Sensitive chemical compass and quantum criticality at finite temperature

Physics Letters A 378 (2014) 1481–1486

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Physics Letters A www.elsevier.com/locate/pla

Sensitive chemical compass and quantum criticality at finite temperature Da-Wei Luo, Jing-Bo Xu ∗ Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 26 October 2013 Received in revised form 17 March 2014 Accepted 8 April 2014 Available online 15 April 2014 Communicated by P.R. Holland Keywords: Quantum phase transitions Spin chain models

a b s t r a c t We investigate the influence of a flip operation of the central spin on the quantum criticality of a radical pair system by employing the spin echo and its product yield. It is found that with echo control on the central spin, the critical behavior can be described by the product yield at very high temperatures. Moreover, we also study the short and long time behavior of the spin echo, and show that the decay factor of the short time evolution scales linearly. The long time evolution shows different statistics for varying chain lengths, temperature and external parameters of the Hamiltonian. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Quantum phase transition (QPT) has been a topic that has attracted much research interest over the years [1]. QPT causes a qualitative change in the structure of the ground state properties of a quantum many-body system as some external parameters of the Hamiltonian are varied to a certain value, which is defined to be the critical point of the system. The behavior of many-body systems near the critical point is strongly influenced by the existence of a QPT, and it is associated with the divergence of correlation length of two-point correlation functions and the vanishing of the gap in the exciton spectrum. This change of phase is caused solely by quantum fluctuations, which exist at absolute zero temperature. In practice, we must work at low but finite temperatures, as close to the absolute zero as possible, and the finite-temperature QPT is important theoretically and experimentally [2–4]. The spin chain model has been a popular candidate for the study of QPTs due to the fact that many types of spin chains can be implemented experimentally and is soluble analytically [1,2,5–10]. Recently, the chain with spin chain environment have attracted much attention in the study of quantum phase transition [8,11,12] and quantum information processing [13]. The Loschmidt echo was introduced in NMR experiments to describe the hypersensitivity of the time evolution to the environmental effects [8]. It has been pointed out that nuclear environments surrounding electron spins in the radical pairs is crucial to the magnetic sensitivity of the chemical reaction, and the chemical product yield in cryptochrome

*

Corresponding author. E-mail address: [email protected] (J.-B. Xu).

http://dx.doi.org/10.1016/j.physleta.2014.04.010 0375-9601/© 2014 Elsevier B.V. All rights reserved.

is believed to affect the visual function of animals [14], which is given by the Laplace time-integral transform of the Loschmidt echo [9]. Moreover, the Loschmidt echo is shown to obey different probability distributions for different system setups [3,15], and the decaying behavior for short time evolution scales linearly with system size [16]. In this Letter we study how the flip operation of the central spin affects the quantum phase transition behavior of radical pair systems by means of the spin echoes and its product yield. It is shown that at very low temperatures, the product yield of the Loschmidt echo is able to pinpoint the critical points of the quantum phase transitions, but as the temperature gets higher, the Loschmidt echo fails to capture the critical points of quantum phase transitions due to thermal fluctuations. However, with echo control on the central spin, the critical behavior can be observed even at very high temperatures. Moreover, we also study the statistical distribution of the spin echo as well as the short and long time behavior of the spin echoes, and find that the product yield has very little dependence on the temperature, meaning the proposed echo control is very effective at removing the thermal fluctuations. It is demonstrated that the decay factor of the short time evolution scales linearly, and the echoes obey different short-time evolution behaviors. The long time evolution shows different statistics for varying chain lengths, temperature and external parameters of the Hamiltonian. 2. Loschmidt echo and the product yield at finite temperature In a radical pair, there are two electrons, each of which is uniformly coupled to its own environment [9]. In this Letter, we made a generalization of the spin environment to allow anisotropic

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coupling of the environmental spins. The environment under an oblique external magnetic field can be described as

HE = J

N   1+γ

2

l =1

σlx σlx+1 +

1−γ 2

y σl y σl+ + λ cos θ σlz 1



where γ is the anisotropic factor, λ is the external magnetic field strength, θ the oblique angle and σ is the Pauli operator. The central electron couples to this spin bath according to

H I = Ω sin θ τ x + Ω cos θ τ z − J g τ z

N 

σlz

l =1

where τ refers to the Pauli operator of the electron, Ω is the electronic Zeeman energy splitting and J g is the coupling strength. Similar to Ref. [9], we can obtain an effective Hamiltonian by making use of the Born–Oppenheimer approximation by first treating the environment operators in H I as c-numbers as H = |++| H + +  |−−| H − , where H ± = H E ∓ J g cos θ lN=1 σlz . This Hamiltonian, which only introduces dephasing into the system, can be diagonalized using the Jordan–Wigner transform followed by a Fourier transform [7,17]. Under the basis spanned by the electron state {|+, |−}, the diagonal elements of the electron’s density matrix does not evolve with time, while the off-diagonal element decays according to ρ+− (t ) = ρ+− (0) Tr[U (+) ρ E U (−)† ], where ρ E is (±) the initial state of the environment and U t = exp[−i H ± t ]. In the study of this decoherence effect, the Loschmidt echo is a very useful tool. The Loschmidt echo was introduced in NMR experiments to describe the hypersensitivity of the time evolution to the environmental effects and is defined as the squared modulus of the decay factor,

 

L (t ) = Tr U t

(+)



2 ρ E U t(−)†  .

(1)

At finite temperature T , the initial state of the environment is chosen to be the Gibbs equilibrium state, ρ E = exp[−β H − ]/ Z , where β = 1/kT is the inverse temperature and Z = Tr(exp[−β H − ]) is the partition function. At zero temperature, this definition reduces

Fig. 1. (Color online.) Product yield of the Loschmidt echo at very low temperature β = 200 (Panel a) and high temperature β = 0.2 (Panel b) as a function of the angle θ and the magnetic field λ. As can be seen, along the critical points, the Loschmidt echo suffers a sudden drop at relatively low temperature, and the critical point is dependent on the magnetic field’s strength as well as the angle. At higher temperatures, the sudden drop cannot be observed due to thermal fluctuations.

to L (t ) = |G |U − U + |G |2 , and has been studied extensively [8,15, 16]. The Loschmidt echo, with the Hamiltonian diagonalized, can be readily calculated as [3]

Recently, it has been pointed out that the Loschmidt echo is closely related to the sensitive chemical compass [9] by means of the product yield Φ(t ) given by

2     L (t ) =  lk (t )/ Z k  ,

Φ(t ) =

†

(2)

k

where



(−) 

;  (−)   (+)   (−)    lk (t ) = 1 + cosh βΛk cos t Λk cos t Λk  (+)  (+)   (−)  (−)  , sin t Λk sin t Λk + cos θk − θk   (−)   (+)    (+) (−)   lk (t ) = cos θk − θk cos t Λk sin t Λk  (+)   (−)   (−)  sinh βΛk ; − cos t Λk sin t Λk Z k = 1 + cosh βΛk



k(±) = cos(2π k/ N ) − (λ ∓ g ) cos θ;

Λk(±) = 2 k(±)2 + γ 2 sin2 (2π k/ N );

 (±) −1 γ sin(2π k/ N ) , θk = tan (±)

k

where  and  refer to the real and imaginary part, respectively. The detailed steps needed for deriving the equations above are given in Appendix A.

1 2

+

s

t

2

L (t )e −st dt ,

(3)

0

where s is the recombination rate. The ultimate product yield Φ(∞) in cryptochrome is believed to affect the visual function of animals [14]. In order to study the temperature effects, we plot the product yield at low and high temperatures as a function of the magnetic field strength and angle in Fig. 1 with N = 80, g = 0.05, s = 0.1 and γ = 0.3. It can be easily seen that while the product yield of the Loschmidt echo suffers a sudden drop along the critical points of QPT and is able to indicate the critical points at very low temperature, this ability is lost when the environment’s temperature get higher due to high thermal fluctuations. Now we study how the oblique angle affect the long-time and short-time evolution of the Loschmidt echo. In order to investigate the long-time evolution, we regard the values of the Loschmidt echo over a long observation time interval [0, T ] as a random variable [3,15] and investigate its statistical properties by calculating its probability distribution. We can obtain the probability distribution by discretely sampling the spin echo over a long period of time [0, 1010 ] and plot the statistical distribution of this sample. It is found that at low temperatures, the probability distribution of the echo is approximately exponential for long chains regardless of whether the bath is critical or not (Fig. 2), while for short

D.-W. Luo, J.-B. Xu / Physics Letters A 378 (2014) 1481–1486

Fig. 2. (Color online.) Distributions of the Loschmidt echo of long chains at low temperatures (β = 200) away the critical point (Panel a) with λ = 0.5, γ = 0.3, N = 80, g = 0.2 and θ = π /6 along with the distribution at the critical point (Panel b) with λ = λc and g = 0.05. For long chains, the distribution is approximately exponential whether the environment is critical or not.

chains, the distribution is double peaked for critical environment and normal otherwise (Fig. 3). This result is in good agreement with Refs. [3,15], and with the difference that the critical point is now determined by the magnetic field’s strength and oblique angle according to λc = 1/ cos(θ) − g. At high temperatures, this behavior is not observed. The short-time evolution is also sensitive to this oblique angle. It has been pointed out in Ref. [16] that the decaying behavior of the Loschmidt echo for short time scales is exponential: L (t ) ≈ exp(−Γ t 2 ) for small t, and Γ scales linearly with system size. In Fig. 4, the decaying factor Γ is plotted as a function of the environment size N for different parameters. As is shown, the decaying factor’s scales linearly with the system size, the slope of the decaying factor against the system size N is dependent on the oblique angle and the environmental temperature. This result is in good agreement with Ref. [16]. 3. Controlled spin echo and the product yield at finite temperature As demonstrated in the previous section, the Loschmidt echo generally fails to capture the critical behavior of the system at higher temperatures due to thermal fluctuations. Recently, it has been shown that the echo control over a spin probe can remove the thermal fluctuation effects and hence reveals the quantum fluctuation effects [18]. Adopting this control scheme, we flip the central electron spin at some t /2 according to (|+ ↔ |−), and the measurement is done at time t. The controlled spin echo is then given by

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Fig. 3. (Color online.) Distributions of the Loschmidt echo of short chains at low temperatures (β = 200) away the critical point with λ = 1.5, γ = 0.3, N = 20, g = 0.05 and θ = π /6 (Panel a) is Gaussian and double-peaked at the critical point (Panel b) with λ = λc .

Fig. 4. (Color online.) Scaling behavior of the decay factor of the Loschmidt echo at the critical point for short time for different system setups. The decaying factor is shown to scale linearly with the environment size and the slope is determined by the external parameters of the Hamiltonian along with the oblique angle of the external magnetic field. The parameters are chosen to be β = 200 and θ = 0 (blue line marked with +); β = 200 and θ = π /4 (red line marked with o) and β = 0.2, θ = π /3 (green line marked by V).

 

(+)† 2

SE(t ) = Tr U t /2 U t /2 ρ E U t /2 U t /2 (−)

(+)

(−)†

,

(4)

where we denote the spin echo after the flip control as SE(t ). With the diagonalized Hamiltonian, we can now obtain the explicit form of the spin echo analytically. Replacing the Loschmidt echo in Eq. (3), we plot the product gain of the controlled spin echo in Fig. 5 as a function of the magnetic field strength and oblique angle at low and high temperature. It can be seen that with the probe electron flipped at t /2, we can effectively remove the ther-

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Fig. 7. (Color online.) Contour plot of the product gain of the controlled spin echo as a function of control error E and the magnetic field λ at high temperature β = 0.2. The vertical dashed line signifies the timing error E = ±3% and the horizontal dashed line corresponds to the critical point λc . It can be seen that for small errors in the control timing, the central spin is still well protected enough to be able to indicate the quantum phase transition of the spin chain environment.

Fig. 5. (Color online.) Product yield of the controlled spin echo at very low temperature (Panel a) and high temperature (Panel b) as a function of the angle θ and the magnetic field λ. As can be seen, along the critical points, the spin echo suffers a sudden drop at both low and high temperatures, and the critical point is dependent on the magnetic field’s strength as well as the angle.

Fig. 6. (Color online.) Product gain of the controlled spin echo as a function of inverse temperature β and the magnetic field λ with a fixed angle of θ = π /6. It’s clear that Φ shows very little dependence on temperature for β > 20, meaning that this control is very effect at removing the thermal fluctuations for the system under consideration.

mal fluctuations and observe the sudden drop of the product gain even at high temperatures, and the critical point is shown to be dependent on the magnetic field’s strength as well as the oblique angle. In order to investigate how effective this method is, we plot the product gain as a function of temperature and the magnetic field strength with a fixed angle of θ = π /6 in Fig. 6. We can see from the figure that the product yield of the controlled spin echo actually shows very little dependence on temperature for β > 20, which means that this control is very effect at removing the thermal fluctuations for the system under consideration. Practically, we may not always be able to time the control spin flip operation ideally at t flip = t /2. Introduce a control timing error E so that the spin flip operation happens at t flip = (1 − E )t /2,

where E < 0 means the spin flip operation happens a bit after t /2 and E > 0 means the spin flip operation happens a bit before t /2. We plot the product gain of the controlled spin echo as a function of control error E and the magnetic field λ at high temperature β = 0.2 in Fig. 7, where the vertical dashed line signifies E = ±3% and the horizontal dotted line corresponds to the critical point λc . It can be seen that for small errors around 3% in the control timing, the central spin is still well protected enough to be able to indicate the quantum phase transition of the spin chain environment. We now turn to the long-time and short-time evolution properties of the controlled spin echo. In Figs. 8 and 9 we plot the probability distribution of the controlled spin echo for long and short chains at high temperatures away and at the critical point. It can be seen that while the long chains retain the approximate exponential behavior, the short chains’ spin echo probability is very different from the Loschmidt echo. Away from the critical point, the probability distribution is no longer Gaussian, but at the critical point, the distribution is still double peaked, so we are still able to pinpoint the critical point via the probability distribution of the controlled spin echo, even at high temperatures, which is not clear using the Loschmidt echo. The short time evolution of the controlled spin echo is also different from the Loschmidt echo. For short times, the controlled spin echo scales according to SE(t ) ≈ exp[−Γ t 4 ]. The decaying factor Γ still scales linearly with environment chain length N, and we plot the result in Fig. 10. It can be seen that the scaling behavior is dependent on the external parameters of the Hamiltonian as well as the environmental temperature, and longer chains generally lead to higher decay rates, meaning that under environment modeled by longer chains, the central electron decays a lot faster. 4. Conclusions In this Letter we study the quantum critical behavior of radical pair systems by means of the Loschmidt echo and controlled spin echo and their product yield. It is shown that at very low temperatures, the product yield of the Loschmidt echo is able to pinpoint the critical points of the quantum phase transitions, but as the temperature gets higher, the Loschmidt echo fails to pinpoint the critical points of QPTs due to thermal fluctuations. However, with echo control on the central spin, the thermal fluctuation is removed, and the critical behavior can be observed even at very high temperatures. Moreover, the product yield is shown to have very

D.-W. Luo, J.-B. Xu / Physics Letters A 378 (2014) 1481–1486

Fig. 8. (Color online.) Distributions of the controlled spin echo of long chains at high temperature (β = 0.2) away the critical point with λ = 0.5, γ = 0.3, N = 80, g = 0.2 and θ = π /6 (Panel a) and at the critical point (Panel b) with λ = λc . Both are approximately exponential.

little dependence on the temperature, meaning the proposed echo control is very effective at removing the thermal fluctuations. Experimentally, it may not always be possible to time the spin echo control ideally, and we have studied how an inaccuracy in the echo control will be affect the critical behavior. The control method presented in this Letter is still effective within a reasonable inaccuracy range and the central spin is well protected enough to detect the critical behavior of the spin chain environment. We have also examined the probability distribution of the controlled spin echo for long and short chains, and it is found that the decay factor of the short time evolution scales linearly, although the Loschmidt echo evolves as exp(−Γ t 2 ) and the controlled spin echo follows a different short-time evolution as exp(−Γ t 4 ). The long time evolution shows different statistics at and away from the critical points for varying chain lengths, temperature, and we can use the distribution to detect the critical points of QPTs at any temperature using the distribution of the controlled spin echo, and the distribution of the Loschmidt echo is only effective to determine the critical points at very low temperatures. Our results may provide a way to understand the quantum phase transition at finite temperature.

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Fig. 9. (Color online.) Distributions of the controlled spin echo of short chains at high temperature (β = 0.2) away the critical point with λ = 0.5, γ = 0.3, N = 20, g = 0.1, and θ = π /6 (Panel a) and at the critical point with λ = λc (Panel b), which is double-peaked.

Fig. 10. (Color online.) Scaling behavior of the decay factor of the controlled spin echo at the critical point for short time for different system setups. The decaying factor is shown to scale linearly with the environment size and the slope is determined by the external parameters of the Hamiltonian along with the oblique angle of the external magnetic field. The parameters are chosen to be β = 200 and θ = 0 (blue line marked with +); β = 200 and θ = π /4 (red line marked with o) and β = 0.2, θ = π /3 (green line marked by V).

Acknowledgements This project was supported by the National Natural Science Foundation of China (Grant No. 11274274). Appendix A In this appendix, we briefly show how to obtain the spin echo equation (2) from the Hamiltonian of the radical pair system. Due

to the τx term in H I , the whole Hamiltonian can only be solved using some approximation. Usually the central electron spin evolve faster than the environmental nuclear spins [9]. As a result, we can first regard the environmental nuclear spins σ as complex numN z z bers and formally diagonalize the H I . Denoting l=1 σl = s , we can rewrite H I in the space spanned by {|↑, |↓} of the central spin as

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H I = Ω sin θ τ x + Ω cos θ τ z − J g τ z s z

Ω sin θ Ω cos θ − J gs z −Ω cos θ + J gs z Ω sin θ 

 Ω sin θ , = − Ω sin θ



=

(5)

where  = Ω cos θ − J gs z . We can formally  diagonalize H I as H I =

E + |++| + E − |−−| where E ± = ± 2 + Ω 2 sin2 θ . For Ω J g, we can expand the eigen-energy E ± as

 E ± = ± Ω − J g cos θ

N 

 z l

σ

(6)

.

l =1

Therefore, the total Hamiltonian H = H E + H I can be written as

H = H E + E + |++| + E − |−−|

= |++|( H E + E + ) + |−−|( H E + E − ) = |++| H + + |−−| H − ,

(7)

N

where we denote H ± = H E + E ± = H E ∓ J g cos θ l=1 σlz , dropping the constant Ω in E ± since it does not contribute to the dynamics of the whole system. The Born–Oppenheimer approximation is applicable for this class of systems [9] because the slowly varying environmental nuclear spins do not generally induce a coherent transition in the faster varying central electronic spin, which provides an effective potential for H ± . Now H ± are anisotropic X Y spin chain models under different magnetic fields and can be readily diagonalized using the Jordan–Wigner transform followed by a Fourier transform and finally a Bogoliubov rotation. To calculate the corresponding Loschmidt spin echo at finite temperature, we consider the anisotropic X Y spin chain Hamiltonian under a magnetic field h along the z-axis in√the momentum space after the Fourier transform al = k ck e ilk / N where al is the fermion annihilation operator of the Jordan–Wigner transform and ck is the fermion annihilation operator  in the momentum space. The Hilbert space factorizes as H = k Hk , and Hk ⊗ H−k = span{|00k,−k , |11k,−k , |01k,−k , |10k,−k }, where the first (last) two vectors span the even (odd) parity sector. For the k-th mode of momentum space, the Hamiltonian acts trivially in the odd parity sector spanned by {|01k , |10k }, and in the even parity sector spanned by {|00k , |11k } the Hamiltonian reads

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