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Quantum Magic Bullets via Entanglement

Seth Lloyd, Je?rey H. Shapiro, and N. C. Wong

Research Laboratory of Electronics, Massachusetts Institute of Technology Cambridge, Massachusetts 02139-4307 Two particles that are entangled with respect to continuous variables such as position and momentum exhibit a variety of nonclassical features. First, measurement of one particle projects the other particle into the state that is the complex conjugate of the state of the ?rst particle, i.e., measurement of one particle projects the other particle into the time-reversed state. Second, continuous-variable entanglement can be used to implement a quantum “magic bullet”: when one particle manages to pass through a scattering potential, then no matter how low the probability of this event, the second particle will also pass through a related scattering potential with probability one. This phenomenon is investigated in terms of the original EPR state, and experimental realizations are suggested in terms of entangled photon states.

arXiv:quant-ph/0009115v1 28 Sep 2000

I. INTRODUCTION

Entanglement is a peculiar quantum phenomenon in which two quantum systems exhibit a greater degree of correlation than is permitted classically. Entanglement has been shown to be a highly useful e?ect for quantum computation and quantum communications, see [1] for a recent review. One of the earliest and most striking pictures of entanglement was given by Einstein, Podolsky, and Rosen (EPR) in their original paper on entangled states [2]. In contrast with the majority of subsequent work, the original EPR paper concentrated on entanglement of continuous variables, position and momentum. Although most recent work on entanglement has focused on discrete systems such as quantum bits or qubits, experimental demonstration of continuous-variable teleportation via entanglement [3], theoretical constructions of analog quantum error-correcting codes [4,5,6], and methods for universal quantum computation over continuous variables [7] suggest that the original EPR state is well-worth revisiting. This paper shows that entanglement of continuous variables exhibits several signi?cant properties in the context of scattering theory. Consider two particles that are entangled in terms of continuous variables such as position and momentum. Measurement of one particle can be shown to project the other particle into the complex conjugate, or time-reversed state. This fact has the following consequence. Consider a potential barrier put in the path of the ?rst particle that re?ects some states and transmits other states with probability one (such a barrier can be thought of as a generalized ?lter). Then there is a related potential barrier for the second particle—corresponding to the time-reversed scattering matrix for the ?rst potential—such that if the ?rst particle is transmitted through its barrier, the second particle will be transmitted through its barrier with probability one. We call this e?ect, a quantum “magic bullet.” This paper presents a theoretical exposition of the magic bullet e?ect both in terms of the original EPR state and in terms of entangled photons, and proposes experimental realizations of quantum magic bullets.

II. EPR-STATE MAGIC BULLETS

First, let us revisit the original Einstein-Podolsky-Rosen state. Suppose that a particle with zero momentum decays into at time t = 0. The particles have position eigenstates |x j , and momentum eigenstates √ two particles ∞ ? |p j = (1/ 2π ? h) ?∞ eipx/h |x j dx, where j = 1, 2. (Like EPR, we restrict our attention to a one-dimensional system: the multi-dimensional generalization is straightforward.) Immediately after the initial particle’s decay, the joint state of the two new particles is

∞ ∞

|ψ

EPR

=

?∞

|x 1 |x

2 dx = ?∞

|p 1 | ? p

2

dp.

(2.1)

These two particles are thus perfectly correlated in position, and also perfectly anticorrelated in momentum. It was this dual correlation that Einstein, Podolsky and Rosen suggested was incompatible with the classical notion of reality: the EPR state seems to allow each particle to be in an eigenstate of two noncommuting observables. Irrespective of interpretations of quantum mechanics (which are notoriously slippery), the EPR state exhibits the following property. We expand |ψ EPR in terms of an arbitrary orthonormal basis {|φz }, indexed by a continuous parameter z , obtaining |φz j = φz (x)|x j dx, where φz (x) = x|φz . Then, recalling that |x ? = |x , we have

1

|ψ

EPR

=

|φz 1 |φz

? 2

dz.

(2.2)

Suppose that a measurement of an operator with eigenstates |φz 1 is made on the ?rst particle. It is immediately seen that if this measurement reveals the ?rst particle to be in the state |φz 1 , the second particle is then in the complex conjugate state |φz ? 2. Complex conjugation in quantum mechanics is equivalent to time reversal. If a particle evolves according to a ? h ? ? , so that an initial state |φ at t = 0 evolves into the state |φ(t) = e?iHt/ Hamiltonian H |φ for t > 0, then taking ? ? iH t/h ? ? the complex conjugate gives the state e |φ , viz., the complex conjugate of the evolved state is the complex ? ?. conjugate of the initial state evolved backward in time (t → ?t) according to the time-reversed Hamiltonian H ? is real, i.e., time-reversal invariant (as is the case for almost all fundamental Hamiltonians, with the notable If H ? can be taken to be real, because H ? |E = E |E exception of the K 0 meson), then the energy eigenfunctions of H ? ? ? implies H |E = E |E . Suppose that both particles in the EPR state are subjected to a time-reversal invariant ? . Over time t, the state |ψ EPR evolves into Hamiltonian H |ψ (t)

EPR

≡

? ? e?iHt/h |φz 1 e?iHt/h |φz

?

?

? 2

dz =

|φz (t) 1 |φz (?t)

? 2

dz.

(2.3)

The state of the ?rst particle at time t is thus perfectly correlated with a state that is the complex conjugate of that particle’s state at time ?t: the second particle is in the time-reversed state of the ?rst. The EPR state also exhibits interesting scattering properties. Suppose that the ?rst particle is subjected to a time? h ? ?(t) = e?iHt/ ?? (t) = S ?? (t) = reversal invariant Hamiltonian corresponding to a unitary scattering matrix S . Note that S ?1 ? S (t). After the particle is scattered, let us make a projective measurement on it, corresponding to the Hermitian ? ? ?2 ?= operator O o oPo , where o is real and Po = Po is the projection operator on the eigenspace corresponding to o. ?? (t), and that the conjugate Now suppose that the second particle is subjected to the inverse scattering operation S ? ? ? ? measurement O = o oPo is made on this particle. Then the results of the two measurements will be perfectly correlated: a result of o for the ?rst particle will be accompanied by a result o for the second particle. That is, if the ?rst particle passes through a given scattering potential, the second particle passes through the time-reversed potential with probability one, regardless of how unlikely it was for the ?rst particle to have breached its barrier. We call this e?ect a quantum magic bullet. The magic bullet e?ect can be implemented in other ways as well. For example, suppose that it is possible to conjugate the phase of the second particle (for example, by sending it o? a phase-conjugate mirror) at time t. The phase conjugation e?ectively performs a spin echo on the particle, resulting in the state |φz (t) 1 |φz (?t) 2 dz . The second particle now performs the same dynamics as the ?rst particle, but with a time lag of 2t. In particular, if the ?rst particle manages to pass through a generalized ?lter, then the second particle will pass through the same ?lter with probability one 2t seconds later. Quantum magic bullets can have many manifestations. We now turn to examples of quantum magic bullets that can be constructed using nonlinear optics.

III. FIELD-QUADRATURE MAGIC BULLETS

Parametric interactions in χ(2) crystals have proved to be rich sources of quantum light-beam phenomena, see [8,9] for a uni?ed treatment of a wide variety of such e?ects, including quadrature-noise squeezing, nonclassical twin-beam production, nonclassical fourth-order interference, and polarization-entangled photon-pair production. All these phenomena originate from the same fundamental physics: in a χ(2) material pumped by a strong beam at frequency ωP and wave vector kP , a single pump photon is converted into a pair of photons—one signal (S ) and one idler (I )—subject to the energy- and momentum-conservation conditions, i.e., ωS + ωI = ωP and kS + kI = kP , respectively. We shall use the continuous-wave, type-II phase matched, doubly-resonant optical parametric ampli?er (OPA)— with vacuum-state signal and idler inputs—as the basis for all of the optical magic-bullet realizations to be discussed. This OPA arrangement, shown schematically in Fig. 1, produces signal and idler outputs with orthogonal polarizations, well-de?ned spatial modes, and ?uorescence bandwidths in the MHz to GHz range. As in [9], we shall assume that the signal and idler linewidths are identical, that there are no losses in the OPA cavity, and that there is no depletion of nor excess noise on the pump beam. The positive-frequency, photon-units ?eld operators for the excited output ?S (t) and E ?I (t), are then conveniently expressed in terms of their respective polarizations of the signal and idler, E ?j (t) = A ?j (t)e?iωj t , for j = S, I . The ?uorescence center frequencies, ωS and ωI , and their complex envelopes via, E 2

full, multi-mode, joint state of these output signal and idler ?elds is known to be a stationary, entangled, Gaussian pure state that is completely characterized by the following normally-ordered (?uorescence) and phase-sensitive spectra [9]:

∞

S (n) (ω ) ≡ =

?∞ ∞ ?∞ ∞

?? (t + τ )A ?S (t) e?iωτ dτ A S ?? (t + τ )A ?I (t) e?iωτ dτ = A I ?S (t + τ )A ?I (t) e?iωτ dτ = A 2G 2 1 ? G ? (ω/Γ)2 ? 2iω/Γ

2

,

(3.1)

S (p) (ω ) ≡

?∞

2G[1 + G2 + (ω/Γ)2 ] . |1 ? G2 ? (ω/Γ)2 ? 2iω/Γ|2

(3.2)

Here, G2 is the OPA pump power, normalized to the threshold power for oscillation, and Γ is the cavity-loss rate. To establish an analogy between the multi-mode signal and idler ?elds and the two-particle EPR state, let us consider an arbitrary pair of entangled single-frequency modes, namely, the signal beam at frequency ωS + ?ω and the idler beam at frequency ωI ? ?ω . The joint signal×idler state for these two modes has the number-ket representation,

∞

|ψ

SI

=

n=0

?n N |n S |n I , ? + 1)n+1 (N

(3.3)

? = S (n) (?ω ) is the average number of photons per mode. Individually, each mode (signal and idler) is in a where N chaotic (Bose-Einstein) state, but their photon numbers are perfectly correlated. We shall return to this photon-pair property in the sections to follow. Our present course is to connect this signal×idler state to the EPR state. For that purpose we need the ?eld-quadrature representation for |ψ SI . The real and imaginary parts of a photon annihilation operator, a ?, i.e, its quadrature components a ?1 ≡ Re(? a) and a ?2 ≡ Im(? a), behave like normalized versions of position and momentum. In particular, the a ? eigenkets, | α , are 1 1 √ ∞ related to the a ?2 eigenkets, |α2 , by Fourier transformation, |α2 = (1/ π ) ?∞ e2iα2 α1 |α1 dα1 . The joint signal×idler state given in Eq. 3.3 takes the following form, when written in the ?eld-quadrature representation generated by the ?I1 , eigenkets of a ?S1 and a

∞ ∞ ?∞

|ψ with

SI

=

?∞

ψ (αS1 , αI1 )|αS1

S |αI1 I

dαS1 αI1 ,

(3.4)

? )α2 + 4 N ? )α2 / π/2. ? (N ? + 1) αS1 αI1 ? (1 + 2N ψ (αS1 , αI1 ) ≡ exp ?(1 + 2N S1 I1

(3.5)

Equations 3.4 and 3.5 are not identical to the EPR state, Eq. 2.1, but they do embody a nonclassical continuous?I1 measurements. When variable correlation. Optical homodyne detection [10] can be used to perform the a ?S1 and a ?I1 observations such measurements are made on the state |ψ SI , the unconditional (marginal) statistics for the a ?S1 and a ? )/4. are identical: their individual outcomes are Gaussian random variables, each with mean zero and variance (1 + 2N It is the joint statistics of these two measurements that reveals pure quantum behavior. In particular, if we are ?I1 measurement remain given that the outcome of the a ?S1 measurement is αS1 , then the conditional statistics of the a ? ). Thus, when ? (N ? + 1)/(1 + 2N ? )]αS1 and conditional variance 1/4(1 + 2N Gaussian, but with conditional mean [ 4N ? N ≥ 1 there is a strong sub-shot-noise correlation between these signal-beam and idler-beam homodyne measurements: ? ?→ ∞, the conditional variance is substantially below that coherent-state (shot-noise) level of 1/4. Moreover, as N the state |ψ SI approaches a normalized version of the EPR state, as can be seen from rewriting ψ (αS1 , αI1 ) as follows: ? ?2 ? ? ? ? 4N (N + 1) ? ? ? ) ?αI1 ? α2 αS1 ? ? exp??(1 + 2N S1 ? exp ? 1 + 2 N ? 1 + 2N ψ (αS1 , αI1 ) = (3.6) ? )/2]1/4 ? )]1/4 [π (1 + 2N [π/2(1 + 2N ?→ 1 δ (αI1 ? αS1 ), ? (π N )1/4 ? ?→ ∞. as N 3 (3.7)

Experimental demonstrations of this signal/idler homodyne correlation have already been reported [11,12], although for optical parametric oscillators (OPOs)—in which the strong mean ?elds arising from above-threshold OPA operation act as homodyne-detection local oscillators—rather than for OPAs. An example of such data, obtained from a triplyresonant optical parametric oscillator [13], is shown in Fig. 2. The upper trace shows the shot-noise level and the lower trace shows the signal-minus-idler intensity di?erence for frequency detunings, ?ω/2π , ranging from 2 to 6 MHz. The > 5 dB noise reduction in the signal-minus-idler intensity di?erence is a manifestation of the nonclassical correlation cited above for the OPA ?eld quadratures.

IV. PHOTON-PAIR MAGIC BULLETS

Most entanglement experiments that rely on parametric optical interactions draw upon the photon-pair property exhibited in Eq. 3.3. Moreover, these experiments—which employ non-resonant, parametric downconverters rather than doubly-resonant optical parametric ampli?ers—are carried out at extremely low photon ?uxes. In this regime, a T -sec-long photon-counting measurement (on either the signal or idler beam) will yield zero counts with near-unity probability 1 ? p, and one count with probability p; the probability of multiple counts at such low ?uxes is negligible. Photon-pair creation within the χ(2) medium is nearly instantaneous, but, for our doubly-resonant OPA, the time correlation between signal and idler photons in the output beams is smeared out to several cavity lifetimes. Thus, to ?S (t) and E ?I (t)—on a photon-counting represent the OPA version of low-?ux photon-pair generation, we decompose E interval [0, T ], where ΓT ? 1—into operator-valued Fourier series whose coe?cients are the photon annihilation operators,

T

a ? Sn ≡ a ?In ≡

0 T 0

+ 2πn/T )t] ?S (t) exp[i(ωS√ dt, E T ? 2πn/T )t] ?I (t) exp[i(ωI √ dt. E T

(4.1)

(4.2)

When there is one photon pair present in [0, T ], its joint state then has the entangled multi-mode number-ket expansion, |ψ

SI

=

n

ψn |1

Sn |1 In ,

(4.3)

in this representation, where |ψn |2 ∝ S (n) (2πn/T ), and our Fourier-decomposition sign convention has forced there to ?Sn photon occurs. From this photon-pair state we can exhibit the conjugatebe an a ?In photon present whenever an a state projection property of the magic bullet e?ect. Suppose that we make a measurement that projects the signal photon onto the state |φ S = n φn |1 Sn . When that measurement yields a non-zero result, it is easy to see that the idler photon is left in the state, ψn φ? n |1

In

|ψ

I

=

n

.

(4.4)

n

|ψn |2 |φn |2

If the ψn are approximately constant over the n values for which |φn | di?ers signi?cantly from zero, then (except for a physically insigni?cant absolute phase) we get |ψ

I

≈

n

φ? n |1

In ,

(4.5)

i.e., the projective measurement on the signal photon has placed the idler photon in the conjugate state. To our knowledge, the preceding conjugate-state projection property of entangled photon pairs has not been observed experimentally. It does, however, have an important application in quantum communications. The recently proposed system for long-distance entanglement transmission (through standard telecommunication ?ber) and long-duration optical storage (in trapped-atom quantum memories) [14] implicitly uses this e?ect. Let us make that cavity e?ect explicit. Consider two high-Q, initially unexcited, single-ended optical cavities—resonant at frequencies ωS +?ω and ωI ??ω , respectively—that have no excess losses. Suppose that these cavities are illuminated by the signal and idler ?elds 4

?S (t) and E ?I (t) for a Tc -sec-long time interval, after which photon-counting measurements n E ?S ≡ a ?? aS (Tc ) and S (Tc )? ? n ?I ≡ a ?I (Tc )? aI (Tc ) are performed on the resulting cavity ?elds. The intracavity photon annihilation operators at time Tc , a ?S (Tc ) and a ?I (Tc ), are related to the initial (vacuum-state) cavity operators, a ?S (0) and a ?I (0), and the input signal and idler ?elds via, a ?S (Tc ) = a ?S (0)e?(Γc +i?ω)Tc +

0 Tc

dt

Tc

?S (t), 2Γc e?(Γc +i?ω)(Tc ?t)+iωS t E ?I (t), 2Γc e?(Γc ?i?ω)(Tc ?t)+iωI t E

(4.6)

a ?I (Tc ) = a ?I (0)e?(Γc ?i?ω)Tc +

0

dt

(4.7)

where Γc is the measurement-cavity linewidth. The OPA statistics from Eqs. 3.1 and 3.2 can now be used to evaluate 2 the normalized photocount-di?erence variance, σn ≡ (? nS ? n ? I )2 /( n ?S + n ? I ), from which the presence of the magic-bullet e?ect can be deduced. The magic-bullet e?ect occurs in the narrowband measurement regime (Γc ? Γ), when the cavity-loading time 2 Tc is long enough for statistical steady-state to be reached (Γc Tc ? 1). In Fig. 3 we have plotted σn vs. Γc /Γ, 2 for several values of the normalized detuning, ?ω/Γ, with G = 0.01 and Γc Tc ? 1. Figure 3 clearly shows the magic bullet e?ect as Γc /Γ ?→ 0. The low photon-?uxes of the signal and idler beams imply that the n ? S and n ?I measurements each yield outcomes that are either zero or one. For there to be a strongly sub-shot-noise value of the 2 normalized photocount-di?erence variance (σSI ? 1), it must be that every signal-cavity count is accompanied by an idler-cavity count, even though the probability that a signal photon will make it into its cavity becomes very low as Γc /Γ decreases. Note that Γc /Γ ? 1 makes the signal and idler ?uorescence spectra approximately constant over their respective measurement-cavity linewidths, in keeping with the general conjugate-state projection requirement that the ψn be approximately constant over the n values for which |φn | di?ers signi?cantly from zero.

V. MAGIC-BULLET PENETRATION OF OPTICAL FILTERS

The cavity-loading conjugate-state projection example that we have just seen extracts single-mode measurements— ?S (t) and E ?I (t). Our ?nal example of the photon number in each cavity—from the multi-mode illumination ?elds, E the quantum magic-bullet e?ect will examine multi-mode measurements of these multi-mode ?elds. Suppose that the signal and idler beams from the OPA illuminate a pair of narrowband optical-transmission ?lters, and that the outputs from these ?lters, in turn, illuminate a pair of unity quantum e?ciency photodetectors. As shown in Fig. 4, we will assume that these ?lters are symmetrically displaced from the center frequencies of the signal and idler ?uorescence spectra, so as to select frequency pairs that are entangled. A magic-bullet e?ect, if it exists in this framework, would involve photon counting over T -sec-long time intervals satisfying ωc T ? 1, where ωc is the ?lter ?S and N ?I to denote the signal and idler count measurements over these intervals, bandwidth. In particular, using N low-?ux OPA operation implies that these measurements each yield either zero counts (a high-probability event) or one count (a low probability event). Moreover, the probability that any particular signal photon will successfully pass through the narrowband signal-beam ?lter will be very low when ωc ? Γ. Thus, if there is a strongly sub-shot-noise 2 ?S ? N ?I )2 /( N ?S + N ?I ) ? 1, it must be signal-minus-idler normalized photocount-di?erence variance, σN ≡ (N that whenever a signal photon is transmitted by the signal-beam ?lter, there is an accompanying idler photon that is transmitted by its ?lter. This signal-transmission/idler-transmission pairing is the magic-bullet e?ect: it occurs 2 whenever σN ? 1, regardless of how unlikely it is for a signal photon to pass through the signal-beam ?lter. Magic-bullet ?lter penetration is intrinsically a multi-mode e?ect, because of the large time-bandwidth product, ωc T ? 1, that is involved. To analyze this situation, let us assume that the signal and idler ?lters have no excess losses, and that their intensity transmissions have K th-order Butterworth shapes given by, |HS (ω )|2 = |HI (ω )|2 = 1 , 1 + [(ω ? ωS ? ?ω )/ωc ]2K 1 . 1 + [(ω ? ωI + ?ω )/ωc ]2K (5.1)

(5.2)

2 The OPA statistics from [9] can be combined with the analysis techniques from [8] to show that σN ≈ 1/2K , for ωc /Γ ? 1 when ωc T ? 1. Evidently, there is a magic bullet e?ect here, but it requires the use of steep-skirted optical ?lters, i.e., K ? 1.

5

VI. DISCUSSION

In this paper we have laid out the basic properties of quantum magic bullets. Starting from the continuous-variable entanglement considered by Einstein, Podolsky, and Rosen, we have shown that a projective measurement on one particle of an entangled pair projects the other into the conjugate (time-reversed) state. Conjugate-state projection, in turn, permitted us to show that when one particle successfully negotiates a scattering potential, its entangled companion will pass through a related scattering potential with probability one, no matter how unlikely the ?rst event was. This is the quantum magic bullet. In seeking optical realizations of magic bullets, we ?rst showed that the ?eldquadrature entanglement that exists between appropriately-paired signal and idler frequencies from optical parametric interactions approximates the EPR-state. The sub-shot-noise levels seen in OPA quadrature-squeezing experiments [15] implicitly con?rm this behavior. The photon-twins behavior seen in OPO intensity-di?erence measurements [11,12,13], provide a direct demonstration of the nonclassical correlation between the signal and idler frequencies. The full, joint Bose-Einstein state has also been seen, in the output from a parametric downconverter, via quantum-state tomography [16]. The ?eld-quadrature form of optical magic bullets is an asymptotic e?ect that is strongly nonclassical only when the average photon number per mode is high. Furthermore, it requires the use of homodyne or self-homodyne measurements. Photon-pair counting measurements in the low-?ux operating regime provide a more attractive optical magic-bullet scenario, although they do not represent the perfect analog of the EPR-state position-momentum en? ?→ ∞ ?eld-quadrature state does. The single-mode realization of the photon-pair magic tanglement that the N bullet, based on intracavity photon-counting, has yet to be demonstrated experimentally. Nevertheless, it is intrinsic to the coupling of polarization-entangled photons from an OPA pair [9] into a trapped-atom quantum memory [17] for the purpose of long-distance transmission and long-duration storage of qubits [14]. The multi-mode realization of the photon-pair magic bullet requires that steep-skirted, low-excess-loss ?lters be used. A possible experimental realization might use a grating/lens/pinhole system in which the frequency components of the signal and idler beams were ?rst dispersed in angle, then focused to spatially separate them on a detector plane, where a pinhole would provide steep-skirted frequency selection prior to photodetection.

ACKNOWLEDGMENTS

This research was supported by the National Reconnaissance O?ce under contract NRO000-00-C-0032.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

C. H. Bennett and P. W. Shor, IEEE Trans. Inform. Theory IT-44, 2724 (1998). A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). A. Furusawa, J. L. S?rensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, Science 282, 706 (1998). S. L. Braunstein, Phys. Rev. Lett. 80, 4084 (1998). S. L. Braunstein, Nature 394, 47 (1998). S. Lloyd and J.-J. E. Slotine, Phys. Rev. Lett. 80, 4088 (1998). S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82, 1784 (1999). J. H. Shapiro and K.-X. Sun, J. Opt. Soc. Am. B 11, 1130 (1994). J. H. Shapiro and N. C. Wong, J. Opt. B: Quantum Semiclass. Opt. 2, L1 (2000). H. P. Yuen and J. H. Shapiro, IEEE Trans. Inform. Theory IT-26, 78 (1980). S. Reynaud, C. Fabre, and E. Giacobino, J. Opt. Soc. Am. B 4, 1520 (1987). K. W. Leong, N. C. Wong, and J. H. Shapiro, Opt. Lett. 15, 1058 (1990). J. Teja and N. C. Wong, Opt. Express 2, 65 (1998). J. H. Shapiro, “Long-distance high-?delity teleportation using singlet states,” to appear in Proc. Fifth International Conf. on Quantum Communication, Measurement, and Computing, Capri, 2000, edited by O. Hirota and P. Tombesi. [15] L.-A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986). [16] M. Vasilyev, S. K. Choi, P. Kumar, and G. M. D’Ariano, Phys. Rev. Lett. 84, 2354 (2000). [17] S. Lloyd, M. S. Shahriar, and P. R. Hemmer, “Teleportation and the quantum Internet,” submitted to Phys. Rev. A. (quant-ph/003147)

6

Pump

Idler

c

(2)

Signal

FIG. 1. Schematic of a doubly-resonant optical parametric ampli?er.

-64

(a)

Noise Power (dBm)

-66 -68 -70 -72 -74 -76 2 3 4 5 Frequency (MHz) 6

(b)

FIG. 2. Shot-noise level (a), and signal-minus-idler intensity di?erence (b), from a KTP optical parametric oscillator.

7

1

2 G = 0.01 Normalized Photocount-Difference Variance

0.1

0.01

Detuning = 0 Detuning = 1

0.001

Detuning = 2 Detuning = 5 Detuning = 10

0.0001 0.001

0.01

0.1

1

Measurement-Cavity Linewidth/Source-Cavity Linewidth

2 FIG. 3. Normalized photocount-di?erence variance, σn , vs. the ratio of measurement-cavity linewidth to source-cavity 2 linewidth, Γc /Γ, for 1% OPA pumping (G = 0.01) and various values of the normalized detuning, ?ω/Γ.

8

Signal

Signal filter BW

Signal linewidth

Idler linewidth Idler Idler filter BW

FIG. 4. Schematic for ?lter-penetration optical magic bullets. BW: bandwidth.

9

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