GRK Workshop Hildesheim 2016 Institute for Theoretical Physics Leibniz University Hannover 08.02.2016
Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations
Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations
Dissipative systems Dissipation: Controlled preparation of many-body states Strongly interacting Rydberg gases exhibit a natural dissipative element Experimental progress by tuning dissipation and interaction Theoretical understanding still limited Here: Variational formulation
Rydberg atoms Highly excited states Natural, controllable decay Strong interaction due to weakly bound electrons One/two photon excitation: ground state couples to Rydberg state
Quantum master equation in Rydberg gases Spin 1/2 representation, : ground state, : Rydberg state Quantum master equation in Lindblad form d ρ = i[h, ρ] + γ (c i ρc i 12 ) dt }{{} {c ic i, ρ} coherent part i }{{} dissipative part Jump operators c i = σ (i) : up spin flipping into down spin, γ: decay rate Hamiltonian with nearest neighbour interaction H = g 2 i σ (i) x + h 2 i σ (i) z + V 4 ij σ z (i) σ z (j) with the transverse field g, longitudinal field h and interaction strength V
Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations
Variational principle Calculation of the dynamics via a variational principle Completely generic Variational minimization of a norm
Variational approach for steady state Variational parametrization of ρ Example: product state ρ = Π i ρ i ρ i = 1 2 + αvar σ Steady state: ρ = 0 Approximation: ρ min Upper bound for product states: ρ ij ρ ij Trace norm x = Tr{ x }
Steady state phase transition n 0.5 0.4 0.3 0.2 0.1 0 ρ p ρ c QME 3 3 QME 4 4 0 2 4 6 8 10 g/γ Rydberg density n for the steady state (h = 0, V = 5γ) H. Weimer, Phys. Rev. Lett. 114, 040402 (2015). Variational results including (ρ c ) and excluding (ρ p ) correlations Comparison with solution of the full Quantum master equation
Integration of the Master equation Most simple: Euler ρ(t + τ) = ρ(t) + τ ρ(t) + O(τ 2 ) ρ(t + τ) ρ(t) τ ρ(t) = 0 ρ var (t + τ) ρ(t) τ ρ(t) min see also: C. V. Kraus, and T. J. Osborne, Phys. Rev. A. 86, 062115 (2012). Midpoint: D ρ var (t + τ) ρ(t) τ 2 [ ρ(t) + ρvar (t + τ) ] min }{{} O(τ 3 )
Integration of the Master equation Upper bound for the concrete calculation For product states: D ij ρ var ij (t + τ) ρ ij (t) τ 2 [ ρ ij(t) + ρ var ij (t + τ)] Translationally invariant system single two-site problem Correlations: three-site problem
Time evolution 0.06 0.04 QME 4 4 ρ p ρ c n 0.02 0 0 2 4 6 8 10 γt Rydberg density n for g = 1γ, V = 2γ and h = 0γ Variational results including (ρ c ) and excluding (ρ p ) correlations Comparison with solution of the full Quantum master equation
Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations
Canonical form of the dynamics Local dynamics in canonical form d d dt ρ 2 1 ( ij = i[h(t), ρ ij ]+ γ k (t) L k (t)ρ ij L k (t) 1 ) 2 {L k(t)l k (t), ρ ij} k=1 Tr{L k (t)} = 0, Tr{L j (t)l k (t)} = δ jk. M. Hall et al., Phys. Rev. A. 89, 042120 (2014). Decoherence rates γ k (t) < 0 for at least one k: dynamics is non-markovian Measure for non-markovianity f (t) = 1 d 2 1 [ γ k (t) γ k (t)]. 2 k=1
Calculation of γ k Unique result for γ k (t) dynamics of a full set of states ρ mn ρ 00 = G 00 ; ρ mn = 1/4 + G mn G mn = σ m σ n ; m, n {0, x, y, z}. Calculation of the dynamics via variational principle initial state: ρ ij (t) ρ mn 0.06 0.04 environment: ρ c (t) 0 0 2 4 6 8 10 0.02 Trace over the contribution of the environment of ρ ijk : itr l {[H int, ρ ijkl ]}
Results for non-markovianity non-markovianity f(t)/γ 0.2 0.15 0.1 0.05 0.5γ 1γ 0 0 2 4 6 8 10 γt For V = 0.5γ, 1γ: Non-Markovian effects f (t ) > 0: Eternal non-markovianity M. Hall et al., Phys. Rev. A. 89, 042120 (2014).
Non-Markovianity and QLMI Quantum linear mutual information with the linear entropy I = S(ρ i ρ j ) S(ρ ij ) S(ρ) = 1 Tr{ρ 2 }. R.M. Angelo et al., Physica A 338, (2004). measure of the strength of correlations ρ c (t) includes correlations arising from the interaction Interaction correlations non-markovianity non-markovianity 0.08 0.06 0.04 0.02 f(v)/γ I(V)/γ 0 0 0.2 0.4 0.6 0.8 1 1.2 V/γ x 10 3 6 4 2 0 QLMI
Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations
Dissipative Ising model Hamilton operator H = J ij σ x (i) σ x (j) + i σ (i) z Single photon transition: g and h couple to r Interaction between two excited atoms leads to σ x σ x term
Symmetries Ising model: global Z 2 symmetry H = J ij σ i zσ j z + i σ i x Symmetry is broken in a low-temperature ferromagnetic phase Our Hamiltonian: Z 2 symmetry is present in the master equation can be broken in an ordered phase ferromagnetic phase paramagnetic phase
Phase transitions Once again: steady state ρ ij min Expansion in order parameter φ σ x ρ ij = 2J + u 2 φ 2 + u 4 φ 4 + u 6 φ 6 + O(φ 8 ) Paramagnetic phase: φ = 0; ferromagnetic phase: φ > 0 First order transition 2u 2 1 2 u 4 2 /u 6 = 0, u 4 < 0. Second order transition u 2 = 0, u 4 > 0. Tricritical point u 2 = 0 and u 4 = 0
Phase diagram 3 d=3 PM γ Jz 2 FM 1 second order transition first order transition Tricritical point 0 0 0.2 0.4 0.6 0.8 1 Jz Second order transition as in equilibrium, first order due to dissipation Second and first order line meet at the tricritical point
Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations
First order jump For large dimensions: mean-field result Jump of the order parameter goes like δφ 1 z ; z = 2d δφ 0.4 0.3 0.2 ln(b) 0 δφ = b( c ) 0.5 1 ln(b) 2 1 2 ln(z)+0.03 3 0 2 4 ln(z) z=4 z=6 0.1 z=20 z=200 0 0 0.05 0.1 0.15 0.2 0.25 TC z
Upper critical dimension Fluctuations [φ Φ] 2 estimated via Ginzburg criterion, Φ = φ Critical exponent of 1/4 along the u 4 = 0 line Relative deviation [φ Φ]2 Φ 2 Φ ( ( ) 1/4 u2 3u 6 ) d/2 3/2 u 2 3u 6 Upper critical dimension of 3 at the tricritical point(u 2 0)
Summary Calculation of the dynamics of Rydberg gases via a variational approach Taking correlations into account improves the results Correlations lead to non-markovian effects QLMI and non Markovianity show a similar dependence on interaction strength V Variational steady state analysis concerning critical behaviour We found multicritical behaviour V. R. Overbeck and H. Weimer, Phys. Rev. A. 93, 012106 (2016).