Dynamics of the Energy Relaxation and Decoherence of a Photon-Atom Bound State in an Anisotropic Photonic Crystal

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Volume 2013, Article ID 980698,9pages http://dx.doi.org/10.1155/2013/980698

Research Article

Dynamics of the Energy Relaxation and Decoherence of

a Photon-Atom Bound State in an Anisotropic Photonic Crystal

Jing-Nuo Wu,

1

Wen-Feng Hsieh,

2

Hsin-Chien Huang,

1

and Szu-Cheng Cheng

1

1_{Department of Physics, Chinese Culture University, Taipei 11114, Taiwan}

2_{Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu 71150, Taiwan}

Correspondence should be addressed to Jing-Nuo Wu; [email protected] and Szu-Cheng Cheng; [email protected]

Received 21 May 2013; Accepted 16 September 2013 Academic Editor: J¨org Fink

Copyright © 2013 Jing-Nuo Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An atom embedded inside photonic crystals can form a photon-atom bound state if the emission frequency of the excited atom is lying inside the photonic-band gap of photonic crystals. We studied the dynamics of the energy relaxation and decoherence of a QPAB, qubit made by a photon-atom bound state in photonic crystals. Dynamics of these measurements are solved analytically through the fractional calculus which has been shown to be appropriate mathematical method for the optical systems with non-Markovian dynamics. From these dynamics, we find that the losses of energy, coherence, and information of a QPAB are inhibited. As compared with those qubits without forming photon-atom bound states, the energy relaxation and decoherence rates of these QPABs are strongly suppressed. Other systems suitable for realizing these properties are discussed.

1. Introduction

A central tenet in quantum computing is to create and manipulate qubits in a controllable way. Many novel methods have been proposed to generate controllable qubit states by changing the surrounding environment of the qubits [1– 3]. When a qubit made by a two-level atom is embedded inside a structured reservoir with the photon density of state (DOS) different from that of free space, the dynamics of the qubit states are affected by the environment and exhibit non-Markovian behavior [4–6]. When strong cou-pling between the qubit and the structured reservoir is concerned, these non-Markovian dynamics are characterized by the memory effect that the information of the system is preserved for a long-time period. This characterization has recently constructed a number of experimental measures for exploring non-Markovianity in a variety of different system-environment models [7–11].

However, it is the interaction between a qubit and its sur-rounding environment that causes the energy relaxation and decoherence of the qubit. Since the quantum technologies rely on controlling quantum states with long-time coherence, it has become increasingly important to effectively suppress

the energy relaxation and decoherence rates of quantum states. To achieve this goal, various schemes have been used to store quantum states with high fidelity through effectively controlling the energy relaxation and decoherence rates [12– 16]. Here we introduce a structured reservoir made by a photonic crystal (PhC) whose photonic density of states (DOS) possesses a photonic-band gap (PBG). An excited atom embedded inside this reservoir is possible to form a photon-atom bound state [5] if the emitted frequency is lying inside the PBG. A qubit made by a photon-atom bound state in PhCs, which we abbreviate as a QPAB, exhibits some non-Markovian dynamics of the energy relaxation and decoherence of the QPAB. Systems with the forbidden bands in the structured reservoirs such as omnidirectional

waveg-uides [17], tunable artificial crystals [18,19], and frequency

dispersive media [20] are also suitable for the realization of these non-Markovian dynamics.

In this paper, we shall show that the energy relaxation and decoherence rates of the QPAB are strongly suppressed and the long-time period of the coherence and information preservation of a qubit can be achieved via this structured reservoir. The steady states, shown by the polarization and von Neumann entropy of the QPAB, exhibit nonzero values

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for a very long-time period, indicating that the losses of energy and coherence of the QPAB are suppressed in this structured reservoir. The corresponding rates of energy relax-ation and decoherence exhibit a fast decaying behavior that further manifests this suppressing phenomenon. This kind of structured reservoir provides qubits with a controllable environment that can preserve the qubit information for a long time.

The PhCs are ordered crystals with periodic variation of refractive index that show forbidden bands in their disper-sion relation. Light-matter interactions in these periodical structures are drastically enhanced because they provide the photons with a reservoir with a threshold-like DOS near

the forbidden band edges [21, 22]. As an excited atom is

embedded (doped) in a PhC, the emitted photon in this structured reservoir behaves very differently from that in free space. The appearance of photon-atom bound states [23–26], enhanced quantum interference [27], and non-Markovian effects [28–30] exist especially in this optical system. With the enhanced light-matter interactions, PhCs have been

demonstrated theoretically and experimentally [31,32] that

they can be effectively used to inhibit spontaneous emission (SE) because of the existence of the photon-atom bound states.

In the studies on the photon-atom bound states in the doped PhCs, Singh controlled the SE cancellation of the PhCs doped with five-level nanoparticles by moving the resonance energies between the energy band and energy gap where bound states are formed [25]. Generalizing the properties of the PhCs to the dispersive media, Rupasov and Singh studied the quantum electrodynamics of a single two-level atom placed in a frequency dispersive medium (DM) and found a polariton-atom bound state with an eigenfrequency

lying within the gap of the polariton spectrum [20,23]. This

generalization arises the intriguing question about whether the quantum optical phenomena observed in PhCs can also be expected in dispersive media and vice versa. We discuss this question through comparing our findings with the results

found in the dispersive media inSection 2.3.

When a practical three-dimensional (3D) PhC is con-sidered, the dispersion relation near the forbidden band edge exhibits a directional-dependent property and thus has been expressed as a vector form by the effective-mass approximation [33]. This 3D material with the anisotropic band structure possesses a DOS different from the free-space

one and proportional to √𝜔 − 𝜔_𝑐Θ(𝜔 − 𝜔_𝑐), where 𝜔_𝑐 is

the band-edge frequency and Θ(𝜔 − 𝜔_𝑐) is the Heaviside

step function. States with frequencies lying below the band

edge𝜔_𝑐are forbidden. In this forbidden region, the emission

of a photon from a qubit, constructed by an excited two-level atom, is inhibited, so a photon-atom bound state [5] is formed when the qubit interacts with its surrounding reservoir strongly.

This paper is organized as follows. In Section 2, the

theoretical model for the system of a qubit in an anisotropic PhC is depicted through the Hamiltonian and the wave function. The kinetic equation of the qubit can be writ-ten as a fractional Langevin equation and solved by the fractional calculus, a mathematical method appropriate for

𝜔10 |1⟩ |0⟩ (a) 𝜔 𝜔c k0 k (b) 𝜔 𝜔c 𝜌(𝜔) (c)

Figure 1: (a) A qubit with excited state|1⟩ and ground state |0⟩. The transition frequency𝜔₁₀is nearly resonant with the frequency range of the PhC reservoir. (b) Directional dependent dispersion relations near band edge expressed by the effective-mass approximation with the edge frequency𝜔_𝑐as solid and dashed curves. (c) Photon DOS 𝜌(𝜔) of the anisotropic PhC reservoir exhibiting forbidden photon mode below the edge frequency𝜔_𝑐.

solving the system with non-Markovian dynamics [6,28], to

construct the reduced density matrix of the qubit. Dynamics of the energy relaxation rate, polarization, and von Neumann entropy are calculated and analyzed based on the elements of this density matrix. Systems of omnidirectional waveguides, tunable artificial crystals, and dispersive media suitable for realizing this property are briefly discussed at the end of this

section. Finally, we summarize our results inSection 3.

2. Dynamics of a Qubit in an Anisotropic PhC

When we consider a system of a qubit formed by an excited atom embedded inside a structured reservoir of an

ani-sotropic PhC shown in Figure 1, the Hamiltonian can be

expressed as 𝐻 = ℏ𝜔₁₀|1⟩ ⟨1| + ∑ ⃗𝑘 ℏ𝜔_⃗𝑘𝑎+_⃗𝑘𝑎_⃗𝑘 + 𝑖ℏ∑ ⃗𝑘 𝑔_⃗𝑘(|0⟩ ⟨1| ⊗ 𝑎+_⃗𝑘 + h.c.) . (1)

Here 𝜔₁₀ is the qubit transition frequency between the

excited state|1⟩ and ground state |0⟩. The photon mode with

frequency 𝜔_⃗𝑘 is created and annihilated by the operators

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the photon mode with frequency 𝜔_⃗𝑘 is specified by 𝑔_⃗𝑘 =

(𝜔₁₀𝑑₁₀/ℏ)[ℏ/2𝜖₀𝜔_⃗𝑘𝑉]1/2_̂𝑒

⃗𝑘⋅ ̂𝑢𝑑 with the fixed qubit dipole

moment ⃗𝑑₁₀ = 𝑑₁₀̂𝑢_𝑑, sample volume𝑉, dielectric constant

𝜖0, and polarization unit vector ̂𝑒⃗𝑘 of the photon mode

with frequency 𝜔_⃗𝑘. If we use the coordinate (𝜃, 𝜙) on the

Bloch sphere to parameterize the state of the qubit with (0, 𝜙) for the excited state and (𝜋, 𝜙) for the ground state, then the initial state of the total system can be written as

|𝜓(0)⟩ = [𝑒𝑖𝜙0cos(𝜃

0/2)|1⟩ + sin(𝜃0/2)|0⟩] ⊗ |0⃗𝑘⟩ with the

initial coordinate (𝜃₀, 𝜙₀) on the Bloch sphere. Here the

vacuum state and one-excitation state of the PhC

reser-voir are expressed as |0_⃗𝑘⟩ = |0₁, 0₂, . . . , 0_⃗𝑘, 0_⃗𝑘+1, . . .⟩ and

|1_⃗𝑘⟩ = |0₁, 0₂, . . . 0_⃗𝑘−1, 1_⃗𝑘, 0_⃗𝑘+1, . . .⟩, respectively. As the

system evolves, the quantum state of the system in the single photon sector can be written as

󵄨󵄨󵄨󵄨𝜓(𝑡)⟩ = [ 𝑢𝑝(𝑡) 𝑒−𝑖𝜔10𝑡𝑒𝑖𝜙0cos(𝜃₂0) |1⟩ + 𝑢_𝑑(𝑡) sin (𝜃0 2) |0⟩] ⊗ 󵄨󵄨󵄨󵄨0⃗𝑘⟩ + ∑ ⃗𝑘 𝐶_⃗𝑘(𝑡) 𝑒−𝑖𝜔⃗𝑘𝑡_{|0⟩ ⊗ 󵄨󵄨󵄨󵄨1} ⃗𝑘⟩ (2)

with initial condition𝑢_𝑝(0) = 1, 𝑢_𝑑(0) = 1, and 𝐶_⃗𝑘(0) = 0.

Here𝑢_𝑝(𝑡) and 𝑢_𝑑(𝑡) stand for the excited-state and

ground-state probability amplitudes of the qubit with the vacuum

state of the PhC reservoir while𝐶_⃗𝑘(𝑡) is the probability

ampli-tude of the photonic state at ⃗𝑘. This quantum state can also be

expressed as|𝜓(𝑡)⟩ = {𝑒𝑖𝜙(𝑡)cos[𝜃(𝑡)/2]|1⟩ + sin[𝜃(𝑡)/2]|0⟩} ⊗

|0_⃗𝑘⟩ + ∑_⃗𝑘𝐶_⃗𝑘(𝑡)|0⟩ ⊗ |1_⃗𝑘⟩ with the time-dependent Bloch

parameters[𝜃(𝑡), 𝜙(𝑡)] when we discuss the change of the

qubit’s state on the Bloch sphere.

By substituting the quantum state in (2) for the time-dependent Schr¨odinger equation, we obtain the equations of motion for the amplitudes as

𝑑 𝑑𝑡𝑢𝑝(𝑡) = −_𝑒𝑖𝜙0_cos1(𝜃₀/2)∑ ⃗𝑘 𝑔_⃗𝑘𝐶_⃗𝑘(𝑡) 𝑒−𝑖Ω⃗𝑘𝑡_, (3a) 𝑑 𝑑𝑡𝐶⃗𝑘(𝑡) = 𝑔⃗𝑘𝑢𝑝(𝑡) 𝑒𝑖𝜙0cos(𝜃₂0) 𝑒𝑖Ω⃗𝑘𝑡, (3b) 𝑑 𝑑𝑡𝑢𝑑(𝑡) = 0 (3c)

with detuning frequencyΩ_⃗𝑘= 𝜔_⃗𝑘− 𝜔₁₀. The last equation of

motion yields𝑢_𝑑(𝑡) = 𝑢_𝑑(0) = 1 meaning that the

ground-state probability of a qubit will not evolve with time, that is, a constant of time. The other two equations can be combined as

(𝑑/𝑑𝑡)𝑢_𝑝(𝑡) = − ∫₀𝑡𝐺(𝑡 − 𝜏)𝑢_𝑝(𝜏)𝑑𝜏 with the memory kernel

𝐺(𝑡 − 𝜏) = ∑_⃗𝑘𝑔2

⃗𝑘𝑒−𝑖Ω⃗𝑘(𝑡−𝜏). This equation of motion reveals

that the time dependence of the qubit state is determined by all previous states, called the long-time memory effect or non-Markovian behavior, through the memory kernel. As the qubit is put in the free space, the memory kernel becomes a

Dirac delta function𝐺(𝑡 − 𝜏) ∝ 𝛿_𝑑(𝑡 − 𝜏) corresponding to

continuous photon DOS. In this case, the reservoir manifests

its memory effect only at an instant time𝜏 = 𝑡 which leads

to the excited amplitude of the qubit decaying exponentially with time. This Markovian result in free space manifests that the qubit loses all memory of its past and decays quickly to its ground state.

For the anisotropic PhC reservoir discussed here, the memory kernel manifests its memory effect within the entire

time interval(0, 𝑡) through 𝐺(𝑡 − 𝜏) = ((𝛽1/2/𝑓3/2)/(√𝜋(𝑡 −

𝜏)3/2))𝑒−𝑖[3𝜋/4−𝛿(𝑡−𝜏)] with the coupling constant 𝛽1/2 =

(𝜔2

10𝑑210√𝜔𝑐)/(16𝜋𝜖0ℏ𝑐3) and the detuning frequency 𝛿 =

𝜔₁₀ − 𝜔_𝑐 of the qubit transition frequency 𝜔₁₀ from the

band edge frequency𝜔_𝑐 [6]. This memory kernel is derived

from the anisotropic dispersion relation of a practical

three-dimensional PhC. Near the band edge frequency𝜔_𝑐, the

dis-persion relation has a vector form and could be expressed by

the effective-mass approximation as [34]𝜔_⃗𝑘≈ 𝜔_𝑐+𝐴( ⃗𝑘 − ⃗𝑘_𝑐)2,

where the curvature𝐴 ≅ 𝑓𝜔_𝑐/𝑘2_𝑐 = 𝑓𝑐2/𝜔_𝑐signifies its

differ-ent values in differdiffer-ent directions through the scaling factor 𝑓. This anisotropic dispersion relation leads to the memory

kernel expressed by the cut-off photon DOS𝜌(𝜔) through

𝐺(𝑡 − 𝜏) = (𝜔2

10𝑑210/4𝜖0ℏ) ∫₀∞𝑑𝜔(𝜌(𝜔)/𝜔)𝑒−𝑖(𝜔−𝜔10)(𝑡−𝜏) with

𝜌(𝜔) = (1/4𝜋2_{)√((𝜔 − 𝜔}

𝑐)/𝐴3)Θ(𝜔 − 𝜔𝑐) and the Heaviside

step functionΘ(𝜔 − 𝜔_𝑐) charactering the cut-off behavior.

With this memory kernel for the anisotropic PhC reservoir

and making the transformation 𝑢_𝑝(𝑡) = 𝑒𝑖𝛿𝑡𝑈_𝑝(𝑡), the

equation of motion becomes 𝑑 𝑑𝑡𝑈𝑝(𝑡) + 𝑖𝛿𝑈𝑝(𝑡) = 𝛽1/2𝑒𝑖𝜋/4 √𝜋𝑓3/2 ∫ 𝑡 0 𝑈_𝑝(𝜏) (𝑡 − 𝜏)3/2𝑑𝜏. (4)

Conventionally, this integrodifferential equation is solved through Laplace transform which leads to the fractal phe-nomenon of the system from this memory kernel in Laplace image [35]. This fractal phenomenon would result in the stochastic nature of the dynamical behavior of the system which appears as the Markovian dynamics. These non-Markovian dynamics can be solved accurately through frac-tional calculus which we have shown its appropriateness by comparing the obtained results with the experimental ones

and finding their consistence [6,28].

In the following, we shall use fractional calculus to solve the non-Markovian dynamics of the system analytically. In particular, the fractional time derivative, one of the operators of fractional calculus, is used to express the integral term of the kinetic equation (4). In the well-known Riemann-Liouville definition, the fractional time derivative operator

𝑑]/𝑑𝑡]is expressed as [36] 𝑑] 𝑑𝑡]𝑓 (𝑡) = _{Γ (𝑛 − ])}1 𝑑 𝑛 𝑑𝑡𝑛∫ 𝑡 𝑎 𝑓 (𝜏) (𝑡 − 𝜏)]−𝑛+1𝑑𝜏 (5)

for𝑛−1 ≤ ] < 𝑛 and Γ(𝑥) being the gamma function. Through

expressing the right-hand-side term of (4) as the fractional

time derivative operator with order𝑛 = 1/2 and applying the

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arrive at the fractional kinetic equation of this qubit-reservoir interacting system as 𝑑1/2 𝑑𝑡1/2𝑈𝑝(𝑡) + 𝑖𝛿𝑑 −1/2 𝑑𝑡−1/2𝑈𝑝(𝑡) +2𝛽 1/2_𝑒𝑖𝜋/4 𝑓3/2 𝑈𝑝(𝑡) = 𝑡 −1/2 √𝜋 . (6) The fractional time derivative in this fractional differential equation indicates a subordinated stochastic process direct-ing to a stable probability distribution [37]. The evolution equation governing the future of the qubit is expressed as the form of a fractional Langevin equation because of its interaction with the PhC reservoir. Our aim of studying the quantum dynamics of relaxation, decoherence, and entropy for the qubit system can be achieved by solving this equation through the Laplace transform for the fractional operators. The basic formula used here is

𝐿 {𝑑] 𝑑𝑡]𝑓 (𝑡)} ≡ ∫ ∞ 0 𝑒 −𝑠𝑡 𝑑] 𝑑𝑡]𝑓 (𝑡) 𝑑𝑡 = 𝑠]𝐿 {𝑓 (𝑡)} −𝑛−1∑ 𝑚=0 𝑠𝑚[𝑑]−𝑚−1 𝑑𝑡]−𝑚−1𝑓 (𝑡)] 𝑡=0 (7)

with𝑠 denoting the Laplace variable. The procedure of

per-forming the Laplace transform on the fractional Langevin

equation leads to ̃𝑈_𝑝(𝑠) = 1/(𝑠 + 𝑖𝛿 + 2𝛽1/2𝑒𝑖𝜋/4𝑠1/2/𝑓3/2),

where ̃𝑈_𝑝(𝑠) is the Laplace transform of 𝑈_𝑝(𝑡). This equation

can be further expressed as a sum of partial fractions as the

roots of the indicial equation𝑌2+ 2𝛽1/2𝑒𝑖𝜋/4𝑌/𝑓3/2 + 𝑖𝛿 =

0 are found with the variable 𝑠1/2 _{having been converted}

into 𝑌. As 𝛽/𝑓3 ̸= 𝛿, we have the different roots of the

indicial equation which leads to the partial-fractional form

of ̃𝑈_𝑝(𝑠) = [(1/(√𝑠 − 𝑌₁)) − (1/(√𝑠 − 𝑌₂))](1/(𝑌₁ − 𝑌₂))

with 𝑌₁ = 𝑒𝑖𝜋/4(−(𝛽1/2/𝑓3/2) + √(𝛽/𝑓3) − 𝛿) and 𝑌₂ =

𝑒𝑖𝜋/4(−(𝛽1/2/𝑓3/2) − √(𝛽/𝑓3_{) − 𝛿). For 𝛽/𝑓}3 _{= 𝛿, the}

indi-cial equation has degenerate root leading to ̃𝑈_𝑝(𝑠) = 1/(√𝑠 +

(𝛽1/2_𝑒𝑖𝜋/4_/𝑓3/2₎₎2_.

The inverse Laplace transform of these partial-fractional forms cannot be found in the conventional mathematical

table because the power of the variable𝑠 is not an integer.

For the fractional power of𝑠, we can find its inverse Laplace

transform in the book of fractional calculus [38] as𝐿−1{1/(𝑠]−

𝑎)} = ∑𝑞_𝑗=1𝑎𝑗−1𝐸_𝑡(𝑗] − 1, 𝑎𝑞) and 𝐿−1{1/(𝑠] − 𝑎)2} =

∑𝑞_𝑗=1∑𝑞_𝑚=1𝑎𝑗+𝑚−2_{𝑡𝐸

𝑡((𝑗 + 𝑚)] − 2, 𝑎𝑞) − [(𝑗 + 𝑚)] − 2]𝐸𝑡((𝑗 +

𝑚)] − 1, 𝑎𝑞_{)} with ] = 1, 1/2, 1/3, . . . and 𝑞 = 1, 2, . . . , 1/].}

Here we name the two-parameter function 𝐸_𝑡(𝛼, 𝑎) as the

fractional exponential function with variable𝑡, order 𝛼, and

constant 𝑎. It is defined as the fractional derivative of an

ordinary exponential function𝐸_𝑡(𝛼, 𝑎) ≡ (𝑑−𝛼/𝑑𝑡−𝛼)𝑒𝑎𝑡 =

𝑡𝛼_∑∞

𝑛=0((𝑎𝑡)𝑛/Γ(𝛼 + 𝑛 + 1)) with the derivative formula

(𝑑𝜇_/𝑑𝑡𝜇_)𝐸

𝑡(𝛼, 𝑎) = 𝐸𝑡(𝛼 − 𝜇, 𝑎). The functional form of

this fractional exponential function 𝐸_𝑡(𝛼, 𝑎) will not be

changed after being performed on derivative operator with fractional or integral order. A linear combination of the fractional exponential functions is a potential solution of

the fractional differential equation. By applying these inverse

Laplace transforms to the partial-fractional forms of ̃𝑈_𝑝(𝑠),

we obtain the analytical solution of the fractional differential equation as 𝑈_𝑝(𝑡) = 1 2𝑒𝑖𝜋/4_√𝛽/𝑓3_{− 𝛿} × [𝑌₁2𝐸𝑡(1/2, 𝑌12) − 𝑌22𝐸𝑡(1/2, 𝑌22) + 𝑌₁𝑒𝑌2 1𝑡− 𝑌 2𝑒𝑌 2 2𝑡] , (8a) 𝑈_𝑝(𝑡) = −2𝛽3/2𝑒𝑖3𝜋/4 𝑓9/2 𝑡𝐸𝑡(1₂,_𝑓𝑖𝛽3) −𝛽1/2𝑒𝑖𝜋/4 𝑓3/2 𝐸𝑡( 1 2, 𝑖𝛽 𝑓3) + (1 + 2𝑖𝑡𝛽 𝑓3 ) 𝑒𝑖𝛽𝑡/𝑓 3 − 2𝛽1/2𝑒𝑖𝜋/4 𝑓3/2_√𝜋𝑡1/2 (8b)

for𝛽/𝑓3 ̸= 𝛿 and 𝛽/𝑓3= 𝛿, respectively. Here we have applied

the recursion relation of the fractional exponential function

𝐸_𝑡(], 𝑎) = 𝑎𝐸_𝑡(] + 1, 𝑎) + 𝑡]_{/Γ(] + 1) to these analytical}

expressions. The time evolution of the wave function is thus obtained as 󵄨󵄨󵄨󵄨𝜓(𝑡)⟩ = [ 𝑢_𝑝(𝑡) 𝑒𝑖𝜙0_cos(𝜃0 2) |1⟩ + 𝑢𝑑(𝑡) sin ( 𝜃₀ 2) |0⟩] ⊗ 󵄨󵄨󵄨󵄨0⃗𝑘⟩ + ∑ ⃗𝑘 𝐶_⃗𝑘_{(𝑡) |0⟩ ⊗ 󵄨󵄨󵄨󵄨1}_⃗𝑘⟩ (9)

with 𝑢_𝑝(𝑡) = 𝑒𝑖𝛿𝑡𝑈_𝑝(𝑡) expressed through (8a) and (8b),

𝑢_𝑑(𝑡) = 𝑢_𝑑(0) = 1, and ∑_⃗𝑘|𝐶_⃗𝑘(𝑡)|2 _{= 1 − [𝑢}

𝑝(𝑡) cos(𝜃0/2)]2−

[sin(𝜃₀/2)]2_.

By definition, the reduced density matrix of the qubit can be directly obtained from this wave function through tracing over the reservoir degrees of freedom. It gives

̂𝜌(𝑡) = (󵄨󵄨󵄨󵄨󵄨𝑢𝑝(𝑡)󵄨󵄨󵄨󵄨󵄨 2 cos2(𝜃0 2) 1 2𝑢𝑝∗(𝑡) 𝑒−𝑖𝜙0sin(𝜃0) 1 2𝑢𝑝(𝑡) 𝑒𝑖𝜙0sin(𝜃0) 1 − 󵄨󵄨󵄨󵄨󵄨𝑢𝑝(𝑡)󵄨󵄨󵄨󵄨󵄨 2 cos2(𝜃0 2) ) ≡ (𝜌11(𝑡) 𝜌10(𝑡) 𝜌₀₁(𝑡) 𝜌₀₀(𝑡)) (10) with the initial one

̂𝜌(0) = ( cos2(𝜃0/2) (1/2) 𝑒−𝑖𝜙0sin(𝜃0)

(1/2) 𝑒𝑖𝜙0_sin(𝜃

0) sin2(𝜃0/2) ) . (11)

The elements in this matrix are associated with the polariza-tion and probabilities of the qubit. In the following, we will

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study the quantum measurements of the energy relaxation, decoherence, and von Neumann entropy based on these elements.

2.1. Energy-Relaxation Rate. As the qubit spontaneously emits a photon, the probability amplitude of the qubit under-goes a quantum damping and becomes small. This ampli-tude damping associated with the energy dissipation of the qubit leads to the flow of energy from the qubit into the environment through the emitted photon. How the energy flows from the initially prepared qubit can be observed from

the excited-state probability of the qubit𝑃(𝑡) through 𝑃(𝑡) =

|𝑢_𝑝(𝑡)|2_cos2_(𝜃

0/2) = |𝑈𝑝(𝑡)|2(𝜃0 = 0) shown inFigure 2(a).

The probability amplitude of a usual qubit[𝛿/𝛽 = (𝜔₁₀ −

𝜔_𝑐)/𝛽 > 0] decays to zero quickly and the qubit has a

short-life time. For the QPAB[𝛿/𝛽 < 0], the probability dynamics

exhibit decay and oscillatory behavior before reaching a steady nonzero value, indicating that the energy flow from the QPAB to the PhC reservoir is inhibited and a photon-atom bound state is formed. The rate of this energy flow can

be measured through the energy-relaxation rateΓrelax.(𝑡) =

− ̇𝜌₁₁(𝑡)/𝜌₁₁(𝑡) = −[( ̇𝑈_𝑝(𝑡)/𝑈_𝑝(𝑡))+( ̇𝑈_𝑝∗(𝑡)/𝑈_𝑝∗(𝑡))] with 𝑈_𝑝∗(𝑡)

being the complex conjugate of 𝑈_𝑝(𝑡) in (8a) and (8b). In

Figure 2(b), we find that the energy-relaxation rates of QPABs and usual qubits decay to zero very quickly and slowly, respectively.

Combining the dynamical behavior of the probability amplitude and energy-relaxation rate of a qubit, we find that the QPAB releases and preserves some of its energy at the very beginning and at the end, respectively. The quick stop of releasing energy results from the formation of the bound state between the qubit and the localized photon. The remaining energy continuously excites the atom leading to the nondecaying dynamics of the probability amplitude

inFigure 2(a) and quick decaying of the energy-relaxation

rate in Figure 2(b). Note that, as the emitted frequency

of the QPAB is detuned deep inside the forbidden band region (𝛿/𝛽 = −5 and −10), the energy-relaxation rates exhibit negative values in the initial period of relaxation. This negative energy-relaxation rate reveals that the qubit can regain its energy in the PhC reservoir. That is, we observe the released energy from the QPAB flowing back to the QPAB during a certain time interval. This character of the long-time memory effect also manifests the controllable relaxation of a qubit inside the structured reservoir. On the other hand, for a usual qubit, the energy-relaxation rate decays slowly and

remains a nonzero value for a long time inFigure 2(b). A

usual qubit releases its energy continuously and the emitted photon from the qubit carries the energy and information away.

When we use a Bloch vector in the Bloch sphere to express the qubit state, the amplitude damping will perform a transformation on the components of the Bloch vector with the values related to the probability of losing the emitted photon [39]. For the Markovian system of an usual qubit, the large probability of losing the emitted photon will convert the Bloch vector toward the ground-state point on the Bloch sphere. The small probability of losing the emitted photon

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 11 Prob abi lit y P(t) 𝛿/𝛽 = −10 𝛿/𝛽 = −5 𝛿/𝛽 = −1𝛿/𝛽 = 2 Time (𝛽t) (a) 0 1 2 3 4 5 6 7 8 9 10 11 𝛿/𝛽 = −10 𝛿/𝛽 = −5 𝛿/𝛽 = −1 𝛿/𝛽 = 2 3.0 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 Relaxa tio n ra te Γrelax. Time (𝛽t) (t ) (b)

Figure 2: Dynamics of (a) the qubit’s excited-state probability𝑃(𝑡) and (b) relaxation rateΓ_relax.(𝑡) of the qubit with different detuning frequencies𝛿/𝛽 = (𝜔₁₀− 𝜔_𝑐)/𝛽 from the band edge frequency 𝜔_𝑐of the PhC reservoir.

in the non-Markovian system of a QPAB will convert the Bloch vector pointing towards the nonzero surface area near the ground-state point on the Bloch sphere. The contracting effect of the amplitude damping on the Bloch vector is strongly suppressed in the non-Markovian system.

2.2. Decoherence and von Neumann Entropy. As a qubit cou-ples with the reservoir, the polarization of the qubit is ran-domized due to the reservoir. The coherence phase of a qubit determined by the qubit polarization is also randomized and leads to the quantum decoherence. As the qubit under-goes the polarization randomization through this quantum

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0 1 2 3 4 5 6 7 8 9 10 11 𝛿/𝛽 = −10 𝛿/𝛽 = −5 𝛿/𝛽 = −1𝛿/𝛽 = 2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8 Po la riza tio n Pz (t ) Time (𝛽t) (a) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 1 2 3 4 5 6 7 8 9 10 11 𝛿/𝛽 = −10 𝛿/𝛽 = −5 𝛿/𝛽 = −1𝛿/𝛽 = 2 Time (𝛽t) S(t ) vo n N euma nn en tr op y (b)

Figure 3: (a) Polarization 𝑃𝑧(𝑡) = 𝜌₁₀(𝑡) + 𝜌₀₁(𝑡) and (b) von Neumann entropy 𝑆(𝑡) = − Tr[̂𝜌(𝑡) log ̂𝜌(𝑡)] = −𝜆₊log𝜆₊ − 𝜆−log𝜆−of the non-Markovian (𝛿/𝛽 < 0) and Markovian (𝛿/𝛽 = 2)

systems.

decoherence, the off-diagonal elements of the reduced den-sity matrix will decay with time. The completely loss of the quantum coherence of one qubit will lead to the occurrence of entanglement sudden death (ESD) in the multiple-qubit system. That is, the time period in which the entanglement could be usefully exploited in the multiqubit system is limited. This limitation is related to the dynamics of the qubit

polarization shown inFigure 3(a)by defining the polarization

of a qubit as𝑃𝑧(𝑡) = 𝜌₁₀(𝑡) + 𝜌₀₁(𝑡) = Re[𝑒𝑖𝛿𝑡𝑈_𝑝(𝑡)] (if 𝜙₀= 0,

𝜃₀= 𝜋/2).

For a usual qubit with 𝛿/𝛽 = 2, the polarization of

the qubit exhibits a fast damping behavior implying that the qubit loses all its polarization in the initial period of time

with 𝛽𝑡 < 4. This polarization damping results from the

qubit scattered inelastically by incoherent background fields from the reservoir. The inelastic scattering randomizes the orientation of the qubit polarization and leads to the decay of the off-diagonal elements of the density matrix. The fast damping of the qubit polarization reveals that the probability of the qubit being scattered by incoherent photons of the reservoir is large in the Markovian system. The photons from

the allowed band with a continuous DOS (seeFigure 1(c))

provide the free-space-like environment to destroy the qubit. On the other hand, the polarization of a QPAB exhibits a nondecaying oscillation. The QPAB loses some of its polarization at the very beginning and then preserves the remaining polarization shown by the steady oscillation of its polarization. This steady polarization ensures that the probability of the QPAB scattered by incoherent photons of the reservoir is greatly lowered in the non-Markovian system. The PBG shown in the photon DOS of the PhC reservoir (see Figure 1(c)) is expected to lower this scattering probability and to lead to the preservation of phase information of the qubit through the steady polarization. The QPAB with a larger amplitude of polarization has less loss of coherence information to the reservoir.

Entropy, a key concept of quantum information theory, measures how much information exists in a state of a physical system. The amount of information is changed with respect to the correlation of a state and the reservoir. The correlation between the environment and a state will transform the initially purified state into a finally mixed state where the state and its environment are entangled. We can thus estimate the correlation between the qubit and reservoir through the entropy of the quantum system, that is, von Neumann entropy. The von Neumann entropy for a quantum state with

̂𝜌(𝑡) is defined as 𝑆(𝑡) = − Tr[̂𝜌(𝑡) log ̂𝜌(𝑡)] = − ∑𝑖𝜆𝑖log𝜆𝑖

with𝜆_𝑖being the eigenvalues of the matrix̂𝜌(𝑡). The von

Neu-mann entropy predicts the upper bound of the information that we will gain after we measure a quantum state. This gain of information after measurement corresponds to the amount of information of a state. The value of this entropy stands for the degree of entanglement between the qubit and the

reservoir. It has a maximal value𝑆_max = log(𝑑), 𝑑 being the

dimension of the matrix ̂𝜌(𝑡), for the maximally mixed state

and is zero for the pure state.

With the eigenvalues 𝜆_± = (1/2){1 ±

√1 − 4cos4_(𝜃

0/2)[|𝑢𝑝(𝑡)|2− |𝑢𝑝(𝑡)|4]} of the matrix in

(10), we show the von Neumann entropy in Figure 3(b)

for the initially excited qubit (𝜃₀ = 0). The entropy has its

minimal value zero at𝑡 = 0 and reaches its maximal value

log2 = 0.693 instantly. After a time on the order of the decay

rate, the entropy becomes steady and its value is nonzero for a QPAB in the non-Markovian system (𝛿/𝛽 < 0) and zero for a usual qubit in the Markovian system (𝛿/𝛽 = 2). These results show that the initially purified qubit becomes maximally mixed with the reservoir instantly. As the qubit equilibrates with the PhC reservoir, the system becomes less mixed. A usual qubit returns to its initially purified state as it quickly disentangles from the reservoir. However, for a QPAB, the long-time correlation with the PhC reservoir

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leads to the preservation of entanglement between the QPAB and its reservoir.

Combining the results of Figures3(a)and3(b), we infer

that the coherence information of a qubit is partially lost and preserved at the very beginning and on the long run, respec-tively. In the QPAB of a non-Markovian case, the interaction with the PhC reservoir does not result in the total loss of the coherence but in the preservation of partial coherence instead. The structured reservoir of a PhC reveals its unique property in controlling the coherence of a qubit. A usual qubit in PhCs exhibits a short-time coherence. But a QPAB shows a property with strongly suppressed decoherence through the steady oscillation of polarization and long-time preservation of the mixed state. The greatly lowered probability of a QPAB scattered by incoherent photons of the reservoir preserves the coherence and information of a QPAB.

When we perform a multistage computation on the avail-able information through the Markovian chain, the data pro-cessing inequality of von Neumann entropy, a basic inequality of information theory, states that the information about the output of the computation will decrease with time if the qubit storing the information is correlated with the Markovian environment [39]. This statement agrees with the result we obtained here. In the Markovian system, the newly produced mixed state by the correlation between the qubit and reservoir recovers its initially pure state and loses the information stored in the qubit. The pieces of information yielded by the multistage computations will thus be independent of each other. On the other hand, in the non-Markovian system, the newly produced mixed state will preserve the information about its output of the previous stage. The memory effect of the PhC reservoir on the qubit’s previous state leads to the preservation of the stored information in the mixed-state qubit. Further computation operations on this mixed state can be used to increase the amount of mutual information between the outputs of the operations and the previous-stage information about the qubit state.

2.3. Reservoirs with Forbidden Bands. Now we discuss the systems suitable for realizing the behavior of strongly sup-pressed relaxation and decoherence. Besides the anisotropic PhC discussed in this section, reservoirs with forbidden bands in their dispersion relation or threshold energy in photonic DOS can be applied to exhibit these properties. Here we briefly introduce the other three systems with this kind of reservoirs including the omnidirectional waveguides (ODWGs), tunable artificial crystals, and frequency disper-sive media.

An ODWG, a waveguide structure based on the com-plete reflection of one-dimensional (1D) PhCs, consists of a

dielectric (or air) layer of lower index𝑛_𝑎 sandwiched by a

1D PhC with refractive indices𝑛₂and𝑛₁ > 𝑛_𝑎and variable

thicknesses ℎ₂, ℎ₁, and ℎ_𝑎 [17, 40, 41]. This structure can

exhibit complete reflection of radiation in a given frequency range for all incident angles and polarizations which have been proven both theoretically and experimentally [42–44]. Within the frequency range of omnidirectional reflection, where the dispersion curve lies inside the photonic band gap

of the PhC, the electromagnetic (EM) wave is allowed to propagate in this waveguide structure with high transmission. On the other hand, if the EM wave has its frequency lying outside this frequency range of omnidirectional reflection, it will find no propagation mode so that it cannot transmit freely through this waveguide. With an embedded qubit with frequency lying inside this forbidden range of the ODWG, the system will exhibit the properties of strongly suppressed relaxation and decoherence.

The second reservoir we introduce, a tunable artificial crystal, is constructed by periodic arrays of circuit elements of Josephson junction [45]. As the artificial crystals are built, the interaction of the EM wave with these crystals can produce a

forbidden region in the band structure and DOS [19,46]. The

frequency range of this forbidden band is tunable through the array parameters and applying external flux. When a qubit is placed in the middle of such an array of artificial crystals with its frequency lying inside the forbidden band where no traveling modes are available, this system will show the properties of strongly suppressed relaxation and decoherence.

In contrast to the PhCs, frequency dispersive media such as semiconductors and dielectrics have energy gaps caused by photon couplings to a medium excitation, for example, an exciton, optical photon, and so forth [47]. The spectrum of these media consists of two branches of allowed states separated by a gap in which propagating polariton modes are completely forbidden. Studying the quantum electrody-namics of a two-level atom placed in this medium in [20, 23], Rupasov and Singh found that if the atomic resonance frequency lies within the gap, the spectrum of the system con-tains a polariton-atom bound state with an eigenfrequency lying within the gap. The radiation and medium polarization of the bound state are localized in the vicinity of the atom. They predicted that the SE of this polariton-atom system is significantly suppressed due to the presence of the bound state [20]. These results about the polariton-atom bound state in the dispersive media are similar to the photon-atom bound state occurring in the PhCs. The suppression of relaxation and decoherence rates of a QPAB is also valid for a qubit with its frequency lying inside the polariton gap in a dispersive medium.

3. Conclusion

A structured reservoir of an anisotropic PhC is introduced for suppressing the relaxation and decoherence of an embedded qubit. The dynamics of the qubit are obtained by applying fractional calculus to solve the fractional Langevin equation of the system analytically. There is a long-time memory effect being characterized by the occurrence of a negative value of the energy-relaxation rate on the dynamics of the relaxation and decoherence of a qubit. The energy-relaxation rates of QPABs and usual qubits decay to zero very quickly and slowly, respectively. The polarization and entropy of a QPAB are oscillating and decaying to a finite value, respectively. The fast decaying of the energy-relaxation rate reveals the relaxation property of a qubit being suppressed while the polarization

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oscillation and nonzero steady values of entropy manifest the decoherence being suppressed. When the dynamical behav-ior of the qubit formed by a photon-atom bound state is compared with that formed without forming a bound state, the suppressing effect on the relaxation and decoherence of a QPAB is enhanced. Physical reasons for the suppressing effect of the structured reservoir are discussed. Other systems suit-able for the realization of this kind of structured reservoirs are briefly illustrated, such as the omnidirectional waveguides, tunable artificial crystals, and frequency dispersive media.

Acknowledgments

The authors acknowledge the financial support from the National Science Council (NSC) of the Republic of China under Contract nos. NSC102-2112-M-034-001-MY3 and NSC102-2112-M-009-016-MY3. S.-C. Cheng thanks the sup-port of the National Center for Theoretical Sciences of Taiwan during visiting the center.

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Dynamics of the Energy Relaxation and Decoherence of a Photon-Atom Bound State in an Anisotropic Photonic Crystal

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