Introduction In this lecture will continue our study of parametric - PDF document

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.453 Quantum Optical Communication Lecture Number 13 Fall 2016 Jeffrey H. Shapiro � c 2006, 2008, 2010, 2012, 2014 Date: Tuesday, October 25, 2016 The two-mode parametric amplifier and photon twins. The dual parametric amplifier and polarization entanglement. Introduction In this lecture will continue our study of parametric amplification. We will pick up where we left off last time, by examining the entangled state that is produced by a two- mode parametric amplifier when its input modes are in their vacuum states. Despite the joint state of the output modes being non-classical—because entanglement is a non-classical property—we’ll see that the individual modes are in classical states. We will develop the number-ket representations for both the joint and the individual states, and show that the former exhibits photon-twinning behavior, which leads to a non-classical signature in differenced direct detection. By appropriately ganging together a pair of two-mode parametric amplifiers, and going to the low-gain limit, we will show how to create a pair of polarization-entangled photons. Polarization entanglement provides the basis for qubit teleportation, which we will mention briefly at the end of today’s lecture and treat in detail next time. The Two-Mode Parametric Amplifier From Lecture 12 we have the following two-mode Bogoliubov transformation relating the annihilation operators for the x - and y -polarized input and output modes, where √ √ we have chosen to specialize the general case to G = µ > 0 and G − 1 = ν > 0: √ √ √ √ ˆ † ˆ † ˆ out x = a G a ˆ in x + G − 1 a and a ˆ out y = G ˆ a in y + G − 1 a in x . (1) in y We know that this transformation preserves annihilation operator commutator brack- ets, and that if we regard a ˆ in x and a ˆ out x as input and output with a ˆ in y in its vacuum state, the first equality in (1) implies that we have a phase-insensitive linear amplifier relation between the input and output modes. The same behavior occurs—i.e., we have a phase-insensitive linear amplifier relationship—if we take a ˆ in y and a ˆ out y as in- ˆ in x in its vacuum state. In the ± 45 ◦ basis, however, (1) reduces put and output with a to a pair of single-mode Bogoliubov transformations, so that a coherent-state input 1

in the a ˆ in + ( a ˆ in ) mode will yield a squeezed-state output in the a ˆ out + ( a ˆ out ) mode. − − Thus, in the ± 45 ◦ basis, the two-mode parametric amplifier specified by (1) is a pair of independent phase-sensitive amplifiers, one for the +45 ◦ -polarized mode and the other for the − 45 ◦ -polarized mode. In Lecture 12, we used characteristic functions to derive the complete statistics for the a ˆ out x and a ˆ out y modes and found the following result for the anti-normally ordered characteristic function associated with their joint density operator ρ ˆ out : χ ρ out ( ζ ∗ , ζ ∗ , ζ , ζ ) = χ ρ in ( ξ ∗ , ξ ∗ , ξ , ξ ) e − ( | ζ x | + | ζ 2 | ) / 2 , 2 (2) y x y x y x y x y A W where √ √ √ √ G − 1 ζ ∗ ∗ . ξ x = G ζ x − and ξ y = G ζ y − G − 1 ζ x (3) y Our interest lies in the special case in which the two input modes are in their vacuum states, so that the preceding anti-normally ordered characteristic function reduces to | ζ y | )+2 √ χ ρ out 2 2 ∗ , ζ y ∗ y = e − G ( | ζ x | + G ( G − 1) Re( ζ x ζ y ) , ( ζ x , ζ x , ζ ) (4) A from which the following anti-normally ordered characteristic functions for ρ ˆ out x and ρ ˆ out y readily follow: ∗ , 0 , ζ x , 0) = e − G | ζ x | , ρ ρ 2 ( ζ ∗ χ out x x , ζ x ) = χ out ( ζ x (5) A A and | . ρ ρ out 2 ∗ , ζ y ) = χ A ∗ , 0 , ζ y ) = e − G ζ | out y χ ( ζ y (0 , ζ y (6) y A Thus, because ρ ρ χ ρ out ∗ , ζ y ∗ , ζ x , ζ y ) � = χ ∗ , ζ x ) χ ∗ , ζ y ) , out y ( ζ x out x ( ζ x ( ζ y (7) A A A we have ρ ˆ out � = ρ ˆ out x ⊗ ρ ˆ out y , which means that ρ ˆ out is an entangled state if it is a pure state. To show that ρ ˆ out is a pure state we could argue that (1) is a unitary transformation, so that its output state must be a pure state when its input state—in this case the vacuum state of both the a ˆ in x and a ˆ in y modes—is a pure state. We shall take a more explicit route to showing that ρ ˆ out implied by (4) is a pure state, viz., we ˆ 2 shall verify that tr( ρ out ) = 1. Using the operator-valued inverse Fourier transform relation � d 2 ζ x 2 ζ y � d † † χ ρ out ( ζ ∗ , ζ , ζ , ζ ) e − ζ x a ˆ out − ζ y a ˆ ∗ a ∗ a ∗ y e ζ x ˆ out x + ζ y ˆ out y , ρ ˆ out = (8) out x x y A x y π π 2

we find that � d 2 α x 2 α y � d ˆ 2 2 tr( ρ out ) = � α y | x � α x | ρ ˆ out | α x � x | α y � y (9) y π π � d 2 ζ x � d 2 ζ y † ˆ † ˆ out e − ζ x a ˆ out − ζ y a ∗ a ∗ a χ ρ out ∗ , ζ y ∗ , ζ x , ζ y )tr( ρ out y e ζ x ˆ out x + ζ y ˆ out y ) (10) = ( ζ x x A π π � d 2 ζ x 2 ζ y χ ρ out � d ∗ , ζ x , ζ y ) χ ρ out ( ζ ∗ ∗ , − ζ y ∗ , − ζ x , − ζ y ) = x , ζ y N ( − ζ x (11) A π π � d 2 ζ x 2 ζ y χ ρ out � d W ( ζ ∗ ∗ , ζ x , ζ y ) χ ρ out ∗ , − ζ y ∗ , − ζ x , − ζ y ) = x , ζ y W ( − ζ x (12) π π � d 2 ζ x 2 ζ y � d | χ ρ out ∗ , ζ y ∗ , ζ x , ζ y ) | 2 . = W ( ζ x (13) π π Here: the trace is evaluated in the first equality using the coherent-state bases {| α x � x } ρ out ρ out and {| α y � y } ; χ N and χ W are the normally-ordered and Wigner characteristic func- tions of the output state; the fourth equality makes use of the Baker-Campbell- Hausdorff theorem; and the last equality follows from χ ρ out being an Hermitian func- W tion of its arguments. Substituting χ ρ χ ρ out 2 2 ∗ , ζ y ∗ , ζ x , ζ ) ∗ , ζ ∗ , ζ , ζ ) e ( | ζ x | + | ζ y | ) / 2 W ( ζ x out = ( ζ x (14) y x y A y e − ( G − 1 / 2)( | ζ x | 2 + | ζ y | 2 )+2 √ G ( G − 1) Re( ζ x ζ y ) , = (15) into (13) we get � d 2 ζ x � d 2 ζ y e − (2 G − 1)( | ζ x | 2 + | ζ y | 2 )+4 √ G ( G − 1) Re( ζ x ζ y ) = 1 , ˆ 2 tr( ρ out ) = (16) π π where the second equality follows from the normalization integral for a 4-D Gaussian probability density function, 1 proving that the output state of the two-mode para- metric amplifier is a pure state when its input modes are in their vacuum states. So, because there must be a | ψ � out on the joint state space of the a ˆ out x and a ˆ out y modes such that ρ ˆ out = | ψ � outout � ψ | , (7) implies that there are no pure states | ψ � out x and | ψ � out y for the individual modes which give | ψ � out = | ψ � out x ⊗ | ψ � out y . In short, the output modes from the two-mode parametric amplifier are entangled when their input modes are in their vacuum states because if their joint state is a product state—i.e., unentangled—and pure, then the individual states must also be pure. Let us delve deeper into the individual and joint states whose characteristic func- tions we’ve just determined. First consider the individual states. Our work in Lec- ture 12 on the phase-insensitive amplifier immediately tells us that the a ˆ out x and a ˆ out y 1 See (24), below, for the necessary formula. 3