Mode-mixing quantum gates and entanglement without particle creation in periodically accelerated cavities

We first consider the simplified case of a cavity in (1 + 1)-dimensional Minkowski spacetime. The cavity is assumed mechanically rigid, maintaining constant length L in its instantaneous rest frame. The proper acceleration at the centre of the cavity is denoted by a(τ), where τ is the proper time. To maintain rigidity, the acceleration must be bounded by |a(τ)|L/c² < 2 [11]. From now on we set c = ℏ = 1.

The cavity contains a real scalar field ϕ of mass μ₀ ⩾ 0, with Dirichlet boundary conditions. We assume that the cavity is initially inertial, and we denote by u_n, n = 1,2,..., a standard basis of cavity field modes that are of positive frequency $\omega _n =\sqrt {\mu _0^2 + (\pi n/L)^2}$ with respect to the cavity's proper time before the acceleration. We also assume that the cavity's final state is inertial, and we denote by $\tilde u_n$, n = 1,2,..., a standard basis of cavity field modes that are of positive frequency ω_n with respect to the cavity's proper time after the acceleration. Because of the acceleration at intermediate times, the two sets of modes need not coincide, but the completeness of each set allows the sets to be related by the Bogoliubov transformation [12, 13]

$$\begin{equation} \tilde u_m = \sum_n \,(\alpha_{mn} u_n + \beta_{mn} u^*_n), \end{equation} \tag{ 1 }$$

where the star denotes complex conjugation. The Bogoliubov coefficient matrices α and β are determined by solving the field equation in the cavity during the acceleration.

In the initial and final inertial regions the field operator ϕ has the respective expansions $\phi = \sum _n(a_n u_n + a_n^{\dagger } u^{*}_n)$ and $\phi = \sum _n(\tilde a_n \tilde u_n + {\tilde a}_n^{\dagger } {\tilde u}^{*}_n)$, where the non-vanishing commutators of the early (respectively late) time creation and annihilation operators are [a_n,a^†_m] = δ_nm $([\tilde a_n, {\tilde a}_m^{\dagger }] = \delta _{nm})$. The early and late time creation and annihilation operators need not coincide, but it follows from (1) that they can be expressed in terms of each other in terms of the Bogoliubov coefficients [12, 13]. In particular, the transformation mixes creation and annihilation operators if and only if some of the β-coefficients are non-vanishing.

Now, working in the Heisenberg picture, the quantum state of the field in the cavity does not change in time. However, given a state |Ψ〉, we interpret its particle content at early times in terms of the early time vacuum |0〉, which satisfies a_n|0〉 = 0, and the early time excitations created by a^†_n. At late times, we similarly interpret the particle content of |Ψ〉 in terms of the late time vacuum $|\tilde 0\rangle$, which satisfies $\tilde a_n|\tilde 0\rangle =0$, and the late time excitations created by ${\tilde a}_n^{\dagger }$. The acceleration hence affects the particle content of the cavity whenever the Bogoliubov transformation (1) differs from the identity transformation. The β-coefficients are responsible for creation and annihilation of particles, while the α-coefficients are responsible for mode mixing. In particular, the vacua |0〉 and $|\tilde 0\rangle$ coincide if and only if all the β-coefficients vanish [12, 13].

We shall express the Bogoliubov coefficients as a time-ordered integral, allowing both the magnitude and the time-dependence of the acceleration to remain general within the rigidity bound |a(τ)|L < 2.

We encode α and β into the matrix $\scriptsize U = \left(\begin{array}{@{}ll@{}}\alpha & \beta \\ \beta^{*} & \alpha^{*}\end{array}\!\right)$, so that the composition of Bogoliubov transformations amounts to matrix multiplication of the corresponding U-matrices. The Bogoliubov identities [12] are then encoded in the matrix equation $\scriptsize \left(\begin{array}{@{}ll@{}}1 & 0 \\ 0 & -1\end{array}\!\right) =U \left(\begin{array}{@{}ll@{}}1& 0 \\ 0 & -1\end{array}\!\right )U^{\dagger}$.

When the acceleration between the initial and final inertial regions is uniform and lasts for proper time  $\bar \tau$, we have [11] $U_h(\bar \tau ) = K_h^{-1} \tilde Z_h(\bar \tau ) K_h$, where $\scriptsize K_h =\left(\begin{array}{@{}ll@{}}{{}_{\rm{o}}\alpha}_h & {{}_{\rm{o}}\beta }_h \\{{}_{\rm{o}}\beta }_h & {{}_{\rm{o}}\alpha }_h\end{array}\!\right)$, $\scriptsize \tilde Z_h(\bar \tau ) =\left(\begin{array}{@{}lr@{}}{Z}_h(\bar{\tau}) & 0 \\ 0 & {Z}_h^{*}(\bar \tau )\end{array}\!\right )$, ${Z}_h(\bar \tau ) ={\mathrm { diag}} \bigl (\mathrm {e}^{\mathrm {i}\Omega _1(h)\bar \tau }, \mathrm {e}^{\mathrm {i}\Omega _2(h)\bar \tau },\cdots \bigr )$, Ω_n(h) are the angular frequencies during the acceleration, _oα_h and _oβ_h are the Bogoliubov coefficient matrices from the initial inertial segment to the uniformly accelerated segment, and the acceleration has been encoded in the dimensionless parameter h = aL. The field modes and the angular frequencies during the acceleration have elementary expressions for μ₀ = 0 and are given in terms of modified Bessel functions for μ₀ > 0. The coefficients encoded in _oα_h and _oβ_h do not have elementary expressions, but they can be written as integrals involving the inertial and accelerated mode functions over the constant time surface where the acceleration begins.

For accelerations that may vary arbitrarily between the initial time τ₀ and final time τ, U(τ,τ₀) is given by the limit of $U_{h_N}(\bar \tau _N) U_{h_{N-1}}(\bar \tau _{N-1})\cdots U _{h_2}(\bar \tau _2) U_{h_1}(\bar \tau _1)$ as N → ∞, such that $\tau -\tau _0 = \sum _{k=1}^N \bar \tau _{k}$ is fixed and each $\bar \tau _k\to 0$. As an infinitesimal increase in τ amounts to multiplying U(τ,τ₀) from the left by $U_h(\bar \tau )$ with infinitesimal  $\bar \tau$, U(τ,τ₀) satisfies the differential equation

$$\begin{equation} {\dot U}(\tau,\tau_0) = \mathrm{i} K_{h(\tau)}^{-1} \tilde{\Omega}_{h(\tau)} K_{h(\tau)} {U}(\tau,\tau_0) , \end{equation} \tag{ 2 }$$

where $\scriptsize \tilde \Omega _{h(\tau )}= \left(\begin{array}{@{}lr@{}}{\Omega }_{h(\tau)}& 0 \\ 0 & -{\Omega }_{h(\tau )}\end{array}\!\right)$, ${\Omega }_{h(\tau )}={\mathrm { diag}} \bigl (\Omega _1\bigl (h(\tau )\bigr ),\Omega _2\bigl (h(\tau )\bigr ),\ldots \bigr )$, and the overdot denotes derivative with respect to τ. The solution is

$$\begin{equation} {U}(\tau_{\mathrm{f}},\tau_0) = T \exp\left(\mathrm{i}\int_{\tau_0}^{\tau_{\mathrm{f}}} K_{h(\tau)}^{-1} \tilde\Omega_{h(\tau)} K_{h(\tau)} \, \mathrm{d}\tau \right), \end{equation} \tag{ 3 }$$

where τ_f denotes the moment at which the acceleration ends and T denotes the time-ordered exponential.

To summarize: the Bogoliubov transformation between the inertial initial segment ending at proper time τ₀ and the final inertial segment starting at proper time τ_f is given by (3). h(τ) may vary arbitrarily for τ₀ ⩽ τ ⩽ τ_f, within the rigidity constraint |h(τ)| < 2: in particular, no small acceleration approximation has been made. For piecewise constant h(τ), (3) reduces to a product of the matrices $U_h(\bar \tau _k)$ from each constant h segment.

A direct consequence of (3) is that the Bogoliubov coefficients evolve by pure phases over any time interval in which h is constant. Particles in the cavity are hence created by changes in the acceleration, not by acceleration itself, as can be argued on general adiabaticity grounds [14, 15]. The cavity is in this respect similar to a single accelerating mirror, which excites photons from the vacuum only when its acceleration is non-uniform [6].

At small accelerations, Ω_n(h), _oα_h and _oβ_h have the expansions [11]

$$\begin{equation} \Omega_n = \omega_n + O(h^2) , \quad n=1,2,\ldots, \end{equation} \tag{ 4a }$$

$$\begin{equation} {{}_{\rm{o}}\alpha}_h = 1 + h {\hat\alpha} + O(h^2) , \quad {{}_{\rm{o}}\beta}_h = h {\hat\beta} + O(h^2) , \end{equation} \tag{ 4b }$$

$$\begin{equation} {\hat\alpha}_{nn} =0,\ \end{equation} \tag{ 5a }$$

$$\begin{equation} {\hat\alpha}_{mn} = \frac{\pi^2\,m n \bigl(-1+{(-1)}^{m+n}\bigr)}{L^4\left(\omega_m-\omega_n\right)^3\sqrt{\omega_m \omega_n}} \quad \hbox{for} {\rm m} \ne {\rm n}, \end{equation} \tag{ 5b }$$

$$\begin{equation} {\hat\beta}_{mn} = \frac{\pi^2\,m n \bigl(1-{(-1)}^{m+n}\bigr)}{L^4\left(\omega_m + \omega_n\right)^3\sqrt{\omega_m \omega_n}}. \end{equation} \tag{ 5c }$$

${\hat \alpha }_{mn}$

${\hat \beta }_{mn}$

We seek U(τ_f,τ₀) in the form

$$\begin{equation} \alpha = \mathrm{e}^{\mathrm{i}\boldsymbol{\omega} (\tau_{\mathrm{f}}-\tau_0)} \bigl(1 + \skew3\hat{A} + O(h^2) \bigr), \end{equation} \tag{ 6a }$$

$$\begin{equation} \beta = \mathrm{e}^{\mathrm{i}\boldsymbol{\omega} (\tau_{\mathrm{f}}-\tau_0)} \skew3\hat{B} + O(h^2), \end{equation} \tag{ 6b }$$

$\skew3\hat{A}$

$\skew3\hat{B}$

$$\begin{equation} \skew3\hat{A}_{mn} = \mathrm{i} (\omega_m-\omega_n) {\hat\alpha}_{mn} \int_{\tau_0}^{\tau_{\mathrm{f}}} \mathrm{e}^{-\mathrm{i}({\omega}_m-{\omega}_n) (\tau-\tau_0)} \, h(\tau) \, \mathrm{d}\tau, \end{equation} \tag{ 7a }$$

$$\begin{equation} \skew3\hat{B}_{mn} = \mathrm{i} (\omega_m+\omega_n) {\hat\beta}_{mn} \int_{\tau_0}^{\tau_{\mathrm{f}}} \mathrm{e}^{-\mathrm{i}({\omega}_m + {\omega}_n) (\tau-\tau_0)} \, h(\tau)\,\mathrm{d}\tau. \end{equation} \tag{ 7b }$$

Two comments are in order. Firstly, while the perturbative solution (7) assumes the acceleration to be so small that |h| ≪ 1, the velocities, travel times and travel distances remain unrestricted, and the solution remains valid even when the velocities are relativistic. Our perturbative treatment is hence complementary to the small distance approximations often considered in the DCE literature [3, 4], while of course overlapping in the common domain of validity.

Secondly, $\skew3\hat{A}_{mn}$ and $\skew3\hat{B}_{mn}$ scale linearly in h, but their magnitudes depend also crucially on whether h changes slowly or rapidly compared with the oscillating integral kernels in (7). In the limit of slowly varying h both $\skew5\hat{A}_{mn}$ and $\skew3\hat{B}_{mn}$ vanish, in agreement with the adiabaticity arguments of [14, 15]. In the limit of piecewise constant h, the changes in the magnitudes of $\skew3\hat{A}_{mn}$ and $\skew3\hat{B}_{mn}$ come entirely from the discontinuous jumps in h [11, 16, 17]. The limit of piecewise constant h may be difficult to realize experimentally with a material cavity, and we emphasize that no such rapid changes in the acceleration are involved in the experimental scenario considered in section 4. This limit can however be simulated by a cavity whose walls are mechanically static dc SQUIDs undergoing electric modulation [8, 18].

Suppose now that |h| ≪ 1, so that the solutions (6) and (7) are valid. Suppose that h is sinusoidal with angular frequency ω_c. For generic values of ω_c the integrals in (7) are oscillatory and have no net growth as τ_f increases. However, when ω_c equals the angular frequency of an oscillating integral kernel in (7), there is a resonance and the corresponding Bogoliubov coefficient grows linearly in τ_f. These resonance conditions read

$$\begin{equation} \skew3\hat{A}_{mn}: \hspace{1ex} \omega_{\mathrm{c}} = |\omega_m-\omega_n|, \end{equation} \tag{ 8a }$$

$$\begin{equation} \skew3\hat{B}_{mn}: \hspace{1ex} \omega_{\mathrm{c}} = \omega_m + \omega_n, \end{equation} \tag{ 8b }$$

The particle creation resonance (8b) is well known in the DCE literature [3, 4, 7, 10, 19–27]. The mode-mixing resonance (8a) has been noted [10, 19–24] but seems to have received attention mainly in situations where it happens to coincide with a particle creation resonance. As the case of interest in the experimental scenario of section 4 will be mode mixing without significant particle creation, we recall here some relevant properties of mode mixing in quantum optics.

Mode mixing without particle creation is known in quantum optics as a passive transformation [28], implemented experimentally by passive optical elements such as beam splitters and phase plates. While mode mixing is present already in classical wave optics, its significance in quantum optics is that the mixing can be harnessed to quantum information tasks. The entangling power of passive transformations is well understood: for example, the mixing generates entanglement from an initial Gaussian state only if this state is squeezed [17, 29–31].

These entanglement considerations are directly applicable to mode mixing in our cavity, and the entanglement can be determined experimentally by measurements on quanta that are allowed to escape from the cavity [32–35]. We emphasize that while the particle creation resonance (8b) can be used to implement two-mode squeezing gates [30, 31, 36] and other multipartite gates [37], the mode-mixing resonance (8a) can be used to implement mode-mixing gates even when no particle creation is present.

Specifically, the oscillating cavity can be tuned to act as a beam splitter—a well-studied quantum gate in continuous variable systems [38]. In the following we apply our results to the special class of Gaussian states, characterized by positive Wigner functions, which allow elegant and powerful analytical results [39]. Gaussian states, including coherent and squeezed states, are routinely prepared in the laboratory. For example, two-mode squeezed states are commonly produced by parametric down conversion [40].

In order to calculate the entanglement generated by the mode-mixing gate we have introduced, we adopt the covariance matrix formalism. We consider a family of harmonic oscillators with position and momentum operators q_i,p_i, where $i = 1,2,\ldots$. We collect these operators in the vector $\mathbb {X}=(q_1,p_1,q_2,p_2,\ldots )$. The canonical commutation relations take the form $[\mathbb {X}_i,\mathbb {X}_j]=\mathrm {i}\Omega _{ij}$, where the only non-vanishing components of the symplectic form Ω are Ω_2i−1,2i = −Ω_2i,2i−1 = 1. The covariance matrix is defined by

$$\begin{equation} \sigma_{ij}=\frac{1}{2}\langle\mathbb{X}_i\mathbb{X}_j+\mathbb{X}_j\mathbb{X}_i\rangle -\langle\mathbb{X}_i\rangle\langle\mathbb{X}_j\rangle. \end{equation} \tag{ 9 }$$

This formalism is suitable for Gaussian states since all the relevant information about the state can be encoded in the first moment $\langle \mathbb {X}_i\rangle$ and the covariance matrix. In fact, the quantification of Gaussian state entanglement requires only its covariance matrix [39].

An initial pure state remains pure to first order in the acceleration [36]. This implies that the reduced state σ_red of two modes will depend only on the Bogoliubov coefficients that mix these two modes. The contribution of coefficients mixing with other frequencies is negligible to linear order in the acceleration.

From now on we specialize to two-mode Gaussian states for which σ_red is symmetric. In the covariance matrix formalism, the entanglement for such states is fully quantified by the smallest symplectic eigenvalue of the partial transpose of the covariance matrix [39]. The partial transpose is given by $\tilde {\sigma }_{\rm{red}}=\mathbf {P}\sigma _{\rm{red}}\mathbf {P}$, where P = diag(1,1,1,−1), and its symplectic eigenvalues are the eigenvalues of $\mathrm {i}\boldsymbol{\Omega }\tilde {\sigma }_{\rm{red}}$. The eigenvalue set has the form $\{-\tilde {\nu }_-,\tilde {\nu }_-,-\tilde {\nu }_+,\tilde {\nu }_+\}$, where $0\leqslant \tilde {\nu }_- \leqslant \tilde {\nu }_+$, and therefore the quantity characterizing the entanglement is  $\tilde {\nu }_-$. Within our perturbative small acceleration expansion, it is shown in [30] that $\tilde {\nu }_-=1-\tilde {\nu }_-^{(1)} + O(h^2)$, where $\tilde {\nu }_-^{(1)}$ is linear in the acceleration.

While the von Neumann entropy would be a natural measure of entanglement for the reduced two-mode state σ_red when the initial state is pure, its small acceleration expansion does not take the form of a power series because of the logarithms involved in its definition [36, 38]. Measures of entanglement based on the partial transpose criterion however do have a power series expansion in the acceleration. We consider the negativity [41], which is in the covariance matrix formalism given by $\mathcal {N}=\max \{0, \frac{1}{2} ( \tilde {\nu }_-^{-1} -1)\}$ [39], and which hence has the small acceleration expansion $\mathcal {N}=\max \{0, \frac{1}{2} \tilde {\nu }_-^{(1)}\} + O(h^2)$.

Suppose now that the state is a separable state of two modes, m and n, in which each mode is squeezed with the same squeezing parameter s > 0. When $\skew3\hat{B}_{mn}$ is negligible compared with $\skew3\hat{A}_{mn}$, it follows from the expression of $\tilde {\nu }_-^{(1)}$ given in [30] that the leading order contribution to the negativity is linear in the acceleration and given by

$$\begin{equation} \mathcal{N} = \bigl| {\rm Im} \bigl( \skew3\hat{A}_{mn} \bigr) \bigr| \sinh s. \end{equation} \tag{ 10 }$$

Figure 1 shows a plot of the negativity (10) as a function of the total oscillation time Δτ = τ_f − τ₀ and the acceleration angular frequency ω_c, assuming purely sinusoidal acceleration with phase chosen so that h(τ) is proportional to $\cos \bigl (\omega _{\mathrm {c}}(\tau -\tau _0)\bigr )$, for fixed ω_r = |ω_m − ω_n| and squeezing parameter s. The linear growth of $\mathcal{N}$ at the resonance, ω_c = ω_r, is evident from the plot. The scale of the vertical axis depends on m, n and s but also on μ₀L, in a way that we shall address in section 4.

Zoom In Zoom Out Reset image size