Unveiling and veiling an entangled light-matter quantum state from the vacuum

The ground state of an atom interacting with the electromagnetic ﬁeld in the ultrastrong coupling regime is composed of virtual photons entangled with the atom. We propose a method to promote to real the entire photonic state, while preserving the entanglement with the atom. The process can be reversed, and the entangled state can be restored in the vacuum. We consider a four-level atom, with two of these levels ultrastrongly coupled to a cavity mode. The process is obtained by making use of either an ideal ultrafast pulse or a more realistic multitone π pulse that drives only the atom. An experimental realization of this proposal will not only enable the investigation of the exotic phenomena of emission of particles from the vacuum, but will also prove that quantumsuperpositionstates can be extracted from the vacuum. Moreover, it will allow one to inspect the ground state in the ultrastrong coupling regime, and to generate on-demand entangled states for quantum information processing.


I. INTRODUCTION
One of the most fascinating features of quantum mechanics is that the quantum vacuum is filled with virtual particles [1].In the presence of intense fields, the quantum vacuum can become a pure entangled state.For example, at the horizon of a black hole, where the gravitational field is very intense, the quantum vacuum is predicted to be a two-mode squeezed entangled state.Outside the event horizon, the virtual particles forming this entangled vacuum state are promoted to real at the expense of the gravitational field [2].
accessible directly and it is not useful for quantum information tasks.
Here we propose a protocol in which the entire entangled cat state forming the ground state in the ultrastrong coupling regime is converted from "virtual" to "real."This is done by swapping the entanglement with the cavity field from the atomic states that ultrastrongly interacts with the cavity to other ones that do not.The swap is implemented by driving the atom with an ideal ultrafast monotone pulse or with a multitone π pulse [35][36][37][38][39].Note that we never drive the cavity.After the pulse, the cat state remains in a noninteracting sector of the Hilbert space, thereby the quantum information of this quantum state now is accessible and can be directly measured.Using an identical pulse the process can be inverted, and the entangled cat state can be restored in the system ground state, becoming again nonmeasurable.

II. MODEL
Our system consists of a four-level atom, {|g , |e , |g , |e }, and a cavity mode with frequency ω c .We call A the subsystem {|g , |e }, that is ultrastrongly coupled to the cavity field B, and C is the subsystem {|g , |e } which does not interact [see Fig. 1(a)].The general procedure, valid for both the ultrafast and the multitone pulse, is as follows [see Fig. 1(b)]: At t 0 , we prepare the system in its ground state.Now, because of the large energy coupling, the atomic subspace A is maximally entangled with the cavity field B, forming an entangled light-matter cat state.Because the system is in its dressed ground state, it is not possible to make a measurement in order to acquire information about The colored disks represent the entanglement between the subsystem A-B, and B-C at different moments.At t 0 the subsystem A is entangled with B, but this entanglement is not directly accessible.After a pulse at t 1 , the cavity field B is entangled with C, and now this entanglement can be measured.After a pulse at t 2 , the cavity field B is entangled again with A. this entangled state.At t 1 , a proper π pulse, sent to the atom, swaps the atomic states from the ones that interact, A, to the ones that do not interact, and that forms the subsystem C. With this process we have a swap of the entanglement from the subsystem A to the subsystem C. The entire cat state, that previously was not accessible, can now be measured.At t 2 , a second pulse, equal to the first, restores the cat state to the ground state.
The Hamiltonian describing the total system is Ĥ = ĤR + Ĥ + D, where the Rabi Hamiltonian

A. Swapping using an ultrafast pulse
We now present the swapping procedure in which the entanglement in the dressed light-matter ground state is mapped into entanglement in the noninteracting sector, i.e., for large coupling λ/ω q > 1, where |± = 1/ √ 2(|g ± |e ).The swap is realized by inducing atomic transitions {|g , |e } → {|e , |g }, using an ultrafast ideal pulse sent to the atom [Fig.3(a)].For the experimental realization, we show that also a multitone π pulse can efficiently lead to the same result [Fig.3(b)].Figure 3 displays the dynamical evolution of the Fock state populations for the states |g , n and |e , n (solid lines), which can be compared with the Fock states associated with |g, n and |e, n (dotted lines) representing the bare populations of the state |C − .
The ultrafast pulse is given by with σmn = |m n|, ε = 1/2, A p = 6 × 10 −3 /ω q , and t i (with i = 1, 2) is the center of the Gaussian pulse.In this simulation no dissipation is taken into account.At t = 0, we prepare the system in the dressed state |C − .In Fig. 3(a), at t1 = 1 (up red arrow), with t = ω q t/2π , an ultrafast pulse depopulates state |C − and populates the Fock states associated with |g and |e (solid lines).The latter states match exactly the ones associated with |g and |e (dotted lines) before the pulse was sent.This proves that the cat state now is an entangled state within the subspace {|g , |e }.The photonic contribution before and after the pulse remains almost unchanged, as shown in Eqs. ( 2) and (3).The cat state FIG. 3. Dynamics of the Fock state populations associated with the atomic states |g and |e (solid lines) when the atom is initially prepared in |C − and is subject to an ultrafast pulse (a), or a multitone pulse (b).These can be compared with the Fock states associated with |g and |e (dotted lines) when the state of the system is |C − .Here, ω g e = 3ω q , ω e g = 2ω q , λ = 1.34ω q , and N = 7.The inset in (a) shows the population for the |C − state (red curve) and the fidelity F for the |C − state (blue curve).The inset in (b) shows the same quantities, but the fidelity F is now calculated by eliminating the relative phases because the revival of the fidelity, which is the same as in the inset in (a), produces very rapid oscillations in a longer simulation, making these difficult to resolve.
is now visible to an external observer, because the photons are real and can be detected.After the first pulse is sent at t1 , superpositions within the state experience a phase shift [41,42] [see inset Fig. 3(a)].At t2 = t1 + td (down red arrow), with td = ω q /|ω e g − ω c |, the state returns in phase with the corresponding light-matter ground state |C − and a second pulse equal to the first brings the cat state back to its original "virtual form" encoded in the ground state of the Rabi Hamiltonian |C − .Figure 3(b) shows the dynamical evolution of the system when a multitone π pulse is sent to the atom and the state of the system is initially |C − .Here, we consider more experimentally realistic conditions, taking into account also the transition |g ↔ |e and |g ↔ |e in the Hamiltonian of the pulse, and including environmental effects [27].The pulse is described by Ĥp = ε p N i=0 cos[ω p i (t − t s )]( σeg + σge + σge + σg e + H.c.), (6) where ε = 1, A p = 50/ω q , ω p i = ω C − − ω i , ω i are the frequencies of |g , 2i + 1 and |e , 2 i , and ω C − is the ground state energy.We note that the transitions described in the pulse Hamiltonian satisfy the optical selection rules for a flux qubit in its optimal point (see Appendix A).

B. Swapping using a multitone pulse
At t1 = 150 (up red arrow), the first pulse is sent to the atom, gradually populating the noninteracting sector until the full entanglement is swapped.At t2 = 500 (down red arrow), a second pulse equal to the first restores (not perfectly because of dissipation) the original |C − state.We note that, because the state |C − is the system ground state, dissipation would automatically restore the initial state within the relaxation time associated with the atom.The inset in Fig. 3(b) shows the population (red curve) of the |C − state and the fidelity F between the state of the system and the state |C ideal = ( σe g + σg e )|C − .

IV. PERFORMANCE OF THE PROTOCOL USING THE MULTITONE PULSE
To quantify the performance of this protocol, we now calculate the amount of entanglement that can be transferred from the initial state |C − to the state |C − .In Figure 4(a), we show the von Neumann entropy [43] calculated after sending a multitone π pulse with seven (green dots) and nine (blue dots) frequencies, and after postselecting onto the {|g , |e } subspace.This approach is valid only when the populations in the |g and |e states are zero after the swapping procedure.However, when there is a small residual population, it is still a good indicator of the amount of entanglement transferred from virtual to real [see inset in Fig. 4(a)].
These results can be compared with the entropy calculated for the subsystem {|g , |e }, when the state of the system is |C − (red dots).Note that for couplings in the range 0.5 < λ/ω q < 0.8, going from seven to nine pulses does not lead to improvements.However, for λ/ω q > 0.8, the number of photons in the ground state increases, and using nine frequencies in the pulse outperforms the seven-frequencies case.
Figure 4(b) displays the fidelity for the state of the system calculated with respect to a target entangled state with α = λ/ω c [27], after sending the multitone π pulse with seven (green dots) and nine (blue dots) frequencies.This result can be compared with the fidelity for the state |C − (red dots).As before, for large coupling, increasing the number of frequencies in the pulse corresponds to an increase in fidelity.
Our proposal can be realized using a flux qubit coupled to a waveguide or LC resonator (see Appendix A).When the reduced external flux that threads the flux qubit is f = 0.5 [44,45], the allowed transitions become |g ↔ |e , |g ↔ |e , FIG. 4. (a) Entanglement entropy S between the field and the atomic subspace {|g , |e } after a multitone π pulse with seven frequencies (green dots) and nine frequencies (blue dots) is sent to the atom.The atom is initially prepared in the |C − state.It is possible to compare this to the entropy relative to the subspace {|g , |e }, when the state of the system is |C − (red dots).The inset shows the trace of the density matrix ρ{|g ,|e } for the subspace {|g , |e }, after a multitone π pulse with seven (green dots) and nine (blue dots) frequencies is sent.(b) Fidelity F (eliminating relative phases) between an ideal cat state entangled with the atom and the system state after a π pulse with seven (green dots) and nine frequencies (blue dots) is sent to the atom.The ideal cat state is generated α = λ/ω c .In (a) and (b) we consider the interacting states {|g , |e } at higher energy with respect to {|g , |e }.Here ω g = 9 ω q , ω c = ω q /2, and ω e g = ω q .
|e ↔ |g , and |g ↔ |e .These optical selection rules are the ones that we consider when we define the multitone pulse.

V. CONCLUSIONS
In this work, we proposed a protocol to unveil the virtual entangled ground state of a two-level system ultrastrongly coupled to a cavity mode.To unveil and make the state visible and measurable, the entangled partner is swapped using an ultrafast pulse or a multitone π pulse.With this proposal, not only is it possible to explore the ground state of the Rabi Hamiltonian in the ultrastrong coupling regime, but it can also lead to an efficient way to generate photonic cat states on demand, which can be an important resource for quantum computation and other quantum technologies [46,47].In the case of a black-hole horizon, after the promotion from virtual to real of particles, the quantum information of the vacuum state is apparently lost, giving rise to the quantum information paradox [48].Understanding and solving this paradox is crucial to have a theory of quantum mechanics consistent with Einstein general relativity.Our proposal could potentially enable experiments using superconducting qubits which mimic the vacuum state in the proximity of a black hole and the particle emission from its horizon, giving the chance to investigate, in laboratory conditions, the black-hole information paradox [49][50][51].
FIG. 6. Fidelity as a function of the normalized light-matter coupling λ/ω q , between the dressed ground state and the state generated after sending the multimode pulse with eight frequencies.Here we used ω eg = ω q = 1, ω g e = 18 ω q , ω e g = 9.1 ω q , ω le = 13 ω q , M eg = 0.83, M e g = 0.5, M e g = 0.137, M g e = 0.116, M le = 0.22, and M le = 0.3.For this simulation we adopted a Hilbert space with eight photons.
eigenstates of a flux qubit with charging energy E C = 2.8 GHz, Josephson energy E J = 40E C , flux qubit charge ratio α = 0.8, and a reduced external flux f = 0.5.With these parameters, the transition |g ↔ |e has frequency ω q ≈ 1 GHz, ω g e ≈ 18ω q , and ω e g ≈ 9.If the cavity frequency is tuned to the frequency transition |g ↔ |e , this transition can achieve the ultrastrong coupling regime.The states |g and |e can represent the atomic noninteracting sector for two reasons: (i) their allowed atomic transitions are much detuned with respect to the cavity; and (ii) the corresponding matrix elements M i j are smaller than M eg .
Using this configuration for a flux qubit, in Fig. 6 is shown the numerically calculated fidelity between the dressed Rabi ground state and the state generated using a multimode pulse for different couplings.The procedure is the same as in the main text.Notice that in these numerical results we take into account all the possible interactions among the states |g , |e , |g , |e , and |l .Here the system Hamiltonian is Ĥ = i, j=g,e,g ,e ,l ω ig |i i| + λ i j |i j|(â + â † ) (A1) with coupling λ i j = λM i j /M eg , such that λ eg = λ.Frequencies and transition matrix elements correspond to the one calculated above for a flux qubit.Using these parameters, the Hilbert space becomes too large to numerically compute the system dynamics.For this reason we reduced the dimension of the photonic subspace, now taking into account eight photons (instead of the 25 considered in the main text).

FIG. 1 .
FIG. 1. (a)The system consists of a two-level system A and a cavity field B interacting deep in the ultrastrong light-matter coupling regime.The subsystem C represents the noninteracting atomic states.(b) The colored disks represent the entanglement between the subsystem A-B, and B-C at different moments.At t 0 the subsystem A is entangled with B, but this entanglement is not directly accessible.After a pulse at t 1 , the cavity field B is entangled with C, and now this entanglement can be measured.After a pulse at t 2 , the cavity field B is entangled again with A.
and Ĥ = ω e g |e e | describe the dynamics of the interacting and noninteracting states, respectively ( h = 1).The energy difference between the latter states is defined by D = ω g (|e e | + |g g |), σx = |e g| + |g e| is a Pauli operator, â is the annihilation operator for the cavity mode, and ω mn = ω m − ω n .We set ω c = ω q and λ = 1.34 ω q .Figure 2(b) shows the eigenstates of the total Hamiltonian Ĥ .For λ 0.5 ω c , the atomic states {|g , |e } hybridize with the photonic states, and the ground state becomes an entangled cat state, |C − ≈ 1/ √ 2(|+ | − α − |− | + α ) (red solid line).Here, |± = 1/ √ 2(|e ± |g ) are the eigenstates of σx , and | ± α = D(±α)|0 are photonic coherent states of light with positive and negative displacement α [40].The colored lines in Fig. 2(b) are Fock states associated with the noninteracting subspace for when the atomic system is in the states |g and |e .For clarity, other dressed eigenstates of the total Hamiltonian are not shown in Fig. 2(b).

FIG. 2 .
FIG. 2. (a) Four-level atomic system: states |g and |e interact ultrastrongly with a cavity mode.(b) Noninteracting states (colored lines) and dressed ground state |C − (red line) of the total Hamiltonian Ĥ .The parameters are chosen so that the atomic states |g and even number of photons are degenerate with the atomic states |e and odd number of photons.
1ω q .Optical selection rules allow the transitions |g ↔ |e , |g ↔ |e , |e ↔ |g , and |g ↔ |e .The transitions are determined by the matrix elements M i j = i| sin(2π f + 2ϕ − )| j , where ϕ − is the difference between the fluxes crossing the Josephson junctions.The calculated matrix elements have values M eg = 0.83, M e g = 0.5, M g e = 0.14, and M e g = 0.12.The transitions |e ↔ |l and |e ↔ |l are also allowed with transition energy ω le ≈ 13 ω q , and matrix transition elements M el = 0.22 and M e l = 0.3.