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Photon Blockade

  • Jonathan D. Pritchard
Chapter
First Online:
Part of the Springer Theses book series (Springer Theses)

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 Trap 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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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.

8.1 The Quantised Electric Field

A full derivation of the quantisation of the electromagnetic field can be found in many standard quantum optics textbooks e.g. [1], and here only the final results are detailed. The electric field is quantised in a finite volume \(V\) to obtain a set of spatial modes described by wavevector \({\varvec{\mathrm{ k} }}\), each of which has two transverse polarisations \(\lambda \) defined by the polarisation unit vector, \(\hat{\varvec{e}}_{{\varvec{\mathrm{ k} }},\lambda }\). Each mode represents a quantum harmonic oscillator with a ladder of energies separated by \(\hbar \omega _{\varvec{\mathrm{ k} }}\), where \(\omega _{\varvec{\mathrm{ k} }}=c \vert {\varvec{\mathrm{ k} }}\vert \). In this picture, a photon corresponds to a single excitation of the oscillator mode. Photons are added or removed from the mode using the creation (\(\hat{a}^\dagger _{{\varvec{\mathrm{ k} }},\lambda }\)) and annihilation (\(\hat{a}_{{\varvec{\mathrm{ k} }},\lambda }\)) operators which act on the wavefunction \({\vert n_{{\varvec{\mathrm{ k} }},\lambda } \rangle }\) representing the number of photons in mode \({\varvec{\mathrm{ k} }}\) as follows,
$$\begin{aligned} \hat{a}_{{\varvec{\mathrm{ k} }},\lambda }{\vert n_{{\varvec{\mathrm{ k} }},\lambda } \rangle }=\sqrt{n_{{\varvec{\mathrm{ k} }},\lambda }}{\vert n_{{\varvec{\mathrm{ k} }},\lambda }-1 \rangle },\quad \hat{a}_{{\varvec{\mathrm{ k} }},\lambda }^\dagger {\vert n_{{\varvec{\mathrm{ k} }},\lambda } \rangle }=\sqrt{n_{{\varvec{\mathrm{ k} }},\lambda }+1}{\vert n_{{\varvec{\mathrm{ k} }},\lambda }+1 \rangle }. \end{aligned}$$
(8.1)
These combine to give the photon number operator, \(\hat{n}_{{\varvec{\mathrm{ k} }},\lambda }=\hat{a}_{{\varvec{\mathrm{ k} }},\lambda }^\dagger \hat{a}_{{\varvec{\mathrm{ k} }},\lambda }\). Eigenstates of this operator are known as Fock states, with exactly \(n\) photons in the mode.
As a consequence of the quantum nature of light, the electric field amplitude and phase can no longer be known simultaneously. This is because they are conjugate variables of the field, analogous to position and momentum, which are therefore constrained by the Heisenberg uncertainty principle.1 Instead, the electric field is represented by the operator [1]
$$\begin{aligned} \hat{\varvec{E}}({\varvec{\mathrm{ r} }},t)&= {\mathrm{ i} }\sum _{{\varvec{\mathrm{ k} }},\lambda }\sqrt{\frac{\hbar \omega _{\varvec{\mathrm{ k} }}}{2\epsilon _0V}}\hat{{\varvec{\mathrm{ e} }}}_{{\varvec{\mathrm{ k} }},\lambda }(\hat{a}_{{\varvec{\mathrm{ k} }},\lambda }(t){\mathrm{ e} }^{-{\mathrm{ i} }\omega _{\varvec{\mathrm{ k} }}t+{\mathrm{ i} }{\varvec{\mathrm{ k} }}\cdot {\varvec{\mathrm{ r} }}}-\hat{a}^\dagger _{{\varvec{\mathrm{ k} }},\lambda }(t){\mathrm{ e} }^{{\mathrm{ i} }\omega _{\varvec{\mathrm{ k} }}t-{\mathrm{ i} }{\varvec{\mathrm{ k} }}\cdot {\varvec{\mathrm{ r} }}}), \nonumber \\&=\hat{\varvec{E}}^{(+)}({\varvec{\mathrm{ r} }},t)+\hat{\varvec{E}}^{(-)}({\varvec{\mathrm{ r} }},t), \end{aligned}$$
(8.2)
where \(\hat{{\varvec{\mathrm{ e} }}}_{{\varvec{\mathrm{ k} }},\lambda }\) is the polarisation unit vector. The operator is separated into the positive and negative frequency components such that \(\hat{\varvec{E}}^{(+)}\) contains only annihilation operators and \(\hat{\varvec{E}}^{(-)}\) contains creation operators, with \([\hat{\varvec{E}}^{(+)}]^\dagger =\hat{\varvec{E}}^{(-)}\).

8.1.1 Coherent States

The electric field emitted by a laser above threshold is described by a superposition of Fock states, known as a coherent state \({\vert \alpha \rangle }\) [1]. The coherent state is defined as
$$\begin{aligned} {\vert \alpha \rangle } = \sum _n\frac{\alpha ^n}{\sqrt{n!}}{\mathrm{ e} }^{-\vert \alpha \vert ^2/2}{\vert n \rangle }, \end{aligned}$$
(8.3)
which is an eigenstate of the creation and annihilation operators with eigenvalues of
$$\begin{aligned} \hat{a}{\vert \alpha \rangle }=\alpha {\vert \alpha \rangle },\hat{a}^\dagger {\vert \alpha \rangle }=\alpha ^*{\vert \alpha \rangle }, \end{aligned}$$
(8.4)
where \(\alpha \) is the complex amplitude of the state. The probability of observing \(n\) photons in this coherent state is
$$\begin{aligned} P_\alpha (n) = \vert \langle n\vert \alpha \rangle \vert ^2=\frac{\alpha ^{2n}}{n!}{\mathrm{ e} }^{-\vert \alpha \vert ^2}, \end{aligned}$$
(8.5)
which is a Poissonian distribution with a mean-photon number of \(\bar{n}=\vert \alpha \vert ^2\), and a fractional uncertainty \(\Delta n/\bar{n}=1/\sqrt{\bar{n}}\). The coherent states are minimum uncertainty states with equal uncertainty in phase and amplitude, thus for \(\bar{n}\gg 1\) the coherent state electric field \({\langle \alpha \vert }\hat{{E}}{\vert \alpha \rangle }={E}_0 \cos (\omega t)\) [1], equivalent to the semi-classical laser field used previously.

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.

The photon statistics can be quantified using the second-order correlation function, also known as the intensity correlation function. For a pair of detectors at positions \({\varvec{\mathrm{ r} }}_1\) and \({\varvec{\mathrm{ r} }}_2\), the normalised second-order correlation function is defined as
$$\begin{aligned} g^{(2)}({\varvec{\mathrm{ r} }}_1,{\varvec{\mathrm{ r} }}_2,t,t^{\prime }) = \frac{\langle \hat{E}^{(-)}({\varvec{\mathrm{ r} }}_1,t)\hat{E}^{(-)}({\varvec{\mathrm{ r} }}_2,t^{\prime })\hat{E}^{(+)}({\varvec{\mathrm{ r} }}_2,t^{\prime })\hat{E}^{(+)}({\varvec{\mathrm{ r} }}_1,t)\rangle }{{\langle \hat{E}^{(-)}({\varvec{\mathrm{ r} }}_1,t)\hat{E}^{(+)}({\varvec{\mathrm{ r} }}_1,t)\rangle \langle \hat{E}^{(-)}({\varvec{\mathrm{ r} }}_2,t^{\prime })\hat{E}^{(+)}({\varvec{\mathrm{ r} }}_2,t^{\prime })\rangle }}, \end{aligned}$$
(8.6)
which describes the correlations between the field at time \(t\) and \(t^{\prime }\). If a continuous light source is used, the relative time \(t^{\prime }\) can be related to a delay \(\tau \) using \(t^{\prime }=t+\tau \), reducing this to the evaluation of \(g^{(2)}(\tau )\).
For a classical field, the correlation function is bounded by the Cauchy–Schwarz inequality [4]
$$\begin{aligned} g^{(2)}(0)\ge 1, \end{aligned}$$
(8.7)
however for a quantum field \(0\le g^{(2)}(0)\le \infty \). The \(g^{(2)}\) function can therefore be used as evidence of a quantum or non-classical light field if \(g^{(2)}(0)<1\).
If the electric-field is single mode, the correlation function can be written in terms of creation and annihilation operators to simplify evaluation of the correlations,
$$\begin{aligned} g^{(2)}(\tau ) = \frac{\langle a^\dagger (t)a^\dagger (t+\tau )a(t+\tau )a(t) \rangle }{\langle a^\dagger (t)a(t) \rangle \langle a^\dagger (t+\tau )a(t+\tau ) \rangle }. \end{aligned}$$
(8.8)
For a coherent state, \(g^{(2)}(\tau )=1\) for all time independent of \(\alpha \), as expected for a classical plane wave field. For a Fock state \({\vert n \rangle }\), \(g^{(2)}(0)=1-1/n\). Thus for the single photon state \(g^{(2)}(0)=0\), violating the classical inequality as expected for this purely quantum state of light. The physical interpretation of this result is that as there is only a single photon, it cannot be simultaneously observed by both detectors. This effect is known as 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.
Fig. 8.1

Hanbury Brown Twiss interferometer. The input light is separated onto a pair of single photon counters using a 50:50 beam-splitter, and coincidence counts are detected as a function of delay \(\tau \) to build up the correlation function

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.

Fig. 8.2

Photon blockade. a Placing an atom at the centre of an optical cavity causes the cavity modes to be detuned by \(\pm \sqrt{n}g_0\), where \(g_0\) is the coupling constant, preventing more than a single photon entering the cavity. (States are labelled \({\vert n,\pm \rangle }\) to denote the photon number \(n\) and dressed state of the atom) b For the EIT system with no interactions, all photons form the dark state \({\vert D \rangle }\), so for two photons there are two dark states \({\vert 2,D^2 \rangle }\). Dipole–dipole interactions detune this state, breaking the EIT condition for the second photon and causing it to couple to the intermediate excited state \({\vert 2,De \rangle }\). This state decays at rate \(\,\Gamma _e\), scattering the photon into a different mode so only a single photon remains in the probe beam

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

To aide the interpretation of a single-photon dark state, it is useful to introduce the concept of a 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]
$$\begin{aligned} \Psi (z,t) = \cos \vartheta \mathcal{ E} _{\mathrm{ p} }(z,t)-\sin \vartheta \sqrt{\rho }\sigma _{g,r}(z,t){\mathrm{ e} }^{\,{\mathrm{ i} }\Delta kz}, \end{aligned}$$
(8.9)
where \(\mathcal{ E} _{\mathrm{ p} }={E}_{\mathrm{ p} }/\sqrt{\hbar \omega _{\mathrm{ p} }/2\epsilon _0}\) is the normalised probe amplitude, \(\Delta k=k_{\mathrm{ c} }-k_{\mathrm{ p} }\) is the wave-vector mismatch, and the mixing angle \(\vartheta \) is related to the group index,
$$\begin{aligned} \tan ^2\vartheta =\frac{6\pi c\rho \,\Gamma _e}{k_{\mathrm{ p} }^2\,\Omega _{\mathrm{ c} }^2}={\mathrm{ n} }_{\mathrm{ gr} }. \end{aligned}$$
(8.10)
The polariton represents a coherent superposition of the electromagnetic field and atomic excitation, denoted by the coherence \(\sigma _{rg}\). For a small group index, the mixing angle is small and the electromagnetic component of the polariton dominates, with a group velocity around \(c\). For a large group index however, the energy from the probe field is transferred to the atomic excitation, giving the field ‘mass’ and enabling slow propagation at speed \(v_{\mathrm{ gr} }\ll c\). At the edge of the medium, the excitation is converted back into an electromagnetic field without loss due to the perfect transmission achieved in EIT. Treatment of the probe as a quantised field yields equivalent results, with the electric field replaced by the electric field operator from Eq. 8.2 [12].

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 \(g^{(2)}(\tau )\)

As a first attempt at predicting the photon statistics of the probe beam output from the blockade region, a simple model of the blockade mechanism is developed. Consider an ensemble of \(\,{\mathcal{ N} }{}\)-atoms confined within a sphere of diameter \(R_{\mathrm{ b} }\) to ensure all atoms meet the blockade condition. This is probed by a tightly focused laser beam with a 1/e\(^2\) radius of \(w_0<R_{\mathrm{ b} }/2\) such that the probe beam is completely contained within the interaction volume to enable complete absorption of the probe beam. As mentioned above, the probe laser can be represented as a coherent state \({\vert \alpha \rangle }\) with a Poissonian distribution of photon numbers, however it is necessary to determine the mean photon number for \({\vert \alpha \rangle }\). To do this, a quantisation volume must be defined, which is trivial for a cavity but not for light in free-space. The purpose of the model is to determine the coincidences of photon arrival times at a detector, so quantisation can be achieved by defining a time window \(\Delta t\) in which photons are binned. In this time light travels a distance \(c\Delta t\), so the probe beam can be quantised by introducing a cylindrical volume \(V=\pi w_0^2c\Delta t\) as illustrated in Fig. 8.3a, where the cylinder is assumed to have a radius equal to the beam waist. For a probe of power \(P\), the mean photon number can then be calculated using Eq.  C.5,
$$\begin{aligned} \bar{n} = \frac{2P\Delta t}{\hbar \omega }. \end{aligned}$$
(8.11)
From the mean photon number, a random input photon train \(C(t)\) is generated for \(10^6\) time-bins of width \(\Delta t\), with the photon number in each time bin determined from the Poissonian distribution of Eq. 8.5. To model the effect of the photon blockade, two output modes are defined—a forward channel \(C_{\mathrm{ f} }(t)\), which represents the probe light on the other side of the atomic ensemble, and a scattered mode \(C_{\mathrm{ s} }(t)\), which represents all other modes in which photons can be scattered by the interaction with the medium. Starting at \(t=0\), the first photon to arrive in the medium is placed in the forward channel and all photons arriving within a period of \(\tau _{\mathrm{ b} }\) are put in the scattered channel. This process is repeated across the entire photon train, as illustrated schematically in Fig. 8.3b.
Fig. 8.3

Simple \(g^{(2)}\) model. a The probe laser field is quantised in a cylinder of length \(c\Delta t\). The probe power can then be converted to \(\bar{n}\) to randomly generate photons in each window \(\Delta t\). b The blockade mechanism is simulated by dividing the input photon train \(C(t)\) into the forward mode of the probe beam \(C_{\mathrm{ f} }(t)\), with only one photon passing through in a period \(\tau _{\mathrm{ b} }\), and a scattered channel \(C_{\mathrm{ s} }(t)\) for the remaining photons. Each box represents a time period \(\Delta t\), with dots showing photon number in each window

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 \({\mathrm{ OD} }=-\log _{\mathrm{ e} }(T)\).

Finally, the second order correlation function is calculated for each of the two output modes. This is achieved by taking the photon train \(C_i(t)\) and simulating the effect of a beam-splitter to separate it into the counts detected by a pair of detectors \(D_1\) and \(D_2\), equivalent to the HBT interferometer in Fig. 8.1. If there are \(n\) photons in a given time-bin, the probability of detecting \(m\) photons at detector \(D_1\) can be found using the binomial distribution
$$\begin{aligned} P(m) = \frac{n!}{m!(n-m)!}p^m(1-p)^{(n-m)}, \end{aligned}$$
(8.12)
where \(p\) is the probability of success which for a 50:50 beam-splitter is 0.5. This distribution allows the beam-splitter to be modelled efficiently to obtain the counts arriving at the first detector in each time window, \(D_1(t)\), from which \(D_2(t)=C_i(t)-D_1(t)\). The normalised correlation function is then calculated using the Weiner-Khintchine theorem [14] as
$$\begin{aligned} g^{(2)}(\tau ) = {\mathrm{ Re} }{}\left\{ \frac{\mathcal{ F} ^{-1}[\mathcal{ F} [D_1(t)]\mathcal{ F} [D_2(t)]^*]}{\sum D_1(t)\sum D_2(t)} \right\} , \end{aligned}$$
(8.13)
where \(\mathcal{ F} \) and \(\mathcal{ F} ^{-1}\) denote the Fourier transform and its inverse.
Having introduced the \(g^{(2)}\) model for the correlations, it is useful to explicitly define the optical depth in terms of physical parameters. From the definition of transmission in Eq.  4.23a, \({\mathrm{ OD} }=-\log _{\mathrm{ e} }(T)=k_{\mathrm{ p} }\ell \chi _{\mathrm{ I} }\). Taking \(\ell =R_{\mathrm{ b} }\) and using the weak-probe limit for the probe-only susceptibility \(\chi _{\mathrm{ I} }\) from Eq.  4.20, the optical depth is given by
$$\begin{aligned} {\mathrm{ OD} } = k_{\mathrm{ p} }\cdot \frac{2\rho d_{eg}^2}{\epsilon _0\hbar \,\Gamma _e}\cdot R_{\mathrm{ b} } = \frac{6\pi R_{\mathrm{ b} }\rho }{k_{\mathrm{ p} }^2} = \frac{36\,{\mathcal{ N} }{}}{k_{\mathrm{ p} }^2R_{\mathrm{ b} }^2}, \end{aligned}$$
(8.14)
where Eq.  5.3 has been used to eliminate \(d_{eg}^2\) and a uniform density approximation used to obtain \(\rho =6\,{\mathcal{ N} }{}/\pi R^3_{\mathrm{ b} }\). It is then trivial to re-scale the group index, and hence velocity, of the dark state polariton from Eq. 8.10 in terms of \({\mathrm{ OD} }\) as
$$\begin{aligned} {\mathrm{ n} }_{\mathrm{ gr} }=\frac{{\mathrm{ OD} }c \,\Gamma _e}{R_{\mathrm{ b} }\,\Omega _{\mathrm{ c} }^2},\quad v_{\mathrm{ gr} }=\frac{c}{1+{\mathrm{ n} }_{\mathrm{ gr} }}\simeq \frac{R_{\mathrm{ b} }\,\Omega _{\mathrm{ c} }^2}{{\mathrm{ OD} }\,\Gamma _e}. \end{aligned}$$
(8.15)
This results in the following simple relation for the blockade time,
$$\begin{aligned} \tau _{\mathrm{ b} }=\frac{R_{\mathrm{ b} }}{v_{\mathrm{ gr} }}=\frac{{\mathrm{ OD} }\,\Gamma _e}{\,\Omega _{\mathrm{ c} }^2}. \end{aligned}$$
(8.16)
Combining these relations together, we consider the case of \(\mathcal{ N} =200\) confined within a blockade radius of \(R_{\mathrm{ b} }\sim 5\,\upmu \)m, corresponding to a density of \(\rho \sim 3\times 10^{12}\) cm\(^{-3}\). This is two orders of magnitude larger than the MOT density, however this is achievable using an optical dipole trap, as will be discussed in Sect.  9.2. The optical depth for this case is \({\mathrm{ OD} }=4.4\), resulting in 99% probability for scattering photons out of the probe beam. Taking \(\,\Omega _{\mathrm{ c} }=\,\Gamma _e\) (consistent with a 5 \(\upmu \)m blockade radius for 60\(S_{1/2}\)) the corresponding blockade time is \(\tau _{\mathrm{ b} }=120\) ns. The probe laser is assumed to be focused to a waist of \(w_0=1\,\upmu \)m to satisfy \(w_0<R_{\mathrm{ b} }/2\), and the model is run for probe powers of 500 fW and 10 pW, equivalent to \(\,\Omega _{\mathrm{ p} }/2\pi =0.6\) and 2.6 MHz respectively.
Fig. 8.4

\(g^{(2)}\) Model results. Correlation function for \({\mathrm{ OD} }=4.4\) calculated for a\(P=500\) fW and b 10 pW, clearly showing a strong anti-bunching for \(\tau <\tau _{\mathrm{ b} }\). c Effect of varying \({\mathrm{ OD} }\) for the forward mode \(C_{\mathrm{ f} }\) at 10 pW, showing at \({\mathrm{ OD} }\sim 2\) the anti-bunching is heavily suppressed. d Changing the number of blockaded ensembles \(\mathcal{ M} _{\mathrm{ b} }\) in the probe beam for \(C_{\mathrm{ f} }\) at 10 pW, \({\mathrm{ OD} }=4.4\) shows even two blockade regions significantly reduces the visibility of the anti-bunching

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.

For an ensemble of \(\mathcal{ N} \)-atoms located at positions \({\varvec{\mathrm{ r} }}_i\), the source-field term in the far field (\(k\vert {\varvec{\mathrm{ r} }}-{\varvec{\mathrm{ r} }}_i\vert \gg 1\) for all \(i\)) is given by [4, 18]3
$$\begin{aligned} \hat{\varvec{E}}^{(+)}_{\mathrm{ sf} }({\varvec{\mathrm{ r} }},t) = - \frac{k^2({\varvec{\mathrm{ d} }}_{\mathrm{ eg} }\times \hat{{\varvec{\mathrm{ r} }}})\times \hat{{\varvec{\mathrm{ r} }}}}{4\pi \varepsilon _0 r} \sum _i^{\,{\mathcal{ N} }{}}{\mathrm{ e} }^{-{\mathrm{ i} }k\hat{{\varvec{\mathrm{ r} }}}\cdot {\varvec{\mathrm{ r} }}_i}\hat{\pi }_i^-\left(t-r/c\right), \end{aligned}$$
(8.17)
which is equivalent to the classical dipole radiation field of Eq.  5.2 with the dipole moment replaced with operator \({\mathrm{ d} }_{eg}\hat{\pi }^-\).
The source-field expression therefore relates the scattered electric field to the properties of the atomic system. If we consider only positions off-axis with respect to the probe and coupling lasers, the incident field \(\hat{\varvec{E}}_{\mathrm{ f} }^{(+)}({\varvec{\mathrm{ r} }},t)\) vanishes, and the electric field reduces to a sum over the dipole operators for the system. Absorbing the geometric factors into the function \(f({\varvec{\mathrm{ r} }})\), the scattered electric field is \(\hat{\varvec{E}}^{(\pm )}({\varvec{\mathrm{ r} }},t) = f({\varvec{\mathrm{ r} }})\hat{\Pi }^\mp \left({\varvec{\mathrm{ r} }},t-r/c\right)\), where \(\hat{\Pi }^\pm \) are the combined raising and lowering operators for the system,
$$\begin{aligned} \hat{\Pi }^\pm ({\varvec{\mathrm{ r} }},t) =\sum _i^{\,{\mathcal{ N} }{}}{\mathrm{ e} }^{\pm {\mathrm{ i} }k\hat{{\varvec{\mathrm{ r} }}}\cdot {\varvec{\mathrm{ r} }}_i}\hat{\pi }_i^\pm (t). \end{aligned}$$
(8.18)

8.5.2 Correlation Function

Using this definition of the electric field, the second order correlation function of Eq. 8.6 can be written as
$$\begin{aligned} g^{(2)}(\tau ) = \frac{\langle \hat{\Pi }^+(t)\hat{\Pi }^+(t+\tau )\hat{\Pi }^-(t+\tau )\hat{\Pi }^-(t)\rangle }{\langle \hat{\Pi }^+(t)\hat{\Pi }^-(t)\rangle \langle \hat{\Pi }^+(t+\tau )\hat{\Pi }^-(t+\tau )\rangle }, \end{aligned}$$
(8.19)
where \(\,\langle \ldots \rangle \) denotes a trace over the density matrix for the atomic system. The correlation function is calculated using the quantum regression theorem which gives [19, 20]
$$\begin{aligned} G(t;t^{\prime }) = \langle \hat{A}(t^{\prime }) \hat{B}(t) \rangle = {\mathrm{ Tr} }\{\hat{A}\sigma _{\mathrm{ cond} }(t;t^{\prime })\} \qquad (t^{\prime }\ge t) \end{aligned}$$
(8.20)
where the \(\sigma _{\mathrm{ cond} }\) is the conditional density matrix defined at time \(t\) as \(\sigma _{\mathrm{ cond} }(t;t)=\hat{B}\sigma (t)\), which represents the state of the system after the action of \(\hat{B}\) is applied.
Applying this theorem to Eq. 8.19 allows the steady-state density matrix \(\sigma ^{\mathrm{ ss} }\) for the \(\mathcal{ N} \)-atom system to be calculated from the optical Bloch equations derived in Sect.  5.4. The conditional density matrix is evaluated using
$$\begin{aligned} \sigma _{\mathrm{ cond} } (0) = \hat{\Pi }^-\sigma ^{\mathrm{ ss} }\hat{\Pi }^+, \end{aligned}$$
(8.21)
which describes the state of the system after a photon has been emitted. The conditional density matrix is then re-normalised and used as an initial condition for the same optical Bloch equations, which are integrated until time \(\tau \) to obtain \(\sigma _{\mathrm{ cond} }(\tau )\). Finally, the second-order correlation function is
$$\begin{aligned} g^{(2)}(\tau ) = \frac{\,{\mathrm{ Tr} }\{\hat{\Pi }^-\sigma _{\mathrm{ cond} }(\tau )\hat{\Pi }^+\}}{\,{\mathrm{ Tr} }\{\sigma _{\mathrm{ cond} }(0)\}}. \end{aligned}$$
(8.22)
For large \(\tau \), the conditional density matrix will evolve back to the steady-state \(\sigma ^{\mathrm{ ss} }\), resulting in \(g^{(2)}(\tau \gg 1)=1\) as required. The fluorescence correlations therefore arise from the dynamic evolution of the system back to the steady-state after emitting the first photon.

8.5.3 Cooperative Emission from Incoherent Atoms

As a first approximation, the atoms are assumed to be incoherent emitters such that the cross-phase factors average to zero, for example due to atomic motion. In this case, the combined operators become separable to give
$$\begin{aligned} \hat{\Pi }^-\sigma \hat{\Pi }^+ = \sum _i^{\,{\mathcal{ N} }{}} \hat{\pi }_i^-\sigma \hat{\pi }_i^+. \end{aligned}$$
(8.23)
The resonance fluorescence correlations of independent two-level atoms for \(\,\Omega _{\mathrm{ p} }\,{=}\,\,\Gamma _e/5\) is shown in Fig. 8.5a, showing anti-bunching with \(g^{(2)}(0)=1-1/\,{\mathcal{ N} }{}\) as each atom can emit a single photon at a random time, with the possibility to observe two photons at zero delay from two atoms but with a non-Poissonian probability. In (b) the correlation function for the EIT system with \(\,\Omega _{\mathrm{ c} }=\,\Gamma _e\) and \(V(R_{ij})=2\,\Gamma _e\) is plotted. The \(\,{\mathcal{ N} }{}=1\) trace is anti-bunched at \(\tau =0\), and then increases to give \(g^{(2)}(\tau )\gg 1\) at \(\tau \sim 1/\,\Omega _{\mathrm{ p} }\). This occurs because in the resonant EIT condition, the emission of a photon projects the atom out of the dark state, requiring another photon to be emitted at a later time to allow the atom to return to the dark state. Assuming a perfect laser system, this would not be observable in an experiment as the probability to emit the initial photon is vanishing due to the EIT condition. This very small probability for emission of the first photon leads to an an anomalously large correlation for the emission of the second photon.
Fig. 8.5

\(\,{\mathcal{ N} }{}\)-atom fluorescence correlations. a Independent two-level atoms for \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e/5\) displaying anti-bunching. b–d Interacting EIT system with \(\,\Omega _{\mathrm{ c} }=\,\Gamma _e\), \(V(R_{ij})=2\,\Gamma _e\) and \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e/5\), \(\,\Gamma _e/2\) and \(\,\Gamma _e\) respectively shows blockade causes bunching, which becomes anti-bunched if the strong probe violates the blockade condition

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

Making a further assumption that the fluorescence emitted by each atom is distinguishable (for example in spatially separated dipole traps as in the experiments in Orsay [21] and Madison [22]) it is possible to also calculate the self- and cross-correlations between atoms \(i\) and \(j\) using
$$\begin{aligned} g_{ij}^{(2)}(\tau ) = \frac{\langle \hat{\pi }_i^+(t)\hat{\pi }_j^+(t+\tau )\hat{\pi }_j^-(t+\tau )\hat{\pi }_i^-(t)\rangle }{\langle \hat{\pi }_i^+(t)\hat{\pi }_i^-(t)\rangle \langle \hat{\pi }_j^+(t+\tau )\hat{\pi }_j^-(t+\tau )\rangle }, \end{aligned}$$
(8.24)
which is evaluated in exactly the same way as for \(g^{(2)}\) except \(\hat{\Pi }^\pm \) is replaced by the single-atom dipole operators. The cross-correlation provides an insight into whether the emission from one atom is related to emission of a neighbouring atom.
Fig. 8.6

Self- and cross-correlations for \(\,{\mathcal{ N} }{}=2\). a \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e/5\). Two-level atoms have an independent cross-correlation as the atoms are non-interacting. b–d Interacting EIT system with \(\,\Omega _{\mathrm{ c} }=\,\Gamma _e\), \(V(R_{ij})=2\,\Gamma _e\) and \(\,\Omega _{\mathrm{ p} }=\,\Gamma _e/5\), \(\,\Gamma _e/2\) and \(\,\Gamma _e\) respectively. This shows the bunching arises from the strong cross-correlation between the atoms, which are correlated by the dipole blockade. For a strong probe, this cross-correlation is suppressed as the system is no longer blockaded

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.

Fig. 8.7

Correlation function for coherent emission for different separations \(R\), where \(m\) is an integer. a Atoms aligned parallel to the probe beam are bunched for all separations. b The perpendicular configuration shows anti-bunching due to destructive interference for \(R=m\lambda +\lambda /4,3\lambda /4\)

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

  1. 1.

    An important consequence of the uncertainty principle is spontaneous emission, which arises due to the coupling between an atom in the excited state and the vacuum fluctuations for the \({\vert 0 \rangle }\) state of each mode [2, 3].

  2. 2.

    This sharp-edged correlation function is similar to that predicted for a \(p-i-n\) junction in which the Coulomb blockade prevents more than a single photon emission [15], however the output coupling efficiency for such devices is too weak to measure the correlations [16].

  3. 3.

    In [4] the atom is assumed to be at the origin, however for a finite displacement an additional phase-factor is required which can be found in Eq.  7.13 of [18].

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  1. 1.Durham UniversityDurhamUK

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