NON-MARKOVIAN CONTINUOUS QUANTUM MEASUREMENT OF RETARDED OVSERVABLES - PDF document

1 NON-MARKOVIAN CONTINUOUS QUANTUM MEASUREMENT OF RETARDED OVSERVABLES Lajos Di´ osi, Budapest Contrary to longstanding doubts, diffusive non-Markovian quantum tra- jectories are single system trajectories and correspond to the true con- tinuous measurement of a certain retarded potential. CONTENTS: • Continuous Quantum Measurement • Stochastic Unravelling • Non-Markovian Measurement Device • Continuous Read-Out • Stochastic Schr¨ odinger Eq. • Conclusions PEOPLE: • Gisin 1984, Belavkin, D. (1988) • Strunz 1996 • Gambetta+Wiseman (2002-3) • Chou+Su+Hao+Yu (1985)

2 Continuous Quantum Measurement Markovian continuous measurement of ˆ x t : stochastic Schr¨ odinger equa- tion (SSE) of the collapsing state vector depending on the history of the read-outs: ψ t [ x ] where x = { x τ ; τ ∈ [0 , t ] } . Formal extension for the non-Markovian (even relativistic) case (1990). Only ψ ∞ [ x ] and p ∞ [ x ] were given. The concept of continuous read-out was missing. Smart non-Markovian quantum trajectories (1996) and their non- Markovian SSE (1997). Doubts: non-Markov quantum trajectory is mathematical fiction (2002). The present work comes to the positive conclusion: the non-Markovian trajectories are measurable single system trajectories. The equations concerning the measurement of ˆ x t must be reparametrized in terms of a given ˆ z t which is a retarded function of ˆ x t . Then we are continuously reading out ˆ z t instead of ˆ x t .

3 Stochastic Unravelling Suppose openness is caused by continuous measurement: ρ t = M t ˆ ˆ ρ 0 ρ t is density matrix, M t is completely positive map (superopera- where ˆ tor). Simplest non-Markovian: � t � t � − 1 � M t = T exp x τ, ∆ α ( τ − σ )ˆ d τ d σ ˆ x σ, ∆ 2 0 0 x τ, ∆ ˆ where α ( τ − σ ) is real positive kernel. Superoperator notation: ˆ O = x τ , ˆ O ] for any ˆ O ; T is time-ordering for all Heisenberg (super)operators . [ˆ The decohered quantity is ˆ x t but the measured quantity may be different, say ˆ z t . Meausured state: either ψ t [ x ], or, e.g.: ψ t [ z ] where x or z are two differ- ent read-outs of the detectors. They must equally unravel the ensemble evolution: ρ t = M ψ t [ x ] ψ † t [ x ] = M ψ t [ z ] ψ † ˆ t [ z ] . It was hard to find a non-Markovian unravelling. Yet, in 1990 i got some ψ t [ x ] and in 1996 Strunz got some ψ t [ z ], in 1997 we found a linear SSE for the latter: � t dΨ t [ z ] α ( t − τ ) δ Ψ t [ z ] x t Ψ t [ z ] − 2ˆ = z t ˆ x t d τ , d t δz τ 0 z τ is a real random variable for τ ∈ [0 , t ]. True state is obtained via normalization: ψ t [ z ] = Ψ t [ z ] / � Ψ t [ z ] � . Probability distribution of z : G [0 ,t ] [ z ] � Ψ t [ z ] � 2 , p t [ z ] = ˜ ˜ G [0 ,t ] [ z ] is a Gaussian distribution defined through α ( τ − σ ). We showed (1997): � t α ( t − σ ) � ˆ x σ � t d σ , M z t = 2 0 � ˆ x σ � t is ˆ x σ ’s quantum expectation value at time t in state ψ t [ z ]. The SSE “measures” the retarded “potential” of ˆ x t rather than ˆ x t itself.

4 Non-Markovian Measurement Device Example: single vonN detector of initial density matrix D 0 ( x ; x ′ ): ρ 0 − → D 0 ( x − ˆ x τ,L ; x − ˆ D 0 ( x ; x )ˆ x τ,R )ˆ ρ 0 . x τ,L ˆ x τ ˆ x τ,R ˆ O = ˆ Superoperator notation: ˆ O = ˆ O and ˆ O ˆ x τ . After read-out of the pointer x : total state goes into the system’s conditional state, depending on the read-out: 1 p ( x ) D 0 ( x − ˆ x τ,L ; x − ˆ ρ ( x ) = ˆ x τ,R )ˆ ρ 0 , p ( x ) = tr D 0 ( x − ˆ x τ,L ; x − ˆ x τ,R )ˆ ρ 0 . Choose discrete time τ = nǫ , n = 0 , ± 1 , ± 2 , . . . . Install an infinite sequence of vonN detectors, labelled by τ = nǫ . Pointer coordinates of the detectors: x τ . The detector of label τ = nǫ measures the Heisenberg operator ˆ x τ of the system via the above mechanism. We switch on the detectors for τ ≥ 0. Assume initially correlated detectors , of intitial wave function: √ � � − ǫ 2 � N exp x τ α ( τ − σ ) x σ φ 0 [ x ] = . τ,σ In continuous (or weak measurement) limit ǫ → 0: � φ 0 [ x ] = G [ x ] . Introduce the characteristic function θ [0 ,t ] of the period [0 , t ]. The total density matrix reads: ρ t [ x ; x ] = T G [ x − θ [0 ,t ] ˆ x c ] M t ˆ ˆ ρ 0 , x c ˆ x, ˆ 2 { ˆ O } . This form guarantees the un- Superoperator notation ˆ O = 1 ρ t = M t ˆ ravelling of the open system dynamics ˆ ρ t .

5 Continuous Read-Out It is crucial to realize that the true time-evolution of the system’s condi- tional state depends on our chosen schedule to reading out the pointers x τ . In fact, we can read out any x τ at any time since all detectors are alaways available. Of course, we better read out the value x τ at a time which is later than the label τ of the detector because the detector will only have coupled to the system at time τ . Hence, a natural schedule is that we read out x τ immediately, i.e., at time τ . As a result, until any given time t > 0 we would read out all pointers x τ for the period [0 , t ] and no others. To calculate the conditional post-measurement state ρ t [ x ] of the system at time t , we trace (integrate) the total density matrix ˆ ∈ [0 , t ]: ρ [ x ; x ] over all x τ with τ / ˆ 1 � � ρ t [ x ] = ˆ ρ t [ x ; x ] ˆ d x τ . p t [ x ] τ / ∈ [0 ,t ] This post-measurement density matrix ˆ ρ t [ x ] of the system depends on the read-outs x τ of τ from [0 , t ] only: 1 p t [ x ] T G [0 ,t ] [ x − ˆ x c ] M t ˆ ρ t [ x ] = ˆ ρ 0 , G [0 ,t ] [ x ] is the marginal distribution of G [ x ]. Instead of reading out the coordinates { x τ ; τ ∈ [0 , t ] } , read out � ∞ α ( τ − σ ) x σ d σ , z τ = 2 −∞ the postmeasurement density matrix becomes: 1 p t [ z ] T ˜ G [0 ,t ] [ z − 2 αθ [0 ,t ] ˆ x c ] M t ˆ ρ t [ z ] = ˆ ρ 0 , where ˜ G [0 ,t ] [ z ] is the marginal distribution of ˜ G [ z ]. This is our ultimate equation for the non-Markovian continuous measurement of the observ- able � t α ( t − σ )ˆ z t = 2 ˆ x σ d σ , 0 which is a sort of retarded potential generated by the Heisenberg variable x τ . ˆ

6 Stochastic Schr¨ odinger Equation Let us find the postmeasurement conditional state 1 p t [ z ] T ˜ G [0 ,t ] [ z − 2 αθ [0 ,t ] ˆ x c ] M t ˆ ρ t [ z ] = ˆ ρ 0 in the form: 1 G [0 ,t ] [ z ]Ψ t [ z ]Ψ † ˜ ρ t [ z ] = ˆ t [ z ] , p t [ z ] where Ψ t [ z ] is the unnormalized conditional state vector of the system. Trace over both sides, norm condition yields: G [0 ,t ] [ z ] � Ψ t [ z ] � 2 , p t [ z ] = ˜ just like for the SSE. Comparing our eqs., they reduce to: 1 Ψ t [ z ]Ψ † x c ] M t ψ 0 ψ † T ˜ G [0 ,t ] [ z − 2 αθ [0 ,t ] ˆ t [ z ] = 0 . ˜ G [0 ,t ] [ z ] The r.h.s. factorizes and we can write equivalently: �� t � t � t � Ψ t [ z ] = T exp x τ d τ − x τ α ( τ − σ )ˆ z τ ˆ d τ d σ ˆ x σ ψ 0 . 0 0 0 This Ψ t [ z ] is the solution of the SSE.

7 Conclusion We proved for the first time that both the formalism of non-Markovian measurement theory (1990) and the non-Markovian SSE (1997) are equivalent with using of correlated von Neumann detectors in the weak- measurement continuous limit, i.e., with the continuous read-out of the values of a given retarded potential of a Heisenberg variable on a singe quantum system. Hint of efficient simulation? Immediate generalizations: complex α ( τ − σ ), indirect measurement on the reservoir.

8 Appendix.– Assume a random time-dependent real variable x τ defined for all time τ and consider the following Gaussian distribution functional of { x τ ; τ ∈ ( −∞ , ∞ ) } : � ∞ � ∞ � � G [ x ] = N exp − 2 d σx τ α ( τ − σ ) x σ d τ , (1) −∞ −∞ where α ( τ − σ ) is a real positive definite kernel. We define its inverse through: � ∞ α − 1 ( τ − s ) α ( s − σ )d s = δ ( τ − σ ) . (2) −∞ We also introduce the normalized functional Fourier transform of G [ x ]: � ∞ � ∞ − 1 � � G [ z ] = ˜ ˜ d σz τ α − 1 ( τ − σ ) z σ N exp d τ . (3) 2 −∞ −∞ � ˜ � Both distributions are normalized: G [ x ]Π τ d x τ = G [ z ]Π τ d z τ = 1. In- stead of their functional distributions G [ x ] , ˜ G [ z ], the statistics of x τ , z τ can equivalently be characterized by their vanishing means M x τ = M z τ = 0 and correlation functions, respectively: 4 α − 1 ( τ − σ ) , M z τ z σ = α ( τ − σ ) . M x τ x σ = (4) 1 We need certain marginal distributions as well, e.g.: � ˜ ˜ � G [0 ,t ] [ z ] = G [ z ] d z τ , (5) ∈ [0 ,t ] τ / and similarly for G [0 ,t ] [ x ]. These marginal distributions are still Gaussian, e.g.: � ∞ � ∞ − 1 � � G [0 ,t ] [ z ] = ˜ ˜ d σz τ α − 1 N [0 ,t ] exp d τ [0 ,t ] ( τ, σ ) z σ , (6) 2 −∞ −∞ where the new kernel is defined by: � t α − 1 [0 ,t ] ( τ, s ) α ( s − σ )d s = δ ( τ − σ ) , τ, σ ∈ [0 , t ] . (7) 0 In most cases, α − 1 [0 ,t ] ( τ, σ ) is a hard nut to calculate explicitly. This work was supported by the Hungarian Scientific Research Fund under Grant No 49384.