Variational analysis of dissipative Ising models

Transcription

1 GRK Workshop Hildesheim 2016 Institute for Theoretical Physics Leibniz University Hannover

2 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations

3 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations

4 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

5 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

6 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

7 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations

8 Variational principle Calculation of the dynamics via a variational principle Completely generic Variational minimization of a norm

9 Variational approach for steady state Variational parametrization of ρ Example: product state ρ = Π i ρ i ρ i = αvar σ Steady state: ρ = 0 Approximation: ρ min Upper bound for product states: ρ ij ρ ij Trace norm x = Tr{ x }

10 Steady state phase transition n ρ p ρ c QME 3 3 QME g/γ Rydberg density n for the steady state (h = 0, V = 5γ) H. Weimer, Phys. Rev. Lett. 114, (2015). Variational results including (ρ c ) and excluding (ρ p ) correlations Comparison with solution of the full Quantum master equation

11 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, (2012). Midpoint: D ρ var (t + τ) ρ(t) τ 2 [ ρ(t) + ρvar (t + τ) ] min }{{} O(τ 3 )

12 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

13 Time evolution QME 4 4 ρ p ρ c n 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

14 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations

15 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, (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

16 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 environment: ρ c (t) Trace over the contribution of the environment of ρ ijk : itr l {[H int, ρ ijkl ]}

17 Results for non-markovianity non-markovianity f(t)/γ γ 1γ γt For V = 0.5γ, 1γ: Non-Markovian effects f (t ) > 0: Eternal non-markovianity M. Hall et al., Phys. Rev. A. 89, (2014).

18 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 f(v)/γ I(V)/γ V/γ x QLMI

19 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations

20 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

21 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

22 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 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

23 Phase diagram 3 d=3 PM γ Jz 2 FM 1 second order transition first order transition Tricritical point Jz Second order transition as in equilibrium, first order due to dissipation Second and first order line meet at the tricritical point

24 Outline 1 Dissipative quantum systems 2 Variational formulation and dynamics 3 Non-Markovianity 4 Steady state phase transitions 5 Fluctuations

25 First order jump For large dimensions: mean-field result Jump of the order parameter goes like δφ 1 z ; z = 2d δφ ln(b) 0 δφ = b( c ) ln(b) ln(z) ln(z) z=4 z=6 0.1 z=20 z= TC z

26 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)

27 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, (2016).