## Abstract

In the preceeding chapters the laser field has been treated in the semi-classical approximation, where the electric field is represented by \(\varvec{E}=\varvec{E}_0\cos (\omega t)\). This assumption is adequate for considering the interaction between strong laser fields with macroscopic ensembles of independent atoms, as in this limit the quantum description of the light-field is indistinguishable from the classical treatment, for reasons that will be discussed below. However, in order to exploit the effect of the cooperative non-linearity at the single-photon level it is necessary to consider the quantised electromagnetic field, without which the concept of a photon becomes meaningless. Importantly, in quantum optics it is not the amplitude of the electric field, but rather the temporal and spatial correlations of the field that reveal the non-classical nature of light. Before considering the cooperative effect, it is necessary to first outline some fundamental ideas of the quantised field.

## Keywords

Coherent State Optical Depth Probe Beam Photon Number Dipole TrapIn the preceeding chapters the laser field has been treated in the semi-classical approximation, where the electric field is represented by \(\varvec{E}=\varvec{E}_0\cos (\omega t)\). This assumption is adequate for considering the interaction between strong laser fields with macroscopic ensembles of independent atoms, as in this limit the quantum description of the light-field is indistinguishable from the classical treatment, for reasons that will be discussed below. However, in order to exploit the effect of the cooperative non-linearity at the single-photon level it is necessary to consider the quantised electromagnetic field, without which the concept of a photon becomes meaningless. Importantly, in quantum optics it is not the amplitude of the electric field, but rather the temporal and spatial correlations of the field that reveal the non-classical nature of light. Before considering the cooperative effect, it is necessary to first outline some fundamental ideas of the quantised field.

## 8.1 The Quantised Electric Field

^{1}Instead, the electric field is represented by the operator [1]

### 8.1.1 Coherent States

Having laid out a framework in which the photon can be defined, it is instructive to consider how to discriminate between a coherent state and non-classical light.

## 8.2 Photon Statistics

For a classical electric field, a photodiode can be used to generate a continuous signal proportional to the intensity of the field. Similarly, for a quantum field, the intensity at a detector is related to the expectation value for the intensity \(\,\langle \hat{I}({\varvec{\mathrm{ r} }},t)\rangle \), where the intensity operator is defined as \(\hat{I}\,{=}\,\hat{\varvec{E}}^{(-)}({\mathrm{ r} },t)\,{\cdot }\,\hat{\varvec{E}}^{(+)}({\mathrm{ r} },t)\). However, this is the same for both a single photon state and a coherent state with \(\bar{n}=1\). It is therefore insufficient to simply measure the field intensity, instead it is necessary to consider the photon statistics of the input field.

*anti*-

*bunching*, applicable to all states with \(g^{(2)}(0)<1\), as photons arrive at well spaced intervals compared to the random distribution of arrival times expected for the coherent state. Similarly, states with \(g^{(2)}(0)>1\) are

*bunched*as photons are more likely to arrive together.

In practise, \(g^{(2)}(\tau )\) is measured with a Hanbury Brown Twiss (HBT) interferometer [5], shown schematically in Fig. 8.1 which uses a 50:50 beam-splitter to separate light onto a pair of photon counters. The correlator records the coincidences between the counters as a function of delay \(\tau \) which can be used to determine the normalised correlation function. The first experimental evidence of anti-bunching was from observation of suppressed correlations at \(\tau =0\) in the resonance fluorescence of a single sodium atom [6], followed by the measurement of \(g^{(2)}(0)=0\) for fluorescence of a single ion [7].

## 8.3 Photon Blockade

As every mode of the quantised electric field is a harmonic oscillator, there is a discrete ladder of energies which for laser light is initially populated with the Poissonian distribution of the coherent state. However, if the harmonicity of this ladder can be broken, it is possible to observe non-classical states of light. An example of this is the interaction between a single atom and the mode inside an optical cavity. The effect of the atom-light interaction is to create an anharmonic energy ladder dependent upon the photon occupation, as illustrated in Fig. 8.2a. The interaction causes the cavity to be shifted off resonance with the probe laser after the first photon is absorbed, preventing another photon from entering the cavity until the first photon leaves. This system allows a coherent state to be filtered into a train of single photons, an effect known as a *photon blockade* [8] or a *photon turnstile*. The resulting anti-bunched output has been observed experimentally for optical cavities [9, 10], and more recently in a superconducting microwave cavity [11], where an artificial atom is used to overcome the limited fidelity in optical cavities due to residual motion of the atom.

For the cooperative non-linearity due to dipole blockade, a similar effect can be realised. In Chap. 5, the suppression of transmission was interpreted as the formation of an entangled state with one atom in the EIT dark state and the remaining atoms resonantly scattering light from the probe beam. Combining this with the concept of a quantised electric field, the formation of the single collective dark state can only involve a single photon from the probe field. Any other photons arriving in the blockaded ensemble now resonantly couple to the excited state of the atoms and are scattered out of the mode of the probe beam, as shown schematically in Fig. 8.2b. This allows only a single photon to pass through un-scattered, resulting in a single photon output in the mode of the probe laser.

An important difference between these two schemes is that for the atom+cavity system, the energy of the optical transition is shifted for \(n>1\) so the cavity completely rejects all but a single photon, ensuring there will never be multiple photons at the cavity output. The Rydberg blockade mechanism, however, does not shift the energy of the optical transition for the probe laser. Instead, the medium changes from being transparent for the first photon, to opaque for \(n>1\). The photon scattering process for \(n>1\) is probabilistic, which may limit the fidelity as a photon turnstile. Nonetheless, it provides a mechanism to generate non-classical states of light from an input field initially in a coherent state without the need for an optical cavity.

### 8.3.1 Dark State Polariton

*dark state polariton*[12]. In the analysis of EIT in Sect. 4.2, the dressed states of the atom were introduced to explain the EIT dark state assuming a constant amplitude driving field. For a weak probe beam, the probe electric field can be coupled to the atomic evolution using the Maxwell-Bloch equations to model the propagation through the medium. The result is that on the two-photon resonance, the system forms a stable, lossless quasi-particle known as a dark state polariton \(\Psi \) [13]

The polariton picture gives two insights relevant to achieving photon blockade. The first is that it shows that a large group index is required in order to transfer the single-photon field into atomic excitation in the medium. Without this, the Rydberg state is not populated and there is no dipole blockade. The second follows on from this, as the requirement of a large group-velocity means the single-photon polariton propagates slowly through the medium. During this propagation time, subsequent photons entering the medium should be scattered, introducing a characteristic delay between photon emission at the output which will be referred to as the blockade time, \(\tau _{\mathrm{ b} }\). In the limit of a strong driving field, this should result in a regular train of single photons separated by time \(\tau _{\mathrm{ b} }\).

Applying the condition for \(\vartheta \) to the experiments presented in Fig. 7.11a for the suppression of transmission for the 60\(S_{1/2}\) state, the weak-probe group index was \({\mathrm{ n} }_{\mathrm{ gr} }\sim 4\times 10^4\), corresponding to a mixing angle of \({\sim }90^\circ \). For these experiments, the polariton is almost entirely composed of atomic excitation, meaning these photon-statistics must play a role in the observed suppression. However, comparing the blockade radius of \(R_{\mathrm{ b} }\sim 5\,\upmu \)m to the 10 \(\upmu \)m 1/e\(^2\) radius of the beam waist shows the probe laser interacts with of order 16 blockade spheres over the beam cross-section. Thus, whilst each blockade region can potentially create a single-photon, the total output can still have as many as 16 photons which makes the direct observation of non-classical light in the probe beam challenging using \(g^{(2)}\), as will be demonstrated below.

This analysis of the photon blockade due to dipole–dipole interactions gives a qualitative description of the mechanism, but a more quantitative approach is required to determine the potential fidelity and parameter range in which photon blockade can be realised. Solving the complete quantum dynamics for a quantised field coupled to an interacting \(\mathcal{ N} \)-atom system is a non-trivial problem, and will not be attempted here. However, it is still possible to gain an insight into the expected correlations for the light output from a single blockaded ensemble. This will be the subject of the following sections.

## 8.4 Simple Model for \(g^{(2)}(\tau )\)

During the propagation of the slow polariton, the other photons are scattered by resonantly coupling on the two-level transition between \({\vert g \rangle }\) and \({\vert e \rangle }\). If the ensemble is not completely optically thick on this probe-only transition, then the efficiency of this scattering process is limited. To account for this effect, photons in the scattered channel can be transferred back into forward channel with a probability equivalent to the probe-only transmission, which is parameterised in terms of the optical depth \({\mathrm{ OD} }=-\log _{\mathrm{ e} }(T)\).

The results for low power are shown in Fig. 8.4a which shows significant anti-bunching up to \(\tau =\tau _{\mathrm{ b} }\) in the forward mode, and bunching for the scattered mode. This occurs because at low power there is a very low probability of observing any photons, so a large proportion of the photons arrive in the medium separated by times \(t>\tau _{\mathrm{ b} }\) and pass through. For the scattered channel, there are now a relatively large fraction of multi-photon events compared to Poissonian statistics as the single photon component is suppressed, giving the observed bunching. For the strong probe results in (b), the bunching of the scattered channel becomes insignificant as most photons are scattered, with only a very slight change in the photon count distribution from the Poissonian input. In the forward channel however, a periodic anti-bunching is observed with strong bunched peaks at harmonics of \(\tau =\tau _{\mathrm{ b} }\). These spikes are asymmetric as it is not possible for photons to arrive closer in time than \(\tau _{\mathrm{ b} }\), but the next photon may arrive at anytime later, smearing out the sharp peak. This also damps the amplitude of peaks at later times. For higher powers, the probability of having at least one photon in each time step \(\Delta t\) tends to unity, causing the \(g^{(2)}\) to look more like a comb of delta-functions.^{2}

To explore the dependence on the optical depth, the model is run for a 10 pW probe with \(\mathcal{ N} =100\) and 400, corresponding to \({\mathrm{ OD} }=2.2\) and 8.8 respectively. The results are shown in (c) compared to \(\mathcal{ N} =200\), with the variation in \(\tau _{\mathrm{ b} }\) clearly visible from the arrival of the first peak. In the inset, the effect of small optical depth is easy to see, as it suppresses the anti-bunching at short times and rapidly damps out the peak visibility. An optical depth of \({\mathrm{ OD} }\gtrsim 4\) is therefore required to observe blockade experimentally.

Another effect that can be added to the model is having more than one blockaded volume in the cross-section of the beam. This is achieved by randomly splitting the input train \(C(t)\) between \(\mathcal{ M} _{\mathrm{ b} }\) blockade regions, and performing the scattering on each blockade independently. The forward scattering from each is then combined, and the correlation of the total output found. Results calculated for the original parameters of \({\mathrm{ OD} }=4.4\) are shown in (d). As more blockade regions are included, the anti-bunching of the output light is suppressed as \(\mathcal{ M} _{\mathrm{ b} }\) photons can propagate through the medium, which for \(\mathcal{ M} _{\mathrm{ b} }\gg \bar{n}\) allows the initial coherent state to be unchanged. Thus the visibility of the anti-bunching of the output light is very small for the experiments of Sect. 7.4 with \(\mathcal{ M} _{\mathrm{ b} }=16\), as mentioned above.

In summary, these results show that the blockaded ensemble can be used to create a regularly spaced, highly correlated train of single photons, analogous to creating ‘hard-edge’ photons in a 1D lattice. The repetition rate of the photon pulses is \(\tau _{\mathrm{ b} }^{-1}\sim \) MHz, which could be used as a semicontinuous single-photon source for quantum information. The fidelity of the single photon output state is limited by the optical depth of the ensemble, however for 400 atoms the model predicts \(g^{(2)}(\tau \,{<}\,\tau _{\mathrm{ b} })\,{<}\,10^{-2}\) which is smaller than the uncertainty in a typical measurement of \(g^{(2)}\) [9, 10]. An implicit assumption of this simple model is that the polariton is formed as soon as the photon is in the medium. However, there may be a finite timescale associated with the formation of a polariton. During this time two-photons could pass through the medium, which would compromise the fidelity. These EIT transients are considered in the next section.

From this simple model, it has been possible to verify the parameter range over which photon blockade can be realised, requiring an optical depth equivalent to several hundred atoms confined within a single blockade volume. This clearly represents a complex system to model rigorously, however if we consider the case of only a few atoms it is possible to calculate the correlations of the scattered field.

## 8.5 Resonance Fluorescence Correlation Functions

### 8.5.1 The Source-Field Expression

In Chap. 5 an \(\,{\mathcal{ N} }{}\)-atom model was developed to calculate the properties of the interacting EIT system. Whilst this model is based on classical driving fields, these optical Bloch equations can be used to calculate the properties of the scattered light field from the atoms using the source-field expression [1]. This states that the electric field operator at position \({\varvec{\mathrm{ r} }}\) is given by \(\hat{\varvec{E}}^{(+)}({\varvec{\mathrm{ r} }},t)=\hat{\varvec{E}}_{\mathrm{ f} }^{(+)}({\varvec{\mathrm{ r} }},t)+\hat{\varvec{E}}_{\mathrm{ sf} }^{(+)}({\varvec{\mathrm{ r} }},t)\), where \(\hat{\varvec{E}}_{\mathrm{ f} }^{(+)}({\varvec{\mathrm{ r} }},t)\) is the incident field and \(\hat{\varvec{E}}_{\mathrm{ sf} }^{(+)}({\varvec{\mathrm{ r} }},t)\) is the radiation field of the atomic dipole, known as the source-field term. This is the quantum analogue of the classical Ewald-Oseen extinction theorem [17], which describes ‘absorption’ as a destructive interference between the incident plane wave and the radiated dipole field.

^{3}

### 8.5.2 Correlation Function

### 8.5.3 Cooperative Emission from Incoherent Atoms

If \(V(R_{ij})=0\), similar curves are obtained for \(\,{\mathcal{ N} }{}>1\), however when \(V(R_{ij})>\gamma _{\mathrm{ EIT} }\), Fig. 8.5b shows that interactions cause the two and three atom system to be very strongly bunched at \(\tau =0\), as expected from the simple model in Sect. 8.4. This bunching can be understood from the analytic EIT dark state for the interacting two-atom system in Eq. 5.21, which has a \({\vert ee \rangle }\) component in place of the \({\vert rr \rangle }\) state expected without interactions. If one of the atoms emits a photon, then \({\vert ee \rangle }\) was populated and correspondingly the other atom must emit a photon within a few spontaneous lifetimes. This is a cooperative emission process mediated by the dipole–dipole interactions. The correlation function therefore verifies that the blockade mechanism scatters multiple photons with very high probability. In (c) and (d) the correlations for \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e/2\) and \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e\) are plotted, showing that for a strong probe field the blockade condition is violated and the light becomes anti-bunched at short times, similar to the correlations for the probe-only system in (a).

Figure 8.5b therefore shows that a Rydberg superatom could be used as a correlated photon source. The directionality of the emission is considered below.

### 8.5.4 Distinguishable Emission

Figure 8.6 shows the results for the two-atom model calculated for the same parameters as before. In the case of two-level atoms, (a), \(g_{21}^{(2)}(\tau )=1\) for all times as the atoms are independent with no correlations between their emission. This is why the self-correlation shows the same correlation function as for \(\,{\mathcal{ N} }{}=1\) in Fig. 8.5a. For the interacting EIT system however, the bunched behaviour is dominated by the cross-correlation, seen from Fig. 8.6b, which is consistent with the interpretation of the bunching as the population of \({\vert ee \rangle }\) discussed above. For (b)–(d), the self-correlation remains approximately constant, whilst the cross-correlations change from being strongly bunched to anti-bunched as the probe power is increased.

These results show that it is possible to not only use the strong Rydberg interactions to generate a single-photon output train, but also to obtain highly correlated fluorescence emission from a pair of atoms. In the current assumption of incoherent phase, the direction of the fluorescence will be uncorrelated, however if the phases are well defined there exist geometries in which the correlations are insensitive to the atomic position.

### 8.5.5 Coherent Emission

To check the effects of the the incoherent assumption from above, the correlation function is evaluated using the \({\mathrm{ e} }^{\pm {\mathrm{ i} }k\hat{{\varvec{\mathrm{ r} }}}\cdot {\varvec{\mathrm{ r} }}_i}\) phase-factors. This also requires a modification of the optical Bloch equations to include the phase of the driving field in the Rabi frequencies as given in Eq. C.4. The correlation function is then calculated for a pair of atoms with the detectors placed orthogonal to the probe wave-vector \({\varvec{\mathrm{ k} }}_{\mathrm{ p} }\) as a function of atomic separation in terms of the probe wavelength \(\lambda \) for \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e/2\), \(\,\Omega _{\mathrm{ c} }=\,\Gamma _e\) and \(V(R)=2\,\Gamma _e\).

The results are plotted in Fig. 8.7 which shows the correlations for atoms aligned parallel (a) and perpendicular (b) to the probe beam. For the parallel geometry in (a), the photons are bunched independent of separation \(R\), whilst for the perpendicular configuration in (b) there is a destructive interference for \(R=m\lambda +\lambda /4,3\lambda /4,\ldots \) resulting in anti-bunching. This suggests the geometry of (a) is more robust for observation of photon blockade, and could be used to generate highly correlated photon pairs. For other detector and atom geometries, the correlation function becomes more sensitive to displacement, resulting in more complex correlation functions.

## 8.6 Summary

In this chapter the concept of a quantised light-field has been introduced, along with its relevance to generating non-classical light-fields using the blockade effect. Rydberg atom interactions ensure only a single dark state polariton can pass lossless through the blockade region, whilst other photons arriving at the medium will be scattered to achieve photon blockade.

A simple model has been used to predict the correlation function arising from this interaction, which shows that a large optical depth in a single blockade sphere is required to obtain a high fidelity single photon output train. Quantitative calculations of the correlations in the scattered light from a few blockaded atoms verify that the blockade causes the atoms to scatter pairs of photons with very high probability, as seen from the strong bunching in the correlation function at short times. These calculations also highlight the importance of geometry in the system, with photon blockade working better for atoms parallel to the probe to avoid sensitivity to atomic position.

The process considered in this chapter is photons scattered out of the probe beam which is destructive. However, this scattering is conditional on whether another photon is in the medium. This conditional behaviour for the case of one or two photons is a first step towards the development of a two-photon quantum gate, as it shows the blockade mechanism is already sufficient to give a non-linearity at the single photon level. Future work should look for ways to use this effect in the dispersive regime to create a phase-shift on the photons. The first challenge though is to create, and probe, a single blockaded ensemble which has a sufficiently large optical depth. Measurement of the anti-bunching from the photon blockade would enable verification of confinement to \(R<R_{\mathrm{ b} }\) before moving on to explore the dispersive regime.

## Footnotes

## References

- 1.R. Loudon,
*The Quantum Theory of Light*, 2nd edn. (OUP, Oxford, 1997)Google Scholar - 2.V. Weisskopf, E. Wigner, Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie. Z. Phys.
**63**(1), 54 (1930)ADSzbMATHCrossRefGoogle Scholar - 3.M.O. Scully, M.S. Zubairy,
*Quantum Optics*(CUP, Cambridge, 2002)Google Scholar - 4.H.C. Carmichael,
*Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations*(Springer, Berlin, 2002)Google Scholar - 5.R. Hanbury Brown, R.Q. Twiss, A test of a new type of stellar interferometer on sirius. Nature
**178**, 1046 (1956)ADSCrossRefGoogle Scholar - 6.H.J. Kimble, M. Dagenais, L. Mandel, Photon antibunching in resonance fluorescence. Phys. Rev. Lett.
**39**(11), 691 (1977)ADSCrossRefGoogle Scholar - 7.F. Diedrich, H. Walther, Nonclassical radiation of a single stored ion. Phys. Rev. Lett.
**58**(3), 203 (1987)ADSCrossRefGoogle Scholar - 8.A. Imamoglu, H. Schmidt, G. Woods, M. Deutsch, Strongly interacting photons in a nonlinear cavity. Phys. Rev. Lett.
**79**(8), 1467 (1997)ADSCrossRefGoogle Scholar - 9.K.M. Birnbaum, A. Boca, R. Miller, A.D. Boozer, T.E. Northup, H.J. Kimble, Photon blockade in an optical cavity with one trapped atom. Nature
**436**, 87 (2005)ADSCrossRefGoogle Scholar - 10.B. Dayan, A.S. Parkins, T. Aoki, E.P. Ostby, K.J. Vahala, H.J. Kimball, A photon turnstile dynamically regulated by one atom. Science
**319**, 1062 (2008)ADSCrossRefGoogle Scholar - 11.C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J.M. Fink, A.A. Abdumalikov, M. Baur, S. Filipp, M.P. da Silva, A. Blais, A. Wallraff, Observation of resonant photon blockade at microwave frequencies using correlation function measurements. Phys. Rev. Lett.
**106**, 243601 (2011)ADSCrossRefGoogle Scholar - 12.M. Fleischhauer, M.D. Lukin, Dark-state polaritons in electromagnetically induced transparency. Phys. Rev. Lett.
**84**(22), 5094 (2000)ADSCrossRefGoogle Scholar - 13.M. Fleischhauer, A. Imamoglu, J. Marangos, Electromagnetically induced transparency: optics in coherent media. Rev. Mod. Phys.
**77**, 633 (2005)ADSCrossRefGoogle Scholar - 14.L. Mandel, E. Wolf,
*Optical Coherence and Quantum Optics*(CUP, Cambridge, 2008)Google Scholar - 15.A. Imamoglu, Y. Yamamoto, Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions. Phys. Rev. Lett.
**72**(2), 210 (1994)ADSCrossRefGoogle Scholar - 16.A.J. Shields, Semiconductor quantum light sources. Nature Photon.
**1**, 215 (2007)Google Scholar - 17.M. Born, E. Wolf,
*Principles of Optics*(CUP, Cambridge, 1999)Google Scholar - 18.G.S. Agarwal, Quantum statistical theories of spontaneous emission and their relation to other approaches. Springer Tracts Mod. Phys.
**70**, 1 (1974)ADSCrossRefGoogle Scholar - 19.K. Mølmer, Correlation functions and the quantum regression theorem. http://owww.phys.au.dk/quantop/kvanteoptik/qrtnote.pdf. Accessed 21 Nov 2010
- 20.K. Mølmer, Y. Castin, Monte Carlo wavefunctions in quantum optics. Quantum Semiclass Opt.
**8**(1), 49 (1996)ADSCrossRefGoogle Scholar - 21.A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, P. Grangier, Observation of collective excitation of two individual atoms in the Rydberg blockade regime. Nat. Phys.
**5**, 115 (2009)CrossRefGoogle Scholar - 22.E. Urban, T.A. Johnson, T. Henage, L. Isenhower, D.D. Yavuz, T.G. Walker, M. Saffman, Observation of Rydberg blockade between two atoms. Nat. Phys.
**5**, 110 (2009)CrossRefGoogle Scholar