MIT OpenCourseWarehttp://ocw.mit.edu 6.453 Quantum Optical Communication Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.453 Quantum Optical Communication Lecture Number 12 Fall 2008 Jeffrey H. Shapiro c�2006, 2008 Date: Tuesday, October 21, 2008 Linear attenuators, phase-insensitive and phase-sensitive linear amplifiers Introduction In this lecture will continue our quantum-mechanical treatment of linear attenua-tors and linear amplifiers. Among other things, we will distinguish between phase-insensitive and phase-sensitive amplifiers. We will also show that the attenuator and the phase-insensitive amplifier preserve classicality, i.e., their outputs are classical states when their inputs are classical states. Finally, we will use the transformation effected by the two-mode parametric amplifier to introduce the notion of entangle-ment. Single-Mode Linear Attenuation and Phase-Insensitive Linear Amplification Slide 3 shows the quantum models for linear attenuation and linear amplification that were presented in Lecture 11. In both cases we are concerned with single-mode quantum fields at the input and output, whose excited modes are as follows,1 Eˆin(t) = aˆin√e−T jωt and Eˆout(t) = aˆout√eT −jωt , for 0 ≤ t ≤ T , (1) where � √L aˆin + √1 − L aˆL, for the attenuator aˆout = (2) √G aˆin + √G − 1 ˆa†for the amplifier, G, with 0 < L < 1 being the attenuator’s transmissivity and G > 1 being the amplifier’s gain. The presence of the auxiliary-mode annihilation operators, ˆaL and ˆaG, in these input-output relations, ensures that [ˆaout, aˆ†] = 1, (3) out1For the sake of brevity, we have omitted the “other terms” that are needed to ensure that these field operators have the appropriate commutators for freely propagating fields. So long as the photodetection measurements that we make are not sensitive to these vacuum-state other modes, there is no loss in generality in using these compact single-mode expressions. 1as is required for the Eˆout(t) expression to be a proper photon-units representation of a single-mode quantum field. Minimum noise is injected by the auxiliary modes when they are in their vacuum states, so, unless otherwise noted, we shall assume that they are indeed in these unexcited states. It is easy to show that the annihilation operator input-output relation, (2), implies the following input-output relation for the θ-quadratures, � √L aˆinθ + √1 − L aˆLθ , for the attenuator aˆoutθ = (4) √G aˆinθ + √G − 1 ˆaG−θ , for the amplifier, where ˆaθ ≡ Re(ˆae−jθ) defines the θ-quadrature of an annihilation operator ˆa. Taking the expectation of these equations, with ˆain being in an arbitrary quantum state, gives, � √L �aˆinθ �, for the attenuator �aˆoutθ � = √G �aˆinθ �, for the amplifier. (5) Because �aˆoutθ �/�aˆinθ � is independent of θ, for both the attenuator and the amplifier, we say that these systems are phase-insensitive, i.e., all the input quadratures undergo the same mean-field attenuation (for the attenuator) or gain (for the amplifier). Output State of the Attenuator In Lecture 11 we derived the means and variances of photon number and quadrature measurements made on the output of the linear attenuator. Today we w ill obtain the complete statistical characterization of this output, and use our result to determine when semiclassical theory can be employed for photodetec tion measurements made on the attenuator’s output. Our route to these results will be through characteristic functions.2 We know that the output mode de nsity operator, ˆρout, is completely characterized by its associated anti-normally ordered characteristic function, χρout ζaˆ†A (ζ∗, ζ) = �e−ζ∗aˆout e out �. (6) Substituting in from (2) and using the fact that the ˆain and ˆaL modes are in a product state, with the latter being in its vacuum state, gives χρout ain+√1−L aˆL) ζ(√L aˆ†+√1−L aˆ†)(ζ∗, ζ) = �e−ζ∗(√L ˆe inL� (7) A ain in aLL= �e−ζ∗√L ˆe ζ√L aˆ†��e−ζ∗√1−L ˆe ζ√1−L aˆ†� (8) χρin 2(1−L)= A (ζ∗√L, ζ√L)e−|ζ|. (9) 2This should not be surprising. We are dealing with a linear quantum transformation. In classical probability theory it is well known that characteristic function techniques are very convenient for dealing with linear classical transformations. So, we are going to see that the same is true in the quantum case. 2� We won’t use the operator-valued inverse transform to find ˆρout from this result, but we will examine what happens when the ˆain mode is in the coherent state |αin�. Here, our known expression for the anti-normally ordered characteristic function of the coherent state leads to χρout (ζ∗, ζ) = e−ζ∗√L αin+ζ√L α∗ in e−|ζ|2 , (10) A outwhich we recognize as being equal to �√L αin|e−ζ∗aˆout eζaˆ†|√L αin�. This shows that a coherent-state input |αin� to the attenuator results in a coherent-state output |√L αin�from the attenuator.3 Moreover, if the input mode is in a classical state, i.e., its density operator has a P -representation ρˆin = d2α Pin(α, α∗)|α��α|, (11) with Pin(α, α∗) being a joint probability density function for α1 = Re(α) and α2 = Im(α), then it follows that the output mode is also in a classical state, with a prop er P -function given by � � Pout(α, α∗) = 1 L Pin α √L , α∗ √L . (12) The derivation of this scaling relation—which coincides with the like result from clas-sical probability theory—is left as an exercise for the reader. The essential message, however, is not the derivation; it is that linear attenuation (with a vacuum-state auxiliary mode) preserves classicality. Output State of the Phase-Insensitive Linear Amplifier Turning to the phase-insensitive linear amplifier, we will determine its output-state behavior by the same characteristic function technique that we just used for the linear attenuator. Substituting in from (2) and using the fact that the ˆain and ˆaG modes are in a product state, with