Entanglement between internal and external degrees of freedom of a driven trapped atom Maryam Roghani, Hanspeter Helm, and Heinz-Peter Breuer Institute of Physics, Albert-Ludwigs University, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany E-mail:
[email protected],
[email protected],
[email protected] Abstract. We study the dynamics of a Λ-shaped trapped atom which is driven by two lasers to conditions approaching electromagnetically-induced transparency (EIT). Solving numerically the full quantum master equation describing the coupled system of electronic and vibrational degrees of freedom, we investigate the dynamics of the entanglement and the quantum mutual information during the cooling process. Far from the Lamb-Dicke limit an intricate behavior of these quantities is found, as well as the emergence of a mixed entangled non-equilibrium stationary state.
PACS numbers: 37.10.De, 03.67.Mn, 42.50.Gy, 42.50.Tx
1. Introduction A trapped atom under conditions approaching electromagnetically-induced transparency (EIT) represents a highly realistic model for current experiments on ion or atom trapping [1, 2]. In such a system the Lamb-Dicke parameter η defined by s
h ¯ k2 (1) 2mω controls the probability of changing the vibrational state n in electronic transitions. In the Lamb-Dicke limit (η → 0), the photon-recoil energy h ¯ 2 k 2 /(2m) is an inconsiderable fraction of the trap vibrational energy h ¯ ω. In this limit only the first sidebands ∆n = ±1 are significant, their probability being proportional to η 2 times that of the carrier transition. However, in many experiments the system is far from the Lamb-Dicke limit and it is important to explore the effect of finite values of η. We have found numerically [3] that even under conditions far from the Lamb-Dicke limit, a rapid removal of vibrational energy of a trapped atom occurs under suitable experimental conditions, and as a result a vibrationally excited atom approaches EIT-like conditions with mean values hni near zero, n being the vibrational quantum number. Moreover, we have demonstrated that the removal of vibrational energy is a consequence of unbalanced sidebands in the spectrum of radiation scattered by the atom from the two laser beams [4]. η=
Entanglement dynamics of a driven trapped atom
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In the present paper we investigate a further important feature of the cooling process, namely the dynamical behavior of the correlations between the external vibrational degree of freedom of the trapped atom and its internal electronic degree of freedom which is described by a Λ-type level structure. From the numerical solution of the corresponding quantum master equation for the laser driven trapped atom we determine the time evolution of the linear quantum entropy, the quantum mutual information and the entanglement between these degrees of freedom during the cooling process. 2. EIT cooling The quantum Markovian master equation describing the underlying model is given by [5, 6] i d ρ(t) = − [H, ρ(t)] + L0 ρ(t), (2) dt h ¯ where ρ(t) denotes the interaction picture density matrix which represents the combined state of vibrational and electronic degrees of freedom of the trapped atom. The total Hamiltonian H = Hcm + Hel + Hint
(3)
includes the harmonic oscillator Hamiltonian describing the vibrational degree of freedom of the tapped atom, Hcm =
∞ X
h ¯ ωn|nihn| +
n=0
h ¯ω p2 1 h ¯ω 2 = + M ω 2 x2 = P + Q2 , 2 2M 2 2
(4)
q
where we have used the harmonic oscillator length a0 = M¯hω to define dimensionless position and momentum operators through Q = x/a0 and P = p a0 /¯h, the Hamiltonian for the internal electronic degree of freedom, Hel = h ¯ ∆1 |1ih1| + h ¯ ∆2 |2ih2|,
(5)
as well as the Hamiltonian representing the interaction with the two counter propagating laser beams, o h ¯ n i~k1 ·~x ~ Hint = g1 e |3ih1| + g2 eik2 ·~x |3ih2| + h.c. . (6) 2 The electronic states of the Λ-configuration are denoted by |ii, i = 1, 2, 3, ∆1 and ∆2 describe the detunings of the laser frequencies from the atomic transition frequencies, and g1 and g2 are the Rabi frequencies corresponding to the laser excitations. Finally, the dissipator of the master equation (2) representing spontaneous emission processes is given by [7] o Γj 1n † † L0 ρ = σjq ρ σjq − σjq σjq , ρ . 2 j=1,2 q=± 2 X X
(7)
Entanglement dynamics of a driven trapped atom
3
We have written this superoperator in the standard Lindblad form involving the Lindblad operators σj− = |jih3|e−ikj x .
σj+ = |jih3|eikj x ,
(8)
0
È8,3\
2
È9,1\ È9,2\
4
È9,3\ 6 8
È i,n \
n vibrational number
Here, we have assumed for simplicity that spontaneous emissions take place with equal probability only parallel or anti-parallel to the harmonic oscillator axis. Hence, the operators σj± describe the spontaneous transitions |3i → |j = 1, 2i corresponding to the emission of a photon parallel or anti-parallel to the harmonic oscillator axis.
1
10
È10,1\ È10,2\ È10,3\
12
È11,1\
0
È11,2\
14
È11,3\
16 0
1
2
3
4
0
5
100
200
300
400
time Ðs
time ms
(a)
(b)
Figure 1: Dynamics of the state populations for the initial product state |i = 2, n = 10i. The gray-scale bands represent the matrix elements hi, n| ρ |i, ni. Parameters (in units of 2π MHz): ω = 0.03, g1 = 1.34, g2 = 0.34, ∆1 = ∆2 = −15, Γ1 = Γ2 = 6, η = 0.1.
20
20
10
10 ` X P \ a0 103
` X Q \ a0 103
The master equation (2) can be solved numerically, representing the vibrational degree of freedom through an appropriately truncated basis of Fock states |ni. Further details of the model and its numerical implementation are discussed in Refs. [3] and [4]. A typical example for the time development of the populations of the electronic states is shown in Figure 1. Here, the system is initially in the product state |i = 2, n = 10i. We observe a rapid approach to the space spanned the electronic
0
-10
-20
0
-10
.03
.1
.3 time ms
(a)
1
3
-20
.03
.1
.3
1
3
time ms
(b)
Figure 2: Time development of the mean atomic position hQi and momentum hPi. Parameters as in Fig. 1.
Entanglement dynamics of a driven trapped atom
4
states |1i and |2i, the excited state hardly being populated, ρ33 < 10−4 . The time evolution of the mean atomic momentum and position is depicted in Fig. 2. We see that a rapid cooling occurs and that the atom settles close to the ground vibrational state (hni=0.015 in this example) within a few milliseconds. 3. Entanglement and quantum mutual information The numerical solution of the master equation (2) provides the complete information on the correlations between the vibrational degree of freedom of the atom and the qutrit of the electronic degree of freedom. We have studied the dynamics of the entanglement between these external and internal degrees of freedom under a variety of EIT conditions using the negativity 1 TA ||ρ || − 1 (9) N (ρ) = 2 as an easily computable entanglement measure [8]. Here, TA denotes the partial transpose of the total system’s density matrix ρ and ||X|| denotes the trace norm of an operator X. 0.5
1.
1.
0.1
0.1
0.01
0.01
10-3
10-3
10-4
10-4
10-5
10-5
SL
0.4
NHΡL
NHΡL
0.3 0.2 0.1 0.0
10-6 .03
.1
.3 time ms
(a)
1
3
.03
.1
.3 time ms
(b)
1
3
10-6
IH1:2L
NHΡL
.03
.1
.3
1
3
time ms
(c)
Figure 3: Dynamics of the negativity, of the mutual information and of the linear entropy. Parameters as in Fig. 1. An example for the dynamical behavior of the negativity (9) is shown in Fig. 3a. Starting from an initial product state, the negativity builds up as the cooling cycle proceeds. After passing over a maximum, the negativity apparently experiences a sudden death and only later, at times when near-stationary-state conditions are reached, a revival of entanglement occurs. A closer inspection of the negativity (see the logarithmic scale in Fig. 3b) shows that it does not really become zero. The time-dependent total system state thus rapidly approaches the region of PPT states (states with positive partial transposition) and is then deflected from it. The closeness of approach to the PPT region strongly depends on the experimental parameters chosen. We find that in the stationary state the negativity is maximal when the intensities of the two counter propagating laser beams are equal to each other. Moreover, the
Entanglement dynamics of a driven trapped atom
5
negativity increases with increasing Lamb-Dicke parameter η and it decreases with increasing decay rate of the excited state. We see from Fig. 3 that under under conditions far from the Lamb-Dicke limit the stationary state of the two-laser trapped-atom system is a mixed state which exhibits substantial entanglement. The atom is thus driven into a mixed entangled nonequilibrium stationary state which results from the combined effects of the dissipative character of the dynamics and of the coupling between internal and external degrees of freedom due to a finite value of the Lamb-Dicke parameter. Our simulation results indicate that the total system is completely relaxing and, hence, the entanglement of the unique invariant state is expected to reveal a certain degree of stability with respect to external perturbations. It is interesting to compare the behavior of the negativity with the dynamics of the quantum mutual information of the electronic and the vibrational state which is defined by (see, e.g., Ref. [9]) I(1 : 2) = S(ρel ) + S(ρvib ) − S(ρ).
(10)
Here, S denotes the von Neumann entropy, S(ρ) = −Tr(ρ Lnρ), while ρel = Trvib ρ and ρvib = Trel ρ represent the reduced states of the electronic and the vibrational subsystem, respectively, given by the corresponding partial traces Trvib and Trel . Figure 3c shows a comparison of the time evolution of the quantum mutual information during the cooling process with that of the negativity and of the linear entropy of the vibrational 2 state, SL = 1 − Tr[ρvib ]. We observe that the dynamics of the mutual information is qualitatively similar to that of the negativity. Both quantities show a pronounced maximum, decrease to very small values, and finally increase again to reach a stationary value. The interpretation of the physical meaning and the origin of this dynamical behavior is a matter of current study. 4. Conclusion We have studied the dynamics of correlations and entanglement of a laser-driven trapped atom near EIT conditions, employing the negativity as an easy computable entanglement measure and the quantum mutual information as a measure for the total correlations between the internal and the external atomic degrees of freedom. The physical relevance of this investigation is given by the fact that the underlying model is based on a rather realistic situation as it can be found in a variety of experiments. Our results suggest several further investigations. The numerical simulations presented here are restricted to pure, uncorrelated product initial states of electronic and vibrational degrees of freedom. Of course, the numerical program allows to study the dynamics of arbitrary pure, mixed and correlated initial states. For example, one could investigate the influence of the initial state on the efficiency of the cooling mechanism, employing, for example, coherent and thermal initial states for the vibrational degree of freedom. Moreover, it will be of great interest to examine in a systematic way the
Entanglement dynamics of a driven trapped atom
6
role of non-factorizing initial conditions and, in particular, to discriminate between the effects of classical and quantum correlations in the initial state. Finally, we remark that our numerical simulations enable of course also the investigation of further novel measures for the quantumness of non-local correlations in composite systems which differ quantitatively and qualitatively from entanglement measures, such as the quantum discord [10, 11] and other related quantities [12, 13] that measure the degree of disturbance of the quantum state induced by local measurements. It is of great interest, both from a fundamental and from a practical point of view, to determine these quantities for the present system and to compare them with the results obtained for the negativity and other entanglement measures. 5. Acknowledgement This research was supported by the Deutsche Forschungsgemeinschaft, grant HE-2525/8. References [1] F. Schmidt-Kaler, J. Eschner, G. Morigi, C. F. Roos, D. Leibfried , A. Mundt, and R. Blatt, Appl. Phys. B 73, 807 (2001). [2] M. Fleischhauer, A. Imamoglu and J. P. Marangos, Rev. Mod. Phys. 77, 633-673 (2005). [3] M. Roghani and H. Helm, Phys. Rev. A. 77, 043418 (2008). [4] M. Roghani, H.-P. Breuer and H. Helm, submitted for publication. [5] M. Orszag, Quantum Optics (Springer, Berlin, 2000) p.276-281. [6] P. Meystre, Atom Optics (Springer, 2001). [7] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2007). [8] G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002). [9] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2002. [10] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2002). [11] L. Henderson and V. Vedral, J. Phys. A 34, 6899 (2001). [12] S. Luo, Phys. Rev. A 77, 022301 (2008); S. Luo, Phys. Rev. A 77, 042303 (2008). [13] A. R. Usha Devi and A. K. Rajagopal, Phys. Rev. Lett. 100, 140502 (2008).
