Implementation of holonomic quantum computation through engineering and manipulating the environment Zhang-qi Yin, Fu-li Li,* and Peng Peng Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China 共Received 3 April 2007; published 18 December 2007兲 We consider an atom-field coupled system in which two pairs of four-level atoms are respectively driven by laser fields and trapped in two distant cavities that are connected by an optical fiber. First, we show that an effective squeezing reservoir can be engineered under appropriate conditions. Then, we show that a two-qubit geometric controlled-PHASE 共CPHASE兲 gate between the atoms in the two cavities can be implemented through adiabatically manipulating the engineered reservoir along a closed loop. This scheme that combines an engineering environment with decoherence-free space and geometric phase quantum computation together has the following remarkable feature: a CPHASE gate with arbitrary phase shift is implemented by simply changing the strength and relative phase of the driving fields. DOI: 10.1103/PhysRevA.76.062311

PACS number共s兲: 03.67.Lx, 03.65.Vf, 03.65.Yz, 42.50.Dv

I. INTRODUCTION

Quantum computation, attracting much current interest since Shor’s algorithm 关1兴 was proposed, depends on two key factors: quantum entanglement and precision control of quantum systems. Unfortunately, quantum systems are inevitably coupled to their environment so that entanglement is too fragile to be retained. This makes the realization of quantum computation extremely difficult in the real world. In order to overcome this difficulty, one proposed the decoherence-free space concept 关2,3兴. It is found that when qubits involved in quantum computation collectively interact with the same environment there exists a “protected” subspace in the entire Hilbert space, in which the qubits are immune to the decoherence effects induced by the environment. This subspace is called decoherence-free space 共DFS兲. To perform quantum computation in a DFS, one has to design the specific Hamiltonian containing controlling parameters, which eigenspace is spanned by DFS states, and the state-unitary manipulation related to the quantum computation goal is implemented by changing the controlling parameters 关4兴. As is well-known, instantaneous eigenstates of a quantum system with the time-dependent Hamiltonian may acquire a geometric phase when the time-dependent parameters adiabatically undergo a closed loop in the parameter space 关5兴. The phase depends only on the swept solid angle by the parameter vector in the parameter space. This feature can be utilized to implement geometric quantum computation 共GQC兲, which is resilient to stochastic control errors 关6–8兴. On combining the DFS approach with the GQC scheme, one may build quantum gates which may be immune to both the environment-induced decoherence effects and the control-led errors 关9兴. In the scheme, quantum logical bits are represented by degenerate eigenstates of the parameterized Hamiltonian. These states have the following features: they belong to DFS, and unitarily evolve in time and acquire a geometric phase when the controlling parameters adiabatically vary and undergo a closed loop.

*[email protected] 1050-2947/2007/76共6兲/062311共5兲

In a recent paper 关10兴, Carollo et al. showed that a cascade three-level atom interacting with a broadband squeezed vacuum bosonic bath can be prepared in a state which is decoupled to the environment. This state depends on the reservoir parameters such as squeezing degree and phase angle. As the squeezing parameters smoothly vary, the atomic state can unitarily evolve in time and always be in the manifold of the DFS. Moreover, after a cyclic evolution of the squeezing parameters, the state acquires a geometric phase. This investigation has been generalized to cases where both quantum systems and manipulated reservoir under consideration are not restricted to cascade three-level atoms and squeezed vacuum 关11兴. These results strongly inspire us that instead of engineering Hamiltonian one may implement the decoherence-free GQC by engineering and manipulating reservoir. In this paper, we propose a scheme in which the quantumreservoir engineering 关12–14兴 is combined with DFS and Berry phase together to realize a two-qubit controlledPHASE 共CPHASE兲 gate 关15兴. We show that atomic states can unitarily evolve in time in a DFS if the change rate of reservoir parameters is much smaller than the characteristic relaxation time of an atom-reservoir coupled system. Moreover, we find that as the reservoir parameters adiabatically change in time along an appropriate closed loop, the atomic state in the DFS acquires a Berry phase and a CPHASE gate with arbitrary phase shift can be realized. To our knowledge, it is the first proposal for the realization of quantum gates by engineering and steering the environment. This paper is organized as follows. In Sec. II, we introduce a cavity-atom coupling model in which two pairs of four-level atoms are respectively trapped in two distant cavities that are connected by an optical fiber. In the model, each of the pairs of the atoms are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. Under large detuning and bad cavity limits, we investigate to engineer an effective broadband squeezing reservoir for the atoms. In Sec. III, we analyze how to realize controlling gates between the atoms trapped in the two cavities by steering the squeezing reservoir. Section IV contains conclusions of our investigations.

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PHYSICAL REVIEW A 76, 062311 共2007兲

YIN, LI, AND PENG

laser

2

laser

H0 =

兺 共rjn兩r jn典具r jn兩 + sjn兩s jn典具s jn兩 + ␦ jn兩e jn典具e jn兩兲

j,n=1

冉兺

冊

2

ﬁber

+

a†nan + b†b ,

n=1 2

(a)

Hd =

|rjn

∆rjn

∆sjn Ωsjn

Ωrjn

r gjn

|sjn

+

s gjn

|ejn

冉

⍀sjn −iLst e jn 兩s jn典具g jn兩 2

冊

⍀rjn −i共Lrt+兲 e jn 兩r jn典具e jn兩 + H.c. , 2

2

Hac =

共grjn兩r jn典具g jn兩an + gsjn兩s jn典具e jn兩an + H.c.兲, 兺 j,n=1

δjn

Hcf = 关b共a†1 + a†2兲 + H.c.兴.

|gjn

(b) FIG. 1. 共Color online兲 共a兲 Atom-field coupling scheme and 共b兲 atomic level configuration for atom j in cavity n. II. ENGINEERING A SQUEEZING ENVIRONMENT AND GENERATING A DECOHERENCE-FREE SUBSPACE

Our scheme is shown in Fig. 1. A pair of four-level atoms are trapped in each of two distant cavities, respectively, which are connected through an optical fiber. In the short fiber limit 关16–18兴, only one fiber mode, b, is excited and coupled to cavity modes a1 and a2 with strength 关19兴. We assume that the cavity modes and the fiber mode have the same frequency . The level scheme of atoms is shown in Fig. 1. Atom j in cavity n is labeled by the index jn with j , n = 1 , 2. The distance between the atoms in the same cavity is assumed to be large enough that there is no direct interaction between the atoms. The levels 兩g jn典 and 兩e jn典 of atom j in cavity n, with j , n = 1 , 2, are stable with a long lifetime. The energy of the level 兩g jn典 is taken to be zero as the energy reference point. The lower lying level 兩e jn典, and upper levels 兩r jn典 and 兩s jn典, have the energy ␦ jn, rjn, and sjn, respectively, in the unit with ប = 1. Transitions 兩g jn典 ↔ 兩s jn典 and 兩e jn典 ↔ 兩r jn典 are driven by laser fields of frequencies Ljns and Ljnr with Rabi frequencies ⍀sjn and ⍀rjn and relative phase , respectively. Transitions 兩g jn典 ↔ 兩r jn典 and 兩e jn典 ↔ 兩s jn典 are coupled to the cavity mode an with the strengths grjn and gsjn, respectively. Here we set ⌬rjn = rjn − = rjn − Ljnr − ␦ jn and ⌬sjn = sjn − − ␦ jn = sjn − Ljns. Under the Markovian approximation, the master equation of the density matrix for the whole system under consideration can be written as 关14兴

˙ T = − i关H, T兴 + Lcav1T + Lcav2T + L fiberT , where H = H0 + Hd + Hac + Hcf with

兺 j,n=1

共1兲

共2兲

Here, H0 is the free energy of atoms and cavity fields, Hd is the interaction energy between the atoms and laser fields, Hac is the interaction energy between the atoms and the cavity fields, and Hcf describes the interaction between the cavity modes and the fiber mode. The last three terms in Eq. 共1兲 describe the relaxation processes of the cavity and fiber modes in the usual vacuum reservoir, taking the forms LcavnT = n共2anTa†n − a†nanT − Ta†nan兲, L fiberT = f 共2bTb† − b†bT − Tb†b兲,

共3兲

where n is the leakage rate of photons from cavity n and f is the decay rate of the fiber mode. Let us introduce collective basis: 兩a典n = 共兩g1n典兩e2n典 兩−1典n = 兩g1n典兩g2n典, 兩0典n = 共兩g1n典兩e2n典 − 兩e1n典兩g2n典兲 / 冑2, + 兩e1n典兩g2n典兲 / 冑2, 兩1典n = 兩e1n典兩e2n典. The states 兩a典n and 兩−1典n are taken as a qubit n for quantum computation. In the large detuning limit, adiabatically eliminating the excited states r gr r gr s gs s gs ⍀1n ⍀2n ⍀1n ⍀2n 1n 2n 1n 2n and setting 2⌬ = 2⌬ = rn and 2⌬ = 2⌬ = sn, from Eq. r r s s 1n 2n 1n 2n 共2兲, we obtain the effective interaction Hamiltonian Hef f = 兺 冑2关an共rneiS+n + snSn兲 + H.c.兴 + Hcf ,

共4兲

n

where S+n = 兩0典nn具−1兩 + 兩1典nn具0兩. In the derivation of Eq. 共4兲, we have assumed the resonant condition ⍀s 2

gr 2

gsjn2

⌬sjn

⍀r 2

具a†nan典 + 4⌬jnr

jn

= 4⌬jns + ⌬jnr 具a†nan典 + ␦⬘jn. In order to satisfy the condition with jn jn the flexible choice of ⍀rjn, ⍀sjn, ⌬rjn, and ⌬sjn, we have introduced additional ac-Stark shifts ␦⬘jn to states 兩g jn典, which can be generated by using a laser field to couple the level 兩g jn典 to an ancillary level. We now introduce three normal modes c and c± with frequencies and ± 冑2 by use of the unitary transformation a1 = 21 共c+ + c− + 冑2c兲, a2 = 21 共c+ + c− − 冑2c兲, b = 冑12 共c+ − c−兲 r s 关17,18兴. In the limit Ⰷ 兩 j 兩 , 兩 j 兩, neglecting the far off-resonant modes c± and setting 1p = −2p =  p with p = r , s, we can approximately write the effective Hamiltonian 共4兲 as

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IMPLEMENTATION OF HOLONOMIC QUANTUM…

Hef f = 共reiS+ + sS兲c + H.c.,

共5兲

where S+ = S+1 + S+2 . Since the modes c± are nearly not excited and decoupled with the resonant mode c, the fiber mode b is mostly in the vacuum state, therefore, L fiberT can be neglected, and Lcav1T + Lcav2T can be approximated as LcavT = 共2cTc − c cT − Tc c兲, †

†

†

兩DF共r, 兲典1 = 兩e1典, cosh r

e2i共tanh r兲2兩e8典 − 兩DF共r, 兲典4 =

sinh r

i

冑cosh 2r 兩e j典 − e 冑cosh 2r 兩e j+4典,

冑

冑

j = 2,3,

2 i e tanh r兩e12典 + 兩e10典 3

2 共tanh r兲 + 共tanh r兲2 + 1 3

d O⌸共0兲, dt 共10兲

where ⌸⬜共0兲 = 1 − ⌸共0兲 and GDF = ⌸共0兲G⌸共0兲. In the limit of r˙ , ˙ Ⰶ ⌫, the last three terms in Eq. 共10兲 can be neglected 关11兴. In this way, Eq. 共10兲 is reduced to d¯DF = i关GDF,¯DF兴. dt

共11兲

Therefore, in the frame dragged adiabatically by the reservoir, the state of the atoms in the DFS unitarily evolves in time. III. REALIZING CONTROLLED PHASE GATES THROUGH MANIPULATING THE SQUEEZING ENVIRONMENT

In this section, we investigate how to realize a CPHASE gate through manipulating the engineered reservoir. Suppose that at the initial time the laser field driving the transition 兩g典 ↔ 兩s典 is switched off but the laser field driving the transition 兩r典 ↔ 兩e典 is switched on and the atoms are in the DFS state 兩⌿共0兲典a = 21 共兩a典1兩a典2 + 兩a典1兩−1典2 + 兩−1典1兩a典2 + 兩−1典1兩−1典2兲 = 兺4j=1兩DF共0 , 0兲典 j / 2. To generate a geometric phase for the atomic state, we smoothly change the parameters of the engineered reservoir along a closed loop, which is divided into the following three steps: 共1兲 From time 0 to T1, hold on = 0, and adiabatically increase the parameter r from 0 to r0; 共2兲 from time T1 to T2, hold on r = r0, and adiabatically change the phase from 0 to 0; 共3兲 from time T2 to T3, hold on = 0, and adiabatically decrease r from r0 to 0. When the cyclic evolution ends, the atomic state becomes 1 兩⌿共T3兲典a = 共兩e1典 + ei1兩e2典 + ei1兩e3典 + ei12兩e10典兲, 共12兲 2

.

4

where geometric phases 1 = −10, 12 = −120 with 共8兲

Let us introduce a unitary transformation O共r , 兲 by 兩i典 12 = 兺i=1 Oij共r , 兲兩e j典, where 兩i典 = 兩DF典i for i = 1 , 2 , 3 , 4. For the transformed density matrix ¯ = O†O, we have d¯ d = i关G,¯兴 + O† O, dt dt

− i⌸共0兲¯⌸⬜共0兲G⌸共0兲 + ⌸共0兲O†

共7兲

= Tr f 共T兲, R = S cosh r + eiS† sinh r, r where 2 2 −1 r 冑 r2 s2 = cosh 共 /  −  兲, and ⌫ = 2共r − s 兲 / . Equation 共7兲 describes the collective interaction of two cascade three-level atoms with the effective squeezed vacuum reservoir 关10兴. The parameters r, s, and are easily changed and controlled at will by varying the strength and phase of the driving lasers 关8兴. We will show that a geometric phase gate can be realized through changing these parameters. The DFS of the atomic system is spanned by the states which satisfy the equation R共r , 兲兩DF共r , 兲典 = 0 关10兴. In terms of basis states 兩e1典 = 兩a典1兩a典2, 兩e2典 = 兩a典1兩−1典2, 兩e3典 = 兩−1典1兩a典2, 兩e4典 = 兩a典1兩0典2, 兩e5典 = 兩0典1兩a典2, 兩e6典 = 兩a典1兩1典2, 兩e7典 = 兩1典1兩a典2, 兩e8典 = 兩1典1兩1典2, 兩e9典 = 冑12 共兩1典1兩0典2 + 兩0典1兩1典2兲, 兩e10典 = 兩−1典1兩−1典2, 兩e11典 = 冑12 共兩0典1兩−1典2 + 兩−1典1兩0典2兲, 兩e12典 1 = 冑6 共兩1典1兩−1典2 + 兩−1典1兩1典2兲 + 冑26 兩0典1兩0典2, the DFS states can be written as

兩DF共r, 兲典 j =

d¯DF = i关GDF,¯DF兴 + i⌸共0兲G⌸⬜共0兲¯⌸共0兲 dt

共6兲

where = 共1 + 2兲 / 2. In the bad cavity limit, Ⰷ , adiabatically eliminating the mode c 关12,14兴 from Eq. 共1兲 with the replacement of the Hamiltonian 共2兲 and the relaxation terms 共3兲 by the effective Hamiltonian 共5兲 and the relaxation term 共6兲, respectively, we can obtain the master equation for the density matrix of the atoms, ⌫ ˙ = − 共R+R + R+R − 2RR+兲, 2

+ 兩e10典具e10兩 onto the DFS. From Eq. 共9兲, we obtain the equation of motion for ¯DF = ⌸共0兲¯⌸共0兲,

共9兲

dO dO To solve where G共r , 兲 = iO† ddtO = iO†关r˙ dr + ˙ d 兴. Eq. 共9兲 in the DFS, let us define the time-independent pro4 O† 兩 i典具i 兩 O = 兺3j=1 兩 e j典具e j 兩 jector ⌸共0兲 = O†⌸共r , 兲O = 兺i=1

2 2 tanh4 r0+ tanh2 r0 3 . By performing 2 0 tanh4 r0+ tanh2 r0+1 3 transformations U1 = e−i1兩−1典11具−1兩 and

sinh2 r

0 1 = sinh2 r +cosh 2r , 0

12 =

local U2 = e−i1兩−1典22具−1兩, the state 共12兲 can be written as 兩⌿⬘共T3兲典a = U1U2兩⌿共T3兲典a = 21 共兩a典1兩a典2 + 兩a典1兩−1典2 + 兩−1典1兩a典2 + ei⌬兩−1典1兩−1典2兲, where ⌬ = 12 − 21 = 共21 − 12兲0. Thus, the CPHASE gate with the phase shift ⌬ is realized. If both the atoms in cavity 1 and the atoms in cavity 2 “see” different environments, 兩12兩 must be equal to 兩21兩 and ⌬ = 0. Therefore, the phase shift ⌬ results from the collective coupling of the atoms in both cavities with the

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YIN, LI, AND PENG 1 0.9995 0.999

Fr

0.9985 T=100/Γ T=200/Γ T=400/Γ

0.998 0.9975 0.997

(a)

0.9965 0

0.2

0.4 r0

0.6

0.8

0.6

0.8

1 0.995 0.99

Fp

0.985 0.98

T = 200/Γ T = 400/Γ T = 1000/Γ

0.975 0.97

(b)

0.965 0

0.2

0.4 r0

FIG. 2. 共Color online兲 共a兲 Fidelity Fr of the atomic state. 共b兲 Fidelity F p of the atomic state.

same engineered environment. If r0 = arctanh共冑冑4 / 3 − 1兲 ⯝ 0.4157, 兩12兩 = 兩1兩. Under this condition with 0 = / 1, the state of the atoms at the time T3 is 兩⌿⬙共T3兲典a = − 21 共−兩a典1兩a典2 + 兩a典1兩−1典2 + 兩−1典1兩a典2 + 兩−1典1兩−1典2兲. In this case, the controlled-Z gate between the two qubits is realized without local transformations. The above results depend on the adiabatical approximation. To check the adiabatical condition, we numerically simulate the following two examples. In the first example, we suppose that at the initial time the atoms are in the state 兩⌿1典a = 关兩a典1兩a典2 + 兩DF共0 , 0兲典2兴 / 冑2 and the laser field driving the transition 兩e典 ↔ 兩r典 are turned on. Then, by slowly switching the laser field driving the transition 兩g典 ↔ 兩s典, we increase the parameter r from 0 to r0 according to the linear function r共t兲 = r0t / T. In the adiabatical limit 共T Ⰷ ⌫−1兲, the atomic state becomes 兩⌿1⬘典a = 关兩a典1兩a典2 + 兩DF共r0 , 0兲典2兴 / 冑2 at the time T. On the other hand, in the Hilbert space spanned by the basis states 兵兩ei典其 for i = 1 , 2 , . . . , 12, we can numerically solve Eq. 共7兲 and obtain the density matrix 1共T兲 of the atoms. Let us define Fr=a具⌿1⬘兩1共T兲兩⌿1⬘典a as the fidelity for this process. As shown in Fig. 2, if T ⬎ 100/ ⌫, Fr is always bigger than 0.997 if r 苸 共0 , 0.8兲, corresponding to the almost perfect evolution. In the second example, we suppose that the atoms are initially in the state 兩⌿2典a = 共兩a典1兩a典2 + 兩DF共r , 0兲典4兲 / 冑2 and all the driving fields are turned on to hold the parameters r = r0 and = 0. By adiabatically changing the phase from 0 to 2 at the rate ˙ = 2 / T, the atomic state at the time T becomes 兩⌿2⬘典a = 共兩a典1兩a典2 + ei12兩DF共r , 2兲典4兲 / 冑2. Let us define the fidelity for this example as F p=a具⌿2⬘兩共T兲兩⌿⬘典a, where 共T兲 is the numerical solution of Eq. 共7兲. As shown in Fig. 2,

F p increases as T increases but decreases as the parameter r0 increases. If T ⬎ 1000/ ⌫, F p is larger than 0.992 for 0 ⬍ r0 ⬍ 0.8. From these two examples, we find that to fulfill the adiabatical condition the time used in the step 2 should be much longer than in the steps 1 and 3. A controlled-Z gate has been numerically simulated by directly solving Eq. 共7兲 with r0 = 0.5, and 0 = / 兩21 − 12兩. In the simulation, we set r˙ = r0 / T1 in the steps 1 and 3, and ˙ = 0 / 共T2 − T1兲 in the step 2 with T1 = 0.05T3 and T2 − T1 = 0.90T3. If T3 ⬎ 1100/ ⌫, we find that the fidelity F=a具⌿共T3兲兩共T3兲兩⌿共T3兲典a is larger than 0.95. For an almost perfect controlled-Z gate with F ⬎ 0.99, we find that T3 must be longer than 6000/ ⌫. Now let us briefly discuss the effects of the atomic spontaneous emission, the fiber mode decay, and cavity photon leakage. For simplicity, but without the loss of generality, we suppose that atomic spontaneous emission rates of the excited levels are equal to ␥. In the large detunnig limit, the characteristic spontaneous emission rate of the atoms is ␥ef f = ␥共⍀2 / 2⌬2兲 关14,16兴 and the effective decay rate of the fiber mode is ef f = f ⍀2g2 / 共4⌬22兲. If f 艋 ␥ and g2 Ⰶ 2, ef f can be much smaller than ␥ef f . Under this condition, the present scheme is feasible if ⌫ Ⰷ ␥ef f . In the current cavity quantum dynamic 共CQED兲 experiment, the parameters 共g , , ␥兲 = 共2000, 10, 10兲 MHz could be available 关20兴. If setting ⍀ / 共2⌬兲 = 冑12 ⫻ 10−3, we have ⌫ ⯝ 4 ⫻ 104␥ef f . The condition is held. In the present scheme, the large cavity decay rate is required to ensure that the cavity modes are in a broadband squeezed vacuum reservoir and then the atoms always “see” the broadband squeezed vacuum reservoir during the dynamic evolution. For an arbitrary small but nonzero value of the squeezing degree of the reservoir, a CPHASE gate with arbitrary high fidelity can always be realized in the represent scheme. The cavity decay does not directly affect the fidelity of the realized CPHASE gates. However, the larger the decay rate is, the longer the operation time of the CPHASE gates is. Thus, we have the condition Ⰷ  , ␥ef f for realizing the reliable CPHASE gates. Based on the parameters quoted above, this condition can be well satisfied. With the parameters of the current CQED experiment, we find that the operation time of the controlled-Z gate, with fidelity larger than 0.95, is about 2.8 ms. It is much shorter than both 1 / ␥ef f and the single-atom trapping time in cavity 关21兴. On the other hand, the present scheme needs a strong coupling between the cavity and the fiber. This could be realized at the current experiment 关22兴. Therefore, the requirement for the realization of the present scheme can be satisfied with the current technology. IV. CONCLUSIONS

We propose a cavity-atom coupled scheme for the realization of quantum controlling gates, in which each of two pairs of four-level atoms in two distant cavities connected by a short optical fiber are simultaneously driven by laser fields and coupled to the local cavity modes through the double Raman transition configuration. We show that an effective squeezing reservoir coupled to the multilevel atoms can be engineered under appropriate driving condition and bad cav-

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ACKNOWLEDGMENTS

ity limit. We find that in the scheme a CPHASE gate with arbitrary phase shift can be implemented through adiabatically changing the strength and phase of driving fields along a closed loop. It is also noticed that the larger the effective coupling strength between the environment and the atoms is, the more reliable the realized CPHASE gate is.

We thank Yun-feng Xiao and Wen-ping He for valuable discussions and suggestions. This work was supported by the Natural Science Foundation of China 共Grants No. 10674106, No. 60778021, and No. 05-06-01兲.

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