Entangling superconducting qubits through an analogue wormhole

We propose an experimental setup to test the role of curved spacetime on entanglement extraction from the vacuum of a quantum field to a pair of artificial atoms. In particular, we consider two superconducting qubits coupled to a dc-SQUID array embedded into an open microwave transmission line, where a suitable external bias is able to mimic a spacetime containing a traversable wormhole. We find that the amount of vacuum entanglement that can be extracted by the superconducting qubits depends on the parameters of the wormhole. At some distances qubits that would remain separable in flat spacetime become entangled due to the presence of the effective wormhole background.

The vacuum of a quantum field is an entangled state [1,2]. Vacuum fluctuations exhibit correlations between different space-time regions, even if they are spacelike separated. This fact underlies many important predictions of Quantum Field Theory in general backgrounds, such as the Dynamical Casimir Effect [3] and Unruh-Hawking radiation [4][5][6][7]. From a more applied viewpoint, it seems natural to ask if these correlations can be exploited as a resource for Quantum Information tasks. This question can be addressed by two alternative approaches. On one hand, moving boundary conditions can turn vacuum fluctuations into real particles via the dynamical Casimir effect, which has recently been experimentally observed [8]. These particles are produced in quantum-correlated pairs [9][10][11][12][13]. On the other hand, entanglement can in principle be swapped to qubits [14][15][16][17] after an interaction with the field. Despite several proposals, the latter possibility has never been confirmed experimentally.
In general, the scenario for extracting vacuum entanglement can be described as follows. At least two qubits are prepared in an uncorrelated state and interact for a finite time with a quantum field initially in the vacuum state. If t is the interaction time, r is the distance between the qubits and v the propagation velocity of the field quanta, entanglement from the vacuum will be swapped to the qubits if their state is entangled after t < r/v. For t > r/v, the qubits might exchange real photons, which might act as an additional source of correlations. An obvious experimental challenge is to achieve the desired interaction time, which requires control of the interaction on timescales that are typically out of reach. However, recent developments in the analysis of quantum information in relativistic scenarios show that the entanglement of relativistic quantum fields is sensitive to acceleration, gravity and the dynamics of spacetime [18]. It seems natural to ask if we can exploit these properties to relax the experimental requirements necessary for extracting vacuum entanglement. Indeed, the extraction of vacuum entanglement in curved spacetimes have been theoretically considered for instance in [19][20][21]. However, for experiments it is necessary to adopt an analogue grav-ity viewpoint [22] and search for experimental platforms where curved spacetimes can be simulated.
Circuit QED [23,24] provides a framework in which the interaction of two-level systems with a quantum field can be naturally considered. The combination of superconducting qubits with transmission lines implement an artificial 1-D matter-radiation interaction, with the advantage of a large experimental accessibility and tunability of the physical parameters. Using these features, fundamental problems in Quantum Field Theory hitherto considered as ideal are now accessible to experiment. The possibility of achieving an ultrastrong coupling regime [25][26][27][28][29] has already been exploited to propose a feasible experimental test of the extraction of vacuum entanglement to a pair of spacelike separated qubits [16]. Moreover, effective spacetime metrics can be implemented as well by means of suitable modulations of the effective speed of light in the electromagnetic medium [30].
One important example of curved spacetime is Ellis metric [31], which represents a spacetime containing a traversable wormhole [32]. We do not have any experimental evidence of the presenceof traversable wormholes in the universe, and observational-based bounds on their abundance have been inferred [33]. Indeed, the existence of traversable wormholes would entail a challenge to the theoretical notion of causality [32,[34][35][36]. This led Hawking to pose the "chronology protection conjecture" [35] in the semiclassical framework of quantum field theory in curved spacetime, according to which quantum effects would preclude the creation of closed timelike curves in spacetimes such as Ellis, thus ruling out the possibility of time travel to the past. This conjecture could only be totally proved or disproved with a full theory of quantum gravity, which remains elusive. From a strictly classical viewpoint, traversable wormholes require exotic energy sources violating the weak energy condition [34]. Moreover, there are also quantum constraints in the form of "quantum inequalities" [37]. However, as unlikely as their existence might look like it is not forbidden on theoretical grounds. On the other hand, it has been suggested that typical phenomena commonly attributed to black holes might be mimicked by Ellis wormholes or other ex-arXiv:2003.13309v1 [quant-ph] 30 Mar 2020 otic objects. In particular, if wormholes existed even the actual identity of the objects in the center of the galaxies could be questioned [38] as well as the source of the observed gravitational waves [39,40]. Furthermore, the existence of closed timelike curves would have an immediate impact in the theory of both classical and quantum computing [41] and wormholes also appear in the celebrated "EPR-ER" conjecture [42]. For all these reasons there is a growing interest both in the theoretical description of wormholes [43][44][45] and in their detection by classical means such as gravitational lensing [46,47], and others [48], including quantum metrology techniques [49,50]. Alternatively, classical simulators [51][52][53][54] and quantum simulators of 1+1 dimensional reductions of Ellis and related wormhole geometries have been proposed, for instance in superconducting circuits [55], Bose-Einstein condensates [56] and trapped ions [45].
In this paper, we consider a setup consisting of two superconducting qubits coupled to a dc-SQUID array embedded into an open transmission line, where a suitable strongly inhomogenous external bias mimics a traversable wormhole background [55]. We show that the amount of vacuum entanglement that can be extracted by the qubits depends on the parameters of the wormhole. The features of this dependence are in turn sensitive to the distance between the qubits. We find a regime of distance where the presence of the wormhole entangles the artificial atoms, which would remain separable if the spacetime were flat.
We now present our model and results. A traversable 1D massless wormhole spacetime can be describe by the following line element [34,55] where the shape function b(r) encodes the wormhole features and is a function of the radius r only. There is a critical value b 0 of r at which b (r = b 0 ) = r = b 0 , which determines the wormhole's throat. The proper radial distance to the throat is defined by [34] l = ± r b0 dr (1 − b(r )/r ) −1/2 , defining two different "universes" or regions within the same universe for l > 0 (as r goes from ∞ to b 0 ) and l < 0 (as the non-monotonic r goes back from b 0 to ∞). Thus, as r → ∞ we have two asymptotically flat spacetime regions l → ±∞ connected by the wormhole throat at l = 0 (r = b 0 ).
The properties of the wormhole will depend on b(r). Of particular interest is the following family of wormholes [31,32,43,44]: for which the proper radial distance to the wormhole throat is simply l 2 (r) = r 2 − b 2 0 . It is shown in [55] that the dynamics of a 1D electromagnetic field in the spacetime given by Eq. (1) is totally equivalent to the one in the following spacetime ds 2 = −c 2 (1 − b(r) r ) dt 2 + dr 2 , which is a spacetime in which the speed of propagation of the electromagnetic field depends on the radius r according to: r ). Furthermore, this effective speed of light can be mimicked for the electromagnetic flux fiedl propagating in a dc-SQUID array embedded in an open transmission line [57][58][59][60], with a suitable strongly inhomogeneous external field bias. In particular, in order to simulate the spacetime given by the equations (1) the profile of the external magnetic flux has the following dependence on the radius: φ ext (r) = φ0 π arccos(1 − b(r) r ). The position x along the transmission line can be related to the coordinate r via |x| = r − b 0 , x ∈ (−∞, ∞). Thus x = 0 at the wormhole's throat r = b 0 and acquires different sign at both sides of the throat. Notice that l 2 = |x|(|x| + 2b 0 ). Thus, we can rewrite the flux profile as a function of . It is shown in [55] that in the particular wormhole spacetimes given by Eq. (2) it is possible to achieve simulated wormhole throat radius in the sub-mm range. Now let us assume that two superconducting qubits are coupled to the SQUID-embedded transmission line described above. The qubit-line coupling strength can be abruptly switched on and off and can reach the ultrastrong coupling regime. In flat spacetime, the extraction of entanglement from the quantum vacuum to a a pair of superconducting qubits was analyzed in [16]. The Hamiltonian, H = H 0 + H I , possesses a free part H 0 for the qubits and the field H 0 = 1 2 Ω(σ z A + σ z B ) + k ω(k)a † k a k and a point-like interaction between them H I ∝ α=A,B σ x A V (χ α ), where V is the quantum fieldwhich can be written in terms of the creation and annihilation operators V (χ) ∝ dk √ N ω k e ikχ a k + H.c. . Here χ A and χ B would be the constant positions of the atoms in a coordinate system in which the spacetime metric is flat. In the flat spacetime case of [16] those are the standard laboratory coordinates {t, x}. In the effective curved spacetime that we are considering here they are {t, l}.
In what follows we choose the initial state |ψ(0) = |eg ⊗ |0 , where qubit A is excited, while qubit B and the field remain in their ground and vacuum states, respectively. In the interaction picture the system evolves into the state T being the time ordering operator. We use the formalism of perturbation theory up to second order and beyond the Rotating Wave Approximation [16] and trace over the field degrees of freedom to obtain the corresponding two-qubit reduced density matrix ρ 12 evaluated at t. The degree of entanglement of this X-state can be characterised by the concurrence, which is given by: , X standing for the amplitude of photon -real and virtual -exchange and k |A 1,k | 2 , k |B 1,k | 2 for the probability of single-photon emission by 1 and 2, respectively. These terms can be computed -following similar techniques as in [16] -as a function of three dimensionless parameters, ξ, K 1 and K 2 . The first one, ξ = c t/ρ, (ρ being the constant distance between the qubits ρ = |χ a −χ b |) allows us to discriminate between two different spacetime regions, namely, the qubits are effectively spacelike separated if ξ < 1 and timelike separated otherwise. The remaining parameters are the dimensionless coupling strengths for qubits 1 and 2: K m = (g m /Ω m ) 2 . We will restrict our analysis to K m Ω m t 1 where our perturbative approach remains valid. For the sake of simplicity, we further consider that g 1 = g 2 = g and Ω 1 = Ω 2 = Ω and thus The results of [16] are directly valid to the coordinate system {t, l}, for which the metric is flat. It is therefore necessary to transform the parameter ξ to the laboratory coordinates. For simplicity, we assume that the qubit position are symmetric with respect to the throat: Then, by using the relation between x and l: where we introduce a dimensionless parameter relating the throat size with the qubit distance in the laboratory coordinates ρ x = 2x B . Thus, from Eq. (5) we get the relation between the qubit distance in free-falling and laboratory coordinates: Moreover, while in the flat coordinate system the time that takes the light to travel between qubits is merely (l B − l A )/c, in the laboratory this time will be given by: Thus, the new parameter: will define the light cone in the laboratory system. If ξ x < 1, light cannot travel between the qubits. By means of Eq.(8), we can relate ξ x with the light cone parameter in free-falling coordinates We get: where we introduce ξ F , which would be the light cone parameter in flat spacetime: Note that in the absence of a wormhole (b 0 = 0), from Eq.(11) we get ξ l = ξ x = ξ F , as expected. By inserting Eqs. (11) and (7) in the flat-spacetime results, we show in Fig. 1 the dependence of the entanglement dynamics on the wormhole parameter b 0 . As expected, it is highly dependent on the qubit distance, since vacuum correlations decay with distance. Indeed, we find three separate regimes. For ultrashort qubit distances ρ x λ -where λ = 2πc/Ω would be the wavelength of the qubit transition in flat spacetime-there is entanglement between qubits both inside and outside the light cone which seems to be independent of the existence and size of a wormhole throat. As ρ x is increased the dynamics of entanglement becomes sensitive to the wormhole. First, the effect is detrimental: at distances at which there is entanglement generation around the light cone in the absence of a wormhole -in agreement with the flatspacetime results [16]-, entanglement vanishes quickly as ε b increases. However, for larger distances ρ x λ some entanglement is generated for timelike separated qubits only if ε b = 0. Therefore, qubits which would remain separable in flat spacetime get entangled due to the presence of the curved background. Since ε x > 1, we cannot say that this is a pure transference of vacuum entanglement, since there could have been photon exchange between the qubits. However, it is still interesting that the effect is a consequence of the presence of an effective curved spacetime. The topological link between the qubits provided by the wormhole enhances photon exchange, making it strong enough to generate quantum correlations.
This entanglement generation between qubits due to an analogue curved spacetime is the main result of this work. We believe that it is within reach of circuit QED technology. Superconducting transmon qubits has been coupled to transmission lines formed by thousands of SQUIDS [61]. In [55] we showed that it is possible to simulate wormholes with b 0 in the sub-mm range by means of an inhomogenuous external magnetic field bias. On the other hand, we need b 0 to be slightly larger than the ρ x = λ/12000 ρ x = λ/8 ρ x = λ wavelength λ, since b 0 /λ b 0 /ρ x = b 0 /(2x b ) = ε b /2 and interesting effects show up for ε b ≥ 5. This means that λ in the sub-mm range is required. For qubits of Ω = 2π × 10 GHz this means propagation speed velocities c for the field of around 10 6 m/s. These speeds has been experimentally reported in SQUID arrays [62].
In summary, we propose a setup of two superconducting qubits coupled to a SQUID array transmission line to test the extraction of entanglement from the vacuum of a quantum field in curved spacetime -in particular, Ellis wormhole metric, simulated by means of a suitable external bias of the SQUID array. We find different regimes in which, according to the distance between the qubits, the presence of the Ellis wormhole can be irrelevant, detrimental to entanglement generation or beneficial. We think that the latter is the most interesting scenario. In particular, we find that, when the distance between the qubits is similar to the wavelength of the qubit transition, there is no entanglement extraction unless the wormhole throat is significantly larger than a certain value (ε b ≥ 5). This means that a pair of qubits which would remain separable in flat spacetime become entangled due to the presence of an effective curved background. We have shown that this would be in principle observable with current circuit QED technology. Therefore, we find that the analysis of analogue quantum simulators of quantum field theory in curved spacetime is not only interesting from the theoretical viewpoint but does have possible experimental applications as a valuable resource for quantum technologies.