Coherence and catalysis in the Jaynes–Cummings model

Before investigating the Jaynes–Cummings evolution, it is worthwhile to pause and consider the simpler, semiclassical Rabi problem in which the quantized field mode is replaced by a classical field [5, 34]. In this model, the single-mode creation and annihilation operators are replaced by a c-number amplitude so that our model Hamiltonian becomes

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{Semiclass}}=-{\rm{i}}{\hslash }g\left(\alpha {\hat{\sigma }}_{+}-{\alpha }^{* }{\hat{\sigma }}_{-}\right).\end{eqnarray} \tag{ 7 }$$

We can write the atomic state in this interaction picture as a superposition of the excited and ground states in the form

$$\begin{eqnarray}&&| {{\rm{\Psi }}}_{\mathrm{Semiclass}}\rangle =G(t)| {\rm{g}}\rangle +E(t)| {\rm{e}}\rangle ,\end{eqnarray} \tag{ 8 }$$

where direct solution of the Schrödinger equation, for an atom prepared in its excited state gives

$$\begin{eqnarray}\begin{array}{rcl}E(t) & = & \cos \left(g| \alpha | t\right),\\ G(t) & = & {{\rm{e}}}^{-{\rm{i}}\arg \alpha }\sin \left(g| \alpha | t\right).\end{array}\end{eqnarray} \tag{ 9 }$$

Note that the phase of the field amplitude, α, has been imprinted onto the phase of the atomic superposition state. If we choose the interaction time such that $g| \alpha | t=\pi /4$ then the atom is left in the equally weighted superposition state $(| {\rm{e}}\rangle +{{\rm{e}}}^{-{\rm{i}}\arg \alpha }| {\rm{g}}\rangle )/\sqrt{2}$. There is no back-action on the field here, as it has been treated classically. In principle this process can be repeated at will with each new atom being transformed into this same superposition state. To treat the coherence as a resource, however, we need to quantize the field mode and keep track of the effects of the interaction both on the atoms and also on the state of the field.

The fully quantum Jaynes–Cummings Hamiltonian (6) conserves the total number of quanta and this feature makes it possible to find an exact solution for the dynamics. In general a pure state of the atom and field mode will be of the form

$$\begin{eqnarray}&&| {\rm{\Psi }}\rangle =\sum _{n}{G}_{n}| {\rm{g}}\rangle | n\rangle +{E}_{n}| {\rm{e}}\rangle | n\rangle .\end{eqnarray} \tag{ 10 }$$

Evolution under the Schrödinger equation with Hamiltonian (6) gives a set of coupled, linear differential equations which are readily solved to give the exact dynamics at any given time:

$$\begin{eqnarray}\begin{array}{rcl}{\dot{G}}_{n} & = & {{gE}}_{n-1}\sqrt{n},\\ {\dot{E}}_{n} & = & -{{gG}}_{n+1}\sqrt{n+1}.\end{array}\end{eqnarray} \tag{ 11 }$$

The evolution of an initially excited atom interacting with an arbitrary cavity state, $| {\rm{\Psi }}(0)\rangle ={\sum }_{n}{c}_{n}| n\rangle | {\rm{e}}\rangle$, is thus given by

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle =\sum _{n=0}^{\infty }{c}_{n}\left[\cos (\sqrt{n+1}{gt})| n\rangle | {\rm{e}}\rangle +\sin (\sqrt{n+1}{gt})| n+1\rangle | {\rm{g}}\rangle \right].\end{eqnarray} \tag{ 12 }$$

This may be understood as the initial excitation of the atom oscillating between the cavity and the atom. Note that the frequency of this oscillation is different in each constant energy subspace, and depends on the total excitation number n + 1: thus as time progresses the oscillations for different total energy drift in and out of phase, giving rise to the famous collapses and revivals of the Jaynes–Cummings model [32–35]. Our interest, however, is in interaction times that are much shorter than the collapse and revival times.

A key feature of the evolution is that at any given time (after the initial time) the atom and field mode will be in an entangled state and it is this entanglement that encapsulates the back action on the state of the field mode. As discussed, we are concerned with a field mode prepared in an initial coherent state interacting with a sequence of atoms, shown schematically in figure 1, each prepared in its excited state, a situation that is reminiscent of the Scully–Lamb theory of the laser [49]. Here, however, we are interested in the combined atom field state that is prepared and the associated effects on the coherence.

Another important effect of large but finite $\bar{n}$ concerns the back-action of the interaction on the field: how is the resource, that is the state of the cavity mode, degraded upon use? Had we chosen to use atoms prepared in the ground state, then the coherence would inevitably be consumed as the light was gradually removed from the cavity by the sequence of atoms. By starting with atoms in excited states, however, this elementary coherence-removing process is avoided and we are left only with the consequences of the fundamental degradation of the quantum coherence.

After a single atom has passed through the cavity, the joint state of cavity and atom is given by equation (13), where for convenience we return to the $| {\rm{g}}\rangle$, $| {\rm{e}}\rangle$ basis of the atom:

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle ={{\rm{e}}}^{-| \alpha {| }^{2}/2}\sum _{n=0}^{\infty }\displaystyle \frac{{\alpha }^{n}}{\sqrt{n!}}\left(\cos (\sqrt{n+1}{{gt}}_{1})| n\rangle | {\rm{e}}\rangle +\displaystyle \frac{\sqrt{n}}{\alpha }\sin (\sqrt{n}{{gt}}_{1})| n\rangle | {\rm{g}}\rangle \right).\end{eqnarray} \tag{ 17 }$$

We consider again the large $\bar{n}$ limit, and for an interaction time t₁ satisfying equation (15), it may be shown that the approximate atom-field state after interaction is given by:

$$\begin{eqnarray}&&| {\rm{\Psi }}({t}_{1})\rangle \approx | {\rm{\Psi }}^{\prime} ({t}_{1})\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(| {\alpha }_{{\rm{e}}}\rangle | {\rm{e}}\rangle +| {\alpha }_{{\rm{g}}}\rangle | {\rm{g}}\rangle \right),\end{eqnarray} \tag{ 18 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}| {\alpha }_{{\rm{e}}}\rangle & = & \left|\sqrt{{\alpha }^{2}-\displaystyle \frac{\pi }{4}}\right\rangle ,\\ | {\alpha }_{{\rm{g}}}\rangle & = & \left|\sqrt{{\alpha }^{2}+1+\displaystyle \frac{\pi }{4}}\right\rangle .\end{array}\end{eqnarray} \tag{ 19 }$$

The approximations leading to this result are straightforward but lengthy, and the full technical details are given in appendix B. The change in the field thus has a clear physical interpretation: an initial coherent state $| \alpha \rangle$ is transformed to a mixture of coherent states, one shifted down in amplitude with probability 1/2 to a state with average photon number α² − π/4, and the other shifted up in amplitude with probability 1/2 to a state with average photon number α² + 1 + π/4. It follows that the average photon number in the cavity field is increased by (approximately) 1/2, as half an excitation is transferred from the atom to the field:

$$\begin{eqnarray}&&{\bar{n}}_{1}=\langle n\rangle \approx \displaystyle \frac{1}{2}\left({\alpha }^{2}-\displaystyle \frac{\pi }{4}+{\alpha }^{2}+1+\displaystyle \frac{\pi }{4}\right)={\alpha }^{2}+\displaystyle \frac{1}{2}.\end{eqnarray} \tag{ 20 }$$

When the process is repeated upon introduction of a second atom to the field, the interaction time is now chosen such that the probability of success is maximal for a coherent state with this increased average photon number, ${\bar{n}}_{1}={\alpha }^{2}+1/2$. The probability of failure, that is of finding the atom in the $| -\rangle$ state, is the average of the probability of failure for each of the two shifted coherent states $| {\alpha }_{{\rm{e}}}\rangle$ and $| {\alpha }_{{\rm{g}}}\rangle$. Remarkably, this second round of atom-field interaction results in a slight increase in fidelity with the desired $| +\rangle$ state in the next round, at order $1/{\bar{n}}^{2}$, due to the increased average photon number. Numerical results for α = 10, corresponding to a mean photon number of $\bar{n}=100$, are shown in table 1.

Table 1.  Probabilities after two atom-field interactions for α = 10 with adjustment of interaction times after the first cycle. Bold digits show the deviation from the corresponding single-atom probabilities for easier comparison.

P(+1)	P(+2)	$P({+}_{2}| {+}_{1})$	P(+2)P(+1)	$P({+}_{2}\,\cap \,{+}_{1})$
0.995909	0.995915	0.995932	0.991841	0.991858
P(−1)	P(−2)	$P({+}_{2}| {-}_{1})$	P(+2)P(−1)	$P({+}_{2}\,\cap \,{-}_{1})$
0.004091	0.004085	0.991631	0.004074	0.004056

The ability of the coherent state cavity field for inducing coherent operations under the Jaynes–Cummings interaction is therefore not degraded upon first use: remarkably, it is slightly improved. It is worthwhile pointing out that although this result may be surprising, it is not unphysical. We should expect, however, that a fixed initial amount of coherence may not be used to produce independent copies of the equal superposition state $| +\rangle$ indefinitely. A complete picture requires us to consider both the many copy limit, and correlations between subsequent atoms from each interaction.

We now wish to derive the evolution of the field under successive interactions with subsequent atoms. We recall that following the first interaction, the cavity state is a mixture of two different coherent states with slightly different amplitudes, as shown in equation (18), and for the second atom we choose an interaction time satisfying $\sqrt{{\bar{n}}_{1}+1}{{gt}}_{1}=\pi /4$, where ${\bar{n}}_{1}={\alpha }^{2}+1/2$. However, for each coherent state $| {\alpha }_{{\rm{e}}}\rangle$, $| {\alpha }_{{\rm{g}}}\rangle$ in the mixture, there is now a slight mis-match between the amplitude squared ${\alpha }_{{\rm{e}}}^{2}$ (${\alpha }_{{\rm{g}}}^{2}$) and ${\bar{n}}_{1}$, used to define the interaction time. If this mis-match is too large, the chosen interaction time will not produce the desired evolution. We begin by considering the consequences of this mis-match on the effect of the interaction on the cavity state in subsequent rounds.

Let us consider an element of the mixture $| {\alpha }^{{\prime} }\rangle$, with

$$\begin{eqnarray*}&&{\alpha }^{{\prime} 2}={\bar{n}}_{r}+\delta ,\end{eqnarray*}$$

where ${\bar{n}}_{r}$ is the average photon number of the field after r interactions, and δ is a real number, introduced to quantify the mis-match between ${\alpha }^{{\prime} 2}$ and ${\bar{n}}_{r}$. After a single interaction, ${\bar{n}}_{1}={\bar{n}}_{0}+1/2$ and $\delta =\pm \left(1/2+\pi /4\right)$. However, we assume in the following only that δ is small compared with α² in order that the analysis can be applied to subsequent rounds. We wish to show that under successive interactions the state of the field remains a mixture of coherent states, parametrized by δ, and to derive the change in δ over time. In this regime, it can be shown (see appendix C) that for an element $| {\alpha }^{{\prime} }\rangle$ of the mixture, after r interactions, the approximate state after one further interaction is indeed given by:

$$\begin{eqnarray}&&| {\alpha }^{{\prime} }\rangle | {\rm{e}}\rangle \to \sqrt{\displaystyle \frac{1}{2}\left(1-\displaystyle \frac{\pi }{4}\displaystyle \frac{\delta }{{\alpha }^{{\prime} 2}}\right)}| {\alpha }_{{\rm{e}}}^{{\prime} }\rangle | {\rm{e}}\rangle +\sqrt{\displaystyle \frac{1}{2}\left(1+\displaystyle \frac{\pi }{4}\displaystyle \frac{\delta }{{\alpha }^{{\prime} 2}}\right)}| {\alpha }_{{\rm{g}}}^{{\prime} }\rangle | {\rm{g}}\rangle ,\end{eqnarray} \tag{ 21 }$$

where

$$\begin{eqnarray}&&| {\alpha }_{{\rm{e}}}^{{\prime} }\rangle =\left|\sqrt{{\alpha }^{{\prime} 2}-\displaystyle \frac{\pi }{4}}\right\rangle ,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&| {\alpha }_{{\rm{g}}}^{{\prime} }\rangle =\left|\sqrt{{\alpha }^{{\prime} 2}+1+\displaystyle \frac{\pi }{4}}\right\rangle .\end{eqnarray} \tag{ 23 }$$

The change in the field thus retains the physical interpretation given for the first round, as analysed in 4.1, but with modified probabilities. An initial coherent state $| {\alpha }^{{\prime} }\rangle$ is transformed to a mixture of coherent states: one is again shifted down in amplitude to a state with average photon number α'²−π/4, but now with probability $1/2(1-\pi \delta /4\alpha {{\prime} }^{2})$, while the other is shifted up in amplitude to a state with average photon number α² + 1 + π/4, but with a modified probability of $1/2(1+\pi \delta /4\alpha {{\prime} }^{2})$. At the end of each round ${\bar{n}}_{r+1}$ for the field as a whole (taking into account all components $| {\alpha }^{{\prime} }\rangle$ of the mixture) increases by 1/2, and δ shifts by $\pm \left(1/2+\pi /4\right)$.

Recall that we have been assuming throughout that δ is small compared with α². We finally note that if δ becomes comparable to α², the scheme breaks down. Indeed equation (14) becomes:

$$\begin{eqnarray}&&{P}_{-}=\displaystyle \frac{{{\rm{e}}}^{-| {\alpha }^{{\prime} }{| }^{2}}}{2}\sum _{n}^{\infty }\displaystyle \frac{{\alpha }^{{\prime} 2n}}{n!}{\left|\cos (\sqrt{{\bar{n}}_{r}+{\rm{\Delta }}n+1}{{gt}}_{1})-\displaystyle \frac{\sqrt{n}}{{\alpha }^{{\prime} }}\sin (\sqrt{{\bar{n}}_{r}+{\rm{\Delta }}n}{{gt}}_{1})\right|}^{2},\end{eqnarray} \tag{ 24 }$$

where ${\rm{\Delta }}n=n-{\bar{n}}_{r}=n-{\alpha }^{{\prime} 2}+\delta$. If δ ∼ O(α'²) then Δn is the same order of magnitude as ${\bar{n}}_{r}$, and it no longer holds that $\sqrt{{\bar{n}}_{r}+{\rm{\Delta }}n+1}{{gt}}_{1}\simeq \sqrt{{\bar{n}}_{r}+1}{{gt}}_{1}=\pi /4$. As a result this probability of failure no longer scales as $O(1/{\bar{n}}^{r})$, and the scheme breaks down.

We now have enough information to understand the evolution of the state of the field under sequential interactions with multiple atoms. At the start of the process, the field is in a coherent state $| \alpha \rangle$, with ${\bar{n}}_{0}=\bar{n}={\alpha }^{2}$, and δ = 0. After a single interaction, derived in detail in the previous subsection and in appendix B, the field is an equal mixture of two coherent states $| {\alpha }_{{\rm{e}}}\rangle$, $| {\alpha }_{{\rm{g}}}\rangle$, $\bar{n}\to {\bar{n}}_{1}={\bar{n}}_{0}+1/2={\alpha }^{2}+1/2$, and $\delta =\pm \left(1/2+\pi /4\right)$. Here, δ ≪ α and thus in each subsequent round δ shifts up or down with (approximately) equal probability, in steps of $\pm \left(1/2+\pi /4\right)$. In other words, δ performs a classical, unbiased random walk with step size $\pm \left(1/2+\pi /4\right)$. After r steps, with r ≪ α, the expectation value of δ is zero, with a spread of ${ \mathcal O }(\sqrt{r})$ [50].

After r ≃ α² steps, the spread of the distribution, and typical values of δ, are therefore ${ \mathcal O }(\alpha )$. At this stage the probabilities of shifting up or down in δ have become slightly asymmetric, as described in the analysis above. For δ > 0, the probability of an increase in δ is slightly bigger than the probability of a decrease. The converse is true for δ < 0. A possible evolution of δ with the number of steps r is shown in figure 3 to illustrate this result. Thus those elements of the mixture $| {\alpha }^{{\prime} }\rangle$ with higher amplitude-squared than the mean photon number tend to shift up further in number, while those with lower amplitude-squared tend to shift down further, leading to a faster spread in the distribution. Let us consider a coherent state $| {\alpha }^{{\prime} }\rangle$ with amplitude near the upper end of the distribution, with δ_u positive and ${\delta }_{u}\simeq { \mathcal O }(\alpha )$, where the subscript u indicates we are considering the upper end of the distribution. At the next interaction,

$$\begin{eqnarray*}\begin{array}{ccl}{\delta }_{u} & \to & {\delta }_{u}+\left(\displaystyle \frac{1}{2}+\displaystyle \frac{\pi }{4}\right)\quad \mathrm{with}\,\mathrm{probability}\quad \displaystyle \frac{1}{2}\left(1+\displaystyle \frac{\pi }{4}\displaystyle \frac{{\delta }_{u}}{{\alpha }^{{\prime} 2}}\right),\\ {\delta }_{u} & \to & {\delta }_{u}-\left(\displaystyle \frac{1}{2}+\displaystyle \frac{\pi }{4}\right)\quad \mathrm{with}\,\mathrm{probability}\quad \displaystyle \frac{1}{2}\left(1-\displaystyle \frac{\pi }{4}\displaystyle \frac{{\delta }_{u}}{{\alpha }^{{\prime} 2}}\right).\end{array}\end{eqnarray*}$$

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Thus the random walk is now biased and the expectation value for ${\delta }_{u}$ increases slightly:

$$\begin{eqnarray}&&\langle {\delta }_{u}\rangle \to {\delta }_{u}\left(1+\displaystyle \frac{\pi }{4}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{\pi }{4}\right)\displaystyle \frac{1}{{\alpha }^{{\prime} 2}}\right).\end{eqnarray} \tag{ 25 }$$

In subsequent rounds, this expectation value increases each time by a multiplicative factor, so that after a further k interactions, we find

$$\begin{eqnarray}&&\langle {\delta }_{u}\rangle \to {\delta }_{u}{\left(1+\displaystyle \frac{\pi }{4}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{\pi }{4}\right)\displaystyle \frac{1}{{\alpha }^{{\prime} 2}}\right)}^{k}.\end{eqnarray} \tag{ 26 }$$

Recalling that the exponential function may be defined by the limit $n\to \infty$ of ${\left(1+x/n\right)}^{n}$, and noting that we are considering α² ≫ 1, it is convenient to express the number of rounds k as a multiple of α², k = l α². In this case we find⁴

$$\begin{eqnarray}&&\langle {\delta }_{u}\rangle \to {\delta }_{u}{\left(1+\displaystyle \frac{\pi }{4}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{\pi }{4}\right)\displaystyle \frac{1}{{\alpha }^{{\prime} 2}}\right)}^{l{\alpha }^{{\prime} 2}}\simeq {\delta }_{u}{\left[\exp \left(\displaystyle \frac{\pi }{4}\left(\displaystyle \frac{1}{2}+\displaystyle \frac{\pi }{4}\right)\right)\right]}^{l}.\end{eqnarray} \tag{ 27 }$$

Thus for $l\simeq { \mathcal O }(\mathrm{log}{\alpha }^{{\prime} })$ and an initial ${\delta }_{u}\simeq { \mathcal O }({\alpha }^{{\prime} })$, then $\langle {\delta }_{u}\rangle \simeq { \mathcal O }\left({\alpha }^{{\prime} 2}\right)$, and the protocol begins to fail for subsequent interactions. We can make a similar argument for δ_l < 0, for those elements $| {\alpha }^{{\prime} }\rangle$ with amplitudes at the lower end of the distribution. Thus we find that after at most ${ \mathcal O }({\alpha }^{{\prime} 2}\mathrm{log}{\alpha }^{{\prime} })$ interactions, the state of the field is no longer useful for enabling coherent operations through the Jaynes–Cummings interaction.

As described above, for ${ \mathcal O }({\alpha }^{{\prime} 2})$ interactions however, the approximations derived for the single interaction case still hold, and we can expect that the atom-field interactions produce ${ \mathcal O }({\alpha }^{{\prime} 2})$ approximate copies of $| +\rangle$ from the initial stream of atoms, each prepared in its excited state $| {\rm{e}}\rangle$.

The ability of the cavity state to approximately induce the desired operation is, however, only part of the picture. Correlations build up between the atom and the field as a result of the interaction, and between subsequent atoms via the field. As shown previously [30], neglecting such correlations can lead us to conclude paradoxical and unphysical effects. In the remainder of this section, we explore the correlations that build up between subsequent atoms in the Jaynes–Cummings interaction.

Conditional probabilities after two rounds, given in table 1 for α = 10, illustrate these correlations between successive atoms. We see that the probabilities after the second atom-field interaction depend strongly on the outcome of the first round: the probability of failure given that the first atom ended up in the state $| -\rangle$ is more than twice as large as if the first atom ended up in the state $| +\rangle$.

To further understand the correlations, we begin by studying the state of the cavity conditioned on the state of the first atom. Conditional on the atom being found in state $| +\rangle$ or $| -\rangle$, the cavity is projected to the state

$$\begin{eqnarray}\begin{array}{rcl}| {{\rm{\Psi }}}_{\mathrm{cav}}{\rangle }_{\pm } & = & \displaystyle \frac{{{\rm{e}}}^{-| \alpha {| }^{2}/2}}{\sqrt{2{P}_{\pm }}}\displaystyle \sum _{n=0}^{\infty }\displaystyle \frac{{\alpha }^{n}}{\sqrt{n!}}\left(\cos (\sqrt{n+1}{{gt}}_{1})\pm \displaystyle \frac{\sqrt{n}}{\alpha }\sin (\sqrt{n}{{gt}}_{1})\right)| n\rangle \\ & \approx & \displaystyle \frac{1}{\sqrt{2{P}_{\pm }}}\left(\left|\sqrt{{\alpha }^{2}-\displaystyle \frac{\pi }{4}}\right\rangle \pm \left|\sqrt{{\alpha }^{2}+1+\displaystyle \frac{\pi }{4}}\right\rangle \right),\end{array}\end{eqnarray} \tag{ 28 }$$

a superposition of coherent states with slightly different average photon number, the phase of which is determined by the observed superposition state of the atom.

Recall that the performance of the scheme for an arbitrary cavity state depends both on the width of the photon number distribution (a narrow distribution implies a unique optimal interaction time) and the overlap of the initial state with a shifted state (one that contains an additional photon in the field). Figure 4 shows the photon number distribution of the cavity field conditional on atomic state $| +\rangle$ or $| -\rangle$ after the first round. Both resultant states have large overlap with the same states shifted up in average photon number, so that they may both be used as a coherence resource for subsequent interactions; a failed interaction does not destroy the reservoir coherence, as found in similar scenarios [30]. Note, however, that the cavity state conditioned on failure for the first atom has a wider distribution in photon number than in the case of success. Moreover, the distribution ${P}_{-}(n)$ has two peaks, suggesting that there is no unique optimal interaction time for the next round. Let us return to equation (28): for the positive phase superposition there is constructive interference for photon numbers near the centre of the distribution P(n), while for the negative superposition this interference is destructive, and only the wings of the distribution remain, as reflected in figure 4. This leads to a larger spread in Rabi frequencies, so that the chosen interaction time is not a good approximation to a quarter-cycle in each constant excitation number subspace: the observed decrease in probability of success seen in table 1 is a result of this effect.

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Again considering the positive phase superposition in equation (28), we observe that the relative photon number variance ${({\rm{\Delta }}n)}^{2}/\langle \hat{n}\rangle$ is decreased when the first atom is successfully produced in the state $| +\rangle$: for our numerical example of α = 10, the variance of the photon number distribution ${P}_{+}(n)$ is given by⁵

$$\begin{eqnarray}&&{\left({\rm{\Delta }}{n}_{+}\right)}^{2}=\langle {\hat{n}}_{+}^{2}\rangle -\langle {\hat{n}}_{+}{\rangle }^{2}=100.211.\end{eqnarray} \tag{ 29 }$$

An analysis of the success probabilities for squeezed cavity states shows that reducing the variance of the number distribution can lead to higher success probabilities up to a certain squeezing strength (see appendix D for details). Considering the effect of distribution width alone (rather than state overlap), the results suggest that after successful interactions the performance of the coherent state cavity field will be enhanced as the distribution narrows, but that this effect only works up to a certain point: when the variance of the cavity state reaches $\sqrt{2\bar{n}/\pi }$, the success probability will peak and can only decrease upon further squeezing.

So far we have considered just two rounds; upon interaction with a sequence of atoms, the cavity field becomes correlated with each subsequent atom, thus also inducing correlations between atoms. Figure 5 shows the conditional success probabilities for preparing the atom in the state $| +\rangle$ for the rth atom after r − 1 successful or unsuccessful attempts with the preceding r − 1 atoms. In the case of success for each atom, as shown in the left-hand plot, an increase of the success probability with the number of atoms is beyond what can be expected due to the increase of the mean photon number in the cavity alone. This supports the idea that the deformation, or squeezing [5, 34], of the cavity state indeed acts to enhance the probability of success.

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To confirm the squeezing of the number variance with each round, the evolution of the number and phase uncertainty during the first three interactions are shown in figure 6. The phase probability distribution, given by [5, 51–53, 34]

$$\begin{eqnarray}&&P(\phi )=\displaystyle \frac{1}{2\pi }\sum _{n,m=0}^{\infty }{{\rm{e}}}^{{\rm{i}}(m-n)\phi }\langle n| \psi \rangle \langle \psi | m\rangle ,\end{eqnarray} \tag{ 30 }$$

shows an increase in variance with each atom: this complements the decrease of the number variance, so that the total uncertainty of the state remains unchanged. Note that when considering the mixed cavity state unconditional upon previous outcomes, both the phase and number variances increase and the total uncertainty ${({\rm{\Delta }}{n}_{\mathrm{tot}})}^{2}{({\rm{\Delta }}{\phi }_{\mathrm{tot}})}^{2}$ increases by around 2% in our example.

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The second graph in figure 5 shows that after initial failure, the success probability for the next interaction is only slightly reduced; an effect which does not change significantly when more unsuccessful outcomes occur. To observe how the probability of successive failure scales with α (or $\bar{n}$) in the Jaynes–Cummings model, a plot of the numerically calculated failure probability as a function of α is shown in figure 7, for the first five interactions which leave the atoms in the state $| -\rangle$. Alongside these plots are shown trend lines, which allow us to estimate the dependence of $P({-}^{r})$ on $\bar{n}$ as

$$\begin{eqnarray}&&P({-}^{r})=P({-}_{r}\,\cap \,{-}_{(r-1)}\,\cap \,...\,\cap \,{-}_{1})\sim \displaystyle \frac{1}{{\bar{n}}^{r}}.\end{eqnarray} \tag{ 31 }$$

Recalling that the probability of failure in the first round is ${ \mathcal O }(\bar{n})$, this $\sim 1/{\bar{n}}^{r}$ dependence indicates that probabilities of multiple failures scales like the product of single-round probabilities, such that subsequent atoms are only weakly correlated.

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Finally, to aid in visualizing the evolution of the cavity state upon successive interactions, figures 8–11 (video online is available at stacks.iop.org/NJP/22/043008/mmedia) show the Husimi Q-function

$$\begin{eqnarray}&&Q(\alpha )=Q(q+{\rm{i}}{p})=\displaystyle \frac{1}{\pi }\langle \alpha | \hat{\rho }| \alpha \rangle \end{eqnarray} \tag{ 32 }$$

in phase-space [34] for the first 9 interactions. Figures 8–10 show the distribution for successful and unsuccessful interactions (i.e. conditioned on $| +\rangle$, $| -\rangle$ respectively), while figure 11 shows the state of the mixed (reduced) density matrix of the cavity when ignoring any information about the atomic states. Each successful interaction (figure 8) moves the centre of the distribution to higher q and therefore higher photon number, while deforming it such that the spread in this direction is squeezed to lower photon number variance. As can be seen in figure 9, failure to produce the state $| +\rangle$ leads to a complete change in the shape of the distribution, with a dip appearing at the centre after one interaction. This is a feature of the fact that the cavity is projected into a negative superposition of two similar states when the first atom is found in the state $| -\rangle$. Repeated failure increases the size of this hole until the outer wing of the distribution is eliminated, and the state is pushed further towards a (squeezed) vacuum state. Interestingly, even after finding an atom in the state $| -\rangle$ once, subsequent successful interactions can still somewhat compensate for the initial breakdown, as they bring the cavity closer to a coherent state with higher photon number again. This behaviour can be seen in figure 10: a successful outcome at any stage in the sequence of success and failure acts to squeeze the distribution and move it towards higher q.

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Returning now to the Åberg scheme; the reservoir-induced operation works identically on each atom, so that the copies produced after each interaction are identical. They are not, however, independent: through the interaction with the reservoir, all of the atoms become correlated, and correlated too with the reservoir itself. Conditional on just one atom being found in state $| -\rangle$, the reservoir collapses into the two-level state

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{{P}_{-}^{L}}}\displaystyle \frac{1}{2}\left(| {\eta }_{L},{l}_{o}\rangle -| {\eta }_{L},{l}_{0}+1\rangle \right)=\displaystyle \frac{1}{\sqrt{2}}(| {l}_{0}\rangle -| {l}_{0}+L+1\rangle ),\end{eqnarray} \tag{ 39 }$$

such that the coherence of the reservoir is almost completely destroyed. Here, ${P}_{-}^{L}$ is the probability for this to happen, which is very small for ladder states of large L, but crucially never zero. This is reflected in the success probabilities of multiple interactions, where it is seen that the probability of multiple failures does not decrease exponentially in the number of systems, but remains proportional to L⁻¹ [30].

By contrast, in the Jaynes–Cummings scheme, the states of subsequent atoms are not identical, however they are (approximately) independent of each other. We have seen that over successive uses the many-copy probability of failure decreases exponentially, scaling like the product of individual failures. Only after ${ \mathcal O }({\alpha }^{2})$ uses does the protocol begin to fail and performance rapidly declines for subsequent uses. Part of the story of the difference between the two schemes is thus the effect of failure on the reservoir of coherence.

In both schemes, after an atom interacts with the cavity the composite state is of the form

$$\begin{eqnarray}&&| {\rm{\Psi }}\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(| {\rm{g}}\rangle | {F}_{{\rm{g}}}\rangle +| {\rm{e}}\rangle | {F}_{{\rm{e}}}\rangle \right),\end{eqnarray} \tag{ 40 }$$

where $| {F}_{{\rm{g}}}\rangle$ and $| {F}_{{\rm{e}}}\rangle$ are the states of the field that couple with the atomic ground and excited states. After a single use, for the Åberg model these are ${\rm{\Delta }}| {\eta }_{L,{l}_{0}}\rangle$ and $| {\eta }_{L,{l}_{0}}\rangle$, while for the Jaynes–Cummings model these are $| {\alpha }_{{\rm{e}}}\rangle$ and $| {\alpha }_{{\rm{g}}}\rangle ;$ for subsequent uses the general expression has the same form. From equation (40), the density matrix of the atom after an interaction is straightforwardly given by

$$\begin{eqnarray}&&{\rho }_{a}=\displaystyle \frac{1}{2}\left[\langle {F}_{{\rm{g}}}| {F}_{{\rm{g}}}\rangle | {\rm{g}}\rangle \langle {\rm{g}}| +\langle {F}_{{\rm{e}}}| {F}_{{\rm{g}}}\rangle | {\rm{g}}\rangle \langle {\rm{e}}| +\langle {F}_{{\rm{g}}}| {F}_{{\rm{e}}}\rangle | {\rm{e}}\rangle \langle {\rm{g}}| +\langle {F}_{{\rm{e}}}| {F}_{{\rm{e}}}\rangle | {\rm{e}}\rangle \langle {\rm{e}}| \right].\end{eqnarray} \tag{ 41 }$$

It follows that the probability of the atom being found in the state $| +\rangle$ after an interaction largely depends upon the inner product $\langle {F}_{{\rm{e}}}| {F}_{{\rm{g}}}\rangle$: if $\mathrm{Re}\left[\langle {F}_{{\rm{e}}}| {F}_{{\rm{g}}}\rangle \right]\gt 0$, then this probability is greater than 1/2. The closer it gets to zero, the more the atom is entangled with the cavity, and therefore its individual state is less well defined. Once it reaches $\mathrm{Re}\left[\langle {F}_{{\rm{e}}}| {F}_{{\rm{g}}}\rangle \right]\lt 0$, the state $| -\rangle$ is more likely to be obtained, where now a larger negative value increases the probability of this outcome. The overlap between the state $| {F}_{{\rm{g}}}\rangle$ and $| {F}_{{\rm{e}}}\rangle$ for the Jaynes–Cummings and Åberg models are shown in figure 12. The coefficient distributions, C_n of the state vectors in the energy basis $| {F}_{{\rm{g}}/{\rm{e}}}\rangle ={\sum }_{n}{C}_{n}| n\rangle$ for the Jaynes–Cummings model are given in the first column of the figure, while those for the Åberg model are displayed in the graphs in the second column. The first row represents $| {F}_{{\rm{g}}}\rangle$ and $| {F}_{{\rm{e}}}\rangle$ in the first interaction. In each case there is a large overlap between these two distributions, leading to a high probability of success for the protocol. The second row shows $| {F}_{{\rm{g}}}\rangle$ and $| {F}_{{\rm{e}}}\rangle$ after a failure in the first round: there is still a large area overlap between these two states in the Jaynes–Cummings case, while in the Åberg scheme the graphs do not overlap at all, and the qubit is found in either state $| +\rangle$ or $| -\rangle$ with equal probability. Finally, the coefficient distributions upon two consecutive failures in the first two interactions are given in the third row. For the Åberg scheme, the overlap of the states $| {F}_{{\rm{g}}}\rangle$ and $| {F}_{{\rm{e}}}\rangle$ is a negative number, corresponding to a higher probability of obtaining the state $| -\rangle$ for the third atom following two failures. By contrast, even in the case of multiple consecutive failures, areas of overlap for the Jaynes–Cummings model are nonzero, but decreasing: the coherent state field may yet be used as a coherent resource, but with slightly decreased efficiency.

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The effect on the cavity state is mirrored in the correlations introduced between subsequent systems, discussed previously. Analytical expressions for the joint probabilities for two consecutive interactions are given in table 2 for the Jaynes–Cummings interaction and the Åberg scheme [29, 30], in terms of $\bar{n}$ and L respectively. The state of the two atomic qubits after the interaction is non-separable in each case, since the joint probabilities show correlations. However, for the Jaynes–Cummings interaction these enter at second order only, such that the probabilities are independent to second order. The single-atom probabilities scale linearly $1/\bar{n}$ in the Jaynes–Cummings case and 1/L in the Åberg scheme. However, the joint probabilities demonstrate the greater robustness of the Jaynes–Cummings model using coherent states against multiple failures: the probability for ending up in the state $| -\rangle$ for two consecutive atoms in this case scales as $1/{\bar{n}}^{2}$ compared to 1/L in Åberg's scheme. This is in agreement with the numerical results of figure 5.

Table 2.  Comparison of probabilities and joint probabilities obtained to ${ \mathcal O }(1/{\bar{n}}^{2})$ using coherent states in the Jaynes–Cummings interaction, and ladder states in the scheme proposed by Åberg [29]. The interaction time has been set such that ${gt}\sqrt{\bar{n}+\mu }=\pi /4$, where μ is an arbitrary real number and $\mu \ll \bar{n}$.

	Coherent states in the Jaynes–Cummings model	Åberg scheme
P(+)	$1-\tfrac{{\left(\pi +2\right)}^{2}}{64\bar{n}}+\tfrac{{\pi }^{4}-4(5-40\mu +16{\mu }^{2}){\pi }^{2}+64(1+2\mu )\pi +16}{4096{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$1-\tfrac{1}{2L}$
P(−)	$\tfrac{{\left(\pi +2\right)}^{2}}{64\bar{n}}-\tfrac{{\pi }^{4}-4(5-40\mu +16{\mu }^{2}){\pi }^{2}+64(1+2\mu )\pi +16}{4096{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{2L}$
P(+) × P(+)	$1-\tfrac{{\left(\pi +2\right)}^{2}}{32\bar{n}}+\tfrac{3{\pi }^{4}+8{\pi }^{3}-4(32{\mu }^{2}-80\mu ){\pi }^{2}+32(11+8\mu )\pi +48}{4096{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$1-\tfrac{1}{L}+\tfrac{1}{4{L}^{2}}$
P(+) × P(−)	$\tfrac{{\left(\pi +2\right)}^{2}}{64\bar{n}}-\tfrac{{\pi }^{4}+4{\pi }^{3}-2(16{\mu }^{2}-40\mu -17){\pi }^{2}+16(9+4\mu )\pi +16}{2048{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{2L}-\tfrac{1}{4{L}^{2}}$
P(−) × P(+)	$\tfrac{{\left(\pi +2\right)}^{2}}{64\bar{n}}-\tfrac{{\pi }^{4}+4{\pi }^{3}-2(16{\mu }^{2}-40\mu +11){\pi }^{2}+16(3+2\mu )\pi +16}{2048{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{2L}-\tfrac{1}{4{L}^{2}}$
P(−) × P(−)	$\tfrac{{\left(\pi +2\right)}^{4}}{4096{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{4{L}^{2}}$
$P(+\,\cap \,+)$	$1-\tfrac{{\left(\pi +2\right)}^{2}}{32\bar{n}}+\tfrac{5{\pi }^{4}-32(4{\mu }^{2}-10\mu -1){\pi }^{2}+64(4\mu +5)\pi +80}{4096{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$1-\tfrac{3}{4L}$
$P(-\,\cap \,+)$	$\tfrac{{\left(\pi +2\right)}^{2}}{64\bar{n}}-\tfrac{{\pi }^{4}-(16{\mu }^{2}-40\mu -13){\pi }^{2}+32(\mu +2)\pi +16}{1024{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{4L}$
$P(+\,\cap \,-)$	$\tfrac{{\left(\pi +2\right)}^{2}}{64\bar{n}}-\tfrac{{\pi }^{4}+4{\pi }^{3}-(16{\mu }^{2}-40\mu -5){\pi }^{2}+32(\mu +1)\pi +16}{1024{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{4L}$
$P(-\,\cap \,-)$	$\tfrac{{\left(\pi +2\right)}^{2}(3{\pi }^{2}+4\pi +12)}{4096{\bar{n}}^{2}}+{ \mathcal O }\left(\tfrac{1}{{\bar{n}}^{3}}\right)$	$\tfrac{1}{4L}$

We conclude with a plausability argument for how many copies one might reasonably expect to be able to extract from a reservoir of coherence. Consider r copies of the equal superposition state $| +\rangle$:

$$\begin{eqnarray}&&| +{\rangle }^{\otimes r}=\displaystyle \frac{1}{\sqrt{{2}^{r}}}{\left(| 0\rangle +| 1\rangle \right)}^{\otimes r}=\displaystyle \frac{1}{\sqrt{{2}^{r}}}\sum _{k=0}^{r}\sqrt{\displaystyle \left(\genfrac{}{}{0em}{}{r}{k}\right)}| r,k\rangle ,\end{eqnarray} \tag{ 42 }$$

where $| r,k\rangle$ denotes the permutation invariant, symmetric r-qubit state containing k zeroes and r − k ones. Note that each term in the superposition corresponds to a different total energy. The binomial distribution has mean k = r/2 and variation r/4. Thus the width of the distribution, the standard deviation, is ${ \mathcal O }(\sqrt{r})$. For large r, the distribution in this range is approximately flat with a steep fall off to zero outside [54], and we find an approximately equal superposition of energy eigenstates containing $\sqrt{r}$ terms, corresponding to those values of k within ${ \mathcal O }(\sqrt{r})$ of r/2.

For the ladder state $| {\eta }_{L,{l}_{0}}\rangle$ in the Åberg scheme, the reservoir state contains exactly L energy eigenstates in equal superposition. Thus, in terms of the coherence properties in the energy basis, r copies of $| +\rangle$ is approximately equivalent to a ladder state of size $\sqrt{r}$. In other words, the ladder state $| {\eta }_{L,{l}_{0}}\rangle$ is aproximately equivalent to L² copies of $| +\rangle$. The scheme as given in [29] may be used to produce an indefinite number of identical mixed-state copies. However, as pointed out above, these are not independent, but rather are in a highly correlated joint state, with the initial coherence of the reservoir distributed across the joint state.

For the quantum optical coherent state the coefficients in the energy eigenbasis $| n\rangle$ are Poisson distributed with a mean and variance both equal to $\bar{n}$. Again in the limit of large $\bar{n}$ this corresponds to an approximately equal superposition of $\sqrt{\bar{n}}$ terms, where each term in the superposition has a different total energy. Thus, in terms of the coherence properties in the energy basis, r copies of $| +\rangle$ is approximately equivalent to a coherent state with average photon number $\bar{n}\simeq r$. Thus we expect a coherent state $| \alpha \rangle$ to be able to create ${ \mathcal O }({\alpha }^{2})$ independent copies of $| +\rangle$ before the protocol starts to break down. The Jaynes–Cummings interaction thus seems to be close to optimal for extracting coherence.

As a final note, if we therefore identify the 'size' of the state in the coherent state case as $\sqrt{\bar{n}}$ and in the Åberg case as L, then we see that the error in the Jaynes–Cummings model scales as the inverse square of the size of the state, $1/\bar{n}$, while in the Åberg state the error scales as the inverse of the size of the resource 1/L. Thus for equivalent resources, the Jaynes–Cummings model produces better copies which are less correlated, and moreover repeated use improves the quality of the copies, for a moderate number of uses.