (direct term) L R bunching anti-bunching L R (exchange term) final state stimulation Bose condensation, superconductivity Pauli exclusion principle Chapter-4

5 ..3 Collision of two non-identical particles Particle is a muon -. Particle is an electron e -. The detector is only sensitive to the charge of the particles, giving no information about their masses. initial state final state in R L f R L R L independent splitting (no interference) (, 0 ) 5% R R L L ( 0, ) 5% R L and L R (, ) 50% Quantum interference disappears even if the actual detector cannot distinguish the two particles. The theoretical possibility of distinguishing the particle and particle is enough to eliminate the quantum interference effect. Chapter-5

6 ..4 Collision of two identical particles with spins (EPR-Bell state) A. Spin singlet state L or (?) R Spin part of the wavefunction (internal DOF) BS symmetrization ti ti postulate lt R L L R anti-symmetric anti-symmetric orbital wavefunction spin wavefunction output particle flux boson symmetric overall wavefunction: boson R L L R anti-symmetric overall wavefunction: fermion fermion fermionic collision (anti-bunching) bosonic collision (bunching) Chapter-6

7 B. Spin triplet states spin part of the wavefunction (internal DOF) symmetrization postulate R L L R symmetric orbital wavefunction symmetric spin wavefunction symmetric overall wavefunction: boson R L L R anti-symmetric orbital wavefunction symmetric spin wavefunction anti-symmetric overall wavefunction: fermion output particle flux is identical to that of spinless particles. boson fermion bosonic collision fermion collision Chapter-7

8 ..5 Bell state analysis A. Linear optics Bell state analyzer H V V H H V V H Experimental set-up for collision of two identical particles V V Coincidence rates C HV ( ) and C HV ( ) depending on the path length difference, for transmission of the state. The constructive interference for the rate C HV enables one to read the information associated with that state (bosonic singlet). Symmetrization/anti-symmetrization y is not required if the two wavepackets do not overlap. distinguishable from detection time Coincidence rates C HV ( ) and C HV ( ) as functions of the path hlength hdifference when the state is transmitted. For perfect timing (=0), constructive interference occurs for C HV, allowing identification of the state sent (bosonic triplet). Linear optics EPR-Bell state analyzer (cannot distinguish + and - states) Chapter-8

9 B. Full Bell State Analyzer (nonlinear quantum circuit) x y H control o target Hadamard gate H Controlled-NOT gate U CNOT IN (out) OUT (in) Chapter-9