key: cord-0622251-p46t4uty
authors: Zhou, Jian-Yong; Zhou, Yue-Hui; Yin, Xian-Li; Huang, Jin-Feng; Liao, Jie-Qiao
title: Quantum entanglement maintained by virtual excitations in an ultrastrongly-coupled-oscillator system
date: 2019-12-24
journal: nan
DOI: nan
sha: 54e322e9256c9874e1bcea5f81153ad539e9059d
doc_id: 622251
cord_uid: p46t4uty

We study the effect of quantum entanglement maintained by virtual excitations in an ultrastrongly-coupled harmonic-oscillator system. Here, the quantum entanglement is caused by the counterrotating interaction terms and hence it is maintained by the virtual excitations. We obtain the analytical expression for the ground state of the system and analyze the relationship between the average excitation numbers and the ground-state entanglement. We also study the entanglement dynamics between the two oscillators in both the closed- and open-system cases. In the latter case, the quantum master equation is microscopically derived in the normal-mode representation of the coupled-oscillator system. This work will open a route to the study of quantum information processing and quantum physics based on virtual excitations.

The ultrastrong coupling (USC) physics [1, 2] has recently attracted much attention from the communities of quantum physics, quantum optics, and condensed matter physics. Great advances have been made in both theory [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] and experiments in various physical platforms, including semiconductor cavity quantum electrodynamical (QED) systems [13] [14] [15] , superconducting circuit-QED systems [16] [17] [18] [19] [20] [21] , coupled photon-2D-electron-gas [22] [23] [24] , light-molecule [25, 26] , and photonmagnon systems [27] . In the USC regime [1, 2] , the coupling strength is comparable to the transition frequencies in the system, and then the rotating-wave approximation (RWA) is invalid, namely the counterrotating (CR) terms should be kept in the interactions. It has been demonstrated that the CR terms could produce some unpredictable physical phenomena [3] and have wide applications in quantum information processing [28, 29] . In particular, the development of the ultrastrong coupling field promotes various studies in quantum optics topics beyond the RWA such as the quantum Rabi model [30] [31] [32] [33] [34] [35] [36] .

One of the interesting effects associated with the CR terms in the USC regime is the generation of virtual excitations. In the presence of the CR terms, the ground states of some typical quantum systems possess virtual excitations. For example, in the quantum Rabi model, it has been shown that virtual photons exist in the ground state [11] . These virtual photons cannot be detected directly even if this absorber is placed inside the cavity, except with very small probability at short times set by the time-energy uncertainty [37] . On the basis of these properties, the ground-state photons in the USC system are considered virtual photons [2] . However, even though these virtual photons cannot be detected directly, there are still ways to probe them. One proposal is to measure the change that they produce in the Lamb shift of an ancillary probe qubit coupled to the cavity [38] . Another proposal is to detect the radiation pressure that they give rise to if the cavity is an op- * jfhuang@hunnu.edu.cn † jqliao@hunnu.edu.cn tomechanical system [39] . These proposals rely on the rapid modulation of either g (light-matter coupling strength) or the atomic frequency. Then the virtual photons can be converted into real ones and extracted from the system [3, 4, 9, [40] [41] [42] [43] [44] [45] .

In this paper, we propose to study another quantum effect, quantum entanglement, associated with the virtual excitations. Here the quantum entanglement is created by the CR terms and hence it is maintained by the virtual excitations. We note that the relationship between quantum entanglement and the CR terms has been previously considered in the quantum Rabi model [8] . In addition, the role of the CR terms in the creation of entanglement between two atoms has been investigated in Ref. [46] . We consider an ultrastrongly-coupled two-harmonic-oscillator system. We study the ground state entanglements of the two oscillators and analyze the average excitation numbers in the system. We also study the entanglement dynamics of the system when it is initially in the zeroexcitation state and hence all the excitations are created by the CR terms. The influence of the environment dissipation on the system is analyzed based on a microscopically derived quantum master equation in the normal-mode representation.

The rest of this paper is organized as follows. In Sec. II, we present the physical model of two coupled harmonic oscillators and the Hamiltonian, we also analyze the property of the parity chain in this system. In Sec. III, we obtain the exact analytical eigensystem of the coupled two-oscillator system. In Sec. IV, the average virtual excitation numbers are calculated analytically and the quantum entanglement of the ground state is analyzed by calculating the logarithmic negativity. In Sec. V, we study the dynamics of the average virtual excitation numbers and quantum entanglement between the two oscillators in both the closed-and open-system cases. Finally, we present a brief conclusion in Sec. VI.

We consider an ultrastrong coupling system, in which two harmonic oscillators are ultrastrongly coupled to each other through the so-called "position-position" type interac- tion. This system is described by the Hamiltonian

where x 1 (x 2 ) and p 1 (p 2 ) are, respectively, the coordinate and momentum operators of the oscillator with the resonance frequency ω 1 (ω 2 ) and mass m, the parameter η is the coupling strength between the two oscillators. By expanding the interaction term, the Hamiltonian (1) can be expressed as

where we introduce the renormalized frequencies and coupling strength as

By introducing the following creation and annihilation operators a = (a † ) † = mω a /(2 )

the Hamiltonian (2) becomes

with C = ( ω a + ω b )/2 being a constant term. Here a † (a) and b † (b) are, respectively, the creation (annihilation) operators of the two oscillators with the corresponding resonance frequencies ω a and ω b . In Eq. (5), the first two terms and the constant term represent the free Hamiltonian of the two oscillators. The parameter g = −ξ/(2m √ ω a ω b ) denotes the coupling strength between the two oscillators. We note that this interaction includes both the rotating-wave and counterrotating (CR) terms. In general, in the case of weak coupling and near resonance, the rotating-wave approximation can be made by discarding the CR terms. In this paper, we consider the ultrastrong-coupling case in which the CR terms cannot be discarded. In the presence of the CR terms, the ground state of the system will include excitations and hence quantum entanglement will exist in the ground state. Note that an ultrastrongly-coupled two-mode system has recently been realized in superconducting circuits [47] .

In this two-oscillator system, we introduce the parity operator as P = (−1) a † a+b † b , which has the standard properties of a parity operator, such as P 2 = I, P † P = I, and P † = P [7, 48] . The Hamiltonian H in Eq. (5) remains invariant under the transformation P † HP = H, based on the relations P † aP = −a, P † a † P = −a † , P † bP = −b, and P † b † P = −b † . The Hilbert space of the system can be divided into two subspaces with different parities: odd and even. The basis states of the oddand even-parity subspaces are, respectively, given by

and

The eigenvalues of the parity operator P corresponding to the odd and even parity states are −1 and 1, respectively.

To study the quantum entanglement of the eigenstates, we need to diagonalize the Hamiltonian H in Eq. (2) . To this end, we introduce the transformation operator [49] 

where the mixing angle θ is defined by

In terms of the transformation, the Hamiltonian in Eq. (2) can be diagonalized as

where the resonance frequencies are defined by

with ξ = −2mg √ ω a ω b . By introducing the annihilation and creation operators

the Hamiltonian (10) can be expressed as

The relations between the operators A (A † ), B (B † ), a (a † ), and b (b † ) can be obtained as

Here the concrete forms of coefficients f i (i = 1, 2, · · · , 7, 8) have been given by

Based on Eq. (13), we know the eigenstates of the system in the representation associated with A † A and B † B as

where the eigenvalues are given by

It is obvious that the ground state of the two-oscillator system is |0 A |0 B . To study the virtual excitations in the system, we need to know the eigenstates which are expressed in the representation associated with a † a and b † b. It implies that we need to diagonalize the HamiltonianH in the representation of a and b. To this end, we express the Hamiltonian (10) with the bosonic creation (annihilation) operators a † (a) and b † (b) as

where we introduce the coupling strengths

To diagonalize Hamiltonian (18), we introduce the squeezing operators S a (r a ) = e r a (a 2 −a †2 )/2 ,

where the two real squeezing parameters are defined by

The transformed Hamiltonian can be written as

where the unitary operator U can be expressed with the operators a and b as

The eigenstates of the Hamiltonian H ′ can be obtained as

where the eigenvalues are defined in Eq. (17) . The eigensystem of the Hamiltonian H can be obtained as

As a result, the ground state of the system can be expressed as

In general, it is hard to write out the ground state in the number state representation. However, we can obtain a numberstate expansion of the ground state in the degenerate twooscillator case [50] , i.e., ω a = ω b . In this case, we have U = exp[−(π/4)(a † b − ab † )] and the ground state becomes

By expanding the squeezing operators, we then have

In terms of the relations we then obtain

It can be seen that the superposition components in the ground state are even parity states. This property can be confirmed because the transform U conserves the excitation number and the squeezing operators change the excitation number two by two, without changing the parity.

In this section, we study the ground-state entanglement in this system by calculating the logarithmic negativity. For the two-oscillator system, if the coupling is sufficiently weak, i.e., g ≪ {ω a , ω b }, the interaction Hamiltonian between the two oscillators can be reduced by the RWA as H I ≈ g(a † b + b † a), which conserves the number of excitations. In this case, the ground state of the system is a trivial direct product of two vacuum states |0 a |0 b , which does not contain excitations. In the presence of the CR terms, the |0 a |0 b is not an eigenstate of the system and the ground state will possess excitations. Below, we use numerical method to obtain the ground state of the Hamiltonian (5) and calculate the ground state entanglement between the two oscillators. In the presence of the CR terms, the ground state of the two-oscillator system can be expressed as where these superposition coefficients are given by C m,n = a m| b n||G , which should be solved numerically. The G|a † a|G 0 and G|b † b|G 0 reveal that the ground state of the system contains excitations. These excitations in the ground state are called virtual excitations because these excitations cannot be extracted from the system. The effect of the virtual excitations can be seen from the probability amplitudes in the ground state. The distribution of these probability amplitudes can also exhibit the parity of the ground state. As the ground state is an even parity state, and hence these probability amplitudes associated with the odd parity basis states will disappear. In Fig. 2 , we show the absolute values of these probability amplitudes |C m,n |. Here we can see that the values of |C m,n | decrease with the increase of m and n and that there is a symmetric relation |C m,n | = |C n,m |. In addition, the values of these odd-parity probability amplitudes C m,n with m+n being an odd number are zero, which is a consequence of the fact that the ground state is an even-parity state.

We also calculate the average excitation numbers a † a and b † b in the ground state |G as

where we have used the formula,

S † a (r a )aS a (r a ) =a cosh r a − a † sinh r a , (33c)

In Fig. 3(a) , we show the average excitation numbers a † a and b † b in the ground state |G as functions of the scaled coupling strength g/ω r in the degenerate oscillator case ω a = ω b = ω r . The results show that the average excitation numbers of the two modes are identical (two curves overlap each other). This is because the corotating terms conserve the excitations and the CR terms create simultaneously the excitations in the two modes. The average excitation numbers increase with the coupling strength since a larger coupling strength corresponds to a faster excitation creation.

The degree of entanglement between the two oscillators a and b in the ground state of the system can be obtained by calculating the logarithmic negativity. Combining with Eq. (31), the density matrix of the ground state can be written as

The degree of entanglement of the ground state can be quantized by calculating the logarithmic negativity [51, 52] . For a bipartite system described by the density matrix ρ, the logarithmic negativity can be defined by

where T b denotes the partial transpose of the density matrix ρ of the system with respect to the oscillator b, and the trace norm ρ T b 1 is defined by

Using Eqs. (34) , (35) , and (36), the logarithmic negativity of ground state of the two coupled oscillators can be obtained. In Fig. 3(b) , we show the logarithmic negativity N as a function of the coupling parameter g/ω r . The curve shows that the degree of entanglement between the two oscillators in the ground state monotonically increases over the entire range of g. This is because the CR terms in Hamiltonian (5) cause the virtual excitations in the ground state of the system and maintain the quantum entanglement between the two oscillators. If the CR terms are discarded, then the ground state of the system becomes a separate state |0 a |0 b .

The phenomenon of quantum entanglement accompanied with virtual excitations can also be seen by analyzing the entanglement dynamics of the system. We consider the case in which the system is initially in the zero-excitation state |0 a |0 b . In the closed-system case, a general state of the system can be written as

By substituting Eqs. (5) and (37) into the Schrödinger equation, the equations of motion for these probability amplitudes A m,n (t) are obtained aṡ

For the initial state |0 a |0 b , the initial condition of these probability amplitudes reads A m,n (0) = δ m,0 δ n,0 . By numerically solving Eq. (38) under this initial condition, the evolution of these probability amplitudes can be obtained. Using Eqs. (35) , (36) , and (37), we can calculate numerically the average excitation numbers a † a and b † b and the logarithmic negativity of the state |ψ(t) . In Fig. 4(a) , we show the time evolution of the average excitation numbers a † a and b † b in modes a and b. Here we can see that, similar to the ground state case, the average excitation numbers in the two modes are identical (the two curves overlap each other). In addition, the average excitation numbers experience a periodic oscillation. In Fig. 4(b) , we show the time dependence of the logarithmic negativity N(t) of the state |ψ(t) . The curve shows that logarithmic negativity between the two oscillators also experiences a periodic oscillation. Here we choose the initial state of the system as |0 a |0 b , the existence of the CR terms still causes the appearance of virtual excitations, which leads to entanglement between the two oscillators. This result is different from that in the RWA case in which the CR terms are discarded in the two oscillators under the same initial state. When we discard the CR terms and choose the initial state as |0 a |0 b , which is the eigenstate of the corotating interaction term g(a † b + b † a), the system will stay this state. Then there are no virtual excitations in the system and no quantum entanglement between the two oscillators.

We also study the influence of the environment dissipations on the dynamics of the system. As we consider the ultrastrong-coupling regime of the coupled system, we derive the quantum master equation in the normal-mode representation of these two coupled oscillators. We employ the standard Born-Markov approximation under the condition of weak system-bath couplings and short bath correlation times to derive the quantum master equation. The secular approximation is made by discarding these high-frequency oscillating terms including exp(±iω A t), exp(±iω B t), and exp[±i(ω A ±ω B )t]. The quantum master equation in the normal-mode representation of Hamiltonian (5) can be written aṡ 

where

is the standard Lindblad superoperator that describes the dampings of the oscillators. The parameters γ a and γ b are the decay rates relating to the heat bath in contact with the oscillators a and b, respectively. Here we consider the zero temperature environments such that the thermal excitation effect can be excluded. 

For an initial state |0 a |0 b , the nonzero density matrix element is ρ 0,0,0,0 (0) = 1. By numerically solving Eq. initial condition, the time evolution of the density matrix ρ s (t) can be obtained. Below we study the dynamics of the average excitation numbers and quantum entanglement in this system. Based on Eq. (39), the expressions of the average excitation numbers a † a(t) and b † b(t) can be expanded as

Therefore, the average excitation numbers a † a(t) and b † b(t) can be obtained by solving the equations of motion for these density matrix elements in the number-state representation. In Fig. 5(a) , the dynamics of the average excitation numbers a † a(t) and b † b(t) is shown in the open-system case with different time t. We observe that the two excitation numbers a † a(t) and b † b(t) overlap each other and initially experience a large oscillation. With the increase of time t, the oscillation amplitudes of the average excitation numbers decrease gradually. In the long-time limit t ≫ 1/γ a,b , the average excitation numbers will reach steady values due to the dissipations.

The entanglement of the density matrix ρ s (t) can be quantified by calculating the logarithmic negativity. In terms of Eqs. (35) , (39) , and (42) , the logarithmic negativity of the state ρ s (t) can be obtained numerically. In Fig. 5(b) , we show the logarithmic negativity N(t) of the density matrix ρ s (t) versus the time t. The result shows that the logarithmic negativity oscillates very fast due to the free evolution of the system. We also find that the envelope of the logarithmic negativity converges gradually with the evolution time t and eventually reaches a stable value due to the dissipations. The time scale of the oscillation-pattern decay for the logarithmic negativity is very similar to that of the excitations created by the CR interaction terms. In particular, we find that there exists steadystate entanglement due to the presence of the CR interaction terms in this system.

In conclusion, we have studied quantum entanglement in an ultrastrongly-coupled two-harmonic-oscillator system. Concretely, we have studied the ground-state entanglement by cal-culating the logarithmic negativity of the ground state. Here, the quantum entanglement is maintained by the virtual excitations generated by the CR terms and bounded in the ground state. We have also studied the dynamics of quantum entanglement of the system. By microscopically deriving a quantum master equation in the normal-mode representation of the two oscillators, we analyzed the influence of the dissipations on the entanglement dynamics and found that there exists steady-state entanglement in this system.

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