0 excitations 1 excitation 2 excitations
3% 91% 6%
Occurrences
Counts during detection window
0 5 10 15 20 25 30 35 40 2 4 6 8 10 12 14 16 18 20 0
Figure 9.3: Measuring the number of excitations on the two-atom entangled state | i by flu- orescence rates. The ground state |gi is shelved outside of the cooling repump manifold with a narrowband ⇡ pulse (see text). The subsequent fluorescence rate under cooling and repump beams corresponds to the total number of excitations in state| i. The ideal| ihas only a single excitation.
ground-state cooling significantly reduces this e↵ect, but sideband cooling reduces the source repetition rate by a factor of 100 and the repetition rate of our measurement drops commensurately.
9.2
Spatial interference
Now that we have prepared our entangled state, we may investigate its optical properties. We scatter a second photon from the entangled pair by a second excitation pulse, step 4 in Fig. 9.2. This time our excitation probability can high and we choose pe= 0.80±0.02. After the second scattering event, the joint atom-photon system is
0↵= p1
2
⇣
|0,1i+ei( 0)|1,0i⌘⌦|s, si , (9.5)
where the photon states |0,1i and |1,0i correspond to a photon in the emission mode of atoms A and B, respectively, and 0 depends on the exciting laser phase and path length di↵erence 0 =⇣ 0LB 0LA⌘+⇣ 0DB 0DA⌘. Detecting a photon in a mode where atoms A and B are indistinguishable projects the state| 0i to
0 p ↵ = p1 2 ⇣ 1 +ei 0⌘⌦|s, si. (9.6)
The photon detection probability is therefore
P / ⌦ p0 0p ↵2
= 1 + cos 0 . (9.7)
The probability of detecting a photon scattered from| i in a common mode depends on the entanglement phase and the observation phase 0. At an observation point a distance
d ,rB rAfrom the atom pair the spatial modes are indistinguishable. The atom-atom
entanglement therefore causes a spatial interference pattern in the detection probability of photons scattered from| i, enhancing the detection probability when = 0= 0
1.0 1.2 1.4 1.6 1.8 2.0 2.2 0.6 0.8 1.0 1.2 1.4 0.0 0.5 1.0 1.5 2.0 2.5 Phase di↵erence, /⇡ Det . prob., P ⇥ 10 3 Enhanceme n t, R Entangled state| Separable state|⇠ Separable state|⇣
Figure 9.4: Interference fringe in the spatial mode of single photons scattered from the entangled state| i(black), the single-excitation separable state|⇣i(orange) and the maximum-interference separable state|⇠i(purple). | iand |⇠iare shown with interference fringe fits (solid lines). The dashed grey line is the mean photon detection probability of|⇣i, and defines regions of enhanced (red) and suppressed (blue) emission. The dashed black line is the expected interference of | i given the measured state preparation fidelity, and corresponds to the theoretical predictionV =C. Measurement by Gabriel Araneda.
and suppressing it when =⇡. In particular the visibility of the interference fringes in radiation from a pair of two-level emitters with a single excitation should be equal to the concurrence of the bipartite quantum state [292, 293]. The concurrence is an entanglement monotone of the two-qubit state. Like the bipartite pure state entanglement
E(| i) = Tr⇢log2⇢, (9.8)
where ⇢ is a partial trace over one of the subspaces, the concurrence is a measure of entanglement that varies from zero for a separable state to one for a perfectly entangled state.
This spatial interference pattern has not been measured before this work, the photon detection probability is simply too low to image the pattern directly. However, we can efficiently sample the same distribution with our near-confocal lens apparatus. By moving the mirror position as shown in Fig. 9.1 we tune the path length di↵erence between atoms A and B in the common detection mode, equivalent to changing the position of detector in the spatial interference pattern.
Because the atom-mirror path length also appears in the Cabrillo entanglement phase , we must displace the mirror between the entangling and measurement steps, step 3 in Fig. 9.2. Assuming that the interatomic distance is fixed, the interference term is = k d where d = d0 d is the path length change between entanglement and measurement. After receiving an entanglement herald we displace the mirror with a piezo in a time that depends on . For /⇡ = 2.5 (the largest displacement we measure) the displacement time is ⌧ = 220µs.
Figure 9.4 shows the detection probabilityPfor several two-atom states as a function of the path length phase di↵erence . The maximum and minimum measured probabilities for the entangled state | i are P( = 0) = (2.10 ± 0.07)⇥10 3 and P( = ⇡) = (1.17 ±0.12)⇥10 3. Fitting an interference fringe amplitude to the data gives a visibility of V = 0.27±0.03, consistent with the concurrenceC = 0.31±0.10 calculated from our parity-reconstructed state and the prediction V =C [292, 293].
§9.3 Summary 137
For comparison, we measure the interference pattern of two separable states. These states are prepared by a combination of optical pumping, shelving to the 5 D5/2, mj = –52 Zeeman state with the 1.76 µm quadrupole transition beam, and global RF pulses on the |gi $|si transition. The state |⇣i=|e, gi is the separable single-excitation state, corresponding to an excitation on a definite atom,A or B.
Scattering a photon under the same conditions used for | i shows no dependence on the path length di↵erence. The interference visibility is ⇡ 0, consistent with the
C(|⇣i) = 0 for a separable state. The single-photon detection probabilityP(|⇣i) is used to define a relative detection probability R =P/P(|⇣i), the right-side axis in Fig. 9.4. The constructive (destructive) interference enhancement (suppression) factor of the entangled state is Rsup = 1.29 (Rsub = 0.72). Because the states | i and |⇣i each contain a single excitation, we expect their mean detection probability over the entire emission mode to be equal, and indeed, the mean of the R(| i) interference fringe is 0.99±0.08.
Separable states with more than one excitation can also produce interference patterns. Single photons scattered from the most general bipartite state
| i=a|g, si+b|s, gi+c|g, gi+d|ssi , (9.9) interfere with visibility
V| i= 2|ab|
|a|2+|b|2+ 2|c|2 , (9.10)
even though the state has concurrence C| i = 2|cd ab|. From Eqn. 9.10 we can see that even separable bipartite states with a mean excitation greater than one have an interference pattern; the rule V =C holds only for d= 0. However the visibility of even general separable states is bounded by V 1/2. We prepare the separable state
|⇠i= 1
2(|gi+|si)⌦(|gi+|si) (a=b=c=d= 1
2), (9.11) using global RF pulses. |⇠i maximizes the interference over the set of separable states. We observe an interference fringe visibility of V|⇠i = 0.15±0.08, also shown in Fig. 9.4.
As expected, this is approximately half of V| i.
9.3
Summary
We have demonstrated how interference between the entangled components of an atomic ensemble can selectively enhance or suppress single-photon emission into free-space modes. We prepared a bipartite entangled state by coupling two trapped ions to a common optical mode. The spatial interference pattern of our two-atom ensemble varies depending on the mean excitation, the degree of entanglement, and the path length di↵erence between the two atoms and an observer. We detect photons from the ensemble with high aperture lens objectives and tune the path length di↵erence using a remote, piezo-mounted mirror to measure the continuous interference pattern.
For bipartite states with a single excitation the interference visibility is equal to the concurrence. An interference measurement of this sort is therefore an entanglement witness for states with no|ssicomponent. In Part III we extend this theory to consider excitations that are distributed amongst many more than two atoms. As the number of atoms in- creases, emission from the entangled ensemble becomes increasingly directed [294]. In the many atom limit, such collective states are a means of efficiently and reversibly coupling
optical qubits into atoms, even without high aperture optics.
As with the previous chapters on our trapped-atom single-photon source, the success probabilities of both our photonic entanglement gate and second photon scattering event are limited dramatically by the collection efficiency of our high-aperture lens atom-light couplers. In the following chapters we turn our attention to a new strategy for efficiently coupling trapped atoms with single-pass optics and demonstrate some of the necessary components.
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Precise hemispheric mirrors
I am a plain, old-fashioned mirror from a bygone age, made of good white metal that stays clear without being polished... I am going to discuss serious matters now. Pay close attention, everyone.
– Unknown, Okagami, the Great Mirror (1119)
Useful quantum networks, be they large scale communication networks or distributed entanglement computing networks, require the efficient interaction of optical fields with material quantum systems. To reach the required efficiency, free-space atom-light cou- plers require optics with larger apertures than the lens objectives used in Chaps. 6 to 8. High precision, high numerical aperture mirrors are one means of mediating sufficiently strong atom-light coupling. In Chap. 4 we saw that the ideal mode converter for a point source emitting spherical waves is a deep parabolic mirror [118, 131]. This single-pass coupling approach requires high numerical aperture (NA) reflectors with sub-wavelength surface precision [111]. However the fabrication of high-NA mirrors with sufficient surface smoothness and form precision remains a technological challenge [295–297].
Whereas a parabolic mirror is the desired reflector for converting spherical waves into plane waves, a hemispheric mirror maximizes the self-interaction of a source by returning spherical waves from the source to their origin. The hemispheric mirror is an intriguing special case for high-NA atom-light couplers that bridges the gap between single-pass optics and cavity quantum electrodynamics. Like an optical resonator, a single hemispheric mirror may enhance atom-light interactions by shaping the vacuum mode density around an emitter, but unlike an optical resonator the hemisphere-mediated atom-light interaction is single-pass. It has been predicted [298] that the spontaneous emission rate of an atomic electron at the centre of curvature (CoC) of a spherical mirror may be suppressed or enhanced depending on the radius of the mirror, even when the mirror radius is much larger than the atomic wavelength. A previous attempt to measure such an e↵ect with spherical optics measured emission rate fluctuations of 1% [255]. An ideal hemisphere that covers exactly half of the atomic emission solid angle will achieve the greatest possible modification, enhancing the spontaneous emission rate by a factor of two when the radius is
R=n2+4 where n is a positive integer and is the transition wavelength, and completely suppressing spontaneous emission when the radius isR=n2. Because deviations from an ideal hemisphere reduce the degree of suppression and enhancement, demonstrating this e↵ect requires the fabrication of hemispheric mirrors with great surface precision.
The fabrication of ideal reference optics is also of interest to the broader optics com- munity. The hemisphere retro-reflects an incoming spherical wavefront and could be used as a reference null for characterizing high-NA focusing optics [115]. Optical-reference mi- croscopy requires reference surfaces considerably better than the surface to be measured.