Squeezed displaced entangled states in the quantum Rabi model
The quantum Rabi model accepts analytical solutions in the so-called degenerate qubit and relativistic regimes with discrete and continuous spectrum, in that order. We show that solutions in the laboratory frame are the superposition of even and odd displaced number states, in the former, and infinitely squeezed coherent states, in the later, of the boson field correlated to the internal states of the qubit. We propose a single parameter model that interpolates between these discrete and continuous spectrum regimes to study the spectral statistics for first and second neighbor differences before the so-called spectral collapse. We find two central first neighbor differences that interweave and fluctuate keeping a constant second neighbor separation.
The quantum Rabi model (QRM) is the lowest dimensionality Hamiltonian describing matter-light interaction,
It describes a boson field, with frequency and represented by the annihilation (creation) operator (), interacting with a qubit, with frequency and represented by Pauli matrices with . Trapped ions Lv et al. (2018) and superconducting circuits Mezzacapo et al. (2014) provide highly controllable experimental platforms for the quantum simulation of the model in the different interaction regimes defined by the coupling strength to field frequency ratio Casanova et al. (2010); Yoshihara et al. (2016).
Through its history, the QRM has motivated the development of computational tools for both spectral and dynamic calculations Wolf et al. (2012); Maciejewski et al. (2014). The model conserves parity and is solvable Braak (2011); Moroz (2013). It is possible to diagonalize it in the qubit basis Moroz (2014, 2016). In the so-called adiabatic approximation, it is possible to estimate the spectrum and its eigenstates Shen et al. (2016); Xie et al. (2019). Generalizations that account for asymmetry between so-called rotating and counter-rotating terms as well as driving showed the existence of degeneracies in the spectrum Li and Batchelor (2015). Extension for more than one qubit Chilingaryan and Rodríguez-Lara (2013); Wang et al. (2014); Zhang and Chen (2015) or field Travěnec (2012); Huerta Alderete and Rodríguez-Lara (2016) have been constructed. The latter was used for the simulation of para-particles in trapped-ion setups Huerta Alderete and Rodríguez-Lara (2017); Huerta Alderete et al. (2017); Huerta Alderete and Rodríguez-Lara (2018).
The QRM accepts analytic solutions in the so-called degenerate qubit, , and relativistic, , regimes Pedernales et al. (2015). After diagonalization in the qubit basis, the first is reduced to two decoupled harmonic oscillators with discrete spectrum and the second to a Dirac equation in (1+1)D with continuous spectrum. In the following, we introduce a single-parameter QRM that interpolates between these two regimes. It is engineered to show spectral collapse Ng et al. (1999); Duan et al. (2016); Penna et al. (2017). We show that, in the degenerate qubit regime, the eigenstates are superposition of even and odd displaced number states correlated to the ground and excited state of the qubit, in that order. In the relativistic regime, the eigenstates are the unbalanced superposition of position states that can be written as infinitely squeezed coherent states correlated to the internal states of the qubit. These obvious in hindsight results are straightforward to calculate. We numerically explore the statistics of the first and second neighbors spectral differences before the transition to the continuous spectrum regime. Their histograms show the interweaving of two central separations for nearest neighbors that keep a constant second neighbor spectral separation. Finally, we use Hussimi Q-function to visually explore the displacement and squeezing of the ground state of our model as the control parameter takes us close to the continuous spectrum regime.
We propose a single-parameter QRM,
that interpolates between the so-called degenerate qubit, , and relativistic, , regimes for the extremal values of the control parameter . We recover the QRM for . A trapped ion quantum simulation Pedernales et al. (2015), provides the desired level of control through blue (red) driving sidebands, () where the parameter () is the frequency of the driving laser detuned to the blue (red) of the ion transition frequency , and the center of mass motion frequency is given by . The effective coupling strength is related to the Lamb-Dicke parameter of the trap and the amplitude of the driving field.
Strictly speaking, our model is solvable Braak (2011); Guan et al. (2018). We focus on the regimes with analytic closed form solution and favour the Fulton-Gouterman procedure to diagonalize it in the qubit basis Moroz (2013),
where the dynamics in the boson sector,
include the boson parity operator, . At this point, we deviate from the standard Bargmann approach and discuss the closed form analytic solutions in the extremal regimes. In addition, we provide a numerical study for the transition from discrete to continuous spectrum, Fig. 1.
Iii Degenerate qubit regime
In the degenerate qubit regime, , the boson sector Hamiltonian reduces to a driven harmonic oscillator,
that is diagonalized by a displacement,
The eigenstates are displaced number states, , with equally spaced spectrum . We used the displacement operator with parameter . The eigenstates in the laboratory frame,
have the familiar form of a maximally entangled boson-qubit state between the unnormalized even (odd) displaced number states,
and the excited (ground) qubit state. This is not a Schödinger cat state Schrödinger, E. (1935); Brune et al. (1992); Monroe et al. (1996). The latter needs semi-classical states in the boson sector. We have highly non-classical boson states that belong to different parity sectors. Quantum entangled states with parity properties are useful to detect weak forces Gilchrist et al. (2004).
For the sake of visualization, we calculate Hussimi Q-function Husimi (1940) for the reduced boson sector states,
where the probability of finding excitations,
is given in terms of the Tricomi function Villanueva Vergara and Rodríguez-Lara (2015) and the function () yields the larger (smaller) value between and . Figure 2 shows the Husimi Q-function for the ground state, , fifth, and tenth, , excited states. The mean expectation value for the boson oscillator quadratures in optical phase space, and , is zero and their squared standard deviations are not,
One dispersion is always larger, , as the displacement parameter is never zero.
Iv Relativistic regime
Here the QRM reduces to a D Dirac equation Gerritsma et al. (2010); Noh et al. (2013); Rodríguez-Lara and Moya-Cessa (2014); Gutiérrez-Jáuregui and Carmichael (2018) and yields an effective boson sector Hamiltonian,
that can be mapped into a simpler form in optical phase space,
It is not bounded from below, has a continuous spectrum, and the solution is Dirac delta normalizable. Its eigenstates in the laboratory frame,
include boson sector eigenstates,
that are the superposition of squeezed coherent states Satyanarayana (1985); Nieto and Truax (1993); Bishop and Vourdas (1994) withe symmetric displacement with respect to the origin, infinite squeezing, but different probability amplitude, , and . Squeezed displaced states are interesting from a fundamental point of view Satyanarayana (1985); Bishop and Vourdas (1994); Nieto and Truax (1993). They are a resource for high precision measurements Huang et al. (2015); Kienzler et al. (2015); Lo et al. (2015) as they increase the signal-to-noise ratio Korobko et al. (2017).
In order to obtain this result, we start from the effective boson sector Hamiltonian and construct the eigenvalues for its square, . From their square roots, we construct the eigenstates,
We use the map of Fock states to optical phase space, , in terms of Hermite polynomials Lebedev and Silverman (1972), and translate them into operator form, ,
Considering the canonical pair, and , where the parameters of the physical oscillator are the mass of the ion and the natural frequency of the ion trap, provides all the scaling factors. The nature of the solution does not allow us to provide reliable numerics for the exact parameter value .
We can write Husimi Q-function for the reduced density matrix of the boson field,
where we construct a closed form for the weight factor Satyanarayana (1985),
in terms of Gauss hypergeometric function Lebedev and Silverman (1972). The only eigenstate that can be calculated exactly is the bosonic vacuum, and .
V Numerical analysis
Calculating the spectrum is an important task. Once the spectrum is known, the intricacies of the system are revealed; for example, its is straightforward to calculate its time evolution, find its phase space configurations Rodriguez et al. (2018), or even model its interaction with an environment González-Gutiérrez et al., 2017. The QRM is one of a few systems that can be solved Braak (2011). Here, we use numerical diagonalization to observe the changes in the spectrum of our single parameter model as it gets closer to the continuous spectrum regime. In our simulations we use a Fock basis of dimension for the bosons on resonance with the qubit, , an ultra-strong coupling parameter, , and work only in the bosonic sector related to the excited qubit state in the Foulton-Gouterman reference frame. This is equivalent to work in the positive parity subspace of the model in the laboratory reference frame.
We focus on the statistic of the normalized energy separation between -th nearest neighbors, . Figure 3 shows the histogram for the first nearest neighbor probability distribution, . In the degenerate qubit regime, the energy levels are equidistant and a peak at the value appears, Fig. 3(a). As the control parameter increases, the histogram displaces towards the origin, two well-defined peaks appear, and the height decreases. For the standard QRM on resonance, and , the histogram is centered at the value , Fig. 3(b). As the control parameter increases, the double peak displaces towards the origin. At some critical value of the control parameter, the peak separation starts decreasing and the height increases, Fig. 3(c). As we get closer to the relativistic regime, where the spectrum is continuous, we expect a single peak at the value , Fig. 3(d).
An interesting feature arises for second nearest neighbors, , where the spectral separation seems constant. It starts in the degenerate qubit regime as single peak at the value , Fig. 4(a), that displaces towards the origin at almost constant height, Fig. 4(b) to Fig. 4(c), that should become a single peak at as we get closer to the relativistic regime, Fig. 4(d). First neighbor separation points to the existence of two central separations, one short and one large, that interweave and fluctuate keeping a constant second neighbor separation.
We proposed a single parameter QRM that interpolates between the degenerate qubit and the relativistic regimes. We provided the analytic solutions in these regimes. The spectrum of the former is discrete and its eigenstates are odd and even displaced number states of the boson field correlated to the excited and ground state of the qubit, in that order. The latter has continuum spectrum and its eigenstates are the unbalanced superposition of infinitely squeezed coherent states correlated to the qubit states.
We conducted a numerical statistical analysis below the transition from discrete to continuous spectrum regimes. We focused on the energy gap between first and second neighbors of the bosonic sector after Foulton-Guterman diagonalization in the qubit basis. We found a structure that points to a distribution of first neighbors separation centered around two peaks with seemingly constant spectral separation for second neighbors. We used Husimi Q-function to follow the process that displaces and squeezes the ground state from the degenerate qubit into the relativistic regime for reasonable control parameter values.
Acknowledgements.F.H.M.-V. acknowledges funding from CONACYT Cátedra Grupal # 551. C.H.A. acknowledeges funding from CONACYT doctoral grant No. 455378. B.M.R.-L. acknowledges funding from CONACYT CB-2015-01-255230 grant and the Marcos Moshinsky Foundation Research Chair 2018.
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