quantum synchronization as a local signature of super and ... · gian luca giorgi (ifisc) bruno...

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GIAN LUCA GIORGI (IFISC)

BRUNO BELLOMO (UTINAM- BESANÇON) MASSIMO PALMA (UNIPALERMO) ROBERTA ZAMBRINI (IFISC)

Quantum synchronization as a local signature of super and sub-radiance

B Bellomo, GL Giorgi, G M Palma, R Zambrini, arXiv:1612.07134 and PRA (in press)

https://arxiv.org/abs/1612.07134

*Introduction

● The simplest case: two atoms in the vacuum

● Liouvillian formalism

● Quantum synchronization induced by a common environment

● Super- and subradiance

● Results

*Two qubits in a (partially) common environment

"!1 !2

HI

=X

i

�x

i

X

k

gik

(ak

+ a†k

)

HS =!1

2�z1 +

!2

2�z2

⇢ = �i [HS +HLS , ⇢] +X

ij

�ij

✓��i ⇢�

+j � 1

2

��+j �

�i , ⇢

◆Lindblad master equation

*Synchronization

Pikovsky, Rosenblum, Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (2001)

*Dissipation-induced quantum synchronization

● Pair of harmonic oscillators (common-bath sync)

● Network of harmonic oscillators (common-bath sync)

● Pair of spins (common-bath sync, detrimental role of dephasing)

● Pair of spins (separate baths, unbalanced dissipation rates, sync as a probe)

● A review on quantum synchronization measures

Previous works

GL Giorgi, F Galve, G Manzano, P Colet, R Zambrini, PRA 85 052101 (2012)

G Manzano, F Galve, GL Giorgi, E Hernández-García, R Zambrini, Sci. Rep. 3, 1439 (2013)

GL Giorgi, F Plastina, G. Francica, R Zambrini, PRA 88 042115 (2013)

GL Giorgi, F Galve, R Zambrini, PRA 94 052121 (2016)

F Galve, GL Giorgi, R Zambrini, arXiv:1610.05060


● Synchronization can be identified with the existence of a common phase in the oscillatory dynamics of local observables (“classical measure”)

● In such systems, synchronization is induced by the presence of a thermal environment

● In the absence of driving (relaxation towards equilibrium), synchronization is observed during a transient (pre-asymptotic limit)

● Fundamental ingredient to observe synchronization: time-scale separation in the spectrum (one eigenvalue much “slower” than anyone else)

CA1(t),A2(t)(�t) =

R t+�tt [A1(t)� A1][A2(t)� A2]dt0qQ2

i=1

R t+�tt [Ai(t)� Ai]2dt0


Local observable to monitor:

The corresponding matrix elements belong to one of the Liouvillian blocks

Eigenvalues

Synchronization measure

measures the difference between the two smaller real parts and can be taken as a measure of the ability of the system to develop monochromatic oscillations

h�x

i

i

�1

2[3�0 ± V ⇤]� i!0

�1

2[�0 ± V ]� i!0

V =

s

(�12 + i 2 s12)2 +

✓�1 � �2

2+ i�

◆2

Re[V ]

*An example

⟨σ1x⟩t⟨σ2x⟩t

0 2 4 6 8-1

-0.5

0

1

0.5

γ0t

2 4 6 8γ0t

-1-0.5

10.5

C⟨σ1x ⟩t ,⟨σ2x ⟩t (Δt)

*Super- and sub radiance

… [Dicke] superradiance is a phenomenon that occurs when a group of N emitters, such as excited atoms, interact with a common light field. If the wavelength of the light is much greater than the separation of the emitters, then the emitters interact with the light in a collective and coherent fashion

Total radiation rate

R H Dicke, Phys. Rev. 93, 99 (1954) M Gross and S Haroche, Phys. Rep. 93, 301 (1982)

I(t) =X

i,j

�ijh�+i �

�j it

Introduction: superradiance

Superradiance is the collective decay of an initially excited system of N atoms due to spontaneous emission of photons. For large N, the emission vs time has the form of a very

sudden peak (‘flash’) that occurs on very short time-scale ∝ 1/N and has a maximum ∝ N2. The phenomenon of superradiance was predicted by Dicke in 1954

Fig. 1. Comparison between ordinary fluorescence and superradiance experiments.

(a) Ordinary spontaneous emission is essentially isotropic with an exponentially decaying intensity (tsp).

(b) Superradiance is anisotropic with an emission occurring in a short burst of duration tsp /N.

*

Here are the “evolution” of the symmetric and antisymmetric states dressed both by the interaction (Lamb shift) and by the dissipation rates. A close system of equaitions of motion can be written for these states

Eigenvalues of the Liouvillian block governing the radiation rate

|SLi, |ALi

0; �2�0; ��0 ± iIm[V ]; ��0 ± Re[V ]

Super- and sub radiance

*

|E⟩ |AR⟩ |A⟩ |Aδ⟩

IA |SR⟩ |S⟩ |Sδ⟩

0 2 4 6 810-4

0.001

0.010

0.100

1

γ0t

I(t)/2

γ 0

slowest rate

*

● The slowest decay rate corresponds to the equivalent of the subradiant state (as expected)

● The synchronization-bringing mode is the coherence between such a subradiant state and the ground state

● Synchronization and subradiant are different physical manifestations of the same microscopic term appearing in the dissipative dynamics

● We have used a classical indicator to measure synchronization. Nevertheless, the connection to subradiance shows that it is a purely quantum effect

Subradiance and synchronization

*

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quantum synchronization as a local signature of super and ... · gian luca giorgi (ifisc) bruno...

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