Large deviation theory of Dario Cilluffo Salvatore Lorenzo quantum - - PowerPoint PPT Presentation
XXIV Convegno Nazionale di Fisica Statistica e dei Sistemi Complessi Large deviation theory of Dario Cilluffo Salvatore Lorenzo quantum jump statistics in a Francesco Ciccarello Massimo Palma chiral waveguide DIPARTIMENTO DI FISICA E
XXIV Convegno Nazionale di Fisica Statistica e dei Sistemi Complessi
Dario Cilluffo Salvatore Lorenzo Francesco Ciccarello Massimo Palma
DIPARTIMENTO DI FISICA E CHIMICA -EMILIO SEGRE’ UNIVERSITA’ DI PALERMO
U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>number of identical subsistems, dubbed “ancillas”, initially all in the same state
takes place via a sequence of “collisions” between the system and the ancillas i.e. via pairwise short unitary interactions lasting a short time
approximation
coherence time for the environment
D(ˆ L) ρ = ˆ L ρ ˆ L† 1
2 (ˆ
L† ˆ Lρ + ρ ˆ L† ˆ L) ,
master equation in Lindbad form jump operator
U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>to a master equation giving information on the average behaviour of the system
each step the system undergoes a series of jumps
an average over the unraveling, i.e. as an average ove different histories
different unraveling describing the same master equation
U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>U
<latexit sha1_base64="rgMADh/Zt5OgR7I2F6JQfynz6Y=">AB83icbVBNSwMxFHzxs9avqkcvwSJ4KrtV0GPRi8cKblvoLiWbZtvQbHZJskJZ+je8eFDEq3/Gm/GbLsHbR0IDPv8SYTpoJr4zjfaG19Y3Nru7JT3d3bPzisHR13dJIpyjyaiET1QqKZ4J5hvBeqliJA4F64aTu8LvPjGleSIfzTRlQUxGkecEmMl34+JGVMicm9WHdTqTsOZA68StyR1KNEe1L78YUKzmElDBdG67zqpCXKiDKeCzap+plK6ISMWN9SWKmg3yeYbPrTLEUaLskwbP1d8bOYm1nsahnSwy6mWvEP/z+pmJboKcyzQzTNLFoSgT2CS4KAPuWLUiKklhCpus2I6JopQY2sqSnCXv7xKOs2Ge9loPlzVW7dlHRU4hTO4ABeuoQX30AYPKTwDK/whjL0gt7Rx2J0DZU7J/AH6PMHyKORgw=</latexit>k
k
Thermodynamics of Quantum Jump Trajectories
Juan P. Garrahan and Igor Lesanovsky
School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom (Received 17 November 2009; revised manuscript received 2 March 2010; published 19 April 2010)
PRL 104, 160601 (2010) Selected for a Viewpoint in Physics P H Y S I C A L R E V I E W L E T T E R S
week ending 23 APRIL 2010
ð Þ PtðKÞ ¼ Tr½ðKÞðtÞ et’ðK=tÞ: (2) The ‘‘large-deviation’’ function ’ðkÞ (k K=t) contains all information about the probability of K at long times [2]. Alternatively, we can describe the statistics of K via the generating function, which also has a LD form [2], ZtðsÞ X
1 K¼0
PtðKÞesK etðsÞ: (3)
½ Derivatives of ðsÞ give moments of the photon number
photons is k0 hKi=t ¼ 0ð0Þ, and the Mandel parame- ter, Q0 ðhK2i hKi2Þ=hKi 1 ¼ 00ð0Þ=0ð0Þ. The LD function around s ¼ 0 encodes the information about fluctuations of typical trajectories [4,10].
(A) Laser driven 2-level system coupled to a T ¼ 0 bath. (B) Large-deviation function ðsÞ. Dynamical trajectories go from more active to less active as s, the conjugate field to the number of emitted photons K, is increased, as shown by the average photon rate kðsÞ hKis=t ¼ 0ðsÞ. The Mandel parameter QðsÞ ¼ 2=3 for all s, indicating that for ¼ 4 trajectories display a form of scale invariance. (C) The photon count probability is obtained from (5) by a Legendre transform: PtðKÞ et’ðK=tÞ with ’ðkÞ ¼ 3½k lnðk=k0Þ ðk k0Þ. It is a ¼ 3 Conway-Maxwell-Poisson distribution [20], PtðKÞ / ½PoissonðK; tÞ3. (D) Representative trajectories from subensem- bles with different average k.
(A) Laser driven 3-level system. Here 1 ¼ 41 and 2 ¼ 1=10. (B, C) The LD function ðsÞ and dynamical order parameter kðsÞ display crossover behavior near s ¼ 0 between active and inactive dynamical regimes. The active side is antibunched, Q < 0. The inactive side is non- fluctuating Q ¼ 0. The peak in Q near s ¼ 0 signals the dy- namical crossover. (D) The fat tail for k < k0 in PtðKÞ is a manifestation of the inactive regime; the thin tail for k > k0 is a manifestation of the active regime. (E) Representative trajecto- ries from inactive and active subensembles. At s ¼ 0 there is (mesoscopic) coexistence of the two dynamical regimes and typical trajectories are intermittent or ‘‘blinking.’’
a r X i v : 1 9 6 . 1 7 v 1 [ q u a n t
h ] 3 J u n 2 1 9
Quantum jump statistics with a shifted jump
Dario Cilluffo,1 Salvatore Lorenzo,1 G. Massimo Palma,1,2 and Francesco Ciccarello1,2
˙ ρ = i [ ˆ H, ρ] + D(ˆ L) ρ ,
Resonance Fluorescence D(ˆ L) ρ = ˆ L ρ ˆ L† 1
2 (ˆ
L† ˆ Lρ + ρ ˆ L† ˆ L) ,
where the driving Hamiltonian ˆ H and jump operator ˆ L are respectively given by ˆ H = Ω (ˆ σ+ + ˆ σ−) , ˆ L = pγ ˆ σ− . (3) ME (1) can be ”unraveled” in terms of (3) by expressing its solution at time t as ρt = Rtρ0 +
∞
X
K=1
Z t dtK · · · Z t2 dt1 Rt−tKJ RtK−tK−1 · · · J Rt2−t1J Rt1ρ0 , (4) where we defined the superoperators Rt ρ = e
−i ✓ ˆ H− i 2 ˆ L† ˆ L ◆ tρ e i ✓ ˆ H− i 2 ˆ L† ˆ L ◆ t ,
J ρ = ˆ L ρ ˆ L† . (5)
detector
atom
Ω − √γ α
chiral waveguide
Figure 1. Waveguide-QED setup to implement unraveling (8) with shifted jump
A two-level atom is coupled to a chiral waveguide, sustaining a one- dimensional field that propagates in one direction only. The atom is driven by two continuous-wave coherent beams: one with Rabi frequency Ω − √γ α propagating out
Photons are counted at one end of the waveguide: the detector senses light emitted from the atom superposed with the coherent wave α.
A class of unravelings for the resonance fluorescence ME (3) is defined by ˆ H =
2
pγ α
σ+ + H.c. , ˆ L = pγ ˆ σ− + iα ,
(a) (b)
(c) (d)
← ← ← ←
Figure 2. Mandel Q parameter Q(s) as a function of α for Ω = 0.1 (a), Ω = 0.25 (b), Ω = 1 (c) and Ω = 5 (d) (we set γ = 1). Each curve was obtained through a mesh of the s-axis and numerical diagonalization of (13), from which θ(s) was inferred by selecting the largest eigenvalue. The s-dependent Mandel Q parameter Q(s) was determined using Eqs. (16) after applying the finite difference method to compute the derivatives of θ(s).