A DETAILED DERIVATION OF THE ZENO HAMILTONIAN FOR THE Z PUMPING

# Dissipative preparation of steady Greenberger-Horne-Zeilinger states for Rydberg atoms with quantum Zeno dynamics

## Abstract

Inspired by a recent work [Reiter, Reeb, and Sørensen, \colorbluePhys. Rev. Lett. 117, 040501 (2016)], we present a simplified proposal for dissipatively preparing a Greenberger-Horne-Zeilinger (GHZ) state of three Rydberg atoms in a cavity. The pumping is implemented under the action of the spontaneous emission of -type atoms and the quantum Zeno dynamics induced by strong continuous coupling. In the meantime, a dissipative Rydberg pumping breaks up the stability of the state in the process of pumping, making be the unique steady state of system. Compared with the former scheme, the number of driving fields acting on atoms is greatly reduced and only a single-mode cavity is required. The numerical simulation of the full master equation reveals that a high fidelity can be obtained with the currently achievable parameters in the Rydberg-atom-cavity system.

###### pacs:
03.67.Bg,32.80.Ee,42.50.Dv,42.50.Pq

## I Introduction

Neutral atoms have shown great potential as matter qubits that possess high-lying Rydberg states and state-dependent interaction. These properties make it possible to implement quantum information processing since the entangling operations can be readily realized by the Rydberg blockade or antiblockade interaction Jaksch et al. (2000); Ates et al. (2007); Urban et al. (2009); Gaëtan et al. (2009); Amthor et al. (2010); Saffman et al. (2010). There is currently great interest in generation of entangled states of Rydberg atoms using time-dependent unitary method. Theoretically, the multipartite entanglements were produced through stimulated Raman adiabatic passage Møller et al. (2008) and asymmetric Rydberg blockade Saffman and Mølmer (2009), respectively, and a spatial cat state for a pair of atom clouds was created via the mechanism of Rydberg dressing Möbius et al. (2013). Experimentally, significant achievements have been obtained towards this field, e.g., using identical Rb atoms and Cs atoms, the deterministic Bell states with fidelities of and were demonstrated Wilk et al. (2010); Zhang et al. (2010); Jau et al. (2016). For non-identical particles, the entanglement between a Rb atom and a Rb atom via Rydberg blockade was reported as well Zeng et al. (2017).

The reservoir-engineering approaches to entanglement generation have attracted much attention in recent years. In such methods, a detrimental source of noise can be converted into a resource, and the target state is the unique steady state of the open quantum system, which means there needs no state initialization. Since the novel concept of “quantum computation by dissipation” was proposed by Verstraete et al. Verstraete et al. (2009), the steady entangled states of two particles have been carried out numerously in various physical systems, including cavity QED systems Kastoryano et al. (2011); Lin et al. (2013), ion trap systems Bentley et al. (2014), optomechanical systems Wang and Clerk (2013), superconducting systems Leghtas et al. (2013); Shankar et al. (2013), and neutral atom systems Carr and Saffman (2013); Rao and Mølmer (2013); Shao et al. (2014), etc. Nevertheless, it remains a challenge to prepare steady multipartite entanglement in a dissipative way. Recently, Morigi et al. put forward a protocol for dissipative quantum control of a spin chain, where an entangled antiferromagnetic state of many-body system was stabilized on the basis of spectral resolution, engineered dissipation, and feedback Morigi et al. (2015). Subsequently, Reiter et al. present a scalable way for dissipative preparation of multipartite GHZ state without feedback Reiter et al. (2016). In their scheme, a “ pumping” and a “ pumping” constitute two crucial operations during the quantum-state preparation, and both of them require an independent harmonic oscillator mode and classical multitone driving fields operated on atoms. In particular, the preparation of steady GHZ state for particles has to involve driving tones in the pumping and driving tones in the pumping altogether. It may therefore consume many resources in terms of experimental realization.

In this work, we concentrate on the dissipative generation of tripartite GHZ state in a composite system based on Rydberg atoms and an optical cavity. The interaction between Rydberg atoms and cavity have been extensively studied before, e.g., a Rydberg-blocked atomic ensemble has a collective enhancement coupling strength compared to the single atom as placed in an optical high-finesse cavity Guerlin et al. (2010), and the Rydberg polaritons (a kind of quasiparticle with photons stored in the highly excited collective states) enable people to find new mechanisms of interaction in quantum optics Zhang et al. (2013); Maxwell et al. (2013). The diagram of the protocol for preparing the GHZ state is illustrated in Fig. 1. Similar to the process of Ref. Reiter et al. (2016), there are two operations to accomplish the goal, one is the pumping that transforms one and two atoms in state into , and the other is the dissipative Rydberg pumping which induces a resonant transition between states and , and then rules out the steady population of state . In what follows, we will discuss in detail the feasibility of realization of the above operations in a Rydberg-atom-cavity system, and it shows that our scheme can greatly reduce the complexities of experimental operations.

## Ii physical system

We consider three four-level atoms of double configuration interact with an optical cavity, and are simultaneously driven by classical laser fields, as shown in Fig. 2. The two stable ground states and are used to be encoded quantum bits. The transition between states and is coupled to a quantized cavity mode with strength , while the transition between states and is coupled to the classical field with Rabi frequency . In the meantime, the ground states and can also be pumped upwards to the excited Rydberg state via two independent classical fields with the same Rabi frequency (generally accomplished by a two-photon process or a direct single-photon process, see Refs. Tong et al. (2004); Saßmannshausen et al. (2013); Hankin et al. (2014) for details), and detuning . Although it is not necessary, for the sake of convenience we have assumed the excited state  () can spontaneously decay downwards to and with the same rate  , respectively. In addition, the atom-dependent light shift of state is introduced so as to break the symmetry of ground states during the pumping.

Under the assumption of Markovian approximation, the decay channels for atoms and cavity are independent, thus the master equation describing the interaction between quantum systems and external environment can be modeled by the Lindblad form

 ˙ρ = −i[HI,ρ]+γe23∑j=1{D[|0⟩jj⟨e|]ρ+D[|1⟩jj⟨e|]ρ} (1) +γr23∑j=1{D[|0⟩jj⟨r|]ρ+D[|1⟩jj⟨r|]ρ}+κD[a]ρ,

where denotes the leaky rate of photon from the optical cavity, represents the superoperator characterizing decay of system, and the corresponding Hamiltonian reads ()

 HI = Hk+Hr, (2) Hk = 3∑i=1(Ω|e⟩ii⟨1|+g|e⟩ii⟨0|a+H.c.+δi|1⟩ii⟨1|), Hr = 3∑i=1(Ωr|r⟩ii⟨0|+Ωr|r⟩ii⟨1|+H.c.−Δ|r⟩ii⟨r|) +∑i≠jUij|rr⟩ij⟨rr|.

It is worth pointing out that there are many ways to implement the atom-dependent light shifts . For example, these terms can be considered as an extra Stark shift of level via introducing other auxiliary levels (an inverse method adopted generally for canceling the Stark shifts), or an energy difference in a rotating frame through replacing the detuning parameters of the classical field driving the transition and the classical field driving the transition by and , respectively. The Rydberg-mediated interaction originates from the dipole-dipole potential of the scale between two atoms located at position and , and is the angle between the vector and the dipole moment aligned parallel to the axis, , with the Bohr radius, the electron charge, and the principle quantum number Saffman et al. (2010); Pupillo et al. (2010).

## Iii Simplified Z-pumping process

Let us first investigate the realization of the full -pumping process by the spontaneous emission of excited state combined with Hamiltonian . To make an analogy with the standard quantum Zeno dynamics of Ref. Facchi and Pascazio (2002, 2008), we divide the Hamiltonian into two parts, i.e., , where is the interaction between atoms and classical fields, and is the interaction between atoms and cavity. In the limit of , the requirement of quantum Zeno dynamics is fulfilled, and the Hamiltonian is reduced to with the orthogonal projection corresponding to the eigenvalue of . In the Zeno subspace of , it is reasonable to neglect the high-frequency oscillatory terms and only keep the near-resonant transitions, then we have the effective Hamiltonian as follows Facchi and Pascazio (2002); Shao et al. (2009); Lin et al. (2016)

 Heffk = (Hsk+Hbk)⊗|0c⟩⟨0c|, (3)

with

 Hsk = Ω[|001⟩(1√6⟨D1|−1√2⟨D2|)e−iδt] (4) +Ω[|100⟩(1√6⟨D1|+1√2⟨D2|)e−iδt] −2Ω√6|010⟩⟨D1|e2iδt+H.c.,

and

 Hbk = −Ω√2[|011⟩(⟨D4|e−iδt+⟨D5|e2iδt)] (5) −Ω√2[|101⟩(⟨D3|e−iδt−⟨D5|e−iδt)] +Ω√2[|110⟩(⟨D3|e2iδt+⟨D4|e−iδt)]+H.c..

In the above expressions we have assumed , and this setting will induce additional light shift for each ground state except and . The cavity mode is frozen to its vacuum state in this subspace, thus the process of pumping is robust against the cavity decay. The qubit basis , as well as the quantum states , , , , are the dark states of the atom-cavity interacting Hamiltonian. After discarding the symbol of the cavity field, we obtain the effective Markovian master equation describing the -pumping process

 ˙ρ=−i[Heffk,ρ]+16∑j=1LjρL†j−12(L†jLjρ+ρL†jLj), (6)

where the Lindblad operator , , , , , , , , , , , , , , , .

The simplified -pumping process of our scheme is shown in Fig. 3. To be more specific, suppose a quantum state is initialized in , it can be first driven into the excited state with the weak coupling strength , as governed by the Hamiltonian of Eq. (5). The excited state then spontaneously decay back to the subspace with two atoms in state , i.e. and with the same emitting rate respectively, or to the ground state with the emitting rate , i.e., a quantum state with one atom in state . Consider this quantum state as a new initial state and repeat a similar pumping and decaying process, the whole system will be finally stabilized into the state . In general, starting from an arbitrary quantum state with one or two atoms in state , the steady state is always achievable. As for the ground state , it is not affected by the above dissipative dynamics because the limit of quantum Zeno dynamics contributes an interaction strength at the order of magnitude , which is much smaller than . Now we finish the process of pumping with only one classical field acting on atoms and a single-mode cavity. What is more, the cavity mode is not populated throughout the process, making it insensitive to the leakage of photon from the cavity.

In Fig. 4, we numerically simulate the -pumping operation with the full Hamiltonian in Eq. (2). The initial state is chosen as a fully mixed state in the basis of quantum bits: , and the corresponding parameters are set as , , , , and . The population of state (solid line) is invariant and the population of state is stabilized at 0.875 (dash-dotted line) after a relaxation time . At this stage, we are able to prepare a steady GHZ state by a subsequent quantum feedback operation Cho et al. (2011); Morigi et al. (2015). A parity check performed on the system can inform us whether the quantum state is () or (). If the target state is supposed to be but we acquire a signal of , a operation applied to one of the qubits will change the state into the target state . In this sense, the -pumping operation combined with the parity measurements makes the current proposal deterministic. In the inset of Fig. 4, we study the evolutions of populations of states and in the presence of a large cavity decay (). Compared with the ideal case , although the population of target state is decreased, it remains . To sum up, we have implemented a robust -pumping operation.

## Iv dissipative Rydberg pumping

Next we turn to the realization of the dissipative Rydberg pumping. To see this process clearly, we rewrite the Hamiltonian , where we have introduced and assumed . Now this model is equivalent to three two-level Rydberg atoms with ground state and excited state collectively driven by a classical field of Rabi frequency . Using the basis , we can reduce the matrix to a matrix,

 Hr=⎡⎢ ⎢ ⎢ ⎢⎣0√6Ωr00√6Ωr−Δ2√2Ωr002√2ΩrU−2Δ√6Ωr00√6Ωr3U−3Δ⎤⎥ ⎥ ⎥ ⎥⎦. (7)

In this subspace, a general wave function of quantum system is described by , where is short for the symmetric state with atoms in . The equations of motion for the probability amplitudes can be derived from the Schrödinger equation to be

 i˙c0 = √6Ωrc1, (8) i˙c1 = √6Ωrc0+2√2Ωrc2−Δc1, (9) i˙c2 = √6Ωrc1+2√2Ωrc3−Δc2, (10) i˙c3 = √6Ωrc2, (11)

and we have set . In the limit of , and are slowly varying functions of , thus it is reasonable to assume that and , and acquire the values of these coefficients as

 c1 = √6ΩrΔc0+2√2ΩrΔc2, (12) c2 = √6ΩrΔc1+2√2ΩrΔc3. (13)

By substituting the above results into Eqs. (8) and (11), we have a pair of coupled equations characterizing the interaction between states and , i.e.,

 i˙c0 = (6Ω2rΔc0+12√2Ω3rΔ2c3)/(1−8Ω2rΔ2), (14) i˙c3 = (6Ω2rΔc3+12√2Ω3rΔ2c0)/(1−8Ω2rΔ2), (15)

which just correspond to the effective Hamiltonian

 Heffr=12√2Ω3rΔ2|+++⟩⟨rrr|+H.c.. (16)

The Stark-shift terms have been disregarded in this process since they can be canceled by introducing ancillary levels, and the order of is ignored too. Under the action of the Rydberg pumping of Eq. (16) and the spontaneous emission of excited Rydberg states , the state is no longer stable, and it will be pumped and independently decay to the “bare” ground states. In fact, engineering the coupling between and any component of , such as , can achieve the same effect. In other words, pumping the whole state to the excited state is not necessary Reiter et al. (2016).

## V experimental feasibility

In experiment, we may employ Rb atoms in our proposal. The range of the coupling strength between the atomic transition and the cavity mode is measured from the weak-coupling regime  MHz to the strong-coupling regime  MHz Mücke et al. (2010); Brennecke et al. (2007); Murch et al. (2008); Larson et al. (2008); Volz et al. (2011); Gehr et al. (2010). Specifically, a single atom cavity coupling strength is Kumar et al. (2016)

 g=μ√ωc2ϵ0V[Lc,Rcλ], (17)

where is the atomic transition dipole moment, is the mode volume of the cavity, is the frequency of cavity, is the radius of curvature of the mirrors, is the cavity length, and is the wavelength of the cavity mode. Thus this strength is adjustable by modulating the relevant cavity parameters. The Rabi frequency can be tuned continuously between  MHz (e.g., a red- and a blue-detuned lasers on the and transitions). The fidelity of the steady state is calculated as , where is the density matrix of target state. For a pure target state (), the definition of the fidelity can be proved to be , which is the square root of the population.

The experiment of cavity QED with a Bose-Einstein condensate provides us the following parameters  MHz Brennecke et al. (2007). For this group of parameters, we can choose , , , , , and adopt the Rydberg state with decay rate  MHz. By substituting these parameters into the original master equation of Eq. (1), we obtain the steady-state fidelity , and this value can be further improved to using another group of experimental parameters  MHz Murch et al. (2008), , . In Ref. Volz et al. (2011), a fibre-based high-finesse cavity also offers us a set of strong coupling parameters  MHz. In this condition, the parameter values , , , and  MHz ( Rydberg states, see e.g., Beterov et al. (2009)) guarantees a high fidelity . To see clearly how fast the system approaches to the steady state from an arbitrary initial state, we investigate the dependence of the steady-state population on in Fig. 5. The solid line, the dashed line and the dash-dotted line are simulated by the effective master equation, the full master equation without and with considering the cavity decay, respectively. These three lines are in excellent agreement with each other under the given parameters, which confirms the efficiency of our scheme again. It should be noted that the assumption of identical atom-cavity coupling strength made throughout the text is only for the discussion convenience. In fact, the fluctuations of result in little variation in the target-state fidelity. For example, the parameters listed in Fig. 5 corresponds to a steady-state fidelity . If we replace  MHz with  MHz,  MHz, and  MHz or 55 MHz, the fidelity is still no less than .

## Vi summary

In summary, we have proposed an efficient mechanism for dissipative generation of the tripartite GHZ state in a Rydberg-atom-cavity QED system. This scheme actively exploits the spontaneous emission of atoms and coherently driving offered by the quantum Zeno dynamics and the Rydberg pumping, which make it robust against the loss of cavity and the fluctuation of atom-cavity couplings. Although the current model is not scalable, it enables us to reduces the operation complexity of the experiment substantially, and a high fidelity is available through the strictly numerical simulation of the full master equation without any approximation. We hope that our proposal may open a new venue for the experimental realization of the multipartite entanglement in the near future.

## Vii Acknowledgements

This work is supported by the Natural Science Foundation of China under Grants No. 11647308, No. 11674049, No. 11534002, and No. 61475033, No. 11774047, and by Fundamental Research Funds for the Central Universities under Grant No. 2412016KJ004.

*

## Appendix A Detailed Derivation of the Zeno Hamiltonian for the Z Pumping

In this appendix, we give the detailed derivation of the effective Hamiltonian of Eq. (4). For the qubit states with one atom in state , we can obtain a closed subspace , , , , , , in the absence of dissipation. Now we expand the original Hamiltonian in Eq. (2) with the above basis and have

 Hap0 = Ω[|001⟩⟨00e|+|010⟩⟨0e0|+|100⟩⟨e00|+H.c. (18) −δ(|1⟩11⟨1|−2|1⟩22⟨1|+|1⟩33⟨1|)]|0c⟩⟨0c|,

and

 Hapg=g[(|00e⟩+|0e0⟩+|e00⟩)⟨000|]|0c⟩⟨1c|+H.c., (19)

where and represent the interactions between atoms and classical fields, and atoms and cavity, respectively. According to the Zeno dynamics Facchi and Pascazio (2002, 2008), we should first find the eigenprojections of . After a straightforward calculation, we get four eigenstates of as

 |E1⟩=1√6(|e00⟩+|00e⟩−2|0e0⟩)|0c⟩, (20)
 |E2⟩=1√2(|e00⟩−|00e⟩)|0c⟩, (21)
 |E3⟩=1√6(|e00⟩+|0e0⟩+|00e⟩)|0c⟩+1√2|000⟩|1c⟩, (22)
 |E4⟩=1√6(|e00⟩+|0e0⟩+|00e⟩)|0c⟩−1√2|000⟩|1c⟩, (23)

corresponding to eigenvalues , , , and , respectively. Remember that the qubit states , , are also the dark states for because of , therefore there are total three Zeno subspaces, i.e.,

 Hp0 = span{|001⟩|0c⟩,|010⟩|0c⟩,|100⟩|0c⟩,|E1⟩,|E2⟩}, Hp1 = span{|E3⟩},          Hp2=span{|E4⟩}. (24)

Now we rewrite in the eigenbasis of as

 Hapk = 2∑m,n=0(PmH0Pn+gηnPn) (25) = Ω{|001⟩|0c⟩[1√6(⟨E1|+⟨E3|+⟨E4|) −1√2⟨E2|]}+Ω{|100⟩|0c⟩[1√6(⟨E1|+⟨E3| +⟨E4|)+1√2⟨E2|]}−Ω|010⟩|0c⟩[1√6(2⟨E1| −⟨E3|−⟨E4|)]+H.c.−δ(|100⟩⟨100| −2|010⟩⟨010|+|001⟩⟨001|)|0c⟩⟨0c| +√3g|E3⟩⟨E3|−√3g|E4⟩⟨E4|.

In order to see the Zeno dynamics clearly, we move into a rotating frame with respect to and obtain

 Hapk = Ω{|001⟩|0c⟩[(1√6⟨E1|−1√2⟨E2|)e−iδt (26) +1√6(⟨E3|e−i(√3g+δ)t+⟨E4|ei(√3g−δ)t)]} +Ω{|100⟩|0c⟩[(1√6⟨E1|+1√2⟨E2|)e−iδt +1√6(⟨E3|e−i(√3g+δ)t+⟨E4|ei(√3g−δ)t)]} −Ω|010⟩|0c⟩[1√6(2⟨E1|e2iδt−⟨E3|e−i(√3g−2δ)t −⟨E4|ei(√3g+2δ)t)]+H.c..

In the limit of Zeno requirement , the high-frequency oscillating terms proportional to can be safely neglected and only the near-resonant terms are preserved. Then we can recover the effective Hamiltonian of Eq. (4) from Eq. (26). The effective Hamiltonian of Eq. (5) can be derived in the same way, where the two-excitation states with two atoms in state are disregarded since the Rabi frequency of the classical fields is weak. In Fig. 6, we check the effectiveness of Eq. (3) by plotting the populations for quantum states in Fig. 6(b) and Fig. 6(d), and comparing the corresponding results obtained from the full Hamiltonian of Eq. (2) in Fig. 6(a) and Fig. 6(c), which shows that they are in excellent agreement with each other under the given parameters.

### Footnotes

1. shaoxq644@nenu.edu.cn
2. gllong@mail.tsinghua.edu.cn

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