Duality and quantum state engineering in cavity arrays

TL;DR

This paper explores a duality between two coupled cavities and an array of cavities, enabling controlled photon transfer and high-fidelity generation of entangled states without populating intermediate cavities.

Contribution

It introduces a duality framework that allows precise control of photon transfer and entangled state generation in cavity arrays, advancing quantum state engineering techniques.

Findings

Achieves perfect photon transfer between any two cavities.

Establishes conditions for high-fidelity NOON state generation.

Demonstrates duality as a tool for quantum state control.

Abstract

A system of two coupled cavities with $N-1$ photons is shown to be dynamically equivalent to an array of $N$ coupled cavities containing one photon. Every transition in the two cavity system has a dual phenomenon in terms of photon transport in the cavity array. This duality is employed to arrive at the required coupling strengths and nonlinearities in the cavity array so that controlled photon transfer is possible between any two cavities. This transfer of photons between two of the cavities in the array is effected without populating the other cavities. The condition for perfect transport enables perfect state transfer between any two cavities in the array. Further, possibility of high fidelity generation of generalized NOON states in two coupled cavities, which are dual to the Bell states of the photon in the cavity array, is established.

Equations

$$H_A={\omega }_1{a}_1^{†}{a}_1+{\omega }_2{a}_2^{†}{a}_2+J\left[a_1^{†}{a}_2+a_1{a}_2^{†}\right].$$

$$H_A=$$

$$+\sum _{n=0}^{N-2}{J}_{n+1}(|n+1⟩⟨n+2|+|n+2⟩⟨n+1|),$$

$$H_B=\sum _{l=1}^N{\tilde{\omega }}_l{b}_l^{†}{b}_l+\sum _{l=1}^{N-1}{\tilde{J}}_l(b_l^{†}{b}_{l+1}+b_l{b}_{l+1}^{†}),$$

$$H_B=\sum _{l=1}^N\widetilde{{\omega }_l}|l⟩⟩⟨⟨l|+\sum _{l=1}^{N-1}{\tilde{J}}_l(|l⟩⟩⟨⟨l+1|+|l+1⟩⟩⟨⟨l|).$$

$$e^{-i{H}_{A}t}|N-1-n,n⟩=$$

$$(\cos Jt{)}^{N-1-(k+l)}(-i\sin Jt{)}^{k+l}$$

$$\sqrt{\frac{{}^{(N-1)}{C}_n}{{}^{(N-1)}{C}_{n+k-l}}}$$

$$⟨n_j{⟩}_t=⟨b_j^{†}{b}_j{⟩}_t=\sum _{l=1}^N|G_{jl}{|}^2⟨b_l^{†}{b}_l{⟩}_0.$$

$$G_{jl}=$$

$$\sin \left(\frac{j\pi k}{N+1}\right)\sin \left(\frac{l\pi k}{N+1}\right).$$

$$⟨n_N{⟩}_t=⟨b_N^{†}{b}_N{⟩}_t=|G_{N1}{|}^2,$$

$$H_A^{\prime }={\omega }_1{a}_1^{†}{a}_1+{\omega }_2{a}_2^{†}{a}_2+{\chi }_1{a}_1^{†2}{a}_1^2+{\chi }_2{a}_2^{†2}{a}_2^2+J\left[a_1^{†}{a}_2+a_1{a}_2^{†}\right],$$

$$\Delta =\frac{(p(p-1)-m(m-1)){\chi }_1+(q(q-1)-n(n-1)){\chi }_2}{m-p}.$$

$$⟨m,n|H_A^{\prime }|m,n⟩=⟨p,q|H_A^{\prime }|p,q⟩.$$

$$H_B^{\prime }=$$

$$+\sum _{l=1}^{N-1}\sqrt{l(N-l)}J(b_l^{†}{b}_{l+1}+b_l{b}_{l+1}^{†}),$$

$${\tilde{\chi }}_{k+1}+{\tilde{\omega }}_{k+1}$$

$$+[(N-1-k)(N-2-k)+k(k-1)]\chi$$

$$|\psi ⟩$$

Full text

I Introduction

Quantum theory provides for fundamentally newer ways of realizing secure communication, faster computation and precision metrology. Quantum networks are basic to implementing these ideas. Physical systems such as spin chains, cavity arrays, Joesphson junction arrays, quantum dots, etc., have been investigated to explore their potential for network implementation.

Coupled-cavity arrays have been used extensively in generating nonclassical states of the electromagnetic field and quantum information processing Har ; Not . The technology has matured to such an extent that precise control of the cavity field dynamics is possible . Suitable tailoring of inter-cavity coupling has been put to use in entanglement generation Miry , quantum state preparation Lar ; Almeida , state transfer between spatially separated cavities Cir ; Verm , exhibition of quantum interference Nil ; Nil2 , Rabi oscillation Yoshi , to mention a few. Consideration of cavities filled with nonlinear media has led to exploration of other quantum phenomena DEC , such as photon blockade Ima ; Birn ; Ada and localization-delocalization related to bunching and antiubunching of photons Nil ; Hen ; Sch . Interplay between intra-cavity nonlinearity and inter-cavity coupling strength has been exploited to control the photon statistics in cavity Sara . Engineering of inter-cavity coupling in arrays offers the possibility of simulating effects of disorder Seg , phase transitions Hart ; Hart2 ; Bran , etc. in condensed matter physics.

The identical and indistinguishable nature of photons require that the number of photons and the number of levels to be occcupied by them are considered together. For example, on the consideration of blackbody radiation, Planck distribution is obtained for $N$ photons to be distributed in $g$ levels when the number of possible ways of distributing as $(N+g-1)!/N!(g-1)!$. It is interesting to note that the result is the same if there are $g-1$ photons and $N+1$ levels. This possibility of interchanging the roles of the number of particles and the number of levels is a duality. Another simple example of duality is in Euler characteristic $V-E+F=2$, where $V,E$ and $F$ refer to the number of vertices, edges and faces respectively of a convex solid. In this expression the roles of $V$ and $F$ are interchangeable. Such duality relationships are very much sought after in physics, which often facilitates understanding of nontrivial aspects of one system in terms of easily accessible features of the other atiyah . In this article, a duality is established between two dynamical systems, namely, one photon in an array of $N$ cavities and $N-1$ photons in two coupled cavities, considering both linear and Kerr-nonlinear cavities. Every transition in the two cavity system has a dual phenomenon in terms of photon transport in the cavity array. This feature helps to identify the conditions required in the linear cavity array for a perfect transport of photon between two cavities equidistant from the respective ends. The result is generalized to nonlinear cavities which allows perfect transport between any two cavities in the array. Another prospect that makes this study interesting in the context of information transfer is the possibility of perfect state transfer from one cavity to another. Our results also point to the possibility of generating NOON states, which have found diverse applications and relate to a Bell state of the dual system Dor ; Keb ; Sean ; Marcel .

Consider a system of two linearly coupled cavities described by the Hamiltonian

[TABLE]

Here $\omega_{1}$ and $\omega_{2}$ are the resonance frequencies of the respective cavities and $J$ is the coupling strength. Suffix $A$ has been used to refer to this system of two coupled cavities. Here $a_{1(2)}$ and $a_{1(2)}^{\dagger}$ are the annihilation and creation operators for the first(respectively, second) cavity. Let $|n+1\rangle$ represent the bipartite state $|N-1-n,n\rangle$ of the two cavities corresponding to $N-n-1$ quanta in the first cavity and $n$ quanta in the second cavity. The total number of quanta in the two cavities is $N-1$. If the number of photons is fixed to be $N-1$, the Hamiltonian is expressed as

[TABLE]

where $\Omega_{n+1}=[(N-1-n)\omega_{1}+n\omega_{2}]$ and $J_{n+1}=\sqrt{(n+1)(N-1-n)}J$.

Now consider a system of $N$ linearly coupled cavities, described by

[TABLE]

where $\tilde{\omega}_{l}$ is the cavity resonance frequency for the $l$th cavity, $b_{l}$ and $b_{l}^{\dagger}$ the annihilation and creation operators for the $l$th cavity. The strength of coupling between the $l$ and $(l+1)$ cavities in the array is $\tilde{J}_{l}$. This form of the Hamitonian can be mapped to that of a spin network which has been studied in the context of state transfer and entanglement generation bose . For a single quantum in the system, the possible states are $|l\rangle\rangle$, which represents one photon in the $l$th cavity while the other cavities are in their respective vacuua. Then the Hamiltonian is

[TABLE]

Duality of the two systems described by $H_{A}$ and $H_{B}$ respectively is identified if $\tilde{J}_{l}=\sqrt{l(N-l)}J$, $\tilde{\omega_{l}}=[(N-l)\omega_{1}+(l-1)\omega_{2}]$ and $l=n+1$. The transition $\ket{N-1-n,n}\rightarrow\ket{N-2-n,n+1}$ in the system of two cavities corresponds to photon transport from $|n+1\rangle\rangle\rightarrow|n+2\rangle\rangle$ in the array. In essence, transitions in the two-cavity system are equivalent to transport of a photon across the cavities in the array.

If the initial state of the two cavity system at resonance $(\Delta=\omega_{1}-\omega_{2}=0)$ is $\ket{N-1-n,n}$, it evolves to

[TABLE]

at time $t$. It is worth noting that the time-evolved state is an atomic coherent state Kim .

At $t=\pi/2J$ the time-evolved state is $\ket{n,N-1-n}$, corresponding to swapping the number of photons in the cavities. Time evolution of the respective probabilities for $\ket{N-1,0}$ to become $\ket{0,N-1}$ corresponding to $N=2,4$ and $6$ are shown in Fig. 1. Complete transfer of photons between the end cavities of the array corresponds to $|N-1,0\rangle\rightarrow|0,N-1\rangle$ transition in the coupled cavity system. By the duality between $H_{A}$ and $H_{B}$, these profiles also represent the probability of transferring a photon from one end to the other in an array of $2,4$ and $6$ cavities respectively. It may be noted that the probabilities attain their peak value of unity, corresponding to complete transport of a quantum between the end cavities, when $t=\pi/2J$.

It is essential that $J_{l}$ are related to $J$ in the specified manner. It is of interest to note that the requirement for such inhomegeneous couplings in linear quantum spin networks, optical waveguide arrays has been explored Chris ; Yogesh . If the coupling strengths $J_{l}$ are equal and all the cavities are identical, the average number of quanta at time $t$ in the $j$-th cavity is given by

[TABLE]

where,

[TABLE]

Here $\tilde{\omega}$ is the resonance frequency of the cavities in the array. If the quantum is initially in the first cavity, that is, $\langle b_{l}^{\dagger}b_{l}\rangle_{0}=\delta_{1,l}$, then

[TABLE]

is the average photon number in the last cavity. For large $N$, $\sin(Nk\pi/N+1)\approx\sin(k\pi)=0$ and $G_{N1}$ tends to zero. This indicates that complete transfer is not possible. Time evolution of the average number of quantum $\langle n_{N}\rangle$ in the end cavity for arrays with $N=$3, 4, 5 and 10 cavities respectively are shown in Fig. 2. From the figure, it is clear that complete transfer occurs if the array has three cavities. Maximum of $|G_{N1}|^{2}$ decreases with increasing the number of cavities. Hence, complete transfer does not occur if the homegeneously coupled array has more than three cavities whereas inhomogeneous coupling achieves complete transfer in shorter time Chris ; Chris2 .

It is to be further noted that complete transition is possible only between the states $\ket{N-1-n,n}$ and $\ket{n,N-1-n}$ of the coupled cavities. Analogously complete transfer of a single photon can occur only in between $(n+1)$th and $(N-n)$th cavities in the cavity array. With linear coupling, it is not possible to achieve complete transfer between two arbitrary cavities in the array.

To see if nonlinearity helps in steering the evolution of states to achieve perfect transfer, we consider the Kerr-type nonlinearity. We present the analysis of two coupled nonlinear cavities. The required Hamiltonian is

[TABLE]

which describes two linearly coupled Kerr cavities.

If it is required to evolve from $|m,n\rangle$ to $|p,q\rangle$, consider the superposition $\ket{X_{\pm}}=1/\sqrt{2}(\ket{m,n}\pm\ket{p,q})$. These two states become approximate eigenstates of the $H^{\prime}_{A}$ if $J<<\chi_{1},\chi_{2},\omega_{1},\omega_{2}$, and

[TABLE]

This condition is equivalent to

[TABLE]

This equality of average energy in the two states is another way of stating the requirement that the states $\ket{X_{\pm}}$ are approximate eigenstates of $H^{\prime}_{A}$. In the discussion that follows it is assumed that $\chi_{1}=\chi_{2}=\chi>0$ and the condition simplifies to $\Delta=2\chi(n-p)$.

If the initial state is $\ket{m,n}=1/\sqrt{2}(\ket{X_{+}}+\ket{X_{-}})$, the state of the system at a later time is, $\ket{\psi(t)}\approx\left[\cos\left(\theta_{a}t\right)\ket{m,n}-i\sin\left(\theta_{a}t\right)\ket{p,q}\right],$ with $\theta_{a}=(\lambda_{s}^{a}-\lambda_{n}^{a})/{2}$. Here $\lambda_{s}^{a}$ and $\lambda_{n}^{a}$ are the eigenvalues of $H^{\prime}_{A}$ corresponding to the approximate eigenvectors $|X_{+}\rangle$ and $|X_{-}\rangle$ respectively. At $t=\pi/(\lambda_{s}^{a}-\lambda_{n}^{a})$, the time-evolved state is $|p,q\rangle$. This is the minimum time required to switch from $|m,n\rangle$ to $|p,q\rangle$. Thus, the state switching (SS) condition given in Eq. 10 ensures that there is complete transfer from the initial state $|m,n\rangle$ to the desired final state $|p,q\rangle$.

It is immediate that detuning $\Delta$ and nonlinear coupling strength $\chi$ can be properly chosen for a given value for $n-p$. As the value of $n$ is specified in the initial state $|m,n\rangle$, the two parameters $\Delta$ and $\chi$ fix the number of quanta say $s$, that can be transferred and the target state becomes $\ket{p=m\pm s,q=n\mp s}$. It needs to be emphasized that for a given $\Delta$ and $\chi$ satisfying the SS condition, no more than two states can have their average energies equal as shown in Fig. 3. Once these parameters are fixed, probability of transition to any state other than the target state is negligible. Hence, $\Delta$ and $\chi$ provide control to steer the system from the initial state $|m,n\rangle$ to the final state $|p,q\rangle$.

In Fig. 3, $\bra{m,n}H^{\prime}_{A}\ket{m,n}$ is plotted as a function of $m$, keeping $m+n=28$ fixed. From Fig. 3(a), it is seen that every state has only one partner state with equal average energy. So, SS can occur between these partner states. It is observed from Fig. 3(b) that not every state has a partner state with equal average energy. Essentially, states without partner states are approximate eigenstates of $H^{\prime}_{A}$ and, therefore, do not evolve. This brings out another control aspect available in the system, namely, the possibility of inhibiting evolution of certain states with properly chosen values of the control parameters $\chi$ and $\Delta$.

Consider the initial state of the coupled cavity system to be $\ket{50}$. In Fig. 4, the probability of detecting the system in the state $\ket{14}$ at later times is shown when the required SS condition is satisfied. The values have been generated from the approximate evolved state $\ket{\psi(t)}$ and also by exact numerical solution of the evolution corresponding to $H^{\prime}_{A}$. It is seen that the quanta are exchanged periodically driving the system between $\ket{14}$ and $\ket{50}$ and transfer to other states is insignificant.

In order to effect transition to other states, the value of $\Delta$ can be chosen properly. The maximum probabilities of detecting the target state $\ket{p,q}$ with $p=1,2,3,4,5$ and $p+q=5$ from $\ket{50}$ are shown in Fig. 5 as a function of $\Delta$. The value of $\chi$ has been chosen to be 0.2. Depending on the value of detuning, exchange of quanta is precisely controlled to different target states.

A duality relation of the two cavity system with the cavity array system is possible in this nonlinear case too. Consider the nonlinear cavity array Hamiltonian

[TABLE]

which includes Kerr nonlinearity in each cavity of the array. This is dual to $H^{\prime}_{A}$ if $\tilde{\omega}_{k+1}+\tilde{\chi}_{k+1}=(N-1-k)\omega_{1}+k\omega_{2}+[(N-1-k)(N-2-k)+k(k-1)]\chi$. With this identification, transitions among levels in the two Kerr cavity system can be mapped to transfer of photon in the Kerr cavity array.

In particular, transition from $\ket{N-1-n,n}$ to $\ket{N-1-q,q}$ in the coupled cavities corresponds to transferring a photon between $(n+1)$-cavity to $(q+1)$-cavity in the cavity array. The condition to realize this transfer is $\langle\langle n+1|H^{\prime}_{B}|n+1\rangle\rangle=\langle\langle q+1|H^{\prime}_{B}|q+1\rangle\rangle$, whose dual relation for the coupled cavities is given in Eq. 11. For the Hamiltonian $H^{\prime}_{B}$, this condition yields,

[TABLE]

to realize complete transfer of photon occurs between the cavities. On employing cavity-dependent nonlinearity $\tilde{\chi}_{l}$, controlled transfer of photons between selected cavities is achievable. Such site-dependent nonlinearity has been realized recently by embedding quantum dots in each photonic crystal cavities Calic ; Hugg ; Surr .

In the limit of weak coupling strength $J$, $\frac{1}{\sqrt{2}}(\ket{n+1}\rangle\pm\ket{q+1}\rangle)$ are eigenstates of $H^{\prime}_{B}$ and the corresponding eigenvalues are denoted by $\lambda_{s}^{b}$ and $\lambda_{n}^{b}$. The initial state $\ket{n+1}\rangle$ evolves under $H^{\prime}_{B}$ to $\ket{\psi(t)}\rangle\approx\cos\left(\theta_{b}t\right)\ket{n+1}\rangle-i\sin\left(\theta_{b}t\right)\ket{q+1}\rangle,$ where $\theta_{b}=(\lambda_{s}^{b}-\lambda_{n}^{b})/{2}$. It is seen that the photon is exchanged periodically between the cavities. An important feature of this process is that the other cavities in the array are not populated to any appreciable extent during the evolution. This conclusion is based on the observation that the states other than $\ket{n+1}\rangle$ and $\ket{q+1}\rangle$ do not contribute appreciably to $\ket{\psi(t)}\rangle$.

If the coupling term in the Hamiltonian $H^{\prime}_{A}$ is taken to be $J\left[e^{i\eta}a_{1}^{\dagger}a_{2}+e^{-i\eta}a_{1}a_{2}^{\dagger}\right]$ making the coupling constants are complex, then the initial state $\ket{m,n}$, evolves to $\ket{\psi(t)}\approx\left[\cos\left(\theta_{a}t\right)\ket{m,n}-ie^{-i(q-n)\eta}\sin\left(\theta_{a}t\right)\ket{p,q}\right].$ These states are of the form $\ket{\psi}=\cos\theta\ket{m,n}+e^{i\phi}\sin\theta\ket{p,q}$, if $\theta=\theta_{a}t$ and $\phi=-(\pi/2+(q-n)\eta)$.

If $m\neq p$ and $\theta\neq 0,\pm\pi/2$, then $\ket{\psi}$ is entangled. Additionally, if $m,q=N$ and $\theta=\pi/4$, the resultant state is

[TABLE]

the generalized NOON state. In the case of cavity array this is equivalent to generating the Bell state $\ket{\psi}\rangle=\cos\theta\ket{n+1}\rangle+e^{i\phi}\sin\theta\ket{q+1}\rangle$.

Another important outcome of the complex coupling in the context of single photon in cavity array is the possibility of state transfer between any two cavities. Consider the initial state of the cavity array to be $\alpha|\mbox{vac}\rangle\rangle+\beta\ket{n+1}\rangle$, which corresponds to the $(n+1)$-th cavity in the superposition $\alpha|0\rangle+\beta|1\rangle$ and the other cavities are in their respective vacuua. If the SS condition is satisfied, the time-evolved state is $\alpha|\mbox{vac}\rangle\rangle+\beta e^{-i\lambda t}(\cos\theta_{b}t\ket{n+1}\rangle-ie^{-i\eta(q-n)}\sin\theta_{b}t\ket{q+1}\rangle)$, where $\lambda=(\lambda_{s}^{b}+\lambda_{n}^{b})/2$. At $2\theta_{b}t=\pi$, then the state of the $(q+1)$-th cavity is the superposition $\alpha|0\rangle+\beta|1\rangle$ and the other cavities in their respective vacuua for the suitable value of $\eta$. Thus, the SS condition ensures the state of the field in the $(n+1)$-th cavity is transferred to the $(q+1)$-th cavity.

To summarize, transfer of a photon in an array of $N$ cavities is dynamically equivalent or dual to the problem of sharing of ($N-1$) photons between two coupled cavities, provided the parameters relevant to the systems are chosen properly. This duality is extendable even if the cavities are of Kerr-type, which, in turn, requires the couplings to be inhomogeneous. Duality between the two systems has made it transparent to identify the correct combination of the coupling strengths and local nonlinearities in the array for complete photon transfer between any two cavities in the array. In the linear case, perfect transport is possible only between the cavities which are symmetrically located from the end cavities of the array. Kerr nonliearity allows perfect transport between any two cavities without any restriction whatsoever. Importantly, this transfer is effected without populating the other cavities in the array, so that the transfer cannot be viewed as a continuous hopping of photons from one cavity to the other. In the presence of Kerr nonliearity and complex coupling strengths among the cavities, perfect state transfer between any two cavities is achieved. This feature is important in the context of encoding and transfer of information. High fidelity generation of entangled states of the form $\cos\theta\ket{m,n}+e^{i\phi}\sin\theta\ket{p,q}$ in coupled cavities is possible in the presence of Kerr nonlinearity in the cavities. Equivalently, Bell states of the cavity array are also achievable with high fidelity. The results of this work are important for designing suitable experiments for controlled transfer to photons by quantum switching.

One of the authors (NM) acknowledges the Department of Atomic Energy of the Government of India for a senior research fellowship.